# Heat engine A **heat engine** is a device that takes [[Heat|heat]] from a hot reservoir, converts part of it into [[Work_(thermodynamics)|work]], and rejects the remainder to a cold one. The rejection is not a design fault to be engineered away: the [[Second_law_of_thermodynamics|second law]] requires it, and the fraction that can be converted is capped by the two reservoir temperatures alone, with no reference to the working fluid, the mechanism or the century. For a reversible engine that cap is `eta = 1 - T_L/T_H` with both temperatures in kelvin, and no real machine has ever reached it.[^yan-eta] In the microsim below one slider does two jobs. Moving the cold-reservoir temperature T_L from −40 °C to 40 °C against a fixed T_H of 250 °C draws the [[Carnot_cycle|Carnot]] ceiling, with a hatched "impossible" zone above it, the 35 % marker of an actual [[Rankine_cycle|Rankine]] plant below it, and the temperature rise of the cooling [[Water|water]] from `dT = Q_L/(mdot c_p)` — 7.5 °C for a 500 MW plant discharging into a river flowing at 20 m³/s.[^yan-lake] A second panel re-reads the same slider as outdoor air for a [[Heat_pump|heat pump]] delivering into a 24 °C house: `COP_HP = T_H/(T_H - T_L)` falls as the outside gets colder, and a unit rated at a [[Coefficient_of_performance|coefficient of performance]] of 5.5 crosses into the forbidden zone below about −30 °C (derived).[^yan-cop] The T–S rectangle drawn alongside makes the arithmetic visible: its area is the net work.[^yan-ts] On the [[Energy]] flagship this article is the root of Part V — Transformation, section *Heat engines and the Carnot limit*. Everything downstream of it — [[Thermodynamic_cycle|the individual cycles]], the [[Thermal_power_station|power station]], the [[Internal_combustion_engine|engine]] in a car — is a particular way of filling in the rectangle, and every one of them is measured against the same ceiling. ## Overview The engine is defined by its boundary rather than its hardware. Across that boundary pass three quantities: heat Q_H in from the hot reservoir, heat Q_L out to the cold one, and net work W. Over a complete [[Thermodynamic_cycle|cycle]] the working fluid returns to its initial state, so its [[Internal_energy|internal energy]] is unchanged and the [[First_law_of_thermodynamics|first law]] reduces to W = Q_H − Q_L. Thermal efficiency is the ratio of what is wanted to what is paid for, η = W/Q_H = 1 − Q_L/Q_H.[^yan-eta] The second law converts that identity into a limit. For a reversible engine the heats are in the ratio of the absolute temperatures, Q_H/Q_L = T_H/T_L, so η = 1 − T_L/T_H, and this depends on nothing but the reservoirs.[^yan-eta] Two consequences follow immediately and both appear in the sim. A real engine always sits below the line, because every irreversibility — friction, unrestrained expansion, mixing, [[Heat_transfer|heat transfer]] across a finite temperature difference, electrical resistance, inelastic deformation, chemical reaction — destroys available work.[^yan-irrev] And an engine plotted above the line is not merely unusually good; it is impossible, and the claim is a sufficient reason to look for an error.[^yan-eta] The cold reservoir is usually the environment, which fixes T_L near 290 K and leaves T_H as the only variable a designer controls. That is why large steam plants push their steam to 300–600 °C, and why the history of the subject is largely a history of raising the top temperature until materials give out.[^yan-steam] Konstantin Likharev's statement of the same result is worth quoting for its scope: the bound applies to every engine, "quantum proposals included", and the internal-combustion engine's advantage over the steam engine is nothing more subtle than a higher T_H — about 1,500 K against a few hundred, which is a Carnot ceiling of 0.8 against a reservoir at 300 K (derived).[^likharev-carnot] ## Examples Almost every device that produces mechanical power from a temperature difference is a heat engine, and the families below are organised by what the working substance does rather than by what the machine looks like. Two families are not treated separately here: liquid-only cycles, which exploit the thermal expansion of a liquid without a phase change and are of low power density, and evaporative engines, in which a liquid's vapour pressure does the work directly. ### Everyday examples The commonest heat engine is the [[Internal_combustion_engine|internal combustion engine]], in which combustion heats the working gas inside the cylinder rather than through a wall. The second commonest is the [[Steam_turbine|steam turbine]], which drives most of the world's electricity whether the heat comes from coal, gas, [[Nuclear_power|fission]] or a [[Solar_thermal_energy|solar]] field. A domestic refrigerator is the same machinery run backwards, and the turbine in a [[Jet_engine|jet engine]] is a heat engine whose useful output is a jet rather than a shaft. ### Earth's heat engine The atmosphere and ocean form a heat engine of enormous power and derisory efficiency. Its hot reservoir is the tropical surface, warmed by absorbed sunlight; its cold reservoir is the polar regions and the upper troposphere, which radiate to space; and its work output is the kinetic energy of winds and currents, dissipated as friction almost as fast as it is made. The temperature difference is small, so the Carnot ceiling is only a few per cent, which is why a system driven by roughly 10¹⁷ W of solar input sustains winds of only tens of metres per second. The thermal accounting behind the reservoirs is set out in [[Earth's_energy_budget|Earth's energy budget]], and the planet's own [[Convection|convective]] overturn is the engine's mechanism. ### Phase-change cycles Boiling and condensing a fluid lets a cycle take in and reject heat at nearly constant temperature, which is exactly what the Carnot construction asks for, and it makes the pump work small because a liquid is nearly incompressible. The [[Rankine_cycle|Rankine cycle]] — pump, boiler, turbine, condenser — is the result and is the basis of essentially every thermal power station. Its practical efficiency of around 35 % is well below the 44 % a Carnot engine would reach between 250 °C and an 18 °C river (derived), and the shortfall is the sim's most instructive number.[^yan-lake] ### Gas-only cycles Where no phase change occurs the working fluid is a gas throughout, and the cycle's shape is drawn by compression and expansion alone: [[Otto_cycle|Otto]], [[Diesel_cycle|Diesel]], [[Brayton_cycle|Brayton]] and [[Stirling_cycle|Stirling]]. Gas cycles can reach much higher peak temperatures than steam, because nothing has to stay below a critical point, but they reject heat over a falling temperature rather than at a plateau, which costs them part of the advantage. The comparison between the four is the subject of [[Thermodynamic_cycle|thermodynamic cycle]]. ### Electron cycles A thermoelectric generator has no moving parts and no working fluid, yet it obeys the same ceiling: the [[Thermoelectric_effect|Seebeck effect]] produces a voltage from a temperature difference, S₁ − S₂ = dV₁₂/dT₁₂, and the device's quality is measured by the figure of merit Z = S²σ/κ.[^mitofsky-seebeck] Andrea Mitofsky notes that the Carnot limit covers heat-to-anything converters of this kind, though not photovoltaic or piezoelectric devices, which are not heat engines at all.[^mitofsky-carnot] The materials are poor by engine standards — lead telluride reaches about 400 µV/K, most materials less than 1 µV/K — and practical devices stay under 10 %.[^mitofsky-seebeck][^mitofsky-eta] A [[Radioisotope_thermoelectric_generator|radioisotope thermoelectric generator]] converting 2 kW of decay heat into 120 W of electricity runs at about 6 %, which is unimpressive until one notices that it does so unattended for decades.[^mitofsky-rtg] ### Magnetic cycles A magnetocaloric material warms when magnetised and cools when the field is removed, because the field orders its magnetic moments and so lowers their entropy. Cycling the field while alternately connecting the material to two reservoirs gives a magnetic refrigerator, and running the same cycle in the engine direction gives work. The devices are compact and use no volatile refrigerant; their limitation is that the useful temperature span per stage is small, so practical machines cascade many stages. ### Cycles used for refrigeration Reversing an engine turns it into a refrigerator or a heat pump, and the figure of merit is no longer bounded by one. For the reversible case `COP_R = T_L/(T_H - T_L)` and `COP_HP = T_H/(T_H - T_L)`, the two differing by exactly one because the heat delivered exceeds the heat taken by the work put in.[^yan-cop] The numbers are large when the span is small: at 300 K with a 10 K lift the ideal cooling COP is about 30, against 3–4 for real air conditioning.[^likharev-cop] The sim's worked case is the book's: 20 kW delivered into a 24 °C house from 0 °C outside needs 3.64 kW of electricity at the rated COP of 5.5, saving 16.36 kW against resistance heating, while the reversible COP of 12.38 would need only 1.62 kW (computed).[^yan-hp] Because COP_HP falls as T_L falls, heat pumps are at their best in mild winters and at their worst exactly when heat is most needed.[^yan-hp] ### Mesoscopic heat engines Engines built from a few degrees of freedom — a single trapped ion, a colloidal particle in an optical trap, a quantum dot between two leads — have been operated as Carnot and Otto cycles. At that scale work and heat are fluctuating quantities and the efficiency of a single cycle can momentarily exceed the Carnot value; the average over many cycles cannot. The bound survives because it is a statement about ensembles, and Likharev is explicit that quantum proposals do not escape it.[^likharev-carnot] ## Efficiency The efficiency panel is the heart of the microsim, and it is built from one division. With T_H fixed at 250 °C = 523.15 K, sliding T_L from 40 °C to −40 °C raises the Carnot ceiling from 40.1 % to 55.4 % (derived). At the book's reference condition, an 18 °C lake, the ceiling is 44.3 % and the actual Rankine plant is marked at 35 %.[^yan-lake] The gap between those two markers is where the cooling water comes in. A 500 MW plant at the Carnot efficiency would draw Q_H = 1,127.5 MW and reject Q_L = 627.5 MW; the same plant at 35 % draws 1,428.6 MW and rejects 928.6 MW (computed). Discharged into a flow of 20 m³/s at c_p = 4.181 kJ/(kg·K), those rejected powers warm the water by 7.50 °C and 11.10 °C respectively — the second being, in the book's words, "a much higher temperature rise".[^yan-lake] Raising T_H to 300 °C instead lifts the ceiling to 49.2 % and drops the ideal rise to 6.17 °C (computed). ILLUSTRATIVE: the printed results of the book's two worked examples were lost in text extraction, so these figures are recomputed from the book's stated inputs with a supplied water density of 1,000 kg/m³; the book's qualitative conclusions are unaffected.[^yan-lake] Two cross-checks from other Portal Books put the 35 % marker in context. A thermal plant is commonly said to reject 2.5 to 3 times its electrical output, which corresponds to η between 25 % and 29 % (computed), and a fission plant turning 2.5 GW of thermal power into 1 GW of electricity runs at 40 %.[^kerlin-reject][^murphy-fission] Real plants therefore populate a band from the high twenties to about forty per cent, and the sim's hatched zone above the Carnot line has never been entered. ### Endo-reversible heat-engines A Carnot engine reaches its ceiling only in the limit of infinitely slow operation, because heat must cross each boundary under a vanishing temperature difference; at the ceiling the power output is zero. The endo-reversible model keeps the cycle internally [[Reversible_process_(thermodynamics)|reversible]] but charges for finite-rate heat transfer at the two boundaries, and maximising power rather than efficiency then gives η = 1 − √(T_L/T_H), the Curzon–Ahlborn result.[^curzon1975] For 250 °C and 18 °C that is 25.4 % (derived) — below the observed 35 %, so the model is a floor of a kind rather than a prediction, but it explains why real plants sit far below Carnot without any appeal to poor engineering. ## History The engine preceded the theory by a century. Thomas Savery's *The Miner's Friend* of 1702 describes a fire engine for raising water by condensing steam, and Thomas Newcomen's atmospheric engines of the 1710s made the idea commercially useful.[^savery1702] James Watt's patent of 1769 for a separate condenser removed the enormous loss of reheating the cylinder every stroke and roughly doubled the fuel economy.[^watt1769] [[Nicolas_Léonard_Sadi_Carnot|Sadi Carnot]] asked the general question in 1824: how much work can be got from a given quantity of heat, and does the answer depend on the substance used? His *Réflexions sur la puissance motrice du feu* answered that it depends only on the two temperatures, and did so while still holding the caloric theory, which makes the achievement stranger rather than smaller.[^carnot1824] [[Rudolf_Clausius|Rudolf Clausius]] and William Thomson rebuilt the argument on the mechanical theory of heat in 1850–1851, and Clausius's introduction of [[Entropy|entropy]] in 1865 turned Carnot's ratio into a state function.[^clausius1865][^thomson1851] William Rankine's *Manual of the Steam Engine* of 1859 carried the new thermodynamics into engineering practice and gave the vapour-power cycle its name.[^rankine1859] ## Enhancements Because the ceiling depends only on the reservoirs, almost every real improvement either raises T_H, lowers T_L, or recovers something from the rejected stream. Raising T_H is limited by metallurgy, which is why gas-turbine blade alloys and cooling schemes are as much a part of efficiency as the cycle is. Lowering T_L is limited by the environment and, at a power station, by the cooling-water permit. Recovery is the third route: a combined-cycle plant uses the exhaust of a [[Brayton_cycle|Brayton]] gas turbine as the heat source of a Rankine steam cycle, so the same fuel passes two engines in series. [[Cogeneration|Cogeneration]] takes the opposite view and gives up shaft work to deliver the [[Waste_heat|rejected heat]] at a useful temperature, which raises the total energy recovered while lowering the thermal efficiency as conventionally defined — a reminder that efficiency is a ratio whose numerator is a choice. ## Heat engine processes A cycle is assembled from a small number of idealised [[Thermodynamic_process|processes]], each holding one property fixed. The Carnot cycle uses four of them in a fixed order: [[Isothermal_process|isothermal]] heat addition at T_H, [[Adiabatic_process|adiabatic]] expansion down to T_L, isothermal heat rejection at T_L, and adiabatic compression back to the start.[^yan-cycle] Because the two isothermals are horizontal lines and the two adiabats are vertical ones on a temperature–entropy diagram, the Carnot cycle is a rectangle there, and the enclosed area is the net work — which is why the sim draws it that way.[^yan-ts] Other cycles substitute different processes: constant volume and constant pressure for Otto and Diesel, constant pressure with adiabatic compression and expansion for Brayton, isothermal plus constant volume with a regenerator for [[Stirling_engine|Stirling]]. The substitutions change the shape of the loop and therefore the efficiency, but never the ceiling, because the ceiling is fixed by the extreme temperatures the loop touches. ## See also - [[Carnot_cycle]] - [[Heat_pump]] - [[Coefficient_of_performance]] - [[Thermoelectric_effect]] - [[Radioisotope_thermoelectric_generator]] - [[Thermodynamic_cycle]] - [[Rankine_cycle]] - [[Second_law_of_thermodynamics]] - [[Exergy]] ## References [^yan-eta]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*. Chapter 6 "Entropy and the Second Law of Thermodynamics", p. 272 (η = 1 − Q_L/Q_H = 1 − T_L/T_H and Q_H/Q_L = T_H/T_L; an actual engine lies below the Carnot value and a device above it is "impossible"; the limit depends only on the absolute reservoir temperatures). Portal Book 115, https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics — the equation displays were lost in text extraction and are supplied in standard form by sub-manual 10 §2.3. [^yan-irrev]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, p. 269 (the list of irreversibilities: friction, unrestrained expansion, mixing, heat transfer across a finite ΔT, electrical resistance, inelastic deformation, chemical reaction). Portal Book 115. [^yan-cycle]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, p. 270 (the Carnot cycle's four processes: A→B isothermal heat addition at T_H, B→C adiabatic expansion, C→D isothermal heat rejection at T_L, D→A adiabatic compression). Portal Book 115. [^yan-ts]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, p. 271 (the Carnot cycle is a rectangle on a T–S diagram). Portal Book 115. [^yan-steam]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, p. 273 (large steam plants use 300–600 °C steam to raise T_H; the actual plant is framed as a Rankine cycle, not a Carnot one). Portal Book 115. [^yan-lake]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, pp. 273–276 (the 500 MW plant, an 18 °C lake at 20 m³/s, Q_L = Q_H − W and ΔT = Q_L/(ṁ·c_p) with c_p = 4.181 kJ/(kg·K); raising T_H raises η and the actual cycle gives "a much higher temperature rise"). Portal Book 115. The printed results were lost in text extraction; the values quoted here — η 44.3 %, Q_H 1,127.5 MW, Q_L 627.5 MW, ΔT 7.50 °C at T_H = 250 °C; η 49.2 % and ΔT 6.17 °C at 300 °C; Q_H 1,428.6 MW, Q_L 928.6 MW, ΔT 11.10 °C at the actual 35 % — are computed in sub-manual 10 §2.3 from the book's stated inputs with ρ = 1,000 kg/m³ supplied. [^yan-cop]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, p. 277 (COP_R = T_L/(T_H − T_L) and COP_HP = T_H/(T_H − T_L); both depend only on the reservoir temperatures). Portal Book 115. [^yan-hp]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, pp. 278–280 (20 kW delivered at 24 °C indoors and 0 °C outdoors with an actual COP of 5.5; the book prints a saving of 16.354 kW, which recomputes to 16.364 kW with W = 3.636 kW; the Carnot COP is 12.38 and W = 1.62 kW, both computed in sub-manual 10 §2.3; the COP falls as T_L falls, so heat pumps "are preferably used in mild winter conditions"). Portal Book 115. [^likharev-carnot]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 1 "Review of Thermodynamics", pp. 22–26 (Q_H = Q_L + W and −W = ∮P dV; Q_H/Q_L = T_H/T_L for the Carnot cycle; η = 1 − T_L/T_H bounds any engine, "quantum proposals included"; internal combustion runs at T_H ≈ 1,500 K against a few hundred kelvin for steam engines, giving a ceiling of 0.8 against 300 K as a derived check). Portal Book 075, https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^likharev-cop]: Likharev (2013), *Part SM: Statistical Mechanics*, Chapter 1, pp. 24–25 (COP_c = T_L/(T_H − T_L) and COP_h = T_H/(T_H − T_L); at T_L ≈ 300 K with ΔT ≈ 10 K the ideal cooling COP is about 30 against 3–4 for real HVAC; heat pumps reach COP_h ≈ 4 in summer and ≈ 2 in winter; COP_c exceeds 1 whenever T_H < 2T_L). Portal Book 075. [^mitofsky-seebeck]: Mitofsky, Andrea M. (2018). *Direct Energy*. Chapter "Thermoelectrics", pp. 191–196 (Seebeck coefficient S₁ − S₂ = dV₁₂/dT₁₂; Peltier coefficient Π = S·T; figure of merit Z = S²σ/κ; PbTe ≈ 400 µV/K, (Bi₀.₇Sb₀.₃)₂Te₃ ≈ 230 µV/K, most materials below 1 µV/K). Portal Book 055, https://open.umn.edu/opentextbooks/textbooks/direct-energy [^mitofsky-carnot]: Mitofsky (2018), *Direct Energy*, pp. 199–201 (η = 1 − T_c/T_h in kelvin; the limit covers heat-to-other-energy converters but not photovoltaic or piezoelectric devices; worked cases 295/309 K → 4.5 % and 266/295 K → 9.8 %). Portal Book 055. [^mitofsky-eta]: Mitofsky (2018), *Direct Energy*, pp. 197–198 (PbTe melts at 924 °C and Bi₂Te₃ at 580 °C; practical thermoelectric devices reach less than 10 %). Portal Book 055. [^mitofsky-rtg]: Mitofsky (2018), *Direct Energy*, pp. 202–203 (a radioisotope thermoelectric generator converting 2 kW of heat to 120 W of electricity, about 6 %; a copper/solder junction gives ≈ 2 µV/K). Portal Book 055. [^kerlin-reject]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, p. 185 (a thermal plant rejects 2.5–3 times its electrical output, which corresponds to η ≈ 25–29 %, computed in sub-manual 10 §2.3). Portal Book 048, https://open.umn.edu/opentextbooks/textbooks/future-energy-opportunities-challenges [^murphy-fission]: Murphy, Thomas W. (2021). *Energy and Human Ambitions on a Finite Planet*, p. 276 (fission plants run at about one-third thermal efficiency; a 2.5 GW-thermal, 1 GW-electric plant is 40 %). Portal Book 097, https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^curzon1975]: Curzon, F. L.; Ahlborn, B. (1975). "Efficiency of a Carnot engine at maximum power output." *American Journal of Physics* 43 (1): 22–24. [^savery1702]: Savery, Thomas (1702). *The Miner's Friend; or, an Engine to Raise Water by Fire*. London: S. Crouch. [^watt1769]: Watt, James. "A New Invented Method of Lessening the Consumption of Steam and Fuel in Fire Engines." British Patent No. 913, 1769. [^carnot1824]: Carnot, Sadi (1824). *Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance*. Paris: Bachelier. [^clausius1865]: Clausius, Rudolf (1865). "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie." *Annalen der Physik und Chemie* 125 (7): 353–400. [^thomson1851]: Thomson, William (1851). "On the dynamical theory of heat." *Transactions of the Royal Society of Edinburgh* 20: 261–288. [^rankine1859]: Rankine, William John Macquorn (1859). *A Manual of the Steam Engine and Other Prime Movers*. London: Griffin. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Heat_engine.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Heat engine* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Heat_engine.html" data-title="Heat engine"></div> *Built from `MICROSIM_GUIDE/specs/sims/Heat_engine.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/Heat_engine) : [Wikitube](https://en.wikitube.io/wiki/Heat_engine) · pinned revision [1371080355](https://en.wikipedia.org/w/index.php?oldid=1371080355) · 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 E32 · sim pending (matter/Heat_engine).*