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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 = 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

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

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

p v^κ = constant         (6)

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