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