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# 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)

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)

## Related Topics

• Fluid Mechanics - The study of fluids - liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.

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