Choked flow
\section{Geometric Choking}
\noindent For steady-state nozzle flow, the massflow is obtained as \\
\begin{equation} \dot{m}=\rho uA=const \label{eq:massflow:a} \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\\
\begin{equation} \dot{m}=\rho^* u^* A^* \label{eq:massflow:a:b} \end{equation}\\
\noindent By definition $u^*=a^*$ and thus\\
\begin{equation} \dot{m}=\rho^* a^* A^* \label{eq:massflow:b} \end{equation}\\
\noindent $\rho^*$ and $a^*$ can be obtained using the ratios $\rho^*/\rho_o$ and $a^*/a_o$\\
\begin{equation} \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} \end{equation}\\
\begin{equation} 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} \end{equation}\\
\noindent Eqns. \ref{eq:as} and \ref{eq:rhos} in Eqn. \ref{eq:massflow:b} gives\\
\begin{equation} \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^* \label{eq:massflow:c} \end{equation}\\
\noindent which can be rewritten as\\
\begin{equation} \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} \end{equation}\\
\noindent Eqn. \ref{eq:massflow:c} valid for:\\
\begin{itemize} \item quasi-one-dimensional flow \item steady state \item inviscid flow \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.
