# Rankine cycle The **Rankine cycle** is the idealized [[Thermodynamic_cycle|thermodynamic cycle]] behind almost every [[Thermal_power_station|steam power station]]: a condensable working fluid, nearly always [[Water|water]], is pumped as a liquid to high [[Pressure|pressure]], boiled and superheated in a [[Heat_exchanger|boiler]], expanded through a [[Steam_turbine|turbine]] to make work, and condensed back to liquid so the pump can raise it again. Its defining trick is that the fluid changes phase. Compressing a liquid costs almost nothing, while expanding a vapour pays a great deal, so the cycle returns a large net work for a small back-work penalty — the reason the [[Heat_engine|heat engine]] that runs the world's coal, [[Nuclear_power|nuclear]], [[Biomass|biomass]], [[Geothermal_power|geothermal]] and [[Concentrated_solar_power|solar-thermal]] plants is this one and not a gas cycle. In the microsim below the reader sets boiler pressure from 0.5 to 20 MPa, turbine-inlet [[Temperature|temperature]] from 300 to 600 °C and condenser pressure, and watches states 1 to 4 slide around a T–s diagram whose vapour dome is drawn from water's saturation curve. Three readouts answer the question. Thermal efficiency is `eta = (h3 - h4)/(h3 - h2)`, the work the turbine gives up divided by the heat the boiler puts in. Pump work is `w_p = v·dp`, small because the specific volume v of liquid water is about a thousand times smaller than steam's. Turbine-exit quality x4 — the mass fraction that is still vapour after expansion — carries the wet-turbine warning, because a cycle can be made efficient on paper and destructive in metal. A reheat toggle splits the expansion in two.[^spec-e34] On the [[Energy]] flagship this article serves *The steam cycle* in Part V — Transformation, the section where the abstract [[Heat_engine|heat engine]] of Part IV acquires a boiler, a condenser and a bill for the [[Waste_heat|heat it throws away]]. ## Description The Rankine cycle is what a [[Carnot_cycle|Carnot cycle]] becomes when engineers insist on building it. Carnot's rectangle on a T–S diagram demands isothermal heat addition and rejection, which a boiling fluid supplies for free — water at 3 MPa boils at 233.9 °C and stays there until the last drop is gone — but it also demands adiabatic compression of a two-phase mixture, which no real pump will do. The Rankine cycle solves this by condensing all the way to saturated liquid before pumping, and by superheating the vapour after boiling. Both changes cost efficiency against Carnot and buy machines that exist.[^yan-ch6][^yan-272] Claire Yan's *Introduction to Engineering Thermodynamics* frames the working power station explicitly this way: the Carnot limit sets the ceiling, and the actual plant is a Rankine cycle operating below it.[^yan-273] The book notes that large steam plants push turbine-inlet steam to 300–600 °C precisely to raise the mean temperature at which heat is added, since that is the only lever the [[Second_law_of_thermodynamics|second law]] leaves.[^yan-273] The [[Entropy|entropy]] penalty for everything else — friction, unrestrained expansion, mixing, [[Heat_transfer|heat transfer]] across a finite temperature difference, electrical resistance, inelastic deformation and chemical reaction — is paid in the real cycle below.[^yan-269] ## The four processes in the Rankine cycle Four devices in a loop, each treated as a steady-flow [[Control_volume|control volume]] with one inlet and one outlet, give four processes numbered by the states between them.[^yan-ch5] Process 1→2 is pumping. Saturated liquid leaves the condenser at state 1 and is compressed isentropically to boiler pressure. Because liquid water is nearly incompressible, the work is simply v·(P2 − P1); for a condenser at 10 kPa and a boiler at 3 MPa that is 0.00101 × 2,990 = 3.0 kJ/kg, against roughly 980 kJ/kg the turbine will return. The back-work ratio is about 0.3 %, where a gas turbine's compressor typically eats half its own output.[^derived-rc] Process 2→3 is heat addition at constant pressure. The subcooled liquid is warmed to saturation, boiled at constant temperature through the whole [[Enthalpy_of_vaporization|latent heat]], and then superheated. This is where almost all the heat goes in — 2,920 kJ/kg for the example above — and where the fuel is burned or the reactor's [[Thermal_conduction|conduction]] path ends.[^derived-rc] Process 3→4 is expansion through the [[Turbine|turbine]], ideally isentropic. Pressure, temperature and [[Enthalpy|enthalpy]] all fall; part of the steam condenses on the way down. The enthalpy drop h3 − h4 is the work, and the shaft carries it to an [[Electric_generator|electric generator]]. Process 4→1 is condensation at constant pressure and temperature, rejecting the [[Latent_heat|latent heat]] to cooling water, air or a lake. It is not waste in the sense of carelessness: it is the price the second law charges for closing the loop, and it is large. Thomas Kerlin puts the rejection of a thermal plant at 2.5 to 3 times its electrical output, which corresponds to a net efficiency of about 25 to 29 %.[^kerlin-185] ## Variables The cycle is written in specific quantities — per kilogram of working fluid — so that the same numbers describe a 5 kW organic unit and a 1 GW turbine hall. The four states carry enthalpy h in kJ/kg, [[Entropy|specific entropy]] s in kJ/(kg·K), specific volume v in m³/kg, pressure P and temperature T; state 1 is the condenser outlet, 2 the pump outlet, 3 the turbine inlet and 4 the turbine exhaust. Work and heat appear as w_t (turbine), w_p (pump), q_in (boiler) and q_out (condenser), all in kJ/kg, and become powers when multiplied by the mass flow rate ṁ in kg/s. Inside the vapour dome the [[Phase_transition|two-phase]] state needs one more number, the quality x, the vapour mass fraction, which interpolates every other property between the saturated-liquid value (subscript f) and the saturated-vapour value (subscript g): h = h_f + x·h_fg and s = s_f + x·s_fg. Isentropic efficiencies η_t and η_p, both fractions below 1, measure how far the real turbine and pump fall short of the ideal ones.[^yan-ch5][^yan-appA] ## Equations Each device is a steady-flow energy balance with kinetic and potential terms dropped, which reduces the [[First_law_of_thermodynamics|first law]] to a difference of enthalpies.[^yan-ch5] The turbine, adiabatic and doing work, gives `w_t = h3 - h4`. The condenser, rejecting heat and doing none, gives `q_out = h4 - h1`. The pump gives `w_p = h2 - h1`, and because the liquid is nearly incompressible this equals `v1·(P2 - P1)` without any table lookup. The boiler gives `q_in = h3 - h2`. Thermal efficiency is net work over heat in: `eta = (w_t - w_p)/q_in = [(h3 - h4) - (h2 - h1)]/(h3 - h2)` For the ideal cycle sketched above — 3 MPa and 350 °C at the turbine inlet, 10 kPa in the condenser — the [[Enthalpy|enthalpies]] are h1 = 191.8, h2 = 194.9, h3 = 3,115.3 and h4 = 2,135.7 kJ/kg, giving w_t = 979.6, w_p = 3.0, q_in = 2,920.5 and η = 33.4 %.[^derived-rc] Isentropic expansion fixes h4 through s4 = s3: the turbine exhaust is wet, with quality x4 = (6.7428 − 0.6493)/7.5009 = 0.812.[^derived-rc] The Carnot efficiency between the same extreme temperatures, 623.15 K and 318.96 K, is 48.8 %, so the ideal Rankine cycle already forfeits a third of the ceiling — not to friction, but to the fact that most of its heat is added well below the peak temperature.[^derived-rc][^yan-272] ## Real Rankine cycle (non-ideal) The ideal cycle has two vertical lines on the T–s diagram; the real one has neither. Expansion through a turbine generates entropy, so state 4 moves right and up, h4 rises, and the work falls by the isentropic efficiency η_t, typically in the high eighties to low nineties for a large machine. The pump suffers the same way, but since w_p is under half a percent of w_t, its inefficiency barely registers. Pressure drops through the boiler tubes and condenser mean the two "constant-pressure" legs slope. Heat leaks from casings and piping. Every one of these is an [[Irreversible_process|irreversibility]] of the kind Yan enumerates, and each widens the gap between the cycle on the diagram and the meter on the generator.[^yan-269][^yan-ch6] The number the microsim watches most closely is not efficiency but exit quality. Below roughly x = 0.88 the droplet load in the last turbine stages erodes the blades, in the same way [[Cavitation|cavitation]] pits a pump impeller, and the wet steam also drags on the rotor. The example's x4 = 0.812 is therefore unbuildable as it stands; the fix is not a better turbine but a hotter, or a reheated, cycle.[^derived-rc][^spec-e34] Scale fixes the numbers. Yan's worked lake example takes a 500 MW net plant against cooling water at 18 °C flowing at 20 m³/s. A Carnot cycle at 250 °C would reach 44.3 % and warm the lake by 7.50 °C; at 300 °C it would reach 49.2 % and warm it 6.17 °C. A real 35 % cycle rejects 928.6 MW instead of 627.5 MW and warms the same flow by 11.10 °C — a much larger temperature rise, which is the thermal-discharge problem in one line.[^yan-carnot-ex][^manual10] Thomas Murphy's accounting agrees on the order: fission plants run at about one-third, and a 2.5 GW-thermal reactor paired with a 1 GW-electric turbine hall is a 40 % machine.[^murphy-fission] ## Variations of the basic Rankine cycle Both classical improvements attack the same weakness — that heat is added over a wide temperature range, much of it low — rather than the turbine itself. ### Rankine cycle with reheat Reheat splits the expansion. Steam leaves the high-pressure turbine partway down, returns to the boiler to be reheated at that intermediate pressure, and finishes in a low-pressure turbine. Take the same 3 MPa, 350 °C cycle and reheat at 0.6 MPa back to 350 °C: turbine work rises from 979.6 to 1,139.5 kJ/kg, heat input rises from 2,920.5 to 3,336.7 kJ/kg, and efficiency improves only from 33.4 % to 34.1 %.[^derived-rc] The efficiency gain is almost an afterthought. The real prize is the exhaust: quality climbs from 0.812 to 0.920, comfortably clear of the erosion threshold.[^derived-rc] Reheat is what lets a designer raise boiler pressure — which does help efficiency — without driving the last stages into wet steam. In the microsim the reheat toggle is the control that moves the quality readout most and the efficiency readout least.[^spec-e34] ### Regenerative Rankine cycle Regeneration attacks the low-temperature end of heat addition instead. Steam is bled from the turbine partway down and mixed with, or used to warm, the feedwater on its way to the boiler, so the coldest part of the heating curve is served by steam that has already done work rather than by flame. One open feedwater heater at 0.6 MPa in the example cycle extracts a fraction y = 0.186 of the flow, raises the feedwater from 194.9 to 673.2 kJ/kg, and cuts the boiler's job from 2,920.5 to 2,442.1 kJ/kg. Net work falls to 860.2 kJ/kg, but heat input falls faster, and efficiency rises from 33.4 % to 35.2 % — more than reheat buys, from a device with no moving parts.[^derived-rc] Large stations stack six to eight such heaters in series, pushing the mean temperature of heat addition steadily toward the boiler temperature and the cycle steadily toward its Carnot rectangle.[^yan-ch6] ## Organic Rankine cycle An organic Rankine cycle replaces water with a fluid that boils at a much lower temperature — a refrigerant, a hydrocarbon or ammonia — so that the cycle can run on a heat source too cool for steam. Yan's appendices tabulate exactly the candidates: ammonia, R134a and carbon dioxide each get a full property table beside water's.[^yan-appBCD] The physics is unchanged; only the saturation curve moves. Below about 200 °C, water's low [[Vapor_pressure|vapour pressure]] makes steam so voluminous that turbines and condensers become absurdly large, while a refrigerant at the same temperature is at a few megapascals and passes through a compact machine. The arithmetic is unforgiving. A source at 150 °C rejecting to 30 °C has a Carnot ceiling of 1 − 303/423 = 28.4 %, and a real organic unit reaches perhaps a third to a half of that.[^derived-rc][^yan-272] That is why these machines are built where the heat is already paid for: [[Geothermal_energy|geothermal]] brine, engine exhaust, [[Cogeneration|industrial waste heat]], and [[Solar_thermal_energy|solar-thermal]] collectors that never reach steam temperatures. The figure of merit is not efficiency but [[Exergy|exergy]] recovered from a stream that would otherwise be dumped. ## Supercritical Rankine cycle Above water's [[Critical_point_(thermodynamics)|critical point]] — 22.06 MPa and 647.1 K, or 373.95 °C — there is no boiling and no vapour dome to cross.[^yan-crit] A [[Supercritical_fluid|supercritical]] Rankine cycle pumps liquid straight past that pressure and heats it continuously from dense fluid to something steam-like, with no latent-heat plateau at all. The heating curve becomes a smooth rise rather than a flat step, which raises the mean temperature of heat addition and therefore the efficiency, and the boiler needs no steam drum to separate liquid from vapour because there is nothing to separate. The cost is metallurgy. Pressures of 25 to 30 MPa and metal temperatures at the top of Yan's stated 300–600 °C band demand alloy steels that resist [[Creep_(deformation)|creep]] and [[Corrosion|oxidation]] for decades, and every increment of temperature is bought from a materials supplier rather than a thermodynamicist.[^yan-273][^derived-rc] In the microsim, pushing boiler pressure past 22.06 MPa is the moment the cycle's left-hand leg stops touching the dome: the two-phase readouts go blank, the quality warning disappears, and the efficiency readout keeps climbing with nothing in the physics to stop it — which is precisely why the limit is written in steel and not in equations.[^spec-e34] Supercritical practice also feeds the bottom of a [[Combined-cycle_power_plant|combined cycle]], where a gas turbine's [[Brayton_cycle|Brayton]] exhaust becomes this cycle's boiler heat, and the two together beat either alone. ## See also - [[Steam_engine]] — the reciprocating ancestor, and the machine Rankine was theorizing - [[Steam_turbine]] — the expansion device of process 3→4 - [[Thermal_power_station]] — the plant the cycle describes - [[Combined-cycle_power_plant]] — a Brayton cycle feeding a Rankine cycle - [[Carnot_cycle]] — the ceiling - [[Heat_engine]] — the parent abstraction - [[Thermodynamic_cycle]] - [[Exergy]] ## References [^yan-ch5]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics.* Chapter 5, "The First Law of Thermodynamics for a Control Volume" (pp. 187–238): steady-flow energy balances for turbines, pumps, boilers and condensers as control volumes (page to pin). https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics [^yan-ch6]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics.* Chapter 6, "Entropy and the Second Law of Thermodynamics" (pp. 239–348): the Rankine cycle, reheat and regeneration (page to pin). https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics [^yan-272]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*, Chapter 6, p. 272: Carnot efficiency `eta = 1 - T_L/T_H`; an actual engine falls below it and one above it is "impossible". The equation display was lost in extraction and the standard form is supplied. [^yan-273]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*, Chapter 6, p. 273: large steam plants use 300–600 °C steam to raise T_H, and the actual plant is framed as a Rankine cycle rather than a Carnot cycle. [^yan-269]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*, Chapter 6, p. 269: the list of irreversibilities — friction, unrestrained expansion, mixing, heat transfer across a finite temperature difference, electrical resistance, inelastic deformation and chemical reaction. [^yan-carnot-ex]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*, Chapter 6, pp. 273–276, the warmed-lake example: 500 MW net, lake at 18 °C and 20 m³/s, c_p = 4.181 kJ/(kg·K). The printed results were lost in extraction; η 44.3 % and 49.2 %, and ΔT 7.50, 6.17 and 11.10 °C, are computed from the book's stated inputs with ρ = 1,000 kg/m³ supplied as a standard value. [^yan-crit]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*, Chapter 3, p. 122: water's critical properties, P_crit = 22.06 MPa and T_crit = 647.1 K. [^yan-appA]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*, Appendix A, "Thermodynamic Properties of Water" (pp. 349–368): the saturated- and superheated-steam tables from which h_f, h_fg, s_f, s_fg and v_f are read (individual rows to pin). [^yan-appBCD]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*, Appendix B, "Thermodynamic Properties of Ammonia" (pp. 369–386); Appendix C, "Thermodynamic Properties of R134a" (pp. 387–398); Appendix D, "Thermodynamic Properties of Carbon Dioxide" (pp. 399–407) — the organic-cycle working fluids tabulated alongside water. [^kerlin-185]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, Chapter 3, p. 185: a thermal plant rejects 2.5 to 3 times its electrical output, which corresponds to a net efficiency of about 25 to 29 % (computed). https://open.umn.edu/opentextbooks/textbooks/future-energy-opportunities-challenges [^murphy-fission]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*, Chapter 6 "Alternative Energy" (pp. 183–322), p. 276: fission plants run at about one-third thermal efficiency, and a 2.5 GW-thermal to 1 GW-electric plant is 40 %. https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^manual10]: Wikitube MICROSIM_GUIDE sub-manual 10, *Earth, Energy and Environment*, §2.3 (Carnot limits) and §A: book 115's equation displays for Chapter 6 were lost in text extraction, so the Carnot and COP forms used here are supplied standard forms to be verified against the PDF pages, and the lake example's results are computed rather than printed. [^derived-rc]: Computed for this article from the balances on this page with standard saturated- and superheated-steam properties of the kind tabulated in [^yan-appA]: at 10 kPa, h_f = 191.83 kJ/kg, h_fg = 2,392.8 kJ/kg, s_f = 0.6493, s_fg = 7.5009 kJ/(kg·K), v_f = 0.001010 m³/kg, T_sat = 45.81 °C; at 0.6 MPa, h_f = 670.56, h_fg = 2,086.3, s_f = 1.9312, s_fg = 4.8288, v_f = 0.001101; at 3 MPa, T_sat = 233.9 °C, and at 3 MPa/350 °C, h = 3,115.3 kJ/kg and s = 6.7428 kJ/(kg·K); at 0.6 MPa/350 °C, h = 3,165.7 and s = 7.5464. From these: the ideal cycle (η 33.4 %, x4 0.812, back-work ratio 0.31 %), the Carnot comparison (48.8 %), the reheat cycle (η 34.1 %, x4 0.920), the single open feedwater heater (y 0.186, η 35.2 %) and the 150/30 °C organic-cycle ceiling (28.4 %). The individual table rows are to pin against Appendix A. [^spec-e34]: Matter & Energy Cluster contract, `_registry/plans/ENERGY_SECTIONS.md` row E34: sim concept (`steam.sat`, `thermo.rankine`), controls boiler pressure 0.5–20 MPa, turbine-inlet temperature 300–600 °C and condenser pressure; states 1–4 on a T–s diagram whose dome comes from `matter.water.vaporPressure`; readouts `eta = (h3 - h4)/(h3 - h2)`, pump work `v dp` and exit quality, with a reheat toggle and a 64-row saturated-steam lookup table. Superheat is modelled as `c_p dT` and is marked ILLUSTRATIVE. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Rankine_cycle.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Rankine cycle* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Rankine_cycle.html" data-title="Rankine cycle"></div> *Built from `MICROSIM_GUIDE/specs/sims/Rankine_cycle.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Rankine_cycle) : [Wikitube](https://en.wikitube.io/wiki/Rankine_cycle) · pinned revision [1360731359](https://en.wikipedia.org/w/index.php?oldid=1360731359) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E34 · sim pending (matter/Rankine_cycle).*