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)

Temperature ^oC
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