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)


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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.

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

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.

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)

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


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.

  • $\begingroup$ Thanks. Your concluding line sums it all well. However, as a hydraulic engineer, I understand that flow across an orifice depends (directly proportional to) on only two factors. 1: the Difference in pressure across the orifice, and 2: the diameter of the orifice. So, in a way, for a given propellant, I am looking for relation between throat dia, and the pressure diff. across the throat (being directly proportional to the temp in the chamber - other properties remaining same for given propellant), which would create such amount of mass flow, that the flow Speed would exceed M1. $\endgroup$
    – Niranjan
    Dec 2, 2021 at 11:52
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    $\begingroup$ @Niranjan And here I thought that the very concept of "choked flow" was that the downsteam pressure is irrelevant, as long as it is low enough to allow the choked flow to establish. $\endgroup$ Dec 4, 2021 at 20:53
  • $\begingroup$ @CuteKitty: Massflow through a CD nozzle for "compressible fluids" is quite perplexing. Since mass cannot be created / destroyed, velocity of mass flowing through an orifice increases as its diameter decreases, (converging nozzle) to let the same mass pass through, in the same amount of time. But for this increase in speed, we need to push the mass with more force (increase pressure prior to the orifice - i.e. increase differential pressure) However, if we increase the pressure difference to infinity, will we get infinite speed at the orifice? NO. Refer next comment to continue: $\endgroup$
    – Niranjan
    Dec 7, 2021 at 1:07
  • $\begingroup$ @CuteKitty: In continuation to my last comment: There is a limit on the mass flow through an orifice. The flow rate cannot increase beyond a value REGARDLESS of the increase in differential pressure. This is dictated by the Coefficient of Discharge of the orifice. Which means there would be a limit to velocity of mass at the exit of orifice. But this is for "non-compressible" fluids. In case of compressible fluids, this exit velocity INCREASES further, "WITHOUT INCREASING MASS FLOW", if the mass flow velocity has reached Mach 1 at the throat. This helps us in creating more "Momentum"... $\endgroup$
    – Niranjan
    Dec 7, 2021 at 1:17
  • $\begingroup$ @CuteKitty: Continued from last comment.. Such increase in momentum (Mass X Velocity) comes at the cost of pressure. Thrust of an engine would depend on the momentum of exhaust gases. Exhausting gases through the throat, with higher velocity, gives us higher momentum, & thus higher thrust. (off course it again depends on the differential pressure between that of the diverging nozzle exit & outside atmospheric pressure, where the exhaust is released. This is to the best of my understanding. Hope you find this informative & useful. $\endgroup$
    – Niranjan
    Dec 7, 2021 at 1:27

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