Isentropic Flow

The second law of thermodynamics requires that the adiabatic and frictionless flow of any fluid results in no change in entropy. Constant entropy flow is called isentropic flow.

Based on the equation of entropy in a compressible flow :

ds = c_v ln(T₂ / T₁) + R ln(ρ₁ / ρ₂)

= c_p ln(T₂ / T₁) - R ln(p₂ / p₁)

= 0                                            (1)

where

ds = change in specific entropy ( kJ/kgK)

c_v = specific heat at constant volume process (kJ/kg K)

c_p = specific heat at constant pressure process (kJ/kg K)

T = absolute temperature (K)

R = individual gas constant (kJ/kg K)

ρ = density of gas (kg/m³)

p = absolute pressure (Pa, N/m²)

Using

κ = c_p / c_v                              (2)

where

κ = specific heat ratio

(1) can be transformed to:

(T₂ / T₁)^{(κ / (κ-1))} = (ρ₂ / ρ₁)^κ

= (p₂ / p₁)                                (3)

(3) express the useful relationship between temperature, density and pressure for an isentropic flow of an ideal gas.

From (3) the relationship between pressure and temperature can be concluded:

p / ρ^κ = constant                          (4)

Density can be expressed:

ρ = 1 / v                         (5)

where

v = specific volume (m³/kg)

Using (5) in combination with (4) transforms to a common expression:

p v^κ = constant                     (6)