Choked flow: Difference between revisions

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\section{Geometric Choking}
=== Geometric Choking ===


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


\begin{equation}
<math display="block">
\dot{m}=\rho uA=const
\dot{m}=\rho uA=const
\label{eq:massflow:a}
</math>
\end{equation}\\


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


\begin{equation}
<math display="block">
\dot{m}=\rho^* u^* A^*
\dot{m}=\rho^* u^* A^*
\label{eq:massflow:a:b}
</math>
\end{equation}\\


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


\begin{equation}
<math display="block">
\dot{m}=\rho^* a^* A^*
\dot{m}=\rho^* a^* A^*
\label{eq:massflow:b}
</math>
\end{equation}\\


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


\begin{equation}
<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)}
\rho^*=\left(\frac{\rho^*}{\rho_o}\right)\rho_o=\frac{p_o}{RT_o}\left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)}
\label{eq:rhos}
</math>
\end{equation}\\


\begin{equation}
<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}
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}
\label{eq:as}
</math>
\end{equation}\\


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


\begin{equation}
<math display="block">
\dot{m}=\frac{p_o}{RT_o}\left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)}  
\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}
\sqrt{\gamma RT_o}\left(\frac{2}{\gamma+1}\right)^{1/2}
A^*
A^*
\label{eq:massflow:c}
</math>
\end{equation}\\


\noindent which can be rewritten as\\
which can be rewritten as


\begin{equation}
<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)}}
\dot{m}=\frac{p_o A^*}{\sqrt{T_o}}\sqrt{\frac{\gamma}{R}\left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}}
\label{eq:massflow:c}
</math>
\end{equation}\\


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


\begin{itemize}
* quasi-one-dimensional flow
\item quasi-one-dimensional flow
* steady state
\item steady state
* inviscid flow
\item inviscid flow
* calorically perfect gas
\item calorically perfect gas
\end{itemize}


\noindent 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.
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.
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