GATE 2022 AE – Question 55
A convergent-divergent nozzle with adiabatic walls is designed for an exit Mach number of 2.3. It is discharging air to atmosphere under the conditions indicated in the figure. Flow through the nozzle is inviscid, the characteristic gas constant for air is 287 J/(kg-K), and γ=1.4. When the reservoir pressure is 25 bar (absolute), and temperature is 300 K, Prandtl- Meyer expansion waves appear at the nozzle exit as shown. The minimum percentage change in the reservoir pressure required to eliminate the wave system at the nozzle exit under steady state is _______%.
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Correct answer: 49.73 to 50.23
Explanation
**Condition for no waves at the exit.** The nozzle is designed for $M_e=2.3$. Expansion (Prandtl-Meyer) waves appear at the exit when the exit pressure is **higher than ambient**... (the jet is under-expanded). The waves vanish when the exit pressure equals the ambient pressure, i.e. when the nozzle is perfectly expanded.
**Required reservoir pressure.** For isentropic flow to $M_e=2.3$ the pressure ratio is
$$\frac{p_0}{p_e}=\left(1+\frac{\gamma-1}{2}M_e^2\right)^{\gamma/(\gamma-1)}=(1+0.2\times2.3^2)^{3.5}=(2.058)^{3.5}=12.50 .$$
With $p_e=p_a$ (1 bar here), the reservoir pressure must be $p_0=12.50\times1=12.50$ bar.
**Change in reservoir pressure:**
$$\frac{25-12.50}{25}\times100=\mathbf{49.98\%\ (a\ reduction)}.$$