Example Problem with Complete Solution

7D-2 : Calculating ΔS from Ideal Gas Tables and from Ideal Gas Heat Capacities 5 pts
Hydrogen (H2) gas is compressed from 4.8 bar and 320K to 15.4 bar and 1300K. Determine the change in the specific entropy of the H2, in kJ/kg, assuming the H2 behaves as an ideal gas. Use…
a.) The Shomate Heat Capacity Equation
     
   
b.) The Ideal Gas Entropy Function
     
   
c.) with constant Heat Capacity, CP, determined at 810K and 10.1 bar.
     
   
 
Read : All equations given are for the molar change in entropy. Make sure and divide your final answer by the molecular weight of hydrogen to obtain a final answer as the change in specific entropy, in kJ/kg-K.
Given: m 1 kg Hydrogen
T1 320 K T2 1300 K
P1 4.8 bar P2 15.4 bar
c.) TC 810 K PC 10.1 bar
Find: Part (a) - (c) DS ??? kJ/(kg K)
Diagram:
Assumptions: 1 - The system consists of one kg of hydrogen, which behaves as an ideal gas.
Equations / Data / Solve:
Part a.) Here, we use the equation given
in the problem statement:
Eqn 1
The heat capacity is determined from the Shomate Equation.
Eqn 2
The values of the constants in the Shomate Equation
for hydrogen are obtained from the NIST WebBook:
T (K) 298 - 1500
A 33.1078
B
-11.508
C 11.6093
D -2.8444
E -0.15967
Substituting Eqn 2 into Eqn 1 and integrating yields:
Eqn 3
We will need the value of the Universal Gas Constant and the molecular weight to determine the change in the specific entropy.
R 8.314 J/mole-K MW 2.016 g/mol
Now, we can substitute values into Eqn 3 to complete part (a) :
41.564 J/mole-K
9.6921 J/mole-K
Now, we can plug values into Eqn 3 : DS 31.872 J/mole-K
DS 15.810 kJ/kg-K
Part b.) In this part of the problem, we use the equation given in the problem statement:
Eqn 4
Properties are determined from Ideal Gas Entropy Tables: At T1: SoT1 1.0139 kJ/kg-K
At T2: SoT2 21.631 kJ/kg-K
Now, we can plug values into Eqn 4 : DS 15.809 kJ/kg-K
Part c.) Once again, we will use the equation given in the problem statement:
Eqn 5
Heat capacity is determined from NIST WebBook: At 810 K: CoP 29.679 J/(mol K)
41.604 J/mole-K
Now, we can plug values into Eqn 5 : DS 31.912 J/mole-K
DS 15.830 kJ/kg-K
Verify: The ideal gas assumption needs to be verified.
We need to determine the specific volume at each state and check if :
(hydrogen is a diatomic gas).
Solving the Ideal Gas EOS for molar volume yields :
Conversion Factors: 1 L = 0.001 m3
1 bar = 100000 N/m2
1 J = 1 N-m
Plugging in values gives us : V1 5.54 L/mol
V2 7.02 L/mol
The specific volume at each state is greater than 5 L/mol and therefore the ideal gas assumption is reasonable.
Answers : a.) DS 15.81 kJ/kg-K
b.) DS 15.81 kJ/kg-K
c.) DS 15.83 kJ/kg-K
Comparison:              
                 
The results in parts (a) and (b) are identical.  This is not a surprise, assuming we integrated the Shomate Equation correctly !
                 
The error in DS associated with the ideal gas assumption in this problem is: 0.13%  
                 
We expected the error to be less than 1% since the molar volumes are greater than 5 L/mole.