Thermodynamic Cycles
Ideal cycles model real engines and power plants. Learn the Otto (petrol), Diesel, Brayton (gas turbine) and Rankine (steam) cycles, their P–v behaviour, efficiency formulas and the design levers that improve them.
A cycle is a repeating sequence of processes that returns the working fluid to its starting state. Real engines are complicated, but air-standard and vapour cycles capture their essential thermodynamics and show which parameters matter.
Petrol engines approximate the Otto cycle, diesels the Diesel cycle, jet engines and gas turbines the Brayton cycle, and steam power plants the Rankine cycle.
An analogy
A cycle is like a round trip: the fluid always comes home, and the net work is the 'area' enclosed by the journey on a P–v diagram.
Otto cycle (spark ignition)
Isentropic compression → constant-volume heat addition (spark) → isentropic expansion (power stroke) → constant-volume heat rejection. Efficiency rises with compression ratio, but knock limits petrol engines to r ≈ 9–13.
Diesel cycle
Heat is added at constant pressure as fuel is injected into hot compressed air. No knock limit, so r = 15–22 is used, giving higher real efficiency despite lower efficiency at equal r compared with Otto.
Brayton cycle (gas turbine)
Compressor → combustor (constant pressure) → turbine → exhaust. Efficiency depends on pressure ratio; the compressor consumes 40–60 % of turbine work (high back-work ratio), so component efficiencies are critical. Regeneration, intercooling and reheat improve it.
Rankine cycle (steam)
Pump → boiler → turbine → condenser. Pumping a liquid takes tiny work, so BWR is ~1 %. Superheat, higher boiler pressure, lower condenser pressure, reheat and regenerative feedwater heating raise efficiency to 40–47 % in modern plants.
Combined and advanced cycles
Brayton topping + Rankine bottoming cycles (CCGT) exceed 60 %. Supercritical CO₂ Brayton cycles, Kalina and organic Rankine cycles target nuclear, solar-thermal and waste-heat applications. Real engine analysis uses fuel–air cycles, heat-release models and CFD.
Formulas
- Otto:
η = 1 − 1 / r^(γ−1) - Diesel:
η = 1 − (1/r^(γ−1)) · (r_c^γ − 1) / (γ(r_c − 1)) - Brayton:
η = 1 − 1 / r_p^((γ−1)/γ) - Rankine:
η = (w_turbine − w_pump) / q_boiler
Worked example: Otto and Diesel
Given: Otto r = 10; Diesel r = 18, r_c = 2; γ = 1.4
Find: Ideal efficiencies.
- Otto: 1 − 10^(−0.4) = 60.2 %
- Diesel: 1 − 18^(−0.4) × (2^1.4 − 1)/(1.4 × 1) = 63.2 %
Answer: Otto 60.2 %, Diesel 63.2 %
Worked example: Simple Rankine cycle
Given: Boiler 3 MPa, 400 °C (h = 3231.7); Condenser 10 kPa (h_f = 191.83); Isentropic turbine exit h = 2193.3 kJ/kg
Find: Efficiency.
- w_pump = v Δp = 0.00101 × 2990 = 3.0 kJ/kg
- q_in = 3231.7 − 194.9 = 3036.9 kJ/kg
- w_net = 1038.4 − 3.0 = 1035.4 kJ/kg
- η = 1035.4/3036.9 = 34.1 %
Answer: ≈ 34.1 %
Common mistakes
- Wrong: Assuming air-standard efficiency is what real engines achieve. — Right: Real engines reach roughly half to two-thirds of it due to losses, heat transfer and real-gas effects.
- Wrong: Forgetting pump/compressor work in net work. — Right: Net work = expansion work − compression work; critical for Brayton.
Applications
- Automotive engines: Otto/Atkinson cycles in petrol and hybrid cars; Diesel in trucks, ships and generators.
- Aviation & power: Brayton cycles in jet engines and gas-turbine power plants.
- Thermal power: Rankine cycles in coal, nuclear, biomass, geothermal and solar-thermal plants.
Key takeaways
- Otto: 1 − r^(1−γ); Diesel adds cut-off penalty.
- Brayton: pressure ratio sets efficiency.
- Rankine: tiny pump work, superheat/reheat/regeneration help.
- Net work = enclosed P–v (or T–s) area.
Going further
- Atkinson/Miller cycles: Expansion ratio larger than compression ratio improves efficiency; used in hybrid-car engines via late intake-valve closing.
- Regenerative Brayton: Exhaust heat preheats compressed air — effective at low pressure ratios (microturbines).
- Supercritical steam: Above 22.1 MPa, no boiling occurs; ultra-supercritical plants (~30 MPa, 600+ °C) reach ~47 % efficiency.