3.7: Validating Ideal Gas Assumption

3.7: Validating Ideal Gas Assumption#

Consider $1 k g$ superheated steam at a temperature of $T = 200^{\circ} C$ and a pressure of $P = 50 k P a$ . 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 $P = 1.5 M P a$ . 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,

$P V_{a} = m R T$

so,

$V_{a} = m R T / P$

#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 $P = 1.5 M P a$ .

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 %