View source for Choked flow

[[Category:Compressible flow]]
[[Category:Quasi-one-dimensional flow]]
[[Category:Inviscid flow]]
<noinclude>
[[Category:Compressible flow:Topic]]
</noinclude>

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=== Geometric Choking ===

For steady-state nozzle flow, the massflow is obtained as

<math display="block">
\dot{m}=\rho uA=const
</math>

Eqn. \ref{eq:massflow:a} can be evaluated at any location inside the nozzle and if evaluated at sonic conditions we get

<math display="block">
\dot{m}=\rho^* u^* A^*
</math>

By definition <math>u^*=a^*</math> and thus

<math display="block">
\dot{m}=\rho^* a^* A^*
</math>

<math>\rho^*</math> and <math>a^*</math> can be obtained using the ratios <math>\rho^*/\rho_o</math> and <math>a^*/a_o</math>

<math display="block">
\rho^*=\left(\frac{\rho^*}{\rho_o}\right)\rho_o=\frac{p_o}{RT_o}\left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)}
</math>

<math display="block">
a^*=\left(\frac{a^*}{a_o}\right)a_o=a_o\left(\frac{2}{\gamma+1}\right)^{1/2}=\sqrt{\gamma RT_o}\left(\frac{2}{\gamma+1}\right)^{1/2}
</math>

Eqns. \ref{eq:as} and \ref{eq:rhos} in Eqn. \ref{eq:massflow:b} gives

<math display="block">
\dot{m}=\frac{p_o}{RT_o}\left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)} 
\sqrt{\gamma RT_o}\left(\frac{2}{\gamma+1}\right)^{1/2}
A^*
</math>

which can be rewritten as

<math display="block">
\dot{m}=\frac{p_o A^*}{\sqrt{T_o}}\sqrt{\frac{\gamma}{R}\left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}}
</math>

Eqn. \ref{eq:massflow:c} valid for:

* quasi-one-dimensional flow
* steady state
* inviscid flow
* calorically perfect gas

It should be noted that the choked massflow can be calculated using Eqn. \ref{eq:massflow:c} even for cases with shocks downstream of the throat.