Question

Consider a one-dimensional inviscid supersonic flow in a diverging duct with heat addition ($Q_{in}$). Which of the following statement(s) is/are always TRUE? 

• A. Mach number, $M_2 > M_1$
• B. Stagnation pressure, $P_1^0 > P_2^0$
• C. Static pressure, $P_2 > P_1$
• D. Stagnation temperature, $T_1^0 < T_2^0$

Hint:

For Rayleigh flow (heat addition in ducts), remember: in supersonic flow, Mach decreases, static pressure and stagnation temperature increase, while stagnation pressure decreases.

Answer 1

The Correct Option is B, C, D

Step 1: Concept of Rayleigh flow.
When heat is added to a supersonic flow (Rayleigh flow conditions), the Mach number decreases toward 1 (thermal choking). Hence: \[ M_2 < M_1 \] Therefore, (A) is false.

Step 2: Stagnation pressure.
In real heat addition (non-isentropic process), stagnation pressure always decreases because of irreversibility: \[ P_2^0 < P_1^0 \] Hence (B) is true.

Step 3: Static pressure.
For supersonic flow with heat addition, static pressure increases as Mach number decreases: \[ P_2 > P_1 \] So (C) is true.

Step 4: Stagnation temperature.
Since heat is added to the system, stagnation temperature must increase: \[ T_2^0 > T_1^0 \] Thus (D) is true.

Step 5: Final check.
- (A) false (Mach number decreases).
- (B) true (stagnation pressure decreases).
- (C) true (static pressure increases).
- (D) true (stagnation temperature increases). \[ \boxed{\text{Correct statements: (B), (C), and (D)}} \]