Choked flow - Revision history

Nian at 13:39, 1 April 2026

2026-04-01T13:39:13Z

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Nian at 11:12, 1 April 2026

2026-04-01T11:12:42Z

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	=== Geometric Choking ===		=== Geometric Choking ===

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

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

	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

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

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

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

	<math ~~display="block"~~>		{{NumEqn\|<math>
	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}
	</math>		</math>}}

	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

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\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^*
	</math>		</math>}}

	which can be rewritten as		which can be rewritten as

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\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)}}
	</math>		</math>}}

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

Nian at 20:56, 21 March 2026

2026-03-21T20:56:02Z

← Older revision		Revision as of 20:56, 21 March 2026
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	__TOC__		__TOC__

	~~\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.

Nian at 19:52, 20 March 2026

2026-03-20T19:52:14Z

← Older revision		Revision as of 19:52, 20 March 2026
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	[[Category:Quasi-one-dimensional flow]]		[[Category:Quasi-one-dimensional flow]]
	[[Category:Inviscid flow]]		[[Category:Inviscid flow]]
			<noinclude>
			[[Category:Compressible flow:Topic]]
			</noinclude>

	__TOC__		__TOC__

Nian: Created page with "Category:Compressible flow Category:Quasi-one-dimensional flow Category:Inviscid flow TOC \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:mass..."

2026-03-16T15:07:44Z

Created page with "Category:Compressible flow Category:Quasi-one-dimensional flow Category:Inviscid flow __TOC__ \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:mass..."

New page

[[Category:Compressible flow]]
[[Category:Quasi-one-dimensional flow]]
[[Category:Inviscid flow]]

__TOC__

\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.