Compressible Gas Flow - Entropy

The entropy change in a compressible gas flow can be expressed as

ds = c_v ln(T₂/ T₁ ) + R ln(ρ₁ / ρ₂) (1)

or

ds = c_p ln(T₂/ T₁ ) - R ln(p₂/ p₁ ) (2)

where

ds = change in entropy (kJ)

c_v = specific heat capacity at a constant volume process (kJ/kgK)

c_p = 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:

p₂v₁ = R T₂ (5)

or transformed to express the final temperature:

T₂ = p₂v₁ / R (6)

Combining (5) and (6):

T₂ = p₂(R T₁ / p₁ ) / R

= p₂T₁ / 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 = c_p 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)