DESIGN AND CONSTRUCTION OF HEAT ENGINES PUBLISHERS OF BOOKS Coal Age ^ Electric Railway Journal Electrical World * Engineering News-Record American Machinist v Ingenierfa Internacional Engineering 8 Mining Journal ^ Power Chemical S Metallurgical Engineering Electrical Merchandising DESIGN AND CONSTRUCTION OF HEAT ENGINES BY WM. E. NINDE, M. E. ASSOCIATE PROFESSOR OP MECHANICAL ENGINEERING, SYRACUSE UNIVERSITY MEMBER OP AMERICAN SOCIETY OF MECHANICAL ENGINEERS FIRST EDITION McGRAW-HILL BOOK COMPANY, INC. NEW YORK: 239 WEST 39TH STREET LONDON: 6 & 8 BOUVERIE ST., E. C. 4 1920 Engineering? Library COPYRIGHT, 1920, BY THE MCGRAW-HILL BOOK COMPANY, INC. THE MAPLE PKE S S YORK DEDICATION To the man who esteems excellence in his calling of greater value than his remuneration 414821 PREFACE The object of this book is to supply in one volume the material most essential to the well-equipped, independent designer of heat engines, and to give this material in the form most convenient for use in class room and practical work, by a separate treatment of the different phases of the subject. Contributory to the contents are the author's note books on engine design covering his practice and observation for over twenty years; revisions and additions necessary to adapt the notes to his teaching work for the past ten years; material from the best technical books and peri- odicals necessary to fill the gaps in first-hand information, and to add breadth and character to the work; and data and drawings from some of the best designers and builders of heat engines in the United States- giving a practical, commercial touch. In the derivation of working formulas, elaborate and abstruse methods have been avoided as much as possible, but no important element which permits of practical treatment has been omitted; also, while reverting to fundamental principles and making the formulas as rational as possible, their limitations are pointed out, and they are usually brought to a form suitable for direct application. Illustrations of none but the most excellent designs are provided. These are necessarily limited on account of space, but qualitative design, and certain details and auxiliaries may be studied in books of a more descriptive character, and in builders' catalogs which are always obtainable. The numbering of formulas begins with each chapter, and manu- facturers' material is usually placed in the latter part of the chapter to facilitate the revisions necessary to keep the work up to date. The nota- tion for a chapter is listed at its beginning except for short, scattered discussions, when it is given only in the text. Steam tables and other tables found in all handbooks are omitted; they are more convenient to use with the formulas of this book if in a separate cover. A few references are given at the ends of chapters to extend the scope of the work. vii viii PREFACE The process of absorption by contact with associates in office, shop and college, and habitual gleaning from technical periodicals, books and catalogs long before the writing of a book was thought of, renders the placing of credit difficult, but when possible it has been done. The author is very grateful to the manufacturers and engineers who have furnished illustrations and data, and to authors and publishers who have permitted the use of material from their publications. Thanks are also due my wife, Luella V. Ninde, whose assistance and encouragement have made the book possible. W. E. N. SYRACUSE UNIVERSITY, January, 1920. CONTENTS PAGE PREFACE * . . i vii Part 1. The Heat Engine CHAPTER I STATUS OP THE HEAT ENGINE .....;. 1 Furnishes bulk of power Water power inadequate Adaptability of dif- ferent types. CHAPTER II THE POWER PLANT '3 Steam cycle not complete in engine Cycle of steam operation Internal- combustion engine. CHAPTER III THE STEAM ENGINE 5 Mechanism Principle of operation Function of valve gear Indicator diagrams Types and classification The uniflow engine. CHAPTER IV THE STEAM TURBINE 23 Of early origin Impulse and reaction Commercial classification Simple impulse turbine Compound impulse turbine Velocity stage Pressure stage Reaction turbine Combinations Governing CHAPTER V THE INTERNAL-COMBUSTION ENGINE 32 Classification and cycles Gas engines Light-oil engines Heavy-oil engines The Diesel engine Governing Starting Cylinder cooling and mufflers Practical data. Part 2. Thermodynamics CHAPTER VI GENERAL POWER FORMULAS AND GASES 51 Power of heat engines Mean effective pressure Mechanical efficiency Practical cycles for using gas The constant-volume cycle The constant- ix X CONTENTS PAGE pressure cycle Volumetric efficiency Temperature rise due to combustion Conventional indicator diagrams. CHAPTER VII STEAM . ; . . 65 Formation under constant pressure Relation of pressure and temperature Supply of heat during formation Flow of steam Effect of friction on flow. CHAPTER VIII ECONOMY 70 The steam plant The prime mover I.h.p. and b.h.p. The heat factor Internal-combustion engines Comparative economy. CHAPTER IX CYLINDER EFFICIENCY 78 The steam engine cylinder Condensation and re-evaporation Compression and clearance Factors affecting cylinder condensation The internal- combustion engine cylinder. Part 3. Friction and Lubrication CHAPTER X MECHANICAL EFFICIENCY 97 Friction horsepower Effect of different methods of lubrication Variation of efficiency with load Effect of maximum steam or gas pressure. CHAPTER XI LUBRICATION 103 General principles Lubricants and their application Clearance Oil grooves Bearing metal Bearing proportions Thrust bearings Eccentric bearing Force-feed systems of lubrication Cylinder lubrication. Part 4. Power and Thrust CHAPTER XII THE SIMPLE STEAM ENGINE 127 Indicator diagrams Mean effective pressure Diagram factors Governing Piston thrust Indicated horsepower Theoretical steam consumption Compression Standard engines Application of formulas. CONTENTS xi PAGE CHAPTER XIII THE COMPOUND STEAM ENGINE 152 Indicator diagrams Condensing engines Influence of the receiver The tandem compound The cross compound Governing Maximum thrust Indicated horsepower Diagram factors Theoretical steam consumption Standard engines Application of formulas. CHAPTER XIV THE INTERNAL-COMBUSTION ENGINE 185 Mean effective pressure Horsepower Practical constants Simple formulas Effect of various factors upon capacity Compression Govern- ing The quality method The quatity method Combined methods Rating Indicator diagrams and maximum piston thrust Maximum indi- cator diagram Application of formulas. CHAPTER XV THE STEAM TURBINE 206 The general method Nozzles and other passages Effect of friction Practical notes on nozzles and other passages Impulse blading Practical notes on impulse blading Velocity-stage impulse turbine Practical notes on velocity-stage turbine Intermediate pressures Effect of heat factor Peabody's direct method Superheated steam Velocity due to reheating General dimensions Number of equal stages Unequal wheel diameters Conservation of residual velocity Pressure-velocity-stage impulse turbine Trajectory and lead The reaction turbine Reaction blading Work division Practical notes on reaction turbines Factors influencing turbine operation The condenser Steam passages Governing Rating and over- load Application of formulas. Part 5. Mechanics CHAPTER XVI THE SLIDER CRANK . 283 Properties of the connecting rod Piston velocity Forces due to pressure in cylinder Efficiency of the slider crank Forces due to gravity Forces due to acceleration The reciprocating parts The connecting rod The effect of angular acceleration Reaction on the guide Combined indicator and inertia diagram Steam engine Internal-combustion engine Reversal of thrust and effect of compression Total turning effort Combined turning effort diagrams Reaction on the frame Application to steam and internal- combustion engines. CHAPTER XVII BALANCING 334 Simple, or primary balancing Reciprocating parts Secondary balance Multi - cylinder engines The connecting rod Turning effort Balance weights. Xll CONTENTS PAGE CHAPTER XVIII REGULATION DURING THE CYCLE. FLYWHEELS 350 The control of speed fluctuation The control of displacement Com- parison of methods Application to practice. CHAPTER XIX REGULATION DURING CHANGE OF LOAD. GOVERNORS 367 Independent method Coordinate method Governor types Centrifugal governors Stability Isochronism Inertia, or centrifugal - inertia gover- nors Relay governors Safety, or over speed governors Dashpots Speed adjustments Speed fluctuation Speed variation General equations for centrifugal governors Equations for conical governors Determination of speed Equations for spring governors Springs Practical considerations Gravity balance Constants of regulation Machine design Illustrations from practice. CHAPTER XX VALVES AND VALVE GEARS 401 Port area Allowable fluid velocity Classification The single-valve gear Lap, lead and angular advance Effect of rod angularity The Bilgram dia- gram Double-ported valves Shifting and swinging eccentrics Rectangular valve diagrams Corliss releasing gear Releasing gear operation High- speed Corliss gears Mclntosh and Seymour gear Cam - and - eccentric gears Reversing gears for steam engines Equivalent eccentric for Stephen- son and Walschsert gears Variable lead for Walschsert gear Hackworth gear Marshall gear Joy gear Doble gear Internal-combustion engine gears Multi-cylinder gears Eccentric-and-cam gears Valve springs Two-cycle engines Reversing gears for internal-combustion engines The sleeve motor gear Details from practice. Part 6. Machine Design CHAPTER XXI GENERAL CONSIDERATIONS 463 Rational machine design Working stresses and factors of safety Static stress Repeated stress Reversed stress Suddenly applied load and shock Mechanical properties of materials Selection of formulas Cylinder walls Combined bending and twisting Rectangular section in torsion Struts Clearances and tolerances Basis of design. CHAPTER XXII CYLINDERS 499 The cylinder Counterbore Inlet and exhaust passages Cylinder heads Stuffing box Cylinder lagging Designs from practice. CONTENTS xiii PAGE CHAPTER XXIII PISTONS 514 Material Strength calculations Piston rings Eccentric rings Rings of uniform section Number of rings Designs from practice. CHAPTER XXIV PISTON RODS .' .'', 530 Piston rod formulas The crosshead end The piston end Numerical values The piston rod as a strut Pressed fits Designs from practice. CHAPTER XXV CONNECTING RODS 544 Body of rod A general formula for direct calculation of dimensions Numerical values Circular section Rectangular section I section Con- necting rod ends Wedge bolts Designs from practice. CHAPTER XXVI CROSSHEADS 562 The crosshead Crosshead pin Crosshead shoe Application of formulas Designs from practice. CHAPTER XXVII CRANKS , . 572 Crank pin Crank arm Hubs Crank designs. CHAPTER XXVIII SHAFTS 586 Shaft types The side-crank shaft The center-crank shaft Methods of calculation General formulas Special formulas Application of formulas to steam engine and gas engine cranks Designs from practice. CHAPTER XXIX FRAMES 611 Stresses in engine frames Side-crank frames Main bearings Wedge ad- justing bolts Cap bolts Designs from practice. CHAPTER XXX FLYWHEELS 623 Hoop stress Unwin's formulas Unwin's method modified Forces pro- ducing bending stresses in arms Number of arms carrying load Bending XIV CONTENTS PAGE moment in arms Working stresses Rim and arm sections Hubs Hub bolts Shear bolts Rim bolts and links Keys Methods of construction Application of formulas Designs from practice. * CHAPTER XXXI TURBINE WHEELS. . 4 ......... 649 Stresses in wheels Material Blading design Application of formulas Designs from practice. CHAPTER XXXII TURBINE SHAFTS .664 Nature of the problem Critical speed Deflection Application of formulas. CHAPTER XXXIII TURBINE CASINGS AND DETAILS 674 DeLaval turbine Ridgway turbine. CHAPTER XXXIV GENERAL ARRANGEMENTS AND FOUNDATIONS 679 General arrangements Foundations Vibration problems Material Bolts and washers Construction Grouting The soil Specification for foundation. CHAPTER XXXV DESIGN METHODS ...................... 687 Personal efficiency Order Thoroughness Study The slide rule Stand- ards Partial-assembly sketches. APPENDIX 1 MOMENT OP INERTIA OF IRREGULAR SECTIONS . , . . 693 APPENDIX 2 UNITED STATES STANDARD BOLTS AND NUTS 695 INDEX. . 697 DESIGN AND CONSTRUCTION OF HEAT ENGINES PART I-THE HEAT ENGINE CHAPTER I STATUS OF THE HEAT ENGINE It is roughly estimated that 85 per cent, of the power utilized in the United States for manufacturing plants, central power plants and elec- tric-railway power stations is furnished by heat engines, the remaining 15 per cent, being water power. This does not include locomotives, steamships or any form of self-propelled vehicles, which would increase the heat-engine percentage still more. It has also been estimated that the total available water power in the United States approximates the present output of steam power. How- ever, by the time this power is developed it is probable that the demand will have so increased that the power developed by the heat engine in its varied forms will be as greatly in excess as at the present time. The importance of heat engines is then obvious and their manufacture will continue to be an important industry until our fuel supply is exhausted or some radical discovery displaces them. Heat engines may be divided into three important classes which are commercially successful today, viz.: 1. The steam engine; 2, the steam turbine; and 3, the internal-combustion engine. The last named may properly be divided into two commercial forms: (a) The gas engine and (6) the oil engine, the former using gaseous, the latter liquid fuel. The gas turbine is not yet recognized in the commercial field and will not be further mentioned. Heat engines may also be classified as: 1. Prime movers; and 2, reversed heat engines, such as the compression refrigerating machine and the air compressor. Prime movers only will be considered. The relative importance of the different classes is an open question 1 2 DESIGN AND COti&PtfUCTlON OF HEAT ENGINES but it is probable that each has its particular field of usefulness which in some cases cannot be as well provided for by the others. Statements were current in the technical press a few years ago that little was to be expected in the way of improved economy for the recip- rocating steam engine, and its passing was predicted. That the limit of improvement had been nearly reached may, at that time have been true across the sea, but since then in this country, the application of superheated steam, the uniflow principle and the locomobile, together with improvements in materials and some details of construction which have made possible higher piston and rotative speeds in the larger units, have greatly improved the steam engine, so that it is holding its own in all-around efficiency and reliability with other forms. The steam loco- motive, that once proverbially wasteful machine, now equals in economic performance some of the best stationary steam plants of a few years ago. But the reciprocating engine has limits in the size of its units, and beyond this the steam turbine is best adapted. The turbine, with its high rotative speed and uniform rotation is well fitted to drive alternating- current generators of large capacity. The floor space is also less than for a reciprocating engine of the same power. The internal-combustion engine has attained the highest efficiency of all heat engines, and this is probably responsible for the prediction that the days of the steam engine were numbered. Notwithstanding all this, the internal-combustion engine furnishes at the present time but a small percentage of the output of stationary power plants, but there seems little doubt but that this ratio will be increased. It is a reciprocat- ing engine, and in this particular may be classed with the steam engine. Power, in the issue of Nov. 17, 1914, contains the statement that the total power of automobile engines manufactured the previous year was equivalent to twice the potential power of Niagara Falls, or 13,500,000 horsepower. This, coupled with the fact that, notwithstanding the increasing demands for large turbine units, the reciprocating engine is still largely in excess, makes it probable that the passing of the reciprocat- ing engine is not imminent. It is not intended to compare the merits of the different types of prime movers, or the different designs of a given type, as practically all when well designed, constructed and operated give good results in their respective fields. The determination of the best machine for a given set of conditions is often much involved and is outside the scope of this work. It is undoubtedly true that the periodic popularity of the various machines is not always based upon a profound knowledge of their actual merits. CHAPTER II THE POWER PLANT The Power Plant includes the prime mover and all other appliances and accessories necessary for the production of power, for manufacturing plants, central stations, ship propulsion, locomotives and automobiles, or for any form of activity requiring power. The development of the power plant has been such that the selection and arrangement of its various apparatus has become a highly specialized profession. The work of the power-plant designer and of the designers of the plant apparatus is inter- dependent, the latter making possible great advancement in plant efficiency, while on the other hand the demands of the plant designer have been an incentive to the designer of power apparatus. Power-plant design is not within the range of this book, and as the development of power apparatus is in a state of continual progress, de- scriptions of different designs will not be attempted, the task of keeping pace with the substantial progress in heat-engine design being thought sufficient; however, the steam cycle is not completed in the engine or turbine, and in view of future references to the different phases of the cycle, the path of the steam through the plant will be traced. Steam is generated in the boiler by the combustion of fuel in the boiler furnace. From thence it is conducted through piping to the turbine or engine where it does work. It is then exhausted, sometimes into the atmosphere, which is very wasteful; sometimes into pipes for heating purposes or for some industrial process, and it is often exhausted into a condenser. There it comes into contact directly, as in the jet condenser, or indirectly, as in the surface condenser, with cold water. The heat is withdrawn from the steam and it condenses back to water, and due to the great difference between the volume of a given weight of water and of steam, a partial vacuum is formed, often approaching within three-quarters of a pound of a perfect vacuum. With either type of condenser the water is pumped out, together with any air which entered with the steam or water. Water must be pumped into the boiler, either by the boiler feed pump or the injector, to replace the water used by the engine, and this completes the cycle. In most large plants the water is passed through a feed-water heater before going to the boiler. This is heated by the 3 4 DESIGN AND CONSTRUCTION OF HEAT ENGINES exhaust from the auxiliaries or main engine, or sometimes by the hot gases of the flue leading from the furnace to the chimney; in the latter case it is called an economizer by common usage. If a surface condenser is used, the condensed steam may be pumped back into the boiler, and barring leakage losses, a nearly continuous cycle with the same water may be effected, approaching the theoretical cycle. With the internal-combustion engine the thermal cycle is complete in the engine cylinder. There are, however, numerous accessories. For the gas engine there must be the gas producer, and the scrubbers for cleaning the gas; with the oil engine, apparatus for handling the oil must be provided; air compressors are also necessary for some types and vapor- izers or carbureters for others. These may be considered as part of the power-plant equipment, but are not part of the engine proper, although in many cases they are furnished by the engine builder. CHAPTER III THE STEAM ENGINE 1. The Steam Engine. The history of the steam engine is full of interest, and should form a part of the reading of every steam engineer. In the popular mind the profession of mechanical engineering is closely associated with the steam engine, and in fact owes much to it as an impor- tant factor in the separation of this branch of science from the general field of engineering, or civil engineering, which formerly included the several now distinct branches. In order to reserve space for present-day practice, it was thought best to refer to the works of Rankine, Thurston, Ewing and others, and to eliminate from this text practically all historical matter. 2. Mechanism. With the exception of comparatively small pumps, all practical reciprocating engines employ for the conversion of heat energy into useful work, the simple slider-crank mechanism. This mech- anism also resolves reciprocating into rotary motion, or vice versa in the case of reversed heat engines such as compressors. For engine purposes the slider-crank mechanism is the simplest and most economical from the standpoint of both construction and operation, and although attempts have been made to improve upon it, they have been fruitless. Neglecting friction, which in some cases is as low as 4 per cent, of the entire power developed in the cylinder, the efficiency is 100 per cent., as proven in Par. 100, Chap. XVI. A diagram of the slider-crank mechanism as applied to the steam engine is shown in Fig. 1. The mechanism proper consists of the frame (/), the crank (e), the connecting rod (d) and the crosshead (c), sliding upon the guides which restrict its movement to a straight line. The piston (a) and piston rod (6) may be considered as an extension to the crosshead and serve to transmit the pressure of the steam to it. Ports for the admission and outlet of steam are shown at 1, 2, 3 and 4. In some engines the same ports are used for inlet and outlet. The con- trol of the steam passing into and from the cylinder is effected by valves which are actuated by an eccentric, a form of crank, which is located on the engine shaft. In Fig. 1, valves 1 and 2 are inlet 'valves, while 3 and 4 are outlet or exhaust valves. To further illustrate the action of the steam engine as a whole, Figs. 5 DESIGN AND CONSTRUCTION OF HEAT ENGINES 2 to 9 are given. Contrary to the usual practice of introducing valve action with a, single-valve engine, it is thought that less confusion will accompany the conception of a separate valve controlling each operation concerned with the handling of steam during the engine cycle, which FIG. 1 with the steam engine is effected in one revolution of the crank; conse- quently, a four-valve engine of the nonreleasing type is shown. Figure 2 is the side of the engine showing the crank and connecting rod. Figure 3 shows the valve-gear side with the flywheel removed. 1? Qb ^y '^~ O J FIG. 2. Figure 4 shows the valve-gear side with the cylinder in section, the rods and levers operating the gear being shown by dotted lines, as are also the crank and connecting rod. Figures 5, and 7 to 9 show valves, with center lines only of wrist FIG. 3. plate arms, levers, crank and eccentric, in order to show the different positions more clearly. This forms a valve diagram, treated more fully in Chap. XX. In practice, with this type of gear, the eccentric changes position in relation to the crank, under the control of the governor located on the THE STEAM ENGINE shaft, as the load on the engine Is varied, but for simplicity, this is neg- lected in the present treatment, and the eccentric is assumed in a fixed position on the shaft. All of these details, and the exact action of the valves, are fully treated in Chap. XX. In Fig. 3, (a) is the eccentric, which by means of the eccentric strap (6) and eccentric rod (c) connects with the rocker or carrier (d). The reach rod (e) connects to the wrist plate (/) . Links ( 00 CO l> 00 TH T^H CO 00 O (N O iO iO r-l r-l CO CO j j j j ^i t> t>- t> 00 00 t T 7 ? CO CO CO CO rt< CO CO CO T}H CO CO CO CO CO \N \eq \eq r-K i-K r-i\ COX W\ \oo \oo r- COX W\ TjH TjH 1C IO O CO CO rH rH rH CO CO r-K ^ O rH (N O5 O5 rH rH rH SO O IO IO O O l> t^ CO CO '-'O C^l C 1 ! THE INTERNAL-COMBUSTION ENGINE 45 Indicator diagrams from a Type "FH" engine is shown in Fig. 36. The fuel consumption of a 20 by 34^ in. DeLa Vergne Type "FH" engine is given in Fig. 37. This engine is rated at 140 b.h.p. at 165r.p.m. The Mclntosh and Seymour Diesel Type Oil Engines are built by the Full Load FIG. 36. Indicator diagrams from a De La Vergne type "FH" engine. Mclntosh and Seymour Corporation, Auburn, N. Y. They are typical vertical, 4-cycle Diesel engines having the fuel injected under air pressure. They are built in two styles, Type A having four cylinders and individual frames for each cylinder, bolted to a common base; and Type B having from one to six cylinders bolted to a single box frame. Both types have a multi-stage air compresser' for fuel- injection and starting air, driven di- rectly by a crank on the end of the main shaft. The speed of the Type A engines ranges from 135 to 170 r.p.m.; and of the Type B, from 120 to 280 r.p.m. A test was published of a Type A engine having a cylinder diameter of 18% in. and a stroke of 28% in., and designed to develop 500 b.h.p. at 164 r.p.m. Fig. 38 shows an indicator diagram taken from this engine, and Fig. 39 a curve of fuel consumption. The Busch-Sulzer Bros. Diesel engine is a typical vertical Diesel engine Lb. Fuel per B.H.P.-HI O ro i O> c > s^ ^ * *^. ?0 40 60 80 100 120 140 160 B.H.P. FIG. 37. Fuel consumption of a De La Vergne type "FH" engine. 46 DESIGN AND CONSTRUCTION OF HEAT ENGINES and is built by the Busch-Sulzer Bros. Diesel Engine Co. of St. Louis. One working cylinder and the two-stage air compressor are shown in sec- tion in Fig. 40, and an end section through the cylinder in Fig. 41. These drawings are to scale and a good idea of general proportions may be obtained from them. Continental Motors are built by the Continental Motors Corporation, Detroit, Mich. This company, which makes motors only, has several types, suitable for automobiles and trucks. They are all water-cooled 4-cycle engines. Model 7W motor has a bore of 3 34 in. and a stroke of 4^ in. Fig. 42 gives a capacity curve at different speeds, showing that the maximum of 42 b.h.p. is at 2140 r.p.m. R.P.M./6S FIG. 38. Indicator diagram from a Mclntosh & Seymour Diesel engine. _b. Fuel per B.H.P.-HF O fvj 4* 0> o X ^ 100 200 300 400 B.H.P. 500 600 FIG. 39. Fuel consumption of Mc- lntosh & Seymour Diesel engine. Some of the most interesting data for this engine are given below: Ratio of clearance to total cylinder volume ....................... . 225 Firing order ....... .'..... ............................... 1-5-3-6-2-4 Total weight ........ < .......................... , ............ 575. Ib. Flywheel, 16 in. diameter ......... '....' ........................ 62. Ib. Compression pressure-gage .................................. 61 . 5 Ib. Crank shaft: 3 bearing 2^ in. diam. Bushings bronze, babbitt lined. Front bearing 2% 2 by 2%e in.. Center bearing 2 Y by 3 in. Rear bearing 2J^ 2 by 3^2 in. Cam shaft: 3 bearing lin. diam. Front bearings 1% by 2*^2 in. Center bearing 1 l K e by 2% in. Rear bearing 1 l K 6 by 2 > in. Connecting rod 8% (3.78 cranks) long. Crank-end bearing 1 2 %2 by 2 in. Piston pin bearing % by 1^ in. Piston 3)^ in. long. Three rings, %6 in. wide. Model E7 is a four-cylinder truck engine with two cylinders cast en bloc and an aluminum crank case. The cylinders are 4% in. diameter with a S^-in. stroke. Fig. 43 is a capacity curve for a Model E7 engine. TILE INTERNAL-COMBUSTION ENGINE 47 Fio. 40. Busch-Sulzer Bros. Diesel engine. 48 DESIGN AND CONSTRUCTION OF HEAT ENGINES FIQ. 41. Busch-Sulzer Bros. Diesel Engine. THE INTERNAL-COMBUSTION ENGINE 49 The maximum power is 44 b.h.p. at 1320 r.p.m. Some of the data for Model E7 is as follows : Ratio of clearance volume to total cylinder volume . 239 Compression pressure-gage 55 Ib. Firing order 1-3-4-2 Total weight 660 Ib. Flywheel, 17M in- diam., weight 105 Ib. 40 30 20 10 400 800 1200 1600 2000 2400 R.P.M. FIG. 42. Power curve for model 7W Continental engine. 9V 40 0.' a: 30 CO ZO I0> ^ + ~ pN / / / > / / / / / 400 600 800 JOOO '1200 1400 I60G R.P.M. FIG. 43. Power curve for model E7 Continental engine. Crank shaft 3 bearing 2*4 in. diam. Bushings bronze, babbitted. Front bearing 2K 6 by 3 in. Center bearing 2% 2 by 3% in. Rear bearing 2 y by 41 % 6 in. Cam shaft 3 bearing 1% in. diam. Front bearing 2> by 2{ 6 in. Center bearing in. Rear bearing 2^ by 1 % in. 50 DESIGN AND CONSTRUCTION OF HEAT ENGINES Connecting rod 11 in. (4 cranks) long. Crank end bearing 2} by 3 in. Piston end bearing 1 J{ 6 by 2^ in. Piston, 5^ long. Three rings, KG wide. References The Eight-cylinder Automobile Engine. The Gas Engine, March and April, 1915. Aeroplane Motors Used by the United States Naval Aeronautic Corps. The Gas Engine, Aug., 1915. Diesel Engines Applied to Marine Purposes. The Gas Engine, Jan., 1916. Twelve-cylinder Engines. The Gas Engine, Jan., 1916. The Problem of Airplane Engines. The Gas Engine, June, 1917. The Design of Motor Truck Engines for Long Life. The Gas Engine, June, 1917. Practical Operation of Gas Engines Using Blast-furnace Gas as Fuel. Trans. A.S.M.E.,vol. 35, p. 151. Heavy Oil Engines. Jour. A.S.M.E., Oct., 1916. An American Racing Engine. The Gas Engine, Nov., 1916. PART II-THERMQDYNAMICS CHAPTER VI GENERAL POWER FORMULAS AND GASES 23. Power of Heat Engines. Heat-engine cycles are usually complete on one side of the piston, and if the pressures acting on that side as the piston travels back and forth are plotted, a diagram is produced which is a measure of the work done during the cycle. This diagram is traced for actual engines by an instrument called an indicator, and all such dia- grams, either actual or for theoretical cycles such as the Carnot cycle, are called indicator diagrams, or perhaps more commonly, indicator cards. If the cylinder is double-acting, two cycles, on opposite sides of the piston, are accomplished at the same time, although corresponding strokes are not simultaneous. Should the cycle depend, for the performance of different functions, upon the two sides of the piston, or upon a main and auxiliary piston (as in 2-cycle gas engines), indicator diagrams must be obtained from these separately and the net work computed. However, in such a case the real work of the cycle is done on one side of the piston and it is this side which will be considered as the working cylinder-end. In practice the indicator diagram is mostly used to determine the mean effective pressure (m.e.p.), which, when engine dimensions, speed and the cycle on which the engine operates are known, may be used for the determination of power, or conversely, the determination of cylinder dimensions necessary for a given power. In all practical engines, all strokes are equal, so let v 8 denote the volume swept through by the piston during a complete single stroke, in cu. ft. Let pi, pz, Pz and p be mean pressures in Ib. per sq. ft. acting during four consecutive strokes of a 4-stroke-cycle engine, on one side of the piston. On all strokes toward the cylinder-end considered, the work- ing substance resists the movement of the piston and the work is negative, The pressure in the working cylinder-end may be considered as abso- lute. The pressure on the opposite side of the piston may be ignored as it does not affect the indicated work of the cycle, 51 52 DESIGN AND CONSTRUCTION OF HEAT ENGINES The work of the cycle, in ft. lb., for one cylinder-end is: W = piv s - p 2 v a + p 3 v s - p 4 v s = (pi pz + PS- Pt)v s = 144P M v s (1) where P M is the m.e.p. in lb. per sq. in., of the whole cycle, referred to one stroke. Any other cycle would give the same result. Then: This m.e.p., exerted throughout one stroke of the piston, would do the work of the cycle for the working substance on one side of the piston. The indicator diagram is drawn with the length representing the length of stroke; then the area divided by the length gives the m.e.p. when dimensions are in terms of pressure and volume, and in (2), TF/144 indicates that pressure is in lb. per sq. in. The work in terms of heat units is: w = & - A where Qi and Q 2 are heat quantities received and rejected per cycle, respectively, and A is the reciprocal of Joule's equivalent (= Substituting this in (2) gives: v s v s = 54 pB-t.u. converted into work perl /^\ Leu. ft. of piston displacement J Formulas (2) and (3) are general expressions of m.e.p. for all heat engine cycles, theoretical or practical. Power is work done in a unit of time; then as 33,000 ft. lb. per minute is the unit of horsepower, the work done per minute divided by 33,000 gives the horsepower. Power determined by means of an indicator dia- gram, which is the power developed in the engine cylinder, is called indi- cated horsepower (i.h.p.). Let N c = the number of cycles per minute; then: ihp = J^ = 144P.jWVe () 33,000 33,000 With the usual slider-crank mechanism employed for reciprocating engines there are two strokes per revolution; then the number of strokes per min. is 2N, where N is the number of revolutions of the crank per minute (r.p.m.). If m denotes the number of strokes per cycle, then the number of cycles per min. per cylinder-end is: AT 2N NC = - m GENERAL POWER FORMULAS AND GASES 53 Then for one working cylinder-end (4) becomes: 2 X 144 P M v s N i.h.p. 33,000 m (5) The maximum difference in pressure on tne two sides of a piston during a cycle is called the maximum unbalanced pressure, and it is this which determines the required strength and consequent weight of the working parts of the engine. If this pressure is high relative to the m.e.p., the engine will be heavy for the work done, and the friction may more than offset the advantage of an economical heat cycle if this is carried too far, as will be presently shown. Brake Horsepower (b.h.p.) is the power delivered at the engine wheel, or net power, as measured by some form of friction brake or dynamometer. It is the difference between the i.h.p. and the friction horsepower (f.h. P; ). Figure 44 is a diagram of a simple form of friction brake. The band consists of blocks of wood held together by band iron, with adjustment at (F). The engine wheel rim sliding inside this band is resisted by friction. The product of this resistance in Ib. and the distance traveled by the rim in ft. per min. is the work in ft. Ib. per min. absorbed by the brake. This divided by 33,000 gives the b.h.p. It is usually inconvenient to measure the resistance right at the wheel rim, so a brake arm of length I is made long enough to rest upon a pedes- tal on the scale platform. If P is the effective force in Ib. after deducting the weight of pedestal and unbalanced weight of brake arm, I is the length of the arm in ft. and N the r.p.m., then: FlG 44 b.h.p. = 2irlNP INP 33,000 = ~ 5252 (6) The b.h.p. measured in this way may be applied by belting to machinery, by direct-connecting to electric generators or other machinery, or thiough gears, and is probably not an exact measure of the net work, as bearing friction must be affected differently in the various cases of -application. For electrical machinery the output is measured at the switchboard in kilowatts, which may be reduced to electrical horsepower (e.h.p.); or, e.h.p. = 1.34 X kw. 54 DESIGN AND CONSTRUCTION OF HEAT ENGINES This is the net power after deducting the friction of engine and generator. If the friction of the generator is known the b.h.p. may be found. In pumping engines pump horsepower is the indicated work done in the pump cylinder, the difference between this and the i.h.p. being the f.h.p. Mechanical efficiency is given by the ratio: _ i.h.p. f.h.p. _ b.h.p. ( . * M = i.h.p. = Lh^l The friction of an engine does not vary greatly for varying loads, so that if the i.h.p. is small relative to the size of the engine, e M is small. Then e M decreases as i.h.p. (which is directly proportional to P M at constant speed) decreases, the limit being zero when the i.h.p. is just sufficient to run the engine (see Chap. X). A diagram of the Carnot cycle for air is plotted to scale in Fig. 45. The m.e.p. of such a diagram is very small when compared with the max- imum pressure, and even though Carnot's cycle were practically pos- sible, an engine employing it would probably be able to do little more than overcome its own friction if gas were the working substance. This would F IG . 45. n t be true of steam, the m.e.p. in one case being about four times that for air when the same pressure and temperature limits were assumed. Neglecting friction, if a cylinder 20 in. in diameter were used on a steam engine, the air engine with the same mean piston speed, developing the same power would require a cylinder about 40 in. in diameter. It is obvious therefore that the utility of Carnot's cycle for gas is limited to what it contributes to the science of thermodynamics. 24. Practical Cycles for Engines Using Gas. In practical efficient engines using gas (air and fuel gas or vapor) as a working substance, the combustion of fuel takes place in the engine cylinder. This necessitates a fresh supply of air for every cycle, with a consequent exhausting of the burnt gases. This may be accomplished with a 2-stroke cycle by the use of auxiliary cylinders; or suction and exhaust strokes may be added, mak- ing a 4-stroke cycle. The constant-volume and constant-pressure cycles are the two cycles now employed for internal-combustion engines, and while the theoretical efficiencies are greatly in excess of those actually attainable, due to the impracticability of utilizing such high temperature, the practical GENERAL POWER FORMULAS AND GASES 55 results are satisfactory when compared with the use of steam as a working substance. Both of these cycles were originally intended for 4-stroke cycles, but have been modified to operate with two strokes. In their theoretical consideration this makes no difference, and heat transfer to and from a given weight of gas may be assumed. 25. The Constant-volume Cycle. Figure 46 is the indicator diagram for the theoretical, 4-stroke, constant-volume cycle, commonly called the Otto cycle. Let it be assumed that the clearance space v\ contains 1 Ib. of air at atmospheric pressure. In practice this is usually a mixture of air and burnt gas, mostly the latter unless some special device is employed for clearing out all the products of combustion, which is known as scavenging. While there are but four piston strokes per cycle, there are six dis- tinct steps as follows: 1. Starting from the position shown by the full lines in Fig. 46, with the inlet valve open and exhaust valve closed as in Fig. 29-A, Chap. V, the piston moves through the distance v s representing the volume of stroke, to the position given by the dotted lines (Fig. 46). This is the suction stroke, during which the charge of fuel and air (necessary for its combustion) is taken into the cylinder. In some engines using the heavy oils with the constant-volume cycle, only air is admitted to the cylinder proper during the suction stroke, the fuel being admitted into a hot bulb forming part of the combustion chamber, or sprayed against a hot plate or into a hot bulb at the end of the compression stroke. With medium compression (280 to 300 Ib. per sq. in. gage), the latter are known as semi-Diesel engines. The suction stroke is simply a mechanical step and plays no direct part in the thermodynamics FIG. 46. Constant-volume cycle. of the cycle. It affects the power by using time that might have been employed for work. The pressure during the suction stroke is less than that of the atmosphere and any loss resulting is charged to engine friction. This pressure difference will be neglected in the discussion of the theoretical cycle. 2. The charge is next compressed adiabatically along line 2-3 during the compression stroke while both inlet and exhaust valves are closed, as in Fig. 29-B, Chap. V. 56 DESIGN AND CONSTRUCTION OF HEAT ENGINES 3. Ignition is now effected, usually by an electric spark except in oil engines using the hot bulb or plate. The addition of heat at constant volume along line 3-4 is assumed, with rise in temperature and pressure. The heat added is: Q! = c v (T* - T$ (8) in which T is absolute temperature in degrees F., and c v the specific heat at constant volume. 4. Adiabatic expansion occurs during the working stroke along line 4-5, both valves remaining closed as in Fig. 29-C, Chap. V. 5. In practical engines the exhaust valve opens at or near the end of the expansion stroke, and the drop in pressure along line 5-2 is accompa- nied by expansion, the temperature T% being indefinite. In the theoret- ical cycle it is assumed that the valves remain closed and that heat is withdrawn along line 5-2, the resulting pressure and temperature being the same as at the end of the suction stroke. This completes the thermal cycle. The heat rejected during the step is: Q,=c.(r,-r,) 0) 6. The exhaust stroke 2-1 is one of the two preparatory steps for the next thermal cycle, removing the burnt gas. The exhausjb valve is open during this stroke as in Fig. 29-D, Chap. V, and the pressure in actual engines is slightly above atmospheric. The loss is charged to engine friction as with the suction stroke, and the pressure difference is neglected for the theoretical cycle. This completes the practical cycle. The efficiency of the constant-volume cycle is : _ Qi-Qa _ c v (T, - T,) - c v (T, - T,) T^- T 2 ~ Taking exponent n = k for adiabatic changes (in which k = c p /c v , c p being specific heat at constant pressure) : or, T 5 T* T, - T T, T, T, - T Then letting r Vz/vi and remembering that: (10) may be written: e GENERAL POWER FORMULAS AND GASES 57 FIG. 47. It is apparent from (11) that efficiency is improved by increasing the compression pressure, which is true in practice with certain limitations (see Chap. IX). The expansion and compression lines of actual indicator diagrams are not adiabatic, but are approximated by the equation: pv n = constant; where n is some value usually less than k. However, with any value other than k ( = c p /c v ), Equation (11) does not apply; because the values of n which are found approximately from actual diagrams are greater or less than k, indicating that heat is added or subtracted as the work- ing substance changes volume. These quantities of heat must be added to or subtracted from Qi and Q 2 in (10). This results in a clumsy formula with assumed values of n at best, and it is better to use (11) for compari- son with actual efficiencies. Practical values of n must be used in deter- mining the clearance volume of actual engine cylinders. Constant-volume heat transfer involves instantaneous combustion and exhaust; this is impossible in practice, causing the explosion line to lean. This, together with the fact that the valves do not open and close exactly at the ends of the stroke, causes rounded corners on the diagram. The theoretical maximum temperature is not per- missible due to the burning and warping of the metal, and the impossibility of lubrication, so approximately 30 per cent, of the heat of combustion must be removed by the circulation of water around the cylinder. This lowers the pressure and reduces the area of the diagram. An actual indicator diagram from an engine operating on the 4-stroke Otto cycle is shown in Fig. 47. A critical examination of the Otto cycle is given in Giildner's Internal-combustion Engines. The T (temperature-entropy) diagram for the Otto cycle is shown in Fig. 48, the numerals corresponding to the same points on the pv diagram. The reception of heat at constant volume is along the line 3-4 and adiabatic expansion along 4-5. Rejection of heat at constant volume is along line 5-2 and adiabatic com- pression along 2-3. From general thermodynamics, line 3-4 is drawn by means of the equation: FIG. 48. 58 DESIGN AND CONSTRUCTION OF HEAT ENGINES T x A0 = C v \0g e TfT i 3 and line 2-5 from: A< = c v log c T< where T x is any temperature above jP 3 or TV 26. The Constant -pressure Cycle. The Diesel cycle is the best known and most widely used of this class. The theoretical difference between this cycle and the Otto cycle is the combustion of fuel at constant pres- sure instead of at constant volume. Diesel's original purpose was to approach the Carnot cycle by applying isother- mal combustion, but practical results with the later engines employing constant-pressure combustion show improvement over the former method, and the cycle is now generally known as the constant-pressure cycle. The theoretical indicator diagram for the Diesel 4-stroke cycle is shown in Fig. 49. FIG. 49. -constant-pressure The suct i O n and exhaust strokes, like those of the Otto cycle, are purely mechanical, and the operation of -the inlet and exhaust valves are the same as for the Otto cycle. Some form of liquid fuel, like crude petroleum, has thus far been the only fuel used in the Diesel engine. The six steps of the cycle are: 1. The suction stroke, line 1-2, during which fresh air only is drawn into the cylinder. 2. The compression stroke, along line 2-3, compresses the air adiabatically to a pressure of about 500 Ib. per sq. in. gage, and a tem- perature high enough to positively ignite the fuel, although none has been supplied up to this time. 3. The fuel valve is now opened (slightly before the end of the stroke), and as the piston starts upon its stroke, oil is injected in the form of spray at a rate which allows the pressure to remain practically constant as the fuel burns along line 3-^. Ignition is effected by the heat of com- pression and no electric spark is required. The amount of fuel in- jected is controlled by the governor and depends upon the load on the engine. The air used for fuel injection is furnished at a pressure from 50 to 100 per cent, in excess of the engine compression pressure, by either an independent compressor or by a compressor connected to the engine. GENERAL POWER FORMULAS AND GASES 59 Sometimes the fuel is pumped directly into the cylinder without the use of compressed air. The heat received is: Q l = c p (T t - Tj (12) 4. The remainder of the working stroke is occupied with adiabatic expansion along line 4-5. 5. The rejection of heat is assumed at constant volume along line 5-2, completing the thermal cycle. The heat rejected is: Q 2 = c v (T, - T 2 ) (13) 6. The exhaust stroke removes the burned charge from the cylinder, completing the practical cycle. The efficiency of the Diesel cycle is: = Qi-2 And from Charles' law, Then: rji fjj * rji J. 2 J. 4 J. 3 Substituting in (15) and letting vz/vi = r, as for the constant-volume cycle, and v*/v\ = e, (15) becomes: e=l- ^ ~ ^ (16) This may be written: If the quantity in brackets is omitted (17) is the same as (11), and as this quantity is always greater than unity it is obvious that for the 60 DESIGN AND CONSTRUCTION OF HEAT ENGINES same value of r, the efficiency of the constant-volume cycle is greater than that of the Diesel. The greater value of r in the Diesel cycle, allow- able because of the absence of fuel during compression, accounts for the superior economy usually attained by the Diesel engine. An actual indicator diagram from a Diesel engine is shown in Fig. 50. Theoretical and actual efficiencies may be obtained for both the con- stant-volume and constant-pressure cycles in terms of heat quantities based upon fuel supplied, in which case: e = _ 2545 B.t.u. per horsepower per hr. The T diagram for the Diesel cycle is shown in Fig. 51. It is similar to Fig. 48 except that the reception of heat is at constant pressure along line 3-4, the equation being: A0 = c The rejection of heat along line 5-2 is at con- stant volume. As Tz/Ts is not equal to T 5 /Tt, the relation of the efficiency formula to the diagram is not ap- parent, as it is with the constant-volume cycle. 27. Volumetric Efficiency. The volumetric efficiency of an internal- combustion engine is the ratio of the volume of the charge (fresh air and fuel) v c at some standard pressure and temperature, to the volume of stroke v a . This is equivalent to the ratio of the actual charge weight w c to the weight, at some standard pressure and temperature, of a volume of the same substance equal to v,, the volume of stroke; let this weight be w a . The weight w cj at actual pressure and temperature will occupy the volume qv s , where q is a factor depending upon the relative volume of fresh charge to total cylinder contents. In theoretical computations it is usually assumed that q is unity. With poor valve design and setting q may be less than unity, and with exceptionally good valve design and setting it may be greater. , With complete scavenging, when all the burnt gas is expelled from the cylinder: FIG. 50. r 1 GENERAL POWER FORMULAS AND GASES 61 The general equation for gas is: pv = wRT in which R is a constant ( = 53.35 for air). Letting the subscript 2 refer to actual cylinder conditions, and absence of subscripts to a standard condition, we may then write: Pz Q^s i P^s Then: . = . v s w s * pT 2 The factor q is difficult of practical determination but is of considera- ble importance in connection with economy and capacity. Aside from affecting the volume of charge it affects the density by its influence upon TV, and any measure which may be taken to increase it by proper valve setting will increase both economy and capacity. 28. Temperature rise due to combustion must be known in order to determine theoretical efficiencies of internal-combustion-engine cycles. Let h = the heating value per cu. ft. of gaseous fuel or per Ib. of liq- uid fuel. For gas this must be at some standard pressure and temperature. a = cu. ft. of air supplied per cu. ft. of gaseous fuel or per Ib. of liquid fuel at standard pressure and temperature. i(Ci - hi). Then the total B.t.u. per horsepower-hour will be: w(Ci - h*) + wi(Ci - hi). The efficiency will be: = 2545 . } ~~ w(Ci - h 2 ) + Wl (d - h^ As steam jackets are not very commonly used, and their use is decreas- ing with the use of superheated steam, they may ordinarily be neglected. Then (3) becomes: / A\ " w(C 1 - A,) The water rate of Rankine's cycle for complete expansion is: 2545 ,.. Wr = ^ W 72 DESIGN AND CONSTRUCTION OF HEAT ENGINES And for incomplete expansion : - c, 5.4 The subscript c refers to terminal pressure. Formula (5) is useful in steam-turbine design. 40. I.H.P. or B.H.P. For an intelligent conception of thermal efficiency and economy the term horsepower must be defined. This is commonly .taken as i.h.p. or as b.h.p. Then (2) may be written: e = 2545 B.t.u. per i.h.p.-hr. or: 2545 B.t.u. per b.h.p.-hr. Formula (7) is commonly applied to steam engines, but (8), sometimes called the thermal efficiency at brake, or economic efficiency, is a more correct measure of economical performance for any heat engine. For engines of the same type, when the mechanical efficiency is -about the same for the usual or rated load, (7) gives a fair measure for comparison, but for widely varying types, or where the ratio of maximum to mean pressure differs greatly, the only true measure of economy must include the friction of the engine. There is no indicated horsepower for the steam turbine, although by assuming mechanical efficiency the so-called turbine horsepower is some- times calculated. For any type of prime mover driving electrical machinery the term electrical horsepower (e.h.p. = 1.34 X kilowatts) is used. This allows for the combined friction of engine and generator. Economy may then be expressed in B.t.u. per e.h.p.-hr. The most common method of expressing economy of steam engines is in Ib. of water per i.h.p.-, or per kw.-hr. This may be used for the com- parison of simple noncondensing engines using saturated steam, but is generally unsatisfactory and is being replaced. by the heat-unit method. To show the discrepancy of basing economy upon water rate alone, as- sume engine A working with 140 Ib. absolute pressure and exhausting to atmosphere with a steam consumption of 20 Ib. per i.h.p.-hr.; while engine B, with a steam pressure of 165 Ib. absolute and 2 Ib. absolute back pressure uses 12 Ib. Assuming initially dry saturated steam in both cases, the B.t.u. per h.p.-hr. for A is: 20 X(1188 - 180) = 20,160 ECONOMY 73 and for B: 12 X(1193 - 94) = 13,188. The ratio of steam consumption of A to that of B is : and of heat consumption: 20,160 _ 13,188 which is nearly 14 per cent. less. 41. The Heat Factor. It is obvious from the T$ diagram, that if the thermal efficiency of the Carnot cycle were to equal unity the tem- perature of exhaust would be zero absolute. There are natural limits which make this physically impossible, and even with the best of our heat engines cause the efficiency to appear pitiably small. As the greater part of this seeming deficiency is not attributable to faulty design or construction, it is fair to take as a standard of comparison a cycle which is as near perfection as these natural limits will allow. This standard is usually the Rankine cycle with complete expansion. The discarding of adiabatic compression is perhaps not necessitated by natural physical conditions, but there is sufficient practical difficulty attending its at- tempted application to warrant its rejection. The Rankine cycle with incomplete expansion is sometimes taken as a standard when the terminal pressure of the actual engine is known. This seems like a reasonable practice when it is remembered that both capacity and economy demand a certain amount of terminal drop; moreover, it is the true measure of cylinder efficiency. Denoting the efficiency of the Rankine cycle by e R , the ratio of effi- ciencies, variously known as the Rankine cycle ratio, efficiency ratio, and heat factor (the last term being used in this book), is: F = - = ** (9) e R w In applying (9) it is customary to take e as the indicated thermal efficiency, and in transferring an indicator diagram to the T$ diagram (see Berry's Temperature-entropy Diagrams) it is necessarily so assumed, but there is no reason in practical work why e should not be based upon net work performed, or b.h.p., for steam engines as well as for internal- combustion engines. Practical values of F range from 0.5 to 0.75. Cycle design consists in selecting values of initial pressure, superheat, vacuum and ratio of expansion which will give the highest cycle efficiency while remaining within commercially practical limits. In 74 DESIGN AND CONSTRUCTION OF HEAT ENGINES designing the engine, proper values of piston speed, rotative speed and cylinder dimensions must be selected and all details so carefully pro- portioned that both the indicated and brake efficiencies of the engine shall approach as nearly as possible the efficiency of the theoretical cycle. A high cycle efficiency combined with a high heat factor insures an economical engine, as the product of the two equals the thermal efficiency, which after all, though uncomplimentary, is the absolute measure of economy when based upon the brake horsepower. This statement will no doubt be challenged, as -it disregards interest on invest- ment, depreciation and other fixed charges. Exception must of course be made for temporary installations, and when the fuel is all refuse, as from wood-working machinery, there being often a surplus to dispose of. In this case the engine should be of the simplest type. However, in general, true economy must always tend toward the conservation of those natural resources which are in greatest danger of depletion; from this viewpoint, thermal efficiency will be the criterion of economy. In the efficiency of the Rankine cycle the feed-water temperature is assumed as that due to exhaust pressure. The cycle for a noncondensing plant in which the feed-water temperature is lower than that of the exhaust steam, should only be applied as the ideal cycle when the boiler plant is included, as the temperature of the feed water has no influence on the economy of the engine. Then, neglecting the steam jacket or reheating receiver, the heat factor is: For complete expansion: e 2545 f F = * = 5(C^C5 For incomplete expansion : 2545 , (Pc - The heat factor for complete expansion is especially useful in steam- turbine design, the steam consumption being reduced to turbine horse- power. From what has preceded it is clear that in reporting economy or effi- ciency it must be plainly stated whether or not the auxiliaries are included, and whether it is based upon i.h.p. or b.h.p. In comparing engines oper- ating under different conditions, as condensing with noncondensing, or those using superheat with those using saturated steam, the heat con- sumption of auxiliaries required for any given mode of operation should properly be included. This may be found in the same manner as for the main engine. ECONOMY 75 Should a portion of the exhaust or receiver steam be utilized for heating or industrial purposes, its total heat content may be deducted from that charged to engine and auxiliaries on the ground that this steam must otherwise have been furnished by the boiler. Should all of the exhaust steam be so utilized the thermal efficiency would then equal the heat factor. On the other hand it may be argued that the manufacturing plant is utilizing waste heat from the engine. At any rate, such an arrange- ment fosters economy and is often a deciding factor in favor of the steam engine. Should all of the exhaust steam be required, a simple type of engine with small ratio of expansion could be employed. This would deliver a greater weight of dryer steam and possess numerous practical advantages if large overload capacity were not required. The heat factor based upon the Rankine cycle with incomplete expansion would probably be greater, and this is the measure of economy for an engine working under these conditions. Cylinder condensation and radiation losses should be reduced to a minimum, and the maximum amount of heat should pass out with the exhaust. Such an engine would be wasteful only upon condition that the exhaust steam were wasted. In some cases even the use of super- heated steam would be well-advised, should this provide the maximum heat content with a minimum of fuel consumption. Theoretically, the heat consumption of auxiliaries is an insignificant fraction of that required by the main engines, but may practically exert considerable influence upon plant economy. The use of uneconomical auxiliaries is never justified except in refuse-burning plants, even though all of their exhaust steam is required for heating feed water. With the expection of that which refers specifically to the steam engine, the preceding paragraphs of this chapter apply as well to the steam turbine, remembering that expansion should always be complete in the turbine. 42. Internal Combustion Engines. A common method of expressing the economy of gas- and oil engines is in cu. ft. of. gas or Ib. of oil per i.h.p.- or per b.h.p.-hr. For producer-gas engines Ib. of coal per h.p.-hr. is usually given in stating engine performance. This practice is useful only in comparing engines using the same fuel, but if the heating value of the fuel is known the B.t.u. per h.p.-hr. may be found. If the fuel is gas the heating value of the gas must be based upon the temperature of the gas as it passes through the meter, or both volume and heating value be reduced to some standard temperature. In the A.S.M.E. Code the higher heating value of the fuel is used as a 76 DESIGN AND CONSTRUCTION OF HEAT ENGINES basis for computing the heat charged to the engine, while the Vereines Deutscher Ingenieure employs the lower heating value. In view of this difference of opinion, even among American engineers, all test reports should state whether the high or low value is used. The use of the lower value may be justified upon similar ground that the heat factor of the steam engine is based upon the Rankine rather than the Carnot cycle, in that the lowering of the exhaust temperature to a point where the latent heat of the water vapor is available is a physical impossibility. On the other hand, as opinions are at variance concerning the proper deduction to be made in determining the lower heating value, greater uniformity may be obtained by the use of the higher value until there is a more general agreement in the determination of the lower value, which, after all, must be more or less arbitrary. The heating value being determined, the heat consumption will be: Lb. fuel per hr. X heating value per Ib. . . B.t.u. per h.p.-hr. = - (12) horsepower or: Cu. ft. fuel per hr. X heating value per cu. ft. , . B.t.u. per h.p.-hr. = - r- (13) horsepower Then, as with steam engines and turbines, the thermal efficiency is: 2545 ( . = B.t.u. per h.p.-hr. Either indicated- or brake horsepower may be taken in (12) to (14), b.h.p. being the more correct, as explained already. Both values are sometimes given in test reports. In this connection it must be remembered that indicator diagrams are not obtainable from engines with very high speed, and that a brake test of very large engines is not always feasible. 43. Efficiency Ratio of Internal-combustion Engines. Actual effi- ciency, either indicated- or brake- may be compared with the theoretical cycle efficiency. This is given by (11) and (17), Chap. VI, for constant- volume and constant-pressure cycles respectively. In these formulas it is usual to assume air as the working substance, with the corresponding value of the ratio of specific heats. The ratio for gas mixture theoreti- cally required for complete combustion is slightly different. The ratio of actual to theoretical efficiency may be called the heat factor as with steam cycles. It is not used for design as with steam turbines nor is it commonly given in reporting tests as with steam engines and turbines. This may in part be due to lack of uniformity in assuming conditions for the theoretical cycle, a difficulty not experienced with the ideal steam cycle. 44. Comparative Economy. Any extensive collection of test data, or ECONOMY 77 an elaborate discussion of the relative economy of the different heat engines is not within the scope of this book. While a great many tests have been recorded which include about every form of heat engine over a wide range of power, the results for any given class vary considerably among themselves, and the reduction of all conditions to an absolute standard by which the different classes may be compared with perfect fairness is nearly impossible. Best efficiencies differ to a considerable extent from the average in all types, and tables may easily be compiled exhibiting the excellence of a favorite type to the disparagement of its rival. It may be said for the steam engine and the steam turbine, that under practically the same working conditions for machines of the same grade and power, their maximum thermal efficiencies are the same, and the selection of one or the other must be governed by other conditions for which one may be better adapted than the other. The internal-combustion engine undoubtedly has a higher maximum thermal efficiency than steam machines. There is less difference in the efficiencies of large and small powers than in steam engines and turbines, giving the advantage to the internal-combustion engine for small and intermediate powers. Some data of heat-engine performance are given in Chaps. Ill to V. Economy under changing loads is also considered in Chaps. XII to XV. Fuel economy is not the only factor in the selection of an engine. In- terest on investment, rental, depreciation and other items having a purely financial bearing are sometimes opposed to the highest fuel economy and govern the selection, not only of the type, but the grade of engine within that type. This condition will probably continue until there is a more acute public conscience, or a new era of economic conditions forces upon us a more careful consideration of the fuel question. CHAPTER IX CYLINDER EFFICIENCY 45. Cylinder efficiency implies losses due in some manner to the cylin- der, and suggests practice as opposed to theory. But all practice has its theory even though it be unrecognized; or, due to its complexity we are unable to follow it through its various ramifications. All theory is not so simple, or so easily checked by experiment as that dealing with the effect of heat upon gases and vapors; and while the passage of heat through metal and other substances has been theorized and experimented upon, the effect of the metal walls of a cylinder upon a vapor or gas which is going through a certain cycle within it involving changes of pressure and temperature; and with the- duration of this cycle varying from two seconds to less than one-tenth of a second; is indeed difficult to analyze. Considering the conductivity of the working substance and of the metal of the cylinder, and the fact that the cylinder is jacketed with some heat-resisting material, steam or water; also the impossibility of in- stantaneous temperature measurements, and the general un-get-at- able-ness of the whole thing for reliable observations; about the best we can do is to cover the entire effect with the heat factor, diagram factor, or the thermal efficiency found from tests. It seems worth while, however, by the application of simple laws to various phases of engine construction and operation, to study the ques- tion of cylinder losses with a view to reducing them. 46. The Steam Engine Cylinder. A comparison of the actual with the theoretical steam consumption of a steam engine shows that much more steam enters the cylinder than is required to do the work, therefore some of it must be condensed in its passage through the cylinder. Initial Condensation. Scientific investigations, notably those of Him, of the behavior of saturated steam in an engine cylinder, have established the fact that nearly all of the condensation occurs as the steam enters the cylinder, the heat being withdrawn by the walls which are at a temperature much below that of the steam. This is known as initial condensation, and continues up to the point of cut-off. Condensation and Re-evaporation. This term really includes initial condensation, but it will be discussed here in its relation to the remainder 78 CYLINDER EFFICIENCY 70 of the cycle, beginning at cut-off. Condensation continues, partly due to the cooling effect of the cylinder walls, and partly to the conversion of heat into work which would occur if expansion were adiabatic. This double condensation causes the expansion curve to fall below the adia- batic for a ways as may be seen in Fig. 54. The transfer of heat since the beginning of the stroke has raised the temperature of the cylinder walls, so that the temperature of the steam as it expands eventually reaches a point below that of the walls. As heat must always flow from a higher to a lower temperature, the steam now begins to receive heat from the cyl- inder walls. This begins to check further condensation and in some cases will re-evaporate condensed steam to such an extent that the steam is dryer at the end of expansion than at cut-off. This has the effect of raising the expansion line above FIG. 54. the adiabatic as shown in Fig. 54. As the exhaust valve opens, the drop in pressure due to free expansion causes further drying. The cylinder walls continue to give up heat to the steam during the exhaust stroke until the closure of the exhaust valve. Compression and Clearance. The cylinder walls at this end of the cylinder are now at their minimum temperature, and the steam enclosed in the clearance space is usually assumed to be dry. Some experiments, however, indicate that clearance steam is not free from moisture at the beginning of compression; this is probably true of small engines, with which most of the compression experiments were made. Completing the stroke compresses the steam, raising its pressure and temperature. Rankine's cycle with clearance shows that if compression raises the pressure in a nonconducting cylinder, so that it equals the initial pressure, and if expansion is complete, clearance does not effect the steam consump- tion; but expansion is not complete in practice, and clearance influences the ratio of expansion considerably, the effect varying with changing cut-off. The effect of clearance and compression is also dependent upon other factors. The whole effect of condensation and re-evaporation so far has been to cool the cylinder walls, so that the incoming steam for the next cycle will give up part of its heat by condensation. Re-evaporation during the latter part of the stroke increases the work done slightly, but is equivalent 80 DESIGN AND CONSTRUCTION OF HEAT ENGINES to heat added at a temperature less than the maximum and is therefore inefficient. Assuming no compression, and that the exhaust valve closes at the end of the stroke enclosing dry steam in the clearance space, initial condensation will be due to the comparatively cool walls at the next admission of steam. It is probable that during engine operation, the inside surface of the cylinder walls never reaches a temperature as high as that of the incoming steam, or as low as that of the exhaust steam, so that with a small amount of compression the temperature of the steam in the clearance space will not be as high as the cylinder- wall temperature; there will then be no further loss of heat, and the clearance steam may be slightly superheated. This condition, with the saving of the clearance steam, is conducive to economy. 55 On the other hand, a higher com- pression may raise the temperature of the steam above that of the walls, and the surface of the walls being large in proportion to the weight of steam in contact with them will cause the steam to condense, lowering the pressure as shown in Fig. 55. It is well known that the presence of water will cause steam to condense much more rapidly than contact with metal of the same temperature; therefore if compression causes condensation, initial condensation for the next cycle will be increased. There has been much discussion about compression and some note- worthy experiments, the range of which, however, has not been comprehensive enough to formulate any rules governing the selection of compression pressures. Mr. Robert R. Fisher in Power, July 29th, 1913, gives the results of tests on a 10 by 30 in. simple, noncondensing Corliss engine, which showed that up to about 45 per cent, of the absolute initial pressure, compression was beneficial to economy. The effect of speed will be considered presently. It has been shown by certain recent experiments that the percentage of clearance volume has little effect on economy. The large clearance, usually considered so wasteful, holds a larger weight of steam propor- tionate to the cooling surface of the walls in a well-designed cylinder, tending to reduce initial condensation. An earlier compression is also necessary to get a given cushion effect; this encloses a greater weight of steam which better resists condensation as the compression temperature rises above that of the walls. The influence of clearance in reducing relative cooling surface is more noticeable at short cut-off, as may be seen CYLINDER EFFICIENCY 81 in Formula 1, Par. 47. The effect of clearance and compression and their relation to cut-off in connection with theoretical steam consumption is discussed in Chap. XII, Par. 62. The cycle just completed in a rather roundabout way was for one end of a cylinder and applies to a single-acting engine. Most steam engines are double-acting, the inlet and expansion stroke for one end being simul- taneous with the exhaust stroke for the other end, thus complicating things. From a consideration of the foregoing, it is obvious that the quantitative prediction of results by any mathematical theory however complex and apparently complete, is futile beyond any peradventure. Hirns analysis, and later the temperature-entropy diagram have been used to study the effect of cylinder walls on saturated steam, and these may be found in works on thermodynamics and the temperature- entropy diagram. 47. Factors Affecting Cylinder Condensation. These include practi- cally everything that must be considered in the design of an engine of a given power, and with few exceptions enter into the design of all steam engines. These exceptions involve special construction or some special application of heat with the express purpose of improving economy. Factors affecting cyl- inder condensation Common to all engines Speed. Size of cylinder. Design of cylinder. Range of pressure and temperature. Ratio oi" expansion. [ Steam jacket. Special < Compounding. { Superheated steam. Speed. Tests of a given engine run at different speeds have shown that better economy is obtained at the higher speeds. At high speed there is less time for heat interchange between the cylinder walls and the steam, and the minimum temperature is probably never so low. While speed is a factor which must usually be considered in the design of the engine, the economy of existing engines may sometimes be im- proved by increasing the speed if the design of the engine will permit of the resulting increase of stress, and the necessary reduction of the m.e.p. to maintain the same power does not necessitate too short a cut-off, the effect of which will be explained presently. If the engine is belted to a jack shaft, the pulley on this shaft must be increased in diameter in the same ratio that the engine speed is increased, to retain the same speed of jack shaft. Should the increased speed cause a greater velocity of the 82 DESIGN AND CONSTRUCTION OF HEAT ENGINES fly-wheel rim than is permissible about 1 mile per minute a smaller fly-wheel must be used, retaining the original jack-shaft pulley. The diameter of the new wheel must be such that the product of the diameter and r.p.m. will be the same as before the speed change. Speed change is sometimes limited by the area of ports or steam pas- sages, which if too small may cause excessive wire-drawing and offset any gain due to increase of speed. It is maintained by some engine builders that piston speed has a greater influence on economy than rotative speed, and it may be that the recent practice of higher piston speed is not all attributable to the demand for increased capacity. This will be further mentioned under cylinder design. Size of Cylinder. For cylinders of similar form the volume varies directly as the cube of the diameter, while the surface varies as the square of the diameter; this may be seen in Table 2. Therefore, there is less cooling surface per unit volume of steam in large cylinders than in small, and consequently less relative condensation in a given time. This in part accounts for the superior economy usually realized in large engines even though the rotative speed is low. The piston speed, however, is usually fairly high, which goes to support the suggestion in the preceding paragraph; there have also been exceptional records made by large engines when the piston speed was unusually low. For a given power requirement there is apt to be a conflict between high piston speed and the large unit idea, especially when the nature of the ser- vice fixes the number of units. A compromise will usually be effected, governed mostly by financial considerations. Design of Cylinder. This includes the clearance volume, ratio of stroke to diameter of cylinder, and the jacket or covering. With the exception of special designs, such as the uniflow engine, it is probably safe to state that the clearance volume should be kept as small as possible. This depends upon the clearance distance between piston and cylinder heads, the type of valves used and the general arrangement of steam chest and ports, which must all be carefully worked out on the drawing board. The ratio of stroke to cylinder diameter is probably more often deter- mined from general considerations of design and construction than from the standpoint of economy. A large ratio, or relatively long stroke gives a smaller relative amount of cooling surface for the same cut-off and cylin- der diameter, as may be seen from Table 2, and permits of a greater piston speed when the rotative speed is limited as in large pumping engines, or is fixed by direct connection to electrical machinery. It CYLINDER EFFICIENCY 83 also reduces the percentage of clearance volume for the same piston speed and diameter of cylinder, the design of the ports being the same for a given piston speed. For engines of the same power and rotative speed a long stroke makes possible a greater piston speed as just stated, but a smaller cylinder, the latter requiring engine parts of smaller cross-sectional area; and although the long stroke necessitates a longer engine, the smaller diameter of pins and bearings, and the lighter parts, will reduce the friction and in- crease the mechanical efficiency. The relation between power, cylinder diameter, rotative and piston speed, may be readily seen in the equations of Chap. XII, from which a bet- ter understanding may be obtained of the preceding discussion. From these equations it is apparent that for an engine of a given power, assum- ing the.m.e.p. to be the same in any case, a high piston speed either means a high rotative speed or a large ratio of stroke to cylinder diameter; or that both rotative speed and ratio may be greater than with a lower piston speed. Although the relative importance of these two factors may not be definitely stated, they are both conducive to economy and the advantage of higher piston speed may be thus explained. As the ratio of surface to volume of steam undoubtedly greatly in- fluences condensation, a clearer understanding of how this ratio is affected by cylinder diameter, length of stroke and cut-off will be of advantage, so a simple formula will be derived. Let: R = the ratio of inside cylinder surface in one end of a cylinder up to cut-off, to the volume enclosed. D = the diameter of the cylinder in in. q = the ratio of length of stroke to diameter of cylinder. I = ratio of the portion of stroke up to cut-off to the entire stroke. k = ratio of clearance at one end to volume "of stroke, neglecting counterbore and ports. Sarface = 2 - + vD(l + k)qD TrD 2 Volume = - (I + k)qD = surface 2(2q(l + k) + 1] " volume q(l + k)D Assuming a clearance of 4 per cent. Table 2 has been computed from (1), giving values of R for values of D, / and q over range enough to show the effect of each. The influence of piston speed upon the design of engines with a limited or fixed rotative speed is not apparent, as the greater piston speed neces- 84 DESIGN AND CONSTRUCTION OF HEAT ENGINES TABLE 2 Values of R D in. 9=1 Q = 2 l = K i = y* i-H l = K z = K J = K 10 1.610 1.090 0.772 1.010 0.746 0.586 20 0.805 0.544 0.386 0.505 0.373 0.293 30 0.536 0.363 0.257 0.336 0.248 0.195 40 0.402 0.272 0.193 0.252 0.186 0.146 sitates a smaller cylinder and a greater ratio of stroke to diameter, which have opposite effects upon the value of R. To better compare these effects examples will be given of two engines of different power assumed to have the same m.e.p. and cut-off, but each having several different piston speeds. The rotative speed will be fixed in each case and suited to the power of the engine. Cylinder diameters were calculated from Formula t (25), Chap. XII, from which the following notation was taken. P = m.e.p. N= r.p.m. S = piston speed in ft. per min. H = horsepower. L = length of stroke in in. I and q are used as in Formula (1), from which R is calculated. Engine No. 1. P = 70, 1 + k = Y, H = 200 and N = 200. S = 600, L = 18", D = 14J4", q = 1 .265, R = 0.725 S = 800, L = 24", D = 12^4", q = 1 .960, R = 0.660 S = 1000, L = 30", D = 11", q = 2.730, R as' 0.630 S = 1200, L = 36", D = 10", = 3.600, R = 0.622 Engine No. 2. P = 70, 1 - f k = 34 H -- = 1000 and N = 100. S = 600, L = 36", D - 31%", q = 1 .135, R = 0.348 S = 800, L = 48", D = 27J^", q = 1.745, R = 0.313 .S = 1000, L = 60", D - 24}^", q = 2.450, R S 0.297 S - 1200, L = 72", D = 22^", q = 3.200, R SB 0.289 It is obvious that the gain due to long stroke more than offsets the loss due to smaller cylinder as the piston speed becomes greater, giving a decrease in the ratio R. The gain per hundred feet of piston speed de- creases with the increase of piston speed, suggesting a practical limit for a given rotative speed, especially if it leads to objectionable construction, CYLINDER EFFICIENCY 85 A comparison of the values of R for engines 1 and 2 does not mean that the relative condensation per Ib. of steam used is indicated thereby, because this is counteracted in part by the higher rotative speed of No. 1. R is but one of the factors influencing condensation, and it may be that its effect is proportional to some power of this factor less than unity. The cross-sectional area of the engine parts is less for the smaller cylinder diameter as previously stated, assuming the maximum un- balanced pressure to be the same in all cases; and while this is offset by increased lengths of such parts as are affected by the stroke, making the weight of these parts practically the same in either case, such parts as the piston, crosshead, connecting rod ends, crank pin and shaft, are only affected by the cylinder diameter, resulting in a lighter engine as stated. Most engine cylinders are of cast iron, which is a good conductor of heat, therefore to reduce heat loss by conduction, radiation and convection, the cylinder must be covered with some heat-resisting material, called a nonconductor. Wcl was formerly used, but as higher pressures and temperatures were used the wood was replaced by asbestos or magnesia in the form of plaster, which is more easily applied and more satis- factory. This is covered with a jacket, or lagging, usually of sheet steel, which fulfils the purpose of a finish and also forms a dead air space which aids in retaining the heat. Steam jackets will be considered presently. Range of Pressure and Temperature. If the difference between initial and back pressure is great, there will be a correspondingly great tempera- ture difference between the cylinder walls and the steam at the moment of admission. As this is responsible for initial condensation, it follows that the greater this range of pressure the greater will be the condensa- tion. This will cause loss of thermal efficiency unless offset by some opposing factor. For example, increasing the steam pressure in a certain cylinder increases the condensation, but with a proper cut-off the gain in work done due to the increased pressure may be such that the work per given weight of steam is more than with the lower pressure. Like- wise the increase of pressure range by lowered back pressure due to con- necting a condenser may produce a similar result; in this case the economy is practically always better. Ratio of Expansion. The ratio of expansion depends upon the clear- ance and the cut-off. For an engine already built the clearance is fixed, and any change in the ratio of expansion is determined by the cut-off. For the Rankine cycle of maximum economy expansion is complete, the 86 DESIGN AND CONSTRUCTION OF HEAT ENGINES FIG. 56. terminal pressure equalling the back pressure. Lowered efficiency re- sults if the ratio is decreased or increased, the latter resulting in a loop in the diagram, giving negative work as in Fig. 56. Complete expansion would also give the highest efficiency in actual engines if it were not for cylinder condensation and friction. Neglecting the latter for the present, it is obvious from Table 2 that if the rate of condensation of a given weight of steam depends upon the surface to which it is exposed other things being equal the relative condensation is greater at short cut-off. Thus, a cut-off must be chosen practically always greater than will give complete expansion, and such that the sum of the loss due to incomplete expansion and that due to condensation is a minimum. This cannot be definitely predetermined, and for a given engine may only be found by test ; but it is usually from J to }, depending upon the various factors already discussed and to follow. Obviously this most economical cut-off must be such as to give the highest thermal efficiency at brake, thus providing for friction. It is also desirable that it correspond to the load to be carried by the engine most of the time, or to the rated horsepower, and this is usually aimed at in steam-engine design. Then by change of cut-off the engine can adapt itself to changes of load from zero brake horsepower to 50 per cent, over- load, and in some cases as high as 100 per cent., the economy decreasing as the load varies either side of the rated load, assuming that this was correctly chosen. A typical steam consumption diagram is shown in Fig. 57. The effect of initial ^ -i condensation shows in a marked degree at the smaller powers. Steam Jacket. A form of cylinder construction in which steam surrounds the cylinder barrel and heads is sometimes used to improve economy. The steam is usually supplied from the steam main which supplies the engine, the condensed steam being piped to a trap. The object of this steam jacket is to keep the cylinder walls at a higher tem- perature throughout the cycle and thus reduce initial condensation. The jacket steam replaces part of the heat carried out with the exhaust due to 01 *" 1_ I 15 3 I0 3 X ^ v ^> ""- _ , - "' 40 50 60 TO 80 90 100 IIC I.H.P. FIG. 57. CYLINDER EFFICIENCY 87 re-evaporation, and supplies that carried away by radiation, which would otherwise be taken from the working steam. While the consumption of steam in the cylinder is reduced by the application of the steam jacket, the jacket consumption partly offsets this result, so that the increase in economy is much less than would other- wise be possible. In some instances there is no gain whatever and some- times an actual loss, so that the steam supply to the jackets is discontinued. It is obvious that engines in which the design or working conditions tend to increase condensation are more greatly benefited by steam jackets, the gain in some instances being as high as 30 per cent. Thus, engines at light load with short cut-off show considerable gain, with less, or no gain at a more economical cut-off, and sometimes a loss at overloads. Comparatively few engines are equipped with steam jackets, their most usual application being to large pumping engines and marine engines of large power. Their diminishing use is probably partly due to the increasing use of superheated steam. Compounding. The general principle of compound engine operation is explained in Par. 3, Chap. III. After expansion is effected in the high- pressure cylinder, the steam is exhausted to a receiver, from whence it is admitted to a low-pressure cyl- inder and again expanded, ex- hausting finally to the atmos- phere or a condenser. In triple- or quadruple-expansion engines this operation is con- tinued through one or two more cylinders. The advantage of compounding may be explained with two-stage expansion and the following discussion will be so confined. Assuming a compound engine to operate on the Rankine cycle with complete expansion, and that for the sake of simplicity the receiver is indefinitely large, so that the transfer of steam to and from the receiver does not change the pressure, the pressure-volume diagram is shown in Fig. 58. Pursuant with current practice let the receiver pressure divide the diagram into two equal parts, equally dividing the work between the two cylinders. This may be done by dividing the difference between the heat content at pi and p 3 at the same entropy, by two, subtracting it from the heat content at pi and finding from an entropy chart or table the pressure corresponding to the resulting heat content. FIG. 58. 88 DESIGN AND CONSTRUCTION OF HEAT ENGINES For example, assume adiabatic expansion between 151 Ib. per sq. in. absolute to 14.7 Ib. absolute, at 1.52 entropy. From table or chart, Table 3 may be constructed. TABLE 3 Pressure Temp., F. Heat content Specific volunne Pl = 151.0 p 2 = 50.8 p, = 14.7 359 282 212 1153.3 1072.2 991.1 vi = 2.859 v, = 7.465 w 3 = 22.39 Thus for a simple engine operating between the same pressures, the temperature range from admission to exhaust is 147 degrees F., while for the compound it is 77 and 70 degrees in the high- and low-pressure cylinders respectively; or approximately one-half of what it would be in the single cylinder, thus reducing the initial condensation. Referring to Fig. 58, the ratio of expansion for a simple engine would be vi/vi, or 7.84. For the compound engine it is v*/vi, or 2.61 in the high- pressure cylinder, and v z /v 2 , or 3 in the low-pressure. For the simple diagram assumed, the cut-offs are the reciprocals of these numbers. Thus, while the total ratio of expansion is the same in both simple and compound engine, and theoretically the most economical, in the actual engine the longer cut-offs in the cylinders of the compound would result in less initial condensation than the shorter cut-off in the simple engine. For the Rankine cycle, assuming as it does a' nonconducting cylinder, there would be no advantage in compounding from a thermodynamic standpoint; but it serves to illustrate the effect of compounding upon temperature range and ratio of expansion, two very important factors influencing condensation. Clearance, compression, the receiver, and the fact that the expansion is not adiabatic, alter the values of Table 3 to some extent when applied to actual engines; these will be considered in Chap. XIII. Cylinder Ratio and Terminal Drop. These quantities are interdepend- ent when a nearly equal division of work is desired, and the question of their proper values to insure the best economy has been considerably discussed. A knowledge of the principles of Chap. XIII is necessary for an intelligent consideration of this subject, and a previous study of the same is assumed. Then it is apparent that within reasonable limits, the power of a compound engine working with a given ratio of expansion is mainly dependent upon the size of the low-pressure cylinder, the high-pressure cylinder, or the ratio of the volume of stroke of the low-pressure to that CYLINDER EFFICIENCY 89 of the high-pressure cylinder, having little influence. That this ratio does affect the terminal pressure in the high-pressure cylinder is plainly seen in Fig. 59, in which V 2 is the volume of stroke of the low-pressure and Vi of the high-pressure cylinder. Volume V\ is for a low cylinder ratio, or value of Vz/Vi, while TV, shown by the dotted lines, is for a high cylinder ratio. Terminal drop p is for the low ratio and p' the high ratio. As elsewhere explained, a certain amount of terminal drop is necessary for practical, economic engine operation, and this is provided for in the low-pressure cylinder by the assumption of a total ratio of expansion which is expected to give the maximum economy at the rated load. Although engines with high cylinder ratios have been built for a good many years, the lower ratios are in the majority, a very common value of the ratio being four. Though there have been high-economy records with both low and high ratios, advo- cates of the high ratio claim superior economy when working conditions are the same in both cases. In Fig. 59 it will be noticed that the total area is some less for the high ratio. On the other hand the cut-off, especially in the high-pressure cylinder, is longer. With a high terminal drop the steam will be dryer as it enters the receiver than with low drop, if the quality is the same at terminal pressure. For the purpose of comparison, let three engines be considered, each to have a rated indicated horsepower of 500, neglecting the diagram factor; a piston speed of 1000 ft. per min., and the same stroke and r.p.m. Also let the initial and back pressures be 140 and 2 Ib. absolute, clearance 4 per cent., compression 0.8 stroke in the low-pressure cylinder, and in the high-pressure cylinder such that the two compression curves lie on the same curve. Assume the curves of expansion and compression to be hyperbolas and that the total ratio of expansion is 30 in all cases. The cylinder ratios are 4, 5.55 and 7.56; cylinder diameters were computed by the formulas of Chap. XIII as was also the theoretical water rate. The temperature range was taken from steam tables for the FIG. 59. DESIGN AND CONSTRUCTION OF HEAT ENGINES pressures used in each case. The quality is given to show the condition the steam may be in at terminal pressure in the high-pressure cylinder to become dry due to free expansion by the time it reaches the receiver pressure. These values are given in Table 4. The cylinder ratios for engines 2 and 3 given in the table are not those used in the calculation, but are the result of rounding up the calculated diameter to even inches. The ratio R was found by Formula (1). TABLE 4 Cyl. diam. Cut-off Temp, range Ratio R Relative effect Cyl. Qual- Water ratio ity rate H.p., in. L.p., in. H.p. L.p. H.p. L.p. H.p. L.p. H.p. L.p. 4.00 16 32 0.0985 0.231 135 105 0.653 0.331 0.999 7.76 1.000 1.000 5.55 14.0 33 0.14170.233 136 104 0.591 0.329 0.994 7.78 1 . 024 1.025 7.56 12.0 33 0.1980 0.247 138 102 0.567 0.316 0.986 7.97 1.063 1.067 By a study of Table 4 and the preceding paragraphs of this chapter it is evident that the economy is affected inversely as the ratio R, the temperature range, the quality and the theoretical water rate. Assuming the product of a constant and the reciprocals of these quantities as unity for each cylinder of engine 1, their relative combined influence may be found and is given in the last two columns of Table 4. This indicates an average gain in economy of the highest over the lowest ratio of 6.5 per cent. The influence of the receiver and piping upon temperature range and condensation have been neglected, it being assumed the same in each case. Lack of comprehensive experimental data, and the complex nature of the problem make impossible the construction of a formula including these tabulated quantities which may be used to predict economy, but it seems reasonable to believe that they give the direction in which im- proved economy may be looked for. Superheated Steam. 'When superheated steam enters the cylinder of a steam engine, heat is transferred to the walls as with saturated steam, but instead of immediate condensation upon contact, which still further augments condensation, the superheat must first be withdrawn. Mean- while, the cylinder walls have increased in temperature, so that, should the steam reach the saturation point before cut-off occurs, the tendency to condensation is greatly reduced. If superheating is carried far enough condensation will not begin until after cut-off, and even be delayed until expansion is partly completed. Engineers differ as to the degree of superheat which may be used to advantage, some maintaining that CYLINDER EFFICIENCY 91 economy increases with the amount of superheat, while a number con- servatively place the limit at about 100 degrees F., having consideration of the damage to castings and the difficulty of lubrication attending the use of exceedingly high temperatures. Even a small amount of superheat is of great advantage, for though some condensation begins at the walls before the point of cut-off is reached, superheated steam is a much poorer conductor of heat than saturated steam, so that the bulk of the steam is but little influenced. The effect of superheat on economy is largely indirect, the actual gain often being twice that given by the Rankine cycle. In addition to the reduction of initial condensation, increased specific volume in part accounts for steam economy, a cubic foot of steam at 140 Ib. absolute pressure and 100 degrees superheat weighing but 86 per cent, of saturated steam at the same pressure. This is in part offset by the fact that the expansion curve falls below tha-t of saturated steam, necessitating a larger volume at cut-off to do the same work. Superheat probably affects several of the factors previously discussed, as it offsets the influence of initial condensation; however, the features of design best adapted to the economical use of saturated steam will prob- ably produce the best results with superheated steam. It seems probable, however, that due to the reduction of condensation at short cut-off, the most economical cut-off will be shorter, making it possible to carry a desired overload without the usual long cut-off and loss of expansive work. On the other hand, as just stated, the expansion curve for super- heated steam is said to drop more rapidly than that of saturated steam, necessitating a longer cut-off to obtain the same terminal pressure. This effect is perhaps more noticeable with high degrees of superheat; in fact, the author has failed to notice much difference in this respect in any super- heated-steam diagrams that have come to his notice, these having been mostly for a moderate degree of superheat. The range of pressure may be increased with superheat, making the condensing simple engine more desirable than with saturated steam. The tendency has been, however, where high pressure has been employed with saturated steam, as in locomotives, to reduce the pressure when superheat is employed. While there are engines specially built for superheated steam, it is advantageously applied to existing engines with various types of valves. This is especially true of locomotives, the addition of the superheater increasing capacity as well as economy. In the Journal of the A.S.M.E., Jan., 1916, Mr. Robt. Cramer points out the theoretical advantage of using steam of high pressure with mod- 92 DESIGN AND CONSTRUCTION OF HEAT ENGINES erate superheat, over lower pressure with greater superheat. He makes his comparison between pressures of 200 and 600 Ib. per square inch, limiting the temperature in both cases to 600 degrees F. This allows superheat of 218 and 113 degrees respectively. A common pressure for compound engines and steam turbines is 150 Ib. gage, and a common range of superheat is from 100 to 150 degrees. Using the nearest values without interpolation from Peabody's entropy table, comparison with a gage pressure of 250 Ib. is given in Table 5, assuming a vacuum of 28 in. of mercury and nearly equal maximum temperature. Efficiency is that of the Rankine cycle, given by Formula (8), Chap. VII. TABLE 5 Absolute pressure 164 800 264 300 Maximum, temperature . . .... 508 000 503 900 Superheat 142 000 97 900 Entropy 1 650 1 590 Efficiency 292 328 The higher pressure shows a gain of 12 per cent., and while this may not be practically realized, there probably would be a substantial increase in efficiency, as the superheat is sufficient to greatly reduce if not prevent all condensation. The gain is obviously due to the reception of a larger percentage of heat at high temperature. There is no practical difficulty in the use of 250 Ib. steam from the standpoint of either boiler or engine, and it is probable that much higher pressures may be practical. This question has been much discussed since the appearance of Mr. Cramer's paper. High pressure shows greater theoretical gain than high superheat, which may be readily understood by a study of the entropy diagram; it is probable, however, that where extremely high pressures are used, high superheat will be a desirable, if not a necessary accompaniment, especially in steam turbine operation, as otherwise, condensation will begin too early. High steam pressures are no novelty, 500 Ib. having been used on naval vessels and as high as 1000 Ib. in automobiles. A steam car has recently been developed to carry 600 Ib. pressure with no superheat. The advisability of carrying such pressures in stationary steam plants is problematical and has a number of noted authorities as its advocates. The question is probably a financial one. If gain in thermal efficiency and capacity offsets interest on investment and insurance, high pressures will be used, the matter of safety having no more bearing than with high- tension electric currents or munition factories. CYLINDER EFFICIENCY 93 While no definite rules have been given for obtaining maximum cylinder efficiency, a careful consideration of the foregoing principles, combined with a reasonable amount of practical experience and common sense, should aid the designer in his compromise of conditions to secure good working results. 48. The Internal-combustion Engine Cylinder. Gas- and oil-engine cycles are discussed in Chaps. V and VI. Due to the high maximum temperature and great temperature range, the theoretical efficiency is high compared to that of the steam engine and turbine. The cumula- tive effect of the high temperature upon the metal walls of the cylinder is such that the metal is heated to a point where it will not be abl to withstand the pressure if some means are not employed to withdraw part of the heat as fast as it is generated. This is most commonly accom- plished by the water jacket, but sometimes air cooling is used with small engines. Heat passes from the working gases through the cylinder walls into the cooling medium, and as the rate of heat flow depends upon tem- perature difference, more heat is withdrawn at combustion and during the early part of expansion than during the remainder of the cycle, and this lowers the maximum temperature. Unlike the steam engine, the temperature of the working fluid during exhaust never becomes as low as that of the cylinder walls. The next charge then receives from the walls a portion of this heat of combustion at much less than the maximum temperature. This addition of heat to the charge causes a proportionately higher temperature during the com- pression stroke, so that for a given compression pressure, the high tem- perature at the time of ignition causes a higher temperature along the combustion line with its correspondingly greater heat transfer. The heat added to the charge, by increasing its specific volume, reduces its weight; this decreases the capacity of the engine, increasing the relative friction, and the ratio of cooling surface to weight of charge also promot- ing heat transfer. It is commonly stated that the quantity of heat withdrawn from the engine cylinder should be the smallest possible amount consistent with practical operation; but while the theoretical efficiency of an internal- combustion engine is independent of the initial temperature of the charge, it is not probable that actual brake efficiency would be improved by entire absence of cooling, even though lubrication and the strength of the metal were not impaired. Cylinder dimensions, speed, etc., affect heat transfer to cylinder walls in the same way as in the steam engine, but the problem is an entirely different one. Prevention of heat transfer is the aim in the steam engine, 94 DESIGN AND CONSTRUCTION OF HEAT ENGINES while in the internal-combustion engine a compromise between thermal efficiency, capacity and lubrication must be effected. It is obvious that the cooling system should be flexible, so that it may be adjusted to give the best operating results. In certain engines the temperature of the cooling water is controlled by a thermostat which may be set at the proper temperature, which Js found by experience. As previously stated, cooling the cylinder has little influence upon economy within reasonable limits. If the heat is not taken out in the cooling water, combustion is suppressed by dissociation as extremely high temperature is reached and the heat is thrown out in the exhaust. As engines increase in size the ratio of cooling surface to volume of cylinder decreases, the natural result being a larger proportion of heat in the exhaust. This is shown by the first three engines in Table 6, taken from Peabody's Thermodynamics. The last engine used blast-furnace gas and was liberally cooled with water. The amount of cooling water required may be found as follows : TABLE 6 Distribution of heat Engine size, in. Work Jacket Exhaust 6.75 X 13.70 0.16 0.52 0.32 9.50 X 18.00 0.22 0.44 0.35 26.00 X 36.00 0.28 0.24 0.48 51.20 X 55.13 0.28 0.52 0.20 Let h = heating value per cubic feet of gaseous fuel or per pound of liquid fuel in B.t.u. w = cubic feet of gaseous fuel or pound of liquid fuel per b.h.p.-hr. W = pound of cooling water per b.h.p.-hr. G = gallons of cooling water per b.h.p.-hr. T R = Temperature rise in water in degrees F. q = fraction of total heat supply withdrawn by cooling water. e B = thermal. efficiency at brake. Equating the heat withdrawn with the heat taken by the water: qwh = WT R Also from (8), Chap. VIII: 2545 7 2545 e B = wh e B Substituting gives: 2545g e B CYLINDER EFFICIENCY 95 or: W = and = If q = 0.35, e B = 0.20 and T R = 70, then: W = 61 and G = 7.5. As a practical suggestion it has been stated that if jacket water is run through the jackets for some time after the engine is shut down, it will prevent the heating of the water remaining in the jacket to the point where salts held in solution will be precipitated, thus forming deposits on the walls, causing overheating. It will also prevent the evaporation of oil from the piston pin. Lack of agreement among authorities concerning combustion in the cylinders of internal-combustion engines, the properties of gases at high temperatures and the effect of this upon combustion makes the discussion of these subjects seem out of place in a book of this character. Much of value of this nature may be found in Giildner's Internal-combustion Engines, to which the reader is referred. Guldner suggests a greater ratio of stroke to diameter than has been customary, and greater piston speeds, both of which are being adopted in present-day practice, especially in automobile engines. Their effect upon capacity and weight is perhaps greater than upon economy of operation. High piston speed involves either large valves or high gas velocities. Guldner recommends a gas velocity of 4500 ft. per min. witn a maximum of 6000 ft. under the most favorable conditions. With a piston speed of 1000 ft. per min. and a valve lift of J its diameter which is given as a maximum by Guldner the valve diameter would be about 0.55 and 0.47 of the cylinder diameter for the two values of gas velocity respectively. This exceeds usual practice at the present time. The piston speed of Gtild- ner's engine given in his book is 730 ft. per min. At maximum valve lift given by him and the valve diameter scaled from his reproduction of a working drawing of the engine, the gas velocity would be 8000 ft. per min. A discussion of this subject with examples from present practice is given in Chap. XX. As in the steam engine, the tendency is toward higher rotary and piston speeds. A piston speed of 2000 ft. per min. is not uncommon in automobile engines and 3200 ft. has been used in racing engines. Rotary speeds from 1500 to 3000 r.p.m. are also used. Although gas velocities are undoubtedly increased by these high speeds, with a consequent loss, there is still a gain in capacity. Valve timing plays an important part in both capacity and efficiency and is discussed in Chap. XX. The influence of compression is con- sidered in Par. 76, Chap. XIV. PART III FRICTION AND LUBRICATION CHAPTER X MECHANICAL EFFICIENCY Notation. H B = brake horsepower (b.h.p.) in general. H BR = b.h.p. at rated load. HI = indicated horsepower (i.h.p.) in general. HIR = i.h.p. at rated load. H f = friction horsepower (f.h.p.). e = mechanical efficiency at any power. e R = the same at rated load. P = maximum unbalanced pressure in engine cylinder in pounds per square inch. P M = mean effective pressure (m.e.p.) in pounds per square inch. fji = coefficient of friction of engine if P M = P- 49. A general expression for mechanical efficiency of engines is given in Formula (7), Chap. VI. The mechanical efficiency changes with the load- ing and is obviously zero when all external load is removed and the engine is only overcoming its own friction. The horsepower developed under these conditions is known as the friction horsepower, and the m.e.p., the friction m.e.p. Friction depends upon many factors, such as the condition of the rubbing surfaces, the system of lubrication and the lubricant used, the relation of work done to the weight of the moving parts; and, other things being equal, it is usually relatively less for a given type of engine as the size is increased. It is therefore apparent that no fixed rule can be given by means of which friction may be determined with accuracy in all cases, but due to the very conditions named, it is convenient for the purpose of arranging and studying the results of tests, to derive formulas giving relations between indicated, brake and friction horsepowers for different condi- tions of loading. To be intelligible, mechanical efficiency must always be stated with reference to some given load; this may be the maximum load or fraction 7 97 98 DESIGN AND CONSTRUCTION OF HEAT ENGINES thereof, or any rated load. The efficiency will then be different for any other load. The friction horsepower usually increases with increase of load; the increase is often slight, however, and for practical purposes the f.h.p. may be assumed constant. This assumption will be made in the present discussion and the results compared with actual values from practice. In what follows, H will denote horsepower and e mechanical efficiency. The subscripts B and / denote brake and indicated horsepower respec- tively. The subscript R refers to rated power; with steam engines this is usually the most economical power, allowing for from 50 to 100 per cent, overload, depending upon the type of valve gear used. For steam turbines and internal-combustion engines, though at or near the load giving best economy, the rated load is more nearly the maximum; in most cases, however, from 10 to 20 per cent, overload may be carried. When subscript R is not used in conjunction with B and /, any other than the rated power is meant. The f.h.p. is assumed constant for a given engine. For convenience of expression let : ^ = fc (1) and --q (2) The expression for mechanical efficiency at any rated load may be written : HBR H B R 1 HIR H BR + H F .. i Hf From (3) : (3) - = - - 1 = l -^ (4) BR CR 6 R Efficiency at any load in terms of b.h.p. is: H B 1 P H, (5) From (1), (4) and (5): 1_ _1_ 6 jj- 1 />\ i J- g * i 4. 1 ~ e W h kH BK ke R Dividing (1) by (2) gives: k HIR H B HB HJR _ 6 q Hj HBR HI HBR VR p from which k = q - (7) MECHANICAL EFFICIENCY 99 To express efficiency in terms of i.h.p., substituting (7) in (6) and solving for e: Should e be known at some other than the rated load, e R may be found from (8); or: e R = 1 - q(l - e} (9) Formulas (6) and (8) give the efficiency for any load when it is known at the rated load, when the fraction of the load is given in terms of b.h.p. and i.h.p. respectively. Results of tests are given in the following tables. Tables 7 to 9 are from Goodman's Mechanics 'Applied to Engineering, and are from tests on an experimental steam engine with different methods of lubrication. The data in Tables 10 to 12 are for Westinghouse vertical 4-cycle, 3-cylinder gas engines using natural gas, and were taken from Giildner's Internal-combustion Engines. TABLE 7. SYPHON LUBRICATION Hj 2.75 9.250 10.230 11.140 12.340 13.950 14.290 H B 0.00 5.630 7.500 7.660 9.090 11.090 11.250 H F 2.75 3.630 2.730 3.480 3.250 2.860 3.040 e 0.00 0.608 0.733 0.688 0.738 0.795 0.788 TABLE 8. SYPHON AND PAD LUBRICATION Hz 2.48 5.160 6.830 8.300 11.500 13.840 17.020 22.300 H B 0.00 2.350 3.940 5.610 8.700 10.820 13.890 19.090 fo 2.48 2.810 2.890 2.690 2.800 3.020 3.130 3.210 e 0.00 0.455 0.578 0.676 0.756 0.783 0.806 0.857 TABLE 9. FORCED LUBRICATION ffl 49.800 102.700 147.600 193.600 217.500 H B 44.500 97.000 140.600 186.000 209 . 500 H F 5.300 5.700 6.500 7.600 8.000 e 0.912 0.945 0.955 . 0.962 0.964 TABLE 10. 25- BY 30-iN. GAS ENGINE e Nominal load Hi HB HF N k Actual Gale. H 262.5 207.5 55.0 151.7 0.375 0.791 0.754 H 449.3 383.8 65.5 150.0 0.693 0.855 0.850 Rated 621.4 553.1 68.3 149.2 1.000 0.891 0.891 Max. 676.7 605.6 71.1 146.7 0.095 0.895 0.894 100 DESIGN AND CONSTRUCTION OF HEAT ENGINES TABLE 11. 19- BY 22-iN. GAS ENGINE Nominal load Hi HB HF A r k e Actual Calc. K 147.2 121.3 25.9 206.3 0.507 0.825 0.780 Rated 273.5 239.0 34.5 203 . 1.000 0.875 0.875 IX 365.5 325.3 40.2 198.0 1.360 0.890 0.862 TABLE 12. 13- BY 14-iN. GAS ENGINE e Nominal load Hi HB HF N k Actual Calc. 18.57 0.00 18.57 265.5 0.000 0.000 0.000 H 53.69 30.81 22.88 264.0 0.273 0.574 0.534 H 85.55 64.37 21.18 264.0 0.570 0.752 0.706 H 114.69 90.25 24.44 260.0 0.800 0.787 0.770 Rated 139.73 112.86 26.87 258.0 1.000 0.808 0.808 Max. 164.22 143.17 21.05 256.8 1.270 0.873 0.842 TABLE 13. HORNSBY AKROYD OIL ENGINE e Nominal load HI HB HF N m Actual Calc. H 13.10 9. CO 4.10 203.0 0.336 0.687 0.677 H 22.40 17.96 4.44 202.4 0.672 0.802 0.807 Rated 32.15 27.74 4.41 202.6 1.000 0.862 0.862 TABLE 14. GULDNER GAS ENGINE Nominal load m HB HF h e Actual Calc. H 18.85 11.15 7.70 0.317 0.592 0.550 H 22.20 13.30 8.90 0.378 0.600 0.594 y* 26.40 17 . 65. 8.25 0.502 0.668 0.660 H 31.10 21.50 9.60 0.612 0.692 0.704 H 37.40 25.90 11.50 0.737 0.692 0.741 Rated 44.20 35.10 9.10 1.000 0.795 0.795 Table 13 contains data given by Gtildner for a Hornsby-Akroyd oil engine. Table 14 is for a Giildner gas engine. From these data it may be seen that for practical use in designing, formulas (6) and (8) may be used. It may also be seen that the value of MECHANICAL EFFICIENCY 101 e for a given load increases with the size of the engine, both for steam and gas; but it is considerably greater for the steam engine in proportion to the power. It is less for Diesel engines that for gas engines; W. H. Adams, Trans. A.S.M.E., vol. 37, p. 460, gives 0.75 for 4-cycle engines and 0.70 for 2-cycle engines. The Hornsby-Akroyd oil engine, Table 13, has a rated load efficiency of 0.862, although developing only 18 b.h.p., but the compression was only about 45 to 50 lb., and the explosion pres- sure only 180 to 200 lb. There seems to be some relation between pressure and mechanical efficiency for engines having about the same m.e.p. When the ratio of maximum unbalanced pressure to m.e.p. is high, the engine parts must be heavier, and while theoretically, friction work depends upon mean rather than maximum pressure, high maximum pressures seem to inter- fere with lubrication; at any rate, the friction m.e.p. increases with the ratio of maximum to mean pressure. The formula f OP mechanical efficiency may be written : An empirical expression, considering the pressure ratio just mentioned, may be written: Hp P H B /i 1 \ = M + m - in which P is maximum unbalanced pressure and P M the m.e.p. The coefficient of friction /* may be assumed as the ratio of friction to useful work if the pressure were uniform throughout the stroke, or if P = P M ; its value may be taken from 0.02 to 0.06. H B may be taken as the b.h.p. of one cylinder end and m a constant depending upon the type of engine. Assume for average values, the following: P P For simple steam engines ............................ p = 2 and /z p = . 08. P P For compound steam engines ........................ p = 2.5 and /z p = 0. 10. P P For gas or other constant-vol. engines ........... .... p- = 4 and /x p = 0. 16. P P For Diesel engines ................................. p- = 6 and ^ p = 0. 24. Values of m for steam engines and 4-cycle internal-combustion engines are given in Table 15. These are rather arbitrary and from 0.05 to 0.15 may be added for 2-cycle engines, the larger values being for small, high- speed engines. 102 DESIGN AND CONSTRUCTION OF HEAT ENGINES Formulas (10) and (11) may be used with judgment for obtaining a fair approximation to ordinary values found in practice, but of course wide deviations will be found. A wider study of data would probably result in more accurate constants, or perhaps a better formula. TABLE 15. VALUES OF m Type of engine Steam-small, with unbalanced valve Steam-compound Steam-locomotive 2-cylinder, single-acting, internal-combustion 3-cylinder, single-acting, internal-combustion 4- and 8-cylinder, single-acting, internal-combustion 6- and 1 2-cylinder, single-acting, internal-combustion 1-cylinder, double-acting, internal-combustion 2-cylinder, double-acting, twin internal-combustion ........ 2-cylinder, double-acting, tandem internal-combustion 4-cylinder, double-acting, twin-tandem internal-combustion Small engines with 1 or 2 cylinders, poorly attended 0.07 0.02 0.08 0.02 0.03 0.04 0.06 0.02 0.04 0.05 0.06 0.04 Reference Mechanics applied to engineering, Prof. Goodman. CHAPTER XI LUBRICATION Notation. d = diameter of bearing in inches. I = length of bearing in inches. k = ratio of length to diameter = l/d. D s = diameter of standard simple engine cylinder, designed for some standard pressure. (See Par. 63, Chap. XII and Par. 72, Chap. XIII.) a = area of rubbing surface of bearing in square inches; this is the projected area (Id) for cylindrical bearings. p = total mean load on bearing in pounds. p mean pressure in pounds per square inch; for cylindrical bear- ings, per square inch of projected area. p M = maximum pressure in pounds per square inch. S = shearing stress in turbine shaft in pounds per square inch. V = velocity of rubbing surface in feet per minute. N = r.p.m. M = speed in knots, or nautical miles per hour. H = horsepower of engine (i.h.p.). h = heat in B.t.u. per hour developed in bearing. C, K and ra are constants in formulas. 50. General Principles. Friction, lubrication and the proportioning of bearing surfaces are so intimately related in connection with the design of heat engines that they may be considered together in one chapter. The preceding chapter dealt with the effect of friction upon power, and later chapters will give the general construction of bearings, while in the present chapter a study of some of the factors influencing friction, and means of reducing it will be attempted. The laws of friction are exceedingly complicated, and while much valuable experimental work has been done, and most lubricating problems are practically handled, no comprehensive mathematical treatment has yet been presented. References to valuable material are given at the end of this chapter. Proper lubrication is effected when a thin film of any lubricant lies between two surfaces in such a way that there is no metallic contact. There is then no appreciable wear. Without lubrication, there is wearing 103 104 DESIGN AND CONSTRUCTION OF HEAT ENGINES of the surfaces, and with great unit pressures the surfaces may u sieze, " or become welded together, so that if further movement is forced the surfaces become torn and "scored." There may be all stages between no lubrication and perfect lubrica- tion, with behavior varying accordingly; for this reason it is- well to consider the laws of dry as well as of lubricated surfaces. The laws of friction of dry surfaces credited to Morin have been revised in the light of more recent experiments and are given by Goodman in Mechanics Applied to Engineering, in parallel with the laws of lubricated surfaces, and they will be given here. DRY SURFACES 1. The f rictional resistance is nearly proportional to the normal pressure between the two surfaces. 2. The f rictional resistance is nearly independent of the speed for low pressures. For high pressures it tends to decrease as the speed in- creases. 3. The frictional resistance is not greatly affected by temperature. 4. The frictional resistance de- pends largely upon the nature of the material of which the rubbing sur- faces are composed. LUBRICATED SURFACES 1. The frictional resistance is almost independent of the pressure with bath lubrication so long as the oil film is not ruptured, and approaches the behavior of dry surfaces as the lubrication becomes meager. 2. The frictional resistance of a flooded bearing, when the temperature is artificially controlled, increases (except at very low speeds) nearly as the speed, but when the temperature is not controlled the friction does not appear to follow any definite law. It is high at low speeds of rub- bing, decreases as the speed increases, reaches a minimum at a speed dependent upon the temperature and the intensity of pressure; at higher speed it appears to increase as the square root of the speed ; and finally, at speeds of over 3000 feet per minute, some believe that it remains con- stant. 3. The frictional resistance depends largely upon the temperature of the bearing, partly due to the viscosity of the oil, and partly to the fact that the diameter of the bearing increases with a rise of temperature more rapidly than the diameter of the shaft, and thereby relieves the bearing of side pressure. 4. The frictional resistance with a flooded bearing depends but slightly upon the nature of the material of which the surfaces are composed, but as the lubrication becomes meager, the friction follows much the same laws as in the case of dry surfaces. LUBRICATION 105 DRY SURFACES 5. The friction of rest is slightly greater than the friction of motion. 6. When the pressures between the surfaces become excessive, seizing occurs. 7. The frictional resistance is greatest at first, and rapidly decreases with the time after the two surfaces are brought together, probably due to the polishing of the surfaces. 8. The frictional resistance is always greater immediately after re- versal of direction of sliding. LUBRICATED SURFACES 5. The friction of rest is enormously greater than the friction of motion, espe- cially if thin lubricants be used, probably due to their being squeezed out when standing. 6. When the pressures between the surfaces become excessive, which is at a much higher pressure than with dry sur- faces, the lubricant is squeezed out and seizing occurs. The pressure at which this occurs depends upon the viscosity of the lubricant. 7. The frictional resistance is least at first, and rapidly increases with the time after the two surfaces are brought together, probably due to the partial squeezing out of the lubricant. 8. Same as in the case of dry surfaces. 15 50 75 100 115 r50 175 200 Rubbing Speed -Ff. per Sec. FIG. 60. IIS The way in which friction varies with velocity for lubricated surfaces is shown in Fig. 60 taken from Goodman. A comparison of curves A and B also shows the influence of temperature as stated in Law 3 for lubricated surfaces. A comparison of curves C and Z), and of curves E and F also shows the effect of pressure. It is usually considered that friction and wear are so closely related as to be nearly synonymous terms, but experiments at low pressures show that for dry surfaces the friction is less than for a bearing flooded with oil, but it is certain the wear must be greater. The friction of a lubricated 106 DESIGN AND CONSTRUCTION OF HEAT ENGINES bearing is due to the shearing of the lubricant, which does not damage the bearing although heat is produced with a rise in temperature. From Law 3 of lubricated surfaces, a hot bearing, within reasonable limits, is more efficient than a cool one, as the thinning of the lubricant due to temperature reduces the friction; this is also shown by curves A and 5, Fig. 60. Martin gives an instance of a steam turbine bearing in which the heat generated at 90 degrees F. was double that generated at 165 degrees, show- ing less friction at the high temperature. He also states that certain steam turbine bearings running at a temperature of 195 degrees F. have given no trouble. High temperature is normal for a properly lubricated high- speed bearing, but the oil must be kept sufficiently viscous to keep the rubbing surfaces apart. The greater the normal viscosity of the lubricant, the higher will the temperature be, due to greater shearing resistance, and the greater the allowable bearing pressure at a given speed. It is not advisable to run a bearing at its least friction, involving as it does high temperature with a less viscous condition of the lubricant, as an increase in temperature from any cause may still further decrease the viscosity so that the oil film will not be dragged completely around the bearing. To control the temperature, some bearings are constructed so that cold water may be circulated around them, and in steam turbine practice with forced lubrication, the oil is passed through a cooler before being fed to the bearings. The temperature of steam turbine bearings is not all due to the shear- ing of the oil, but in part to the transmission of heat of the steam through the shaft, so a bearing temperature of 180 degrees F. is probably perfectly normal for a turbine if there is no foaming of the oil. A. G. Christie, Trans. A.S.M.E., vol. 34, p. 435, I I I J states that if the bearings of a steam turbine are -* flooded with oil at 100 degrees F. so that its tempera- ture upon leaving is about 125 degrees, the life of the oil is much longer than when very hot oil is used, and the wear of the bearings absolutely pre- vented. He further says that the use of water- FIG. 61. cooled bearings is to be discouraged, water being much better employed in an oil cooler. A light oil, with ample continuous feed, properly filtered and cooled will probably give minimum friction with maximum life of oil and bearings. With oil-bath lubrication as in Fig. 61, oil is drawn up around the bear- ing as shown, the layer being thicker on the "on" side than on the "off" LUBRICATION 107 side. This has been shown to be so by actual experiment by Goodman, and that in case of wear, it was greater on the off side. Experiments by Tower also show that the oil pressure varies, being a maximum nearly midway between the on and off sides, and greater on the off side than on the on side. He also shows that to feed oil at the center where the pressure is greatest causes the oil to flow out rather than in. The results of some of Tower's tests, given by Goodman (Mechanics Applied to Engineering), are given in Table 16, which shows the seizing pressure and coefficient of friction for different methods of supplying lubricants, and for different kinds of bearings. Bath lubrication gives the best results and may be taken as the standard of lubrication. Equiva- lent methods are those in which the bearing is flooded with oil, or fias stream feed, such as some of the forced feed systems. TABLE 16 Form of bearing O i M, J^ Seizing load, Ib. per sq. in. Coefficient of 100 150 370 550 600 200 90 friction 0.01 0.01 0.006 0.006 0.001 .01 0.035 TABLE 17 Mode of lubrication Relative friction Tower Goodman Bath 1.00 1.00 1.32 2.21 4.20 Saturated pad Ordinary pad 6.48 7.06 Syphon. . Pad lubrication, giving next best results, is accomplished by having the journal in contact with oil-soaked pads. Waste in railway journal boxes is used for this purpose. A comparison between methods of applying lubricant is given in Table 17, from Goodman. 51. Lubricants and Their Application. As acid is produced by the decomposition of animal and vegetable oils, they are little used for heat engine lubrication, except as the former may be combined in small 108 DESIGN AND CONSTRUCTION OF HEAT ENGINES proportions with mineral oil to form cylinder oil for saturated steam, promoting better retention of the oil on the surfaces. Mineral oils may then be considered as the lubricants for heat engines, and a few important characteristics will be mentioned. Further information may be found in Gill's Engine Room Chemistry and in treatises on Testing and Ex- perimental Engineering. Viscosity, or "body, " is the degree of fluidity or internal friction of an oil. This varies with the temperature and must be tested at some standard temperature, commonly 70 degrees F. for engine oils and 212 degrees F. for cylinder oils. The least viscous oil which will stay in place and do its work should be used, as it absorbs less power in friction. Viscosimeters for testing the viscosity of oils are commonly constructed upon the principle of the rate of flow through a standard orifice, and the results are given in seconds. The longer it takes for a given quantity of oil to flow through the orifice, the greater the viscosity. When viscosity is given, the viscosimeter should be named. Density or specific gravity of mineral oils is usually measured with the Baume hydrometer, and expressed in degrees Baume (degrees B.). The specific gravity is found by the formula : 144.3 Specific gravity == ^3 + defr B ; The cold test is the temperature at which the oil will just flow. In stationary power plants this is of little importance, but for exposed work in cold climates it must not be overlooked. The flash point is the temperature at which oil gives off vapors in sufficient quantity to explode when mixed with air, but the oil will not continue to burn. The fire test or burning point is the temperature at which vapor enough is given off to burn continuously. The gumming test is to ascertain the amount of change taking place in an oil when in use. Gumming may seriously interfere with the distri- bution of oil to bearing surfaces. Acid Test. Acid in mineral oils is usually sulphuric, used in the refin- ing process and not entirely washed out. An acid content exceeding 0.3 per cent, is considered harmful. The friction test, to determine the coefficient of friction is made with a machine specially devised for this purpose. It must be made at stand- ard temperature and with a definite method of applying the lubricant. There is considerable variation in the characteristics of oils as given by different authorities, but a few values are given in Table 18. LUBRICATION 109 TABLE 18 Kind of oil Gravity, deg. B Flash point deg. F. High-pressure cylinder oil 26 . to 28 General cylinder oil 24 . to 27 Heavy engine oil j 25 . 5 to 28 Gas engine cylinder oil 26 . Automobile engine oil j 29 . 5 Air compressor oil j 550 to 600 530 to 560 385 to 410 410 to 500 430 500 to 530 The cold test is usually 30 or 32 degrees F. A turbine builder recommends a pure mineral oil of 600 degrees test. Charles G. Sampson, Trans. A.S.M.E., vol. 35, p. 151, says that for a blast furnace gas engine main bearing, crosshead, crank pin and guides, an oil having a specific gravity of 0.888, and a flash point of 435 degrees F. gave good results. For the cylinders, when the piston speed was 600 ft. per min., an oil having a specific gravity of 0.902, and a flash point of 380 degrees F. was satisfactory, but was not at 850 ft. per min.; for this speed a specific gravity of 0.92 and flash point of 502 degrees F. were used. He states that with the slower speed and lighter oil the wear was excessive. Mathot says that the flash point of gas engine cylinder oil is negli- gible; that it should have a viscosity similar to bearing oil; in fact the same oil is used for bearings and cylinder in the best practice. Greases are composed of oils and fats mixed with soap, forming a more or. less solid mass. They are used for heavy pressures or in places difficult to lubricate with oil; or, where the movement is comparatively small, as in valve gears. For extra heavy pressures, solid lubricants, such as soapstone, mica or graphite are mixed with the grease. Systems. The individual system of lubrication is where each bearing is provided with an oil cup or grease cup, with restricted feed, or with an oil hole with intermittent feed. The latter is fortunately passing out of date. The forced feed or continuous system may be direct, in which oil under pressure of from 3 to 20 Ib. per sq. in. is forced directly to the bearings, returned through a strainer or filter and sometimes through a cooler to the pump, where it again starts upon its circuit; or it may be a gravity system, in which the oil is pumped to an elevated tank located upon the wall or on the engine frame, and flows by gravity to the bearings. In this system the filter may be in the basement, on the ground floor, or elevated with the storage tank, with which it is sometimes integral. 110 DESIGN AND CONSTRUCTION OF HEAT ENGINES The forced feed system may have restricted feed, the supply being throttled by a valve, or the wearing parts may be continuously " flooded" with oil, with stream feed, being a form of oil bath, and this method is gaining favor for heat engine lubrication. The oil pump may be a plunger pump operated from some moving part of the engine as the rocker arm or valve lever, or it may be a centrifugal pump (see Fig. 252, Chap. XIX) operated by some rotating part. With steam turbines, a gear pump is often driven by an extension of the governor spindle. In large units, or in a plant in which there are a number of engines of tur- FIG. 62. Sight-feed valve. bines, a separate pump driven by a steam cylinder, turbine or motor supplies oil for the entire plant. Manifold oilers, sometimes having a common pump and sometimes a pump for each line which is led from it to a bearing, are often used. Sometimes a separate compartment con- tains cylinder oil, the manifold oiler thus furnishing complete lubrication. The splash system of lubrication, used mostly on the smaller engines, consists of an enclosed crank case containing oil at such a level that the crank or counterbalance enters the surface at each revolution, splashing the oil upon the bearing surfaces. In single-acting engines of the trunk- LUBRICATION 111 piston type, this also lubricates the piston, which is lubricated separately in other systems, sometimes with a different grade of oil. Mat-hot depreciates the splash system, in that it gives impurities no time to settle; they are thus brought continuously to the wearing surfaces. A combination of the forced-feed and splash system is used in some instances, notably with automobile engines. The oil is forced into the shaft bearing by a gear pump, from thence to the crank pin through holes in the shaft from which it is thrown. It is claimed that the rapid motion of the cranks (which do not dip in the oil in the bottom of the Female Shank ~~Mcrfe Shank FIG. 63. "Clear vision" sight-feed oiler. oil pan) whip the oil into a fine spray, providing ample lubrication for the cylinders. In feeding oil directly to the bearing, either with an oil cup or with the forced-feed system, a sight feed is generally employed; a sight-feed valve is shown in Fig. 62. The valve may be placed on top of the bearing or it may be a part of an oil cup. The flow of oil in both cases may be regulated by a valve. A special sight-feed oiler is shown in Fig. 63. This is made by the Richardson-Phenix Co. In some cases the sight-feed valve is one of several in a manifold oiler located on some part of the engine, or turbine frame, with pipes leading to the different bearings; or it may be located near the bearing, but in the ICturn pipe. In either case it shows whether the oil is flowing. 112 DESIGN AND CONSTRUCTION OF HEAT ENGINES When the individual system is used on engines which are shut down once or twice a day, moving parts such as crosshead pins, crank pins and eccentrics may be supplied with oil cups, but as these cannot be filled while in motion, special devices must be resorted to for continuous opera- tion. For the crosshead, the most common is the wiper and wiper cup, shown in Fig. 64. Oil drops upon the wick from a sight-feed cup or valve and is wiped off into the cup near the end of the stroke. Eccen- trics are also oiled in this manner. An alternative in both cases is the telescopic oiler (also called trombone oiler). For the crank pin a cen- trifugal oiler is used. A con- venient form, especially for the individual system, is the Nugent oiler. The weight keeps the cup in an upright position at the center line of the shaft. Sometimes a stand, fastened to the floor or engine frame replaces the hanging weight, especially in the forced-feed system. The telescope oiler is sometimes used for the crank pin in center-crank engines. Many oiling devices, as well as system layouts, may be seen in the catalogs referred to at the end of FIG. 64. Crosshead this chapter. Oil guards, enclosing the crank and connecting rod, and sometimes the eccentrics, are often used, and may be seen in engine catalogs. All of these appliances are also used with the continuous system. In some installations, oil cups are provided with each bearing, which, when filled with oil through the system piping, are shut off by a valve; in case the oiling system is shut down for cleaning or repairs, the cup valves are opened and lubrication provided. Grease Cups. Two types of grease cups are used, one a screw-feed cup, the other an automatic cup. In the latter, when the handle is screwed up to the top of the stem, the spring exerts pressure upon the grease, the feed being regulated by a screw which forms a plug cock. Self-oiling bearings have an oil pocket in the bottom from which oil is carried to the top of the shaft by rings or chains which are supported on the shaft. A gage glass on the outside of the bearing indicates the amount of oil in the pocket. They are used on the smaller steam turbines and sometimes in connection with a forced-feed system; they are also LUBRICATION 113 used on outer bearings of engines and sometimes for main bearings of large engines. Giildner objects to them for internal-combustion engines as they reduce the wearing surface and weaken the bearing. Bearings are shown in Chaps. XXIX and XXXIII. FIG. 65. Gas engine main bearing grooves. FIG. 66. Steam engine main bear- ing grooves. Clearance. For proper lubrication there must be some clearance between journal and bearing. No hard-and-fast rule may be made' and standards differ with different builders. Goodman says that for ordinary lubrication the clearance should be 0.001 of the journal diameter, FIG. 67. Crank- and crosshead box grooves. and slightly more for flooded bearings. Martin says the clearance for steam turbine bearings should be from 0.006 to 0.008 in. Oil Grooves. It is sometimes claimed that oil grooves are unnecessary, and for the uniform conditions under which most bearing tests are made, FIG. 68. Crosshead shoe grooves. this may be true; but judging from the practice of most builders, it is apparent that there is a general feeling in favor of them. Their form and arrangement have been the subject of considerable discussion, and no attempt of a settlement of the question will be made here; but a number 8 114 DESIGN AND CONSTRUCTION OF HEAT ENGINES of forms given by leading authorities and used by successful builders will be given. The grooves are shown as though laid out on a flat surface equal in length to the projection of the circular arc which they represent, and of a width equal to the length of the bearing. There is probably no doubt but that bearings should be chamfered at the edges, and the corners of chamfer and grooves rounded as phown in Fig. 69. Bearing Metal. It has long been considered that proper lubrication cannot be secured be- tween surfaces of the same material, and while this does not hold for cylinders and valves, at least, it is nevertheless true that in nearly all Fl ' 69 ~h^mfer S VeS *** cases > the J ournals and bearings of heat engines are of different materials. The journals are practically always of steel, while the bearing boxes are either of some form of bronze, or are lined with a soft metal such as babbitt or some other " anti-friction " metal, the latter method being predominant. From Law 4 for lubricated surfaces, it makes little difference what the bearing metal is if the bearing is amply lubricated, as the journal is not in contact with the bearing; but bearings are not always perfectly lubri- cated, so the selection of the material is important, especially for heavy loads or high speeds. For heavy loads subject to blows, bronze is usually employed, although the crank-pin boxes of most gas engines are lined with babbitt. While all bearings should be properly fitted, this is especially true of bronze, but if carefully fitted and well lubricated, it will probably outwear babbitt. Goodman gives the advantages and disadvantages of soft white metal for bearings as follows : Advantages. (a) The friction is much lower than with hard bronzes, cast iron, etc., hence it is less liable to heat. (b) The wear is very small indeed after the bearing has once got well bedded (see disadvantages). (c) It rarely scores the shaft, even if the bearing heats. (d) It absorbs any grit that may get into the bearing, instead of allowing it to churn round and round, and so cause damage. Disadvantages. (a) Will not stand the hammering action that some shafts are subjected to. (6) The wear is very rapid at first if the shaft is at all rough; the action resembles that of a new file on lead. At first the file cuts rapidly, but it soon clogs, and then ceases to act as a file. (c) It is liable to melt out if the bearing runs hot. (d) If made of unsuitable material it is liable to corrode LUBRICATION 115 The crosshead pins of nearly all steam- and internal-combustion engines, having comparatively small wearing motion, are provided with bronze bearings, as are some small valve-gear pins, but the main and outer bearings, the crank-pin box and the crosshead shoes are babbitted. Sometimes steam engine pistons have rings or strips of babbitt, but this is not the rule. Steam turbine shaft bearings are usually babbitt lined. The term babbitt is used in a rather general way; there are a number of anti-friction metals on the market which are used for the same purpose. The design of the various bearings is considered in connection with the different engine details of which they form a part. 52. Bearing Proportions. The bearing formulas in common use are empirical and are not based upon the laws of friction. The most common expression is : P V = C (1) in which p is the pressure in pounds per square inch of projected area, V the velocity of rubbing surfaces in feet per minute, and C a constant which differs for various bearings. The mean pressure is assumed, which may equal the maximum in some cases. In steam- and gas-engine work it is the resultant of all forces acting upon the bearing surface, which, for main and outer bearings includes weight of shaft, wheel, crank, etc., pull of belt and piston thrust; the latter, including inertia, is explained in Par. 103, Chap. XVI. The piston thrust is often the chief factor in internal-combustion engines, especially in small and medium powers, and consists of the mean pressure due to gas pressure and inertia of reciprocating parts. This will be further treated in Chap. XVI. A maximum pressure p M is also prescribed, beyond which it is best not to go. Strength considerations, the most important from the stand- point of safety, are discussed in the chapters dealing with the details involved. It is apparent that preliminary assumptions must be made, and the results checked by other limiting conditions. Let P be the total mean load on the bearing, I the length and d the diameter of the bearing in inches, and N the r.p.m. Then : and: (3) 116 DESIGN AND CONSTRUCTION OF HEAT ENGINES Substituting in (1) gives: 12C ~ y 'T C (4) Table 19 gives values of p M and C which have been used in practice. TABLE 19 Bearing PM C V Main bearing Crosshead pin 125 to 200 1200 to 1500 50,000 "8 a W Crosshead pin locomotive Crank pin 3500 to 4800 1000 to 1200 200 000 s 1 Crankpin locomotive 1200 to 1700 2 02 Crosshead shoe 35 to 75 50 000 Piston 25 30000 H Main bearing Crosshead or piston pin. 350 to 400 1000 to 1800 42,000 ,O a Crank pin 1000 to 1700 90,000 6 "Orosshead shoe 35 to 45 a Trunk piston 21 The values of C for the steam engine were given by James Christie in the American Machinist, Dec. 15, 1898, and have been used as a check by the author since that time. For light loads, high speeds and efficient lubrication, it was stated that the values may be doubled. The values for internal-combustion engines are those given by Giildner, and coincide with those given by other authorities. In all cases it is assumed that steel journals and bearings lined with babbitt are used. Exception is made for crosshead boxes which are usually of bronze- and crossheads and pistons. Where the force acting upon a bearing is rapidly alternating, better lubrication is possible, as time is given for the oil to flow between the surfaces when the pressure is removed. This obtains to some extent with all cylindrical bearings of a reciprocating engine, but especially for the connecting-rod bearings; this accounts for the larger value of p M and C given for crank and crosshead pins. These bearings also have better air circulation due to their motion. A little computation will show that under the same conditions of operation regarding temperature, lubrication, etc., Formula (1) gives absurd results for widely varying surface speeds unless C is varied with the speed. A formula credited to the late Edwin Reynolds has been LUBRICATION 117 used for the main bearings of engines, and has also been used by the author lor heavy-duty mill bearings; it is: pVV = K (5) the notation being the same as for (1). For extremes of speed the con- stant K must vary, but a given value covers a wider range of conditions than does C in Formula (1). Table 20 from: the author's note book will illustrate. TABLE 20 Service K Main bearing horizontal engine . Main bearing vertical engine . Sugar mill gear drive Sugar mill roll shaft Freight car journal 3000 FIG. 91. FIG. 92. In numerical computation, either V H or V L may be considered as unity, the other values of v being in proportion. For simplicity, re- lease and admission are assumed to occur at dead center, and the expan- sion and compression curves are hyperbolas. The diagram employed to show the relation of events consists of a curve of piston displacement plotted on a rectified crank circle, as in Fig. 91. FIG. 93. If angularity of connecting rod is to be taken into account, this may be done as in Fig. 92. This is neglected in the diagrams which follow. In order to avoid confusion, the relation of cylinder and crank is the same as in Chap. XX, the cylinder being at the left of the crank as shown in Fig. 93. The crank rotates clockwise and crank positions are num- bered from the head-end dead center of the high-pressure engine. For the THE COMPOUND STEAM ENGINE 161 tandem compound this coincides with the low-pressure engine and may be shown in the form given in Un win's Machine Design, which is also in- troductory to the method to be employed. Tandem Compound. Fig. 94 produces the head-end high-pressure diagram and the crank-end low-pressure diagram. Dividing V R into equal parts to represent a rectified crank circle, the receiver pressure may be plotted, representing, however, the pressure changes for but one stroke of the pistons. FIG. 94. Beginning with the head end of both diagrams, operations may be traced. From to 1 on the high-pressure diagram, steam is admitted to the cylinder. At 1, expansion of all steam including clearance steam, begins in the high-pressure cylinder; at the same time steam is being forced out of the low-pressure cylinder until 14 is reached, when compres- sion begins and is finished at 15. At the same time high-pressure ex- pansion ends at 2. High-pressure release and low-pressure admission are now assumed to occur simultaneously. Pressure p i5 in the low-pres- sure clearance, pressure p 2 in the receiver and p b in the high-pressure cylinder are now all changed to p s (= pio), as both cylinders are now open to the receiver. The piston now starts on the return stroke from right to left, and as the low-pressure piston displaces volume as it takes steam from the receiver, more rapidly than the high-pressure piston as it exhausts into 11 162 DESIGN AND CONSTRUCTION OF HEAT ENGINES the receiver, the pressure falls until low-pressure cut-off is reached at 11. Low-pressure expansion now begins, and at the corresponding pqint 4 of the high-pressure diagram, compression of steam into the receiver is begun by the high-pressure piston. This continues until the high- pressure exhaust valve closes at 5, shutting off the communication with the receiver. Compression in the high-pressure cylinder is complete at 6, and at the same time low-pressure expansion is complete. Fig. 95 is a modification of this diagram, from which indicator dia- grams may be traced for both ends of each cylinder. The diagrams used in forming the equations are in full lines. With this method the receiver volume may not be drawn to scale, but all shaded areas extending into FIG. 95. the central square indicate communication with the receiver. All such areas join a diagonal line as shown, and if shaded areas from two cylinder ends meet, they are both in communication with the receiver. The upper end of the diagonal corresponds to head-end dead center of the high- pressure crank, and following this line to its lower extremity, all events of the cycle may be traced. The indicator diagrams of Figs. 95, 97, 100 and 103 are for a small re- ceiver, exaggerating the pressure variation. The volume of the low- pressure cylinder is drawn one-half the computed volume to save space. No special work division is aimed at, the diagrams being used to illustrate the method. Figs. 96, 99, 102 and 105 are for condensing engines with a THE COMPOUND STEAM ENGINE 163 cylinder ratio of 6.42, a back pressure of 2 Ib. absolute and a receiver volume equal to the volume of the low-pressure cylinder. From Fig. 95 the following equations may be written. p&i = P& 2 (a) ptv 2 + p&R + p u c L = Pa(v z + V R + CL) (b) V R + C L ) = p 4 (*>4 + V R + Vu) (c) p 3 2>4 = Pn (e) p 4 (t>4 + VR) = p b (v 5 + V R ) (/) PbV$ = P&CH (g) pnVn = pizVn (h) From (a), (6), (c) and (i): P*(V4 + VR + Vn) = piVi + Pl4^14 + P&R (j) and from (/), Then from 0') and (fc): (I) From (e), (I) and (i), an equation containing only known quantities is derived; or: / _t_ /n. .t. , -TV <) The equations following give all other points on the diagram. From (i) : () From (*), p u - (o) From (/), P6 = a (P) From (g), p 6 - (g) From (a), p 2 = (r) From (c) and (d), p z = pw V 2 + Vn) , , V* + tfc + Cx, If Z L = low-pressure cut-off: V 4 = 2 - IL.VH (t) Combined diagrams for the head end are shown in Fig. 96, in which the receiver volume equals the volume of the low-pressure cylinder. The dotted lines show the indefinitely large receiver assumed in Par. 66. Except for the back-pressure line of the high-pressure diagram and the steam line of the low-pressure diagram the lines are practically identical. The larger the receiver the more nearly will the lines mentioned approach the dotted lines, 164 DESIGN AND CONSTRUCTION OF HEAT ENGINES It is plain that the influence of the receiver is to increase the maximum pressure in the low-pressure cylinder, increase the work done in this cylinder and decrease the work of the high-pressure cylinder. FIG. 96. FIG. 97. Cross Compound. Diagrams for cross-compound engines will be drawn with the high-pressure crank leading the low-pressure crank by THE COMPOUND STEAM ENGINE 165 90 degrees. They would be identical with those in which the low-pressure crank leads so long as angularity of the connecting rod were neglected. There are three sets of diagrams required to show conditions commonly found in practice due to the action of the governor upon the valve gear. Case 1. In which low-pressure cut-off occurs after compression in one end of the high-pressure cylinder and before release in the other end. The diagram is shown in Fig. 97 and is similar to Fig. 94 except that the piston displacement curves for the low-pressure cylinder are not in phase with the high-pressure curves. In Fig. 98, V* = I' 5 FIG. 98. and vn is determined by the position of the low-pressure crank when the high-pressure crank is in the compression position as in Fig. 98. Then : versin A V H B = 90 - A and VL L D i = - versm B + C L . Or, if x n the fraction of stroke up to high-pressure compression : Vn = t'zX0.5 - \/X H - X H 2 ) + C L . If #11 > viz, use Case 2. From Fig. 97 the following equations may be written : Pivi = pzVz (a) p 2 v 2 + pizV R = Pz(v 2 + V R ) (b) P3 (02 + VR) = P 4 (V 4 + VR) (c) Pity* + VR) + p^C L = p 5 (>4 + VR + CL) (d) Ps(v* + V R + V L ) = p 6 (v & + V R + Vu) (e) #10 = Po (/) Pll = P G (g) ?>6>6 = P&H W PG(VR + Vn) = piz(v R + 0i 2 ) (i)' pizVn = pnVis (j) 166 DESIGN AND CONSTRUCTION OP HEAT ENGINES From (a) to (e): ffl FIG. 99. FIG. 100. From (i)j multiplying both sides of the equation by V R : VR (m) THE COMPOUND STEAM ENGINE 167 Substituting (k) and (wi) in (I), solving for p 6 and equating with (g) gives an equation containing only known quantities: = Pn = +.. + VR + (n) The equations which follow locate all other points on the diagram. From (h), p 7 = P^? ( ) From (z), pi 2 = 6 .. g , - (p) C H From (j), From (e) and (/), From (k), Pefye + VR - V4 4- V R 4 (r) w From (d) and (k), p* From (a), p 2 = ^ v* + V R (u) From (c), (0 Combined diagrams for the head end are shown in Fig. 99, using the same data as in Fig. 96. The dotted lines, as before, show an indefinitely large receiver, and the variation of pressure and work due to a receiver of practical size may be seen. Case 2. In which low-pressure cut-off occurs before compression in the high-pressure cylinder. This is shown in Fig. 100, which is in other respects like Fig. 97. In Fig. 100, 168 DESIGN AND CONSTRUCTION OF HEAT ENGINES and VQ is determined by the position of the high-pressure crank when the low-pressure crank is in the cut-off position, as in Fig. 101. Then: versin B = ~^ . A = 90 - B and v 6 = ^versin A + c a . VL Or, if 1 L = fraction of stroke up to low-pressure cut-off : V & = ^(0.5 \^1 L 1 L 2 ) + C H . If v G < v 7} use Case 1. From Fig. 100, the following equations may be written: (a) p&z + p 7 v R = p 3 (v 2 + V R ) (b\ V B ) = VR) + p U C L = P 5 (V 4 + V R + C L ) (d) V R + CL) = p 6 (>6 + VR + v n ) (e) Pio = Pb (/) Pii = P& (g) Pe(v& + V R ) = p 7 (v 7 + V R ) (h) p 7 v 7 = p 8 c H (i) piivu = pizVi* (j) puVu = pibC L (k) From (a) to (e), and (k), PG(VG + V R + Vn) = piVi + p 7 v R + p u v u (I) From (h), multiplying both terms of the equation by V R and solving for p 7 v R , , . Substituting (m) in (2), solving for p 6 and equating with (g) gives an equation containing only known quantities. (r, + ... + u) - VJ I VR All other points are located by the following equations: From (h), p 7 = ~ (o). From (i), p% = -^ (p) From (j), piz = ^ n (#). From (k), p i6 = / \ i //\ P&(VQ ~\~ V R -f- fii) From (e) and (/), p 5 = pi = ^ +T~ From (d) and (*), P< = p ' ( ^ + " B + ^ ~ P " y " (0 f 4 T v From (a), p 2 = *& (). From (c), p 3 = ^f^- () THE COMPOUND STEAM ENGINE 169 Combined diagrams for the head end are shown in Fig. 102. Case 3. In which high-pressure release occurs before low-pressure cut-off, which is greater than one-half stroke. This is shown in Fig. 103. FIG. 102. From Fig. 103, FIG. 103. V H , TT H" CB 170 DESIGN AND CONSTRUCTION OF HEAT ENGINES and v 4 is determined by the high-pressure crank position at the time of low-pressure cut-off as shown by Fig. 104. Then: 2(wi6 - flu). D . cos A = and v t = (1 + cos B) + e.. Or, if 1 L = fraction of stroke up to low-pressure cut-off: v, = 1^(0.5 + Vl L - k 2 ) + Ca . . The value of v\\ is determined by the low-pressure crank position at the time of high-pressure compression and is found as for Fig. 101, re- placing VQ by Vf t or: C L .p. .5 V X H.P. FIG. 104. in which X H = fraction of stroke up to compression in the high-pressure cylinder. From Fig. 103, the following equations may be written: (g) PQ From (a) to (c), From (j) to (Q, C L ) = = Pi 0') + ww) (0 = pi S c L (n) v fl + v u ) = piVi + p\*(v R + + v u ) = p 7 (v R + v n ) (p) (m) Substituting (p) in (o) and solving for p 4 gives: p* = ^7 + THE COMPOUND STEAM ENGINE From (/) to (h), and (ri): r V R -f Vn) 171 v< t r, (r) Equating (q), (r) and (7) and solving for p 7 gives an equation con- taining only known quantities. piVi(v* + VR) + pnVn(v* ~\- VR - (0 w The following equations locate all other points. Pl(Vl + VR ~ From (h) and (i), From (k), p 8 = From ( S ) and (), ?6 = PlO = V6 T t'B (w). From (Z), + VR FIG. 105. = PeC^s + ^g + CL) Pi7^n Vs + V R From (w), PIS = - (a;). From (e) and (/), i From (m), p 16 = Pl4 ^ 14 . From (c) and (d) , p 3 = y P u = () () (z) (B) Combined diagrams for the head end are shown in Fig. 105. The foregoing discussion covers all practical cases of tandem and cross- compound engines, the latter with cranks 90 degrees apart. It is obvious that the same analysis may be applied to any arrangement of cylinders 172 DESIGN AND CONSTRUCTION OF HEAT ENGINES and cranks, to 3-cylinder compounds, or to the large variety of cases found in triple- and quadruple-expansion engines; but due to the large number of formulas involved and their infrequent application, it was thought best to limit the treatment to the 2-cylinder compound engine. The method of this paragraph is helpful in the study of compound- engine diagrams, the ability to rightly interpret which being impossible without the knowledge due to such an analysis. For instance, the sloping line of the low-pressure diagram has often been attributed to wire- drawing, and in one instance to the author's knowledge, the point in the steam line of the low-pressure diagram of a cross-compound engine caused by high-pressure exhaust closure was thought by a consulting engineer to be the point of cut-off. The distorted diagrams sometimes taken from triple-expansion engines are not the result of improper valve setting or too small ports, but are due to compression and expansion of receiver steam. The sharp corners do not appear on actual diagrams, and with large receivers, both high- and low-pressure diagrams often closely resemble those from simple engines. If plotted to different scales for combining, however, the general form is more like the conventional diagrams. 68. Governing. The speed regulation of compound engines is accomplished in two ways: (1) by direct governor control of both cylin- ders; (2) by direct control of the high-pressure cylinder only. In method (1) the increased or de- creased weight of steam exhausted into the receiver is provided for in the low-pressure cylinder by a lengthened or shortened cut-off, thus maintaining a nearly con- stant receiver pressure, as shown I FIG. 106. in Figs. 99, 102 and 105, in which the receiver pressure at the time of low-pressure cut-off is the same. Assuming an indefinitely large receiver, the receiver pressure would be constant for this method of governing if the low-pressure cut-off were rightly selected, as shown in Fig. 106. Both cylinders then respond to the governor, and the regulation should be practically as sensitive as with a simple engine. In method (2) the low-pressure cut-off is fixed, and for different loads on the engine, equilibrium requires different pressures as shown in Fig. 107, in which, for convenience, the compression curves are shown on THE COMPOUND STEAM ENGINE 173 FIG. 107. the same hyperbolas for all loads, necessitating different points of high- pressure exhaust closure. If these receiver pressures were attained instantly upon change of high-pressure cut-off, the control would probably be as sensitive as with method (1) and perhaps more sensitive, as the high-pressure terminal drop being more uniform, the change of total diagram area is greater for a given change of high-pressure cut-off; but a number of strokes are required to raise or lower the receiver pressure to meet the new load, dur- ing which time the change of high- pressure cut-off must be greater than for method (1) with a con- sequent increased speed fluctu- ation. As the receiver pressure changes, the cut-off gradually changes to that required for equilibrium at the new load. It is obvious that the larger the receiver the longer the time re- quired for pressure adjustment, which, however, means a more uniform receiver pressure, an advantage gained by a large receiver in any case. Slightly better economy is obtained by method (2) according to some engine builders. Were it not for the great labor of determining the changing high- pressure cut-off during receiver-pressure adjustments to correspond to a new load, the equations of the previous paragraph could be used for this purpose. Some idea of the pressure changes may be given by assuming a constant new cut-off. Then taking the data from which Fig. 96 for a tandem compound is plotted, the pressure at low-pressure cut-off is : 7? 4 = PU = 20.2 Ib. The maximum receiver pressure is: p b = 22.15lb. Assuming the low-pressure cut-off as given and the high-pressure cut-off increased to 0.5 stroke, the new value of pn for equilibrium is, from (m) : p 4 = pu= 43.3lb. Then finding p b from (k) and p 4 again from (j) by substituting the value of p 5 just found, and so on, alternating between (j) and (A;), p 4 grad- ually rises and is shown in Fig. 108, in which ordinates are changes of pres- 174 DESIGN AND CONSTRUCTION OF HEAT ENGINES sure from 20.2 to 43.3, and abscissas are the number of strokes since the change of high-pressure cut-off. The calculations were made with a slide rule, but it is obvious from the nature of the operation that theore- tically, the curve would never touch the upper line. Under the assumptions of constant high-pressure cut-off, the engine load must gradually increase as the receiver pressure is built up. For constant load, the high-pressure cut-off would gradually shorten, length- ening the time required to regain equilibrium, as less steam is exhausted into the receiver with each stroke. On engines governing by method (2) there is usually a hand adjust- ment for the low-pressure cut-off. If there is a considerable load change, being fairly constant for quite a period during the day, the cut-off may then be adjusted so as to keep a more uniform receiver pressure and a more even division of work between the cylinders. 43.3 20.2 4 6 8 Number of Strokes FIG. 108. \Z 14 In starting a compound engine, the initial pressure in the receiver is approximately that of the atmosphere. The pressure is gradually built up as in the case of changing load, but requires a longer time. To hasten pressure increase, the low-pressure cut-off may be shortened by the hand adjustment, and gradually lengthened again as the pressure rises, giving a better work distribution between the cylinders. In engines governing by method (1) there is usually a provision for ad- justing the relative high-pressure and low-pressure cut-offs, making it possible to obtain any desired work division. On some engines there is also an independent adjustment for starting, by means of which the low-pressure gear connections may be temporarily broken and a short low-pressure cut-off obtained; then when equilibrium is attained the con- nection is again made and both cylinders controlled by the governor. These various appliances are illustrated in Chap. XX, THE COMPOUND STEAM ENGINE 175 The relative merits of the two methods depend somewhat upon con- ditions. Both are used on high-grade engines. For engines with large overload capacity, especially when the overload may come on unexpect- edly and remain for an appreciable period, method (1) is probably pref- erable. For comparatively small ranges of load, it is simpler to govern only the high-pressure cylinder. With a large receiver and a sensitive governor this method will give good results even with fluctuating loads*. For the rated power, the terminal pressure in the low-pressure cylinder is usually selected, presumably so as to give maximum economy. This may vary from 2 to 4 Ib. above back pressure, but should not be below 5 Ib. absolute. 69.. Maximum Thrust. In Figs. 96 and 99 it may be seen that the maximum pressure in the low-pressure cylinder is greater than the pres- sure at cut-off, which, returning to the notation used previous to Par. 68, is denoted by P 2 . Let the ratio of the maximum absolute pressure to PI be denoted by w, then the maximum piston thrust may be determined. Tandem Compound. The thrust of both high-pressure and low-pres- sure pistons is taken by the main piston rod and transmitted to the other engine parts. Let this be denoted by P x ' } then, neglecting the effect of inertia of reciprocating parts: Px = -i - rnP z ) + (mP 2 - P.) (34 ) Cross-compound. The maximum thrust of the high-pressure piston is, from Par. 66: Px = (Pi - P,) ^ X35) The influence of the receiver is to reduce this somewhat, as may be seen in Figs. 99 and 102, but may be neglected on the side of safety. The maximum low-pressure thrust is: Px = (wiP - P 3 ) (36) The influence of the receiver may not safely be neglected and is provided for by the factor m, to be derived presently. The parts of the high- and low-pressure engines are alike, therefore, the greater value given by Formulas (35) and (36) must be used. Except for small values of R or when very large receivers are employed, the maximum value of P x is given by (36), and should this be much in excess, the high- pressure piston rod may be determined to resist the thrust given by (35). 176 DESIGN AND CONSTRUCTION OF HEAT ENGINES With tandem engines with the low-pressure cylinder next the frame the more usual arrangement (35) may be used for the high-pressure rod; should the high-pressure cylinder be next the frame, the low- pressure rod must be found from (36) and the main rod from (34). The value of m for a tandem engine may be found by taking the ratio P3/P4, Fig. 95, from (s). Substituting the notation of Par. 66 and letting q be the ratio of the receiver volume to volume of stroke of the low-pres- sure cylinder: m (37) R It is obvious from (37) that m increases with / 2 but not in the same ratio; the maximum low-pressure cut-off must therefore be assumed. Table 26 is computed from (37) for y 2 cut-off, assuming that ki = k 2 = 0.04, and with values of q ranging from 0.25 to 1.5, and R from 3 to 8. Table 27 is for. a cut-off. TABLE 26 0.25 0.5 0.75 i 1.5 3 1.53 1.38 .30 1.24 1.18 4 1.69 .47 .36 1.29 1.21 5 1.81 .54 .41 1.32 1.23 6 1.90 .59 .44 1.35 1.25 7 .1.98 .63 .46 1.37 1.26 8 2.05 .67 .48 1.38 1.27 TABLE 27 Q 0.25 0.5 0.75 i 1.5 3 1.80 1.57 1.45 .36 .27 4 2.04 1.71 .54 .44 .32 5 2.22 1.81 .62 .48 .35 6 2.35 1.89 .66 .53 .38 7 2.47 1.95 .69 .56 .39 8 2.58 1.99 .72 .57 .41 THE COMPOUND STEAM ENGINE 177 For a cross-compound engine the maximum value of m is found under Case 1, by the ratio of (s) to (p), which gives, when I* ^ 0.5: 1 + k l - Xi ..I R(q 0.5 + 1 \(q + A* + + 0.5 R (38) Comparing Fig. 97 with Fig. 103, it is obvious that the maximum value of m, neglecting angularity of the connecting rod, occurs when l z = 0.5; substituting this in (38) and assuming that ki = k z = 0.04, and that Xi = 0.8, Table 28 may be computed for different values of q and R. TABLE 28 Q R 0.25 0.5 0.75 1 1.5 3 2.03 1.63 1.45 1.36 1.25 4 2.15 1.69 .49 1.38 1.27 5 2.23 1.73 .52 1.40 1.28 6 2.30 1.76 .54 1.41 1.29 7 2.35 1.78 .55 1.42 1.29 8 2.38 1.80 .56 1.43 1.30 The slight rise in pressure shown in the cross-compound low-pressure dia- grams at the beginning of the stroke may be neglected. This rise is due to the fact that the high-pressure piston is near mid-stroke, traveling at about maximum velocity while the low-pressure piston is leaving the end of the stroke; the high-pressure piston is displacing volume more rapidly for a short time than the low-pressure piston, causing the rise in pressure, usually not more than 1 pound. The effect of inertia of the reciprocating parts has the same effect as with the simple engine. In the latter it was neglected when maxi- mum thrust was considered as explained in Par. 59, Chap. XII. In the compound it tends to offset the effect of m, but except in high-speed engines it may be neglected, which is usually on the side of safety. In any engine, the inertia of the piston only should be deducted from the effect on the piston rod, the inertia of piston and rod from the effect on the crosshead and so on. In view of this, and the possibility of more or less erratic action of a compound engine, inertia may usually be neglected except in its effect upon turning effort. Formulas (34) to (36) may be used in determining the engine parts 12 178 DESIGN AND CONSTRUCTION OF HEAT ENGINES of compound engines, selecting ra from Tables 26, 27 or 28; should R lie between two tabular values the next higher may be taken. It is sometimes said that a tandem compound requires no receiver other than the connecting piping; a comparison of Tables 26 and 27 with 28 will show that if extreme maximum thrust is to be avoided, the require- ments are practically the same as for a cross-compound engine, especially for long-range cut-off. 70. Indicated Horsepower. If the power is to be determined from a test, general Formula (17), Chap. XII may be applied to each cylinder, the sum being the total power. For the purpose of design, (22) to (25), Chap. XII may be applied. Case 1. Single-acting, 2-cylinder compound. = P H D^S P L D L *S 84,000 "" 84,000 ; \ . ': -( + ') ^ From which: (40) From (32): D, Case 2. Double-acting, 2-cylinder compound. D L = 213 __ _ (43) From which: D H may be found from (41). From (22), Chap. XII: - ; ; - ;' i *= ' ,: * ' v;,. (44) where L is the stroke in inches. If P H and P L are from theoretical diagrams they should be multiplied by a diagram factor. In Par. 57, Chap. XII, a quotation from Kent's Mechanical Engineers' Pocket Book is given, with a portion of a table of diagram factors from Seaton. The remainder of the table, applying to marine engines is as follows: * THE COMPOUND STEAM ENGINE 179 Particulars of engine Factor Compound engines with expansion valve to h.-p. cylinder, jacketed, with large ports, etc. . 9 to . 92 Compound engines with ordinary slide valves, -cylinders jacketed, good ports, etc ................................................... 0.8 to 0.85 Compound engines as in early practice in the merchant service, with- out jackets and expansion valves ............. ..................... 0.7 to 0.8 Fast running engines of the type and design usually fitted in war ships 0.6 to . 8. Creighton gives the following diagram factors for stationary com- pound engines: Kind of engine Factor High speed, short stroke, unjacketed ........................... 0.60 to 0.80 Slow rotative speed, unjacketed ................................ 0.70 to 0.85 Slow rotative speed, jacketed .................................. 0.85 to 0.90 Corliss ....................................... ............... 0.85 to 0.9Q Triple-expansion .......................... . .................. 0.60 to 0.70 Bauer and Robertson give for marine compounds: Kind of engine Factor Large engines up to 100 r.p.m .......................... . ....... 0. 60 to 0. 67 Small engines greater than 100 r.p.m ........................... 0.55 to 0.60 Triple-expansion war vessels with high r.p.m .................. 0.53 to 0.54 Triple-expansion merchant vessels up to 100 r.p.m ............. 0.56 to 0.61 71. Theoretical steam consumption may be calculated as in Par. 61, Chap. XII. The weight of steam is measured by the high-pressure cylin- der; then, referring the low-pressure m.e.p. to the high-pressure cylinder, (26), Chap. XII becomes: W = p+p L MIi + fc) - fc(l + *i - *0] (45) where W is the water rate in pounds per horsepower-hour, w, the weight per cubic foot of steam at initial pressure and W B its weight at back pressure. 72. Standard Engines. Standard engine parts may be selected for compound engines in the same manner as for simple engines working under different pressures, as explained in Par. 63, Chap. XII, by equating tiie maximum thrust of the standard engine with that of the compound. For the tandem compound: From which: MP __ -=KD L (46) 180 DESIGN AND CONSTRUCTION OF HEAT ENGINES For the cross-compound : High-pressure cylinder \P^ - I r 2 from (35), D * = D * \~~~ = KL>L Low-pressure cylinder ~ ~ /mP 2 P 3 T ^^ from (36), D * = D *y ~ p = KD ^ As explained in Par. 69, the greater value must be used. Values of K may be tabulated for given pressure ranges, receiver ratios, etc., and it will be found that cylinder ratio has little effect on power if within practical limits. 73. Application of the formulas of this chapter to design will be shown by a number of examples. As noncondensing compounds may be designed by the simple method of Par. 65, the examples will be confined tto the more important condensing compound. Example 1. Let PI = 165, P 3 = 2, P K = 4, p 2 = 6, #1 = 0.9, z 2 = 0.8 and fci = k 2 = 0.04. From (16): /165 \ 4 From (17): R K 2 = 82.5. From (18): P ~ 165 "041b JT 2 , . ~ rr jiU.rr ID. \/6.42 X 82.5 From (19): ,0.14 2X0.24_ " 6.42 20.4 From (20) : 20.4 r 2 = ^- = 3.4. From (21): 1 04 1 2 = ^ - 0.04 = 0.266. From (22) : 20.4 X 6.42 X 0.304 165 From (23) : 1.04 " 0.24 ~ From (24) : 0.201. 165 _ _ r T = = 27.5 THE COMPOUND STEAM ENGINE 181 From (27) : - 14 - 3.5. From (28) : _0.24 Tcz ~ 0.04 = From (29) : P H 165[(0.24 X 2.458) - 0.04] - 20.4[0.9 + (0.04 X 3.5 X 1.2528)] = 69.4 Ib. From (30): P L = 20.4[(0.306 X 2.224) - 0.04] - 2[0.8 + (0.04 X 6 X 1.791)] = 10.6 Ib. The high-pressure terminal drop given by (31) is: di = ^| - 20.4 = 18 Ib. 4.3 The total m.e.p. referred to the low-pressure cylinder is: If H = 1000, S = 800 and the diagram factor is 0.85, (43) gives: d L = 56", and from (41), D n = 22". If L be taken as 48", N = 100. The size of the engine is then: 22" and 56" by 48" - 100. The ratio of low-pressure to high-pressure work is given by (33), and is: The cylinder ratio in Example 1 is above average practice. High ratios have been objected to as lacking overload capacity. Assuming a cut-off of Y hi each cylinder, P 2 , found from (26), is 24.54. Substituting these values in (29) and (30), gives: P H = 133.6 and P L = 21.34. The ratio of low- to high-pressure work is: H L _ 21.34 X 6.42 _ H~ H = 133.6 The m.e.p. referred to the low-pressure cylinder is: + 21.34 = 42.14 182 DESIGN AND CONSTRUCTION OF HEAT ENGINES Then: and the overload capacity is 97 per cent., which seems ample. Assuming a receiver volume equal to 0.5 the volume of stroke of the low-pressure cylinder, the maximum value of m from Table 28 is 1.78 for a cross-compound engine. Then the standard engine having the same parts as the compound, has from (47) for the high-pressure side, a diameter of: D s = ft,..pg~ = 1.0751)* if p S} the standard pressure is 125. As D " = ^A2 Ds = ' 423Di - For the low-pressure side, from (48) : This is larger and should be used. A small receiver was taken for safety, as such a receiver might be used; but for special cases, if standard com- pound engine tables are not to be used, actual values of q may be taken. D - '''" This value is still larger than for the high-pressure cylinder and should be used in this case if q is no smaller than 1. For the tandem compound with a maximum cut-off of % stroke and q = 0.5, m = 1.95; and from (46): - = 0.6770. If q = 1, D s = 0.636D L . Example 2. Assume the same data as before except P K , which is 3 the minimum extreme given. Without taking step by step as was done in Example 1, the principal results are: R = 7.42, P 2 = 19.45, 1 2 = 0.281, Zi = 0.237, P H = 77.7, P L = 10.34, Also: ~ = 0.988 ~ + H = 20.82 d* = 24.55. HH K> For q = 0.5, D^ = 0.513D L for cross-compound, and 0.656D L for tandem. THE COMPOUND STEAM ENGINE 183 Example 3. The same as 1 and 2 except P K is 6, the maximum ex- treme given. Then: ; R = 5.25, P z = 21.8, 1 2 = 0.246, h = 0.161, P a = 57.6, P L = 10.82. Also: 77 Prr ^ = 0.986. V + ^ = 21 - 77 - dl = 10 - 1 - /i^ /t For q = 0.5, D s = 0.54D L for cross-compound, and 0.72Z) L for tandem. If q is 1.5 for this cylinder ratio, the high-pressure cylinder gives a greater thrust than the low-pressure; it is safe to try both (47) and (48) for cross-compound engines. It may be seen that throughout the entire range, Formula (18) gives very close results, and that the maximum variation of the total m.e.p. referred to the low-pressure cylinder is 0.95 lb., or 4.5 per cent, of the smallest value. Due to reasons explained in Par. 47, Chap. IX, the actual variation would probably be even less; this shows that the power of a compound engine depends mainly upon the low-pressure cylinder, the cylinder ratio affecting it but little. The m.e.p.'s of the examples given must be multiplied by a diagram factor before using in the power formula. Taking D L = 56 as just calculated, q = 0.5, and the nearest safe whole number for D s , Table 29 is given so that comparison may be easily made. TABLE 29 Ds Example PK R DH DL C.C. Tandem 2 3 7.42 20 56 29 37 1 4 6.42 22 56 30 38 3 6 5.25 24 56 31 41 D s increases as the value of R decreases, and this' is a measure of the weight and cost of the engine; it also probably has some influence on the friction. A slightly greater maximum power may be obtained from the lower ratio when the low-pressure is of the same diameter in all cases, but ample overload may be carried by engines of higher ratio as shown in Example 1. Example 4. A comparison will now be made with a Nordberg engine referred to in their Bulletin. It is a 19 and 44 by 42 in. cross-compound engine, and the cylinder ratio from (32) is: 184 DESIGN AND CONSTRUCTION OF HEAT ENGINES For the particular test mentioned: Pi = 169.18, P 3 = 1.91 and the receiver pressure, which would be somewhere between P 2 and raP 2 , was 20.75 Ib. Actual m.e.p. 's were fiP H = 52.605, and/ 2 P L = 9.557. The total m.e.p. referred to the low-pressure cylinder was 19.352. Clearance and compression are not given, and it will be assumed that ki = 0.07, & 2 = 0.05, xi = 0.9, and z 2 = 0.8; also that p 2 = 6 Ib. With the above data, the results from the formulas of Par. 66 are: P 2 = 21.7, Z 2 = 0.24, h = 0.14, P H = 61.4 and P L = 11.21. Using a diagram factor of 0.85 : fiP H = 52.2 and / 2 P L = 9.55. The total m.e.p. referred to the low-pressure cylinder is 19.28, differing from the test value by less than 0.5 per cent. Assuming a receiver volume 1.5 times the volume of stroke of the low- pressure cylinder, (38) gives: m = 1.028. The maximum receiver pres- sure is then : 1.028 X 21.7 = 22.3. This is 1.55 Ib. greater than the actual receiver pressure given, a result to be expected due to shrinkage of the diagram as shown by the diagram factor. The data of Example 1 is plotted in Figs. 96 and 99, the dotted lines for an indefinitely large receiver and the full lines showing the effect of a receiver of practical volume. CHAPTER XIV THE INTERNAL-COMBUSTION ENGINE 74. Introduction. In the treatment of this chapter a knowledge of the contents of Chaps. V and VI is assumed. A thorough understanding of the principles of operation of the internal-combustion engine can not be had without following through the power formulas based upon heat quantities; but certain practical formulas have value and are derived in this chapter; their constants are also given in terms of the quantities in the more theoretical formulas. Notation. h = heating value (high or low) in B.t.u. per pound of liquid fuel, or per cubic foot of gaseous fuel at standard temperature and pressure. a = actual air supply in cubic feet per pound of liquid fuel, or per cubic foot of gaseous fuel, at standard temperature and pressure. e M = mechanical efficiency of engine. 6 = indicated thermal efficiency of engine. e B = thermal efficiency referred to brake horsepower; called by Gtildner, economic efficiency. e v = volumetric efficiency the ratio of the volume of the charge (fresh air and fuel) at standard temperature and pressure, to the volume of stroke. See Chap. VI, Par. 27. P M = mean effective pressure in pounds per square inch. p unbalanced pressure in pounds per square inch at any point of stroke. p x = the maximum value of p. P = total unbalanced pressure at any point of stroke. P x = maximum value of P. D = diameter of cylinder bore in inches. L = length of piston stroke in inches. N = r.p.m. S = mean piston speed in feet per minute, ra = number of strokes per cycle. H horsepower in general. 185 186 DESIGN AND CONSTRUCTION OF HEAT ENGINES HI = indicated horsepower per working cylinder end. H B '-= brake horsepower per working cylinder end. H M = maximum horsepower, indicated or brake. H R = rated horsepower, indicated or brake. = JT" HR = liquid fuel used in pounds per hour. = gaseous fuel used in pounds per hour, at standard temperature and ressure. W F V F and pressure. 75. Mean Effective Pressure. The absence of accurately predeter- mined pressures possible in the steam engine makes the study of indicator diagrams for the internal-combustion engine more difficult, and a good collection of diagrams more desirable. These diagrams are useful for determining the proper yalve setting, timing of ignition, best fuel mixture, and for finding the output and efficiency of the engine. However, the uncertainty of limiting pressures, the lack of uniformity of assumptions for a theoretical diagram and the wide deviation of the actual from such a reference diagram, makes the use of the diagram factor (see Chap. XII, Par. 57) for determining the m.e.p. less satisfactory than for the steam engine and other methods of calculation are more in favor. In Par. 29, Chap. VI, the following formulas are derived from a more general discussion in Par. 23 of the same chapter. For gaseous fuel P M = 5Aee v , . (1) For liquid fuel. P M = 5Aee v - (2) These apply to both 2-stroke and 4-stroke cycles, the difference in the values of P M for the two cycles being usually due to the different values of e and e v , the indicated thermal efficiency and volumetric effi- ciency respectively. As explained in Chap. VI, this is the mean pressure which, if exerted throughout one stroke of the piston, would do the work of the cycle for one working cylinder end. 76. Horsepower. In Chap. VI, Par. 23, the following general formula for indicated horsepower is derived for one cylinder end: _ 2 X 144 P M v a N ,ox : 33,000 ~^T where N is the r.p.m., m the number of strokes per cycle, and v s the volume of stroke in cubic feet. THE INTERNAL-COMBUSTION ENGINE 187 If L is the length of stroke in inches, and D the diameter of the cylin- der bore in inches: 7T D 2 L Vs ~ 4 * 144 " 12 Substituting in (3) gives: P M D*LN P M D*S 1 252,100m 42,000m where S is the mean piston speed in feet per minute, which is: 7 P^-f'ISS (5) The brake horsepower is: e M P M D 2 LN MM ... euHl ~ 252,100m == 42,000m For gaseous fuel, substituting (1) in (4) and (6) gives: h D 2 LN h D 2 S ' F ^~TT ' 46,700 m = ev a + 1 ' 7790 m And as e Af e = e B : ft D*LN h D 2 S ,_. #* = ^^^'46j6^ = ^ r ^n-779^ From (7) and (8) : D = 88.25 J mHl(a + l) = 88.25 > g ^ a + D (9) \ ee F /i \ e B e v Sh For liquid fuel, substituting (2) in (4) and (6) gives: Ri = eey h. D 2 LN = eev h D 2 S h D*LN h D 2 S HB=eBev a'WlW^ = Bev a'^^ (1]) From (10) and (11): Z)= 88.25^g| = 88.25 Although derived in a different manner, these equations give results identical with the formulas of Giildner, which are in terms of brake horsepower. Formulas (9) and (12), being used for design, are in terms of piston speed rather than stroke and revolutions. Some discussion of piston speed is given in Par. 48, Chap. IX; with proper design there seems to be no restrictions within reasonable limits, other than those of gas velocity, which will be discussed in Chap. XX. 188 DESIGN AND CONSTRUCTION OF HEAT ENGINES The relation of S, L and N may be adjusted by means of (5) when the cylinder diameter has been determined; if N is fixed by reason of service considerations, L may be found. If not, the question is one of the ratio of stroke to diameter; this ranges from 1 to 2, Diesel engines commonly having the ratio 1.5. Some discussion of this ratio is given in Par. 47, Chap. IX, under Design of Cylinders, and insofar as it concerns the weight of the engine parts it applies to internal-combustion engines. For an extensive discussion of the subject, the reader is referred to Giild- ner's Internal-combustion Engines. The heating value of fuel and theoretical air supply are given in Table 30. Also see Par. 42, Chap. VIII. The actual air supply must be greater than the theoretical to insure perfect combustion; aside from this there should be an excess of air to provide for an increase of fuel supply for overloads. Even if an engine is rated at its maximum capacity there are enough uncertain factors in its design to make it unsafe to allow too small a margin over the air theoretically required for perfect combustion. Table 30 is abridged from Guldner's Internal-combustion Engines, and TABLE 30 Fuel h (low) per a (theoreti- cal) per a (actual) per e B for H B per cylinder end of Cu. ft. Lb. Cu. ft. Lb. Cu. ft. Lb. 5 10 25 50 100 and over Illuminating gas lean 505 5.5 7.5 0.20 0.22 0.24 0.26 0.27 Illuminating gas rich 675 6.5 10.0 0.20 0.22 0.24 0.26 0.27 1.1 Producer gas anthracite. . . . 141 0.85 to 0.17 0.19 0.21 0.23 0.24 1.4 1.0 Blast-furnace gas 106 0.75 to 0.18 0.20 0.22 0.24 1.2 Coke-oven gas 505 5.3 7.0 . 0.17 0.19 0.21 0.23 257 Kerosene 18,900 185.0 to 0.11 0.12 0.13 353 288 Crude oil Diesel 18,000 176.0 to 0.25 0.26 0.27 0.30 0.315 323 240 Gasoline 18,500 176.0 to 0.19 0.21 0.23 323 128 Alcohol (90 per cent, vol.) 10,300 96.5 to 0.22 0.24 0.26 193 may be used as a guide in the selection of h, a and e B . Values of H R are for one cylinder end. Both theoretical and practical values of air supply THE INTERNAL-COMBUSTION ENGINE 189 are given, the latter being based upon an overload capacity of from 15 to 20 per cent. Volumetric efficiency is discussed in Par. 27, Chap. VI; some practical values from Gtildner are given in Table 31. The values from Guldner's TABLE 31 Type of engine e v Slow-speed, mechanically operated inlet valve . . Slow-speed, automatic inlet valve High-speed, mechanically operated inlet valve. . High-speed, automatic inlet valve 0.88 to 0.93 0.80 to 0.87 0.78 to 0.85 0.65 to 0.75 tables are based upon a standard pressure of 14.7 Ib. per square inch and 32 degrees F., although he recommends 28.92 in. of mercury and 59 degrees F. The A.S.M.E. Code adopts 30 in. of mercury and 60 degrees F. The mechanical efficiency e M varies from 0.6 to 0.9 and is discussed in Chap. X. Simple Formulas. Thus far the formulas have been of a general character and these are necessary for a comprehensive study of engine power; but by making substitution of experimental values of P M for rated power in (6), or of h, a, e v and e B in (8) and (11), a simple formula may be derived. E. W. Roberts, in The Gas Engine Handbook gives (the notation being changed) : ., ., in which the constant K differs for different fuel and with the type of en- gine. This may be written: Then: D = 0.41^^i (15) Comparing (14) with (6) and (8), the value of K may be expressed in terms of the quantities in these formulas; or: K = MI^ (16) also: K = 46 > 7QQm ( a + 1) / 17) e B e v h As a -f 1 is but slightly different from a for liquid fuels, (17) may be applied to all internal-combustion engines. 190 DESIGN AND CONSTRUCTION OF HEAT ENGINES Roberts gives for the average size of 4-cycle engine: For natural gas K = 16,000; For gasoline K = 14,000; stating that K becomes smaller as the cylinder size increases. He cites a case of an automobile engine with exceptionally large valves and ball bearings in which K was 11,500. Smaller values have been obtained notably in airplane engines (see Table 32). For 2-cycle engines Roberts gives: For gas K = 12,000; For gasoline K = 10,000; For large blowing engines K = 8,400. It is claimed that these values are approximate. The Bruce-Macbeth Engine Co. published a formula for 4-cycle engines which gives : v 1,000,000 K - p*- in which P B varies as follows: For natural gas, P B = 75 lb., and K = 13,330; For producer gas, P B = 60 lb., and K = 16,660; For gasoline, P B = 60 lb., and K = 16,660. A comparison of this formula with (16) shows that P B is practically equal to e M PM, the brake m.e.p. The indicated m.e.p. is then equal to P B divided by e M > Comparing the constants of 4-cycle and 2-cycle engines given by Roberts, the 2-cycle has but 40 per cent, greater capacity than the 4- cycle. Substituting the respective values* of K in (17), assuming a and h the same for both engines, the value of the product of e B e v for the 4- cycle engine is 43 per cent, greater than for the 2-cycle. This difference is probably in part due to e B , but largely to e v . Comparing the value of K for the large 2-cycle blowing engines mentioned by Roberts with that of the Bruce-Macbeth producer-gas engine gives practically equal values of the product e^v- This is no doubt due to the greater value of e v made possible by the separate charging cylinders of the large 2-cycle engine; this is probably greater than for the 4-cycle engine, while e B is smaller. Taking K from engine ratings or tests, and assuming e B , e v and h (or taking them from Table 30 or from tests), the air supply may be determined from (17) ; or: Ke B e v h 1 " 46/700^ - X THE INTERNAL-COMBUSTION ENGINE 191 A comparison of engine ratings for a given fuel may thus be made; the smaller the air excess the nearer to the maximum power is the engine rated, remembering that there must be some excess air at maximum power. Also by assuming e M , the value of P M may be found from (16). The influence of e v upon capacity, and undoubtedly upon efficiency, is considerable, and every practical means available for increasing this should be employed. Ample valve openings, easy passages and proper timing are probably the chief factors in a high value of e v . From Par. 27, Chap. VI, it is obvious that the temperature of the charge directly affects capacity by determining in part the value of e v ; the other factor is the pressure of the charge, which depends upon valves, passages, etc. as just stated. The value of e B may be kept a maximum by proper compression, carburetion, ignition and lubrication, in addition to the factors just mentioned. Table 32 contains some data pertaining to engines of different type and capacity, taken from builders catalogues, published results of tests, and information furnished the author directly. These may be used for comparison. TABLE 32 Type m Number of cyl. HB per cyl. D (in.) L (in.) L/D N s K CM PM Stationary Stationary Stationary 4 4 4 1 1 1 1.5 5.00 7.00 3M 5M 6 4 7 8K .14 .27 .41 550 425 375 366 496 532 18,000 18,000 16,400 0.80 0.80 0.80 70 70 78 Stationary 4 1 15.00 8% 14 .62 250 584 16,400 80 78 Automobile 4 4 10.00 3H 5K .61 2,800 2,450 15,500 80 81 Automobile . . . 4 6 5.00 3% 4 .10 1,500 1,000 15,750 78 83 Automobile Automobile 4 4 6 8 7.50 8.75 3% 3% 4M 5M .38 .64 2,600 2,400 1,950 2,050 17,500 13,700 0.78 80 74 92 Boat ? 1 7.00 *H 4 .84 750 500 9,700 74 70 Boat ? 1 12.00 5H 5 .87 675 563 9,300 76 71 Boat Boat 4 4 4 6 7.50 8.34 4^ 3% 5H 5K .27 .40 1,000 1,300 875 1,140 11,900 11,550 0.80 0.78 106 112 '3 1 Airplane Hot bulb Hot bulb Diesel 4 4 4 4 6 1 1 3 20.00 100.00 140.00 150 00 4^ 17 20 21 6K 27K 34 H 30 .37 .62 .72 .43 1,350 200 165 180 1,465 917 950 900 9,900 15,900 16,250 15,900 0.78 0.85 0.85 75 130 75 73 91 Diesel 4 4 125.00 18% 28% .50 164 775 13,200 73 105 Diesel ? 4 200.00 20 36 .80 95 570 6,840 70 105 O Natural gas Natural gas (D.A.) Blast-furnace gas (D.A.) 4 4 4 4 2 2 37.50 112.00 500.00 12K 20 45 14 36 60 .14 .80 1.33 275 125 70 640 750 700 15,400 16,100 19,000 0.80 0.81 0.84 82 78 63 Values of K and P M have been calculated, the latter by assuming a mechanical efficiency determined by Formulas (10) and (11), Chap. X. 192 DESIGN AND CONSTRUCTION OF HEAT ENGINES Until recently but little information was available concerning the Diesel engine. Much valuable information is contained in publications of the A.S.M.E., a few notes from the paper mentioned at the end of Chap. V being given at this place : Comparing the 4-cycle and 2-cycle Diesel, the latter gives from 70 to 80 per cent, more power for the same cylinder dimensions and speed. The 4-cycle has about 10 per cent, better efficiency and the mean tem- perature is lower. In this country, the 4-cycle engine is built in sizes up to from 700 to 1000 b. h.p., and above that the 2-cycle is used. The Diesel engines built in the United States run from 150 to 300 r.p.m., with piston speeds from 600 to 900 ft. per minute. Some high- speed marine engines run as high as 480 r.p.m. In Europe the highest commercial speed is from 350 to 400 r.p.m., with as high as 550 for sub- marines. Practically all Diesel engines are single-acting, and nearly all in this country have trunk pistons without crossheads. The mechanical efficiency at full load is about 75 per cent, for 4-cycle and 70 per cent, for 2-cycle engines, i Some Diesel engine data are given in Table 32. The m.e.p. of internal-combustion engines of small size and high speed is seldom known; it is sometimes calculated by assuming the mechanical efficiency. Some values of P Mj also of h are given in connection with com- pression pressures in Table 35, Par. 77, which may be compared with those already given in Table 32. E. W. Roberts, in The Gas Engine of October, 1917, gives values of e M PM for a number of engine types, which represent recent practice in the United States. These values are given in Table 33; values of e M are assumed, from which the values of P M and K are com- puted and given in the table. TABLE 33 Type *M P M e M PM K Hot bulb 2-cycle 33 0.70 47 15,300 Producer gas 57 0.80 71 17,700 Gasoline 2-cycle 60 0.75 80 8,400 Modern high-speed gasoline 80 0.80 100 12,600 Modern airplane Diesel. 4-cycle 105 75 0.80 0.73 130 103 9,600 13,500 It is to be assumed that the engines are 4-cycle unless otherwise stated. The mechanical efficiency of the airplane engine is perhaps too small; indications are that a much better value has been obtained. THE INTERNAL-COMBUSTION ENGINE 193 From the various data given it is obvious that while there is a general agreement, there is no definite standard of engine rating, and if there were it is not likely that capacities found from tests would always agree with the assumptions. With new work it is well to determine D from (9) or (12), using data from Tables 30 and 31. This result may be checked by (15), taking K from Table 32 or 33; or by (6), taking P M from Table 32, 33 or 35, and e M from (10) and (11), Chap. X, or from Table 32 or 33. With assumed values of e B , e v and h, the theoretical maximum capacity limit; is when a is the theoretical air supply required for perfect combustion. The actual maximum power will be obtained with a value of a greater than this, the power always being less than the theoretical maximum. It is apparent that the more perfect the mixture and ignition, the smaller will be the excess of air required. Should Formula (6) or (15) be used for determining cylinder dimen- sions, the air supply should be checked by (18), and if equal to, or less than the theoretical minimum, the engine will not develop the power desired assuming that e B} e v and h have been rightly chosen. The effect of changing the standard temperature from 32 to 60 degrees F., and in using higher instead of lower heating value of the fuel is easily found. From this it is clear that h jh - , 1 - e v and - e v a + 1 a do not change with change of standard temperature, and that at any standard temperature, eh or e B h is constant for a given fuel whether higher or lower value is taken as the standard. At very high speeds the power is limited by wire-drawing as discussed in Par. 60, Chap. XII, relative to steam engines; and in like manner the range over which power is proportional to piston speed may be increased by large valves, free passages, proper valve setting, etc. Maximum horse- power is reached at a certain speed, beyond which the power decreases. This does not mean, of course, that the power is not available at speeds much beyond that giving maximum power, as the power requirements may be much less than the maximum, as in automobile propulsion with good roads and other favorable conditions. Power and speed curves are given in Par. 22, Chap. V. Fuel Consumption. From equations (12) to (14), Chap. VIII, the fol- lowing formulas are derived: let WF = the weight in pounds of liquid fuel used per hour. V F = the volume in cubic feet at standard temperature and pressure, of gaseous fuel used per hour, 13 194 DESIGN AND CONSTRUCTION OF HEAT ENGINES h = the heating value per pound of liquid fuel or per cubic feet of gaseous fuel at standard temperature and pressure. Then: 2545# eh and: V F = eh (19) (20) If H is i.h.p. or b.h.p., e must be the corresponding efficiency. If h is the higher or lower value, e must be based upon it. To find the fuel per h.p.-hr., divide by H. 77. Compression. In Chap. VI, Formulas(ll) and (17) give the effi- ciencies of the constant- volume and constant-pressure cycles respectively. From these it is apparent that the higher the compression the greater the efficiency. There are, however, practical limits. In engines com- pressing the mixture of fuel and air, ignition will occur prematurely from the heat generated thereby if compression is carried too far. The igni- tion point varies with different fuels and with different mixtures. Within certain limits compression may be increased to advantage by employing a lean mixture poor in fuel; but above a certain limit, the m.e.p. de- creases too rapidly, reducing the mechanical efficiency as explained in Chap. X. The theoretical thermal efficiency increases but slowly beyond a certain increase in compression, and a point is reached where the product of mechanical and thermal efficiency will decrease if compression is carried higher. This product, as already shown, is the thermal efficiency at brake, or true efficiency. This is shown in Fig. 109, taken from Gtildner. THE INTERNAL-COMBUSTION ENGINE 195 It is not well to approach too near this limit. The friction load is increased by high compression and unless there are positive advantages to be gained, moderation should be practised. Engines with moderate compression have sometimes shown remarkable economy, showing that other features of design offset this seeming advantage. With the medium- and high-compression oil engines, compression is carried high enough to insure complete combustion. The mechanical efficiency of these engines is low, but is more than offset by the attain- ment of high compression without in any way sacrificing the charge weight. In the methods of power determination employed in the preceding paragraph theoretical efficiency is not considered, therefore compression pressure does not enter into the discussion. By avoiding extremes, it is probable that considerable leeway may be allowed in the selection of compression pressure, but in the use of the formulas of Par. 75 it is assumed that a pressure suitable to the conditions of operation is employed. The same engine may sometimes be used for several different fuels, but better efficiency or capacity might be obtained if the compression were changed. Table 34 gives compression pressures in pounds per square inch, absolute, according to the authorities noted, while Table 35 was given by R. E. Mathot in Power, April 11, 1916. The m.e.p.'s in the latter table are maximum figures from well-designed engines working under favorable conditions with an overload capacity of from 10 to 15 per cent. Table 35 gives values from actual practice covering over 600 tests by Mr. Mathot, on about 40 different makes of European and American engines. TABLE 34 Type of engine Abs. comp. pressure Authority Gasoline 45 to 95 Lucke Kerosene (hot bulb) . Natural gas 30 to 75 75 to 130 Lucke Lucke Citv cas 60 to 100 Lucke Producer gas 100 to 160 Lucke Blast-furnace gas 120 to 190 Lucke Natural gas 95 to 110 Roberts Illuminating gas 105 Roberts Gasoline 95 Roberts Kerosene 75 Roberts Blast-furnace gas 142 Roberts Producer gas 115 to 130 Roberts 196 DESIGN AND CONSTRUCTION OF HEAT ENGINES TABLE 35 Kind of fuel h (high) Comp. press, absolute PM Blast-furnace gas Producer gas . 100 135 185 to 215 155 to 185 70 75 Natural gas 900 155 to 165 90 Coke-oven gas 550 135 to 145 80 Illuminating gas (coal gas) Kerosene .... ... 650 20,000 125 to 155 55 to 85 85 60 Kerosene (with water injection) 20,000 85 to 100 60 Benzol (industrial engine) 17,000 85 to 100 70 Benzol (automobile engine) 17,000 100 to 115 90 Alcohol (industrial engine) 13 000 115 to 145 85 Alcohol (automobile engine) 13 000 125 to 175 95 Crude oil (Diesel engine) 18,500 515 105 Crude oil (2-cycle hot bulb engine) 18500 155 to 175 45 Crude oil (4-cycle, hot bulb engine) Tar oil (Diesel engine) 18,500 18,000 275 to 315 615 75 100 78. Governing. There are two general methods employed to regulate the speed of internal-combustion engines. The first is the intermittent impulse system in which the working cycle is always the same, but at loads less than the maximum the number of cycles per minute is reduced by skipping. This is known as hit-and-miss governing. It is used on many small engines and is economical, as the combustion of each charge takes place under the best conditions. Due to the irregularity of its impulses, especially at light loads, it does not regulate closely; this may be in part offset by a very heavy flywheel, and hit-and-miss engines are sometimes used to operate small dynamos, but for larger powers where close regulation is required, and for driving alternators for parallel operation, they are impracticable. The other general method is the variable impulse system. In this system the m.e.p. is changed as in the steam engine, but no cycles are skipped. This system has two subdivisions: the quality method, in which the proportion of fuel to air is varied; and the quantity method, in which a charge of constant quality is admitted during a portion of the suction stroke, or more commonly, is throttled during the entire stroke. Sometimes a combination of the quality and quantity methods is used. The Quality Method. In this method as applied to gas and light-oil engines, the air inlet valve admits a full charge; the fuel supply is under the control of the governor and is throttled at light loads. The com- pression pressure is constant, which is considered a theoretical advantage. THE INTERNAL-COMBUSTION ENGINE 197 Fig. 110 is an indicator diagram from a gas engine using quality regulation, the full lines showing full load and the dotted lines lighter loads. From medium to full load this method gives good results. For very light loads the mixture is so lean as to be difficult to ignite and slow burning, leading to the possibility of skipping and poor economy. FIG. 110. Quality regulation. FIG. 111. The quality method is used for the heavy-oil engine, but in this, igni- tion depends upon the high temperature of the highly compressed air, and all fuel delivered to the cylinder, no matter how small the quantity, is perfectly consumed. As theoretically indicated by Formula (17), Chap. VI, the shorter the combustion line sometimes called the cut-off the greater the thermal efficiency, and this is true in practice. This is, of course, the indicated efficiency, the reduced mechanical efficiency at very light loads offsetting this gain. Fig. Ill is an indicator diagram for a Diesel engine, the full lines being for rated load and the dotted lines for a lighter load. The Quantity Method. This method with variable admission, as usually employed, admits a charge of uniform quality during a portion of the stroke, when the valve is quickly closed by a trip mechanism similar in principle to the gear of a Corliss steam engine. Assuming a full-load diagram similar to Fig. 110, a light-load diagram (exagger- ated for the purpose of illustration) is shown in Fig. 112. The charge is ad- mitted along the suction stroke from a to b when the inlet valve suddenly closes. FlG - 1 12. Quantity method with _. . , , . , - . , , . variable admission. During the remainder 01 the suction stroke the charge is expanded from b to c, and theoretically, at least, passes over the same curve on the portion of the compression stroke from c to b. There is then no fluid-friction loss during this part of the cycle, the only loss of this character being the usual suction and back-pressure 198 DESIGN AND CONSTRUCTION OF HEAT ENGINES FIG. 113. Quantity method throttling governor. with losses caused by the normal valve openings as shown by the shaded area of Fig. 112. In the quantity method with throttling governor, a charge of uniform quality is throttled throughout the suction stroke. This is best shown by another exaggerated diagram in Fig. 113, assuming the same weight of charge or the same pressure at the end of the suction stroke. The fluid-friction loss is indicated by the dotted area. In reality, the shaded areas in both Fig. 112 and Fig. 113 are relatively small, and while the cut-off method might be expected to give slightly better economy, the throttling method is more often used, due no doubt to the simpler mechanism required. The usual argument against quantity governing is the low compression pres- sure at light loads; this tends to reduce the rate of combustion, but with a properly proportioned mixture this is not so noticeable. Combined Methods. By employing as lean a mixture as ^will give reasonable economy at the minimum load, and using the quantity method between this load and the lightest load obtained with maximum compression; then if the quality method be employed for loads greater than this, it is probable that better results may be obtained over a wider range of power. According to Power, this method is used with the 5000 horsepower gas engine of the Ford Motor Company. The hit-and-miss method might and probably has been applied to the lower pressure range, combined with either the quality or quantity method, and when the closest regulation is not required should give good economy over a wide range of power. This arrangement might be applicable to rolling-mill engines and should give as good regulation, surely, as the single-eccentric Corliss engines formerly used for such service. More will be said about hit-and-miss governing in Chap. XVIII. The apparatus for effecting the different methods of governing will be ex- plained in Chaps. XIX and XX. 79. Rating. Internal-combustion engine rating is not as uniform as that of the steam engine. With the larger gas engines and oil engines, there is usually some provision for overload. As previously stated, THE INTERNAL-COMBUSTION ENGINE 199 Giildner allows from 15 to 20 per cent. A number of manufacturers of Diesel engines allow 10 per cent, overload for a given time such as two hours, and state that much larger loads have been carried for a limited time. It is apparent that some small engines are rated at their maximum capacity, or at the maximum load at which they will operate continuously without giving trouble. While desirable, it is not essential to have absolute uniformity in rat- ing, so long as the conditions of rating are plainly stated. All internal- combustion engines have a maximum capacity at which they will run continuously without heating, and in most cases may exceed this capacity for a limited time. This condition may just fill the bill for certain work, and manufacturers should determine these limits as accurately as possible and use them conservatively in their specifications. The minimum ca- pacity of satisfactory operation should also be known. With these data at hand, and a carefu^y plotted economy curve, the mill owner, con- sulting engineer, and tho engine manufacturer himself will be able to handle power problems with accuracy and meet all guarantees. The lack of overload capacity is often mentioned as an objection to the internal-combustion engine, and indeed, when the Diesel engine is rated to give 10 per cent, overload, and it is known that a steam engine with long range cut-off will run continuously and satisfactorily barring economy with an overload of from 50 to 100 per cent., the flexibility of the steam engine is certainly attractive for certain kinds of work. Take for example, a 200-h.p. steam engine designed for 50 per cent, overload; a 200-h.p. Diesel engine as usually rated will carry a maximum load of 220 h.p., while the steam engine will carry 300 h.p. Is this a fair comparison? Both engines operate with maximum economy at 200 h.p., at least approximately. A Diesel engine rated at 273 h.p. would be required to develop a maximum power of 300 h.p.; then at 200 h.p., the rated power of the steam engine, the Diesel carries about 73 per cent, of its rating. But how does the economy at this load compare with the economy of the steam engine at the same load? In Fig. 114 are plotted two curves showing B.t.u. per minute at different powers for a Diesel engine and a noncondensing, uniflow steam engine. Some assumptions had to be made in reducing the data so that a comparison might be made, and the chart may be considered as only suggestive. If 200 h.p. is assumed as the rated power of the Diesel, it will not be the most economical load, but it will give better economy than the steam engine" at this load, and will have the same overload capacity 200 DESIGN AND CONSTRUCTION OF HEAT ENGINES If the maximum power that may be developed continuously by an internal-combustion engine is denoted by H M and the rated power by H R , the ratio of the maximum to the rated load is: q = ^. H R The fraction of the maximum load to give a certain rated load is: 1 - -E M - 400, .350 50 100 150 ZOO 250 300 350 I.H.R FIG. 114. or: The overload capacity in per cent, is : Overload = 100(0 - 1) Per cent, overload . (21) Table 36 gives the required percentage of the maximum capacity required for the rated load, to allow a given overload capacity. TABLE 36 Overload in per cent, of rated = 100 (q - I).. 10.0 91.0 20.0 83.3 30.0 76.9 40.0 71.4 50.0 66.6 60.0 62.5 70.0 58.8 80.0 55.5 90.0 52.6 100.0 50.0 Rated load in per cent, of maxi- mum = lOO/^ THE INTERNAL-COMBUSTION ENGINE 201 Then with a curve such as Fig. 115, showing the fuel consumption at different percentages of the maximum load, the rated-load economy for any overload requirement may be determined, and guarantees intelli- gently made. Any means of increasing the economical load range, such as suggested in the preceding paragraph, will increase the possibility of providing for overloads with good economy for the rated and lighter loads. 80. Indicator Diagrams and Maximum Piston Thrust. Actual indi- cator diagrams are of great value to the designer, and some knowledge of them absolutely essential; but, as with the steam engine, it is of great convenience to be able to plot conventional diagrams which approximate actual diagrams, and these may often be used as a basis of design. tr-o |.6 CQ ^ ,' -o \ X --. ^ 1^ .1 .Z .3 .4 .5 .6 .7 .8 .9 1.0 Fraction of Max. Load FIG. 115. In Par. 30, Chap. VI, a method is given of plotting diagrams for the constant-volume and constant-pressure cycles. P M may be found from Formulas (1) or (2), or assumed; then if the compression pressure of the constant-volume cycle is assumed the explosion pressure may be found by a simple calculation. The compression pressure must also be assumed for the constant-pressure cycle in order to determine the clearance volume; the volume at the end of combustion may then be found by trial and error. The exponent of the curve must be chosen so as to give a curve as near the actual as possible ; then in most cases with the constant- volume cycle, the clearance volume will be the practically required volume. The value of n varies in practice, and is not usually the same for both compression and expansion curves, although so assumed in order to simplify calculation. The value is usually taken from 1.3 to 1.35, but is sometimes as high as 1.5 for part of the curve. It is not constant in most cases but an average value may be assumed which will give close approximations. Roberts gives 1.3 as the exponent of the compression curve and 1.35 for the expansion curve. An actual constant-volume 202 DESIGN AND CONSTRUCTION OF HEAT ENGINES diagram is given in Fig. 116, with a conventional diagram giving the same m.e.p. The diagrams are drawn separately as part of the outline coin- cides, but are to the same scale. A pencil tracing may be made of one, and by laying it over the other a comparison may be made. The m.e.p. of the actual diagram is 112 Ib. Then from (36), Chap. VI, the pressure rise was found to be 344 Ib., assuming the exponent to be 1.3; the compression pressure is 100 Ib. per square inch gage and r is found from (31), Chap. VI. The suction pressure was taken as 14.7 Ib. absolute, giving a ratio of clear- ance volume to volume of stroke of 0.26 (see Formula (33), Chap. VI). Table 37 gives ratios of clearance to piston displacement for different gage compression pressures and ab- solute suction pressures, taken from a FlG - 116< data card by the Bruce-Macbeth Engine Co. It is stated that 14.1 suction pressure is to be used from sea level to an altitude of 1000 ft.; 13.5 Ib. from 1000 to 2000 ft.; 13 Ib. from 2000 to 3000 ft., and 12 Ib. from 4000 to 5000 ft. The value of the exponent is 1.3. TABLE 37 Ratio of clearance to piston displacement for suction of: VJCl^U JJ1 COQU.J. C 111 IU. 14.1 13.5 113 12 t 80 0.302 0.291 0.282 0.264 90 0.274 0.263 0.255 0.239 100 0.250 0.241 0.233 0.218 110 0.231 0.223 0.215 0.202 120 0.215 0.207 0.200 0.187 130 0.201 0.194 0.187 0.175 140 0.189 0.182 0.176 0.165 150 0.178 0.172 0.166 0.156 160 0.169 0.163 0.158 0.148 170 0.161 0.155 0.150 0.141 180 0.153 0.148 0.143 0.134 190 0.146 0.141 0.137 0.128 200 0.140 0.135 0.132 0.123 210 0.135 0.130 0.126 0.118 220 0.130 0.125 0.122 0.114 THE INTERNAL-COMBUSTION ENGINE 203 FIG. 117. The conventional diagram of Fig. 116 was first drawn with a vertical explosion line; then the corners were rounded, the explosion line made slightly leaning, and from where it meets the maximum pressure line a new expansion curve is drawn with the same exponent. This increases the area, offsetting the rounded corners. The modified diagram is shown in heavy lines. If desired, the explosion line maybe left vertical and connected to the expansion line by a curve of small radius, and the maximum pressure retained for safety, as this diagram is mainly used for computing the forces acting on engine parts due to gas pressure. Fig. 117 is a full-load diagram for a Diesel engine, and a con- ventional diagram. The m.e.p. is 86 lb., but at this rating the en- gine will carry an overload of 35 per cent. While it is sometimes stated that the exponent of Diesel curves may be as high as 1.5, the conventional diagram of Fig. 117 has rectangular hyperbolas for expansion and compression curves, and it appears that an exponent less than unity would give a somewhat closer agreement with the expansion curve. Although this diagram agrees with the actual diagram better than any with an exponent greater than unity, the equation of the hyperbola could not be used to determine the clearance volume, because the compression curve is more like an exponential curve. Giildner says that for usual loads the value of c (Par 30, Chap. VI) is 2.5, and the ratio of the "cut-off" (so called because it appears like the cut-off of a steam-engine diagram) to the entire stroke is 0.1. This gives a compression ratio of 16, and with a maximum compression pressure of ___ _ 500 lb. gage and a suction pressure of 14.7 lb. absolute, the exponent for the compression curve is 1.28. Using this also for the expansion curve, the m.e.p. from (41), Chap. VI, is 119 lb. As this is from a conventional diagram it must be multiplied by a diagram factor. If this is 0.88, the m.e.p. is 105 lb., a value often found. The actual compression curve of Fig. 117 is reproduced in Fig. 118 FIG. 118. 204 DESIGN AND CONSTRUCTION OF HEAT ENGINES and compared with a curve having an exponent of 1.28, shown dotted. For a force diagram it is likely that any of these curves will give accurate enough results. In using (41), Chap. VI, the value of e may best be found by assuming some value near 2.5 and solving for P M - If this is greater than the re- quired value, decrease to 2.4 and so on; a few trials will give a value near enough for practical use. If the hyperbola is used, the formulas in Par. 56, with Fig. 82 of Chap. XII may be used, in which x will be zero. Maximum Diagram. In design, the chief value of the indicator dia- gram is in the determination of pressure at different parts of the stroke; therefore, the maximum diagram for a given class should be determined. Lucke, in his Gas Engine Design uses 450 Ib. per square inch gage as the maximum pressure in the cylinder; Giildner uses 356 Ib. gage with such a factor of safety that the stresses may not be excessive should the pres- sure be 425 Ib. With the maximum^ pressure of 356 Ib., Giildner con- structs a diagram which he uses in subsequent calculations. The diagram is for a producer-gas engine with a compression pressure of 128 Ib. gage and m.e.p. of 96.5 Ib. This diagram is given in Fig. 119 for comparison. The maximum pressure in the Diesel engine is predetermined and is usually about 500 Ib. per square inch above the atmosphere, although sometimes as high as 600 and as low as 450 Ib. To allow for rounding corners the cut-off should be taken a little later than found by the actual m.e.p. In fact, with both constant-pressure and constant-volume diagrams the actual m.e.p. may be divided by a diagram factor before substituting in (36) or (41) of Chap. VI; the diagram factor may be about 0.9. In cylinder design the maximum gage pressure must be employed. For other FlG 119 engine parts the maximum unbalanced pressure is required; this may include the effect of inertia, but maximum stresses should also be checked for maximum gas pressure, neglecting inertia. Letting p x denote the maximum unbalanced pressure per square inch with or withDUt inertia of the reciprocating parts, and P x the total maximum unbalanced pressure; then: P, = (22) THE INTERNAL-COMBUSTION ENGINE 205 For the total unbalanced pressure at any other part of the stroke: P = .'-^ (23) This value may be used in checking some of the combined stresses when the maximum may not occur at the point of the maximum direct thrust. 81. Application of Formulas. This will be given by examples. Example 1. Design a single-acting, 4-cylinder, 4-cycle gas engine using anthracite producer gas, to develop a rated b.h.p. of 500 at a piston speed of 750 ft. per minute. From Table 30, h = 141 B.t.u., a = 1.4 cu. ft. (the maximum) and e B = 0.24; and from Table 31, e v may be taken as 0.82. From (9), for one cylinder: r>_o S2 n / 4 X 125 X 2.4 ' ?8 - 25 Vo24X 0.82X750X141 ~ 2L2 m " This may be taken anywhere from 21 to 22 inches. Comparing with (15), taking K from Table 33: D = 0.41 ^5 700 X 125 750 The engine size may be: 22" X 30" - 150. Example 2. Design a single-acting, 3-cylinder, 4-cycle Diesel engine to develop a rated b.h.p. of 300 at a piston speed of 800 ft. per minute. From Table 30, h = 18,000, a = 323 (maximum) and e B = 0.315; and from Table 31, e v = 0.82. Then from (12) : n 9* /~ 4 X 100 X323~ * 8 ' 25 \0.315 X 0.82 X 800 X 18,000 = 16 ' 45 m ' Comparing with (15), taking K from Table 33: D = 0.41 J 13,500 X 100 800 The size may be taken as: 17" X 26" - 185. It is obvious that the values of K in Table 33 for these two cases are conservative, as they correspond with the maximum air supply given in Table 30. Reference: The references at the end of Chap. V may be used in con- nection with this chapter. CHAPTER XV THE STEAM TURBINE 82. Introduction. Chapter IV may be considered as introductory to the present chapter, in which turbine parts will be treated only in their relation to capacity and economy. A working knowledge of Thermo- dynamics is also essential, the flow of steam having direct application. Aside from general treatment no originality is claimed. The excellent works of Peabody, Martin, Jude and Stodola have been studied, as well as catalogues and other material furnished by some of the leading Ameri- can manufacturers of steam turbines. The prospective steam turbine designer is advised to fill in the outline here given by a study of the works mentioned. Notation. A = the reciprocal of Joule's equivalent = M?8- k = the thickness factor the fraction of the arc containing nozzles, which is available after deducting blade thickness, etc. m = the ratio of blade length to pitch diameter of wheel or dia- phragm. y = fraction of kinetic energy used to overcome friction. q = velocity coefficient the ratio of actual velocity to what it would be without friction. z fraction of residual energy available for kinetic energy in next nozzles, /i = fraction of increase of kinetic energy due to conservation of residual energy. A = change of. i = fraction of heat drop per stage utilized in reaction turbine. j = fraction of exit energy from blade or guide of a reaction tur- bine utilized to produce kinetic energy in next guide or blade. e = "indicated" thermal efficiency. e M = mechanical efficiency. e D = diagram efficiency. e B blade efficiency of a reaction turbine. 206 THE STEAM TURBINE 207 F = heat factor ratio of actual " indicated" thermal efficiency to Rankine efficiency. Sometimes referred to as " total" heat factor in multistage turbines. The "overall" heat factor is equal to e M F, in which e M may include turbine and generator. F s = heat factor per stage, or hydraulic efficiency. F R = reheat factor = F/F S . F D = distribution factor. R = distribution ratio. V = velocity in feet per second in general; also denotes flow from nozzles or guides. V N = relative velocity of entrance. V x = relative velocity of exit. V R = residual velocity. Vw = velocity of whirl. V F = velocity of flow. S = velocity at pitch line of blades in feet per second. W = steam consumption in pounds per horsepower-hour. G = total steam consumption in pounds per hour. w = weight of steam in pounds per second. , a = nozzle area in square inches. v = specific volume of steam of any condition. P = pressure in pounds per square inch absolute. f w = tangential force in pounds acting upon blades. f F = axial thrust upon blades in pounds. E. = energy in foot-pounds. t = temperature in degrees F. C = heat content of steam in B.t.u. above 32 degrees F., for any condition. { AC = adiabatic heat drop; any portion of Ci-C 2 along the same entropy as that of Ci. h = heat of the liquid. L = latent heat of steam. x = dryness fraction. H = turbine horsepower. kw = kilowatts. D = diameter of pitch circle of wheel in inches. d = radial length of blade in inches. c tip clearance in inches. n B = number of rows of moving blades. n v = number of velocity stages. n P number of pressure stages. 208 DESIGN AND CONSTRUCTION OF HEAT ENGINES N = r.p.m. a = angle of nozzle or exit of guide. 6 = angle of entrance to moving blade. = angle of exit from moving blade. Also entropy. 5 = angle due to flow of residual steam; also angle of entrance to guide. 83. The General Method. A critical study of steam flow through a turbine, other than that based upon rather broad general principles, involves problems of great complexity and will not be attempted in this book. A few simple principles combined with the broader and better- known coefficients of performance, and a few of the simpler refinements to give a better understanding of the principles involved, will be the basis of procedure. If in determining the power of a steam turbine a method parallel to that of the steam engine were employed, that is: if the sum of the work done upon each blade were determined in detail, there would enter into the calculation so many uncertainties that the accumulation of errors would render the determination useless. It is obvious that the large surface of the engine piston acted upon by accurately known steam pres- sures offers a different problem than the turbine with its multitude of blades, even though the mechanical principles involved in the latter are simple. A more general method is therefore best adapted to the turbine and will be outlined. As it is impossible to take indicator diagrams from a turbine, there can be no indicated horsepower. An equivalent to this, the power developed by the action of the steam upon the blades, is called the turbine horsepower and will be denoted by H. The thermal efficiency is denoted by e; C is the heat content for steam of any condition and h the heat of the liquid as in Chap. VII. The steam consumption per hour is denoted by G and the consumption per h.p.-hr. by W. Letting subscript 1 refer to initial absolute pressure and 2 to absolute exhaust pressure being that of the atmosphere or condenser Equation (4), Chap. VIII may be written: W _ 2545 (Ci-fe) From (9), Chap. VIII, and (8), Chap. VII: e = F^=-^ (2) GI ft 2 where F is the heat factor, or the ratio of actual "indicated" efficiency to that of the Rankine cycle. THE STEAM TURBINE 209 From (1) and (2) : W _G_ _2545_ " H F(d - d) Or: TT Crr (O i ^ 2 / //i\ ^54^ This formula also applies to the steam engine but is not of practical use in power determination. Ci C 2 is always the expression for heat drop due to adiabatic expansion, therefore Ci and C 2 must always have the same entropy. The heat factor F, sometimes called the efficiency ratio, is the product of several factors, such as blade efficiency due to form, and factors due to friction of nozzles, blades and discs, and to radiation losses. Experi- ment and experience have fixed some of these factors approximately, but their product must always check with F as found from test; the value of F found from tests is therefore more reliable. As many steam turbines are employed for driving electric generators, they are often rated in kilowatts; all calculations in this book will be made in turbine horsepower which is obtained from kilowatt rating thus: (5) e M where e M is the mechanical efficiency of turbine and generator. Steam consumption per kw.-hr. is: G^ = 1.34TF (6) kw e M As complete expansion is always assumed in turbine operation, F must be based upon the Rankine cycle with complete expansion, given by (10), Chap. VIII, which is: 2545 ( . = W(d - C 2 ) Equating (4) and (5) gives: 3410 kw = d - C 2 ' G ' This is sometimes known as the "overall" heat factor. The heat factor is fairly well known for turbines of given power, and ranges from 0.45 to 0.75, and may be higher. Formula (3) may be written: 2545ff ,, - F(d - C.) Assuming F, finding Ci and C 2 from entropy table or chart for the selected 14 210 DESIGN AND CONSTRUCTION OF HEAT ENGINES pressures and quality, (8) gives the weight of steam per hour which must be passed through a turbine of given power. It now remains to propor- tion the various turbine parts to handle this steam with a minimum loss. 84. Nozzles and Other Passages. The general formula for friction- less, adiabatic flow of one pound of steam is given by (12), Chap. VII, and is: = 223.7 C-C2 (9) where V is the velocity in feet per second when the initial velocity is zero. The heat content C may be calculated from Formulas (2), (3) and (4) of Chap. VII, for dry saturated, wet, and superheated steam respectively, with the help of an ordinary steam table; but may be more conveniently taken from Peabody's entropy table or a Molier diagram. In Equation (9) and in what follows, the subscript 2 does not neces- sarily refer to the exhaust pressure as assumed in Par. 83, but may indicate any pressure less than pi at which calculations are required. The weight of steam in pounds per second which will flow through an orifice is, taking the area of the orifice in square inches: ' The specific volume v may be found in Peabody's entropy table. Both V and v are the values corresponding to the area considered. It may be shown mathematically that when a maximum weight of gas passes through an orifice there is a certain definite ratio between the initial pressure and the pressure in the orifice; and that the latter does not decrease nor the rate of flow increase, however much the pressure against which the orifice discharges is lowered. The calculated ratio for air is 0.5274. Experiments on saturated steam give a value of about 0.58, sometimes taken as 0.6. There are a number of empirical formulas for flow through an orifice and the agreement with (9) is very close. These formulas may be found in engineering handbooks and will not be given here. Formula (10) may be written: (11) It is well known that if any fluid flows at high velocity through an orifice with sharp edges as in Fig. 120, the cross-sectional area of the jet will be less than the area of the orifice, forming a vena contracta. If the orifice THE STEAM TURBINE 211 is formed as in Fig. 121 the jet will fill the orifice, in which case Formula (11) applies, and indicates that the velocity increases at a higher rate than the volume, the ratio being higher at the entrance. Fig. 121 is known as a converging nozzle and is in general the form used for the entrance of FIG. 120. FIG. 121. all machined nozzles. Nozzles in lower stages of compound turbines, sometimes called guide vanes, are often formed by "casting in" steel plates as in Fig. 122. Assuming the radial width uniform, the nozzle converges whether 1-2 or 1-3 be considered the width of entrance. The I FIG. 122. distance 4-5 is the minimum width and this may continue a short distance to 6-7 ; if this distance is made too great, frictional loss will result. The exit of the nozzle is at 6-7, the jet being formed by the time it reaches this cross section; and if the pressure against which it flows is not less than 0.58 of the pressure at entrance, it will retain its sectional form, dimensions and direction after it leaves the nozzle. Neglecting friction, the velocity of flow will be that given by (9), the heat content C 2 being that obtained at a pressure 0.58 of the absolute pressure of the entering steam. If the pressure against which the jet flows is appreciable less than 0.58 of the initial pressure, it will have a tendency to scatter and its kinetic energy will be dissipated if a converging nozzle is used, this effect being more marked the lower the discharge pressure; this makes it useless for turbine propulsion. Adding a diverging passage to the converging nozzle as in Fig. 123 overcomes this difficulty by con- FIG. 123. 212 DESIGN AND CONSTRUCTION OF HEAT ENGINES trolling the direction of flow until the pressure and specific volume cor- respond to the pressure against which the nozzle discharges; the result is a compact jet of the same cross-section as the nozzle exit. From (11), this indicates that in the diverging portion the specific volume increases more rapidly than the velocity. The relative rate of this increase has been the subject of theoretical speculation, and curves of various form have been tried, connecting the smallest area, or throat to the exit; but the straight line, commercially the simplest and cheapest, seems to give practically as good results as any other and is generally used. The throat area is determined as for the exit of the converging nozzle, taking C and v corresponding to 0.58 of the initial pressure; the exit area is found by using C and v for the exit pressure. This calculation may be made in two stages, using the throat pressure for the initial pressure of the diverging portion, and adding the velocity so obtained to the throat velocity, but it is not so satisfactory. Example. Design a converging-diverging nozzle to expand adiabat- ically y Ib. of steam per sec. at 150 Ib. gage, to atmospheric pressure. Taking the nearest absolute pressures from Peabody's entropy table, Ci is 1193.3 for an absolute pressure of 164.8 (entropy 1.56). At 96.1 Ib. absolute approximately 0.58 of 164.8 C 2 is 1149.8. Then from (9) the velocity at throat is: V = 223.7\/43^5 = 1476 ft. per sec. The specific volume at throat from entropy table is 4.412. Then from (11): 144 X0.25 X 4.412 AQ a = 1476 = - 108 sq ' m ' The corresponding diameter for a round section is nearly % in. At exit, C 2 = 1018 at 14.7 Ib. absolute and entropy 1.56. Then from (9): V = 223.7\/175.3 = 2960ft. per sec. At 14.7, v = 23.13, and from (11): 144 X 0.25 X 23.13 "296^ = ' 281 Sq ' in ' For a round nozzle this is between !% 2 an d /-g m - diameter. If the nozzle is for a condensing turbine with 28-in. vacuum, the exit pressure nearest to this in the table is 1.005. 2 = 871.1. Then from (9) : V = 223.7 \3222. = 4015 ft. per sec. THE STEAM TURBINE 213 At this pressure, v = 256.8, and from (11): 144 X 0.25 X 256.8 4015 = 2.31 sq. in. For a round nozzle this is between l l ^{ Q and 1% in. diameter. Divergent nozzles formed by cast-in plates are like Fig. 122, but the portion between 4-5 and 6-7 is made longer; the area at 4-5 is the throat, and the diverg- ence is made by increasing the radial width at the exit 6-7. This is usually done in a" straight line. The form is not one of great accuracy and the sides formed by the casting are rough. Fig. 124 gives some idea of the general form. In Formula (11), a may be taken as the total nozzle area, being the product of the number of nozzles and the area of a single nozzle. For nozzles of the type shown in Figs. 122 and 124, it is often more convenient to take the total effective opening in the diaphragm. In Fig. 125 let I be the length, measured at the pitch line, of the opening containing cast-in plates, forming nozzles. This is the entire circumference in some cases. The effective area of exit of each nozzle is bd, and b is equal to 61 sin a. Let: Z&i = kl where k is less than unity and allows for the thickness of the plate; k may be t/(b + t), or it may also allow for bridges used to strengthen the diaphragm when the nozzles extend around the entire circumference. Then the effective area with all dimensions in inches, is: a = kdl sin a (12) This also applies at the throat by using d' (Fig. 124) in place of d. If I = irD, where D is the diameter of nozzle pitch circle in inches, and the ratio d/D be denoted by m, then (12) becomes: FIG. 125. a = irkmD 2 - sin a (13) Effect of Friction. In the form of nozzle shown in Fig. 121, Prof. Rateau found the actual flow practically equal to the theoretical when the exit pressure was 0.58 of the initial ; but when discharging into higher 214 DESIGN AND CONSTRUCTION OF HEAT ENGINES pressures than this the flow decreased, the discrepancy between theoret- ical and actual being greater the less the pressure drop. Martin (Design and Construction of Steam Turbines) says that this was probably due to the formation of a vena contracta, reducing the effective area. In diverging nozzles, the increased surface causes friction, which retards the flow. Various factors combine to make the actual discharge less than the theoretical, and it is well to provide for this. Let y be the fraction of the theoretically available heat energy dissipated by friction and radiation; also let it include any loss due to vena contracta or any influence resulting from an improper form of passage, since it is probably true in part that a correction of one will .also reduce the other. Then the fraction of the heat drop available is 1 y, and the equation for flow is given by (13), Chap. VII, which is: = 223.7 (l -y)(Ci -C 2 ) (14) The expression 1 y is known as the nozzle efficiency. If the loss were wholly due to friction (which, for the usual com- mercial nozzles is probably an accurate enough assumption for the present purpose), a quantity of heat y(C C 2 ) would be returned to the steam during its flow through the nozzle, increasing its entropy and volume. Formulas (14), (18) and (20) of Chap. VII give the resulting quality of steam at pressure p 2 when the heat quantity y(Ci C 2 ) is added to the heat content resulting from adiabatic expansion to that pressure. The specific volume for this new value may be substituted in (11) and the area determined. The entropy table may be readily used for such problems. Example. Using the same data as in the previous example, determine throat and exit areas when the values of y are 0.05 and 0.15 respectively. From the entropy table Ci C 2 for the throat is 43.5 as before; then from (14) : V = 223.7 V0.95 X 4375 =1437. The resulting heat content at 96.1 Ib. is: C 2 + y(Ci- C 2 ) = 1149.8 + 2.175 = 1151.975 B.t.u. This occurs between the entropy 1.56 and 1.57 and we must interpolate to find the specific volume as follows: 1157.7 1151.97 4.451 4.4120 1149.8 1149.80 4.412 0.0107 7.9 is to 2.17 as 0.039 is to 0.0107 v = 4.4227 THE STEAM TURBINE 215 Then from (11): 144 X 0.25 X 4.4227 a = 1437 = o- 111 sq- m - This is less than 3 per cent, greater than before; the increase due to drying the steam is about 0.5 per cent. At exit: V = 223.7 V0.85 X 175.3 = 2750 The resulting heat content at 14.7 Ib. is: 1018 + 26.3 = 1044.3 B.t.u. The heat content at entropy 1.6 is 1044.9, which will be assumed as close enough. Then v = 23.88, and from (11): 144 X 0.25 X 23.88 -2750- = 0.312 sq. in. This is an increase of 11 per cent, over the area found by neglecting friction; about 3.3 per cent, of this is due to increased specific volume, the remainder to decreased velocity. Using ordinary steam tables for determining v, X N , the new dryness factor due to the degradation of heat may be found as follows: The value of # 2 at 14.7 Ib. due to adiabatic expansion from dry steam, using the nearest value of initial pressure to 164.8, may be found by Formulas (5) and (6), Chap. VII, and is: 0.524 + 1.037 - 0.3118 -L4447- Then from (14), Chap. VII: .,- 0.865 ^P- 3 = 0.892 From (16), Chap. VII: v 2 = 0.017 + (0.892 X 26.773) = 23.817 cu. ft. as compared with 23.88 found by the entropy table. All calculations were performed on the slide rule, and with the exception of the determina- tion of volume at throat pressure from the entropy table, no interpolations were made. The ratio of velocity reduced by friction to that due to adiabatic expansion is: 1 ~ y = VU85 = 0.922. 216 DESIGN AND CONSTRUCTION OF HEAT ENGINES Formula (21), Chap. VII gives the heat returned to the steam in terms of this velocity ratio, which is there denoted by q } and is: Heat returned = 296Q2 , ( n 1 ~ ' 85) = 26.3 oUjUUU which is exactly the value found before. The factor q is known as the velocity coefficient and its relation to y is: q = Vl-y (15) or: V = 1 - q 2 (16) Other Passages Used as Nozzles. The fixed vanes, or guides, of a FIG. 12G. reaction turbine serve as convergent nozzles and are shown in Fig. 126, from Martin. This design is called the normal Parsons blading, and the shortest distance between the blades is made equal to % the pitch. Ac- cording to Martin (Design and Construction of Steam Turbines), the discharge angle varies as the pitch is changed and cannot be determined by the tangent to the curve on the concave side, as the back of the adja- cent blade is also curved and has a different tangent. In blades % in. wide (axial width), the pitch of the blades in the casing (guides) is Y in. and on the rotor the pitch is ^f g in. By experiment, the discharge angle of the guides is between 17 and 18 degrees, and for the blades on the rotor, between 18 and 19 degrees. Another form of blade, called the wing blade is used where large discharge angles are required and is shown THE STEAM TURBINE 217 in Fig. 127. The discharge angle of wing blades is from 40 to 50 degrees. Semi-wing blades are normal blades specially spaced and set to have a discharge angle of from 28 to 30 degrees. Due to the rounded backs of Parsons blades, the stream lines unite as shown by the dotted lines of Fig. 126, and according to Martin, no i. ^ FIG. 127. deduction is needed for thickness. Then, as these guides extend around the entire periphery, k in (12) and (13) is unity. Peabody says the necessary allowance for blade thickness is indefinite. Guide blades of velocity-stage impulse turbines are used to change the direction of steam at constant pressure and are not considered as nozzles. The simplest form of guide is symmetrical with inlet and dis- B C FIG. 128. charge angles equal, as in Fig. 128. The width of the passage is constant and neglecting friction, the radial height d is constant. If friction is considered, there will be a slight increase in radial height on the discharge side as shown in Fig. 128(7. Assuming q the velocity coefficient, the heat returned by friction is given by (21), Chap. VII. Putting this in place of y(Ci (7 2 ) in (14), Chap. VII (assuming wet steam), gives a^; then find v 218 DESIGN AND CONSTRUCTION OF HEAT ENGINES from (16), Chap. VII and substitute in (11) of this chapter, using qV for Fo, the relative velocity at entrance. If the subscript N refers to the increased area: a* _ v N _ #A- a qv qx (17) as, neglecting the volume of water in the steam, the volume is directly proportional to the dryness factor. Then : (18) x 50,OOOzL where L is the latent heat at the pressure in the guide. By similar sub- stitution, conditions for superheated steam may be found from (18) and (20), Chap. VII, the latter being for initially wet steam becoming super- heated. When the entropy table (Peabody's) is used a more direct way FIG. 129. may be to add the heat returned as given by (21), Chap. VII to the heat content at the pressure considered, and then find x. The heat returned is : Q R = TV(1 - i 50,000 Unsymmetrical guides are shown in Fig. 129. In this case the passage becomes narrower toward the exit, and for constant area of passage, the height d must increase so that the product of width and height is constant. This is approximated by making: or: 61 r~ 6 2 = M 2 _ sin A sin B THE STEAM TURBINE 219 Then the connecting line is made straight. This must now be multiplied by (17) if friction is to be considered. Then: , X N bi , a AT sin A d>2 = =- i = ; 5 ai (19) qx 6 2 qx sin B In some cases the ratio d z /di, as given by (19) is excessive and would lead to impracticable blade dimensions; this may be remedied by in- creasing di after d z has been determined. This is also discussed at the end of Par. 88. In applying (19) the angle A should be the angle deter- mined by the velocity diagram and not the actual blade angle if increased as in Fig. 135 (to be explained later). These equations may also apply to moving blades of impulse turbines when q refers to relative velocity. The form of guide in Fig. 129 is sometimes used for convergent nozzles, in which case d% = di and the area b z d 2 must be determined from (11) as for any nozzle. 85. Practical Notes on Nozzles and Other Passages. Nozzles should be used only with the range of pressure for which they are designed. Over-expansion, or carrying the expansion in the nozzle to a pressure below that into which the steam flows, causes a disturbance in the nozzle, with a loss of efficiency which increases rapidly with increase of back pressure. Experiments have shown that a slight amount of under-expansion in the nozzle to a pressure slightly greater than the back pressure is benefi- cial to blade efficiency. This may be accomplished by slightly reducing the exit area; possibly the neglect of the increase of specific volume due to friction would provide the right allowance, in which case all formulas for nozzle design are contained in this paragraph, and no reference to Chap. VII need be made. The decreased velocity given by (14) must not be overlooked. Martin says that the entrance to the throat of a nozzle must be "very easy and well rounded" (referring to a round nozzle) or the discharge may be only 0.8 or 0.9 of the theoretical. Jude says that the inlet end should not have a large radius, but a small rounding off is advantageous. He further states that a round nozzle reasonably correct in shape, and not working with pressures entirely unsuitable, should give a velocity coefficient q of 0.95, but that undoubtedly q is a little lower for square or rectangular nozzles "on account of the natural internal instability of the jet." Also, "an appendage to a circular nozzle, to change the circular jet into a square or rectangular jet, also involves a loss of efficiency, rarely less than another 3 per cent, (velocity), and may often amount to 10 or 15 per cent., according to the way it is made, its continuity and the condition of the surfaces. " 220 DESIGN AND CONSTRUCTION OF HEAT ENGINES Martin says that curvature has an adverse effect on efficiency, es- pecially when the curves leading to the throat are abrupt. For experiments upon straight convergent-divergent nozzles with easy well-rounded entrance and taper not too rapid, Martin gives a formula for loss of energy, which in our notation is: 6(Cl _ C2 _ 45 ) 10,000 He says that experiments upon nozzles with curved entrance and in which a round section changed to rectangular (nearly square) at exit, gave values fully twice as great. For the example of nozzle design pre- viously given, Ci - C 2 = 175.3. Then from (20), y = 0.078, or 1 - y = 0.922. From (15), q = 0.96; or the loss of velocity is 4 per cent. Twice the energy loss gives 1 y = 0.844, or q = 0.918; or the velocity loss is 8.2 per cent. These figures may be compared with those given by Jude. Martin says that round nozzles lead to inefficiency in turbine practice. This is probably due to the entrance of a circular jet into the rectangular opening between the blades. However, round nozzles are much used, especially in the first stages of certain types. No definite rules can be given for taper of nozzles. Too small a taper gives long nozzles with resulting friction; while if too large, the effect upon the jet is detrimental. A total taper (change of diameter) of from 1 in 6 to 1 in 12 is used for straight round nozzles. A smaller taper is sometimes used, especially when the increase of area in the diver- gent portion is small. Practical construction may sometimes call for a much .greater taper than 1 in 6 for such nozzles as shown in Fig. 124, and in all cases judgment must be used. From various experiments, Martin gives some practical values of y and q as follows: When V = 800 to 1200, q = 0.92 and y = 0.15. For nozzles similar to Figs. 122 and 124, the values were lower. The maxi- mum values found for these were with a superheat of 130 degrees F. and with V = 1900 to 2100; then q = 0.95 and y = 0.098. When the super- heat was but 4 degrees F., and V = 1770, q = 0.915 and y = 0.16. For Figs. 126 and 127 no direct experiments were made, but computed veloci- ties from turbine tests indicate that for V = 200 to 300, q = 0.95 and y = 0.1. In all curved passages, centrifugal force results, causing variation in pressure, the maximum being at the concave surface. If unconfined laterally the jet spreads, reducing the efficiency. Experiments indicate that the value of q for the passages between impulse blades for theo- THE STEAM TURBINE 221 retical steam velocities of from 200 to 2600 ft. per sec., when the jet is unconfined laterally, is given by the formula: ' When the jet is confined by shrouding as in Fig. 130, the value of q may be much increased. From (21) the velocity coefficient for unshrouded guides and blades is greater at high steam velocities, which is contrary to the usual effect of friction. Peabody gives for flow through guides and blades: V ~ 10,000 which gives: (22) This is no doubt intended for unworn, shrouded blades made by machining or drop forging. Jude says that che average maximum velocity co- efficient for the best form of closed-vane passage is about 0.955, but that the average value of the better-known turbines appears to be about 0.92. However, various FIQ 13Q phenomena due to the steam surrounding the vanes and the transfer of steam from one passage to another give an equivalent to the reduction of this value, sometimes to as low as 0.6 or 0.7. Martin gives from 0.68 to 0.72, while Rateau gives 0.75 as a fair average. Jude states that he has confirmed Rateau's results, but that with special blades, he has obtained values as high as 0.8. As we have taken q as the ratio of actual to theoretical velocity at the exit of the passage, and as the relative velocity in one vane passage of an impulse turbine is theoretically constant up to the entrance of the next vane passage, the intervening space with its losses may be considered as a part of the passage and these latter values of q as applying to this case, instead of the higher values found by experiment upon single rows of blades. These higher values may only be used with other factors which separately account for other losses, a detailed study of which may be found in the works cited. Jude says, however, that for velocicy-stage blading it is probably more correct to take a higher value of q, such as 0.92, as an average in ascertaining the diagram efficiency. It is possible that as high as 0.85 may be taken for q in the case of velocity-stage tur- bines, but why it should be greater than for a simple impulse wheel is not altogether clear. 222 DESIGN AND CONSTRUCTION OF HEAT ENGINES The constants for blade passages of the reaction turbine may best be treated under reaction blading. A number of practical nozzle designs are shown in Chap. XXXIII. In guide vanes and nozzles formed by casting in plates the pitch should be as large as is consistent with properly directing the stream. Methods of drawing. nozzles, guides and blading are given in Chaps. XXXI and XXXIII. 86. Impulse Blading. A simple arrangement of nozzles and blading for an impulse turbine is shown in Fig. 460, Chap. XXXIII. It will be seen that steam from the nozzle will sometimes impinge upon several blades at one time, but in our conception of the operation we may assume one pound acting upon a single blade. FIG. 131. The means of determining the jet velocity has already been given by (14). The change of velocity and the consequent change of kinetic energy which the jet undergoes in its passage between the blades may best be explained by the velocity diagram, the fundamental form of which may be easily understood by the crude apparatus shown in Fig. 131, which needs little explanation. A paper may be tacked to the board A which slides upon B . The sliding bar G holds a pencil in its lower end, and another pencil is held by H which is attached to B, but made removable. C and D may be grooved pulleys fastened together, the ratio of diameters being the ratio of jet to blade velocity. To draw a diagram, disconnect the cord E and remove H] then turn the pulleys so as to slide bar G and draw line 1-2. This represents the direction and velocity of the jet from the nozzle. Connect cord E and THE STEAM TURBINE 223 disconnect cord F after replacing G to its original position and removing the pencil. Place H so that its pencil point is at 2 and turn the wheel through the same angle as before, drawing line 2-3. This represents the direction of motion and velocity of the turbine blade. Move A back so the pencil in H is at 2 again, than remove H, connect cords E and F, place pencil of G on paper at 1 and turn wheel through same angle as the two previous times. The relation of the movements of G and A will be that of the jet to the moving blades, and the line 1-3 will be drawn on the paper. This is the direction of the pencil in G with reference to the paper, or, it represents the direction of the jet and its velocity in relation to the moving blade at the time the steam begins to enter the blade passage, and is known as relative velocity. If friction were absent and there were no other forces tending to change the velocity (such as expan- sion in the blades of the reaction turbine), the relative velocity would be constant, but its direction would change due to the influence of the blade passage which now controls its flow. The lines 1-2 and 2-3 represent velocities with reference to a fixed object upon the earth's surface and are known as absolute velocities. As in marine service the turbine is in motion, the fixed parts of the turbine, such as the casing, may be considered the zero of motion from which absolute velocity is measured. By changing the angle of the rod G, or the relative diameters of C and Z>, velocity diagrams of different form may be drawn; it is simpler, however, to abandon the apparatus, which was only mentioned with the thought of making the matter plain to some who might not easily grasp the principle. A simple velocity diagram, including the diagram for exit from the blade, is shown in Fig. 132. The concave surface of the blade against which the driving force is exerted is shown by the heavy line, the back of the blade being omitted. The blade is shown tangent to V N , the relative velocity at entrance, so that the jet may enter without shock; this is the usual treatment, but more will be said upon the subject presently. The blade curve must be tangent to line V x for impulse blades, to insure the direction of flow. The blade shown in Fig. 132 is symmetrical, angles and being equal; and as friction is neglected, Vx, the relative velocity at exit, is \ \ FIG. 132. 224 DESIGN AND CONSTRUCTION OF HEAT ENGINES equal to V N . S is the velocity of the blade (at the pitch circle) and must be equal in both entrance and exit diagrams. If in Fig. 131 bar G were changed to the angle 5 of Fig. 132 and the line V R drawn with the paper stationary instead of the line 1-2 of Fig. 131, the movement of both bar and paper would draw the line V x for the line 1-3, which must then be relative velocity as already stated. This shows that line V R of Fig. 132 represents absolute velocity, and it is known as residual velocity. The reduction of the velocity of the steam in its pas- sage through the blades is the difference between V and V R , both being absolute velocities. The kinetic energy due to residual velocicy V R is the energy lost, and should be kept as small as practicable. The absolute entrance velocity V of Fig. 132 may be resolved into two components, one parallel to the shaft center and represented by line 1-2, 133 called the velocity of flow; the other in the direction of blade motion by line 23, called the velocity of whirl. Likewise the absolute exit velocity, or residual velocity may be re- solved into 4-6, the exit velocity of flow, and 5-6, the exit velocity of whirl. The latter, in the diagram given is negative, being opposite in direction to the blade movement ; but the momentum due to it drives the blade forward so is considered positive. A more general treatment may be given by considering friction and assuming an unsymmetrical blade; such a diagram is Fig. 133, to which the same general definitions apply. The kinetic energy supplied to the blade is due to the jet from the nozzle at velocity V, and is, in B.t.u. for 1 Ib. ; V 2 V A ^g = 778 X 64.32 V223.' The rejected energy due to residual velocity is: / V \223.77 and the energy lost by friction is: /-- \223.77 223.7 THE STEAM TURBINE 225 or: The efficiency of the blades, or diagram efficiency e D , is then given by: ^^^'^_v'-v*-jM-*> ; (23) The force of impact is the product of mass and velocity change per second. It is obvious that the change of tangential velocity of the steam produced by its impingement upon the turbine blade, is the difference be- tween the tangential velocity, or velocity of whirl, at entrance and exit; which is: V cos a V R cos d. The direction of blade motion having the plus sign, the value of V R cos d is negative. To obviate the necessity of considering which sign this value has, we may, in view of the fact that the tangential component of V x is always negative, and S always positive, write: V R cos d = V x cos H- S Then the tangential velocity change is: V cos a - ( - V x cos + S) or: V cos a + V x cos S Then, substituting qV N for V x , the tangential force due the flow of w Ib. of steam per sec. is: f w = -(V cos a + qV N cos - S) = - V w .(24) y y This is the load on all blades receiving the steam; to find the load per blade divided by the number of blades. Similarly, the axial pressure on the blades is due to the difference of the velocity of flow at entrance and exit, and is given by the expression: f p = (V sin a - qV N sin 0) = - V f (25) */ *7 This is known as axial thrust and must be provided for by a thrust bearing. If f F is negative the thrust is in the opposite direction from the velocity of flow. Formulas (24) and (25) are general and apply to reac- tion blades also. 15 226 DESIGN AND CONSTRUCTION OF HEAT ENGINES A graphical construction for the parenthetical quantities of (24) and (25) is given in Fig. 134, which may be drawn to scale and then measured as an alternative. Fig. 134 is a reproduction of Fig. 132 with the line V x (= qVir) produced on the upper side of the center line of blades. The useful work per second done upon the turbine wheel isfwS ft. lb.; and as wV 2 /2g is the ft. lb. of energy supplied by the nozzles per second, the diagram efficiency is: e D ~ (V cos a + qV cos - S) (26) The quantity in parenthesis may be calculated, or measured from the diagram, Fig. 134. Formulas (23) and (26) give the same result, the latter being more convenient. FIG. 134. Formula (26) may also be written: e D From Fig. 133: 2 y (cos a + q -^ cos -y V N cos 6 = V cos a S s v cose Substituting in Formula for e D and rearranging gives: S S Also V N sin 6 = V sin a THE STEAM TURBINE 227 or, \VN _ sin a ~V ~ sin e Equating with the value of V N /V just given: tan 6 sin a S COS a =. (28) When S/ V and a are known K may be found, and cos may be deter- mined from (28). Table 38 gives values of K and cos for different values of a and S/V; then by assuming different values of q and , e D may be quickly determined. Within reasonable limits it is apparent that decreasing increases the efficiency; and that for a given value of cos 0/cos and q the efficiency varies directly as K. Then it may be seen that decreasing the angle a increases the efficiency, and for maximum efficiency such a value of S/V must be chosen which will make K a maxi- mum. This may be found by equating the first derivation of S S 2 ^- T COS a v^. to zero, which gives : cos a (29) TABLE 38 a S/V 0.3 0.4 0.5 0.6 K cos & K I cos K cos K e cos e 15 20 25 0.400 0.383 0.364 21-13' 26-53' 34-58' 0.932 0.892 0.819 0.453 0.432 0.405 24-37' 32-25' 39-54' 0.929 0.844 0.767 0.466 0.439 0.406 28-54' 37-55' 46-8 / 0.875 0.789 0.693 0.440 0.407 0.366 35-ll' 45-51' 54-5' 0.817 0.696 0.586 This will give the maximum value of K for any given value of a. Then S is one-half the velocity of whirl, and neglecting friction, Formula (27) for symmetrical blades reduces to: e D = cos 2 a or with unsymmetrical blades: e D = COS' which may give a greater efficiency and tend to reduce the effect of fric- tion. 228 DESIGN AND CONSTRUCTION OF HEAT ENGINES 87. Practical Notes on Impulse Blades. The velocity coefficiencs for impulse blades and guides are given at the latter end of the preceding paragraph; these are for shrouded blades, which are practically always used with impulse turbines. While the nozzle exit is reduced slightly from the area theoretically indicated, as stated in Par. 84, the blade and guide passages must not be restricted as this may lead to throttling. It is probably well to consider all effects of friction and use Formula (17) or (19) in determining exit areas. As the clearance space between nozzles and blades probably gives rise to friction loss with a possible slight spreading of the jet, the radial length of the blades may be slightly greater than the radial width of the nozzles; possibly if this were made equal to what the nozzle width would be by making allowance for the drying effect of friction, the condition would be properly met. Some designers increase the blade length beyond this rather arbitrarily to avoid any possibility of choking (see also velocity-stage blading); with round nozzles, how- ever, the blade areas would exceed nozzle areas and choking would not be likely. As stated earlier in this paragraph, the entrance angle of the blade is usually assumed equal to the angle 6 at which the steam enters the blade; the idea is that with such a construction the steam slides into the blade passage without shock, and that friction is thus reduced to a minimum . According to Martin, this is an error, and better practical results have been obtained by making the blade angle some 5 to 15 degrees greater than the angle 6, Fig. 133. This is shown in Fig. 135. If this is not done, the decreasing relative velocity due to friction changes the direc- tion in which the steam attempts to flow, exerting pressure on the blade surface form- ing the back of the passage and tends to retard the motion of the wheel. The tra- jectory diagrams, Fig. 150, are drawn with the entrance angle of the moving blades greater than 6, and it may be seen that the stream from the nozzle is deflected backward slightly upon joining the tra- jectory curves, showing a forward impulse. Had the entrance angle been equal to 0, the opposite would have been true, as friction is taken into account in these diagrams. This is explained in Par, 92. The advisability of making . less than must be left to the designer's FIG. 135. THE STEAM TURBINE 229 judgment; it is usually done and no doubt increases the efficiency if not carried too far. Jude says that the drying of the steam due to blade friction is improbable, as any water present is thrown to the outside of the curved path by centrifugal force and is difficult to reevaporate. No definite rule controls the blade length. For reaction turbines Martin gives a minimum length of ^5 the drum diameter for stationary turbines and 1/75 for marine turbines, the latter, however involving a loss of efficiency. The maximum length for the low-pressure end is given as 1/5 the pitch diameter. These ratios m may be used as a guide and should be as reliable for impulse turbines. There seems to be no rational method of determining the pitch of impulse blading, which in practice ranges from 48 to 68 per cent, of the axial width. It would seem reasonable to make the pitch as large as possible and still properly direct the steam, thus reducing the friction surface, and it is probable that the greater the radius of curvature of the blade, the greater the pitch may be. A method of drawing blades is given in Chap. XXXI, by which the blades of Figs. 149 and 150 were drawn. The speed at the pitch circle of simple impulse turbines such as the De- Laval ranges from 500 to 1400 ft. per second. For pressure-stage impulse turbines a range from 300 to 650 ft. is used, some large modern turbines having a blade-tip velocity as high as 950 ft. per sec. ; in this case the blade length of the last wheel exceeds J4 of the pitch diameter, or m is greater than J^. The higher velocities necessitate great care in the designing of wheels and shafts which is discussed in Chaps. XXXI and XXXII. 88. Velocity-stage Impulse Turbine. For the purpose of studying the principle of speed reduction of the velocity-stage turbine, frictionless, symmetrical blades and guides will first be considered. A dia- gram for a single stage with zero exit velocity of whirl is given in Fig. 136. In this diagram, 6 = $, and it is obvious that the triangle formed by the exit diagram is equal to that formed by the dotted lines at the left of the en- trance diagram. These are shown together in Fig. 137. Retaining the zero exit velocity of whirl, assume that the blade speed S is to be made one-half that shown in Fig. 137. This is done by dividing the base line FIG. 136. 230 DESIGN AND CONSTRUCTION OF HEAT ENGINES into two parts, forming four triangles as shown in Fig. 138. Disposing these triangles in a manner similar to Fig. 136 gives Fig. 139, which is a double diagram, having two rows of moving blades and one row of guides, which, except that there is no expansion in them, serve as nozzles for the FIG. 137. FIG. 138. Subscripts 1 and 2 are for the first and second blades of the second wheel. stages respectively. From Fig. 139 it may be seen that for the assumption made, the blade speed varies inversely as the number of velocity stages operated from one set of nozzles. In practice there is the ever-present friction, and the blades and guides are not always symmetrical, especially the latter; as when angle 2 is re- duced sometimes to equal 6, in 2-stage wheels the blades are not so flat in form. Fig. 140 is a diagram for a 2-stage wheel with these modifications. FIG. 140. Three or more rows of moving blades may be treated in the same manner, the ratio S/V being reduced correspondingly. The tangential impulse on the blades may be found for each moving row by (24), or THE STEAM TURBINE 231 graphically from Fig. 134, using subscripts 1, 2, 3, etc., for the successive stages. The total energy is given by: S (fwi + fw2 + fws, etc.) Letting V w = V cos a + qV N cos S, the parenthetical quantity of (24), the total energy is: and as the energy of the jet from the nozzles is wVi 2 /2g, the diagram effi- ciency is : rtCf * = (Wi + v + v "*> etc -) From the equations of paragraph 86 it would seem that whatever the velocity coefficients were, maximum efficiency would be expected when the diagram is constructed so that : Q _ V cos a ~2~ for the last stage, if this does not necessitate too small a value of a. The tangential pressure on the blades may be found from (24), and this gives the proportion of work done by each set of moving blades. The axial thrust is the sum of the thrust on all sets of moving blades, and may be found from (25), using the proper subscripts. In Fig. 139 it may be seen by inspection that V\ cos i = 4S, and V N i cos diagram, or Molier's diagram shows that be- tween two given temperatures the difference of heat content is greater at .a greater entropy. An example is given in Table 39, the values being taken from Peabody's entropy table. An inspection of the entropy table also shows that at a certain entropy the heat content increases at a nearly uniform rate for considerable intervals of 20 or even 40 degrees; or TABLE 39 Entropy 1.52 Entropy 1.59 Temperature Heat content Temperature Heat content 228 181 1010.2 953.0 228 181 1037.7 978.6 57.2 59.1 AC/A is practically constant for this interval and is more accurate for the wider range. We may then find the rate at which the available heat increases with the entropy as steam expands between two pressures in a turbine with a certain heat factor F. The calculation may be made at some intermediate point between Cj and C 2 , where the heat content is C A , all being taken at THE STEAM TURBINE 239 the same entropy cj> A . It is preferable for the sake of accuracy to have: C A = ^^ as nearly as possible without interpolation, but its exact value must be used in the following formulas. The increase of entropy in expanding from PI to P A will be due to the addition of the heat quantity : a En tn r< \ r ) (l^i L> A ) The heat content C* at entropy 0* (at constant pressure P,) will be: C* = C A + (1 - F) (C, - C,) V33) from which <* may be found. Let AC, be the change of heat content for the temperature interval AT 7 at constant entropy A , and AC* the change for the same interval at entropy 0*; then the ratio of heat change is: AC* AT = AC* AC, ~ AC, AT The temperature interval should be equally divided on either side of T A . The rate of increase of available heat between Ci and C 2 over that due to adiabatic expansion, computed for heat content C A is: AC* _ AC, " _ Ci- C 2 /AC* \ ,_, n r<~ ~ r 7*~ \A? r " / ' 1^1 ^A ^l ^A ^^^A I Cl C 2 If this gives the same value when C A is taken at any temperature between Ci and C 2 , the variation is in a straight line for equal increments of C; as this is very nearly true for saturated steam, the fraction of increase of available heat between d and C 2 will be one-half the value given by (34) . Then the ratio of the available heat to that available due to_a single adiabatic expansion from PI to P 2 is: This is only true for an infinite number of pressure stages, for which R is equal to the reheat factor F R ; but it may be used for constructing gram which is generally applicable. 240 DESIGN AND CONSTRUCTION OF HEAT ENGINES To make allowance for this increase of available heat during expansion, the adiabatic heat assignments, whether equal or in any desired propor- tion, must decrease from C\ to C 2 at the same rate that the available heat increases, but their sum must not change. This is best illustrated by the diagram of Fig. 144. The vertical height represents Ci C 2 , and AC the desired heat assignment per stage, being a fraction of Ci C 2 which is the adiabatically available heat from pressure PI to P 2 . It is obvious from (34) and (35) that the difference between the hori- zontal dimensions R and 2 R is equal to the rate of change of available heat, and that their average is unity. The dimensions F D , to the same *0 ? f- f- f- vL \t -------- ,2-R H FIG. 144. scale as R, are the distribution factors, each one bisecting the distance AC; multiplying AC by corresponding values of F D gives the heat quan- tities which, beginning with the first stage, should be successively sub- tracted at constant entropy (corresponding to Ci), in order to determine the heat content at each stage, in the same manner that equal heat quantities (Ci Cz)/n P were subtracted to determine the pressures for Fig. 143. An inspection of Fig. 144 will show that no matter how the heat is distributed, the sum of the -products of F D and AC is equal to the sum of AC. For equal divisions, Fm, being the ratio of the available heat to that which would be available without heat degradation, is also equal to the ratio F/F S which is the reheat factor previously mentioned. It is obvious that F m ( = F R ) increases with the number of stages, its limit being the distribution ratio R for an infinite number of stages, and this value, found by another method, is used by Martin as the reheat factor. THE STEAM TURBINE 241 p (35a) s it is obvious that F s decreases as the number of stages increases, and this may be seen in Fig. 143. It then follows that F R increases and F s de- creases as AC becomes smaller, which indicates that when the stages are unequal, F R and F s do not have the same value for each stage. From Fig. 144, as F Di = F R , it may be seen that for equal stages : F a = 1 + l - ~~ (R - 1} (36) It may then be assumed that for any work division, the reheat factor for the heat drop AC is given by (36). As an example of application, the pressures of Fig. 143 will now be determined with the same data as for Fig. 141 and a heat factor of 0.6. Then when PI = 164.8 and = 1.56, d = 1193.3; and when P 2 = 14.7 and = 1.56, C 2 = 1018.0. The mean heat content is: At entropy 1.56 the nearest value in Peabody's entropy table is at 53.6 Ib. and 285 degrees, for which C = 1105.2 and this will be taken as C A , at entropy A (= 1.56). From (33): C B = 1105.2 + [0.4 X (1193.3 - 1105.2)] = 1140.44 B.t.u. At temperature 285 degrees, the entropy B at which this value is found is between 1.60 and 1.91; interpolation gives B = 1.607. At < A = 1.56, taking AT at 40 degrees, 20 degrees on either side of 285 degrees: AC A = 1127.7 - 1082.1 = 45.6 At B 1.61, for 20 degrees either side of 285 degrees. AC* = 1163.88 - 1116.36 = 47.52 Then: AC. _ 47.52 _ AC A 45.60 " The ratio R is given by (35), and is, when C A is the mean between Ci and C 2 : R = 4 = 1.041 or 1.04 nearly. AC A For practical application a portion of Fig. 144 may be omitted; then 16 242 DESIGN AND CONSTRUCTION OF HEAT ENGINES Fig. 145 is the diagram for four equal stages with the data given and determined. The quantity AC is the same for each stage ; or : C, 1193 - 3 1018 AC = 4 TABLE 40 = 43.825 1 Stage 2 Fz>-AC r 3 C'i-S(FD-AC') at = 1.56 4 P (abs.) 5 Ci-SCF-AC) 6 <*> 7 Actual heat drop 8 V Steam 1993.30 ' 1193.30 1193.30 chest 45.15 164.80 26.30 1.560 1148.15 45.15 1148.15 1167.00 1167.00 1 45.15 44.25 94.15 26.30 1.584 1121.85 4.590 45.15 1103.90 1140.70 1140.70 2 44.25 43.40 52.33 26.30 1.609 1095.26 7.852 45.44 1060.50 1114.40 1114.40 3 43.40 42.50 28.21 26.30 1.636 1069.17 14.306 45.23 4 42.50 1018.00 14.70 1088.10 1.665 25.070 1.04- F OI -I.03 Table 40 gives the steps in determining intermediate pressures, and a check calculation showing actual heat drops per stage. In column 5, F-AC subtracted from the heat content of each stage gives the amount of heat available for adiabatic expansion in the following stage. The entropy, column 6, is that corresponding to the pressure (column 4) of the same line. The subtrahend in column 7 is found by following down the entropy given in the same line of the table, to the pressure given in the line below. The quantities in the table were taken from Peabody's entropy table by interpolation and the work done on a slide rule. The heat drop occurs in the nozzles connecting the compart- ments, and which really belong to the stage related to the compartment into which they discharge, FIG. 145. THE STEAM TURBINE 243 An alternative method which possesses some advantages, makes use of the reheat factor instead of the distribution factor. This is given in Table 41. In column 2, F R -&C, for stage n + 1, is subtracted from the heat content; the resulting heat content is placed hi the line below, and at the original entropy the corresponding pressure is found sometimes by interpolation. The unused heat of this stage (1 F S )F R -AC (= F R - AC F-AC) is now added, and at the pressure just found the new entropy is taken. F R -AC of the following stage is now subtracted and the re- sulting heat content down this new entropy line locates the next pressure, and so on. TABLE 41 Steam chest 1193.30 45.15 1148.15 18.06 1166.21 45.15 1121.06 18.06 1139.12 45.15 1093.97 18.06 1112.03 45.15 1066.88 164.80 94.15 1.5600 1.5840 52.24 27.83 14.478 1.6087 1.6343 The specific volume for any pressure used for determining nozzle areas, must be found at the entropy of the stage above, as the entropy of each stage shows fche increase due to heat degradation in that stage, and this occurs after the steam leaves the nozzles, with the exception of a slight amount due to nozzle friction. This applies to both methods. The check of this latter method is the lowest pressure, and had F B - AC, 244 DESIGN AND CONSTRUCTION OF HEAT ENGINES been more accurately taken as 45.14, the resulting pressure in Table 41 would have been more nearly 14.7. The value of R is found as before from (35), and F R easily calculated from (36). The necessity of determining many values of F" D for multi- stage turbines is obviated and much labor saved. Approximate values of R are given in Table 42 for various pressure ranges and heat factors, for saturated steam, taken from Peabody. TABLE 42 Heat factor 150 Ib. gage to 28 in. vacuum 150 Ib. gage to atmos- phere Atmosphere to 28 in. vacuum 0.55 .09 1.050 1.040 0.60 .08 . 1.045 1.035 0.65 .07 1.035 1.030 0.70 .06 1.030 1.025 0.75 .05 1.025 1.020 Superheated Steam. The direct method of pressure distribution just given will not give as accurate results when applied to superheated steam; an inspection of a T$ Diagram shows this. Assuming the nozzles to be designed for saturated steam, and the same heat factor in each case, the nozzles, from (11), will accommodate less superheated steam, but from (3), less superheated steam is required for the same power. The entropy diagram for superheated steam is given in Fig. 146. The dotted lines, as before, show assumed entropy changes due to nozzle friction, while the shaded areas are available adiabatic heat drops per stage. With less initial superheat, the lower stages would be like those of Fig. 143. For any pressures, heat drops may be determined as by the method given for columns 5 to 7 of Table 40. This assumes that the nozzle areas are determined for these pressures, heat drops and specific volumes, and that the actual heat factor is the value used in the calculations. Velocity Due to Reheating. Early in the present paragraph it was assumed in Formula (31) that the heat available for producing velocity is (Ci C 2 )/n P . Later it was shown that heat drop along the new entropy due to reheating is the available heat; and while a smaller percentage of this is converted into work (as discussed under Effect of Heat Factor), there seems to be no reason why it should not be available for producing velocity if we assume that all of the energy of residual velocity is used for reheating. With this assumption (Ci C 2 )/n P should be multiplied THE STEAM TURBINE 245 by F D , or, more generally, each part of Ci C 2 , or AC, for any distribu- tion, should be multiplied by the reheat factor F B for that stage, the quan- tity FR-&C being the heat drop along the entropy found for a given stage. The quantity AC is the desired fraction of the total heat drop before being multiplied by the distribution factor FD. For saturated steam F R may be found from (36) for any work division. The formula for velocity then becomes: V = 223.7V '(1 ~ yW*'&C (37) The sum of the products F R -AC is greater than Ci 2, but the unused heat in the stages of a multi-stage turbine is available to cause flow intothe following stages, while in the single-stage turbine the surplus energy due to residual velocity is discharged at the lo vver pressure limit and is no longer available. The actual work done, however, is the product of Ci C 2 and F, so that apparently the velocity increase is not available for increased work; in view of this it may perhaps be permissible to ignore F R in drawing velocity dia- grams and in area calculations, or to make some allowance for it in choosing the friction factor y. General Dimensions. Equat- ing (11) and (13) gives: kmD 2 sin a = 144w from which: D = 6.77 wv sn a (38) (39) FIG. 146. where D is the diameter in inches at pitch circle of blades, guides or noz- zles. In determining conditions, or analyzing, it may also be desirable to solve for other quantities in (38). 246 DESIGN AND CONSTRUCTION OF HEAT ENGINES With impulse turbines the value of m for maximum ratio of blade length to diameter of pitch circle will give the minimum wheel diameter for the last stage; this is found by multiplying this ratio by the ratio di/dz found from (19). Then V and v are for the discharge from the last nozzles. F may depend upon S, which has been previously fixed, or, vice versa, their relation being given by the velocity diagram. Then N, the r.p.m., is given by: irDN S 12 X 60 = 9 D Some limiting values of m were given in Par. 86. According to Martin, in a pressure-stage turbine with single velocity stages, with a 28-in. vacuum, D 2 should not be less that 0.57 X output in kw. The relation between diameter, speed, number of stages and efficiency is mentioned in Par. 95. Multiplying (38) by S and substituting the value of w in terms of (3) gives: 3 D = 19.5 3 kmNF(Ci - C 8 ) sin a kwV V kme M NF(C l -C 2 )sma The value of v (or x, from which v may be computed) may be found at the pressure within the stage considered, and at the value of heat content equal to : C x = Cn + (1 - F) (Ci - Cn) - (1 - F s )F R 'AnC (40a) The subscript n refers to any stage; Cn is the heat content'at the pressure considered, at the same entropy as Ci, and AnC is the heat drop for the stage, along the same entropy. It is accurate enough for most purposes, especially with reaction turbines, to take F instead of F s and omit F B ; then C x = Cn + (1 - F) (Ci - Cn - AnC) (406) Formula (40) is convenient, as the effect of the various factors may be readily seen. Number of Equal Stages. As calculations to determine the number of stages seldom results in a whole number, the influence of reheating will be neglected and the simpler formulas used Two factors influence the number of pressure stages: first, the form of nozzles, and second, the THE STEAM TURBINE 247 peripheral speed. If convergent nozzles are to be used, the heat drop AC in any stage must be no greater than that due to a pressure drop to 0.58 of the pressure in the preceding stage; then: Gl G 2 f n\ nr ? ^- (41) With less pressure stages and a greater heat drop, divergent (con- vergent-divergent) nozzles must be used. If S is the limiting factor, V may be found after the velocity diagram is designed; then solving for n P in (31) gives: - OU,UUU /., The actual value of AC, or of V and S may then be determined if n P is taken at the nearest whole number which will not exceed the limits imposed. . Unequal wheel diameters are sometimes employed to avoid extreme blade lengths. This leads to unequal stages and sometimes to the reduc- tion of the number of stages. If the same form of velocity diagram is used in each stage, V is proportional to S, which is proportional in turn to D; then if nozzle friction is assumed the same, the heat drop is propor- tional to D, and for any stage, with these assumptions : d-C 2 D may be determined by (39) for the last stage by assuming V, and the design worked along stage by stage toward high-pressure end, re- ducing wheel diameters at discretion and checking by (38) for blade lengths. Assuming D and S, V may be calculated tentatively, and heat drop AC may be found from (37), neglecting F R ; then from (36), F R may be found, from which V, S and D may be corrected, using (37) for calculating V. The corrected relation bewteen heat drop and pitch diameter would then be, for similar velocity diagrams: F R (l - ?/)AC D*_ 2[F R (l ~ 2/)AC] SZ) 2 but the error is slight. If convergent nozzles are to be used, divide the absolute condenser pressure by 0.58 to obtain the absolute pressure in the preceding stage; the heat drop AC between these pressures will be that for the last stage, and V may be found from (37) instead of assumed. This will determine S, which may be greater or less than desired and a compromise must be made. 248 DESIGN AND CONSTRUCTION OF HEAT ENGINES Another method is to start from the high-pressure end with the assumption of equal stages and equal diameters. If toward the low-pres- sure end the blades are too long, the wheel diameter may be increased, V increasing in proportion, until m in (38) is not too large; if the form of velocity diagram is not changed, D/V is constant and m varies inversely as D 3 . A greater heat drop will accompany an increase in diameter and the remaining stages must reduce the heat content to exactly Cz. The values of AC should check with (43) for similar velocity diagrams; they may then be laid off on Fig. 144 and the design completed. In some of the modern large multi-stage turbines the wheels increase in diameter from the first stage to the last. There is a single velocity stage in each pressure stage, with full peripheral admission. The work per stage necessarily increases toward the low-pressure end of the turbine. Conservation of Residual Velocity. The higher stages of most pressure- stage turbines have partial admission; that is, steam is admitted through but a portion of the periphery. The reason for this is, that otherwise extremely short blades are required to keep the nozzle area to the required amount. The blades are then kept an appreciable distance from the next set of nozzles in order to allow the steam to spread out and fill the nozzles. With admission around the entire periphery, equal wheel diameters, and clearance on the discharge side reduced to about one-half the blade width, a good percentage of the residual velocity is available as initial velocity of entrance to the nozzles. For any stage thus situated the available energy is : ^C (44) where z is the fraction of residual energy available as kinetic energy, and V R is the residual velocity of the preceding stage. The value of F should be nearly 1 + /* times its value without con- servation of residual energy and this slightly reduces F R , making it difficult to arrive at exact values. The fraction of increase: .C (45) ^223.7y may be determined approximately by taking: AC = LZ_ THE STEAM TURBINE 249 or dividing unequally along the adiabatic as previously described, as V R 2 will have the same ratio to F R -AC as when the correction is made. In order to have V the same in each stage the available energy must be the same. For the stages in which there is no conservation the stages with partial admission let: This must equal the value given by (44) ; as the stages are so nearly equal, F' R may be taken as equal to F R , and y may be assumed the same for each stage. Then : A'C = (1 + /i) AC (46) If UP and n'p denote the number of stages with and without conserva- tion respectively, it is obvious that: Then: \n i z , An ^ AC = TT\ i \~~i 0*7) ri P (l + fj.) + n P Taking A'C from (46), a diagram similar to Fig. 144 may be constructed, and the proper assignments per stage made. For the value of z in (44), Peabody gives 0.9; Martin gives 0.4 to 0.5 when the clearance between the wheels and diaphragms is reduced to a minimum, and the entrances to the guide blades are properly formed to receive the discharge from the preceding wheel. When the stages are not nearly equal, or part of them contain velocity stages, the calculation becomes involved; trial and error must be resorted to and check calcula- tions made. The foregoing discussion will give some idea of the conservation of the energy of discharge; aside from increasing the heat factor between 5 and 10 per cent., it is probably permissible to arrange blading to take advan- tage of conservation but neglect it in calculation. 91. Pressure-velocity-stage impulse turbines have two or more velocity stages in one or more of the pressure stages. The simplest case is the stationary Curtis turbine built for small powers by the General Electric Co. There are usually two pressure stages, with two, and some- times three velocity stages in each. In this turbine the rim velocity varies inversely as the number of velocity stages and as the square root of the number of pressure stages. Due to fewer pressure stages the heat 250 DESIGN AND CONSTRUCTION OF HEAT ENGINES drop per stage is so large that diverging nozzles are required. The velocity diagram for each stage is similar to Fig. 140 for two velocity stages, from which V may be measured or calculated for a desired value of S. Then AC for each stage may be calculated from (37), and the required number of pressure stages from (41) ; the design may then proceed as for any pressure-stage turbine with equal pressure stages. A not uncommon arrangement is to place two velocity stages in the first pressure stage only, reducing the pressure and the number of stages. The velocity diagrams may be designed and the values of V obtained. Sometimes S may be the same for the one-stage and two-stage diagrams, but often the two-stage wheels are made smaller though sometimes larger in diameter; in any case V is greater for the first stage and the heat drop AC will be greater. In general : ' (48) C l - C 2 the subscript n denoting any stage. 27 2 may be solved for, and sub- tracting successively V\ V 2 Z) etc., the number of stages may be deter- mined. The distribution factors may be neglected for the present, the pressures, specific volumes and areas being approximate; then the blade lengths may be checked by (38) as .the calculating proceeds toward the low-pressure end, and an increase in diameter may be desirable. It may be necessary to increase the diameter of the last one or two wheels to have SF 2 entirely accounted for. This is equivalent to the second method under unequal wheel diameters in the preceding paragraph, and is more general. As previously stated, in finding AC from (37), F R may first be neglected, but from the value so found, F R may be obtained from (36) and AC recalculated by (37) with sufficient accuracy. As with Formula (43), (48) is more correctly stated: F B (1 - y)knC] Vn 2 The method just outlined is applicable to any arrangement, such for instance, as the Curtis marine turbine referred to in Par. 12, Chap. IV. When the approximate calculation is complete and the number and pitch diameter of the wheels determined, the values of AC may be laid off on a diagram such as Fig. 144 and the nozzle and blade dimensions checked. In the preliminary calculations, nozzle exits only need be determined. 92. Trajectory and Lead. The actual path of the steam in its pas- sage through the moving vanes of a turbine is known as the trajectory. The* relative velocity of steam V N in impulse blading is constant if we neglect the effect of friction, which for simplicity will be done in the present discussion. THE STEAM TURBINE 251 Let the concave surface of a blade in Fig. 147 be divided into a number of parts and lettered a, b, c, etc. When the steam of velocity V N has passed over the distance ab, the blade has moved the distance W due to its velocity ; then: 66' = ab - cc' = (ab + be) ~ V N FIG. 147. FIG. 148. and so on until: ee r ae _ V N (49) is the distance the blade travels in the time it takes a particle of steam to pass over its surface. The distances ab, ae, etc., must be measured along the curve; a flexible scale may be used for this, or a close approxima- tion may be made by small divisions on a large scale drawing. Trajectories are shown in Fig. 148 for a number of ratios of S to V. The convex side of the blade is different in form and gives another path; by tracing the trajectories of both sides as in Fig. 149, some idea may be obtained of the path of the stream as it passes between the FIG 14Q blades. An example of a simple wheel in which a = 20 and S = N cos a/2, is given in Fig. 149, and for a 2-velocity-stage wheel in which S = V cos a/5 in Fig. 150. Lead. In pressure stages with partial admission in which there is considerable space between the wheel and the next set of nozzles, the residual energy of the jet is dissipated and there is little or no initial velo- city of entrance, so it makes little difference where the succeeding noz- 252 DESIGN AND CONSTRUCTION OF HEAT ENGINES zles are located; but if the clearance is small, so that part of this velocity might be utilized, the nozzles should be placed so as to receive this re- sidual steam with as little loss as possible. This may be done by finding the distance passed over as the steam passes through the blades as just described, and placing the next set of nozzles so as to receive this dis- charge; as this set usually subtends a larger arc than those preceding, but little difficulty is experienced. Where there are two or more velocity stages with partial admission, it is absolutely necessary for the guides to FIG. 150. be so placed that all of the steam discharged from one wheel will be directed to the next; it is well in this case to provide one or two extra guides at each end of the row. For marine turbines run at variable speeds, enough additional guides must be provided for the extremes of speed. In using formulas for finding the trajectory, the average velocity would probably be more correct. This may be found by assuming the friction loss uniform throughout the passage. Between entrance and exit the fraction of loss is 1 #; at the point b it is: ab f , - (1 0) ae THE STEAM TURBINE 253 in which ae is the total length of the passage measured along the surface of the blade. The velocity coefficient between a and b is then: and the velocity at b is: - -'' '* The average velocity between a and b is FIG. 151. Then: (50) The distance cc may be found by using ac instead of ab, and in like manner other points on the curve may be found. At exit, where ab is changed to ae: Formula (50a) may be used for finding the lead. Fig. 151 shows the arrangement of nozzles and guide blades for two velocity stages, the dotted lines showing the theoretical limits of the steam path. The lines from nozzles to blades follow the direction of the stream from the nozzles, while from moving blades to guides the direction is tangent to the trajec- tory curves at exit. 254 DESIGN AND CONSTRUCTION OF HEAT ENGINES Due to variable velocity of steam in reaction blading, it is not practi- cable to find the trajectory; and as the passages form convergent nozzles in which expansion occurs, and there is full admission, it is not important. It is desirable to draw trajectory curves for both front and back sur- faces of the passage for all moving blades. If the curves are not smooth, and free from too sudden changes, the blade form should be modified if the best efficiency is desired. If friction is neglected, (50) may be written : __ ab V* Then it is obvious from Fig. 133 that for the convex side of the blade the trajectory along the straight portion at entrance will be parallel to the jet from the nozzle. If friction is considered, (50) may be written: bV_ = _ S ab /. ab 1 Then in order that the trajectory at entrance may be parallel to the jet so as not to have a retarding influence on the back of the blade, W must be greater, making the entrance angle greater than 6 as shown in Fig. 135. 93. The Reaction Turbine. The essential feature of the reaction turbine is the ex- pansion of steam in the mov- ing blades. The guides cor- respond to the nozzles between stages in a pres- sure-stage impulse turbine and present no new feature. Although the principle is simple, the large number of stages makes a detailed analysis difficult due to the resulting accumulation of error if correct coefficients are not used. It has been stated that the design of reaction turbines is based upon empirical rules fixed by experience, to a larger extent than for other turbines. Be that as it may, considerable deviation from the conditions indicated by theory is practised in the interest of simplified construction, with appar- ently no deleterious effect upon efficiency. The same general method will be used as in the study of the impulse turbine, with certain modi- fications found desirable for this type. FIG. 152. THE STEAM TURBINE 255 Reaction Bidding. The velocity diagram is similar to that of the impulse turbine as shown in Fig. 152. V x is always greater than V N due to expansion between the blades caused by heat drop AC. The effect of friction may not well be shown on the diagram. If Fig. 152 is for the first stage, V is due to the heat drop only, but if for any following stage it is due to the heat drop and also the residual ve- locity V R from the preceding row of blades. Likewise the exit velocity V x from the moving blades is due to the relative velocity V N at entrance, and the expansion due to the heat drop AC. As already stated in Par. 84, the angle of discharge of both blades and guides depends upon the pitch and setting of the blades and is not obvious from the drawing, but that has no special bearing upon our calculations, as the blades are assumed to be set to secure the angles desired, .and shown upon the velocity diagram. The forces acting upon the blades are determined as in Par. 86 for the impulse turbine, and are: and: an f w = -(V cos a + V x cos - S) (51) t7 fr = (V sin a V x sin 0) (52) For any but the first set of guides which are really the first set of nozzles let A^C be the heat drop, and let A B C be the heat drop for the following row of blades including reheating; then equating energy: and: where i and j are coefficients making allowance for losses. It is convenient to consider one row each of guides and blades as a stage; then for one stage, the available heat is: F -AC=A r i AC- . * R AC h AcC 50,000 i It is usual to assume that V x = V, and V R = V N ; it then follows that = a, and <5 = 0. Then (53) becomes : which is the heat drop per stage (for one row each of guides and blades). 256 DESIGN AND CONSTRUCTION OF HEAT ENGINES Also (51) and (52) become: fw=- (2V cos *-S) (55) and: f,= From Fig. 152: TV = V 2 sin 2 a + (V cos a - S) 2 (66a) Then the energy supply in foot-pounds per second received by each stage is: E s = 778^-AC = l-jl--2cosa- (56) As mentioned in connection with impulse turbines, F R may be neglec- ted in preliminary calculations, and it is not altogether clear whether values of i and j given by Martin were based upon F R -AC or AC; if the latter, F R may be neglected. The work done in foot-pounds per second, per stage is: E w = f w S= (2 cos a - ) (57) The diagram efficiency is then: S E w (58) More general values of f w and AC given by (51) and (53) may be used in (57) and (56) for determining e D , but (58) gives a good idea of the relative effects of different values of a and S/V. Martin gives i = 0.9 and j = 0.52, which were found by analyzing the performance of a large high-pressure marine turbine, but he adopts the values: i = 0.89 and j = 0.5. As many reaction turbines have open-ended blades and guides, there is tip leakage, and only a portion of the steam does useful work. Fig. 153 is a diagram showing the arrangement of blades and guides, d being THE STEAM TURBINE 257 the blade length and c the tip clearance. The flow through passages of this kind presents a very complicated problem and the fraction of the steam passed through which may be assumed to be effectively applied is not obvious; but Martin gives it as: d - c d+\c / = 15.25 in. Then: S a = 157 S z = 239 S L = 361 V H = 278 7, =398 V L = 602 Neglecting wing blades on the low-pressure cylinder, the heat drop oer row of moving blades (or per stage) is: A#C = 3.08, A 7 C = 6.34, A L C = 14.5. It now remains to distribute the work among the three cylinders. This is rather arbitrary, but we will take three barrels per cylinder and designate the barrels by subscripts 1, 2, 3, etc. By taking D H = 10.75 and re-arranging the data, the following results are obtained, in. which n a , n I} and n L denote the number of rows of moving blades, or the number of stages on the respective cylinders, and n\, n 2 , etc. the stages per barrel. D L = 23" S L = 361 'V L = 602 n L = 10 A L , . v--" FIG. 173. In Fig. 173, ceb and oea are similar triangles; then: li = K Z 2 R Substituting in (7) gives: F=F,|- If the scale of velocity is such that V T is represented by R, then V is given FIG. 174. by K, which may be measured from a diagram and plotted on the piston path or crank circle, the latter being sometimes rectified. Fig. 174 shows the velocity plotted on the crosshead path and crank circle. From an inspection of Fig. 173 it may be seen that: K = R sin e R cos tan 0. The plus sign applies to the half of the crank circle toward the cylinder and the minus sign to the opposite half. Then : K 5 = sin cos e tan <(>, THE SLIDER CRANK 289 Also : From Fig. 173: From which: Let then: and: Then: and: Then: LO/U ti COS L sin = R sin 0. # . sin = ~Y sin 0. L L ~R = H ' sin0 n 1 /fs\rt fl\ 2 . /^ n , l-j /Sill C7\ cos

n 1 f Sm *V N 1 \ n ) K T, cos 9 1 . = 1 + sin 9 fi n L /sin 9\ 2 L \ V n / J ,,,.,.. V = V ^ : : = F r (sin cos tan ) = F.Fl C Se Ln fU/1 _ / sm \ n (8) (9) 101. Forces Due to Pressure in Cylinder. The forces acting on the slider-crank mechanism may also be determined by the method of instant centers and by the use of Fig. 175, in which P is the total unbalanced pressure on the piston (see notation). The tangential force (also known as turning effort and crank effort) due to P, denoted by P T} may be found by taking moments about instant center c; or: Pli = PT It. 19 290 DESIGN AND CONSTRUCTION OF HEAT ENGINES From which, p_ =P^. k By similar triangles as before : Then: li = K 1 2 '' R P T =P.- = P(sin cos 6 tan cos e = PI + L /sm ttA/1 (- \ V n sin 6\ 2 = sin 6 (10) FIG. 175. P T is a positive turning effort when the pressure P acts in the same di- rection as the movement of the piston; if in the opposite direction it is negative. The force due to P transmitted along the connecting rod, denoted by P L , is found by taking moments about shaft center o and substituting (10). Then: P L lo = PrR = PK or: By similar triangles : K- L lo n THE SLIDER CRANK 291 Then: P -P K PL ~ P T O (11) P L is the maximum force due to P producing bending moment on crank * pin and shaft at any time, and equals P at dead center. The normal force on guide due to P is denoted by P N , and is also found by taking moments about shaft center o and substituting (10). Then: P N x = P T R or: .... By similar triangles : 1 K y R sin 6 x VL 2 -y 2 Then: Psinfl L /sin n V 1 ~ (I The direction of action of P N is shown in Fig. 204 and Table 56. Force P is transmitted as an equal and parallel force to the crank pin, which in turn is transmitted to the main bearing. Two couples then hold the rod in equilibrium and may be equated as follows: P N L cos (f> = PL sin or: P N = P tan 0. It is obvious from Fig. 175 that K/x = tan 0, so that the value of P N is the same as that given by (12). The radial force on the crank due to P is denoted by P Rj and is found by taking moments about instant center a; or: PZ 3 = Prh and: P R =P T l ^= P T cot (0 + <*>). 292 DESIGN AND CONSTRUCTION OF HEAT ENGINES But: PT= PL = PL Sin (0 Then: P L cos (e + 0) p cos (e + ) = p cos e - sin 2 _ /sin 0\ \ n / This is zero when 6 + < equals 90 when crank and connecting rod are at right angles. Efficiency of the Slider Crank. From (9) and (10) it is clear that: P T V T = PV (14) It is shown in mechanics that PV is the rate of work done at any instant, and since (14) holds good for every position occupied by the mechanism, the work done at the crank pin equals that done in the engine cylinder if friction is neglected ; or the efficiency is : = (15) When it is considered that in high-grade steam engines the entire loss by friction is sometimes less than 5 per cent., the futility of trying to replace the slider crank with mechanisms for which great saving is claimed is apparent. 102. Forces Due to Gravity. Horizontal Engine. The connecting; rod of a horizontal engine has a negative effect upon the turning effort when the crank is in the half of the crank circle toward the cylinder, and. positive when on the opposite side. TABLE 50 Quadrant Force Formula 1 2 3 4 i G T (16) + -h C T (18) + + THE SLIDER CRANK 293 Let G B be the reaction of the rod of weight W at the crank pin. The turning effort may be found by taking moments about shaft center o; then : G T R = G B R cos 6 or: G T = G B COS 6 = L f W COS 8 (16) Li The force G B also acts as an equal and parallel force at the center of the main bearing. The force due to W acting on the guide is: G A = W -G B (17) Let C be the weight of the unbalanced part of the crank, or the weight of the crank pin; or if a counterbalance is used it may be the unbalanced __ --M // - - L "'" \ /' -- __^ Jjf i FIG. 176. part of this weight. Let r c be the distance from the center of the shaft to the center of gravity of the weight C. The algebraic sum of these weights referred to the crank pin center may be expressed by: 8") Then the turning effort due to these is: C T = 2 (^ C ) cos (18) The signs for G T and C T for the different quadrants in Fig. 176 are given in Table 50 under the corresponding quadrant numbers. The plus sign indicates a positive turning effort that the force acts in the direction of motion; while the minus sign is for a negative effect. If the overbalance due to C is on the crank side the signs are as given in Table 50; if opposite the crank the signs for C T must be reversed. Aside from friction the weight of other parts does not affect the turn- ing effort. 294 DESIGN AND CONSTRUCTION OF HEAT ENGINES Vertical Engine. The weight of the piston, piston rod and cross- head obviously affect the turning effort, but as it acts in the same line with the pressure it may be accounted for in drawing the indicator diagram. The effect of the connecting rod is to increase the turning effort on the down stroke and decrease it on the up stroke; its weight is sometimes added to that of the strictly reciprocating parts but it is more correctly treated by itself and this will be done, leaving the approximate method to the judgment of the designer. The rod exerts a downward force G and a lateral force G N as shown in Fig. 177. It must be assumed that the entire weight is supported by the crank pin; then: G = W. The weight of the rod acting at the center of gravity and resisted by an equal force at the crank pin forms a couple equal to: G(L L G ) sin 0. This is balanced by the couple: G N L cos or: G N L cos = W(L L ) sin . From which: T,~ sin th GN ~- (19) wn - sin FIG. 177. /sn 0\ ~ hr) Taking moments about crank center o for forces G } G N and C, the corresponding turning efforts are found. The equations are: G XT R = GR sin G NT R = G N R cos C T R = THE SLIDER CRANK 295 Then by cancellation and substitution the turning efforts are: GXT = W sin 6 L G \ sin 26 GNT =G N COSe=w(l- G } ^C Bin* The total turning effort due to the gravity of the rod is: G T = GXT + GNT TABLE 51 (20) (21) (22) (23) Quadrant Force Formula 1 2 3 4 GXT (20) + + GNT (21) -f + CT (22) + + The signs for the quantities GXT, GNT and C T for crank positions in the different quadrants of Fig. 177 are given in Table 51, the plus sign for forces acting in the direction of turning as . before; the sign for C T assumes the over- ,..--" balance on the pin side of the crank. The t A forces G and G N also act at the center of the main bearing. If the engine center line is inclined from the vertical 5 degrees as shown in Fig. 178, modification of the reactions must be made as follows : G B = W - W(l - ) sin 5 (24) (25) GN =G(l - -^) tan cos 5 The effect on the turning effort is: G T = G B sin (0 - d) G NT = G N COS (0 6) C T = s FIG. 178. (26) (27) (28) 296 DESIGN AND CONSTRUCTION OF PI EAT ENGINES The signs for the different portions of the crank pin path depend upon the angle 5 and may be determined by inspection for a particular case. Formulas (24) to (28) reduce to those already given for a horizontal or vertical engine. 103. Forces Due to Acceleration. Practically all the inertia forces of the slider-crank mechanism may be accounted for by considering the acceleration of any point in the connecting rod in the direction x, parallel to the crosshead path, and the direction y normal to the crosshead path. In the formulas, which will be given without derivation, co is the angular velocity in radians per second, and may be expressed in terms of r.p.m. thus: 2irN irN "GOT =30 (2 ) The acceleration in ft. per sec. per sec. in the direction x, Fig. 179 is: 2 a x on . 4- cos 26 + - and in the direction y: a '=I sin 6 (30) (31) FIG 179. As force is the product of mass and acceleration, the general formula is : (32) in which a may be either a x or a Y , and if W is the total weight, a must be the acceleration of the mass center. Formulas (30) and (31) give acceleration in any point in the rod; and for any mass assumed as concen- trated at that point, (32) gives the force. An approximation to (30) is given by (48), the latter being much more simple, and for most practical applications accurate enough. Hereafter F Y will denote the inertia of the rod normal to the line of stroke, F YA its effect at the crosshead and F YB at the crank. F x will denote the inertia of the rod along the line of stroke, referred to its mass center. When I is zero, a x is the acceleration of the piston and other purely reciprocating parts. The inertia of these parts will be denoted by F P . The force exerted by the inertia of a mass upon the engine parts in a THE SLIDER CRANK 297 horizontal direction is toward the cylinder when the crank is on this side of the crank circle, and in the opposite direction when the crank is on the side away from the cylinder; an exception is when the crank is in the vicinity of 90 degrees from the engine center line, and this may be seen by plotting (30) as in Fig. 180, in which a H and a c are accelerations at head and crank ends of the stroke respectively. The signs in Formula (30) are used simply to obtain numerical results, and for this purpose may be taken as plus when angles 6 and 26 are in the half of the crank circle toward the cylinder. This has a negative effect upon the turning effort on the stroke from head to crank end when the numerical result has a positive sign, and a posi- tive result when the sign is minus. On the return stroke, however, the effect is just the reverse. This may be shown by Fig. 181, which gives the acceleration diagrams for the two strokes, the positive sign de- noting that the inertia of the moving part would assist the turning effort and the negative sign that it would have a retarding effect. The Reciprocating Parts. If I equals L in (30), a x is the acceleration of the crosshead and piston. At the dead centers (30) becomes : a x = R<. The minus sign gives the numerical value for the head end and the plus sign for the crank end. Then the inertia of the reciprocating parts for head and crank ends may be computed, and other points found graphi- cally by Klein's method shown in Fig. 182. From (32) : (33) A\JO\J \ 'I If / and: 2930 298 DESIGN AND CONSTRUCTION OF HEAT ENGINES Fig. 182 is an exact representation of (30) and proof is given in Dalby's Balancing of Engines previously referred to. The lower part of the con- struction is added for the convenience of plotting to any scale. After deciding upon a force scale, F H and F c may be laid off on the diagrams as shown ; the lines ab and cd are located by making the construction at the dead center only one of these being necessary. The Connecting Rod. If in (30), I equals L G , the distance to the mass center, the effect of acceleration of the rod parallel to the line of stroke may be found. If we assume n an imaginary value of n, and substitute in (30): n n -- FIG. 182. then using this value in (33) and (34), values of F x for the rod may be found graphically. The inertia of the reciprocating parts may be combined with the indicator diagram of total pressure before determining turning effort, and often the connecting rod is added to the reciprocating parts to simplify calculation, probably with sufficient accuracy in most cases; but to be more exact the rod requires a separate treatment, and F X) the inertia of the rod assumed concentrated at the mass center may be treated as W in connection with the gravity effect of the rod for vertical engines. Then for the component parallel to line of stroke, comparing with (20): F XT = Fx sin (35) For the component normal to line of stroke, comparing with (21): L \ sin 20 (36) THE SLIDER CRANK 299 Fig. 183 shows the effect of rod inertia; it may also be used as a diagram for the effect of inertia of the reciprocating parts, taking F P only, if it is desired to treat it separate from the indicator diagram. The total effect of Fx on the turning effort is: (37) XT XNT TABLE 52 Sector of circle T? 1 1 2 3 4 5 6 Fxr (35) + + + FXNT (36) + + + The signs of FXT and FXNT are given in Table 52 for different sectors of the circle as shown in Fig. 183. A change occurs when the inertia FIG. 183. curve crosses the line in the case of FXNT, and again when the crank is at right angles to the center line of the engine. The effect of angular acceleration is best treated by finding the moment of inertia as described in Par. 99. To derive the formulas, assume the rod divided into a number of parts of weight w, each a distance I ft. from the center of the crosshead pin. The force due to acceleration of the rod normal to the line of stroke acts away from the center line of engine, and 300 DESIGN AND CONSTRUCTION OF HEAT ENGINES its effect on turning effort is shown in Fig. 184. The force exerted by each section of the rod is : F Y = a Y = ' T -Rw*sm 9 (38) g g L The reaction of this force on the crank pin in a direction normal to the line of stroke is : The total reaction is: F YB = 2F YN = r 2wZ 2 sin 9 CO T gnL / sin 9 FIG. 184. The effect upon the turning effort is: FYT = FYB COS 6 ..2 '1 7 sin 29 2gnL N* 5865nL . 7 sin 29 (39) (40) The signs for FYT for the different quadrants are given in Fig. 184. The total turning effect of the rod inertia is : FT FXT ~\- FXNT + FYT (41) The reaction on the guide due to F Y is found by taking moments about the center of percussion, the location of which is given by (6). If F YA is this reaction: THE SLIDER CRANK 301 From which: (42) The reaction on the guide due to F x is found as for W in (19); or: sin 6 F XN /sin 0\ " br) The total guide pressure due to acceleration of the rod is: FA FYA ~\~ FXN (43) (44) The forces FXN, F x and FYB acting at the crank pin, also act as equal and parallel forces at the center of the shaft. 104. Combined Indicator and Inertia Diagrams. As the pressure on the piston and inertia of the reciprocating parts are both applied to the crosshead pin, they may be combined and their turning effort con- sidered together. It is first necessary to construct a stroke diagram giving the unbalanced steam or gas pressure; that is, the difference between the pressure on the two sides of the piston. This is found by plotting the distance between the forward pressure of one diagram and the back pressure of a diagram for the other side of the piston taken at the same time. Steam engine diagrams will be first considered. A pair of indi- cator diagrams is shown in Fig. 185, and it is best that they should be diagrams of total pressure as this allows for the piston rod. The shaded area represents the stroke- diagram for the head end, giving the pressures acting on the piston during the stroke from head to crank end of FIG. 185. cylinder. The pressure is zero at a and becomes negative after that. For convenience this may be plotted from a straight line, and taking pressures above the plotting line as driving the crank in the direction of motion (with the plus sign), and below the line as tending to retard the motion of the crank (with the minus sign), Fig. 186 is plotted, with inertia diagram shown below, and finally the combined diagrams. A convenient method of plotting the combined diagrams is to invert the inertia diagram on the stroke diagram as shown in dotted lines, thus 302 DESIGN AND CONSTRUCTION OF HEAT ENGINES performing graphical addition and subtraction; then the measurements may be transferred to a diagram plotted from a straight line, or may be taken directly to plot a crank effort diagram. This applies especially to steam engine diagrams; for the internal-combustion engine confusion may be avoided by following the method shown by the full lines, adding the pressures algebraically. If the engine is vertical the weight of the reciprocating parts should be added to the down-stroke diagram and subtracted from the up-stroke FIG. 186. diagram as shown in dotted lines; Fig. 186 shows the addition made to the head-end diagram, which would be the case if the cylinder were above the crank the most common arrangement. If the center line of the engine leans 8 degrees from the vertical as in Fig. 178: P w *= W cos 5 (45) This quantity is added to the head end if the cylinder is higher than the shaft, and subtracted from the crank end. THE SLIDER CRANK 303 Internal-combustion Engine Diagrams. The conventional stroke dia- gram for a 4-cycle constant- volume engine is shown in Fig. 187 with inertia diagram and combined diagram below. The diagram is for an automobile engine running 1500 r.p.m. The connecting rod is not included in the inertia diagram, so that the combined diagram shows only the forces applied at the wrist pin. The inertia of the rod must be included in the forces acting at the crank, but this is best given a separate treatment. Data for the diagrams are given in Table 55. Accurate results cannot be obtained by including the rod with the reciprocating parts, especially if the center of gravity is near the crank pin, which is true of this type. In modern well-designed engines, suction and exhaust pressures differ but little and both are assumed as atmospheric in this chapter. FIG. 187. If the engine is not horizontal, P w (shown dotted) must be taken into account as in Fig. 186 and Formula (45). In finding values of P for different crank positions, the spacing around the crank circle should be equal; the corresponding piston positions are therefore not equally spaced and the spaces are greater at the head end than at the crank end of the stroke. These points in the piston path may be found as in Fig. 188, then marked on a piece of tracing cloth and transferred to diagrams such as Figs. 186 and 187 with a pricker point, placing the tracing cloth end-for-end for the return strokes in order that letters H and C may fall on the same letters on the diagram. Figs. 186 and 187 show that pressure is constantly exerted against the bearings, absorbing work by friction. By plotting the forces of such 304 DESIGN AND CONSTRUCTION OF HEAT ENGINES diagrams upon a rectified crank circle the mean pressure upon crank pin and shaft may be obtained for the cycle. An approximation may be made by including the connecting rod with the reciprocating parts. Then the mean pressure from such diagrams is a measure of the work absorbed by friction; this is not the mean effective pressure or the alge- FIG. 188. braic sum of the pressures, but the mean of the actual pressure against the bearings during the cycle, regardless of signs. For the main bearing a resultant should be taken, including weight of wheel, shaft, etc., pull of belt or reaction due to gears, or any other forces acting on the bearing. 105. Reversal of Thrust and Effect of Compression. It has been generally accepted that compression is necessary to smoothness of operation of a steam engine, but in few instances is it probable that the necessary compression has been fixed by calculation, or if so, the inertia of the reciprocating parts has probably been neglected. An examination of this is best made by inverting the crank side of the upper dia- gram of Fig. 186. The pressures above the plotting line are those acting toward the crank, while those below the line act toward the head end. This is shown in Fig. 189. Neglecting inertia, reversal of thrust occurs when the pressure line crosses the plotting line, which, for the dia- grams shown gives a gradual re- versal, occurring before the end of the stroke. When inertia is con- sidered it occurs where the pressure line crosses the inertia curve; this curve may be considered the virtual plotting line for the remaining dia- grams of this paragraph. The effect of inertia is to cause reversal to occur later in the stroke, but in Fig. 189 it is still gradual*; if the diagrams were plotted on a rectj- FIG. 189. THE SLIDER CRANK 305 fied crank circle a more correct way of determining the relative time of reversal it would appear still more gradual. With higher speed assume the inertia curve to be the dotted line A. In any case reversal occurs where the inertia curve crosses the pressure curve, therefore for curve A the reversal is at initial pressure for the head end and a little less at the crank end. The reversal at head end is gradual, and at crank end the reversal force is the pressure represented by ran, and while it appears to occur abruptly, it is probable that the crank travels over an appreciable angle while the pressure is being applied and there would be little or no tendency to " pound, " even with some play in the pin bearings. However, if the inertia is less and the curve does not cut the pressure curve (due to compression) an appreciable distance before the dead center, a pound may be heard if the pin bearings are not properly adjusted. With inertia curve B, reversal at head end occurs at o, after the stroke has begun, but is gradual and would probably cause no pound; at crank end it occurs at dead center but is gradual. If there were no compression the pressure curve would be as shown clotted; then for the inertia curve shown in full lines the force due to- re versal would change at dead center from ql to In; for curve A it is from qm to mn, and for curve B from qn to zero, the reversal force being gradually applied. It is obvious that if reversal does not occur before the end of the stroke, the force ql, qm or qn has no influence upon smooth- ness of running, but that this depends upon the magnitude of the force starting the piston on its return stroke, such as In or ran; if this is large a " knock" or " pound" may be the result, and it is clearly an advantage when inertia is small, to have the pressure curve cross the inertia curve before the end of the stroke. A serious obstacle to this is that without a high compression, the terminal pressure at large loads is higher than the compression pressure and the pressure line does not cross the plotting line; this is especially true with single-eccentric, non-releasing gears, which reduce the compression as the terminal pressure is increased. The uniflow engine has some advantage in this respect, but with heavy loads reversal may occur but little, if any before the end of the stroke. A study of the diagrams makes clear the fact that increasing the weight of the reciprocating parts has sometimes aided in smoothness of operation; it may also account for the non-analytical discussions in technical papers from time to time by operating engineers, as to the necessity of compression, with fervent adherents to both sides of the question, which may only be satisfactorily explained by the use of re- versal diagrams. 20 306 DESIGN AND CONSTRUCTION OF HEAT ENGINES As the piston speed S = 4RN, Formulas (33) and (34) may be written: This shows the influence of weight, piston speed and rotative speed upon inertia. To obtain the reversal pressure at the crosshead pin, the reciprocating parts proper must be considered alone; for the effect at the crank pin, it will be approximately correct to add the weight of the con- necting rod to W, but more accurate to determine the inertia of the rod separately as explained in Par. 103. When an engine is running idle, as a locomotive in "coasting," reversal curves are due to inertia only, and are shown in Fig. 190. Rever- sal occurs near mid-stroke and is very gradual. Indicator diagrams for a single-acting engine are also shown in Fig. 190 with dotted lines. As the pressure line does not cross the inertia curve at the crank end of the stroke, there is no reversal of thrust at this place; on the return stroke reversal J FIG. 190. occurs when the compression curve crosses the inertia curve, and unless inertia is nearly equal to or greater than initial pressure, there is advan- tage in having the compression pressure greater than the inertia. It should be noted that the reversal which would have occurred at the crank end of the stroke in a double-acting engine, is deferred until the inertia curve crosses the plotting line on the return stroke a short time before the head-end reversal. It may be seen from Fig. 189 that if compression pressure is high relative to inertia, the effect of inertia is more largely applied at the cylinder end of the frame; whereas with high inertia pressure and small compression the retarding effect comes on the bearing. In Fig. 189, if the cut-off is long, as shown dotted, the turning effort will be as uniform as possible if the inertia effect is negligible; the shorter the cut-off the less uniform it becomes. If the inertia effect is increased for the short cut-off the effective pressure becomes more uniform through- out the stroke. A locomotive may thus be started with a maximum cut-off, and by bringing back the reverse lever gradually as the engine gains speed, the maximum uniformity may be obtained at all times. Failure to do this results in discomfort to passengers and increased wear and tear on the mechanism. THE SLIDER CRANK 307 A steam automobile will take a hill more smoothly when running slow, if the cut-off be lengthened and the pressure throttled. A single-acting, constant- volume internal-combustion engine diagram is shown in Fig. 191. For the full lines there is no reversal except where the inertia curve crosses the plotting line during the exhaust and suction strokes. Should inertia at the head end of the stroke be high enough to cross the compression line as shown dotted, reversal occurs gradually EXPANSION EXHAUST SUCTION FIG. 191. COMPRESSION where the lines cross at n, and again at the end of the stroke. The mag- nitude of the reversing force depends upon the difference between inertia and the maximum gas pressure; if inertia is higher as in curve A, Fig. 191, reversal occurs after the expansion stroke has begun, and is gradual. A reversal diagram of a 4-cycle, double-acting internal-combustion engine is shown in Fig. 192, and with a different order of firing in Fig. 193. In Fig. 192, for the values assumed, reversal occurs at the begin- ning of the head-end and crank-end expansion strokes, and again, gradu- H.EXP. C. COM P. H.SUCT. C. EXH. H. COM P. C.SUCT.. FIG. 192. ally, during the head-end suction stroke. In Fi5. 193, abrupt reversal occurs only at the beginning of the head-end expansion stroke, and gradual reversal during the head-end exhaust stroke. Increasing the inertia to the dotted lines improves conditions for Fig. 192 by making the reversal gradual near the end of head-end compression, and reducing the reversal pressure at the beginning of crank-end expansion. 308 DESIGN AND CONSTRUCTION OF HEAT ENGINES s 1 10 c CO 2 CO ^ o T-H r- 1 r T i O O O o c d 1 1 1 1 C^ CO ' 10 s 05" s. t> 00 8 8 8 T-H o O O o o C 3 o 1 1 1 00 o 00 g So g s 8 CO oo g * i b g 00 8 g O5 8 s (N 2 oo CO c 1 ) t CN s s So So g s? i ' i T-H CO g Jo S i g c 5 T 1 d d o i ^ d o o T-H T- 4 T 1 10 a S g 00 8 i i 8 CO T-H CO s s 1 CO T-H tf 5 4 CO r - i H o o o T-H t* 4 ~ 4 1C > CO ^ 10 CO * o CO W 3 CO r M ~l *s 1 t ^ \ ^" i- *SK ^ -~* \ ^ - .^, 1. ^ s -M Si 5 I * '53 v ^ ^ S ^L - X + 1 'a 1 H H > * L J ^ ^- 1 ^ i i j v^l - II * ! 9 v Q> _g 3 g a w ~ ^ S H i THE SLIDER CRANK 309 to ? ^ C^j CO o l-H l-H l-H !"H rH o O O o o o ^ *~I * TH ^- i-H CO tt a & o oo 8 i 8 s rH CO o i 1 co rH 1C > i CO CO o o o o T-H T-H l-H o o o rH r- H __ on oo o O5 O5 8 8 CO O5 to Jo 8 i-H o i g a rH ^ (N CO C o o o b b c ) b 05 s 8 S5 & 8 i i O5 CO S c; i (N GQ ^ OS 00 o l-H o i-H & i TH rH ,_, 8 a CO ^ g > i rH C-l C o b b b c > 1 1 O5 cp ^ ^ to to 8 s So O5 cr s i 8 S 05 rH CO TH 00 9 i S 1 1 CO 00 s S5 s ? ' * 1 1 B t^ (N CO CO rH rH "s cl 5 s W c o C > 1 1 OS CO 05 to rH 1 1 00 c- O5 00 S C O | 1 "* g 00 oo s s i i 8 CO 3 58 s i-H i-H i g I~ b B c^ o o T i-H O C > o 1 1 to (N CO co to a 00 8 8 8 o rH (N c3 8 to g > (N 00 T-H o rH T i-H C > 1 1 r <> C1 * *s N <3s * -?> G S ** a 5 g o "m -~ " s +1 *f. ^ ~s H q -1 > * (N 2 = 109. Other data for the rod found from methods in Par. 99 are: I = 78,654, r 2 = 59, L G = 6.32, L P = 8.07. / 7 \ ^ --., x K y / -^ x ^ [\ J / \ / / \ \ Of Z .3 FIG. 194. The weight of crank referred to pin = 500 Ib. Weight of counter- balance referred to pin = 2225 Ib. The inertia of the reciprocating parts are, from (33) and (34): F H = 12,250 Ib., and F c = 8750 Ib. Other points in the curve were found from Klein's construction, Fig. 182. The connecting rod is not included with the reciprocating parts but is treated separately. Total pressure P includes the effect of inertia of the recip- rocating parts proper. The counterbalance overbalances the crank by 1725 Ib., referred to the crank-pin center. The signs for C T are then opposite to those given in Table 50. If the area of Fig. 194 is found by a planimeter and divided by the length the mean ordinate may be found, and a line representing the mean turning effort drawn as shown. The fluctuation of turning effort may thus be plainly seen, and the areas between the curve and this line repre- sent energy fluctuations. The ratio of the maximum area thus enclosed to the area of the rectangle formed by the mean line and the plotting line (for one stroke, one revolution or one cycle, as desired) is called the coef- ficient of the fluctuation of energy. Referred to one stroke, this quantity for Fig. 194 is 0.333. The mean line must always be found from the diagram for the entire cycle. 312 DESIGN AND CONSTRUCTION OF HEAT ENGINES o c oo" H I gj o i CO 8 8 8 00 So THE SLIDER CRANK 313 c-; o o 8 CO o" rH O g rS O o" CO go o co oo iO CO iO (N a 8 id rH 1 id g S | CO 1 .2 1 'i CO co CO ^H ^j 00 CO c5 i r H *^* *"*" 1 00 CO rH t> iO ^1 C^l 9 1 rH 1C 1 1 % CO I 00 o 3 8 00 rH (N iO rH 1 rH 1 CO IS T rH i s g g s rH (N ^ ~f CO 1 7 10 1 05 s i 8 rH (N rH rH t^ CO i 00 O5 IO oo N ( c^ CO iO rH O5 J>^ 1 co 1 ^f C^ C^ l> >. 1 1 I So co^ 1 o" rH 2J. rH 9) ft| H ^ ** fc, *** Ct, *" ^ ** ^ h fc. THE SLIDER CRANK 317 oo 06 06 06 S IS > ^ 55 9 1 1 8 0 10 10 8 i-H c^ CO T^i C^ TT T^ fH 1 cq CO s 00 10 . iH TH 1 1 T-H ; 1 TH 1 1 1 g T-H T-H CO CO 00 oo 05 00 CO 3 a TH 1 I K =F P 1 tein 0' (49) The signs for P assume the force to be acting in the direction of piston motion; when the reverse is true the signs are reversed. This may be determined from diagrams similar to those given in Figs. 186 and 187. Fig. 204 shows the numbering of the sectors, the meaning of the signs, and the action of the different forces, while Table 56 gives the signs. TABLE 56 Force Formula Sector of circle 1 2 3 4 5 6 f PN (12) __ __ "DA FPN (49) + + + - PA FXN (43) + + + 1 FYA (42) + + + PN (12) + + -t- + + ,+ FPN (49) + + + PBY ' FXN (43) + + + FYB (39) + + + FCY (50) + + + - r F P (32) + + + + PBX [FX (32) - + + + + - 1 F cx (51) + + If the revolving parts are not balanced, or are overbalanced due to a portion of the reciprocating parts being balanced at the crank, the cen- trifugal force of this weight will react at the main bearing. Normal to the line of stroke this force is : F CY = S (^ -) flco 2 sin e \H l (50) This forms a part of p BY , the sign being as in Table 56 if the crank and pin THE SLIDER CRANK 325 are under-balanced ; if overbalanced the signs are reversed. The effect parallel to line of stroke is: This is added to p BX , the sign being as in Table 56 for the unbalanced crank and pin. The forces of Fig. 204 and Table 56 may be plotted or tabulated sepa- rately, similar to Table 57, and combined if desired. They do not all act in the same plane, and may produce other forces and couples. Their effect may be taken up within the frame itself in some cases, so as to have little or no effect upon the foundation or other support, especially in multi-cylindered engines. Some of the forces are insignificant in some cases, but with different types, sizes, speeds, etc., it is well to prove this by calculation; the formulas are simple and easily applied with the aid of Table 53 and the diagrams, and their use may account for various things giving trouble to engine builders. The forcesFx (due to the connecting rod) andFp(due to the reciprocat- ing parts) act along, or parallel to the line of stroke, but may act on the frame either at the cylinder end or the bearing end, or may be divided between them, depending upon the steam or gas pressure acting upon the piston. It must be borne in mind, however, that if the restraint of the foundation bolts is neglected, the forces acting upon the frame in the line of stroke are limited by the steam or gas pressure in the cylinder; this must be determined by the indicator diagram alone, inertia having no effect. If inertia is great, part of it may be balanced by compression, in which case it is transmitted to the foundation from the cylinder end of the engine; any excess inertia is transmitted to the frame and thus to the foundation at the shaft end of the engine. The inertia effects normal to the line of stroke cause bending and torsional stresses in the frame, but these are influenced by the attachment to the foundation. Divisions 2 and 5 in Fig. 204 are not quite the same for F P and F x , as the inertia curves do not cross the line at the same point. The action of the forces p B x and p BY along the crank in a line from the center of the shaft to the center of the crank pin is: PR = PBX COS + PBY sin 6 (52) All or part of the forces of which these are composed may be taken in finding p R . Formula (52) is convenient as practically all balancing is done along the crank center line. The forces of Fig. 204, added algebraically are : 326 DESIGN AND CONSTRUCTION OF HEAT ENGINES PA = PN + F PN + F XN + F YA PBX = F P + Fx + Fcx PBY = PN + F PN + FXN + ^V* -f (53) (54) (55) Formula numbers for the various forces are given in Table 56. It may be convenient at this place to state the meaning of the notation contained in Formulas (53) to (55). F P FP F x 3 /Oil , = normal component of force P due to steam or gas pressure. If P includes the inertia of the reciprocating parts, P N includes F PN . Should the inertia of the connecting rod be included as in rough calculations. P N includes FXN- = the inertia of the reciprocating parts. = the normal component of F P . = the inertia of the connecting rod parallel to line of stroke, as- sumed as concentrated at mass center of rod. FIG. 206. FXN = normal component of F*. FYA effect at crosshead pin of inertia of connecting rod normal to line of stroke. FYB = same at crank pin. Fcx = effect of centrifugal force of crank or counterbalance parallel to line of stroke. FCY = same normal to line of stroke. In vertical engines G N , the gravity effect of the connecting rod, should THE SLIDER CRANK 327 be added to p A and P B Y', its value is given by (19) and the signs are the same as for P N in Table 56. \ XI X X \ \ \ XJ FIG. 207. With the data for the Corliss engine already considered, and by the use of Fig. 204 and Table 56, Table 57 has been computed. The method of determining the counterbalance will be given in the next chapter. The values of p A are plotted along the crosshead path in Fig. 205. Values 328 DESIGN AND CONSTRUCTION OF HEAT ENGINES o o o o o t^ ~H ^ 00 I-H OO O (O CO CO CO ^ ^ (M 1 1 I I CO O CO O CO O OS CO U5 CO ^ I I ,4'f o oo o o oo rH (N CO * N S CO OS CO | OO TfH N | ' . -5 rH 1 I O 00 CO 00 # (N r-l 1C N O O CO CN O 00 N CO 00" 00" CO" 00 COCO'HOSOOCOOOrHOOO t^-OS'O COoOCO O>OO OiOOO CDOCOOO-OOiOOt>'00'-i (N t>- M 5 O CO -O I I w" IN" oo" co" co" ^ 1 11 o" o" i-H r-l I ^ CO M 4 OS CO i-H i-H t^ CO o I I I (N * I I "N t O IN os q I I oo to co -H IN OS CO IN CO to_ * oo os co * to o M" (N N CO* tO* t^ CO CO | CO CO T(T 1-1 fff 1 1 1 S 8 S S 8 S fc i g CO CO CO 1 1 1 IO" I I CO T of 00* III* 00* 00 CO* II o o II *" co 88 1 a, a, fc, ft. 330 DESIGN AND CONSTRUCTION OF HEAT ENGINES of p BY and F C Y (for counterbalance) are plotted on a rectified crank circle in Fig. 206, and of p BX and F C x in Fig. 207. The resultant p s of forces PBX and p B v may best be found graphically from Figs. 206 and 207. This has been done, taking the counterbalance into consideration, and plotted on a crank circle in Fig. 208, giving both direction and magnitude of the force acting during the revolution due to the moving parts of the FIG. 209. slider-crank mechanism. This force acts on the main bearing. To ob- tain the force acting at the crank pin, that due to the crank and counter- balance (F C Y and FCX) must be omitted. This is shown in Fig. 209. The actual force on the bearing is somewhat different as it is not in the same plane; neither do the forces due to crank and counterbalance act in the same plane as the forces at the center of the crank pin, but for most calculations for strength and wear this may be neglected. FIG. 210. To obtain the total force acting on the crank pin or shaft at any point, the resultant of p s and the steam or gas pressure must be obtained; when bearing pressure is desired the effect of gravity must be included (con- necting rod, crank, shaft, wheel, etc.). The force due to steam or gas pressure may be found by the use of a stroke diagram such as Fig. 186 or 187, neglecting inertia. In Fig. 210 it may be seen that the force THE SLIDER CRANK 331 P R acts at an angle -\- = a, from the point on the pin which is nearest the cylinder when the crank is on the head-end center, and that it moves around the pin in a counter-clockwise direction. This is also true of the shaft. FIG. 211. Fig. 208 or 209 may be combined with Fig. 210; then to find the forces acting on shaft or pin, these may be drawn in the position of head-end dead center; by revolving the resultant diagram on the same center in a counter-clockwise direction, the resultant forces may be transferred to FIG. 213. FIG. 214. the pin or shaft in such a manner that they pass through the center if produced. Figs. 209 and 210 for the crank pin are combined in Fig. 211 and transferred to the pin in Fig. 212. The latter shows the normal force acting on all sides of the crank pin by means of which the maximum reversed or repeated loads may be determined. 332 DESIGN AND CONSTRUCTION OF HEAT ENGINES Figs. 208 and 210 for the shaft are combined in Figs. 213 and 214 in the same way. These diagrams are for the 20 by 48 in. Corliss engine previously men- tioned, and it must be remembered that they are not strictly accurate due to the forces not being in the same plane. In all of these diagrams except Figs. 212 and 214, the force acts from the point on the circle from which the line starts, and the effect of gravity of the parts is neglected. In Figs. 212 and 214 the force curve is all drawn outside the circle which represents pin or shaft, and acts toward the circle; the radial force lines are numbered in order showing the direction of force change during the cycle. Reversed and repeated loads may be taken from these digarams (Figs. 212 and 214), the reversed loads being diametrically opposite. If two forces are more than 90 degrees apart they produce a certain amount of reversal; this usually produces bending. As the stress measured from any neutral axis is proportional to the distance from the axis, the inten- sity and kind of stress may be found at any point on the surface for forces inclined at any angle. Various couples are produced by the constituents of p A and P B Y, some of which tend to neutralize the others. Their effect is also influenced by the counterbalance if any is used. The attachment of the frame to the foundation or other support in- fluences the effect of these forces which might be determined by taking moments about certain points; but the origin of moments is so indefinite, and the behavior of the supports under active forces so uncertain that but little satisfaction is obtained by an attempt at anything like accurate analysis. For large engines, as careful a consideration as possible of the proportion of the reciprocating parts to be balanced, and for small light engines, a careful application of the best balancing methods, will bring the most satisfactory results. In all that precedes, an absolutely uniform motion of the crank pin has been assumed. This is not strictly the case, but in most cases there is no great error in the assumption, and an attempt to account for this would be useless. The treatment of the mechanics of the slider crank given here, while simple in principle, is rather complicated, and for most design problems only the direct forces due to gas or steam pressure and to heavy weights such as the flywheel, are used. However, no one is absolutely sure of the wisdom of neglecting certain forces until he has worked through a number of problems, and for turning effort diagrams used in flywheel design, the neglect of the effect of acceleration results in a method but little short of a rough rule of thumb. Also, when inertia is combined with the stroke THE SLIDER CRANK 333 diagram of Fig. 187, the resulting force diagram differs considerably from the original diagram due to gas pressure only; were the connecting rod included, giving the forces acting on the crank pin, the difference would be still greater. Perfect balance is claimed for certain engines of ordinary construction ; there is no such thing, and the statement either shows lack of knowledge or an attempt at a "snappy" piece of advertising. The heat engine designer cannot afford to be ignorant of the simple principles of this chapter even though he uses them but little. If the forces are carefully calculated the factor of judgment may be reduced in selecting the factor of safety; if, as is usually the case, it is considered unnecessary to make such elaborate calculations, the factor of judg- ment should be such as to cover discrepancies. This will be further discussed in Par. 166, Chap. XXI and in connection with the various details in later chapters. Reference Balancing of engines W. E. Dalby. CHAPTER XVII BALANCING 108. Introduction. The subject of the balancing of reciprocating engines is well covered by Prof. W. E. Dalby in his excellent work by this title, and it is not possible to claim a complete treatment in a short chap- ter; however, an attempt is made to bring the essentials of balancing into simple practical form, so that they may be used to solve all balancing problems connected with the usual forms of reciprocating engines. It might be more correct to say that the lack of balance may be determined by these principles with a view to keeping it a minimum. Notation. W = weight in general in pounds. W B = weight to be balanced referred to the crank pin. WP = weight of reciprocating parts in pounds. W = unit weight in pounds. C = weight of crank and pin in pounds referred to pin center. P = force in general in pounds. Cp = radial inertia (centrifugal force) of a revolving weight in pounds. Q = a couple. M = moment in pound -feet or pound-inches of weights to be balanced. A B = area in square inches or square feet of face of counterweight. t = thickness of "counterweight in inches or feet. I = dimensions in general in inches or feet. r = radii in general. L = length of connecting rod from center to center in feet. R = crank radius in feet. w = angular velocity in radians per second. N = r.p.m. n = L/R. g = 32.16. 109. Simple, or Primary Balancing. Centrifugal force tends to dis- place a revolving weight in a radial direction from the axis about which 334 BALANCING 335 it revolves. The magnitude of this force is equal to the product of its mass and radial acceleration; or: W W /TrN\ 2 r* E> 9 z? / i / 1 \ OF = /vo> 2 = ti I -JTJT- 1 (1) Had W been taken as an indefinitely small division of the weight and R its distance from the axis, the sum of these products would equal the total weight multiplied by the distance of its center of gravity from the axis. It then follows that two weights diametrically opposite, revolving about an axis, will be in balance if the product WR is the same for each, FIG. 215. and this is the general principle of practical counterbalancing of machine parts. The balancing of parts revolving in a circle is simple, and practically consists in suitably placing the balance weights so as to make a neat design. However, for over-hung cranks there is an unbalanced couple due to the impossibility of placing the counterbalance in the same plane as the portion of the weight to be balanced. This is shown in Fig. 215. So far as forces are concerned, balance is obtained if the connecting rod is disconnected when: = Will + WR (2) 336 DESIGN AND CONSTRUCTION OF HEAT ENGINES where W is the weight of the portion of the pin in the bearing, Wi the weight of the crank and W 2 the weight of the counterbalance. The crank and counterbalance, as shown, are in the same plane, but the pin is not in the plane with the portion of the weight which balances it, and the couple : W RuPm g is formed, which has a tendency to cause rotation in a plane normal to the plane of rotation. In practice this is usually small relatively and is ig- nored. With the center-crank engine the counterbalance is divided between the two crank arms and the couples balance. Reciprocating Parts. Let the effect of angularity of the connecting rod be temporarily neglected and its weight assumed added to the recip- rocating parts. These parts must be brought*to rest at the end of each stroke, the force required being the product of their mass and the radial acceleration of the crank. If C be the weight of crank and pin referred to the center of the crank pin, and W p the weight of the reciprocating parts, the force acting at the crank pin at dead center is: g If the crank and pin only are balanced, then at the dead center, W p is unbalanced, and this is true in a lesser degree for every other position of the crank except when it is at right angles to the line of stroke. If W P is also balanced by a revolving weight, the system is in balance only at the dead centers, the counterbalance being too heavy by the amount W P when the crank is at right angles to the line of stroke. A compromise is usually made by balancing all of the revolving parts and a certain per- centage of the reciprocating parts; at dead center the engine is under- balanced and at right angles, overbalanced. The percentage of reciprocating parts to be balanced depends upon the type of engine and location. For locomotives, a special formula is used in which the total engine weight is a factor, but the limits are usually between 55 and 65 per cent. In all engines with but one rod connecting with the crank pin this state of unbalance must exist, even though the system as a whole may be balanced in a multi-cylinder engine. In many small engines with slow or moderate speed and heavy frames, the counterbalance is omitted, or if added by the use of a disc crank, no calculations are made in proportioning it ; but with large engines or high BALANCING 337 speeds, counterbalance is of advantage, not only for smoothness of opera- tion but for relieving undue wear and strain. 110. Secondary Balance. If, in addition to balancing the revolving parts, a weight be placed opposite the crank so that the product of weight and distance to mass center equals the product of crank radius and weight of reciprocating parts, the radial inertia of this weight at any point in its path is equal to that given by (1). From (33) and (34), Chap. XVI, the radial inertia of the reciprocating parts is greater than this at the head end of the stroke and less at the crank end. If the rod were infinitely long, or a Scotch yoke were used, there would be perfect balance at the dead centers, and this is known as primary balance. The balance along the line of stroke which would be necessary to compensate for rod angu- larity is known as secondary balance. It does not include the inertia of the rod itself, but only such masses as may be considered concentrated at the crosshead pin. From (32) and (48) of Chap. XVI, taking I as zero in the latter, the inertia of the reciprocating parts along the line of stroke is: tffcos 6 + y cos 26 1 (3) L L j This is the approximate formula, but is sufficiently accurate for the pres- ent purpose. The negative signs are omitted, but cos 8 and cos 20 have signs as explained in connection with Formula (30), Chap. XVI. Removing the brackets and multiplying and dividing the term con- taining cos 26 by 4, (3) may be written : W W J? F = u*R cos + (2w)- jt- R cos 26 y y rtL/ = primary force + secondary force (4) This shows the following: 1. The primary force along the line of stroke is equivalent to that which would be produced if the reciprocating parts were concentrated at the crank-pin center and revolved at the same speed as the crank. 2. The secondary force along the line of stroke is equivalent to that produced by a mass equal to the reciprocating parts concentrated at a radius equal to the fraction R/4L of the crank radius, and whose rotative speed is twice that of the crank. When the crank is at either dead center, this mass is always at the head-end dead center. The relation of the crank with this imaginary crank is shown in Fig. 216. A graphical expression of (3) is given in Fig. 217, the radius of the imaginary crank in this case being R/n. The algebraic sum of the dis- 338 DESIGN AND CONSTRUCTION OF HEAT ENGINES tances from the vertical center line to the projections of the two cranks on the horizontal center line gives the inertia to the same scale that R gives the primary force at dead center. It is obvious that for engines with a single cylinder, the balancing of the secondary force is impracticable, therefore the reciprocating parts do not permit of perfect balance with revolving weights. FIG. 216. Complete balance of the reciprocating parts by a revolving weight is usually undesirable, so that the importance of secondary balance is only seen in certain combinations of multi-cylinder engines. It should not be inferred from what has preceded that revolving weights at crank pin and imaginary crank-pin centers would produce the same forces in all directions as those produced by the reciprocating parts; the FIG. 217. effect is identical only along the line of stroke, the reciprocating parts having no other influence upon the frame except through the normal component F PN , given by (49), Chap. XVI. 111. Multi-cylinder Engines. In this paragraph a number of com- mon arrangements will be discussed relative to the balancing of revolving parts and purely reciprocating parts along the line of stroke, leaving the BALANCING 339 connecting rod and the normal components of cylinder pressure and inertia to later paragraphs. Complete balance includes the balance of forces and couples. Un- balanced forces may be determined by taking the algebraic sum of the components of all forces normal to a given diameter, assuming them to act in the same plane normal to the axis of revolu- tion, as in Fig. 218. Perfect balance is obtained if Pi + P 2 + Pa = 0. Unbalanced axial couples may be deter- mined by taking moments about some axis cutting the shaft center at right angles, of the components of all forces in their re- spective planes of revolution, normal to a plane containing both this axis and the shaft center line. This is shown in Fig. 219. The center of gravity of the upward forces is given by: FIG. 218. lu = and for the downward forcei: + The couple tending to cause rotation of the shaft in an axial plane is the product of the distance between the center of gravity and the lesser of the two forces, PI + P* or P 2 -j- PS. Let this be denoted by P ; then: Q = Po(l D ~ lu) (7) t=5=ii r if i I 1 h j FIG. 219. the subscripts D and U denoting down and up forces respectively. It is often necessary to find a resultant couple due to the forces p B x and p BY of Fig. 204, Chap. XVI; this is given by: Q = Vo^Tos (8) 340 DESIGN AND CONSTRUCTION OF HEAT ENGINES r FIG. 220. If the forces are not in balance, there will be a force and a couple act- ing on the engine. The force may form a couple with some other force due to the reaction of the supports, which may act in the same or opposite direction of the first couple. * If all forces and couples are not balanced in the engine itself considered as a free body, it is practically impossible, due to the uncertainty in locating the origin of moments, to determine the couple acting on the sys- tem; but usually this is not necessary. In some engines it is difficult to determine the maximum couple with- out plotting values for the entire cycle. Two-crank engines with cranks in the same phase are treated the same as engines with a single cylinder so far as inertia is concerned. With cranks at 180 degrees, the primary force is balanced, but the small imaginary crank is always at the head-end dead center when the actual crank is at either center; consequently this gives an unbalanced effect equal to twice the secondary force. Referring to Fig. 220, if PI is at the crank-end dead center it is smaller than P 2 , and : Q = Pi(h ~ W W If the crank and pin are not balanced, their centrifugal force, referred to the crank-pin center, should be added to PI in (9). There is no lateral component in the dead-center position, which gives a maximum couple. To obtain the best results the cylinders should be placed as close together as possible, and the cranks (and perhaps a portion of the reciprocating parts) should be balanced. Four-crank engines such as f 3 y _ ? Fig. 219, with cranks in the same plane, equal spacing of cylinders and weight of re- ciprocating parts, are in bal- ance for the primary force but not the secondary force; both primary and secondary couples are balanced. The system as a whole is in no better balance if the revolving parts are balanced, but bending strains in frame and shaft, and pressure on bearings are relieved thereby. Three-cylinder engines with cranks opposed as in Fig. 221 have a primary force error unless the reciprocating parts of the center engine FIG. 221. BALANCING 341 are equal to the weight of the other two, which is not practical. There is a secondary force error, but if the cranks are evenly spaced there is no couple error. Angle Engine. This is shown in diagram in Fig. 222, with centers at an angle of 90 degrees. If the reciprocating parts are the same weight in both engines, and all the reciprocating and revolving parts are balanced by a revolving weight placed opposite the crank, the primary force is balanced. There still remains the unbalanced secondary force, which acts along a line inclined 45 degrees from the line of stroke as shown at 1-2, Fig. 222, and is a maximum as the crank passes each dead center, which is four times a revolution. As both connecting rods engage with the same crank pin the unbalanced axial couple is small. This type of FIG. 222. FIG. 223. engine is the best balanced of any type of 2-cylinder engine of usual construction. Three-crank engines with cranks at 120 degrees, cylinders in the same plane and on the same side of the shaft, have no force error due to purely reciprocating and revolving parts. It has been shown that the effect of the reciprocating parts along the line of stroke is the same as if their weight were attached to the crank to produce the primary force. Then assume three cranks at 120 degrees as shown in Fig. 223 with such weights attached Assume the cranks to move from a symmetrical position through the angle A. Then from trigonometry: cos (A -f- B) = cos A cos B sin A sin B. and: cos (A B) = cos A cos B -\- sin A sin B. 342 DESIGN AND CONSTRUCTION OF HEAT ENGINES Adding together gives: cos (A + B) + cos (A - B) = 2 cos A cos B. lfB = 60, 2 cos B = 1; and: cos (A -f B) + cos (A - B) = cos A. This shows that the projection of centrifugal force upon the center line of engine, or any other diameter, of three equal weights revolving at equal radii with the same angular velocity, are in equilibrium in all posi- tions. As the secondary force along the line of stroke is equivalent to the projection of the centrifugal force due to the mass of the reciprocating parts concentrated at the center of an imaginary crank pin traveling on a circle R/4L times that of the engine crank, and at a velocity twice as great, it follows that these imaginary cranks are always 120 degrees apart on the crank circle. According to Fig. 223, they are then in equilibrium among themselves. This is true of any revolving weights, such as the cranks, if they are of equal weight and their mass centers are the same distance from the shaft center. As previously stated, the force due to the reciprocating parts is not caused by a revolving weight, and is equivalent to a revolving system only along the line of stroke. The normal component F PN , due to inertia of the reciprocating parts is not in balance in a 3-cylinder engine; this will be considered later. There is a couple error, which is a maximum for the primary force when the center crank is at right angles to the line of stroke. For the reciprocating parts this is equal to the product of the inertia, the cosine of 30 degrees and the distance between the center of the first and third cylinders. If the cranks are not balanced, the radial inertia of crank and pin, referred to the pin center, must be added to the inertia. The actual couple due to reciprocating parts is some less due to the secondary effect. Six-crank engines with cylinders in the same plane and on the same side of the shaft are balanced along the line of stroke for both primary and secondary forces, the same as the 3-crank engine. It is practically the same as two 3-cylinder engines placed end to end as shown in Fig. 224. Applying Formulas (5), (6) and (7) shows that if the arrangement of BALANCING 343 cylinders about the line midway between the two center cylinders is symmetrical and the reciprocating parts and cranks weigh the same for each cylinder, there is no couple error, either primary or secondary. The only advantage of balance weights in this engine is to relieve frame and shaft strains and bearing pressures, and to reduce this to a minimum, part of the reciprocating parts should be balanced. The effect of the forces acting on shaft and bearing, with and without counterbalance may be seen in Figs. 208 and 209, Chap. XVI. 112. The Connecting Rod. Forces due to the mass of the rod were discussed in Chap. XVI, and their effect upon the frame may be sepa- rately considered if desired. In computing counterbalance it has commonly been assumed that if the rod is divided between the reciprocating and revolving parts inversely as the mass center divides the rod, it is correctly disposed of. As the pro- portion of reciprocating parts to be balanced is either fixed experimentally as in locomotive practice, or arbitrarily assumed, such a rule is probably accurate enough. It is, in fact, correct along the line of stroke, as may be seen by adding La W I, L G \W -Y- a, toll - 7^) a 2 L g \ L I g where W is the weight of the rod, ai = Ru 2 cos 6, the acceleration of the crank pin, and a 2 is the acceleration of the crosshead pin as given by (30), Chap. XVI, when I is zero. The result is identical with (30) and (32), Chap. XVI which gives the inertia of the rod along the line of stroke when I = L Q . Normal to the line of stroke, the force due to the rod is the algebraic sum of FYB and F X N given by (39) and (43), Chap. XVI, the signs being taken from Table 55 of the same chapter. For a rod of uniform section, five cranks long, this gives: while the divided rod method gives: 0.5 g an error of 42 per cent. Neglecting the balance of the crank, let it be assumed that 60 per cent. of the reciprocating parts is to be balanced; then the balance weight required to balance this according to the usual method is: W 344 DESIGN AND CONSTRUCTION OF HEAT ENGINES where WP is the weight of the reciprocating parts. If W = kW P , the constant radial inertia (centrifugal force) of the weight, assumed to have its center of gravity on the crank circle is : The inertia of the reciprocating parts at the head end of the stroke, from (33), Chap. XVI is: -, 9 W P Ra* r H = 1.Z - 9 At the head end, from (34) : The inertia of the rod along the line of stroke at the head-end dead center is, from (30) and (32), Chap. XVI: , =1. Q g At the crank end : The total head-end inertia is: and at the crank end: (0.8 + 0.9/b) WpR< ** Comparing with the counterbalance, the amount unbalanced at ends of stroke is : For head end, (0.6 + 0.3fc) Effli' c/ For crank end, (0.2 + 0.1 A;) g Average for two ends, (0.4 + 0.2k) - J/ Normal to the line of stroke, when the crank is at right angles to the line of stroke, the inertia as just given, is: 7 y = 0.353^=0.353*^. .".-". The overbalance in this position is: (0.6 + 0.447/b) WpRu *. BALANCING 345 With the divided rod method it was assumed that g 9 This would give an overbalance of : (0.6 + 0.3*) y which is less than the actual. The ratio of the actual to the assumed is: actual overbalance 0.6 + 0.447& assumed overbalance 0.6 + 0.3& If k = 1, the ratio is 1.162. For the Corliss engine which is being carried through the book, k = 0.86, and the ratio is 1.15, an error of 15 per cent. It may be observed that the method is more correct the nearer the mass center of the rod is to the crank pin. This also permits the rod to be more nearly balanced by a revolving weight and reduces the force and couple errors due to the rod mass. The rod cannot be perfectly balanced in engines of usual form, even in 6-crank engines. 113. Turning Effort. The turning force is transmitted to the engine frame at the guide (or cylinder, when a trunk piston is used), giving the force p A . This is plotted for the 2*0- by 48- in. Corliss engine in Fig. 206, Chap. XVI. It forms a couple equal to the turning couple, but may be offset, at least for part of the cycle, by the effect of counterbalance and other forces normal to the line of stroke acting at the shaft center. This may be seen in Fig. 206, Chap. XVI, which is drawn to the same force scale as Fig. 205. These forces are periodic and tend to cause vibration normal to a plane containing the engine and shaft center lines ; they may not well be balanced. In many engines with massive frames and heavy foundations the effect of turning effort is unimportant and is seldom given any thought. In ships and automobiles, in which the engines are made as light as prac- ticable, vibration may be considerable, due to this cause, but is offset in part by arranging cylinders and cranks so as to obtain as uniform a turning effort as possible; that is, to provide more impulses per cycle. The vibrating forces are then more frequent but vary less from the mean force applied. With uniform turning effort as given by steam turbines and electric motors, the couple is constant, and this condition is approximated in multi-cylinder engines. As previously stated, the nature of supports or foundation determines to a great extent the effect of unbalanced forces. If the periodic motion is in synchronism with the natural period of vibration of the supports the 346 DESIGN AND CONSTRUCTION OF HEAT ENGINES consequences may be serious; but if not, a small amount of unbalanced force will usually have no serious effect. 114. Balance weights are of various forms, some of which are shown in Chaps. XXVII and XXVIII. The required balance referred to the crank pin may be determined, then when the general form of the counter- weight is selected, it may be drawn to scale with assumed dimensions. With certain simple forms the center of gravity may be calculated ; then the area may be calculated or found by a planimeter and the moment determined. Some designers prefer to draw the weight to scale, then cut it out of card board or thin wood and either balance it on a point, or hang it from two different points so that it is free, allowing a plumb line to hang from the point of suspension; the point where the lines cross in the two posi- tions is the center of gravity. Sometimes the first trial will suffice, any deviation from the required weight being provided for by varying the thickness. When there is not room to accommodate the required weight, the casting is sometimes cored and filled with lead. The segment of a circular ring is a form rather commonly used, and this lends itself to simple means of finding dimensions. Fig. 225 shows a crank with a counterweight of this kind. The moment of the counter- weight about the shaft center must equal the sum of the moments of all weights to be balanced ; or : wA B tl B = S(TFZ). = M (10) where w is unit weight, A B the area of the segment forming the counter- balance, t its thickness and 1 B the distance of its center of gravity from the shaft center. All dimensions must be either in feet or inches; if the latter, w is the weight per cu. in. From (10): Without taking space for the derivation of the formula: sin r + r It is plain that: Then letting r /V = q: A B 1 B = ?(r 3 -r 3 )sin| = |r 3 (g 3 - 1) sin | (12) BALANCING For a plain balance with no limits for r , 347 = 3 1 + 3A B 1 B 2r 3 sin 1 + (13) FIG. 225. Then: r = qr (14) If a disc crank is used which fixes both r and 7*0, and if t is assumed: sin ^ = If 6 is assumed: - 1) t = 3M - 1 sn (15) (16) With a disc crank, the thickness of the disc should not be counted as part of either arm or balance thickness. 348 DESIGN AND CONSTRUCTION OF HEAT ENGINES In the Corliss engine for which data is tabulated in Chap. XVI, the values of p R from Table 57 were taken at crank positions 6, 12, 18 and 24, and averaged. This method is arbitrary, and in this case gave slightly smaller weight than by taking 0.6 of the reciprocating parts added to the revolving parts, the rod being divided inversely as the mass center. The average gives the radial inertia required of the counterweight and is : 10,366 + 20,580 + 4126 + 25,385 From (1), solving for W (W B in this case) gives the weight referred to the crank pin; or: _gC P _ 32.16X15,114 _ R- 2 X 109 Then from (10), in pound-inches. M = 2225 X 24 = 53,400 From (13), more conveniently taking dimensions in inches, and assum- ing a cast-iron crank: Then: a = si 1 , 3 X 53,400 \ r 2 X 0.26 X 6.375 X 12 3 X 0.866 r = 3.22 X 12 = 38.75 in. This assumes the thickness of balance as 6^6 in., which is the same as the arm thickness for the crank designed for the 20-in. Corliss engine in Chap. XXVII. The angle d was taken as 120 degrees. The crescent and the segment of a circle are forms used in locomotive balancing. Fig. 226 shows a crescent, the distance be- tween the centers of the two arcs being denoted by l c . A B is the area of the crescent and A the area of the segment between this and the chord. A simple exact formula for 1 B was published by the author in the American Machinist some years ago and is : A 7 (17) FIG. 226. or: Al c = A B 1 B = (18) BALANCING 349 If the counterbalance is a segment, A = and l c = ; then the formula fails; then: or: A.JL, - (20) CHAPTER XVIII REGULATION DURING THE CYCLE. FLYWHEELS 115. Introduction. It is evident from the turning-effort diagrams of Par. 106, Chap. XVI that the force applied to the crank pin in the direction of motion varies considerably during the engine cycle, even when a number of cylinders are employed. Unless the resistance varies in identically the same way, which it never does, a change of velocity must occur during the cycle, the number of fluctuations depending upon the number of times the turning effort varies from the mean; also the displacement of the crank pin from the place it would occupy at any instant with per- fectly uniform motion, occurs. The function of the flywheel is to limit speed fluctuation and dis- placement to an amount found by practice to give satisfactory operating results. The method of controlling speed fluctuation is simpler and is much more commonly used for designing wheels for all purposes. The dis- placement method is more difficult and is used only when engines are designed to drive alternating-current generators which are to operate in parallel. The relation between the two methods will be shown in Par. 118. In applying either of these methods to practical work certain assump- tions are made and refinements neglected, as the varying condition of operation of any type of engine does not permit of great accuracy. At- tention is called to these approximations in the proper places. Notation. D = diameter of wheel in feet. R = radius of crank circle in feet. r = radius of gyration in general, in feet. w = weight in pounds in general. W R = weight of wheel rim in pounds. W w = total weight of wheel in pounds. W = weight in pounds concentrated at crank-pin center, which would give the same effect as the weight of the entire wheel concen- trated at its center of gyration. 350 FLY WHEELS 351 M = mass assumed concentrated at center of crank pin, corresponding to W ( = W/g). F = turning effort in pounds, above or below the mean effort. E = mean energy per piston stroke in foot pounds. AE = fluctuation of energy above or below the mean; usually the maximum = eE. H = indicated horsepower. V = mean velocity of rim in feet per second. Vi = maximum velocity of rim. Vz = minimum velocity of rim. v velocity of crank pin in feet per second. N = r.p.m. a = linear acceleration of crank pin in feet per second per second. t = time of one engine cycle in seconds. s = displacement in feet of the crank pin from its mean position, c = coefficient of fluctuation of energy. d = coefficient of fluctuation of velocity. A = change of. a = displacement of crank pin from mean position in electrical degrees. c = number of cycles of alternator per second. p = number of poles. m = time scale = number of seconds per inch. n = force scale = number of pounds per inch. q = scale of Mv = number of units of Mv per inch. k = scale of Ms number of units on Ms per inch. x, y and I = measurements on diagrams in inches. C and K = constants in Formula (8) and Table 60. g = 32.16. 116. The Control of Speed Fluctuation. This method may be applied where the resistance varies, so long as the cycle is constant at a given load. In most cases where such calculations are made the resistance is assumed to be uniform for a given load, so that the turning effort dia- gram shown in Fig. 194, Chap. XVI will be used to illustrate the methods of determining flywheel weight in this chapter. This is reproduced in Fig. 227. As it is the variation from mean which causes change of speed, the areas above and below the mean effort line are the areas considered, and these are shaded. Each one of these lobes bounded by the curve and the mean line represents the fluctuation of energy as the crank pin passes between two intersections of these lines, and is denoted by AJ. The 352 DESIGN AND CONSTRUCTION OF HEAT ENGINES crank pin has a certain velocity at A (and the flywheel rim a proportion- ate velocity). In passing from A to B energy is taken from the wheel equivalent to the area of the lobe between A and B, and its velocity is reduced. At B the velocity begins to increase and continues to do so until C is reached, the kinetic energy stored in the wheel during this time, as the crank pin travels from B to C being represented by the lobe between B and C. Kinetic energy varies as the square of the velocity; then it is obvious that the maximum fluctuation of velocity is caused by the maximum fluctuation of energy, and the largest lobe AJ57 must be used in the follow- ing calculations. Let E be the energy of the cycle in ft. Ib. divided by the number of strokes, or the mean energy per stroke. Then the coefficient of the fluc- tuation of energy is : e E (1) FIG. 227. Basing c upon the energy per stroke has advantages over energy per cycle in simplifying the final equations." If N = r.p.m., there are 2N strokes per minute; the i.h.p. then is: 2NE NE ~ 33,000 ~ 16,500* Then for all engines the mean energy per stroke is : N As just stated, the change of kinetic energy of the rim due to the maximum fluctuation of velocity from V% to V\ ft. per sec., or vice versa, is equivalent to the maximum fluctuation of energy along the crank- pin path; then from (1): ( Vl 2 _ y 2 2) W R (Vi - F 2 )(Fi + F 2 ) &E = eE = (3) 2g 2g If V is the mean velocity and 8 the coefficient of the fluctuation of speed: (4) FLY WHEELS Also, nearly enough for practical purposes: 353 or: 2 60 = 7rDN 30 where D is the wheel diameter in ft. Substituting (2), (4) and (5) in (3) gives: 16,500e# (5) N Solving for W R gives : 60 X 30 X 20 W R = 194,000,000 (6) Stresses in flywheels will be treated in Chap. XXX, but to limit these to safe values the rim speed is not allowed to exceed a certain limit; a com- mon speed allowed for cast-iron wheels is one mile (5280 ft.) per minute. Then: irDN = 5280 and DN = 1680. Substituting this in (6) gives : Formula (7) is convenient in estimating when it is likely that the wheel will run up to the limiting speed; but it gives a lighter rim than does (6) should the rim velocity be reduced, and this must be kept in mind. If values of e and 5 are known, (6) and (7) may be written: W - C H - K rtrt - D2 N 3 ~ * N The value of e in Fig. 227 is 0.33. This was plotted for the 20- by 48- in. Corliss engine designed in Chap. XII with a J^ cut-off. Goodman gives the values of e for double-acting steam engines in Table 58. TABLE 58 Cut-off Single cylinder Two- cylinder cranks at 90 Three- cylinder cranks at 120 0.1 0.35 0.088 0.040 0.2 0.33 0.082 0.037 04 0.31 0.078 0.034 0.6 0.29 0.072 0.032 0.8 0.28 0.070 0.031 Full 0.27 0.068 0.030 354 DESIGN AND CONSTRUCTION OF HEAT ENGINES It will be seen that e decreases when the cut-off, and therefore the load increases; but not in the same ratio. It would seem from this that a value of H near the maximum should be used. It is &E (which varies as eH) that causes the fluctuation of speed, and not e ( = -g-j For 4-cycle internal-combustion engines working on the Otto cycle, Goodman gives the values in Table 59. TABLE 59 Exploding at Values of e for internal-combustion engines Single-acting Double-acting 1-cyl. 2-cyl. 4-cyl. 1-cyl. 2-cyl. Every cycle Alternate cycles . . . 3.7to4.5 8.5 to 9.8 1 . 5 to 1 . 8 2 . 5 to 3 . 0.3 to 0.4 2.3 to 2.8 0.3 to 0.4 TABLE 60 Service e 3 C K Pumping . . 0.4 Ho 2,320,000,000 824 Machine shop Textile paper and flour mills 0.4 4 Ks Ho 2,710,000,000 3 100 000 000 960 1 100 Spinning machinery 0.4 Moo 7,740,000,000 2,750 Gas compression 0.4 Moo 7,740,000,000 2750 Rolling mill 2 Ho 11620000 000 4 110 Cable railway 0.4 M20 9,300,000,000 3,300 Electrical machinery (belted) 4 Mso 11,620 000 000 4 110 Direct-connected alternator 4 y$v 387 500 OOOp 140p Sugar mill grinder Sugar mill crusher 0.4 0.4 H a >73 3,200,000,000 5,600,000,000 r 1,160 2,010 In Chap. XIV, Par. 78, reference was made to hit-and-miss governing at light loads. From (6) and (7) it may be seen that the speed fluctuation of a wheel of given weight varies as the product eH. If every alternate impulse is missed the value of H is one-half full load. Then taking values from Table 59, to give the same regulation at half load a single-cylinder gas engine would need a wheel 10 per cent, heavier, and a two-cylinder engine 83 per cent, as heavy as for full load. This may not hold true for lighter loads, but by making a wheel on a hit-and-miss engine some 20 per cent, heavier than the formula indicates, fair regulation should be 23 FLY WHEELS 355 obtained for quite a wide range of load. If combination governing were used, the hit-and-miss principle would not be applied except at the lower range; if at one-half load, one-quarter load could be carried by missing every alternate impulse. Many tables of 6 have been published and there is considerable variation. Table 60, for double-acting, single-cylinder steam engines is given as an example in tabulating C and K of Formula (8), also to give some values of d which have been taken from various sources. The value for the direct-connected alternator will be further explained in Par. 3 ; p denotes the number of poles. The table has been used by the author for several years and part of the data are from his practice. It must be used with judgment; in some cases the values of C and K may give too small values to enable the proper construction of a wheel for strength, especially if the rim speed is near the maximum limit. The value of c is taken a little high to provide for discrepancies. From Table 58 it will be seen that the value of e varies inversely as the square of the number of alternations of turning effort. This would mean that a cross-compound steam engine requires a wheel but one-quarter as heavy as a single-cylinder engine of the same power. But with compound engines, load distribution between high- and low-pressure sides is con- siderably affected by different valve settings, and this in turn affects the form of the turning effort diagram, often increasing the value of AE. Then too, a compound engine does not respond to the governor quite as quickly as a simple engine, especially if the high-pressure cylinder only is directly governed. It therefore seems that some allowance should be made in using values of e found from carefully designed compound dia- grams. With large units which are to have frequent applications of the indicator this need not be very much, possibly an increase of 25 per cent. In a good many cases the author has determined wheel weights by the use of Table 60 for a simple engine, and taken from 75 to 100 per cent, of this weight for cross-compound engines of the same power, and while this may seem excessive, it has been satisfactory. For internal-combustion engines with more than one cylinder it is probably not necessary to make much allowance, although there may be a small difference in the indicator diagrams of the different cylinders; increasing the value of e about 10 per cent, over the value found with equal indicator diagrams will probably give ample allowance. For strict accuracy the diameter D used in the formulas should be equal to twice the radius of gyration of the entire wheel, or, more strictly, of all parts revolving on the shaft, and WR should include the entire weight of wheel and other revolving parts. Jf w is the weight of a part 356 DESIGN AND CONSTRUCTION OF HEAT ENGINES of the wheel or other revolving part and r its radius of gyration about the shaft center, then: W a (f) 2 : S(w 2 ) (9) The calculation of wr 2 requires much time and is usually an unnec- essary refinement. If D is taken as the outside diameter of the wheel, the weight of the wheel rim found by (6), (7) or (8), and the weight of arms, etc. neglected in the calculation, the energy of the wheel will be equal to or greater than that required if it is a belt wheel; for a heavy flywheel with a thick rim (radially) the weight may fall a little short if calculated in this way, but increasing the weight so found by 10 per cent, will make ample provision. The value of e in Table 60 for simple steam engines will cover this. In estimating it is convenient to know the weight of the entire wheel. After finding rim weight W R from (6), (7) or (8), sufficient metal will be allowed for a well-built wheel of the more common designs if the total weight W is as follows : For belt or rope wheels. .... FV = 1.7 to 1.8TF (10) For heavy-rimmed flywheels . . W w = 1.5tol.6W R (11) 117. The Control of Displacement. Alternators operating in par- allel must be in synchronism to give good results. They must not only have the same r.p.m. but the angular distance between corresponding poles (in electrical degrees) must not exceed a certain limit or cross currents will be set up, lowering the average voltage and causing loss of effectiveness and efficiency. The general effect of irregularity of turning effort upon speed fluctua- tion during the cycle has been discussed in the preceding paragraph; it now remains to study the effect upon the displacement of the wheel from some mean position it would occupy if revolving with a perfectly uniform velocity. To try and make this clear before proceeding with a mathematical determination of the displacement curve, a rather imaginary piece of apparatus shown in Fig. 228 will be examined. Assume two wheels running side by side with the same r.p.m. Wheel No. 1 runs with absolute uniformity of speed and has on it two paper drums operated by gears with a uniform motion. The arrow shows the direction the paper moves. Wheel No. 2 is an engine flywheel subject to the variation of turning effort. Arm A, containing a pencil is fastened to a spoke of this wheel. During the revolution the relative angular position of the two wheels changes, causing the pencil to move across the spoke of wheel FLY WHEELS 357 1 and draw a curve as the paper moves at right angles to the pencil move- ment. This is a displacement curve and with a constant load on the engine it should be exactly alike for each cycle of the engine. A line drawn in the direction of paper motion in such a way that the algebraic sum of the areas enclosed between it and the curve for a com- plete cycle is zero, is the mean line; this is shown dotted, and if the pencil of wheel 2 is brought over to this line it will show the mean position of wheel 2 relative to wheel 1 as the two wheels revolve. FIG. 228. A similar curve may be drawn from the crank-effort diagram of any engine. When the turning effort is above or below the mean it causes positive or negative acceleration of the wheel. It is convenient to refer all force and motion to the crank-pin center. Then if M is the mass ( = W/g, where W is the weight in Ib.) giving the same effect as the wheel concentrated on the crank circle, a the linear acceleration of the crank pin and F the turning effort in Ib. above or below the mean : F = Ma or: a= M (12) Then a is directly proportional to F, and Fig. 227 may be converted into an acceleration diagram by changing the scale. 358 DESIGN AND CONSTRUCTION OF HEAT ENGINES In practice the displacement is very slight, so that with the usual spacing of the crank circle, if it be assumed that equal increments of time are accompanied by equal increments of space as the crank moves around the circle, the degree of accuracy will be in keeping with that of the whole method. Then by changing the scale, space may be changed to time; then the time of one cycle in seconds which is represented by the length of the diagram is: For a 2-stroke cycle, t For a 4-stroke cycle, t 60 N 120 N (13) Fig. 227 is reproduced in Fig. 229 as an acceleration-time diagram. The dimensions x, y and I are always in inches, and areas in sq. in. MEAN LINE PLOT LINE \0 I 2 3 4 S 6 FIG. 229. Acceleration-time diagram. A general expression for acceleration is: dv dt a = -T. or: dv = a-dt (14) It is then obvious that by integrating the acceleration-time curve above and below the mean line, a curve may be plotted with change of velocity as ordinates and time as abscissas a velocity-time diagram. A general expression for velocity is: ds dt V = -r. or: ds = v dt (15) After finding the mean line for the velocity-time curve, this curve may be integrated and a displacement-time curve plotted. From the mean line found for this curve the maximum displacement from mean may be found. FLY WHEELS 359 All integrations will be made with a planimeter, the mathematical ex- pressions being used to make the operations clear. Fig. 230 is a velocity-time curve and Fig. 231 a displacement-time .'iirve. To make practical use of these diagrams scales must be used for the different quantities, and it avoids much confusion if these are determined a^ the work proceeds. MEAN LINE PLOT LINE FIG. 230. Velocity-time diagram. For all curves: 1 in. = m seconds = j, where t is found from (13). Then: dt = mdx where dx is in inches. Let: 1 in. = n Ib. = n units of F (= Ma). Or: 1 in. = - units of a = - ft. per sec. 2 (16) FIG. 231. Displacement-time diagram. n has already been determined for plotting Fig. 227. Then: (17) To integrate the acceleration-time curve to obtain change of velocity, the mathematical expression may be reduced by substituting (16) and (17) in (14) as follows: w r v -u* (18) 360 DESIGN AND CONSTRUCTION OF HEAT ENGINES The quantities n and m are known. If it is desired to find the dis- placement of a wheel already designed or built, M is also known. The quantity in the integration sign is the area in sq. in. between any two ordinates found by the planimeter. If this is the first space of Fig. 229, shown shaded, it is below the mean line and must be laid off at point 1 of Fig. 230 from and below the plotting line, the mean line not having been determined. The mean line is found by taking the difference of the areas above and below the mean line and dividing by the length in inches. If the area above the line is greater, the mean line will be drawn above the plotting line, or vice versa. If y\ is the distance from mean to plotting line and 3/2 from plotting line to curve, this may be expressed algebraically thus: y\ = I y^'dx + I (19) To fix the scale for the velocity-time curve, determine the total height that it is desired to make it, then add together the maximum and mini- mum values of Az; (not algebraically) and divide by the desired height in inches and round up to some convenient figure. In most design problems M is unknown and may be carried through the calculation as an unknown quantity. We may then write (18) as follows: fx t M-kv = nm I y dx (20) Jxi Then let: 1 in. = q units of Mv; 1 in. = -^units of v = ~ ft. per sec. Then: To integrate the velocity-time curve to obtain displacement the ex- pression is found by substituting (16) and (21) in (15), thus: f*tt f*X2 m /X2 As = vdt = Ifyn-dx = 5? ydx (22) Jti Jxi MJxi If M is unknown: f* M-As = qm I ydx (23) Jxi The scale is found as for the velocity-time curve ; then let : 1 in. = k units of Ms; FLY WHEELS 361 or: k * t 1 in. = -=-= units of 5. M The mean line is found as before from (19). The maximum ordinate y M from the mean line must now be found and this corresponds to the greatest displacement S M in feet. Then: "'Jf-V ; (24) If M is unknown and to be determined : M-^i ! . (25) SM The value of S M is determined from electrical conditions. There are 360 electrical degrees between two poles of the same sign. If the alter- nator has p poles, the number of electrical degrees in the entire circle is : 360-| = 180p. If the allowable displacement either side of mean is a electrical degrees (degrees of phase), the maximum allowable displacement of the crank pin in feet is: ' There seems to be a scarcity of data regarding a in electrical engineer- ing hand books, but 2.5 is given in papers from the Trans. A.S.M.E. referred to at the end of this chapter. If the number of electrical cycles per second is denoted by c, the rela- tion between the cycle, number of poles, and speed is given by: 120c It is convenient to tabulate data as the work proceeds as shown in Table 61. The minus sign indicates that the larger part of the area up to that number is below the mean line; when it is above the mean line the sign is plus. Obviously the final value should be zero, bringing the curve back again to the plotting line to the point from which it started. In determining the areas for the table, the planimeter may be started at the beginning of the curve each time if the diagram is not too large; or, each division may be measured and added algebraically. From (6), the weight varies inversely as the square of the diameter 362 DESIGN AND CONSTRUCTION OF HEAT ENGINES O 00 T-H o ^ - ..1 Tfl IO b d ^ 1 1 co o CO CO 1 1 TH *8 1 1 CO 00 Jg 1 7 7 ec 83 d ^ 1 iO t> CO TH CO 1 1 00 / co 10 oo C^l iO 00 1 1 00 co^oS^ 1-1 rH O ^f 1 1 CO o o s TH' 1 | CO 00 O "* 1C 1 1 2 CO O O5 C^ d co t^ 1 1 ~ co J2 os oo " d k ** ^ | , TH CO ^1 (^ CO l> T^ iO CO o oo 1 1 TH TJH 1 1 03 fl o 03 : 03 FLY WHEELS 363 at which it is applied ; then if the weight ( = gM ) assumed concentrated at the radius R of the crank circle is referred to diameter D, we have : or: W=w(^y (28) The general notes in reference to weight of arms, etc. of the preceding paragraph apply to wheels designed by the displacement method also. It would be a farce, however, to take the trouble to calculate a wheel by this method for a simple engine, to use on a cross-compound engine. 118. Comparison of Methods. For a given engine with certain condi- tions of speed, steam pressure, valve setting, etc., Formula (6) may be written : WR= constant (2g) Also from (25), (26) and (28): W R = g~ (30) constant Equating (29) and (30) we may write: pd = constant or: 1 p X constant This is the form for the value of 5 for alternators in Table 60, and if the displacement in electrical degrees is to be the same for all alternators running at a given speed, 5 must vary inversely as the number of poles, or as the number of cycles per second. This would indicate that the arbi- trary use of the usual values of 6 for all sorts of electrical machinery is in error. If the constant in (31) is carefully chosen (and it depends upon the crank-effort diagram and is as reliable as e), the simple method of wheel weight determination in Par. 116 is reliable and saves much time; but in new work it is usually desirable to investigate by more elaborate means, and errors may sometimes be caught in this way; it is for this reason that the displacement method is given in this book. 119. Application to Practice. A wheel will be designed to regulate an alternating-current generator, neglecting any flywheel effect of the gen- erator. The engine is the 20 by 48 in. Corliss engine of Chap. XII. The turning-effort diagram calculated in Chap. XVI has been reproduced in 364 DESIGN AND CONSTRUCTION OF HEAT ENGINES Fig. 227 of this chapter and is the basis of calculation for the methods of both Pars. 116 and 117. For a rim speed of 1 mile per minute the wheel diameter could be 16.8 ft. Assume it to be 16 ft. in diameter. While the maximum power at which the alternator is expected to give the best service should probably be taken, the diagrams have all been worked out for the rated horse- power of 450, and this will serve as an application of the methods. The diagram factor was omitted in plotting the indicator diagrams, but this is probably on the safe side. Weights will first be determined from the method of Par. 116, using Formula (8) with the constant C; as the speed is less than 5280 ft. per min., K can not be used. The value C in line 8 of Table 60 has been used for electric generators for direct or alternating currents, whether belt-driven or direct connected. This gives: _ 11,620,000,000 X 450 16 2 X 100 3 2 ' 5C Line 9 for a 25-cycle (30-pole) generator gives: 30 X 387,500,000 X 450 16 2 X 100" 2 ' 5C For a 60-cycle (72-pole) generator, line 9 gives: 72X387,500,000X450 16 2 X 100 3 From this it is apparent that if line 8 had been used for a 60-cycle machine, there might have been trouble. The displacement method is given in Par. 1 17. The areas and values of Ms and Mv are given in Table 61. The length of the diagrams as origi- nally plotted was 12 in. and it was intended to have the height about 3 in. As the steam engine is a 2-cycle engine, the time of one cycle is, from (13): then: 0.6 n _ 12 = - 05 ' As already determined for plotting the turning-effort diagram : n = 10,000 Ib. From Table 61, the maximum plus value of Mv is 980, the maximum minus value 340, and their sum (not algebraic) 1320. For a height of 3 in. the scale would be: q = - - = 440; or, take it as 500. FLY WHEELS 365 In the same manner the scale of Ms is: k = 30. The maximum ordinate measured from the mean line of Fig. 231 is: y M = 1.72 in. From (27), for a 25-cycle alternator: Taking a = 2.5, and as R = 2, (26) gives: From (25), at the crank pin: 30 V 1 72 M = AMEQOA = 885 and W = 32 ' 16 X 885 = 285,000 Ib. (J.OOooZO From (28), the weight referred to the rim is: W R = 285,000 X ^- 2 = 17,800 Ib. ' In the same manner for a 60-cycle machine we have : 20 X 60 ~TOO~ Of) y 1 70 120 X 60 _ 2.5X2 P " ~TOO~ SM 28.65 X 72 = 21,250 and F = 32.16 X 21,250 = 683,000 Ib. W R = 683,000 X - := 42,750 Ib. Tabulating the results for comparison, taking W w = about 1.6 W B , gives Table 62. TABLE 62 Method WR Ww Formula (8), line 8 of Table 60 20 500 33 000 Line 9 of Table 60 30 poles 20 500 33 000 Line 9 of Table 60 72 poles Displacement method 30 poles 49,000 17,800 80,000 29 000 Displacement method 72 poles 42,750 70000 No allowance having been made in the displacement method for possible discrepancies, the values are some smaller than by the simpler method, which in the present problem may be said to give equally good results. Allowance is made for assuming the radius of gyration as the radius of the outside of the rim. 366 DESIGN AND CONSTRUCTION OF HEAT ENGINES In Fig. 230 the height of the diagram is 2.7 in. and the scale is q/M. Then the speed fluctuation of the crank pin in ft. per sec. is: For the 25-cycle machine, 2.7X500 ni _ v *- Vl * "8850- =0 - 153 ' For the 60-cycle machine, 2.7 X 500 2 " l -2172-50 The mean velocity of the crank pin is: _ 2irRN _ TT X 2 X 100 _ 01 "60" "30" Then the coefficient of speed fluctuation is : For the 25-cycle, 5 = 0.0637 1 For the 60-cycle, d = ^- Equating these with (31) gives: 4.58p It is very clear that 5 must depend upon the number of poles when the engine is to drive alternators in parallel. In Par. 106, Chap. XVI, attention was called to the inequality of crank effort for the two ends of the cylinder due to the reduction of effective piston area by the piston rod, although these diagrams were drawn for equal cut-off. In some small steam engines with simple gear, the head- end cut-off is longer than the crank-end, increasing the inequality. When no provision is made for equalizing cut-off, this should be taken into account in plotting diagrams. With gears which will permit it, mak- ing the crank-end cut-off longer will probably give a more uniform turning effort; but most erecting and operating engineers would probably try to obtain equal cut-offs, so it is safer to ignore such a possible improve- ment in drawing the diagrams. A flywheel for this problem will be designed in Chap. XXX. References Determination of flywheels to keep the angular variation of an engine within a fixed limit. Trans. A.S.M.E., vol. 22, p. 955. Flywheel capacity for engine-driven alternators. Trans. A.S.M.E., vol. 24, p. 98. CHAPTER XIX REGULATION DURING CHANGE OF LOAD. GOVERNORS 120. Introduction. The governor, or regulator, is a device for con- trolling the speed of heat engines and other motors. It is therefore closely allied with the valve-gear mechanism and it is sometimes a little difficult to locate the dividing line between the two. Commercially, some small throttling governors are all there is of valve gear as well as the valve, and needs only to be connected with the steam line and belted to the engine shaft. In general the parts directly associated with the forces which are in equilibrium at a certain speed (called the tachometer by Zeuner), and the stand or wheel to which they are attached, comprise the governor; the same governor may be applied to various governor problems and in different ways. There are two general methods of applying the governor to the control of speed; 1. The independent method, in which the only connection with the engine to be governed is that required to turn the governor shaft. This method may be direct, when there is a positive connection sometimes links and levers between the governor and the valve controlling the working fluid or fuel ; or indirect, when the governor directly controls the supply of fluid to a steam or hydraulic cylinder or the current to sole- noids, which in turn operate the main controlling valve or valves. 2. The coordinate method, in which the governor coordinates with valve gear operated by 'eccentrics or cams which receive their motion from the engine shaft and bear a positively timed relation to it. The governor and all parts connected with the independent method will be treated in this chapter; the valve gear and its relation to the governor will be considered in Chap. XX. Due to more or less contradiction of terms in governor nomenclature, and the fact that different names are given to the same thing by different builders and writers, the schedule of notation will take the form of defi- nition to some extent, making reference easy. Notation. C = centrifugal force in pounds of governor weights when governor is in perfect equilibrium at a certain speed, neglecting friction or any 367 368 DESIGN AND CONSTRUCTION OF HEAT ENGINES externally applied force. This IB balanced by the centripetal force caused by springs or dead weights belonging to the governor proper. F = external resistance in pounds due to valve-gear adjusting mech- anism, opposing change of position of governor parts in either direction, thereby increasing or decreasing the effective centri- petal force as the speed is increased or decreased respectively. F is sometimes called the speed regulating force. It is measured at the point of attachment to the valve-gear controlling mechan- ism, as the sliding sleeve of the fly-ball governor. F may also be assumed to include the friction of the governor bearings, ra = the total movement in inches of the part on which the force F acts. It is sometimes called the stroke or the sleeve lift. Fm the measure in inch-pounds of the ability of the governor to overcome external resistance, or to do work on the valve-gear adjusting mechanism with a given speed variation. This assumes the mean value of F and is called the work capacity with a given speed variation. It is also known as the power of the governor but this is a misnomer as power involves the time element. AC = change of centrifugal force from any value C, required to over- come resistance F and allow the governor to assume another posi- tion corresponding to a new load. In any governor with inertia effect this aids in overcoming F. e = the ratio of AC to C ( = AC/C) The ratio 'F/e is sometimes known as the energy of the governor (this is a misnomer as F/e is a force), and Fm/e as the total work capacity in inch-pounds. F/e is the effect on the point where F is measured, of the total centrifugal force of the revolving weights. N = r.p.m. of the governor weights in any position of equilibrium corresponding to C. A7V = change of N corresponding to AC. 8 = the ratio of AN toN (= AN/N). This is called the speed variation by some makers of governors. It is also sometimes known as the coefficient of sensitiveness. F and 8 are interdependent, but as F in a working governor is the actual resistance to be overcome, and d must depend upon it. As 8 measures either increase or decrease in speed, the total range of variation or fluctuation due to F is the sum of 8 in both directions, or practically 25, as demon- strated in Par. 130. Ni = minimum value of N, for maximum load on engine. A load caus- ing a lower speed is out of control of the governor. REGULATION DURING CHANGE OF LOAD 369 Nz = maximum value of N, for minimum load on engine. A limit is sometimes assumed at zero load. N M = mean of Ni and N z (= (N t + JVi)/2). v = coefficient of fluctuation of speed ( = (N 2 Ni)/N). This is sometimes called the coefficient of speed regulation, and by some writers is considered as the measure of sensitiveness. If v is too small, "it will react with the variation of turning effort within the cycle. This leads to a restless governor play known as 'hunt- ing." 1 This is also true of 6. The value of v depends upon governor proportions only. a = the coefficient of speed fluctuation for the engine cycle dependent upon flywheel regulation, for any given load. The value of this coefficient depends upon the variation of turning effort and the kinetic energy of the flywheel. W = weight of centrifugal weight in pounds. This may be a single weight or may be divided into two or more parts. r = the distance in inches from the axis of rotation to the center of gravity of the weight W for any position. This should include, referred to the center of W } all parts having a centrifugal action. If subscripts 1 and 2 are used, they correspond to the same numbers applied to N. M c = centrifugal moment without friction or external resistance. M T = centripetal moment without friction or external resistance. M P = moment of the force F. P = force on spring in pounds. For other notation see diagrams. 121. Governor Types. The names most commonly applied to gover- nors used on modern heat engines are: Centrifugal governors. Inertia, or centrifugal-inertia governors. Relay governors. Safety, or over-speed governors. These will be defined in a general way in Pars. 122 to 127. A mathe- matical treatment will be given centrifugal governors in Pars. 128 to 133, these being of the only type capable of satisfactory analysis. Governors of the different types selected from practice will be illus- trated in Pars. 135 to 139. 122. Centrifugal Governors. In these governors the centrifugal force of revolving weights is balanced at a certain speed by centripetal force due to dead weights or springs, or both. Two simple forms are shown in 24 370 DESIGN AND CONSTRUCTION OF HEAT ENGINES Fig. 232. In Fig. 232-A, if W is the weight and C the centrifugal force of one ball, the centrifugal moment about pivot x is Ch ; the centripetal moment about the same point is Wr. These moments must be equal to be in equilibrium, so for some definite speed. Likewise for Fig. 233-5. Ch = Wr. Cb = Pa where P is the spring load balanced by one weight. A more extended treatment will presently be given. Any change in the load on the engine tends to change the speed ; this changes the centrifugal force of the revolving weights, causing it to bal- ance the centripetal force in a new position, at a slightly different speed. This new position adjusts the valve gear (including the mechanism of the independent method) to the new load. FIG. 232. Stability. A governor is stable if a change of position necessitates a change of speed in order to maintain equilibrium, an increase of speed always being accompanied by an increase of centrifugal moment, which, in turn, involves an increase in the radius of rotation of the revolving weights. Isochronism. In an isochronous governor the centrifugal and centri- petal forces are in equilibrium for all positions of the governor weights at the same speed. If the load changes, the slightest hint at change of speed causes the revolving weights to fly to a new position (if the external resistance to be overcome is relatively small), adjusting the valve gear to suit the new load, at the original speed. On account of friction, strict isochronism is not possible, and a governor theoretically designed for such is usually unsatisfactory and unreliable. A certain amount of stability is desirable even for service requiring the closest regulation. REGULATION DURING CHANGE OF WAD 371 Isochronism is the limit of stability. If this limit is passed an engine would run faster with heavy than with light load. One engine builder states that their governor has regulated satisfactorily when so adjusted; but it is of the centrifugal-inertia type and supplied with a dashpot, without which such adjustment would be impossible. A conical pendulum governor is shown in diagram in Fig. 232-A. This is sometimes called the Watt governor. If a central weight is added as shown dotted, it is called a loaded governor, or Porter governor. The pendulum governor must always be a vertical-spindle governor. Some- times a spring is used instead of the central weight, but the gravity effect of the revolving weights is still an important factor, so a vertical spindle must be used. A spring governor is shown in diagram in Fig. 232-B. This particular form is sometimes known as the Hartnell, or Wilson Hartnell governor. The gravity effect of the weights is small compared with the spring load and this type may be placed in any position. In fact, when in a hori- zontal position there is absolutely no gravity effect, while for the vertical position there is a slight gravity moment except when the weights are exactly in line vertically with the pivots. There are different forms of spring governors, some of which will be shown in later paragraphs. Shaft, or flywheel governors are located on the engine shaft, and must always be spring governors. They may have either one or two weights. 123. Inertia, or Centrifugal-inertia Governors. About the earliest example of this type to be practically applied was the Rites governor, which has been adopted by various builders and modified in some cases to suit their requirements. They are usually, though not always, shaft governors. Revolving masses tend to maintain a uniform speed unless acted upon by some external force. If a weight is caused to revolve around a governor shaft, being connected with the shaft by a system of linkage in such a way that a change of angular relation between shaft and weight operates upon the valve gear, it is obvious that any change in speed of the shaft will, due to the inertia of the weight, effect changes of adjust- ment. As force is the product of mass and acceleration, the ability of the revolving weight to overcome the resistance due to the valve gear and change the position of the gear depends upon its weight and the time in which the change of angular relation between shaft and weight occurs. As the inertia weight fixes no speed limits, but acts only upon sudden change of speed, it is always used in conjunction with a centrifugal governor. 372 DESIGN AND CONSTRUCTION OF HEAT ENGINES The principle of the Rites inertia governor may be illustrated by Fig. 233. A bar containing a weight at each end is pivoted on the governor wheel at A. The eccentric rod is attached at B and swings through the arc of a circle as the governor makes adjustments. The right end of the weight is heavier than the other, being practically in gravity balance about the pivot A. The center of gravity is therefore at F As the wheel revolves, the centrifugal force C of the entire weight produces a centrifugal moment Cb about A. A spring connects the fixed point E on the wheel with point D on the weight, producing a centripetal moment Pa in an opposite direction (clockwise) to Cb, exactly as in the centrifugal governor described in Par. 122, and at a certain speed: Cb = Pa. FIG. 233. These forces are not great enough in some governors of this type to overcome the resistance of the valve gear without too great speed varia- tion, but the necessary force is furnished by inertia I of the weight. If the wheel is turning clockwise as shown by the arrow, and should suddenly increase in speed as load is thrown off the engine, the wheel would plunge ahead. The weight, due to its inertia, would tend to revolve uniformly, the inertia / acting counterclockwise as indicated in Fig. 233. The in- crease in centrifugal force due to increase in speed would tend to move the weight in the same way about pivot A. Should the wheel decrease in speed, both inertia and centrifugal force would act to move the weight in a clockwise direction relative to the wheel. There is some inertia effect at some positions in many centrifugal gov- REGULATION DURING CHANGE OF LOAD 373 ernors, especially of the shaft-governor type. When present the work capacity of the governor is determined by the sum of the centrifugal and inertia effects if they are in the same direction, and by their difference if opposed. This may be illustrated by Fig. 234. Assume a wheel to be rotating in the direction R (clockwise) shown by the full-line arrow. Let a weight be pivoted at x, and assume that the centrifugal force C of the weight is just balanced by the spring, holding the weight in the position shown, at a certain speed. Now assume the wheel to suddenly increase in speed. The centrifugal force of the weight will be increased; the weight will fly out until the tension of the spring balances it in a new posi- tion. During the sudden change of speed the weight "hung back" due to its inertia, exerting the force /. While the centrifugal force exerted a positive or clockwise moment about pivot x, the inertia exerted a FIG. 234. FIG, 235. negative or counterclockwise moment. The total moment is then: M = Cr c - Ir u If the centrifugal moment is greater the weight would go outward, making the proper adjustment of the valve gear; but if the inertia moment is greater the reverse would be true. Suppose the governor weight to be pivoted at y as shown dotted. This causes the moment of / to act in the same direction as that of (7; then: M = Cr c + Ir H If the wheel turns counterclockwise the moments are in the same direction for the full-line position of the weight, and opposed for the dotted position. It is clear that the effect of / will be zero when the lines connecting weight center with wheel center and with pivot form an angle of 90 degrees as shown in Fig. 235. It is known from geometry that in 374 DESIGN AND CONSTRUCTION OF HEAT ENGINES this position the weight center will fall on a circle whose diameter is a line joining the wheel center and pivot. Then from what has preceded, if the effect of inertia and centrifugal force are to work together in gear adjustments, the weight should always remain inside the circle for clock- wise motion of the wheel, and outside the circle for counterclockwise motion. This is shown in Fig. 236. In some governors the inertia effect is nearly negligible and such are known as centrifugal governors. If the inertia effect is great they are called inertia, of centrifugal-inertia governors. Inertia governors act quickly and have been found to give exception- ally good regulation under widely varying and suddenly changing loads, although there are some prominent engine builders who do not favor much inertia effect. It is obvious that when inertia is present in any governor, its moment must be of the same sign as the centrifugal moment to obtain the best results. However, all governors are not so built. FIG. 236. Due to the impossibility of predicting the relative angular accelera- tion of shaft and inertia weight during change of load, the design of in- ertia governors is dependent upon the results of practice, and no attempt will be made to treat it mathematically. 124. Relay Governors. When considerable force is required to make adjustments of the valve gear a hydraulic cylinder, usually operated under oil pressure, is employed. The controlling valve of the oil cylinder is a small balanced piston valve requiring a very small force to move it; this in turn is controlled by a governor of ordinary capacity. 125. Safety, or Over-speed Governors. These are auxiliary gover- nors used in addition to the regular governor, more usually on the steam turbine. They consist of weights and spring so proportioned that at a certain excess of speed the weight will overcome the spring tension and operate a trigger, release a heavy weight or strong spring which shuts the throttle valve and stops the turbine. In some cases the relay principle REGULATION DURING CHANGE OF LOAD 375 is used, the trip mechanism admitting or releasing steam from an auxiliary cylinder whose piston is connected with the valve. The safety governor is a simple centrifugal governor and is adjustable. 126. Dashpots. Under certain load conditions, especially with ex- tremely sensitive governors, there is a tendency toward a restless movement of the governor weights, sometimes periodic and sometimes accompanying load changes. This action is called hunting, and is sometimes prevented by the use of dashpots. A dashpot consists of a cylinder containing a fluid such as oil, and supplied with a piston which allows the fluid to pass between it and the cylinder from one side of the piston to the other as displacement occurs. The piston rod is attached to some moving part such as the governor-sleeve crosshead, and yields readily under gradual changes, but offers considerable resistance to a sudden change. Sometimes less clearance is allowed between the piston and cylinder walls, the connection from one side of the piston to the other being through a small pipe; a valve is placed in the pipe so that the resistance may be adjusted. Dashpots may be seen on some of the governors illustrated in later paragraphs. 127. Speed Adjustments. The speed of a governor, and conse- quently the speed of the engine, may be changed by changing the weight (except in the simple Watt governor) or spring. With some governors this may be done while the engine is running. Where several engines are direct-connected to alternators running in parallel, small motors operated from the switch-board sometimes automatically are in some cases attached to the governors in such a way that the tension of the spring may be varied and the alternators brought into synchronism. 128. Speed Fluctuation. By mean speed, the speed at rated load is usually meant, and this is usually assumed to be : - , .:! ^-*+^ ' (1) in which N 2 and N\ are maximum and minimum r.p.m. respectively, the engine is designed to run under the control of the governor possibly from no load to maximum load. The coefficient of the fluctuation of speed is: fa ' . | = ^^> = ? 2 cos 2 cos Substituting these values in (18) gives: 1 tan or simplifying: F 2q - =W tan* (19) from which W, F or e may be found if the others are known or assumed- In practice k may be taken from 10 to 15, and q ranges from 1 to 2, a common value being from 1.2 to 1.35. If = 6 for all positions the equation is much simplified, but advan- tage is taken of a difference in the values to reduce the speed fluctuation (see Par. 128) . The value of F depends upon the type of gear and size of engine. It may sometimes be determined on an engine already built by hooking on a spring balance and moving the parts. For a small or me- dium-sized Corliss engine it may be 3 or 4 Ib. As previously stated, e ( = 26) ranges from 0.02 to 0.05, depending upon the desired sensitive- ness; the smaller the value of e the more sensitive is the governor. From given values of F and e, it may be seen from (19) that W will have the greatest value when tan 0/tan 6 (or /Q is maximum. Determination of Speed. Solving for N in (5) gives: N =- 187.7^ (20) From (13), (16) and Fig. 238: Ch P = Wr P from which: Wr ' - (21) REGULATION DURING CHANGE OF LOAD 381 Observing that r = h tan 0, and substituting the values previously found in (21), and (21) in (20), gives: N = 187.7 (22) FIG. 239. It is obvious that N increases as h decreases; if tan a 25 > a (37) v ^ 26 Weight of Links. When heavy weights or strong springs are employed, the weight of arms and links may usually be neglected. It is more accurate to include them and this may be easily done. In getting their centripetal moment, their weight may be assumed concentrated at their center of gravity but for their centrifugal moment this would not be correct. For the upper link, dividing the link into a number of parts, the centrifugal moment is: M - s wrN * tttt Mc " S 35^30' a REGULATION DURING CHANGE OF LOAD 387 where w is the weight of one part (see Fig. 245). For the lower link, the effect must be transferred to the upper link at the juncture of the two; then: wrN 2 x 35,230 'T (39) Gravity Balance. As far as possible, all parts of a governor should be in gravity balance for all parts of the revolution. This applies especially to governors with horizontal shafts. Selection of Constants of Regulation. Goodman says: "If a governor be required to work over a very wide range of power, such as all the load suddenly thrown off, a sensitive, almost isochronous governor with dash-pots gives the best results; but if very fine governing be required over small vari- ations of load, a slightly less sensitive governor without dash-pots will be the best." He further says: " Nearly all governor failures are due to their lack of power." The word power as used here is a misnomer, meaning the product Fm t the work capacity with a given speed variation. It is therefore safe to assume F amply large; on the other hand, if the actual value falls too far short of the assumed value, d is reduced, and the rela- tions given by (37) may not obtain. Machine Design. Failure of a governor due to broken parts or loosened bolts is exceedingly serious, and great care should be exercised in design and construction. Friction of joints and bearings should be eliminated as much as possible and ample lubrication provided for. All governors should be thoroughly tested and adjusted before leaving the factory. ILLUSTRATIONS FROM PRACTICE 134. Weight-loaded Conical Governors. A good example of the loaded governor of the Porter type is given in Fig. 246, which is used on the Bass Corliss engine built by the Bass Foundry and Machine Co., Fort Wayne, Ind. In this governor, part of the center weight is hollow to receive shot for adjusting the weight. A sliding weight is also placed on the bell crank to allow for further adjustment. A dashpot is furnished to prevent wide fluctuation when the load is changed. A safety pawl is provided which prevents the governor falling to its lowest position when starting the engine. It is so constructed that when the engine gains sufficient speed to lift the sleeve, the pawl falls by grav- 388 DESIGN AND CONSTRUCTION OF HEAT ENGINES ity; then should the governor belt break or slip, the governor upon coming to rest may sink to its lowest position, in which it engages the safety cams and prevents the engine from taking steam. The so-called safety collar which must be turned to the safety position by hand after the engine has started, is more convenient when starting the engine, but is only a partial safety device; neglect to set the collar may result in disaster. FIG. 246. Bass-Corliss governor. The Bass engine is sometimes fitted with an idler pulley running on the governor belt. This connects with the pawl, and in case of belt accident, falls, pulling the pawl out of position. This has the advantage of easy starting and stopping. This device is shown on a governor about to be described. A governor of the Proll type, built by the Nordberg Manufacturing Co. for use on their Corliss engines is shown in section in Fig. 247- A, and REGULATION DURING CHANGE OF LOAD 389 a side elevation in Fig. 247-.B. Fig. 247-J5 is arranged for a simple engine of moderate size having the Nordberg long-range cut-off gear. It is provided with a dashpot and has an idler pulley which operates the safety device. In a portion of the connection between the governor crosshead and the double bell crank, a spring may be seen holding two parts to- gether. Between these two parts is a separating strut with a projecting lever containing a pin which slides in a slot of the rod which is fastened A B FIG. 247. Nordberg governor. to the idler arm. Should the belt fail or run off, the idler would drop; the end of the slot would strike the pin, remove the strut and allow the spring to raise the bell-crank lever, which for this type of gear prevents steam admission, and the engine would stop. A Nordberg governor of the Porter type is shown in Fig. 248. This is employed when the engine is used to drive an alternator in parallel. It is also designed for the long-range gear, but the details of design are different. A small motor, shown at the right, is provided; this is operated 390 DESIGN AND CONSTRUCTION OF HEAT ENGINES from the switch board, and by driving a lead screw, shifts a counter- weight, varying the speed and bringing the engine into synchronism. A spring is interposed between the gear-operating lever and the dashpot, and the safety latch releases a weight if the belt breaks, which forces the levers in the safety position. FIG. 248. Nordberg governor. 135. Spring-loaded Flyball Governors. Fig. 249 shows the simple spring governor used on the smaller sizes of ''Economy" turbines built by the Kerr Turbine Co., Wellsville, N. Y. It is located on the end of the turbine shaft, its connection with the governor throttle valve being shown in Fig. 250. The governor weights 408 are of semi-annular cross- REGULATION DURING CHANGE OF LOAD 391 ,,401 27 FIG. 249. Kerr turbine governor. FIG. 250. Kerr governor on turbine. 392 DESIGN AND CONSTRUCTION OF HEAT ENGINES section, fitted with hardened tool-steel knife edges 402 which are securely fitted into the governor cup 400. The governor weights have rolling contacts shown by the dotted lines, and act against the governor collar 404, transmitting their motion to it. The governor collar in turn trans- mits its motion to the governor spindle 411 through the spindle pin 406. The spindle has a sliding clearance in the reamed hole provided for it in the end of the turbine shaft. A governor spring 410 adjusted by nut 415 works against the governor weights through the spring sleeve 409 and the governor collar. A stop nut 412 screwed up to such a position that the governor spring sleeve comes out against it at its extreme outer travel, prevents the weights from throwing out of their seats. The guard 401 is fastened to the governor cup, serving to protect the weights and supplementing the stop nut in preventing over-travel of the governor weights. Any movement of the governor is transmitted to the governor valve through the pivot block 424 and the governor lever 425. The thrust comes on a race of ball bearings 428. Lock nut 427 holds the ball race in place. Lubrication is supplied by an oil cup 426. Small cord pack- ing keeps the oil from running out along the spindle. This packing is kept in place by gland 430 and nut 431. In some governors of the Jahns type, the movement of the sliding sleeve is effected by bell cranks en- gaging with the sleeves through roller- bearing connections, with races set at such angles that the movement of the weights due to centrifugal force is transmitted in such a manner that the force F exerted by the sleeve is practically constant for each position of its travel. This was mentioned in Par. 130, and may be illustrated by Fig. 251 which is somewhat exaggerated for the purpose. Using the same notation as in Par. 132, remembering that lever arm a is coincident with b, and taking d as the apparent arm for F, i't is clear fhat : <.. J FIG. 251. CL 2 But due to the slanting roller race, arm a is shorter for all positions REGULATION DURING CHANGE OF LOAD 393 by the constant amount c. It is then clear that the ratio of arm a to arm a is greater in position 2 than in position 1; or: FIG. 252. Kerr turbine governor. To have e and F constant, or c vary as F, the following relation should hold: a z (di - c) Pi 394 or:- DESIGN AND CONSTRUCTION OF HEAT ENGINES a 2 P^2 (40) which is the same as (30). This governor is used to quite an extent on steam turbines and inter- nal-combustion engines. Fig. 252 shows the governor used on the larger turbines built by the Kerr Turbine Co., and is used in connection with an oil relay system of governing. It is driven from the turbine shaft by spiral gears. Below the speed governor the safety governor A is located; this will be de- FIG. 253. Mclntosh & Seymour governor. scribed later. At the lower end of the governor is a gear pump which supplies the oil relay cylinder and the lubricating system. The supply line is connected to the pump discharge line ahead of a 30-lb. relief valve. The oil supplied to the bearings is taken from between the discharge from the 30-lb. relief valve and a 3-lb. relief valve. This provides 30 Ib. pressure for the relay cylinder and 3 Ib. for the lubricating system. Except in detail of design this governor is practically the same as the Kerr governor described at length earlier in this paragraph. The pivot block B which operates the governor lever is at the top of the spindle. The governor yoke C fitted with hardened tool-steel knife-edge blocks carries on it two governor weight-arms D. These compress the spring by means of two pivot struts. REGULATION DURING CHANGE OF LOAD 395 136. Centrifugal Shaft Governor. Fig. 253 shows the shaft governor built by the Mclntosh and Seymour Corporation, Auburn, N. Y. and used on their steam engines. It is of the centrifugal type. To quote from Bulletin No. 52 of their Type F engines: "The governor is so designed that the com- bined inertia effect of the different parts due to change of speed, while it acts in the proper direction to assist the governor, is very slight, as a considerable inertia effect has proved to result in instability and unsatisfactory governing." It will be noticed that this is contrary to the views of other builders who use inertia governors. The governing eccentric is placed upon a fixed eccentric in such a way that the radius of the eccentric path and the angular position of the eccentric relative to the crank are both changed as the governor weight changes position. A leaf spring is used to balance the centrifugal force of the weight at the desired speed. 137. Inertia, or Centrifugal-inertia Governors. Fig. 254 shows the centrifu- gal-inertia governor of the Lentz engine built by the Erie City Iron Works, Erie, Pa. The hub D is keyed to the lay shaft which is geared to the engine shaft. The centrifugal weights A are pivoted to this hub. They are also connected by links E to the inertia weight B. The spring C connects the inertia weight with an arm on hub D. The governor as shown in Fig. 254 turns counterclockwise. Should the load decrease and the engine speed up, the hub D tends to accelerate in a counterclockwise direction. Cen- trifugal force tends to throw the weights out; the inertia weight lags and has the same action on the weight levers. 396 DESIGN AND CONSTRUCTION OF HEAT ENGINES A hand wheel on the end of the shaft moves the rod G, on the end of which is a taper. This raises or lowers the rod F which presses upon the spring near its support and changes the tension. Thus the speed of the engine may be changed while it is running. This may be operated by a small motor from the switchboard when the engine is driving an alternator in parallel. FIG. 255. Kerr turbine relay governor. 138. Relay Governors. The Kerr Turbine Co. relay governor is shown in Fig. 255. The pilot valve is shown at 591. Oil under pressure enters at the center of the valve chamber and is admitted to one end or the other of oil cylinder 539. The exhaust from the oil cylinder passes out near the ends of the valve chamber into the space surrounding the driv- REGULATION DURING CHANGE OF LOAD . 397 ing gears. The governor is shown in its extreme highest position with the governor valve 556 closed. This valve is a double-seated balanced valve and opens downward. Upon starting it is in its extreme lowest position. The following description is taken from Bulletin 26 of the Kerr Turbine Co. : "As the turbine is started up the weights are held in position by the governor spring until centrifugal force begins to compress the spring. As the speed increases the weights swing outward in a larger and larger circle trans- mitting their motion to the governor lever and giving it an upward motion. This causes an upward motion of the relay pilot valve 591 and uncovers thp ports so as to release the pressure on the upper side of the oil relay piston 540 and admit pressure on the bottom side. This causes an upward travel of the piston and decreases the steam valve opening. At the same time the motion of the valve is communicated to the governor lever 560 through the starting lever 548, which is pivoted at 549 and the starting lever link rod 541. This downward travel of the lever at its outer end brings the oil relay pilot valve back toward its original position, gradually decreasing the flow of oil, and finally a position is reached where the steam valve opening admits the proper amount of steam for the load to be handled. The pilot valve will then be in a central position and the pressure on the top and bottom of the relay piston will be equal. When the speed decreases, the governor weights travel in toward the governor shaft, causing a downward motion of the governor lever and pilot valve which opens up the chamber below the piston to exhaust and the chamber above the piston to admission. This causing downward travel of the steam valve, admitting more steam, at the same time causing an upward movement, of the governor lever, which brings the pilot valve back toward its original position until the speed and load become settled again. The pilot valve will now be in a central position and the pressure on the 1 top and bottom of the relay piston will be equal. "To receive close regulation the starting lever link rod 541 is moved toward the fulcrum 549 of the starting lever; and for wide speed regulation and great stability it is pivoted toward the outer end of the starting lever. The relay pilot valve will always run with the ports all closed as soon as the load is settled ; in other words, the relay pilot valve is the fulcrum of the governor mechanism. A speed variation of from 2 to 5 per cent, can be secured while the turbine is running by shortening or lengthening the starting lever link rod 541, which has right- and left-hand threads on the ends. This speed variation can also be ob- tained on all turbines driving alternators by means of the synchronizer. This consists of a synchronizer spring case 1582 to which is attached the upper and lower covers. The upper one has a clearance hole through which the stem, which has a collar attached, can move. The upper end is connected to a continu- ation of the governor lever. By means of the adjusting wheel the spring may be 398 DESIGN AND CONSTRUCTION OF HEAT ENGINES *- To H.P.Lim FIG. 256. Kerr turbine mixed-pressure valve. REGULATION DURING CHANGE OF LOAD 399 given any degree of compression desired, and locked there by means of the locking wheel. The greater the compression of the spring, the greater the speed of the turbine. A speed of about 5 per cent, above that given by the regular governor can be obtained." The steam valve shown in Fig. 255 is the regular high-pressure valve. A mixed-pressure valve is shown in Fig. 256. The starting lever fulcrum is shown at H. The top valve is the low-pressure valve admitting steam to the low-pressure element on the turbine shaft. The bottom valve admits steam to the high-pressure element. When there is sufficient FIG. 257. Allis-Chalmers safety governor. low-pressure steam to carry the load, the operation is exactly as described for the regular high-pressure turbine; the valve stem then slides through the high-pressure valve without engaging with it. Should the low-pres- sure steam not be ample for the load, the slight decrease in speed causes the governor weights to take a new position toward the center, lowering the pilot valve and admitting pressure to the top side of the oil piston. The nut B then engages with the high-pressure valve, forcing it open against the spring, which forces the valve again to its seat when the stem ascends. The turbine now operates as a mixed-pressure turbine, or if there is no low-pressure steam, as a high-pressure turbine. A check valve in the low-pressure steam line prevents an outflow of steam from the turbine, should the pressure exceed that in the low-pressure main. A small piston on the lower end of the valve stem is connected with the high-pressure steam line for the purpose of balancing the valves, oil piston and stem. 400 DESIGN AND CONSTRUCTION OF HEAT ENGINES 139. Safety, or Over-speed Governors. Fig. 257 shows the safety governor of the Allis-Chalmers Co. It is located in the end of the tur- bine shaft and does not depend upon gears. It is a simple spring gover- nor, which, when some predetermined speed is reached, compresses the spring, forces out the spindle and releases the trigger A . This releases the rod B, to the end of which a heavy weight is hung. This weight in turn is attached to a lever which, when the weight drops, closes the main throttle valve and stops the turbine. FIG. 258. Bruce-Macbeth mixing valve. 140. Gas engine governors are usually of the spring-loaded type. In some cases they coordinate with the valve gear, and such arrangements will be treated in Chap. XX. More commonly they operate the throttle valve, which is usually a small butterfly valve in constant-volume oil engines. In gas engines the throttle valve is known as a governor valve or mixing valve. Such a valve, used on the Bruce-Macbeth engine for natural and illuminating gas is shown in Fig. 258. CHAPTER XX VALVES AND VALVE GEARS 141. Introduction. The design of valve gears is essentially a drawing board proposition. The division of mechanics most used is kinematics, It is true that gear parts must have sufficient strength and wearing surf ace, and the analysis to determine forces acting is not usually very difficult; but the force required to move the valve is uncertain in many cases and assumptions must be made which makes the method largely a rule of thumb. Corliss valves in large engines, though unbalanced, are easily moved by hand so long as they are kept in motion, showing that the film of steam between valve and seat partly balances them. By assuming poor lubrication, and taking the force required to move the valve after it has stood a while under pressure, forces acting on all parts of the gear are easily found. There is no force theoretically required to move balanced valves if friction is neglected; they may be computed the same as unbalanced valves, but as one of the advantages of balanced valves is the reduction in weight of the gear, a certain proportion such as one-half of the force may be used. The details illustrated later in this chapter will be given as a guide, but the computations will be omitted in most cases. The dimensions are usually the result of experience, and the stresses and bearing pressures may be found when the gear layout is complete. The problem of treating the subject of valve gears in a single chapter is difficult and may only be done with any degree of satisfaction by dis- cussing with reasonable thoroughness a few of the gears in common use which involve the principles common to other gears, illustrations of the latter being given with little or no discussion. A number of gears are in use today on engines built some years ago, but these gears are not manufactured today; such gears are then practic- ally obsolete from the standpoint of the designer, and little or no attention will be given to them. It may be thought by some that the Corliss gear is entering this class, but as this was practically the first to give a shortened valve travel with quick opening and is still used in both the 26 401 402 DESIGN AND CONSTRUCTION OF HEAT ENGINES older and modified forms, with and without the releasing mechanism, it is treated in a fairly thorough manner. No special valve diagram is championed; those selected will enable nearly eveiy gear problem to be handled. The Zeuner diagram, while elegant, may be used for no problem which may not be better solved by one of the other diagrams given; neither does it clarify the principle of valve action better; therefore it is omitted. For a larger collection of gear designs and a more thorough treatment of some phases of the subject the reader is referred to the excellent books mentioned at the end of this chapter. Notation. a = port area. S = piston speed. Also stress, and on valve diagrams, steam lap. A = piston area. V = steam or gas velocity. D = cylinder diameter. w = width of port. I = length of port = kD. Also lead on valve diagrams. d = poppet valve diameter. h = poppet valve lift. r = V C /V F . q = d/D. $, P, E, T, B, W and I are dimensions on diagrams. A, Q, D, L, C, R and M are angles on diagrams. 142. Port Area. About the first requirement in valve gear design is the port area. This is determined by assuming the port wide open throughout the stroke, and that the steam or gas is of constant specific volume, although this is really far from true, especially during the exhaust stroke. But with this assumption the areas of port and cylinder bore are inversely proportional to mean velocities of steam or gas and piston; or: aV = AS (1) in which a and A are areas in sq. in. of port and cylinder bore respectively, S is the mean piston speed in ft. per min. and V the nominal mean gas or steam velocity in ft. per min. Then : AS , . a = T The length of the port in steam engines (which will be taken as its greater dimension) varies from about 0.8 the cylinder diameter in small, VALVES AND VALVE GEARS 403 slow-speed steam engines to somewhat greater than the cylinder diameter for locomotives with flat valves. Corliss engines usually have ports equal in length to the cylinder diameter. For piston valves the length is much greater, while in poppet valves the area is equal to the product of lift and effective circumference. As a general guide, it is considered desirable to have the port opening at any part of the piston stroke proportional to the piston velocity at that point, the maximum port opening corresponding to the maximum piston speed.; it is obvious, however, that this could be possible only with engines in which the valve is open during the entire stroke unless opening and closing is instantaneous, but it gives a basis of comparison which will be used in the following paragraphs. Allowable Fluid Velocity V. For steam, the following formulas may be used as a guide. These are: Steam port, V = 7000 ^l~ (3) Exhaust port, V = 5000 ^l~ (4) By varying the numerical coefficient these formulas may be used for the ports of internal-combustion engines. Values for these engines will presently be given in connection with poppet valves. If double ports are used each opening may be taken one-half of the value found by (2). By the term port opening, the amount that the valve uncovers the port is meant. In single-valve engines where the same ports are used for inlet and exhaust, the port is made large enough to accommodate the exhaust steam; then the inlet edge of the valve does not usually uncover the entire port. Also, due to certain characteristics of the valve motion the exhaust edge of the valve usually has considerable over-travel. The terms steam and exhaust are used in common parlance for the incoming and exhaust steam respectively. Slide valves may be used to include all those having sliding surfaces. The relative proportions of such valves and their seats are determined by practice with the various kinds. As an example of design, double-ported Corliss valves will be taken. Fig. 259 shows sections of steam and exhaust valves at one end of the cylinder. By experience with the layout of Corliss valve diagrams the following proportions were found to give good results: w = i (5) & = o.5d (6) wi = 0.233d (7) bi = 0.6d (8) c = 0.27rf (9) 404 DESIGN AND CONSTRUCTION OF HEAT ENGINES Certain other dimensions are given which may be used in cylinder design. The values of b and 61 were made ample to provide sufficient seal when the valves are closed; this may often be made less if desired and may be determined or checked by means of the valve diagram given later. As separate valves are used, the ports may be entirely uncovered, and there is no necessity of excessive over-travel of the exhaust valve. Some 1 Jfeam Passage Exhaust Passage FIG. 259. Corliss double-ported valves. over- travel is advantageous as it gives a quicker opening and allows for adjustment. The valves in Fig. 259 are shown wide open (the steam valve moves in a counterclockwise direction in opening, and the exhaust valve clockwise), with some over-travel, also some clearance at the heel of the valve to insure against partly closing the second port in case of adjustment; this does not give an ideal steam passage, but is apt to give VALVES AND VALVE GEARS 405 better practical results than too little allowance. The operation of this valve will be better understood from a study of the gear, the layout of which is also determined experimentally and in connection with valve and port design; there is no distinctly logical order in proceeding. The length of the port is usually equal to the cylinder diameter in Corliss engines, but a fraction of this length is taken up by ribs which connect the bridge with the two parts of the cylinder for strength and stiffness and to insure against distortion due to heat. The length may then be taken: I = kD where k may be from 0.85 to 0.9; or it may be any desired fraction for special designs. Then the area of the steam port in Fig. 259 is: a = 2wl = 2kwD The piston area is: A " V Substituting these values in (2) gives : Then from (5) : If in (11), V is taken as the inlet velocity, the correct velocity will be obtained through the exhaust port, as w\ from (7) is proportioned so that: Vw -8kV Having determined d, other dimensions may be found by Formulas (6) to (9). As V is greater for larger cylinders, the valve diameters are relatively smaller. In some large low-pressure cylinders the diameter of the steam valves is less than of the exhaust valves, the relative proportions being the same. Other types of valves may be treated in the same general way. Single-ported poppet valves are used almost entirely for internal-com- bustion engines. Some have flat seats as in Fig. 260- A, but usually the seat is conical with an angle of 45 degrees as in Fig. 260-#. The area of opening of the flat-seated valve is: a F = irdh (12) 406 DESIGN AND CONSTRUCTION OF HEAT ENGINES The area through a conical seat is: i = ird 7= + 2V2 The subscripts F and C denote flat and conical seats respectively. Let: Vc a P 1.414 r = VF ac 1+0.5^ a (13) (14) A 6 FIG. 260. Single-ported poppet valves. In Table 63 are values of r for the complete range of h/d found in practice. TABLE 63 h/d ,. 0.040 1.386 0.060 1.373 0.080 1.360 0.100 1.345 0.120 1.333 0.140 1.321 0.160 1.310 0.180 1.297 0.200 1.285 0.220 1.273 0.240 1.262 0.250 1.257 When h/d = 0.12, near the center of the range, r = %. For this value (14) gives: dp 3 3 j, /ir\ dc == = 7 Q>F = 77Ttt/l (Id) r 4 4 It may be assumed that this gives the area of opening for all poppet valves. It is on the side of safety for all values of h/d greater than 0.12, and these values are most common in modern engines. The error is slight for smaller values. The theoretical opening is greater in flat- seated valves, but on account of the obstruction to flow offered by sharp corners, it is doubtful if the capacity is greater than for conical valves having the same lift. The conical seat is usually much narrower than shown in Fig. 260, the opening approaching that of the flat seat for high lifts. The lift of a flat-seated valve to give full opening of the port is J4 the diameter of the port, neglecting the stem. For a conical seat it is higher, but d/4 is the maximum value of h given in treatises on inter- nal-combustion engines, and this is seldom used. Dropping the subscript in (15) and substituting in (2) gives: 3 4 8. VALVES AND VALVE GEARS Dividing through by d* gives: r r h 407 Let h/d = m and d/D = q; then: V S 1 (16) 3mq 2 From (16) Table 64 has been calculated. This table will be found useful in preliminary calculations, and for any assumed or measured values of m and q the value of V/S is seen. This factor multiplied by S gives the nominal gas velocity. TABLE 64 Q. Values of V/S for m = 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.25 0.25 89.00 66.60 48.00 44.50 38.00 33.30 29.40 26.70 24.30 21.40 0.30 61.80 46.30 37.00 30.90 26.50 23.10 20.60 18.50 16.80 14.80 0.35 45.40 34.00 24.50 22.70 19.40 17.00 15.10 13.70 12.40 10.90 0.40 34.70 26.00 18.80 17.40 14.90 13.00 11.60 10.40 9.44 8.34 0.45 27.40 20.60 14.80 13.70 11.80 10.30 9.14 8.23 7.49 6.59 0.50 22.20 16.70 12.00 11.10 9.50 8.34 7.40 6.66 6.06 5.34 0.55 18.40 13.80 11.00 9.18 7.85 6.89 6.12 5.50 5.01 4.40 0.60 15.50 11.60 9.26 7.71 6.60 5.78 5.14 4.63 4.20 3.70 0.65 13.20 9.88 7.90 6.57 5.63 4.93 4.38 3.94 3.59 3.16 0.70 11.40 8.52 6.80 5.66 4.85 4.25 3.78 3.40 3.09 2.72 0.75 9.88 7.42 5.92 4.94 4.22 3.70 3.29 2.96 2.69 2.37 0.80 8.70 6.52 5.21 4.34 3.73 3.25 2.89 2.60 2.37 2.09 0.85 7.70 5.78 4.61 3.84 3.29 2.88 2.56 2.30 2.01 0.90 6.86 5.15 4.12 3.43 2.94 2.57 2.29 2.05 0.95 6.15 4.62 3.69 3.08 2.64 2.31 2.05 1.00 5.55 4.17 3.33 2.78 2.38 2.08 Various values of V are given by different authorities. Giildner recommends 4500 ft. per min., with 6000 as a maximum under favorable conditions; but these values do not seem feasible for modern high-speed engines. It is true that the lower the value of V/S can be kept the greater will be the capacity of the engine, therefore as large values of m and q should be selected as is practicable. A few data of representative engines are given in Table 65. The dimensions of the airplane engine were scaled from a drawing and may not be strictly correct. If the piston speed were 2000 ft. per min., V would be 12,900. The low value of V/S in airplane engines is no doubt an important factor in enabling them to obtain as high an m.e.p. as 130 Ib. 408 DESIGN AND CONSTRUCTION OF HEAT ENGINES TABLE 65 Engine Valves m q v/s a V 4-cylinder truck inlet 156 448 10 60 1 196 12 600 4-cy Under truck . . exh 171 448 9 70 1 196 11 530 4-cylinder auto inlet 170 440 10 10 1 660 16 800 4-cylinder auto exh. 208 440 8 25 1 660 13 700 6-cylinder auto both 200 433 8 95 1 565 14 000 6-cylinder auto inlet 182 440 9 42 1 750 16 500 6-cylinder auto exh. 222 440 7 70 1 750 13500 8-cylinder airplane both 227 477 6 45 ? ? Double-ported poppet valves are used for superheated-steam engines and uniflow engines. The port area may be taken twice that of the single valve, or: 3 " (17) a = Trdh F _L s = Also: Table 64 may be used by taking one-half the tabular value. 143. Classification. The term valve gears is practically always as- FIG. 261. sociated with reciprocating engines, as in these there is always a complete cycle of operations. This is not true of the steam turbine; the gear is not in time relation with the wheels, but is closely connected with the governor, so is treated in Chap. XIX. A classification of the gears of reciprocating engines is difficult and not very satisfactory, but lists will be given for steam and internal com- bustion engines; by dividing into a number of headings a general idea of methods, valves and gears may be had, and these terms will be used in paragraphs which follow. By the term eccentric, a sort of enlarged crank pin is usually meant, VALVES AND VALVE GEARS 409 Valve system Gear drive Valve control the diameter of which is such that the shaft diameter lies within it; this is so for the reason that the eccentric is a casting which surrounds the shaft and is fastened to it by key or set screw or both. This is shown in Fig. 261-.A. The center of the eccentric often lies within the shaft circle as shown. On certain locomotive valve gears and some shaft governors, the gear is not driven by such an eccentric, but by a small pin as shown in Fig. 261-5. This is also known as the eccentric and will be so included when the term is used in this book. (1) Steam engines. Single valve Double valves (expansion valve) Binary or separate valves Uniflow Eccentric Eccentric and wrist plate nonreversing Eccentric and cam Eccentric Eccentric and crosshead reversing Connecting rod Shifting eccentric Swinging eccentric Releasing gear Reversing link motion r TT , . , I Plain D-valve I Unbalanced <-,.. , Flat J { Gridiron valve [ Balanced. Single- to triple-ported Piston. Balanced, single- or double-ported Rocking. Unbalanced, single- to quadruple-ported Poppet. Balanced, double-ported Piston acting as valve (2) Internal-combustion engines. ( 4-cycle. Mechanically operated inlet and exhaust 4-cycle. Automatic inlet valve Valve system < . , | 4-cycle. Sleeve valve [ 2-cycle. 2-port and 3-port f Cam Gear drive.. . { Cam and eccentric [ Eccentric ^i -j.1. jv j.j.v / Inlet valve Constant gear motion with throttling governor < . . Releasing gear for inlet or fuel valve Hit-and-miss Fuel valve or pump control f Poppet Valves j Sleeve [ Piston acting as valve Valves. Valve control 410 DESIGN AND CONSTRUCTION OF HEAT ENGINES 144. Single -valve Gear. This is the simplest form of gear and it sometimes seems the most difficult to understand, as all events are accomplished for both ends of the cylinder by a single valve. The subject is usually approached by a consideration of the valve without lap or lead, and with no angular advance of the eccentric; these terms will be ex- plained presently. Angularity of connecting rod and eccentric rod will also be neglected, it being assumed that the Scotch yoke is used for each; then the displacement of the valve and piston is measured by the projec- tion upon the line of stroke of the eccentric center and crank pin center respectively. A sectional diagram is shown in Fig. 262 of such a gear. A general description of steam engine operation is given in Chap. Ill, in which the cycle is traced through for a 4-valve engine, so it will not be necessary to go #A I Kfc FIG. 262. into details here. In Fig. 262, a is the steam chest and b the exhaust passage leading from the cylinder. The crank and eccentric circles are shown separately to avoid confusion. The piston is at the head end of the stroke and the crank at the head- end dead center. The eccentric is at right angles with the crank, causing the valve to be in the center of its travel, just covering both ports. As the eccentric leads the crank, a movement in a clockwise direction will open the head-end port to the steam chest a, and the crank-end port to the exhaust passage b. If the movement is continued until the crank is at the crank-end dead center, the valve will again be at its central position with both ports closed. Then admission, cut-off, release and compression all occur at the end of the stroke and the indicator diagram is a rectangle. The arrangement just described in which the high-pressure steam is in a on the outside of the valve, gives what is known as outside admission. If steam entered at b (necessitating some device to keep the valve on the VALVES AND VALVE GEARS 411 seat) and left past the outer edges of the valve, entering a and thence to the atmosphere, the arrangement is known as inside admission; then to open the valve at the head end to high-pressure steam, the crank and eccentric must rotate in a counterclockwise direction, and this is the principle of operation of the steering-gear engine used on steam ships. Lap, Lead and Angular Advance. Still assuming the crank to be at the head-end dead center and that there is outside admission, let us arbitrarily add steam lap S and exhaust lap E to the valve as in Fig. 263-A, and study the effect. The port is now closed, and the valve must be moved to the right the distance >S (the steam lap) before it begins to open, and the FIG. 263. eccentric must move through the angle Q (the lap angle) to bring the valve to the opening position, or admission. If the eccentric and crank are still at right angles as in Fig. 262, the crank would travel the angle Q from dead center before admission would occur, which is not permissible. It then becomes necessary to advance the eccentric through the angle Q while the crank is still at the head-end dead center; this angle is now denoted by A, as it is the angular advance. In some books the angular advance is the angle between crank and eccentric measured from the crank in the direction of motion. This seems logical, but it is American practice to consider angular advance the angle between eccentric and crank minus 90 degrees, and it will be so considered here. If the eccentric radius were such as to just open the valve to steam 412 DESIGN AND CONSTRUCTION OF HEAT ENGINES before steam lap were added, the opening would now be P S, in which P is the width of the port. It is usual to determine this opening as in Par. 142; then denoting the steam port opening by P s , the eccentric radius must be made equal to : \ = S + P s (18) where T is the valve travel, which is twice the eccentric radius when the eccentric drives the valve directly as assumed in this discussion. For- mula (18) is correct in any event, as it contains only valve measurements. Admission position is shown in Fig. 263-5. As explained in Chap. Ill, it is customary to open the valve to steam a little before the end of the stroke is reached ; the amount the valve is open when the crank is on dead center is called lead, and is denoted by I. The angle the eccentric and crank must pass through to move the valve the distance I is called the lead angle. The angular advance is the algebraic sum of the lap angle and lead angle] or: A = Q + L (19) Lead is shown in Fig. 263-C, the crank (not shown) being at the head-end dead center. It now remains to follow the crank from admission through a complete revolution, tracing events and determining the indicator diagram pro- duced. This is done in Fig. 264 for the head-end port. Arrows show direction of rotation, direction of piston travel (on indicator diagrams) and direction of valve travel. Full valve sections are shown at the top with valve in central position (the position when lap is measured) and at the extreme position with head-end steam port and crank-end exhaust port open. In the crank-eccentric diagram, crank and eccentric are shown connected by a light line to avoid confusion. The center line of the valve is directly under the eccentric center in each position. The following description accompanies Fig. 264: Position I. Admission. Valve displacement = S = steam lap. Steam edge of port just opening. Position II. Dead Center. Valve displacement = S + I = steam lap + lead. Valve open amount of lead (Z). Position III. Maximum Steam Opening. Valve displacement = T/2 = half of valve travel. Valve open to steam T/2 S. Position IV. Cut-off. Valve displacement = S = steam lap. Steam edge of valve just closing. Position V. Valve Central. Valve displacement = zero. Steam lap S and ex- haust lap E are measured in this position. Position VI. Release. Valve displacement = E = exhaust lap. Exhaust edge of valve just opening. VALVES AND VALVE GEARS 413 Position VII. Maximum Exhaust Opening. Valve displacement = T/2 = half travel of valve. Valve open to exhaust T/2 E. Position VIIL Compression. Valve displacement = E = exhaust lap. Ex- haust edge of valve just closing. From the valve section at the top of Fig. 264 the following equations are derived: < Engine El. Max. Steam Opening IP. Cutoff yn.Max.Exhausf Opening Y///////X Y///////X YlH. Compression FIG. 264. (usuaUy) (20) (21) (22) Formula (22) is arbitrary but may be used as a guide. 414 DESIGN AND CONSTRUCTION OF HEAT ENGINES For minimum value of W (as PI > P) : or: ?~ E - B (23) Turning the indicator- and valve-section diagrams end for end, and the crank-eccentric diagrams through an angle of 180 degrees would show l*L. Engine the crank-end events, but this is unnecessary when the angularity of the connecting rod is neglected. When familiarity with Fig. 264 is attained the crank-eccentric dia- gram may be used alone to show any valve setting. In following through the events, valve dimensions were assumed; in practice the events of the stroke are assumed, or it may be said that the indicator diagram is designed. In Fig. 265 a crank-eccentric diagram is shown for the head end, for an indicator diagram drawn above it. Any crank circle may be taken, angles and ratios being independent of this. A valve travel may be assumed, then transferred as shown to any VALVES AND VALVE GEARS 415 other value by graphical proportion when one dimension (usually P s ) is known. It is necessary with this diagram to assume a lead as a certain fraction of the port opening, as the actual valve travel may be different from that assumed. The following procedure may be used : Crank positions are found by projection from the desired indicator diagram as shown. The angle between crank positions for admission and cut-off is obviously equal to the angle between eccentric positions for the same events. The projec- tion of these eccentric positions upon the horizontal center line of the valve travel circle (the eccentric circle) must be at the same point, as the valve must be in the same position for both events. Then connecting the points where the valve-travel circle cuts the crank at 1 and 2, and swinging this line around tangent to an arc of radius S into position 3-4 parallel to the vertical center line as shown, the eccentric positions for admission and cut-off, the steam lap-angle Q and the steam lap S are determined. The exhaust events may be first assumed, the line 5-6 being swung around to position 7-8 tangent to the arc of radius E. The angular advance may thus be determined by admission and cut-off (the steam events), or by release and compression (the exhaust events). If deter- mined from the steam events, the exhaust lap is found by adjusting a proper relation between release and compression for the angular relation between crank and eccentric so found. If determined from the exhaust events (which is unusual except for engines with a releasing gear), the steam lap would be found from the best relation of admission to cut-off possible with this angular relation. Sometimes, especially in locomotives, exhaust lap is zero or may even be negative, when the valve is open to exhaust at both ends when in the central position; it is then known as exhaust clearance. Effect of Rod Angularity. In Fig. 266- A for the Scotch yoke, the horizontal distance between crank pin 1 and crosshead pin 2 is the same for all positions of the crank pin. Let this distance L be the length of the connecting rod of a slider-crank mechanism; then for the same angular movement of the crank from the head-end dead center, the cross- head pin of Fig. 266-A moves the distance a, while that of Fig. 266-B moves the distance 6. This is due to the angularity of the rod and it is obvious that the crosshead and piston of Fig. 266-5 will always be nearer to the crank than that of Fig. 266-A except in the dead-center positions. In valve diagrams it is convenient to consider piston displacement in connection with the crank circle. Then in Fig. 266-5, with the center of the crosshead pin as a center, draw an arc of radius L cutting the center 416 DESIGN AND CONSTRUCTION OF HEAT ENGINES line of engine; this marks off the distance 6 which is the piston displace- ment from the head end of the stroke. Thus for any position of the crank pin the corresponding piston position may be shown on the crank pin circle diameter. The same is true for valve displacement and the eccentric circle, but the maximum angularity of the eccentric rod is usually so small that it is neglected. It is usual to neglect angularity in drawing valve diagrams, at least for preliminary work; the events used are then called nominal cut-off, compression, etc. To find the actual events, the method just given is used and this is shown in Fig. 267. It will be noticed that all events of FIG. 266. the stroke from head to crank end occur later in the stroke than the nominal events, while those from crank to head end occur earlier. Actual indicator diagrams are as shown in Fig. 267, if, with the piston at the head end of the stroke, the pencil of the indicator is to the right of the paper clips as at A, Fig. 268, the drum moving as indicated by the arrow as the cord is unwound; if for this piston position the arrangement is as at B, Fig. 268, the indicator diagrams will be changed end for end. The diagram of Fig. 265 has been given because it shows the actual relation of crank and eccentric, and the principles of valve motion may more clearly be seen than with any other diagram. Moreover, it is general in its application and may be used for any eccentric-driven gear in conjunction with other diagrams showing the motion of wrist plate, linkage or cams. For the single valve it has limitations except as the VALVES AND VALVE GEARS 417 trial-and-error method is used; this is also true of the Reuleaux and Zeuner diagrams. Therefore no further use will be made of this diagram for the single valve except to explain principles, but a diagram having k__ 1 Head End Diagrams Crank End Diagrams FIG. 267. all the advantages and none of the disadvantages of the other diagrams for the single valve will now be explained. The Bilgram Diagram. Starting with the crank at the head-end dead center and with angular advance A, Fig. 269, move crank and eccentric through angle 6. Distance 1-2, perpendicular to vertical center Line, represents valve displacement from central position by the method of Fig. 265. Produce the new crank position as shown, forming angle below the horizontal center line and to the right of the vertical center line. Now lay off angle A (the angular advance) above the horizontal center line as shown by the heavy line. The intersection of this line with the valve-travel circle is the eccentric center of the Bilgram diagram, being a fixed point for any given valve, setting. Drop the perpendicular 3-4 to the new position of the crank produced. Then triangles 1-2-0 and 3-4-0 are equal, as each contains equal sides 1-0 and 3-0 (being radii of the same circle), one right angle, and the angle A + 9. Then line 3-4 equals line 1-2, the dis- placement of the valve from the central position. This is true for all 27 FIG. 268. 418 DESIGN AND CONSTRUCTION OF HEAT ENGINES positions of the crank, the perpendicular 3-4 sometimes falling on the center line of the crank and sometimes on this line produced beyond the center of the circle. .C.L. Engine V I Crank Circle = Stroke of Piston - FIG. 269 FIG. 270. The Bilgram diagram. For convenience of application the lap circles are drawn about the eccentric center of the Bilgram diagram, the radius of the circle equalling the lap. This is shown in Fig. 270, drawn with the same data as Fig. VALVES AND VALVE GEARS 419 264, with which it may be compared. With the same numbers for crank positions the following description will make the diagram clear: I. Admission. Valve displacement (1-2) = S = steam lap. Steam edge of valve just about to open. II. Dead Center. Valve displacement (1-3) = S + 1 = steam lap + lead. Valve open amount of lead I. III. Maximum Steam Opening. Valve displacement (1-0) = T/2 half of valve travel. Valve open to steam (0-4) = P s = T/2 - S. IV. Cut-off. Valve displacement (1-5) = S steam lap. Steam edge of valve just closing. V. Valve Central. Valve displacement = zero. VI. Release. Valve displacement (1-6) = E = exhaust lap. Exhaust edge of valve just opening. VII. Maximum Exhaust Opening. Valve displacement (10) = T/2 = half travel of valve. Valve open to exhaust (0-7) = PE = T/2 E. VIII. Compression. Valve displacement (1-8) = E = exhaust lap. Exhaust edge of valve just closing. Fig. 270 is the head-end diagram; for the crank-end diagram the location of the lap circles and the crank positions for the various events are diametrically opposite. In applying the diagram the cut-off, maximum steam port opening P s and lead I are usually assumed. The cut-off position of the crank, lead line and arc of radius P s are drawn as shown; then the steam lap circle is drawn so as to come tangent to these three lines, fixing the center of eccentric, angular advance and valve travel. The compression is the exhaust event usually assumed; producing this line as shown dotted, the exhaust lap circle is drawn tangent to it from the eccentric center, then the release position is tangent to the other side. Should release and compression exchange positions, release would occur before com- pression (in fraction of stroke); exhaust lap would then be negative (the valve would be open to exhaust at both ends when valve is at center of travel, giving exhaust clearance) and the circle would be a dotted line. The effect of angularity of connecting rod is found as in Fig. 267. Double-ported Valves. There are various designs of multi-ported valves, but the principle of all is to increase the valve opening with a given travel. The principle of operation of a double-ported, balanced slide valve is shown in Fig. 271. An adjustable balance plate (or pressure plate) has depressions to match the ports, the pressure being kept from all portions of the valve not opposite the ports by the valve seat and balance plate. At A the valve is displaced to the right by the amount of the steam lap and is just about to open the head-end port. At B it has moved from the edge of the port a distance equal to the width of the 420 DESIGN AND CONSTRUCTION OF HEAT ENGINES auxiliary port through the valve; up to this time the opening has been double that indicated by the movement, as steam enters at two edges, but any further movement up to the position shown at C will not increase the opening, as the port through the valve is closed as fast as the cylinder port is opened. Sometimes the port opening equals the width of the auxiliary port before the valve reaches the position B (where the auxiliary port begins to close) ; then the opening is as for a single port till position B is reached and constant port opening begins. The valve is at the FIG. 271. Double-ported slide valve. extreme left of its travel at D, uncovering the exhaust port; a double- ported effect at the exhaust edge is not necessary with a single valve as the exhaust lap is small and there is usually a wide opening. Some double-ported valves give a double opening throughout their entire travel, as the Corliss valve of Fig. 259. Shifting and Swinging Eccentric. In Chap. XIX, illustrations are given of governors in which the angular position and radius of the eccentric are changed when the load changes. If the eccentric moves in a straight line it is called a shifting eccentric, and if it swings from a pivot in the arc of a circle, a swinging eccentric, VALVES AND VALVE GEARS 421 The diagram of Fig. 265 will first be used to show the changes, which are given for different methods in Fig. 272. At A is shown an older style in which the eccentric is rotated about the shaft. The angular advance is increased with an increasing lead, but the valve travel is not changed. Full lines show the setting for maximum cut-off and dotted lines for a shorter cut-off. A shifting eccentric is shown at B in which both valve travel and angu- lar advance are changed, as is also the case with the swinging eccentric shown at C. The pivot for Fig. 272-C was located at the crank side -of the vertical center line, but it is sometimes at the right. The pivot was located to give zero lead when the angular advance is 90 degrees. The lead is constant for B. These diagrams are not convenient for the actual design of the changing eccentrics, so the same three cases are shown for the Bilgram diagram in Fig. 273. Crank positions for all events of the stroke (for the head end) are shown by the full lines, and for the shortened cut-off by the dotted lines; the decreased valve- travel circle for the short cut-off is omitted as it serves no useful purpose. The path of the eccen- 422 DESIGN AND CONSTRUCTION OF HEAT ENGINES trie change is drawn as though the crank pin were at the bottom of the circle and rotates counterclockwise. It will be noticed that the maximum steam port opening P s (see Fig. 265) has decreased considerably for the shorter cut-off of B and C; it may easily be shown that for one-quarter cut-off the opening would be extremely small unless the valve travel and steam lap were very large. It is for this reason that double- and triple-ported valves are used, giving double or treble the opening indicated by the Bilgram diagram, which gives the nominal port opening. Rectangular valve diagrams are diagrams in which ordinates represent valve displacement, and abscissas either piston or crank displacement; if the latter, the diagram is plotted on a rectified crank circle. As dis- placement is the sum of lap and port opening, such diagrams may be used to show port opening during the cycle, and may show the point on piston path or crank circle at which the valve event occurs. To illustrate a rectangular diagram on the piston path, together with a double-ported valve, shifting eccentric and some of the principles of Par. 142, the design of a high-speed engine will be assumed with data as follows: Cylinder diameter 12 in., stroke 18 in. and r.p.m. 200; initial gage steam pressure 125 lb., back pressure 15 Ib. absolute and clearance 6 per cent. The nominal steam velocities from (3) and (4) are: V s = 7250 and VE = 5180. The ports in the cylinder must be determined from V E ', then (2) gives: 113 X 600 1Q t aE= -5180- =13.1sq.m. Assuming the length of the port equal to the cylinder diameter, the width is: .10, = 1.09, say The maximum steam port opening will be determined from V s , which from (2) gives: 113 X 600 as= -7250- =9 ' 38 and Ps= ^= .78. ' -, - On account of the double-ported valve and the desirability of having sufficient port opening at short cut-offs, assume the nominal port opening which will be used for the Bilgram diagram at maximum cut-off to be VALVES AND VALVE GEARS 423 equal to W E or 1% in. Also assume the maximum cut-off to be % stroke and the nominal lead (that used on the Bilgram diagram) to be constant and equal to H 6 in. We are now ready to start the Bilgram diagram and this is shown in Fig. 274 at the top. 424 DESIGN AND CONSTRUCTION OF HEAT ENGINES We must assume compression or release to find the exhaust lap. This is rather arbitrary and is done in various ways. It is sometimes determined by assuming that nominal compression raises the compression pressure to initial pressure when the cut-off is very short. With the clearance and compression assumed, initial pressure is reached with a nominal compression of 0.56 stroke (see Chap. XII). Let us assume that with 0.1 cut-off, nominal compression is 0.5 stroke. Laying this off on the Bilgram diagram the exhaust lap is found as shown. The following data are obtained from the diagram: Valve travel, 4% in.; steam lap, iKe in.; exhaust lap, % 2 m - The displacement curve on the piston path is called the valve ellipse; this is sometimes drawn by neglecting the angularity of the connecting rod, but this is taken into account in Fig. 274, the rod being 5 cranks long. The crank circle is usually divided into a number of equal parts (they need not be equal but it is usually more convenient) from which the crank positions are drawn; the correct piston position is found by drawing the arc to the center line as shown for the crank position marked Ko cut-off, and this is produced to the center of the valve ellipse (not a true ellipse) from which valve displacements are measured. The lap lines are then drawn in, the subscripts H and C denoting head and crank end; should exhaust lap be negative it is drawn on the same side of the center as the steam lap, as it is on the same side of its respective port edge. Ordinates measured from these lap lines to the ellipse give valve openings for a single-ported valve at any point; it is then easy to find the part of the stroke at which the valve is opened and closed. The ellipse is tangent at the ends at the ordinate S H -f I- The upper valve ellipse is for % cut-off (the maximum) and the lower for y cut-off. In both ellipses head-end cut-off occurs at 6, head-end release at d and head-end compression at e] the subscript 1 denotes the same for the crank end. Admission is so near the ends of the stroke that it may not be well shown in this diagram unless drawn to a large scale. The diagram may be easily modified for the double-ported valve without changing in any way the timing of the events; the only difference is the increased steam valve opening. The valve is shown to scale in Fig. 271. There are different methods of proportioning the auxiliary port P A through the valve; in this case it has been taken as one-half of P s . The port in the cylinder at the valve face must be equal to B v + 2P A (= B v + PS) as shown in Fig. 271. To show the effect of the double port draw lines on the ellipses a dis- tance P A from the steam lap lines. During this displacement the opening VALVES AND VALVE GEARS 425 is twice that due to the displacement, the piston travelling up to the point where the line crosses the ellipse; after this the opening is constant as explained in connection with Fig. 271. When the ellipse again crosses this line the valve begins to close again, closing at b as before the double port was added. The opening diagram now has the form afgb. For the one-quarter cut-off P s is less than P A , so that the double opening obtains all the time the valve is open as shown at h. Piston-velocity curves drawn with a crank-circle radius equal to P s for the % cut-off are shown plotted on the steam and exhaust lap lines for the long cut-off in dotted lines, and for the short cut-off on the steam-lap line as far as the valve is open. It will be seen that in all cases the valve-opening curve rises quicker than the velocity curve at I 3 4- 5 6 7 8 9 10 // 12 first, and for the long cut-off lies above till maximum steam port opening has been reached. The exhaust opening curve is above it for three- quarters of the stroke. For the short cut-off, the velocity line is crossed early by the valve-opening line. The effect upon the steam pressure depends upon the value of nominal steam velocity V assumed in deter- mining the ports, but it is obvious that for any type of gear in which the valve closes gradually, the ratio of port opening to piston velocity de- creases very rapidly beyond a certain point, and the relative steam velo- city increases; if this velocity is very high the pressure head increases and wire-drawing occurs a very common experience. Gradual closing of the exhaust port gives the effect of an early compression and is not objectionable, as allowance may be made for it. The method of plotting a piston displacement curve on a rectified crank circle, accounting for angularity of the connecting rod, is shown in Fig. 92, Chap. XIII This forms part of a rectangular valve diagram which is shown in Fig. 275. The valve displacement curve may be plotted in the same way, although the angularity of the eccentric rod is usually 426 DESIGN AND CONSTRUCTION OF HEAT ENGINES neglected. By placing this curve in correct phase relation with the piston displacement curve the crank position for any given event may be deter- mined, from whence the piston position may be found. This is traced through for head-end release by dotted lines, the letters denoting the same as in Fig. 274. Fig. 275 is for the single-ported valve with % cut- off in Fig. 274. This diagram is very valuable for complicated gears difficult to handle in any other way. One or more valve curves may be used, drawn on separate pieces of tracing cloth so that their relative positions may be changed. It is sometimes convenient to draw the lap lines parallel to the curves, especially when an auxiliary valve is used. This diagram will be referred to in connection with sleeve valves for internal-combustion engines. Valve Stems. Let d be the diameter of the stem, a the unbalanced area of the valve, fj, the coefficient of friction, p the unbalanced steam presures in Ib. per sq. in. and S the stress in the stem. Then: -j S = fj,pa or: d = J^ (24) \ 7TO For practical results, a high coefficient of friction and low stress must be used. Assume M = 0.25, S = 5000 and p = 125. 145. Corliss Releasing Gear. It is probable that, strictly speaking, the Corliss gear is a releasing gear with rocking valves and a wrist plate operated by a single eccentric; however, the name has been applied to engines with two eccentrics and a double wrist plate, and even when a steam wrist plate is omitted. It is also applied to nonreleasing gears with shaft governor and swinging eccentric, although this usage is opposed by some engineers. In Fig. 276-A, a valve is shown at the extreme left of its travel, fully opening the port at the right; as it uncovers the port at the left it travels the distance T. At B, the central arm of a double bell crank travels the same horizontal distance. It is so connected to the two valves that it begins to open the port at the same point in its horizontal projec- tion that A does, and opens the valve the same width at the end of its travel. The distance traveled by the valve is T\, much less than T. The valve starts to open slowly due to the dead-center effect of the bell crank arm, but opens quickly when the port edge is reached. This is the effect of the wrist plate. If another bell crank is used as at C, and VALVES AND VALVE GEARS 427 the movement is made from the full to dotted position, this effect is still more marked due to the more rapid increase of angle a as the valve moves to the right. This effect is also obtained in angle 5 of the main bell crank. The relative positions of the bell cranks and the change of angu- larity of the rod connecting them has influence, and advantage is taken of these principles in different ways in Coiliss and other gears. In designing the Corliss gear two diagrams are employed, one for the wrist plate and valve levers and another for the crank and eccentric. The latter is sometimes omitted, the laps being taken from a table, the values of which are usually for the old, slow-speed type with very late compression and are not generally applicable. Some idea of these diagrams is given in Chap. III. 7M W7//b7777777? i-^-H FIG. 276. The crank-eccentric diagram is that of Fig. 265. The cut-off is effected by the governor and trip mechanism, so the cut-off may be neg- lected in the diagram; it may be found as in Fig. 265, neglecting lead, and gives the maximum cut-off should the load be too great for the engine, and the governor fail to trip the gear. The distances S and E will be used for convenience but are not the laps; starting with the wrist plate at the center of its travel, they give the positions of the eccentrics which will move the valves by the amount of their laps (lap plus lead in the case of the steam valves). In most diagrams given in text-books, very late compression is as- sumed, giving a small angular advance; the resulting maximum cut-off under governor control is therefore about 0.375 to 0.4 stroke and presents no difficulties for reasonable overloads. In laying out the diagram 428 DESIGN AND CONSTRUCTION OF HEAT ENGINES in this paragraph, the indicator diagram of the Corliss example (20 X 48 100) of Chap. XII is used. The problem calls for a long-range cut- off, but the single-eccentric layout will be first taken up. This shows the limit of capacity when a rather early compression (0.8 stroke) is used, and shows why so late a compression and release as are practicable are generally used; also, in conjunction with the diagrams of Chap. XIII, why double eccentrics with separate steam and exhaust wrist plates were FIG. 277. Used on the low-pressure cylinders of compound Corliss engines even though the high-pressure cut-off gave ample overload capacity with a single wrist plate. The single-wrist-plate diagram is shown in Fig. 277 and the crank-eccentric diagram in Fig. 278. These are for the head end, and as the gear is so adjustable and cut-off depends upon governor con- nections, angularity of the connecting rod is neglected in this paragraph. Procedure. The diagram for wrist plate and eccentric may be first drawn. Until considerable experience is had, this may require a number of trials, but for a giyen compression and release, formulas may be made VALVES AND VALVE GEARS 429 which greatly reduce labor. The method by which Figs. 277 and 278 were drawn has been used by the author for several years with satisfac- tion and dispatch. The dimensions given are those practically required FIG. 278. with the exception of the angles, which are omitted. For standard gears already designed, and usually intended for several cylinder dia- meters, r s and r E are fixed. For new work their value is more or less 430 DESIGN AND CONSTRUCTION OF HEAT ENGINES arbitrary, the minimum depending upon the detail design of the mechan- ism. In Fig. 277 they are taken equal to d, the valve diameter, which in this case was computed by the formulas of Par. 142. R s was taken asr,s + 1.5 in. and R E = l.IRs- The radius R T of the main wrist- plate pin depends upon the design of the reach-rod releasing mech- anism, but its horizontal travel is T. Draw arcs of radii r s and r E from valve centers; from center of wrist plate draw arcs of radii R s , R E and R T . Draw lines from center of wrist plate tangent to arcs of r s and r E . For new work draw valve levers normal to these lines. For old designs the levers must be so located that the travel of the dashpot pin at radius r A will be equally divided by the horizontal center line, but the normal line may be used tentatively. Now locate a point on arc R s below the tangent line, and on arc R E above the tangent line. This is arbitrary, but is made about 11.5 degrees in Fig. 277. Old practice made this point on the line and for very late compression this is permissible; but the method shown reduces the travel when the valve is closed. It necessi- tates a greater angle of wrist-plate motion 0, which is 69 degrees in this design. Now connect this point with the valve lever position already found; locate other extreme wrist-plate positions by angle 0, which is arbitrary and found by trial; and bisect, giving the position when the wrist plate is at the center of its travel. With the length of connecting link found for the first extreme position laid off with a beam compass, find the valve lever positions to correspond with the other wrist-plate positions. For single-ported valves the dimensions would be different, but these are not much used. Draw in one of the ports in any position, the location of the second port being found later. Show the steam valve edge dotted in the lead position which may be taken from KG to J^ of the port width; the latter was assumed. Now draw the crank-eccentric diagram for the compression and release desired; the latter is taken arbitrarily and not far above the center line as shown; in Fig. 278 it is ]/ in. on a 5 in. circle. The angular advance may be found as described for Fig. 265. Now draw the crank on the head-end dead center. The eccentric circle may be made equal to T if this is greater than T E \ its value is unimportant. For the steam valve, cut the angular advance position with arc of radius T s /2; this gives the virtual eccentric radius to give the steam wrist-plate arm the travel T s . Lay off S on Fig. 277, giving the position of the wrist-plate pin when the crank is on the dead center. Find the corresponding valve lever position; this is shown dotted. This position corresponds with VALVES AND VALVE GEARS 431 the lead position of the valve; taking this distance along the valve circle on a divider, the three positions (two extreme and central) of the valve may be found. The position corresponding to the central position of the wrist plate gives the steam lap. It will be noticed that when the valve is wide open there is some over- travel; this gives a little quicker valve opening. At B draw the valve edge in its extreme position; this is some greater than that indicated at A due to the wrist-plate arm crossing its dead center, and besides this some clearance must be allowed the latch hook; but this is usually small and may be overlooked if ample allowances are made. When in position B, the back edge of the valve must have suffi- cient seal to prevent leakage; this was made % in. in Fig. 277, but may be greater for large engines and less for smaller engines. This determines one valve face. Now move this face to its extreme open position as at C. It is well to allow a small amount for adjustment, and discrepancies in the cylinder casting as shown. This determines the bridge width. Draw a center line through the bridge as shown and locate on the vertical center line of the valve chamber as at D. The valve design may be now completed. The exhaust valve may be treated in the same way, the dotted posi- tion corresponding to compression, when the valve is " line-and-line " (just closing). Dimensions may be determined by BI, C\ and DI as for the steam valve. The exhaust valve is sometimes made symmetrical as with the steam valve. The dotted lines across the valves in D and DI show the slot for the tee-head of the valve stem; it is located so that it is vertical when the valve edge is halfway between line-and-line and full closed positions, thus allowing for wear. Valve-opening diagrams for steam and exhaust valves are shown above and below the crank-eccentric diagram respectively in Fig. 278. The steam valves open quickly, the more so because of the over-travel. A piston velocity curve is shown dotted, but the opening curve is above it until past the point of maximum cut-off. The theoretical point of maximum cut-off is found by placing the eccentric to the extreme right of its travel; this places the wrist plate in its extreme position, and if cut-off does not occur before this time the valve is not unhooked and the opening curve is that shown by the full line. But past the point when the eccentric and wrist plate are at the extreme right the governor does not control, so that this gives the limit of valve release. However, it takes some appreciable time for the dashpot to close the valve, so that the real cut-off is later. 432 DESIGN AND CONSTRUCTION OF HEAT ENGINES For the early compression assumed, the angular advance is so great that the maximum cut-off under governor control is but Y stroke; this is at b, the valve being released at this point. Furman says that a good dashpot will close the valve in Jfe sec -> which for 100 r.p.m. would be about 0.1 of a revolution. Assuming the valve to be entirely closed in this time, the piston would travel the horizontal distance between b and c; further assuming the closing curve between these two points to be a straight line, the actual displacement curve would be abcic and the opening of the port would follow the line afghjh. It will be noticed that the port is wide open at 0. 1 stroke. The actual closing for maximum cut- off begins at about 0.29 and ends at 0.43 stroke; a relatively slower dash- pot would give a later cut-off, and if the speed were high, would show signs of wire-drawing. In Fig. 277, the exhaust valve shows no over-travel but the opening is quick; it is not likely that the opening would be less than shown as the compression is not apt to be earlier than assumed, but if some over-travel is desired, R E may be increased, thus increasing the angle of valve travel. The displacement curve needs no modification as there is no releasing mechanism for the exhaust valve. Lang-range Cut-off. These gears were formerly furnished with two eccentrics and a double wrist plate, the steam and exhaust portions working independently. For the design considered the exhaust wrist plate may be exactly as determined, so will not be redrawn. A wrist plate is sometimes used for the steam valves, but the travel of the valve from the central wrist-plate position to full closed position is nearly the same as to the full open position, therefore, the wrist-plate pin must be located nearer the top and the angular motion much reduced. This led to abandoning the wrist plate for the steam valve and this will be as- sumed; then in Fig. 279 the wrist plate is not shown. Fig. 280 is the crank-eccentric diagram. As in this problem a % cut-off under governor control is desired, the crank must be in this position as the wrist plate and eccentric are at the extreme right; moving the crank back to head-end dead center brings the eccentric back of the vertical center, giving negative angular advance. As the valve is open the amount of the lead in this position, it is obvious that when the wrist plate is at the center of its travel the valve is open, giving negative steam lap, or steam clearance. The design is completed at A, B, C and D as before, taking S on the valve lever, which travels equally either side of the vertical center; a rod connects this with the crank-end lever which is longer, and receives the reach rod. The bridge is made the same as for the short-cut-off gear but the valve faces are less. VALVES AND VALVE GEARS 433 The displacement diagram is practically the same as for a single- valve engine, but due to a negative angular advance, is tilted in the opposite direction; if the gear fails to trip, cut-off will not occur until well on the return stroke. f // Steam Clearance 5 * FIG. 279. Assuming the valve to close in 0.1 of a revolution as before, release of the gear would occur at 6 to give a % cut-off; the actual maximum cut- off would therefore be much greater, nearly full stroke. The. actual opening curve would be afghjh. The opening is not quite as quick as : O.I7S5troke ~ FIG. 280. before, but while there is a large over-travel, the maximum valve travel is not as great as with the short-cut-off gear, the travel when the valve is closed and under pressure being much less. The large over-travel is necessary for a quick port opening and may even be increased if desired. 2S 434 DESIGN AND CONSTRUCTION OF HEAT ENGINES Releasing Gear Operation. There are various designs of releasing gears used by different builders, but the principle is practically the same in all, with one exception (see Nordberg Bulletin). A simple sketch (not an actual design) will be used to illustrate the principle and this is given in Fig. 281, which is similar to the gear used on the Fishkill Corliss engine. This type is used as the operation is clearly seen. The valve lever or steam crank a is keyed to the valve stem. To it is attached the dashpot rod 6 which pulls the lever down and closes the valve when it is released. The bell crank (or steam rocker) c receives a constant motion from rod d which connects with the wrist plate. At the upper end of the bell crank is hung the latch e, hav- ing a hardened steel block at its lower end, which engages with a like block on the end of the valve lever. The latch is secured to a pin which passes through a hub at the end of the bell crank, and at the other end is attached a lever, sometimes called the knock-off lever, which carries the roller /; if this roller is forced away from the valve stem the latch is unhooked. The cam collar g, sometimes called the governor toe collar, moves independently of the other mechanism and is attached to the governor bell crank i>y the rod h. The cam collar carries two rollers, i being the knock-off cam and j the safety cam; as bell crank c oscillates, the latch roller / may come in contact with one of the cam rollers, depending on the position of the governor when the latch is unhooked. For simplicity it will be assumed that the latch is unhooked when the centers of the rollers are on the same radial line from the center of the valve stem. Fig. 282 is a diagram showing governor and cam roller in the three important positions. A is the safety position, B the starting position and C the zero cut-off position. These will be taken up separately. The latch roller has a constant oscillation through angle B. The safety position A. The governor is in its lowest position. The latch is held away from the valve lever by the safety cam and will not "hook up." The valves remain shut and the engine will not start, or if running, will stop. To start the engine the governor must be lifted FIG. 281. VALVES AND VALVE GEARS 435 so that it rests on the high part of the safety collar, or more safely, on a safety pawl which drops out of the way when the engine gets up speed and the governor lifts. The starting position B. The valves will hook up and will not un- hook, as both cut-off and safety cams are out of reach of the latch roller. Cut-off will be late, depending on the angular advance. When the engine speeds up and the governor begins to take control, it lifts from the safety pawl (shown by separate sketch), which drops out of the way, allowing ut-Off Cam on Cam Collar Rollar on Latch ^Safety Cam on Cam Collar (Motneccessary on low pressure cylinder of compound eng/ne) Head-End Valve Diagram 6- Full Angular Movement of Valve Lever FIG. 282. the governor to sink to its safety position should the belt break. The so-called safety collar which must be turned to the safety position by hand after the engine is started is more convenient when stopping the engine, but is only a partial safety device. Zero cut-off position C. The latch is held away from the valve lever by the knock-off cam and will not hook up. This is a limiting position and is never reached unless the governor " hunts, " as the cut-off must at least be long enough to carry the friction load. The governor in action gives any cut-off between zero and the maxi- mum cut-off under governor control. Actual cam positions may be found 436 DESIGN AND CONSTRUCTION OF HEAT ENGINES for a given cut-off which is to be the same at head and crank ends. For equal cut-offs, the knock-off and safety cams would be closer together at the head than at the crank end if it were necessary to have the safety cam in contact at the extreme lowest position of the latch roller; for, as the piston moves farther on its stroke from head to crank end than on the return stroke for the same movement of the eccentric, cut-off would occur later in the head-end than in the crank-end stroke if latch roller and knock-off cam bore the same relation at both ends of the cylinder. Then to equalize a given grade of cut-off, the head-end cam must be nearer to the latch roller, or the crank-end cam farther away, or both. The safety cam should always keep the latch from hooking up when the governor is in the lowest position, but the engagement with the latch roller need not be at the lowest position; the valve lever may move a certain distance before the valve begins to open, and the release of the latch may occur at any point within this distance, which is much less for the long-range cut-off than for the single- wrist-plate gear as may be seen by comparing Figs. 277 and 279. Comparing the long- and short-range gears, the same governor motion (and consequently the same speed fluctuation) is used for both; then for the same range of load it is obvious that regulation will be closer for the long-range gear; this is true of any type of valve gear. In cross-compound engines the low-pressure cut-off may either be adjusted by hand to any fixed position, or be under the control of the governor. In the latter case the receiver pressure is kept more nearly constant under change of load, avoiding very low pressures with light loads and excessively high pressures with heavy loads. When the low- pressure cut-off is controlled by the governor, the fulcrum pin of the con- trolling lever is lengthened into a cross shaft having one bearing on the governor stand, which is usually located on the high-pressure engine frame, and the other on a stand occupying a similar position on the low- pressure frame and sometimes called the " dummy." A lever is placed on the low-pressure end of the shaft, with rods connecting it with the cam collars of the low-pressure gear. This lever is sometimes constructed so that it may be adjusted while the engine is running, dividing the load between the two cylinders as desired. It is sometimes arranged so that the governor may be temporarily disconnected and hand adjustment used for starting as explained in Chap. XIII. Figs. 247 and 248, Chap. XIX show safety devices used in connection with Corliss and other trip gears, and are explained in the text. Corliss Valve Stems. With certain assumptions the following formula VALVES AND VALVE GEARS 437 has been derived, where D is the cylinder diameter, d the valve diameter and di the diameter of the stem at the smallest part. di = 0.03 \ffi~D (25) The maximum steam pres- sure p may be taken as 100 Ib. per sq. in. for low- pressure cylinders to give practical results. 146. High-speed Cor- liss Gears. This term is commonly applied to non- releasing gears when rock- ing valves and a shaft governor are employed. The diagrams are much the same as for the releasing gear, these giving the maxi- mum cut-off, which may be about % stroke; then for the shorter cut-offs the eccentric swings or shifts, giving smaller valve travel and greater angular ad- vance. The steam distri- bution is practically the same as for a single valve except that the opening is a little quicker. Assuming the same diagram as Figs. 277 and 278, the curve m is the valve opening curve for J cut-off; it indicates that greater over-travel for the maximum cut-off would be desirable. 147. Mclntosh and Seymour Gear. Formerly the gridiron-valve engine built by Mclntosh and Seymour had expansion frfcEml g 438 DESIGN AND CONSTRUCTION OF HEAT ENGINES valves on the steam valves, but in the Type F engines these are omitted, greatly simplifying the gear and still giving good results. End and side elevations of this gear are shown in Fig. 283. The motion for the steam valve is equivalent to one wrist plate driving another. The valve move- ment is very small. The mechanism is operated from rocking lay shafts FIG. 284. Lentz valve gear. which receive their motion from eccentrics. The steam eccentric is attached to the governor and is offset in its connection with the lay shaft for the purpose of obtaining equal cut-offs. 148. Cam and Eccentric Gears. In these gears the cam is used in place of the wrist plate and for the same purpose. They are used only with poppet valves, their advantage being that they are free to move VALVES AND VALVE GEARS 439 after the valve is seated without interfering with it. The crank-eccentric diagram of Fig. 265 may be used and the cam designed to give as quick an opening as practicable. As with the Corliss gear, if the lead is ignored in drawing the crank-eccentric diagram, the cut-off will be some longer than the diagram indicates; a slightly shorter cut-off may be taken to offset this if desired. Lentz Gear. The eccentrics of this gear are on a lay shaft driven by miter gears. The steam eccentrics (one for each end) are connected with the governor, being shifting eccentrics with constant lead. An end section showing the general arrangement is given in Fig. 284. In laying out the crank-eccentric diagram for the steam valves, the line of stroke would be assumed as a line connecting the lay shaft center with the mean position of the cam arm; the dead -center crank position for correspond- ing ends of the cylinder would lie along this line toward the cam. For the exhaust valves, the dead-center position would be on the side opposite the cam. If the engine is right hand and runs over, the rotation for Fig. 284 is counterclockwise. 149. Reversing Gears. Space will not permit an extended treatment of this subject, but a brief description of a few of the most important gears will be given, with some explanation of the principles involved; for a description of other types, or a fuller treatment of any given type the reader is referred to the works named at the end of the chapter. Most of the illustrations are diagrammatic. Reversing gears are usually associated with the locomotive and ship engine; they are also used for hoisting, rolling mills and for some other stationary engines. They are sometimes used in connection with the Corliss gear, in which case their function is reversing only; but in most cases they are also used for varying the cut-off. This latter function was probably discovered accidentally. As first applied to locomotives, reversing gears consisted of two eccentrics, each with its own eccentric rod with sort of a crab-claw hook on the other end; either one of these* could be hooked onto the rocker pin by a double bell crank operated from the cab; this gave full gear forward or backward. It is obvious that the reversing operation could not well be done when the locomotive was under much motion, and it was probably to obviate this defect that the sliding link of the Stephenson gear and the sliding block of the Walschaert gear were devised, both of which were invented about the year 1844. Most reversing gears give an effect similar to a single shifting or swinging eccentric, and when the path of change can be approximately determined, the Bilgram diagram may be used for preliminary work if a single valve is employed. 440 DESIGN AND CONSTRUCTION OF HEAT ENGINES Stephenson Reversing Gear. This is perhaps the best known and the most difficult to lay out, as there are many variables which effect the results. Fig. 285 shows a link which has been used on stationary engines; the proportional figures are based upon the diameter d of the saddle pin, which may be made 0.1 the cylinder diameter (D s , Chaps. XII and XIII) for unbalanced valves and % of this for balanced valves. A is the saddle with pin forged integral, by which the link is lifted and lowered; while B is the link block which is drilled and counter bored for the pin which is fitted to rocker or valve-rod crosshead. For solid-ended link as shown, one flange of the link block is loose, and fastened to the block with rivets or countersunk screws. 00 25- \ 1.25 < FIG. 285. Stephenson link. . Fig. 286 shows the gear in neutral position (mid-gear) at A, forward gear at B and backward gear (reverse) at C, as arranged for a locomotive with outside admission and the link block attached directly to the valve-rod crosshead. Should there be inside admission, or a rocker which reverses the valve motion, the crank would be on the opposite dead center. As it is locomotive drafting room practice to show the cylinder at the right, this has been assumed in Fig. 286, although for all previous work it has been at the left. Equivalent Eccentric. The link radius is usually equal to the distance from the shaft center to the center of the link block at mid travel. In VALVES AND VALVE GEARS 441 Fig. 287 assume the link to be straight and the eccentric rods so long that their angularity has no effect; further assume that the valve-rod cross- head is raised and lowered instead of the link. Let the valve rod be Valve Rod FIG. 286. moved to position B, % of the distance to the lower end of the link. Now let the crank and eccentric be rotated clockwise through a small angle, moving the end of the link the distance a; the valve rod would then Valve flod IB Valve Pod FIG. 287. be moved the distance a/2. If the two eccentrics are connected by a line, it is clear that a point located Y of the distance from top to bottom of this line would travel a horizontal distance of a/2. The movement of 442 DESIGN AND CONSTRUCTION OF HEAT ENGINES the valve rod due to two eccentrics and a link is then equivalent to that produced by a single eccentric which shifts along this connecting line in the same ratio that the link block is shifted. Open and Crossed Rods. If the rods are open when the eccentrics are both on the side toward the link, they are known as open rods. If crossed when in this position they are known as crossed rods; this is shown in Open Rods Crossed Rods FIG. 288. Fig. 288. Placing the eccentrics in the opposite position indicated by the dotted lines gives the center of the link its maximum travel 6; this is the valve travel for mid-gear, or when the equivalent single eccentric has an angular advance of 90 degrees. From Fig. 288 it is obvious that due to the effect of angularity, b is greater than a for the open rods and less than a for the crossed rods; it may also be seen that when the crank is on the dead center in each case the valve displacement for full gear is a/2 and for mid-gear it is 6/2. Assuming the gear to be so designed that full- Open Rods 'Crossed Rods FIG. 289. and mid-gear lead are the same at both ends of the stroke, this may be shown on the Bilgram diagram as in Fig. 289. The curve of eccentric change is unknown but has been assumed as the arc of a circle. It will be noticed that for open rods the lead increases from full to mid-gear; this is always used for locomotives and is ideal for the service as it gives a greater port opening when the gear is "notched up" to give short cut-off at high speed. Equalization of lead is also more important at mid-gear than at full gear for the same reason. VALVES AND VALVE GEARS 443 Crossed rods give a decreasing lead which may be zero or even negative at mid-gear; this is sometimes used for hoisting engines and tractors when it is desired to control the engines by the reverse lever; if placed in mid-gear no steam is admitted. Lead FIG. 290. Walschaert gear. - The most complete practical treatment of the design of the Stephenson link motion with which the author is acquainted is given in Link and Valve Motion by W. S. Auchincloss. Walschaert Reversing Gear. This gear has largely replaced the Steph- enson gear in locomotive construction in the United States. This is JO Main Rod Crossheadl Arm ~~7 * i \ Inside Admfssion ionL'mk FIG. 291. Walschaert gear for inside admission. largely due to constructional purposes, although it possesses some other advantages. Fig. 290 shows a general view of the gear with the crank on crank-end dead center and the radius rod in mid-gear position. This is for outside admission which is used for flat valves. Inside ad- 444 DESIGN AND CONSTRUCTION OF HEAT ENGINES mission is used with piston valves and requires a different arrangement. The radius rod is dropped to the lowest position for full forward gear; this divides the force required to move the valve between the link trunnions and the eccentric rod. For back gear the radius rod is above the trunnions, which carry the combined loads of radius rod and eccentric rod, which, with the exception of switching engines is for a small proportion of the time. Figs. 291 and 292 are diagrams showing the arrangement for inside and outside admission respectively, with the crank on the head- end dead center. Equivalent Eccentric. It may be seen that the eccentric has no angular advance; the effect of angular advance is obtained by the combination FIG. 292. Walschaert gear for outside admission. lever, which, when the radius rod is in mid position moves the valve twice the sum of lap and lead. The method of proportioning this lever is obvious. It is also plain that the effect of the crosshead might be ob- tained by an eccentric with an angular ad vance of 90 degrees for an outside- admission gear as shown at B, Fig. 293. The eccentric which operates the link is at A. Connect these two points by a straight line; on this line a point C will be located which depends upon the proportions of the combination lever, which in turn is determined by the lap. Then C is the equivalent eccentric which would give the same results as the gear. When the reverse lever is " notched up" the radius of eccentric A (which has no angular advance) is virtually decreased say to OD. As the lead is constant, C must travel parallel to line OD to the new position ; at any VALVES AND VALVE GEARS 445 rate the ratio AC/AB is constant, so C must move in a vertical line. As C moves to the center line, eccentric A has no effect on it; the angular advance of the equivalent eccentric is 90 degrees and the valve travel is twice the sum of lap and lead. As C crosses the line toward A\, the gear is reversed. This may be shown on the Bilgram diagram as in Fig. 294. FIG. 293. FIG. 294. Variable lead is sometimes used on passenger and fast freight locomo- tives equipped with the Walschaert gear. It is obtained at the expense of too great a lead in back gear, but this is not much used on these engines. Maximum mid-gear lead is determined, which may be greater if the lead is to be variable. The eccentric is then set back for forward gear by \ FIG. 295. swinging the eccentric crank out or in, depending upon whether it leads or follows the crank, thus reducing the full-gear-forward lead and in- creasing the full-gear-backward lead. This may be shown on a Bilgram diagram as in Fig. 295. This also gives a longer cut-off and later com- pression in forward gear upon starting, but shorter in back gear. 446 DESIGN AND CONSTRUCTION OF HEAT ENGINES Hackworth and Marshall Gears. A simple diagram of the Hackworth gear is shown in Fig. 296. A block slides in the straight link which may be turned through angle A, which reverses the engine. If placed in a vertical position the link has no effect and the valve travels twice the sum of lap and lead. The link and slid- ing block are equivalent to an eccen- tric fixed at right angles to the crank (with no angular advance); there is therefore a similarity of principle to the Walschaert gear. In the Marshall gear the link and block are replaced by a swing link as shown in Fig. 297. Figs. 296 and 297 are for outside admission; for inside admission the eccentric must be on the same side of the circle as the ciank, or the eccentric rod must be FIG. 296. Hackworth gear. produced beyond the link block and the rod attached to this extension. The Hackworth gear is used on the Case steam tractor. The Marshall gear has been used on reversing Corliss rolling mill engines of large power. Line of Stroke FIG. 297. Marshall gear. Joy Reversing Gear. In this gear no eccentric is used, but the motion is taken from the connecting rod. A link is attached to the rod at A and another to the frame at B] these connect at C. To the link AC VALVES AND VALVE GEARS 447 another link is pivoted at D; this is equivalent to the eccentric of the Marshall gear, the remainder of the gear being much the same. The arrangement is equivalent to having the eccentric on the same side of the Valve Poof FIG. 298. Joy gear. FIG. 299. Doble gear. shaft as the crank, therefore for outside admission the reversing link must be between the point D and the valve rod. The Hackworth, Marshall and Joy gears have constant lead; but variable lead could be obtained as with the Walschaert gear by causing the angle between the center line of action of the eccentric rod and the 448 DESIGN AND CONSTRUCTION OF HEAT ENGINES line of stroke to be slightly different from 90 degrees. The Bilgram dia- gram for these gears is the same as for the Walschaert gear. The Doble reversing gear, used on the Doble steam car, is like the Joy gear with links AC and CB omitted, the eccentric rod being attached at A on the connecting rod. The link block slides in a straight line, being like the Hackworth gear in this respect; it is practically the same as the Hackworth gear with the eccentric replaced by the connecting rod attach- ment at A, Fig. 299. The valve has inside admission. The Doble engine is a uniflow engine, so that the gear controls the steam valves only. INTERNAL-COMBUSTION ENGINES 150. Cam Gears. General arrangements of cam gears may be seen in Chap. V. The design of cams in general may be found in books on mech- anism, but application will be made to two cases. The first thing to be Cylinder v 1 H \ - FIG. 300. determined is the valve timing, and Fig. 300 gives the position of the crank, assuming a horizontal engine. Fig. 301 shows the timing laid off on rectified crank circles, two revolutions being required for the 4- cycle engine. The letters H and C denote head and crank ends of the stroke respectively. Fig. 302 shows a timing diagram referred to the cam shaft, being curves of cam rise on a rectified zero circle of the cam. VALVES AND VALVE GEARS 449 As the cam makes one revolution per cycle, the angles e and i are one- half the angles E and / respectively. It may be assumed that the cam Expansion >/ > 90 ' Zo* FIG. 302. L /on ' Compress/or C n H 270 36 roller travels in the direction of the arrow, the letters H and C indicating the corresponding positions of the crank. Table 66 gives values of E and / taken from various sources. The FIG. 303. TABLE 66 Case Eo EC Io Ic Roberts' Handbook 45 3 10.3 14.7 35 4 Large, slow-speed 25 5 10 Small, medium- or slow-speed 25.0 0.0 3.0 4.0 Small, high-speed 35 2 6.0 15 Bruce-Macbeth 45 15 10 45 Continental 4-cylinder truck Continental 6-cylinder auto 42.6 55 8.3 12.0 17.9 12.0 29.4 45.0 Racing engine 50 10 10 50 Low-speed (Power, Oct. 13, 1914) High-speed (Power, Oct. 13, 1914) 30.0 40.0 5.0 10.0 10.0 15.0 10.0 to 15.0 30.0 to 35.0 29 450 DESIGN AND CONSTRUCTION OF HEAT ENGINES negative sign indicates that the event occurs after the dead center at which it is theoretically supposed to occur. The method of drawing the piston velocity curve is given in Chap. XVI ; the radius of the crank circle may be taken equal to the rise of the cam roller, giving a curve of proper scale to be transferred to the cam layout for comparison. This is shown in Fig. 303. Using Fig. 302, as a guide, Fig. 304 is drawn for the exhaust cam with roller. The cam rise is denoted by h, the radius of the base circle of the cam by R and the radius of the roller by r. The following proportions are used in Fig. 304: R = 2h, r = 0.6# = l.2h. It may be assumed that the cam roller rotates clockwise while the cam is stationary. The engine crank passes through positions H and C FIG. 304 Exhaust cam with roller. FIG. 305. Inlet cam with roller. (Fig. 303) coincident with the passing of the center of the cam roller through positions so marked on Fig. 304. If a separate spring is used to return the push rod and other mechanism transmitting motion to the valve stem, the cam roller rides on the cam body of radius R, the clearance k being between the push rod and valve-stem lever. According to G. W. Meunch in Power, Oct. 13, 1914, k should be about ^2 i n -> an d somewhat more for large engines; the clearance should be adjusted when the engine is hot. For small engines with very high speed the clearance is some- times made less. In drawing the cams the center of the roller is placed at the opening and closing angles as shown; the cam-rise lines are then drawn tangent to the base circle and roller circle as shown. The circular arc forming the crown of the cam is then drawn, and joined to the tangent lines with a radius. This curve may be transferred to the roller-center circle as VALVES AND VALVE GEARS 451 shown. The velocity curve may be laid off on this circle, care being taken to place the crank-end of the curve (the low end) at the point marked C. If desired the velocity curve may be transferred to the base circle as shown by heavy dotted line. Curves A are the velocity curves. It may be seen that the clearance k influences the form of the cam, a small clearance causing the valve to open and- close more slowly. Fig. 305 is the inlet cam drawn in the same manner. Due to the opening of the valve after dead center is passed, the cam curve lies below the velocity curve at first. The cam curve B tends to cor- rect this, but the curve shown at D is the curve used, as the cost of production is less. A type of cam much used for small engines has a push rod with mush-room head with a hardened steel face. Ex- haust and inlet cams are drawn together in Fig. 306. The form of the cam is quite different, and the opening and closing are more rapid. The push rod is held against the cam by a spring, the clearance being between the push rod and valve stem or lever. Multi-cylinder Gears. The usual order of firing for multi-cylinder en- gines is given in Par. 106, Chap. XVI. There is some difference of opinion regarding the 8-cylinder engine. Considering cylinder No. 1. at the front of the car and the driver's right as R, the usual order of firing is: 1-L, 2-#, 3-L, l-R, 4-L, 3-R, 2-L and 4-R. The Gas Engine, April, 1915, gives for the "Ferro:" 1-L, 3-R, 3-L, 4-R, 4-L, 2-R, 2-L and l-R; it quotes the reason as follows: "(1) Because two successive impulses acting on one crank pin has substantially the effect of a prolonged impulse with the average thrust in a vertical line, and since all shaft and crank-pin bearings are, or should be split in a horizontal plane, it is right that the thrust should be as nearly as possible at right angles to the path of the split. (2) If the spark is advanced too far, and No. 1 left fires too early after No. 1 right, it is apparent that this will set up no crank-shaft strains whatever, but if No. 4 left followed No. 1 right, a too early ignition means an opposed torque tendency at the opposite ends of the crank shaft. " FIG. 306. Inlet and exhaust cams mushroom type. 452 DESIGN AND CONSTRUCTION OF HEAT ENGINES 151. Eccentric-and-cam Gear. Trial and error must be used in determining the best form of cam and the proper lift. The diagram of Fig. 265 may be employed for the crank and eccentric as seen in Fig. 307. The eccentric circle may be assumed, and the angles determined; then the proper dimensions may be found by trial. The movement of the eccentric rod required to give full valve lift is equal to h when the proper scale is found. H-0 is a reference line, or center line of an imaginary crank, revolving at the speed of the lay shaft (H the speed of the engine shaft), and passing through points marked H and C as the engine crank passes head- and crank-end dead centers respectively, thus locating the eccentric position for any position of the engine crank. The diameter H-H lies on a line joining the lay-shaft center with the center of motion Exhaust Inlef- FIG. 307. of the cam end of the eccentric rod, the side on which h lies being away from the cam if there is tension in the eccentric rod in opening the valve. 152. Valve springs may be calculated by the chart and formulas of Par. 132, Chap. XIX, using PI for the spring tension when the valve is seated, P% when wide open and m for the valve lift h, or the spring de- flection under any circumstances. Further let: z = weight of valve and other parts to be moved by spring. N = r.p.m. of engine. r T = ratio of time required to close valve, to time required per stroke. Then the force required to accelerate the valve is: irhN , . Pl ~ VALVES AND VALVE GEARS 453 The force required to hold the valve against suction is: (27) where D is the cylinder diameter in inches and p the maximum suction in Ib. per sq. in., and which may be from 5 to 9 Ib. The larger of the two values of PI should be used. A good value of Pz/Pi is about 1.15, but this may be increased when it gives a larger number of coils than is convenient. 153. Two-cycle Engines. The piston forms the valve with this type, and as the ports are opened only near the end of the stroke, the same principles do not apply to timing and port opening. It has been stated that the width (measured along the stroke) should be from 9 to 13 per cent, of the stroke and the exhaust port from 20 to 22 per cent., and each should extend around 90 degrees of the circumference. This was given especially for heavy oil engines, but probably applies as well to gasoline engines. 154. Reversing Gears. There are various reversing gears used on internal-combustion engines, especially on Diesel engines used for ship FIG. 308. Diesel engine reversing gear. propulsion. Fig. 308 shows one type used on Diesel engines, in which there are two cam shafts, either of which may be in gear. 155. Ignition and Fuel Regulation. High-tension is largely used. In this type the sparking and timing apparatus is not usually made by the engine builder so space will not be taken to describe it. With low-tension or make-and-break ignition there are moving parts connected with the engine, so a brief description will be given. This type is used for large gas engines and sometimes for small kerosene engines. 454 DESIGN AND CONSTRUCTION OF HEAT ENGINES An igniter block from The Gas Engine is shown in Fig. 309 which is almost self explanatory. This must be operated by some form of trip mechanism which may be properly timed. Diagrams of two methods are given in Fig. 310. Timing may be changed in the upper method by raising or lowering the roller a and this may be under governor control if desired. In the lower figure the cam may be advanced or re- tarded. The crank 6 and cam c run at half the engine speed. Fuel regulation of gas engines and light-oil engines is commonly done by throttling; for the latter the carbureter is used, and this is not usually designed or manufactured by the engine builder. The subjects of ignition and FIG. 309. Igniter block. carburetion are too extensive to attempt a treatment ill a book of this kind and are omitted. For gas engines the regulation usually depends upon a mixing valve controlled by the governor and this is shown in Chap. XIX. FIG. 310. The Jacobson Gas Engine Co. use a form of trip gear and this is shown in Fig. 311. Oil fuel regulation is accomplished in several ways. Figs. 312 to 314, taken from Journal of A.S.M.E. show three methods used with Diesel VALVES AND VALVE GEARS 455 FIG. 311. Jacobson releasing gear. FIG. 312. By-passing to suction chamber. FIG. 313. Controlling length of suction stroke. 456 DESIGN AND CONSTRUCTION OF HEAT ENGINES engines. In all cases the governor rises for light loads when less fuel is required; the principle of operation may be easily seen. A regular Diesel spray nozzle is shown in Fig. 315. Oil is delivered under pressure between perforated plates by a small passage; when the fuel valve is opened, high-pressure air enters at A and sprays the oil into the cylinder. An open nozzle for a Diesel engine is shown in Fig. 316. Oil is pumped in during the suction stroke against no pressure; it is then sprayed into the cylinder when the fuel valve is opened. It is claimed by some that this type of nozzle will not foul as easily as the closed nozzle. 156. The Sleeve Motor Gear. The best diagram for designing this gear is doubtless the rectangular diagram on the rectified ciank circle. Two revolutions of the crank are required. The author does not have the measurements for this engine, so the diagram of Fig. 317 is only assumed. The piston displacement curve is given above, assuming a connecting rod 4 cranks long. The valve-displacement curves neglect the angularity of the eccentric rods, but this may easily be accounted for if desired. Considerable "cut- and-try" is necessary, and it is best to draw the valve curves separately on tracing cloth. Valve- opening curves are compared with piston velocity curves (dotted), laid off on a rectified crank circle. FIG. 314. Changing stroke of pump. FIG. 315. Diesel fuel nozzle and valve. Different arrangements may be made to give the same valve timing; by trial, an arrangement may be obtained which will give the best results, VALVES AND VALVE GEARS 457 and this has doubtless been done in the Willys-Overland engines although this method may not have been used. Fig. 317 is given to show its utility in problems of this kind. FIG. 316. Open type of fuel nozzle. Exhaust > In I FIG. 317. Rectangular diagram for sleeve motor. 157. Details from Practice. Some idea of valve gears may be had from Chaps. Ill and V, and their connection with the governor in Chap. XIX, but a few details will be given in this paragraph. 458 DESIGN AND CONSTRUCTION OF HEAT ENGINES FIG. 318. Eccentric and strap. FIG. 319. Corliss valves and stems. FIG. 320. Bonnets for Corliss valves. VALVES AND VALVE GEARS 459 FIG. 321. Bass-Corliss dashpot. FIG. 322. Lentz poppet valve. 460 DESIGN AND CONSTRUCTION OF HEAT ENGINES Fig. 318 is a simple fixed eccentric; the eccentric is split for convenience in placing on the shaft, and the strap is necessarily split for placing over the eccentric, and it also serves for making adjustment. i ?.'5 3 3.'5 4 4.5 5 55 Cast Iron 4 5 - i I 8 9 19 II 1 ! Loads of Same Sign Repeated Load c c 2 CJ ro LO CO CTi 1 | X. Reversed Load 75 C s s | Loads of Opposite Sign [ rn ^s. s"~ vD CO ^v X X FIG. 326. machines, such as automobile and airplane engines, the working stresses are often higher than in heavy rolling-mill engines. No one would think of using a J^-in. bolt in a place where strength is required in the latter, even though calculation showed it ample, while for the automobile engine a J^-in. bolt is a large bolt relatively. Machines to be handled by unskilled labor should have a high factor of judgment; they should be as near "fool proof " as possible without removing all profit. 160. Selection of Formulas. In the more simple applications of load there is little divergence of opinion regarding the formulas applying to calculations for stiffness and strength, but for loading involving complicated straining actions, different formulas are sometimes used for the same purpose, giving results often widely at variance. This is to a lack of agreement with tests or to a lack of test data in the case 476 DESIGN AND CONSTRUCTION OF HEAT ENGINES of the more theoretical formulas, and in some cases to the impossibility of a satisfactory rational treatment, leading to the use of empirical formulas. The formulas most commonly subjected to a variety of treatment are for: 1. Cylinder walls. 2. Combined bending and twisting. 3. Rectangular sections in torsion. 4. Struts. The relative importance of maximum stress and maximum strain (deformation) is involved in the first two. Let S = tensile or compressive unit stress. S s = shearing unit stress. E = modulus of elasticity (Young's mod- ulus) . Es = modulus of transverse elasticity (in shear) . e = deformation (extension or compres- sion) . m = the reciprocal of Poisson's ratio. All stresses are in pounds per square inch. Let the parallelepiped in Fig. 327 be acted upon by three forces, producing stresses, Si, $ 2 and $ 3 - It is shown in any good treatise on mechanics of materials or applied mechanics that the stress Si produces a deformation in the direction of action equal to: Si 61 = E The stresses S 2 and $ 3 each produce a deformation in a plane at right angles to their direction of action equal to: ^2 1 j &3 1 ei = W'm ei = E'm The fraction 1/m is known as Poisson's ratio and is usually between % and y. If 82 or $ 3 is a compressive stress, elongation occurs; if tensile, the result is compression in the direction of stress Si. The total deformation in this direction is the sum of these, or the resultant deformation is: $1 , Sz . 83 E mE mE Arranging so that compressive stress may be given the minus sign, and GENERAL CONSIDERATIONS 477 letting S R be the resultant simple stress which would produce the elonga- tion e R , gives: S R = Ee R = 8,- ^-^ (5) m The same treatment may be made in the direction of $2 or $3, but it is assumed that Si is the greatest. If S 2 and $3 are compressive and Si tensile, or vice versa, it is obvious that S R is greater than Si. The "maximum strain theory" limits S R to the safe tensile or compressive stress. In some applications S$ is zero, the equation becoming: S K = Ee R = S l - ^ (6) According to (6), if Si and S 2 are both tension, S R is less than Si, in- dicating an increase in tensile strength due to the effect of S 2 . Prof. Cotterill says: "An addition to the tenacity of a material, consequent on the application of a lateral tension, can, however, hardly be considered as intrinsically probable, and such direct experimental evidence as exists is against the supposition." However, other prominent authorities use the maximum strain theory in its entirety, the equivalent stresses being called "true stresses" by Merriman. Wherever applicable, the maximum strain theory will be used in this book but only when Si is of different sign from S 2 and $3 (tensile if the others are compression and vice versa); otherwise the effect of these secondary stresses will be neglected, insuring results always on the safe side. If, in (6), $ 2 is equal to Si, but is of opposite sign, it may be shown that a shearing stress S s of equal intensity is produced. Then letting Si = S s and S 2 = S S) (6) becomes: or: In other words, a given shear stress is accompanied by an equivalent direct stress of greater intensity than the shear stress. This indicates that if the maximum strain theory is correct for stresses of opposite sign, the ratio of the resistance to shearing to that of tension or compression is equal to: m m + l 478 DESIGN AND CONSTRUCTION OF HEAT ENGINES If S R and S s are known, (7) gives : m = 1 SR Sg Some values of m are given in Table 79. TABLE 79 m Material Goodman Johnson Average Cast iron 3 to 4 7 3 8 Wrought iron 3 6 3 6 Steel 3 6 to 4 6 3 72 3 8 Brass 3 1 to 3 3 3 06 3 2 Copper . . 2 . 9 to 3 3 06 3 A safe value of m for most of the materials used in heat engine construc- tion is 3^, and this will be used in this book. This gives: = fS = 0.77S* (8) a result agreeing well with values commonly given for ductile materials, although some authorities claim that the ratio S S /S R is much less. Hereafter when S R is the only stress appearing in the working formula, the subscript will be omitted, S denoting the maximum allowable, or the actual tensile or compressive stress. If E s is the modulus of transverse elasticity, or coefficient of rigidity, it may also be shown that: And for m = 2(m + 1) E (9) (10) 161. Cylinder Walls. There are several formulas for determining the thickness of a cylinder wall, some of which were compared by Alfred Petterson in the American Machinist of Feb. 15, 1900. The one most generally accepted, and probably the most accurate, is that of Lame, which will be given here. Let: GENERAL CONSIDERATIONS 479 p = internal pressure in pounds per square inch. S T = tangential, or hoop stress at inner surface, as given by the original formula of Lame. S P = radial stress at the inner surface, due directly to the pressure and equal to it. S = equivalent simple tensile stress due to S T and S P . m = the reciprocal of Poisson's ratio. r = internal radius of cylinder in inches. TI = external radius of cylinder in inches. D = internal diameter in inches. t = thickness of cylinder wall in inches. Let: 7*1 T + -. 1+r < The original Lame formula gives : ri 2 + r >. n*+l (ID where S T is obviously a tensile stress. The radial, or normal stress is compressive, and is: n 2 - 1 From Formula (6), Par. 2, the equivalent simple tensile stress is: (12) + S = S T - - F = m + - m (13) FIG. 328. Formula (13), known as Birnie's formula, assumes no longitudinal force due to reaction of cylinder heads, the condition being shown by Fig. 328, in which the heads are supported independently of the cylinder. This condition is found only in engines fitted with separate cylinder liners, but as longitudinal stress, being tensile, supposes a reduction of S, or a greater allowable working pressure, it is neglected, as stated in Par. 480 DESIGN AND CONSTRUCTION OF HEAT ENGINES 160. If longitudinal stress is considered, the formula is known as Claverino's formula.* Taking m as 3^, (13) becomes: 1.3n 2 + 0.7 and P n 2 - 1 rc 2 1 1.3n 2 + 0.7* S + 0.7p = \S - 1.3p = n r = r(n 1) = (14) (15) (16) (17) r V j V \ \ \ \ \ X^ .025 .0 FIG 5 .0" 329. 5 .IZ Formula (16) may be written: n = The chart in Fig. 329 will facilitate calculation. If S and p are known, (n 1)/2 may be found and used in (17) ; or if t and D are known : n - I t_ 2 = D and from the chart S/p may be found. When the pressure is small relative to the working stress, a simple for- mula, such as is used for steam boilers, will give results which are usually con- IZS sidered sufficiently accurate. Assuming a pressure of 150 Ib. per sq. in. gage, an allowable stress of 1500 Ib., a common condition for stream engine cylinders, the thin cyl- inder formula is in error 9 per cent, as compard with (17). If radial stress is considered in the thin cylinder formula the error is still 6 per cent. Should the pressure be 500 Ib. and the stress 3500, as for a Diesel oil en- gine, the error is nearly 12 per cent., and with radial stress, nearly 8 per cent. * In hydraulic cylinders for high pressure, Claverino 's formula gives results more nearly agreeing with practice than Birnie's formula. The formula for n is: + 0.4 p., r^r- nm 3H Tn en t may be found from (17). GENERAL CONSIDERATIONS 481 162. Combined Bending and Twisting. Let: S = simple tensile or compressive stress produced by bending, or more generally, by any simple application of load. S s = simple shearing stress produced by twisting or by a simple shear- ing load. S R = The equivalent simple tensile or compressive stress which would produce the same deformation as that resulting from the joint action of S and 83. This is taken as the actual stress under a given load, or maximum allowable working stress. M = the bending moment in inch-pounds. M s = the twisting moment in inch-pounds. M R = a bending moment which, with a given section modulus, would produce S R . z = modulus of section in bending. z s = modulus of section in torsion. m = the reciprocal of Poisson's ratio. The maximum direct stress, tensile or compressive, produced by the combination of a simple direct stress with a simple shearing stress is given by the equation : This is called the major principle stress and is sometimes the only one considered ; but for the maximum strain theory, the minor principle stress, of opposite sign and normal to S\, must be considered, and is given by the equation : Then from (6) , Par. 160, the equivalent simple stress is : and when m = 3~: o S R = 0.35S + 0.65 VS 2 + W (20) Formulas (19) and (20) are general for any case where S and Sa may be found for the same spot, the location giving the maximum value of S R being the weakest place in any machine part. For circular sections only, the surface shearing stress due to torsion is uniform, and a maximum at all points for a given twisting moment, and is : Ms Os = Zs 31 482 DESIGN AND CONSTRUCTION OF HEAT ENGINES The maximum bending stress is also at the surface and is: For all sections formed by concentric circles: Z S = 22 Then: Substituting these values of S and S s in (19) gives: (21) and form = 3-: M B = 0.35M + 0.65VM 2 + M s 2 (22) To reduce the use of large numerical quantities, it is convenient to take: 8 8 k , _ : =fc then: - = (23) M Then (20) becomes: S R = [0.35 + 0.65 And (22) becomes: M R = M[0.35 + 0.65 The chart in Fig. 330 will aid in the use of (23) and (24). The difference between the use of (18) and (20) may be illustrated by an example : Find the resultant stress in a shaft 10 in. in diameter when the simple stress due to bending is 5000 Ib. and the ratio of twisting to bending moment is unity. By (18), Si = 6035, and by (20), S R = 6350, the former involving an error of nearly 5 per cent. if the elongation theory is correct. One other theory has received some consideration and has been in- corporated in a number of text books. It assumes the resultant shearing stress as the limiting stress, the resultant bending stress being ignored. It therefore naturally reduces to a twisting moment formula L5A 6B FIG. 330. GENERAL CONSIDERATIONS 483 equated to the modulus of section for torsion. To avoid confusion, the formula will not be given, but the resultant shearing stress for the ex- ample just given is S SR = 3535, which indicates that the shaft might safely be smaller. If the maximum strain theory is assumed, a corre- sponding direct stress given by (7) is: S R 4600; this, when compared with resultant stress 6350 given by (20), is in error 27 per cent. The error increases with small values of k, and unless increasing factors of safety are employed, undue tensile stress may be produced. This "maximum shear theory" is sometimes known as Guest's Law, although not resulting in Guest's formula, which is empirical. Letting the bending stress S equal zero in (19) gives: m + l SR = ~^T' Ss which is the same as given by (7), Par. 160. If the bending moment is zero, (21) may be written: 77? -^ . S a X 2z = S s z s =- M s (25) which is the equation for simple twisting; or if the twisting moment is zero, M R = M. It therefore appeals that general equations (19) and (21) give a satisfactory solution of combined bending and twisting for most problems occurring in heat engine design, and in their simplified form given by (23) and (24) are not difficult to apply. From (7) : m ti s Then: m- 1 SR_ 2m 2S S and , _i_ -1 CY m -\- \ _ Pa 2m 2S S ' Values of S R /S 8 are given by some experimenters, differing considerably from those given here, and no attempt is made to relate them to m. If desired, these values may be substituted in (21), giving safer results, especially for large values of M a /M. If m = 1, (20) is the same as Guest's formula, although the latter is not derived in this way. This is the safest of all formulas for combined bending and twisting moment of round shafts, but perhaps is unnecessarily so. Recent values of S E /Sss for various grades of steel range from 1.45 to 2, a value for mild carbon steel being 1.546. This would change (24) to: MR = M[0.23 + 484 DESIGN AND CONSTRUCTION OF HEAT ENGINES Assuming the theoretical relation given by (7), this gives: m = 1.84. That there is difference of opinion concerning combined bending and torsion is well known, but until the relation between direct and shearing stresses is better known, greater refinement seems unwarranted. Since the above was written, an interesting article on combined stresses by Prof. A. Lewis Jenkins has appeared in the August number of the Journal of the A.S.M.E., p. 694; it is also in the Transactions, vol. 39, p. 929. 163. Rectangular Section in Torsion. The torsional modulus of section which applies to circular sections does not express the stress relations in a rectangular section. The stress in the extreme fiber of the latter, instead of being a maximum, is zero, the maximum stress being at the surface of the center of the long side. It may probably be assumed that the stress at the center of the short side bears the same ratio to that of the long side as their distances from the center of the section, inversely. These stresses reduce to zero at the corners in a manner represented by a curved line, the exact form of wjiich is probably unknown. The best expression for the modulus of section of a rectangular section is an empiri- cal formula given by St. Venant as follows: b = the length of the short side. h = the length of the long side. Ss = maximum shearing stress, at center of long side. SBI = shearing stress at center of short side. M s = the twisting moment. z s = modulus of section in torsion. ' -& > Let: 6 = then: M s = z 8 S s = 3 _^\ 8x h*S s = ~ - S a (27) where c = 3 + 1.8s x 2 Then: rr fi M-S ( f )'>L\ S S = C-^ (28) GENERAL CONSIDERATIONS 485 where S s is the maximum stress, which is at the center of the long side, as just stated : Then the stress at the center of the surface of the short side may be taken as: o b Q r MX o.si = -r o s = Cz Let: Then: Cx /l 3 3 + l:8s (29) \ \ The chart in Fig. 331 may be used to find C and 164. Struts. Two of the most important engine parts the piston rod and connecting rod are struts. Therefore the selection of a suitable . strut formula, which shall be safe while not demanding an excess of material, is essential to rational de- sign. As already stated, this need not be a rational formula; this re- lieves an embarrassing situation, as the only formula which rests on any satisfactory rational basis is the long column formula of Euler, which, with few exceptions, is outside the range of ratios of length to radius of gyra- tion found in practice. Although this formula, for prac- tical conditions, is inferior to any one of several empirical formulas, it is surprising to note that it is still used by some authorities. How- .ever, for exceedingly long struts which fail by buckling, and in which the direct unit load is small, Euler's formula gives results agreeing well with experiment; and as it serves to fix a limit for an empirical formula to be discussed later, it is given here. P = ultimate load on strut. p = ultimate unit load (pounds per square inch) . A = area in square inches of section of maximum stress. This is at the center when both end conditions are the same. .3 A FIG. 331. 486 DESIGN AND CONSTRUCTION OF HEAT ENGINES I = actual length of strut in inches. li = the distance between two consecutive points of contrary flexure, sometimes called the effective length. It is this length which deter- mines the strength of the strut. r = the least radius of gyration, in inches, of section A. E = The modulus of elasticity of the material (Young's modulus). n = a factor depending upon the condition of the ends of the strut. Then: = P _ ir*E Tr 2 nE (30) From (30) it may be seen that n is used so that the formula may be given in terms of actual length; then: l l \ 2 n -- (rj Table 80 gives the theoretical conditions usually considered in strut discussions. They assume accurate dimensions, homogeneity of material, and the application of load at the exact gravity axis of the strut. The actual conditions are never accurately known in practice, and intermediate values of n are sometimes used which give values agreeing approximately with tests with such end conditions as are practically attainable, and will be mentioned later. Of the several strut formulas in common use, the most satisfactory, in the author's opinion, is an em- pirical formula given by the late Prof. J. B. Johnson (Modern Framed Structures, and Materials of Con- struction), a brief discussion of which will now be given. The formula is: P = K-q(Y (31) _^ which is the equation of the para- bola, K and q being constants. Prof. Johnson assumed that: K = S E (see Par. 159, Law 8), and q is to be determined so as to bring a parabola tangent to Euler's curve. This of course occurs when the slope of the two curves is the same. For convenience let p = y, and 1/r = x (see Fig. 332). FIG. 332. GENERAL CONSIDERATIONS TABLE 80 487 Case I, End conditions 1 i 2Z H One end fixed; the other end free and unguided. IV 2 1 i i Both ends pivoted. * * 3 I 0.7Z 2 (nearly) One end fixed; the other pivoted, but guided in the direction of axis of fixed end. 4 1 0.5Z Both ends fixed, with common axis. 488 DESIGN AND CONSTRUCTION OF HEAT ENGINES Then Euler's formula becomes: And Johnson's is: y = S E qx 2 . The slope of each curve is: T-I i dy 2ir 2 nE Euler, ~ = dx x* Johnson, ~.= 2qx. When the curves are tangent to each other, y and dy/dx are the same for both; then: 7r ^- = S E - qx 2 (32) and: '-*' ..-; (33) Dividing (32) by (33) gives: 8, x = x qx or: :*-* From which: Substituting (34) in (32) gives: S E - ^ From which: 9 = iS < 35) Johnson's formula then becomes: .Sir, 2 / 7 \ 2 (36) The limit of application of this formula may be found from (34), which gives: (37) 2q or: GENERAL CONSIDERATIONS 489 For any greater value of l/r, Euler's formula is to be used, which is: 'jjjf^^^S; p = ! ^ (38) The value of q may be found which gives close agreement with tests under certain practical conditions and n may be found from (35). This value of n may be assumed to be the same for different materials tested with the same end conditions. The ordinary test conditions are for: 1. Round ends. 2. Pin ends. When the pins are of substantial diameter, friction resists buckling, making the strut stronger than for a round or pivot- ended strut. 3. Flat ends, which are stronger than pin ends, but not as strong as fixed ends. For the purpose of this book the following values of n will be assumed, although they are some smaller than those used by Johnson: 1. Round ends, n = 0.90. 2. Pin ends, n = 1.45. 3. Flat ends, n = 2.00. ; . That they agree with tests in some cases at least may be seen from Fig. 333. These actual end conditions do not occur in engine design, but are given here to assist the designer in determining a value of n which most nearly represents his practical problem. Special cases may arise in which the theoretical values of n in Table 75 may be safely used. Figure 333 is an ultimate unit load curve for four formulas, with round ends, with the values of n just given. Curve A represents Euler's long column formula, and B, Johnson's parabolic formula. These curves are tangent to each other and together form the adopted curve. Curve C represents Ritter's formula as given by Goodman (Mechanics Applied to Engineering), in which the limiting stress at failure when l/r is zero is taken as the "crushing strength of a short specimen," or practically, the ultimate tensile strength. Curve D is also Ritter's formula as given by Kimball and Barr (Machine Design), in which the limiting stress at failure is taken as the elastic limit. With these curves is plotted, values from tables in the Pencoyd handbook (curve E) which are the average of many tests of struts composed of medium steel structural shapes. To obtain a fair comparison, the physical constants used in the formulas are the same as for the material used for the Pencoyd tests. From a study of the curves it may be observed: 490 DESIGN AND CONSTRUCTION OF HEAT ENGINES 1. That Euler's formula has little application for values of l/r ordinarily found in practice. 2. That Hitter's formula (Goodman) gives results considerably above the tests when S is the ultimate tensile strength. 3. That Hitter's formula (Kimball and Barr) gives values far below the test values when S equals the elastic limit. It is safe, but safety is not the only important consideration in good design. 70000 4. That Johnson's formula follows the tests very closely excepting for small values of l/r. Ritter's formula is often given in text-books on machine design and it is no doubt a good safe formula for general design. It is generally men- tioned as a rational formula, but as it gives a higher stress for a given unit load when the elastic limit is higher, this can not be true. In a certain gasoline engine connecting rod the unit load is 7100 Ib. Ac- cording to Ritter's formula the stress produced in the rod is 10,850 ft>. when the elastic limit is 38,000; but when the elastic limit is 120,000, GENERAL CONSIDERATIONS 491 the stress in the rod as given by the formula is 18,850, being 74 per cent, greater. This of course is inconsistent. It is generally conceded that long struts fail when or before the elastic limit of the material is reached. If Johnson's parabola is made tangent to Euler's curve, it seems reasonable to assume that any failure occur- ring when the unit load lies on or below these combined curves must occur at or within the elastic limit. We may then employ factors of safety with the elastic limit as a basis for any value of l/r, which would not be feasible for formulas employing higher values for the crushing strength of short struts, even though correct. It is probable, as stated in the Pencoyd handbook, that higher ultimate loads would obtain for round or square sections, due perhaps in part to absence of thin, unsupported flanges, and partly to the ability to obtain better end conditions. On the contrary, the elastic limit of the material decreases as the thickness increases, and this may more than offset the gain by symmetry of section. Johnson's formula will be employed in this book, and using the phy- sical constants given in Table 73, the following special formulas may be derived, which are conservative for materials commonly employed in engine construction. For cast iron there is no marked elastic limit. The value of K in Formula (31) is taken as 60,000, this giving results agreeing with tests. Round ends Steel. When l/r < 117: p = 38,000 - 1.4 ~ (39) Wrought iron. When l/r ^126: p = 27,000 - 0.79 (-)' (40) Cast iron. When l/r ^ 59: p = 60,000 - 8.5 (-Y (41) Pin ends Steel. When l/r < 147: p = 38,000 - 0.87 (-)* (42) Wrought iron. When l/r ^166: p = 27,000 - 0.49^-) * (43) Flat ends Steel. When l/r < 173: p = 38,000 - 0.63 (-Y ( 44 ) Wrought iron. When l/r ^ 195: p = 27,000 - 0.355 (-) * (45) Cast iron. When l/r ^ 88: p = 60,000 - 0.38 (-\ * (46) 492 DESIGN AND CONSTRUCTION OF HEAT ENGINES When a strut is subjected to bending stress, from eccentric loading or otherwise, the sum of the bending stress at center of strut and direct unit load should not exceed the unit load allowed by the strut formula, using least radius of gyration. Should the end conditions not be the same in all planes, as in an engine connecting rod, the maximum value of q/r must be used, as it is obvious from (31) that this will give the mini- mum value of the ultimate load. 40000 30000 20000 10000 100 ZOO 300 FIG. 334. To facilitate calculations of steel struts, the chart in Fig. 334 for round, pin and flat ends is given, being plotted from special equations (39), (42) and (44). 165. Clearances and Tolerances. In the various machinists' hand- books and books on general machine design there are many tables giving clearances and tolerances for different kinds of work and also much other GENERAL CONSIDERATIONS 493 data relating specifically to the shop. While it is necessary for the de- signer to have such data, it seems to be outside of the scope of this book, so comparatively little is said on the subject. No absolute stand- ard has been adopted, but many concerns have their own standards. These should be in the hands of all designers in a particular office so that there may be uniformity of practice, at least for that office. The stand- ards should be adopted by the cooperation of engineering department and machine shops, and should not be changed unless experience shows that alterations are desirable. 166. Basis of Design. The forces acting on engine parts are com- monly computed for the maximum steam or gas pressure in the cylinder. If these alone may be considered, it is a simple matter to determine stresses in both magnitude and kind. For large, slow-speed engines, these FIG. 335. forces, with those produced by heavy weights such as the flywheel, are all that are practically necessary to consider; but for high-speed engines, the inertia forces may approach those due to steam or gas pressure, and may not safely be overlooked. The forces acting normal to the line of stroke are comparatively small for most engine parts, so that by the use of combined indicator and inertia diagrams a comparatively simple method of selecting factors of safety may be devised which will cover ordinary cases. Aside from attachment to foundation or other supports, the frame is not affected by inertia forces along the line of stroke; indicator diagrams only are required to determine the forces. This is also true of the cylin- der. Practically all other parts are affected by inertia. Single-acting Engines. Fig. 335 shows stroke diagrams combined with inertia diagrams for single-acting steam and internal-combustion engines. Reference may also be made to Chap. XVI. Whether the resulting stresses are repeated or reversed will depend largely upon the amount of inertia. The inertia diagram may include only the recipro- cating parts in determining the forces acting upon certain parts, or for some cases the connecting rod may be included. 494 DESIGN AND CONSTRUCTION OF HEAT ENGINES The following formulas may assist in obtaining factors. From (33) and (34) , Chap. XVI, if n is the ratio of connecting rod to crank length : z? i i r H _ n -f- 1 F^ ~ n - 1* Let: P 2 F H and Pi = k. The notation is taken from Fig. 335. Then: T l =P l -F H = P,(l - k) and: , n I Let the maximum value of T be T M ' } then: (47) (48) (49) This may be called the pressure factor and may be a part of the factor of judgment. FIG. 336. In the single-acting steam engine, compression may offset part or all of the inertia at the head end as shown in the dotted lines. Double-acting Engines. In these engines, the greater range of T is found in the crank-end diagrams, and these are shown for both steam- and internal-combustion engines in Fig. 336. The equations for this case are : T,=P,- Fa -= Pi (l - fc ^-j) (50) and: T 2 = P 2 + F H = P 1 (q + k) (51) In the steam engine, compression may reduce P 2 and T 2 , making them negative in some cases. As an aid in determining factors Table 81 is given. As the single- acting steam engine is little used it is omitted. GENERAL CONSIDERATIONS 495 I ( (C ( &- 1 < 000=5 J EM ^ 32^ 34^ Clearance . ... 3/ /8 H Y For larger sizes the clearance and allowance for counterbore may be taken the same as the maximum values in the table. In internal-combustion engines and the uniflow steam engine, clear- ance is determined by the required compression pressure. 168. Inlet and exhaust passages in steam cylinders are dependent upon the allowable nominal steam velocities; the area of a passage is given by Formula (2), Chap. XX, and is: S P A ,,, a = -y- in which V will be taken in ft. per min. in this chapter. The values of V are given by Formulas (3) and (4), Chap. XX. In steam engines the steam and exhaust connections are for standard round piping; then we may write: a s a E 504 DESIGN AND CONSTRUCTION OF HEAT ENGINES where subscripts S and E denote steam and exhaust respectively. Then (5) becomes: and: D ' = (6) (7) The next larger diameters of standard wrought pipes may be used if D s and D E are intermediate values. Values of V s and V E may be found from the curves of Fig. 339, which were plotted from (3) and (4), Chap. XX. 9000 4000 40 50 D FIG. 339. 60 Formula (5) may also be used for internal-combustion engines by using the proper value of V. This also applies to (6) and (7) when the passages are of circular section. Lucke says that V should not exceed 3000. Measurements of the cylinder of a well-known automobile engine gives for the entrance to the cylinder a velocity of 8400 ft. per min. and for the exit 6800 ft., for a piston speed of 1000 ft. per min. A modern vertical gas engine gives a gas velocity through the inlet opening of the cylinder head of 5380 ft. per min., and through the exhaust opening, 7850 ft., with a piston speed of 640 ft. These velocities are all nominal velocities as explained in Par. 142, Chap. XX. Velocities through ports and valves of both steam and internal-combustion engines are discussed in Chap. XX. 169. Cylinder heads are largely of empirical design as there are no satisfactory rational formulas for determining the thickness . of metal. In shallow heads which set into the cylinder but a small distance, a flat circular plate formula may be used as a check. For deep heads, such as used on Corliss cylinders, the strength depends largely upon the ribs, which form sort of a truss. Such a head is shown in Fig. 340. As a guide the following dimensions are given: ti = 1. It t z = 0.6h CYLINDERS 505 It will be noticed that the head has a snug fit in the cylinder for but a short distance, from % to 1 in., depending on the cylinder diameter, after which it is cut away J-^2 m -> or sometimes less. It was once common to have the entire flange surface fit tight against the cylinder, with paper or some other packing between ; but now an annular space only comes in contact with the cylinder; this sur- face, about ^4 in. wide, permits a greatly increased pressure per sq. in. and is more easily kept steam-tight. This surface is sometimes ground in place, but this is not necessary for a steam-tight joint. It is well to keep the stud circle as near the inner edge as possible without danger of breaking out ; good results are obtained if this distance is about 1.2 times the stud diameter, and this same distance may be taken to the outer edge of the head. Table 84 gives the dimensions of high-pressure and low-pressure heads, TABLE 84 FIG. 340. High-pressure heads Low-pressure heads D, in. DC, in. DH, in. Ds, in. ds. in. No. studs A for p= 100 D, in. DC, in. DH, in. I>fi, in. ds, in. No. studs s, for p= 100 10 10% 14% 12% % 8 3260 34 34^ 40^ 37H 1% 20 5090 12 12% 16% 14K % 8 3370 36 36H 42^ 39M 1% 22 5180 14 14% 18% 16K % 10 3670 38 38M 44^ 41H 1% 22 5780 16 16% 21% 18% 1 10 3660 40 40^ 46M 43M 1H 24 5890 18 18% 23M 21 1 12 3870 42 42H 48M 45M 1% 24 6470 20 20% 25K 23 1 14 4080 44 44^ 50K 47H 1% 26 6560 22 22% 27K 25 1 16 4320 46 46M 52^ 49H 1% 28 6680 24 24% 30 27% 1H 16 4080 48 48 54M 51M 1% 30 6760 26 26K 32 29% 1H 18 4250 50 50^ 57 53% IK 30 6240 28 28K 34 31% i 20 4470 52 52K 59 55% 1% 30 6740 30 30K 36H 33K 1% 20 3980 54 54K 61 57% 1% 32 6800 32 32 Y* 38K 35K 1M 22 4110 56 56H 63 59% IH 32 7330 34 34 Y Z 40K 37 K l>i 24 4240 58 58H 65 61% m 34 7380 36 36K 42> 39K IK 26 4390 60 60H 67M 64 IK 34 6440 38 38K 45 41% 1% 26 4100 62 62% 69K 66 IK 34 6870 40 40M 47K 44 1H 24 4060 64 64% 71M 68 IK 36 6920 42 42M 49K 46 1M 24 4470 66 66% 74M 70H 1% 36 6280 44 44 K 51 K 48 1H 26 4530 68 68% 76H 72H 1% 36 6670 46 46K 53 H 50 1M 28 4610 70 70% 78H 74M 1% 36 7070 48 48K 56> 52 K 1% 28 4260 72 72% 80% 76% 1H 38 7080 50 50K 58> 54K 1% 28 4650 74 74% 82% 78% 1% 38 7490 76 76% 85^ 81 1% 38 6860 78 78% 87K 83 1% 38 7220 80 80% 89% 85 1% 38 7500 506 DESIGN AND CONSTRUCTION OF HEAT ENGINES with the number of studs, and the stress at the root of the thread for a pressure of 100 Ib. per sq. in. The stress for any other pressure may be easily found as it is proportional to the pressure. The notation is as on Fig. 340. For Corliss high-pressure cylinders, a depth of head equal to the diam- eter of the valve is satisfactory; the writer has used, for high-pressure heads : Depth = Q.2D + 1.75 (8) and for low-pressure heads: Depth = 0.17D + 1.8 (9) There is no such similarity between the designs of heads for internal- combustion engines, but the same general principles apply. The load on cylinder-head bolts or studs is practically a dead load, and the stand- ard factor of safety could be taken as 2 if the stress due to screwing up were known. It is better, however, to take the repeated load factor 3, based upon the total steam pressure and divided by the total bolt area at root of threads, which is the assumption in Table 84; this may be called the nominal stress. Small bolts are much more liable to be damaged by a wrench, so the factor of judgment may be: /s = 1 + V (10) a s The factor of safety is then: in which d s is the stud diameter. The safe stress for machinery steel, from Table 73, Chap. XXI is: 38,000 / High-grade steel is sometimes used, especially for internal-combustion engines. With cylinder-head castings for internal-combustion engines, great care must be exercised in design and in the foundry work. There must be an even distribution of metal with no abrupt changes from thin to thick metal. Ribs must not be placed so that they make the casting too rigid and prevent the uneven expansion due to difference in tempera- ture. Cored passages should be as easy as possible. Stuffing Box. If any special form of metallic packing is to be used, the stuffing box should be designed accordingly, but for the numerous soft CYLINDERS 507 packings on the market, Fig. 341 gives good results if proportioned according to Tables 85 and 86, and the following formulas: A = 0.15D* B = 2d s C = l.5d s E = D B + 2Ad s F = E + 2Ad s K = D B - d FIG. 341. TABLE 85 d, in. D, in. L, in. Jf, in. 3Hor<5 ^5 d+iy 2 d+1% d+2 4K 5 6 3K 4 4K TABLE 86 d, in. ds, in. . . No. studs 3 % - 2 4 1 2 5 DS 2 6 1 3 7 1 3 8 1H 3 9 1^ 3 The stuffing-box flange is made thick to offset the weakening effect of the box, which cuts away the ribs at the center. 170. Cylinder Lagging. Steam cylinders are covered with about 1J^ in. of some heat-resisting material in the form of plaster, such as asbestos or magnesia. Outside of this is the lagging, usually of sheet steel, and provision should be made for fastening this to the cylinder. This may be seen in some of the designs which follow. 508 DESIGN AND CONSTRUCTION OF HEAT ENGINES A water jacket is required for internal-combustion engines, except for very small cylinders, which may be air-cooled. No lagging is required. Besides the illustrations of the following paragraphs, some features of design may be found in Chaps. Ill and V. The foundry and machine shop should be kept in mind with a view to reducing the cost of production, and no design should be used which has been known to give trouble in operation. DESIGNS FROM PRACTICE 171. Steam Engine Cylinders. The Corliss engine cylinder of Figs. 337 and 338, while not designed for an actual engine, is substantially the FIG. 342. Bruce-Macbeth gas engine cylinder. design used by the author for several years and shows the characteristic features of Corliss cylinders. The cylinder foot is included; openings are provided for the vacuum dashpots, but in some cases these are secured to the foundation independently. The cylinder foot is usually bolted to the foundation. CYLINDERS 509 172. Internal-combustion Cylinders. A vertical gas engine cylinder is shown in Fig. 342. This is the design of the Bruce-Macbeth Engine Co., Cleveland, Ohio. With a stress of 3500 Ib. and a gage pressure of 400 Ib., Fig. 328 and Formula (17) of Chap. XXI give a wall thickness of practically % in. The actual thickness being 1 in. makes k in (1) equal to Y in. The jacket wall is made equal to one-half the cylinder-wall thickness. With a gage pressure of 400 Ib., the nominal stress in the cylinder- head bolts at root of thread is 8850 Ib. If the elastic limit of the bolt FIG. 343. Bruce-Macbeth cylinder heads. material is 38,000, the factor of safety is 4.3. Taking the repeated-load standard factor as 3, gives a factor of judgment of 1.43 instead of 2, as given by (10); the former is no doubt ample. The head for this cylinder is shown in Fig. 343. The valves and valve cages are shown in Chap. XX. Fig. 344 shows the air-cooled cylinder of the Franklin Automobile Engine Steel ribs are cast in the cylinder as shown. These are sur- rounded by a jacket and air is drawn through the passages between the ribs. 510 DESIGN AND CONSTRUCTION OF HEAT ENGINES The cylinder wall, measured from the bottom of the ribs is thick. With a gage pressure of 350 Ib. and no allowance for wear, etc., Formula (1) of this chapter gives a stress of 3325 Ib., while a pressure of 400 Ib. gives 3800 Ib. With the standard factor of 6 for repeated load for cast iron, neglecting the factor of judgment, these stresses give nec- 5ectionA-A FIG. 344. Cylinder of Franklin automobile engine. essary ultimate strengths of 20,000 and 22,800 Ib. respectively. The latter is no doubt conservative, as the better grades of gray iron have a tensile strength as high as 30,000 Ib. There are four %-in. bolts holding the cylinder to the frame. By the S.A.E. standard, they have 24 threads per inch. The stress at root of thread for direct thrust only is 9000 Ib. for a maximum explosion pres- sure of 350-lb. gage, and 10,300 for 400 Ib. pressure. This gives a factor CYLINDERS 511 of 4.2 and 3.7 respectively for the conservative value of the elastic limit given in Table 73, Chap. XXI. Both of these are greater than the stand- ard factor 3 for repeated load, giving factors of judgment of 1.4 and 1.23. This would, no doubt, provide for the side thrust due to the connecting rod. The maximum pull on the bolts occurs when the product P N a (Fig. 345) is a maximum; as before, neglecting the stress due to screwing up, the stress is due to a repeated load. If n is t he ratio of the length of connecting rod to length of crank, (12), Chap. XVI, gives: . P sin 6 [1 0\ 2 The pull on the bolts for the arrangement of bolts in Fig. 344 is: FIG. 345. Assuming the diagram of Fig. 274, Chap. XX ap- plies to this engine which is not strictly true the values of P B were calculated for all positions covering that giving the maximum value; corresponding pressures were taken from the indi- cator diagram in the same figure, and the maximum combined stress from these is 11,300 Ib. per sq. in., and occurs at position 1, or very near head-end dead center. This does not include the inertia of the rod, which might have modified the result somewhat. From the same dia- gram, the stress produced by maximum gas pressure is 8620 Ib. The stress due to P N is transmitted through the rod, so inertia forces must be included in the value of P used (this should properly be the p A of Fig. 204, Chap. XVI, instead of P N ) ; but in obtaining the direct load on the bolts in line of stroke, inertia has no effect and gas pressure only must be used. For the value of mild steel in Table 73, Chap. XXI, the maximum stress just found gives a factor of safety of 3.36, but S.A.E. standard bolts have an elastic limit of 60,000 Ib., giving a factor of 5.3, or a factor of judgment of 1.77. In a steam engine, the maximum steam pressure continues to a point beyond that at which P B is a maximum, so that the stress in the bolts (neglecting initial stress) is greater than that produced by direct steam pressure in the line of stroke. Fig. 347 shows a side elevation and plan of a pair of cylinders of the 512 DESIGN AND CONSTRUCTION OF HEAT ENGINES O ^ FIG. 346. Sections of Sturtevant cylinder. FIG. 347. Sturtevant airplane engine cylinder. FIG 348. Sturtevant cylinder heads. CYLINDERS 513 Sturtevant airplane engine, without liners. Fig. 346 shows a section without sleeve, and a section with the sleeve in place. The cylinders are attached to the crank case by bolts extending through lugs as shown at section, and through the cylinder head. Section D-D FIG. 349. Sturtevant cylinder heads. Fig. 348 gives outside views of two cylinder heads cast enbloc, with rocker arm fulcrums; the fulcrums were formerly cast separately and bolted to the head. Fig. 349 shows such a head, which also better gives the detail design. 33 CHAPTER XXIII PISTONS Notation. D = diameter of cylinder bore in inches. F = length of piston, or piston face, in inches. L = distance between the centers of gravity of the two semi-circles, in inches. , a = area of half circle in square inches. r = radius of cylinder bore in inches. 7*1 = radius of outside of piston ring when free, assumed constant for an eccentric ring. t = thickness of ring in inches, at any section. t M = maximum thickness of ring opposite cut. w = width of ring in inches. 7 = moment of inertia of ring section. z = modulus of section of piston section. M = bending moment at any section of ring. E = modulus of elasticity. S = bending stress in piston; also at any section of ring when in cylinder, exerting pressure p N . ks= stress developed in stretching ring over piston to place it in groove. SM= maximum bending stress when ring is in cylinder (at section of thickness t M ). p = maximum unbalanced pressure per square inch in cylinder. p N = normal pressure in pound^ per square inch of ring upon cylinder wall. W= total maximum pressure in pounds, upon engine piston. 173. Pistons are made in several different styles, the most common for steam engine use being the box piston, cast in one piece as in Figs. 361 and 363, and the built-up, or bull-ring piston shown in Fig. 360. In the former, ribs are usually provided for strengthening the piston; the holes in one side of the piston necessitated by the core, are tapped and plugged. The cores between the ribs are sometimes connected, leaving holes in 514 PISTONS 515 the ribs. This type of piston has given some trouble, due to the forma- tion of cracks where the walls join hub and rim; this may be due to poorly distributed metal or improper cooling of the casting. This design is not much used on very large engines. The bull-ring piston is composed of the spider, or body, the bull ring (sometimes called junk ring), the follower, and, in common with the solid piston, one or more packing rings, which prevent the leakage of steam past the piston. The stiffening ribs contain bosses into which are tapped the follower bolts. A circular rib connects the radial ribs and is provided with set screws and lock nuts, by means of which the bull ring may be ad- justed to take up wear. If wear is excessive the cylinder may be rebored and a new bull ring provided. Trouble is sometimes experienced by the breaking of follower bolts, and as this has been attributed to expansion, long studs, tapped into the farther side of the piston have been used by some builders, as shown in Fig. 360; the stud is thus allowed to stretch and bend slightly without undue stress. The wearing surface of the piston is sometimes provided with rings or strips of Babbitt metal, or some other anti-friction metal; this is more common in large pistons. Cast iron upon cast iron usually gives the best wear, but an occasional exception .arises, when a change to a babbitted bull ring gives better results, due, no doubt, to some defect in the metal of cylinder or ring. When a single packing ring is used and made in sections, or where two " snap rings " are used, the bull ring is made in one piece, but for the usual single ring it is in two parts, as shown in Fig. 360. When lightness is required in a large piston, perhaps largely for the purpose of balancing, as in marine engines, conical pistons are sometimes used, as in Fig. 362. Sometimes the box piston is made conical. Single-acting steam engines and internal-combustion engines have trunk pistons, which also serve as crossheads, examples of which are shown in Figs. 364 and 365. Trunk pistons should be designed so that the thrust due to the angularity of the connecting rod may be transmitted from cylinder wall to wrist pin without distorting the piston; the proper placing of ribs provides for this in very light pistons, as seen in Fig. 365. The material used in pistons is usually cast iron, but there is no reason, aside from expense, why the spider and follower of a bull -ring piston may not be steel castings. Pistons for automobile engines are sometimes made of an aluminum alloy, but it has probably not been tried sufficiently long to insure success. Morley, in his Strength of Materials, says that under repeated stress, pure aluminum tends to " creep," or gradually fail. Aluminum also has a high coefficient of expansion more than twice as 516 DESIGN AND CONSTRUCTION OF HEAT ENGINES great as for iron. If enough aluminum is used to greatly lighten the pis- ton, some trouble may possibly ensue. According to the American Machinists' Handbook, aluminum becomes granular and easily broken at about 1000 F. The working stress may be taken as 1200 Ib. for cast iron pistons, and 5000 Ib. for steel castings. From Tables 73 and 74, Chap. XXI, this gives a factor of judgment over the standard reversed stress factor of 1.11 for cast iron, and zero for steel. There is no strictly rational method of determining the stress. The flat plate formulas would apply only to the simple plate piston, which is u 5 FIG. 350. rarely used. The piston has been considered as a beam supported at the centers of gravity of the two parts on either side of a diameter, by one of the leading manufacturers, with the maximum piston load applied at the center. Possible rupture would occur along a diameter, which would be resisted by the weakest section. This would be through the cored holes of the box piston, the plugs not adding to the strength. This method may be used as a check and will be given. Fig. 350 is self-explanatory. From mensuration: L = D* 2 12a and as: a = I = 37T PISTONS 517 From the beam equation: WL. Sz = - - > and as W = 4 4 Sz 12 (1) FIG. 351. /Si may be found for a given design, or the necessary value of z for a given stress. In most cases the section may be put into a simple form as in Fig. 351, omitting the follower and bull ring from the built-up piston. The center of gravity may be found by calculation, locating the neutral axis xx] the moment of inertia may then be found and divided by the distance of the extreme fiber from the neutral axis, giving the minimum value of z, and this must be used in (1). For the conical piston, Fig. 352, an empirical rule, based in part upon a flat plate formula, is: / ^ (2) \ For cast iron, if S = 1200 and T = 0. For steel casting, if S = 5000: T = 0.023 Vp5 T l may equal 0.5 T. Using the factor 7.2 for cast iron, as mentioned in connection with Table 72, Chap. XXI, the working stress may be 2220 Ib. when the ultimate strength is 16,000 Ib. The hub into which the piston rod is fitted will be considered under the subject of piston rods. Trunk pistons will be discussed under cross-heads. The length of piston F must be such as to allow of sufficient strength 518 DESIGN AND CONSTRUCTION OF HEAT ENGINES and to give ample wearing surface. For constructional reasons, F is proportionally larger in small than in large engines. For steam engines, an empirical formula giving satisfactory results is: F = f + 2" (3) 174. Piston Rings. Numerous varieties of piston rings are on the market, but the most common is some form of spring ring, so called because it is made larger in diameter than the cylinder bore, and when in place, exerts a pressure upon the cylinder surface. In bull-ring pistons this pressure is augmented by springs placed between the packing ring and bull ring at intervals, as shown in Fig. 360. When flat springs are used, lugs are cast on the ring to keep them in place. Such rings are usually cut at an angle of 45 degrees and a "keeper" provided to prevent leakage, as shown in Fig. 353. \ FIG. 353. FIG. 354. Rings providing all of their own spring are called "snap rings;" these are either made of uniform thickness or eccentric, the latter ap- proximating a form which will remain circular under uniform pressure at any diameter which does not strain the ring beyond the elastic limit. Two methods of cutting these rings are shown in Fig. 354. Some authorities claim that the ring of uniform thickness is superior to the eccentric ring; they are more common, possibly because cheaper to construct. The design of the snap ring is often empirical, but a fairly satisfactory rational method is not difficult and will be given. The eccentric ring best lends itself to such a discussion. The ring must be designed so that it will not be stressed above its elastic limit when in place in the cylinder, or while sprung over the edge of the piston while placing in the groove. The latter process is done but seldom and the stress may be higher than the working stress, which, however, is constant and may be high. Figure 35.5 shows the ring in place, Fig. 356 shows it free, while PISTON* 519 Fig. 357 shows it being placed in the groove. A somewhat smaller opening would suffice for the last, but the assumption made is safe and simplifies calculation. FIG. 355. FIG. 356. In Fig. 355, the force P (which is the normal pressure per sq. in. multiplied by the projected area in angle 2 N v\^ : XV x'''^ * K P-- 1C >1 \\ 1 \\ ll \\ N\ // FIG. 357. 20 degrees until the maximum thickness at 180 degrees is reached. In each case P acts midway between the cut and the section x. In this dis- cussion the radius will be taken to the outside of the ring instead of 520 DESIGN AND CONSTRUCTION OF HEAT ENGINES at the neutral axis of the sections, involving a small error; the bent beam theory is also ignored. In Fig. 355: P = p N w- 2r sina = 2p N wr sin a (4) Taking moments about x (any section) : M = Pr sin a = 2p N wr 2 sin 2 a (5) Also at any section: V * and: . "Z From the general equation for change of curvature from free radius n to radius of cylinder bore r, substituting M and / from (3) and (4) : 1 1 M 2S ^~^ = EI = Ei = C nstant (8) That is, the same relation must hold for every section. Assuming the stress developed by placing the ring in the groove (Fig. 357) as kS; : \ ~ rTfc = = constant : y (9) Taking S and t at the maximum section (opposite the cut) and equating values of 1/ri, from (8) and (9) gives: = Kr (10) Equating (5) and (6) when 2a = 180, t = t M and S = S M , and solv- ing for p N , which is constant, gives: 2 /-,i\ Value of t at any Section. Substituting (5) and (7) in (8) when 2a = 180, and again when it equals any angle, gives: 1 l_ 24p N r* 24p N r* sin 2 a r r, ~ Et M * Et* from which: t = t M \/^^ (13) Table 87 is given to facilitate calculation. PISTONS TABLE 87 521 2a sin a sin 2 a \/sin 2 a 20 0.17365 0.0302 0.3112 40 0.34202 0.1172 0.4892 60 0.50000 0.2500 0.6300 80 0.64279 0.4130 0.7450 100 0.76604 0.5870 0.8375 120 0.86603 0.7500 0.9085 140 0.93969 0.8830 0.9595 160 0.98481 0.9700 0.9900 180 1.00000 1.0000 1.0000 It is usually accurate enough to draw a circle touching as many of the points found by (13) as possible. From (8), conveniently taking S = S M and t = t M , the radius of the ring when free is: r , = _ = - = mr (i4\ 1 *? ^f 9 ^f / ~r~~ W M ~ ~KE Length of Cut. Referring to Fig. 354, the length cut out of the ring after it is turned to radius TI is: 2?r(ri r) This allows it to fit in the cylinder when finished, after which, a small amount c may be cut out; the total cut is then: I = 2rr(ri - r) + c = 27rr(m - l) + c (15) The amount c cut out after the ring is turned or ground to exactly fit the cylinder may be taken as: c = 0.006D + 0.02" (16) which is an empirical formula. Maximum Stress. For a ring of given dimensions the maximum stress in place is, from (8) : KM ft ti (17) And when being placed in the groove: I 1 (18) To assist in calculation, or to determine different relations approximately, Table 88 has been calculated for eccentric rings. The stresses are high, but for the grade of cast iron used for rings, are not excessive. 522 DESIGN AND CONSTRUCTION OF HEAT ENGINES TABLE 88 a+*>? ' K If E = 12,000,000 and AC = 1 .2 SM icSAf PN m 0.000915 0.0436 5,000 6,000 0.79 .0194 0.001095 0.0480 6,000 7,200 1.15 .0212 0.001280 0.0519 7,000 8,400 1.57 .0228 0.001465 0.0555 8,000 9,600 2.05 .0246 0.001650 0.0592 9,000 10,300 2.63 .0258 0.00183 0.0623 10,000 12,000 3.23 .0270 0.00201 0.0654 11,000 13,200 3.92 .0290 0.00219 0.0684 12,000 14,400 4.68 1.0300 0.00238 0.0714 13,000 15,600 5.52 1.0310 0.00256 0.0741 14,000 16,800 6.41 1.0325 0.00274 0.0767 15,000 18,000 7.35 1.0335 0.00293 0.0795 16,000 19,200 8.43 1.0347 0.00311 0.0819 ( 17,000 20,400 9.51 1.0361 Rings of Uniform Section, although cut, sprung together, and turned to fit the cylinder, do not exert a uniform pressure. To do so, r, p, E and t must be constant, and (12) gives: 1 1 Et* a - q snv 1 - 24 PAT sin 2 a K*E - 1 - sin 2 a (19) KE The value of TI varies at every point, but Unwin gives an approximate method for finding the form of the free ring by assuming 7*1 the same over an angle of 30 degrees. Fig. 358 shows the method, in which ab f = aa' b'b" = a' a", etc. The distance ca, = r\ for the arc ab', c'b' for the arc b'b", c"b" for the arc b"b'" ', etc. The method is exact if these arcs are indefinitely small, so that greater theoretical accuracy would be obtained by taking a larger number of divisions; but the errors of draftsmanship increase with too small divisions however, offsetting the theoretical advantage. Large rings are sometimes hammered to this form, but more com- monly it is neglected. All rings, however, should be turned or ground to the diameter of the cylinder when sprung together. A common method of making small rings, whether eccentric or of uniform thickness, is to cast a pipe and cut the rings from it. A more refined way, and one which will not cut away the best metal, which is always nearest the skin, is to PISTONS 523 cast the rings separately, allowing but slight finish, then grind to size. Number of Rings. An empirical formula for width or ring is: w = 0.02D + 0.2" (20) which may be used as a guide. As the width increases but slowly with the cylinder diameter, it is customary to retain the same width for several diameters, advancing by sixteenths or even eighths of an inch for the larger sizes. The chart of Fig. 359 is plotted from (20) with the changes mentioned. "4 I" 4 ? Trim 1 i" 4 I I 10" 10" 30 ' FIG. 359. 4 50 If p is the maximum unbalanced pressure per square inch in the cylinder, the following relation may be of assistance in fixing conditions: p N wn = 40 from which: = JL (21) This is an empirical formula in which n is the number of rings. There is 524 DESIGN AND CONSTRUCTION OF HEAT ENGINES no definite rule for determining the number of rings. If p N is assumed, n may be found; if n is assumed, p N may be found, and from Table 88, K and m may be taken for determining the thickness and free diameter. It is obvious that great refinement in these selections is not necessary. It may be considered that n is 2 for steam engines, and should a single ring be used the width may be twice as great; otherwise the thickness will be excessive. As already stated, single rings are provided with springs, so that a thinner ring may be used, giving greater flexibility and a more uniform pressure against the cylinder wall. Internal-combustion pistons have from 3 to 10 rings, 3 being a com- mon number for small engines. Many Diesel engines have 6 rings. Rings must be fitted into the grooves so that they are free to adjust themselves. F. A. Halsey, in the Handbook for Machine Designers, says that a little side play in piston rings is desirable, and that the steam pressure will force the ring against the side of the groove, preventing leakage. DESIGNS FROM PRACTICE 175. Steam Engine Pistons. Fig. 360 shows a built-up piston used on the Bass-Corliss engine. A modification is made in the original design FIG. 360. Bass-Corliss piston. by making the follower bolts as described in Par. 173. While Babbitt metal is shown in the bull ring, this is not standard. Using the method of stress determination described in connection PISTONS 525 with Fig. 351, the stress from (1) is 2400 Ib. with 100 Ib. steam pressure. With 125 Ib. the stress is 2980 Ib. It is probable that the actual stress is not so high ; at -any rate this piston has been successfully used with a pressure of 125 Ib. But it is more satisfactory to have (1) give a lower stress, and for this reason Formula (3) was devised by the author and gives a greater depth. The change makes but little difference, however, in this particular design, still giving 2875 Ib. stress for 125 Ib. pressure. With the ultimate strength of cast iron 16,000 Ib., this gives a factor of safety of but 5.57, less than that suggested for repeated stress. The Cyl. Bore less JJM _ ' Nominal 5ore 20" FIG. 361. Mclntosh and Seymour piston. possible factor for reversed stress in cast iron suggested in connection with Table 72, Chap. XXI, is 7.2. With this as a basis the factor of judgment is 0.77, or less than unity, which, in view of the fact that the follower plate and bull ring were ignored in the calculation, may be reasonable. If the piston spider were a steel casting the factor of judg- ment is 1.74; it would probably equal at least unity for semi-steel or some of the better grades of cast iron often used. Unwin gives 3000 Ib. as the working stress in pistons. Fig. 361 shows a piston designed for Type F gridiron- valve engines built by the ]\JcIntosh and Seymour Corporation, Auburn, N. Y. It is a 526 DESIGN AND CONSTRUCTION OF HEAT ENGINES box piston containing no ribs, but having a wide face and rather thick walls. The holes through which the cores were held are drilled out and FIG. 362. Locomotive piston. tapped with a 2-in. pipe tap, plugged, and then these plugs are faced flush with the piston as it is finished in the lathe. FIG. 363. Locomotive piston. For strength calculations, this piston is composed of two flat plates; one of these plates has a central load, and this is resisted ai the edge by PISTONS 527 a portion of the pressure on the other plate transmitted through the cylindrical portion of the piston. The other plate, exposed to a uniform steam pressure has a central load, and at the edge a load caused by the resistance of the first plate to bending. Morley's Strength of Materials contains the elements of an analysis of this condition, and it may be H2 1 -4 pi T 1 i I 1 I 1 1 i i 1 I 1 1 Ol .i (L II ij 1 1 i 1 1 1 1 1 1 1 1 -M 1| P 1 i L,. ,__^..._ 9-- - >J V-.OOS'Smaller than Sore of : Cylinder - FIG. 364. Bruce-Macbeth gas engine piston. possible to combine them; it is probably not practical, however, and the design must be mostly empirical. Figure 362 shows a locomotive piston designed by the American Loco- motive Co. It is a conical piston with cast steel body riveted to a cast iron bull ring. With a pressure of 200 lb., a working stress of 5000 Ib. 528 DESIGN AND CONSTRUCTION OF HEAT ENGINES 5ech'onA-A FIG. 365. Franklin automobile engine piston. FIG. 366. Water-cooled gas engine piston. PISTONS 529 and using the mean angle (90 - 20.5), T equals 1% in. This is but J in. more than the dimension given. Another style of locomotive piston by the same builders is shown in Fig. 363. This is a box piston with ribs, the outer ends of which are cut away, allowing the core to extend around the piston. The core holes are filled by driving in rivets which are countersunk, and faced in finishing the casting. This piston is cast iron. From (1), the stress with 200 Ib. pressure is 1340 Ib. Using the standard factor 7.2, the factor of judg- ment is 1.66. If the standard factor 12 is used (see Chap. XXI), the factor of judgment is unity. 176. Internal-combustion Engine Pistons. A gas engine piston used by the Bruce-Macbeth Engine Co., Cleveland, Ohio, is shown in Fig. 364. There is little opportunity for strength calculation in such pistons, and the determination of their dimensions is largely a matter of judgment based on experience. The bearing surface must be sufficient to keep the bearing pressure within certain limits and this is considered in Chap. XXVI. In the larger engines running at slow and medium speeds, the inertia effect is not great, and sufficient thickness may be used so that ribs are not necessary. Figure 365 is the piston of the automobile engine built by the Franklin Automobile Co., Syracuse, N. Y. It is very fight, and ribbed for strength and stiffness. A helical groove is turned upon the surface for oil dis- tribution. There are practically no strength calculations which may be made for this piston. A water-cooled piston for a double-acting 'gas engine is shown in Fig. 366. CHAPTER XXIV PISTON RODS Notation. D = diameter of cylinder bore in inches. d = diameter of body of piston rod in inches. r = radius of gyration of rod section. ~* ti = thickness of wall of crosshead neck, considered as a thick cylinder. to = ditto for piston hub. p = maximum unbalanced pressure per square inch in cylinder. Pi = normal pressure per square inch in rod fit, due to taper or forcing. P = total maximum unbalanced pressure on piston. Also used for total ram pressure in making pressed fit. PA = portion of total piston load taken by taper when rod is in compression. P F portion taken by friction due to fit. P B = portion taken by shoulder. S T = tensile stress through key eye at diameter d z . S R = tensile stress at root of thread. S c = compressive stress in portion of key in rod, and in rod, due to key. SK = compressive stress at shoulder. $1 = hoop stress in neck of crosshead due to taper fit. S = hoop stress in hub of piston due to taper, or forced straight fit. Z = length of piston rod in inches, taken from center of piston face to center of crosshead pin. n = taper from axis in inches per foot. p = coefficient of friction. / = factor of safety. m = number of threads per inch. T = tons per inch of diameter, per inch of length, required to make a pressed fit the maximum, as fit is complete. TI = tons per inch of diameter required to make a pressed fit. 530 PTSTON RODS 531 177. Piston Rod Formulas. There are various formulas for deter- mining the diameter of piston rods, most of which take no account of the method of attachment, which is usually the weakest part of the rod. Some formulas consider the length, and while it is well to check the rod as a strut, in few instances will it be found weak in this respect when end fastenings have been properly provided for. Too elaborate an analysis may not be necessary, but from an examination of a number of piston rod failures, it is evident that certain important points are sometimes overlooked. There are many methods of fastening the rod to crosshead and piston. The best crosshead connection in the author's opinion is the taper fit and key when properly designed and constructed, with not too steep a taper, but one which will usually allow the removal of the rod without a hydraulic press. This also applies to the piston end when piston design permits of its convenient application, but as this is seldom so, the taper fit and nut may be used. The nut may be circular, partially or wholly concealed within the piston, and tightened by a special wrench, but the plain nut is simplest, and while it may increase the clearance volume slightly, the effect upon economy is negligible. Standard thread is too coarse for large diameters, so to facilitate tightening, a finer thread is usually employed; five or six threads per inch gives good results. Steep taper with no shoulder is sometimes employed, but the shoulder is preferable, as it definitely fixes the length of the rod. In some designs, as in Fig. 371, the end of the rod " bottoms" in the crosshead, an excel- lent design but more difficult to fit. A straight fit with shoulder and nut is sometimes used for the piston end, and is good practice; but a pressed fit with rod end cold-riveted is to be condemned as unreliable. As the formulas derived will cover most forms of attachment, the analysis first employed will be for a rod having a taper fit and key at the crosshead end, and a taper fit and nut at the piston end. It is assumed that when the rod is in tension the entire load is carried by the smallest rod section cut by the key at the crosshead end, and by the section at the root of the thread at the piston end. In compression, at both ends, the load is in part taken by the wedging action of the taper which is resisted by the hub considered as thick cylinder; by the friction of the wedging action; and the remainder by the shoulder. The crosshead end is shown by Fig. 367, all dimensions being in inches. The area wd 2 , subjected to crushing, is not more that one-half of the area 2wb, in shear; therefore, crushing will be considered and shear ig- nored in the present discussion. With low compressive stress, shearing is not likely to begin. 532 DESIGN AND CONSTRUCTION OF HEAT ENGINES For equal strength in tension and compression, for the most econom ical dimensions: or: Also: from which: w = S T S p(S FIG. 367. Substituting in the expression for w gives: From Fig. 367: irg 4 = d, Ifb r 6 v _di 1 - nx Substituting the value of d z just found, gives: J _ D lp(Sr + S c ) nx\ S T Sc -f (2) If a thrust P A will produce an allowable normal unit pressure pi: PA = -T-dilipi. PISTON RODS 533 The additional thrust which may be resisted by friction under pressure pi is: P F = Tr Then: PA + PF = irdilipi ( y^ + /*) If this is equal to P, no shoulder is required, but if less, the remainder, P B , must be carried by the shoulder. Letting n/12 -f- M = k: P B = p - (p A + p F ) = ^(D*p - 4diZifcpi). Also: Equating these two values of P B and solving for d gives: It will be found convenient to determine PI for a given value of /Si, the allowable stress in the crosshead neck. From (16) and (17), Chap. XXI, the thick cylinder formula for free ends is: + 0.7pi , - 13pi or: Pi =- 2/ - Si = gSi (5) 1.3(-^ + 1) + 0.7 Substituting (5) in (3) gives: d . p 2 ?> - y.t.*ga. + dl . (6 ) \ ^>X If \/p (16) di = 0.0177>\/P (17) w = 0.00524Z)y^ (18) li = 0.046Z)Vp (19) The minimum value of e may be O.Gdi, but l\ must not be decreased without checking by (6). The distance e + 6 must not be greater than the value used to determine di and d 2 . It is well to make b ^ di. From (12) : Zo_>S i _2500 li ~ So "4000 " If 1 ^ 0.625/1, no collar will be required, and d may equal di as in Fig. 369, if this value is not less than that given by (14). Then from (14) : d T > d - - 0.125 = d ~ + 0.125) (20) Also from (13) : d T ? O.OllD-v/P + 0.216 (21) The greater value of d T must be used. It is usually preferable to have: d ^ di The piston rod as a strut may be checked by Formula (36), Chap. XXI. The rod is not fixed at the ends as this would involve the binding of crosshead and piston between guides and cylinder walls, but the large bearing surfaces tend to keep the ends of the rod in line, and for safety it may be considered as a pin-ended strut; it probably is between a pin end and a flat end in strength. Formula (42), Chap. XXI may then be used as the special formula, taking I from the center of the piston face to the center of the crosshead pin. Solving for d gives (where d = 4r) : ' ~~ The factor of safety /is made up of the standard factor for reversed load, and a factor of judgment of from 1.4 to 1.6, due to the possibility of PISTON RODS 537 some eccentricity from adjustment of the bull ring. Taking / as 9 for double-acting steam engines, (22) becomes: (23) If this value is greater than that given by (6) or (16), it must be used; but usually it will give a smaller value and may be used when lightness is especially desirable, or for short-stroke engines, in which case the rod ends must be proportioned as already explained, resulting in a collar at each end. The factor f^ T for double-acting steam engines, for parts loaded as the piston rod body, is also given as 6 in Par. 166, Chap. XXI, making the factor of judgment 1.5 for the value just assumed. Example. As an example of design, a rod will be proportioned for the 20 by 48 in. Corliss engine of Chap. XII, the unbalanced steam pres- sure being 125 Ib. per sq. in. From (16), d = 4.225, say 4^"- From (17), di = 3.96, say 4". From (18), w = 1.162, say 1% 6 ". From (19), Zi = 10.29, say 10^". Also e = 3.175, say 3", and b = 4". In rounding up fractions it is safe never to make di less than the value given by the formula, while e + b must never be greater. If it is desired to save space, e may be made % of the calculated value, especially with a steel crosshead. From Chap. XXIII, the piston face is 7 in. (= 1 ), which is greater than 0.625Zi, and no collar will be required. Then d = di. From (20), d T = 3.29, say 3% in. From (13), d T = 2.716 in., so that 3J4 in. from (20) is safe. Following tentatively through the design of the details involved, the length of the rod I is about 86 in.; then from (23), d = 3.82 in., showing that the rod is safe as a strut. The above calculations are for open hearth hammered steel with an elastic limit of 38,000 Ib.; for other steels the general formulas must be used, and working stress substituted according to the judgment of the designer. 178. Pressed Fits. Straight fits are sometimes used on one or both ends of the rod. These vary from a sliding fit to a forced fit capable of withstanding the entire load, and in some cases, with the addition of riveting over the end, the load is so carried. The riveting, however, of cold steel adds but little to the safety, and in general the method is 538 DESIGN AND CONSTRUCTION OF HEAT ENGINES untrustworthy. If supplemented by nut or key, straight fits of various degrees of tightness are satisfactory when once assembled, but are difficult to take apart if forced. A simple method of computing the stresses due to forced fits, and perhaps as reliable as any, is to assume a coefficient of friction, and determine the relation between ram pressure, thickness of hub wall, normal pressure, and stress resulting therefrom. The method was in part covered by the taper fit analysis already given, and the notation already applied to the crosshead end will be used, di being the uniform diameter of the fit. P will also be used for total ram pressure and T for tons per inch of diameter per inch of length of the fit, due to P. Then : P = 20007Wi = irdilipm (24) From which: T = P = ^^ (2V 2MOdili 2000 From (5) and (24) : P 20007 7 , _ N pi = -yy- = - - = qSi (26) TTttiti/i 7TJU in which, from (8) : *- '" 1 < - (27) +0.7 From (26) : p 1= 2000T = P q Si ' ' Then from (9) : di These equations give the relations mentioned and are general; they will be referred to in other chapters. Taking P as the piston load and writing di = KD, (25) may be written : T - Dp (30) ' 2550KI, Or, in tons per inch of fit diameter: T, = TL = P (31) f\ p e f\ TS~ vf * / From (31) and (26) : PISTON RODS 539 It is obvious that TI is not the measure of stress in the hub, but it is nearly always the quantity given in limiting pressed fits. If li is constant, TI varies directly as the cylinder diameter, but this is usually not the case, Zi usually changing with the cylinder diameter, although sometimes not at the same rate. While it is proper to use a pressed fit for piston rods, it is safer not to count on their holding power, but to provide a key or nut, with a collar to take the entire load. An example will illustrate. Assume a 20 in. engine with 125 Ib. steam pressure, and for the piston end, changing sub- scripts as in the previous paragraph, let d = 4 in. and 1 = 7 in. Assume a stress of 4000 Ib. in the piston hub and a coefficient of friction of 0.25. Let the outside diameter of the piston hub be 8 in. From (27), q is 0.509; from (26), p l = 2036. From (25), T = 0.8 and from (24), P is 44,800 Ib., the ram pressure required. The maximum piston load is 39,250 Ib. The ram thrust is some greater, but as this value is allowed some fluctuation, it may be assumed as about the same in this particular case. If the fit is to carry the entire load there must be a factor of safety and the increase of stress will be proportional thereto. While the piston walls and ribs add to the strength of the hub, the uneven distribution of metal and the possibility of shrinkage strains may result in cracks if the stress is assumed much greater than would be permissible in the plain hub if there were no ribs. With a factor of safety of only 2, the stress in the hub would be 8000 Ib., which is excessive for cast iron. To retain the stress at 4000 Ib., T being 1.6, and from (26), q being 1.018, the thick cylinder formula fails, showing that a solid piston would not permit as low a stress. The situation is not much relieved by assuming as high a coefficient of friction as 0.4, which is unwarranted, as the surface is always lubricated. It looks as if the margin of safety must be provided by undue stress and cold riveting, when no other fastening is used. It seems more rational to allow a reasonably snug fit and not to depend upon it for axial load; then d c (Fig. 368) may be determined from (11), in which k is zero for a straight fit. The diameter of the threaded portion may be determined from (14), taking n as zero and checking by (12a) and (13). The rod is sometimes fastened to the piston with a straight fit and nut, and screwed into the crosshead, where it is held by a lock nut or clamped by splitting the crosshead neck. As the ends are usually safe for this design if the rod is designed as a strut, Formula (23) may be used, and if it is desired that the diameter shall be the same for all strokes, the maxi- mum stroke may be assumed. As an example, let a rod be designed for the 20 by 48 in. engine of the 540 DESIGN AND CONSTRUCTION OF HEAT ENGINES preceding paragraph. Using the same data as far as it applies, I = 86 and d = 3.82, u say 3J in. Calling this d c in (11), letting k be zero and solving for d 0) the diameter of fit, gives: d = 3.162, say 3J in. The threaded end may be 3 in., which is greater than the value given by (13). As all the load is carried by the shoulder and nut, a driving fit, or even a snug hand fit will be ample. It may be considered advisable to make the fit larger and use a collar. If the fit is the same diameter as the rod, the diameter of the collar may be found from (11), or, d c = 4.47, say 4J<2 in. The thread may be 3% in., and may be the same at both ends of the rod. TIG. 369. A small amount may usually be removed from the rods just designed, for the purpose of truing up when worn out of round or scored, so it may be advisable to add from J{g to }/ in. depending upon the rod diameter to the diameter of the body of the rod for this purpose, es- pecially if the calculated diameter is not greater than given by (23). In any case, the shoulders must not be reduced by turning, to values less than those given by (6) and (11). 179. Designs from Practice. Fig. 369 shows the crosshead end covered by Formulas (16) to (19), while Fig. 370 shows the screwed end. These have been used by the author on Corliss engines for a number of years and have been entirely satisfactory. In determining the diameter of the screwed-end rod, Formula (16) was used for the sake of uniformity; this usually gives a larger rod than is necessary for the PISTON RODS 541 strength of the crosshead threaded end, and if the screwed end is to be standard construction, (22) may be used, and if it is desired to use the same diameter of rod for all strokes, it may be computed for the maximum stroke as previously stated. FIG. 370. With the smaller rod it may be necessary in some cases to use the collar at the piston end, shown in Fig. 368, in order that d s may not be too small. This may readily be checked by (11), and the diameter of the threaded end by (21). FIG. 371. Locomotive piston rod. Fig. 371 shows a locomotive rod used by the American Locomotive Co. A stress of 9500 Ib. in tension is allowed at the least area through the key way. Considered as a static stress, this gives a factor of judg- ment of 2 if the elastic limit is 38,000 Ib. Instead of a shoulder, this rod FIG. 372. Mclntosh and Seymour piston rod. ''bottoms" in the crosshead. A tail rod is provided which connects to a small crosshead whose guide is a projection from the cylinder head. The purpose of the tail rod is to relieve the pressure of the piston due to its weight, and prevent wear in the cylinder. 542 DESIGN AND CONSTRUCTION OF HEAT ENGINES A rod used by the Mclntosh and Seymour Corporation on their Type F, Gridiron-valve engine is shown in Fig. 372. The rod is screwed into the crosshead and held by a lock nut. The crosshead for the same engine is shown in Chap. XXVI, and the piston in Chap. XXIII. The fit in the latter is a straight hand fit, a clearance of 0.002 in. being allowed. This fit only extends a short distance from shoulder and nut, the interme- diate portion being turned slightly smaller. A special nut is used, and \ I n 1 1 o| H FIG. 373. Mclntosh and Seymour piston rod details. is sunk into the piston and tightened by a special wrench. The wrench, nut and locking collar are shown in Fig. 373. The collar is split, and expanded into the counterbore of the piston by a taper set screw. The rod shown is for a 20 in. engine; its dimensions may be compared with those of the examples given, the length being practically the same. Rod and nut are steel forgings, the wrench a wrought iron forging and the lock- ing plate cast iron. A tandem compound rod is very similar to the rod shown in Fig. 371. PISTON RODS 543 It may be designed by the foregoing methods, the main rod taking the maximum thrust of both pistons, and the tandem portion the thrust of the piston farthest from the frame. The low-pressure cylinder is com- monly next the frame, but not always; by this arrangement, the main rod affects the effective piston area less. The strut length may be taken as before for the main rod, and from center to center of piston faces for the extension. If an intermediate crosshead is used, measurements will be to the center of its pin. The rod may also be screwed into the crosshead, and straight piston fits may be used. With some large tandem steam engines, and with most large gas engines, an intermediate crosshead is used. For gas engines the rod is made hollow for water cooling, but the same general principles of design ajjply as already given. CHAPTER XXV CONNECTING RODS Notation. D = diameter of engine cylinder in inches. d = diameter of round rod or depth of section in inches, in plane of vibration. 6 = width of section in inches. I = length of rod from center to center of pins in inches. R = radius of crank circle in feet. L = stroke of piston in inches. A = area of rod section in square inches. / = moment of inertia of rod section about axis perpendicular to plane of vibration. r = radius of gyration of rod section in inches. This is the radius of gyration used in the strut formula, and does not necessarily correspond to the value of / just given (see Formulas (6) and (9)). c = distance from neutral axis to extreme fiber (= d/2). W = weight of rod in pounds. w = weight per cubic inch of rod material. Wi = weight per inch of length of rod. M = bending moment in pound-inches. F force in pounds, due to angular vibration, tending to bend rod. P = total force exerted by piston in pounds. p = unbalanced pressure per square inch in cylinder. p s = ultimate strength of rod as a strut. S E = elastic limit of rod material. S = stress in rod due to bending. S R = stress resulting from bending and direct load. a = acceleration at x normal to center line of engine. V tangential velocity of crank pin in feet per second. .AT = revolutions per minute. n = ratio of length of connecting rod to radius of crank circle (= 1/12R). q = a constant in strut formula which varies with end conditions (see Chap. XXI, Par. 164). 544 CONNECTING RODS 545 f s = factor of safety for angular vibration. f p = factor of safety for strut load. e = ratio of width to depth of rectangular section ( = b/d). 180. Body of Rod. Empirical formulas are often used to determine the dimensions of the body of the connecting rod. These formulas are suitable within a certain range of conditions under which they have proven satisfactory in practice, or have been checked by a more complete analysis. While too elaborate an analysis is neither necessary or desirable, a safe rational formula must take account of the combined effects of the strut load and the vibration, or whipping action. While methods of checking for stresses so produced are given in certain texts, the direct solution is omitted. By using Johnson's strut formula discussed in Chap. XXI, the diameter or depth of the rod maybe solved for, and by numerical substitutions a reasonably convenient working formula derived. In the derivation it is assumed that the maximum sum of the unit load at or near the center of the rod due to the piston thrust, and the compressive bending stress due to whipping, must not be greater than the safe unit load allowed by the strut formula. This may not be strictly scientific, but it seems reasonable, and checks well with more exact analyses of combined compression and bending. It is further assumed that the section is uniform throughout the length of the rod. This is no doubt always on the safe side and it greatly simplifies matters. v*- FIG. 374. Fig. 374 is a sketch of a simple slider-crank mechanism, showing the outline of the rod and certain notation used in the discussion. From Formula (31), Chap. XVI, the acceleration at right angles to center line of engine, of any point, is given by: V 2 x a --g- jane (1) The forces resulting from this acceleration will be assumed to produce a bending moment in the rod, but will be applied to the actual rod length and not to its projection upon the center line. No error of practical 35 546 DESIGN AND CONSTRUCTION OF HEAT ENGINES significance is involved by this assumption. Then the force acting per inch of length is: MI V 2 x . . ' -D- ' T ' sin 0. g R I As Wij the weight per unit length is constant, the total force for a given value of B is : sin = WV 2 2gR sin 6 (2) -,--J 1, J From (1) it is obvious that the load F due to inertia is distri- buted along the rod directly proportional to the distance from the center of the crosshead pin, as shown by the load dia- gram in Fig. 375. The distance of the center of gravity from the center of the crosshead pin is 2/31', this is the center of oscil- lation. The reaction on the crosshead pin is F/3, and on the crank pin 2F/3. More exact values may be obtained by the method of Chap. XVI. The general formula for bending moment is : M = moment of reaction S moment of loads. Then for the crosshead pin reaction, taking moments about any section distance x from the crosshead pin : FIG. 375. F _ wi x 3 ' x " its maximum value; or a somewhat larger value may be used to cover all cases. The unit load on the rod as a strut is: mP _ irmpD 2 The unit stress due to bending is: s = *f- c . . ;' Johnson's strut formula for ultimate unit load from (31), Chap. XXI is: Ps ^E ~ The whipping action always produces reversed stress with a factor of safety f s ; while for direct load, the stress is partly reversed with single- and double-acting engines. With internal-combustion engines the pressure factor affects the factor of safety and this is explained in Par. 166, Chap. XXI, which it is well to consult. It is then well that a separate factor of safety f P be used for piston thrust, making it more convenient to use ultimate loads and stresses in the general equation, which, ac- cording to our assumption is: Ps > f s S + fptn -j (5) From Table 82, Chap. XXI, it may be assumed that for ordinary cases the factor of safety for double-acting engines, both steam and gas, may 548 DESIGN AND CONSTRUCTION OF HEAT ENGINES be 6, and for single-acting engines 3. This factor 3, however, is the product of the standard f actor f A and pressure f actor f T of Chap. XXI, and it is based upon the maximum steam or gas pressure being used to deter- mine the load. Should determinations be made for rod dimensions at different crank angles for internal-combustion engines, using actual pressures (including inertia of reciprocating parts), the proper standard factor f A (Chap. XXI) should be used instead of / P , and this will depend largely upon the inertia as may be seen in Table 16, Chap. XXI. For ordinary work/ P may be taken as 3, and P as the maximum total pressure, determining the dimensions when the crank is 90 degrees from line of stroke, as with the steam engine, in which maximum thrust and maximum whipping action occur when in this position if cut-off is as late as one- half stroke, and engines are usually so designed for over-load purposes. Substituting in (5) the values already given, we have: I WV 2 c irmpD 2 '' For any : form of section, Where d is the diameter or depth of section, we may write: r = ad, A = 5d*, I = <7d 4 , c = g- Also : L 7r 2 #W 2 LW 2 7 , nL w 1A wdnLd 2 '' == 12nR - > F := wlA - 520 Substituting in (6) and solving for d gives: Trmf P pD nL ^ ^^ " (7) Where: = 17,440,0005^ SU The moment of inertia / should be taken with the axis at right angles to the plane of vibration. From the strut formula, it is obvious that the resistance to failure is least when : ^ = maximum (9) If r is about an axis in the plane of vibration, the end conditions may be for a flat-ended strut in high-grade work, but more safely for pin ends. If r is about an axis at right angles to the plane of vibration, a raund- ended strut should be taken, as the pins are in rotation and of no assist- CONNECTING RODS 549 ance in guiding the strut; in fact, large pins, poorly lubricated, may produce a bending stress. The length of connecting rods ranges from 4 to 6 times the crank length, the smaller values being used for gasoline engines, and the larger for horizontal stationary steam engines. Numerical Values. Usual practice is to make the connecting rod of an open hearth steel forging; then S E = 38,000 and w = 0.284. Since the whipping action may be estimated with reasonable accuracy, the factor of judgment may be unity; then f s = 6. For double-acting engines, f p = 6 as already stated, and for single acting engines f P = 3 for usual work. To provide for a slight eccentric load due to movement of pins in bearings, we may make m = 1.1, which is larger than any value of I/cos found in practice. Then by substituting values of a, d and 2 = l2 550 Then: or: DESIGN AND CONSTRUCTION OF HEAT ENGINES 1.4 = 0.63 d 2 = b 2 e = , = 0.67. a Then if e < 0.67, the rod would fail about axis yy, and as this would usually be true: a = 4= and q = 0.63. V12 As before, substituting in (7) and (8), special formulas for rectangular sections may be derived. Then : K = 32,400,000,000 And: 43,900e (13) For an I section, definite proportions must be assumed in order to de- rive a special formula. By trial, a section may be selected which will give a theoretical maximum of strength per Ib. weight, but too light a sec- tion may have a tendency to buckle or twist, even though the usual beam and strut formulas may indicate sufficient strength. Thin flanges reduce the value of r about the y axis, which must be the axis used for this -..-^^.^ section in selecting q and r. A rather rugged section is given in Fig. 377, for wh^ch a special formula will be derived. For this section : d = 0.375, o- = 0.039, a =0.125, q = 0.63. Then: And: Fig. 378 gives a number of connecting rod / sections used on locomo- tives. Applications of Formulas (11), (13) and (15) will be made in the follow- ing examples: (1) Find the diameter at center of a round rod for a 20 in. by 48 in. Corliss engine, running 100 r.p.m., the rod being 6 cranks long. The CONNECTING RODS 551 maximum unbalanced steam pressure is 125 Ib. per sq. in. From (10): 6 2 X 1002 X 48 3 K = From (11): d = 1.63 +A/1.< 24,300,000,000 1.63- , (6 X 125 X 400) + (5.1 X 36 X 48*) a _ . = 6.5 m. 34,500 Trooien's Formula, Bulletin of University of Wisconsin, No. 252, taking no account of speed gives 5 in. For 75 r.p.m., a more usual speed for the older Corliss engines of the same size, (11) gives 5% in. diameter. The above shows that formulas based upon former practice, taking no account of speed, are not generally applicable. It is also apparent that for higher speeds a rod of round section is a little bulky. (<--- ST T I N. I I Sack End Front End Main Roof FIG. 378. (2) Find a rectangular section for the same data, taking e = 0.45. From (12) : From (13): d = 1.23 + 6 2 X 100 2 X 48 3 32,400,000,000 (6 X 125 X 400) + 2.18 X 36 X 48 2 0.45 7.33 in. 43,900 X 0.45 This gives a section 73 percent of the area of the round one, but still rather heavy. (3) Find the depth of an I section for the same data, of the form of Fig. 377. From (14) : 2 X 100 2 X 48 3 K From (15): d = 0.985 40,400,000,000 = 0.985- . QS , 2 , (6 X 125 X 400) + (4.37 X 36 X 48*) _ . ' 985 16,500 552 DESIGN AND CONSTRUCTION OF HEAT ENGINES This gives a section about 56 per cent, of the area of the round section and 77 per cent, of the rectangular section. (4) Find the section at center of a gasoline engine rod. The engine is 3M by 4 in., running 1500 r.p.m. and the value of n is 4. The section is similar to Fig. 377, but with the following values: a = 0.224, 5 = 0.379, a = 0.0295, q = 1.4. The combined diagram of Fig. 187, Chap. XVI is used to determine p, and d will be calculated for several positions from (14) and (15). As actual pressures are used, the pressure factor of Par. 166, Chap. XXI is not re- quired. According to the formulas of this paragraph, the standard factor for this case is 3.5, and this is taken as/ P in the formulas. The ratio g/r 2 of the strut formula is greater about the x axis, so is used. Maximum pressure occurs at position 1 of the crank, which is 15 degrees from the head-end dead center. The depth of section for this position is 0.828 in.; at position 2, d is 0.739; at 3, it is 0.669. It is therefore a maximum in this case where the pressure is a maximum, the whipping action having less effect than the decrease in pressure as the piston moves away from the end of the stroke. The actual depth of a rod for an engine of the same size and speed is 0.794 in., the calculated value being a trifle over 4 per cent, greater. Taking the approximate method previously mentioned, with the method of Par. 166, Chap. XXI, and using the factor of safety given in Table 82 of the same chapter; then assuming the maximum whipping action with the crank 90 degrees from line of stroke, (14) and (15) give d = 0.883 in. This is on the side of safety and a little over 11 per cent. greater than the depth of the actual rod. In this method it was assumed that p = 335, the maximum gage pressure, and //> = 3. While being fairly rational and providing for the principle straining actions, Formula (7) gives results agreeing with practice. 181. Connecting Rod Ends. Where the body of the rod joins the "stub end" there is direct repeated or reversed stress as at the center of the rod. There is also bending stress due to vibration and this may be more than is commonly supposed, although often neglected. Formula (2) may be written: Were the rod of uniform section, one-third of this would react approxi- mately normal to the rod at the crosshead end, and two-thirds at the crank end. The reaction, especially 'at the crank end, may be greater CONNECTING RODS 553 than this, although in part due to a heavy stub end which is partially balanced and may exert but little bending moment. It is presumably safe to assume a uniform section unless the desire for extreme lightness necessitates greater refinement, in which case the prin- ciples of Chap. XVI must be resorted to. The weight is then: W = 0.142rcLA (17) where A is the area of section at the center of the rod. Taking Fig. 375 as a load diagram and using the notation of Fig. 379, the bending moment at dimension di for the crosshead end is given by: At the crank end it is: 2Fhr. 1 1( h.-t = -i Vi 5} 18 5 6 6 6 20 5K *x 6H &y 2 22 6 iy* 7M 7M 24 W 8 8 8 26 7 8H 8>i 8>i 28 7H 9 9 9 30 8H 10 . 10 10 The pressure p ( = 125) may be considered the standard pressure acting in standard cylinders as explained in Par. 63, Chap. XII. From (3), d = Q.313D S . The dimensions of the crosshead pin are also given, based on a pressure of 1200 Ib. per sq. in. of projected area. The length is taken equal to that of the pin, as explained in Chap. XXVI; the diameter is then % the diameter of the crank pin. Figure 395 and Table 92 are for standard crank pins used by the author for Corliss engines. The notation only applies to the figure and table should it disagree with that used elsewhere in this chapter. While these tables were used for several years on successful engines, they are given mainly to illustrate convenient methods of arranging de- sign data. The selection of dimensions for other steam pressures is ex- plained in Par. 63 ; Chap. XII. CRANKS TABLE 92 577 d, in. di, in. 6, in. dz, in. 2i, in. c, in. ds, in. Oil holes 3K 4% ^ 1 IK IK K i-K" 4 5K % 1 IK IK H i-K' r 4K 5% K IK 1M l^ % i-K" 5H 7K H IK 1% 2 % i-K" 6 7% i IK 1% 2K H i-% r/ 6K 8K IK 1% 2K 2^ i-%- 7K 9% IK IH 2K 2^ ^ i-%" 8 10H IK IK 2K 3 H i-K" 8K 11 iH 1 2K 3K K i-K" 9 n*i IK 1% 2K 3% K 2-%" 10 13 1% i 2^ 3^ K 2-%" The method of finding the mean pressure per sq. in. of piston area, P M acting on the pin is described in Par. 104, Chap. XVI. For steam Shaft Side ofP/n on Dead Center FIG. 395. engines this is practically the m.e.p. For engines with long-range cut-off it approximates the maximum pressure in the cylinder and may be safely taken as such. Adopting the notation of this chapter, Formula (1), Chap. XI may be written: T\9r> jAr (9) The value of C is taken from Table 93 of the same chapter. Taking this value and solving for JV, the corresponding rotative speed may be found. 37 578 DESIGN AND CONSTRUCTION OF HEAT ENGINES With p = P M = 125, P = 1000 and k = 1; the values used for Table 91 substituted in (3) and the value of d so found substituted in (9), we have: _ 4.3 C fkp _ 2430 D s L V T- -rxA / T-k (10) TABLE 93 Di, in. N Di, in. N Di, in. .V 2 1220 12 202 32 76 3 810 14 174 34 71 4 610 16 152 36 67 5 487 18 135 38 64 6 405 20 122 40 61 7 347 22 110 42 58 8 304 24 101 44 55 9 270 26 93 46 53 10 243 28 87 48 50 11 221 30 81 50 48 It is interesting to note these results in Table 93, which compare favor- ably with conservative speeds often used. Higher speeds than those given in the table have been used for large engines, but it is stated in Par. 52, Chap. XI, that the value of C, which was taken as 200,000, may be doubled with excellent design and conditions of operation. Internal-combustion Engine. For these, the mean pressure on the pin must be used for P M in (10), and this depends upon the indicator and inertia diagrams as already explained. Gtildner assumes for his standard diagram 128 Ib. for the expansion stroke, 18 Ib. for each idle stroke (exhaust and suction) and 32 Ib. for the compression stroke. The mean of these is 49 Ib. The value of C from Table 19, Chap. XI is 90,000. Taking k = 1, P = 1400 and p = 400, (10) becomes: " 4220 D s Then from (3) for the same assumptions: d = OA73D S (12) Guldner takes S as 12,000. For the factor of safety already used S e must be 50,000 if / 3 = 1, or 63,000 if f 3 = 1.25. With other data as be- fore, this gives, from (8) : rs = W~: ,000 X 1400 = 1.3 nearly. CRANKS 579 and from (3) : d = QA15D a (13) This increases the allowable value of A 7 " by 14 per cent. From (6), the value of S when d is as given by (12) is 7150 practically the same as in Table 90 when / 3 is 1.25. The best value of S may be determined only when the material is known. It is better to take the value of P M from diagrams drawn for a particular design at least for a particular type. By using a greater value of P the ratio k is reduced, decreasing the moment arm for the crank and shaft. If the product kP is constant, (3) shows that the pin diameter is not changed. It may also be seen from (8) FIG. 396. that this makes PS constant, and these relations may be convenient in determining conditions. In all pin design there should be a fillet where the pin changes diame- ter. If the pin is cast integral with the crank there should be a fillet where they join. Crank Arm. A crank arm of substantial design is shown in Fig. 396. This is a plain crank having all the requisites to transmit power from the crank pin to the shaft. Sometimes a counterbalance is added, and sometimes a disc crank is made by adding rim and web, but these should not be depended upon to add to the strength of the crank and may be omitted for the present discussion. In a crank of this general design the greatest liability to failure is 580 DESIGN AND CONSTRUCTION OF HEAT ENGINES through the pin hole when the crank is on one of the dead centers, and this will first be considered. Assuming a perfectly rectangular section, the bending stress is: The direct stress is: SD = 6f These act at the same time at the dead center, P P being the total force in- cluding steam or gas pressure and all inertia effects. The sign is plus for tensile stress and minus for compressive. The total stress is : At the beginning of the stroke both signs are minus on the pin side, or face of the crank, and for the back side the first sign is plus and the second, minus. At the end of the stroke both signs are plus on the face, while on the back the first is minus and the second plus. From the equations of Par. 166, Chap. XXI, the worst condition for given indicator and inertia diagrams may be determined. Cast Iron Crank. For the conditions assumed in Table 82, Chap. XXI, for the factor 5 for steam engine cranks there is nearly full reversal, and the factor 7 may be assumed for cast iron as previously explained. There is an initial stress in the metal around the pin presumably higher than the liveload stress. The factor for this may be 4 as for static stress. Due to this initial stress and the fact that the conditions assumed in determining the factor 7 only obtain for very large overloads; also because the result- ing dimensions are greater than those of many actual cranks, the factor of safety may be taken as 5. This factor will be applied by making P P in (14) equal to P x and taking both signs positive. Then the general formula is: The diagrams of Fig. 186, Chap. XVI, drawn to scale for the 20 in. Corliss engine previously referred to, when analyzed by the method of Par. 166, Chap. XXI give a factor of 4.3 for % cut-off. For % cut-off the factor is 5.55. These are both for cast iron. Then in view of the initial stress in the crank-pin eye due to forced fit, the factor 5 seems safe. It is likely that the iron used for cranks would usually have a tensile strength greater than 16,000 lb., but this value will be used here. An old rule was to make the length of crank fit % of the shaft diame- CRANKS 581 ter. Another rule was to make the diameter of the shaft ^ the cylinder diameter. This makes the length of the crank % the cylinder diameter. This may be done satisfactorily in steam engine practice by taking the cylinder diameter as that of the standard cylinder D s carrying some stand- ard pressure. For any other pressure (and this may be used for internal- combustion engines if desired) the length of the fit may be found as for other standard dimensions as explained in Par. 63, Chap. XII, or for compound engines in Par. 72, Chap. XIII. If h, the arm depth, is made 0.32Z) (S , there is room for an oil groove on the hub, even on the smallest sizes. Then taking P x as referred to the standard cylinder D s , and substituting the value of h in (15), we have: which is a special formula when h = 0.32D,s. As it is well to confine the use of cast iron cranks to steam engines not carrying high pressures, another special formula may be given, based upon the pin dimensions of Table 91. Then let p = 125, S = 3200 and 1 = 0.32As, for which: 6 = 0.672> s = 2.ld (17) This dimension may be obtained as shown in Fig. 396, and while rupture might not occur in line with the pin diameter, the method has proven satisfactory in practice. Steel Cranks. For steam engines carrying high pressures and for gas engines, cranks should be steel castings As dimensions are not so exces- sive there is no difficulty in applying the formula with the factors already assumed in the introduction, and used for the pin. Formulas (14) to (16) also apply to steel cranks From Table 73, Chap. XXI, the elastic limit is given as 30,000 Ib. Taking S as 6000 and other data as before, a special formula for steam engines is: b = 0.36As = I.12d (18) For gas engines, calculations show a massive construction when the pin is to be forced in. It is no doubt the best practice to cast the pin with the crank. As an example, the proportions of (12) will be assumed in which k = 1. Let a standard pressure p be 400; then S is 7150. Let it be assumed that the length of the hub fit is 0.5D S and that h = QA2D S . Then as d = QA73D S , 1 may be taken as QA5D S . Substituting these in (16) gives: b = 1.3D S = 2.7d (19) which is excessive. f x-4 ( _ 582 DESIGN AND CONSTRUCTION OF HEAT ENGINES The crank arm is sometimes cut away as shown in Fig. 398. This is permissible, especially in steel castings, but is not logical design. It gives a poor section for torsional stiffness when the crank center is normal to line of stroke; it also necessitates joining a light and heavy section if the walls are made thin. It may lighten the crank if the stroke is long, and it reduces the moment arm by moving the neutral axis toward the pin ; but it also reduces the section modulus and should be carefully checked for tension on the side away from the pin. To find the stress in this section the general formula is : S--=P X 1 (20) c where 7 is the moment of inertia about axis XX and c is the distance to the extreme fiber as shown in Fig. 397. Well designed cranks seldom need check- FlG< 397 ' ing when in a position normal to line of stroke. They are then subjected to torsion and bending. The bending moment is: M B = P P l x where l x is the distance from the pin center to any section as shown in Fig. 396. The modulus of section for this case is about axis YY. The twisting moment is : M T = Pplo where 1 is taken from Fig. 396. If the section is rectangular the maximum shearing stress is at the center of the surface of the long side and may be found from Par. 163, Chap. XXI. At the center of the short side the stress is proportional inversely to the stress at the long side, as the lengths of the sides of the rectangle. These stresses may be combined by (20) , Chap. XXI. As stated in Par. 166, Chap. XXI, the factor of safety for this position may be taken as 3 for ductile materials and 6 for brittle material, such as cast iron. 190. Hubs. The subject of pressed fits is treated in Par. 178, Chap. XXIV, so the discussion will not be repeated. Let T be tons per inch of diameter per inch of length required to com- plete the fit; let /i be the coefficient of friction and t the thickness of hub. Then from (28), Chap. XXIV: and from (29), same chapter: CRANKS The outside diameter is then: d a = d + 2t 583 (23) For cast iron cranks, taking /* = 0.25, T = 0.8 and S = 4000, as- suming static loading: q = 0.51 and t = 0.5d. Then: d H = 2d (24) The allowable static stress in a steel casting is 15,000 Ib. If T is 1.25, the outside diameter could be less than 1.3d. If the stress is taken at 6250 when T is 1.25, d = 2d as just found for cast iron. There are FIG. 398. Nordberg crank and pin. other considerations besides strength in crank design, such as the necessity of a diameter sufficient to cover the bearing, but the formulas may be used as a check. Crank pins are forced in and riveted over cold as shown in Fig. 396. There is no tendency for the pin to turn so no key is needed, but as all the power of the engine is transmitted through the shaft fit, the shaft is keyed to the crank with one or two keys as a precaution against slipping. In Fig. 396, if the fit were not tight enough to keep the crank from slip- ping on the shaft, the surface ma of the keyway must sustain the entire load. If S c is the crushing stress in the key, the moment of resistance of the key is approximately. d 584 DESIGN AND CONSTRUCTION OF HEAT ENGINES This must equal the twisting moment FT?, where r is the radius of the crank circle in inches. Then: Should the key take the entire load the standard factor of safety would be 3 for repeated load. It is probable in most cases that the fit would hold without a key, so if desired, a factor of judgment less than unity may be applied. For steam engines the maximum value of P T may be taken as P x - For internal-combustion engines, Giildner finds that the maximum P T is 0.5P* for his standard reference diagram. FIG. 399. Bass-Corliss disc crank. FIG. 400. Plain crank with counterbalance. By assuming values in terms of D s , a special formula may be derived. In doing so it is best to retain d and >S C . For the steam engine, as already assumed, a = Q.375D S . Also let r = l.2D s which covers most cases. Assume p as 125. Then: m = , d S c (26) For the internal-combustion engine we may take: a = 0.5D, r = 0.8Z) s andp = 400. Then: (27) m These two special formulas may be used for preliminary calculations. If two keys are used they are placed 90 degrees apart. The value of m for each key may be taken as one-half that for a single key, or perhaps CRANKS 585 some greater. This arrangement weakens the shaft less than a single key. The factor of judgment may be taken as 0.75, making a total factor of 2.25, which, with an elastic limit of 38,000 gives S c = 17,000. 191. Crank Designs. Fig. 398 shows a crank used on steam engines built by the Nordberg Manufacturing Co., Milwaukee, Wis. The hub projection next to the bearing is omitted, except that necessary for facing. A small amount may be placed on the opposite side so long as it does not interfere with the connecting rod. This arrangement either shortens the moment arm used in determining the shaft strength, or makes the crank arm thicker, or both. A thicker arm reduces the difficulties en- countered in determining b. FIG. 401. Gas engine crank. The oil groove must be omitted in this design, but this matters little if oil shields are used. Figure 399 is the design of a disc crank used by the Bass Foundry and Machine Co., Fort Wayne, Ind. The disc portion is considered by some to have a more finished appearance, the rim usually being polished. A simple arm crank with counterbalance is shown in Fig. 400. It is proportioned for the 20 by 48 in. Corliss engine designed through the book. The crank of Fig. 396 is for the same engine and is calculated for cast iron, but Fig. 400 is for steel. Figure 401 shows a gas engine crank with pin cast integral, and is therefore of steel casting. CHAPTER XXVIII SHAFTS Notation. D = diameter of cylinder in inches. D s = diameter of cylinder when some standard pressure is used (see Par. 63, Chap. XII and Par. 72, Chap. XIII). d = diameter of pin or shaft determined for stress relations. Used especially for shaft fit in hub. dp = diameter of crank pin. d s = diameter of shaft at any part considered. h = thickness of crank arm, measured axially. 6 = width of crank arm. r = radius of crank circle in inches. 1 moment arm of crank pin in inches. 1 A = moment arm of crank arm in inches. 1 M = moment arm of main journal in inches, IP = length of crank pin in inches. l s = length of main journal in inches. p = maximum unbalanced steam or gas pressure in pounds per square inch. P = maximum pressure on crank pin per square inch of projected area. P M = mean pressure per square inch of piston area, including inertia, per cycle. Px maximum total unbalanced steam or gas pressure in pounds. P P = total unbalanced force, including inertia, acting in line of stroke P L = thrust along connecting rod in pounds, including inertia of re- ciprocating parts and of rod itself. P T maximum turning effort in pounds. P R = force in pounds acting along crank, toward shaft center. P B = maximum belt pull in pounds. R = reaction on bearings in pounds. Subscripts correspond with those of force Px, PT, etc. R M = mean reaction on bearing during cycle, in pounds. W = weight of flywheel in pounds. 586 SHAFTS 587 M B = bending moment. M T = twisting moment. M R = equivalent bending moment. A = total deflection. 6 = deflection per inch of length. / = moment of inertia. E = modulus of elasticity. S = stress in general, direct or combined. S P = stress in crank pin. S A = stress in crank arm. S M = stress in main journal. S E = elastic limit. S s = shearing stress. S D = direct stress in crank arm. / = factor of safety. C and K are constants in wear formulas in Chap. XI. Other notation on figures. All dimensions are in inches, and forces in pounds. 192. Shaft Types. Two general types of crank shaft are in use; the side-crank, shown in Fig. 402 and the center-crank, shown in Fig. 403. FIG. 402. Side-crank shaft. FIG. 403. Center-crank shaft. Center-crank shafts are also constructed with no outer bearing for small engines, the wheel being overhung and close to the bearing. Sometimes two wheels -are employed, one containing the governor. 588 DESIGN AND CONSTRUCTION OF HEAT ENGINES In this country the use of the center-crank is mostly confined to engines of small size and to multi-cylinder engines, being a necessity for the latter. Large engines, both steam and internal-combustion, are nearly always provided with the side-crank shaft. On the continent, both large and small internal-combustion engines are built with the center-crank shaft. The advantage claimed for the center-crank is an even distribution of pressure on two bearings, and the advantage from the standpoint of strength of a beam supported at both ends, over a cantilever. It is obvious that an outer bearing is necessary with a side-crank shaft. For many classes of service it is also n'ecessary with a center-crank shaft, especially for large sizes with heavy wheels, in which case there are three bearings to keep in alinement. Failure to do so results in excessive strain, so that engineers in this country are inclined to use the side- crank shaft where feasible, and although heavier, stresses may be cal- culated with greater accuracy. In all cases of design, the point in each bearing from which to measure lengths for bending moment calculations is indeterminate, and certain assumptions have to be made. For the side-crank shaft with two bear- ings, calculations are comparatively simple, as is also the case with center- crank shaft with no outer bearing. For three bearings, and for multi- throw shafts, the calculations become much involved if an attempt at rigid analysis is made. The Theorem of Three Moments (Clapyron's formula) is sometimes applied to this case and referred to as an exact method, but the laws of bending of center-crank shafts are very complex ; in fact, part of the deflection is due to torsion of the crank arms. Were these complications due to shaft form taken into account, assumptions must still be made due to the resistance to bending offered by the bear- ings, and the nature and exact location of the reactions. It is therefore certain that any such method is far from exact, and in all probability gives no better results than the simple methods employed with discretion, and some consideration of successful practice. For side-crank shafts an analysis which has given satisfactory results in practice has been employed, and this is checked for various other factors which have a limiting influence on design. For two-bearing center-crank shafts a simple analysis is also made for crank pin, arm and shaft journal. For center-crank shafts with outer bearing the portion of the shaft between main and outer bearings is designed as for a side-crank engine, while the cranks are designed as for a two-bearing, center-crank shaft, ignoring the remainder of the shaft. The bending moment due to wheel SHAFTS 589 load would be practically zero at the point of maximum bending due to pis- * ton thrust, and it is unlikely that two maximum values are found simul- taneously at the same point. It may be that wheel load, etc. will require a larger diameter of the main journal than given by the simple center- crank analysis; should this be greater than the required crank pin diameter, they should be made equal. With multi-throw crank shafts, the crank next to the delivery end of the shaft is designed as a single-throw shaft, with the addition of any turning effect from the other cylinders. This is best determined from a combined turning-effort diagram, omitting the end cylinder diagram. It is usually comparatively small, especially if of the 4-cycle type, and may be pro- vided for by multiplying the moments found by the simple method by a factor a little greater than unity possibly 1.1. On account of greater uncertainty respecting strains in center-crank shafts, lengths for determining moments are measured more nearly from the centers of bearings; at the crank pin, however, the load due to con- necting rod thrust is more symmetrical about the center of the pin, and may be assumed as a uniform load for a portion of the pin near the center. Material Shafts for large engines are often open hearth steel forg- ings, the elastic limit of which may conservatively be taken as 38,000, as given in Table 73, Chap. XXI. Higher grades of steel are sometimes used, and properties of some of the alloy steels are tabulated in Chap. XXI. Allowable bearing pressures sometimes limit the stress, so that high elastic limits are not required unless desired for their higher factor of safety. In some designs however, especially where few bearings are used for a multi-throw shaft, the ratio of bearing pressure to stress is low, so that higher permissible stresses are a great advantage, enabling the use of smaller diameters. The side-crank shaft will be first treated, both for steam and internal- combustion engines, after which the center-crank shaft will be analyzed, largely in connection with 4-cycle internal-combustion engines, where it is most used. As for crank pins, the factor of safety may be determined by the aid of Par. 166, Chap. XXI. From Table 82 of Chap. XXI, the factor for single- acting engines may be taken for usual conditions as 3.45; for double- acting steam engines as 5 and for double-acting internal-combustion engines as 4.2. These factors are based upon maximum steam or gas pressure only, and apply to crank positions on or near dead center. As explained in Par. 166, Chap. XXI, the stresses are practically repeated for crank positions giving large turning efforts, and the factor may be taken 590 DESIGN AND CONSTRUCTION OF HEAT ENGINES as 3, with perhaps a factor of judgment of from 1.1 to 1.2. In order to reduce deflection it is better to use a higher factor for shafts with con- siderable distance between bearings, so in such cases the factor will be as in Table 82, Chap. XXI. 193. Side-crank Shaft. Except when very heavy wheels and long shafts are used, the diameter of a side-crank shaft may often be deter- mined by combining the bending and twisting moments due to the piston thrust. From Formula (11) and Table . 53 of Chap. XVI it may be seen that when the connecting rod is but four cranks long, the pressure along the rod is but little more than 3 per cent, greater than the piston thrust; if fric- tion were considered the difference would be still less. Then for this discussion it may be considered suf- ficiently accurate to neglect the angu- larity of the connecting rod. The older rules assumed the mo- ment arm 1 of Fig. 404 to extend to the center of the bearing. For long lines of shafting, or where ball-and-socket bearings are used, this method is probably correct; but when ball joints are used in engine bearings, they do not allow freedom in the line of thrust. It is then probable that the point of maximum bending is near the crank end of the bearing. The author has assumed for many years that it is at the point where the shaft enters the crank. The shaft is practically always reduced to form a shoulder here; then if this reduced diameter be solved for at this point, and the diameter in the bearing increased % to Y in. to form a shoulder, the equivalent moment arm for the larger diameter will extend a short distance along the bearing. By this method it is not necessary to assume stress greater than that indicated by the laws of fatigue, which was necessary with the old assumption. Steam Engine Shaft. For engines cutting off steam later than one- half stroke, which is now usual when overloads are being carried, the maximum bending and twisting moments occur when the crank is normal to the line of stroke. The bending moment is then: FIG. 404. and the twisting moment: M B = P X 1 M T P x r. SHAFTS 591 The value to be used in these formulas is not strictly P x , but P L , the force acting in the direction of the connecting rod. As this differs little from P P , the thrust in line of stroke (including inertia), and it is desired to base the factor of safety upon P x , the latter may be used. From (22), Chap. XXI, these may be combined thus: M K = 0.35M* + QM\M B * + M T * (1) Equating with the modulus of section of the shaft gives: _ 3/32M; d V~^S~ For the standard cylinder diameter D s carrying standard unbalanced unit pressure p, the length and diameter of the crank pin was taken as 0.321)5 in Chap. XXVII. The crank hub was taken as 0.375D 6 , Then for the design shown in Fig. 404: 1 = + 0.375D 5 = 0.535Z> a (3) Let r = 1.2D 8 , and as p = 125 (Chap. XXVII), and / = 5, by sub- stituting these values in (1) and (2), d may be found in terms of D s , or: M R = 98.6ZV and (^ (4) Taking the elastic limit S E as 38,000, S = 7600 and: d = QMD S Formulas (4) and (5) are special and apply for the assumptions made. Their use, however, will give good practical results. Between the bearings there is combined bending and twisting. The twisting moment is the same as for the case just considered, and the bending may be caused by weight of wheel and shaft, weight of gear or armature, belt pull, or magnetic pull of arm- ature. Any of these forces may be referred to the bearing by taking moments about the other bearing; then a resultant may be found graphically as in Fig. 405, in which R w is the reaction due to the weight and R B the reaction due to the belt pull. Total reaction R may include all forces acting be- tween the bearings. The central part of the shaft is usually enlarged to take the wheel or any other desired mountings, and the smaller diameter only requires cal- 592 DESIGN AND CONSTRUCTION OF HEAT ENGINES culation for strength and stiffness. Referring to Fig. 406, the moment arm 1 may be taken to the center of the bearing in this case, as there is sometimes freedom of movement of the bearing in this plane. The bending moment is: M B = Rio From (1) the equivalent bending moment is: M R = 0.65 J(R1 ) + (6) If, after adding the shoulder to d found from (2), the value of M R from (6) gives a greater result in (2) than this, it must be used. FIG. 406. The deflection may be taken as for a cantilever of length 1 . Rio* S.SRlp* = 5lo == 3EI ~ Ed s * '' 12 V Then: (7) where d is the deflection per ft. of length. Table 73, Chap. XXI gives: E = 29,000,000. A value of 5 sometimes given is 0.003. Then from (7) : d a = All three of these methods should be used and the largest diameter adopted. By using a lower stress in the first method, a diameter will be obtained which may be ample for nearly all conditions which may arise, but it gives a larger shaft than is required in a good many cases, increasing the cost. A further check should be made by the formulas of Par. 52, Chap. XI, and by the bearing pressure in Table 19 of the same chapter. The weight of crank and part of connecting rod should properly be included, also the SHAFTS 593 mean piston thrust. The resultant mean pressure on the bearing may be found for each stroke of the cycle and the average of these taken as the value of P in Chap. XI. It is not claimed that these values give limits beyond which it is never permissible to go, but it is well to keep within or near these figures if possible. A common ratio of length to diameter of steam engine bearings is 2. A bearing length may sometimes be made twice the diameter of shaft by the first method even though the actual diameter must be greater accord- ing to the second or third methods, providing the check on bearing pres- sure and wear by Chap. XI is satisfactory. In some cases the shaft is enlarged immediately outside the bearing. The crank fit and bearing may then be proportioned for the piston load, and the considerations of friction and wear treated in Par. 52, Chap. XI. VilS Vip Y//////V////A Y///////////////A//////////////////////////A , 4 ^ ~ ~ FIG. 407. As an example of application, assume the 20 by 48 in. Corliss engine mentioned in Chap. XII and elsewhere through the book, to drive a 300 kw. railway generator. The data for this generator was taken from an old table, but will answer for the problem. The weight of the arma- ture is 20,500 Ib. The wheel, determined by the method of Chap. XVIII, weighs 33,000 Ib. The two eccentric hubs, governor pulleys, wheel hub, and limiting lines of generator are shown in Fig. 407 As / = 6 between bearings, S = 6330. The reaction of the wheel at center of main bearing is: = 33,000 X 94.5 = lo / Of the generator: Rg= 20,500 X 61.5 AO I 38 594 DESIGN AND CONSTRUCTION OF HEAT ENGINES The weight of shaft may be neglected at first. The bending moment due to total reaction R (= 31,950, or say 32,000 Ib.) is: M B = 32,000 X 31.5 = 1,010,000 Ib. The maximum turning effort is 39,250 (approximately). The twisting moment is: M T = 39,250 X 24 = 945,000 Ib. From (1) or (6) : M R = 1,380,000. This is more conveniently found from (24), Chap. XXI. Now from (2) : /3 ~\ 2 X 1,380,000 TT X 6330 This gives some idea of the necessary diameter, but we know it will be some greater. In the shaft fit (5) gives: d = 0.51 X 20 = 10.2 in. With a M-in. shoulder the diameter of bearing would be : d = 10.5 in. With this diameter as a basis let k, the ratio of length to diameter, be 2. The length may be taken as 21 in. Checking for deflection by (8), with reaction R gives: d = 0.175v / 32,000v / 31.5 = 13.2 in. As the weight of the shaft was not included it is better to take the diam- eter between bearing and hub fit as 13.5 in. It is desirable when taper keys are used in the flywheel hub, to increase the diameter so that the key will clear the smaller portion of the shaft. If straight keys are set in the shaft this is not necessary. In this case we will make the diameter through wheel and generator fit 16 in., an increase of 2^ m - From the generator table, the range of shaft diameter is from 13 to 16 in., so this is allowable. The reaction of the shaft at the main bearing is 4000 Ib. The reaction of the crank is 3700 and of the connecting rod 675 Ib. The total reaction producing bending moment is due to wheel, generator and shaft, the effect of the latter being safely taken as a concentrated load at the center; this reaction is nearly: R = 36,000 Ib. The reaction causing bearing friction is the resultant of this plus crank and rod effect, the mean load due to piston thrust and inertia. The latter at maximum cut-off is 32,500, and as this acts normal to the other, the mean reaction causing wear is : RM = V40,400 2 + 32,500 2 = 51,800 Ib. SHAFTS 595 The main bearing may have any dimensions from 10.5 by 21 in. to 13.5 by 27 in., as far as strength is concerned. By trying several values by the formulas of Par. 52, Chap. XI, we will take 12 by 24 in. and check. This does not change the reactions perceptibly, but lengthens the moment arm 1 to 33 in. For strength: M B = 36,000 X 33 = 1,190,000 M T = 39,250 X 24 = 945,000. From (1) or (6) : M R = 1,520,000. From (2) , = 3/32 X 1,520,000 = 13 45 in \ TT X 6330 which is safe, as 13.5 in. was taken. For deflection: From (8) : d s = 0.175v / 36,000\/33 = 13.9 in. This is a little larger than the dimension taken, but the moment arm is probably less than 33 in., and there is some variation allowed in the deflection factor used, which was a mean value. Bearing pressure. = 51,800 12 X 24 Wear. From (4), Chap. XI: C = Q ' 262 X 5 *f *Jgg = 56,300. From (6), Chap. XI: v 0.512 X 51,800 /100 - - : The values of C and K are greater than the tabular values, but these factors are more or less arbitrary and may be increased under good con- ditions. In determining P, C and K, the maximum load pressure was assumed in the cylinder. The shaft is drawn to scale in Fig. 407. There must be fillets in all places where the shaft changes diameter. A smaller diameter might have been used if stronger material had been assumed, but the modulus of elasticity is practically the same for all different steels, and this is what influences the deflection. The eccentrics and straps, governor pulley, etc. were neglected. The 596 DESIGN AND CONSTRUCTION OF HEAT ENGINES shaft, crank and connecting rod are often omitted; if bearing pressure and the constants for wear are based upon such omissions it matters little. The 10 in. of length between the outer bearing and the swell could no doubt be reduced, equalizing the load more nearly between the two bearings. It is possible that a more detailed study would show the same for the 21 in. at the crank end, shortening the shaft slightly. Internal-combustion Engine Shafts. The maximum bending and tor- sional strains do not occur at the same time in these shafts, and d should be computed for maximum bending stress at dead center, and for combined bending and twisting at the position of maximum turning effort. At the dead center (1) gives: M B = M B . Then this may be used in (2) . At the position of greatest turning effort : M B = P L 1 and M T = P T r where P T is the turning effort and P L the thrust along the rod. P L will be used for this part of the discussion, although the numerical value of P P may actually be used with small error. The position of P L and the value of PT may be found from the crank-effort diagram. Giildner finds this position 40 degrees from the head-end dead center for his reference diagram, with a value of P L = 0.7P*, and P T = 0.5P*. In Fig. 197, Chap. XVI, the maximum turning effort is at 45 degrees. P L = 0.53P* and PT = 0.43P*. PX is the total maximum gas pressure, while P P is the total pressure at any point including the effect of inertia. The value of P L and PT taken from Chap. XVI include the inertia, while Giildner 's values do not; they would apply at starting and are safer. Combining the twisting and bending moments in (1), d may be found from (2). The analysis for stress, deflection and wear between the bearings are the same as for the steam engine shaft. Assuming standard pressure, proportions of crank pin and length of shaft fit, a special formula may be derived as was done for the steam engine. In deriving a special formula for the internal-combustion engine side-crank in Chap. XXVII, it was assumed that the length of crank fit a was 0.5D 5 ; then taking k as unity, (12) of Chap. XXVII was derived; or, letting d P be the pin diameter (and also the length for this case) : d P = 1 P = QA73D S . Then: 1 = 0.737D.S SHAFTS 597 Further assume that P = 400, / = 4.2 and S = 9000. At dead center: Px = ^D 2 X 400 and TT X 400 X 0.737 This is the only moment acting in this position. Taking Gtildner's values at the position of maximum turning effort: 7T P L = 0.7 X D/ X 400 = 220D s < P T = 0.5 X jlV X 400 = 157IV. Then: M B = 162ZV and M T = 126ZV From (1) : M R = 182ZV. This is less than M B for the dead-center position, so the latter must be used; then from (2): d = 0.64As (9) As this is much greater than the value given by (5) for the steam engine there is less likelihood of the computations between bearings giving a greater diameter than with the steam engine, although the wheel of the latter may not be as heavy. As mentioned in connection with steam engine shafts, the diameter may be increased outside the bearing if a larger diameter is required for the wheel load, etc. Comparing with the problem of the steam engine, a 20-in. double- acting gas engine might have a 32-in. stroke. With the same piston speed it would run 150 r.p.m. and develop approximately one-half the power. A tandem engine would develop the same power as the steam engine, and aside from the increased inertia effects would produce no greater strains than the single-cylinder gas engine. The wheel would probably be no heavier as it runs at higher speed. From (9), the diameter of the shaft fit is: do = 0.64 X 20 = 12.8; say 12% or 13 in. In the bearing the diameter could be 13H m - Otherwise it is probable that about the same dimensions could be used. The generator from the tables used before weighs but 17,000 Ib. and the greatest shaft fit is 14 in. 598 DESIGN AND CONSTRUCTION OF HEAT ENGINES It is likely that the shaft might be turned to this diameter at the genera- tor end and still have sufficient stiffness. 194. Center-crank Shafts. For either steam or gas engines the center-crank shaft can have no advantage in a problem like that of the preceding paragraph. If a center-crank is specified, the pin diameter may be determined in a manner which will be given later. The arms may be checked for bending on dead center, and for tension and bending when normal to the line of stroke. In this latter position, due to the uncertain effect of the extra bearing, it is safe to ignore it, as it can only cause torsion of the pin by undue distortion and bearing friction. In multi-cylinder marine steam engines taking power from one end of the shaft say at engine No. 1. The bending moment on the pin of No. 1 caused by the maximum turning effort of this engine, may be com- bined with the twisting moment of all the other engines^ For the end journal of No. 1, the bending moment caused by the thrust of this engine must be combined with the maximum turning moment of all engines. This can be determined accurately only by combined turning-effort diagrams at maximum load. The formulas developed for the multi-throw crank for internal- combustion engines may be applied generally to steam engines, the chief difference being in the ratio of twisting to bending, and the position of the crank when principal calculations for the different portions are made. Internal-combustion Engines. Calculations show that the crank pin is subject to maximum stress intensity when on dead center, and that forces other than the direct piston thrust affect it but little. In this position the moment arm of the main journal is short, so that it receives its maxi- mum stress at or near the position of maximum twisting moment, the stress being combined bending and torsional stress. The maximum stress in the arms is at dead center position usually, but they may be checked for combined bending and twisting in the posi- tion of maximum turning moment. The following analysis will cover these points in as simple a manner as possible, and are considered as reliable as a more complicated method. In treating the subject, a number of general formulas are given for the various straining actions; these may be used as a check upon design after the proportions are assumed. The more important formulas are marke(J *, and from these, design formulas will be derived in terms of factors which may be assumed, or taken from practice. These are usually sufficient for design if proper values are assigned. Provision is made in the more general formulas for an overhung wheel and pull of belt. Their resultant P G may be resolved into any required SHAFTS 599 components, the sign of which may be easily determined. Should there be an outer bearing, the effect of P G may be taken as one-half. To get a better idea of the forces and reactions dealt with in the general formulas, the skeleton drawing of Fig. 408 is given. The reac- tions are all shown acting at the same point in each bearing, although i FIG. 408 this need not always be so assumed; the point is often indeterminate and will be given in what follows without reference to the length of the bearing. The twisting moment due to other cylinders is also given in the FIG. 409. general formulas, although for most multi-cylinder internal-combustion engines this is not great. It may be studied from the combined crank- effort diagrams of Par. 176, Chap. XVI. In order to give further notation, Fig. 409 is drawn. The thrust of the connecting rod P L is assumed in Fig. 409 as a load distributed over 600 DESIGN AND CONSTRUCTION OF HEAT ENGINES the fraction q of the central portion of the crank pin, in which q may be any desired value from zero to unity. Both Figs. 408 and 409 contain notation and must be consulted in the following formulas. To make it more convenient to use P G in certain equations, it may be referred to bearing No. 1 as an equivalent reaction R E which acts in an opposite direction from the actual reaction R G i. This force will give the same moment at a given section actually given by P . From Fig. 408 : (10) The distances A and B may be taken to the center of the bearing or to some assumed point given by 1 of Fig. 409. At any section between R E and the connecting rod load : M B = R M (B - lo) = P A( - B -^ n Then: P G A(B - 1 ) This may now be combined with R T , R R i or R L i, and 1 M or 1 A may be used instead of 1 where moments are desired at points from which these measurements are taken. The added subscript N refers to components of R G or R E normal to the crank, and P refers to their component parallel to the crank. The crank considered is either for a single cylinder, or for the engine nearest to where the load is applied, and will be referred to as No. 1. The force P L acting along the rod will be used in the equations, but as previously shown P P may be taken as its numerical value without serious error. For P L , P T and P R see Par. 99, Chap. XVI, the notation for these quan- tities being the same. Crank at Dead Center P L = P x and R L = R x . Also f = 3.45 for single-acting engines and 4.2 for double-acting engines. Crank pin. From P x : M* = R*ilo-?'& (12)* From P : M B = R E 1 (13) The resultant of these may be found graphically and used for the bending moment. For vertical engines with cylinder over crank, (12) and (13) are of opposite sign and in line. From P G : M T = R 02N r (14) SHAFTS 601 From the turning effort of all cranks when No. 1 is on dead center: M T = 2P T r * (15) The sum of (14) and (15) may be combined with the resultant of (12) and (13) by means of (1); then: MR=1 ^ (16) * Crank arm. From P x : M B = R X1 1 A (17)* S = From the turning effort of all cranks when No. 1 is on dead center : M B = 2P T (r-^) (19) o 6M fi /ork x 8 = ~w (20) The moment given by (19) is always normal to that given by (17). From P G : M B = R GZN (r-^) (21) 8 ~ ^ (22} ' hb* (22) Also from P : M B = R EP 1 A (23) From turning effort of all cylinders when No. 1 is on dead center : M T = 2P T 1 A (24a) From P G : M T = RG^A (25) For the algebraic sum of (24a) and (25), assuming that h is less than b, the stress at center of the shorter side from Par. 163, Chap. XXI is: Ss = M T (Zb + 1.8ft) (26) At the center of the long side: c M T (3b + 1.8ft) ,_ * = -fttjjr- (27) For the direct load due to P x : S D = % (28)* 602 DESIGN AND CONSTRUCTION OF HEAT ENGINES This is not of great importance, but is combined with (17) in the design formula, so is marked. In combining arm stresses three combinations may be made: (1) The algebraic sum of (18), (20), (22), (24) and (28). (2) Combine the algebraic sum of (18) and (24) with (27), by the use of Formula (20), Chap. XXI. (3) Combine the algebraic sum of (20) and (22) with (26), by (20), Chap. XXI. Each of these values must come within the allowable stress. Main journal, at juncture with arm. FromP*: M B = R xl l M (29) From P : M B = R E 1 M (30) From turning effort of all cranks when No. 1 is on dead center: M T = SP r r (31) The resultant of _(29) and (30) may be combined with (31) by means of (1); then S or d s may be found from (16). Crank in Position of Maximum Turning Effort. Factor of safety may be 3.3 for both single- and double-acting engines except when wheel is overhung, then 4.2 should be used. For the main journal, on account of uncertainty of the point of application of load, a factor of judgment of about 1.5 may be used. Crank pin. From P L : M B = R Ll l -^'& (32) From P \ M B = R E 1 (33) From turning effort of all cranks but No. 1, when No. 1 is at its position of maximum turning effort: M T = SP r r (33a) From component of P normal to crank : M T = R G2N r (34) The resultant of (32) and (33) may be combined with the algebraic sum of (33a) and (34) by (1); then S or d P may be found from (16). Crank arm. From P R , the component of P L : M B = R R1 1 A (35) SHAFTS 603 For the effect of turning effort, No. 1 should be omitted. It will be in .the position of its maximum effort, however. With this difference, (18) and (27) apply to the arms in this position. Formula (28) applies by sub- stituting R R i for RXI. Then by taking (35) instead of (17), the analysis for crank on dead center applies to this case for the arms. Main journal at juncture with arm. From P L : M B = R L1 1 M (36)* From P : M B = R E 1 M (37) From the turning effort of all cylinders when crank No. 1 is in posi- tion of maximum turning effort: M T = SP r r (38)* FIG. 410. The resultant of (36) and (37) may be combined with (38) by means of (1) ; then S or d s may be found from (16). Some of the moments counteract the others, especially due to P G . This will depend upon the direction of belt pull, etc., and if this is to be taken into account in designing stock engines, the worst condition should be taken. Bearing pressures and wear may be checked by Par. 52, Chap. XL The analysis is much the same for the arrangement shown in Fig. 410. The reaction Ri will be greater for crank No. 1; therefore the bending moments will be greater. There might be advantage in using higher grade material for a crank of this design to keep dimensions smaller. If the positions of the bearings may be located with reference to cylin- der No. 1, reactions RXI and R X z may be found and calculations made as 604 DESIGN AND CONSTRUCTION OF HEAT ENGINES for a symmetrical crank, special formulas for which will now be derived. Then RXI, instead of one-half of P x would be a larger fraction. As this. relation would not likely be standard for engines of different size, special formulas would perhaps not be worth while in this case. Special Design Formulas for Symmetrical Crank. Referring to Fig. 409, let IP = kd P , h = xd P , b = yd P and 1 M = md P . Then: 7 71 / I X \ J h = IM + 2 = ( + g j d and Also P*=^ ; (40) Crank Pin. Using only (12) for crank at dead center: <> Substituting in (12) gives: (43) We may solve for d P from (16) and (43), or for S if dimensions are known; but it is more convenient to consider first the relation of S to P, as a compromise must sometimes be made between them. Then taking S P from (16) and P from (42) : - ,...* (44)* By assuming k, x, m, and = 180 - (a + 0). From the properties of triangles: 7*2 7*i 7*i sin a ~ siiT^ ~ sin [180 (a + 0)] ~ For the very small angles involved : a sin a r<> sin (a + 0) ~~ sin 5* sin 5 (20) If the rim and hub are rigid, the bending moments at the two ends of the arm are proportional to these angles; then for equal bending stress at the two ends: where Z A is the modulus of section of the arm. If the arm sections at the two ends are similar and h A is the depth of section in inches in the di- rection of motion : (22) Table 98 shows this relation. TABLE 98 ri/r 2 1.500 1.750 2.000 2.500 3.000 4.000 5.000 6.000 1 145 1 205 1 260 1 358 1 442 1 587 1 710 1 820 A rule sometimes used for wheel arms is to make the section modulus at the hub twice that at the rim, and this usually gives a well-appearing design. With these proportions, when bending only is considered and when TI/TZ is greater than 2 which is nearly always the case the rim end is stronger than the hub end. It is always safe to take z A2 /z A i as 1.5, the minimum value of Table 98, and this gives a good appearance. It is obvious that there is a point of contra-flexure at which the bend- ing moment is zero, and that this moves out toward the rim as the ratio a/ d, and therefore r 2 /ri is made smaller. The fact that the rim is not rigid, and that the arm is more flexible toward the rim due to the reduced FLYWHEELS 631 depth, makes it apparent that the point of zero bending moment is still nearer the rim than (20) indicates. It is therefore obviously safe to as- sume the arm as a cantilever loaded at the end, which may be considered as at the inside of the rim at radius r\. The modulus of section may be taken at the radius r 2 , which is at the outside of the hub. The moment arm may be taken to the rim center for (18) and to the outside for (17). The bending moment, taking P from (17) or (18) is then: M A = 12(R - r 2 ) ~ (23) l\v The maximum value of P/N V must be used. For a heavy rim with belt try both (17) and (18). For heavy-rimmed balance wheel (18) is used. The values of P and N v may be found from Pars. 203 and 204 respec- tively, and R in (23) taken as R } R or r\ as desired. For the arm at the hub: S BA z A2 = M A (24) From this either S BA or z A2 may be determined. Values of z A2 will be given in Par. 207. 206. Working Stresses. A flywheel may be considered as subject to repeated loads which never have as great a range as from zero to full load ; or if reversed, the percentage of maximum load is small, and cast iron is much stronger in compression as explained in Par. 159 , Chap . XXI . Safety decreases, however, as speed increases, so if a repeat factor is taken at a comparatively low speed, the stress at this speed may be considered the maximum limit. The factor may then be increased, or the allowable stress decreased gradually up to the maximum allowable speed. For cast iron, which is most commonly used, let the minimum factor be 5; if the ultimate strength is taken as 16,000 lb., this makes S equal to 3200. Let the velocity at stress 3200 be 1000 ft. per min. and assume the stress to vary as the cube root of the velocity inversely. Then: 32,000 /OCN ,-,: * " s = -yr ' (25) This is an entirely empirical formula and may be taken as the maximum limit of allowable stress in rim or arms for velocities at either center or outside of rim from 1000 to 6000 ft. per min. where the materia is cast iron. Subtracting from this the hoop stress given by (13), or approxi- mately and safely by (5), leaves the allowable bending stress in the rim which limits the value found by (14). Subtracting the value given by (11) from that of (25) gives the allowable bending stress in the arm near the hub. The rim section is determined by consideration of weight, size of belt, 632 DESIGN AND CONSTRUCTION OF HEAT ENGINES or the stze and number of ropes. In determining arm dimensions there are various empirical formulas, but they are not suitable for general application. If a number of assumptions are made in (10), a simple formula may be derived which will assist in determining arm dimensions, and if desired, these may be more carefully checked. Then let A A /A R = 0.25, R/L = 1.25, L = 0.8R and r l + r 2 = 1.125fl. If z A2 = 2z Al for similar sections, A A2 = 1.29 A A . For cast iron, w = 0.26. Then: and To facilitate calculation Table 99 is given. TABLE 99 (26) (27) V s S FZ N = 6 N = 8 AT =10 N = 12 4000 2015 448 435 415 405 4500 1940 570 550 525 510 5000 1870 700 680 650 630 5280 1840 785 760 725 705 5500 1810 850 820 785 765 6000 1760 1050 980 940 910 It is convenient and safe to take V at the outside of the rim in finding S and SFZ. It is probably safer not to design wheels for a velocity much less than 4000 ft. per min. even though they are to run slower than this; this is especially applicable to large wheels. 207. Rim and Arm Sections. The form of rim section is determined by the service required. The faces of belt wheels are crowned as shown in Fig. 437, which is made for four belts of different widths. There is no fixed rule for the height of the crown, although formulas are sometimes given more usually for small shop pulleys. A formula given in Leut- wiler's Machine Design and credited to Mr. C. G. Barth is: y = 32 (28) where b R is width of face and y the height of crown in inches. Formulas for the ratio of wheel face to belt width are given for small pulleys, but are generally unsatisfactory for large wheels. From % to 1H in. may be allowed, depending upon the size of the wheel and condi- tion of service. FLYWHEELS 633 There are a number of designs of grooves used on rope wheels, but they are usually in agreement on the angle of groov3, which is 45 degrees. The sides against which the rope bears are usually straight, but sometimes the sides are curved; it is claimed for the latter that the rope is more apt to turn in the groove, causing more even wear and prolonging its life. The multiple system is mostly used for main drives and is therefore of most in- FIG. 432. Engineers' standard groove. FIG. 433. Cresson standard groove. terest to engine designers. Figure 432 shows a groove known as the Engineers' Standard, designed by the Jones and Laughlin Co. Table 100 gives dimensions for ropes of different size . TABLE 100 [8 1 IK IK 2 2% 2^6 3% 3K 4 3K 3^6 K i* IK 2% 5 /16 IK 2H i IK IK .2 KG K KG 1 IK IK In the G. V. Cresson Co. catalogue of 1907, their standard for the American, or continuous system has a groove angle of 45 degrees, while for the English, or multiple system the angle is 30 degrees. The groove for the multiple system is shown in Fig. 433. The ratio of the tension on the tight side to that on the loose side is greater than when the 634 DESIGN AND CONSTRUCTION OF HEAT ENGINES angle is 45 degrees; the power transmitted would therefore be greater for the same maximum tension. With the curved groove, an old rope, well fitted to the groove would settle deeper in the groove where the angle is greater, while a new rope will ride the groove at a smaller angle, thus tending to divide the work. The author used a groove much like the Cresson standard for a number of years, and so far as he knows it has always been entirely satisfactory, and especially well adapted to heavy work and large overloads. As the rims of belt and rope wheels are relatively thin, flanges, and sometimes ribs are used to increase strength, as shown in Fig. 437. The modulus of section is for a section similar to a T; in taking: ,.-! .:;.--;:' (29) c must be taken from the neutral axis to the edge of the ribs toward the wheel center. The section should be taken near the arm, as the maximum bending moment is at this point. The modulus of section of a rope wheel rim may be found by the graphical method of Appendix 1. The section between two rope centers may be taken, the value of Z R being for the sum of these sections. Heavy flywheels usually have a section so nearly rectangular that the section modulus may be that of a rectangle; or, The weight of the rim is determined by one of the methods of Chap. XVIII. If this is W R : 2ir X 12Rb R h R w = W R . If 6* = k R h R : ~W I VF ' h * = 24wwb R R = \24irJLfl R may be taken to give a certain velocity; then the outside diameter is: D = 2R + h R (31) If D is to have a certain value, R may be assumed, a tentative value of h R found, and adjustment made by trial and error. In using (30a), some standard value of k R may be taken, Halsey preferring %. To check the weight when adjustments are made: W R = ir(l2D - h R )h R b R w (32) FLYWHEELS 635 For cast iron wheels, w = 0.26, and (30a) becomes: lfk = %: l '"' ; ' '""'" (34) In some very heavy rims, when the diameter is limited by a gear shaft, as was the case with Fig. 436, a more efficient wheel is obtained by making b R greater than h R , so long as the latter is large enough to prop- erly accommodate the links. In determining the thickness of belt wheel rims, the first form of (30a) may be used, b R having been determined from the required belt width; R may be taken as the outside radius. The value of h R so found would give less than the required weight, and crowning reduces this still more by reducing the radius to the center of gravity; but the addition of ribs and flanges will probably compensate for this. If greater accuracy is desired, the weight may be carefully calculated after the section is designed. In finding the moment of inertia of the section it is best to neglect the crown, using only the thickness h R found from (30a), and the ribs and flanges. The depth of rim flanges in inches may be made about 0.3D, where D is outside diameter in feet. The thickness of the flange may be about 0.2 times the depth of the flange if this is not greater than the rim thickness. Arm sections are usually approximate ellipses drawn by circular arcs. A theoretically exact section modulus may be found graphically by the method of Appendix 1. It is usually sufficiently accurate to assume an ellipse and this simplifies calculation. It will be assumed in this discus- sion that: z A2 = 1.5z Al (35) Also let the sections taken along the arm be assumed similar. Then if b A = k A h A ; Trb A2 h 2 A2 Trk A h 3 A2 irk A h 3 A1 , . Z ^ = "32" ~32" " L5 "32~ From this: h A i = kiYDP ' 875 ^ 2 (37) Then: LA2 = 4 = Q.7S5k A h A2 (38) 636 DESIGN AND CONSTRUCTION OF HEAT ENGINES And: Also : A Al = 0.765A A2 (40) To find the mean area: and: A AS = Then from (7), the mean area is: A A = 0.707k A h A2 * (41) A good arm proportion is when k A = 0.5; then the following special formulas may be obtained: At hub: z A2 = 0.0492/* 3 A2 (42) At hub: A A2 = 0.392/^ 2 (43) At rim: A A1 = 0.392^1 (44) The mean area is : A A = 0.353/i 2 A2 (45) Equating (42) with the bending moment given by (23) gives: h = 2.73 (46) \ &BA S BA may be obtained from (27). 208. Hubs. The analysis of hubs given in Par. 3, Chap. XXVII may be used for wheel hubs and will not be repeated. The result given by (24) of that chapter, making the outside hub diameter twice the bore, is quite common practice. With this ratio, combining (21) and (22) of Chap. XXVII, gives: T - ** S (47} IDL "1250 where T DL is tons per inch of diameter per inch of length required to com- plete the fit if the hub were forced on, S the stress and n the coefficient of friction. This formula will be used in connection with the discussion of hub bolts. If I is the effective length of fit on the shaft and T D is tons per inch of diameter: T D = T DL l (48) FLYWHEELS 637 To simplify equations, especially when I is not well known, T D is some- times used; when I is later determined, S may be checked by (47). 209. Hub Bolts. Let it be assumed that the bolts are to hold the hub to the shaft with a pressure equivalent to a forced fit requiring T D tons per inch of diameter to complete. Then if d is the diameter of the fit, d T the diameter of the bolt at root of thread, Si the bolt stress and n the number of bolts : TTfl Or: Taking T D = 5, p = 0.25, n = 4 and Si = 10,000: d T = 0.637\/d (50) The bolts may be selected from the bolt table in Appendix 2. The resultant of the force F 2 acts on the hub bolts and reduces the value of T D ] this resultant is: P F = F 2 2 sin (51) where is the angle made by the arm with the diameter at right angles with the bolts Table 101 gives values of S sin 0. TABLE 101 1 i X 6 8 10 2 sin e 2.000 2.704 3.240 Then in reality the equation is: ird T 2 nSi _ ~~ The value of T D is then: T ^ nrdr'nffi p 1 TD ~ 20005 L~T~ Formula (52) may be used as a check, in which T D must be a positive quantity capable of holding the wheel firmly to the shaft. The original value of T D used in (49) should be used for determining stress in (47) and (48), as the effect of the arms does not reduce the hub stress. Hub bolts are usually steel bolts having a nut on each end (commonly called studs). 638 DESIGN AND CONSTRUCTION OF HEAT ENGINES Shear bolts are sometimes used in large wheels, the hubs being forced on the shaft. Such a construction is shown in Figs. 436 and 437. This style of fastening cannot be designed satisfactorily without drawing to scale, and may require more than one trial. An algebraic solution is clumsy and liable to error; therefore a simi-graphic method is easier and better. Figure 434 shows a diagram for one arm. The force P/N V at the rim causes a direct force equally divided among the three bolts, and acting in the same direction, parallel to P. The force F% may be obtained from (10) and the resultant of these two forces found graphically as shown. Unless x is much greater than y, the bolt for which the diagram is drawn receives the maximum load. The force P Y acting on this bolt may be found as follows: FIG. 434. Equating moments about the geometrical center of the bolts: Also: Px = Pv x y or: .. ...,. Substituting gives: (53) This may be combined graphically with the resultant already found, FLYWHEELS 639 giving the final resultant P R . This is resisted by the bolt in double shear. Then: _ Or: The length of fit in either hub or arm must be : 0.8 (54) This construction requires a driving fit of turned bolts in reamed holes. As with the arms, the maximum assumed stress is probably seldom reached, so whether the stress is repeated or reversed, it is probably through a small range and of an intensity much less than the maximum. A factor of safety of 3 may be assumed, which, if the elastic limit in shear is taken as 29,000 as in Table 73, Chap. XXI, gives S 8 = 10,000, nearly. S c may be taken as 12,000, but with the usual construction this will not be reached. Shear bolts are also usually stud bolts. 210. Rim Bolts and Links. These must hold the rim tension T given by (12), or more simply and safely, by (1); then: A T = ^ (56) Oi where A T is the total area of the link sections, or the area of bolts at root of thread. It may also be taken as the bearing area of the link head in which case Si = S c , the compressive stress. Neglecting the effect of arms: A - WA * V * (57) 9650S! As the stress distribution in links is probably not uniform, it is safer to use a low stress. If this is taken as 5000 and the rim is cast iron, (57) becomes: A 172 A R (W\ ~ 186,000,000 The link may be proportioned for the maximum speed, or when V = 5280, and the dimensions retained for all speeds; then: AT = 0.154* (58a) With / links, the bearing pressure may be twice as great as the tensile stress just assumed; the area for this is then: A c = - = 0.0754* (59) 640 DESIGN AND CONSTRUCTION OF HEAT ENGINES It must be remembered that (58) and (59) give the area of both links at any section. In providing for A Ct allowance must be made for clearance in the link pocket, as the projection at the link ends will not bear on the surface nearest the link due to this clearance. For rim bolts: ird T 2 n where n is the number of bolts and d T the diameter at root of thread. Then (57) becomes: fi . lwA R far\\ UT - o7\/-o Iwi; or \ Oin For a cast iron wheel with a rim velocity of 5280 ft. per min., if >Si = 7000: (61) \ n If Si = 10,000: ^ (62) n Bolts may be selected from Appendix 2. As with the hub bolts, stud bolts are generally used, having a nut at each end. 211. Keys. Standard keys are often used, a given width of key always being used with a given shaft diameter. While such keys are usually ample they may be much stronger than necessary and weaken the shaft unnecessarily. It is always well to check a key in engine work. With notation already used in this chapter, taking m as the mean depth of the key way in either shaft or hub, the equation of moments gives : S c ml -| = 12P-^- From which: The hub is sometimes cored for a part of its length, through which the key does not bear, therefore I must be taken as the actual bearing length. S c may be taken as about 10,000 Ib. 212. Methods of Construction. Wheels are commonly made in one of the following ways: 1. Cast in One Piece. Small wheels are made in this way. Some- times the hub is split to relieve shrinkage strains; after boring to a snug fit it is clamped to the shaft by the hub bolts. FLYWHEELS 641 2. Cast in One Piece and Split Apart. This method is sometimes used with small and medium-sized wheels. It cannot be considered very good engineering. Arms sometimes part from the rim when the wheel is split, which leads one to believe that large internal stress may. exist. 3. Cast in Halves with Planed Joints. This is a good construction if properly designed and cast, and is used for large wheels. Figure 435 is an example of this design. For belt and rope wheels it is probably better to have the split at the arms, making a double arm. Such wheels are much less apt to be broken in transportation, and are no doubt stronger when in use. 4. With One or Two Arms Cast with a Segment of the Rim. This type usually has the arms fastened to the hub with shear bolts. A design with one arm cast with a segment is shown in Fig. 436. 5. Built up from Separate Hubs, Arms and Rim Segments. Such a wheel is shown in Fig. 437. It is likely that this construction gives a mini- mum of internal stress when properly machined. It also permits ease of shipment. A series of oft-quoted experiments were made by Prof. Benjamin on very small wheels of different design to determine their relative strengths, in which the one-piece wheel proved strongest. It does not seem as if this applies to large wheels, and it is very doubtful if the wheel of Fig. 437 would be as strong cast in one piece as it is at present, even though the feat were practicable and shipment possible. The relative area of arm and rim section should doubtless be in- fluenced by the mode of construction. With a wheel made by method (4), and even more by (5), the arm section may be as much smaller than the rim as desired if computation shows it strong enough. With the other methods this is not so, due to internal stress. While cast iron is the most common material for wheels, semi-steel and sometimes steel castings are used. If the latter were more common, report of flywheel " explosions" would be more uncommon. Special wheels are built up of steel plates, and some are wound with steel wire. A wheel was built by the Mesta Machine Co. a few years ago for a rim velocity of 10,000 ft. per min. Neglecting arm influence, the hoop stress from (5) would be 2700 Ib. The wheel was cast from air-furnace iron with a tensile strength of 30,000 Ib. 213. Application of Formulas. The weight of a wheel for a 20 by 48 in. Corliss engine was determined in Chap. XVIII. It was to run 100 r.p.m. and the rim weighed 20,500 Ib. With an outside diameter of 16 feet, the formulas of Par. 207 gives a section 15 in. deep and 10 in. wide. The latter measurement gives a weight a little in excess, and will be 41 642 DESIGN AND CONSTRUCTION OF HEAT ENGINES modified slightly in the finished design as shown in Fig. 435. The sh#ft bore of 16 in. was determined in Chap. XXVIII, and the outside diam- eter of the hub is 32 in. The length of hub was taken as 18 in. This might perhaps be 24 in., but 18 will be used. The number of arms will be taken as 8. From the above data, as many additional data will be taken as possi- ble, and are: R = 7.37; L = 5.42; r l = 6.75; r 2 = 1.33; V = 4620; t = 3. From (19), Case 3: N v = 7. A R = 150, and from (30), Z R = 375. From (18): 20,500 X 4620 _ r 1930 X 3 1MUU 1D> M 12 X 6.04 X 16,400 M A = = = 170,000. 32,000 s = From (23) : From (25) : From (26) : S F2 = 573 Ib. From (27) : S BA = 1247 Ib. From (46) : 3 170,000 t A . = 14 in. From (42) : z A2 = 135. From (43) : A AZ = 76.5. From (37) : h A i = 12.25 in. From (44) : A A1 = 58.7. From (45): A A = 69. From (8) : = 0-26 X 69 X 150 X 4620 2 17.37 2275(552 + 940). \5.42 ~ From (9) : SFI = ' 7 = 350. Oo./ FLYWHEELS 643 From (10) : 0.26 X 69 X 4620 2 r 150 5.42 X 8.08] ~2275~ ~ LI492 + 8^T7^J = 32 > 8C From (11), the actual value of S F z is: e 32,800 , OQ S " 2 = ~76^~ 428 lb " From (12): 86,000 - 24,900 = 61,100 lb. From (13): From (14): 6 X 20,600 X 7.37 X 0.132 - ~~375~ From (24), the actual value of S BA is: 170,000 = 135 This is slightly greater than the assumed value. From (15), the total rim stress is: S B = 408 + 320 = 728 lb. From (16), the total arm stress is. S A = 428 + 1260 = 1688 lb. This is about 87 per cent, of the allowed stress from (25). Hub Bolts. From (50): d T = 0.637\/16 = 2.55 in. The nearest safe standard bolt diameter from the bolt table of Appendix 2 is 3 in. The actual diameter at root of thread is 2.629 in. From (51) and Table 101 : P F = 32,800 X 2.704 = 89,000 lb. From (52) the effective value of T D is: T X 0.25 r^X 2.629* X 4 X 10,000 __ 1 _ TD ~ 2000 X 161 ~4~ This is tons per inch of diameter. The total force is: 16 X 2.975 = 47.5 tons, 644 DESIGN AND CONSTRUCTION OF HEAT ENGINES which far exceeds the weight of the wheel, and would therefore hold it if the shaft were vertical. From (48) : 2 975 T DL = ' = 0.248 tons per in. of diameter per in. of length. \.2i From (47), the hub stress due to this force is: It must be assumed that the total hub stress is 4000 lb., the value used in determining hub thickness, the difference being due to P F . Rim Links. Assuming the links to take the entire load, (58) gives: 150 X 462Q2 _ = 186,000,000 : The area of one link section is 8.6 sq. in. and the section may be made 2J4 by 3% in. As the bearing pressure is twice as great, the head may be 5% in. wide; allowing Y in. on each side for clearance, the head width may be 6J^ in. wide. This reduces the rim section by the area: 2 X 6.75 X 2.25 = 30.3 sq. in. The original rim area has been assumed as 150 sq. in., but is really about 135 sq. in. The actual hoop stress is then about 30 per cent, greater than the computed hoop stress, but the bending stress is affected but little as the metal is removed from the center. The bending moment midway between the arms is one-half as great as at the arms. The bending stress may be assumed as about 0.6$^, or 190 lb.; the hoop stress may be taken as 30 per cent, greater than S T , or 530 lb., the sum being 720 lb., which is practically the same as S R previously found. If there were two rim bolts, and they were assumed to carry the entire load, their diameter would be 3 in. In this type of wheel the bolts are used for convenience of construction, so they may be made smaller than this, perhaps 2 in. in diameter. Keys. It has been assumed that the hub is cored out at the middle third of its length, giving an effective length of 12 in. If a single key is used, (63) gives for the mean depth of key way: 12 X 16,400 X 16 m ='- 10,000X12X16 = This may be made \% in. and the width of the key 3J^ in. If two keys spaced 90 degrees apart are used, they may be half as large, or, as the FLYWHEELS 645 author has usually done, they may be made from % to % as large say \Y deep by 2Ji in. wide. Total Weight. The weight of the wheel is approximately 32,675 Ib. Before this example was computed, Table 62, Chap. XVIII was computed, giving 33,000 Ib. as the weight; this is 1.6 times the rim weight. Figure 435 is a scale drawing of a portion of the wheel, showing con- struction. The method of gradually enlarging the arm as it joins the rim has been used by the author on many heavy wheels, although not original with him. Holes (not shown) are cored in the rim are for jacking the FIG. 435. wheel around for the purpose of valve setting or to get the crank from dead center. They are spaced about 6 in. center to center. 214. Designs from Practice. Figure 436 shows a wheel built by the Bass Foundry & Machine Co. It is 18^ ft. in diameter and weighs 140,000 Ib. It runs 75 r.p.m., giving a rim speed of only 4350 ft. per min. It was built for a geared rolling mill drive and its diameter was limited by the countershaft carrying the gear. To increase its efficiency, the rim was made wide. It is an example of an arm and rim section being cast together, the arm being fastened to the hub by shear bolts. Both links and bolts are used at the rim, the former being designed to carry the entire load. 646 DESIGN AND CONSTRUCTION OF HEAT ENGINES 1 _J E Q o II o oj FIG. 436. Bass-Corliss rolling-mill engine wheel. FLYWHEELS 647 648 DESIGN AND CONSTRUCTION OF HEAT ENGINES Figure 437 shows another Bass wheel. It is a double wheel, 24 ft. in diameter with a 10 ft. face crowned for four belts. It is a good example of a built-up wheel, each arm and segment being cast separately. The wheel weighs about 200,000 Ib. and runs 80 r.p.m., giving a rim speed of 6000 ft. per min. The wheel is driven by a 28 and 56 by 60 in. cross-compound Corliss engine. The bearings are 18 by 36 in. and the shaft, made of crucible steel by the fluid compressed process, has a hole 8 in. in diam- eter through its entire length. Reference Halsey's handbook for machine designers. CHAPTER XXXI TURBINE WHEELS Notation. d = diameter of wheel bore in inches. Do = diameter of disc in inches measured to center of gravity of rim and blades. r = radius of same = D /2. r = radius in inches at any point. t = thickness of disc at radius r . t = thickness of disc at radius r. W R = weight of rim in pounds. W B = weight of blades, shrouding, etc. in pounds. W = weight of rim, blades, etc. carried by disc. P = load per inch of periphery due to W. F = tangential force in pounds at any radius, due to transmission of power. E = modulus of elasticity. S = stress in general, in pounds per square inch. So = assumed tangential stress where disc and rim join. SB = equivalent simple tangential stress where disc and rim join. S L = limiting tangential stress in rim, giving same strain as S in disc. S c = radial stress where disc joins rim, due to centrifugal force of W. S D = shearing stress produced by F at any radius r at which F is calculated. S H = hoop stress in thin revolving ring of diameter D . S T = total tangential stress at any radius r. S R = total radial stress at any radius r. f w = tangential force due to jet velocity. f p = axial force due to jet velocity. T M = bending moment on blades. Z = modulus of section of blades. n = number of blades receiving steam at one time. Also exponent. I = moment arm of blade in inches. m = the reciprocal of Poisson's ratio. 649 650 DESIGN AND CONSTRUCTION OF HEAT ENGINES N = revolutions per minute. V = velocity in feet per second. a = area in square inches. H = horsepower developed in blading of wheel. q = fraction of the rim carried by itself. 215. In all impulse turbines the blades are carried upon disc wheels which must be designed so that the maximum stress will not exceed certain limits at the intended speed of rotation. The hoop stress in a revolving ring is given by Formula (4), Chap. XXX. Taking the diameter in inches at the center of gravity of rim and blades, the formula may be written for steel: (1) 107 VIOO For rim velocities ordinarily used for flywheels, the value of S H is not excessive, but a velocity of 500 ft. per sec., not uncommon in turbine operation, gives: S H = 26,500. This does not include the effect of the blades, which, when the radial thickness of the rim is small compared to the diameter, may be found by dividing the centrifugal force of the blades by 2?ra, where a is the area of the rim section in sq. in. If we consider a ring with an axial thickness t , and S c the radial stress due to some external load such as the blades, the resulting tangential stress is: 2o This is only strictly true if the radial thickness of the ring is very small relative to DO, but as stated in Ch&p. XXX, may be assumed cor- rect for usual flywheel pro- portions. Assume for example that S c = 16,000, To = 1, and D /2a = 5. Then for a rim velocity of 500 ft. per sec., the tangential stress due to rim load would be 80,000 Ib. It is obvious that such a stress is too great, and the radial thickness of the ring must be increased. The strength does not increase, however, in proportion to the added area, and the axial thickness must often be in- creased toward the center of the wheel as shown in Fig. 438. The thickness to is determined from the radial load due to blading and FIG. 438. TURBINE WHEELS 651 rim. The thickness at any other radius, for discs having a central hole, is commonly found from the exponential equation : (2) The principal stresses produced by the centrifugal force of the blading and by the rotation of the disc itself .are radial and tangential, the latter being the most important. The effect of forcing the wheel upon the shaft may probably be safely neglected. This produces an initial stress, probably nearly static, and not increased by rotation; on* the other hand, it probably decreases the range of stress due to starting and stopping. Then the radial stress due to rim loading varies from Sc at the radius TO to zero at the surface of the bore; and due to the disc itself it increases from zero at radius r to a maximum, and back to zero at the bore. The tangential stress is always greater than zero and is best shown by the chart of Fig. 440. Probably no exact mathematical analysis has been devised for the case of the rotating disc; those which are used are greatly involved and the formulas are awkward and cumbersome, necessitating extensive use of trial and error. After a vain attempt at simplification it was decided to use a set of curves, the values of which were derived from another set of curves in Martin's excellent work. The mathematical treatment of the subject may be found in treatises on the steam turbine by Stodola and Jude, and in Morley's Strength of Materials. 216. It may be assumed that the blades, and a portion of the shaded area of the rim in Fig. 439, must be carried by the disc, involving one or more trial calculations, or, safely, the weight of the entire rim maybe included. In either case the rim weight will be denoted by W R and the weight of blades, shrouding, spacers, etc. by W B - Taking r at the center of gravity of rim and blades is entirely arbitrary, but safe. It may be taken at the thin- nest section of the disc where it joins the rim. Were the rim revolving free it would have the stress S H , and a correspondingly great tangential strain; but the strain in the rim must equal that in the disc at the point of attachment, and the stress will therefore be reduced to S L . The radial stress in the rim may be considered as zero, but in the disc it has the value FIG. 439. 652 DESIGN AND CONSTRUCTION OF HEAT ENGINES S c . If S is the assumed tangential stress at thife point, the simple equivalent stress from Formula (6), Chap. XXI is: S = S -^- m For equal strains in rim and disc : SL _ S E _ I /o Sc where m is the reciprocal of Poisson's ratio. Then: & = So - (3) When So is finally determined (it will equal S T ), m _ C4) rr o v*/ This may be used as a check by comparing with values of m in Table 79, Chap. XXI. If So = S c and m = 3%, S L = Q.7S. The fraction of the rim carried by itself under stress S L is : & S H S H The remainder, 1 q, must be carried by the disc. Substituting the value of S H from (1) gives: f . 107 \100 Then the entire extraneous weight carried by the disc is: W = (1 - q)W R + W B (6) From the first statement of Formula (3), Chap. XXX, the load per inch of periphery is found to be: 22 V100 The required thickness at radius r is then : UOO And from (2) : at any radius r. The total tangential stress at any radius is : S T = k T S H + C T S C (9) TURBINE WHEELS 653 And the total radial stress: S K = k K S u + c R S c (10) where, as given by (1): Stress S T is generally the greater and is usually a maximum at the bore; but in some cases it is well to try for greater values of r/r , especially for larger values of n. An inspection of the general stress equations makes it clear that for similar discs the stress varies directly as the square of the peripheral ve- locity (or, as D 2 N 2 ) ; then the factors k T , C T , k R and C R are suitable for any disc, and are given in Figs. 440 and 441 for three ratios of d to D , and for various ratios of r to r . Values of k and c were taken from Martin's curves for n 0, 1 and 2, and a smooth curve drawn between these points for several values of r/r ; other values may be found by inter- polation. It is probable that as great accuracy as the problem demands, or as is consistent with the practical accuracy of the analysis, may be obtained by the use of these curves. In applying the equations and the curves, solve for (5), (6), (7) and (11), and decide upon suitable values of S and S c . Take a trial value of n, select k T and C T at the bore and solve for S T in (9). This should usu- ally be equal to S c , and different values of n should be tried until the right value of ST is found. It may be safe to check the value when r/r is 0.9; then ra may be checked by (4), which will be nearly as assumed if S T for r/r = 1 is practically equal to S . If all is satisfactory, determine enough values of t from (8) to outline the wheel section. For a disc without a central hole, such as the wheel of the DeLaval simple impulse turbine, the formula is : i/jLyrA\_>tt t = i o s c \ioo) lAioJ \w) J (12) where e = 2.718, the base of the Naperian system of logarithms. Then t may be found from (7) as before. The stress is assumed to be uniform throughout the wheel, the tan- gential and radial stresses being equal. Jude says the reasoning upon which Formula (12) is based is open to grave doubt; however, practical results are obtained by its use. If holes are bored in a disc for the purpose of equalizing pressure on the two sides, it may be assumed that the stress at tbe radius at which the 654 DESIGN AND CONSTRUCTION OF HEAT ENGINES OJ - c TURBINE VHEELS 655 i o 7 - C 656 DESIGN AND CONSTRUCTION OF HEAT ENGINES holes are bored is increased in the same ratio that the circumferential section is decreased. If the rim is wide, as when two or more rows of blades are carried, it is probable that the stress at the edges is greater than S L , though still less than S H . An extreme case is the drum of a reaction turbine with web connections to the spindle. The axial length between the webs or sets of arms may be so great as to receive practically no support therefrom, the stress being nearly equal to S H due to the rotation of the drum itself. To this must be added the stress due to the centrifugal force of the blad- ing, which may be found by dividing the centrifugal force by 2?ra, as previously stated. The area a may be taken as the product of the thick- ness of the drum and the distance from center to center of the two adja- cent rows of moving blades. As the rim velocity of the drum type is usually less than 360 ft. per sec., there is little difficulty in keeping the stresses within practical limits. 217. Material for turbine wheels may range from ordinary open hearth machinery steel, to the alloy steels, and may be selected from Tables 73 to 78, Chap. XXI. Working stress may be determined by selecting a factor of safety from Par. 159, Chap. XXI, when the nature of the loading is determined. Changing the notation of Formula (141), Chap. VI, and taking the dis- tance from the center to where the load is applied in inches instead of feet, gives: FrN 63,000 or, p = ?3W (13) where F is the force in Ib. at any radius, required to transmit H at N r.p.m. A shearing stress SD is produced over the area, a = 2irrt or, _ F _ MyXXKff a ~ rHN It will be found that S D is insignificant compared to ST, so that the live load due to driving may be neglected. The stresses due to rotation may be practically considered as static stresses and a standard factor of safety of 2 employed. Due to possible slight vibration, a factor of judg- ment of from 1.15 to 1.35 may be employed, an average giving a total factor of 2.5. In an example of disc design given by Martin, a working TURBINE WHEELS 657 stress of 16,000 is assumed for mild steel; a factor of 2.5 gives an elastic limit of 40,000 lb., which seems reasonable. For high-speed wheels the use of high stresses is necessary, and it may be possible that the theory of rotating discs is more accurate with high stresses; at any rate low values of stress in the formulas result in absurd dimensions which are never found in practice. There can be little doubt that in a rapidly increasing thickness of metal the stress distribution is not that indicated by any theory that has any semblance of simplicity. However, judged by the usual standards of machine design, the methods used to determine the dimensions of steam tur- bine parts must be satisfactory when it is con- sidered that proportionate to the period of its practical application, no other type of prime mover has probably had so few accidents. 218. Blading Design. The process of de- termining blade angles and lengths was discussed in Chap. XV. The determination of width is rather arbitrary and depends upon the length. The matter of angles and width being fixed, the radius and location of center may be found. The known quantities in Fig. 442 are the angles A and B, and the width a + b. It is seen from the figure that: a = R cos A, and b R cos B. Then: a _ cos A b = r FIG. 442. or, Then: Or: And cos B cos A - 5 cos B cos b = a + b 1 + cos A cos B R = cos B (15) (16) The blade pitch and form of the back depend upon the radius r. As stated in Chap. XV, as large a pitch as practicable probably tends to reduce friction, and this is obtained by making r small. In impulse 42 658 DESIGN AND CONSTRUCTION OF HEAT ENGINES blading the center of radius r usually lies within the blade section, some- times on the curve of radius R. In Fig. 443, p is the maximum pitch which will at least theoretically direct the discharge steam at the angle B relative to the blade. It may be easily shown that: A +B cos p = ^_ x cos B Or: cos A + B p = x cos B (17) FIG. 443. Good results will usually be obtained when x = R\ then: cos A + B b- cos A + B p = cos B cos 2 B a + b 2 cos B- cos A + B (18) Formula (18) may be used to find the pitch tentatively. If all spaces are to be equal a slight change may be necessary. There is no definite rule for spacing and it is probable that more or less latitude is permissible. TURBINE WHEELS 659 Strength of Blades. Formula (24), Chap. XV, gives the tangential force f w which must be divided among the blades taking steam at one time. For full peripheral admission this would be all the blades on a wheel. Formula (25) Chap. XV, gives the axial force f F) this being usually small. If I is the distance from the center of the blade to the weakest section in inches, and n is the number of blades receiving steam, the bending moment is: M = f - - I (19) n where / refers to either f w or f F . If Z is the section modulus of the section considered and S is the stress: & = ~ (20) Should the section be irregular, as the blade section, Z may be found by the graphical method given in Appendix 1. Stress of the same sign (either tensile or c6mpressive) may be determined by (20) at point of maximum stress by using both f w and f F , which act at right angles to each other; the sum will be the maximum stress. Reducing S c by making t greater, reduces the value of n, but the ratio, will give greater values of t at all values of r, so that nothing is gained. If certain values of t are desired at hub and rim, n may be found from (8), S c from (7) and S T from (9). Factor of safety and material may then be chosen. Unless an elaborate mathematical analysis is used, it is probably safer to cause intermediate values of t to lie on the curve plotted from (8) . This is one of the cases where the addition of metal may be a source of weakness rather than strength. 219. Application of Formulas. Assume a wheel similar to Fig. 444. Let the weight of the blades be 35 Ib. (W B ) and the total weight of the rim 74 Ib. (W K ). Also let N = 5000, D = 25.125 in. and r ? = 12.562 in. Let the material be nickel steel with an elastic limit of 50,000 Ib.; then with a factor of safety of 2.5, S = S c = 20,000. Then from (5), q = 0.445. From (6), W = 76 and from (7), to = 0.73. Assume that d = 0.2D . From (1), S H = 31,500. By several trials, solving for stress at diameter d, n was taken as 1.5. Then (9) and the charts give: S T = (0.355 X 31,500) + (0.425 X 20,000) = 19,700. For r/r = 0.9, S T = 17,510 and for r/r = 1, S T = 18,380. At the outside of the hub where r/r = 0.24, S T = 18,400. As the hub strength- 660 DESIGN AND CONSTRUCTION OF HEAT ENGINES ens the disc in this region, n might have been less. The thickness at the hub where r = 3.5 in. is given by (8), and is 3.68 in. Values of t for various sections may be found by (8) and the curve drawn. Taking S = S T , and S L = 0.7 S , (4) gives: m = 3.63. This is some larger than the value assumed, and the value of S L was based upon the other value, but the result is probably as nearly correct as the method in general. 220. Design from Practice. Wheels are made of various forms, from a disc of uniform thickness to the disc of uniform strength of the DeLaval Defiail of Rim FIG. 444. Southwark-Rateau turbine wheel. Section B~B Class A turbine. The latter is designed by Formulas (7) and (12). A disc of uniform thickness is expressed by (2) when n = 0. In some cases t is found from (7); then t at the hub from (8), so that S T from (9) is within limits; then a straight line connects these two dimensions, making a disc easy to machine. Figure 444 is a wheel designed to run 5000 r.p.m. by the Southwark Foundry and Machine Co., Philadelphia, Pa. The curve of this wheel does not check with (8) ; an approximation was probably made to simplify construction. TURBINE WHEELS 661 FIG. 445. DeLaval Class C turbine wheel fastening. FIG. 446. DeLaval multi-stage turbine wheel fastening. 662 DESIGN AND CONSTRUCTION OF HEAT ENGINES Figure 445 shows the method of attaching the wheel of the DeLaval velocity-stage turbine to the shaft. Each wheel is held by a Woodruff key, and the wheels are spaced and centered by rings. All are held against -609 FIG. 447. Kerr turbine wheel fastening. FIG. 448. Rim of Curtis turbine wheel. a shoulder on the shaft by a nut. Fig. 446 shows the method of attaching the wheels of the DeLaval pressure-stage turbine to the shaft. Tapered sleeves are keyed to the shaft, the wheels being drawn onto these by a nut. FIG. 449. Ridgway turbine blading. The labyrinth packing between the diaphragms and wheel is also shown. Figure 447 shows the method of fastening the disc to the turbine shaft TURBINE WHEELS 663 used on the Kerr Economy turbine of the smaller sizes. The hubs are prevented from turning by dowels. The discs are of uniform thickness. In the larger sizes the wheels are made in one piece and held to the shaft by keys. A section of the rim of a Curtis turbine wheel, built by the General Electric Co., is shown in Fig. 448. This is a 2- velocity stage wheel used in the first pressure stage. The blades shown are drop-forged with dis- tance piece integral with blade. A projection on the blade passes through a shroud ring and is riveted. This strengthens the blading and forms one side of the steam passage through the blades. FIG. 450. Ridgway turbine blade. The method of fastening the blading of the Ridgway-Rateau-Smoot turbine is shown in Fig. 449, and a detail of the blade in Fig. 450. In the Ridgway blades no fillers or shrouds are used, as these are formed by the blades themselves. These blades are machined all over and usually made of monel metal. An advantage of the bulb-end type of blades is that in case of accident to any of the blades, any one may be removed without disturbing the rest. References Dynamic balance A.S.M.E. Journal, August, 1916. Blade fastening for steam turbines Power, July 20, 1915. Strength on materials Prof. Arthur Morley. Design and construction of steam turbines. .H. M. Martin. CHAPTER XXXII TURBINE SHAFTS Notation. d = diameter of shaft in inches. di = inside diameter of hollow shaft. d B = diameter of journal. I = length in inches from center to center of bearings. x = distance to any point on shaft in inches; used in derivation of formulas. a = length of section of shaft to be taken as isolated load = q-Ax. Ax = actual measurement in inches on bending moment diagram, corresponding to a. h pole distance in inches on vector diagram for bending moments. k = same for deflection diagram. z = actual measurement in inches corresponding to bending moment on diagram. y = same for deflection. p = weight scale = pounds per inch. q = distance scale = inches per inch. m = units of z-Ax/d* per inch. 5 = actual deflection in inches under any load. W = weight of isolated load in pounds. w = weight per inch of length. S = shearing stress in journal due to torque. E = modulus of elasticity. / = moment of inertia of shaft section. M = bending moment at any point. M M = mean bending moment in length a of shaft, or in length Ax of diagram. N = normal r.p.m. of turbine. N c = r.p.m. at critical speed.' co = angular velocity at critical speed, in radians. V = linear velocity in feet per second of isolated load W revolv- ing at radius 5. H = horsepower. 664 TURBINE SHAFTS 665 221. Nature of the Problem. The rotative speed of the reciprocating- engine shaft is usually so low that vibration needs little or no considera- tion. In small gasoline engines with very high rotative speed the shaft is so well supported by bearings that vibration is not usually considered. With the shaft of the reaction turbine the drum construction is such that the required stiffness is attained easily for the speeds at which this type is operated. For the multi-stage impulse turbine with disc wheels, the shaft receives comparatively little stiffening from the hubs and this is neglected in calculation so that the shaft must maintain stability by its own stiffness. It must not run at a speed anywhere near its critical speed. The determination of the shaft diameter is therefore an entirely differ- ent problem from that of the engine shaft, the latter being one of bending and twisting. These are both insignificant in the turbine shaft except at or near a certain speed the critical speed. For preliminary calculations it is convenient to have some formula to assist in fixing dimensions; then these may be checked by the methods of the following paragraphs. If the shaft be assumed to transmit power, the smallest diameter, which may be taken at the journal, will be given by the formula: where H is the horsepower, S the shearing stress and N the r.p.m. The stress S must be taken low 2000 to 2500 to give practical results. Length of bearing may be determined from Par. 52, Chap. XI, and any necessary compromises made. The shaft between the bearings is larger, sometimes having a number of different diameters increasing toward the center. A tentative determination of an equivalent straight-sided shaft may be made by assuming a uniform load composed of all the wheels and a shaft of a greater diameter than that given by (1) ; then by the general principles of Par. 222 the following formula may be derived: 1850 lEI 1850 \EIl /0 , Nc " ~ : -- where E is the modulus of elasticity, / the moment of inertia, w the load per in. of length including shaft, W the total weight in Ib. and I the length of shaft in inches between bearing centers. For a solid shaft, taking # = 30,000,000: N c = 2,250,000 2 (3) 666 DESIGN AND CONSTRUCTION OF HEAT ENGINES NO should be much greater or much less than the actual shaft speed, N. By assuming the number of r.p.m., (2) may be written: (4) JH-7CFV/.LV Or from (3) *- l 1490 As actual conditions are usually not so favorable to stiffness, the actual critical speed will probably be less than that given by (2) or (3) . A good deal of information about the principles underlying critical speed maybe found in Morley's Strength of Materials, to which the author is largely indebted for the ideas of the next two paragraphs. An abso- lutely correct solution is probably impossible from the practical stand- point, due to various influences which may not easily be taken into account, but a method will be given which is accurate in principle to within narrow limits; if this is carefully applied, practical results may be expected. 222. Critical Speed. If a body is acted upon by a periodic force having the same frequency as the natural vibration of the body, the total energy will be increased at each application, increasing the strain energy and therefore the stress. If this continues until the elastic limit of the material is reached, distortion of the body will result. The frequency causing this is called the critical frequency. When a body vibrates, the energy is of two kinds, kinetic and poten- tial, their sum being constant. In the mean position the velocity is a maximum and the energy is all kinetic; at the extremes it is all potential. The motion of natural vibration is simple harmonic, and therefore displacements may be treated as projections on the plane of vibration, of a body revolving with a constant angular velocity. Assume the body in question to be a turbine shaft, deflected under its own weight and that of the disc wheels; with carefully balanced shafts this weight may be considered as the periodic force, the period being that of rotation. The force acts in but one direction, but the rotation of the shaft presents opposite sides alternately to the force, producing the same effect as forced vibration. If the rotative speed is the same as the natu- ral period of vibration it is called the critical speed; then the potential energy required to force the shaft to its deflected position is equal to the kinetic energy of the entire mass, should its axis whirl on a surface of revolution developed by the center of the deflected shaft, the speed of, whirling being the critical speed. TURBINE SHAFTS 667 If the system is not in perfect balance there is a tendency to whirl at all speeds below the critical speed, the latter probably being lower on this account. As the speed is increased beyond this point the tendency to whirl is reduced, and the shaft tends to revolve around the axis of gravity of the system. It is more convenient, and sufficiently accurate for practical purposes to assume each wheel or portion of shaft as an isolated load. Let these loads be TFi, W 2 , etc., and the deflections under the loads 5i, S 2 , etc. re- spectively. These deflections are radii of the circles the loads are as- sumed to revolve upon at the instant the critical speed is reached. The potential energy of these loads in ft.-lb. is: The kinetic energy is : 900 X 2880 Equating (6) and (7) gives: T *C 2880 2 (W S 2 ) (7) 900 X 2880 24 From which: If in (8), load w per unit length were used in place of W, the expres- sion would be: Then from the equation of the elastic curve, if the value of d in terms of x be substituted, the result for a shaft of uniform section, with uni- form load, and for ends supported, would be given by Formula (2) . 223. Deflection. Before Formula (8) can be applied it is necessary to determine the deflection under each load W; this may best be done graphi- cally. The underlying principles of what follows are thoroughly treated in Morley's Strength of Materials, but an attempt will be made to ex- plain their practical use in such a way that they may be applied if this fundamental knowledge is lacking. 668 DESIGN AND CONSTRUCTION OF HEAT ENGINES The general equation of the elastic curve, when the curvature is slight, is: dx 2== Kl ( 9) Integrating gives: dv CM i-jm : This gives the slope at a certain point in the deflection curve; another integration gives y, the deflection from some reference point. Another fundamental beam equation is : Ax ^T '' ~Ef~ (16) Mrftb K-a,-> <- a r ^--ffj-f r-a 4 -> t-a f + t-e ff .++a.. FIG. 452. The quantity Az is in actual inches on the diagram. quantities in brackets are variable. For solid shafts: In (16) the 64 where d is the diameter of the shaft in inches. Should the shaft be hol- low, d 4 may be replaced by d 4 di 4 , where d is the inside diameter. As most shafts are solid, (16) becomes: M M a _ 64pq z h/ ~wr *E v (17) The quantity in brackets may be plotted. As the product z-Ax is small, it may be better to include part of the constants in the brackets in TURBINE SHAFTS 671 some cases, especially if d is large. With a shaft of uniform diameter between bearings, d is constant and may be taken out of the brackets. When it is decided what shall be plotted, the quantities may be laid off in succession as shown in Fig. 452C ; a pole may be located at any point as shown, the rays drawn and the corresponding links laid off parallel to them on B. The quantities y and k are in actual inches on the dia- grams; the former must be multiplied by the distance scale q and the latter by the scale for the quantity plotted ; or : . . ., , z-Az 1 in. = m units of n - d 4 In addition to this the constants of (17) must be used, as the entire quantity: M M a El must be used to determine the deflection d. The vertical intercepts of Fig. 4525 are proportional to the deflection; then: ''" ' i.y^.y-Ky ' * : (18) Should the wheels have offset hubs, and not be located at the centers of the steps, separate moment diagrams may be drawn, one for the shaft and one for the wheels, taking the loads at the center of gravity of wheel and step in each case. Should the steps be long they may be divided into two or more loads. These two diagrams may be combined ; then dividing into suitable lengths, each one of which covers a portion of the shaft of constant diameter, the deflection curve may be plotted and the deflection under each load found. The shaft steps and wheels may be kept separate if desired in determining S(TFS) and 2(W d 2 ). The deflection curve may be drawn more correctly by an inscribed curve touching the sides of Fig. 4525. The dimensions h and k should be chosen so that the moment and de- flection curves will not be too flat, as it is easier to draw a deeper dia- gram accurately. 224. Application of Formulas. As an example a 13-stage turbine has been taken. The data was partly assumed and not very accurate. The diagrams were similar to Figs. 451 and 452 and will not be repeated. In plotting the deflection curve, a was taken instead of Az; this re- moved one q from the constant in (18). The scales were: q = 6; p = 300; m = 0.04. Also: h = 3 and k = 3. From (18), K = 0.00264. The length from center to center of bearing is 67 in. and the total weight 1850 Ib. The remainder of the data is placed in Table 102. 672 DESIGN AND CONSTRUCTION OF HEAT ENGINES TABLE 102 No. w a d z az d* y d W8 TFS2 1 80 10.80 4.30 1.42 0.0440 1.44 0.0038 0.304 0.00116 2 170 5.12 4.40 2.49 0.0220 2.20 0.0056 0.950 0.00532 3 100 2.80 5.30 3.40 0.0117 2.57 0.0068 0.680 0.00463 4 100 2.97 5.52 3.33 0.0106 2.74 0.0072 0.720 0.00520 5 145 2.97 5.67 3.64 0.0104 2.92 0.0077 1.120 0.00863 6 145 2.97 5.80 3.83 0.0100 3.02 0.0079 1.145 0.00905 7 145 2.97 5.93 3.94 0.0094 3.10 0.0082 .188 0.00975 8 145 2.97 5.93 3.96 0.0095 3.12 0.0084 .220 0.01025 9 145 3.24 5.80 3.90 0.0112 3.11 0.0083 .205 0.01000 10 175 3.91 5.40 3.73 0.0172 3.04 0.0080 .400 0.01120 11 200 4.60 5.27 3.43 0.0204 2.86 0.0075 .500 0.01125 12 250 5.40 4.85 2.82 0.0275 2.48 0.0065 1.622 0.01060 13 80 10.80 4.30 1.52 0.0480 1.48 0.0039 0.312 0.00111 Taking S(TF6) and S(TF5 2 ), Formula (8) gives for the critical speed: If an equivalent diameter of 5J^ in. be assumed for a straight shaft, and the total weight taken as a uniform load, (3) gives: N c 2800. The determination of N c by either (8) or (3) should be greatly different from the actual turbine speed for safety probably at least 100 per cent. greater, or 50 per cent, as great. It is practically impossible to consider all factors which enter into the problem, but with due allowance the method may be considered safe if the shaft is taken as a beam supported at the ends, and only loads between supports be considered. Jude says that "the chief concern is whether the critical speed is (e.g.) 300 or 3000 r.p.m.; not whether it is 300 or 310." Martin says it is not uncommon to run turbines of the disc type above the critical speed. "In that case, however, the efficiency of the turbine may be expected to diminish with time, since on every occasion on which the turbine is started up or stopped, the rotor has to pass through its critical speed and the consequent vibration gradually enlarges the fine clearances used where the shaft passes through the high-pressure diaphragms." Jude further says: "It is found that if the speed be increased quickly, so that the critical velocity is of only passing influence, the whirl quiets down so much so, in fact, that the stability of the system is in general greater than at speeds below the critical." He also states that the normal TURBINE SHAFTS 673 speed should be as remote from the critical speed as possible, and that properly it should be above rather than below. In the single-stage DeLaval turbine the speed is far above the critical speed ; concerning the multi-stage turbine, however, it is said in Catalogue D: "The use in a multi-stage turbine of a shaft running above or near its critical speed is an error, not only on the grounds of safety and freedom from vibration, but also because any whipping or eccentric rotation of the shaft will require enlarged clearances where the shaft passes through the diaphragms. In other words, the leakage areas will need to be consider- ably increased. It therefore follows that the total leakage is reduced by using a shaft sufficiently large and stiff to suppress such vibration entirely, thereby permitting the radial clearance to be correspondingly reduced, although the larger shaft has a greater circumference." The use of material for wheels which permit the use of high stresses, will both lighten the hub and shorten the fit, making possible a shorter shaft. A straight shaft is stiffer than a stepped shaft of the same diam- eter at center, so if some method of mounting wheels be used which permits a straight shaft, it will increase the critical speed. For turbines of the drum type the shafts are stiffened by the drum, and the critical speed is probably never reached. But high speeds are not possible with this type as may be seen from Chap. XXXI. References The design and construction of steam turbines H. M. Martin. Strength of materials Prof. Arthur Morley. CHAPTER XXXIII TURBINE CASINGS AND DETAILS 225. There is very little calculation involved in the design of casings, which are practically the frames of turbines; therefore this chapter will be comprised of illustrations and descriptions of some of the details of several makes of turbine. One of the principal problems in connection with casings is to provide suitable clearances for uneven expansion of casing and rotor. This be- comes more of a problem in large turbines, so it has been customary to divide the turbine into high-pressure and low-pressure elements, some- times on separate shafts, forming cross-compound turbines, and some- times on the same shaft as tandem-compound turbines. Recently very large turbines have been placed in a single casing. An accident to such a turbine, doubtless due to expansion, is described in Power, March 19, 1918. Some idea of turbine casings may be obtained from the illustrations of Chap. IV, and these may supplement those of this chapter. But few dimensioned drawings are given ; the treatment is therefore more quali- tative than quantitative. In some cases assembly drawings are given, the relations of the parts being better seen from these. No special order is adhered to, the parts of a given make being given together. 226. DeLaval Turbine. A general view of the casing of the Class B turbine is shown in Fig. 453. The lettered parts are known as: A. Wheel case. B. Wheel case cover. C. Turbine wheel. D. Inner packing bushing bracket. E. Inner packing bushing. F. Outer packing bushing bracket. G. Outer packing bushing. H. Outer bearing bracket. K. Insulating cover. L. Exhaust flange. M. Governor. V. Turbine shaft. Figure 454 shows a partial section of a velocity-stage Class C turbine. The steel retaining ring completely covers the wheels and serves to protect the cast iron casing in case of accident to the wheels. 674 TURBINE CASINGS AND DETAILS 675 Figure 455 shows the nozzles for this turbine, A being for the high- pressure condensing turbine, therefore having a great divergence; B being FIG. 453. DeLaval Class B turbine casing. FIG. . 454. Section of DeLaval Class C turbine. for low-pressure condensing, or high-pressure noncoudensing turbines, has less divergence. 676 DESIGN AND CONSTRUCTION OF HEAT ENGINES FIG. 455. Nozzles for DeLaval Class C turbine. FIG. 456. Outboard bearing bracket for DeLaval Class C turbine. I FIG. 457. Radial and thrust bearing of DeLaval Class C turbine, TURBINE CASINGS AND DETAILS 677 FIG, 458. Retaining rings of DeLaval multi-stage turbine. FIG. 459. Carbon packing of DeLaval multi-stage turbine. 678 DESIGN AND CONSTRUCTION OF HEAT ENGINES Figure 456 shows a sectional elevation of the outboard bearing bracket and flexible coupling of the Class C turbine. The bearings are ring-oiling; they are entirely separate from the stuffing boxes, so that no steam or water can reach them or enter the oil reservoir. Figure 457 shows the combined radial and thrust bearing and governor drive of the DeLaval velocity-stage turbine. Figure 458 shows the steel retaining rings of the DeLaval pressure- stage turbine. FIG. 460. Clearances of Ridgway turbine. The carbon packing at the low-pressure end of the DeLaval pressure- stage turbine is shown in Fig. 459. Provision is made for introducing "live steam at a reduced pressure between the second and third rings, so that any leakage into the turbine will be of steam and not of air. The labyrinth packing between the diaphragms and hubs of the wheels is shown in Fig. 446, Chap. XXXI. 227. Ridgway Turbine. Figure 460 shows sections of the first four stages of a Ridgway turbine. This shows the comparatively large blade clearances used in these turbines. The clearance A is % Q in. or more; B is % m - r more, and C is J in. or more. CHAPTER XXXIV GENERAL ARRANGEMENTS AND FOUNDATIONS Notation. h = depth of foundation in feet. d = diameter of foundation bolts in inches. D s = diameter of standard steam engine cylinder in in. (see Chapters XII and XIII). Wp = weight of foundation in pounds. WE = weight of engine in pounds. v = volume of foundation in cubic feet. N = revolutions per minute. 228. General Arrangements. For ordinary standard engines general arrangement drawings are not usually required. For special work or for some special arrangement of piping they are required. In any case, especially with large engines of the cross-compound, twin or duplex types, such drawings are a useful check and are valuable for the erecting engineer. A simple layout of a Bass-Corliss engine is shown in Fig. 461. The engine is a 26 and 54 by 48 in. cross-compound, provided with a rope wheel. The wheel is in four parts, each having its own hub and arms and bolted together at the flanges Two wheels carry eight 2 in. ropes each and the others each carry nine 1J^ in. ropes. The receiver and piping are under the floor, and so arranged that the connecting rod of either engine may be disconnected and the other engine continued in operation. Should the low-pressure engine carry the load, live steam which has been passed through a reducing valve is admitted to the receiver. Engines so built by the Bass Foundry and Machine Co. are heavier than the ordinary compound engine; the long-range cut-off gear is used so that one engine may carry the normal load continuously with a reasonable overload in case of accident to the other side. This is desirable in plants with few engines perhaps only one in which an acci- dent requiring a complete shut-down would be serious. 229. Foundations are provided to secure the engine and to absorb vibration. The mass required for the latter depends upon a number of factors, but aside from cost, it is better to have too great than too small 679 680 DESIGN AND CONSTRUCTION OF HEAT ENGINES a mass. In fact, greater cost in building the foundation may sometimes prevent far greater cost in corrections made necessary by excessive vibration. A formula for extreme foundation depth devised by the author some years ago for Corliss engines is : (1) FIG. 461. General arrangement of Bass-Corliss engine. where h is the depth in feet and D s the diameter of the standard cylinder used in Chaps. XII and XIII. This formula is entirely empirical and in- tended for ordinary conditions of soil, etc. It has been used for many large Corliss engines and so far as is known has proved satisfactory. This may be due to the fact that the proportions are rather massive, but concrete is comparatively cheap. GENERAL ARRANGEMENTS AND FOUNDATIONS 681 A formula of a more scientific form is given by E. W. Roberts in his Gas Engine Handbook, which, with change of notation is: W F = KW (2) W F is the required weight of foundation in lb., W E the weight of the engine and N the r.p.m. The value of K in the seventh edition of the handbook is 0.35. After consulting with ten leading builders of stationary gas engines in this country, Mr. Roberts calculated values of K from their data for various types of engines. These values, with much other valuable matter relating to foundations was published in The Gas Engine for September, 1916. The volume in cu. ft. of a concrete foundation is: v = kW (3) Mr. Roberts does not consider these investigations final, but at present they are the best effort that has been made toward the solution of the in- ternal-combustion-engine foundation problem. The values of K and k proposed by him are given in Table 103. TABLE 103 Type of engine K k 4-cylinder vertical gas engine 0.130 . 000975 3-cylinder vertical gas engine. . . 150 0.001130 2-cylinder vertical gas engine 175 001310 4-cylinder vertical Diesel engine 177 001330 Single-crank, double-acting tandem 0.320 0.002400 Twin-crank, double-acting tandem 0.190 0.001430 Single-cylinder horizontal semi- Diesel 300 002250 2-cylinder horizontal semi- Diesel 240 001800 3-cylinder, horizontal semi- Diesel 4-cylinder horizontal semi- Diesel 0.230 0.225 0.001730 0.001690 2-cycle horizontal semi- Diesel . . 0.230 0.001730 Material Foundations were formerly made of brick but concrete is now mostly used. A mixture of 1 part of fresh Portland cement, 3 parts of sharp clean sand and 5 parts of gravel or crushed stone is commonly used. In some cases a 1, 2, 4 mixture is used; this is stronger and may be better under certain conditions. Bolts and Washers. The determination of the size of foundation bolts G82 DESIGN AND CONSTRUCTION OF HEAT ENGINES is arbitrary; however, if too small they break, causing a great deal of trouble. For Corliss engines the author devised the formula: ,7 ft -J. 1 tA\ d = TF + o W 3*r PTI where d is the bolt diameter in inches and D s the diameter of the standard cylinder as before. The number of bolts properly influences the diameter, but in (4) it is assumed that the usual number and arrangement of bolts are used. Square nuts are used on the lower ends of the bolts as these have a greater dis- tance across corners, and do not require such small clearances in the lugs of the washer. Hexagon nuts are used at the "upper end as they require less space and are more easily handled with a wrench. A simple foundation washer is shown in Fig. 462, for which the following formulas apply; the notation is given on the figure and d is the diameter of the bolt as before. o FIG. 462. = d ^+4 B = d D = l.W+^tol|) ; . The dimensions are given in Table 104. Sometimes the lugs which TABLE 104 d, in. B, in. C, in. Z), in. E, in. G 1 , in. IK 2K 2% 3 3M 3K 3% 9 LO 12 13 15 16 18 19 21 22 24 4 4% 4% 5% IK 2 2K 4K 6 6% 7% 8 IOK 11 IK 2 2K 2% 3 3K 4 K 1 IK IK 2 2K GENERAL ARRANGEMENTS AND FOUNDATIONS 683 hold the nut are made deeper and are connected, forming a pocket. Then should a bolt break at the thread at the upper end it may be unscrewed from the lower nut and replaced. However, should it break well down in the foundation, the pocket would be a detriment as will be shown presently. Construction. Foundation plans are furnished by the engine builder. In some instances drawings are furnished of wooden templets which are located over the foundation and hold the bolts in position while the founda- tion is under construction. The foundation should be below the frost line if exposed, but they are often in a basement. In small foundations the bolts and washers are sometimes embedded in the concrete; but it is better practice to have the bolt in a pipe, or in a box formed by four boards. This box is smaller at the bottom than at the top and may be removed after the foundation is finished if desired. The pipe or box allows the bolt to be moved in any direction to allow for discrepancies in the engine bed holes. In large foundations the washers are placed in pockets. Tunnels, through which a man may crawl, lead to these pockets. Such a founda- tion by the Bass Foundry and Machine Co. is shown in Fig. 463. This was built of brick for a 22 by 42 in. Corliss engine designed to run 100 r.p.m. The depth was made according to (1), and the foundation bolts of the frame by (4). The boxes for the bolts are not shown, but a note on the drawing calls for holes through brickwork 4 in. square. Should a bolt break the lower end will drop down in the pocket in the brickwork if no pocket is used on the washer. The nut may then be unscrewed by a man in the tunnel, and an ingenious mill-wright or engineer will find some means of removing it. By removing the washer a small chain may be let down from the top and fastened around the thread at the lower end. Grouting. A certain amount of space commonly 1 in. is left be- tween the top of the foundation and the engine frame for grouting. The engine is supported on wedges by means of which it may be raised or lowered until it is properly leveled ; then the grouting is poured in. This may be one of several substances. Sulphur has been used for small engines; and for heavy mill engines, iron and steel turnings are rusted together with sal-ammoniac. Cement is now the most common sub- stance used. When this is well set, the wedges are removed and the nuts on the foundation bolts tightened. The Soil. Ketchem, in his Structural Engineers' Handbook, gives the following values of maximum allowable pressure on the soil in tons per sq. ft. : 684 DESIGN AND CONSTRUCTION OF HEAT ENGINES GENERAL ARRANGEMENTS AND FOUNDATIONS 685 Ordinary clay, and dry sand mixed with clay 2 Dry sand and dry clay 3 Hard clay and firm, close sand 4 Firm, coarse sand and gravel , . . 5 Shale rock 8 Hard rock 20 All inferior soils such as loam 1 While rock is an ideal foundation bed in many ways, vibration may be transmitted through it to the walls of the building. It has been advised to make a pocket in the rock, in which a layer of sand is placed before starting the foundation footings. Vibration may be transmitted through poor spongy soil, and if the plant is in a residence district this may be very objectionable. Increasing the weight of the foundation will help, but this will not always suffice. Balancing the engine seems to be the best remedy for this; in fact, much lighter foundations may be used with well-balanced engines in any case. Horizontal vibration is usually the most troublesome. In a horizontal engine this may be remedied largely by balancing all of the reciprocating parts. In a vertical engine the revolving parts only should be balanced ; but there remains the vibration due to the turning effort which can not be balanced. It has usually been considered bad practice to build an engine founda- tion in contact with the walls of the building; but Roberts, in the article referred to, tells of a consulting engineer in Chicago who does this, his idea being to increase the mass that must be set in motion by the unbal- anced forces of the engine. He further claimed that the results were entirely satisfactory. The opposite practice of isolating the foundation by a bed of sand has already been referred to. Roberts tells of the use of cork for this pur- pose, and a cork sheeting is being used in this country. A concrete pit with a heavy bottom is first made; the sides and walls are lined with this sheet cork; then into this the foundation proper is poured. The following foundation specification was furnished by the Buckeye Engine Co. for a 15J4 by 18 in. vertical steam engine. The depth is 4 ft., and about 15 in. margin is left around the bed casting. There are eight 1% in. bolts and the allowance for grouting is 1 in. SPECIFICATION FOR FOUNDATION Build the foundation of a good concrete mixture as follows: 5 parts clean gravel or broken stone; 3 parts clean sharp sand, and 1 part of good grade Portland cement. Mix thoroughly and ram in. In building, 686 DESIGN AND CONSTRUCTION OF HEAT ENGINES leave room around each bolt hole so that bolts can be moved ^ in. in any direction. This is best done by placing a pipe or a wooden box around bolts, which can be taken out after foundation is finished, or lifted up as the building of the foundation progresses. The drawings represent a sufficient amount of masonry to give the proper weight, but it must be on solid ground. If sufficiently solid ground is not obtained at the depth shown by drawing, the excavation must be continued deeper; the width must also be increased. The additional depth is to be filled with the same concrete mixture. In many cases however, sufficient solidity may be had by a thorough ramming of the ground. The copings can be made of the same material or cut stone, and the whole should have at least two weeks to set before engine is placed. Observe closely the drawing of "templet" for placing foundation bolts. References American foundation practice The Gas Engine, Sept., 1916. Exhaust vibration TheGas Engine, Oct., 1916. CHAPTER XXXV DESIGN METHODS 230. Personal efficiency is one of the first requisites of the designing engineer. Many men are able to promote the business of others to per- fection but are exceedingly negligent of their own. A designer should have a well-organized system of his own whether the concern by whom he is employed has such a system or not. He must be orderly, consistent and thorough, and must not take things for granted. His own records should be well kept as well as those he keeps for the company. He should have note and data books for different phases of his work. He should have the best handbooks and design manuals and be familiar with them. His working formulas should be well selected. His drafting instruments should be kept in good condition, and al- though he attains to a position in which he is required to do little or no drafting, he should never feel above working over the board occasionally, for unless he has traversed this path he is hardly fit to direct the efforts of those who do this kind of work. The designing engineer should never get beyond study. He should keep abreast of the times in his special work and should strive to broaden out some. Technical periodicals help in this, as does also membership in an engineering society notably one of the national societies such as the American Society of Mechanical Engineers. He should be ready to lead in thought as far as he is permitted to do so and must not simply drift along in the regular routine of the office. He should use a good slide rule such as the Log Log Duplex; he can then do more work with less fatigue and therefore of improved quality. Some firms require important calculations to be kept in note books which are the property of the company. This is good practice, and if required should be attended to conscientiously. If not required it is well for the designer to keep such data himself. If in charge of the department and responsible for shop orders, parts most needed or requiring the most time to obtain should be ordered first ; this includes large or difficult castings, large forgings, and sometimes steel castings. It is sometimes difficult to do this as some of these parts are logically the last to be designed, but with proper system in designing this can usually be managed, so that the work may be started before the 687 688 DESIGN AND CONSTRUCTION OF HEAT ENGINES design is complete. This is not the most desirable practice from the de- signer's standpoint, but the writer is sure that it may often be done to great advantage. Printed lists are usually furnished for the parts built by different departments. If this is not done, a list may easily be made on tracing cloth with spots blocked for drawing or sketch numbers. In this way nothing will be overlooked. A list of Corliss engine parts prepared for the forge shop is given in Table 105. Space is left for the machine shop draw- ing showing the finished part as a matter of convenience to the office. TABLE 105. THE BASS FOUNDRY AND MACHINE COMPANY, ENGINEERING DEPARTMENT Subject FORGINGS Order Answering SMITH-SHOP Fort Wayne, Ind Name of part Drawing No. Sketch No. Pieces Crank shaft Connecting rod Eccentric rod Reach rod Piston rod Valve stems Crank pin Crosshead pin Crosshead key Hook bolts, main bearing T-head bolts, main bearing Hook bolts, outer bearing T-head bolts, outer bearing Extension bed links Girder bolts Flywheel hub bolts Flywheel rim bolts Flywheel links bolts Flywheel keys Rope wheel hub bolts Rope wheel rim bolts Rope wheel tie rods Rope wheel keys Gear bolts Gear links Gear keys Carrier shaft Foundation bolts Eccentric strap bolts Conn, rod wedge bolts DESIGN METHODS 689 The smith shop sketch numbers, number of pieces and the date of order are also given. A duplicate copy should be kept in the office, and when a piece is ordered, this may be signed by the smith shop clerk when the sketch is delivered. Similar lists may be made for other departments. Standards. Much work in heat engine design may be standardized, and this has been shown through the chapters on machine design. The application of standard tables to other pressures and to compound engines is explained in Chaps. XII and XIII. The main dimensions may be in- TABLE 106. STANDARD ENGINES Name of Part Drawing No. Cylinder Girder Guide barrel No. Main bearing No. Outer bearing Cylinder heads Cylinder foot Piston Crosshead Crank . Size of engine Class of engine Main bearing diam , length. ~ Crosshead { Screw & Locknut I Taper fit & key (Arm (Disk Wrist plate Carrier Center foot Cent, of cyl. to cent, of shaft = Cent, of eng. to bottom of castings Remarks : I Double { Single t Double Eccentric Valves Bonnets Valve gear Dashpots Release rig Carrier pin stub Valve gear pin stub Governor Connecting rod. . . . Eccen. & reach rods Piston rod Valve diagram .... Si 1 Double (Solid } gtrap f Single ]_ , . Double 690 DESIGN AND CONSTRUCTION OF HEAT ENGINES eluded in one table for engines designed for a standard pressure, making it convenient in furnishing data for specifications. Horsepower tables may also be computed, but such calculations are so quickly made with a slide rule that this may be of little use. As soon as an engine of any size has been standardized it is of great convenience to enter the drawing numbers in a blank similar to Table 106, which was prepared for Corliss engines. This greatly facilitates getting orders into the shop. In furnishing date to salesmen for work not included in standard tables, the blank shown in Table 107 has been found useful. TABLE 107 Subject Order Answering Fort Wayne, Ind. Maximum . . Regular Piston speed Steam pressure Ibs. R.p.m. Minimum Economical horsepower with cut-off in h.p. cyl Maximum horsepower with cut-off in h.p. cyl Run over or under Belt forward or back Diam. of wheel Width of face for belt No. of ropes Diam Weight of wheel Ibs. Main bearing Outer bearing Counter shaft bearing Main shaft diam swelled to Count, shaft diam swelled to Guide barrel Bed Diam. steam inlet to cyl. h.p l.p ~ , . . f H.p ~ A , . , , Face of piston i T Diam. steam outlet to cyl. h.p l.p I L.p , J H.p ~ , , . 1 Diam , { Diam Diam. piston rod i T Crosshead pin j T ,, Crank pin \ , A . ( L.p I Length I Length Connecting rod diam. at neck At center Arranged so that h.p. or l.p. engine may run alone and carry whole load Further data : Partial-assembly Sketches. Sketches of groups of parts are of great convenience in design. They are also helpful in checking existing design, as any interference or other discrepancy is readily seen. These may be made on tracing cloth and a set of blueprints blocked for inserting di- mensions used for each case. These may be kept on file for reference. If certain dimensions are not required a dash may be used. These sketches, with the standard tables previously referred to are useful in getting out large castings such as the frame much sooner than can be done by the ordinary process of design. Figs. 464 and 465 are part of a DESIGN METHODS 691 FIG. 464. Fio. 465. 692 DESIGN AND CONSTRUCTION OF HEAT ENGINES set of such sketches used by the author in Corliss engine design. They are self-explanatory. The above suggestions are given as a few examples of how work may not only be expedited, but the liability of error lessened. A good system will be found in a large number of places, but if not, the suggestions may be helpful. The lists given are more applicable to smaller plants, but the same principle will apply to plants of any size. If data is looked up for a given problem, it is well to put it in a data book in good form for future reference. The same may be said of formulas derived, care being taken to give all notation in the proper units. They are then readily available should such problems again arise. Many other ways will suggest themselves if one starts to systemize, but all system should be as simple as possible so that it may not become a burden. APPENDIX 1 MOMENT OF INERTIA OF IRREGULAR SECTIONS A graphical method of finding the moment of inertia is given in Mor- ley's " Strength of Materials, " with a full explanation of the theory. The method will be briefly described here without proof. In Fig. 466, the outside irregular line bounds the section of area A, whose moment of inertia is required. Draw any axis XX below the figure normal to the plane of bending (parallel to the neutral axis); also draw line SS any distance above the figure. Across the figure at any point draw line PQ parallel to XX. Where line PQ cuts the outline N N, M/M FIG. 466. of area A, erect perpendiculars PN and QM as shown. Connect points N and M to any point in line XX. Lines NO and MO cut line PQ at PI and Qi, and these are points in an interior area AI. By drawing a number of lines PQ, enough points may be found through which to draw the curve enclosing area A\. By erecting perpendiculars from PI and Qi and connecting points NI and MI with 0, points P 2 and (? 2 are found, locating points in a third area A 2 . Having found the three areas, and knowing the distance d between lines SS and XX, we may proceed as follows: The distance of the center of gravity of the section (of area A) from line XX is: (D 693 694 DESIGN AND CONSTRUCTION OF HEAT ENGINES The moment of inertia about axis XX is: Ix = A z d* (2) And about neutral axis GG: I Q = I x - Ay* (3) The modulus of section is: . z=^ ^:v: (4) where c may be either c s or c x ; if the elastic limit in tension and com- pression is the same, the larger value of c must be used. If M is the bend- ing moment in Ib. in., and S the stress in Ib. per sq. in.: M = * (5) C and Me , . b = -j- (6) J G The areas may be found by a planimeter. If not drawn actual size and one inch on the drawing represents n actual inches, d must be multi- plied by n, and all areas found by the planimeter by n 2 ; then these values may be used in the formulas. APPENDIX 695 I I 1 1 Z, Q 5 s * 2 s-. ! ^ o V w ^ 5 11 |z 5-5 I *(5 * | 1 c^ &13 Q M4J 8.2 cox t-K i-N \co w\ \oo \IM i N O* CO C0\ rH \ r-K ^ <* N O \<;o -rj- \w v rH COX \5O rH\ > \ \00 rH\ N rH \ CO rH\ jvj ' >* ^e \ Tj< to \ \ec VH \M \ to Oi\ ^-N n\ \oc i-N v-i \T)< rH rH Cq t-\ rH\ r-l\ to H O>\ r> 1 ri- CO' iii-^iOiOiOO5COO c< o r- t^C^^i itOrHi (tOI>(NOi T cf i>" co" oT o o o o o o o o o o o o *-! I-H TH T-I co co -^ o co i^ as O O O O O O O o O O O (N \ rH\ CO\ \rH r-K 1QK CO\ l-K K VH \O \CO \00 \N NflO K rH\ VH VH \^0 V0 V VH Nj-l rH iH t-\ C4 C4 rH i-Hv \ O5\ COK rH\ IO\ N CO COK t^\ 5\ t^\ rH\ UJ\ OS\ Nrf< VH \oo VH \e* VH \PO v* \oo \oo V* \oo \^ \oo VJ( \oo r-K "5\ COK l-K i-K 0>\ >0\ COK t-K r-K i-K COK rH\ >CK COK bX INDEX Absolute pressure, 10 Acceleration, forces due to, 296 of reciprocating parts, 297 Klein's method, 298 Adiabatic expansion of gas, 56, 59 of steam, 67 Admission, 8 Alloy steel see steel. Aluminum, 474 Angular acceleration of connecting rod, 299 Angular advance, 411 Angularity of connecting rod, 415 Area of ports, 402 Automatic cut-off engine, 17, 139 Axial thrust, 225 Bearings, proportions, 115 steam turbine, 117 thrust, 119 Belt forward or back, 15 Bending and twisting, combined, 481 Bilgram diagram, 417 Blade efficiency, 257 Blades of steam turbines, 24, 216, 217, 218, 222, 255, 271, 277, 657 Blading, impulse practical notes on, 228 Blast-furnace gas, 38 Bleeder turbine, 30 Bolts and nuts, U. S. Standard, 695 Bolts, strength of, 695 Brake horsepower, 53, 98, 187 Bruce- Macbeth gas engine, 43, 190 Busch-Sulzer Bros. Diesel engine, 45 Back pressure, 11, 131 Balancing, 334 angle engine, 341 connecting rod, 343 four-crank engine, 340 multi-cylinder engine, 338 primary, 334 reciprocating parts, 336 secondary, 337 six-crank engine, 342 three-crank engine, 340, 341 turning effort, 345 two-crank engine, 340 Balance weights, 346 Basis of design, 493 Bearing cap bolts, 619 Bearing wedge bolts, 617 Bearings, clearance, 113 eccentric, 120 main, 617 metal, 114 oil grooves, 113 Cam valve gears, 448 Capacity curves of Continental engines, 49 Cap bolts, 619 Card factor, 135 Carnot's cycle, 54 Cast iron, 469 Center of gravity, 285 of irregular section, 693 Classification of internal-combustion engines, 32 of steam engines, 13 of steam turbines, 24 Clearance, 79, 131, 202, 503 in bearings, 113 in internal-combustion engines, 202 Collars for shafts, 609 Column formulas, 485 Combined bending and twisting, 481 Combined indicator and inertia diagrams, 301, 303 Combustion, temperature rise due to, 61 697 698 INDEX Compound steam engines, 87, 152, 154 condensing, 155 cross-compound, 20 tandem, 20 Compound turbines, 30 Compression, 11, 79, 131, 145, 194, 196, 304 effect upon capacity of steam engine, 145 effect upon economy of steam engine, 79, 145 effect upon smoothness of operation, 304 in internal-combustion engines, 194 Compression pressure, steam engine, 10, 131, 135 internal-combustion engine, 194 Condensation, cylinder, 81 and re-evaporation, 78 affected by compounding, 87 cylinder ratio, 88 design of cylinder, 82 range of pressure and temperature, 85 ratio of expansion, 85 size of cylinder, 82 speed, 81 steam jacket, 86 superheated steam, 90 Condenser, 265 Condensing engine, 155 Connecting rod, 285, 343, 544 body, 545 center of percussion, 286 ends, 552 moment of inertia of, 287 properties of, 285 radius of gyration of, 285 wedge bolts, 555 Conservation of residual velocity, 248 Constant-pressure cycle, 33, 58 Constant-volume cycle, 33, 55 Continental motors, 46 Conventional indicator diagrams, com- pound engine, 153 internal-combustion engine, 63, 201 simple steam engine, 130 Cooling of cylinder, 37, 41, 94 Cooling water, 94 Corliss long-range cut-off, 432 Corliss valve gear, 426 Counterbore, 503 Cranks, 572 Crank arm, 579 effort, 289 hub, 582 pin, 573, 589 Crankshaft, 586 Crosshead, 562 pin, 563 shoe, 565 Critical speed of turbine shaft, 666 Curve of fuel consumption, 43, 45, 46 of steam consumption, 22, 86 Cushion steam, 133 Cut-off, commercial, 129 long range, 432 to find from m.e.p., 133 Cycle, Carnot's, 54 of internal-combustion engine, 32, 54, 55, 58 Rankine's, complete expansion, 67 incomplete expansion, 68 of steam engine, 3, 6 of steam plant, 3 Cylinder, 499 condensation in, 78, 81 diameter, compound engine, 178 internal-combustion engine, 187, 189 simple steam engine, 143 efficiency, internal-combustion en- gine, 93 steam engine, 78 feed, 133, 157 heads, 504 internal-combustion engine, 509 lagging, 507 lubrication, 122 ratio in compound engines, 88, 154, 155, 157 steam engine, 508 walls, strength of, 478, 499 D Dashpots, 375, 459 Dead center, 8 INDEX DeLaVergne oil engine, 43 Design methods, 687 Diagram efficiency, 225, 226 256 Diagram factor, 33, 34, 40, 58, 192 Dimensions and data for Continental engines, 46, 49 for Bruce- Macbeth gas engine, 44 Double-ported valves, 419 Dry saturated steam, 66 E Eccentrics, 408 shifting and swinging, 420 Economy, 70 comparative, 76 internal-combustion engine, 75 prime mover, 71 steam engine, 71 steam plant, 70 steam turbine, 71 Efficiency, brake, 185 cylinder, 78 Diesel cycle, 59 economic, 185 hydraulic, 237 mechanical, 54, 97, 185, 191, 192 Otto cycle, 56 Rankine cycle, 67, 68 ratio of internal-combustion engine, 76 thermal, 56, 59 volumetric, 60, 189 Electrical horsepower, 53, 209 Engine classification, 13, 32 Entropy diagram, 57, 60, 236, 237, 245 Entropy of steam, 67 Equivalent eccentric, 440, 444 Exhaust muffler, 41 Expansion of steam, 9 Flywheel, 350, 623 arms, bending of, 628, 629 virtual number, 629 constants, 354 hubs, 636 hub bolts, 637 hub shear bolts, 638 methods of construction, 640 rim bolts and links, 639 rim and arm sections, 633 strength of, 623 Unwin's formulas for, 626 weight, comparison of methods, 363 displacement method, 356 velocity method, 351 Formation of steam under constant pres- sure, 65 Formulas, selection of, 475 Forced fits, 537 Foundation, 679 bolts and washers, 681 construction, 683 depth, 680 grouting, 683 material, 681 specification, 685 weight and volume, 681 Four-cycle engine, 35 Frame, 611 center-crank, 611 reactions on, 321 side-crank, 613 stresses in, 611 Friction, 104 effect on flow of steam, 68, 213 laws for dry and lubricated sur- faces, 104 Friction horsepower, 53, 98 Fuel consumption, 43, 45, 46, 193 Fuel nozzle, 456 Factor of judgment, 486 Factor of safety, 464, 493, 497 Filtration of lubricating oil, 122 Flat plate formulas, 501 Flow of steam, 68 G Gas consumption see fuel consumption Gas engine, 33, 38 cycles, 54 Gas velocity, 95, 407 General arrangements, 679 700 INDEX Governing, compound engines, 172 Diesel engines, 454 hit-and-miss, 40, 196 internal-combustion engines, 40, 196, 400, 454 quality, 41, 196 quantity, 41, 197 steam engines, 16, 138 steam turbines, 31, 267 Governor, 367 centrifugal, 369, 377, 378, 390, 395 conical, 371, 378 dashpots, 375 examples from practice, 387 formulas, centrifugal governors, 377 conical governors, 378 speed fluctuation, 375 speed variation, 376 spring governors, 382 gravity balance, 387 hunting, 386 inertia, 371, 395 isochronous, 370 machine design, 387 practical considerations, 386 relay, 374, 396 safety, 374, 400 selection of constants of regulation, 387 speed adjustment, 375 spring, 371, 382, 387, 390, 395 stable, 370 Grease, 109 cups, 112 Guide, pressure on, 300 H Heat engine, commercially successful, 1 important, 1 Heat factor, 73, 209, 237 per stage, 237 Heating value of fuel, 188, 196 higher and lower, 196 Heat of the liquid, 66 Heat supplied in formation of steam, 66 Heavy-oil engine, 33, 39 High steam pressure, 91 Hit-and-miss governing, 40, 196, 354 Horsepower, brake, 53, 187, 189 compound engine, 178 electrical, 53, 209 friction, 53 limit of, 144, 193 indicated, 52, 141, 178, 187 internal-combustion engine, 186 simple formulas for, 189 locomotive, 144 simple steam engine, 141 steam turbine, 208 Hoop stress, 625 Hot bulb engine, 39 Hot plate engine, 39 Hubs, 582, 636 Hub bolts, 637, 638 Hydraulic efficiency, 237 Ignition, 34, 453 Impulse, 23 turbine, 24, 26, 222, 268 Indicated horsepower, 52, 98, 141, 178, 186 compound engine, 178 internal-combustion engine, 186 simple steam engine, 141 Indicator diagrams, 51 compound engine, 153-173 internal-combustion engine, 34, 43-46, 55-60, 197-204 simple steam engine, 10-21, 128 Inertia, 296 diagrams, 297 and indicator diagrams combined, 301 Initial condensation, 78 Initial pressure, 10 Inlet and exhaust passages, 503 Intake muffler, 42 Internal-combustion engine, 32, 185 2-cylinder, 319 3-cylinder, 320 4-cylinder, 321 6-cylinder, 322 8-cylinder, 323 12-cylinder, 323 2-cycle, 35, 37, 453 INDEX 701 Internal-combustion engine, 4-cycle, 33 cycles, 32, 54 data, 43, 191 Diesel, 34 gas, 38 heavy-oil, 39 hot bulb, or plate, 39 light-oil, 38 starting, 41 Irregular section, center of gravity of, 693 moment of inertia of, 693 modulus of section of, 693 Jacket water, 37, 94 practical suggestions, 95 Keys, for cranks, 583 for flywheels, 640 Kilowatts, 209 Lagging, cylinder, 507 Lap, 411 Latent heat of saturated steam, 67 Lead, 8, 411 in steam turbines, 250 variable, 445 Light-oil engines, 33, 38 Limits of horsepower, 144, 193 Lubricants, 107 acid test, 108 cold test, 108 density, 108 fire test, 108 flash point, 108 friction test, 108 grease, 109- gumming test, 108 viscosity, 108 Lubricating systems, 109 forced feed or continuous, 109, 121 individual, 109 intermittent feed, 109 restricted feed, 109 splash system, 110 Lubrication, cylinder, 123 forced, 99 syphon, 99 and pad, 99 M Machine design, 463 Materials, properties of, 470 Maximum horsepower, 144, 193 Maximum strain theory, 477 Mclntosh and Seymour Diesel engine, 45 Mean effective pressure, 52 compound engine, 155, 158 gas and oil engines, 63, 186, 192, 190 Rankine cycle, 67 simple steam engine, 130 Mechanical efficiency, 54, 97, 191, 192 Mechanics, 283 of slider crank, 283 Mechanism of steam engine, 5 Methods of design, 687 Modulus of section of irregular section, 694 Moment of inertia of connecting rod, 286 of irregular section, 693 Mufflers, 41, 42 N Noncondensing steam engine, 20 Nozzles, fuel, 456 and other passages, 210 practical notes on, 219 steam turbine, 210, 219, 274-276 O Operation of steam engine, 6 Oil grooves, 113 Oil guards, 112 Otto cycle, 33 Peabody's direct method, 238 Piston, 515 acceleration of, 296, 298 702 INDEX Piston, box, 516, 525, 526 built-up, 524 conical, 517, 526 internal-combustion engine, 529 rings, 518 rod, 530 formulas, 531 speed, 82, 83, 95, 143, 144, 191 thrust, compound engine, 175 internal-combustion engine, 201 simple steam engine, 140 trunk, 527, 528 velocity of, 287 Poisson's ratio, 476 Port area, 402 Power of heat engines, 51 internal-combustion engine, 49, 186 steam engine, 208 steam turbine, 208 Power plant, 3 Practical cycles using gas, 54 Pressed fits, 537 Pressure and temperature of saturated steam, 66 Producer gas, 38 Properties of materials, 470 Q Quality governing, 41, 196 Quantity governing, 41, 197 Radius of gyration, 285, 286 Rankine cycle for complete expansion, 67 for incomplete expansion, 68 Rating of internal-combustion engines, 198 of steam turbines, 267 Ratio of expansion, 10, 85 Ratio of stroke to cylinder diameter, 82, 95, 143, 192 Reaction, 23 Reaction turbine, 27, 254, 279 practical notes on, 260 Receiver, 19 influence of, 159 Reciprocating engine passing not im- minent, 2 Rectangular section in torsion, 484 Rectangular valve diagram, 422, 425, 457 Re-evaporation, 78 Reheat factor, 237 Release, 11 Releasing gear, 434, 453 Reversal of thrust, 304 Reversing gears, 439 Doble, 448 Hackworth, 446 Internal-combustion engine, 453 Joy, 446 Marshall, 446 Stephenson, 440 Walshaert, 443 Right-hand and left-hand engine, 14 Rim bolts and links, 639 Rope wheel grooves, 633 Run over or under, 15 S Saturated steam, 66 Self -oiling bearing, 112 Shaft, 568 application of formulas, 605 at dead center, 600 at maximum turning effort, 602 center-crank, 587, 598 collars, 609 internal-combustion engine, 596, 598 material, 589 multi-throw, 589 side-crank, 587, 590 special formulas for, 604 steam engine, 590, 598 steam turbine, 664 Shifting eccentric, 420 Shock, 468 Shrouded blades, 28, 257 Side-crank and center-crank shafts, 13 Sight feed, 111 Simple impulse turbine, 24 Simple steam engine, 18 INDEX 703 Single-acting and double-acting engines, 13 Sleeve motor gear, 456 Slider crank, 5, 283 efficiency of, 292 Speed, 81, 143, 191 fluctuation of, 351, 376 Spring governors see governors. Springs, 384, 452 Standard engines, 149, 179 Standards, 689 Starting internal-combustion engines, 41 Steam, 65 flow of, 68 formation of, 65 high pressure, 91 jacket, 86 passages, 266, 503 superheated, 66, 90, 244 Steam consumption, 22, 144, 179, 208 of Rankine cycle, 71, 72 Steam cycle, 3 Steam engine, 5, 127 compound, 18, 152 condensing, 20 noncondensing, 20 operation of, 6 simple, 18, 127 uniflow, 21, 135 Steam turbine, 23, 206 bearings, 117 bleeder, 30 casing and details, 674 combinations, 29, 260 compound, 30 condensing, 31 conservation of residual velocity in, 248 dimensions, 245 double-flow, 29 effect of condenser on, 265 elements of, 24 factors influencing operation of, 264 impulse-and-reaction, 30 low-pressure, 30 mixed-pressure, 30 noncondensing, 31 number of stages, 246 pressure-stage impulse, 26, 234, 268 Steam turbine, pressure-velocity-stage impulse, 29, 249, 268 reaction, 27, 254, 279 repeated-flow, 29 shaft, 664 simple impulse, 24, 222 steam passages in, 266 superheated steam in, 244 unequal wheel diameters in, 247 velocity-stage impulse, 25, 229 practical notes on, 231 wheels, 649 Steam velocity, 403, 504 Steel, casting, 471 chromium-vanadium, 474 machinery, 471, 472 nickel, 472 nickel-chromium, 472 Strength of materials, 470 Stress, repeated, 466 reversed, 466 static, 466 suddenly-applied, 468 working, 464 Stroke diagram, 140 Strut formula, 485 Stuffing box, 506 Suddenly-applied load, 468 Superheated steam, 66, 90, 244 Swinging eccentric, 420 Temperature rise due to combustion, 61 Terminal drop, 88, 134 Terminal pressure, 10 Three-port engine, 37 Throttling engine, 16, 138 Thrust bearing, 119 Thrust of piston, 140, 170, 201 Thrust, reversal of, 304 Torsion of rectangular shaft, 484 Total heat of saturated steam, 67 Total ratio of expansion, 19 Trajectory, 250, 278 Turbine, bearings, 117 bearing temperature, 106 blade design, 657 reaction, 216, 217 704 INDEX Turbine, casings, 674 details, 674 horsepower, 72, 208 kilowatts, 209 shaft, 664 critical speed of, 666 nature of problem, 665 wheels, 694 application of formulas, 659 material, 656 radial stress, 655 tangential stress, 654 Turning effort, 289, 310, 314 diagram, internal-combustion en- gine, 315, 319-322 steam engine, 311, 314 U Uniflow engine, 21, 135 U. S. Standard bolts and nuts, 695 Unwin's flywheel formulas, 626 Valve, mixed-pressure, 398 double-ported, 419 operation, 8 poppet, 405, 408, 459 slide, 403 springs, 452 stems, Corliss, 436 slide valve, 426 Valve diagram, 6 Bilgram, 417 Valve, Corliss, 426 long-range, 432 rectangular, 422, 457 Valve gear, 401 classification, 408 cam, 448 cam-and-eccentric, 438, 452 Corliss, 426, 436 details, 457 high-speed Corliss, 437 internal-combustion engine, 448 Lentz, 439 Mclntosh and Seymour, 457 reversing see reversing gears single-valve, 410 sleeve motor, 456 Variable lead, 445 Velocity diagrams, 222, 229, 254, 269, 270, 278 Velocity due to reheating, 244 Velocity of gas, 95, 407 of piston, 287 of steam, 403 relative and absolute, 223 Volumetric efficiency, 60, 189 W Water power, inadequate, 1 Water rate see steam consumption Wedge bolts, 555, 617 Wet steam, 67 Wheels, engine, 350, 623 turbine, 649 Work of expansion when pv n = constant, 64, 128 UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last datejsj#mBedJt>elow. INSURING DEC 13 1948 LD 21-100m-9,'47(A5702sl6)476 VC 33320 414821 5 UNIVERSITY OF CALIFORNIA LIBRARY