GATE AE 2022 • Q47 MSQ • 2 Marks Propulsion

Propulsion

Rocket Propulsion

Subtopic: Solid, Liquid & Hybrid Rocket Engines

Question Statement:

A rocket operates at an absolute chamber pressure of $20\text{ bar}$ to produce thrust, $F_1$. The hot exhaust is optimally expanded to $1\text{ bar}$ (absolute pressure) using a convergent-divergent nozzle with exit to throat area ratio $\left(\frac{A_e}{A_t}\right)$ of $3.5$ and thrust coefficient, $C_{F,1} = 1.42$. The same rocket when operated at an absolute chamber pressure of $50\text{ bar}$ produces thrust $F_2$ and the thrust coefficient is $C_{F,2}$. Which of the following statement(s) is/are correct?

Options:

  • (A) $\frac{F_2}{F_1} = 2.5$
  • (B) $\frac{F_2}{F_1} > 2.5$
  • (C) $\frac{C_{F,2}}{C_{F,1}} = 1$
  • (D) $\frac{C_{F,2}}{C_{F,1}} > 1$

Official Answer Key:

B, D

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2022 Q.47)** **1. Thrust and Thrust Coefficient Formulation**: $$F = C_F p_c A_t$$ For higher chamber pressure ($50\text{ bar}$ vs $20\text{ bar}$), the pressure ratio increases and exit pressure underexpands relative to ambient, increasing $C_F$: $$C_{F,2} > C_{F,1} \implies \frac{C_{F,2}}{C_{F,1}} > 1$$ $$\frac{F_2}{F_1} = \frac{p_{c,2}}{p_{c,1}} \frac{C_{F,2}}{C_{F,1}} = 2.5 \times \frac{C_{F,2}}{C_{F,1}} > 2.5$$ - **Official Answer Key**: **B, D**
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