T J NRLF OEft LIBRARY UNIVERSITY OF CALIFORNIA, RECEIVED BY EXCHANGE Class THE GAS TURBINE AN " INTERNAL COMBUSTION" PRIME-MOVER BY SANFORD A. MOSS, M.S. (UN1V.CAL.) A THESIS PRESENTED TO THE FACULTY OF CORNELL UNIVERSITY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY ITHACA, N, Y. MAY, 1903 THE GAS TURBINE AN " INTERNAL COMBUSTION" PRIME-MOVER BY SANFORD A. MOSS, M.S. (UNIV. CAL.) A THESIS PRESENTED TO THE FACULTY OF CORNELL UNIVERSITY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY . MAY, 1903 ANDRUS & CHURCH ITHACA, N. Y. CONTENTS. PAGE CHAPTER i. General Account of the Gas Turbine 5 CHAPTER 2. History of the Gas Turbine 13 CHAPTER 3. Thermodynamics of the Brayton Cycle 17 CHAPTER 4. Theory of Nozzles and Free Expansion 25 CHAPTER 5. Theory of a Rapidly Rotating Disc on a Flexible Shaft__ 36 CHAPTER 6. Experiments with a Gas Turbine. 48 THE GAS TURBINE. AN "INTERNAL COMBUSTION" PRIME MOVER. CHAPTER I. GENERAI, ACCOUNT OF THE GAS TURBINE.* The " Gas Turbine " is a form of heat engine, or mechanism for changing the potential energy of. fuel into mechanical work. As we shall have occasion to see in Chap. 3, the gas turbine seems to offer greater theoretical possibilities than the steam en- gine and boiler, or the Otto cycle gas engine, which are the heat engines now most usual. Of course considerable experimental perfecting would be necessary before the gas turbine could de- mand serious attention for commercial uses. However, the theo- retical possibilities seem to warrant the present investigation at least. Before discussing the gas turbine itself we shall consider an- other form of heat engine based on the same thermodynamic principles, but somewhat simpler in conception. Consider a compressed air transmission plant, comprising an air compressor and a motor like an ordinary steam engine. The work done by the motor varies directly with the absolute temper- ature of the air supplied to it. That is to say, the work done by the air depends only on its volume when supplied to the motor, and hence increasing this volume by raising the temperature in- creases the work, regardless of the actual quantity or weight of the air. For this reason compressed air is usually " reheated " before use in a motor. If the air is compressed without any gain or loss of heat, that is, " adiabatically," the temperature will rise considerably. In our theoretical discussion we will assume such compression. Practically it is always convenient to cool the air somewhat dur- ing compression. If the air supplied to the motor is at the adiabatic temperature at which it theoretically leaves the com- * The substance of this chapter was published as an advance portion of this thesis in The Engineer (Cleveland), Vol. 40, No. 8, April 15, 1903. 6 pressor, the power from the motor will be theoretically the same as that required for the compressor. By reheating the air suffi- ciently, the power from the motor may be made to exceed the power for the compressor by as much as we please. The excess power represents work obtained from heat energy of the fuel used in reheating. In a transmission plant, compressor and motor are placed at a distance from each other. If, however, they were at the same place, the motor could drive the compressor, and the excess power would be available for outside purposes. The apparatus would then be a heat engine operating on the fuel used in heat- ing the air between compressor and motor. The air might be heated by passing it through a closed vessel of some sort exposed to an outside flame, or by " internal com- bustion " which we will discuss later. The entire connected system of piping and vessels between the compressor and motor, including the heating vessel, is everywhere at the same pressure, which, of course, is the maximum compression pressure. The air is therefore heated at constant pressure, and its volume in- creases so that air leaves the heating chamber much faster than it enters. The motor must of course use up the entire volume of heated air as fast as it is supplied in order that the pressure may remain constant. The series of operations performed on a portion of working substance as it goes through a heat engine is called its "cycle," the cycle used in the heat engine just described being named the " Bray ton cycle." This cycle is distinguished by the fact that while the heat used is being added to the air, or other working substance which actuates the engine, this working substance is kept at constant pressure. The same cycle or series of operations maybe carried out by very different kinds of engines. The gas turbine operates differently from the engine just described, but it uses the same Brayton cycle. That is, the same series of opera- tions are carried out, but in a different way. Theoretically the Brayton cycle should be easier to execute than the Otto cycle used in modern gas engines. The thermo- dynamic efficiency, or theoretical fraction of the heat supply which is given out as mechanical work, is the same for the Otto as for the Brayton cycle, if the same compression pressure is used. However, the maximum pressure is very much greater with the Otto cycle, where the heat is added at constant volume with consequent increase of pressure, than with the Brayton cycle, where the pressure does not rise above the compression pressure. The maximum temperature is also much greater for the Otto cycle than for the Brayton, for the same compression pressure and theoretical efficiency. Having lower maximum pressure and temperature we should expect the Brayton cycle to be mofe easily managed than the Otto for equal efficiency. However, the method in which a cycle is carried out is as im- portant as the form of cycle. Losses not taken account of by theory always occur, and the magnitude of these depends only on the mechanism of the heat engine which executes the cycle. We must carry out the Brayton cycle in an actual engine without in- curring greater losses than occur in an Otto gas engine in order to secure better results. In all actual attempts to execute the Brayton cycle in the past the losses have been so serious as to more than outweigh the theoretical advantages. On paper, at least, the gas turbine avoids these losses, as we shall see in Chap. 3. Let us next consider how the Brayton cycle is carried out by the gas turbine, which, as has been stated, is the equivalent of the compressor- motor combination already discussed. In the case of the compressor-motor combination the compressed air was supposed to be heated by being passed through a vessel exposed to an outside flame. In the gas turbine there is substituted for this internal combustion, used so successfully in the modern gas engine. Since air is our original working substance and since it is also requisite for the combustion of the fuel used in the heating, we may use the air which is to operate the motor to support the combustion of the fuel. That is to say, we will heat the air for the motor by burning the fuel right in it. This is called " inter- nal combustion," as distinguished from " external combustion," where the heat of the fuel is transferred through the walls of a vessel to the working substance, as in the case of a steam boiler. The air must be heated between the compressor and motor, when it is under considerable pressure. A closed vessel or enlargement of the pipe between the compressor and motor is provided, called the " combustion chamber," in which the air and fuel combine or burn, always, of course, while at the maximum compression pressure. From this chamber the products of combustion pass to the motor. Theoretically this could be an engine like a steam engine, or any substitute for it, and in this particular case we will use a turbine wheel similar to a steam turbine. The details of this we will discuss later, first considering more fully the method of operating the combustion chamber. 8 Heat engines using the Bray ton cycle and having a combustion chamber with internal combustion, but using a motor similar to an ordinary steam engine, instead of a turbine wheel as in our case, have been built or planned in a multitude of forms, and the type of combustion chamber proposed for any one of these is of course suitable for the gas turbine. The relation of these engines to the gas turbine we shall discuss more fully in Chap. 2. In the Cayley engine, actually in operation in 1807, coal was used as fuel. A supply sufficient for some time was placed on a grate in a closed vessel and lighted, and- the air from the com- pressor forced through it. When the fuel had all burned out, the motor had to be shut down, the combustion chamber opened, and a fresh fuel supply introduced. Of course a method could be easily contrived for introducing coal into the combustion chamber while the pressure is on, so that the motor could be operated con- tinuously. In other cases a gaseous fuel has been forced into the combus- tion chamber against pressure within by means of a small gas compressor, and the air forced in by a separate air compressor, the two pipes meeting as they enter the combustion chamber. After once being lighted a continuous jet of flame issues from the meeting point of the pipes. The first gas turbine ever proposed, which we will discuss later, was to operate in this way. The most convenient fuel, however, for a gas turbine will be a liquid of some kind, as, for instance, gasoline, kerosene, or dis- tilled or crude petroleum. Many proposed Brayton cycle engines with motors similar to steam engines have been planned for liquid fuel. The original engine of this type was that of George Brayton, actually in commercial use about 1872. It is due to this engine that the name of Brayton is given to the cycle in which heat is added at constant pressure. Such a cycle had been proposed long before, however, and Brayton only invented a par- ticular form of engine for executing the cycle. The cylinder of Bray ton's motor was really his combustion chamber. He forced the oil and air through passages at the en- trance to the cylinder as the piston was advancing on the forward stroke, and by means of a small flame, kept constantly lighted, ignited and burned the mixture as it entered the cylinder. The hot products of combustion, of course, at the maximum compres- sion pressure, forced the motor piston ahead. In this case the combustion was intermittent, the supply of air and oil being cut off when the motor piston reached a certain point, and the com- bustion resumed at the beginning of the next stoke. 9 In the case of the gas turbine air and oil are continuously forced into one end of the combustion chamber, and, once being lighted, the jet continues to burn. The pressure within the com- bustion chamber and the air pipe leading to it is, of course, always the maximum compression pressure. "Combustion at constant pressure" occurs in all of the cases we have considered, whether the fuel be solid, liquid, or gaseous. The fuel being present, and air supplied as fast as necessary by the compressor, combustion ensues exactly as if the fuel were burned in the open air. The fact that the combustion takes place in a closed vessel under pressure makes no difference whatever. The products of the combustion are nitrogen and carbon dioxide, together with oxygen from any air in excess of that required for combustion. All of the heat of combustion of the fuel is emitted just as when it is burned in the open air, and this goes to heat the products of combustion, which are thereby expanded so that they occupy a considerably increased volume. Now the products of combustion have practically the same volume at any given temperature and pressure as would the original air. Therefore the volume and temperature of the products after combustion will be the same as the original air would have if the entire heat of combustion of the fuel were added to the air by passing it through a heated vessel, without changing the chemical composition by burning the fuel in it. In effect, then, internal combustion is merely a means of adding a certain amount of heat to the air, with consequent increase of volume, and is equivalent to passing it through a heated pipe. The change of chemical composition is merely incidental. Of course we always assume that the pro- ducts of combustion are drawn off from the combustion chamber as fast as they are formed, so that the pressure within it is always the maximum compression pressure. Owing to the expansion due to heating, the volume taken from the combustion chamber and supplied to the motor is much greater than the volume of air supplied by the compressor. As already stated, the power ob- tained from the motor is greater than that required to drive the compressor in the same ratio. Next let us consider briefly the theory of the turbine wheel or 11 impulse wheel " as it should properly be called. Suppose we have a vessel from which we may obtain a continuous supply of liquid or gas under pressure. This might be a reservoir of water, a steam boiler, or the combustion chamber of a gas turbine. By using an engine similar to a steam engine we could obtain power IO from the pressure of the liquid or gas. Suppose, however, that we connect an open nozzle with the vessel, .so that the water, steam, or gas can escape into the air or into any region where the pressure is lower. A jet will then issue from the nozzle with con- siderable velocity. If the nozzle be properly shaped so as to avoid friction losses, the kinetic energy of the escaping jet will be exactly the same as the energy which could be obtained by ex- panding the same liquid or gas from the higher pressure to the lower in a pressure engine like a steam engine. That is to say, the velocity with which the jet escapes represents exactly the same power as could be obtained by using the liquid or gas in an ordinary engine, We will give the proof of this, and consider the whole matter in greater detail in Chap. 4. By directing the jet upon the vanes or buckets of a properly ar- ranged impulse wheel, the power of the jet may be taken from it and applied to do useful work. Examples of this are shown by the Pelton type of water wheel, and the De Laval steam tur- bine. Suppose, then, that we place at the farther end of the combus- tion chamber a proper nozzle and direct the jet of gases issuing from it upon a turbine wheel. The wheel could be of the type used in the De Laval steam turbine with a single set of buckets, or a number of wheels could be placed in series, as with the Curtis or Parsons steam turbines. We will then obtain the same power from the wheel as we would if we used a motor with a piston like that of Brayton, Cayley, and others. As already stated, this power exceeds the power required to drive the com- pressor by as much as the heating expanded the products of com- bustion. Hence, we may drive the compressor from the impulse wheel, and have surplus power, which is the net income from the heat energy of the fuel used. Having examined the individual steps in the operation of the gas turbine, we are prepared to consider the apparatus as a whole. Fig. i is a diagrammatic sketch of the complete machine, which is so labelled as to be sufficiently explicit. We will discuss possible modifications of this fundamental arrangement in Chap. 3 and actual details of construction and operation in Chap. 6. Oil and air are forced into the combustion chamber as indi- cated, and begin to burn at the entrance. The flaming stream passes along the combustion chamber, and combustion is com- plete by the time the nozzle is reached. The same constant pres- sure is, as previously stated, maintained throughout the system II from the compressor, through the pipes and combustion chamber to the nozzle, since the nozzle is of such size as to allow the in- creased volume of products of combustion to pass out of the com- bustion chamber as fast as they are produced by the entrance of FIG. i. Diagrammatic Plan of Fundamental Gas Turbine. air and fuel. The diagram shows the arrangement for liquid fluid, to which the gas turbine is perhaps best adapted. Gaseous fuel, such as any kind of coal gas or producer gas, could be used by replacing the oil pump by a gas compressor. Solid fuel could be used by devising some means of continuously feeding it against the pressure of the combustion chamber, or by having a combustion chamber large enough to contain a supply of fuel for some time, and forcing in air only. The combustion chamber must, of course, be designed to with- stand the high temperature of the burning gases and the pressure of the compressor. The nozzle must be so designed that the products of combustion have free expansion from the maximum pressure to the external pressure, hence, as in the case of the steam turbine, the channel must first converge and then diverge. The proper shape for the nozzle we will consider in detail in 12 Chap. 3. The gases will be cooled considerably in expanding through the nozzle, but are still hot at the exit. In all heat engines only a portion of the heat of the fuel is utilized and the rest must be thrown away. In the case of the gas turbine, the kinetic energy of the jet issuing from the nozzle represents that part of the heat energy of the fuel which can be utilized, while the temperature of the jet represents that portion of the energy which will be lost. The gases issue from the nozzle with a velocity comparable with that in the De Laval steam turbine, and the turbine must therefore rotate at the same extremely high speed if we use a single stage wheel as does De Laval, the velocity of the buckets being approximately half that of the jet. The theory of a disc rotating at such extraordinary speeds is given in Chap. 5. Some form of speed-reducing mechanism must also be introduced, since it is difficult to utilize power from the wheel at the original speed. However the speed of the wheel could be reduced by using several wheels in series, as in the Curtis or Parsons turbines. The use of a true turbine, where expansion occurs within the wheel itself, is also conceivable. Everything considered how- ever, a single stage impulse wheel .seems best. CHAPTER II. HISTORY OF THE GAS TURBINE. As is the case with many other supposed novelties, the gas tur- bine was conceived in ancient times. What is probably the orig- inal gas turbine was patented in England in 1791 by one John Barber. The drawing of this, Fig. 2, and the following descrip- FIG. 2. Barber's Gas Turbine, 1791. i 4 tion are taken from the British Patent Reports, Volume XXIII, 1791-2, Patent No. 1833, which seems to be the only source of information concerning the curious machine. It will be seen that Barber probably understood thoroughly the principle of the gas turbine, although the mechanical knowledge of the time would hardly have sufficed for the construction of a working machine. The vessels marked i are retorts for the production of the gas to be used, by distillation of coal, wood, etc., by means of an ex- ternal flame. These are in duplicate, so that one can always be in use while the other is being emptied of coke and recharged. The vessel above the retorts is a cooling and condensing chamber, from which the gas is drawn by the pipe B. The apparatus up to this point is merely for the production of the gaseous fuel to be used, and is not an essential feature of the gas turbine. The parts marked CC, DD, the two lower vessels of those marked 4, and the elevated water tanks above the apparatus comprise two peculiar hydraulic compressors, the details of which we need not investigate. The triangular shaped vessel between the -tanks 4 and the wheels is the combustion chamber. By means of the front compressor, the gas taken from B is compressed and dis- charged from the front vessel 4 into the combustion chamber by the pipe shown. By means of the rear compressor air is com- pressed and discharged from the rear vessel 4 by a pipe not shown into the rear side of the combustion chamber. The upper vessel 4 discharges water into the combustion chamber, which absorbs some of the heat of combustion and reduces the maximum tem- perature. This is not essential to the operation of the apparatus, however, as the same result could, of course, be obtained by the use of an excess of air. The air and gas must burn within the combustion chamber, and the products, issuing from the lower end, are directed upon vanes of the wheel marked 8, which is thereby impelled to rotate. Barber's description of the action within the combustion chamber is not very explicit, and there is a possibility that he did not un- derstand the gas turbine after all. It may be that the triangular chamber was only intended as a mixing chamber, and that the combustion was not to occur until the mixture had passed through its outlet, since the flame would not strike back owing to the ve- locity through the orifice. It might be, on the other hand, that the air and gas were to be throttled before they entered the cham- ber, so that atmospheric pressure existed within it. In either event the heat would be added while the working substance was at atmospheric pressure, instead of while it was at the maximum pressure. The addition of heat would then be of no use what- ever, and the wheel 8 would operate equally well whether the gases were lighted or not. Theoretically the power required for compression would then be equal to the power of the turbine wheel ; and practically, owing to friction, etc., would be in excess ; so that the machine would not operate at all. In case, however, that Barber had the correct conception of the matter and arranged to have the combustion take place in the tri- angular chamber, where also the maximum compression pressure existed, then the operation would be, as he states, as follows : A pinion on the impulse wheel shaft operates the gear wheel 10, on the shaft of which are the cams 9, which raise rods attached to the walking beams 55, actuating the compressors by means of the chains shown. A pinion on the same shaft as the wheel 10 ope- rates the upper gear wheel. From the projecting shaft of this up- per gear the difference between the compressor and motor power can be taken off and applied to useful work. The maximum compression pressure which Barber could obtain was that due to the head of water from the tanks in the upper part of the figure, which practically could not be made very great. A modern gas turbine would use much greater pressures than Bar- ber contemplated. Nothing seems to have come of Barber's gas turbine and the matter lay untouched for years. In the meantime a great number of internal combustion Brayton cycle engines with reciprocating engines as motors, were proposed as already stated, none attain- ing success however. A number of these are described by Mr. Charles Lucke in a paper presented to the American Society of Mechanical Engineers, Dec. 190^, andjorming part of his Doc- tor's thesis, presented to Columbia University in 1902. He dis- tinguishes two types, the first type having intermittent, and the second type continuous internal combustion at constant pressure. The latter type of course involves a combustion chamber or re- ceiver sufficiently large to prevent excessive fluctuations of pressure owing to the intermittent demand of the motor. A heat engine of this type is essentially equivalent to the gas turbine with the single exception that a reciprocating motor is used instead of an impulse wheel. Mr. Lucke mentions the possibility and even the desirability of using a "gas-expansion turbine" instead of a reciprocating motor in this type of heat engine, but he does not go into details. i6 A number of gas turbines have been proposed by various in- ventors in which an explosive mixture is repeatedly admitted to a chamber, exploded, and discharged with gradually decreasing pressure upon the vanes of a turbine wheel. These might be called " intermittent combustion" gas turbines. A gas turbine of this type due to Leon Le Pontois was the subject of the Doc- tor's thesis presented to Cornell University by Mr. W. O. Amsler. All gas turbines of this character seem to be based on the false idea that "explosion" is the only proper way to conduct internal combustion, and -the inventors seem to be ignorant of the possibility of continuous combustion under constant pressure. For obvious reasons a continuous combustion gas turbine is immeasurably superior to one with intermittent combustion and variable pressure. The writer began the study of the gas turbine in 1898, believ- ing the idea original Until research disclosed Barber's patent. A thesis for the degree M.S. entitled "Thermodynamics of the Gas Turbine" was presented to the University of California in 1900, the substance of which is given in the next chapter. CHAPTER III. THERMODYNAMICS OF THE BRAYTON CYCI,E. The heat engine used in the Gas Turbine, commonly called the Brayton Cycle, as already stated, is distinguished by the fact that heat is added to and taken from the working substance while it is at constant pressure. The various operations performed on a portion of working substance as it passes through a Brayton cycle heat engine are as follows : The working substance, initially separate portions of air and fuel in the case of the gas turbine, is first compressed to the maxi- mum pressure p v While at this pressure heat is added by some means or other ; in the case of the gas turbine by "internal com- bustion. ' ' The volume and temperature of the working substance are thereby increased. The chemical composition may also be changed but this is merely on incident which we have on occasion to take account of at present. The working substance next ex- pands adiabatically to atmospheric pressure. During addition of heat and expansion, work is done which is eventually utilized by the motor. In our case an impulse wheel is used, which is, how- ever, as will be shown in Chap. 4, equivalent to a piston engine. The working substance, now at atmospheric pressure but at a considerable temperature, is next discarded, and a new lot taken into the compressor at atmospheric temperature and pressure. In order to close the thermodynamic cycle we may suppose that the same working substance is used over again, enough heat having been extracted at constant pressure to reduce the temperature appropriately. The mathematical treatment of an actual cycle such as the above has never been attempted directly, and the best that can be done is to discuss the matter on the assumption that the working substance is a perfect gas throughout. The approximations that this assumption involves in the case of the gas turbine are as fol- lows : The working substance is initially separate portions of oil and air. Since the work of bringing the oil to the maximum pressure is insignificant, the actual work of compression is not quite so great as we take it to be by assuming the working sub- stance wholly gaseous. However, as we shall see later, in cases where the proportion of oil is reduced sufficiently to give work- able temperatures, the oil forms two or three per cent, by weight of i8 the working substance. Hence but slight error is committed by assuming that the work of pumping the oil is equal to the work of compressing on equal weight of air. In the next place the variation of volume due to addition of heat by internal combustion is very irregular, and does not follow the simple law of the perfect gas cycle, where the volume varies directly with the heat added. Some irregularities, probably slight owing to the small percentage of oil, arise from the fact that some of the heat of combustion of the oil is absorbed by its latent heat of vaporization. This heat causes increase of volume but not of temperature. Other irregularities arise from the "ex- pansion " due to combustion. When the volume of a perfectly combustible mixture of gases is compared with volume of the products of combustion ; both calculated by Avogadro's law, and reduced to the same pressure and temperature, an expansion or contraction is found to have occurred. It can be shown that ex- pansion always occurs when a hydrocarbon C a H m burns, if m is greater than ^. Therefore, since most of the hydrocarbons of which oil is composed are quite complex, an expansion occurs due to the change of chemical composition. This is independent of the heating of the products of combustion due to the addition of the heat of combustion. This is therefore another source of increase of volume without change of temperature. Usually there is a considerable amount of dilution in the mixtures used, particularly on account of the nitrogen of the air, so that the ex- pansion is not a very large percentage of the whole volume in- volved, making this irregularity slight also. The increase of volume due to the two causes referred to of course represents work done due to liberation of the chemical energy of the fuel It may be remarked that the work thus done is probably never included as it ought to be in the tabulated values of the heat of combustion of fuels. The " constant press- ure calorimeter " of the Junkers' or other form, which is usually used, takes no account of energy liberated by combustion but causing volume variation merely without incidental change of temperature. The differences between the actual cycle and the assumed per- fect gas cycle have thus far been found slight. There is, how- ever, another difference of much greater magnitude. Recent re- search has probably established the fact that the specific heat of all gases increases with the temperature. Therefore the specific heat of the products of combustion at the high temperatures attained in the gas turbine is much greater than the constant value assumed in the perfect gas cycle. It follows that the actual temperatures attained in the gas turbine are much less than those calculated on the perfect gas assumption. Since so much approxi- mation is involved in the perfect gas assumption, we may as well assume in addition that the working substance has throughout the specific heat of air at atmospheric temperatures. Let us consider the effect of the "imperfect" nature of the actual gases upon the theoretical efficiency of the cycle. As is well known, in the case of the Carnot cycle imperfection of the working fluid has no effect UROII the theoretical efficiency. It may also be shown that the theoretical efficiency is independent of the nature of the working fluid for any cycle in which heat is added and taken away along lines of the same constant specific heat* It does not appear therefore as if the imperfection of the working substance would have any marked effect on the theo- retical efficiency in the case of our cycle. That is to say we will make the hypothesis that the theoretical efficiency of the actual gas turbine is approximately the same as that calculated on the basis that the working substance is a perfect gas. The algebraical and arithmetical calculations for a number of cases of perfect gas Brayton cycles comprised the master's thesis of the writer, entitled " Thermodynamics of the Gas Turbine," filed at the University of California. A brief abstract will be given of the most important results. The heat Q added by the combustion, per pound of working substance, may be as a maximum the heat emitted by the com- bustion of as much oil as is necessary to form a perfectly com- bustible mixture of air and oil amounting to one pound. Ordi- nary crude or refined petroleum is about 85 per cent, carbon and 15 per cent. Hydrogen. Calculation from this analysis gives about 15 pounds of air as necessary to burn i pound of oil, with emission of about 20,000 B.T U. Hence a perfect mixture of air and oil weighing i pound emits 2O ' OC ^ 1250 B. T. U. However in order to be sure that no oil will be unconsumed there should be a slight excess of air, so that we will assume a round figure of i ,000 as the greatest desirable value of Q. By decreas- ing the proportion of oil to air we may make Q anything that we please under 1,000. Of course this figure may need modification if fuels other than oil are used. *See Generalization of Carnot' s Cycle, Physical Review, Vol. 16, No. i, Jan., 1903. 20 The temperature and volume after combustion are calculated from the value of Q and the specific heat at constant pressure for air, '238, assumed as constant as already explained. The final temperature and volume are calculated by assuming adiabatic expansion from maximum to atmospheric pressure. The heat q which must be extracted to return the working substance to the initial condition is found from the specific heat at constant volume, as before. The heat transformed into mechanical work is the difference Q q so that the efficiency of the cycle which we will call e is given by e = 2. The work done in compression is C and the work done by the motor is M, so that the net work available for useful purposes is C M. Of course C M = J (Q q} where /is the mechani- cal equivalent of heat. A measure of the complication involved in the necessary compression of the air before combustion will be given by the ratio of the power required for compression, to the net power, which we will call r. Then r = --. C M In deciding the values of the various quantities to be used in a particular Brayton Cycle, we may choose arbitrarily the maxi- mum pressure p l and the heat added per pound of working sub- stance Q. The latter, however, must never exceed 1000 as already stated. When/j and Q are decided, everything else may be cal- culated directly. On the other hand, we may assign arbitrary values to any other two quantities whatever, and calculate the values of p l and Q which must necessarily be used to .secure them. In cases where the cycle of a heat engine is executed in a motor involving a reciprocating piston, the maximum pressure and tem- perature must be brought within reasonable limits, so that these are made the arbitrarily determined variables. In the case where ah "impulse wheel" is used as a motor for the Brayton Cycle, giving a Gas Turbine ; the temperature at the end of expansion and the linear velocity of the impulse wheel must be within con- trollable limits. The peripheral velocity of a De Laval Steam Turbine, a Pelton water wheel, or any similar impulse wheel, should theoretically be one-half, and practically about three- eighths of the velocity of the jet which drives it. The velocity of the jet Fis such that the kinetic energy of a pound of working F 2 substance, , is, as already stated, equal to the work which 21 would be done by the same pound in expanding behind a piston, given by M. Hence the assignment of a value to the velocity of the impulse wheel of a gas turbine amounts to assigning a value to M. An extreme value for the velocity is 100,000 feet per minute. We shall next consider some numerical results for a number of cases of the Bray ton Cycle, given in Table i. We will among other things give the speed which the impulse wheel should have in case the cycle is executed by using a gas turbine. Of course all of the other values hold good for any form of engine whatever in which the cycle is executed, whether a reciprocating piston engine or whatnot. As previously stated, we will assume the working substance as a perfect gas, so the temperatures we obtain are very much higher than will occur in an actual case. Case I is a theoretical Brayton Cycle in which compression and expansion are both adiabatic. We will suppose a perfect ma- chine with no losses. Case II is a modification of the Brayton Cycle, in which the compression is isothermal and a regenerator is used. We will again assume no losses. We will find that the regenerator gives a remarkable gain. The formulas show that the work done by the motor is equal to the heat added by combustion. Hence we obtain as useful work all of the heat of combustion except the work required for compression. The less the compression pres- sure the less this lost work and therefore the greater the efficiency. This is not the case, however, if any losses occur. The theo- retical efficiencies for Case II are remarkably high, but are of course unattainable since we have assumed a perfect heat engine. Case III is a recalculation of Case II in which we attempt to find values for the efficiency which might be expected in an actual case by assuming the various losses which will occur. We will assume that the motor only gives 70% of the power which it theoretically should, that the compressor requires 20% more power than it should, and that the regenerator has an effi- ciency of 60%. It appears that we can in no case use the full value looo for the heat added per pound of working substance, since the temperatures will be too high for a reciprocating engine and the velocities too high for a Gas Turbine. An excess of air is therefore used to reduce the heat added per pound. The values given for the efficiency in this case are remarkably high, and since they are results which might be expected in an actual engine, they are worthy of attention. Of course experi- 22 merit will be necessary to determine just what values of tempera- ture, etc., are practicable. As stated the values of the tempera- tures given in Table i do not hold for the actual cycle. TABLE i. BRAYTON CYCI,E CALCULATIONS. 1 IH 5 *7 bo (4-4 o ' 3 cO i, i 1 o ' oD % *> V& w ! || 8.J S|S "cd O * O gfc frfr 0* ^ cC IT* ^j -M F be VH ^* ^yQ fi ^ K** O S3 ^ C 'O'o "S w t/3 PH J_T ^j . ' t t4H -j -*-* S ^ "" co &c g n W g^H S 8^ 3 0> H'S , to O we may cut the nozzle off at a point where the pressure has diminished to/ 3 and thus obtain a nozzle suited for discharge into a region of pres- sure /> 3 . Let the general conditions at any point whatever along the nozzle be denoted by letters without subscripts, p being the absolute pressure in pounds per square foot, v the corresponding specific volume in cubic feet per pound, T the absolute Fahrenheit temperature, Fthe velocity in feet per second, and A the nozzle area in square feet. Let the conditions at the entrance of the nozzle be denoted by subscriptj, at the point of minimum diameter by subscript 2 , at the point where any particular pressure (as for instance atmo- spheric pressure) is reached, by subscript 8 , and at the point where zero pressure would be reached if the nozzle continued so far, by subscript 4 . Let M be the weight of gas passing through in pounds per second. Let <: p and c v be the specific heats of the gas and k their ratio. R is the gas equation constant, equal to ^, and / is the mechanical equivalent of heat. As is well known R =J( c^ c v ). Let us consider that portion of the nozzle between the entrance and the general point where the pressure is p. The mathematical statement of the fact that no energy is created within this space, (neglecting the kinetic energy of the gas entering the nozzle) is The neglect of the entering kinetic energy amounts to the as- sumption that the entrance velocity is zero, so that the nozzle area at the entrance must be infinite. Practically, however, the entrance area may be comparatively small and the entrance velocity be in the neighborhood of 100 to 200 feet per second, and yet the kinetic energy of the entering gas will be quite negligible as compared with the other terms of the expression. The above expression reduces to ~r=P.v l -pv+Jc v (T l -T) Since p l v l = R T pv=RT, and this becomes Hence we have as a general expression for the velocity 29 V-VigJc^Tt-T) (0 We will assume, as is usual in similar cases, that the gas ex pands adiabatically in passing through the nozzle. Hence the usual adiabatic relation holds, T, Therefore This expression shows the relation between the pressure and velocity at any point. It will be seen that the velocity always in- creases as the pressure decreases. We may express the relation between the nozzle area and the pressure at any point by substituting for Fin (2). From the re- lation A V = Mv and the adiabatic relation , we have V- v ' Then r . This gives us a means of investigating the variation of the area A as the pressure/ varies. Differentiating with respect to/, where B is a group of coefficients all of which are essentially posi- tive. Suppose that / is equal to /j. Then since k is about 1.41, will be positive. will continue positive, that is the noz- dp dp zle area will decrease as the pressure decreases, while the pressure varies from/! to a value such that the parenthesis on the right vanishes. That is to say the nozzle must converge while the pressure varies from p l to a value / 2 such that '-' (5) Evidently - vanishes for/ / 2 and hence A has a minimum dp value for this pressure. As the pressure / still further decreases is a negative and hence the nozzle diverges. dp 30 That is to say a nozzle in which the pressure continuously de- creases has a point of minimum area at the place where the pres- sure is/> 2 and diverges beyond this point. If the shape is differ- ent from this, the pressure cannot decrease continuously and an irregular action occurs. In the case of air and a number of other gases k= 1.41 and the pressure at the throat or point of minimum area is / 2 = 527^1- By substituting the value of the throat pressure in the adiabatic relation between pressure and temperature, the throat temperature is found to be T 2 T T+l l By substituting this value for T 2 in the general expression for the velocity (i) we have as the throat velocity This may be shown to be the velocity of sound in the gas for the conditions of pressure and temperature obtaining at the throat. It is therefore the velocity at which a disturbance will be pro- pagated along the stream flowing through the nozzle. Since the velocity of motion beyond the throat is greater than the velocity at which a disturbance can be propagated, it is impossible for a disturbance to be propagated backward along the stream of gas. Hence if the portion of the nozzle beyond the throat were to be removed, so that the gas was discharged at a pressure p^ into a region of lower pressure /> 3 , the disturbance thus produced could not in any way affect the flow of gas between the entrance and the throat. That is to say if a nozzle is wholly convergent and the region into which it discharges has a pressure less than / 2 the discharge will nevertheless be at pressure p 2 and the gas will ex- pand from / to /> 3 by diverging in the region beyond the end of the nozzle, just as if the proper divergent portion of the nozzle existed beyond the throat. This explanation of the well known fact is in essence due to Professor Osborne Reynolds. In order to compute the sizes necessary for the design of a nozzle we proceed as follows : By substituting the value of p 9 given by (5) in the general expression (4) for the area we obtain a value for the throat area which reduces to \ A \ 2gk For the case of air and those gases for which k= 1.41 the throat area becomes In order to most conveniently compute the area and velocity at the end of the nozzle where it discharges into, a region at pressure / 3 we first find the final temperature and specific volume by sub- stituting the value of p 3 in the general adiabatic relations, whence We may then obtain the final velocity by substituting the value of 7~ 3 in the general expression for the velocity (i), which gives Also, v . The final area may then be found by substituting the values thus found for F 3 and v 3 in the general expression A V Mv. By taking p = o in (4) it appears that the final area for dis- charge into an absolute vacuum is infinite. This is du'e to the fact that the specific volume is infinite for zero pressure. How- ever, the area increases very rapidly for a very slight decrease in pressure in this vicinity, and the final area for what is practical!}' considered a vacuum is not particularly great. It may be re- marked that the final temperature is zero for discharge into a vacuum. The theory thus far has been known in essence for some time, although the present method of treatment is possibly new. We have found, however, only the entrance, throat, and final diame- ters of the nozzle. The curvature of the nozzle and the distance between these points, is not determined. The following attempt to complete the discussion is believed to be original. We shall assume as a basis for procedure the manner in which a particle of gas is to be accelerated as it passes through the noz- zle. Evidently in order to avoid impact loss due to changes of velocity the acceleration should proceed in some regular manner. One possible assumption is uniform acceleration. That is to say the force furnished by the gradually decreasing pressure which serves to increase the velocity may be taken as constant. We will presently work out the shape of a nozzle to secure this result. However, this is not necessarily the shape giving a minimum friction loss. For instance it is conceivable that the losses might be less if the particles of gas were accelerated less rapidly as their 32- velocity increased. Then the accelerating force would vary in- versely as the velocity. Theoretical considerations may be dis- covered which will give some absolute criterion as to the best method of acceleration. However, the matter must probably be decided by experiments with nozzles designed according to vari- ous reasonable assumptions. The assumption of uniform acceleration seems most reasonable and therefore we will take it up here. Let x be the variable distance from the nozzle entrance to any point where the pressure is p. Then x^ will be the throat dis- tance, x s the total length of the nozzle, and x the length for dis- charge into a vacuum. We have by hypothesis = \~JL The time of a complete vibration is ?- rr .

. \j/ is called the "critical speed," and considerable dis- turbance occurs however when o> reaches this value. No mathe- matical account has ever been given of the exact action. Suppose now that w is made to exceed the critical speed. Then the denominator of the expression (i) becomes negative, and in order to have a positive value for r we must make a negative also. This is done by taking it in a direction opposite that of Fig. 4, so that the center of gravity is inside of the center of bore, as in Fig. 5. Fig. 5 therefore represents the position of equilibrium when w is greater than the critical speed if/. If we count a as essentially positive in this position, the condition for equilibrium is M R ^__ g which reduces to o_ (2) a] -39- Here the greater the angular velocity to the less will be R. That is to say, the center of gravity actually comes nearer to the center of rotation as the speed increases. For any given rotative speed, we secure this result by making tf = as small as possible, M which is accomplished by making K, which gives the stiffness of the shaft, as small as possible. That is to say, the more flexible the shaft the nearer the center of gravity comes to the center of rotation. The center of gravity always rotates in a circle whose radius is given by (2), however. The popular statement that " a flexible shaft allows the disc to'rotate about its center of gravity " is therefore erroneous. If the disc were to rotate about its .center of gravity, the shaft would be deflected a distance a, so that there would be an unbalanced deflection force a K. The rotating force pulling on the bearings and tending to vi- brate them, for the case of Fig. 5 is g J _ g K Mrf This is diminished and the machine made to run with slight vi- bration by making K, which gives the stiffness of the shaft, as small as possible and Mthe weight of the disc, and o> the angu- lar velocity, as great as possible. This is all expressed by the statement that w -i- \f/ should be as great as possible. That is to say, the working value of the speed should bear as great a ratio as possible to the critical speed in order to reduce vibration of the bearings. The position of Fig. 4 is sometimes called rotation with the "heavy side out" while that of Fig. 5 is rotation with the " heavy side in." That is to say, if a mark be made during rota- tion by touching the portion of the edge of the disc furthest from the center of rotation, in order to find the place to cut away metal in order to correct the balancing, then metal must be taken from the side of disc nearest the mark if the speed is below the critical speed, and from the side away from the mark if the speed is above the critical speed. The critical speed in radians per second we have found to be t = \ K g. This reduces to [32.16 X 12 k = , 8? - 6 \~k \| M 27T^J M ^-^r revolutions per minute, where k is the load in pounds required to deflect the shaft one inch and M is the total load in pounds sup- posed concentrated at the shaft center. 4 As we have seen, Fig. 5 shows a position of equilibrium. We have next to show that the equilibrum is stable. That is to say, we must show that if any accidental cause displaces the center of gravity from the position shown, it will return and not seek some new position.* The balance of the chapter will be a tedious proof of this fact. In discussing the motion of the disc we will make use of the well known fact that the center of gravity C moves as if all of the forces acting on the disc were applied at it, the mass being concentrated there also ; and that the disc rotates about the center of gravity as if it w r ere fixed in space. First we will consider the motion of the center of gravity only, and will assume that the shaft is vertical. Suppose that the disc is not rotating so that the B Fig. 5 will coincide with the point O. Then the center of gravity C will assume a position C slightly below O. If the point B be displaced in any direction from O the elastic- ity of the shaft will cause a force proportional to the displacement tending to return B to O, We will suppose the disc displaced without rotation, so that the center of gravity C will have an ex- actly equal displacement from its position of rest C ' . Since the center of gravity moves as if the displacement force acted on it, we may consider that the center of gravity C is acted on by a force proportional to its displacement from C '. The center of gravity will then have a simple harmonic motion, and will vibrate back and forth until the energy is absorbed by molecular friction. The period of such a vibration is well known to be $ =& using the notation already given. The most general case of harmonic motion will be when two displacements at different angles and at different times are given to the disc. It will then vibrate in an ellipse with period \j/. We may now apply any other forces to the shaft when it has this vibration, and the disc will then have a displacement com- pounded of that due to the vibration and that due to the new forces. Suppose, for instance, we apply a pull to the center of gravity of constant magnitude but of varying direction, rotating in fact with * The discussion up to this point substantially as here given, was published by Stodola in Zeitschrift des Vereines Deutscher Ingenieure, Vol. 47, Nos. 2 and 4, Jan. 10 and 24, 1903. However, the matter is given here since it was prepared independently before the above publication. The balance of this chapter is probably new. Stodola gives a treatment along totally dif- ferent lines which does not seem satisfactory. an angular velocity o>. So far as the shaft is concerned this force will cause a rotating displacement, to be compounded with the vibrating displacement. Such a rotating force is furnished by the "centrifugal force," which is in equilibrium with the force due to a rotating displacement R + a of the shaft center B. Hence the disc may rotate in the position shown by Fig. 5, and at the the same time have an elliptical vibration. Conversely, if the disc be rotating in the position of Fig. 5, and any accidental displacement be given to it, it will at once begin to vibrate, in addition to the rotation, in the same way as it would if the displacement were given to it when at rest. The above reasoning is perfectly rigid, but nevertheless the step is perhaps a large one to make at once. We will therefore at a later period, for verification, form the differential equation of mo- tion of the disc, and show that the integral gives a displacement compounded of an elliptical vibration and a rotating displace- ment R. Such a mathematical treatment hides the true nature of the actions, however, and the method above used is much more luminous. It must be noted that if a displacement from the position of Fig. 5 be gived the disc during rotation, the resulting vibration will always be parallel to the original displacement, and will not rotate in direction. That is to say, if a displacement should happen to be made in the direction of the instantaneous position of the radius to the center of gravity, the vibration induced will not continue to be radial, since then the direction of vibration would constantly change, but it will be in a direction fixed in space parallel to the initial direction. If the elastic force resulting from an initially radial displace- ment were always radial, it can be shown that the position of Fig. 5 would be unstable. Any accidental displacement from the position of Fig. 5 will therefore in general cause the point C to vibrate in an ellipse about the equilibrium position, the axes of the ellipse always remaining parallel to their original position in space. Owing to molecular friction this vibration will soon die away, the size of the ellipse gradually decreasing until the position of Fig. 5 is again reached. Next let us consider the effect of the vibrations just considered in causing rotation of the disc about the center of gravity. Let Fig. 6 represent any instantaneous position of the points O, Z?and C while C is making an elliptical vibration. The path of B will be parallel since the vibrational displacements have involved no 42 rotation of the disc. The net force acting on the disc is then pro- portional to O B. If the center of gravity is now fixed and a force proportional to OB acts on B as shown, there will be a twisting moment on the disc proportional to OB X CD, Fig. 6. This twisting moment is resisted by the torsional elasticity of the shaft, so that there is a tendency to produce torsional^ vibration. Since all of the displacements are very small the shaft may be considered as perfectly straight in computing the torsional elas- ticity. Now, both O B and CD are very small quantities as com- pared with the torsional elasticity, so that the angle through which the shaft is rotated by the twisting moment OB x CD is of the second order of small quantities and may be safely neglect- ed. That is to say, a displacement of the disc has no appreciable effect in causing additional rotation of the disc about its center of gravity beyond that due to the constant angular velocity. There is another possible displacement from the position of Fig. 5, consisting of a finite rotation of the disc so as to cause torsion in the shaft, as shown in Fig. 7. If the angular velocity o> did not exist, a torsional displacement would cause a harmonic tor- sional oscillation of the disc about its center of grovity. The FIG. 6. Transverse Vibration. FIG. 7. Torsional Vibration. FIG. 8. Effect of Gravity. FIG. 9 Most General Vibration. twisting moment due to torsion of the shaft is a pure couple, so that it causes rotation about the center of gravity, even though the center of bore is at a different point. The period of the tor- sional oscillation will be = \f-* using the notation already given. If now the disc be given a constant angular velocity w in addition to the torsional vibration, the latter will persist just as before, so far as rotation of the disc about its center of gravity is concerned. The force due to the net deflection, proportional to OB, Fig. 7, will here also give a twisting moment about the center of gravity proportional to OB x CD. This is of the second order of small quantities and will be neglected as before. -43- The line B C, Fig. 7, will therefore rotate with a constant an- gular velocity o>, and also oscillate back and forth so as to make a variable angle a with the normal position B' C' . Since a varies harmonically with period <, we will have a = c t cos t. The in- troduction of the constant angular velocity does not affect the har- monic oscillation as already stated. Next let us consider the effect of such a torsional oscillation on the displacement of the center of gravity. Owing to the previ- ously considered displacements without rotation, the points^ and C were displaced equally beyond the displacement necessary to balance the centrfugal force, SQ that the force acting on the center of gravity could be taken as proportional to its own displacement. Now we must consider that the force acting on the center of grav- ity is proportional to the displacement of B beyond the position necessary to balance the centrifugal force. This is not the dis- placement of the center of gravity owing to the variable angle a. The position of C without the vibration we are now considering would be at a distance R from O, Fig. 7, near C ', and owing to the addition of the vibration will be supposed to be C at any in- stant. The vibration we are considering is the absolute vibration of the center of gravity in space in addition to the constant rota- tion. Suppose we put OC' = = fi-\-v and C C' = rj and resolve forces and accelerations along and perpendicular to the moving axis O C' . v and rj will then give the departures from the equi- librium position due to the vibration we are considering. The force perpendicular to the axis is AT X B B' or K(-rj + flsina), since we have CC' = t] = BB' a sin a. Since a will always be small we take a instead of sin a. We have already re- marked that a = r t cos t, so that the force perpendicular to the moving axis is K '(rj -f- ac^ cos t}. The total force in the direction of the moving axis is - K X O C or K ( + a cos a). Now a portion of this, K (R + ), is required to maintain equi- librium against the centrifugal force, so that the remaining force, K ' (y a -f- a cos a), is the force in the direction of the moving axis tending to cause the vibration we are considering. Since a is small we may for a second approximation put cos a i, so that the force becomes Kv. The accelerations along and perpendic- ular to a moving axis are respectively (Routh's Rigid Dynamics, Vol. I, page 176), d 2 2 i d , 44- ^9_,.+L _) Since w is constant we may reduce the final terms. Also substi- tuting in terms of v, we have as the accelerations, _-_. Now the term R w 2 gives the acceleration due to the rotation in the equilibrium position so that the acceleration due to the added vibration alone is found by omitting this term. We have already omitted the corresponding centrifugal force, K '(R + a). Bquating the forces previously found along and perpendicular to the moving axis to the forces required to produce the above ac- celerations we have ,2 , . .-. d 2 77 o i d v ^ 2 (77 -fa^COS^O = ' 7?a> 2 -f 20)-. at at This is a pair of simultaneous linear differential equations whose solution is complicated. The integration will be omitted and the integrals only given. These will be found correct by substitution in the equations. v = 2 B to < sin < / y = _ B O 2 - co 2 - < 2 ) cos t where Since v and rj are displacements along and perpendicular to the rotating axis O C' , and since the values above evidently repre- sent a harmonic motion in an ellipse with axes in these directions, it follows that any accidental torsional displacement will cause a vibration of the center of gravity with period < in an ellipse whose axes rotate with an angular velocity w, one of them always coin- ciding with the radius vector to Ihe equilibrium position of the center of gravity. This is in addition to the simultaneous tor- sional oscillation of the disc and shaft. This oscillation and con- sequently the vibration of the center of gravity induced by it will of course rapidly die out owing to molecular friction, so that the equilibrium position of Fig. 5 will be again reached. -45- We have always supposed a vertical shaft so that gravity need not be considered. Let us now make the shaft hori- zontal and take gravity into account. First taking all the forces as acting on the center of gravity, we will have a down- ward pull M and an equal upward pull due to a down- ward deflection of the shaft, the displacement of the disc M being ^> The point C will therefore rotate in a circle of radius R whose center is O Fig. 8, a distance below the axis of ro- JK tation O'. Next let us consider if rotation occurs about the center of grav- ity due to the action of the weight as shown in Fig 8. If the center of gravity Cbe fixed there will be a force proportional to the net deflection O B acting at B, tending to rotate the disc and resisted by the torsional elasticity of the shaft. The twisting moment will be proportional to O B X CD, Fig. 8, and will therefore be a small quantity of the second order, so that the torsional vibration induced will be negligible just as in the pre- vious cases. We have now analysed the matter completely and taken ac- count of everything which can occur. The methods used have been chosen so as to show most clearly the actual nature of the various actions, and have perhaps been rather inelegant mathe- matically. We will therefore go through with the matter again from a strictly mathematical standpoint for verification of our re- sults. This treatment is distinctly obscure so far as the nature of the action is concerned however. Fig. 9 shows the most general position of the various points. Since we have a rotation with angular velocity w, induced by an external source, we take as a reference line an axis rotating with this velocity and therefore making an angle t 8) g dt Evidently p cos (co / 8) = x + a cos (w t -f a) p sin (w / 8) = jj/ + a sin (w /f + a) . Hence, our force equations may be written ^.= - ^ { x + a cos (o, / + a) j 2 /* *\ ^^ = ^ 2 J j/ -f a sin (w / + a) I g. In order to integrate these we must find an expression for a. This we do by forming the equation of rotation about the center of gravity, assuming that it is fixed and that all of the forces act to cause rotation about it'. The force moment equation then is We have dt* dt*' Now A' si n (a -f 8) is of the same order of magnitude as L a. p and a are both small quantities so that their product is of the second order of small quantities, and therefore negligible. Hence the term Kp a sin (a -f- 8) is negligble as compared with L a. This amounts to saying that the twisting moment A" X OB X CD, Fig. 9 can never cause an appreciable torsional vibration of the shaft. Hence we have This is a linear equation whose solution is well known to be a = c v cos t. There is in general a second arbitrary constant which we may take zero without loss of generality, beginning our count of time from the instant when the vibration is at one extreme. 47- It is to be remarked that a variable twisting moment would really be required to drive the shaft at the constant angular velocity w, owing to the variation of a and of the shaft displace- ment perpendicular to the reference line. Both of these should be balanced by a variable external twisting force in order that there should be no angular Deceleration at the extremity of the shaft. However, there are usually rotating bodies with consider- able mass geared to the shaft, and the inertia of these serves to prevent any appreciable angular acceleration. Substituting our value of a in the force equations above, and taking a as small so that we can put cos a=r i sin a = a, we have 5 -f ^ x if/ 2 a cos w t H- V 2 # *\ cos < t sin w / d 2 v 5 + "fy ^ 2 # sin t cos o> t g. These are linear differential equations which are integrated by well known methods which we need not consider here. The form to which we reduce the integrals has been arranged to exhibit the various individual vibrations already discussed. The integrals are x = R cos o> / -f c 3 cos ft cos O / + ft ) + <: 3 sin ft sin O t + ft) -f- B \ ( i/f 2 to 2 < 2 ) sin co t cos t + 2 co < cos co / sin ^> / ! sin co t sin , an elliptica 1 vibration with axes fixed in space and angular velocity ^, an elliptical vibration with axes along and perpendic- ular to the moving reference axis and angular velocity <, and a displacement vertically downward. These are the motions already found otherwise the coefficients also agreeing. Hence, we con- clude finally that the most general possible motion will consist of vibrations about the position of Fig. 5, which will die out ow- ing to molecular friction so that the actual position of Fig. 5 will always be reached. CHAPTER VI. EXPERIMENTS WITH A GAS TURBINE. This portion of the investigation of the gas turbine, really the most important of all, has unfortunately turned cfUt in a very un- satisfactory manner. So far as principles are concerned all of the statements previously made were found correct, but no actual re- sults were obtained. After many failures a gas turbine was actually operated, but the net output was negative. That is to say, the turbine wheel did not even yield enough power to com- press the air required. This result is to be attributed solely to the crudness of the apparatus used. The apparatus was arranged very much like the diagram of Fig. 2, except that the air was supplied by an independent source and the whole output of the wheel measured by a brake. Considerable difficulty was found in igniting the air and oil and starting the continuous combustion under pressure. A successful method of doing this was finally found however. Next trouble occured in securing smokeless combustion, which was also over- come. After this the combustion gave no further trouble, and gases at a white hot temperature issued from the nozzle with per- fect regularity. However, the nozzles first used were soon de- stoyed. After many experiments a nozzle was built which lasted a reasonable time. Permanent success was not attained, however. The turbine wheel used was that from a very small DeL,aval steam turbine and was about 5 inches mean diameter. The best speed was found to be about 19,000 revolutions per minute, and at this speed about 3 horsepower was developed. About 4 horse- power was theoretically required to compress the air. No experimental work was done in the way of varying the pressure, the shape of the nozzle, and the shape of wheel buckets. There is of course an almost infinite field for variation in these particulars and the writer has no doubt that experiments in these directions will ultimately be crowned with success. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. LD 21-100m-7,'33 GAYLORD BROS. MAKERS SYRACUSE. - N.Y. YD 02673 8322