Governing equations on integral form: Difference between revisions

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{{quote|Energy can be neither created nor destroyed; it can only change in form}}
{{quote|Energy can be neither created nor destroyed; it can only change in form}}


\[E_1+E_2=E_3\]\\
<math display="block">
E_1+E_2=E_3
</math>


\begin{itemize}
;<math>E_1</math> Rate of heat added to the fluid in <math>\Omega</math> from the surroundings
\item[$E_1$:] Rate of heat added to the fluid in $\Omega$ from the surroundings
: heat transfer
\begin{itemize}
: radiation
\item heat transfer
;<math>E_2</math> Rate of work done on the fluid in <math>\Omega</math>
\item radiation
;<math>E_3</math> Rate of change of energy of the fluid as it flows through <math>\Omega</math>
\end{itemize}
\item[$E_2$:] Rate of work done on the fluid in $\Omega$
\item[$E_3$:] Rate of change of energy of the fluid as it flows through $\Omega$
\end{itemize}


\[E_1=\iiint_{\Omega} \dot{q}\rho d\mathscr{V}\]\\
<math display="block">
E_1=\iiint_{\Omega} \dot{q}\rho dV
</math>


\noindent where $\dot{q}$ is the rate of heat added per unit mass\\
where <math>\dot{q}</math> is the rate of heat added per unit mass


\noindent The rate of work done on the fluid in $\Omega$ due to pressure forces is obtained from the pressure force term in the momentum equation.\\
The rate of work done on the fluid in $\Omega$ due to pressure forces is obtained from the pressure force term in the momentum equation.


\[E_{2_{pressure}}=-\oiint_{\partial \Omega}(p\mathbf{n}dS)\cdot\mathbf{v}=-\oiint_{\partial \Omega} p\mathbf{v}\cdot\mathbf{n}dS\]\\
<math display="block">
E_{2_{pressure}}=-\iint_{\partial \Omega}(p\mathbf{n}dS)\cdot\mathbf{v}=-\iint_{\partial \Omega} p\mathbf{v}\cdot\mathbf{n}dS
</math>


\noindent The rate of work done on the fluid in $\Omega$ due to body forces is\\
The rate of work done on the fluid in $\Omega$ due to body forces is


\[E_{2_{body\ forces}}=\iiint_{\Omega}(\rho\mathbf{f}d\mathscr{V})\cdot\mathbf{v}=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}d\mathscr{V}\]\\
<math display="block">
E_{2_{body\ forces}}=\iiint_{\Omega}(\rho\mathbf{f}dV)\cdot\mathbf{v}=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV
</math>


\[E_2=E_{2_{pressure}}+E_{2_{body\ forces}}=-\oiint_{\partial \Omega} p\mathbf{v}\cdot\mathbf{n}dS+\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}d\mathscr{V}\]\\
<math display="block">
E_2=E_{2_{pressure}}+E_{2_{body\ forces}}=-\iint_{\partial \Omega} p\mathbf{v}\cdot\mathbf{n}dS+\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV
</math>


\noindent The energy of the fluid per unit mass is the sum of internal energy $e$ (molecular energy) and the kinetic energy $V^2/2$ and the net energy flux over the control volume surface is calculated by the following integral\\
The energy of the fluid per unit mass is the sum of internal energy <math>e</math> (molecular energy) and the kinetic energy <math>V^2/2</math> and the net energy flux over the control volume surface is calculated by the following integral


\[\oiint_{\partial \Omega}(\rho \mathbf{v}\cdot\mathbf{n}dS)\left(e+\frac{V^2}{2}\right)\]\\
<math display="block">
\iint_{\partial \Omega}(\rho \mathbf{v}\cdot\mathbf{n}dS)\left(e+\frac{V^2}{2}\right)
</math>


\noindent Analogous to mass and momentum, the total amount of energy of the fluid in $\Omega$ is calculated as\\
Analogous to mass and momentum, the total amount of energy of the fluid in <math>\Omega</math> is calculated as


\[\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)d\mathscr{V}\]\\
<math display="block">
\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)dV
</math>


\noindent The time rate of change of the energy of the fluid in $\Omega$ is obtained as\\
The time rate of change of the energy of the fluid in <math>\Omega</math> is obtained as


\[\frac{d}{dt}\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)d\mathscr{V}\]\\
<math display="block">
\frac{d}{dt}\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)dV
</math>


\noindent Now, $E_3$ is obtained as the sum of the time rate of change of energy of the fluid in $\Omega$ and the net flux of energy carried by fluid passing the control volume surface.\\
Now, <math>E_3</math> is obtained as the sum of the time rate of change of energy of the fluid in $\Omega$ and the net flux of energy carried by fluid passing the control volume surface.


\[E_3=\frac{d}{dt}\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)d\mathscr{V}+\oiint_{\partial \Omega}(\rho \mathbf{v}\cdot\mathbf{n}dS)\left(e+\frac{V^2}{2}\right)\]\\
<math display="block">
E_3=\frac{d}{dt}\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)dV+\iint_{\partial \Omega}(\rho \mathbf{v}\cdot\mathbf{n}dS)\left(e+\frac{V^2}{2}\right)
</math>


\noindent With all elements of the energy equation defined, we are now ready to finally compile the full equation\\
With all elements of the energy equation defined, we are now ready to finally compile the full equation


\begin{equation}
<math display="block">
\frac{d}{dt}\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)d\mathscr{V}+\oiint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}d\mathscr{V}+\iiint_{\Omega} \dot{q}\rho d\mathscr{V}
\frac{d}{dt}\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)dV+\iint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV
\label{eq:governing:integral:energy}
</math>
\end{equation}\\


\noindent The surface integral in the energy equation may be rewritten as\\
The surface integral in the energy equation may be rewritten as


\[\oiint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=\oiint_{\partial \Omega}\rho\left[e+\frac{p}{\rho}+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS\]\\
<math display="block">
\iint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=\iint_{\partial \Omega}\rho\left[e+\frac{p}{\rho}+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS
</math>


\noindent and with the definition of enthalpy $h=e+p/\rho$, we get\\
and with the definition of enthalpy <math>h=e+p/\rho</math>, we get


\[\oiint_{\partial \Omega}\rho\left[h+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS\]\\
<math display="block">
\iint_{\partial \Omega}\rho\left[h+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS
</math>


\noindent Furthermore, introducing total internal energy $e_o$ and total enthalpy $h_o$ defined as\\
Furthermore, introducing total internal energy <math>e_o</math> and total enthalpy <math>h_o</math> defined as


\[e_o=e+\frac{1}{2}V^2\]\\
<math display="block">
e_o=e+\frac{1}{2}V^2
</math>


and\\
and


\[h_o=h+\frac{1}{2}V^2\]\\
<math display="block">
h_o=h+\frac{1}{2}V^2
</math>


\noindent the energy equation is written as\\
the energy equation is written as


\begin{equation}
<math display="block">
\frac{d}{dt}\iiint_{\Omega}\rho e_o d\mathscr{V}+\oiint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}d\mathscr{V}+\iiint_{\Omega} \dot{q}\rho d\mathscr{V}
\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV
\label{eq:governing:integral:energy:b}
</math>
\end{equation}\\


==== Summary ====
==== Summary ====
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