### **Detailed Step-by-Step Solution (GATE AE 2026 Q.14)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to identify the four thermodynamic processes constituting the ideal air-standard Stirling heat engine cycle.
**2. Step-by-Step Mathematical Derivation**:
1. **Thermodynamic Definition of the Stirling Cycle**:
The ideal Stirling cycle operates between two thermal reservoirs ($T_H$ and $T_C$) with ideal regenerative heat exchange. Its four closed-system processes are:
- **Process 1-2**: Isothermal compression at low temperature $T_C$ (heat rejection to cold reservoir: $Q_{\text{out}}$).
- **Process 2-3**: Isochoric (constant volume) regenerative heating from $T_C$ to $T_H$ (absorbing internal regenerative heat $Q_{\text{reg}}$).
- **Process 3-4**: Isothermal expansion at high temperature $T_H$ (heat addition from source: $Q_{\text{in}}$).
- **Process 4-1**: Isochoric (constant volume) regenerative cooling from $T_H$ to $T_C$ (depositing heat back into the regenerator matrix).
2. **Key Distinctions from Other Gas Cycles**:
- **Carnot Cycle**: Two isotherms + two reversible adiabatics (isentropics).
- **Ericsson Cycle**: Two isotherms + two isobars (constant pressure).
- **Otto Cycle**: Two isentropics + two isochors.
- **Brayton Cycle**: Two isentropics + two isobars.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) Two reversible isobars, and two reversible adiabatics**: ❌ **INCORRECT**. This defines the ideal Brayton (Joule) cycle.
- **(B) Two reversible isotherms, and two reversible isobars**: ❌ **INCORRECT**. This defines the ideal Ericsson cycle.
- **(C) Two reversible isotherms, and two reversible isochors**: ✅ **CORRECT**. The Stirling cycle consists of two reversible isothermal processes and two reversible constant-volume (isochoric) regenerative processes.
- **(D) Isentropic compression, constant volume heat addition...**: ❌ **INCORRECT**. This defines the internal-combustion Otto cycle.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Remember the memory mnemonic: Stirling has Constant Volume (isochoric) regeneration, whereas Ericsson has Constant Pressure (isobaric) regeneration. Both achieve theoretical Carnot efficiency with ideal regenerators.
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