A liquid propellant rocket engine operates at a chamber stagnation pressure $p_c = 6.0\text{ MPa}$ with a propellant mass flow rate $\dot{m} = 15.0\text{ kg/s}$. The nozzle throat diameter is $d_t = 80\text{ mm}$. If the rocket produces a total sea-level thrust of $F = 32.0\text{ kN}$ at an ambient pressure $p_a = 0.10\text{ MPa}$, the thrust coefficient $C_F$ of the nozzle is closest to:

(A) 1.06

(B) 1.42

(C) 0.85

(D) 1.78

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles In rocket propulsion, the thrust coefficient $C_F$ characterizes the amplification of thrust achieved by expanding high-pressure chamber gases through a converging-diverging de Laval nozzle: $$C_F = \frac{F}{p_c A_t}$$ where: - $F$ is the total engine thrust - $p_c$ is the combustion chamber stagnation pressure - $A_t = \frac{\pi d_t^2}{4}$ is the nozzle throat cross-sectional area. ### 2. Analytical Formulation Given throat diameter $d_t = 80\text{ mm} = 0.080\text{ m}$: $$A_t = \frac{\pi}{4} (0.080)^2 = \frac{\pi}{4} \times 0.0064 = 0.0016 \pi \approx 5.0265 \times 10^{-3}\text{ m}^2$$ Chamber pressure is $p_c = 6.0\text{ MPa} = 6.0 \times 10^6\text{ Pa}$. Product of $p_c A_t$: $$p_c A_t = (6.0 \times 10^6) \times (5.0265 \times 10^{-3}) = 30{,}159.3\text{ N} \approx 30.16\text{ kN}$$ ### 3. Step-by-Step Calculation Total thrust is $F = 32.0\text{ kN} = 32{,}000\text{ N}$. Evaluating $C_F$: $$C_F = \frac{F}{p_c A_t} = \frac{32{,}000}{30{,}159.3} \approx 1.061$$ ### 4. Distractor Forensics - **Option A (1.06):** Correct calculation. - **Option B (1.42):** Arises if candidate uses nozzle exit area $A_e$ instead of throat area $A_t$, or assumes vacuum thrust. - **Option C (0.85):** Inverted the ratio or used radius instead of diameter. - **Option D (1.78):** High altitude / vacuum thrust coefficient assumption. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Always use throat area $A_t$ (never exit area $A_e$) in the denominator for thrust coefficient $C_F = F / (p_c A_t)$. - **Standard Reference:** Sutton, G.P. & Biblarz, O., *Rocket Propulsion Elements*, 9th Edition, Chapter 3 (Nozzle Theory and Thermodynamic Relations).

← Prev Question Next Question →