# 7.26 Pipe Partially Filled with Flowing Water

Given a cast iron pipe is 2/3 filled with a steady, uniform flow of water at 60 deg F. The pipe has an inside diameter of 24” and a slope of .75”/foot

Find the flow rate in gallons/minute.

    [1]:

    from fluids.units import *
    from math import pi
    Di = 24*u.inch
    depth = 2/3*Di
    S = 0.75*u.inch/u.ft
    rho = 62.364*u.lb/u.ft**3
    mu = 1.1*u.cP
    eD = 0.00036 # roughness not given in problem, only relative roughness

    [2]:

    # Use the TANK class to make the geometry easier
    tank = TANK(D=Di, L=1e-100*u.m, horizontal=True)
    A = tank.SA_from_h(depth)/2 # wetted surface area of a paper thin tank - divide by two as there is only one side

    tank = TANK(D=Di, L=1e100*u.m, horizontal=True)
    P = tank.SA_from_h(depth)/tank.L

    Rh = A/P
    Dh = 4*Rh
    A, P, Rh, Dh

    [2]:

    (0.20670024467339204 <Unit('meter ** 2')>,
     1.1647220208174018 <Unit('meter')>,
     0.1774674480081778 <Unit('meter')>,
     0.7098697920327112 <Unit('meter')>)

    [3]:

    # 10 meter length basis assumed
    L = 10.0*u.m
    dH = L*S
    dP = dH*rho*u.gravity

    fd = .0155 # initial guess
    for i in range(5):
        K = K_from_f(fd=fd, L=L, D=Dh)
        v = (2*dP/(K*rho))**0.5
        Q = A*v
        # update friction factor
        Re = Reynolds(D=Dh, rho=rho, mu=mu, V=v)
        fd = friction_factor(Re=Re, eD=eD)
    Q.to(u.gal/u.min)

    [3]:

24410.948055443227 gallon/minute

The solution given in Crane is 24500 gpm without iteration.