# Compressible Gas Flow - Entropy

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

ds = c_{ v }ln(T_{ 2 }/ T_{ 1 }) + R ln(ρ_{ 1 }/ ρ_{ 2 })(1)

or

ds = c_{ p }ln(T_{ 2 }/ T_{ 1 }) - R ln(p_{ 2 }/ p_{ 1 })(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/m^{ 3 })

p= absolute pressure (Pa, N/m^{ 2 })

### Example - Entropy Change in an Air Heating Process

Air - * 10 kg * - is heated at constant volume from temperature * 20 ^{ o } C * and

*101325 N/m*to a final pressure of

^{ 2 }*405300 N/m*.

^{ 2 }The final temperature in the heated air can be calculated with the ideal gas equation :

p v = R T(3)

where

v= volume (m^{ 3 })

The ideal gas equation * (3) * can be transformed to express the volume before heating:

v_{ 1 }= R T_{ 1 }/p_{ 1 }(4)

Since * v _{ 1 } = v _{ 2 } * the ideal gas equation

*(3)*after heating can be expressed as:

p_{ 2 }v_{ 1 }= R T_{ 2 }(5)

or transformed to express the final temperature:

T_{ 2 }=p_{ 2 }v_{ 1 }/ R(6)

Combining (5) and (6):

T_{ 2 }=p_{ 2 }(R T_{ 1 }/p_{ 1 }) / R

= p_{ 2 }T_{ 1 }/p_{ 1 }(7)

=(405300 N/m^{ 2 }) (273 K + 20 K) / (101325 N/m^{ 2 })

= 1172 K - the final gas temperature

The change in entropy can be expressed by * (2) *

ds = c_{ p }ln(T_{ 2 }/ T_{ 1 }) - R ln(p_{ 2 }/ p_{ 1 })

ds =(1.05 kJ/kgK) ln((1172 K) / (293 K)) - (0.33 kJ/kgK) ln((405300 N/m^{ 2 }) / (101325 N/m^{ 2 }))

= 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.

## Related Documents

### 2nd Law of Thermodynamics

Entropy and disorder.

### Steam Entropy

Basic steam thermodynamics - entropy diagram.

### Superheated Steam - Entropy

The entropy of steam superheated to temperatures above saturation points.

### Wet Bulb Globe Temperature (WBGT)

The Wet Bulb Globe Temperature can be used to measure the general Heat-Stress index.