Isentropic Flow Relationships

Related Resources: thermodynamics

Isentropic Flow Relationships

Fluid Flow Engineering and Design
Thermodynamics

A isentropic flow is when a fluid flow is both adiabatic and reversible. That is, no heat is added to the flow, and no energy transformations occur due to friction or dissipative effects. For an isentropic flow of a perfect gas, several relations can be derived to define the pressure, density and temperature along a streamline.

In an ideal gas for an isentropic process, the following relationships exist between static properties at any two points in the flow:

P₂ / P₁ = ( T₂ / T₁ )^{[ k / ( k-1) ]} = ( ρ₂ / ρ₁ )^k

where

P = static pressure
ρ = static density
T = static temperature
P and T are in absolute terms.

The stagnation temperature, T_o, at a point in the flow is related to the static temperature, as follows:

T_o = T + V²₂ / ( 2 c_p )

The energy relation between two points is:

h₁ + V₁² = h₂ + V₂² / 2

The relationship between the static and stagnation properties ( T₀, P₀, and r₀ ) at any point in the flow can be expressed as a function of the Mach number M:

T₀ / T = 1 + ( k - 1 ) / 2 M²

P₀ / P = ( T₀ / T ) ^{( k / ( k - 1 ) )} = ( 1 + ( k - 1 ) / 2 M² ) ^{( 1 / ( k-1) )}

ρ₀ / ρ_{= ( T₀ / T ) ^{( k / ( k - 1 ) )} = ( 1 + ( k - 1 ) / 2 M² ) ^{( 1 / ( k-1) )}}

Compressible flows are often accelerated or decelerated through a nozzle or diffuser. The point at which the Mach number is sonic is called the throat and its area is represented by the variable, A^*. The following area ratio holds for any Mach number.

A / A^* = 1 / M [ ( 1 + ( 1 / 2 ) ( k - 1 ) / 2 M² ) / ( 1 / 2 ) ( k + 1 ) ] ^{( k + 1 ) / ( 2 ( k - 1 ) )}

where
A = area (length2)
A^* = area at the sonic point (M = 1.0)

Source

American Gas Association and the National Fire Protection Association. Used by permission