One-dimensional flow with heat addition - Revision history

Nian: /* Thermal Choking */

2026-04-01T18:25:07Z

Thermal Choking

← Older revision		Revision as of 18:25, 1 April 2026
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	When the heat addition reaches $q^\ast$ the flow becomes sonic and the flow is said to thermally choked. Thermal choking is illustrated in Figure~\ref{fig:TSPV:d}, where the curve representing the energy equation (the blue line in the <math>p\nu</math>-diagram) is tangent to the Rayleigh line and if more heat is added the blue line will move to the right of the Rayleigh line and thus there are no solutions for <math>q>q^\ast</math>. So what happens if more heat is added to the flow after thermal choking is reached. The answer is different if the flow is subsonic or supersonic. For a subsonic flow, the upstream flow will be adjusted such that the slope of the Rayleigh line changes and the energy equation curve becomes tangent to the Rayleigh line. This means that the massflow per unit area (<math>C</math>) is reduced and <math>q^\ast</math> is increased such that <math>q^\ast</math> equals the heat added to the flow. Note that the upstream total conditions will not be changed in this process (see Figure~\ref{fig:thermal:choking:sub}).		When the heat addition reaches $q^\ast$ the flow becomes sonic and the flow is said to thermally choked. Thermal choking is illustrated in Figure~\ref{fig:TSPV:d}, where the curve representing the energy equation (the blue line in the <math>p\nu</math>-diagram) is tangent to the Rayleigh line and if more heat is added the blue line will move to the right of the Rayleigh line and thus there are no solutions for <math>q>q^\ast</math>. So what happens if more heat is added to the flow after thermal choking is reached. The answer is different if the flow is subsonic or supersonic. For a subsonic flow, the upstream flow will be adjusted such that the slope of the Rayleigh line changes and the energy equation curve becomes tangent to the Rayleigh line. This means that the massflow per unit area (<math>C</math>) is reduced and <math>q^\ast</math> is increased such that <math>q^\ast</math> equals the heat added to the flow. Note that the upstream total conditions will not be changed in this process (see Figure~\ref{fig:thermal:choking:sub}).

	{{~~NumEqn~~\|<math>		{{InfoBox\|<math>
	M_{1^\prime} = f(q^\ast)		M_{1^\prime} = f(q^\ast)
	</math><br><br><math>		</math><br><br><math>
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	</math><br><br><math>		</math><br><br><math>
	u_{1^\prime} = M_{1^\prime}a_{1^\prime}		u_{1^\prime} = M_{1^\prime}a_{1^\prime}
	</math>~~\|nonumber=1\|infobox=1~~}}		</math>}}

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Nian at 08:30, 1 April 2026

2026-04-01T08:30:14Z

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Nian at 19:37, 20 March 2026

2026-03-20T19:37:54Z

← Older revision		Revision as of 19:37, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
	~~[[Category:Topic]]~~
	[[Category:One-dimensional flow]]		[[Category:One-dimensional flow]]
	[[Category:Inviscid flow]]		[[Category:Inviscid flow]]
	[[Category:Continuous solution]]		[[Category:Continuous solution]]
			<noinclude>
			[[Category:Compressible flow:Topic]]
			</noinclude>

	__TOC__		__TOC__

Nian at 17:09, 20 March 2026

2026-03-20T17:09:55Z

← Older revision		Revision as of 17:09, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
			[[Category:Topic]]
	[[Category:One-dimensional flow]]		[[Category:One-dimensional flow]]
	[[Category:Inviscid flow]]		[[Category:Inviscid flow]]

Nian: /* Thermal Choking */

2026-03-17T10:28:01Z

Thermal Choking

← Older revision		Revision as of 10:28, 17 March 2026
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	==== Thermal Choking ====		==== Thermal Choking ====

	~~\noindent~~ When the heat addition reaches $q^\ast$ the flow becomes sonic and the flow is said to thermally choked. Thermal choking is illustrated in Figure~\ref{fig:TSPV:d}, where the curve representing the energy equation (the blue line in the $p\nu$-diagram) is tangent to the Rayleigh line and if more heat is added the blue line will move to the right of the Rayleigh line and thus there are no solutions for $q>q^\ast$. So what happens if more heat is added to the flow after thermal choking is reached. The answer is different if the flow is subsonic or supersonic. For a subsonic flow, the upstream flow will be adjusted such that the slope of the Rayleigh line changes and the energy equation curve becomes tangent to the Rayleigh line. This means that the massflow per unit area ($C$) is reduced and $q^\ast$ is increased such that $q^\ast$ equals the heat added to the flow. Note that the upstream total conditions will not be changed in this process (see Figure~\ref{fig:thermal:choking:sub}).		When the heat addition reaches $q^\ast$ the flow becomes sonic and the flow is said to thermally choked. Thermal choking is illustrated in Figure~\ref{fig:TSPV:d}, where the curve representing the energy equation (the blue line in the <math>p\nu</math>-diagram) is tangent to the Rayleigh line and if more heat is added the blue line will move to the right of the Rayleigh line and thus there are no solutions for <math>q>q^\ast</math>. So what happens if more heat is added to the flow after thermal choking is reached. The answer is different if the flow is subsonic or supersonic. For a subsonic flow, the upstream flow will be adjusted such that the slope of the Rayleigh line changes and the energy equation curve becomes tangent to the Rayleigh line. This means that the massflow per unit area (<math>C</math>) is reduced and <math>q^\ast</math> is increased such that <math>q^\ast</math> equals the heat added to the flow. Note that the upstream total conditions will not be changed in this process (see Figure~\ref{fig:thermal:choking:sub}).

	\[		<math display="block">
	~~\begin{aligned}~~		M_{1^\prime} = f(q^\ast)
	M_{1'} & = f(q^\ast)\\		</math>
	~~T_{1'} & = f(T_o, M_{1'})\\~~
	~~p_{1'} & = f(p_o, M_{1'})\\~~
	~~\rho_{1'} & = f(p_{1'}, T_{1'})\\~~
	~~a_{1'} & = f(T_{1'})\\~~
	~~u_{1'} & = M_{1'}a_{1'}\\~~
	~~\end{aligned}~~
	\]

			<math display="block">
			T_{1^\prime} = f(T_o,\ M_{1^\prime})
			</math>

			<math display="block">
			p_{1^\prime} = f(p_o,\ M_{1^\prime})
			</math>

			<math display="block">
			\rho_{1^\prime} = f(p_{1^\prime},\ T_{1^\prime})
			</math>

			<math display="block">
			a_{1^\prime} = f(T_{1^\prime})
			</math>

			<math display="block">
			u_{1^\prime} = M_{1^\prime}a_{1^\prime}
			</math>

			<!--
	\begin{figure}[ht!]		\begin{figure}[ht!]
	\begin{subfigure}[b]{0.5\textwidth}		\begin{subfigure}[b]{0.5\textwidth}
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	\label{fig:thermal:choking:sub}		\label{fig:thermal:choking:sub}
	\end{figure}		\end{figure}
			-->

	~~\noindent~~ In a choked supersonic flow, there is no possibility for pressure waves to travel upstream in the flow and thus the upstream flow conditions can not be changed as in the subsonic case. Moreover, since a normal shock is an adiabatic process (a jump between two points on the same Rayleigh line), the total temperature is not changed over a chock. From before we have		In a choked supersonic flow, there is no possibility for pressure waves to travel upstream in the flow and thus the upstream flow conditions can not be changed as in the subsonic case. Moreover, since a normal shock is an adiabatic process (a jump between two points on the same Rayleigh line), the total temperature is not changed over a chock. From before we have

	\[\dfrac{T_o}{T_o^\ast}=\dfrac{(\gamma+1)M_1^2}{(1+\gamma M_1^2)^2}(2+(\gamma-1)M_1^2)\]		<math display="block">
			\dfrac{T_o}{T_o^\ast}=\dfrac{(\gamma+1)M_1^2}{(1+\gamma M_1^2)^2}(2+(\gamma-1)M_1^2)
			</math>

	~~\noindent~~ Inserting the normal shock relation		Inserting the normal shock relation

	\[M_2^2=\dfrac{2+(\gamma-1)M_1^2}{2\gamma M_1^2-(\gamma-1)}\]		<math display="block">
			M_2^2=\dfrac{2+(\gamma-1)M_1^2}{2\gamma M_1^2-(\gamma-1)}
			</math>

	~~\noindent~~ one can show that		one can show that

	\[\dfrac{T_o}{T_o^\ast}=\dfrac{(\gamma+1)M_2^2}{(1+\gamma M_2^2)^2}(2+(\gamma-1)M_2^2)\]		<math display="block">
			\dfrac{T_o}{T_o^\ast}=\dfrac{(\gamma+1)M_2^2}{(1+\gamma M_2^2)^2}(2+(\gamma-1)M_2^2)
			</math>

	~~\noindent~~ and thus $T_o^\ast$ is not changed by the normal shock and consequently $q^\ast$ is unchanged if there is a normal shock between station 1 and 2. So, it is not possible to change the upstream static flow conditions and a normal shock will not make it possible to add more heat. The only possible solution is a normal shock upstream of station 1 and thus subsonic flow through the heat addition process.		and thus <math>T_o^\ast</math> is not changed by the normal shock and consequently <math>q^\ast</math> is unchanged if there is a normal shock between station 1 and 2. So, it is not possible to change the upstream static flow conditions and a normal shock will not make it possible to add more heat. The only possible solution is a normal shock upstream of station 1 and thus subsonic flow through the heat addition process.

Nian: /* Rayleigh Line */

2026-03-17T10:18:03Z

Rayleigh Line

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Nian: /* Heat Addition Process */

2026-03-17T10:12:56Z

Heat Addition Process

← Older revision		Revision as of 10:12, 17 March 2026
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	</math>		</math>

	Form Eqn.~\ref{eq:governing:dT:diff:mod:a} we can make the following observation\\		Form Eqn.~\ref{eq:governing:dT:diff:mod:a} we can make the following observation

	<math display="block">		<math display="block">

Nian: /* Heat Addition Process */

2026-03-17T10:12:39Z

Heat Addition Process

← Older revision		Revision as of 10:12, 17 March 2026
Line 263:		Line 263:
	</math>		</math>

	Now, by specifying an initial subsonic flow state and dividing the heat addition corresponding to choked flow, <math>q^\ast</math>, into small portions $\delta q$, one can perform integration as indicated in Figure~\ref{fig:dq}. The result is presented in the in Figure~\ref{fig:TS:closeup}. The subsonic process corresponds to the upper line. As heat is added the Mach number is increased and at <math>M=\gamma^{-1/2}</math> the maximum temperature is reached. Adding more heat will reduce the temperature and increase the Mach number until sonic conditions are reached (<math>M=1.0</math>). As can be seen in Figure~\ref{fig:TS:closeup}, the lean of the subsonic branch of the Rayleigh line is lower than the isobars (gray lines), which means the increasing heat will reduce pressure. The lower part of the blue line in Figure~\ref{fig:TS:closeup} is the supersonic branch of the Rayleigh line, which is obtained in the same way starting from a supersonic flow condition. A flow state resulting in the same sonic conditions as for the subsonic case is calculated and used as a starting state. The corresponding $q^\ast$ is calculated and the same calculation of consecutive flow states in a step-wise manner is performed. As can be seen in Figure~\ref{fig:TS:closeup}, the lean of the supersonic part of the Rayleigh curve is steeper than the isobars (gray lines), which means that pressure increases as heat is added to the flow. As we saw from Eqn.~\ref{eq:governing:dT:diff:mod:b}, <math>dT/dT_o</math> becomes infinite when the flow approaches the sonic the sonic state. After the sonic state is reached, further heat addition is impossible without changing the upstream flow conditions. This will be made clearer in the next section.		Now, by specifying an initial subsonic flow state and dividing the heat addition corresponding to choked flow, <math>q^\ast</math>, into small portions <math>\delta q</math>, one can perform integration as indicated in Figure~\ref{fig:dq}. The result is presented in the in Figure~\ref{fig:TS:closeup}. The subsonic process corresponds to the upper line. As heat is added the Mach number is increased and at <math>M=\gamma^{-1/2}</math> the maximum temperature is reached. Adding more heat will reduce the temperature and increase the Mach number until sonic conditions are reached (<math>M=1.0</math>). As can be seen in Figure~\ref{fig:TS:closeup}, the lean of the subsonic branch of the Rayleigh line is lower than the isobars (gray lines), which means the increasing heat will reduce pressure. The lower part of the blue line in Figure~\ref{fig:TS:closeup} is the supersonic branch of the Rayleigh line, which is obtained in the same way starting from a supersonic flow condition. A flow state resulting in the same sonic conditions as for the subsonic case is calculated and used as a starting state. The corresponding $q^\ast$ is calculated and the same calculation of consecutive flow states in a step-wise manner is performed. As can be seen in Figure~\ref{fig:TS:closeup}, the lean of the supersonic part of the Rayleigh curve is steeper than the isobars (gray lines), which means that pressure increases as heat is added to the flow. As we saw from Eqn.~\ref{eq:governing:dT:diff:mod:b}, <math>dT/dT_o</math> becomes infinite when the flow approaches the sonic the sonic state. After the sonic state is reached, further heat addition is impossible without changing the upstream flow conditions. This will be made clearer in the next section.

	Using the differential relations above, we can get a good picture of the development of flow variables as heat is continuously added to the flow (see Figure~\ref{fig:rayleigh:trends}).		Using the differential relations above, we can get a good picture of the development of flow variables as heat is continuously added to the flow (see Figure~\ref{fig:rayleigh:trends}).

Nian: /* Heat Addition Process */

2026-03-17T10:12:14Z

Heat Addition Process

← Older revision		Revision as of 10:12, 17 March 2026
Line 225:		Line 225:
	==== Heat Addition Process ====		==== Heat Addition Process ====

	~~\noindent~~ With the differential relations in place, we can now study the continuous change in flow quantities from the initial flow state to the flow state after the heat addition process by dividing the total amount of heat added to the flow, $q$, into small portions, $\delta q$, and calculate the change in flow properties for each of these heat additions, see Figure~\ref{fig:dq}.\\		With the differential relations in place, we can now study the continuous change in flow quantities from the initial flow state to the flow state after the heat addition process by dividing the total amount of heat added to the flow, <math>q</math>, into small portions, <math>\delta q</math>, and calculate the change in flow properties for each of these heat additions, see Figure~\ref{fig:dq}.

			<!--
	\begin{figure}[ht!]		\begin{figure}[ht!]
	\begin{center}		\begin{center}
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	\label{fig:dq}		\label{fig:dq}
	\end{figure}		\end{figure}
			-->

	~~\noindent~~ Let's first examine the temperature change by rewriting Eqn.~\ref{eq:governing:dT:diff:final} as\\		Let's first examine the temperature change by rewriting Eqn.~\ref{eq:governing:dT:diff:final} as

	~~\begin{equation}~~		<math display="block">
	dT=\dfrac{1-\gamma M^2}{1-M^2}dT_o\Leftrightarrow \dfrac{dT}{dT_o}=\dfrac{1-\gamma M^2}{1-M^2}		dT=\dfrac{1-\gamma M^2}{1-M^2}dT_o\Leftrightarrow \dfrac{dT}{dT_o}=\dfrac{1-\gamma M^2}{1-M^2}
	~~\label{eq:governing:dT:diff:mod:a}~~		</math>
	~~\end{equation}\\~~

	~~\noindent~~ which is equivalent to\\		which is equivalent to

	~~\begin{equation}~~		<math display="block">
	\dfrac{dh}{\delta q}=\dfrac{1-\gamma M^2}{1-M^2}		\dfrac{dh}{\delta q}=\dfrac{1-\gamma M^2}{1-M^2}
	~~\label{eq:governing:dT:diff:mod:b}~~		</math>
	~~\end{equation}\\~~

	~~\noindent~~ Form Eqn.~\ref{eq:governing:dT:diff:mod:a} we can make the following observation\\		Form Eqn.~\ref{eq:governing:dT:diff:mod:a} we can make the following observation\\

	\[\dfrac{dT}{dT_o}=0\Rightarrow \gamma M^2=1\Rightarrow M=\sqrt{1/\gamma}~~\]\\~~		<math display="block">
			\dfrac{dT}{dT_o}=0\Rightarrow \gamma M^2=1\Rightarrow M=\sqrt{1/\gamma}
			</math>

	~~\noindent~~ which means that the maximum temperature will be reached when the Mach number is $\sqrt{1/\gamma}$. Since $\gamma$ is a number ~~grreter~~ than one for all gases, this implies that the maximum temperature can only be reached if the flow is subsonic. For air, this the maximum temperature will be reached at $M=0.845$.\\		which means that the maximum temperature will be reached when the Mach number is <math>\sqrt{1/\gamma}</math>. Since <math>\gamma</math> is a number greater than one for all gases, this implies that the maximum temperature can only be reached if the flow is subsonic. For air, this the maximum temperature will be reached at <math>M=0.845</math>.

	~~\noindent~~ If we evaluate Eqn.~\ref{eq:governing:dT:diff:mod:a} for sonic flow ($M=1$), we see that the derivative becomes infinite.\\		If we evaluate Eqn.~\ref{eq:governing:dT:diff:mod:a} for sonic flow (<math>M=1</math>), we see that the derivative becomes infinite.

	\[\|M\|\rightarrow 1.0 \Rightarrow \dfrac{dT}{dT_o}\rightarrow \pm \infty~~\]\\~~		<math display="block">
			\|M\|\rightarrow 1.0 \Rightarrow \dfrac{dT}{dT_o}\rightarrow \pm \infty
			</math>

	~~\noindent~~ Now, by specifying an initial subsonic flow state and dividing the heat addition corresponding to choked flow, $q^\ast$, into small portions $\delta q$, one can perform integration as indicated in Figure~\ref{fig:dq}. The result is presented in the in Figure~\ref{fig:TS:closeup}. The subsonic process corresponds to the upper line. As heat is added the Mach number is increased and at $M=\gamma^{-1/2}$ the maximum temperature is reached. Adding more heat will reduce the temperature and increase the Mach number until sonic conditions are reached ($M=1.0$). As can be seen in Figure~\ref{fig:TS:closeup}, the lean of the subsonic branch of the Rayleigh line is lower than the isobars (gray lines), which means the increasing heat will reduce pressure. The lower part of the blue line in Figure~\ref{fig:TS:closeup} is the supersonic branch of the Rayleigh line, which is obtained in the same way starting from a supersonic flow condition. A flow state resulting in the same sonic conditions as for the subsonic case is calculated and used as a starting state. The corresponding $q^\ast$ is calculated and the same calculation of consecutive flow states in a step-wise manner is performed. As can be seen in Figure~\ref{fig:TS:closeup}, the lean of the supersonic part of the Rayleigh curve is steeper than the isobars (gray lines), which means that pressure increases as heat is added to the flow. As we saw from Eqn.~\ref{eq:governing:dT:diff:mod:b}, $dT/dT_o$ becomes infinite when the flow approaches the sonic the sonic state. After the sonic state is reached, further heat addition is impossible without changing the upstream flow conditions. This will be made clearer in the next section.\\		Now, by specifying an initial subsonic flow state and dividing the heat addition corresponding to choked flow, <math>q^\ast</math>, into small portions $\delta q$, one can perform integration as indicated in Figure~\ref{fig:dq}. The result is presented in the in Figure~\ref{fig:TS:closeup}. The subsonic process corresponds to the upper line. As heat is added the Mach number is increased and at <math>M=\gamma^{-1/2}</math> the maximum temperature is reached. Adding more heat will reduce the temperature and increase the Mach number until sonic conditions are reached (<math>M=1.0</math>). As can be seen in Figure~\ref{fig:TS:closeup}, the lean of the subsonic branch of the Rayleigh line is lower than the isobars (gray lines), which means the increasing heat will reduce pressure. The lower part of the blue line in Figure~\ref{fig:TS:closeup} is the supersonic branch of the Rayleigh line, which is obtained in the same way starting from a supersonic flow condition. A flow state resulting in the same sonic conditions as for the subsonic case is calculated and used as a starting state. The corresponding $q^\ast$ is calculated and the same calculation of consecutive flow states in a step-wise manner is performed. As can be seen in Figure~\ref{fig:TS:closeup}, the lean of the supersonic part of the Rayleigh curve is steeper than the isobars (gray lines), which means that pressure increases as heat is added to the flow. As we saw from Eqn.~\ref{eq:governing:dT:diff:mod:b}, <math>dT/dT_o</math> becomes infinite when the flow approaches the sonic the sonic state. After the sonic state is reached, further heat addition is impossible without changing the upstream flow conditions. This will be made clearer in the next section.

	~~\noindent~~ Using the differential relations above, we can get a good picture of the development of flow variables as heat is continuously added to the flow (see Figure~\ref{fig:rayleigh:trends}).		Using the differential relations above, we can get a good picture of the development of flow variables as heat is continuously added to the flow (see Figure~\ref{fig:rayleigh:trends}).

			<!--
	\begin{figure}[ht!]		\begin{figure}[ht!]
	\begin{subfigure}[b]{0.5\textwidth}		\begin{subfigure}[b]{0.5\textwidth}
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	\label{fig:rayleigh:trends}		\label{fig:rayleigh:trends}
	\end{figure}		\end{figure}
			-->
			<!--
	\begin{figure}[ht!]		\begin{figure}[ht!]
	\centering		\centering
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	\label{fig:TS:closeup}		\label{fig:TS:closeup}
	\end{figure}		\end{figure}
			-->

	==== Rayleigh Line ====		==== Rayleigh Line ====

Nian: /* Differential Relations */

2026-03-17T10:08:00Z

Differential Relations

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