# Combustion Chamber Back Pressure due to gas injection

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.

$$P_{res}/P_{a} = \left(\frac{\gamma+1}{2}\right)^{\frac{\gamma}{\gamma-1}}$$
with $$P_{res}$$ the tank pressure, $$P_a$$ the ambient pressure, $$\gamma = \frac{c_p}{c_v}$$ the ratio of heat capacities.