One-dimensional flow with heat addition: Difference between revisions

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\noindent The aim is to derive relations for pressure ratio and temperature ratio as a function of Mach numbers. We will do that starting from the momentum equation.\\
The aim is to derive relations for pressure ratio and temperature ratio as a function of Mach numbers. We will do that starting from the momentum equation.


\begin{equation}
<math display="block">
p_2-p_1=\rho_1 u_1^2 - \rho_2 u_2^2  
p_2-p_1=\rho_1 u_1^2 - \rho_2 u_2^2  
\label{eq:governing:mom}
</math>
\end{equation}\\


\noindent Assuming calorically perfect gas\\
Assuming calorically perfect gas


\[\rho u^2=\rho a^2 M^2=\rho \frac{\gamma p}{\rho} M^2=\gamma p M^2\]\\
<math display="block">
\rho u^2=\rho a^2 M^2=\rho \frac{\gamma p}{\rho} M^2=\gamma p M^2
</math>


\noindent which inserted in Eqn. \ref{eq:governing:mom} gives\\
which inserted in Eqn. \ref{eq:governing:mom} gives


\[p_2-p_1=\gamma p_1 M_1^2 - \gamma p_2 M_2^2\]\\
<math display="block">
p_2-p_1=\gamma p_1 M_1^2 - \gamma p_2 M_2^2
</math>


\[p_2(1+\gamma M_2^2)=p_1(1+\gamma M_1^2)\]\\
<math display="block">
p_2(1+\gamma M_2^2)=p_1(1+\gamma M_1^2)
</math>


\noindent and thus\\
and thus


\begin{equation}
<math display="block">
\frac{p_2}{p_1}=\frac{1+\gamma M_1^2}{1+\gamma M_2^2}
\frac{p_2}{p_1}=\frac{1+\gamma M_1^2}{1+\gamma M_2^2}
\label{eq:governing:mom:b}
</math>
\end{equation}\\


\noindent From the equation of state $p=\rho RT$, we get\\
From the equation of state <math>p=\rho RT</math>, we get


\begin{equation}
<math display="block">
\frac{T_2}{T_1}=\frac{p_2}{\rho_2 R}\frac{\rho_1 R}{p_1}=\frac{p_2}{p_1}\frac{\rho_1}{\rho_2}
\frac{T_2}{T_1}=\frac{p_2}{\rho_2 R}\frac{\rho_1 R}{p_1}=\frac{p_2}{p_1}\frac{\rho_1}{\rho_2}
\label{eq:tr:a}
</math>
\end{equation}\\


\noindent Using the continuity equation, we can get $\rho_1/\rho_2$\\
Using the continuity equation, we can get <math>\rho_1/\rho_2</math>


\[\rho_1 u_1=\rho_2 u_2 \Rightarrow \frac{\rho_1}{\rho_2}=\frac{u_2}{u_1}\]\\
<math display="block">
\rho_1 u_1=\rho_2 u_2 \Rightarrow \frac{\rho_1}{\rho_2}=\frac{u_2}{u_1}
</math>


\noindent Inserted in Eqn. \ref{eq:tr:a} gives\\
Inserted in Eqn. \ref{eq:tr:a} gives


\begin{equation}
<math display="block">
\frac{T_2}{T_1}=\frac{p_2}{p_1}\frac{u_2}{u_1}
\frac{T_2}{T_1}=\frac{p_2}{p_1}\frac{u_2}{u_1}
\label{eq:tr:b}
</math>
\end{equation}\\


\begin{equation}
<math display="block">
\frac{u_2}{u_1}=\frac{M_2a_2}{M_1a_1}=\frac{M_2}{M_1}\frac{\sqrt{\gamma RT_2}}{\sqrt{\gamma RT_1}}=\frac{M_2}{M_1}\sqrt{\frac{T_2}{T_1}}
\frac{u_2}{u_1}=\frac{M_2a_2}{M_1a_1}=\frac{M_2}{M_1}\frac{\sqrt{\gamma RT_2}}{\sqrt{\gamma RT_1}}=\frac{M_2}{M_1}\sqrt{\frac{T_2}{T_1}}
\label{eq:tr:c}
</math>
\end{equation}\\


\noindent Eqn. \ref{eq:tr:c} in Eqn. \ref{eq:tr:b} gives\\
Eqn. \ref{eq:tr:c} in Eqn. \ref{eq:tr:b} gives


\begin{equation}
<math display="block">
\sqrt{\frac{T_2}{T_1}}=\frac{p_2}{p_1}\frac{M_2}{M_1}
\sqrt{\frac{T_2}{T_1}}=\frac{p_2}{p_1}\frac{M_2}{M_1}
\label{eq:tr:d}
</math>
\end{equation}\\


\noindent With $p_2/p_1$ from Eqn. \ref{eq:governing:mom:b}, Eqn \ref{eq:tr:d} becomes\\
With <math>p_2/p_1</math> from Eqn. \ref{eq:governing:mom:b}, Eqn \ref{eq:tr:d} becomes


\begin{equation}
<math display="block">
\frac{T_2}{T_1}=\left(\frac{1+\gamma M_1^2}{1+\gamma M_2^2}\right)^2\left(\frac{M_2}{M_1}\right)^2
\frac{T_2}{T_1}=\left(\frac{1+\gamma M_1^2}{1+\gamma M_2^2}\right)^2\left(\frac{M_2}{M_1}\right)^2
\label{eq:tr:e}
</math>
\end{equation}


==== Differential Relations ====
==== Differential Relations ====
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