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3.7: Validating Ideal Gas Assumption

3.7: Validating Ideal Gas Assumption#

Consider \(1\:kg\) superheated steam at a temperature of \(T=200^\circ C\) and a pressure of \(P=50\:kPa\). Calculate the volume of steam at this state based on:

a) ideal gas assumption

b) thermodynamic tables using CoolProp

c) calculate the error percentage

d) Now consider pressurizing steam at the same temperature to a point close to saturation where the pressure is \(P=1.5\:MPa\). Calculate the error percentage for voulme based on the comparison between ideal gas assumption and using thermodynamic tables from CoolProp

Solution Approach for a)#

based on ideal gas assumption,

\(PV_a=mRT\)

so,

\(V_a=mRT/P\)

#define variables
P = 50E+3   #pressure in Pa
m = 1   #mass in kg
T = 200 + 273.15   #temperature in K
R = 461.5   #steam gas constant in J/kg.K

V_a = m * R * T / P   #volume in m3

print('The calculated volume based on ideal gas assumption is:', f"{V_a:.1f}", 'm3')

The calculated volume based on ideal gas assumption is: 4.4 m3

Solution Approach for b)#

Density is used to calculate volume since its an standard output for CoolProp. Then,

\(D=m/V_b\)

so

\(V_b=m/D\)

# import the libraries we'll need
import CoolProp.CoolProp as CP
fluid = "water"  # define the fluid or material of interest

D = CP.PropsSI("D", "T", T, "P", P , fluid)   #fluid density in kg/m3
V_b = m / D   #volume in m3

print('The volume based on thermodynamic tables from CoolProp is:', f"{V_b:.1f}", 'm3')

The volume based on thermodynamic tables from CoolProp is: 4.4 m3

Solution Approach for c)#

Since the value obtained from thermodynamic tables is an experimental value, it is used as a reference to calculate error percentage

\(E=100\times (|V_a-V_b|)/V_b\)

# import the libraries we'll need
import numpy as np

E = 100 * np.absolute(V_a-V_b)/V_b

print('The error percentage is:', f"{E:.3f}", '%')

The error percentage is: 0.253 %

Solution Approach for d)#

the same path is followed for a pressure of \(P=1.5\:MPa\).

P = 1.5E+6   #pressure in Pa
V_a = m * R * T / P   #volume based on ideal gas assumption in m3

D = CP.PropsSI("D", "T", T, "P", P , fluid)   #fluid density in kg/m3
V_b = m / D   #volume based on thermodynamic tables in m3

E = 100 * np.absolute(V_a-V_b)/V_b
print('The error percentage for a higher pressure is:', f"{E:.3f}", '%')

The error percentage for a higher pressure is: 9.905 %
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