# Thermodynamic cycle A **thermodynamic cycle** is a sequence of [[Thermodynamic_process|processes]] that returns a working substance to the [[Thermodynamic_state|state]] it started in. Because every property of the substance is a [[State_function|state function]], the round trip changes none of them: the [[Internal_energy|internal energy]] and the [[Entropy|entropy]] of the fluid are the same at the end as at the beginning. [[Heat|Heat]] and [[Work_(thermodynamics)|work]] are not state functions, so they do not return to zero, and their difference over the loop is what the machine is for. Run one way the cycle is a [[Heat_engine|heat engine]]; run the other it is a refrigerator or a [[Heat_pump|heat pump]]. In the microsim below the reader chooses a cycle from a menu — Carnot, [[Otto_cycle|Otto]], [[Diesel_cycle|Diesel]], [[Brayton_cycle|Brayton]], [[Stirling_cycle|Stirling]] — and sets the compression ratio r, the heat-capacity ratio γ, the cutoff ratio ρ and the pressure ratio r_p. Both loops redraw together: the pressure–volume loop, whose shaded area is the net work, and the temperature–entropy loop, whose area is the net heat. The efficiency readout follows the closed form that belongs to the selected cycle — `eta_Otto = 1 - r^(1-gamma)`, `eta_Diesel = 1 - r^(1-gamma) (rho^gamma - 1)/(gamma (rho - 1))`, `eta_Brayton = 1 - r_p^((1-gamma)/gamma)`, and Stirling with a perfect regenerator equal to Carnot — and the Carnot ceiling between the same two extreme temperatures is drawn on every setting, so the cost of the loop's shape is always visible. On the [[Energy]] flagship this article is the child of Part V — Transformation, section *Cycles: Otto, Diesel, Brayton, Stirling*, and the sibling of [[Heat_engine|heat engine]], which supplies the ceiling this page's cycles are measured against. The relation between the two is the whole lesson: the reservoir temperatures fix what is possible, and the shape of the loop fixes how much of it is collected. ## Heat and work Over one complete cycle the first law reduces to ∮dU = 0, so the net heat added equals the net work done: Q_net = W_net. The sign of that quantity decides which machine one has. A loop traversed clockwise on a pressure–volume diagram encloses positive work and is an engine; the same loop traversed counter-clockwise consumes work and pumps heat up a temperature gradient. Neither heat nor work is a property of the fluid. Both are transfers across the boundary, and both depend on the path taken between two states, which is why a cycle can deliver work at all while returning its working substance unchanged. The bookkeeping that follows is therefore always done on the loop, never on the endpoints. ### Relationship to work The work of a closed system is the integral of pressure over volume, so for a cycle the net work is −W = ∮P dV, the signed area enclosed on the p–V diagram.[^likharev-work] That area is what the microsim shades. Its counterpart on the temperature–entropy diagram is ∮T dS, the net heat, and because the two must be equal over a closed loop the two shaded areas are the same number drawn twice. The two pictures answer different questions. The p–V loop shows what the machine does mechanically: a tall thin loop is a high-pressure, small-displacement engine, a broad flat one the opposite. The T–s loop shows what the machine does thermodynamically, and it is the more revealing of the two, because the Carnot cycle is a rectangle there whatever fluid is used, and every other cycle is a distortion of that rectangle whose lost area is exactly the efficiency it gives up.[^yan-ts] ### A list of thermodynamic processes Cycles are assembled from a short list of idealised processes, each holding one property fixed: [[Isothermal_process|isothermal]] at constant temperature, isobaric at constant pressure, isochoric at constant volume, and [[Adiabatic_process|adiabatic]] with no heat crossing the boundary. An adiabatic process that is also reversible is isentropic, and for an ideal gas it obeys `p V^gamma = const`, with γ the ratio of the specific heats.[^yan-ideal-relations] All four are limiting cases of the [[Polytropic_process|polytropic]] relation pVⁿ = const: n = 0 gives isobaric, n = 1 isothermal, n = γ isentropic and n → ∞ isochoric. The value of γ is an external input rather than a result of the cycle: about 1.4 for air and other diatomic gases, 5/3 for a monatomic gas. The sub-manual notes that the excerpt of the statistical-mechanics text used here does not print γ, so its source must be labelled wherever an adiabat is drawn.[^likharev-gamma] The sim exposes γ as a control for that reason, and the loops visibly steepen as it is raised. ### Example: The Otto cycle The [[Otto_cycle|Otto cycle]] is the idealisation of the spark-ignition [[Internal_combustion_engine|internal combustion engine]] and the cycle the microsim opens on.[^yan-ch1] It has four processes: isentropic compression from volume V₁ to V₂, heat addition at constant volume, isentropic expansion back to V₁, and heat rejection at constant volume. Because both heat exchanges happen at constant volume, the efficiency collapses to a function of the compression ratio r = V₁/V₂ and γ alone: `eta_Otto = 1 - r^(1-gamma)` At r = 8 and γ = 1.4 that is 56.5 %, and at r = 10 it is 60.2 % (derived). Those numbers are well above what any car engine delivers, and the gap is the subject of *Modeling real systems* below. They are also well below the Carnot ceiling drawn alongside them: taking the peak cycle temperature as the 1,500 K quoted for internal combustion and the cold reservoir as 300 K gives a ceiling of 0.8 (derived), so the ideal Otto cycle at r = 8 collects only about seven-tenths of what the temperature extremes would allow, before a single real loss is counted.[^likharev-ice] The reason is visible on the T–s loop. Constant-volume heat addition happens over a rising temperature, not at the top temperature, so the average temperature at which heat enters is lower than T_max; constant-volume rejection happens over a falling temperature, so the average temperature at which heat leaves is higher than T_min. Carnot's rectangle is the case where both averages equal the extremes, and every departure from it is paid for. Raising r pushes the loop toward the rectangle, which is why compression ratio is the single most important number in the design — and why knock, not thermodynamics, is what limits it in practice. ### Power cycles Power cycles are the engine direction, and they divide by what the working fluid does. Gas cycles keep the fluid gaseous throughout: Otto and [[Diesel_cycle|Diesel]] in reciprocating engines, [[Brayton_cycle|Brayton]] in gas turbines. Vapour cycles boil and condense the fluid, which lets heat enter and leave at nearly constant temperature; the [[Rankine_cycle|Rankine cycle]] is the example that drives most of the world's [[Thermal_power_station|power stations]]. Diesel differs from Otto only in adding its heat at constant pressure over a cutoff ratio ρ = V₃/V₂, and its efficiency carries a penalty factor for doing so: η_Diesel = 1 − r^(1−γ)·(ρ^γ − 1)/(γ(ρ − 1)). That factor exceeds one for every ρ > 1 and tends to one as ρ → 1, so at equal compression ratio the Diesel cycle is always the less efficient of the two — at r = 18, ρ = 2 and γ = 1.4 it gives 63.2 % (derived). Diesel engines win in practice because they are not held to equal compression ratio: having no premixed charge to detonate, they run at r near 18 where a spark-ignition engine cannot. Brayton replaces reciprocation with steady flow, and its efficiency depends on the pressure ratio rather than the volume ratio: η_Brayton = 1 − r_p^((1−γ)/γ), which is 48.2 % at r_p = 10 and 62.2 % at r_p = 30 (derived). It is the cycle of the [[Jet_engine|gas turbine]], and the [[Turbojet|turbojet]] is one shaft carrying compressor and turbine with the combustor between them.[^ruijgrok-brayton] ### Heat pump cycles Traversed the other way, a cycle moves heat from cold to hot at the cost of work, and the figure of merit is a [[Coefficient_of_performance|coefficient of performance]] rather than an efficiency: COP_R = T_L/(T_H − T_L) for cooling and COP_HP = T_H/(T_H − T_L) for heating, the two differing by exactly one.[^yan-cop] Values above one are normal and are not a violation of anything, because the output is heat moved rather than energy created. The vapour-compression cycle is the near-universal implementation: compress the refrigerant vapour, condense it while rejecting heat, throttle it through an expansion valve, and evaporate it while absorbing heat. The throttle is deliberately irreversible — it is the one place a working cycle accepts a large entropy penalty in exchange for mechanical simplicity — and replacing it with an expander recovers a few per cent at considerable cost. ## Modeling real systems Every efficiency quoted above belongs to an air-standard analysis: the working fluid is treated as a single ideal gas with constant specific heats, combustion is replaced by heat addition from outside, and there is no friction, no leakage and no pressure drop. The gap between those numbers and a dynamometer is not one effect but a stack of them. The first is the ideal-gas assumption itself. The model holds at high temperature and low pressure in the superheated region, within about 1 % over the region the book shades, and fails near saturation and near the [[Critical_point_(thermodynamics)|critical point]].[^yan-ideal-limits] Where it fails, a compressibility factor Z = Pv/(RT) restores the relation as v = Z·R·T/P, read off a generalised chart against the reduced properties P_r = P/P_crit and T_r = T/T_crit.[^yan-z] The book's worked case is steam at 3 MPa and 350 °C, whose tabulated specific volume is 0.09056 m³/kg: the ideal-gas value 0.09586 m³/kg is 5.9 % high, while Z = 0.94 gives 0.09011 m³/kg, within 0.5 % (computed).[^yan-steam-z] Using the ideal-gas law without checking is, in the book's phrase, "a common mistake".[^yan-ideal-limits] The second is irreversibility: friction, unrestrained expansion, mixing, [[Heat_transfer|heat transfer]] across a finite temperature difference, electrical resistance, inelastic deformation and chemical reaction all destroy available work, and every one of them is present in a running engine.[^yan-irrev] The third is that the working fluid's specific heats rise with temperature, so a constant-γ analysis overstates the expansion work at the top of the loop. Real-cycle analysis therefore keeps the loop's topology and replaces each idealised leg with a measured or modelled one — which is why the microsim is labelled as an ideal-cycle instrument. ILLUSTRATIVE: it computes closed-form air-standard efficiencies with constant γ and no losses, so its numbers are upper bounds on a real machine of the same geometry, not predictions of one. ## Well-known thermodynamic cycles A small number of loops account for nearly all practical machines, and they are best compared at the level of what each does with its heat rather than by hardware. | Cycle | Heat in | Heat out | Ideal efficiency | Typical realisation | |---|---|---|---|---| | Carnot | isothermal at T_H | isothermal at T_L | 1 − T_L/T_H | none (a reference) | | Otto | constant volume | constant volume | 1 − r^(1−γ) | spark-ignition engine | | Diesel | constant pressure | constant volume | 1 − r^(1−γ)(ρ^γ−1)/(γ(ρ−1)) | compression-ignition engine | | Brayton | constant pressure | constant pressure | 1 − r_p^((1−γ)/γ) | gas turbine, turbojet | | Stirling | isothermal at T_H | isothermal at T_L | 1 − T_L/T_H with ideal regeneration | external-combustion engine | | Rankine | boiling at ≈ constant T | condensing at ≈ constant T | below Carnot for the same extremes | steam power station | *Air-standard forms for the gas cycles; the Carnot and Stirling entries are the reversible limits.*[^yan-eta][^yan-ch1] ### Ideal cycle An ideal cycle is one built entirely from reversible processes on a fluid whose equation of state is known exactly. It is a fiction in two senses: no real process is reversible, and no real fluid is an ideal gas across the range a cycle sweeps. Its purpose is comparative. Because an ideal cycle's efficiency depends only on ratios the designer controls — r, ρ, r_p, γ — it separates the part of a machine's performance that follows from its topology from the part that follows from its execution. Two engines with the same ideal efficiency and different measured efficiencies differ in execution; two with different ideal efficiencies differ in kind. ### Carnot cycle The [[Carnot_cycle|Carnot cycle]] is four processes in a fixed order: isothermal heat addition at T_H, adiabatic expansion to T_L, isothermal heat rejection at T_L, and adiabatic compression back to the start.[^yan-cycle] On a T–s diagram it is a rectangle, and its efficiency is 1 − T_L/T_H with no reference to the working substance.[^yan-ts][^yan-eta] No engine has been built to it, for a practical reason: the isothermal legs require heat to cross a boundary at vanishing temperature difference, which takes infinite time, so a Carnot engine at its stated efficiency produces zero power. It survives as the standard against which every other loop is scored, and as the line the microsim draws on every setting. [[Nicolas_Léonard_Sadi_Carnot|Sadi Carnot]] derived it in 1824 while still holding the caloric theory of heat.[^carnot1824] ### Stirling cycle The [[Stirling_cycle|Stirling cycle]] replaces Carnot's adiabats with constant-volume legs and adds a regenerator — a thermal store that absorbs the heat given up on one leg and returns it on the other. With a perfect regenerator the external heat exchanges take place only on the two isothermal legs, so the cycle's efficiency equals Carnot's between the same reservoirs. That is the strongest claim any practical loop can make, and it is why the [[Stirling_engine|Stirling engine]] keeps reappearing: its heat is supplied externally, so it will run on anything hot, and its regenerator recovers what other gas cycles throw away. The difficulty is that a real regenerator has finite [[Heat_exchanger|heat-transfer]] area and a dead volume that never participates in the work-producing expansion. Both penalties grow with engine speed, so Stirling engines are efficient and slow, which is a poor combination for vehicles and an acceptable one for cryocoolers and small generators. ## State functions and entropy The properties of a cycle follow from one fact: internal energy and entropy are state functions, so ∮dU = 0 and ∮dS = 0 for the working fluid over any complete loop, reversible or not. Combining the first and second laws for a simple compressible substance gives the fundamental relation dE = T dS − P dV, from which every cycle's heat and work integrals can be read.[^likharev-fundamental] Because the fluid's entropy returns to its starting value, any entropy generated inside the machine must be exported with the rejected heat. For a reversible cycle the exported entropy exactly equals the imported entropy, Q_H/T_H = Q_L/T_L, which rearranges to the Carnot efficiency.[^yan-eta] For a real cycle, generation inside the boundary forces Q_L/T_L > Q_H/T_H, so more heat must be rejected per unit of heat supplied, and the efficiency falls: the Clausius inequality ∮δQ/T ≤ 0 is the compact statement of it. Entropy is therefore not an extra bookkeeping convention laid on top of the cycle — it is the reason the loop cannot be closed for free, and the T–s diagram is the drawing in which that cost has an area. This is also what makes a cycle's shape, rather than its hardware, the object of study. Two machines whose T–s loops coincide have the same efficiency however differently they are built, and a loop that comes closer to the Carnot rectangle is a better engine in a sense that owes nothing to materials or manufacture. The microsim is built around exactly that comparison. ## See also - [[Otto_cycle]] - [[Diesel_cycle]] - [[Brayton_cycle]] - [[Stirling_cycle]] - [[Stirling_engine]] - [[Internal_combustion_engine]] - [[Turbojet]] - [[Heat_engine]] - [[Carnot_cycle]] - [[Rankine_cycle]] ## 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; a device above the Carnot line 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 in this range were lost in text extraction and are supplied in standard form by sub-manual 10 §2.3. [^yan-cycle]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, p. 270 (the Carnot cycle's four processes in order). 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-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)). Portal Book 115. [^yan-irrev]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, p. 269 (friction, unrestrained expansion, mixing, heat transfer across a finite ΔT, electrical resistance, inelastic deformation and chemical reaction as sources of irreversibility). Portal Book 115. [^yan-ch1]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 1 "Basic Concepts and Definitions", pp. 31–58 (the Otto cycle and the heat-engine vocabulary; page to pin). Portal Book 115. [^yan-ideal-relations]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 4 "The First Law of Thermodynamics for Closed Systems", pp. 127–186 (the isentropic ideal-gas relations, of which `p V^gamma = const` is the one used here; page to pin). Portal Book 115. [^yan-ideal-limits]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 3 "Ideal and Real Gasses", pp. 109–110 (the ideal-gas model holds at high temperature and low pressure in the superheated region, within about 1 % in the shaded region of the water T–v diagram, and fails near saturation and the critical point; applying it without checking is called "a common mistake"). Portal Book 115. [^yan-z]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 3, pp. 116–119 and p. 122 (Z = Pv/(RT) and v = Z·R·T/P; reduced properties P_r = P/P_crit and T_r = T/T_crit; Z → 1 as P → 0; water's P_crit = 22.06 MPa and T_crit = 647.1 K). Portal Book 115. [^yan-steam-z]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 3, pp. 121–123 (steam at 3 MPa and 350 °C: tabulated v = 0.09056 m³/kg and Z = 0.94 giving 0.09011 m³/kg). Portal Book 115. The percentage errors quoted here — ideal +5.9 %, Z-corrected −0.50 %, with P_r = 0.136 and T_r = 0.963 — are computed in sub-manual 10 §2.2. [^likharev-work]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 1 "Review of Thermodynamics", pp. 22–23 (Q_H = Q_L + W and −W = ∮P dV over a cycle). Portal Book 075, https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^likharev-fundamental]: Likharev (2013), *Part SM: Statistical Mechanics*, Chapter 1, pp. 10–12 (dE = T dS − P dV and T = (∂E/∂S)_V). Portal Book 075. [^likharev-ice]: Likharev (2013), *Part SM: Statistical Mechanics*, Chapter 1, p. 24 (internal combustion runs at T_H ≈ 1,500 K against a few hundred kelvin for steam engines; η = 1 − T_L/T_H bounds any engine). The 0.8 ceiling for 1,500 K against 300 K is derived in sub-manual 04 §5.3. Portal Book 075. [^likharev-gamma]: Likharev (2013), *Part SM: Statistical Mechanics*, Chapter 1, pp. 19–20 (P V = N T with T in energy units and C_P − C_V = N). Sub-manual 04 §5.3 records that γ — 5/3 for a monatomic gas — is not printed in the excerpt, so the value used on any adiabat must carry its own source; the 1.4 used here for air is a standard value. Portal Book 075. [^ruijgrok-brayton]: Ruijgrok, Gerrit; Voskuijl, Mark; Varriale, Carmine (2025). *Elements of Airplane Performance*, pp. 176–180 (the gas-turbine cycle as it appears in propulsion analysis; used here as the cross-check on the Brayton efficiencies). Portal Book 008, https://open.umn.edu/opentextbooks/textbooks/elements-of-airplane-performance [^carnot1824]: Carnot, Sadi (1824). *Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance*. Paris: Bachelier. ## Further reading - Yan, *Introduction to Engineering Thermodynamics* (2022), Chapter 6 "Entropy and the Second Law of Thermodynamics", pp. 239–348, and Chapter 3 "Ideal and Real Gasses", pp. 105–126 — Portal Book 115. - Likharev, *Essential Graduate Physics, Part SM: Statistical Mechanics* (2013), Chapter 1 "Review of Thermodynamics" — Portal Book 075. - Mitofsky, *Direct Energy* (2018), for the converters that obey the Carnot ceiling without running a cycle at all — Portal Book 055. - Murphy, *Energy and Human Ambitions on a Finite Planet* (2021), for where cycle efficiency sits in a national energy budget — Portal Book 097. - Carnot, *Réflexions sur la puissance motrice du feu* (1824), the essay that made the loop the object of study. ## External links - [Introduction to Engineering Thermodynamics](https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics), Yan (2022) — Portal Book 115, the cycle chapters used throughout this page - [Part SM: Statistical Mechanics](https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics), Likharev (2013) — Portal Book 075 - [Elements of Airplane Performance](https://open.umn.edu/opentextbooks/textbooks/elements-of-airplane-performance), Ruijgrok, Voskuijl and Varriale (2025) — Portal Book 008, for the gas-turbine cross-check - The Wikipedia pair's external links list further cycle diagrams and property tables <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Thermodynamic_cycle.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Thermodynamic cycle* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Thermodynamic_cycle.html" data-title="Thermodynamic cycle"></div> *Built from `MICROSIM_GUIDE/specs/sims/Thermodynamic_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/Thermodynamic_cycle) : [Wikitube](https://en.wikitube.io/wiki/Thermodynamic_cycle) · pinned revision [1360674752](https://en.wikipedia.org/w/index.php?oldid=1360674752) · 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 E33 · sim pending (matter/Thermodynamic_cycle).*