The GATE Grind

GATE 2023 AE – Question 41

Propulsion · Intakes and nozzles: non-rotating propulsion components · 2 marks · Multiple choice

A scramjet has intake, isolator, combustor and nozzle. Station 3 is combustor entry. With constant stagnation enthalpy from free stream to 3 and specific-heat ratio γ, select $M_3$.

source diagram and its labels are reproduced in the attached image.
  1. $\sqrt{\frac{2}{\gamma-1}(\frac{T_\infty}{T_3}M_\infty-1)}$
  2. $\sqrt{\frac{2}{\gamma-1}\left[\frac{T_\infty}{T_3}(1+\frac{\gamma-1}{2}M_\infty^2)-1\right]}$
  3. $M_\infty T_\infty/T_3$
  4. $\sqrt{\frac{T_\infty}{T_3}\left(\frac{\gamma+1}{2}M_\infty-1\right)}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $\sqrt{\frac{2}{\gamma-1}\left[\frac{T_\infty}{T_3}(1+\frac{\gamma-1}{2}M_\infty^2)-1\right]}$

Explanation

The flow from the free stream to station 3 has **constant stagnation enthalpy**, so for a calorically perfect gas the stagnation temperature is also constant:
$$T_{0,3}=T_{0,\infty} .$$

**Write both stagnation temperatures with the isentropic relation** $T_0=T\left(1+\dfrac{\gamma-1}{2}M^2\right)$:
$$T_3\left(1+\frac{\gamma-1}{2}M_3^2\right)=T_\infty\left(1+\frac{\gamma-1}{2}M_\infty^2\right).$$

**Solve for $M_3$.**
$$1+\frac{\gamma-1}{2}M_3^2=\frac{T_\infty}{T_3}\left(1+\frac{\gamma-1}{2}M_\infty^2\right)$$
$$M_3=\sqrt{\frac{2}{\gamma-1}\left[\frac{T_\infty}{T_3}\left(1+\frac{\gamma-1}{2}M_\infty^2\right)-1\right]}\quad(\text{option B}).$$