Choked flow: Difference between revisions
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=== Geometric Choking === | === Geometric Choking === | ||
For steady-state nozzle flow, the massflow is obtained as | For steady-state nozzle flow, the massflow is obtained as | ||
<math | {{NumEqn|<math> | ||
\dot{m}=\rho uA=const | \dot{m}=\rho uA=const | ||
</math> | </math>}} | ||
Eqn. \ref{eq:massflow:a} can be evaluated at any location inside the nozzle and if evaluated at sonic conditions we get | Eqn. \ref{eq:massflow:a} can be evaluated at any location inside the nozzle and if evaluated at sonic conditions we get | ||
<math | {{NumEqn|<math> | ||
\dot{m}=\rho^* u^* A^* | \dot{m}=\rho^* u^* A^* | ||
</math> | </math>}} | ||
By definition <math>u^*=a^*</math> and thus | By definition <math>u^*=a^*</math> and thus | ||
<math | {{NumEqn|<math> | ||
\dot{m}=\rho^* a^* A^* | \dot{m}=\rho^* a^* A^* | ||
</math> | </math>}} | ||
<math>\rho^*</math> and <math>a^*</math> can be obtained using the ratios <math>\rho^*/\rho_o</math> and <math>a^*/a_o</math> | <math>\rho^*</math> and <math>a^*</math> can be obtained using the ratios <math>\rho^*/\rho_o</math> and <math>a^*/a_o</math> | ||
<math | {{NumEqn|<math> | ||
\rho^*=\left(\frac{\rho^*}{\rho_o}\right)\rho_o=\frac{p_o}{RT_o}\left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)} | \rho^*=\left(\frac{\rho^*}{\rho_o}\right)\rho_o=\frac{p_o}{RT_o}\left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)} | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
a^*=\left(\frac{a^*}{a_o}\right)a_o=a_o\left(\frac{2}{\gamma+1}\right)^{1/2}=\sqrt{\gamma RT_o}\left(\frac{2}{\gamma+1}\right)^{1/2} | a^*=\left(\frac{a^*}{a_o}\right)a_o=a_o\left(\frac{2}{\gamma+1}\right)^{1/2}=\sqrt{\gamma RT_o}\left(\frac{2}{\gamma+1}\right)^{1/2} | ||
</math> | </math>}} | ||
Eqns. \ref{eq:as} and \ref{eq:rhos} in Eqn. \ref{eq:massflow:b} gives | Eqns. \ref{eq:as} and \ref{eq:rhos} in Eqn. \ref{eq:massflow:b} gives | ||
<math | {{NumEqn|<math> | ||
\dot{m}=\frac{p_o}{RT_o}\left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)} | \dot{m}=\frac{p_o}{RT_o}\left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)} | ||
\sqrt{\gamma RT_o}\left(\frac{2}{\gamma+1}\right)^{1/2} | \sqrt{\gamma RT_o}\left(\frac{2}{\gamma+1}\right)^{1/2} | ||
A^* | A^* | ||
</math> | </math>}} | ||
which can be rewritten as | which can be rewritten as | ||
<math | {{NumEqn|<math> | ||
\dot{m}=\frac{p_o A^*}{\sqrt{T_o}}\sqrt{\frac{\gamma}{R}\left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}} | \dot{m}=\frac{p_o A^*}{\sqrt{T_o}}\sqrt{\frac{\gamma}{R}\left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}} | ||
</math> | </math>}} | ||
Eqn. \ref{eq:massflow:c} valid for: | Eqn. \ref{eq:massflow:c} valid for: | ||
