Designing of CD nozzle with respect to Mass flow rate of propellants

Question

How do I calculate the flow rate of exhaust gases required at the throat of a CD nozzle, with given throat diameter of "D", so as to achieve a "Choked flow condition" (velocity of gases exiting the throat equals Mach 1)

Answer 1

Introduction

For a Convergent-Divergent Nozzle to achieve choked flow conditions, a certain pressure-ratio must be achieved:

$$\left(\frac{p_{t}}{p_{c}}\right)_{\text {cr }}=\left(\frac{2}{\gamma+1}\right)^{\left(\frac{\gamma}{\gamma-1}\right)}$$

With $\left(\frac{p_{t}}{p_{c}}\right)_{\text {cr }}$ the critical pressure-ratio, $p$ the pressure, $\gamma$ the specific heat ratio, and subscripts $_t$ and $_c$ referring to throat and chamber conditions respectively.

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We also have the following relations for an isentropic ideal gas: $$ \left(\frac{T_t}{T_{c}}\right)=\left(\frac{p_t}{p_{c}}\right)^{\left(\frac{\gamma-1}{\gamma}\right)}=\left(\frac{\rho_t}{\rho_{c}}\right)^{(\gamma-1)} $$ With $T$ the temperature and $\rho$ the density

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And the continuity equation for mass flow: $$\dot{m}= \rho_t A_t U_t = \rho_c A_c U_c = \rho_t A_t a_t$$ With $\dot{m}$ the mass flow, $A$ the cross-sectional area, $U$ the flow velocity, and $a$ the speed of sound (flow velocity is equal to the speed of sound at the throat for choked flow). Subscripts $_t$ and $_c$ once again refer to throat and chamber conditions.

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The speed of sound is defined as: $$a = \sqrt{\gamma\ R_{sp}\ T}$$ Where $R_{sp}$ is the specific gas constant (not to be confused with the universal gas constant)

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And, finally, the ideal gas law: $$\rho = \frac{p}{R_{sp}\ T}$$

Answer

Combining all of the relations above leads to an equation for the mass flow in a convergent-divergent nozzle assuming isentropic flow (for the derivation specifics see this link on choked flow by NASA) :

$$\dot{m}=\frac{A_t p_c}{\sqrt{R_{sp}\cdot T_c}} \sqrt{\gamma}\bigg(\frac{\gamma+1}{2}\bigg)^{-\frac{\gamma+1}{2(\gamma-1)}}$$

With $A_t$ the throat area, $p_c$ the chamber pressure, $R_{sp}$ the specific gas constant, $T_c$ the chamber temperature, and $\gamma$ the specific heat ratio.

Now I don't like the look of all those $\gamma$'s, so let's replace them with the Vandenkerckhove function $\Gamma$, just to make it look nicer (it's exactly the same otherwise)

$$\dot{m}=\frac{\Gamma\ A_t\ p_c}{\sqrt{R_{sp}\ T_c}}$$

With a certain propellant selected, $\Gamma$ and $R_{sp}$ should be known. If you know the throat diameter $D_t$ you know the throat area $A_t$, but as you can see you should also know the chamber pressure and chamber temperature, $p_c$ and $T_c$, to be able to determine the mass flow in the nozzle.