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Compressible Gas Flow - Entropy
Calculating entropy in compressible gas flow
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Entropy change in compressible gas flow can be expressed as
ds = cv ln(T2 / T1) + R ln(ρ1 / ρ2) (1)
or
ds = cp ln(T2 / T1) - R ln(p2 / p1) (2)
where
ds = entropy change
cv = specific heat capacity at a constant volume process
cp = specific heat capacity at a constant pressure process
ρ = density of gas
Example - Entropy Change in an Air Heating Process
Air - 10 kg - is heated at constant volume from temperature 20 oC and 101325 N/m2 to a final pressure of 405300 N/m2.
The final temperature in the heated air can be found using the ideal gas equation:
p v = R T (3)
where
v = volume
The ideal gas equation (3) can be transformed to express the volume before heating:
v1 = R T1 / p1 (4)
Since v1 = v2 the ideal gas equation (3) after heating can be expressed as:
p2 v1 = R T2 (5)
or transformed to express the final temperature:
T2 = p2 v1 / R (6)
Combining (5) and (6):
T2 = p2 (R T1 / p1) / R
= p2 T1 / p1 (7)
= (405300 N/m2) (273 K + 20 K) / (101325 N/m2)
= 1172 K - the final temperature
The change in entropy can be expressed by (2)
ds = cp ln(T2 / T1) - R ln(p2 / p1)
ds = (1.05 kJ/kg.K) ln(1172 K / 293 K) - (0.33 kJ/kg.K) ln(405300 N/m2 / 101325 N/m2)
= 1 (kJ/kgK)
Total change in entropy:
dS = (1 kJ/kgK) (10 kg)
= 10 (kJ/K)
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