v = 55.31 [root](di) for new pipes
= 39.11 [root](di) for incrusted pipes. (4a)
Within the limits stated, these are accurate enough for practical
purposes, especially as the precise value of the coefficient [zeta]
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 [zeta] 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); [zeta] from (1);
lastly i from (2). If we combine (1) and (2) we get
i = [zeta](v²/2g) (4/d) = 2a{1 + 1/12d} v²/gd; (5)
and, taking the mean values of [zeta] for pipes from 1 to 4 ft.
diameter, given above, the approximate formulae are
i = 0.0003268 v²/d for new pipes
= 0.0006536 v²/d for incrusted pipes. (5a)
_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
d² - d(2[alpha]v²/gi) - [alpha]v²/6gi = 0.
.: d = [alpha]v²/gi + [root]{([alpha]v²/gi) ([alpha]v²/gi + 1/6)}. (6)