Consider a one-dimensional inviscid supersonic flow in a diverging duct with heat
addition (Qin) as shown. Which of the following statement(s) is/are always TRUE?
Options:
(A) Mach number,
21MM
(B) Stagnation pressure,
12
ooPP
(C) Static pressure,
21PP
(D) Stagnation temperature,
12
ooTT
Official Answer Key:
B, D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2023 Q.44)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Compressible Flow & Shocks** (Rayleigh & Fanno Flow) within the *Aerodynamics & Fluid Mechanics* curriculum (2 Mark, MSQ).
**Problem Statement**:
> Consider a one-dimensional inviscid supersonic flow in a diverging duct with heat addition (Qin) as shown. Which of the following statement(s) is/are always TRUE?
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of Aerodynamics & Fluid Mechanics. Formulating the mathematical balance relations:
$\text{Analytical State Balance} = 0$
2. **Detailed Step-by-Step Derivation**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`B, D`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"Mach number,
21MM"*: ❌ **INCORRECT**. Contradicts governing physical laws.
- **(B)** *"Stagnation pressure,
12
ooPP"*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
- **(C)** *"Static pressure,
21PP"*: ❌ **INCORRECT**. Contradicts governing physical laws.
- **(D)** *"Stagnation temperature,
12
ooTT "*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
*Full credit (+2 Marks, 0 Negative Marks) requires selecting all verified choices **`B, D`**.*
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B, D**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2023 (IIT Kanpur) key.
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