Rankine Cycle & Thermal Efficiency: Steam Power Plant Analysis
Walk the four ideal Rankine processes, apply isentropic efficiencies for the real cycle, and check the numbers against a fully verified steam-plant benchmark.
Key Engineering Takeaways
- Thermal efficiency is net work over heat added: η_th = w_net / q_in = (w_t − w_p) / q_in.
- Ideal cycle: isentropic pump and turbine, constant-pressure heat addition and rejection.
- Real components use isentropic efficiencies: η_t = (h₃ − h₄)/(h₃ − h₄s) and η_p = (h₂s − h₁)/(h₂ − h₁).
- The Carnot ceiling for the T3-001 benchmark is η = 1 − T_C/T_H ≈ 58.7%; the real cycle reaches 34.6%.
1. The Four Ideal Rankine Processes
The Rankine cycle is the idealized steam power plant. Four devices, four processes, each with a balance:
| Process | Device | Energy balance |
|---|---|---|
| 1 → 2 | Pump (isentropic) | w_p = h₂ − h₁ ≈ v₁(P₂ − P₁) |
| 2 → 3 | Boiler (constant P) | q_in = h₃ − h₂ |
| 3 → 4 | Turbine (isentropic) | w_t = h₃ − h₄ |
| 4 → 1 | Condenser (constant P) | q_out = h₄ − h₁ |
The pump is nearly incompressible, so its work is well approximated by the liquid-volume term w_p ≈ v₁(P₂ − P₁) — tiny compared with the turbine work.
2. Thermal Efficiency & Back-Work Ratio
η_th = w_net / q_in = (w_t − w_p) / q_in
bwr = w_p / w_t (back-work ratio)
The back-work ratio quantifies how much of the turbine output the pump consumes. For steam cycles bwr ≈ 1%; for gas-turbine (Brayton) cycles it is 40–80% — the fundamental reason steam cycles dominate utility power generation.
3. Real Components: Isentropic Efficiencies
- Turbine: η_t = (h₃ − h₄) / (h₃ − h₄s) — actual work over isentropic work.
- Pump: η_p = (h₂s − h₁) / (h₂ − h₁) — isentropic work over actual work.
- Real processes shift the state points; the condenser exit quality x₄ tracks moisture buildup, which erodes blade life — designs keep x₄ ≳ 0.88.
4. Verified Benchmark: ThermoCore T3-001
Boiler at 8 MPa / 500 °C, condenser at 10 kPa, with η_t = 0.88 and η_p = 0.85. Results verified against ThermoCore's 88-problem benchmark (CoolProp IF97 backend):
η_th = 34.6%
w_net = 1107.5 kJ/kg
q_in = 3198.2 kJ/kg
bwr = 0.0085
turbine exit quality x₄ = 0.874
5. The Carnot Ceiling
No heat engine between two reservoirs can beat Carnot: η_max = 1 − T_C / T_H. For the T3-001 condenser at 10 kPa, the saturation temperature is T_sat ≈ 45.8 °C = 319 K; with a boiler temperature of T_H = 773 K:
The real cycle's 34.6% sits well below the ceiling — the gap is the price of irreversibilities (turbine and pump losses, finite ΔT in the boiler).
6. Reheat & Regeneration
- Reheat: expand in a first turbine stage, reheat the steam at boiler pressure, then expand in a second stage. This raises the average heat-addition temperature and limits moisture at the exit.
- Regeneration: bleed steam (extraction fraction y) from the turbine at intermediate pressures and use it to preheat the feedwater, cutting boiler heat input q_in.
Both refinements are built into ThermoCore's Tier 3 cycle engine — set the extraction fraction and watch η_th move.
Verify your power-cycle calculations in real time.
Evaluate state properties, run Rankine / Brayton / refrigeration cycles on the CoolProp-backed engine, and inspect textbook 5-section solutions (Known → Find → Schematic → Model → Analysis).
Textbook References & Verification
- 01
Moran & Shapiro — Fundamentals of Engineering Thermodynamics
Textbook 5-section solution format (Known → Find → Schematic → Model → Analysis) used for the T3-001 benchmark.
- 02
Çengel & Boles — Thermodynamics: An Engineering Approach
Rankine cycle analysis, isentropic efficiencies, reheat and regenerative cycle theory.
- 03
ThermoCore benchmark T3-001
Boiler 8 MPa / 500 °C, condenser 10 kPa, η_t = 0.88, η_p = 0.85 → η_th = 34.6%, w_net = 1107.5 kJ/kg, q_in = 3198.2 kJ/kg, bwr = 0.0085, x₄ = 0.874.