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Chapter 2

Chapter 2#

Question #11: Pressure Cooker#

A pressure cooker is designed to withstand a maximum gauge pressure of \(P_{max}=3\:atm\).

a) what is the maximum cooking temperature this pressure cooker can reach containing pure water inside?

b) a relief valve is used in this pressure cooker to release pressure at \(P_r=2\:atm\). Calculate the maximum operating temperature in this pressure cooker using the relief valve.

c) assuming an spring with a constant of \(k=10\:kN/m\) is used in the valve to pressurize the cooker. How much the spring should be compressed to withstand the release pressure given the valve is a circular one with a diameter of \(D=10\:mm\).

CH2-Q11.jpg

Solution Approach for a)#

The cooking temperature is the boiling temperature of water at the pressure of the cooker.

#import librarier
import CoolProp.CoolProp as CP

#define variables
P_atm = 101325   #atmospheric pressure in Pa
P_max = 3 * P_atm   #maximum guage pressure in the cooker in Pa
P_max_total = P_max + P_atm   #maximum total pressure in Pa

fluid = "water"  # define the fluid or material of interest

T_max = CP.PropsSI("T", "P", P_max_total, "Q", 0 , fluid)   #max saturation (boiling) temperature in K

print('The maximum cooking temperature is', f"{T_max-273.15:.1f}", 'C')

The maximum cooking temperature is 144.1 C

Solution Approach for b)#

The relief valve releases the pressure beyond \(P_r=2\:atm\), therefore the pressure cooking will operate at this pressure while cooking.

P_r = 2 * P_atm   #operating guage pressure in the cooker in Pa
P_r_total = P_r + P_atm   #total operating pressure using the relief valve in Pa

T_opr = CP.PropsSI("T", "P", P_r_total, "Q", 0 , fluid)   #operational saturation (boiling) temperature in K

print('The operating cooking temperature using the relief valve is', f"{T_opr-273.15:.1f}", 'C')

The operating cooking temperature using the relief valve is 134.0 C

Solution Approach for c)#

The compressed spring should withstand the release pressure, and the pressure is:

\(P=F/A\)

so

\(F=PA\)

where F is the force applied by the spring

\(F=k \times x\)

so

\(x=F/k\)

and A is the cylindrical surface area of the applied force

\(A=\pi D^2/4\)

#import librarier
import numpy as np

#define variables
D = 0.01   #valve diameter in m
k = 10E+3   #spring constant in N/m

A = np.pi * D ** 2 / 4   #vavle area in m2
F = P_r * A   #spring force in N

x = F / k   #spring compression in m

print('The spring compression is', f"{x * 1000:.1f}", 'mm')

The spring compression is 1.6 mm
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