GATE ME 2022 Set 1 — Question 30

For a purely convergent nozzle operating under choked conditions, the Mach number at the exit plane (throat) is exactly unity ($M_e = 1$).The mass flow rate ($\dot{m}$) for a choked flow is given by the expression:$$\dot{m} = \frac{P_0 A_e}{\sqrt{T_0}} \sqrt{\frac{\gamma}{R}} \left( \frac{2}{\gamma+1} \right)^{\frac{\gamma+1}{2(\gamma-1)}}$$where:

• $P_0$ is the inlet stagnation pressure
• $A_e$ is the exit area
• $T_0$ is the inlet stagnation temperature
• $\gamma$ is the ratio of specific heats
• $R$ is the specific gas constant

From this equation, it is evident that the mass flow rate $\dot{m}$ is directly proportional to the inlet pressure $P_0$ (i.e., $\dot{m} \propto P_0$). Therefore, if $P_0$ is increased, the mass flow rate through the nozzle will increase. This makes statement (C) correct and (A) incorrect.Furthermore, once the nozzle is choked, the Mach number at the exit plane remains fixed at $M_e = 1$, regardless of any further increase in the inlet pressure $P_0$, provided the back pressure $P_{back}$ is low enough to maintain the choked condition. Thus, statement (B) is correct. For a purely convergent nozzle, the Mach number cannot exceed unity within the nozzle or at the exit; supersonic flow ($M > 1$) can only be achieved in a divergent section. Therefore, statement (D) is incorrect.