An ideal gas flows steadily through a constant area duct with friction (Fanno flow). As the flow proceeds downstream towards the choking point, the stagnation temperature $T_0$:
Options:
(A) remains constant
(B) increases
(C) decreases
(D) first increases and then decreases
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2021 Q.16)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This problem evaluates **Basic Fluid Mechanics** (Conservation Equations of Mass, Momentum & Energy) within the *Aerodynamics & Fluid Mechanics* syllabus (1 Mark, MCQ).
**Problem Statement**:
> An ideal gas flows steadily through a constant area duct with friction (Fanno flow). As the flow proceeds downstream towards the choking point, the stagnation temperature $T_0$:
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Physical / Analytical Relations**:
Governed by fundamental conservation and constitutive laws under Aerodynamics & Fluid Mechanics. Formulating the analytical equations:
\text{Governing Law} = 0
2. **Step-by-Step Analytical Deduction**:
- 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 **`A`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"remains constant"*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
- **(B)** *"increases"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(C)** *"decreases"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(D)** *"first increases and then decreases"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2021 (IIT Bombay) key.