7.24 Laminar Flow

Objective: Rate an orifice plate in laminar flow

Problem statement: calculate the Reynolds number to determine the type of fluid

Given: S.A.E. 10W oil flows through a 3” schedule 40 pipe. It has a measured delta P of 0.4 psi. The orifice plate has a 2.15” diameter bore, and is a standard sharp-edged orifice. Find the flow rate through the orifice in gallons/minute.

from fluids.units import *
from math import pi

NPS, Di, Do, t = nearest_pipe(NPS=3, schedule='40')
A = 0.25*pi*Di*Di
D2 = 2.15*u.inch
mu = 40*u.cP # given
rho = 53.6*u.lb/u.ft**3

# Assume an absolute pressure of 5 bar.
dP = 0.4*u.psi
P1 = 5*u.bar
P2 = P1-dP
k = 1.3 # assumed

# First calculate the orifice with the standard formula
m = differential_pressure_meter_solver(D=Di, rho=rho, mu=mu, k=k, D2=D2, P1=P1, P2=P2,
                                   m=None, meter_type='ISO 5167 orifice',
                                   taps='corner')
Q = (m/rho).to_base_units()
print('Flow rate is: %s'% Q.to(u.gal/u.min))
v = Q/A
Re = rho*v*Di/mu
Re.to_base_units()

Flow rate is: 88.00974933637421 gallon / minute

1947.5090707193106 dimensionless

Because the flow rate is laminar, outside the range of the ISO formula, we turn to another set of data - a set of CFD results developed for laminar flow by Hollingshead.

# First calculate the orifice with the standard formula
m = differential_pressure_meter_solver(D=Di, rho=rho, mu=mu, k=k, D2=D2, P1=P1, P2=P2,
                                   m=None, meter_type='Hollingshead orifice')
Q = (m/rho).to_base_units()
print('Flow rate is: %s'% Q.to(u.gal/u.min))
v = Q/A
Re = rho*v*Di/mu
Re.to_base_units()

Flow rate is: 77.16025233085325 gallon / minute

1707.4277843809423 dimensionless

The answer given in Crane is that a calibration for the meter must be provided. They assume a C of 0.75. The value of C according to Hollingshead is below.

differential_pressure_meter_C_epsilon(D=Di, D2=D2, m=m, P1=P1, P2=P2, rho=rho, mu=mu, k=k,
                                          meter_type='Hollingshead orifice')[0]

0.7156763185721802 dimensionless