The GATE Grind

GATE 2023 AE – Question 39

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

A perfect gas stored in a large reservoir exhausts into the atmosphere through a convergent duct. The reservoir pressure is P0 and temperature is T0. The jet emerges from the nozzle at choked conditions with average velocity u, Mach number M, pressure p, temperature T, and density ρ. If the reservoir pressure is increased, then

  1. u, M, p, T, and ρ increase
  2. u, p, T, and ρ increase, but M remains the same
  3. u, M, and T remain the same, but p and ρ increase
  4. u, M, T and ρ remain the same, but p increases

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Correct answer: (C) u, M, and T remain the same, but p and ρ increase

Explanation

The nozzle is convergent and **choked**, so the exit Mach number is $M=1$, whatever the reservoir pressure (as long as it stays choked).

**Exit static temperature.** For $M=1$: $T=\dfrac{T_0}{1+(\gamma-1)/2}=\dfrac{2T_0}{\gamma+1}$. It depends only on the reservoir temperature $T_0$, which is unchanged. So **$T$ is unchanged**.

**Exit velocity.** $u=a=\sqrt{\gamma RT}$, which depends only on $T$, so **$u$ is unchanged**.

**Exit pressure.** $p=p_0\left(\dfrac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)}$ is proportional to $p_0$, so **$p$ increases**.

**Density.** $\rho=p/(RT)$, with $T$ fixed and $p$ larger, so **$\rho$ increases**.

So $u$, $M$ and $T$ stay the same while $p$ and $\rho$ increase: option **C**.