A converging-diverging rocket nozzle exhausts combustion gases with specific heat ratio $\gamma = 1.30$. If the nozzle area expansion ratio is $A_e / A_t = 10.0$, which of the following statements regarding the isentropic supersonic exit flow are **CORRECT**?

(A) The exit Mach number $M_e$ satisfies the 1D area-Mach relation $\frac{A_e}{A_t} = \frac{1}{M_e} \left[ \frac{2}{\gamma+1} \left(1 + \frac{\gamma-1}{2} M_e^2\right) \right]^{\frac{\gamma+1}{2(\gamma-1)}}$

(B) Because $A_e / A_t = 10 > 1$, the exit Mach number is supersonic ($M_e > 1$)

(C) Decreasing ambient atmospheric pressure (increasing altitude) changes the nozzle exit Mach number $M_e$ during fully expanded operation

(D) The exit static temperature is given by $T_e = T_c \left(1 + \frac{\gamma-1}{2} M_e^2\right)^{-1}$, where $T_c$ is chamber stagnation temperature

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles 1. **Area-Mach Relation:** For isentropic flow of an ideal gas through a de Laval nozzle, the ratio of any duct cross-section $A$ to the sonic throat $A_t$ ($M=1$) is uniquely determined by local Mach number $M$ and $\gamma$: $$\frac{A}{A_t} = \frac{1}{M} \left[ \frac{2}{\gamma + 1} \left(1 + \frac{\gamma - 1}{2} M^2\right) \right]^{\frac{\gamma + 1}{2(\gamma - 1)}}$$ For $A_e / A_t = 10$, the supersonic branch yields $M_e \approx 3.7$. 2. **Ambient Pressure Independence:** In supersonic nozzle flow with no interior shock waves, disturbances cannot propagate upstream against the supersonic flow ($M_e > 1$). Hence, changes in external ambient pressure $p_a$ have **zero effect** on the internal exit Mach number $M_e$ or exit static pressure $p_e$. 3. **Static Temperature:** Follows directly from adiabatic stagnation temperature conservation $T_{0} = T_c = T_e (1 + \frac{\gamma-1}{2} M_e^2)$. ### 2. Statement Evaluation - **Statements A, B, D:** **CORRECT.** - **Statement C:** **INCORRECT.** Ambient pressure variations do not change exit Mach number $M_e$. ### 3. Exam Trap & Standard Reference - **Trap Alert:** External ambient pressure cannot alter supersonic exit conditions unless a shock wave is forced inside the diverging nozzle bell. - **Standard Reference:** Sutton, G.P. & Biblarz, O., *Rocket Propulsion Elements*, Chapter 3.

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