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.

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

# 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

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

# 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)

24410.948055443227 gallon/minute

The solution given in Crane is 24500 gpm without iteration.