~_n_n n_-r n n, REESE LIBRARY OF THE UNIVERSITY OF CALIFORNIA.' Deceived A/y ,-89 ssion No. / Q 6 3 . C/js.v No. THE ELEMENTS OF WATER SUPPLY ENGINEERING. BY E, SHERMAN GOULD, M. AM. SOC. C. E. NEW YORK: THE ENGINEERING NEWS PUBLISHING CO. 1899. COPYRIGHTED, 1899, BY THE ENGINEERING NEWS PUBLISHING CO, 72% 30 PREFACE. " Practical Hydraulic Formula? " was first published in 1880. The second edition appeared in 189 in a form much extended by notes on THE QUALITY and THE QUANTITY OF WATER and on THE CALCULATION" and THE CONSTRUCTION OF DAMS. In the present work all the matter contained in the first and second editions of "Practical Hydraulic Formulas" is repub- lished ; but, under the heading of " NOTES TO PARTS I. AND II.," it has been supplemented with copious memoranda, elucidating and greatly extending many important points inadequately treated in the previous editions. An entirely new part (PART II I.) has been added. This treats Of the FLOW OF WATER THROUGH MASONRY CONDUITS Oil the basis of Darcy's formulae, with some practical details of aqueduct and tunnel construction. A few paragraphs are devoted to the subject of the FILTRATION" OF PUBLIC WATER SUPPLIES, suffi- cient, it is hoped, to indicate the proper lines of further investi- gation to those who are interested in pursuing them. A some- what fuller treatment is given to the subject of PUMPING EN- GINKS AND DUTY TRIALS. Some pages, have been added on the subject of ARCHES AND ABUTMENTS. As an extended project of water supply frequently embraces the construction of arched masonry aqueducts, it is believed that the brief and practical rules given in this part of the book will prove acceptable to the hydraulic engineer. A set of carefully calculated, labor-saving TABLES, with explanations and examples, terminates the book. It will be seen that the present work covers so wide a field that to retain for it as a whole the title originally given to the first part would be misleading. It is, therefore, called " THE ELEMENTS OF WATER SUPPLY ENGINEERING," which name more truly indicates its scope. While it is believed that every topic connected with water- supply engineering has been at least touched upon, especial pains have been taken to go into very close detail in the matter of the principal dimensions and quantities involved in the designing of hydraulic work. As these are the points which the author has most carefully sought for in his own reading and observation, so , he believes that they are the ones which others may be most in- terested in finding fully treated of in the present volume. E. SHERMAN GOULD. OF THE UNIVERSITY TABLE OF CONTENTS, INTRODUCTION. CHAPTER I. Flow through a short horizontal pipe Effect on velocity of increased length Frictional head Hydraulic grade line Hydrostratic and hy- draulic pressures Piezometric tubes Results of raising a pipe line above the hydraulic grade line Why the water ceases to rise in the upper stories of the houses of a town when the consumption is increased In- fluence of inside surface of pipes upon velocity of flow Darcy's coef- ficients Fundamental equations Length of a pipe line usually deter- mined by its horizontal projection Numerical examples of simple and compound system^ Pages 11-24 CHAPTER II. Calculations are the same for pipes laid horizontally or on a slope Qualification of this statement Pipe of uniform diameter equivalent to compound system General formula Numerical example Use of logarithms (foot note) Numerical example of branch pipe Simplified method Numerical examples Relative discharges through branches variously placed Discharges determined by plotting Caution regarding results obtained by calculation Numerical examples Pages 25-38 CHAPTER III. Numerical example of a system of pipes for the supply of a town Es- tablishment of additional formulae for facilitating such calculations Determination of diameters Pumping and reservoirs Caution regard- ing calculated results Useful approximate formulae Table of 5th powers Preponderating influence of diameter over grade illustrated by example Pages 39-48 CHAPTER IV. Use of formula 14 illustrated by numerical example of compound system combined with branches Comparison of results Rough and 8 TABLE OF CONTENTS. smooth pipes Pipes communicating with three reservoirs Numerical examples under varying conditions Loss of head from other causes than friction Velocity, entrance and exit head Numerical examples and general formula Downward discharge through a vertical pipe- Other minor losses of head Abrupt changes of diameter Partially opened valve Branches and bends Centrifugal force Small import- ance of all losses of head except frictional in the case of long pipes All such covered by " even inches" in the diameter - Pages 49-63 CHAPTER V. Notes on pipelaying - Pages 64-67 APPENDIX. Weight of cast iron pipe Various useful formulae Pages 68-70 SECOND PART. NOTES ON WATER SUPPLY ENGINEERING. Quality of water Quantity of .water Dams, calculation of; con- struction of Reference to other .publications - - Pages 71- 105 Notes to Parts I and II - ..... . . Pages 106-122 THIRD PART. Flow of Water Through Masonry Conduits - Pages 123-125 Some Details of Tunnel and Acqueduct Construction Pages 125-126 Filtration of Public Water Supplies - Pages 127-129 Pumping Engines and Duty Trials - - - Pages 129-137 Arches and Abutments ------- Pages 137-154 Hydraulic Tables * . - * Pages 154-162 Index - - Pages 163-168 INTRODUCTION TO HYDRAULIC FORMULA. The following pages first appeared as a series of articles in the columns of ENGINEERING NEWS. They are now repub- lished with a few corrections and additions. In virtue of the law of gravitation, water tends naturally to pass from a higher to a lower level, and without a difference of level there can be no natural flow. It can be said in all seriousness although the statement may seem to invite the unjust accusation of an ill-timed attempt at pleasantry that the whole science of hydraulics is founded upon the three following homely and unassailable axioms : First. That water always seeks its own lowest level. Second. That, therefore, it always tends to run down hill, and Third. That, other things being equal, the steeper the hill, the faster it runs. In the case of water flowing through long pipes, the hill down which it tends to run is the HYDRAULIC GRADE LINE. If the pipe be of uniform diameter and character, the hydraulic grade line is a straight line joining the water surfaces at its two extremities, provided that the pipe lies wholly below such straight line, and its declivity is measured like that of all hills by the ratio of its height to its length. But if there be any changes whatever in the pipe, either in diameter or in the nature of its inside surface ; or if there be in- (9) 10 INTRODUCTION TO HYDRAULIC FOUMILJE. crease or diminution of the volume of water entering it at its upper extremity by reason of branches leading to or from the main pipe, then the hydraulic grade line becomes broken and distorted to a greater or less extent, so that its declivity is not uniform from end to end, but consists of a series of varying grades some steeper than others though all sloping in the same direction. As regards the third axiom, the proviso " other things being equal" must not be overlooked. For we shall find that a pipe of greater diameter but less hydraulic declivity than another, may give a greater velocity to the water passing through it. Also, of two pipes of the same hydraulic slope and diameter, the one having the smoother inside surface affords the greater velocity. The vertical distance from any point in a pipe to the hy- draulic grade line, constitutes the Piezometric JieigJtt, and meas- ures the hydraulic pressure at that point. It will be seen that the solution of problems relating to the flow of water through pipes, lies in the knowing or ascertaining of the piezometric height at any desired point. In general, it is necessary to es- tablish the piezometric height for every point of change of any kind which occurs throughout the entire length of the conduit. The joining of the upper extremities of these heights gives the complete hydraulic grade line. The object of the following papers is to establish systematic methods for tracing the hydraulic grade line under the different circumstances likely to occur in practice, and generally, to fur- nish solutions for a large number of practical problems, com- mencing with the simplest cases and extending to some rather intricate ones, not usually embraced in our hydraulic manuals. E. S. G. SCRANTON, Pa., May, 1889. HYDRAULIC FORMULAE. CHAPTER I. Flow through a Short Horizontal Pipe Effect on Velocity of Incr, ased Length Frictional Head Hydraulic Grade Line - Hydrostatic and Hydraulic Pres- sures Piezometric Tubes Result of liaising a I 'ipe Line Above the Hydraulic Grade Line Why the Water Ceases to Rise in the Upper Stories of the Houses of a Town when the Consumption is Increased Influence of Inside Surface of Pipes Upon Velocity of Flow Darcy's Coefficients Fundamental Equations- Length of a Pipe Line Usually Determined by its Horizontal Projection Nu- merical Examples of Simple and Compound Systems. Let us suppose a reservoir of large relative area and depth to be tapped near its bottom by a horizontal cylindrical pipe, of which the length is equal to about three times its diameter. If there were no physical resistance to the flow, the velocity of the water issuing from the pipe would be given by the formula for the velocity of falling bodies . in which V = velocity in feet per second, g = the acceleration due to gravity = 32.2 ft., and H = the height, expressed in feet, of the surface of the water in the reservoir above the center of the Pipe- Observation shows, however, that in the case cited the ve- locity of discharge is equal only to that theoretically due to a height of about two-thirds of H:, that is : !?_? = 6.55 yjj. 12 PRACTICAL HYDRAULIC FORMULA. The remaining third of the height is consumed in over- coming the resistance offered to entry by the edges of the orifice to the inflowing vein of water. The head necessary to overcome the resistance to entry is therefore about one-half of that neces- sary to produce the velocity of flow. If the length of the pipe should be increased progressively and indefinitely, the velocity would be found to diminish in- versely as the square root of the length. It would correspond, therefore, to a smaller and smaller percentage of the total head H. The resistance to entry diminishes directly as the velocity, and the. head necessary to overcome it is always equal to about one-half of that necessary to produce the given velocity as cal- culated by the laws of falling bodies. As the length of the pipe (always supposed to remain hori- zontal) increases, and the velocity of discharge diminishes, the sum of these two heads, i. e., one and a half times that necessary to produce the actual velocity, is no longer equal to the total head H, as we have seen to be the case when the length of the pipe is only about three diameters. What, then, becomes of the remainder of #? It is consumed in overcoming the increasing frictional resistances engendered by contact of the Coving water with the inside surface of the pipe. When the pipe is very long, and the velocity therefore relatively low, the sum of the velocity and entrance heads is small, and by far the greater part of the total head is required to force the water through the pipe against the opposition offered by friction to its flow. In such cases, which are those occurring most generally in practice when water is conveyed from a reservoir for the supply of a town, the velocity and entrance heads are commonly ignored, and the total head If is supposed to be available for overcoming the frictional resist- ances. As this occasions, however, an error although generally a very small one in the wrong direction, judgment is required in exercising this latitude. Later on we will revert to this point, PRACTICAL HYDRAULIC FORMULAE. 13 but for the present we will consider only frictional resistances, particularly since and indeed because in practice our assumed data are almost always sufficient to afford an ample margin to cover the neglected factors. In what precedes we have considered a horizontal pipe issuing from a reservoir in which the surface of the water is main- tained at a constant level. In practice these conditions rarely obtain. FIG. i. Suppose a system, such as is shown by Fig. 1, consisting of a reservoir and pipe line of varying and contrary slopes. As the level of the water in the reservoir would be subject to fluctua- tions, and liable at times to be greatly drawn down, it is custom- ary to consider the surface of the water as standing at its lowest possible level, i. e., the mouth of the pipe. In this case, the value of H would be equal to the difference of level of the two extremities a and b of the pipe, and the line ab joining the centers of the two ends would form what is called thv hydraulic grade line, the establishing of which is the first step to be taken in laying out a system of water supply. Suppose that at the points c, d, and e vertical tubes, open at their upper ends, were connected with the pipe. The water, when flowing freely from the end b of the pipe would rise in each of these tubes to about the height of the hydraulic grade line at these points, and, if branches were connected at the points c, d, and e, they would, when closed, sustain a pressure upon their 14 PRACTICAL HYDRAULIC FORMULAE. gates equal to the bead comprised between the gates and the grade line. If the gates were open, the branches would discharge water under heads equal to the difference of level of the hydraulic grade line at the point of embranchment and their remote extremities, less a certain amount depending upon the volume discharged, which will be spoken of hereafter. At d, where the top of the pipe just touches the grade line, there would be no pressure at all when the water was flowing through the pipe, except the very small amount due to the depth of water in the pipe itself. If the end b should be closed so that there was no movement of water in the pipe, the water would rise in the tubes, if they were long enough, until it stood at the same level as the water in the reservoir, and the pressures at c, d, and e would be equal to the head comprised between these points and the level of the water in the reservoir. The latter is called the hydrostatic press- ure, or simply the static pressure, and the former the hydraulic pressure, at these points. The tubes spoken of are known by the name of piezometric tithes. . The importance of correctly establishing the hydraulic grade fine is illustrated by reference to a case such as is shown in Fig. 2, in which the pipe, at the point c, rises above the grade line ab. To explain: It will be readily deduced from what has been al- ready said in reference to horizontal pipes that the velocity of flow, and. consequently the delivery, of a pipe increases with the steep- ness of its slope. In this case the pipe ab is divided into two parts, the one ac with a hydraulic grade line flatter than ab, and the other cl with one steeper than ab. The delivery of the entire system, if the pipe were of the same diameter throughout, would be governed by the flatter portion ac, and the portion cb would be capable, in virtue of its steeper slope, of discharging a greater PRACTICAL HYDRAULIC FORMULA. 15 volume of water than it could receive from ac. Consequently it would act merely as a trough and would never run full, and if a piezometric tube were placed in it at d for instance, no water would rise in the tube, and no pressure be exerted. FIG. 2. It is very important, therefore, in locating a pipe line that the pipe should nowhere rise above the hydraulic grade line. The full amount of water could indeed be carried over the high point c by means of siphonage, but this expedient is not resorted to in practice. Should the nature of the ground require such a location as that shown in Fig. 2, it would be necessary to increase the diam- eter of the pipe between a and c, so that it would deliver the re- quired volume under the reduced head, and to diminish that be- tween c and b, so that it should only deliver the same volume under its increased head, and therefore run full. The calculations necessary to determine the proper diameters will be shortly de- veloped. Should the axis of the pipe coincide exactly with the hy- draulic grade line ab, the pipe would run full (provided the feed were sufficient) but would be under no pressure, and no water would rise in piezometric tubes placed on any part of the pipe. Moreover, as the slope would be the same for any portion of the pipe, the velocity and delivery would be unchanged, whether we 16 PRACTICAL HYDRAULIC FORMULAE. cut the pipe off at a comparatively short length, or extend it in- definitely. As a further and very interesting practical illustration of the effects of a hydraulic grade line of varying steepness, let us con- FIG. 3. sider (Fig. 3) the case of a house supplied with water by a pipe communicating with a reservoir. Suppose the pipe to be sufficiently large to furnish a certain volume of water per hour to the upper story of the house. If now a larger volume were required, it is clear that, unless we in- crease the diameter of the pipe, it would be necessary to increase the steepness of pitch of the grade line, in other words, to in- crease the head, or difference of level between the reservoir and the point of discharge. The increased volume could therefore be only drawn from a lower story.* Or, to put the same conditions under a different form, sup- pose, as before, the pipe to be just large enough to supply the top story of the house, the taps on the lower floors being closed. Should they be opened, it is evident that a greater amount of water would be discharged from them than from the upper one, because they would discharge under a greater head. The result would be a diminished flow or perhaps no flow at all on the top floor, and an increased discharge of water at a lower level. * In other words, if we wish to increase f he volume, the diameter of pipe rem in inj? constant, we must increase the velocity; and the increased veJocit^ can only be obtained by an increased difference ot' level between the two ends of the pipe. If the elevation of the upper end, or surface of water in the reservoir, cannot be increased, that of the 1 jwer end, or point of discharge, must be diminished. PRACTICAL HYDRAULIC FORMULA. ..17 This case shows why the water ceases to rise in the up- per stories of the houses of a town when the consumption in- creases. It has been found by observation that the velocity of water flowing through pipes is greatly affected by the nature of their inside surface, increasing with the smoothness and diminishing with the roughness of the same. By direct experiment, coeffi- cients have been established for different conditions of surface. It has also been found that these coefficients vary slightly with the diameter of the pipe, a pipe of a certain size giving a greater ve- locity than one of the same character of inside surface but of smaller diameter, the differences becoming smaller as the diame- ters increase. The value of this coefficient, which will be designated throughout this paper by C, is given below for a number of dif- ferent diameters and for two classes of pipes, those which are clean and smooth on the inside, and those which are rough and incrusted, the difference being as 2 to 1. As all pipes, after a few years of service, are liable to become more or less roughened and ob- structed by deposits, it is always safer when calculating the proper diameters of a permanent water supply, to assume rough pipes at once, although diameters thus calculated will, for perhaps a number of years, deliver quantities greatly in excess of the de- sired amounts. The coefficients given below are those determined experi- mentally by DARCY. Of course, in the subsequent calculations which will be made, any other values might be substituted for the ones given. It is well to remark, however, in regard to the coefficient, that although this factor is a controlling one in the calculation of the discharge of pipes, it is useless to attempt an excessive refinement in establishing its value, because not only is it difficult to determine this value with exactness for a given diameter and condition of pipe, but this condition, and even the 18 PRACTICAL HYDRAULIC FORMULA. diameter of the pipe, is liable to undergo considerable variation in the same pipe in the course of a few years. TABLE OF COEFFICIKNT3. Diameter in Value of Cfor Value of C for inches. rough pipes. smooth pipes. 3 0.00080 0.00010 4 0.00076 0.00038 6 0.00072 0.00036 8 0.00068 0.00034 10 0.0006S 0.00033 12 0.00066 0.00033 14 0.00065 O.OOOS25 16 0.00064 O.OOT32 24 0.00064 0.00032 30 0.00063 0.000315 36 0.00062 0.00031 48 0.00062 0.00031 In all the following calculations, the coefficient for rough pipes will be used. The two fundamental equations relating to the flow of watefr through long pipes are : DX H = cr' (i) L Q =AV (2) Equation N~o. 2 will generally be written : ,3, by taking the value of Ffrom (1). The first of these has been established by DARGY ; the sec- ond is based upon a self-evident proposition. In these equations : D - diameter of pipe ia feet H = total head L = length of pipe " " C = coefficient V = mean velocity in feet per second Q discharge in cubic feet per second A = area of pipe in square feet = D 2 x 0.785 TJie above two formula solve, directly or indirectly, all prob- PRACTI 'AL HYDRAULIC FORMULA 19 /.V relating to the flow through lung pipe*, and all such prob- lem* muxt be brought into a form admitting of their application, in order to obtain a solution. H It will be observed that is the rise or fall per foot of Lt length of pipe, and is therefore the natural sine of the inclina- tion of the slope to the horizon. This relation is frequently H used under the form / = . Using this notation, (1) would be L written : D i = c r* In long pipes the length is generally taken as being equal to the horizontal distance separating the two ends of the pipe, as the difference between this distance and the actual length of the pipe is relatively insignificant. If, however, a case should present it- self in which this difference was considerable, the actual length of pipe should be taken. Further on, an extreme case of this kind will be given, presenting isome interesting features. Some practical examples of the use of these formulae will now be given. In all that follows, the resistances of entry, exit, and velocity will be neglected, and the total head will be considered as available for overcoming friction. The examination of cases where the above factors are included is reserved for a later portion of this paper, as they are of secondary importance when dealing with long pipes. Example 1. A pipe, 1 ft. in diameter and 1,000 ft. long, has a total fall of 10 ft. What are the velocity and volume of its dis- charge ? Substituting the given values in (I) we have : i x 10 = 0.00066 V* 1000 V = 3.39 ft. per second. 20 PRACTICAL HYDRAULIC FORMULA. Using this value of Fin (2), we Lave : Q = 0.785 X 3.89 Q = 3.065 cu ft. per second. Example 2. Two reservoirs, having a difference of level of water surface of 30 ft., are joined by a pipe 3,000 ft. long. What should be the diameter of the pipe to deliver 16 cu. ft. of water per second from the upper to the lower reservoir ? Eliminating V between (1) and (2) we have : D X H Q* ~LY~C~ ~A*' Observing that A D 2 0.785 ; J)X H Q 2 _ L X C ~ D* X 0.616 Whence D * - Q!^_LX_C H X 0.616 If we knew the proper value of the coefficient C in the above equation, it could be immediately solved, and the value of D ob- tained. But C varies with the diameter, and the diameter, is as yet unknown. We must therefore have recourse to " Trial and Error" for a solution. Suppose it should appear to us, at first sight, that a 12-in. pipe was likely to be of the proper size. We therefore take C = 0.00066, and write : _ 256 X 3000 X 0.00066 30 X 0.616 " D* = 27.70 D = 1.94 ft. From this we see that the pipe should be nearly 2 ft. in diam- eter, and as we have taken too large a coefficient (that for 24 ins. = 0.00064), we are sure that 1.94 is too large. As pipes are never- made of fractional diameters, the above value of/) would be taken = 24 ins., and therefore we would push the calculation no further. If the case had happened to be one requiring minute accuracy, we would recalculate the above equation, using 0.00064 for the value PRACTICAL HYDRAULIC FORMULA. 21 of C. The result would be, D = 1.93 ft. nearly, practically the same as the value already obtained. The above examples (which are those commonly occurring in practice) are very simple, and involve only the direct application of the fundamental formulae. Let us now consider cases of a more complicated character, where they can only be used indirectly, and where a certain amount of judgment and tact is required in the preparation of the data. Example 3. Suppose a reservoir R (Fig. 4) containing a depth of water of 50 ft. above the center of the horizontal pipe A, 1 ft. in diameter and 1,000 ft. long, connected by a reducer with another horizontal pipe B, 2 ft. in diameter and 3,000 ft. Jong. It is required to calculate the piezometric head li at the junction, from which the discharge can be calculated, and the hydraulic .grade line abc established. \ \ \ \ 1000 3000 FIG. i. It is evident that the 24-in. pipe must, under the head h, dis- = 0.1007 D = 0.63 This value of D indicates a practical diameter of 8 ins. In order to check this value, we may write (4) under the form : o = / D * x H x - 6lS V \ Lx C Substituting given values : 1'|Q7 X 50 X 0.616 2.178 Q = 1.193 cu. ft. per second, thus proving the correctness of the work. 28 PRACTICAL HYDRAULIC FORMULA. These calculations can be abridged, and, in many cases, suffi- cient accuracy secured by adopting a mean common value for (7. If we do so in the present case, becomes a common factor, and disappears from the calculation, (5) becoming L I, I, 1 3 - + 1--, etc. (5) bis D B as as d 3 * If this equation be worked out for the above given values, we have : D = 0.64 or 8 ins. as before. It will be observed that this process might have been used with advantage in the previous example, by ascertaining the dis- charge of an equivalent pipe, and then calculating the heads necessary to produce this discharge through the different pipes. In calculating fifth powers and roots, a table of logarithms is almost indispensable. If none is at hand a table of squares and cubes is of some use, remembering that a number can be raised to the fifth power by multiplying together its square and cube. Fifth roots, in the absence of logarithms, can only be extracted by il trial and error/' using the above rule for fifth powers.* Example 4th. A horizontal pipe (Fig. 7), 48 ins. in diameter and 2,000 ft. long, issues from a reservoir in which the surface of the water is maintained at a constant height of 50 ft. above the center of the pipe. Midway, this pipe is tapped by a branch pipe 24 ins. in diameter and 500 ft. long, with a rising grade of 4 ft. in 500. What is the piezometric head li at the junction, and what the discharge from each pipe ?f It is evident that the 48-in. pipe above the junction musts, with the head 50 li, discharge as much water per second as the * All hydraulic calculations are greatly facilitated by the use of logarithms; and those engaged in making such calculations should not fail to familiarize themselves with the us>e of the&e powerful auxiliaries to arithmetical work. t With these lengths and diameters, the ahove system does not properly come under the classification of "long pipes." As the present object is only to exemplify methods of calculation, the example is equally good. PRACTICAL HYDRAULIC FORMULA. g combined discharge of the 48-in. pipe below the branch with the head h, and the 24-in. pipe with the head li 4. From (3), FIG. 7. which in this case will perhaps be the most convenient equation for quantity, though that derived from (4) is frequently useful, we have : 1000 X 0.00062 q = 12.56 4 / L? 1000 X 0.00062 500 X 0.00064 which, put in equation, give : 1000 X 0.00062 12.56.4/ t?. r innn v n n + 3.U ' 2(ft - 4 > 1000 X O.OC062 500 X 0.00064 The coefficients 0.00062 and 0.00064 are so nearly equal that we may, in the following calculations, discard them as common factors. Dividing by 3.14 and striking out also the common factors -j-^oo an( * T! o> we nave simply i 4 Vso - h = 4 I/A" + I/A 4 Squaring 800-16/i = 16A + /i 4 + 8 |/A 4 A which gives: 33 A = 804 -8 VA - 4 A Neglecting, for a first approximate value of li the quantities affected by the radical : 33A = 804 30 PRACTICAL HYDRAULIC FORMULAE. Neglecting decimals : h = 24. Substituting this value for h under the radical : 33A := 804 - 8 V576 - 96 \vliich gives, always neglecting decimals, a second approximate value : h = 19. A third and fourth approximation give respectively // = 20.3 and h = 20. We will take 20.1 as very near the true value.* Substituting 20.1 in place of h in the equations giving the quantities discharged, we have : /4 x 29.9 Q = 12.56 A/ = 174.45 r n ft* /4 X 20.1 12.56 A/ = 143.05 r p. R9. /2 X 16.1 q' = 3.14 |/ = 31.50 " 0.32 We have thus : Q = a + q'. The ahove method gives directly the true value of h; but it involves tedious figuring, even in our example, which happens to admit of many simplifications owing to the number of common factors. It will be easier, and often shorter, to obtain the value of h by first assuming one which we judge likely to be near the truth, calculating what discharge it would give from the two branches, and then calculating the head necessary to discharge the same quantity from the single pipe above the branch. Then, comparing the total height thus obtained with the known height of the water in the reservoir, we can deduce the true value of h by a proportion. Let us apply this method to the above example. We know * The value of h may he obtained directly by using the usual formula for adfected quadratic? ; bur, with the aid of a table of squares and square roots, the above approximate method will generally be the easier and quicker one. PRACTICAL HYDRAULIC FORMULAE. 31 at once that li must be less than 25. because that would be its value if the 24-in. branch were closed. Supposing we judged that 22 ft. would be about correct. We then have to solve the two equations : q = 12.56 \/ = 149.60 " n RO X 22 62 / 2 X 18 q' =--3.14 { - = 33.30 0.32 also, for the equal discharge through the 48-in. pipe above the branch, squaring (3), we have : (182.90) 2 X 0.62 h = = 32.87 (12. 56) 2 X 4 This height, added to 2*2, the assumed value of h, gives a total height of 54.87 ft. as against 50 ft., the actual total height. By proportion we have : h 50 22 54.87 This value of h agrees with that already found. If the 24-in. branch were closed we should have for the dis- charge : 1.24 When the 24-in. branch was open we had a total discharge of 174.73 cu. ft. per second. There is an increase, therefore, of about 9| per cent, by opening the branch. Let us now see what the discharge would be if the branch were placed only 500 ft. from the reservoir, instead of 1,000 ft., all the other conditions remaining the same. We will assume h = 33 ft. and solve the two equations / Tx~33 q = 12.56 4/ --- = 149.5 \ 1.500 X 0.00062 2 X 29 0.32 32 PRACTICAL HYDRAULIC FORMULAE. (191. 8) 2 X 0.31 also h' = 18.07 (12 56, 2 X 4 giving a total height of 51.07 as against 53. Reducing : h 50 33 ~ 51.07 h = 32.3 Using this value, instead of the assumed one, we have : / 4 X 17.7 /4X323 / 2 X 23.3 12.56 1/ r 0.31 r 0.93 17.7 /4X323 /2X 23.3 = 12561/ -Kllti/ I r 0.93 f* 0.32 189.83 = 148.03 + 41.76 very nearly. As compared with the discharge when the 24 in. branch is closed this shows a gain of 19 per cent., just double the gain when the branch was located at the center of the pipe. Supposing now that the branch were placed 1,500 ft. from the reservoir. Assuming 10 ft. as a probable value of h we have : q = 12.56 / - = 142.46 ' 500 X 0.00062 /V X b 137 = X6 19.23 (161.7) 2 X 0.93 also: h' = = 38.53 (12.56) 8 X 4 h 50 By propor which will give a rise of 16 ft. between the ex- tremity and the main junction, and requires an elevation of piezometric head, at this junction, of (176), as shown in the fig- ure. To obtain the proper diameter of pipe for this grade and dis- charge, we have, using (4), and assuming C = 0.00076 as a prob- able value ; A /(^) 2 X 1000 X i = \/ - X LOOO X 0.00076 8 X 0.61 whence D 5 = 0.017304 and D = 0.444. PRACTICAL HYDRAULIC FORMULA. 43 Or, for the next highest even inch : D = 6 inches. As regards the diameter of the pipe on the east side, since the length and discharge are the same as for the west side, and only the heads vary, being respectively 16 and 36 ft., it can be ob- tained by means of (11). Thus : * ,0017304 X16 D' = 96 D' = 0.3777 or, for next highest even inch : D' = 5 inches. The above head of 18 ft. per thousand produces a velocity of flow in a 5 in. pipe of a little over 3 ft. per second, which is somewhat greater than it should be. If the limit of velocity is overstepped to any considerable degree in a system of pipes such as we are considering, it would be best to use a larger pipe and check its flow down to the desired delivery by means of a gate or stop cock placed near its upper end, the effect of which will be to diminish the head. In the present instance the excess of velocity is probably not sufficient to render this precaution necessary. The elevations are such that the above diameters of 6 and 5 ins. are also proper for the side mains G G', F F'. It is now necessary to calculate the diameter of the central main from B to C. This main might be divided into two parts, that between F F' and G G' and that between G G' and H H', but we will calculate it upon the supposition of a uniform diameter, capable of delivering the entire volume of f cu. ft. per second as far as IT H'. Assuming a probable value of C = 0.00066, we have from (4): 16 Z> 5 = - X 1.32 9 44 PRACTICAL HYDRAULIC FORMULA. whence: D* = 0.3817 and: D = 0.826 = 10 ins. Taking now the mains E E' and D D ', and beginning on the west side, assuming as before a grade of 8 ft. per 1,000, we find the length and head equal to those of F F' etc., the only differ- ence being the quantity it is desired to deliver, which is now i cu. ft. as against in F F'. The relation (9) is therefore ap- plicable, and we have: z>=\/o. 017304 X - 16 1 whence: I)' = 0.0097335 and D' = 0.396 or, say, D' = 5 ins. The mains on the east side are determined as before: / 16 0.0097335 X - D' = 0.346 This is not quite 4J ins., but to insure the desired delivery, it will be best to take the next highest even inch, and call it 5 ins. As regards the central main from A toB, we find two grades, the upper one ^-f^ and the lower J^-Q. The lower section must deliver, under a grade of ^o, all the water required for F F', G G', and H H', aggregating 2 cu. ft. per second. Using (4), and taking 0.00066 as a probable value of C, we have : 4 X0.66 jys - -- 6.1 PRACTICAL HYDRAULIC FORMULAE. 45 whence : Z> 5 = 0. 4328 and : D = 0.846 This is very nearly lOi ins., and a 10 in. pipe would answer, though 12 ins. would be better. The upper section must deliver 2.6 cu. ft. per second, under a grade of lTr 5 -y-. Taking the same probable value of (7, we have : 6.25 X 0.66 ^ D* = 3.05 whence : D = 1.237 which we can take as either 15 or 16 ins. This diameter might have been obtained from that of the- lower section, by means of (12). Thus : 10 6.25 D' * = 0.4328 X - X 5 1 D' = 1.287 This last formula might have been used throughout, but (4) is about as short and convenient ; frequently more so. The diameters being thus determined, the quantities should be verified by (3). They will be found somewhat in excess of those proposed, owing to the general increase of the diameters. As regards the height to which the water must be raised, the- data show that 3 cu. ft. per second must be raised to a sufficient height to reach D D' at an elevation of (201) above datum. lf t \v& adopt a grade of YoVo ^ ne proper diameter of the pipe would be : 9 X 0.65 z>* = 2.44 D = 1.32 or, D = 16 ins. If, instead of pumping, the water were collected in a reservoir by damming up the natural flow of some stream, and the dam were of necessity situated at an elevation so great that a clanger- 46 PRACTICAL HYDRAULIC FORMULAE. ous pressure is apprehended, it would be necessary to first receive the water into a distributing reservoir situated at a lower level, or else, as a less advantageous expedient, to reduce the pressure by gates, properly located for the purpose. It should be well understood that all the above assumed data, particularly such as relate to heads, are subjected to considerable variation in actual practice. All the calculations have been based, of necessity, upon the hypothesis that the exact allotted volume per second is being simultaneously drawn from the whole system, This would rarely be the case ; for at any given second, the draught would be liable to fluctuate greatly from the average. Indeed, these calculations should only be regarded as fixing, with some degree of approximation, the proper relative discharges and pressures at the different points supplied. The remaining north and south pipes should be calculated in the same way. Thus, those below F F' on the west side dis- charge 1-6 cubic ft. with a grade of j^Vo"- This would re- quire a 4 in. pipe. The draught from these would somewhat lowerthe piezometric heads at their junctions with the side mains. In a fine calculation, these reductions should be worked out, as was done in the previous example of branch pipes ; in general, however, and in cases where the whole supply is supposed to be carried through to the extremity of the mains, and delivered at the highest elevation, as was done in the present instance, and where a liberal interpretation has been given to the calculation of diameters, this is not indispensable. At the same time, it should be a guiding principle of water-works engineering that a few hours spent in the office, in what may sometimes be considered an over-refinement of calculation, is by no means a waste of time, and frequently enables one to make advantageous and economical modifications in a project of distribution. It may here be noted that (12) admits of being put into a very convenient form for rapid approximations. To do this, we PRACTICAL HYDKAULIC FORMULAE. 47 luive only to calculate the discharge of a pipe 1 ft. in diameter, with a fall of 1 ft. per thousand, and to refer all other discharges with the fall per thousand feet to it, in order to obtain the cor- responding diameter. The quantity discharged by the above pipe is 0.961 cu. ft. per second, and the square of the same is 0.9^4. Equation (12) may then be written : D= A/ Xl.08 or very nearly : D=4/~ (13) we have also very nearly : Q= i'D*xH which may be more conveniently expressed thus : Q = D- V D X H (14 bis) We have, also, r = M D X Hx 1.6 (Uter.) in which V velocity in feet per second. These last formulae, it will be perceived, are based on the fact that, given a certain probable degree of roughness, a pipe 1 ft. in diameter, with a fall of 1 ft. in a thousand, will deliver 1 cu. ft. of water per second. If we desire to apply them to smooth, clean pipes, we have only to Jialve the coefficient for a 12-in. pipe, which will be equivalent to writing the above formula; thus : - (15) 2H 0- = I/7? 8 X 2 H (16) These formulae will be found of very great utility in arriving quickly at approximate results. They can be advantageously used in sketching out a network of pipes such as we have just been con- sidering. To facilitate their use the following table of fifth powers 48 PRACTICAL HYDRAULIC FORMULA. has been calculated. This table indicates, by inspection, the diam- ters in inches corresponding to the fifth roots of the right-hand side of the equations,, expressed in feet. Diameters in incfies. Fifth Powers in feet. Diameters in inches. Fifth Poivers infect. 3 0.000977 22 20.72 4 004115 24 32.00 5 01-256 28 47.75 6 0.03125 28 69.17 8 o.ru? 30 97.66 10 0.4019 32 134 9 12 l.OOOJ 34 182.6 14 2.1615 36 243.0 16 4.214 40 411.5 18 7 594 42 525.2 20 12 86 48 1,021 All the diameters which have been already calculated can be obtained very nearly by the use of (13). Relations (13) and (14) might also have been used in some of the previous examples. Formulae (13) and (14) serve to show the comparatively small influence of grade as affecting the volumes discharged, which point has been already alluded to, and the preponderating influence of diameter. Thus, we see by the above formulae, that for a diameter of 1 ft. and a fall of j^-, the volume of discharge is 1 cu. ft. If we wish to double this discharge by increasing the fall, we must adopt a'grade of i^Vo? * e -> we must quadruple the fall. If, on the other hand, we wish to produce the same result by in- creasing the diameter without changing the grade, we need only adopt a diameter of 1.32 ft. and even a little less, on account of the decrease in the coefficient. That is to say, to double the discharge, we must increase the fall 300 percent., or the diameter 32 per cent. NOTE.- In completion of what has been already said in this chapter (page 41 ). re- garding the limit of veloci'iea for pipes of different diameters, the follow!ng table (founded upon that given by Mr. Fanning) indicates pretty closely the maximum /velocities which it is generally advisable to produce : Diameter in inches Velocity in ft. per sec 2.5 12 3.5 18 4.5 24 5.5 30 6.5 36 7.5 42 8.5 9.5 CHAPTER IV. Use of formula (14) illustrated by numerical example of compound system combined with branches Comparison of results Rough and smooth pipes Pipes com- municating with three reservoirs Numerical examples under varying condi- tions Loss of head from other causes than friction -Velocity, entrance and exit heads Numerical example* and general formulce Downward discharge through a vertical pipe Other minor losses of head Abrupt changes of diam- eterPartially opened valve Branches and bends -Certrifugal force Small importance of all losKes of head except fractional in the case of long pipes All such covered by "even inches" in the diameter. As an illustration of the use of (14) we will calculate by its aid the discharge from a reservoir, tapped at a depth of 50 ft. by a horizontal compound system consisting successively of 2,000 ft. of 12-in. pipe, 2,000 ft. of 24-in. pipe and 2.000 ft. of 12-iiu Each of these three lengths of pipe is tapped midway by a 6-in. pipe, laid horizontally, 'the one nearest the reservoir having a length of 3,000 ft.; the next 1,000 ft., and the last 500 ft. (See Fig. 9, bis.) All the pipes being open, it is desired to find the piezometric heads //, //', //", h" ', k"", at each branch and change of diameter, and the volumes discharged by each branch and section of main pipe. Beginning at the lower end and assuming 6 ft. as an ap- proximate value of //, we have from (14), H always representing thejfall per 1,000: V 6 -f h' = 15.36 4 36 = \ W(h^' - 15.30) h" = 15 65 / l/>.65 / = . _ _ _ V '9. 36 + I/ = V 3-' (h' ' '- 15.65) ' 32 50 PRACTICAL HYDRAULIC FORMULAE. h"'= 16.09 V 14.08 = \' h" " 16.09 h' ' ' ' = 30.17 /10 06 V-14.08 + 1/ = I x h' ""-30.17 r Q5! 32 h' ' " ' = 48.i FIG. 9&. Comparing this value with the given height 50, we may in- crease all the preceding values of h, U', etc., in the proportion of 50 . But in practice we would not wish to reckon on the total 48.82 head, and it would be preferable therefore to let the values stand as they are. We will now calculate the quantities, calling those discharged from the successive sections of main pipe, beginning at the lower end, Q, Q', Q", Q'", Q"", and Q'"", and those discharged by the branches, beginning also at the lower end, 7, q ', q", respectively, using both (3) and (14). The results given by (14) naturally-check exactly, since they depend directly upon the method used in de- termining 7i, h'j etc. By (3) By (14) Q =2.39 2.45 q ---- .56 .61 Q +q = 2.95 3.06 PRACTICAL HYDRAULIC FORMULAE. 51 By (3). By (14). Q' - 2.96 3.06 Q" = 2.99 3.05 q' = .65 .70 Q" -f q' = 3.64 3.75 Q'" = 3.68 3.75 Q"" =3.63 3.75 q" = .52 .56 Q" + q" =4.15 4.31 #'"" = 4.13 4.32 The above example was very favorable to the use of (14), be- cause of the lengths assumed for the different pipes, but in almost all cases it will greatly reduce the volume of calculation, and frequently give sufficiently close results. Indeed, as all these calculations are merely approximations, and as we have taken our coefficients pretty high, it would no doubt often be found, couM the actual discharges be measured, that the apparently less exact formula gave the more correct results. In all the previous examples, the coefficients for rough pipes have been used. It is well to remember that, as is shown by (15) and (16), the discharge of a clean pipe of given diameter is about 41 per cent, greater than that of a rough pipe of the same diameter ; also that the diameter of a clean pipe, discharging an equal volume with a rough one, will be about 88 per cent, of the latter. Be- tween these limits of smoothness and roughness there are, of course, an indefinite number of gradations. A very interesting investigation is that of a system of pipes communicating with two reservoirs, and discharging either freely in the air, or into a third reservoir situated at a lower elevation aa shown in Fig. 10. A B (60) 52 PRACTICAL HYDRAULIC FORMULA. Let us suppose the water surfaces in A and B to be respect- ively 100 and 80 ft. above the water surface in C, and that ulf the pipes shown in the figure are 12 ins. in diameter. Let the- total length of pipe from A to C be 4,000 ft. If communication were shut off from B, the flow would be- direct from A to C; if communication were shut off from (7, it- would be direct from A to />. If A were shut off, the flow would be from B to C. If all the communications were wide open, we desire to know whether the flow would be from A to B and C, or from A and B to C 1 ; and in either case, to know the- piezometric head li, at the junction D, and the volumes dis- charged. First, let the junction D be situated midway in the 4,000-ft. pipe joining A and C, and let the length B D be 1,000 ft. Let us for a moment revert to the supposition that Bis shut off. The flow woul'd then be from A. to C, the hydraulic grade line would be a straight line joining the surfaces A and C, and under our present hypothesis, that the junction Dis in the middle of A C, the piezometric head h would be 50 ft. above the surface of the lower reservoir C. But B is supposed to be 80 ft. above the same, and therefore the flow must be from A and B to C. We might at first sight suppose that the flow from B to C would be in virtue of the head 80 50 30 ft., which is the difference of level between B and the piezometric head at the junction; but just as a branch drawing water from a main pipe lowers the piezometric head at the junction, so does a branch discharging into the main pipe raise it. It is necessary to see what the height h will be in the present case. The quantity discharged into C is equal to the sum of the quantities passing from A and B. All areas and coefficients- being equal, and all reductions made, we have : 1/7- I/ 50 -; 4 - I/ 8 - PRACTICAL HYDRAULIC FORMULAE. 53 whence : h = 65+ y 4000 - 90 h + and, by successive approximations : h = 74 Using this value of li in (3),, we obtain the different dis- charges as follows : =5.88 = 3.18 " = 2.37 This gives a very close agreement in the relation Q = Q' + " Suppose now that the diameter of the branch 5 D be reduced to 6 in., all the other conditions remaining the same. Still regard- ing the coefficients as equal, in order to get rapidly at an ap- proximation, factoring the areas and simplifying, we have : whence : 16.5 h = 840 + 4 VSOOO^ and, by successive approximations : h = 58 This value of k gives the following quantities : =5.21 = 4 43 ' = 1.08 A tolerably close check, but showing that the true value of h is a little greater than the even 58 ft. at which we have placed it- Let us now suppose that the pipe B D is increased to a diam- eter of 30 in., all the other conditions remaining as before. Then: \/? = 4/^13+9 A/sTT^ ' 2 2 V whence : h = 79.90 Giving : = 6.111 = 3.065 }" = 2.816 54 PRACTICAL HYDRAULIC FORMULA. a close approximation ; the true value of h lies between 79.85 and 79.90. As li increases with the diameter of the pipe B D, it might at first seem as though, by indefinitely increasing the diameter, li might be so increased as to cause a flow from A into B. A mo- ment's reflection, however, will show that under the assumed con- ditions the diameter can never be sufficiently increased to cause a flow toward B. For it has been seen that when B is shut off, the piezometric head at D is 50 ft. It is raised by opening the com- munication with B, and allowing water to flow into the main from B. It is evidently, therefore, an essential condition of the increase of piezometric height that the flow should be from, not to, the reservoir B. But the effect will be different if the junction/) be sufficiently advanced toward the reservoir A. Let us suppose the positions of the three reservoirs to remain the same, all the pipe diameters to be 12 ins., and the point of junction of the pipe B D to be placed at 500 ft. from A (Fig. 11). If communication with// were shut off, the piezometric height at D would be 87.5 ft. There would therefore be a flow from A to B and C when. the pipe leading to B was open. But this flow would not take place under the head 87.5, for the draft toward B would lower it. >?soo FIG. 11. To ascertain the true value of li at the point />, we have the- relation : 500 3500 2500 PRACTICAL HYDRAULIC FORMULAE. 55 simplifying ; / / h / h - 80 100 - h = y - + y 47 h = 4960 11.88 V h* - 80 h whence, by successive approximations : h = 82.65 Using this value of h we get : Q = 5.695 Q' = 4.698 Q" = 0.995 When B is shut off, in the above system, the discharge from A to C is 4.33 cu. ft. per second. In all that precedes, only the resistance due to friction has been considered, and the total difference of level between the source of supply and the discharge has been taken as available for overcoming this frictional resistance. In the case of long pipes, where the velocity is comparatively low, this resistance is so greatly in excess of all the others that, in order to simplify calculations, they are neglected. This leads to no material error in cases where the pipe is over 1,000 diameters in length. Attention, however, has been already called to the fact that there are other resistances which require a certain proportion of the total head to overcome them, leaving only the remainder available as against friction. Indeed, it is evident, if we assume all the head to be consumed by frictional resistance alone, the water in the pipe would be in exact equilibrium, and no flow could take place. It will now be proper to show how the total loss of head, from all causes, may be calculated. And first, a word in refer- ence to the phrase " loss of head ''just employed. This term, often met with in treatises on hydraulics, may occasionally prove confusing. It is really little more than a convenient abbrevia- tion. When we speak, for instance, of " the loss of head due to velocity," we mean the head, or fall, theoretically necessary to 56 PRACTICAL HYDRAULIC FORMULA. produce that velocity. Similarly, when we speak of " the loss of head due to resistance to entry," we mean the amount of head, or pressure, necessary to force the fluid vein into the mouth of the pipe or orifice, against the resistance of its edges. This resist- ance, it may be remarked in passing, as well as that due to bends, elbows and branches, shortly to be mentioned, is caused by the fact that water is not a perfect fluid, and therefore changes of di- rection in its flow require a certain amount of force to break or distort the form of the fluid vein as, though to a very much less degree, would be the case with a plastic body under similar cir- cumstances. The property of water which causes these resist- ances is called its viscosity. As applied to long pipes, the principal " loss of head," and fclie only one hitherto considered, is the f fictional. The term thus applied means the height or pressure necessary to overcome the friction of the water passing with a given velocity through a pipe of given length and diameter. Thus, when we speak of the frictional loss of head per 1,000 ft. in reference to a given pipe, we mean the fall per 1,000 ft. necessary to maintain the given or desired velocity, as against friction. We will now investigate this subject by means of the follow- ing problem : Two reservoirs (Fig. 12) containing still water and having a difference of level of 30 ft., are joined by a pipe 12 ins. in diameter and 3,000 ft. long. What is the velocity of dis- charge between the upper and lower reservoirs ? FIG. 12. From what has been already said, it will be seen that, besides the frictional loss of head, there will be the loss of head due to PRACTICAL HYDRAULIC FORMULAE. 57 velocity and that due to entrance. If the pipe discharged freelv in the air at its lower end, at the vertical distance of 30 ft. below the surface of the water in the upper reservoir, these three would be the only losses of head incurred, and their sum would be equal to 30 ft.; but as the discharge takes place in a reservoir, the sur- face of the water in which is supposed to cover the end of the pipe, to a sufficient depth to cause the discharge to take place in still water, there is the further loss of head due to the extinction of the velocity which is dissipated in vortices. This loss consti- tutes what may be called the back pressure of the reservoir. In solving this problem, let us first, as heretofore, neglect all losses except frictional ones. AV r e have then, from (1), using the above data, and the coefficient for rough pipes : i = 0.00066 V* 100 V* = 15 15 V = 3.89ft. per second. The head theoretically necessary to produce this velocity is given by the formula derived from the law of falling bodies, F 2 h 3 -^- by substitution of the above value F. Thus : *ff 15.15 ~&rr h= 0.2352 Besides this, there is the loss of head due to entrance. We have already seen that this is always equal to about half the velocity head. We have then : h h 4- = 0.3528 2 The loss of head from back pressure of the water in the lower reservoir, being that necessary to extinguish the velocity, must be -equal to that necessary tb'produce the same. We have, therefore, for the total losses, outside of friction : 58 PRACTICAL HYDRAULIC FORMULA. And the head available for overcoming friction becomes 30 0.588 = 29.412 We must now recast our original calculation, using 29.4 ft. instead of 30 as available frictionai head. Thus : 29.4 = 0.00066 F 2 3000 F 2 = 14.3 V = 3.35 This is a very small reduction from the velocity already ob- tained. But, in order to see how our previous solution is affected by the change, we will work out new values for the subheads. _ 14.8 ~GIA h = 0.23 h h -f + h = 0.575 2 30-0.575 = 29.425, leaving the previous valu^ practically unchanged. Let us now see, by means of a general formula, what is the amount of error which we commit when we ignore all resistances except friction. Calling Fthe actual mean velocity, that is the actual volume discharged divided by the area of the pipe (3), we have, in the case of discharge between two reservoirs, as shown in Fig. 12, the following subheads, which together make up the total head //: F 2 F 2 F 2 L C F a ~ 20 40 20 D 5 F 2 L C F 2 H = -f- 40 D JJ=0.039F 2 -f Z.CF 3 D That is to say, by using (3), which gives H = L C F 2 PRACTICAL HYDRAULIC FORMULA. 59 we make the error of omitting a height not quite equal to 4 per cent, of the square of the velocity. In long pipes this is a very trifling amount. If the pipe discharged in free air, we would have : v* v* L c v* H= 4 + 20 ig D H = 0.0233 F 2 + L C V* Iii this case we make the still smaller error of omitting %\% of F- In all cases, having obtained V 9 by means of (1), we can easily judge from the nature of the problem whether it is neces- sary to take account of these errors. In designing a system of pipes, where the problem generally is to find the proper diameter for a certain discharge, the practice of taking the next highest even inch will almost always amply suffice to cover all omissions. As has been already stated, in all ordinary circumstances of pipelaying, the horizontal measurement of the pipe is taken in- stead of its actual length. It is only in special cases that this cannot be done. The extreme limit occurs in the case of a ver- tical pipe discharging from the bottom of a reservoir. This con- stitutes a very interesting special case, for should the reservoir be of indefinitely large area, but of relatively shallow depth, the rela- H tion tends toward unity as //, and consequently H, increase. Li The velocity, as determined by (1), tends therefore toward : and remains constant, no matter how greatly L may be increased. If we apply this formula to a 12-in. pipe of indefinite length, using the coefficient for rough pipes, we get, V= 38.9 60 PRACTICAL HYDRAULIC FORMULAE. This is the maximum velocity of discharge in feet per second for a vertical 12-in. pipe under the given circumstances.* There are several minor losses of head, besides those already considered, which are liable to occur from changes of diameter, branches, and bends or elbows. Our experimental knowledge of the effects of these features is very limited, and it is probable that much weight should not be attached to the formulae given for their determination. A brief space will be devoted to their consideration, more with a view to make the present paper com- plete than for any practical value which they possess. When water passes through a pipe of which the diameter is abruptly Changed, at a certain point, to a greater or a smaller one, there is a loss of head due to the eddies formed and the sudden contraction of the fluid vein. In practice such pipes are always joined by a reducer, or special casting, which forms a tapering connection between the two. This greatly diminishes the agita- tion of the water in passing from one pipe to the other. It would seem, however, that the mere change of velocity, independent of such agitation, causes some slight modification of the profile of the hydraulic grade line ; and it will be well, in any event, to give formulae for the different cases which may occur when abrupt changes take place, as these give rise to the maximum retardation. The following formulae are taken from Claudel's Aide Memoir e, ninth edition. First. When the change is from one pipe to another of smaller diameter, we have : whence: * = ' 49 ?7 h = 0.00076 V 2 V being the velocity of the water in the smaller pipe. We have seen, by examples previously given, how this velocity may be obtained. * The same result may be inferred from what has been said in Chapter I. about a pipe la ; d so lhat its axis coincides with the hydraulic parade line. Obviously, a vertical pipe discharging downward is a special case of such coincidence. PRACTICAL HYDRAULIC FORMULA. 61 Second. If the water (Fig. 13), in its passage from the greater to the smaller pipe, passes through an opening in a thin diaphra FIG. is. as in the case of a partially opened stop-cock, we have : 2g \0.62S' in which Fis the velocity in B, the area of cross-section of B f and S' the area of the opening in the diaphragm. Third. When the flow is from one pipe to another of larger diameter : in which F = velocity in small pipe, and V velocity in larger one. When the water passes from a pipe into a reservoir, as in the case lately considered, V becomes zero, and we have, as already established in that case : r a h = 20 Another loss of head is that due to branches (Fig. 14). In. c FIG. 14. this case the water flowing from A, with a velocity F, is split at the junction, part passing on toward B, with a reduced velocity F', and part entering the branch and flowing toward (7, with the velocity V". The loss of head occasioned by perturbations of the- water at the junction has not been satisfactorily investigated. (52 PRACTICAL HYDRAULIC FORMULA. When the branch leaves the main at a right angle, this loss, as determined by a few incomplete experiments, is : 3 v* ~ 20 V" being the velocity in the branch. We have already seen how this velocity may be calculated. If, as is generally the case in practice, the branch is deflected gradually instead of forming an abrupt angle of 90, the. vortices are nearly annulled, and the only loss can be from the difference of the velocities in the three pipes. Thus for B and C, respec- tively, we have : /v- v * h = ( \ 20 ) (V h' = For bends, or elbows, Navier's formula for loss of head is : F2 / \ A h = I 0.0123 + 0.0183 R 1 - 20 \ ' R in which V = velocity of flow, 72 = the radius of the bend, taken along the axis of the pipe, and A =the length of the bend, also measured along the axis. It will readily be seen how very trifling the loss of head from this cause will be in all ordinary cases. The water passing around a bend exercises a radial thrust upon it which may sometimes be so considerable as to require bracing against. The expression for the centrifugal force Fis : M V* F= R in which J/=the mass of the liquid in motion, V= its velocity, and R= the radius of the bend measured on its axis. As an illustration, we will suppose a pipe 24 ins. in diameter, through which the water flows with the velocity of 8 ft. per sec- ond, around a bend of 8 ft. radius. PRACTICAL HYDRAULIC FORMULA. tf3 The mass of the liquid in motion is its weight divided by g. The centrifugal force, therefore, per running foot is : 3.14 v 62.5 8 2 32.2 8 F = 48.721bs. If the bend turns a quarter circumference, its development on the axis will be 12.57 ft., and the total thrust on the bend will be 48.72 x 12.57 = 612.4 Ibs. This would be liable to be intensified by sudden changes in velocity, and if the bend is not well abutted, might tend to draw the joints. FIG. 15. Fig. 15 shows the manner in which such losses of head as we have been just considering modify the the profile of the hydraulic grade line. The dotted line shows the grade as determined by the calculations which we have already made for a line of pipes of varying diameter. The full line, broken at the reservoir and at each change of diameter, shows the hydraulic grade as modified by losses of head due to velocity and changes of diameter. It will be understood, of course, that this is a mere random sketch, with- out reference to proportion. The result of what precedes in reference to all losses of head other than friction shows that in practice, and in the case of long pipes, such losses exercise but a trifling influence. A very small increase in the diameter of the pipe over that obtained by calcula- tion based on fractional head alone, such as would naturally be made to get even inches, will in almost all cases largely cover all losses due to velocity, entrance, branches, bends, etc. , CHAPTER V. XOTES 01* PlPELAYING. It will not be amiss at present to give some hints respecting Pipelaying. Enough has been already said to show how greatly the smoothness or roughness of the interior of a pipe affects the velocity of the flow of water through it. A line of pipes is made up of a great number of separate lengths joined together, generally by the spigot end of each pipe entering into the hub or bell end of the other. Each of these joints occasions more or less friction, and it is essential, not only on this account, but also and more particularly in order to make a substantial and enduring piece of work, that the pipes should be laid as evenly as possible, and the joints well fitted and calked. The alinement should be straight and the grade regular. This latter is the more important of the two, because sags and depressions in the line occasion deposits of impurities in the low points and accumula- tions of air. in the high ones. The line should run straight and even between changes of grade and direction. Each low point, or point from which the grade rises both ways, should be pro- vided with a special ' blow-off " and stop cock, to clear it of sediment by blowing off under pressure, and each summit, or point from which the grade falls both ways, should have a special air vent, or hydrant, to discharge the accumulated air from time to time. When a new line of main is filled for the first time, or When a line is refilled after having been emptied for any cause, all the blow-offs and air cocks should be opened, and the water PRACTICAL HYDRAULIC FORMULAE. 65 admitted very slowly by giving a few turns only to the admission valve. Then, as the pipe gradually fills, the blow-offs should be closed progressively as the water reaches them, and the air cocks also, beginning to close the latter, if possible, from the lower end, and only when they discharge solid water. Changes of horizontal direction should be joined by as easy curves as can be obtained. In sharp curves and large diameters, special curved pipe may be necessary, but, in general, curves are got in with straight pipe, using all short pieces that may be on hand, and, if necessary, cut- ting whole pipe, and joining the straight pieces with sleeves. In a well-laid pipe line, all pipe, particularly all those of 20" diameter and upward, are laid on blocks. These blocks consist of pieces of wood sawed out to regular dimensions, there being two under each pipe, one just behind the hub and the other as near the spigot end as will permit of the joint being easily reached for calking, say about 2 feet. For diameters from 36" to 48", the length of these blocks may be equal to the diameter of the pipe, and about a foot wide and 6" thick. For smaller pipe they may be about two feet longer than the diameter of pipe and pro- portionately lighter than for the larger sizes. The pipe is held in its place on these blocks by means of wooden wedges placed side by side, on opposite sides of the pipe, and driven past each other. For 48" pipe these wedges may be about 18" long, 6" wide and 4" thick at the butt. For smaller pipe they will of course be lighter. The instrumental alinement of the pipe line presents no particular difficulty, because the excavation once correctly started is not likely to deviate to any injurious extent. It is much more difficult, as it is more important, to keep the grade. This is best effected, practically, I think, in the following manner : Let the ordinary marked grade stakes be set for the excavation. Then when the proper depth has been reached, or nearly so, let grade plugs be driven in the bottom of the trench, every 50 feet or 66 PRACTICAL HYDRAULIC FORMULAE. oftener, with their heads exactly to grade. . A line can then be stretched from one to the other, and the blocks laid to it. It is better to bed the blocks a trifle low, say a quarter to half an inch, particularly with heavy pipe and hard bottom, and then raise the pipe to grade by driving in the wedges. It is not necessary to set the pipes with rod and level. If the grade pings have been driven as suggested, a competent foreman will adopt any one of many ways for sighting in the pipe to the proper level. With soft ground and heavy pipes, longitudinal stringers are ad- vantageously employed under the blocks, the spaces between them being well packed with broken stone or other ballast. When pipe have been laid and calked, it is advantageous to cover them as soon as possible by backfilling the trench, to prevent the joints from drawing in consequence of expansion and contraction due to exposure to the changes of temperature of the air. In backfilling the trench after the pipe is laid, be very care- ful that the earth is well tamped in under the pipe, so that it may have a solid bearing throughout its entire length. The earth put in next to the pipe should be clean and free from stones. Be particularly careful that no large stone gets under the pipe, as in case of a sudden jar, such as would be produced by a casual " water hammer," -it might punch a piece out of the pipe, or at least crack it. In leading and calking joints, the specifications generally call for a certain depth of lead. The specifications of the city of New York require 4 ins. of lead for 48-in. pipe. It is a great advantage to have a deep joint, although the necessity is some- times denied upon the ground that in calking it is impossible to "upset" the lead to a depth of more than perhaps half or fof an inch, and that therefore the additional lead is of no advantage. This is, however, a mistake, for an abundant depth must be allowed for cutting off the protruding lead in recalking a joint which has drawn. The lead which is drawn out when a line of PRACTICAL HYDRAULIC FORMULA 67 pipe contracts is not forced back again when the pipe expands ; on the contrary it remains out, and, if the pipe contracts again, a fresh amount will probably be again drawn out, all of which must be cut off when the joint is recalked. In calking joints, yarn or gasket is first driven in until the proper depth is secured, and lead is then poured in, either by the use of the clay dam or *' snake," or by the use of a metallic '"clip." The latter is very useful in laying large pipe, as it en- able the whole joint to be run at one pouring. A refinement in running joints is to employ a lead gasket, for example a piece of lead pipe, hammered flat, and circled around the pipe. It is driven in and calked, and the remainder of the joint is then run and calked in the usual manner. The advantages of this system are that there is no perishable material used, and that the joint is calked on both faces, which is very favorable to making it tight. It is sometimes difficult to get the trench and pipe sufficiently dry to admit of pouring a molten joint. In such cases it is necessary to put the lead in cold, perhaps in the form of a ring of flattened lead pipe, as above, and calk as put in. This makes a very satisfactory joint, only it is slower and more expensive. Mr. Billings, in his excellent treatise, (e Some Details of Water- Works Construction," gives Mr. Dexter Brackett's formula for the average weight of lead in a joint about 2| in. deep, as follows : l = 2d, in which I = pounds of lead per joint, and d = inside diameter of pipe in inches. As the pipes are usually 12 ft. in length, the weight of lead per running foot equals one-sixth of diameter of pipe in inches. If a 4-in. joint is used, we would have I = 3.2d, and the weight per running foot under above assumption of length d d of pipe would be , or, in round numbers, . 3.75 4. APPENDIX. As it is convenient in making estimates to have the correct weight of cast iron pipes of different diameters in handy form, I subjoin a table of weights of pipes made by the Warren Foundry, of Phillipsburg, N. J.: TABLE SHOWING THICKNESS OF METAL AND WEIGHT PER LENGTH FOR DIFFERENT SIZES OF PIPE UNDER VARIOUS HEADS OF WATER. g 50 ft, head. 100 ft. head. 150 ft. head. 200 ft. head, 250 ft. head. 300 ft. head. It IS I P L 11 1. ii 1. 3 1. 1 | e SB || if JS ft sS || *M If ji If |1 If c QJ 33 || 1 2 ~ ir J2 S ~ H ~ 5*0 g- g ~ 2 .294 63 .312 67^ .330 72 .348 76Vfc .366 81 .384 8ft- 3 .314 144 .353 149 .36* 153 .371 157 380 161 .390 166 4 .361 197 .373 204 .385 211 .397 218 .409 226 .421 235 5 .378 254 .393 265 .408 275 .423 286 .438 298 .453 309 6 .393 315 .411 330 .429 345 .447 361 .465 377 j .483 393 8 .422 445 .451)! 475 .474 502 .498 529 .522 557 .546 584 10 .459 eoo .489! 641 .519 682 .549 723 .579 766j .609 808 12 .491 768 .527 826 .563 885 .599 944 .635 1,0 )4| .671 1,064 14 .524 954 .566 1,031 .608 1,111 .650 1,191 .692 1,272; .734 1,352 16 .580 1,215 .604 1,253 .652 1,360 .700 1.463 .748 1,568! .796 1,673 18 .589 1,370 .643 1,500 .697 1,630 .751 1,761 .805 1,894 ; .859 2.02ft; 20 .622 1.603 .6821 1,763 .742 1,924 .802 2,086 .862 2,->48| .922 2.412 24 .687 2.120 .759 2,349 .831 2,580 .903 2,811 .975 3,045! 1.047 3.279' 3'J .785 3,020 .875 3,376 .965 3,735 1.055 4,095 1.145 4.458 1 1.235 4,822 36 .882 4,070 .990 4,581 1.098 5,096 1.206 5,613 1.314! 6,133; 1.422 6,656 42 .989 5,265 1.106 5,9.i8 1.232 6657 1.358 7.360 1.484! 8,070; 1.610 8,804 48 1.078 6616 1.222 7,521 1.366 8431 1.510 9,340 1.654 10,269 1.798 All pipe cast vertically in dry sand, in lengths of 12 ft., except the 2-in., which are cast 9 ft. long. The general formula for weight of cylindrical cast iron pipe of givem thickness is : W = 0.82 (D + T) T X L, (1) in which W - weight in pounds. D = inside diameter in inches. T = thickness of metal in inches. L = length iu inches. PRACTICAL HYDRAULIC FORMULAE. 69 A convenient approximate formula for the weight per foot of cylin- drical part of a cast iron pipe is : W = 10 (D + T) T. (2) In calculating the weight of cast iron pipe, they are always considered as being cylindrical, and eight inches of length are added as the equiva- lent of the hub or bell for all diameters. Thus a pipe measuring 12 feet over all, including the hub, would be considered, as regards weight, as a plain cylindrical pipe 12 feet 8 inches long. For instance, to calculate the weight of a 42-inch pipe, 12 feet over all and 0.980 inch thick, by the above approximate formula (2) we have: W 10 X 12.67 X 42.98 X 0.98 = 336.6 Ibs. This is about \\% in excess of weight given in Warren Foundry iable. If we wish to find the thickness, the length, diameter and weight be- ing given, we have from (1): / w D* n T=\/ - -+ - V Q.S2L 4 2 To find the proper thickness in inches corresponding to the head in feet, H, of pressure for a given diameter in inches, D, we have : T = 0.00006 H D -f 0.0133 D + 0.296. The hydraulic engineer will find it both interesting and useful to work out concise formulse covering frequently recurring cases, and enter them in his notebook for future reference. These should be carefully checked by testing them numerical I y, so that they can be used with con- fidence when wanted. For instance, it is a common practice to allow 100 United States gallons per capita to be consumed in 10 hours, in calculat- ing the proper supply for a town. To find the diameter of a pipe to con- vey this amount we may use the approximate formula : H in which D = diameter in feet. H= fall per 1,000. N '= number of population to be supplied. Also, in estimating theoretical horse power necessary to raise a given volume of water to a certain height and in a certain time, we have 70 PRACTICAL HYDRAULIC FORMULAE. and _ ' 5.7 in which Q = cubic feet per second . Q' = millions of U. S. gallons per 24 hours. H = lift in feet. In calculating the power necessary to pump water into a reservoir situated at certain horizontal and vertical distances from the pumps, and connected with them by a force main of given diameter, we must imagine the water to be raised to a certain elevation above the reservoir such that the difference of level between this elevation and that of the reservoir shall be sufficient to convey the required amount of water to the reservoir against the friction of the force main. Thus, suppose we wished to deliver 1 cubic foot per second through a 12 inch force main to a reser- voir 100 feet above the pumps, and distant 10,000 feet from them. By our approximate formula (13) we know that this requires a fall of one foot in a thousand, so, as the distance is 10,000 feet, we need 10 feet to overcome the friction, and our pumps must therefore be able to raise the given volume a total height of 110 feet. This calls for 12.47 theoretical horse power, by the formula just given. SECOND PART. NOTES ON WATER SUPPLY ENGINEERING. QUALITY OF WATER. The first question in regard to a water supply is, evidently, quality. The solution of. this question, either generally or in any particular case, is by no means a. simple nor any easy one, partic- ularly since it is now fully recognized that chemical analysis, to which we naturally recur in such cases, is far from being a final criterion, but is only one element in a group of data from which we infer the probable quality of a given source of supply. As animal refuse and the waste products of human -industry are the principal sources of menace to a water supply, we com- monly look for a high degree of purity in water drawn from thinly populated districts devoid of manufactories. Such districts, however, are not often to be met with in the vicinity of large towns, and, even when they are, we must expect a gradual en- croachment upon them, from the natural growth of the neighbor- hood. One excellent criterion of quality is the general health of the communities using the water. Also, the order of fishes which find their habitat in the streams of a given district. Little dan- ger, for instance, would be apprehended from the use of a good trout stream. Although all the fresh water used upon the earth reaches it 72 PRACTICAL HYDRAULIC FORMULA. in the form of rain which is first drawn up by evaporation, chief- ly from the sea, there are several different forms under which it presents itself for our use. The first broad classification of these forms would be that which divide.3 the supply into surface water and ground water. By surface water is understood the water of lakes, ponds, rivers and streams, all water which in fact is collected directly from the surface of the earth; and by ground water, that which is derived from wells and filtering galleries, and from springs when taken at or near their source. Each of these classes admits of much subdivision, but the differences will be principally those of degree, and not of kind. For instance, we have the smaller streams, such as creeks, brooks, etc., and also the larger ones, rising to the dignity of rivers. While these certainly do present slightly different char- acters, still their main difference is one of size. Again, though the waters of lakes and ponds differ somewhat from each other, and from those of streams and rivers, still they are only the col- lected products of these latter, which they consequently greatly resemble. Ground water, proceeding directly from the earth, offers more distinctive characteristics, shared, generally, by all its sub- classes. As regards the relative salubrity of water drawn from minor streams and that from large rivers, it would seem that they stood nearly upon a par, their principal difference, as already mentioned, being that of size. At first sight it would appear that the smaller streams, situated near the headwaters of the larger ones, or rivers, would possess a higher degree of purity from the fact of the water being collected from a comparatively wilder and more thinly populated district. This is not, however, of necessity, the case, because the river, although passing through a denser population, affords by its greater volume a greater degree PRACTICAL HYDRAULIC FORMULAE. 73 of dilution. The true criterion in respect to this view of the question would seern to be the density of population per square mile of drainage era, together with the proximity of its center of gravity to the stream itself. The further the hulk of the popula- tion is from the stream or river, the greater the chances of purifi- cation by natural filtration. Large rivers are apt to have towns directly on their banks, which drain all their sewage immediately into the stream. Large rivers, being made up principally of the smaller streams, would appear to form the general average of all their feeders. It must be borne in mind, however, that besides the yield of the smaller streams the river is partly fed by direct surface wash from its immediate banks, thus imparting to it a somewhat modi- fied character, distinct from that of the smaller brooks emptying into it, and causing the water of the large river to be, as a general rule, softer and warmer than that of its small feeders. Generally speaking, however, in the large river you get all the water and all the impurities, thus making, as already stated, a pretty fair general average of the whole, while of the smaller feeders some will have a greater and some a less degree of concen- tration of impurities than the average. There is another point worthy of note as regards the relative quality of the water taken near the mouth of a great river, or from the smaller streams near its source. All impurities enter- ing such streams or rivers have a greater chance of being exter- minated by oxidation, by the lower forms of organic life and by fishes, the longer they remain exposed to these agencies. Hence Hear the moutli of long rivers we have a right to assume that many of the impurities which entered near the headwaters have been destroyed during their long passage toward the mouth. On the other hand, this prolonged sojourn has increased the proba- bility of development of disease germs which have escaped actual destruction. The question then comes up : Had we better take our impurities and disease germs fresh or stale ? And the answer 74 PRACTICAL HYDRAULIC FORMULA. would probably be : Fresh, if we must take them at all, and can- not trust to time for their destruction. There is another point of difference between Che two classes of streams, which, although possessing an engineering rather than a sanitary character, it may not be amiss to refer to here. In the case of the small streams a greater necessity will generally exist for storage, in order to secure a uniform supply, while gravity can be more often counted upon as a motive power than in the case of the large river, where storage is seldom needed, and where pumping is almost invariably necessary ; a notable exception being that of Washington, D. C., where the water of the Potomac flows into the city by gravity. Ground water may be drawn from shallow or deep seated wells the latter of ten improperly called artesian from galleries, or directly from springs at the point where they burst forth from the earth. Shallow wells are supplied by the rain which falls and soaks into the ground in their immediate vicinity. In seasons of much rain the level of saturation is comparatively near the surface ; in seasons of drought the level descends as the water gradually drains off to the nearest valley. While an isolated shallow well may afford water of excellent quality and considerable relative coolness, such wells situated in towns and villages, or even when located near the cesspools of solitary dwellings, constitute what upon the whole must be con- sidered the most objectionable supply in common use. Their hardness and saltness when compared with neighboring springs are a good indication of their relative contamination by human ref- use. These qualities are observed to increase with time, and the growth of the village a striking corroboration of what has been- advanced above. Deep seated wells, and springs, may be fed by rain falling on PRACTICAL HYDRAULIC FORMULAE. 7.-) far distant points. The water from these wells is apt to be im- pregnated with earthy salts, and therefore to be hard, frequently to the extont of unfit ness for domestic uses. The temperature is apt to be higher than that of shallow wells. The supply from deep wells is more abundant and steady than from shallow ones, the volume of supply being more dependent upon depth than diameter ; indeed increased diameter only affords greater storage in any given case. Ground water is frequently obtained from drains or filtering galleries, or lines of pipe with open joints, chiefly located near and parallel to and lower than rivers, which galleries intercept the water flowing through the ground toward the river, and which probably are also fed, to some extent, by the water of the river itself, leaching back to the drain, gallery or pipe line. Water in considerable quantities is sometimes collected from springs, and conveyed away immediately as it bubbles up from the ground. The new water supply of Havana, Cuba, is a nota- ble instance of this, where some four hundred springs, furnishing over five millions of cubic feet per 24 hours, have been collected about ten miles from the city, and the water conveyed in an aqueduct to a distributing reservoir, whence it is delivered to the city in cast iron pipes. Such a supply seems likely to be the purest that can be obtained. Nevertheless, spring water is apt to be somewhat hard from the amount of earthy^ salts frequently held in solution. It will be seen from the above that the question of the rela- tive purity of different classes of water is a very complicated and uncertain one, not admitting of a general solution, but involving the consideration of a great number of special cases. Ordinarily the choice in any given instance is very limited, most towns hav- ing but few sources of supply to select from. The choice is ordinarily further controlled and limited by questions of quan- tity and cost, so that it seems hardly worth while to consider the 76 PRACTICAL HYDRAULIC FORMULAE. subject under its general aspect at all, but simply to make a special study of each special case. QUANTITY OF WATER. Next in importance to quality comes the question of quan- tity. It will be observed that the growing tendency is to increase the amount allotted per capita per diem. It is found io be neces- sary to make abundant provision for the future growth of the town to be supplied, to anticipate an increasing individual use of water, and also to provide for the yearly increase of leakage, con- sequent upon the gradual deterioration of the work and of the house plumbing. This latter is a fruitful though often overlooked cause of a diminishing supply. In general it may be said that a hundred gallons per twenty- four hours per capita, to be consumed in ten hours, with a liberal allowance for future growth of population, is a safe but not ex- travagant estimate. We frequently hear of a town finding that its water supply has become inadequate ; we never hear of one suffering from too great a one. The control and diminution of waste are now occupying a great deal of attention, particularly in England, where the density of the population renders strict econ- omy necessary. Frequently, no doubt, the best and cheapest way of increasing a deficient water supply would be to reduce waste, by the use of meters and other means for securing the co-opera- tion of consumers. From the purely engineering point of view the principal in- terest involved in the hourly supply is its connection with the size of pipes required for its delivery. A hundred gallons per head per twenty-four hours if delivered in ten hours, is at the rate of ten gallons per head per hour, or about 0.00037 cubic foot per second. It has just been stated that quantity is secondary to quality, but in studying a water supply project the first step is to decide upon the quantity necessary or desirable to obtain. This fact PRACTICAL HYDRAULIC FOliMULJE. 77 being settled, the question will naturally follow,, How can we as- certain what the yield of a given stream will be ? One way is by gaging, and this should be always done, choosing both the driest and wettest seasons for the purpose. But it is evident that all this takes time, and even a year's continuous gaging would not be considered as conclusive in any case where the demand nearly approaches the probable supply, because we must calculate on an occasional year or two of very exceptional drought. Another way is to make a survey of the area which drains into the stream under study, above the point at which it is proposed to take the water. This area, combined with the rainfall, known or assumed, aiid a general knowledge of the character of the watershed, fur- nish reliable data for calculating the approximate yield of the stream. Here again, however, we are confronted with the neces- sity of consuming much time, for although the survey can be rapidly made, the records of rainfall require at least as much time as does gaging. Fortunately in many cases we can make pretty close estimates of the amount of water probably derivable from a given area, by using data already collected for neighboring dis- tricts, and at any rate we can always make reasonable assumptions when once we know the number of square miles of territory which drain into our stream the liability of such assumptions to be correct increasing with the area, for a large area is less subject to special variations from local causes than a small one. The average yearly rainfall in the Croton basin, which fur- nishes by far the larger part of the water supply of the city of Xew York, is about 46 inches. Long experience shows that in this basin each square mile of watershed, or drainage area may be safely counted upon, one season with another, to furnish one million of U. S. gallons per twenty-four hours, or 365 millions per year. On the other hand, a precipitation of 46 inches gives very nearly 800 millions per square mile per year. Hence, in the Croton basin about 40$ of the total rainfall is found to be avail- able for water supply. 78 PRACTICAL HYDRAULIC FORMULA. It must be borne in mind that the above yield represents the yearly average, which may easily vary forty times either way for any given shorter period. This fact establishes two important points in regard to water supply. First, the necessity of adequate storage reservoirs to convert this yearly average into a daily one ; and secondly, the necessity, in the interest of safety, to give these reservoirs ample overflows, or spillways, in order to provide free escape to the surplus water which may flow into them in immense volumes during freshets. These considerations bring its naturally to the question of stor- age, a most important and by no means simple one. The amount of storage necessary to insure a regular daily supply varies of course with the extent of the watershed in proportion to the de- mand. The larger the area, the smaller may be the storage. In some exceptional cases the supply may be so great that its abso- lute minimum yield is greater than the maximum demand, and in such cases no storage is necessary. On the other hand cases BO unfavorable may possibly occur when the total yearly average is needed, and this leads to a maximum storage capacity. Let us consider such a case, and suppose a community which requires a supply of 10 million gallons a day from a drainage area of 10 square miles, and follow the course of events through an entire year. The year will be divided into three periods: The period of average flow, the period of drought, and the period of over- supply. The period of average flow will be that in which the daily yield of the stream is exactly equal to the daily draft in our supposititious case 10, 000,000 gallons. The period of drought will be that in which the yield is less than the above, and the period of over-supply, that in which it exceeds it. These last two periods will, of course, vary in intensity, through indefinite gradations. In order that the storage capacity should be ideally perfect, it would be necessary to so proportion it that at the commence- PRACTICAL HYDRAULIC FORMULAE. 79 incut of the period of drought the reservoir should he exactly full and of capacity sufficient to hridge over the interval between the drought and the commencement of the period of average yield. At the commencement of the over-supply or freshet period, the reservoir should he completely empty, and of capacity sufficient to receive and retain all the surplus water until the period of average flow was again reached. Not a drop should or- dinarily escape except through the supply pipes, and an overflow or spillway should he unnecessary except to provide for extraordi- nary contingencies, such as cloudbursts, etc. It is clear that such an ideal state of things is impossible of realization. It would be based upon a regularity of regimen that could never occur except, perhaps, by chance, during a single year. Even in periods of extreme drought (and this extreme is a variable) there would be some water flowing in the stream, and the storage reserve would need to be drawn upon only for the difference between this amount and the daily supply ; while during freshets the amount to be stored would be reduced by the daily supply being drawn off. Moreover, besides the average intensity of droughts and freshets, there come cycles of still greater intensity, all of which circumstances are controlling factors in the problem. lu the oase assumed the only way to secure the total flow, so that none shall pass to waste over the spillway except in the case of a cloudburst or of some other phenomenon, and at the same time to provide for droughts of maximum intensity, is to con- struct a reservoir or reservoirs of capacity to contain the total yearly yield of the stream, and to commence the use of the supply with a full reservoir, so that there shall always be a year's supply ahead. This treatment of the problem is certainly a heroic one, and has probably never been fully carried out in practice, although the city of New York is reaching well on toward it, in the vast 80 PRACTICAL HYDRAULIC FORMULAE. storage works executed and contemplated in the Croton basin. Fortunately so unfavorable a case as the one assumed, when the total yield of the stream is needed, rarely presents itself, and in the majority of cases there is an excess, more or less considerable, of the supply over the demand. The solution of the problem of storage capacity lies between the two extremes above instanced, in one of which no storage is necessary, and, in the other, when it is necessary to have capacity for the yield of the whole year. It is evident that we cannot say, d priori, of any proposed water supply, that storage for so many days will be necessary, or sufficient, without knowing at least approximately the total yield as well as the desired consumption. In cases where close calculation is needed, as when it appears necessary to utilize the greater part of the supposed supply, the proper course to pursue is to ascer- tain, by actual survey, the drainage area; to ascertain, by rational assumption when direct observations are lacking, the average precipitation, and then allow from one-quarter to one-third of the same as available, backing these data by gagings, as com- plete as may be possible, of the stream, and calculate the storage capacity accordingly. I have confined myself in the above to a general view of the principles involved in planning storage reservoirs, nor do I think it wise to enter into more elaborate calculations, as they might lead the inexperienced to suppose that the problem really admitted of a general mathematical solution. Such is not the case, and un- less the known factors point clearly to self-evident assumptions, great caution and much study should be bestowed upon the fixing of the data on which the design oi an economical and satisfac- tory water supply is to be based. One thing is certain : Except for economical reasons there is no danger of having too great storage capacity. I do not happen to recall an instance of a com- munity suffering from the possession of too much stored water, PRACTICAL HYDRAULIC FORMULAE. 81 v.Jle the want of enough of it is proving a serious trouble to cities and towns all over the country. I have already adverted to the matter of adequate spillways for discharging the flood waters of freshets. There is no uni- formity of practice for the dimensions of these all-important ad- juncts, and it is probable that the great majority of those now in existence have been proportioned by guesswork, or, "upon gen- eral principles/ 7 This is all wrong ; and in this, as in all other questions of design, we should first ascertain what conditions our structure will have to fulfill, and then dimension it accordingly. The capacity or open area of a spillway, is made up of its length and height of notch. It must be large enough to pass all the water of extreme floods without danger of over-topping the dam. Forty times the average flow, or 40 million of gallons per square mile and per 24 hours or 64 cubic feet per square mile per second is none too liberal an allowance, particularly for earthen dams. For this amount of water we have the two simple approximate formulae to determine the length and depth of notch of a spillway, the depth being counted from the level of the lip of the dam to the surface of still water in the reservoir : L = 20 V'Z (0 D = 3 \~A C (2) in which L = length in feet, D depth in feet, A = area of watershed in square miles, and C = a certain additional height above the water in the reservoir, depending upon the character and construction of the dam. If we should wish to provide for a different amonntof water, we must generalize formula (2), writing: y^ D = - 6 x*VA + c. O) in which Q cubic feet per second per square mile. DAMS. Large reservoirs are generally formed by building a dam across the valley of the stream furnishing the supply. Naturally, $3 PRACTICAL HYDRAULIC FORMULA. the narrowest point is chosen, but further investigation may prove such point to be not the most favorable one. A solid foundation is the first requisite, and sometimes firm rock is found so much nearer the surface at a point where the valley is wider that a dam built there would be actually shorter than at the nar- rower point, besides saving the extra excavation to get down to solid bottom. It is abundantly worth while to devote considera- ble time in exploring and surveying, before fixing definitely upon the location of the proposed dam. In examining the character of the foundations, I think that test pits furnish the only trustworthy information. At great depths, these would be very expensive, and recourse is generally had to drilling. This furnishes good indications when properly inter- preted, but also has occasioned many expensive misconceptions of the ground. The test pit remains the only sure means of ascer- taining what is below the surface. The character of the ground will determine the class of dam which should be built. If good rock bottom is to be found, a ma- sonry dam will be the best, and perhaps not much more expensive than a properly constructed earthen one. All the elements of a masonry dam are more fixed and precise than those of an earthen dam can be, so there is less necessity for piling up what may in reality be redundant work, to provide for contingencies which we cannot exactly determine qualitatively. Masonry dams may be divided into three classes ; low, medi- um and high. Although the lines of demarcation are somewhat vague, we may class all dams less than thirty feet high as low, those between thirty and sixty as medium, and all those above sixty as high. Before commencing our investigations it will be extremely useful to establish certain data. Calculation shows that the equation of equilibrium of a dam with vertical faces is : Wx* = 20.83 H* t (4) PRACTICAL HYDRAULIC FORMULAE. 83' in which Jr= weight in pounds of a cubic foot of the masonry, H= the height of wall, and .T = its thickness, both in feet. The weight of a cubic foot of water is assumed at 62.50 pounds. From this equation we derive : 4.565 H x = . (5) V W These equations show that the overturning moment varies as the square of the height, and the resisting moment as the square of the thickness, and the square root of the density of the masonry; while the value of x, the thickness, varies as the height, and inversely as the square root of the density, of the wall ; that is to say, from (5) we deduce : X \~W = 4.565, H a constant. If we assign 125 pounds as the unit weight, or weight per cubic .foot, of the masonry, we find : x = 0.41 H. (6) This is the value of x for exact static equilibrium. We may obtain whatever factor of safety we wish by simply multiplying the square of 0.41 by such factor and extracting the square root of the product. Thus, suppose we wish a factor of 2.5. Oper- ating as above, we find : x = 0.648 H. (7) As the assumption of weight is somewhat arbitrary, we may for simplicity write (7) thus : 2 H .= : ,8, that is to say, a " plumb " wall to resist water pressure should be twice as thick as one to resist average earth pressure. However, dams are not built plumb. They generally have vertical backs, toward the water, and battering faces. The readiest way to transform a vertical or plumb wall into a trape- zoidal one of equivalent resisting moment is to follow Vauban's 84 PRACTICAL HYDRAULIC FORMULAE. principle, that all equivalent walls with vertical backs have the- same thickness at one-ninth of their height from the bottom. This rule holds very closely good within wide limits. As an application, let us take the case of a wall to sustain water, 27 feet high. If vertical, its thickness should be 18 feet for a factor of safety of 2.50, and we would have the rectangle A B CD, shown in Fig. 16. Transforming, according to Vau- A E IB K 19.5 20.25 -* FIG. 16. ban's rule, to a trapezoidal section, with top width uf ft feel, we- get the figure A E F C, of which the bottom width O Pis 12>.50 feet. Verifying the comparative stability of the two sections, we find that of the trapezoidal one to be within about 1J# of the rectangle, while its area is nearly 30$ smaller. Such a section as A E F (7 is obviously awkward, presenting a top-heavy appearance, from the redundant thickness of its upper half. Some such section as A E H G C would be pref- PRACTICAL HYDRAULIC FORMULAE. 85 oruble ; it has still less area and not much less stability than the trapezoid A E F C. Passing to dams of medium height, let us take for an example one 54 feet high (Fig. 17). The proper thickness, if vertical, per 40.50 FIG. 17. formula, (8) is 36 feet. Transforming the rectangular section into a triangular one, by Vau ban's rule, we obtain the right- angled triangle A B ' C which possesses certain interesting properties. In the first place, the triangular section has only 86 PRACTICAL HYDRAULIC FORMULAE. about 85$ of the stability of the rectangle. Secondly, its base will always be three-quarters of its height. Thirdly, the result- ant of its weight combined with the thrust of the water (still assuming the specific gravity of the masonry to be 2) will always cut the base about 11$ within its "middle third"; while, of course, the action of the weight alone will cut the base exactly at the inner extremity of the middle third. Evidently such a section is impossible in practice, because it involves a top width of zero. Let us give it a top width of 10 feet, with a face batter of an inch to the foot to the upper part. This batter will always intersect the hypotenuse of the triangle afc a distance from the top equal to one and a half times the top width, whatever that may be. This composite structure has a resisting moment about 8fi less than that of the rectangular one of the same height, and base of 36 feet. It still has a factor of safety of 2.42, with the above relative densities of masonry and water, while the section is about 40# less than the rectangular one. Fig. 17 shows all dimensions, and the triangle of forces. It will be noted how the addition of the upper trapezoid modifies the points of application of the pressures. In regard to all dams, high or low, we may lay down the leading principle that the line of pressure should always pass within the middle third of the base, especially the line which corresponds to a full reservoir ; that is, the line which is the re- sultant between the weight of the dam itself and the thrust of the water. In very high dams it is not sufficient that this condi- tion be fulfilled for the base only : it must hold good also for any horizontal bed between the base and the top, because in such dams, in order to economize material, the face is given the form of a convex curve, and if this convexity be too great it will occur that, while the base may have a satisfactory width, some of the upper beds parallel to it will not. The object in designing a high dam is to give the section such a form that it shall be a "section PRACTICAL HYDRAULIC FORMULA. 7 of equal resistance,' 7 because this is always the secrion of greatest economy. The problem is further complicated in the case of very high dams by the fact that the resistance to overturning is not the only thing to be considered. We must also determine whether the area of the lower beds is sufficient to resist the crushing strain brought upon them by the weight of the superin- cumbent mass. In making this investigation it is obvious that the first step will be to fix a proper limiting unit strain, or ad- missible pressure, per square foot upon the masonry. This limit depends upon the nature of the material used and also upon the views' of the designer. For ordinarily good masonry, 15,000 pounds per square foot would be considered a conservative limit,, being a trifle over 100 pounds per square inch. If the resultant of the pressures cut any bed exactly in the middle, we could ascer- tain the pressure per square foot upon such bed by simply divid- ing the whole weight of the mass resting upon it by its length. But when the resultant moves from the center, the strain is no longer evenly distributed over the entire bed, but is intensified upon that portion comprised between the point where the result- ant cuts the bed and the nearer extremity of the same, reaching its maximum intensity at the extremity itself, or as we should say, at the nearer toe. The investigation of this varying strain, which increases in proportion as the resultant approaches the nearer toe, is somewhat obscure, and rests upon assumptions of somewhat unsatisfactory demonstration. The following two formulas may, however, be accepted as reliable approximations to the truth, within the limits occurring in ordinary practice : 2 w 4 W P = - (L-1.5D), (10) , in which : P= pressure, in pounds, per square foot. 88 PRACTICAL HYDRAULIC FORMULAE. L = length of given bed, in feet. W weight, in pounds, of mass above given bed. D - distance, in feet, from point of intersection of resultant with given bed, to nearer extremity of same. L Formula (9) is used when D is equal to or less than . L 3 Formula (10) is used when D is equal to or greater than . When L 3 D = , either formula gives unit strains eoual to twice the total 3 weight above bed, divided by its length. We have then the three following conditions which the pro- per section of a high masonry dam should fulfill : First, the lines of pressure should lie within the middle third of all beds. Sec- ondly, the maximum unit strains should not exceed a moderate fixed limit. Thirdly, the section should be one of equal or nearly equal resistance. Now then, in the light of what has been already established, let us feel our way toward the proper design for a darn 160 feet high fulfilling the above three conditions. Let us assume a density of masonry double that of water, a limiting unit strain of 15,000 pounds, and, as is usual in such cases, let us consider a length of one foot of dam, so that the area of our section in square feet will represent an equal volume in cubic feet. Knowing that one of the necessary conditions is that the re- sultants shall lie within the middle third of all the horizontal beds which we may suppose to divide the section, we feel sure that we cannot go far wrong in first laying down the right-angled triangle ABC (Fig. 18), of base equal to three-quarters of the height, or 120 feet for the total height of 160 feet. Desiring a top width of say 20 feet, we lay off the same from A, giving the face of this portion of the section a batter of -j^-. This batter we already know will strike the hypotenuse 30 feet vertical from the top. Now as we know that the effect of placing this top story PRACTICAL HYDRAULIC FORMULA. 89 upon our triangle will be to draw the line of vertical downward pressure, due to weight of masonry alone, away from the center of gravity of the triangle A B G> and therefore outside of the " middle thirds " on the water side ; and, also, anticipating a FIG. 18. little the knowledge which we shall presently acquire, we give the back of the section, from a point 80 feet from the top, an outward flare of one to four, which is shown in the figure hy the small triangle D E B. and which increases the total width of base to 140 feet. The section is thus divided into three trapezoids, 90 PRACTICAL HYDRAULIC FORMULA. respectively 30, 50 and, 80 feet high, with corresponding widths of 20, 223, 60 and 140 feet. Next we determine, either graphically or bv simple calcu- lation, based upon the properties of similar triangles, the points where the line passing through the center of gravity of the superincumbent mass cuts the imaginary bed D F, and also where it cuts the same when shoved forward by the thrust uf the water, acting at right angles to it. These points are shown in the figure to be respectively at 19.35 ft. and 25 ft. from D and F. The figure shows also the intensity of the strain in pounds at these points, obtained by multiplying the total area above D F by 125, the assumed weight in pounds per cubic foot of masonry, although the calculations were actually made in units of volume, for the sake of rapidity and ease, counting the weight of a cubic foot of masonry as 1, and that of water as ^. We next proceed in the same way in regard to E C. Here we have first the point where the line passing through the center of gravity of the entire section A E C cuts the base E C, 55.80 ft. from E, and which corresponds to an empty reservoir ; and secondly, the point, 60.32 ft. from C, where the resultant of the weight of the section A E G, plus the weight of water resting upon the inclined surface D E, combined with the forward thrust of the water acting under a head of 160 ft., cuts the same base E C, which point corresponds to a full reservoir. The in- tensities in pounds of all these strains are shown on the figure. Our design now fulfills one of the imposed conditions . The lines of pressure lie well within the middle third, except at D, where the condition is not so binding. It remains to see how it complies with the second one. For this, we recur to our formulae (9) and (10); and first to ascertain the unit strain at D. For this we employ (9), within which the case just falls. Substituting numerical values, we have : 2 X 337500 P = - - - = 11628 Ibs. per sq. ft. 3 X 19.35 PRACTICAL HYDRAULIC FORMULAE. 91 For the strain at F we use (10): 4 X 337500 P = - (60 - 37.5) = 8438 Ibs. per sq. ft. 3600 Passing to E C, we have for maximum strain at E, when reservoir is empty, 4 X 1337500 P = - (UO - 83.70) = 15368 Ib8. per sq. ft. f OF THE 19600 I TJNIVERSI For maximum strain at C, when reservoir is full : x.o . P = I*-""* "x"g = V CXL This shows that V increases with the square root of the diameter and of the head, and diminishes with the square root of the coefficient and length. If diameter d be given in inches, then to obtain Fin feet,, multiply C by 12. Thus : V = , v L Page SO. The general form of an adfected quadratic is: * 2 < a x = b Whence : Page 31. Throughout this book proportions are written in the fractional form, as being the most convenient. The propor- tion - - = -, is read : "h is to 22 as 50 is to 54.87. 54. o7 Page 42. All the relations between the different elements of two long pipes may be expressed in a single equation : D'*.H' .Q*.L. ~ Assuming a mean common value for (7 and C': I? 5 .H' .Q*.L WATER SUPPLY ENGINEERING, 107 The most useful outcome of the above is when two pipes have the same length and head ; that is, when L = L' and H = //'. Then : Q That is, other things being equal, the quantities discharged by two pipes are as the square roots of the fifth powers of their diameters. It is generally easier and more satisfactory to ascertain what we want to know about a given pipe by direct calculation, rather than to deduce it from the known elements of another pipe, by means of the relation (a). Page 47. The great practical importance of the formula} on this page has not been brought into sufficient prominence in the original text. A more thorough elucidation will be given now. Equation (3), page 18, can be written thus : Q= /0.616J'.g r c.x, Consulting the table of coefficients on the same page, we see that those for rough pipes from 8 to 48 in. in diameter vary from 0.00068 to 0.00062. These coefficients do not greatly differ, and, moreover, their square roots only enter the formulas. If, then, for pipes of the above diameters we take a uniform value, G = 0.000616, the above formula reduces to : Q = i/ l JL <6) L H Furthermore, since is a ratio, we may always reduce If to its value when L = 1000 ; that is to say, we can establish the relation : h H 1000 ~ L 108 WATER SUPPLY ENGINEERING. in which h equals the fall, or head, per 1000. Therefore, H h replacing in (b) by its equal , we have : // 1000 Q \' Always recollect that li represents the head per 1000, as against //, which represents the total head in the total length. The relation (c) may be generalized thus : Again, from table, page 18, we see that the value of C, for rough pipes from. 3 to 6 in. in diameter, varies from 0.00080 to 0.00073. Also, equation (3), on same page, may be written : 0.785 X 0.785 /> . H g = |/ - C .L Adopting, therefore, a mean value of 0.000785 for C, for pipes H h of above diameters, and reducing to , as before, we have : L 1000 Q = V 0.785 D 4 h (e) and generalizing, Q 3 = 0.785 (/) hD* For smooth pipes, from 8 to 48 in. diameter, (d) becomes : # 2 = 2 (0) hD* For smooth pipes from 3 to 6 in. diameter, (/) becomes : Q* = 1.57 (h) hD* From the above, we see that the discharge of a smooth pipe is 1.40 times that of a rough one of same diameter, while that of a rough one is 0.70 times that of a smooth one. WATER SUPPLY ENGINEERING, 109 Observing that velocity is always equal to quantity dis- charged divided by area of pipe, we have for the velocity of flow, in feet per second, for rough pipes, from 8 to 48 in. diameter : v= 1.27 v~Dh W For those from 3 to 6 in. diameter : For smooth pipes respectively : V = 1.80 VD~h (fc) and F=1.60 VDh W The relation between the diameters of rough and smooth pipes, for equivalent discharges, may be also deduced, giving : Rough Diameter --- = 1.15 (m) Smooth Diameter That is, the diameter of a rough pipe to give the same dis- charge as a smooth one should be 1.15 times that of the latter. It is to be remarked that if q = U. S. gallons per minute, and d = diameter of pipe in inches, we have : c 2 - = o.8i in) hd* This is an intermediate value between (d) and (/), so that in a great many cases it would be quite safe to use (d) or (/) indif- ferently for feet and seconds, or for gallons, inches and minutes. Comparison with observed velocity through pipes of differ- ent diameters and different degrees of roughness and smooth- ness* seems to establish the fact that the above series of formu- lae, from (a) to (w), both inclusive, cover the whole range of practical cases, and that, apart from entirely abnormal con. ditions, no cast-iron pipe, however smooth, will give a greater discharge than that deducible from ( _. _ __ Duty in mil. ft.-pds. Thus for an engine having a duty of 100 million foot pounds per 100 Ibs. coal : 198 P = = 1.98, 100 or practically 2 Ibs. coal per hour per horse power. Rating 1 Ib. average coal as equal to the evaporation of 10 Ibs. water or the development of 10,000 B. T. IL, we may say that " high duty" implies not less than: 1,000,000 ft. pds. per Ib. coal 100,000 " water. 100 " B. T. U. Duty trials are perhaps most generally made in this country under actual working conditions, that is, the main feed is pumped from the hot well, and the jacket and separator water fed back to the boiler. This is more satisfactory as representing normal con- ditions, but complicates the measurements. The management of a duty trial is a very intricate affair, and cannot be fully described here. The report of the commit- tee on standard method of conducting duty trials of pumping en- gines, of the American Society of Mechanical Engineers, in its revised form, should be consulted in this connection. As illustrating the general outline : of a duty trial, the follow- ing example, condensed and simplified from one given in the above report, will now be instanced: A high duty compound pumping engine is supplied with steam at 135 Ibs. absolute pres- sure. This corresponds to a total heat, above zero Fahrenheit, of 1,220.70 B. T. U. There is a separator on the main steam pipe. After passing through this separator the steam is found to still contain H per cent. -moisture. This moisture affects the latent WATER SUPPLY ENGINEERING. 133 heat of the steam (which at above pressure is 866.60 B. T. U.), so that its total heat above zero is : 1220.70 866.60 X 0.015 = 1207.7, Both cylinders are jacketed, and there is a reheater supplied with boiler steam. Water from jackets, separator, and reheater feed back to boiler. The different supplies of water fed to boiler during trial (10 hours), with their temperatures, are as follows : Main feed, at 215 18,863 Ibs. Low pressure jacket, at 225 615 " High and reheater, at 290 815 " Separator, at 340 ... 210 " Total feed 20,503 Ibs. The total heat furnished by the boiler is therefore : Main feed (1,207.7 215) 18,863 = 18,725,300 Low pressure jacket (1,207. 7 255) 615 = . ... 604,361 High pressure jacket and reheater U.207.7 290) 815 = 747,926 Separator, neglected. Total B. T. U 20,077.587 The net area of pump plunger is 308 sq. in., and the average stroke 3 ft. Number of single strokes during trial 24,000 Pressure by gauge on force main 95.00 Ibs. vacuum gauge on suction main 3.69 " equivalent to difference of level of gauges.. 4.31 " Total pressure .103.00 Ibs. The work done by the pump is therefore 308 x 103 x 3 x 24000 = 2,284,1*8,000 ft. Ibs. 2284128000 Duty = = 113765066. 20077587 Indicated horse power; as determined during trial, 128.15. Pump horse power, as above : Hence : 2284128000 - = 115.36. 10 X 60 X 33000 11536 Efficiency = = 90 per cent. 128 .15 134 WATER SUPPLY ENGINEERING. This is efficiency of the engine as regards work done by the pistons, not as regards heat utilized. Upon this latter basis, the calculation would be as follows : Since 1 B. T. U. = 772 ft. Ibs. and one horse power = 1,980,000 ft. Ibs. per hour, the 1980000 number of B. T. U. per hour per horse power is 772 = 2565. But the total B. T. U. furnished by the boiler, per hour, is 2007758.7. Hence, the theoretical horse power corre- sponding to the heat unit supplied is : < 20077587 Theoretical H. P. = = 782 2565 Therefore : 11536 Efficiency = = 0.1475 782 The above theoretical horse power is, of course, impossible of realization, for it supposes the temperature of the hot well to be reduced to zero. Suppose, in the above case, for round numbers, that the temperature of the hot well were 100 ; the tempera- ture of the steam, at 135 Ibs. absolute pressure, is 350 above zero Fah. and 810 above zero absolute. Then, by Carnot's law, the efficiency of a perfect heat engine working between the given limits is: 350 - 100 810 Then, according to one view of the subject : 1475 Efficiency = = 0.48 nearly. 3086 According to another view, which considers only the latent heat of the steam as that theoretically utilizable, we have for the total theoretically possible work-in foot pounds per pound of steam (or feed water), the following formula : w = T -f 46t) WATER SUPPLY ENGINEERING. 185 In which : W = total passible work in foot pounds per pound of feed water. L H - latent heat of steam at given pressure. T = temperature of steam at given pressure. t = temperature hot well or condenser. T -f 461 = absolute temperature of steam. 772 = Joule's equivalent. In the example : W = 866.6 X ? X 772 = 206510. 810 As there were 20,503 Ibs. of steam furnished, or water fed: 206510 X 20503 = 4234074530 ft. Ibs. Comparing this with the work actually done by the pumps, gives : 2284128000 Efficiency = = 0.54 nearly. 4234074530 The usual English method of conducting a duty trial seems to be to feed the boiler exclusively from a measured tank, wasting the jacket and injection water. The quantity and temperature of the wasted discharge of the air-pump is also measured, and the quantity of heat necessary to raise it from its initial tempera- ture to that of the hot well calculated and recorded as "rejected heat." This quantity of heat, reduced to B. T. U. per minute per indicated horse power, is known as " Donkin's Coefficient/' When, however, the feed, injection, and jacket water are measured separately, a more accurate estimate is possible than when the total discharge of air-pump is used. To this rejected heat is added the heat utilised per minute per indicated horse power, which is 2565 = 42.75 multiplied by the indicated horse power developed 60 during the trial. The rejected and utilized heat added together should equal the total heat reckoned from boiler consumption. It always falls short, and the balance is put down to errors and radiation. 136 WATER SUPPLY ENGINEERING.. As an example of the English method the following is given based upon a trial made by Professor Unwin. The wasted jacket water was measured separately, and a more accurate basis of cal- culation thus established than by the use of Donkin's coefficient. All calculations are made in this example for the total duration of trial and not reduced to " per minute." Duration of trial . . 24 hours. Total water pumped 200,000,000 Ibs. Lift, including friction 50 ft. Work done 10,000,000,000 ft. pds. Feed at 51 108,500 Ibs. Jacket water wasted 16,9001bs. Feed used in work (108,500-16,900) 91,600 " Injection water, at50 3,633,000 " Condensed steam 91,600 " Coalburned 11,000" Temperature of hot well 75 " steam, 75 Ibs. absolute 307 Total heat of above steam 1208 Indicated horse power , 255.50 Pump horse-power 210.44 210.44 Efficiency = 82.3* 255.50 Rejected heat : Injection, 3,633,000 (75-50) 90,825,000 Feed, 91,600 (75 51 > 2,198,400 Jacket water, 16,900 (307-51) 4,326,400 Total rejected heat 97,349,800 Heat units utilized during trial, 255.5 X 2,565 X 24 15,728,580 Total heat accounted for... 113,078,380 Total heat furnished by boilers, 108,500 (1,208-51) = 125,534,500 Heat accounted for = 113,078 380 Deficit (about 102) 12,456,120 The duty, calculated per million heat units furnished to cyl- inders (108500 16900) (1208 51) is : 10000000000 X 1000000 Duty = = 94356358 ft. Ibs, 91600 X 1157 The English practice is to calculate duty per cwt., or 112 Ibs. of coal. Then : 10000000000 X 112 Duty = = 101818102 ft. Ibs. 11000 Apart from minor details, and considering only those of large WATER SUPPLY ENGINEERING. 137 capacity, there are two prominent types of pumping engines in use in the United States ; namely, the direct acting, with or without a high duty attachment, and the rotative, or crank and flywheel engine. The merits of these two types are fully set forth in the descriptive pamphlets of their respective makers; here it will suffice to briefly enumerate their respective claims, as follows : The advocates of the direct-acting type claim a large reduc- tion of weight by replacing (when high duty is required) the fly- wheel and crank shaft by a comparatively light special attach- ment, and a greater security against damage in case of an acci- dent suddenly relieving the engine of its loud from the fact that there is no dangerous momentum stored in heavy moving parts. Also, that high duty is realized through a greater range of developed power, that is, nearly the same economy is claimed when working at reduced, as at full speed. The advocates of the rotative type claim that " its princi- pal advantages are positive action of steam valves and cut-offs, and absolute full stroke of steam pistons and plungers under varying pressures of steam and water. In these engines, there- fore, there can be no increase in the clearance spaces between the steam pistons and cylinder heads causing waste of steam, nor loss of capacity by deficient plunger displacement." It is but simple justice to say that many magnificent speci- mens of both types are to be found doing excellent service all over the country. ARCHES AND ABUTMENTS. Some of the grandest and most interesting examples of hydraulic engineering are to be found in the arched aqueducts of ancient and modern times. A study of the principles of the arch is therefore an essential part of the equipment of the water- works engineer. The span being given, the starting point of all arch calcula- 138 WATER SUPPLY E>GINEERING. tion is thickness or depth of the crown at the key. This dimen- sion is fixed in practice by some one of the various empirical formulae in general use. Although these have been deduced from existing structures, and the most approved ones can be supported by many examples, they exhibit considerable variation in their results. Here are five, with the names of the authorities who- give them: Perronnet. D = I + 0.035 S (1) Croizette-Desnoyers. D = 0.50 + 38 \'R (2) D = 0.50 4 0.33 VR (3) Boix. D = 0.75 VS Rebolledo. D = 1.15 + 0.035 (S V} (4) In these, D = depth of key; S = span; R = radius of cnrva of in trades; V= versine or rise. If the intrados be elliptical, R assumed or approximate radius at crown. All dimensions in feet. Of these formulae (1) is said by Leville to apply to all curves- of intrados, semi-circular, segmental, or elliptical. Formula (2) applies to semi-circular arches, and also to segmental ones, or those formed by an arc of a circle less than a semi-circle, when 8 the rise is more than . If less, (3) applies. This formula is 6 slightly changed from the original. It will be instructive to apply these formulas to a series of arches of different spans, for the purpose of 'comparison. Com- mencing with semi-circular arches of 30, 60 and 100 ft. span, the following table gives the value of D from formula? (1); (2); (4) and (5). SEMI-CIRCULAR ARCHES. S D (1) (2) (0 (5) 30 2.05 1.97 2.33 1.68 50 2.75 2.40 2.76 2.03 100 4.50 3.18 3.48 2.90 WATER SUPPLY ENGINEERING. 139 For segmental arches of 60, of which the radius is equal to the span, and the versine or rise is 0.134 of span or radius, formulae (1), (3), (4) and (5) give: SKGMENTAL ARCHES OF 60. 8 D (1) (3) <4) (5) 30 2.05 2.31 2.33 2.06 50 2.75 2.83 2.76 2.67 100 4.50 3.80 3.48 4.18 These results agree better than the previous ones, but it is evident that we can exercise considerable latitude in the matter of keys and yet have good authority, and still better, good ex- amples to sustain us. These thicknesses of key apply more particularly to railroad and highway bridges. They are sufficient to carry on the arch ring alone and without taking account of the spandril backing, track and trains, with 2 or 3 ft. of earth filling over the extrados. For heavier embankments some authors recommend an addition to the depth of key of 2 per cent, of height of embankment. Thus, if an arch carries a 50-f t. embankment, add one foot to depth of key. In the absence of special empirical formulae for aqueduct arches, the above may be safely taken for this purpose, especially since for aqueducts the spandril backing will generally be carried up level with the extrados at the crown. It is a well-known fact that a semi-circular arch can be car- ried up to a considerable height above the spring line before the voussoirs begin to bear upon the centering, as they are kept in place below this height, by friction. It has been found that the point at which centering becomes necessary to support the vous- soirs of a semi-circular arch occurs at about half the height of the rise, or at 60 measured from the vertical. The joint at this 140 WATER SUPPLY ENGINEERING. point, or the point nearest to it, is known as the " joint of rup- ture," and is one of the most important elements of arch design- ing. It is up to this level that the backing of the haunches should always be carried. The arch proper commences at this joint, and includes 120. All below this joint must be considered as forming part of the abutments. In a well-proportioned voussoir arch, the thickness of the arch ring should increase from the crown toward the haunches. The radial length of any joint between the key and the joint of rupture should be such that its vertical projection, or the cosine of the angle which it makes with the vertical, shall be equal to D the depth, D, of the key. It is, therefore, = , a being the cos. a angle which the joint makes with the vertical. The cosine of 60 (or sine of 30) being ^2, we have for the length, L, of the joint of rupture of a semi-circular arch : L = 2D (6) The length of any joint intermediate between the key and joint of rupture may be found as shown on the right-hand side of Fig. 24, by drawing the horizontal line df&t the distance c d = FIG. 24. D from the springing line A B. Then the length of any joint, a b, is found by drawing c b, and taking e b = R. WATER SUPPLY ENGINEERING, 141 This process can be continued below the joint of rupture, as shown in the figure, when the curve of the extrados rapidly flat- tens, becoming finally an asymptote to the springing line. The above process is due to Dejardin. Dubosque gives a more rapid method, resulting in a somewhat greater thickness, which is shown on the left-hand side of Fig 24, and is described as follows: Join g and li, the exterior extremities of the joint of rupture and the imaginary joint at the crown. Bisect g h in K. Draw Km perpendicular to g h, intersecting the vertical at m. From ?, with radius m g, m h, describe the arc V V For semi-circular arches, this reduced to : T = 1.30 + 0.20 S. (U) Formulae (13) and (14) take no account of the height of the abutment, which is not, contrary to what might be supposed, a very important factor. If the top thickness be determined by the for- mula, the batter of the wall will amply provide for any additional thickness rendered necessary by the greater or less height. A good formula, of, I believe, German or Russian origin, for thickness of abutments of semi-circular arches in which the height, H, enters is : T = 1 + 0.20 S 4- 0.16 H. (15) In ordinary cases this may be reduced to : T = 0.30 s. For piers, Rebolledo gives : T = 2.50 D'+ 0.10 H. In which D = depth of key of arch, and H = height of pier. In wide spans and heavy loads, the weight borne by foot of piers and abutments must be considered with regard to resist- ance to crushing. EXAMPLE. Determine the principal dimensions of a 3-cen- 144 WATER SUPPLY ENGINEERING. tred arch, 30-ft. span, 10-ft. rise, abutments 12 ft. high. See- Fig. 25. From (11) or (12). R = 21.83 ft. r=8.17 ft. Thickness, D, of key from formulae (1) to (5) respectively, 2.05 ; 2.27 ; 2.33 ; 1.85. Take J9 = 2 ft. Joint of rupture oc- V curs at = 6 ft. below crown, and its length, per (8) = 1.80 D 2 = 3.60 ft. The top thickness, or thickness at spring-line of ,30 abutments per (13) = 1.30 + 0.14 x 30i/ = 8.57 ft. 10 The formulae for thickness of abutments give that necessary to sustain the thrust of the arch. No account is taken of the counter thrust of the earth embankment behind the abutment. In many cases the abutments could be lightened by relieving arches or otherwise. When an arch fails, it is generally on account of settling or WATER SUPPLY ENGINEERING. 145 spreading of the abutments, or to bad work or materials. If there is no movement in the supports, it can only fail by direct crushing of the materials a very rare case or by distortion of the arch. Distortion is caused by a sinking at one point and rising at another. The object is, therefore, to get such a sub- stantial and evenly distributed permanent load upon the arch, if possible, before striking the centres, that no irregular strains- can come upon it to cause distortion. Naturally, the lighter the permanent road, the stronger and stiifer must be the arch in order to resist transient and unequal loading. These remarks apply more particularly to viaducts, for an aqueduct arch is sel- dom subjected to unbalanced stresses. As regards the best form of arch, a given opening can be successfully spanned by an arch of any form. Generally speak- ing, the curve of intrados is selected in reference to the special conditions of each case, the amount of head room required, etc^ There are, however, certain forms which, other things being: equal, are best fitted for certain kinds of loading. If an arch is to sustain a single load, concentrated at the centre and heavy as- compared with the weight of the arch itself, the gothic arch is most suitable. If the loading increases gradually from the crown to the haunches, as in the case of an ordinary earthen embank- ment, the curve of pressure would more nearly approach the arc of a circle. Should the loading be much greater at the haunches than at the crown, the curve would approach the elliptical form. In all cases the curve will be found to rise to the pressure ; for the true curve of pressure is always represented, inverted, by a flexible cord, similarly loaded to the arch. Where head room near haunches can be spared, and great economy of material is not essential, the full centred, or semi-circular arch, will generally be preferred, both for its great structural stability and the beauty of its proportions. The "High Bridge" of the old Croton aqueduct may be here instanced. When the full centred arch is inadmissible, the segmental arch, or that formed by the are 146 WATER SUPPLY ENGINEERING. of a circle less than J80 in amplitude, has much to recommend it. Of these, the arc of 60., with span equal to radius and rise 0.134 S seems a very happy selection both as to appear- ance and convenience of dimensions. Should an arch be intended to support a body of water in direct contact with its extrados, the proper theoretical form would be that of the hydrostatic arch, which resembles a cycloid, which in turn resembles an ellipse. In all practical cases, however, such as tanks or aqueducts, a level bottom is necessary, and the spandrils are built up level with the crown. In this case the conditions governing the hydro- static curve do not obtain, and the load from the water is simply an equally distributed one, pressing at all points vertically down- ward, or at least practically so. Although the above empirical formulae suffice to correctly proportion any except very unusual forms, it will be well to de- vote a few words to the more theoretical features of the subject. Suppose it were wished to calculate the curve of pressure in the semi-circular voussoir arch, half of which is shown in Fig. 26. FIG. 26. Assuming it to be of good materials and workmanship, without which all calculation would be impossible, that part of the struc- WATER SUPPLY ENGINEERING. 147 tn re lying above the joint of rupture e/is taken as comprising the arch proper. The section is supposed to be divided into an arbitrary number of fictitious voussoirs a, b, c and d, in the figure, which will give, equally well with the true ones, the form of the curve of pressure. The weight of and upon each of these voussoirs is then estimated, and the position of the line passing through the centre of gravity of each voussoir and its load deter- mined. From these data the position of the vertical line JF pass- ing through the centre of gravity of the entire section above joint of rupture is fixed. The horizontal line If cutting the central joint at one third f ^- J of its length from the top is then drawn to its intersection with the vertical W. From the point of in- tersection the line Z is drawn cutting the joint of rupture at one third t J of its length from the bottom. The weight of the half arch and load above ef is then laid off to scale on W, and from its extremity the horizontal line g h is drawn to Z, completing the triangle of forces. The length of the line g h to the same scale as JF gives the value of horizontal thrust at the crown. On the horizontal line H produced, take aj= g h, and from it extremity lay off/ K = weight of voussoir, a, and load. Join a k. On a Tc produced, lay off b I = a k. From I, lay off lm = weight of voussoir, b, and load. Proceed thus, working the curve down from voussoir to voussoir to the line of rupture. The last resultant should coincide with the oblique line Z, which fact furnishes an excellent check upon the accuracy of the work. The broken line representing the line of pressure can now be harmonized with a curve, drawn by hand and the actual voussoirs laid down on the drawing to show where and at what angle the curve cuts their joints. The line of pressure may be continued down to the spring- ing line and on through the abutment, but this is not generally necessary. 148 WATER SUPPLY ENGINEERING. The reason why the points of application of If and Z at the, crown and joint of rupture are placed at the upper and lower extremities, respectively, of the middle third of these joints is the following: The general tendency of all arches, except those carrying very unusual loads at the haunches, is to sink at the crown and consequently rise at the haunches. When the crown sinks the arch opens at the intrados, in the neighborhood of the key, rotating upon its upper edge, at the extrados. Inversely, the joint of rupture will open at the extrados, rotating around its lower edge, at the intrados. If the joint at the crown does not actually open, there is always a tendency to do so, and a corre- sponding tendency at the joint of rupture ; and at the points around which the voussoirs tend to rotate the compressive stresses are at their maximum, diminishing progressively until reaching the other extremity of the joint, where the tendency is to open. At this point the compressive stresses become -zero. The total compressive stresses on these joints may therefore be represented graphically by the area of a right-angle triangle constructed upon each joint, with its base at the extremity around which rotation tends to take place, and its apex at the extremity which tends to open. The resultant stress passes through the centre of gravity of each triangle, which is at one-third of its height from the base. The heights of the triangles being represented by the length of the joints, the points of application of H and Zmust, to conform with the above theory, be placed as in the figure. In this connection see pages 87 and 88. For a properly proportioned arch, such as would be the out- come of the practical rules already laid down, and is shown in Fig. 26, the amount of horizontal thrust at the crown which is to arch calculation what abutment reaction is to that of girders and bridges can be obtained more readily than in the above ex- ample by the use of Navier's formula : H = P X R (17) WATER SUPPLY ENGINEERING. 149 in which P = pressure or weight per square unit at the crown, and R = radius of the intrados at the crown, expressed in the same linear unit. Thus, in last example, Fig, 26, if the weight of arch and load on or in the immediate vicinity of the key were 1,000 Ibs. per square foot, and the radius 15 ft., the total horizontal thrust would = 15,000 Ibs. In example, Fig. 25, if the unit pressure were the same, the radius at the crown being 21.84 ft , the total horizontal thrust H would equal 21, 840 Ibs. As regards the whole subject of arch design, it may be broadly stated, on the authority of Dejardin, that if the arch is propor- tioned according to the rules already laid down, and is well con- structed with good materials, no calculation whatever is needed to demonstrate its stability. If, however, the engineer should be called upon to discuss an existing structure built upon different lines (and which, nevertheless, might fulfil all the requirements of stability) he should proceed as above directed, making differ- ent assumptions until a line of pressure is found which shall at least lie inside of the arch, at all points, the presumption being that if such a line can exist, the arch will find it. Should no curve be found which did not cut the intrados or extrados at some point or points, it would indicate that rotation would occur around such point or points, with intense local compressive stress. If this result were found in any design under discussion, it would justify the rejection of such design. If it were found in any ex- isting structure, it would prove that the construction and ma- terials of the arch were of a nature to permit it to resist bending moments at such points, or that the assumed data regarding weights and loading were incorrect. Should the line of pressure, while not leaving the limits of the section, approach very near to the intrados or extrados, the amount of compressive stress set up at these points is to be determined according to the rules laid down for masonry dams, considering the compression to be con- centrated upon the area of joint lying between the point of ap- plication of the pressure and the nearer extremity of the joint. 150 WATER SUPPLY ENGINEERING. The chief uncertainty which militates against the value of all arch calculation lies in the fact that we are obliged in most cases to make almost random guesses at the superincumbent loads ac- tually and not theoretically sustained by the arch. As a simple example of this may be cited the case of an arch sustaining a high masonry wall. Theoretically, the whole weight of the wall would be resting upon the arch, and the higher the wall the more danger to the arch. Practically, we know that the higher the wall the less danger would there be of its falling in if we should break an opening through it near the bottom. It will be noticed in all that precedes that the joint of rup- ture has been placed at half the height of the rise. It is found in practice that this assumed position results in a well-propor- tioned and secure arch, particularly if the masonry backing be carried up to this height at the extrados. Strictly speaking, however, we cannot so broadly generalize the problem, and prob- ably it is impossible to tell exactly where the joint of rupture is located in any existing arch. Lame and Olapeyron assert that FIG. 27. the line of rupture a b, Fig. 27,is so situated that the tangent c d, at the point b, and the tangent e/at the crown, will always inter- WATER SUPPLY ENGINEERING. 151 sect at the line g It, passing through the centre of gravity of the mass lying above the line of rupture in such a position that these three lines will all meet at one and the same point, i. To deter- mine the position of the joint of rupture by this rule, it would be necessary to operate by trial and error. It might be used to test the assumed position of the joint in any particular case, but it does not seem to possess sufficient practical utility to render its use recommendable. All that precedes relates to the voussoir arch, the funda- mental principle of which is that it preserves its stability by equilibrium alone, independent of any cohesion of the mortar in which the stones are bedded. Brick and concrete arches belong to a totally different class, as their stability depends principally or wholly upon the cohesion of the mortar. They are monolithic in character, and while the empirical formulae established for voussoir arches can be used to determine their proper dimensions, the cal- culations respecting the line of pressure do not apply. The increased thickness toward the haunches of a brick arch may be given by bonding in additional rings, or, as will generally be found preferable, by giving the arch a uniform thickness throughout, and obtaining the desired increase by adding to the spandril backing, as shown in Fig. 28. In the case of brick arches of small span, ib is best to build them in independent concentric rings, but if the span and radius of curvature are relatively large, it will frequently be found advisable to bond the rings together. Should it be desired to apply calculation to a brick arch in order to ascertain its probable stability, recourse should be had to the old method of Lame and Glapeyron, which is based upon Boistard's experimental researches. Without going into a full description of this method, which can be found in text books, the formula will be given to determine whether a given arch is stable as against sinking at the crown and raising at the haunches, by far the most common case of failure. 152 WATER SUPPLY ENGINEERING. Referring to Fig. 28, supposing the abutments to be unyield- ing, and considering only the arch a b c d e f, exclusive of the backing, the notation is as follows, a b being the springing line and c d the assumed joint of rupture: W = weight of mass ab c def including loading. W = weight of mass c d ef including loading. A = distance from a to line passing through centre of gravity of W. x = distance from d to line passing through centre of gravity W. y vertical distance e g, from d to e. JR = total rise, o e, from springing line to extrados. Then the equation for exact static equilibrium is : and for stability : (18) W A W R > o (19) Thus, in the arch shown in Fig. 28, assuming a span of 30 ft.; a rise of 10 ft., and a depth of key = 2.50 ft., let the data be : > W - 6,000 Ibs. W = 2,250 Ibs. R = 12.50 ft. A = 7.40 ft. x = 5.40 ft. y = 7.50 ft. WATER SUPPLY ENGINEERING. 153 Then from (18) and (19): 5.4 6000 X 7.40 - 2250 X 12.50 X = 24,150 Ibs. 7.5 This would indicate that the moment W A, making for stability, had a factor of safety of nearly 1.85. It will be noted that equations (18) and (19) contain two variables, depending upon the position of the joint of rup- ture. Write (18) in this form: iW A- W x\ R ( - J = o (20) Then, should the arch break, it would be at some joint, a #, W x such as would make ~ maximum. This point can be found y by trial and error. If such maximum value gives a negative result in (20), it would indicate that the portion of the arch below the joint of rupture was too light. It is necessary that the constructing engineer should be familiar with the above processes of calculation, or, at least, be aware of their existence, in order to know what can and cannot be accomplished by figuring. The successful development of arch building having been mainly along purely experimental lines, very little aid is to be hoped for from abstract mathematical reasoning, for the very good reasons among others that the most important data must be assumed, or. in other words, guessed at, and that in the case of voussoir arches the adhesion and cohesion of the rnortar is and must be ignored, although it may really play a very important part, particularly where small materials are used. In the case of concrete arches, when the limit of small and amorphous materials is reached, the strength of the in or tar is everything. The first requisites for an arch are unyielding foundations and abutments, without which fracture and perhaps destruction must ensue. Then come good workmanship and materials, and 154 WATER SUPPLY ENGINEERING. great judgement as to time and manner of striking centres and loading arch. Design has been purposely left to the last, because if all the other requirements are observed, the shape and dimen- sions of the mere arch ring are matters which, though important, are still of lesser moment. For further details of arch design, see " Van Nostrand's Engineering Magazine/' for December, 1883, and February, 1884. EXPLANATION OF THE TABLES. Table 1 gives the areas of circles in square feet, correspond- ing to diameters in inches. Table II contains five columns, headed respectively Z>, F, Q, G and // P, giving various properties of rough cast-iron long pipes of different diameters having falls of one to ten per thousand. D = diameter of pipe in inches; V velocity in feet per second; Q discharge in cubic feet per second; G = dis- charge in U. S. gallons per hour, and II P = theoretical or net horse-power necessary to raise the quantity discharged one foot high. In this table Fis calculated by the formula (1 bis.) or (1 ter.}, Q is calculated by multiplying Fby the area of the pipe, G by multiplying Q by 27,000, and H P either by dividing Q by 8.82 or multiplying G by 42 and pointing off seven decimal places. EXAMPLE. A pipe 40 inches in diameter is laid to a grade of i-oVo- What is the discharge and the net horse-power necessary to lift the volume discharged to a height of 113 ft.? The dis- charge is 947,241 U. S. gallons per hour/and the horse-power for each foot of lift is 3.98. Then 3.98 x 113 = 449.74 is the required, net horse-power. Any of the above data can be obtained for falls per thousand not given in the table by multiplying the values given for falls- of .j-^-y. by the square root of the fall. EXAMPLE. A pipe 20 in. in diameter has a fall of 3.43 per WATER SUPPLY ENGINEERING. 155 1,000. What is the discharge in cubic feet per second ? What the horse power per foot of lift ? I 1 or a fall of ^n> Q = 3.511. Therefore with a fall of 3,43, Q = 3 .5 11 4/3 .43 = 6.50 cu. ft. per second. Again, for 115 Vo> HP= 0.40. Therefore, for .ViS, HP = 0.40 ,y/^43 = 0.74. RI Table III. gives the value of for different values of R, U* from which the mean velocity U can be deduced. EXAMPLE. Fig. 29 represents the cross-section of the old FIG. 29. Croton Aqueduct. When running to within 1.12 ft. of the crown, as shown in the figure, the wet section = 49.19 sq. ft., the wet perimeter 20.72 ft., and the mean hydraulic radius 49.19 R = = 2.37. Let / = 0.00021. Then by table III. : 20.72 U' = 0.00006794. 156 WATER SUPPLY ENGINEERING. Also by the data : 72 / Therefore : = 0.0004977. F 2 0.0004977 = 0.00006794 49770 ~ 6794 U = 2.71. This value agrees quite nearly with that obtained by experi- ments with floats executed under the author's direction in 1884. It will be readily seen how extremely useful this table is in rapidly determining the discharge of conduits. TABLE I. D = inches. A = square feet. D - inches. A = square feet. 1 0.00545 26 3.6868 '2 0.02180 27 3.9760 3 0.0491 28 4.2760 4 0.0872 29 4.5868 5 0.1364 30 4.9087 6 0.1964 31 5.2413 7 0.2672 32 5.5848 8 0.3490 33 5.9394 9 0.4418 34 6.3048 10 0.5454 35 6.6812 11 0.6599 36 7.0686 12 0.7854 37 7.4665 13 0.9217 38 7.8756 14 1.0690 39 8.2957 15 1.2272 40 8.7264 16 1.3962 41 9.1682 17 1.5762 42 9.6211 18 1.7671 43 10.0847 19 1.9689 44 10.5589 20 2,1816 45 11.0440 21 2.4052 46 11.5408 22 2.6397 47 12.0479 23 2 8852 48 12.5660 24 3.1416 49 13.0951 25 3.4087 50 13.6354 WATER SUPPLY ENGINEERING. 157 TABLE II. 1000 D V Q G HP 3 0.56 0.027 729 0.003 4 0.66 0.057 1539 0.007 6 0.83 0.163 4401 0.02 8 0.99 0.344 9288 0.04 10 1.12 0.612 16524 0.07 12 1.23 0.966 26082 0.11 U 1.34 1.432 38664 0,16 16 1.44 2.010 54270 0.23 18 1.53 2.704 73008 0.31 20 1.61 3.511 94797 0.40 22 1.70 4.488 121176 0.51 24 1.77 5.561 150147 0.63 26 1.84 6.784 183168 0.77 28 1.91 8.167 220509 0.93 30 1.99 9.769 263763 1.11 32 2.06 11.505 310635 1.31 34 2.12 13.367 360909 1.52 36 2.20 15.552 419904 1.76 38 2.26 17.800 480600 2.02 40 2.32 20.247 540669 2.30 42 2.38 22.898 618246 2.60 44 2.43 25.658 692766 2.91 46 2.49 28.737 775899 3.26 48 2.54 31.918 861786 3.62 1000 D V Q G HP 3 0.79 0.039 1053 0.005 4 0.93 0.081 2187 0.010 6 1.18 0.231 6237 0.03 8 1.40 0.489 13203 0.06 10 1.59 0.868 23436 0.10 12 1.74 1.366 36882 0.16 14 1.90 2.031 54837 0.23 16 2.05 2.862 77274 0.33 18 2.16 3.817 103059 0.43 20 2.28 4.973 134271 0.56 22 2.40 6.336 171072 0.72 24 2.50 7.855 212085 0.89 26 2.60 9.586 258822 1.09 28 2.70 11.545 311715 1.31 30 2.81 13.794 372438 1.56 32 2.91 16.252 438804 1.84 34 3.00 18.915 510705 2.15 36 3.11 21.985 593595 2.49 38 3.20 25.203 (580481 2.86 40 3.28 28.625 772875 3.25 42 3.37 32.423 875421 3.68 44 3.44 36.323 980721 4.12 46 3.52 40.624 1096848 4.61 48 3.59 45.112 1218024 5.12 158 WATER SUPPLY ENGINEERING. TABLE If. (CONTINUED.) 1000 D V 3 a HP 3 0.97 0.048 1296 0.006 4 1.15 0.100 2700 0.012 6 1.44 0.282 7614 0.03 8 1.72 0.600 16200 0.07 10 195 1.065 28755 0.12 12 2.13 1.672 45144 0.19 14 232 2.480 66960 0.28 16 2.51 3.504 94608 0.40 18 2.65 4.683 126441 0.53 20 279 6.085 16429o 0.69 22 2.93 7.735 208845 0.88 24 3.0o 9H15 259605 1.09 26 3.19 11761 317547 1.33 28 3.31 14.154 382158 1.61 30 3.45 16936 457272 1.92 32 3.57 19.938 538326 2.26 34 367 23.139 624753 2.62 36 3.81 26.933 727191 305 38 3.91 30.795 831465 3.49 40 4.02 35 083 947241 3.98 42 4.12 39639 1070253 4.50 44 4.21 44.453 1200231 5.04 46 4,31 49742 1343034 5.64 48 4.40 55290 1492830 6.27 1000 D V Q a HP 3 1.12 0.055 148) 0.006 4 1.32 O.U5 3iOi 0.013 6 1.67 0.^27 8S29 0.04 8 1.98 u.691 18657 0.08 10 2.25 1.2-*) 33185 0.14 12 2.46 1.931 5213f 0.22 14 2.68 2.005 77355 0.33 16 2.90 4.048 109298 0.46 18 3.06 5.407 145989 0.61 20 3.22 7.023 189621 0.80 22 3.40 8.976 242352 1.02 24 3.54 11.123 300321 1.26 26 3.68 13.568 366336 1.54 28 3.82 16.334 441018 1.85 30 3.98 19.538 527526 2.22 32 4.12 23.010 621270 2.61 34 4.24 26.733 721791 3.03 36 4.40 31.104 8:^9808 3.53 38 4.52 35.600 961200 4.04 40 4.6t 40.493 1093311 4.59 42 4.76 45.796 1236492 5.19 44 4.86 51.317 1385559 5.82 46 4.98 57.474 1551798 6.52 48 5.08 63.835 1723545 7.24 WATER SUPPLY ENGINEERING. 159 TABLE II. (CONTINUHD.l 1000 D V Q G HP 3 1.25 0061 1647 0.007 4 148 0129 3483 0.015 6 1.86 0.365 9855 0.04 8 2.23 0.778 21006 0.09 10 2.51 1.370 36990 0.16 12 2.75 2.159 5*293 0.25 14 3.00 3207 86589 0.36 16 3.24 4.523 122121 0.51 18 3.42 6.043 163161 0.69 20 3.60 7.852 212004 0.89 22 3.79 10.006 270162 1.14 24 3.96 12442 335934 1.41 26 4.11 15.154 409158 1.72 28 4.27 18.259 492993 2.07 30 4.45 21.845 589815 2.48 32 4.61 25747 695169 2.92 34 474 29.886 806922 3.39 36 492 34779 939033 3.94 38 505 39.774 1073898 4.51 40 5.19 45.293 1222911 5.14 42 5.32 51.184 1381968 5.80 44 5.43 57.335 1548045 6.50 46 5.57 64 283 1735641 7.29 48 568 71375 1927125 8.09 1000 D V Q G HP 3 1.37 0.067 1809 0.008 4 1.62 0141 3807 0.016 6 2.04 0400 10800 0.05 8 2.44 0.852 23004 0.10 10 2.75 1.502 40554 017 12 3.01 2.363 63801 0.27 14 3.28 3.51)6 94662 0.40 16 3.55 4.956 133812 0.56 18 3.75 6.626 178902 0.75 20 3.94 8.593 v 32011 O.bS 22 4.15 10.956 295812 1.24 24 4.33 13.605 367335 1.54 26 4.50 16.592 447984 i.bs 28 4.68 20.012 540324 2.27 30 4.87 23.907 645489 2.71 32 5.04 28.148 759996 3.19 34 5.19 32.723 8835J1 3.71 36 5.39 38.102 1028754 4.32 38 5.53 43.554 1175958 4.94 40 5.68 49.c69 1338363 5.62 42 5.83 56.09- 1514430 6.36 44 5.95 62.826 1696302 7.13 46 6.10 70.400 19008' in 7.98 48 6.22 78.161 2110347 8.86 160 WATER SUPPLY ENGINEERING. TABLE II. (CONTINUED.) moo D V Q G HP 3 1.48 0.073 1971 0.008 4 1.75 0.152 4104 0.017 6 2.20 0.431 11637 0.05 8 2.62 0.914 24678 0.10 10 2.97 1.619 43713 0.18 12 3.26 2.559 69093 0.29 14 3.55 3.795 102465 0.43 16 3.84 5.361 144747 061 18 4.05 7.156 193212 0.81 20 4.26 9.295 250965 1.05 22 4.48 11.827 319329 1.34 24 4.68 14.705 397035 1.67 26 4.87 17.956 484812 2.04 28 5.05 21.594 583038 2.45 30 5.27 25.870 698490 2.93 32 5.45 30.438 821826 3.45 34 5.61 35.371 955017 4.01 36 5.82 41.142 1110834 4.67 38 5.98 47.098 1271646 5.34 40 6.14 53.578 1446606 6.08 42 6.30 60.612 1636524 6.87 44 6.43 67.894 1833138 7.70 46 6.58 75933 2050191 8.61 48 6.72 84.444 2279988 9.58 1000 D V Q G HP 3 1.58 0.077 2079 0.009 4 1.87 0.163 4401 0.019 6 2.36 0.463 12501 0.05 8 2.80 0.977 26379 0.11 10 3.18 1.733 46791 0.20 12 3.48 2.732 73764 0.31 14 3.79 4.052 109404 0.46 16 4.10 5.724 154548 0.65 18 4.33 7.651 206577 0.87 20 4.55 9.928 268056 1.13 22 4.79 12.646 341442 1.43 24 5.00 15.710 424170 1.78 26 5.20 19.172 517644 2.17 28 5.40 23.090 623430 2.62 30 5.63 27.638 746226 3.13 32 5.83 32.561 879147 3.69 34 6.00 37.830 1021410 4.29 36 6.22 43.969 1187163 4.99 38 6.39 50.328 1358856 5.71 40 6.56 57.243 1545561 6.49 4? 6.73 64.749 1748223 7.34 44 6.87 72.540 1958580 8.23 46 7.04 81.249 2193723 9.21 48 7.18 90.224 2436048 10.23 WATER SUPPLY ENGINEERING. 161 TABLE II. (CONTINUED.) 9 1000 D F Q O HP 3 1.68 0.082 2214 0.01 4 1.98 0.172 4644 0.02 6 2.50 0.490 13230 0.06 g 2.97 1.037 27999 0.12 10 3.37 1.837 49599 0.21 12 3 69 2.897 78219 33 14 4.02 4.297 116019 0.49 16 4.33 6.045 163215 0.69 18 4.59 8.111 218997 0.92 20 4.83 10 539 284553 1.20 22 5.09 13.438 362836 1.52 24 5 30 16.653 449631 1.89 26 5 52 20.352 549504 2.31 28 5.73 24.501 661527 2.78 30 5.97 29 307 791289 3.32 32 6.18 34.515 931905 3 91 34 6.36 40.100 1082700 4.55 36 6.60 46.655 1259685 5.29 38 6.78 53.400 1441800 6.06 40 6.96 60.733 1639791 6.89 42 7.14 68.694 1854738 7.79 44 729 76.975 2078325 8.73 46 7.47 86.211 2327697 9 78 48 7.62 95.753 2585331 10 86 10 1000 - I) V Q a HP 3 1.77 0.087 2349 0.01 4 2 09 0.182 4914 0.02 6 2.63 0.515 13905 0.06 8 3.16 1.103 29781 0.13 10 3.55 1.935 52245 22 12 3.89 3.054 . 82458 0.35 14 4.24 4.533 122391 0.51 16 4.58 6.394 172638 0.73 18 4.84 8.552 230904 0.97 20 5.09 11.106 299862 1.26 22 5.35 14.124 381348 1.60 24 5.60 17.595 475065 2.00 26 5.82 21.458 579366 2.43 28 6.04 25.823 697221 2.93 30 6.29 30.878 833706 3.50 32 6.51 36.358 981666 4.12 34 6.70 42.244 1140588 4.79 36 6.96 49.200 1328400 5.58 38 7.15 56.313 1520451 6.39 40 7.34 64.049 1729323 7.26 42 7.53 72.446 1956042 8.22 44 7.68 81.093 2189511 9.20 46 7.87 90.828 2452356 10.30 48 8.03 100.905 2724435 11.44 162 WATER SUPPLY ENGINEERING. TABLK III. 12 RI u* R RI u* R RI ~U* 0.05 0.00015800 1.75 0.00006863 4.0 0.00006715 0.10 0.00011200 1.80 0.00006856 4.1 0.00006712 0.15 0.00009667 1.85 0.00006849 4.2 0.00006710 0.20 0.00008900 1.90 0.00006842 43 0.00006707 0.25 0.00008440 1.95 0.00006836 4.4 0.00006705 0.30 0.00008133 2.00 0.00006830 4.5 0.00006702 0.35 0.00007914 2.05 0.00006824 4.6 0.00006700 0.40 0.00007750 2.10 0.00006819 4.7 0.00006698 0.45 0.00007622 2.15 0.00006814 4.8 0.00006696 0.50 0.00007520 2.20 0.00006809 4.9 0.00006694 0.55 0.00007437 2.25 0.00006804 5.0 0.00006692 0.60 0.00007367 2.30 0.00006800 5.1 0.00006690 0.65 0.00007308 2.35 0.00006796 5.2 0.00006689 0.70 0.00007257 2.40 0.00006792 5.3 0.00006687 0.75 0.00007213 2.45 0.00006788 5.4 0.00006685 0.80 0.00007175 2.50 0.00006784 5.5 0.00006684 0.85 0.00007141 2.55 0.00006780 5.6 0.00006682 0.90 0.00007111 2.60 0.00006777 5.7 0.00006681 0.95 0.00007084 2.65 0.00006774 5.8 0.00006679 1.00 0.00007060 2.70 0.00006770 5.9 0.00006678 1.05 0.00007038 2.75 0.00006767 6.0 0.00006677 1.10 0.00007018 2.80 0.00006764 6.1 0.00006675 1.15 0,00007000 2.85 0.00006761 6.2 0.00006874 1.20 O.\)0006983 2.90 0.00006759 6.3 0.00006673 1.25 0.00006968 3.0 0.00006753 6.4 0.00006672 1.30 0.00006954 3.1 0.00006748 6.5 0.00006671 1.35 0.00006941 3.2 0.00006744 7.0 0.00006666 1.40 0.00006929 3.3 0.00006739 7.5 0.00006661 1.45 0.00006917 3.4 0.00006735 8.0 0.00006658 1.50 0.00006907 3.5 0.00006731 8.5 0.00006654 1.55 0.00006897 3.6 0.00006728 9.0 0.00006651 1.60 0.00006888 3.7 0.00006724 9.5 0.00006648 1.65 0.00006879 3.8 0.00006721 10.0 0.00006646 1.70 0.00006871 3.9 0.00006718 INDEX. Alinement (see pipelaying, tunnels) Albuminoid ammonia 114 Arches: Abutments for, formula for calculating 143 Best form of 145 Curve of pressure, calculation of 146-149 Design, general rules governing 149-153 Failure, modes of. 145 Joint of rupture of 139 Thickness, methods of calculating 137-143 Artesian wells, general discussion of. 74, 114 Axioms of hydraulics 9 Backfilling trenches for water pipe 63 Back-pressure, loss of head from 57 Bends and elbows, loss of head by 60 Calking: Directions for 66 Lead, weight required 67 Center walls for earth dams 96 Chlorine 114 Coefficients, smoothness of pipes 17, 18, 51 Concrete, composition for hydraulic work 99 Conduits, masonry: Flow through, compared with pipe lines 124 Horse-shoe section, maximum flow of 125 Large sizes, form of section best for 125 Core walls for earth dams 96 Croton basin, Rainfall available from 77 Curve of pressure in arches 146-149 Dams: Character of, suitable to different conditions 82 Concrete masonry work for 99 Foundation beds, determining suitability of 82 Foundation pits, drainage of 104 Location, conditions governing 82 Dams, earth: Centre walls for 96 Discharge outlets for 98, 122 Embankments for 96, 98 Spillways for 97 Dams, masonry: Base, construction of 95 Classes of 82 Design of high, Example illustrating 88-93, 122 164 INDEX. Formulas for, unreliability of 93 Equilibrium of. Rectangular sections 82 Trapezoidal sections 83 Hydrostatic pressure of water against 119 Line of pressure, location of 86 Plan, conditions governing selection of 94 Overturning, stability against 82-86 Pressure on foundation masonry, safe amount 87, 121 Sliding on base, stability against 117-119 Stone masonry work for , 100 Stability, Vauban's principle 84 Diameter (see pipes, pipe lines, pipe line systems) Discharge outlets for earthen dams 98, 122 Discharge (see pipes, pipe lines, pipe line systems) Drainage of foundation pits 104 Duty trials (see pumping engines) Embankments for earth dams 96, 98 Entry, resistance to Head, loss of due to 66 Short horizontal pipes 12 Evaporation: From water surfaces . . 116 Sources, principal 72 Fifth powers, Tables of 48, 110 Fifth roots, calculation, use of logarithms in 28 Filters, mechanical, Action, method of 128 Cost of operation 128 Filtration, general discussion of 127 Fire service, quantity of water required for 116 Flow: Friction, loss of head from 56 Horse shoe conduits, maximum through 125 Pipe lines, branched, Branches variously placed, effect on 31-38 Calculation, example illustrating , . . 28, 106 Pipe lines, Calculation, example illustrating 19, 21 Fundamental equations for 18, 106 Varying diameters, calculation of 22 Pipe line system, Calculation, abbreviated method 49, 107 Calculation, formulas for 42, 106 Communicating with two reservoirs 51 Through open channels 123 Velocity of, Comparative through masonry conduits and smooth pipes.... 124 INDEX. 165 Grade of pipe lines, effect on 14 Length of pipe effect on 12 Smoothness of pipe, effect on 17 Short horizontal pipes 11 Vertical pipes 59 Formulas: Abutments for arches 143 Diameter of pipe carrying 100 gallons per capita in ten hours. . . 69 Dimensioning spillways 81 Duty of pumping engines 131 Equilibrium of masonry dams 82 Flow in pipe line system 41, 42, 106 Flow through long pipes 18, 106 Flow through open channels 123 Horse power for pumping 69, 112 Safe pressure on bottom courses of masonry dams 87, 121 Spillways, approximate for determining 81 Storage reservoirs, capacity of 117 Thickness of arch keys 138 Velocity of falling bodies 11 Weight of cast iron pipe 168 Weight of lead required for calking 67 Foundation blocks for water pipe 65 Foundations: Dams, testing suitability of 82 Drainage of ; 104 Friction in pipes, head required to overcome 12, 56 Grade (see pipes, pipe lines, pipe line systems, pipelaying, flow) Head: Definition of Horizontal pipe lines 11 Pipe lines with varying grades 13 Height of, Friction in pipes, amount required to overcome 12 Resistance to entry, amount required to overcome 12 Losses due to, Back pressure , 57 Bends and elbows 60 Changes in diameter of pipe 59 Friction of flow 56 Resistance to entry 56 Velocity, height required to produce 56 Loss of Definition of term 55 Hydraulic grade line, effect on 63 Pipe line systems, calculation of 56 Practical importance of 63 166 INDEX. Pressure in pipe lines due to 14 Heart walls for earth dams 96 Hydraulic grade line: Correct determination, importance of 14 Definition of term 9 Determination for pipe lines with varying grades 18 Head, changes caused by loss of 63 Pipe lines, calculation for 21 Pressure in pipe lines, relation to 25 Varying steepness, effect on flow 14 Hydraulic pressure, definition of term 14 Hydraulics, axiomatic truths of 9 Hydrostatic pressure: Against plane surfaces 119 Definition of term 14 Impurities, mineral, different kinds of 114 Joint of rupture, definition of 139 Line of pressure, in high masonry dams '. 86 Masonry dams (see dams, masonry) Masonry, stone, rules for constructing 100 Mechanical filters (see filters, mechanical) Mineral impurities in water. 114 Nitrates and nitrites 114 Piezometric head: Branched pipe lines, calculation for 28, 106 Calculation of, example illustrating 21 Pipe line system, calculation for 49, 107 Pipe line system, communicating with two reservoirs, calcu- lation for 52 Piezometric height, definition of term 10 Piezometric tubes, definition of 14 Pipe, diameter of, having capacity of 100 gallons per capita in ten hours, formula for , 69 Pipelaying: Alinement, methods of 65 Backfilling trench 66 Calking, directions for 66 Calking, weight of lead required 67 Foundation blocks for 65 General discussion 65 Grades, method of preserving 65 Pipe lines: Diameter, Calculation, example illustrating 20 Changes in, loss of head from 59 Flow, Calculation of, example illustrating 19 Fundamental equations for 18, 106 Smoothness of pipe, effect on 17 INDEX. 167 Varying diameter of pipe effect of 22 Friction in, head required to overcome 12, 56 Head of (see head) Hydraulic grade line Calculation, example illustrating 21 Determination for varying grades 13 Length, measurement adapted for calculations 19 Location with respect to hydraulic grade line 14 Piezometric head, calculation of, example illustrating.. 21 Pressure in, General discussion 14 Relation to hydraulic grade line 25 Uniform diameter equivalent to compound system, calcula- tion of 26, 27 Pipe lines, branched, Flow, Branches variously placed 31-38 Calculation of example illustrating 28, 106 Pipe line system: Communicating with two reservoirs 51 Diameter of pipes for, calculation of 39 Flow, Calculation, abbreviated method 49, 107 Formulas for calculating '. 42, 47, 106, 107 Friction in 12, 56 Head, loss of, calculation of 56 Piezometric head, calculation, abbreviated method 49, 107 Pipes: Cast iron, tables of weight and thickness 68 Friction in 12, 56 Long horizontal, velocity of flow in 12 Short horizontal Discharge through, velocity of 11 Resistance of entry to 12 Smoothness, coefficients of 17 Vertical, flow through 59 Pressure (see pipes, pipe lines, pipe line systems) Pumping horse power required, formula for 69, 112 Pumping engines: Classes, description of different 129 Class, selection of, conditions governing 129 Duty of, formulas for 131 Duty trials, Definition of 130 Methods of conducting 130-137 High duty, proper field of use 130 Types of, merits of different 137 Pumps (see pumping engines) Purity (see water supply) 168 INDEX. Rainfall: Amount available from Croton Basin 77 Methods of estimating 77 Per square mile of drainage area 78 Reservoirs, Storage: Capacity required, conditions governing 78 Spillways for, approximate formulas 81 Smoothness (see pipes, pipe lines, pipe line systems) Spillways: Dams, earth 97 Formulas, approximate, for 81 Springs, quality of water supply from 75 Storage, capacity required: Calculation of, example illustrating 78 Formula for calculating 117 Stone masonry, rules regarding construction 100 Streams: Relative purity of large and small 72 Yield, method of estimatng 77 Tables: Cast iron pipe, weight and thickness of 68 Coefficients of smoothness of pipes 18 Explanation of 154 Fifth powers of pipe diameters 48 Tubes, piezometric, definition of 14 Tunnels: Alinement, possible accuracy of 126 Construction of 125 Velocity (see flow, head) Water supply: Artesian wells for, general discussion 74, 114 Evaporation, amount of 72, 116 Filtration of 127 Impurities, common mineral 114 Quality of: Chemical analysis inadequate to determine 71, 113 General criterion of 71 General discussion on 71-76 Large and small streams 72 Springs, general purity of 75 Wells, impurity of 74 Quantity required: Fire service 116 Future growth in population, provisions for 76 Per capita per 24 hours 76 Sources, classification of 72 Storage for, example illustrating 78 Wells: Artesian, general discussion 74, 114 Driven, quality of supply from 74 UNIVERSITY OP CALIFORNIA LIBRARY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW 31 <\ REC'D LD FEE -v- m 15 70 -5PM 30rn-6 ( '14