# 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(T_{ 2 }/ T_{ 1 }) + R ln(ρ_{ 1 }/ ρ_{ 2 })

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

= 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/m^{ 3 })

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

κ = c_{ p }/ c_{ v }(2)

where

* (1) * can be transformed to:

(T_{ 2 }/ T_{ 1 })^{ (κ/(κ-1)) }= (ρ_{ 2 }/ ρ_{ 1 })^{ κ }= (p_{ 2 }/ p_{ 1 })(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 (m^{ 3 }/kg)

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

p v^{ κ }= constant(6)

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