Relief Valve Sizing - Foundations of Spiritual and Physical Safety with Chemical Processes

The above equations can be rearranged to solve for the area of the orifice, $A$ , for a given flow rate and other conditions. Note that the discharge coefficient, $C_d$ , is a function of the geometry of the orifice and the flow conditions. The discharge coefficient is typically determined experimentally with viscosity, downstream piping, backpressure, and other factors influencing the value. Estimates for some of those parameters can be found in Chapter 10 of Crowl and Louvar. For a preliminary estimate, a value of 0.65 is often used for liquid flow and 0.9 for gas flow.

Relief valve sizing in some instances does required verification with actual flow per the AMSE Boiler and Pressure Vessel Code.

In Class Example: Venting CO2 from 2L Bottle¶

A two liter bottle is half filed with water. There is a relief valve on the top designed to relieve at 60 psig with a flow of 61 scfm. What is the flow rate of CO2 through the relieve per the above equations?

import numpy as np

#fist find the Mach number
gamma = 1.3 #specific heat capacity ratio for CO2, estimate
P1 = 60 + 12.5 #initial pressure, estimate, psia
Patm = 12.5 # atmospheric pressure, psia
Ma = min(1, np.sqrt(2/(gamma-1)*((P1/Patm)**((gamma-1)/gamma)-1))) #Mach number

Ma

#now calculate the molar flow rate
R = 8.314 # gas constant, J/(mol*K)
Temp = 295 # temperature, K
Area = np.pi/4*(1/4*0.0254)**2 # area of the pipe, m^2
Cd = 0.9 # discharge coefficient
Mw = 0.04401 # molar mass of CO2, kg/mol
# molar flow rate, mol/s
ndot = P1*6894.76*Area*Cd*np.sqrt(gamma/(R*Temp*Mw))*Ma*(1+(gamma-1)/2*Ma**2)**((gamma+1)/(2-2*gamma)) 
print(f'The molar flow rate is {ndot:0.2f} mol/s')

#now convert that molar flow rate into scfm, PV = nRT
Tstandard = 298.15 # standard temperature, K
Pstandard = 101325 # standard pressure, Pa
scmps = ndot*R*Tstandard/Pstandard #units of m^3/s
scfm = scmps*60*(3.281**3)
print(f'The flow rate in scfm is {scfm:0.1f}')

Why is this number different than the reported flow rate of CO2 through the relief valve? Could the diameter of the relief be slightly larger?

How would you get the generation rate of CO2 from the submlimation of dry ice?

Q = hA(T_{water} - T_{ice}) \\ mdot = Q/\Delta H_{vap}

(1)

Does area ( $A$ ) change? How would you get the heat transfer coefficient ( $h$ )?