Isentropic flow relations

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tracelc

Isentropic flow

Constant entropy flow is called Isentropic flow.

From a consideration of the second law of thermodynamics, a reversible flow maintains a constant value of entropy.

In the theory of streamtubes, isentropic flow is the basis for compressible flow.

The well known incompressible flow does not apply if $M << 1$ is violated. Therin

$M = \frac{u}{a}$

is called the Mach-number. It is the relation between the flow-velocity u (fluid or object) and the speed of sound a in the surrounding medium.

In a caloric ideal gas the speed of sound depends on the absolute temperature T (K), the adiabatic exponent $\gamma$ and the specific gas constant R (J/(kg K))

$a = \sqrt{\gamma R T}$.

In this case all thermodynamic properties (temperature, pressure, density and speed of sound) can be expressed with explicit formulas which are functions of the Mach-number!

In the following all thermodynamic properties are related to their properties at rest (u=0). From the energy equation for a frictionless, adiabatic flow

$\frac{u^2}{2}+h=h_t$

you will get that the enthalpy at rest ht is allways the same regardless of an isentropic or non-isentropic state change. As a matter of fact - $h=c_p T$ - the temperature is it as well. In contrary the pressure depends on how the gas is brougt to rest. The pressure at rest is only obtained if the state change is isentropic. If the entropy changes the pressure at rest changes as well, e.g. when passing a shock.

Pressure

$\frac{p_0}{p} = (\frac{T_0}{T})^\frac{\gamma}{\gamma - 1}$

$\frac{p_0}{p}=(1+\frac{\gamma-1}{2}M^2)^{\frac{\gamma}{\gamma-1}}$

Temperature and speed of sound

$\frac{T_0}{T}= \left( \frac{a_0}{a}\right)^2=1+\frac{\gamma-1}{2}M^2$

Density

$\frac{\rho_0}{\rho}=(1+\frac{\gamma-1}{2}M^2)^{\frac{1}{\gamma-1}}$

Critical values

Values which can be found at M=1 (speed of sound) are sometimes called critical values. They are marked with the superscript *.

They differ from the properties at rest only by a constant value and are therefore used as reference values as well.

In the special case for gases which contain two atoms ($\gamma=1.4$) one gets:

$\frac{a^*}{a_0} = 0.913$

$\frac{p^*}{p_0} = 0.528$

$\frac{\rho^*}{\rho_0} = 0.634$

Critical cross-section

The cross-section $A^*$ in a nozzle where M equals 1 is often used as a reverence, even if M=1 is never reached.

$\frac{A}{A^*}=\frac{1}{M}*(\frac{\gamma+1}{2})^{\frac{-\gamma+1}{2(\gamma-1)}}*(1+\frac{\gamma-1}{2}M^2)^{\frac{\gamma+1}{2(\gamma-1)}}$

$\left( \frac{A}{A^*} \right)^2 = \frac{1}{M^2} \left[ \frac{2}{\gamma+1} \left( 1 + \frac{\gamma-1}{2}M^2\right)\right]^{(\gamma+1)/(\gamma-1)}$

Summary

As one can see all thermodynamic properties decrease with increasing Mach-number.

Literature

Spurk, J.H. (2004), StrÃ¶mungslehre. Eine EinfÃ¼hrung in die Theorie der StrÃ¶mungen, ISBN 3-540-40166-0, 5.th Ed., Springer-Verlag, Berlin..