488 HYDROMECHANICS [HYDRAULICS, Sluice in Cylindrical Pipe (fig. 92). Ratio of height of } opening to dia-> 1-0 I
fi i 1! i i meter of pipe ) 0>1_ i-oo 0-948 856 740 609 46C 315 159 10 ?= o-oo 0-07 0-26 0-81 2-06 5-52 17-0 97-8 Fig. 92. Fig. 93. Cock. in a Cylindrical Pipe (fig. 93). Angle through which cock is turned = 0. 0= 5 10 15 20 25 30 35 Ratio of } cross sec- > 926 850 772 692 613 535 458 tions . . . ) f.- 05 29 75 1-56 3-10 5-47 9-68 = 40 45 50 55 60 65 82 Ratio of J cross sec- > 385 315 250 190 137 091
tions . . . )
- .-
17-3 31-2 52-6 106 206 486 00 Throttle Valve in a Cylindrical Pipe (fig. 94). 8 = c.= 5 24 10 52 15 90 20 1-54 25 2-51 30 3-91 35 6-22 40 10-8 =
45 187 50 32-6 55 58-8 60 118 65 256 70 751 90 00 78. Practical Calculations on the Flow of Water in Pipes. In the following explanations it will be assumed that the ., " pipe is of so great a length that only the loss of head in friction against the sur face of the pipe needs to be considered. In general it is one of the four quan tities d, i. v, or Q which requires to be determined. Fig. 94. For since the loss of head h is given by the relation h = il, this need not be separately considered. There are then three equations (see eq. 4, 69, and 9, 73) for the solution of such problems as arise : where o = 005 for new and =0 01 for incrusted pipes, .. v 2 di Q =i (2). (3). Problem 1. Given the diameter of the pipe and its virtual slope, to find the discharge and velocity of flow. Here d and i are given, and Q and v are required. Find from (1) ; then w from (2) ; lastly Q from (3). This case presents no difficulty. By combining equations (1 ) and (2), v is obtained directly : For new pipes For incrusted pipes For pipes not less than 1, or more than 4 feet in diameter, the mean values of are For new pipes 0-00526 For incrusted pipes 01052. Using these values we get the very simple expressions v = 55 31 /di for new pipes = 39"ll/o^ for incrusted pipes Within the limits stated, these are accurate enough for practical purposes, especially as the precise value of the coefficient cannot be known for each special case. Problem 2. Given the diameter of a pipe and the velocity of flow, to find the virtual slope and discharge. The discharge is given by (3); the proper value of by (1); and the virtual slope by (2). This also presents no special difficulty. Problem 3. Given the diameter of the pipe and the discharge, to find the virtual slope and velocity. Find v from (3) ; from (1) ; lastly i from (2). If we combine (1) and (2) we get 4 /. . 1 * 2 , 5 , - . . . . (0) , and, taking the mean values of f for pipes from 1 to 4 feet diameter, given above, the approximate formula; are i = -0003268 for new pipes ] ( r MI .2 ( vU-) = 0-0006536 for incrusted pipes d J Problem 4. Given the virtual slope and the velocity, to find the diameter of the pipe and the discharge. The diameter is obtained from equations (2) and (1), which give the quadratic expression gi bgi (6). For practical purposes, the approximate equations gi 0-00031 -r + 083 for new pipes = 80-24 = 00062 + 083 for incrusted pipes are sufficiently accurate. Problem 5. Given the virtual slope and the discharge, to find the diameter of the pipe and velocity of flow. This case, which often occurs in designing, is the one which is least easy of direct solution. From equations (2) and (3) we get d If now the value of in (1) is introduced, the equation becomes very cumbrous. Various approximate methods of meeting the difficulty may be used. (a) Taking the mean values of given above for pipes of 1 to 4 feet diameter we get - Y^l 5 /Q! (8) V gir V i 5 /Q2 = 2216 . / -2_ for new pipes / v i 5 /fvi = 0-2541 . /-5-for incrusted pipes ; V i equations which are interesting as showing that when the value of is doubled the diameter of pipe for a given discharge is only in creased by 13 per cent. (b) A second method is to obtain a rough value of d by assuming = a. This value is 5 /onHT 5 / S /Q 2 5 / d = /5S , /a = 0-6319 . / . /a. V g**i V V i V Then a very approximate value of is and a revised value of d, not sensibly differing from the exact value, is S /S20" V . 5 A>- s / f , ,,, f OiP*t, If (Vfi TIQ / > If d = i^i - WTV ! (e) Equation 7 may be put in the form iQ 5 / n ^ 1 s /32aQ d= v / .;, N nirh i + ,V,
(9).
