The point of maximum entropy on a Fanno-curve in a Temperature-Entropy (T-s)
diagram represents the
Options:
(A) maximum flow Mach number
(B) minimum flow Mach number
(C) sonic Mach number
(D) normal shock in the flow
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2022 Q.14)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to identify the physical state represented by the point of maximum entropy on a Fanno curve in a Temperature-Entropy ($T-s$) diagram.
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Equations of Fanno Flow**:
Fanno flow describes 1-D steady, adiabatic flow of an ideal gas through a constant-area duct with wall friction. The governing conservation laws are:
- Continuity: $\rho u = G = \text{const}$
- Energy: $h_0 = c_p T + \frac{u^2}{2} = \text{const}$
- Equation of State: $p = \rho R T$
2. **Entropy Condition ($ds = 0$)**:
Combining the Second Law $T ds = dh - v dp$ with the continuity and energy invariants yields:
$\frac{ds}{du} = \frac{u}{T} \left( \frac{u^2}{a^2} - 1 \right) = \frac{u}{T}(M^2 - 1)$
Setting $\frac{ds}{du} = 0$ for maximum entropy:
$M^2 - 1 = 0 \implies M = 1$
3. **Physical Significance (Choking Limit)**:
- The upper branch ($M < 1$) undergoes frictional heating and accelerates toward $M = 1$ with increasing entropy.
- The lower branch ($M > 1$) undergoes frictional deceleration toward $M = 1$ with increasing entropy.
- The maximum entropy point corresponds strictly to sonic flow ($M = 1$). Duct friction cannot drive flow past this point without changing inlet conditions (Fanno choking).
**3. Comprehensive Options & Distractor Analysis**:
- **(A) stagnation condition**: ❌ **INCORRECT**. Stagnation corresponds to $M = 0$ ($u = 0$, maximum temperature).
- **(B) maximum pressure**: ❌ **INCORRECT**. Maximum pressure occurs at $M = 0$.
- **(C) sonic condition (M = 1)**: ✅ **CORRECT**. At the point of maximum entropy on the Fanno curve, the local Mach number is strictly unity ($M = 1$).
- **(D) maximum temperature**: ❌ **INCORRECT**. Maximum temperature occurs at $M = 0$ ($T = T_0$).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Both the Fanno curve (adiabatic with friction) and Rayleigh curve (frictionless with heat transfer) reach their maximum entropy points at exactly the sonic state $M = 1$.
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