One-dimensional flow with heat addition: Difference between revisions
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Created page with "Category:Compressible flow Category:One-dimensional flow Category:Inviscid flow Category:Continuous solution __TOC__ \section{One-Dimensional Flow with Heat Addition} \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.\\ \begin{equation} p_2-p_1=\rho_1 u_1^2 - \rho_2 u_2^2 \label{eq:governing:mom} \end{equation}\\ \noindent Assuming calo..." |
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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.\\ | \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.\\ | ||
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\end{equation} | \end{equation} | ||
==== Differential Relations ==== | |||
\noindent The equations presented in the previous section gives us the flow state after heat addition but since the heat addition, unlike the normal shock, is a continuous process, it is of interest to study the the heat addition from start to end. In order to do so we will now derive differential relations starting from the governing equations on differential form. We will start with converting the integral equation for conservation of mass for one-dimensional flows to differential form.\\ | \noindent The equations presented in the previous section gives us the flow state after heat addition but since the heat addition, unlike the normal shock, is a continuous process, it is of interest to study the the heat addition from start to end. In order to do so we will now derive differential relations starting from the governing equations on differential form. We will start with converting the integral equation for conservation of mass for one-dimensional flows to differential form.\\ | ||
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\end{equation}\\ | \end{equation}\\ | ||
==== 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}.\\ | \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}.\\ | ||
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\end{figure} | \end{figure} | ||
==== Rayleigh Line ==== | |||
\noindent The continuity equation for steady-state, one-dimensional flow reads | \noindent The continuity equation for steady-state, one-dimensional flow reads | ||
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\end{equation}\\ | \end{equation}\\ | ||
==== 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}). | \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}). | ||
