UC-NRLF PRACTICAL HYDRAULIC FOR THE Distribution of Water Through Long Pipes, E. SHERMAN GOULD, O >- OS 4 - *> REESE LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Accessions No. -4. 3 3. 2-0 Shelf No. PRACTICAL HYDRAULIC FORMULE FOR THE Distribution of Water Through Long Pipes. E. SHERMAN GOULD, M. AM. SOC. < . K. Consult-in (r Engineer to the Scranton Gas and Water Co. XE\V YORK : ENGINEERING NEWS PUBLISHING CO. C COPYRIGHTED, 1889, ENGINEERING NEWS PUBLISHING CO. ^ *> .5 a-a INTRODUCTION. 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, pro- vided that the pipe lies wholly below such straight line, and its de. clivity 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 of dia- meter or in the nature of its inside surface ; or if there be increase or diminution of the volume of water entering it at its upper ex- tremity 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 INTRODUCTION. 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 hydraulic grade line, constitutes the Piezometric height, and measures 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 establish the piezometric height for every point of change of any kind which occurs through- out the entire length of the conduit. The joining of the upper ex- tremities 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 furnish solutions for a large number of practical problems, commencing with the simplest cases and extending to some rather intricate ones, not usually embraced in our hydraulic manuals. E. 8. G. SCRANTON, Pa., May, 1889. TABLE OF CON TENTS. INTRODUCTION. CHAPTER I. Flow through a short horizontal pipe Effect on velocity of increased length Fractional head Hydraulic grade line Hydrostatic and hy- draulic pressures Piezometric tubes Result of raising a pipe line above the hvdraulic 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- ficientsFundamental equations Length of a pipe line usually deter- mined by its horizontal projection Numerical examples of simple and compound systems. 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 regard- ing results obtained by calculation Numerical examples. 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 formula Table of 5th powers Preponderating influence of diameter over grade illustrated by example. CHAPTER IV. Use of formula 14 illustrated by numerical example of compound sys- tem combined with branches Comparison of results Rough and smooth pipes Pipes communicating with three reservoirs N umerical examples under varying conditions Loss of head from other causes than friction Velocity, entrance and exit heads 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. E. Sherman Gould, M. Am. Soc. C. E. Consulting and Constructing Engineer for Water-Works. SCRANTON, PA. HYDRAULIC FORMULyE. CHAPTER I. Flow through a Short Horizontal Pipe' Effect on Velocity of Increased Length Frictional Head Hydraulic Grade Lme Hydrostatic and Hydraulic Preswire* Piezometric Tubes Result of liaising 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 Influence of Inside Surface of Fipes upon Velocity of FlowDarcy's Coefficients Fundamental Equations Length of a Pipe Line usually Determined by its Horizontal Projection Numerical Ex- amples <>f Simple and Compound Systems. Let us suppose a reservoir of large relative area and capacity 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 velocity of discharge is equal only to that theoretically due to a height of about two-thirds of H, that is : 8 PRACTICAL HYDRAULIC FORMULAE. The remaining third of the height is consumed in overcoming the resistance offered to entry by the edges of the orifice to the in- flowing vein of water. The head necessary to overcome the resis- tance to entry is, therefore, about one-half of that necessary 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 inversely as the square root of the length. It would correspond, therefore, lo a smaller and smaller percentage of the total head H. The resis- tance 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 calculated 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 H ? It is consumed in overcoming the frictional resistances en- gendered by contact of the moving 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 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 H is supposed to be available for overcoming the frictional resistances. As this occasions, however, an error al- though generally a very small one in the wrong direction, judg- ment is required in exercising this latitude, Later on we will re- vert to this point, but for the present, we will consider only fric- PRACTICAL HYDRAULIC FORMULAE. 9 tional 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 maintained 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 fluctuations, and liable at times to be greatly drawn down, it is customary to consider the surface of the water as standing at its lowest possible level,i. e., the mouth of the pipe. In Ibis 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 a b joining the centers of the two ends would form what i called the hydraulic grade line, the estab- lishing of which is the first step to be taken in laying out a system or water supply. 10 PRACTICAL HYDRAULIC FORMULAK. 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 gates equal to the head 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 ex- tremities, less a certain amount depending upon the volume dis- charged, which will be spoken of hereafter. At rf, 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. This 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 tubes. The importance of correctly establishing the hydraulic grade line is illustrated by reference to a case such as is shown in Fig. 2, in which ihe pipe, at the point c. rises above the grade line a b, 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 a b is divided into two parts- PRACTICAL HYDRAULIC FORMULAE. 11 the one a c with a hydraulic grade line flatter than a b and the other c b with one steeper than a b. The delivery of the entire system, if the pipe were of the same diameter throughout, would be governed by the flatter portion a c, and the portion c b would be capable, in virtue of its steeper slope, of discharging a greater volume of water than it could receive from a c. 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 syphonage, 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 dia- meter of the pipe between a and c, so that it would deliver the re- 12 PRACTICAL HYDRAULIC FORMULAE. 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 hydraulic grade line ab, the pipe would run full (provided the feed were suffi- cient) 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 cut the pipe off at a comparatively short length, or extended it indefi- nitely. 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. PRACTICAL HYDRAULIC FORMULAE. 13 Suppose the pipe to be just sufficiently large to furnish a cer- tain 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 increase the diameter of the pipe, it would be necessary to in- crease the steepness of pitch of the grade line, in other words, to increase 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 in same conditions under a different form, suppose, as before, the pipe to be just large enough to supply ttie 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. This case shows why the water ceases to rise in the upper stories of the houses of a town when the consumption increases. 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, coeffici- ents have been established for different conditions of surface. It na,s also been found that these coefficients vary slightly with the diameter of the pipe, a pipe of a certain size giving a greater veloc- ity than one of the same character of inside surface but of smaller diameter, the differences becoming smaller as the diameters in- crease. The value of this coefficient, which will be designated through- out this paper by C, is given below for a number of different dia- meters 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- 14 PRACTICAL HYDRAULIC FORMULAE. 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 experiment- ally 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 refine- ment in establishing its value, because not only is it difficult to determine this value with exactness for a given diameter and con- dition of pipe, but this condition, and even the diameter of the pipe, is liable to undergo considerable variation in the same pipe in the course of a few years. Diameter in inches. 3 4 6 8 10 12 14 16 24 30 36 48 TABLE OF COEFFICIENTS. Value of C for rough pipes. 0.00080 OOOT6 0.00072 0.00068 0.00066 0.00066 0.00065 0.00064 0.00064 000063 0.00062 0.00062 Value of C for smooth pipes. 0.00040 0.00038 0.00036 0.00034 0.00033 0.00033 0.000325 0.00032 0.00032 0.000315 0.00031 0.00031 In all the following calculations, the coefficient for rough pipes will be used. PRACTICAL HYDRAULIC FORMULAE. 15 The two fundamental equations relating to the flow of water through long pipes are : D x H = C F a (1) L Q = A V (2; Equation No. 2 will generally be written : /D x H Q = A I/- (3) V C x L by taking the value of V from (1). The first of these has been established by DARCY ; the second is based upon a self-evident proposition. In these equations : D = diameter of pipe in feet H = total head " " L = length of pipe " " C = coefficient V = mean velocity in feet per second Q = disc harge in cubic feet per second A = area of pipe in square feet = D 2 x 0.785 The above two formulae solve, directly or indirectly, all prob- lems relating to the flow through long pipes, and all such prob- lems must 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 length L of pipe, and is therefore the natural sine of the inclination of the slope to the horizon. This relation is frequently used under the H form / = . Using this notation, (1) would be written : L D I = C V? 16 PRACTICAL HYDRAULIC FORMULAE. 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 some 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 (1) we have : 1 x 10 = 0.00066 V 2 1,000 V 3.89 ft. per second. Using this value of Fin (2), we have ; = 0.785 X 3.89 Q = 3.055 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 2 L X G~ ~A* PRACTICAL HYDRAULIC FORMULAE. 17 Observing that A = D* 0.785; D x H 5 whence, suppressing the common factor; C L d h C 2 Z a C 3 ^3 4 ?v - = -- 1 --- 1 --- 1 --- (5) D" di s d 2 5 d 3 s d 4 B The above is the general formula. Substituting the special values of our example : 3300 0.33 0.52 0.96fi 0.432 - X (7= -- 1 --- 1 --- 1- - IP 1 K (4 3) 5 (2 3, s (1-2J B Giving a preliminary approximate value to C of 0.00066, we have 2-178 = 0-33 + 0.123 + 7.335 H 13.824 = 0.1007 = 0.63 This value of D indicates a practical diameter of Sins. in order to check this value, we mav write (4) under the form : Q = |/ J> X If V 0.616 7, X C PRACTICAL HYDRAULIC FORMULAE. 25 Substituting given values : 0.1007 X 50 X 0.61C V ~ y " 2.178 ^ = 1.193 cu . ft. per second. thus proving the correctness of the work. These calculations can be abridged, and, in many cases, suffi- cient accuracy secured by adopting a mean common value for C. If we do so in the present case, C becomes a common factor, and disappears from the calculation, (5) becoming L it h h = 1- 4- etc. (5) bis D* d," dj V 1000 X 0.00*62 +3 14 500 X 0-00064 * With these given lengths and diameters, the above system does not propetly come under the classification of "lone 1 pipes." As the present object i only to exemplify methods of calculation, the example is equally good. PRACTICAL HYDRAULIC FORMULAE. 27 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 fac- tors Tifor and 55^, wr have simply: 4 >/50 h = 4 Vh. + \/h 4 Squaring: 00 16 h = 16 h + h 4 + 8 V h 9 4h which gives ; 33/< =804 8 v/ft 2 4/i Neglecting, for a first approximate value of h the quantities affected by the radical : 33ft = 804 Neglecting decimals : ft = 24. Substituting this value for h under the radical : 33ft = 804 8 V 576 96 which gives, always neglecting decimals, a' second approximate value : ft = 19. A third and fourth approximation give respectively li = 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 : - 12 4 ' 29 - 9 - 12.56 j/ _ * 0.6-2 0.62 =lt -,, 5 We have thus; 28 PRACTICAL HYDRAULIC FORMULAE. The above 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 at once that h 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 : / 4 X 2: = 12.56 /*/ - r n fi2 22 149.60 62 '2X18 0.32 3. 14 1/ =33.30 also, for the equal discharge through the 48-in. pipe above the branch, squaring (3), we have : (182.90) 2 X0.62 h = = 32.87 (12-56) 2 X4 This height, added to 22, the assumed value of h, gives a total height of 54.87 ft. as against 50 ft., the actual total height. By pro- portion we have : h so 22 54.87 This value of h agrees with that already found. If the24-in. branch were closed, we should have for the dis- charge : /4X60 Q = 12.56 \/ = 159.51 f 1.24 PRACTICAL HYDRAULIC FORMULAE. 29 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 9i 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 / 4X33 if = 149.; ' i son v n finnfta q = 12.56 47 = 149.5 1500 X 0.00062 / 2 X 29 0.32 g' =3.14 4 = 42.3 " 30! (191.8)* X 0.31 alSO h' = = 18.07 (12. 56)* X 4 giving a total height of 51.07 as against 50. Reducing : h 50 33 51.07 h. =--32. 3 Using this value, instead of the assumed one, we have : 4X177 4X 32.3 -j - 28.3 12.564 = 12.564 h 3.14 4 0.31 0.93 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 : 4 X 10 32.564 =142.46 POO X 0.00062 fj' =3.14 4 =19.23 0.32 30 PRACTICAL HYDRAULIC FORMULAE. (161- 7) 2 X 0.93 also: h' = =3853 (12.56) 2 X4 h 5.i By proportion 10 4 8 53 // = 10.30 Using this value instead of the assumed one : / 4X39.7 / 4X10.3 / 2 X 6. 12.56.4 -- =12.56 4/ --- h 3.14 A/ - 0.93 0.32 0.32 164.13 =144-57 + 19.68 very nearly. As compared with the discharge when the 24-in. branch is closed this shows a gain of not quite 3 per cent., which is in marked con- trast to the gain when the branch was only 500 ft. from the reser- voir, being less than one-sixth of the gain, in that case. It will be interesting to study a little more in detail the question of relative discharges. We have seen that when there is no branch open on the 48-in. pipe, its discharge is 159.51 cu. ft. per second, The 24-in. branches, wherever placed, increase the total discharge, but diminish that in the 48-in. pipe, below the branch. By com- paring the above quantities, it will be perceived that the flow from the 48-in. pipe is diminished approximately by that proportion of the quantity flowing through the 24-in. branch which is represented by its proportionate distance from the reservoir. Thus, when the branch is 1,500 ft., or three-quarters of the length of the 48-in. pipe from the reservoir, as in the last case, its discharge is 19.62 cu. ft. per second. Three-quarters of this quantity is 14.715, which, sub- tracced from 159.51, leaves 144.795, or very nearly that of the 48-in - pipe below the branch, as determined by calculation. In the same way half of the discharge, when the branch is situ- ated half way from the reservoir, subtracted from 159.51, gives also very nearly the amount discharged below the branch. When the PRACTICAL HYDRAULIC FORMULAE. 31 branch is 500 ft., or one-quarter of the total distance from the reser- voir, one quarter of its discharge taken from 159.51 gives very closely the discharge as calculated for the 48-in. pipe below the branch. Let us now take an extreme position for the branch, and sup- pose it placed close to the reservoir, so that there is practically no portion of the 48-in pipe between it and the reservoir. There will, therefore, be no part of the flow from the branch subtracted from that of the main oipe, and the two will each discharge the same quantity as if the other were not there. That is, the 48-in pipe will discharge 159.51, and the 24-in. 53.24 cu. ft. per second. If we should take another extreme position for the branch, and suppose it placed at the end of the 48 in pipe.it is obvious that, with its assumed rising grade of 4-ft. in 500. it would discharge no water at all. A position could be found by trial where it would just cease to discharge water, but for the object of the present in- vestigation this is not necessary. Pig. 8. If the above results are plotted, as in Fig. 8, a very instructive diagram is obtained. The successive 500 ft. lengths being laid off 32 PRACTICAL HYDRAULIC FORMULAK. as abscissae, and the discharges measured upon the corresponding ordinates, it will be seen that their extremities all lie nearly in the same straight line. If, therefore, the discharges for any two posi- tions of the branch be calculated, and a straight line drawn pass- ing through their extremities, the discharge for any other position of the branch can be obtained by erecting an ordinate at the given point to the straight line, and the flow through the main also ob- tained by subtracting the proper portion of that of the branch. In practice, when making calculations similar to those under consideration, one error must be carefully guarded against namely, the supposing that the actual results will be exactly as calculated. The chief value of these calculations lies in the fact that, they furnish pretty trustworthy relative results, that is, they establish fairly well in practice the fact that if a certain pipe de- livers a certain volume of water in a certain position, it will de- liver a certain greater or less amount in another. The actual amounts, in either case, cannot be surely determined, as they de- pend upon so many varying circumstances about which, even when aware of their existence, we have no exact date. Let us next suppose a system in which the48-in. pipe is tapped every 500 ft, by a 24-in. pipe, 500 ft. long, laid as before with a grade of 4 ft. in 500.- Assuming a height of 9 ft. for the piezometric column h nearest the tree end of the pipe we have : A X 9 . , /2 X 5 12.664 /< y + 8.144/' = 12,56 4/4 (h- 9) \ 0.31 V 0.32 ^3T~ Since the denominators under the radicals are so nearly equal we may cancel them, and making other simplifications, write : V 9 + 7 r 10 = Whence: /* H'C These relations indicate that, other things being equal, the squares of the discharges vary directly as the heads and the fifth powers of the diameters, and inversely as the lengths; and that, other things being equal, the fifth powers of the diameters vary directly as the squares of the discharges and the lengths, and in- versely as the heads. As these relations are generally used for approximations, the coefficients may be dropped, and the equations written in this form : (7) (8) -/*^ PRACTICAL HYDRAULIC FORMULAE. ~ 8 " = rHr -/ Other combinations can be made from these relations. Thus : (12) Commencing now with the west side of the main HH', we have cu. ft. to be delivered at an elevation of (160) above datum. As the pipe will be a comparatively small one, we will assume a grade of y^, which will give a rise of 16 ft. between the extremity and the main junction, and requires an elevation of piezometric head, at this junction, of (176), as shown in the figure To obtain the proper diameter of pipe for this grade and dis- charge, we have, using (4), and assuming C = 0.00076 as a probable value: U) 2 X1000 < 0.00076 1)6 = 8 X 0.61 whence D 5 = 017304 and D = 0.444. 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 obtained by means of (11). Thus: D-= 4 _L^_L^I V 36 If = 0.3777 or, for next highest even inch : D' = 5 inches. 40 PRACTICAL HYDRAULIC FORMULAE. 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 some- what greater than it should be. If the limit of velocity is over- stepped 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 dowja 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 tliis 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 on the supposition of a uniform diameter, capable of delivering the entire volume of | cu. ft. per second as far Assuming a probable value of G = 0.00066, we have from (4) : J6 X 1.32 6.1 whence : y* = o.s47 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 difference being the quantity it is desired to deliver, which is now J cu. ft. as against $ in F F'. 1 he relation (9) is therefore applicable, and we have : 7, - / ! V ~ A / 0.017304 X r 16 i whence : j/ ^ 0.0097335 PRACTICAL HYDRAULIC FORMULAE. 41 and D- ^ 0.395 or, say, D' = 5 ins. The mains on the east side are determined as before : 5 16 " ~ I/ 0.00973*5 X 31 Z)' = 0.346 This is not quite 4^ ins. but to ensure 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 to B, we find two grades, the upper one y^ and the lower !&g a . The lower section must deliver, under a grade of T g(j, 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 : n r, = * X 0.66 6.1 whence : z>~ = o. 4028 and : D = o.846 This is very nearly 10] ins., and a 10 in., pipe would answer, though 12 ins. would be better. The upper section must deliver 2.5 cu. ft. per second, under a grade of TT /fo. Taking the same probable value of C, we have : ^3 __ 6.25 X 0.66 2.05 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 5 4 D' = 1.287 This last formula might have been used throughout, but (4) is abjut as short and convenient; frequently more so. 42 PRACTICAL HYDRAULIC FORMULAE. 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. If we adopt a grade of TTJ %^, the proper diameter of the pipe would be : 9X 0.65 b* = 2.44 D = 1.32 or: 1) = 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 danger- 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 ai^d 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 discharge 1-6 cubic ft. with a grade of VO S 00 . This would require a 4-in pipe. The draught from these would somewhat lower the piezometric PRACTICAL HYDRAULIC FORMULAE. 43 heads at their junctions with the side mains. Ill a fine calcula- tion, 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, as was done in the present instance, and where a liberal interpretation has been given to the calcula- tion 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 con- sidered 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 con- venient form for rapid approximations. To do this, we have only to calculate the discharge of a pipe 1 ft. in diameter, with a fail of 1 ft. per thousand, and to refer all other discharges with the fall per thousand feet to it, in order to obtain the corresponding diam- eter. The quantity discharged by the above pipe is 0.961 cu. ft. per second, and the square of the same is 0.924. Equation (12) may then be written : or very nearly : D=|/ - (13) we have also very nearly : Q, = v ^ x H iu) 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 44 PRACTICAL HYDRAULIC FORMULAE. pipes, we have oiily to halve the co efficient for a 12-in. pipe, which will be equivalent to writing the above formulas thus : 5 /Q 2 D {/ - (15) > Iff Q,= V D* x 1H (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 has been calculated. This table indicates, by inspection, the diam- eters in inches corresponding to the fifth roots of the right-hand side of the equations, expressed in feet. Diameters in inches. Fifth Powers in feet. Diameters in ins. Fijth Powers in feet 3 0.000977 22 20.72 4 0.004115 24 32.00 5 0.01256 26 47.75 6 0.03125 28 69.17 8 0.1317 bO 97.66 10 0.4019 32 134.9 12 1.0000 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 1024.0 All the diameters which have been already calculated can be ob- tained very nearly by the use of (13). Relations (13) and (14) might also have been used in some of the previous examples. Formulas (13) and (14) serve to show the comparatively small in- fluence 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 formulas, that for a diameter of 1 ft. and a fall of 1Ty \ )0 , 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 ^oo, i- e., we must quadruple the fall. If, on the other hand, we wish to produce the same result by increasing the diam- PRACTICAL HYDRAULIC FORMULAE. 45 eter without changing the grade, we Deed only adopt a diameter of 1.32 ft. and even a little less, on account of the decrease in the co- efficient. That is to say, to double the discharge, we must increase the fall 300 per cent., or the diameter 32 per cent. NOTE. In completion of what has been already said in this chapter (pge 37), regarding the limit of velocities for pipes 01 different diameters, the follow- ing table (founded upon that given by Mr. Fanning) indicates pretty closely the maximum velocities which it is generally advisable to produce: Diameter in inches, 6 12 18 24 30 36 42 48 Velocity in ft. per sec., 2.5 3.5 4.5 5-5 6.5 7.5 8.5 9.5 CHAPTER IV. Use of formula U illustrated l>\i numerical examvle of compound system com- bined with branches Comparison of results Rough and smooth pipes Hpes communicating with three reservoirs Numerical examples under varying con- ditions Loss of head from other causes than friction Velocity, entrance and exit heads Numerical examples and general formulce Downward discharge through a vertical pipe, Other minor losses of head Abrupt, changes of dia- meter Partially opened valve Branches and bends Certrifugal force Small importance, of all losses of head except frictional in the case of long pipes- All suclt, covered by "even inches, 1 " 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 horizon- tal 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-in. Each of these three lengths of pipe are themselves tapped midway by a 6-in. pipe, laid horizontally, the one nearest the reservoir having a length of 3,000ft; the next, 1,000 ft., and the last, 500ft. (See Fig. 9, bis) All the pipes being open, it is desired to find the piezometric heads /<, h', U", h" f , h"", 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 assuminging 6 ft. as an approxi- mate value of h, we have from (14), H always representing the fall per 1,000. V ~6~4r ^ = V k' _G h' = 15 36 ^936 = * 32 (k" 15.36) k" = 15.65 1)36 + 15 3 i 5 = 32 ses h'" = 16.09 = 30.17 *' 14 08 + Vl %r = v ' h - 30.17 h = 48.82 PRACTICAL HYDRAULIC FORMULAE. 47 Comparing this value with the given height 50, we may increase all the preceding values of h, h', etc., in the proportion of 4 i^- 2 . But in practice we would not wish to reckon on the total 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"" f , and those discharged by the branches, beginning also at the lower end, q, q' y q" respectively, using both (3) and (14). The results given by (14) naturally check exactly, since they depend directly upon the method used in deter- mining h, h', etc. By (3) By (14) Q = 2 39 2.45 Q = .56 .61 Q _|_ q = 2.95 3.06 Q = 2/96 3.06 Q = 2.99 3.05 Q' = .63 70 + = 79.90 Giving: Q = c.in Q' = 3.065 Q" = 2.816 a close approximation ; the true value of h lies between 79.85 and 79.90. As /i increases with the diameter of the pipe B D, it might at first seem as though, by indefinitely increasing the diameter, h 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 towards 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 commu- nication 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 reser- voir B. But the effect will be different if the junction D be sufficiently advanced towards 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 B were shut off, PRACTICAL HYDRAULIC FORMULAE. 51 the piezometric height at D would be 87.5 ft. There would there- fore 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 draught towards B would lower it. Fig. 11. To ascertain the true value of h at the point D, we have the relation : ''Too /< 2,500 simplifying loo- h / h - v /h- + I/ 47 h = 4060 11.86 V 7i 2 80 h whence, by successive approximations : h = 82.65 Using this value of h we get : Q = 5.695 Q' = 4.698 Q" = .995 When B is shut off, in the above system, the discharge from A to Cis 4.83 cu. ft. per second. In all that precede?, 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 52 PRACTICAL HYDRAULIC FORMULAE. 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 wa- ter 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 reference 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 abbreviation. When we speak, for instance, of " the loss of head due to velocity," we mean the head, or fail, theoretically necessary to 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 resistance, 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 direction in its flow require a cer- tain 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 circumstances. The property of water which causes these resistances is called its viscosity. As applied to long pipes, the principal " loss of head," and the only one hitherto considered, is the frialional. The term thus ap- plied means the height or pressure necessary to overcome the fric- fion of the water passing with a given velocity through a pipe of PRACTICAL HYDRAULIC FORMULAE. 53 given 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 following problem: Two reservoirs (Fig. 12) containing still water and hav- ing a difference of level of 30 ft., are joined by a pipe 12 ins. in diam- eter and 3000 ft. long. What is the velocity of discharge 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 ve- locity, and that due 10 entrance. If the pipe discharged freely 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 surface 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 constituted what may be called the backpressure of the reservoir. In solving this problem, let us first, as heretofore, neglect all 54 PRACTICAL HYDRAULIC FORMULAE. losses except frictional ones. We have then, from (1), using the above data, and the coefficient for rough pipes : i = 0.00066 F a 100 V 2 = 15.15 V = 3 89 ft. per second The head theoretically necessary to produce this velocity is F* given by the formula derived from the law of falling bodies, h 2<7 by substitution of the above value V . Thus : 15.15 h= 64.4 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 velo- city head. We have then : ft ft H = 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 to produce the same. We have therefore for the total losses, outside of friction : h h+ h ft = 0.588 2 And the head available for overcoming friction becomes 30 0.588 = 29.412 We must now recast our original calculation, using 29.4 ft. in- stead of 30, as available frictional head. Thus ; 29.4 = 0.00066 F 2 3000 F 2 = 14.8 F = 3.85 This is a very small reduction from the velocity already ob- PRACTICAL HYDRAULIC FORMULAE. 55 tained. But, in order to see how our previous solution is affected by the change, we will work on new values for the sub-heads. Thus: 14.8 h = 64.4 h = 0-23 ft h -\ \-h = 0.575 2 300.575 = 29.425 leaving the previous value 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 F the actual mean velocity, that is the actual volume discharged divided by the area of the pipe (3), we have, in the case of discharge betweed two reservoirs, as shown in Fig. 12, the following subheads, which together make up the total head H: F 2 F 2 F 2 L C F 2 a = -4- - + -+ - 20 40 20 D 5F 2 LC F 2 40 D H = 0.039 F 2 + L C V- D That is to say, by using (3) which gives, H = L G F 2 D we make the error of omitting a distance 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 : F 2 v L C F 2 // =-+-+- 20 40 D H = 0-233 F 2 + L C V s 1) 56 PRACTICAL HYDRAULIC FORMULAE. In this case we make the still smaller error of omitting 2 % of V*. In all eases, having obtained V^ by means of (1), we can easily judge from the nature of the problem whether it is necessary 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 Dext highest even inch will almost always amply suffice to cover all omissions. As has been already stated, in all ordinary circumstances of pipe laying, the horizontal measurement of the pipe is taken in- stead of its actual length. It is only in special cases that this can- not be done The extreme limit occurs in the case of a vertical pipe discharging from the bottom of a reservoir. This constitutes a very interesting special case, for should the reservoir be of indefi- nitely large area but of relatively shallow depth, the relation H tends towards unity as L. and consequently H increases. The L 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, 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 PRACTICAL HYDRAULIC FORMULAE. 57 much weight should not be attached to the formulae given for their determination. A brief space will be devoted to their considera- tion, more with a view to make the present paper complete than for any practical value which they possess. When water passes through a pipe of which the diameter is ab- ruptly 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 con- nection between the two. This greatly diminishes the agitation 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 hy- draulic grade line: and it will be well, in any event, to give for- mulae 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 Memoire, ninth edition. First. When the change is from one pipe to another of smaller diameter, we have : y 2 // = 0.49 20 whence : /< = O.OOOTG V- V being the velocity of the water in the smaller pipe. We have seen, by examples previously given, how thU velocity may be ob- tained. Fig. 13 Second. If the water (Fig. 13), in its passage from the greater 58 PRACTICAL HYDRAULIC FORMULAE. to the smaller pipe, passes through an opening in a thin diaphragm, as in the case of a partially opened stop-cook, we have : V 2 f 8 \ 2 = ( l I 2 Q \0.62 S' / 20 in which V is the velocity in B, S the area of cross-section of B, and S', the area of the opening in the diaphragm. Third. When the flow is from one pipe to another of larger diameter : _(v- vr- 20 in which V = 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 : F 2 c, Fig. 14 Another loss of head is that due to branches (Fig. 14). In this case the water flowing from A with a velocity F, is split at the junction, part passing on towards B, with a reduced velocity F', and part entering the branch and flowing towards C, with the ve- locity V". The loss of head occasioned by perturbations of the wa- ter at the junction has not been satisfactorily investigated. When PRACTICAL HYDRAULIC FORMULAE. 59 the branch leaves the main at a right angle, this loss, as deter- mined by a few incomplete experiments, is : 3 F' 2 h= 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 respectively, we have : F- For bends, or elbows, Navier's formula for loss of head is : F* / \ A h = I 0.0128 + 0.0186 R I 2flf V ' R in which V = velocity of flow, E = 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 readilv oe 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 'or the centrifugal force Fis : M F 2 R in which M= the mass of the liquid in motion, V = its velocity , and R = the radius of the bend measured on its axis. 60 PRACTICAL HYDRAULIC FORMULAE. As an illustration, we will suppose a pipe 24 ins. in diameter, through which the water flows with the velocity of 8 ft. per second, around a bend of 8 ft. radius. The mass of the liquid in motion is its weight divided by g. The centrifugal force, therefore, per running foot is : 3.14 X 62.5 8 2 F = X 32.2 8 F = 48.72 Ibs. 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 ve- locity, and if the bend is not well abutted, might tend to draw the joints. JB'ig. 15. Fig. 15 shows the manner in which such losses of head as we have been just considering, modify 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 PRACTICAL HYDRAULIC FORMULAE. 61 losses of head due to velocity and changes of diameter. It will be understood, of course, that this is a mere random sketch, without 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 frictlonal 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. THE END. ENGINEERING NEWS is a weekly journal of 60 pages 10 by 14 ins. in size; it publishes each week more than two hundred items of news relating to railroads, water-works and miscellaneous con- tracting intelligence; it is especially a newspaper for engineers, railway officials and contractors, and as such is without a rival in this country. 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