Otto cycle - Wikipedia

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id="toc-Process_0–1_intake_stroke_(blue_shade)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Process_1–2_compression_stroke_(B_on_diagrams)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Process_1–2_compression_stroke_(B_on_diagrams)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Process 1–2 compression stroke (<i>B</i> on diagrams)</span> </div> </a> <ul id="toc-Process_1–2_compression_stroke_(B_on_diagrams)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Process_2–3_ignition_phase_(C_on_diagrams)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Process_2–3_ignition_phase_(C_on_diagrams)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Process 2–3 ignition phase (<i>C</i> on diagrams)</span> </div> </a> <ul id="toc-Process_2–3_ignition_phase_(C_on_diagrams)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Process_3–4_expansion_stroke_(D_on_diagrams)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Process_3–4_expansion_stroke_(D_on_diagrams)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Process 3–4 expansion stroke (<i>D</i> on diagrams)</span> </div> </a> <ul id="toc-Process_3–4_expansion_stroke_(D_on_diagrams)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Process_4–1_idealized_heat_rejection_(A_on_diagrams)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Process_4–1_idealized_heat_rejection_(A_on_diagrams)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Process 4–1 idealized heat rejection (<i>A</i> on diagrams)</span> </div> </a> <ul id="toc-Process_4–1_idealized_heat_rejection_(A_on_diagrams)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Process_1–0_exhaust_stroke" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Process_1–0_exhaust_stroke"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Process 1–0 exhaust stroke</span> </div> </a> <ul id="toc-Process_1–0_exhaust_stroke-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cycle_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cycle_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Cycle analysis</span> </div> </a> <ul id="toc-Cycle_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Power" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Power"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Power</span> </div> </a> <ul id="toc-Power-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Increasing_power_and_efficiency" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Increasing_power_and_efficiency"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Increasing power and efficiency</span> </div> </a> <ul id="toc-Increasing_power_and_efficiency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Otto cycle</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 32 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-32" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">32 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D9%88%D8%B1%D8%A9_%D8%A3%D9%88%D8%AA%D9%88" title="دورة أوتو – Arabic" lang="ar" hreflang="ar" data-title="دورة أوتو" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cicle_d%27Otto" title="Cicle d'Otto – Catalan" lang="ca" hreflang="ca" data-title="Cicle d'Otto" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Ott%C5%AFv_cyklus" title="Ottův cyklus – Czech" lang="cs" hreflang="cs" data-title="Ottův cyklus" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Otto-kredsproces" title="Otto-kredsproces – Danish" lang="da" hreflang="da" data-title="Otto-kredsproces" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Otto-Kreisprozess" title="Otto-Kreisprozess – German" lang="de" hreflang="de" data-title="Otto-Kreisprozess" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%8D%CE%BA%CE%BB%CE%BF%CF%82_%CE%8C%CF%84%CF%84%CE%BF" title="Κύκλος Όττο – Greek" lang="el" hreflang="el" data-title="Κύκλος Όττο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ciclo_Otto" title="Ciclo Otto – Spanish" lang="es" hreflang="es" data-title="Ciclo Otto" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ciklo_de_Otto" title="Ciklo de Otto – Esperanto" lang="eo" hreflang="eo" data-title="Ciklo de Otto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</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%AA%D9%88" title="چرخه اتو – Persian" lang="fa" hreflang="fa" data-title="چرخه اتو" 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_Beau_de_Rochas#Étude_thermodynamique" title="Cycle de Beau de Rochas – French" lang="fr" hreflang="fr" data-title="Cycle de Beau de Rochas" 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_Otto" title="Ciclo Otto – Galician" lang="gl" hreflang="gl" data-title="Ciclo Otto" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A4%ED%86%A0%EC%82%AC%EC%9D%B4%ED%81%B4" title="오토사이클 – Korean" lang="ko" hreflang="ko" data-title="오토사이클" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Siklus_Otto" title="Siklus Otto – Indonesian" lang="id" hreflang="id" data-title="Siklus Otto" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ciclo_Otto" title="Ciclo Otto – Italian" lang="it" hreflang="it" data-title="Ciclo Otto" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Otto-ciklus" title="Otto-ciklus – Hungarian" lang="hu" hreflang="hu" data-title="Otto-ciklus" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%93%E0%B4%9F%E0%B5%8D%E0%B4%9F%E0%B5%8B_%E0%B4%9A%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%82" title="ഓട്ടോ ചക്രം – Malayalam" lang="ml" hreflang="ml" data-title="ഓട്ടോ ചക്രം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Otto-proces" title="Otto-proces – Dutch" lang="nl" hreflang="nl" data-title="Otto-proces" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%85%E0%A4%9F%E0%A5%8D%E0%A4%9F%E0%A5%8B_%E0%A4%9A%E0%A4%95%E0%A5%8D%E0%A4%B0" title="अट्टो चक्र – Nepali" lang="ne" hreflang="ne" data-title="अट्टो चक्र" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AA%E3%83%83%E3%83%88%E3%83%BC%E3%82%B5%E3%82%A4%E3%82%AF%E3%83%AB" title="オットーサイクル – Japanese" lang="ja" hreflang="ja" data-title="オットーサイクル" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Beau_de_Rochas-syklusen" title="Beau de Rochas-syklusen – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Beau de Rochas-syklusen" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Cykl_Otta" title="Cykl Otta – Polish" lang="pl" hreflang="pl" data-title="Cykl Otta" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Ciclo_de_Otto" title="Ciclo de Otto – Portuguese" lang="pt" hreflang="pt" data-title="Ciclo de Otto" 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%9E%D1%82%D1%82%D0%BE" title="Цикл Отто – Russian" lang="ru" hreflang="ru" data-title="Цикл Отто" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/Tziclu_Otto" title="Tziclu Otto – Sardinian" lang="sc" hreflang="sc" data-title="Tziclu Otto" data-language-autonym="Sardu" data-language-local-name="Sardinian" class="interlanguage-link-target"><span>Sardu</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Ottov_cyklus" title="Ottov cyklus – Slovak" lang="sk" hreflang="sk" data-title="Ottov cyklus" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%D9%88%DA%95%DB%8C_%D8%A6%DB%86%D8%AA%DB%86" title="سوڕی ئۆتۆ – Central Kurdish" lang="ckb" hreflang="ckb" data-title="سوڕی ئۆتۆ" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%93%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AF%8B_%E0%AE%9A%E0%AF%81%E0%AE%B4%E0%AE%B1%E0%AF%8D%E0%AE%9A%E0%AE%BF" title="ஓட்டோ சுழற்சி – Tamil" lang="ta" hreflang="ta" data-title="ஓட்டோ சுழற்சி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Sabit_hacim_%C3%A7evrimli_motor" title="Sabit hacim çevrimli motor – Turkish" lang="tr" hreflang="tr" data-title="Sabit hacim çevrimli motor" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</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%9E%D1%82%D1%82%D0%BE" title="Цикл Отто – Ukrainian" lang="uk" hreflang="uk" data-title="Цикл Отто" 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%D9%88%D9%B9%D9%88_%DA%86%DA%A9%D8%B1/%D8%B3%D8%A7%D8%A6%DB%8C%DA%A9%D9%84" title="اوٹو چکر/سائیکل – Urdu" lang="ur" hreflang="ur" data-title="اوٹو چکر/سائیکل" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Chu_tr%C3%ACnh_Otto" title="Chu trình Otto – Vietnamese" lang="vi" hreflang="vi" data-title="Chu trình Otto" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A5%A7%E5%9C%96%E5%BE%AA%E7%92%B0" title="奧圖循環 – Chinese" lang="zh" hreflang="zh" data-title="奧圖循環" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link 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 skin-invert-image" 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" 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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;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws</a></div></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;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div></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, #202122 ); 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;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div></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, #202122 ); 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;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div></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;"> <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;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div></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;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div></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;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>History</li><li>Culture</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); 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;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></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;;color: var(--color-base)"><div class="sidebar-list-title-c">Other</div></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="" 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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> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:P-V_Otto_cycle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/P-V_Otto_cycle.svg/300px-P-V_Otto_cycle.svg.png" decoding="async" width="300" height="296" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/P-V_Otto_cycle.svg/450px-P-V_Otto_cycle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/P-V_Otto_cycle.svg/600px-P-V_Otto_cycle.svg.png 2x" data-file-width="338" data-file-height="333" /></a><figcaption>Otto cycle pressure–volume diagramThe idealized diagrams of a four-stroke Otto cycle <a href="/wiki/Pressure_volume_diagram" class="mw-redirect" title="Pressure volume diagram">Both diagrams</a>: the intake (<i>0-1 and colored green)</i> stroke is performed by an <a href="/wiki/Isobaric_process" title="Isobaric process">isobaric</a> expansion, followed by an <a href="/wiki/Adiabatic_process" title="Adiabatic process">adiabatic</a> compression (<i>1-2 and colored orange</i>) stroke. Through the combustion of fuel, heat is added in a constant volume (<a href="/wiki/Isochoric_process" title="Isochoric process">isochoric process</a>) process (2-3), followed by an adiabatic expansion process power (3-4 <i>and colored red</i>) stroke. The cycle is closed by the exhaust (4-0 <i>and colored blue</i>) stroke, characterized by isochoric cooling and isobaric compression processes. </figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:OttoCycleTSDiagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/OttoCycleTSDiagram.svg/300px-OttoCycleTSDiagram.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/OttoCycleTSDiagram.svg/450px-OttoCycleTSDiagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/OttoCycleTSDiagram.svg/600px-OttoCycleTSDiagram.svg.png 2x" data-file-width="709" data-file-height="709" /></a><figcaption>Temperature-Entropy diagram</figcaption></figure> <p>An <b>Otto cycle</b> is an idealized <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">thermodynamic cycle</a> that describes the functioning of a typical <a href="/wiki/Spark-ignition_engine" title="Spark-ignition engine">spark ignition</a> <a href="/wiki/Piston_engine" class="mw-redirect" title="Piston engine">piston engine</a>. It is the thermodynamic cycle most commonly found in automobile engines.<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 Otto cycle is a description of what happens to a gas as it is subjected to changes of pressure, temperature, volume, addition of heat, and removal of heat. The gas that is subjected to those changes is called the system. The system, in this case, is defined to be the fluid (gas) within the cylinder. Conversely, by describing the changes that take place within the system it also describes the system's effect on the environment. The purpose of the Otto cycle is to study the production of net work from the system that can propel a vehicle and its occupants in the environment. </p><p>The Otto cycle is constructed from: </p> <dl><dd>Top and bottom of the loop: a pair of quasi-parallel and <a href="/wiki/Isentropic_process" title="Isentropic process">isentropic processes</a> (frictionless, <a href="/wiki/Adiabatic" class="mw-redirect" title="Adiabatic">adiabatic</a> reversible).</dd> <dd>Left and right sides of the loop: a pair of parallel <a href="/wiki/Isochoric_process" title="Isochoric process">isochoric processes</a> (constant volume).</dd></dl> <p>The isentropic process of compression or expansion implies that there will be no inefficiency (loss of mechanical energy), and there be no transfer of heat into or out of the system during that process. The cylinder and piston are assumed to be impermeable to heat during that time. Work is performed on the system during the lower isentropic compression process. Heat flows into the Otto cycle through the left pressurizing process and some of it flows back out through the right depressurizing process. The summation of the work added to the system plus the heat added minus the heat removed yields the net mechanical work generated by the system. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Processes">Processes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=1" title="Edit section: Processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The processes are described by:<sup id="cite_ref-Shapiro_2-0" class="reference"><a href="#cite_note-Shapiro-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Process 0–1 a mass of air is drawn into piston/cylinder arrangement at constant pressure.</li> <li>Process 1–2 is an adiabatic (isentropic) compression of the charge as the piston moves from bottom <a href="/wiki/Dead_centre_(engineering)" title="Dead centre (engineering)">dead center</a> (<i>BDC</i>) to top dead center (<i>TDC</i>).</li> <li>Process 2–3 is a constant-volume heat transfer to the working gas from an external source while the piston is at top dead center. This process is intended to represent the ignition of the fuel-air mixture and the subsequent rapid burning.</li> <li>Process 3–4 is an adiabatic (isentropic) expansion (power stroke).</li> <li>Process 4–1 completes the cycle by a constant-volume process in which heat is rejected from the air while the piston is at bottom dead center.</li> <li>Process 1–0 the mass of air is released to the atmosphere in a constant pressure process.</li></ul> <p>The Otto cycle consists of isentropic compression, heat addition at constant volume, isentropic expansion, and rejection of heat at constant volume. In the case of a four-stroke Otto cycle, technically there are two additional processes: one for the exhaust of <a href="/wiki/Waste_heat" title="Waste heat">waste heat</a> and combustion products at constant pressure (isobaric), and one for the intake of cool oxygen-rich air also at constant pressure; however, these are often omitted in a simplified analysis. Even though those two processes are critical to the functioning of a real engine, wherein the details of heat transfer and combustion chemistry are relevant, for the simplified analysis of the thermodynamic cycle, it is more convenient to assume that all of the waste-heat is removed during a single volume change. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The four-stroke engine was first patented by <a href="/wiki/Alphonse_Beau_de_Rochas" title="Alphonse Beau de Rochas">Alphonse Beau de Rochas</a> in 1861.<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> Before, in about 1854–57, two Italians (<a href="/wiki/Eugenio_Barsanti" title="Eugenio Barsanti">Eugenio Barsanti</a> and <a href="/wiki/Felice_Matteucci" title="Felice Matteucci">Felice Matteucci</a>) invented an engine that was rumored to be very similar, but the patent was lost.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2023)">citation needed</span></a></i>]</sup> </p><p>The first person to build a working four-stroke engine, a stationary engine using a coal gas-air mixture for fuel (a <a href="/wiki/Gas_engine" title="Gas engine">gas engine</a>), was <a href="/wiki/Germany" title="Germany">German</a> engineer <a href="/wiki/Nicolaus_Otto" title="Nicolaus Otto">Nicolaus Otto</a>.<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> This is why the four-stroke principle today is commonly known as the Otto cycle and four-stroke engines using <a href="/wiki/Spark_plug" title="Spark plug">spark plugs</a> often are called Otto engines. </p> <div class="mw-heading mw-heading2"><h2 id="Processes_2">Processes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=3" title="Edit section: Processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The cycle has four parts: a mass containing a mixture of fuel and oxygen is drawn into the cylinder by the descending piston, it is compressed by the piston rising, the mass is ignited by a spark releasing energy in the form of heat, the resulting gas is allowed to expand as it pushes the piston down, and finally the mass is exhausted as the piston rises a second time. As the piston is capable of moving along the cylinder, the volume of the gas changes with its position in the cylinder. The compression and expansion processes induced on the gas by the movement of the piston are idealized as reversible, i.e., no useful work is lost through turbulence or friction and no heat is transferred to or from the gas during those two processes. After the expansion is completed in the cylinder, the remaining heat is extracted and finally the gas is exhausted to the environment. Mechanical work is produced during the expansion process and some of that used to compress the air mass of the next cycle. The mechanical work produced minus that used for the compression process is the net work gained and that can be used for propulsion or for driving other machines. Alternatively the net work gained is the difference between the heat produced and the heat removed. </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:OTTO_CYCLE.png" class="mw-file-description" title="Otto Cycle"><img alt="Otto Cycle" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/OTTO_CYCLE.png/600px-OTTO_CYCLE.png" decoding="async" width="600" height="325" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/OTTO_CYCLE.png/900px-OTTO_CYCLE.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/05/OTTO_CYCLE.png 2x" data-file-width="991" data-file-height="536" /></a><figcaption>Otto Cycle</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Process_0–1_intake_stroke_(blue_shade)"><span id="Process_0.E2.80.931_intake_stroke_.28blue_shade.29"></span>Process 0–1 intake stroke (blue shade)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=4" title="Edit section: Process 0–1 intake stroke (blue shade)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A mass of air (working fluid) is drawn into the cylinder, from 0 to 1, at atmospheric pressure (constant pressure) through the open intake valve, while the exhaust valve is closed during this process. The intake valve closes at point 1. </p> <div class="mw-heading mw-heading3"><h3 id="Process_1–2_compression_stroke_(B_on_diagrams)"><span id="Process_1.E2.80.932_compression_stroke_.28B_on_diagrams.29"></span>Process 1–2 compression stroke (<i>B</i> on diagrams)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=5" title="Edit section: Process 1–2 compression stroke (B on diagrams)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Piston moves from crank end (BDC, bottom dead centre and maximum volume) to cylinder head end (<i>TDC</i>, top dead centre and minimum volume) as the working gas with initial state 1 is compressed isentropically to state point 2, through <a href="/wiki/Compression_ratio" title="Compression ratio">compression ratio</a> <span class="texhtml">(<i>V</i><sub>1</sub>/<i>V</i><sub>2</sub>)</span>. Mechanically this is the isentropic compression of the air/fuel mixture in the cylinder, also known as the compression stroke. This isentropic process assumes that no mechanical energy is lost due to friction and no heat is transferred to or from the gas, hence the process is reversible. The compression process requires that mechanical work be added to the working gas. Generally the compression ratio is around 9–10:1 <span class="texhtml">(<i>V</i><sub>1</sub>:<i>V</i><sub>2</sub>)</span> for a typical engine.<sup id="cite_ref-Electropaedia_5-0" class="reference"><a href="#cite_note-Electropaedia-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Process_2–3_ignition_phase_(C_on_diagrams)"><span id="Process_2.E2.80.933_ignition_phase_.28C_on_diagrams.29"></span>Process 2–3 ignition phase (<i>C</i> on diagrams)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=6" title="Edit section: Process 2–3 ignition phase (C on diagrams)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The piston is momentarily at rest at <i>TDC</i>. During this instant, which is known as the ignition phase, the air/fuel mixture remains in a small volume at the top of the compression stroke. Heat is added to the working fluid by the combustion of the injected fuel, with the volume essentially being held constant. The pressure rises and the ratio <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 (P_{3}/P_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P_{3}/P_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fd81afe3efaddb24890612ce6cfbfaee0af5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.065ex; height:2.843ex;" alt="{\displaystyle (P_{3}/P_{2})}"></span> is called the "explosion ratio". </p> <div class="mw-heading mw-heading3"><h3 id="Process_3–4_expansion_stroke_(D_on_diagrams)"><span id="Process_3.E2.80.934_expansion_stroke_.28D_on_diagrams.29"></span>Process 3–4 expansion stroke (<i>D</i> on diagrams)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=7" title="Edit section: Process 3–4 expansion stroke (D on diagrams)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The increased high pressure exerts a force on the piston and pushes it towards the <i>BDC</i>. Expansion of working fluid takes place isentropically and work is done by the system on the piston. The volume ratio <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_{4}/V_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{4}/V_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f23b9eae971e367e8eef9561738b3ff2b8f4ee7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.981ex; height:2.843ex;" alt="{\displaystyle V_{4}/V_{3}}"></span> is called the "isentropic expansion ratio". (For the Otto cycle is the same as the compression ratio <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_{1}/V_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{1}/V_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c985b090662d887a7a67b7e3037c3334f5f366a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.981ex; height:2.843ex;" alt="{\displaystyle V_{1}/V_{2}}"></span>). Mechanically this is the expansion of the hot gaseous mixture in the cylinder known as expansion (power) stroke. </p> <div class="mw-heading mw-heading3"><h3 id="Process_4–1_idealized_heat_rejection_(A_on_diagrams)"><span id="Process_4.E2.80.931_idealized_heat_rejection_.28A_on_diagrams.29"></span>Process 4–1 idealized heat rejection (<i>A</i> on diagrams)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=8" title="Edit section: Process 4–1 idealized heat rejection (A on diagrams)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The piston is momentarily at rest at <i>BDC</i>. The working gas pressure drops instantaneously from point 4 to point 1 during a constant volume process as heat is removed to an idealized external sink that is brought into contact with the cylinder head. In modern internal combustion engines, the heat-sink may be surrounding air (for low powered engines), or a circulating fluid, such as coolant. The gas has returned to state 1. </p> <div class="mw-heading mw-heading3"><h3 id="Process_1–0_exhaust_stroke"><span id="Process_1.E2.80.930_exhaust_stroke"></span>Process 1–0 exhaust stroke</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=9" title="Edit section: Process 1–0 exhaust stroke"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The exhaust valve opens at point 1. As the piston moves from "BDC" (point 1) to "TDC" (point 0) with the exhaust valve opened, the gaseous mixture is vented to the atmosphere and the process starts anew. </p> <div class="mw-heading mw-heading2"><h2 id="Cycle_analysis">Cycle analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=10" title="Edit section: Cycle analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this process 1–2 the piston does work on the gas and in process 3–4 the gas does work on the piston during those isentropic compression and expansion processes, respectively. Processes 2–3 and 4–1 are isochoric processes; heat is transferred into the system from 2—3 and out of the system from 4–1 but no work is done on the system or extracted from the system during those processes. No work is done during an isochoric (constant volume) process because addition or removal of work from a system requires the movement of the boundaries of the system; hence, as the cylinder volume does not change, no shaft work is added to or removed from the system. </p><p>Four different equations are used to describe those four processes. A simplification is made by assuming changes of the kinetic and potential energy that take place in the system (mass of gas) can be neglected and then applying the first law of thermodynamics (energy conservation) to the mass of gas as it changes state as characterized by the gas's temperature, pressure, and volume.<sup id="cite_ref-Shapiro_2-1" class="reference"><a href="#cite_note-Shapiro-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (July 2017)">page needed</span></a></i>]</sup><sup id="cite_ref-Fundamentals_6-0" class="reference"><a href="#cite_note-Fundamentals-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (July 2017)">page needed</span></a></i>]</sup> </p><p>During a complete cycle, the gas returns to its original state of temperature, pressure and volume, hence the net internal energy change of the system (gas) is zero. As a result, the energy (heat or work) added to the system must be offset by energy (heat or work) that leaves the system. In the analysis of thermodynamic systems, the convention is to account energy that enters the system as positive and energy that leaves the system is accounted as negative. </p><p><span class="anchor" id="Equation_1a"></span> Equation 1a. </p><p>During a complete cycle, the net change of energy of the system is zero: </p> <dl><dd><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 \Delta E=E_{\text{in}}-E_{\text{out}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E=E_{\text{in}}-E_{\text{out}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36b9851dc0668496ea50264bb8dece897c42469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.553ex; height:2.509ex;" alt="{\displaystyle \Delta E=E_{\text{in}}-E_{\text{out}}=0}"></span></dd></dl> <p>The above states that the system (the mass of gas) returns to the original thermodynamic state it was in at the start of the cycle. </p><p>Where <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 E_{\text{in}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{in}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30741f4c6236f0fdbc624bbafb59859a1376be1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.319ex; height:2.509ex;" alt="{\displaystyle E_{\text{in}}}"></span> is energy added to the system from 1–2–3 and <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 E_{\text{out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de166566162d8a55cbfc30fd7824d480732e1559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.323ex; height:2.509ex;" alt="{\displaystyle E_{\text{out}}}"></span> is energy removed from the system from 3–4–1. In terms of work and heat added to the system </p><p><span class="anchor" id="Equation_1b"></span> Equation 1b: </p> <dl><dd><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 W_{1-2}+Q_{2-3}+W_{3-4}+Q_{4-1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1-2}+Q_{2-3}+W_{3-4}+Q_{4-1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac157415a7d67d712de0f82c9fb7736dc6562c3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.465ex; height:2.509ex;" alt="{\displaystyle W_{1-2}+Q_{2-3}+W_{3-4}+Q_{4-1}=0}"></span></dd></dl> <p>Each term of the equation can be expressed in terms of the internal energy of the gas at each point in the process: </p> <dl><dd><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 W_{1-2}=U_{2}-U_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1-2}=U_{2}-U_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63c3b9f6ebb86794c44be6b23bc8300876726261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.571ex; height:2.509ex;" alt="{\displaystyle W_{1-2}=U_{2}-U_{1}}"></span></dd></dl> <dl><dd><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 Q_{2-3}=U_{3}-U_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{2-3}=U_{3}-U_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84af9befd32dfc9a30333ff2366417f6bee0b716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.215ex; height:2.509ex;" alt="{\displaystyle Q_{2-3}=U_{3}-U_{2}}"></span></dd></dl> <dl><dd><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 W_{3-4}=U_{4}-U_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{3-4}=U_{4}-U_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/186177c6593691cdb45e82a9fad4c949c9dd94d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.571ex; height:2.509ex;" alt="{\displaystyle W_{3-4}=U_{4}-U_{3}}"></span></dd></dl> <dl><dd><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 Q_{4-1}=U_{1}-U_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{4-1}=U_{1}-U_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bba22434876be6602fc066f968bdd624f456c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.215ex; height:2.509ex;" alt="{\displaystyle Q_{4-1}=U_{1}-U_{4}}"></span></dd></dl> <p>The energy balance Equation 1b becomes </p> <dl><dd><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 W_{1-2}+Q_{2-3}+W_{3-4}+Q_{4-1}=\left(U_{2}-U_{1}\right)+\left(U_{3}-U_{2}\right)+\left(U_{4}-U_{3}\right)+\left(U_{1}-U_{4}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1-2}+Q_{2-3}+W_{3-4}+Q_{4-1}=\left(U_{2}-U_{1}\right)+\left(U_{3}-U_{2}\right)+\left(U_{4}-U_{3}\right)+\left(U_{1}-U_{4}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2215d70932deb2418fe3f67c1e775a9e93b0e3ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:84.817ex; height:2.843ex;" alt="{\displaystyle W_{1-2}+Q_{2-3}+W_{3-4}+Q_{4-1}=\left(U_{2}-U_{1}\right)+\left(U_{3}-U_{2}\right)+\left(U_{4}-U_{3}\right)+\left(U_{1}-U_{4}\right)=0}"></span></dd></dl> <p>To illustrate the example we choose <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="The material near this tag is possibly inaccurate or nonfactual. (August 2022)">dubious</span></a> – <a href="/wiki/Talk:Otto_cycle#Dubious" title="Talk:Otto cycle">discuss</a></i>]</sup> some values to the points in the illustration: </p> <dl><dd><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_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01169a5c106e6c69705d7422d5f256eb04f769e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.903ex; height:2.509ex;" alt="{\displaystyle U_{1}=1}"></span></dd> <dd><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_{2}=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{2}=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3e02bdf82209a6b6375598fe0f9909d8dcce59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.903ex; height:2.509ex;" alt="{\displaystyle U_{2}=5}"></span></dd> <dd><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_{3}=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{3}=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24cbbf4474135a62f9ba84b1cc9add0d76c2332" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.903ex; height:2.509ex;" alt="{\displaystyle U_{3}=9}"></span></dd> <dd><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_{4}=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{4}=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4372835179dd01aaf6d3542dbab218dce0964b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.903ex; height:2.509ex;" alt="{\displaystyle U_{4}=4}"></span></dd></dl> <p>These values are arbitrarily but rationally <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="The material near this tag is possibly inaccurate or nonfactual. (August 2022)">dubious</span></a> – <a href="/wiki/Talk:Otto_cycle#Dubious" title="Talk:Otto cycle">discuss</a></i>]</sup> selected. The work and heat terms can then be calculated. </p><p>The energy added to the system as work during the compression from 1 to 2 is </p> <dl><dd><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 \left(U_{2}-U_{1}\right)=\left(5-1\right)=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(U_{2}-U_{1}\right)=\left(5-1\right)=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecd0108477775c6f4ac0a7c2c8b3d3ca5c60eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.267ex; height:2.843ex;" alt="{\displaystyle \left(U_{2}-U_{1}\right)=\left(5-1\right)=4}"></span></dd></dl> <p>The energy added to the system as heat from point 2 to 3 is </p> <dl><dd><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 \left({U_{3}-U_{2}}\right)=\left(9-5\right)=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>9</mn> <mo>−</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({U_{3}-U_{2}}\right)=\left(9-5\right)=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b97a14add8ba23e38266deb5734856ae8b8a07e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.267ex; height:2.843ex;" alt="{\displaystyle \left({U_{3}-U_{2}}\right)=\left(9-5\right)=4}"></span></dd></dl> <p>The energy removed from the system as work during the expansion from 3 to 4 is </p> <dl><dd><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 \left(U_{4}-U_{3}\right)=\left(4-9\right)=-5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>−</mo> <mn>9</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(U_{4}-U_{3}\right)=\left(4-9\right)=-5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bffaaeed62860e8b1ae842ba5518a99999d3549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.075ex; height:2.843ex;" alt="{\displaystyle \left(U_{4}-U_{3}\right)=\left(4-9\right)=-5}"></span></dd></dl> <p>The energy removed from the system as heat from point 4 to 1 is </p> <dl><dd><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 \left(U_{1}-U_{4}\right)=\left(1-4\right)=-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(U_{1}-U_{4}\right)=\left(1-4\right)=-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e0794713ad22c99d1a53593df60802112795904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.075ex; height:2.843ex;" alt="{\displaystyle \left(U_{1}-U_{4}\right)=\left(1-4\right)=-3}"></span></dd></dl> <p>The energy balance is </p> <dl><dd><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 \Delta E=+4+4-5-3=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mo>=</mo> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mn>5</mn> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E=+4+4-5-3=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c253676d94bfc1ec0ca4b99dd81a5bdb86ca94c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.05ex; height:2.343ex;" alt="{\displaystyle \Delta E=+4+4-5-3=0}"></span></dd></dl> <p>Note that energy added to the system is counted as positive and energy leaving the system is counted as negative and the summation is zero as expected for a complete cycle that returns the system to its original state. </p><p>From the energy balance the work out of the system is: </p> <dl><dd><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 \sum {\text{Work}}=W_{1-2}+W_{3-4}=\left(U_{2}-U_{1}\right)+\left(U_{4}-U_{3}\right)=4-5=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Work</mtext> </mrow> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum {\text{Work}}=W_{1-2}+W_{3-4}=\left(U_{2}-U_{1}\right)+\left(U_{4}-U_{3}\right)=4-5=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62889f386efa02cc540f031ab147092cb7952f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:66.206ex; height:3.843ex;" alt="{\displaystyle \sum {\text{Work}}=W_{1-2}+W_{3-4}=\left(U_{2}-U_{1}\right)+\left(U_{4}-U_{3}\right)=4-5=-1}"></span></dd></dl> <p>The net energy out of the system as work is -1, meaning the system has produced one net unit of energy that leaves the system in the form of work. </p><p>The net heat out of the system is: </p> <dl><dd><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 \sum {\text{Heat}}=Q_{2-3}+Q_{4-1}=\left(U_{3}-U_{2}\right)+\left(U_{1}-U_{4}\right)=4-3=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Heat</mtext> </mrow> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum {\text{Heat}}=Q_{2-3}+Q_{4-1}=\left(U_{3}-U_{2}\right)+\left(U_{1}-U_{4}\right)=4-3=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b858bb4915ec3daced414fc9f76374213c939ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:62.84ex; height:3.843ex;" alt="{\displaystyle \sum {\text{Heat}}=Q_{2-3}+Q_{4-1}=\left(U_{3}-U_{2}\right)+\left(U_{1}-U_{4}\right)=4-3=1}"></span></dd></dl> <p>As energy added to the system as heat is positive. From the above it appears as if the system gained one unit of heat. This matches the energy produced by the system as work out of the system. </p><p><span class="anchor" id="Thermal_efficiency"></span> Thermal efficiency is the quotient of the net work from the system, to the heat added to system. Equation 2: </p> <dl><dd><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 \eta ={\frac {-\left(W_{1-2}+W_{3-4}\right)}{Q_{2-3}}}={\frac {\left(U_{1}-U_{2}\right)+\left(U_{3}-U_{4}\right)}{\left(U_{3}-U_{2}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ={\frac {-\left(W_{1-2}+W_{3-4}\right)}{Q_{2-3}}}={\frac {\left(U_{1}-U_{2}\right)+\left(U_{3}-U_{4}\right)}{\left(U_{3}-U_{2}\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f6cf3d8a07f1e6f8dce203a0bfdb711eaabd9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.287ex; height:6.509ex;" alt="{\displaystyle \eta ={\frac {-\left(W_{1-2}+W_{3-4}\right)}{Q_{2-3}}}={\frac {\left(U_{1}-U_{2}\right)+\left(U_{3}-U_{4}\right)}{\left(U_{3}-U_{2}\right)}}}"></span></dd></dl> <dl><dd><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 \eta =1+{\frac {U_{1}-U_{4}}{\left(U_{3}-U_{2}\right)}}=1+{\frac {(1-4)}{(9-5)}}=0.25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>9</mn> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =1+{\frac {U_{1}-U_{4}}{\left(U_{3}-U_{2}\right)}}=1+{\frac {(1-4)}{(9-5)}}=0.25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f04abb7385d2692c08f8c3a3cdc8e293fef36e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.185ex; height:6.509ex;" alt="{\displaystyle \eta =1+{\frac {U_{1}-U_{4}}{\left(U_{3}-U_{2}\right)}}=1+{\frac {(1-4)}{(9-5)}}=0.25}"></span></dd></dl> <p>Alternatively, thermal efficiency can be derived by strictly heat added and heat rejected. </p> <dl><dd><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 \eta ={\frac {Q_{2-3}+Q_{4-1}}{Q_{2-3}}}=1+{\frac {\left(U_{1}-U_{4}\right)}{\left(U_{3}-U_{2}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ={\frac {Q_{2-3}+Q_{4-1}}{Q_{2-3}}}=1+{\frac {\left(U_{1}-U_{4}\right)}{\left(U_{3}-U_{2}\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d4402363fa2c4f40462d0b4af3398a9ee133a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.801ex; height:6.509ex;" alt="{\displaystyle \eta ={\frac {Q_{2-3}+Q_{4-1}}{Q_{2-3}}}=1+{\frac {\left(U_{1}-U_{4}\right)}{\left(U_{3}-U_{2}\right)}}}"></span></dd></dl> <p>Supplying the fictitious values </p><p><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 \eta =1+{\frac {1-4}{9-5}}=1+{\frac {-3}{4}}=0.25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mn>9</mn> <mo>−</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>3</mn> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>0.25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =1+{\frac {1-4}{9-5}}=1+{\frac {-3}{4}}=0.25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b40f24461d05edb450ffc2dbd53121d16efe10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.413ex; height:5.343ex;" alt="{\displaystyle \eta =1+{\frac {1-4}{9-5}}=1+{\frac {-3}{4}}=0.25}"></span> </p><p>In the Otto cycle, there is no heat transfer during the process 1–2 and 3–4 as they are isentropic processes. Heat is supplied only during the constant volume processes 2–3 and heat is rejected only during the constant volume processes 4–1. </p><p><span class="anchor" id="The_above_values"></span> The above values are absolute values that might, for instance <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="The material near this tag is possibly inaccurate or nonfactual. (August 2022)">dubious</span></a> – <a href="/wiki/Talk:Otto_cycle#Dubious" title="Talk:Otto cycle">discuss</a></i>]</sup>, have units of joules (assuming the MKS system of units are to be used) and would be of use for a particular engine with particular dimensions. In the study of thermodynamic systems the extensive quantities such as energy, volume, or entropy (versus intensive quantities of temperature and pressure) are placed on a unit mass basis, and so too are the calculations, making those more general and therefore of more general use. Hence, each term involving an extensive quantity could be divided by the mass, giving the terms units of joules/kg (<a href="/wiki/Specific_energy" title="Specific energy">specific energy</a>), meters<sup>3</sup>/kg (specific volume), or joules/(kelvin·kg) (specific entropy, heat capacity) etc. and would be represented using lower case letters, u, v, s, etc. </p><p><span class="anchor" id="Equation_1"></span> Equation 1 can now be related to the specific heat equation for constant volume. The <a href="/wiki/Heat_capacity" title="Heat capacity">specific heats</a> are particularly useful for thermodynamic calculations involving the <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a> model. </p> <dl><dd><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_{\text{v}}=\left({\frac {\delta u}{\delta T}}\right)_{\text{v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>v</mtext> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>u</mi> </mrow> <mrow> <mi>δ</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>v</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\text{v}}=\left({\frac {\delta u}{\delta T}}\right)_{\text{v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce672516e84249472545f21d5e568b4f7e1c40f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.903ex; height:6.176ex;" alt="{\displaystyle C_{\text{v}}=\left({\frac {\delta u}{\delta T}}\right)_{\text{v}}}"></span></dd></dl> <p>Rearranging yields: </p> <dl><dd><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 \delta u=(C_{\text{v}})(\delta T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>v</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta u=(C_{\text{v}})(\delta T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376fe2e7318b71a8e9e4b26322e950eb110c20a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.542ex; height:2.843ex;" alt="{\displaystyle \delta u=(C_{\text{v}})(\delta T)}"></span></dd></dl> <p>Inserting the specific heat equation into the thermal efficiency equation (Equation 2) yields. </p> <dl><dd><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 \eta =1-\left({\frac {C_{\text{v}}(T_{4}-T_{1})}{C_{\text{v}}(T_{3}-T_{2})}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>v</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>v</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =1-\left({\frac {C_{\text{v}}(T_{4}-T_{1})}{C_{\text{v}}(T_{3}-T_{2})}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb28418f11d2f125521a75d91004b744afa6006" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.763ex; height:6.509ex;" alt="{\displaystyle \eta =1-\left({\frac {C_{\text{v}}(T_{4}-T_{1})}{C_{\text{v}}(T_{3}-T_{2})}}\right)}"></span></dd></dl> <p>Upon rearrangement: </p> <dl><dd><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 \eta =1-\left({\frac {T_{1}}{T_{2}}}\right)\left({\frac {T_{4}/T_{1}-1}{T_{3}/T_{2}-1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =1-\left({\frac {T_{1}}{T_{2}}}\right)\left({\frac {T_{4}/T_{1}-1}{T_{3}/T_{2}-1}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5854b10c29f11072d3c9bf04ab685c869814700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.573ex; height:6.509ex;" alt="{\displaystyle \eta =1-\left({\frac {T_{1}}{T_{2}}}\right)\left({\frac {T_{4}/T_{1}-1}{T_{3}/T_{2}-1}}\right)}"></span></dd></dl> <p>Next, noting from the diagrams <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_{4}/T_{1}=T_{3}/T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{4}/T_{1}=T_{3}/T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e5382fbca82524fd17397b1f9f3d5351fc9ccf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.071ex; height:2.843ex;" alt="{\displaystyle T_{4}/T_{1}=T_{3}/T_{2}}"></span> (see <a href="/wiki/Isentropic_process" title="Isentropic process">isentropic relations for an ideal gas</a>), thus both of these can be omitted. The equation then reduces to: </p><p><span class="anchor" id="Equation_2"></span> Equation 2: </p> <dl><dd><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 \eta =1-\left({\frac {T_{1}}{T_{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =1-\left({\frac {T_{1}}{T_{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/249a34edc5265d3745857134ff5e2c2537b3884b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.94ex; height:6.176ex;" alt="{\displaystyle \eta =1-\left({\frac {T_{1}}{T_{2}}}\right)}"></span></dd></dl> <p>Since the Otto cycle uses isentropic processes during the compression (process 1 to 2) and expansion (process 3 to 4) the <a href="/wiki/Isentropic_process" title="Isentropic process">isentropic equations</a> of ideal gases and the constant pressure/volume relations can be used to yield Equations 3 & 4.<sup id="cite_ref-Reynolds_7-0" class="reference"><a href="#cite_note-Reynolds-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p><span class="anchor" id="Equation_3"></span> Equation 3: </p> <dl><dd><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 \left({\frac {T_{2}}{T_{1}}}\right)=\left({\frac {p_{2}}{p_{1}}}\right)^{(\gamma -1)/{\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)=\left({\frac {p_{2}}{p_{1}}}\right)^{(\gamma -1)/{\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48aeb29f01ec021828a9bce8d866bc3fa39dea7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.468ex; height:6.676ex;" alt="{\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)=\left({\frac {p_{2}}{p_{1}}}\right)^{(\gamma -1)/{\gamma }}}"></span></dd></dl> <p><span class="anchor" id="Equation_4"></span> Equation 4: </p> <dl><dd><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 \left({\frac {T_{2}}{T_{1}}}\right)=\left({\frac {V_{1}}{V_{2}}}\right)^{(\gamma -1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)=\left({\frac {V_{1}}{V_{2}}}\right)^{(\gamma -1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a436ebc5d29c92a5b92cffa0dc1e732505ca881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.939ex; height:6.676ex;" alt="{\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)=\left({\frac {V_{1}}{V_{2}}}\right)^{(\gamma -1)}}"></span></dd></dl> <dl><dd><dl><dd>where</dd></dl></dd></dl> <dl><dd><dl><dd><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 {\gamma }=\left({\frac {C_{\text{p}}}{C_{\text{v}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>v</mtext> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\gamma }=\left({\frac {C_{\text{p}}}{C_{\text{v}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8937107a1d0bfbfc91e254d62161dfbc1678b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.426ex; height:6.343ex;" alt="{\displaystyle {\gamma }=\left({\frac {C_{\text{p}}}{C_{\text{v}}}}\right)}"></span></dd></dl></dd></dl> <dl><dd><dl><dd><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 {\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5a51bb731eb471125cf853d8c845695870617f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle {\gamma }}"></span> is the specific heat ratio</dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd>The derivation of the previous equations are found by solving these four equations respectively (where <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the <a href="/wiki/Specific_gas_constant" class="mw-redirect" title="Specific gas constant">specific gas constant</a>):</dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><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_{\text{p}}\ln \left({\frac {V_{1}}{V_{2}}}\right)-R\ln \left({\frac {p_{2}}{p_{1}}}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\text{p}}\ln \left({\frac {V_{1}}{V_{2}}}\right)-R\ln \left({\frac {p_{2}}{p_{1}}}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f95c30d278d0ddcb3be826111b719612afee1bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.474ex; height:6.176ex;" alt="{\displaystyle C_{\text{p}}\ln \left({\frac {V_{1}}{V_{2}}}\right)-R\ln \left({\frac {p_{2}}{p_{1}}}\right)=0}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><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_{\text{v}}\ln \left({\frac {T_{2}}{T_{1}}}\right)-R\ln \left({\frac {V_{2}}{V_{1}}}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>v</mtext> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\text{v}}\ln \left({\frac {T_{2}}{T_{1}}}\right)-R\ln \left({\frac {V_{2}}{V_{1}}}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce249a53e452aea2d47a7ad70340e26a3e8f43de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.616ex; height:6.176ex;" alt="{\displaystyle C_{\text{v}}\ln \left({\frac {T_{2}}{T_{1}}}\right)-R\ln \left({\frac {V_{2}}{V_{1}}}\right)=0}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><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_{\text{p}}=\left({\frac {\gamma R}{\gamma -1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mi>R</mi> </mrow> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\text{p}}=\left({\frac {\gamma R}{\gamma -1}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5065ecd3e42498bc45fa1df6aaf9b78a4156687a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.429ex; height:6.176ex;" alt="{\displaystyle C_{\text{p}}=\left({\frac {\gamma R}{\gamma -1}}\right)}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><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_{\text{v}}=\left({\frac {R}{\gamma -1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>v</mtext> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\text{v}}=\left({\frac {R}{\gamma -1}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e5bb7c88cb69fa73b853972815071fbc8e290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.383ex; height:6.176ex;" alt="{\displaystyle C_{\text{v}}=\left({\frac {R}{\gamma -1}}\right)}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>Further simplifying Equation 4, where <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is the compression ratio <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_{1}/V_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V_{1}/V_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/183f20124f526d88c69e502f5fd831c481bf0a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.791ex; height:2.843ex;" alt="{\displaystyle (V_{1}/V_{2})}"></span>: </p><p><span class="anchor" id="Equation_5"></span> Equation 5: </p> <dl><dd><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 \left({\frac {T_{2}}{T_{1}}}\right)=\left({\frac {V_{1}}{V_{2}}}\right)^{(\gamma -1)}=r^{(\gamma -1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)=\left({\frac {V_{1}}{V_{2}}}\right)^{(\gamma -1)}=r^{(\gamma -1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6324c1edaf5a2beb2af3253b3160f8083a527c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.591ex; height:6.676ex;" alt="{\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)=\left({\frac {V_{1}}{V_{2}}}\right)^{(\gamma -1)}=r^{(\gamma -1)}}"></span></dd></dl> <p>From inverting Equation 4 and inserting it into Equation 2 the final thermal efficiency can be expressed as:<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (July 2017)">page needed</span></a></i>]</sup><sup id="cite_ref-Fundamentals_6-1" class="reference"><a href="#cite_note-Fundamentals-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (July 2017)">page needed</span></a></i>]</sup> </p><p><span class="anchor" id="Equation_6"></span> Equation 6: </p> <dl><dd><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 \eta =1-\left({\frac {1}{r^{(\gamma -1)}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =1-\left({\frac {1}{r^{(\gamma -1)}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b5fe634ec9b3a45360e7c6d57012c1b03e34e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.081ex; height:6.176ex;" alt="{\displaystyle \eta =1-\left({\frac {1}{r^{(\gamma -1)}}}\right)}"></span></dd></dl> <p>From analyzing equation 6 it is evident that the Otto cycle efficiency depends directly upon the compression ratio <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>. Since the <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> for air is 1.4, an increase in <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> will produce an increase in <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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span>. However, the <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> for combustion products of the fuel/air mixture is often taken at approximately 1.3. The foregoing discussion implies that it is more efficient to have a high compression ratio. The standard ratio is approximately 10:1 for typical automobiles. Usually this does not increase much because of the possibility of autoignition, or "<a href="/wiki/Engine_knocking" title="Engine knocking">knock</a>", which places an upper limit on the compression ratio.<sup id="cite_ref-Shapiro_2-2" class="reference"><a href="#cite_note-Shapiro-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (July 2017)">page needed</span></a></i>]</sup> During the compression process 1–2 the temperature rises, therefore an increase in the compression ratio causes an increase in temperature. Autoignition occurs when the temperature of the fuel/air mixture becomes too high before it is ignited by the flame front. The compression stroke is intended to compress the products before the flame ignites the mixture. If the compression ratio is increased, the mixture may auto-ignite before the compression stroke is complete, leading to "engine knocking". This can damage engine components and will decrease the brake horsepower of the engine. </p> <div class="mw-heading mw-heading2"><h2 id="Power">Power</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=11" title="Edit section: Power"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The power produced by the Otto cycle is an energy developed per unit of time. The Otto engines are called four-stroke engines. The intake stroke and compression stroke require one rotation of the engine crankshaft. The power stroke and exhaust stroke require another rotation. For two rotations there is one work generating stroke.. </p><p>From the above cycle analysis the net work produced by the system : </p> <dl><dd><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 \sum {\text{ Work}}=W_{1-2}+W_{3-4}=\left(U_{2}-U_{1}\right)+\left(U_{4}-U_{3}\right)=+4-5=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> Work</mtext> </mrow> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum {\text{ Work}}=W_{1-2}+W_{3-4}=\left(U_{2}-U_{1}\right)+\left(U_{4}-U_{3}\right)=+4-5=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353d732979c5b6df123c6199af4ec3f8a56a92aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:68.595ex; height:3.843ex;" alt="{\displaystyle \sum {\text{ Work}}=W_{1-2}+W_{3-4}=\left(U_{2}-U_{1}\right)+\left(U_{4}-U_{3}\right)=+4-5=-1}"></span></dd></dl> <p>(again, using the sign convention, the minus sign implies energy is leaving the system as work) </p><p>If the units used were MKS the cycle would have produced one joule of energy in the form of work. For an engine of a particular displacement, such as one liter, the mass of gas of the system can be calculated assuming the engine is operating at standard temperature (20 °C) and pressure (1 atm). Using the Universal Gas Law the mass of one liter of gas is at room temperature and sea level pressure: </p> <dl><dd><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 M={\frac {PV}{RT}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>V</mi> </mrow> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M={\frac {PV}{RT}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa9ede59e457be15013388a6473efa4a7bab02e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.909ex; height:5.343ex;" alt="{\displaystyle M={\frac {PV}{RT}}}"></span></dd></dl> <dl><dd><i>V</i>=0.001 m<sup>3</sup>, <i>R</i>=0.286 kJ/(kg·K), <i>T</i>=293 K, <i>P</i>=101.3 kN/m<sup>2</sup></dd></dl> <dl><dd><i>M</i>=0.00121 kg</dd></dl> <p>At an engine speed of 3000 RPM there are 1500 work-strokes/minute or 25 work-strokes/second. </p> <dl><dd><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 \sum {\text{ Work}}=1\,{\text{J}}/({\text{kg}}\cdot {\text{stroke}})\times 0.00121\,{\text{kg}}=0.00121\,{\text{J}}/{\text{stroke}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> Work</mtext> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>J</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>kg</mtext> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>stroke</mtext> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mn>0.00121</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>kg</mtext> </mrow> <mo>=</mo> <mn>0.00121</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>J</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>stroke</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum {\text{ Work}}=1\,{\text{J}}/({\text{kg}}\cdot {\text{stroke}})\times 0.00121\,{\text{kg}}=0.00121\,{\text{J}}/{\text{stroke}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc1f484a22f18e4a9178784098b2d37788016bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:61.91ex; height:3.843ex;" alt="{\displaystyle \sum {\text{ Work}}=1\,{\text{J}}/({\text{kg}}\cdot {\text{stroke}})\times 0.00121\,{\text{kg}}=0.00121\,{\text{J}}/{\text{stroke}}}"></span></dd></dl> <p>Power is 25 times that since there are 25 work-strokes/second </p> <dl><dd><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 P=25\times 0.00121=0.0303\,{\text{J}}/{\text{s}}\;{\text{or}}\;{\text{W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>25</mn> <mo>×</mo> <mn>0.00121</mn> <mo>=</mo> <mn>0.0303</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>J</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>W</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=25\times 0.00121=0.0303\,{\text{J}}/{\text{s}}\;{\text{or}}\;{\text{W}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ddc64ce551b37abb19d080ed8f0635344793661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.602ex; height:2.843ex;" alt="{\displaystyle P=25\times 0.00121=0.0303\,{\text{J}}/{\text{s}}\;{\text{or}}\;{\text{W}}}"></span></dd></dl> <p>If the engine uses multiple cylinders with the same displacement, the result would be multiplied by the number of cylinders. These results are the product of the values of the internal energy that were assumed for the four states of the system at the end each of the four strokes (two rotations). They were selected only for the sake of illustration, and are obviously of low value. Substitution of actual values from an actual engine would produce results closer to that of the engine. Whose results would be higher than the actual engine as there are many simplifying assumptions made in the analysis that overlook inefficiencies. Such results would overestimate the power output. </p> <div class="mw-heading mw-heading2"><h2 id="Increasing_power_and_efficiency">Increasing power and efficiency</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_cycle&action=edit&section=12" title="Edit section: Increasing power and efficiency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The difference between the exhaust and intake pressures and temperatures means that some increase in efficiency can be gained by use of a <a href="/wiki/Turbocharger" title="Turbocharger">turbocharger</a>, removing from the exhaust flow some part of the remaining energy and transferring that to the intake flow to increase the intake pressure. A gas turbine can extract useful work energy from the exhaust stream and that can then be used to pressurize the intake air. The pressure and temperature of the exhausting gases would be reduced as they expand through the gas turbine and that work is then applied to the intake gas stream, increasing its pressure and temperature. The transfer of energy amounts to an efficiency improvement and the resulting power density of the engine is also improved. The intake air is typically cooled so as to reduce its volume as the work produced per stroke is a direct function of the amount of mass taken into the cylinder; denser air will produce more work per cycle. Practically speaking the intake air mass temperature must also be reduced to prevent premature ignition in a petrol fueled engine; hence, an <a href="/wiki/Intercooler" title="Intercooler">intercooler</a> is used to remove some energy as heat and so reduce the intake temperature. Such a scheme both increases the engine's efficiency and power. </p><p>The application of a supercharger driven by the crankshaft does increase the power output (power density) but does not increase efficiency as it uses some of the net work produced by the engine to pressurize the intake air and fails to extract otherwise wasted energy associated with the flow of exhaust at high temperature and a pressure to the ambient. </p> <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=Otto_cycle&action=edit&section=13" 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">Wu, Chih.<i> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2kr26tu9MaMC">Thermodynamic Cycles: Computer-aided Design and Optimization</a></i>. New York: M. Dekker, 2004. page 99</span> </li> <li id="cite_note-Shapiro-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Shapiro_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Shapiro_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Shapiro_2-2"><sup><i><b>c</b></i></sup></a></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="CITEREFMoranShapiro2006" class="citation book cs1">Moran, Michael J.; Shapiro, Howard N. (2006). <i>Fundamentals of engineering thermodynamics: SI version, SI units</i> (5th ed.). Chichester: Wiley. p. 376. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-03037-0" title="Special:BookSources/978-0-470-03037-0"><bdi>978-0-470-03037-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+engineering+thermodynamics%3A+SI+version%2C+SI+units&rft.place=Chichester&rft.pages=376&rft.edition=5th&rft.pub=Wiley&rft.date=2006&rft.isbn=978-0-470-03037-0&rft.aulast=Moran&rft.aufirst=Michael+J.&rft.au=Shapiro%2C+Howard+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOtto+cycle" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMike_Busch" class="citation journal cs1">Mike Busch. "150-Year-Old Technology". <i>Sport Aviation</i>: 26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sport+Aviation&rft.atitle=150-Year-Old+Technology&rft.pages=26&rft.au=Mike+Busch&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOtto+cycle" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGunston1999" class="citation book cs1">Gunston, Bill (1999). <i>Development of Piston Aero Engines</i> (2 ed.). Sparkford, UK: Patrick Stephens Ltd. p. 21. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7509-4478-6" title="Special:BookSources/978-0-7509-4478-6"><bdi>978-0-7509-4478-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Development+of+Piston+Aero+Engines&rft.place=Sparkford%2C+UK&rft.pages=21&rft.edition=2&rft.pub=Patrick+Stephens+Ltd&rft.date=1999&rft.isbn=978-0-7509-4478-6&rft.aulast=Gunston&rft.aufirst=Bill&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOtto+cycle" class="Z3988"></span></span> </li> <li id="cite_note-Electropaedia-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Electropaedia_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mpoweruk.com/heat_engines.htm">"Heat Cycles - Electropeaedia"</a>. Woodbank Communications Ltd<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-04-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Heat+Cycles+-+Electropeaedia&rft.pub=Woodbank+Communications+Ltd&rft_id=http%3A%2F%2Fwww.mpoweruk.com%2Fheat_engines.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOtto+cycle" class="Z3988"></span></span> </li> <li id="cite_note-Fundamentals-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fundamentals_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fundamentals_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Gupta, H. N. <i>Fundamentals of Internal Combustion</i>. New Delhi: Prentice-Hall, 2006. Print.</span> </li> <li id="cite_note-Reynolds-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Reynolds_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReynolds_&_Perkins1977" class="citation book cs1">Reynolds & Perkins (1977). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/engineeringtherm00reyn"><i>Engineering Thermodynamics</i></a></span>. McGraw-Hill. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/engineeringtherm00reyn/page/249">249</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-052046-2" title="Special:BookSources/978-0-07-052046-2"><bdi>978-0-07-052046-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Thermodynamics&rft.pages=249&rft.pub=McGraw-Hill&rft.date=1977&rft.isbn=978-0-07-052046-2&rft.au=Reynolds+%26+Perkins&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fengineeringtherm00reyn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOtto+cycle" class="Z3988"></span></span> </li> </ol></div></div> <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_cycles210" 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_cycles210" 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 href="/wiki/Stirling_cycle" title="Stirling cycle">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 class="mw-selflink selflink">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?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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