HYDROMECHANICS HYDRAULICS.] reservoir to the surface of the lower reservoir, that is GQ7t foot-pounds. This is expended in three ways. (1) The head -|- , corresponding to an expenditure of GQ^ foot-pounds of work, is employed in giving energy of motion to the water. This is ultimately wasted in eddying motions in the lower reservoir. (2) A portion of head, which experience shows may be expressed in the form j^- , corre sponding to an expenditure of GQCo^ foot-pounds of work, is em ployed in overcoming the resistance at the entrance to the pipe . (3) As already shown the head expended in overcoming the surface fric- 4L v* 4L v 2 tion of the pipe is ,- ^ corresponding to GQ-^~ -^- foot-pounds of work. Hence - (5). If the entrance to = 1 08 to 1"505. 485 to proceed by approximation. Find an approximate value of d by assuming a probable value for as mentioned below. Then from that value of d find a corrected value for and repeat the calculation. The equation above may be put in the form (6); d from which it is clear that the head expended at the mouthpiece is equivalent to that of a length 4Li> 2 = 8-025 If the pipe is bellmouthed, is about = "08. the pipe is cylindrical, = 6 - 505. Hence In general this is so small compared with that, for practical Cti calculations, it may be neglected ; that is, the losses of head other than the loss in surface friction are left out of the reckoning. It is only in short pipes and at high velocities that it is necessary to take account of the first two terms in the bracket, as well as the third. For instance, in pipes for the supply of turbines, v is usually limited to 2 feet per second, and the pipe is bellrnouthed. Then 1-08 = 067 foot. 2flf v 2 2 to 4i feet per second, and then 1 5 = 1 to 5 foot. In 20 either case this amount of head is small compared with the whole virtual fall in the cases which most commonly occur. When d and v or d and h are given, the equations above are solved quite simply. When v and h are given and d is required, it is better In pipes for towns supply v may range from of the pipe. Putting 1 + = 1 -505 and f = 01, the length of pipe equivalent to the mouthpiece is 37 - 6 d nearly. This may be added to the actual length of the pipe to allow for mouthpiece resistance in approximate calciilations. 72. Coefficient of Friction for Pipes Discharging Water From the average of a large number of experiments, the value of for ordinary iron pipes is C=0 007567 ........ (7). But practical experience shows that no single value can be taken applicable to very different cases. The earlier hydraulicians occupied themselves chiefly with the dependence of on the velocity. Hav ing regard to the difference of the law of resistance at very low and at ordinary velocities, they assumed that might be expressed in the form The following are the best numerical values obtained for so ex pressed : a
Prony (from 51 experiments) D Aubuisson 0-006836 0-00673 0-005493 0-001116 0-001211 0-00143 Eytelwein Weisbach proposed the formula C I * 0- 00*50 l 004289 ... (8) f the coefficien lv Vv The following short table gives Weisbach s values c of friction for different velocities in feet per second :- { = 0-1 0-0686 0-2 0-0527 0-3 0-0457 0-4 0-0415 0-5 0-0387 0-6 0-0365 0-7 0-0349 0-8 0-0336 0-9 0-0325 1 0-0315 H 0-0297 4 0-0284 2 0-0265 3 0-0243 4 0-0230 6 0-0214 8 0-0205 12 0-0193 20 0-0182 73. Darcy s Experiments. All previous experiments on the re sistance of pipes have been superseded by the remarkable researches carried out by the late Inspector-General of the Paris Water Works, M. Darcy. His experiments were carried out on a scale, under a variation of conditions, and with a degree of accuracy which leaves little to be desired, and the results obtained are of very great prac tical importance. These results may be stated thus : (1) For new and clean pipes the friction varies considerably with the nature and polish of the surface of the pipe. For clean cast- iron it is about 1 1 times as great as for cast-iron covered with pitch. (2) The nature of the surface has less influence when the pipes are old and incrusted with deposits, due to the action of the water. Thus old and incrusted pipes give twice as great a frictional resist ance as new and clean pipes. M. Darcy s coefficients were chiefly determined from experiments on new pipes. He doubles these co efficients for old and incrusted pipes, in accordance with the results of a very limited number of experiments on pipes containing incrus tations and deposits. (3) The coefficient of friction may be expressed in the form O C = cH ; but in pipes which have been some time in use it is v sufficiently accurate to take = o : simply, where oj depends on the diameter of the pipe alone, but a and on the other hand depend both on the diameter of the pipe and the nature of its surface. The following are the values of the constants. For pipes which have been some time in use, neglecting the term depending on the velocity ; / n (9). These coefficients may be put in the following very simple form, without sensibly altering their value : For clean pipes = 005( I+T^
For slightly in crusted pipes l( 1 +7775
iM
(9a). Darcy s Value of the Goefficicnt of Friction {for Velocities not less than 4 inches per second. a
For drawn wrought-iron or smooth ) cast-iron pipes ... | 004973 084 For pipes altered by light inerus- ) tations ... ( 00996 084 Diameter ( Diameter < of Pipo in Inches. New Pipes. lacrusted Pipes. of Pipe in Inches. New Pipes. Incrusted Pipes. 2 0-00750 0-01500 18 00528 01056 3 00667 01333 21 00524 01048 4 00625 01250 24 00521 01042 5 00600 01200 27 00519 01037 6 00583 01167 30 00517 01033 7 00571 01143 36 00514 01028 8 00563 01125 42 00512 01024 9 00556 01111 48 00510 01021 12 00542 01083 54 00509 01019 15 00533 01067 These values of are, however, only applicable when the velocity exceeds 4 inches per second. To embrace all cases Darcy proposes the expression 1 4. (10); which is a modification of Coulomb s, including terms expressing the influence of the diameter and of the velocity. For clean pipes Darcy
found these values
