GATE 2022 AE – Question 47
A rocket at absolute chamber pressure 20 bar produces thrust $F_1$. Its hot exhaust is optimally expanded to 1 bar through a convergent-divergent nozzle with $A_e/A_t=3.5$ and thrust coefficient $C_{F,1}=1.42$. At absolute chamber pressure 50 bar, thrust and thrust coefficient are $F_2$ and $C_{F,2}$. Which statements are correct?
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Correct answer: (B) $F_2/F_1>2.5$; (D) $C_{F2}/C_{F1}>1$
Explanation
**Thrust coefficient** $C_F=\dfrac{F}{p_cA_t}$, so $F=C_F\,p_c\,A_t$ for a fixed nozzle.
For a given nozzle (fixed $A_e/A_t$) the thrust coefficient is
$$C_F=\underbrace{\sqrt{\frac{2\gamma^2}{\gamma-1}\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{\gamma-1}}\left[1-\left(\frac{p_e}{p_c}\right)^{\frac{\gamma-1}{\gamma}}\right]}}_{\text{momentum part}}+\left(\frac{p_e-p_a}{p_c}\right)\frac{A_e}{A_t}.$$
When the chamber pressure rises from 20 bar to 50 bar, the pressure ratio $p_e/p_c$ is fixed by $A_e/A_t=3.5$ (so $p_e$ rises in proportion to $p_c$), while the ambient pressure $p_a=1$ bar stays the same.
- The momentum part is the same, because $p_e/p_c$ is unchanged.
- The ambient-pressure correction, $-\dfrac{p_a}{p_c}\dfrac{A_e}{A_t}$, becomes **less negative** as $p_c$ increases.
So $C_F$ rises: **$C_{F2}/C_{F1}>1$** (option D).
**Thrust:** $F_2/F_1=\dfrac{C_{F2}}{C_{F1}}\times\dfrac{p_{c2}}{p_{c1}}>1\times\dfrac{50}{20}=2.5$, so **$F_2/F_1>2.5$** (option B).
Answer **B and D**.