Choked flow

Geometric Choking

For steady-state nozzle flow, the massflow is obtained as

$$\dot{m}=\rho uA=const$$

Eqn. \ref{eq:massflow:a} can be evaluated at any location inside the nozzle and if evaluated at sonic conditions we get

$$\dot{m}={\rho }^{*}{u}^{*}{A}^{*}$$

By definition $u^{*}=a^{*}$ and thus

$$\dot{m}={\rho }^{*}{a}^{*}{A}^{*}$$

${\rho }^{*}$ and $a^{*}$ can be obtained using the ratios ${\rho }^{*}/{\rho }_o$ and $a^{*}/a_o$

$${\rho }^{*}=\left(\frac{{\rho }^{*}}{{\rho }_o}\right){\rho }_o=\frac{p_o}{R{T}_o}{\left(\frac{2}{\gamma +1}\right)}^{1/(\gamma -1)}$$

$$a^{*}=\left(\frac{a^{*}}{a_o}\right)a_o=a_o{\left(\frac{2}{\gamma +1}\right)}^{1/2}=\sqrt{\gamma R{T}_o}{\left(\frac{2}{\gamma +1}\right)}^{1/2}$$

Eqns. \ref{eq:as} and \ref{eq:rhos} in Eqn. \ref{eq:massflow:b} gives

$$\dot{m}=\frac{p_o}{R{T}_o}{\left(\frac{2}{\gamma +1}\right)}^{1/(\gamma -1)}\sqrt{\gamma R{T}_o}{\left(\frac{2}{\gamma +1}\right)}^{1/2}{A}^{*}$$

which can be rewritten as

$$\dot{m}=\frac{p_o{A}^{*}}{\sqrt{T_o}}\sqrt{\frac{\gamma }{R}{\left(\frac{2}{\gamma +1}\right)}^{(\gamma +1)/(\gamma -1)}}$$

Eqn. \ref{eq:massflow:c} valid for:

• quasi-one-dimensional flow
• steady state
• inviscid flow
• calorically perfect gas

It should be noted that the choked massflow can be calculated using Eqn. \ref{eq:massflow:c} even for cases with shocks downstream of the throat.