# Find the combustion chamber pressure of a rocket engine.

Find the combustion chamber pressure of a rocket engine using hydrogen and oxygen as propellants, whose mass flow in combustion chamber is \(300\,kg/s\). The temperature of the combustion chamber is \(3500\,K\). Gas flowing into the engine has a ratio of specific heats of \(1.2\) and a molecular weight of \(16\). The rocket nozzle throat area is \(0.25\,m^{2}\).

Mass flow rate at the throat which is at sonic condition is \[\dot{m}=\rho ^{*}A^{*}V^{*}\]Specific gas constant \[R=\frac{8314}{16}=519.6\,J/\left ( kg\,K \right )\]At sonic condition, \[\frac{\rho ^{*}}{\rho _{0}}=\left ( \frac{2}{\gamma +1} \right )^{\frac{1}{\gamma -1}}

=\left ( \frac{2}{1.2+1} \right )^{\left ( \frac{1}{1.2-1} \right )}=0.621\]\[\rho ^{*}=\left ( \frac{\rho ^{*}}{\rho _{0}} \right )\rho _{0}=0.621\rho _{0}=0.621\left ( \frac{p_{0}}{RT_{0}} \right )

=0.621\left ( \frac{p_{0}}{519.6\times 3500K} \right )=3.415\times 10^{-7}p_{0}\]At sonic condition,

\[\left ( \frac{T^{*}}{T_{0}} \right )=\left ( \frac{2}{\gamma +1} \right )=\left ( \frac{2}{1.2 +1} \right )=0.9091\]\[T^{*}=\left ( \frac{T^{*}}{T_{0}} \right )T_{0}=0.9091\times 3500=3181.85\,K\]Also, at throat\[V^{*}=a^{*}=\sqrt{\gamma RT^{*}}=\sqrt{\left ( 1.2 \right )\left ( 519.6 \right )\left ( 3500 \right )}=1477.27\,m/s\]Since, \(\dot{m}=\rho ^{*}A^{*}V^{*}\)\[\Rightarrow 300=\left ( 3.415\times 10^{-7} p_{0}\right )\times \left ( 0.25 \right )\times 1477.27\]\[\Rightarrow p_{0}=2.3787\times 10^{6}\,N/m^{2}\]