One-dimensional flow with heat addition: Difference between revisions

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==== 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}.


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\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}).


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\label{fig:rayleigh:trends}
\label{fig:rayleigh:trends}
\end{figure}
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\begin{figure}[ht!]
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\label{fig:TS:closeup}
\label{fig:TS:closeup}
\end{figure}
\end{figure}
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==== Rayleigh Line ====
==== Rayleigh Line ====
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