What the Rankine cycle is

The Rankine cycle, devised by William Rankine in 1859, is the reference thermodynamic cycle for electricity production from water steam and is still the operating principle of over 80% of world thermal power plants (coal, gas, oil, biomass, nuclear and concentrated solar). It converts heat into mechanical work through four successive transformations of a working fluid (typically water): liquid-phase compression by the pump, vaporisation in the boiler, steam expansion in the turbine, condensation in the condenser. Compared to the Carnot cycle (unachievable theoretical limit) Rankine is 40-50% less efficient but is technologically feasible because compression happens on the incompressible liquid (little work) instead of on two-phase vapour. Modern supercritical plants reach 45-48% net efficiency (Rankine with double or triple reheat).

The four phases of the Rankine cycle

The T-s (temperature-entropy) diagram of the ideal Rankine cycle shows four transformations:

1. 1 → 2: Pump (isentropic liquid compression) — saturated liquid at low pressure (condenser outlet) is compressed to boiler pressure. Because liquid is almost incompressible, specific work is small: w_pump ≈ v_L·(p_H − p_L), where v_L ≈ 0.001 m³/kg. For a 10 MPa/10 kPa plant: w_pump ≈ 10 kJ/kg, less than 1% of turbine work (900-1400 kJ/kg).
2. 2 → 3: Boiler (isobaric heating) — compressed liquid is heated, vaporised and superheated in the boiler (coal/gas combustion, nuclear reactor, or solar heliostats). Turbine-inlet enthalpy h_3 is typically 3000-3600 kJ/kg for steam at 400-600 °C.
3. 3 → 4: Turbine (isentropic expansion) — superheated steam expands in the turbine yielding work. Temperature and pressure drop; part of the steam condenses forming a mixture of quality x_4. Work w_turb = h_3 − h_4 is the useful portion.
4. 4 → 1: Condenser (isobaric condensation) — the steam-liquid mixture rejects heat to the cold sink (river/sea water or cooling tower) and fully condenses to saturated liquid, closing the cycle. Strong vacuum (5-10 kPa, ~35-45 °C) in the condenser is crucial for efficiency.

The efficiency formula

Applying the first law to the four components gives thermodynamic efficiency:

η_Rankine = W_net / Q_in = (w_turb − w_pump) / (h_3 − h_2) ≈ (h_3 − h_4) / (h_3 − h_1)

Net work is turbine work minus (small) pump work; boiler heat is the enthalpy change from boiler inlet (post-pump) to outlet (h_3). For steam, h_3 depends on pressure and temperature at turbine inlet, h_4 on the outlet quality x_4 (steam fraction, remainder is liquid). Quality is derived from isentropic entropy: x_4 = (s_3 − s_f(T_cond)) / s_fg(T_cond). A good cycle has x_4 = 0.85-0.95: lower values mean too much liquid water in the turbine, causing blade erosion at low pressure. The solution is reheat: after the high-pressure turbine stage the steam is returned to the boiler and reheated to 500-600 °C before completing expansion. A reheat cycle increases η by 4-6 percentage points and significantly improves x_4.

Rankine vs Carnot: why efficiency is lower

Carnot efficiency is the theoretical upper limit for a heat engine: η_Carnot = 1 − T_L/T_H. With T_H = 600 °C (873 K) and T_L = 40 °C (313 K): η_Carnot = 64.1%. A good supercritical Rankine with those temperatures instead reaches 44-47%, about 70% of Carnot. Why the gap?

• Heat is added at variable temperature: the boiler heats liquid from T_1 up to T_boiler and then superheats to T_max. The mean thermodynamic temperature of heat addition is therefore lower than T_max, ~350-450 °C instead of 600 °C, reducing η vs Carnot.
• Condensation is at constant T_L: here matches Carnot, no loss.
• Real turbine losses: friction, misalignment, moisture → isentropic η 0.80-0.90, further 10-15% cut.
• Generator losses: ~98% but still 1-2 percentage points.
• Auxiliary consumption: circulation pumps, boiler fans, cooling tower → 4-7% of gross power.

Net electric efficiency of a modern supercritical coal plant is thus 42-45% (vs Carnot 64%): 55-58% of thermal input is rejected to environment (condenser, stack, tower).

Rankine cycle variants and technological evolution

• Basic saturated Rankine: dry saturated steam at turbine inlet (T_max = T_boiler). Efficiency 25-30%. Used in obsolete plants or biomass mini-plants up to 200 kW.
• Rankine with superheat: adds superheating phase at T_max > T_boiler. Efficiency 30-38%. Standard for medium plants (10-50 MW).
• Rankine with single reheat: turbine in two stages (HP + LP), between which steam returns to boiler. Efficiency 38-42%. Standard for large plants (200-800 MW).
• Rankine with double reheat: three turbine stages. Efficiency 42-45%. Modern supercritical plants.
• Regenerative Rankine: steam extractions from turbine preheat feedwater in open/closed feedwater heaters. Adds 2-4 points to η. Combined with reheat is the technological standard.
• Supercritical and ultra-supercritical (USC) Rankine: boiler pressures over 22.1 MPa (water critical point) and T_max up to 600-620 °C with special steels. Net electric η 45-48%. CO₂ emissions 20% lower than subcritical plants.
• Organic Rankine Cycle (ORC): organic working fluids (toluene, R245fa, siloxanes) replacing water. Optimised for low-temperature sources (80-350 °C): geothermal, biomass, waste heat from engines/turbines, low-concentration solar. Efficiency 8-24%.

Steam properties and IAPWS-IF97 tables

Rigorous Rankine calculation requires knowing steam thermophysical properties (enthalpy h, entropy s, specific volume v, saturation temperature) across the pressure and temperature range. Current international standard is IAPWS-IF97 (International Association for the Properties of Water and Steam, Industrial Formulation 1997), published in 1997 and implemented in all simulation software (Aspen Plus, EES, REFPROP, IAPWS-IF97 free libraries). IF97 divides the p-T plane into 5 regions with region-specific polynomial formulas (up to 43 terms per region), guaranteeing 0.005% accuracy on saturation and 0.03% on enthalpy. For didactic/preliminary calculations, printed tables (Smith-Van Ness, Cengel, Moran-Shapiro) offer interpolated values; for quick manual work Mollier h-s diagrams (allowing direct h_4 reading after isentropic expansion) or T-s nomographs are used. This calculator uses fast polynomial approximations with ~3-8% accuracy vs IF97: fine for pre-sizing and teaching, not for detailed design.

Applications: from thermal power to waste-heat ORC

• Thermal power plants: coal (being phased out in EU, still dominant in China/India), natural gas (in combined cycle with Brayton), oil (residual), biomass.
• Nuclear power: PWR and BWR use Rankine cycle with saturated or slightly superheated steam at 280-300 °C, T_cond 30 °C → net η 32-34%.
• Concentrated solar (CSP): parabolic mirrors or solar tower heat a fluid (oil, molten salts) to 400-560 °C generating steam for Rankine with day/night thermal storage.
• High-enthalpy geothermal: steam at 200-350 °C from underground, direct Rankine (Larderello) or binary (ORC).
• Waste-to-energy (WtE): waste combustion at 850+ °C with energy recovery via Rankine, η 20-27%.
• Cogeneration: Rankine with mid-pressure steam extraction for district heating (combined electric+thermal η > 80%).
• Waste-heat ORC: energy recovery from diesel engine/turbine exhausts (T = 200-400 °C), cement/steel furnaces. Plants from 100 kW to 20 MW electric.
• Small biomass plants (10-500 kWe): ORC on wood chips for farms and local district heating.

How to use the calculator

Enter the boiler saturation temperature T_b in °C (typically 200-370 °C, corresponding to boiler pressures 1.5-22 MPa; water critical point is 374 °C - 22.1 MPa). Enter the turbine inlet temperature T_max in °C (equal to T_b for saturated Rankine; 400-600 °C for superheated Rankine; above 600 needs super-high-alloy steels). Enter the condenser temperature T_c in °C (30-45 °C typical, corresponding to 4-10 kPa vacuum: lower is better efficiency). Enter the turbine isentropic efficiency η_iso (0.80-0.90 for modern axial multi-stage turbines; 0.65-0.75 for small industrial turbines). Enter the steam flow rate ṁ in kg/s (for 100 MW electric typically 80-100 kg/s). The calculator returns Carnot efficiency (upper limit), ideal and real Rankine efficiency, turbine outlet steam quality (warning if x < 0.85), net specific work, net delivered power and an assessment comparing to typical real-plant efficiencies.