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I am trying to calculate prior to testing the expected back pressure resulting from injecting gaseous oxygen through an injector orifice (choked) into a rocket combustion chamber before ignition. I have tried using the Borda-Carnot equation with the assumption that the chamber mean flow is sufficiently low and that the pressure differential between the chamber and atmosphere is sufficient to drive the same mass flow through the nozzle that was injected into the chamber. Calculating the delta P under these assumptions yields an unreasonably small delta P across the nozzle. Is there a hand-calc method available that would provide some insight to this? I know that there are highly localized dynamics in the gas flow that are likely going to cause significant deviation from the analytical values, but I am mostly looking for a first-order estimate.

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I believe that the Borda-Carnot equation is only valid for incompressible flows, while you mention that the injector orifice is choked. While the flow becomes incompressible pretty quickly after the orifice, at this precise point, it is not.

In this case, I think you should use 1D isentropic nozzle theory (wikipedia offers a good start on the topic).

If the injector orifice is choked and there is no shock in the injector channel or downstream of it, then the pressure ratio should be

$$ P_{res}/P_{a} = \left(\frac{\gamma+1}{2}\right)^{\frac{\gamma}{\gamma-1}} $$

with $P_{res}$ the tank pressure, $P_a$ the ambient pressure, $\gamma = \frac{c_p}{c_v}$ the ratio of heat capacities.

That's assuming you are injecting gas into gas. If the fuel in the tank is liquid, then the situation is different because of evaporation.

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