The entropy equation: Difference between revisions
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From the second law of thermodynamics | From the second law of thermodynamics | ||
<math | {{NumEqn|<math> | ||
\frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right) | \frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right) | ||
</math> | </math>}} | ||
From the energy equation on differential non-conservation form internal energy formulation | From the energy equation on differential non-conservation form internal energy formulation | ||
<math | {{NumEqn|<math> | ||
\frac{De}{Dt} = \dot{q} - \frac{p}{\rho}(\nabla\cdot\mathbf{v}) | \frac{De}{Dt} = \dot{q} - \frac{p}{\rho}(\nabla\cdot\mathbf{v}) | ||
</math> | </math>}} | ||
The continuity equation on differential non-conservation form | The continuity equation on differential non-conservation form | ||
<math | {{NumEqn|<math> | ||
\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 \Rightarrow \nabla\cdot\mathbf{v}=-\frac{1}{\rho}\frac{D\rho}{Dt} | \frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 \Rightarrow \nabla\cdot\mathbf{v}=-\frac{1}{\rho}\frac{D\rho}{Dt} | ||
</math> | </math>}} | ||
and thus | and thus | ||
<math | {{NumEqn|<math> | ||
\frac{De}{Dt} = \dot{q} +\frac{p}{\rho^2}\frac{D\rho}{Dt} | \frac{De}{Dt} = \dot{q} +\frac{p}{\rho^2}\frac{D\rho}{Dt} | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
\frac{D\rho}{Dt}=-\frac{1}{\nu^2}\frac{D\nu}{Dt} | \frac{D\rho}{Dt}=-\frac{1}{\nu^2}\frac{D\nu}{Dt} | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
\rho\frac{De}{Dt} = \rho\dot{q} -\frac{p}{\rho\nu^2}\frac{D\nu}{Dt} = \rho\dot{q} -\rho p\frac{D\nu}{Dt} | \rho\frac{De}{Dt} = \rho\dot{q} -\frac{p}{\rho\nu^2}\frac{D\nu}{Dt} = \rho\dot{q} -\rho p\frac{D\nu}{Dt} | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
\rho\left[\frac{De}{Dt}+p\frac{D\nu}{Dt}-\dot{q}\right]=0\Rightarrow\frac{De}{Dt}=\dot{q}-p\frac{D\nu}{Dt} | \rho\left[\frac{De}{Dt}+p\frac{D\nu}{Dt}-\dot{q}\right]=0\Rightarrow\frac{De}{Dt}=\dot{q}-p\frac{D\nu}{Dt} | ||
</math> | </math>}} | ||
Insert <math>De/Dt</math> in Eqn. \ref{eq:second:law} | Insert <math>De/Dt</math> in Eqn. \ref{eq:second:law} | ||
<math | {{NumEqn|<math> | ||
\dot{q}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)\Rightarrow | \dot{q}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)\Rightarrow | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
T\frac{Ds}{Dt}=-\dot{q} | T\frac{Ds}{Dt}=-\dot{q} | ||
</math> | </math>}} | ||
Adiabatic flow: | Adiabatic flow: | ||
<math | {{NumEqn|<math> | ||
T\frac{Ds}{Dt}=0 | T\frac{Ds}{Dt}=0 | ||
</math> | </math>}} | ||
In an adiabatic, steady-state, inviscid flow, entropy is constant along a streamline. | In an adiabatic, steady-state, inviscid flow, entropy is constant along a streamline. | ||
