1. Overview and principle
- Working fluid: Typically air (open cycle) or another gas (closed cycle). In open cycles, combustion products are exhausted; in closed cycles, heat exchangers transfer heat to/from the working fluid.
- Basic processes (ideal cycle):
- 1–2: Isentropic compression in a compressor.
- 2–3: Constant-pressure heat addition in a combustor or heater.
- 3–4: Isentropic expansion in a turbine.
- 4–1: Constant-pressure heat rejection to surroundings (open) or cooler (closed).
- Applications: Aircraft propulsion (turbojets, turbofans), stationary power generation, marine propulsion, and as topping cycles in combined-cycle plants.
2. Ideal Brayton cycle analysis
2.1 Assumptions
- Steady-flow, internally reversible processes.
- Ideal gas with constant specific heats \(c_p, k\).
- Negligible changes in kinetic and potential energy (except in propulsion analysis).
2.2 Temperature–entropy (T–s) diagram
On the T–s diagram, compression (1–2) and expansion (3–4) are vertical lines (isentropic), while heat addition (2–3) and rejection (4–1) are horizontal (constant pressure).
2.3 Pressure ratio and temperature relations
For isentropic compression/expansion of an ideal gas: \[ \frac{T_2}{T_1} = r_p^{\frac{k-1}{k}}, \quad \frac{T_3}{T_4} = r_p^{\frac{k-1}{k}} \] where \(r_p = \frac{p_2}{p_1} = \frac{p_3}{p_4}\) is the pressure ratio.
2.4 Thermal efficiency
For the ideal Brayton cycle: \[ \eta_{\text{th,ideal}} = 1 - \frac{T_1}{T_2} \cdot \frac{T_4 - T_1}{T_3 - T_2} = 1 - \frac{1}{r_p^{\frac{k-1}{k}}} \] Efficiency increases with pressure ratio \(r_p\) for given \(T_1\) and \(T_3\).
2.5 Net work and back work ratio
- Compressor work: \[ w_c = c_p (T_2 - T_1) \]
- Turbine work: \[ w_t = c_p (T_3 - T_4) \]
- Net work: \[ w_{\text{net}} = w_t - w_c \]
- Back work ratio: \[ \text{BWR} = \frac{w_c}{w_t} \] High for gas turbines (often 40–60%), meaning a large fraction of turbine work drives the compressor.
3. Real Brayton cycle
- Compressor and turbine are not isentropic: introduce isentropic efficiencies \(\eta_c, \eta_t\).
- Pressure drops in combustor and heat exchangers reduce effective \(r_p\).
- Non-ideal combustion and heat transfer limit maximum \(T_3\) (turbine inlet temperature).
3.1 Isentropic efficiencies
- Compressor: \[ \eta_c = \frac{T_{2s} - T_1}{T_{2a} - T_1} \]
- Turbine: \[ \eta_t = \frac{T_3 - T_{4a}}{T_3 - T_{4s}} \]
4. Variations to improve performance
4.1 Regeneration (recuperation)
- Use turbine exhaust heat to preheat compressed air before combustion.
- Reduces fuel needed for same \(T_3\), increasing efficiency, especially at low–moderate \(r_p\).
- Effectiveness: \[ \epsilon = \frac{T_{2'} - T_2}{T_4 - T_2} \]
4.2 Intercooling
- In multi-stage compression, cool air between stages to reduce compressor work.
- Increases net work but may reduce efficiency unless combined with regeneration.
4.3 Reheating
- In multi-stage expansion, reheat between turbine stages to increase work output.
- Increases net work but may reduce efficiency unless combined with regeneration.
4.4 Combined modifications
Intercooling + reheat + regeneration can be combined for high specific work and efficiency, at the cost of complexity and capital cost.
5. Applications
- Aircraft propulsion: Turbojets, turbofans, turboprops — Brayton cycle adapted for thrust production.
- Power generation: Simple-cycle gas turbines, combined-cycle plants (Brayton topping + Rankine bottoming).
- Marine propulsion: Naval ships, fast ferries.
- Industrial drives: Large compressors, pumps.
6. Worked examples
Example 1: Ideal cycle efficiency
Given: \(T_1=300\ \text{K}\), \(T_3=1400\ \text{K}\), \(k=1.4\), \(r_p=8\).
- \(T_2 = 300 \times 8^{0.2857} \approx 579.3\ \text{K}\).
- \(T_4 = 1400 / 8^{0.2857} \approx 725.3\ \text{K}\).
- \(\eta_{\text{th}} = 1 - T_1/T_2 = 1 - 300/579.3 \approx 0.482\) (48.2%).
Example 2: Effect of regeneration
Same as above, with perfect regenerator (\(\epsilon=1\)) and \(T_4 > T_2\).