A scramjet engine features an intake, isolator, combustor, and a nozzle, as shown in the schematic. Station 3 indicates the combustor entry point. Assume stagnation enthalpy to be constant between Stations 1 and 3, and air to be a calorically perfect gas with specific heat ratio \(\gamma\). Select the correct expression for Mach number \(M_{3}\) at the inlet to the combustor from the options given.
(Figure: schematic of the engine with Station 1 at the intake entry, where the Free stream \(M_{\infty}, T_{\infty}, P_{\infty}\) arrives, Station 2 at the isolator entry, Station 3 at the combustor entry, Station 4 at the combustor exit and Station 5 at the nozzle exit.)
![M3 equals the square root of (2 over gamma minus 1) times {(T-infinity over T3) times [1 plus ((gamma minus 1) over 2) M-infinity squared] minus 1}](/uploads/question-images/gate-ae-2023/q41-b.png)
GATE AE - 2023,
Answer: B
The correct expression is (B), which follows from conservation of stagnation temperature between the free stream and Station 3.
With constant stagnation enthalpy and a calorically perfect gas, the stagnation temperature is the same at the free stream and at Station 3. Write \(k = (\gamma - 1)/2\):
\[ \begin{aligned} T_{0} &= T_{\infty}(1 + kM_{\infty}^{2}) \\ T_{0} &= T_{3}(1 + kM_{3}^{2}) \end{aligned} \]
Equating the two and solving for \(M_{3}\):
\[ \begin{aligned} kM_{3}^{2} &= \frac{T_{\infty}}{T_{3}}(1 + kM_{\infty}^{2}) - 1 \\ M_{3}^{2} &= \frac{1}{k}\left[\frac{T_{\infty}}{T_{3}}(1 + kM_{\infty}^{2}) - 1\right] \end{aligned} \]
Since \(1/k = 2/(\gamma - 1)\), taking the square root gives exactly expression (B).
Expression (C), \(M_{3} = M_{\infty}\sqrt{T_{\infty}/T_{3}}\), would hold only if the velocity were unchanged through the intake, which ignores the deceleration. Expressions (A) and (D) do not follow from the energy equation; for example, at \(T_{3} = T_{\infty}\) they give \(M_{3} = 0\) or the square root of a negative number, whereas the correct result gives \(M_{3} = M_{\infty}\).
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