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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Idealized_Stirling_cycle_thermodynamics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Idealized_Stirling_cycle_thermodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Idealized Stirling cycle thermodynamics</span> </div> </a> <ul id="toc-Idealized_Stirling_cycle_thermodynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Piston_motion_variations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Piston_motion_variations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Piston motion variations</span> </div> </a> <ul id="toc-Piston_motion_variations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Volume_variations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Volume_variations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Volume variations</span> </div> </a> <ul id="toc-Volume_variations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pressure-versus-volume_graph" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pressure-versus-volume_graph"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Pressure-versus-volume graph</span> </div> </a> <ul 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mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cicle_de_Stirling" title="Cicle de Stirling – Catalan" lang="ca" hreflang="ca" data-title="Cicle de Stirling" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Stirling-Kreisprozess" title="Stirling-Kreisprozess – German" lang="de" hreflang="de" data-title="Stirling-Kreisprozess" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ciclo_Stirling" title="Ciclo Stirling – Spanish" lang="es" hreflang="es" data-title="Ciclo Stirling" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%D8%B1%D8%AE%D9%87_%D8%A7%D8%B3%D8%AA%D8%B1%D9%84%DB%8C%D9%86%DA%AF" title="چرخه استرلینگ – Persian" lang="fa" hreflang="fa" data-title="چرخه استرلینگ" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cycle_de_stirling" title="Cycle de stirling – French" lang="fr" hreflang="fr" data-title="Cycle de stirling" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ciclo_Stirling" title="Ciclo Stirling – Galician" lang="gl" hreflang="gl" data-title="Ciclo Stirling" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Stirlingov_ciklus" title="Stirlingov ciklus – Croatian" lang="hr" hreflang="hr" data-title="Stirlingov ciklus" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ciclo_Stirling" title="Ciclo Stirling – Italian" lang="it" hreflang="it" data-title="Ciclo Stirling" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%92%D7%9C_%D7%A1%D7%98%D7%A8%D7%9C%D7%99%D7%A0%D7%92" title="מעגל סטרלינג – Hebrew" lang="he" hreflang="he" data-title="מעגל סטרלינג" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Stirling-ciklus" title="Stirling-ciklus – Hungarian" lang="hu" hreflang="hu" data-title="Stirling-ciklus" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Ciclo_Stirling" title="Ciclo Stirling – Portuguese" lang="pt" hreflang="pt" data-title="Ciclo Stirling" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A6%D0%B8%D0%BA%D0%BB_%D0%A1%D1%82%D0%B8%D1%80%D0%BB%D0%B8%D0%BD%D0%B3%D0%B0" title="Цикл Стирлинга – Russian" lang="ru" hreflang="ru" data-title="Цикл Стирлинга" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A6%D0%B8%D0%BA%D0%BB_%D0%A1%D1%82%D1%96%D1%80%D0%BB%D1%96%D0%BD%D0%B3%D0%B0" title="Цикл Стірлінга – Ukrainian" lang="uk" hreflang="uk" data-title="Цикл Стірлінга" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D8%B3%D9%B9%D8%B1%D9%84%D9%86%DA%AF_%DA%86%DA%A9%D8%B1" title="اسٹرلنگ چکر – Urdu" lang="ur" hreflang="ur" data-title="اسٹرلنگ چکر" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li> </ul> <div class="after-portlet 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Thermodynamic cycle that includes the basic Stirling engine</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.3em;border-bottom:1px solid #aaa;"><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></th></tr><tr><td class="sidebar-image" style="display:block;margin:0.3em 0 0.4em;"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Carnot_heat_engine" title="Carnot heat engine"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/220px-Carnot_heat_engine_2.svg.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/330px-Carnot_heat_engine_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/440px-Carnot_heat_engine_2.svg.png 2x" data-file-width="840" data-file-height="370" /></a></span><div class="sidebar-caption">The classical <a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamics" title="Thermodynamics">Classical</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Chemical_thermodynamics" title="Chemical thermodynamics">Chemical</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div> <ul><li><a href="/wiki/Equilibrium_thermodynamics" title="Equilibrium thermodynamics">Equilibrium</a> / <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth</a></li> <li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second</a></li> <li><a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">Third</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Closed_system" title="Closed system">Closed system</a></li> <li><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Ideal_gas" title="Ideal gas">Ideal gas</a></li> <li><a href="/wiki/Real_gas" title="Real gas">Real gas</a></li> <li><a href="/wiki/State_of_matter" title="State of matter">State of matter</a></li> <li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li> <li><a href="/wiki/Control_volume" title="Control volume">Control volume</a></li> <li><a href="/wiki/Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal</a></li> <li><a href="/wiki/Adiabatic_process" title="Adiabatic process">Adiabatic</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic</a></li> <li><a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li> <li><a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversibility</a></li> <li><a href="/wiki/Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Heat_engine" title="Heat engine">Heat engines</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li> <li><a href="/wiki/Thermal_efficiency" title="Thermal efficiency">Thermal efficiency</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div><div class="sidebar-list-content mw-collapsible-content"><div style="font-size:90%;padding-bottom:0.2em;border-bottom:1px solid #aaa;">Note: <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables</a> in <i>italics</i></div> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/Reduced_properties" title="Reduced properties">Reduced properties</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Property databases</a></li></ul> <div style="font-size:90%;margin-top:0.4em;border-top:1px solid #aaa;text-align:center;"> <table> <tbody><tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:1.676ex;" alt="{\displaystyle c=}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Compressibility" title="Compressibility">Compressibility</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c042bf1456bd4d2a8caed1f4912820a7ecbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta =-}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc4a48eb2412f08b54fe438b5139c88f9cfa372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \partial p}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92d4583d351f08c1c70985f0c843b2fff1b01e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="{\displaystyle \alpha =}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> </tbody></table></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a></li> <li><a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li> <li><a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul> </div> <ul><li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li> <li><a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li> <li><a href="/wiki/Bridgman%27s_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li> <li><i><a href="/wiki/Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li></ul> </div> <div class="plainlist"><ul><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Internal_energy" title="Internal energy">Internal energy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(S,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f33f9c6551562ec836007b035c2de6323d2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(S,V)}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(S,p)=U+pV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(S,p)=U+pV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6407d78e5f39d07f70e2414a92e08e2e068519f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:2.843ex;" alt="{\displaystyle H(S,p)=U+pV}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(T,V)=U-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(T,V)=U-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e93692f031ba6484d82731c54db83a69daed3f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.867ex; height:2.843ex;" alt="{\displaystyle A(T,V)=U-TS}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(T,p)=H-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(T,p)=H-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd7a8f0b8ae04963da133e3b202432e1b6caed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.614ex; height:2.843ex;" alt="{\displaystyle G(T,p)=H-TS}"></span></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li>History</li><li>Culture</li></ul></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> History</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li></ul> </div> <ul><li><a href="/wiki/History_of_perpetual_motion_machines" title="History of perpetual motion machines">"Perpetual motion" machines</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Philosophy_of_thermal_and_statistical_physics" class="mw-redirect" title="Philosophy of thermal and statistical physics">Philosophy</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li> <li><a href="/wiki/Entropy_and_life" title="Entropy and life">Entropy and life</a></li> <li><a href="/wiki/Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Theories</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul> </div> <ul><li><a href="/wiki/Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size:85%;">("living force")</span></a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Motive power</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/List_of_important_publications_in_physics" title="List of important publications in physics">Key publications</a></th></tr><tr><td class="sidebar-content"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/An_Inquiry_Concerning_the_Source_of_the_Heat_Which_Is_Excited_by_Friction" title="An Inquiry Concerning the Source of the Heat Which Is Excited by Friction">An Inquiry Concerning the<br />Source ... Friction</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<br />Heterogeneous Substances</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the<br />Motive Power of Fire</a></i></div></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Timelines</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Heat engines</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <div class="hlist"><ul><li>Art</li><li>Education</li></ul></div></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Maxwell%27s_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li> <li><a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy as energy dispersal</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li> <li><a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li> <li><a href="/wiki/Th%C3%A9ophile_de_Donder" title="Théophile de Donder">de Donder</a></li> <li><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li> <li><a href="/wiki/Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li> <li><a href="/wiki/Walther_Nernst" title="Walther Nernst">Nernst</a></li> <li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li> <li><a href="/wiki/John_Smeaton" title="John Smeaton">Smeaton</a></li> <li><a href="/wiki/Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li> <li><a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li> <li><a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/John_James_Waterston" title="John James Waterston">Waterston</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Other</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Nucleation" title="Nucleation">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the <a href="/wiki/Adiabatic" class="mw-redirect" title="Adiabatic">adiabatic</a> Stirling cycle. For the idealized Stirling cycle, see <a href="/wiki/Stirling_engine#Theory" title="Stirling engine">Stirling engine § Theory</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Applications_of_the_Stirling_engine" title="Applications of the Stirling engine">Applications of the Stirling engine</a></div> <p>The <b>Stirling cycle</b> is a <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">thermodynamic cycle</a> that describes the general class of Stirling devices. This includes the original <a href="/wiki/Stirling_engine" title="Stirling engine">Stirling engine</a> that was invented, developed and patented in 1816 by <a href="/wiki/Robert_Stirling" title="Robert Stirling">Robert Stirling</a> with help from his brother, an <a href="/wiki/Engineer" title="Engineer">engineer</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The ideal <a href="/wiki/Otto_cycle" title="Otto cycle">Otto</a> and <a href="/wiki/Diesel_cycle" title="Diesel cycle">Diesel</a> cycles are not totally reversible because they involve heat transfer through a finite temperature difference during the irreversible <a href="/wiki/Isochoric_process" title="Isochoric process">isochoric</a>/<a href="/wiki/Isobaric_process" title="Isobaric process">isobaric</a> heat-addition and heat-rejection processes. The irreversibility renders the thermal efficiency of these cycles less than that of a <a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot engine</a> operating within the same limits of temperature. Another cycle that features isothermal heat-addition and heat-rejection processes is the Stirling cycle, which is an altered version of the Carnot cycle in which the two isentropic processes featured in the Carnot cycle are replaced by two constant-volume regeneration processes. </p><p>The cycle is reversible, meaning that if supplied with mechanical power, it can function as a <a href="/wiki/Heat_pump" title="Heat pump">heat pump</a> for heating or <a href="/wiki/Refrigeration" title="Refrigeration">cooling</a>, and even for <a href="/wiki/Cryogenic" class="mw-redirect" title="Cryogenic">cryogenic</a> cooling. The cycle is defined as a closed <a href="/wiki/Regenerative_heat_exchanger" title="Regenerative heat exchanger">regenerative</a> cycle with a <a href="/wiki/Gas" title="Gas">gaseous</a> working fluid. "Closed cycle" means the working fluid is permanently contained within the <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a>. This also categorizes the engine device as an <a href="/wiki/External_combustion_engine" title="External combustion engine">external heat engine</a>. "Regenerative" refers to the use of an internal heat exchanger called a <a href="#Regenerator">regenerator</a> which increases the device's <a href="/wiki/Thermal_efficiency" title="Thermal efficiency">thermal efficiency</a>. </p><p>The cycle is the same as most other heat cycles in that there are four main processes: compression, heat addition, expansion, and heat removal. However, these processes are not discrete, but rather the transitions overlap. </p><p>The Stirling cycle is a highly advanced subject that has defied analysis by many experts for over 190 years. Highly advanced thermodynamics is required to describe the cycle. Professor Israel Urieli writes: "...the various 'ideal' cycles (such as the Schmidt cycle) are neither physically realizable nor representative of the Stirling cycle".<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The analytical problem of the <a href="/wiki/Regenerative_heat_exchanger" title="Regenerative heat exchanger">regenerator</a> (the central heat exchanger in the Stirling cycle) is judged by Jakob to rank "among the most difficult and involved that are encountered in engineering".<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Idealized_Stirling_cycle_thermodynamics">Idealized Stirling cycle thermodynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=1" title="Edit section: Idealized Stirling cycle thermodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Stirling_cycle_pV.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Stirling_cycle_pV.svg/350px-Stirling_cycle_pV.svg.png" decoding="async" width="350" height="279" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Stirling_cycle_pV.svg/525px-Stirling_cycle_pV.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Stirling_cycle_pV.svg/700px-Stirling_cycle_pV.svg.png 2x" data-file-width="281" data-file-height="224" /></a><figcaption>A <a href="/wiki/Pressure_volume_diagram" class="mw-redirect" title="Pressure volume diagram">pressure/volume graph</a> of the <i>idealized</i> Stirling cycle. In real applications of the Stirling cycles (e.g. Stirling engines) this cycle is quasi-elliptical.</figcaption></figure> <p>The <b>idealized</b> Stirling<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> cycle consists of four <a href="/wiki/Thermodynamic_processes" class="mw-redirect" title="Thermodynamic processes">thermodynamic processes</a> acting on the working fluid (See diagram to right): </p> <div><ol style="list-style-type:upper-roman"><li>1→2 Isothermal heat addition (expansion).</li><li>2→3 Isochoric heat removal (constant volume).</li><li>3→4 Isothermal heat removal (compression).</li><li>4→1 Isochoric heat addition (constant volume).</li></ol></div> <div class="mw-heading mw-heading2"><h2 id="Piston_motion_variations">Piston motion variations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=2" title="Edit section: Piston motion variations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Stirling_cycle" title="Special:EditPage/Stirling cycle">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">June 2020</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ideal-stirling-cycle.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Ideal-stirling-cycle.gif/220px-Ideal-stirling-cycle.gif" decoding="async" width="220" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Ideal-stirling-cycle.gif/330px-Ideal-stirling-cycle.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/9/9e/Ideal-stirling-cycle.gif 2x" data-file-width="360" data-file-height="256" /></a><figcaption>A model of a four-phase Stirling cycle</figcaption></figure> <p>Most <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> textbooks describe a highly simplified form of Stirling cycle consisting of four processes. This is known as an "ideal Stirling cycle", because it is an "idealized" model, and not necessarily an optimized cycle. Theoretically, the "ideal cycle" does have high net work output, but it is rarely used in practical applications, in part because other cycles are simpler or reduce peak stresses on bearings and other components. For convenience, the designer may elect to use piston motions dictated by system dynamics, such as mechanical linkage mechanisms. At any rate, the efficiency and cycle <i>power</i> are nearly as good as an actual implementation of the idealized case. A typical piston crank or linkage in a so named "kinematic" design often results in a near-sinusoidal piston motion. Some designs will cause the piston to "dwell" at either extreme of travel. </p><p>Many kinematic linkages, such as the well known "<a href="/w/index.php?title=Ross_yoke&action=edit&redlink=1" class="new" title="Ross yoke (page does not exist)">Ross yoke</a>", will exhibit near-sinusoidal motion. However, other linkages, such as the "<a href="/wiki/Rhombic_drive" title="Rhombic drive">rhombic drive</a>", will exhibit more non-sinusoidal motion. To a lesser extent, the ideal cycle introduces complications, since it would require somewhat higher piston acceleration and higher viscous pumping losses of the working fluid. The material stresses and pumping losses in an optimized engine, however, would only be intolerable when approaching the "ideal cycle" and/or at high cycle rates. Other issues include the time required for heat transfer, particularly for the <a href="/wiki/Isothermal_process" title="Isothermal process">isothermal processes</a>. In an engine with a cycle approaching the "ideal cycle", the cycle rate might have to be reduced to address these issues. </p><p>In the most basic model of a free piston device, the kinematics will result in <a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">simple harmonic motion</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Volume_variations">Volume variations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=3" title="Edit section: Volume variations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In beta and gamma engines, generally the phase angle difference between the piston motions is <i>not</i> the same as the phase angle of the volume variations. However, in the alpha Stirling, they are the same.<sup id="cite_ref-Organ_6-0" class="reference"><a href="#cite_note-Organ-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The rest of the article assumes sinusoidal volume variations, as in an alpha Stirling with co-linear pistons, so named an "opposed piston" alpha device. </p><p>caveat: Among the many inaccuracies in this article, a co-linear alpha configuration is referenced, above. Such a configuration would be beta. Alternatively, it would be an alpha, that has an unacceptably inefficient linkage system. </p> <div class="mw-heading mw-heading2"><h2 id="Pressure-versus-volume_graph">Pressure-versus-volume graph</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=4" title="Edit section: Pressure-versus-volume graph"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This type of plot is used to characterize almost all thermodynamic cycles. The result of sinusoidal volume variations is the quasi-elliptical shaped cycle shown in Figure 1. Compared to the idealized cycle, this cycle is a more realistic representation of most real Stirling engines. The four points in the graph indicate the crank angle in <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p><span typeof="mw:File"><a href="/wiki/File:PV_plot_adiab_sim.png" class="mw-file-description" title="Figure 1: Pressure vs volume plot, with four points labeled in crank angle degrees"><img alt="Figure 1: Pressure vs volume plot, with four points labeled in crank angle degrees" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/PV_plot_adiab_sim.png/650px-PV_plot_adiab_sim.png" decoding="async" width="650" height="451" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/10/PV_plot_adiab_sim.png 1.5x" data-file-width="657" data-file-height="456" /></a></span> </p><p>The <i><a href="/wiki/Adiabatic" class="mw-redirect" title="Adiabatic">adiabatic</a></i> Stirling cycle is similar to the <b>idealized</b> Stirling cycle; however, the four thermodynamic processes are slightly different (see graph above): </p> <ul><li>180° to 270°, pseudo-<a href="/wiki/Isothermal" class="mw-redirect" title="Isothermal">isothermal</a> <a href="/wiki/Thermal_expansion" title="Thermal expansion">expansion</a>. The expansion space is heated externally, and the gas undergoes near-isothermal expansion.</li> <li>270° to 0°, near-constant-volume (or near-<a href="/wiki/Isometric_process" class="mw-redirect" title="Isometric process">isometric</a> or <a href="/wiki/Isochoric_process" title="Isochoric process">isochoric</a>) heat removal. The gas is passed through the <a href="/wiki/Regenerative_heat_exchanger" title="Regenerative heat exchanger">regenerator</a>, thus cooling the gas, and transferring heat to the regenerator for use in the next cycle.</li> <li>0° to 90°, pseudo-<a href="/wiki/Isothermal" class="mw-redirect" title="Isothermal">isothermal</a> <a href="/wiki/Gas_compression" class="mw-redirect" title="Gas compression">compression</a>. The compression space is <a href="/wiki/Intercooler" title="Intercooler">intercooled</a>, so the gas undergoes near-isothermal compression.</li> <li>90° to 180°, near-constant-volume (near-<a href="/wiki/Isometric_process" class="mw-redirect" title="Isometric process">isometric</a> or <a href="/wiki/Isochoric_process" title="Isochoric process">isochoric</a>) heat addition. The compressed air flows back through the regenerator and picks up heat on the way to the heated expansion space.</li></ul> <p>With the exception of a Stirling <a href="/wiki/Thermoacoustic_engine" class="mw-redirect" title="Thermoacoustic engine">thermoacoustic engine</a>, none of the gas particles actually flow through the complete cycle. So this approach is not amenable to further analysis of the cycle. However, it provides an overview and indicates the cycle work. </p> <div class="mw-heading mw-heading2"><h2 id="Particle/mass_motion"><span id="Particle.2Fmass_motion"></span>Particle/mass motion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=5" title="Edit section: Particle/mass motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Figure 2 shows the <a href="/wiki/Streakline" class="mw-redirect" title="Streakline">streaklines</a> which indicate how gas flows through a real Stirling engine. The vertical colored lines delineate the volumes of the engine. From left to right, they are: the volume swept by the expansion (power) piston, the clearance volume (which prevents the piston from contacting the hot heat exchanger), the heater, the regenerator, the cooler, the cooler clearance volume, and the compression volume swept by the compression piston. </p> <table align="center"> <tbody><tr> <td valign="top"><span typeof="mw:File"><a href="/wiki/File:Particle_mass_plot.png" class="mw-file-description" title="Figure 2: Particle mass plot"><img alt="Figure 2" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Particle_mass_plot.png/650px-Particle_mass_plot.png" decoding="async" width="650" height="432" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e0/Particle_mass_plot.png 1.5x" data-file-width="750" data-file-height="499" /></a></span> </td> <td valign="top"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Alpha_Stirling_highres.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Alpha_Stirling_highres.gif/216px-Alpha_Stirling_highres.gif" decoding="async" width="216" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Alpha_Stirling_highres.gif/324px-Alpha_Stirling_highres.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/9/96/Alpha_Stirling_highres.gif 2x" data-file-width="432" data-file-height="396" /></a><figcaption>Alpha type Stirling. Animated version.</figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Heat-exchanger_pressure_drop">Heat-exchanger pressure drop</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=6" title="Edit section: Heat-exchanger pressure drop"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Also referred to as "pumping losses", the pressure drops shown in Figure 3 are caused by viscous flow through the heat exchangers. The red line represents the heater, green is the regenerator, and blue is the cooler. To properly design the heat exchangers, <a href="/wiki/Multivariate_optimization" class="mw-redirect" title="Multivariate optimization">multivariate optimization</a> is required to obtain sufficient heat transfer with acceptable flow losses.<sup id="cite_ref-Organ_6-1" class="reference"><a href="#cite_note-Organ-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The flow losses shown here are relatively low, and they are barely visible in the following image, which will show the overall pressure variations in the cycle. </p><p><span typeof="mw:File"><a href="/wiki/File:Heat_exchanger_pressure_drop.png" class="mw-file-description" title="Figure 3: Heat exchanger pressure drop"><img alt="Figure 3: Heat exchanger pressure drop" src="//upload.wikimedia.org/wikipedia/commons/8/8a/Heat_exchanger_pressure_drop.png" decoding="async" width="650" height="459" class="mw-file-element" data-file-width="636" data-file-height="449" /></a></span> </p> <div class="mw-heading mw-heading2"><h2 id="Pressure_versus_crank_angle">Pressure versus crank angle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=7" title="Edit section: Pressure versus crank angle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Figure 4 shows results from an "adiabatic simulation" with non-ideal heat exchangers. Note that the pressure drop across the regenerator is very low compared to the overall pressure variation in the cycle. </p><p><span typeof="mw:File"><a href="/wiki/File:Pressure_vs_angle.png" class="mw-file-description" title="Figure 4: Pressure vs crank angle plot"><img alt="Figure 4: Pressure vs crank angle plot" src="//upload.wikimedia.org/wikipedia/commons/5/5a/Pressure_vs_angle.png" decoding="async" width="650" height="455" class="mw-file-element" data-file-width="638" data-file-height="447" /></a></span> </p> <div class="mw-heading mw-heading2"><h2 id="Temperature_versus_crank_angle">Temperature versus crank angle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=8" title="Edit section: Temperature versus crank angle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span typeof="mw:File"><a href="/wiki/File:Temperature_vs_angle.png" class="mw-file-description" title="Figure 5: Temperature vs crank angle plot"><img alt="Figure 5: Temperature vs crank angle plot" src="//upload.wikimedia.org/wikipedia/commons/7/71/Temperature_vs_angle.png" decoding="async" width="650" height="462" class="mw-file-element" data-file-width="628" data-file-height="446" /></a></span> </p><p>Figure 5 illustrates the adiabatic properties of a real heat exchanger. The straight lines represent the temperatures of the solid portion of the heat exchanger, and the curves are the gas temperatures of the respective spaces. The gas temperature fluctuations are caused by the effects of compression and expansion in the engine, together with non-ideal heat exchangers which have a limited rate of <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a>. When the gas temperature deviates above and below the heat exchanger temperature, it causes thermodynamic losses known as "heat transfer losses" or "hysteresis losses". However, the heat exchangers still work well enough to allow the real cycle to be effective, even if the actual thermal efficiency of the overall system is only about half of the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">theoretical limit</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Cumulative_heat_and_work_energy">Cumulative heat and work energy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=9" title="Edit section: Cumulative heat and work energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span typeof="mw:File"><a href="/wiki/File:Energy_vs_crank_angle.png" class="mw-file-description" title="Figure 6: Heat and work energy vs crank angle"><img alt="Figure 6: Heat and work energy vs crank angle" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Energy_vs_crank_angle.png/800px-Energy_vs_crank_angle.png" decoding="async" width="800" height="541" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/d/d4/Energy_vs_crank_angle.png 1.5x" data-file-width="1030" data-file-height="696" /></a></span> </p><p>Figure 6 shows a graph of the alpha-type Stirling engine data, where 'Q' denotes heat energy, and 'W' denotes work energy. The blue dotted line shows the <a href="/wiki/Work_output" title="Work output">work output</a> of the compression space. As the trace dips down, work is done on the gas as it is compressed. During the expansion process of the cycle, some work is actually done <i>on</i> the compression piston, as reflected by the upward movement of the trace. At the end of the cycle, this value is negative, indicating that compression piston requires a net input of work. The blue solid line shows the heat flowing out of the cooler heat exchanger. The heat from the cooler and the work from the compression piston have the same cycle energy. This is consistent with the zero-net heat transfer of the regenerator (solid green line). As would be expected, the heater and the expansion space both have positive energy flow. The black dotted line shows the net work output of the cycle. On this trace, the cycle ends higher than it started, indicating that the <a href="/wiki/Heat_engine" title="Heat engine">heat engine</a> converts energy from heat into work. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Pseudo_Stirling_cycle" title="Pseudo Stirling cycle">Pseudo Stirling cycle</a></li> <li><a href="/wiki/Stirling_engine" title="Stirling engine">Stirling engine</a></li> <li><a href="/wiki/Solar-powered_Stirling_engine" title="Solar-powered Stirling engine">Solar-powered Stirling engine</a></li> <li><a href="/wiki/Stirling_radioisotope_generator" title="Stirling radioisotope generator">Stirling radioisotope generator</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRobert_Sier1999" class="citation book cs1">Robert Sier (1999). <i>Hot air caloric and stirling engines. Vol.1, A history</i> (1st Edition (Revised) ed.). L.A. Mair. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-9526417-0-4" title="Special:BookSources/0-9526417-0-4"><bdi>0-9526417-0-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hot+air+caloric+and+stirling+engines.+Vol.1%2C+A+history&rft.edition=1st+Edition+%28Revised%29&rft.pub=L.A.+Mair&rft.date=1999&rft.isbn=0-9526417-0-4&rft.au=Robert+Sier&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+cycle" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Organ, "The Regenerator and the Stirling Engine", p. xxii, Foreword by Urieli</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Organ, "The Regenerator and the Stirling Engine", p. 7</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Jakob, M. (1957) <i>Heat Transfer II</i> John Wiley, New York, USA and Chapman and Hall, London, UK</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">A. Romanelli <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1704.01611">Alternative thermodynamic cycle for the Stirling machine</a>, American Journal of Physics 85, 926 (2017)</span> </li> <li id="cite_note-Organ-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Organ_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Organ_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Organ, "The Regenerator and the Stirling Engine"</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.ent.ohiou.edu/~urieli/stirling/me422.html">Israel Urieli (Dr. Iz), Associate Professor Mechanical Engineering: Stirling Cycle Machine Analysis</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100630062833/http://www.ent.ohiou.edu/~urieli/stirling/me422.html">Archived</a> 2010-06-30 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_cycle&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>I. Urieli <a rel="nofollow" class="external text" href="https://www.ohio.edu/mechanical/stirling/index.html">Stirling Cycle Machine Analysis</a></li> <li>Polytropic cycle inside Stirling engine <a rel="nofollow" class="external text" href="http://www.transformacni-technologie.cz/en_34.html">Stirling engine cycle</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Thermodynamic_cycles" style="padding:3px"><table class="nowraplinks mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamic_cycles" title="Template:Thermodynamic cycles"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamic_cycles" title="Template talk:Thermodynamic cycles"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamic_cycles" title="Special:EditPage/Template:Thermodynamic cycles"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Thermodynamic_cycles" style="font-size:114%;margin:0 4em"><a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Thermodynamic cycles</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/External_combustion_engine" title="External combustion engine">External<br />combustion / thermal</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Without phase change<br />(<a href="/wiki/Hot_air_engine" title="Hot air engine">hot air engines</a>)</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brayton_cycle#Reverse_Brayton_cycle" title="Brayton cycle">Bell Coleman</a></li> <li><a href="/wiki/Brayton_cycle" title="Brayton cycle">Brayton/Joule</a></li> <li><a href="/wiki/Carnot_cycle" title="Carnot cycle">Carnot</a></li> <li><a href="/wiki/Ericsson_cycle" title="Ericsson cycle">Ericsson</a></li> <li><a class="mw-selflink selflink">Stirling</a></li> <li><a href="/wiki/Pseudo_Stirling_cycle" title="Pseudo Stirling cycle">Stirling (pseudo/adiabatic)</a></li> <li><a href="/wiki/Stoddard_cycle" class="mw-redirect" title="Stoddard cycle">Stoddard</a></li> <li><a href="/wiki/Manson_engine" title="Manson engine">Manson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">With phase change</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kalina_cycle" title="Kalina cycle">Kalina</a></li> <li><a href="/wiki/Hygroscopic_cycle" title="Hygroscopic cycle">Hygroscopic</a></li> <li><a href="/wiki/Rankine_cycle" title="Rankine cycle">Rankine</a> (<a href="/wiki/Organic_Rankine_cycle" title="Organic Rankine cycle">Organic Rankine</a>)</li> <li><a href="/wiki/Regenerative_cycle" class="mw-redirect" title="Regenerative cycle">Regenerative</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Internal_combustion_engine" title="Internal combustion engine">Internal <br />combustion / thermal</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atkinson_cycle" title="Atkinson cycle">Atkinson</a></li> <li><a href="/wiki/Brayton_cycle" title="Brayton cycle">Brayton/Joule</a></li> <li><a href="/wiki/Diesel_cycle" title="Diesel cycle">Diesel</a></li> <li><a href="/wiki/Expander_cycle" title="Expander cycle">Expander</a></li> <li><a href="/wiki/Gas-generator_cycle" title="Gas-generator cycle">Gas-generator</a></li> <li><a href="/wiki/Homogeneous_charge_compression_ignition" title="Homogeneous charge compression ignition">Homogeneous charge compression ignition</a></li> <li><a href="/wiki/Humphrey_cycle" title="Humphrey cycle">Humphrey</a></li> <li><a href="/wiki/Lenoir_cycle" title="Lenoir cycle">Lenoir</a></li> <li><a href="/wiki/Miller_cycle" title="Miller cycle">Miller</a></li> <li><a href="/wiki/Otto_cycle" title="Otto cycle">Otto</a></li> <li><a href="/wiki/Scuderi_cycle" title="Scuderi cycle">Scuderi</a></li> <li><a href="/wiki/Staged_combustion_cycle" title="Staged combustion cycle">Staged combustion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combined_cycle" class="mw-redirect" title="Combined cycle">Combined</a></li> <li><a href="/wiki/High-efficiency_hybrid_cycle" title="High-efficiency hybrid cycle">HEHC</a></li> <li><a href="/wiki/Mixed/dual_cycle" title="Mixed/dual cycle">Mixed/dual</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Refrigeration</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hampson%E2%80%93Linde_cycle" title="Hampson–Linde cycle">Hampson–Linde</a></li> <li><a href="/wiki/Kleemenko_cycle" title="Kleemenko cycle">Kleemenko</a></li> <li><a href="/wiki/Pulse_tube_refrigerator" title="Pulse tube refrigerator">Pulse tube</a></li> <li><a href="/wiki/Regenerative_cooling" title="Regenerative cooling">Regenerative cooling</a></li> <li><a href="/wiki/Transcritical_cycle" title="Transcritical cycle">Transcritical</a></li> <li><a href="/wiki/Absorption_refrigerator" title="Absorption refrigerator">Vapor absorption</a></li> <li><a href="/wiki/Vapor-compression_refrigeration" title="Vapor-compression refrigeration">Vapor-compression</a></li> <li><a href="/wiki/Siemens_cycle" title="Siemens cycle">Siemens</a></li> <li><a href="/wiki/Vuilleumier_cycle" title="Vuilleumier cycle">Vuilleumier</a></li> <li><a href="/wiki/Ionocaloric_refrigeration" title="Ionocaloric refrigeration">Ionocaloric</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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