Compressible Gas Flow - Entropy
Calculating the entropy in a compressible gas flow.
The entropy change in a 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 = change in entropy (kJ)
cv = specific heat capacity at a constant volume process (kJ/kgK)
cp = specific heat capacity at a constant pressure process (kJ/kgK)
T = absolute temperature (K)
R = individual gas constant (kJ/kgK)
ρ = density of gas (kg/m3)
p = absolute pressure (Pa, N/m2)
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 calculated with the ideal gas equation:
p v = R T (3)
where
v = volume (m3)
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 gas temperature
The change in entropy can be expressed by (2)
ds = cp ln(T2 / T1) - R ln(p2 / p1)
ds = (1.05 kJ/kgK) ln((1172 K) / (293 K)) - (0.33 kJ/kgK) ln((405300 N/m2) / (101325 N/m2))
= 1 (kJ/kgK)
Total change in entropy:
dS = (1 kJ/kgK) (10 kg)
= 10 (kJ/K)
