Entropy in Compressible Gas Flow

Calculating entropy in compressible gas flow

Entropy change in 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 = entropy change

c_v = specific heat capacity at a constant volume process

c_p = specific heat capacity at a constant pressure process

T = absolute temperature

R = individual gas constant

ρ = density of gas

p = absolute pressure

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 + 20 K) / (101325 N/m²)

    = 1172 K - the final temperature

The change in entropy can be expressed by (2)

ds = c_p 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/kg.K

Total change in entropy:

dS = (1 kJ/kg.K) (10 kg) = 10 kJ/K

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