The figure below depicts two ideal gas turbine cycles, cycle 1-2-3-4-1 and cycle
1-2-3'-4'-1, on a T-s diagram. Which one the following statements is FALSE?
Options:
(A) The thermal efficiency of the two cycles is the same
(B) The specific work of the two cycles is the same
(C) The processes 2-3 and 2-3' are isobaric
(D) The amount of heat added in the combustion process is greater for the cycle
1-2-3'-4'-1
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.18)**
**1. Intuitive Physical Setup & Given Parameters**:
The problem presents two ideal Brayton gas turbine cycles on a temperature-entropy ($T$-$s$) diagram:
- **Cycle 1**: $1 \to 2 \to 3 \to 4 \to 1$
- **Cycle 2**: $1 \to 2 \to 3' \to 4' \to 1$
Both cycles share the same compression process $1 \to 2$ (operating between the same pressure limits $p_1$ and $p_2$, so identical pressure ratio $r_p = p_2/p_1$).
Cycle 2 is heated to a higher turbine inlet temperature $T_{3'} > T_3$.
We are asked to identify which statement is **FALSE**.
**2. Step-by-Step Thermodynamic Derivation**:
1. **Thermal Efficiency Analysis**:
- The air-standard thermal efficiency of an ideal Brayton cycle depends solely on the pressure ratio $r_p$:
$\eta_{\text{th}} = 1 - \frac{1}{r_p^{(\gamma-1)/\gamma}}$
- Since both cycles have identical isentropic compression from $p_1$ to $p_2$, they have the exact same pressure ratio $r_p$.
- Therefore, $\eta_{\text{th}, 1} = \eta_{\text{th}, 2}$. Statement A is **TRUE**.
2. **Combustion Heat Addition Analysis**:
- Heat added in constant-pressure combustion is $q_{\text{in}} = c_p (T_3 - T_2)$ for cycle 1, and $q_{\text{in}}' = c_p (T_{3'} - T_2)$ for cycle 2.
- Since $T_{3'} > T_3$, the heat added in cycle 2 is strictly greater: $q_{\text{in}}' > q_{\text{in}}$. Statement D is **TRUE**.
- Processes $2-3$ and $2-3'$ are standard constant-pressure heat additions along isobar $p_2$. Statement C is **TRUE**.
3. **Specific Work Output Analysis**:
- Specific net work output is:
$w_{\text{net}} = \eta_{\text{th}} \times q_{\text{in}}$
- Since $\eta_{\text{th}}$ is identical for both cycles but $q_{\text{in}}' > q_{\text{in}}$, the net specific work of cycle 2 must be strictly greater:
$w_{\text{net}}' = \eta_{\text{th}} q_{\text{in}}' > w_{\text{net}}$
- Graphically, the enclosed area on the $T$-$s$ diagram represents the net work output per unit mass ($w_{\text{net}} = \oint T ds$). The area enclosed by $1-2-3'-4'-1$ is clearly larger than $1-2-3-4-1$.
- Therefore, claiming that *"The specific work of the two cycles is the same"* is **FALSE**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) The thermal efficiency of the two cycles is the same**: ❌ Valid statement (not false). Both cycles operate at identical $r_p$.
- **(B) The specific work of the two cycles is the same**: ✅ **CORRECT (FALSE STATEMENT)**. Increasing maximum cycle temperature at fixed $r_p$ strictly increases the enclosed $T$-$s$ area and specific work output.
- **(C) The processes 2-3 and 2-3' are isobaric**: ❌ Valid statement (not false). Heat addition in an ideal Brayton cycle is strictly isobaric.
- **(D) The amount of heat added in the combustion process is greater for the cycle 1-2-3'-4'-1**: ❌ Valid statement (not false). $c_p(T_{3'} - T_2) > c_p(T_3 - T_2)$ because $T_{3'} > T_3$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Thermal efficiency of an ideal Brayton cycle depends only on pressure ratio $\eta = 1 - r_p^{-(\gamma-1)/\gamma}$, but specific work output $w_{\text{net}}$ scales directly with peak turbine inlet temperature $T_{\text{max}}$.
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