d = 2[alpha]v²/gi + 1/12 (6a)
= 0.00031 v²/i + .083 for new pipes
= 0.00062 v²/i + .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^5 = 32[zeta]Q²/g[pi]²i. (7)
If now the value of [zeta] 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 [zeta] given above for pipes of 1 to 4
ft. diameter we get
d = [root 5](32[zeta]/g[pi]²) [root 5](Q²/i) (8)
= 0.2216 [root 5](Q²/i) for new pipes
= 0.2541 [root 5](Q²/i) for incrusted pipes;
equations which are interesting as showing that when the value of
[zeta] is doubled the diameter of pipe for a given discharge is only
increased by 13%.
(b) A second method is to obtain a rough value of d by assuming [zeta]
= [alpha]. This value is
d´ = [root 5](32Q²/g[pi]²i) [root 5][alpha]
= 0.6319 [root 5](Q²/i) [root 5][alpha].