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class="vector-toc-numb">2</span> <span>Modern diagram</span> </div> </a> <ul id="toc-Modern_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Carnot_cycle" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Carnot_cycle"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Carnot cycle</span> </div> </a> <ul id="toc-Carnot_cycle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Carnot's_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Carnot's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Carnot's theorem</span> </div> </a> <button aria-controls="toc-Carnot's_theorem-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Carnot's theorem subsection</span> </button> <ul id="toc-Carnot's_theorem-sublist" class="vector-toc-list"> <li id="toc-Explanation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Explanation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Explanation</span> </div> </a> <ul id="toc-Explanation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Efficiency_of_real_heat_engines" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Efficiency_of_real_heat_engines"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Efficiency of real heat engines</span> </div> </a> <ul id="toc-Efficiency_of_real_heat_engines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Carnot_engine_and_Rudolf_Diesel" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_Carnot_engine_and_Rudolf_Diesel"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>The Carnot engine and Rudolf Diesel</span> </div> </a> <ul id="toc-The_Carnot_engine_and_Rudolf_Diesel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_a_macroscopic_construct" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#As_a_macroscopic_construct"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>As a macroscopic construct</span> </div> </a> <ul id="toc-As_a_macroscopic_construct-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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">10</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" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" 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Available in 18 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-18" 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">18 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/%D9%85%D8%AD%D8%B1%D9%83_%D9%83%D8%A7%D8%B1%D9%86%D9%88_%D8%A7%D9%84%D8%AD%D8%B1%D8%A7%D8%B1%D9%8A" title="محرك كارنو الحراري – Arabic" lang="ar" hreflang="ar" data-title="محرك كارنو الحراري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A8%E0%A7%8B%E0%A6%B0_%E0%A6%A4%E0%A6%BE%E0%A6%AA_%E0%A6%87%E0%A6%9E%E0%A7%8D%E0%A6%9C%E0%A6%BF%E0%A6%A8" title="কার্নোর তাপ ইঞ্জিন – Bangla" lang="bn" hreflang="bn" data-title="কার্নোর তাপ ইঞ্জিন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/M%C3%A0quina_de_Carnot" title="Màquina de Carnot – Catalan" lang="ca" hreflang="ca" data-title="Màquina de Carnot" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Carnot-varmluftmotor" title="Carnot-varmluftmotor – Danish" lang="da" hreflang="da" data-title="Carnot-varmluftmotor" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B7%CF%87%CE%B1%CE%BD%CE%AE_%CE%9A%CE%B1%CF%81%CE%BD%CF%8C" title="Μηχανή Καρνό – Greek" lang="el" hreflang="el" data-title="Μηχανή Καρνό" 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/M%C3%A1quina_de_Carnot" title="Máquina de Carnot – Spanish" lang="es" hreflang="es" data-title="Máquina de Carnot" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%88%D8%AA%D9%88%D8%B1_%DA%AF%D8%B1%D9%85%D8%A7%DB%8C%DB%8C_%DA%A9%D8%A7%D8%B1%D9%86%D9%88" title="موتور گرمایی کارنو – Persian" lang="fa" hreflang="fa" data-title="موتور گرمایی کارنو" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B9%B4%EB%A5%B4%EB%85%B8_%EA%B8%B0%EA%B4%80" title="카르노 기관 – Korean" lang="ko" hreflang="ko" data-title="카르노 기관" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A8%E0%A5%8B_%E0%A4%8A%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A4%BE_%E0%A4%87%E0%A4%82%E0%A4%9C%E0%A4%A8" title="कार्नो ऊष्मा इंजन – Hindi" lang="hi" hreflang="hi" data-title="कार्नो ऊष्मा इंजन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Mesin_Carnot" title="Mesin Carnot – Indonesian" lang="id" hreflang="id" data-title="Mesin Carnot" 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/Macchina_di_Carnot" title="Macchina di Carnot – Italian" lang="it" hreflang="it" data-title="Macchina di Carnot" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/M%C3%A1quina_de_Carnot" title="Máquina de Carnot – Portuguese" lang="pt" hreflang="pt" data-title="Máquina de Carnot" 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%A2%D0%B5%D0%BF%D0%BB%D0%BE%D0%B2%D0%B0%D1%8F_%D0%BC%D0%B0%D1%88%D0%B8%D0%BD%D0%B0_%D0%9A%D0%B0%D1%80%D0%BD%D0%BE" title="Тепловая машина Карно – Russian" lang="ru" hreflang="ru" data-title="Тепловая машина Карно" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%87%E0%B8%88%E0%B8%B1%E0%B8%81%E0%B8%A3%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%A3%E0%B9%89%E0%B8%AD%E0%B8%99%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%8C%E0%B9%82%E0%B8%99" title="เครื่องจักรความร้อนการ์โน – Thai" lang="th" hreflang="th" data-title="เครื่องจักรความร้อนการ์โน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Carnot_%C4%B1s%C4%B1_motoru" title="Carnot ısı motoru – Turkish" lang="tr" hreflang="tr" data-title="Carnot ısı motoru" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%D8%A7%D8%B1%D9%86%D9%88_%DA%A9%D8%A7_%D8%AD%D8%B1%D8%A7%D8%B1%D8%AA%DB%8C_%D8%A7%D9%86%D8%AC%D9%86" title="کارنو کا حرارتی انجن – Urdu" lang="ur" hreflang="ur" data-title="کارنو کا حرارتی انجن" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8D%A1%E8%AF%BA%E7%83%AD%E6%9C%BA" title="卡诺热机 – Wu" lang="wuu" hreflang="wuu" data-title="卡诺热机" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%A1%E8%AB%BE%E7%86%B1%E6%A9%9F" title="卡諾熱機 – Chinese" lang="zh" hreflang="zh" data-title="卡諾熱機" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1335241#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Theoretical engine</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Carnot_heat_engine" title="Special:EditPage/Carnot heat engine">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Carnot+heat+engine%22">"Carnot heat engine"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Carnot+heat+engine%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Carnot+heat+engine%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Carnot+heat+engine%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Carnot+heat+engine%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Carnot+heat+engine%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">October 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Carnot-engine-1824-vector.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Carnot-engine-1824-vector.svg/175px-Carnot-engine-1824-vector.svg.png" decoding="async" width="175" height="368" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Carnot-engine-1824-vector.svg/263px-Carnot-engine-1824-vector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Carnot-engine-1824-vector.svg/350px-Carnot-engine-1824-vector.svg.png 2x" data-file-width="354" data-file-height="744" /></a><figcaption><a href="/wiki/Axis_of_rotation" class="mw-redirect" title="Axis of rotation">Axial</a> <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross section</a> of Carnot's heat engine. In this diagram, <i>abgh</i> is a cylindrical vessel, <i>cd</i> is a movable <a href="/wiki/Piston" title="Piston">piston</a>, and <i>A</i> and <i>B</i> are constant–temperature bodies. The vessel may be placed in contact with either body or removed from both (as it is here).<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> </figcaption></figure> <p>A <b>Carnot heat engine</b><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is a theoretical <a href="/wiki/Heat_engine" title="Heat engine">heat engine</a> that operates on the <a href="/wiki/Carnot_cycle" title="Carnot cycle">Carnot cycle</a>. The basic model for this engine was developed by <a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Nicolas Léonard Sadi Carnot</a> in 1824. The Carnot engine model was graphically expanded by <a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Benoît Paul Émile Clapeyron</a> in 1834 and mathematically explored by <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> in 1857, work that led to the fundamental thermodynamic concept of <a href="/wiki/Entropy" title="Entropy">entropy</a>. The <a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a> engine is the most efficient heat engine which is theoretically possible.<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> The efficiency depends only upon the absolute temperatures of the hot and cold heat reservoirs between which it operates. </p><p>A heat engine acts by transferring energy from a warm region to a cool region of space and, in the process, converting some of that energy to <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">mechanical work</a>. The cycle may also be reversed. The system may be worked upon by an external force, and in the process, it can transfer thermal energy from a cooler system to a warmer one, thereby acting as a <a href="/wiki/Refrigerator" title="Refrigerator">refrigerator</a> or <a href="/wiki/Heat_pump" title="Heat pump">heat pump</a> rather than a heat engine. </p><p>Every thermodynamic system exists in a particular <a href="/wiki/Thermodynamic_state" title="Thermodynamic state">state</a>. A <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">thermodynamic cycle</a> occurs when a <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">system</a> is taken through a series of different states, and finally returned to its initial state. In the process of going through this cycle, the system may perform work on its surroundings, thereby acting as a heat engine. </p><p>The Carnot engine is a theoretical construct, useful for exploring the efficiency limits of other heat engines. An actual Carnot engine, however, would be completely impractical to build. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Carnot's_diagram"><span id="Carnot.27s_diagram"></span>Carnot's diagram</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot_heat_engine&action=edit&section=1" title="Edit section: Carnot's diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the adjacent diagram, from Carnot's 1824 work, <i><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the Motive Power of Fire</a></i>,<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> there are "two bodies <i>A</i> and <i>B</i>, kept each at a constant temperature, that of <i>A</i> being higher than that of <i>B</i>. These two bodies to which we can give, or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of <a href="/wiki/Caloric_theory" title="Caloric theory">caloric</a>. We will call the first the furnace and the second the refrigerator."<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Carnot then explains how we can obtain <a href="/wiki/Work_(physics)" title="Work (physics)">motive power</a>, i.e., "work", by carrying a certain quantity of heat from body <i>A</i> to body <i>B</i>. It also acts as a cooler and hence can also act as a refrigerator. </p> <div class="mw-heading mw-heading2"><h2 id="Modern_diagram">Modern diagram</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot_heat_engine&action=edit&section=2" title="Edit section: Modern diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Carnot_heat_engine_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/300px-Carnot_heat_engine_2.svg.png" decoding="async" width="300" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/450px-Carnot_heat_engine_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/600px-Carnot_heat_engine_2.svg.png 2x" data-file-width="840" data-file-height="370" /></a><figcaption>Carnot engine diagram (modern) - where an amount of heat <i>Q</i><sub>H</sub> flows from a high temperature <i>T</i><sub>H</sub> furnace through the fluid of the "working body" (working substance) and the remaining heat <i>Q</i><sub>C</sub> flows into the cold sink <i>T</i><sub>C</sub>, thus forcing the working substance to do <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">mechanical work</a> <i>W</i> on the surroundings, via cycles of contractions and expansions.</figcaption></figure> <p>The previous image shows the original piston-and-cylinder diagram used by Carnot in discussing his ideal engine. The figure at right shows a block diagram of a generic heat engine, such as the Carnot engine. In the diagram, the "working body" (system), a term introduced by Clausius in 1850, can be any fluid or vapor body through which <a href="/wiki/Heat" title="Heat">heat</a> <i>Q</i> can be introduced or transmitted to produce work. Carnot had postulated that the fluid body could be any substance capable of expansion, such as vapor of water, vapor of alcohol, vapor of mercury, a permanent gas, air, etc. Although in those early years, engines came in a number of configurations, typically <i>Q</i><sub>H</sub> was supplied by a boiler, wherein water was boiled over a furnace; <i>Q</i><sub>C</sub> was typically removed by a stream of cold flowing water in the form of a <a href="/wiki/Condenser_(heat_transfer)" title="Condenser (heat transfer)">condenser</a> located on a separate part of the engine. The output work, <i>W</i>, is transmitted by the movement of the piston as it is used to turn a crank-arm, which in turn was typically used to power a pulley so as to lift water out of flooded salt mines. Carnot defined work as "weight lifted through a height". </p> <div class="mw-heading mw-heading2"><h2 id="Carnot_cycle">Carnot cycle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot_heat_engine&action=edit&section=3" title="Edit section: Carnot cycle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Carnot_cycle" title="Carnot cycle">Carnot cycle</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Carnot_cycle_pV_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Carnot_cycle_pV_diagram.svg/350px-Carnot_cycle_pV_diagram.svg.png" decoding="async" width="350" height="273" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Carnot_cycle_pV_diagram.svg/525px-Carnot_cycle_pV_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Carnot_cycle_pV_diagram.svg/700px-Carnot_cycle_pV_diagram.svg.png 2x" data-file-width="281" data-file-height="219" /></a><figcaption>Figure 1: A Carnot cycle illustrated on a <a href="/wiki/PV_diagram" class="mw-redirect" title="PV diagram">PV diagram</a> to illustrate the work done.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Carnot_cycle_ST_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Carnot_cycle_ST_diagram.svg/350px-Carnot_cycle_ST_diagram.svg.png" decoding="async" width="350" height="268" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Carnot_cycle_ST_diagram.svg/525px-Carnot_cycle_ST_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Carnot_cycle_ST_diagram.svg/700px-Carnot_cycle_ST_diagram.svg.png 2x" data-file-width="293" data-file-height="224" /></a><figcaption>Figure 2: A Carnot cycle acting as a heat engine, illustrated on a temperature-entropy diagram. The cycle takes place between a hot reservoir at temperature <span class="texhtml"><var style="padding-right: 1px;">T</var><sub>H</sub></span> and a cold reservoir at temperature <span class="texhtml"><var style="padding-right: 1px;">T</var><sub>C</sub></span>. The vertical axis is temperature, the horizontal axis is entropy.</figcaption></figure> <p>The <b>Carnot cycle</b> when acting as a heat engine consists of the following steps: </p> <ol><li><b>Reversible <a href="/wiki/Isothermal" class="mw-redirect" title="Isothermal">isothermal</a> expansion of the gas at the "hot" temperature, <span class="texhtml"><var style="padding-right: 1px;">T</var><sub>H</sub></span> (isothermal heat addition or absorption).</b> During this step (<span class="texhtml mvar" style="font-style:italic;">A</span> to <span class="texhtml mvar" style="font-style:italic;">B</span>) the gas is allowed to expand and it does work on the surroundings. The temperature of the gas (the system) does not change during the process, and thus the expansion is isothermic. The gas expansion is propelled by absorption of heat energy <span class="texhtml"><var style="padding-right: 1px;">Q</var><sub>H</sub></span> and of entropy <span class="texhtml">Δ<var style="padding-right: 1px;">S</var><sub>H</sub> = <var style="padding-right: 1px;">Q</var><sub>H</sub> / <var style="padding-right: 1px;">T</var><sub>H</sub></span> from the high temperature reservoir.</li> <li><b><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a> (<a href="/wiki/Reversible_adiabatic_process" class="mw-redirect" title="Reversible adiabatic process">reversible adiabatic</a>) expansion of the gas (isentropic work output).</b> For this step (<span class="texhtml mvar" style="font-style:italic;">B</span> to <span class="texhtml mvar" style="font-style:italic;">C</span>) the piston and cylinder are assumed to be thermally insulated, thus they neither gain nor lose heat. The gas continues to expand, doing work on the surroundings, and losing an equivalent amount of internal energy. The gas expansion causes it to cool to the "cold" temperature, <span class="texhtml"><var style="padding-right: 1px;">T</var><sub>C</sub></span>. The entropy remains unchanged.</li> <li><b>Reversible isothermal compression of the gas at the "cold" temperature, <span class="texhtml"><var style="padding-right: 1px;">T</var><sub>C</sub></span> (isothermal heat rejection)</b> (<span class="texhtml mvar" style="font-style:italic;">C</span> to <span class="texhtml mvar" style="font-style:italic;">D</span>). Now the gas is exposed to the cold temperature reservoir while the surroundings do work on the gas by compressing it (such as through the return compression of a piston), while causing an amount of waste heat <span class="texhtml"><var style="padding-right: 1px;">Q</var><sub>C</sub> < 0</span> (with the <a href="/wiki/Heat" title="Heat">standard sign convention for heat</a>) and of entropy <span class="texhtml">Δ<var style="padding-right: 1px;">S</var><sub>C</sub> = <var style="padding-right: 1px;">Q</var><sub>C</sub>/<var style="padding-right: 1px;">T</var><sub>C</sub> < 0</span> to flow out of the gas to the low temperature reservoir. (In magnitude, this is the same amount of entropy absorbed in step 1. The entropy decreases in isothermal compression since the multiplicity of the system decreases with the volume.) In terms of magnitude, the recompression work performed by the surroundings in this step is less than the work performed on the surroundings in step 1 because it occurs at a lower pressure due to the lower temperature (i.e. the resistance to compression is lower under step 3 than the force of expansion under step 1). We can refer to the first law of thermodynamics to explain this behavior: <span class="texhtml">Δ<var style="padding-right: 1px;">U</var>= <var style="padding-right: 1px;">W</var>+<var style="padding-right: 1px;">Q</var> </span>.</li> <li><b>Isentropic compression of the gas (isentropic work input)</b> (<span class="texhtml mvar" style="font-style:italic;">D</span> to <span class="texhtml mvar" style="font-style:italic;">A</span>). Once again the piston and cylinder are assumed to be thermally insulated and the cold temperature reservoir is removed. During this step, the surroundings continue to do work to further compress the gas and both the temperature and pressure rise now that the heat sink has been removed. This additional work increases the internal energy of the gas, compressing it and causing the temperature to rise to <span class="texhtml"><var style="padding-right: 1px;">T</var><sub>H</sub></span>. The entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Carnot's_theorem"><span id="Carnot.27s_theorem"></span>Carnot's theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot_heat_engine&action=edit&section=4" title="Edit section: Carnot's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem (thermodynamics)</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Real_vs_Carnot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Real_vs_Carnot.svg/400px-Real_vs_Carnot.svg.png" decoding="async" width="400" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Real_vs_Carnot.svg/600px-Real_vs_Carnot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Real_vs_Carnot.svg/800px-Real_vs_Carnot.svg.png 2x" data-file-width="900" data-file-height="330" /></a><figcaption>Real ideal engines (left) compared to the Carnot cycle (right). The entropy of a real material changes with temperature. This change is indicated by the curve on a <a href="/wiki/Temperature%E2%80%93entropy_diagram" title="Temperature–entropy diagram">T–S diagram</a>. For this figure, the curve indicates a vapor-liquid equilibrium (<i>See <a href="/wiki/Rankine_cycle" title="Rankine cycle">Rankine cycle</a></i>). Irreversible systems and losses of heat (for example, due to friction) prevent the ideal from taking place at every step.</figcaption></figure> <p><b>Carnot's theorem</b> is a formal statement of this fact: <i>No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs.</i> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{I}={\frac {W}{Q_{\mathrm {H} }}}=1-{\frac {T_{\mathrm {C} }}{T_{\mathrm {H} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{I}={\frac {W}{Q_{\mathrm {H} }}}=1-{\frac {T_{\mathrm {C} }}{T_{\mathrm {H} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4b37a837f64937573ce1b331faa1aa34b6c385" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.214ex; height:5.843ex;" alt="{\displaystyle \eta _{I}={\frac {W}{Q_{\mathrm {H} }}}=1-{\frac {T_{\mathrm {C} }}{T_{\mathrm {H} }}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Explanation">Explanation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot_heat_engine&action=edit&section=5" title="Edit section: Explanation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This maximum efficiency <span class="texhtml"><var style="padding-right: 1px;">η</var><sub>I</sub></span> is defined as above: </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">W</span> is the work done by the system (energy exiting the system as work),</li> <li><span class="texhtml"><var style="padding-right: 1px;">Q</var><sub>H</sub></span> is the heat put into the system (heat energy entering the system),</li> <li><span class="texhtml"><var style="padding-right: 1px;">T</var><sub>C</sub></span> is the <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a> of the cold reservoir, and</li> <li><span class="texhtml"><var style="padding-right: 1px;">T</var><sub>H</sub></span> is the <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a> of the hot reservoir.</li></ul> <p>A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient. </p><p>It is easily shown that the efficiency <span class="texhtml mvar" style="font-style:italic;">η</span> is maximum when the entire cyclic process is a <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible process</a>. This means the total <a href="/wiki/Entropy" title="Entropy">entropy</a> of system and surroundings (the entropies of the hot furnace, the "working fluid" of the heat engine, and the cold sink) remains constant when the "working fluid" completes one cycle and returns to its original state. (In the general and more realistic case of an irreversible process, the total entropy of this combined system would increase.) </p><p>Since the "working fluid" comes back to the same state after one cycle, and entropy of the system is a state function, the change in entropy of the "working fluid" system is 0. Thus, it implies that the total entropy change of the furnace and sink is zero, for the process to be reversible and the efficiency of the engine to be maximum. This derivation is carried out in the next section. </p><p>The <a href="/wiki/Coefficient_of_performance" title="Coefficient of performance">coefficient of performance</a> (COP) of the heat engine is the reciprocal of its efficiency. </p> <div class="mw-heading mw-heading2"><h2 id="Efficiency_of_real_heat_engines">Efficiency of real heat engines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot_heat_engine&action=edit&section=6" title="Edit section: Efficiency of real heat engines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a real heat engine, the total thermodynamic process is generally irreversible. The working fluid is brought back to its initial state after one cycle, and thus the change of entropy of the fluid system is 0, but the sum of the entropy changes in the hot and cold reservoir in this one cyclical process is greater than 0. </p><p>The internal energy of the fluid is also a state variable, so its total change in one cycle is 0. So the total work done by the system <span class="texhtml mvar" style="font-style:italic;">W</span> is equal to the net heat put into the system, the sum of <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_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/215f330aa0c9ca32a10c4a8410224ecfba72b713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.303ex; height:2.509ex;" alt="{\displaystyle Q_{\text{H}}}"></span> > 0 taken up and the waste heat <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_{\text{C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\text{C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869ca8625f345d6d1abea2a1bb280819b07f1c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.257ex; height:2.509ex;" alt="{\displaystyle Q_{\text{C}}}"></span> < 0 given off:<sup id="cite_ref-PlanckBook_6-0" class="reference"><a href="#cite_note-PlanckBook-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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=Q=Q_{\text{H}}+Q_{\text{C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>Q</mi> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=Q=Q_{\text{H}}+Q_{\text{C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b47d58c8975470011fd6fd7b3247674404d3475b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.871ex; height:2.509ex;" alt="{\displaystyle W=Q=Q_{\text{H}}+Q_{\text{C}}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"><table role="presentation" style="border-collapse:collapse; margin:0; border:none; width:100%;"><tbody><tr><td style="border:none; padding:0 0.4ex;" rowspan="2"></td> <td style="width:100%; border:none; padding:0;"></td> <td style="border:none; padding:0 0.4ex;" rowspan="2"></td></tr> <tr><td style="border-left:none; border-top:1px dashed; border-right:none; border-bottom:none; padding:0;"></td></tr></tbody></table></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>For real engines, stages 1 and 3 of the Carnot cycle, in which heat is absorbed by the "working fluid" from the hot reservoir, and released by it to the cold reservoir, respectively, no longer remain ideally reversible, and there is a temperature differential between the temperature of the reservoir and the temperature of the fluid while heat exchange takes place. </p><p>During heat transfer from the hot reservoir at <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_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06d8f87c63ac5f50ca4d8d51d38bc6a3a2c1bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.822ex; height:2.509ex;" alt="{\displaystyle T_{\text{H}}}"></span> to the fluid, the fluid would have a slightly lower temperature than <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_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06d8f87c63ac5f50ca4d8d51d38bc6a3a2c1bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.822ex; height:2.509ex;" alt="{\displaystyle T_{\text{H}}}"></span>, and the process for the fluid may not necessarily remain isothermal. Let <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 S_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e798c9eccc7ec29a9c6a1729aa16d43973186daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.826ex; height:2.509ex;" alt="{\displaystyle \Delta S_{\text{H}}}"></span> be the total entropy change of the fluid in the process of intake of heat. </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 S_{\text{H}}=\int _{Q_{\text{in}}}{\frac {{\text{d}}Q_{\text{H}}}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S_{\text{H}}=\int _{Q_{\text{in}}}{\frac {{\text{d}}Q_{\text{H}}}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc3fe5d50cfed03d294bef9d05f909f540b473a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.845ex; height:6.176ex;" alt="{\displaystyle \Delta S_{\text{H}}=\int _{Q_{\text{in}}}{\frac {{\text{d}}Q_{\text{H}}}{T}}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"><table role="presentation" style="border-collapse:collapse; margin:0; border:none; width:100%;"><tbody><tr><td style="border:none; padding:0 0.4ex;" rowspan="2"></td> <td style="width:100%; border:none; padding:0;"></td> <td style="border:none; padding:0 0.4ex;" rowspan="2"></td></tr> <tr><td style="border-left:none; border-top:1px dashed; border-right:none; border-bottom:none; padding:0;"></td></tr></tbody></table></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <p>where the temperature of the fluid <span class="texhtml mvar" style="font-style:italic;">T</span> is always slightly lesser than <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_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06d8f87c63ac5f50ca4d8d51d38bc6a3a2c1bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.822ex; height:2.509ex;" alt="{\displaystyle T_{\text{H}}}"></span>, in this process. </p><p>So, one would get: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\frac {Q_{\text{H}}}{T_{\text{H}}}}={\frac {\int {\text{d}}Q_{\text{H}}}{T_{\text{H}}}}\leq \Delta S_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>≤</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {Q_{\text{H}}}{T_{\text{H}}}}={\frac {\int {\text{d}}Q_{\text{H}}}{T_{\text{H}}}}\leq \Delta S_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbfca426ba56c1b04033d63a31ae969f2289472e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.399ex; height:6.176ex;" alt="{\displaystyle {\frac {Q_{\text{H}}}{T_{\text{H}}}}={\frac {\int {\text{d}}Q_{\text{H}}}{T_{\text{H}}}}\leq \Delta S_{\text{H}}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"><table role="presentation" style="border-collapse:collapse; margin:0; border:none; width:100%;"><tbody><tr><td style="border:none; padding:0 0.4ex;" rowspan="2"></td> <td style="width:100%; border:none; padding:0;"></td> <td style="border:none; padding:0 0.4ex;" rowspan="2"></td></tr> <tr><td style="border-left:none; border-top:1px dashed; border-right:none; border-bottom:none; padding:0;"></td></tr></tbody></table></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span>)</b></td></tr></tbody></table> <p>Similarly, at the time of heat injection from the fluid to the cold reservoir one would have, for the magnitude of total entropy change <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 S_{\text{C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S_{\text{C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/534034cc40c08e05faa8bfb17c053a1dcd855127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.78ex; height:2.509ex;" alt="{\displaystyle \Delta S_{\text{C}}}"></span>< 0 of the fluid in the process of expelling heat: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 S_{\text{C}}\geqslant {\frac {Q_{\text{C}}}{T_{\text{C}}}}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo>⩾</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mfrac> </mrow> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S_{\text{C}}\geqslant {\frac {Q_{\text{C}}}{T_{\text{C}}}}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34bda2ec49057d914effc262f35dbadf9eca07d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.232ex; height:5.843ex;" alt="{\displaystyle \Delta S_{\text{C}}\geqslant {\frac {Q_{\text{C}}}{T_{\text{C}}}}<0}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"><table role="presentation" style="border-collapse:collapse; margin:0; border:none; width:100%;"><tbody><tr><td style="border:none; padding:0 0.4ex;" rowspan="2"></td> <td style="width:100%; border:none; padding:0;"></td> <td style="border:none; padding:0 0.4ex;" rowspan="2"></td></tr> <tr><td style="border-left:none; border-top:1px dashed; border-right:none; border-bottom:none; padding:0;"></td></tr></tbody></table></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span>)</b></td></tr></tbody></table> <p>where, during this process of transfer of heat to the cold reservoir, the temperature of the fluid <span class="texhtml mvar" style="font-style:italic;">T</span> is always slightly greater than <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_{\text{C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8cceeda7b56dc8324a06365177af4ec1268787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.509ex;" alt="{\displaystyle T_{\text{C}}}"></span>. </p><p>We have only considered the magnitude of the entropy change here. Since the total change of entropy of the fluid system for the cyclic process is 0, we must have </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 S_{\text{H}}+\Delta S_{\text{C}}=\Delta S_{\text{cycle}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>cycle</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S_{\text{H}}+\Delta S_{\text{C}}=\Delta S_{\text{cycle}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b93bc51fe9f9241b70cdfe011203cf260c41e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.913ex; height:2.843ex;" alt="{\displaystyle \Delta S_{\text{H}}+\Delta S_{\text{C}}=\Delta S_{\text{cycle}}=0}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"><table role="presentation" style="border-collapse:collapse; margin:0; border:none; width:100%;"><tbody><tr><td style="border:none; padding:0 0.4ex;" rowspan="2"></td> <td style="width:100%; border:none; padding:0;"></td> <td style="border:none; padding:0 0.4ex;" rowspan="2"></td></tr> <tr><td style="border-left:none; border-top:1px dashed; border-right:none; border-bottom:none; padding:0;"></td></tr></tbody></table></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span>)</b></td></tr></tbody></table> <p>The previous three equations, namely (3), (4), (5), substituted into (6) to give:<sup id="cite_ref-FermiBook_7-0" class="reference"><a href="#cite_note-FermiBook-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 -{\frac {Q_{\text{C}}}{T_{\text{C}}}}\geqslant {\frac {Q_{\text{H}}}{T_{\text{H}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mfrac> </mrow> <mo>⩾</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {Q_{\text{C}}}{T_{\text{C}}}}\geqslant {\frac {Q_{\text{H}}}{T_{\text{H}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c38575acab3ff683c1bdd830137c198ea831d9f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.139ex; height:5.843ex;" alt="{\displaystyle -{\frac {Q_{\text{C}}}{T_{\text{C}}}}\geqslant {\frac {Q_{\text{H}}}{T_{\text{H}}}}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"><table role="presentation" style="border-collapse:collapse; margin:0; border:none; width:100%;"><tbody><tr><td style="border:none; padding:0 0.4ex;" rowspan="2"></td> <td style="width:100%; border:none; padding:0;"></td> <td style="border:none; padding:0 0.4ex;" rowspan="2"></td></tr> <tr><td style="border-left:none; border-top:1px dashed; border-right:none; border-bottom:none; padding:0;"></td></tr></tbody></table></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span>)</b></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Incomprehensible plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Ambox_question.svg/40px-Ambox_question.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Ambox_question.svg/60px-Ambox_question.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Ambox_question.svg/80px-Ambox_question.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>may be very hard to understand</b>.<span class="hide-when-compact"> Please help <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Carnot_heat_engine&action=edit">clarify it</a>.</span> <span class="date-container"><i>(<span class="date">January 2024</span>)</i></span></div></td></tr></tbody></table> <p>For [ΔSh ≥ (Qh/Th)] +[ΔSc ≥ (Qc/Tc)] = 0 </p><p>[ΔSh ≥ <b>(Qh/Th)</b>] = - [ΔSc ≥ (Qc/Tc)] </p><p>= [-ΔSc <i>≤</i> <b>(-Qc/Tc)</b>] </p><p>it is at least <b>(Qh/Th) <i>≤</i> (-Qc/Tc)</b> </p><p>Equations (<b><a href="#math_2">2</a></b>) and (<b><a href="#math_7">7</a></b>) combine to give </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\frac {W}{Q_{\text{H}}}}\leq 1-{\frac {T_{\text{C}}}{T_{\text{H}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>≤</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {W}{Q_{\text{H}}}}\leq 1-{\frac {T_{\text{C}}}{T_{\text{H}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7fa95797190eadb0dda009ca8413543e66a08a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.899ex; height:5.843ex;" alt="{\displaystyle {\frac {W}{Q_{\text{H}}}}\leq 1-{\frac {T_{\text{C}}}{T_{\text{H}}}}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"><table role="presentation" style="border-collapse:collapse; margin:0; border:none; width:100%;"><tbody><tr><td style="border:none; padding:0 0.4ex;" rowspan="2"></td> <td style="width:100%; border:none; padding:0;"></td> <td style="border:none; padding:0 0.4ex;" rowspan="2"></td></tr> <tr><td style="border-left:none; border-top:1px dashed; border-right:none; border-bottom:none; padding:0;"></td></tr></tbody></table></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_8" class="reference nourlexpansion" style="font-weight:bold;">8</span>)</b></td></tr></tbody></table> <p>To derive this step needs two adiabatic processes involved to show an isentropic process property for the ratio of the changing volumes of two isothermal processes are equal. </p><p>Most importantly, since the two adiabatic processes are volume works without heat lost, and since the ratio of volume changes for this two processes are the same, so the works for these two adiabatic processes are the same with opposite direction to each other, namely, one direction is work done by the system and the other is work done on the system; therefore, heat efficiency only concerns the amount of work done by the heat absorbed comparing to the amount of heat absorbed by the system. </p><p>Therefore, (W/Qh) = (Qh - Qc) / Qh </p><p>= 1 - (Qc/Qh) </p><p>= 1 - (Tc/Th) </p><p>And, from (7) </p><p><b>(Qh/Th) <i>≤</i> (-Qc/Tc)</b> here Qc it is less than 0 (release heat) </p><p><b>(Tc/Th) <i>≤</i> (-Qc/Qh)</b> </p><p><b>-(Tc/Th) ≥ (Qc/Qh)</b> </p><p><b><i>1+</i>[-(Tc/Th)] ≥ <i>1+</i>(Qc/Qh)</b> </p><p>1 - (Tc/Th) <b>≥ (Qh + Qc)/Qh</b> here Qc<0, </p><p>1 - (Tc/Th) <b>≥ (Qh - Qc)/Qh</b> </p><p>1 - (Tc/Th) <b>≥ W/Qh</b> </p><p>Hence, </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 \leq \eta _{\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>≤</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta \leq \eta _{\text{I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f36a8ea7bf61bacf36b1a5ef8bfc389e4e5886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.249ex; height:2.509ex;" alt="{\displaystyle \eta \leq \eta _{\text{I}}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"><table role="presentation" style="border-collapse:collapse; margin:0; border:none; width:100%;"><tbody><tr><td style="border:none; padding:0 0.4ex;" rowspan="2"></td> <td style="width:100%; border:none; padding:0;"></td> <td style="border:none; padding:0 0.4ex;" rowspan="2"></td></tr> <tr><td style="border-left:none; border-top:1px dashed; border-right:none; border-bottom:none; padding:0;"></td></tr></tbody></table></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_9" class="reference nourlexpansion" style="font-weight:bold;">9</span>)</b></td></tr></tbody></table> <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 \eta ={\frac {W}{Q_{\text{H}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ={\frac {W}{Q_{\text{H}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a7c2dc6f0b641e4f0abf0c59931dc38b69b510d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.407ex; height:5.676ex;" alt="{\displaystyle \eta ={\frac {W}{Q_{\text{H}}}}}"></span> is the efficiency of the real engine, 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 \eta _{\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\text{I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7460fa0a10dc43d00b0843349f484c90563e0735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.981ex; height:2.176ex;" alt="{\displaystyle \eta _{\text{I}}}"></span> is the efficiency of the Carnot engine working between the same two reservoirs at the temperatures <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_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06d8f87c63ac5f50ca4d8d51d38bc6a3a2c1bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.822ex; height:2.509ex;" alt="{\displaystyle T_{\text{H}}}"></span> 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 T_{\text{C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8cceeda7b56dc8324a06365177af4ec1268787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.509ex;" alt="{\displaystyle T_{\text{C}}}"></span>. For the Carnot engine, the entire process is 'reversible', and Equation (<b><a href="#math_7">7</a></b>) is an equality. Hence, the efficiency of the real engine is always less than the ideal Carnot engine. </p><p>Equation (<span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span>) signifies that the total entropy of system and surroundings (the fluid and the two reservoirs) increases for the real engine, because (in a surroundings-based analysis) the entropy gain of the cold reservoir as <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_{\text{C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\text{C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869ca8625f345d6d1abea2a1bb280819b07f1c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.257ex; height:2.509ex;" alt="{\displaystyle Q_{\text{C}}}"></span> flows into it at the fixed temperature <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_{\text{C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8cceeda7b56dc8324a06365177af4ec1268787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.509ex;" alt="{\displaystyle T_{\text{C}}}"></span>, is greater than the entropy loss of the hot reservoir as <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_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/215f330aa0c9ca32a10c4a8410224ecfba72b713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.303ex; height:2.509ex;" alt="{\displaystyle Q_{\text{H}}}"></span> leaves it at its fixed temperature <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_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06d8f87c63ac5f50ca4d8d51d38bc6a3a2c1bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.822ex; height:2.509ex;" alt="{\displaystyle T_{\text{H}}}"></span>. The inequality in Equation (<b><a href="#math_7">7</a></b>) is essentially the statement of the <a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a>. </p><p>According to the second theorem, "The efficiency of the Carnot engine is independent of the nature of the working substance". </p> <div class="mw-heading mw-heading2"><h2 id="The_Carnot_engine_and_Rudolf_Diesel">The Carnot engine and Rudolf Diesel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot_heat_engine&action=edit&section=7" title="Edit section: The Carnot engine and Rudolf Diesel"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1892 <a href="/wiki/Rudolf_Diesel" title="Rudolf Diesel">Rudolf Diesel</a> patented an <a href="/wiki/Internal_combustion_engine" title="Internal combustion engine">internal combustion engine</a> inspired by the Carnot engine. Diesel knew a Carnot engine is an ideal that cannot be built, but he thought he had invented a working approximation. His principle was unsound, but in his struggle to implement it he developed a practical <a href="/wiki/Diesel_engine" title="Diesel engine">Diesel engine</a>. </p><p>The conceptual problem was how to achieve isothermal expansion in an internal combustion engine, since burning fuel at the highest temperature of the cycle would only raise the temperature further. Diesel's patented solution was: having achieved the highest temperature just by compressing the air, to add a small amount of fuel at a controlled rate, such that heating caused by burning the fuel would be counteracted by cooling caused by air expansion as the piston moved. Hence all the heat from the fuel would be transformed into work during the isothermal expansion, as required by Carnot's theorem. </p><p>For the idea to work a small mass of fuel would have to be burnt in a huge mass of air. Diesel first proposed a working engine that would compress air to 250 atmospheres at 800 °C (1,450 °F), then cycle to one atmosphere at 20 °C (50 °F). However, this was well beyond the technological capabilities of the day, since it implied a compression ratio of 60:1. Such an engine, if it could have been built, would have had an efficiency of 73%. (In contrast, the best steam engines of his day achieved 7%.) </p><p>Accordingly, Diesel sought to compromise. He calculated that, were he to reduce the peak pressure to a less ambitious 90 atmospheres, he would sacrifice only 5% of the <a href="/wiki/Thermal_efficiency" title="Thermal efficiency">thermal efficiency</a>. Seeking financial support, he published the "Theory and Construction of a Rational Heat Engine to Take the Place of the Steam Engine and All Presently Known Combustion Engines" (1893). Endorsed by scientific opinion, including <a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Lord Kelvin</a>, he won the backing of <a href="/wiki/Krupp" title="Krupp">Krupp</a> and <span title="German-language text"><span lang="de" style="font-style: normal;"><a href="/wiki/Maschinenfabrik_Augsburg-N%C3%BCrnberg" class="mw-redirect" title="Maschinenfabrik Augsburg-Nürnberg">Maschinenfabrik Augsburg</a></span></span>. He clung to the Carnot cycle as a symbol. But years of practical work failed to achieve an isothermal combustion engine, nor could have done, since it requires such an enormous quantity of air that it cannot develop enough power to compress it. Furthermore, controlled fuel injection turned out to be no easy matter. </p><p>Even so, the Diesel engine slowly evolved over 25 years to become a practical high-compression air engine, its fuel injected near the end of the compression stroke and ignited by the heat of compression, capable by 1969 of 40% efficiency.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="As_a_macroscopic_construct">As a macroscopic construct</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot_heat_engine&action=edit&section=8" title="Edit section: As a macroscopic construct"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Carnot heat engine is, ultimately, a theoretical construct based on an <i>idealized</i> <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a>. On a practical human-scale level the <a href="/wiki/Carnot_cycle" title="Carnot cycle">Carnot cycle</a> has proven a valuable model, as in advancing the development of the <a href="/wiki/Diesel_engine" title="Diesel engine">diesel engine</a>. However, on a macroscopic scale limitations placed by the model's assumptions prove it impractical, and, ultimately, incapable of doing any <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>.<sup id="cite_ref-hang_9-0" class="reference"><a href="#cite_note-hang-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> As such, per <a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a>, the Carnot engine may be thought as the theoretical limit of macroscopic scale heat engines rather than any practical device that could ever be built.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>For example, for the <a href="/wiki/Isothermal_process" title="Isothermal process">isothermal expansion</a> part of the Carnot cycle, the following <i>infinitesimal</i> conditions must be satisfied simultaneously at every step in the expansion:<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <ul><li>The hot reservoir temperature <i>T<sub>H</sub></i> is infinitesimally higher than the system gas temperature <i>T</i> so heat flow (energy transfer) from the hot reservoir to the gas is made without increasing <i>T</i> (via infinitesimal work on the surroundings by the gas as another energy transfer); if <i>T<sub>H</sub></i> is significantly higher than <i>T</i>, then <i>T</i> may be not uniform through the gas so the system would deviate from thermal equilibrium as well as not being a reversible process (i.e. not a Carnot cycle) or <i>T</i> might increase noticeably so it would not be an isothermal process.</li> <li>The force externally applied on the piston (opposite to the internal force on the piston by the gas) needs to be infinitesimally reduced externally. Without this assistance, it would not be possible to follow a gas PV (Pressure-Volume) curve downward at a constant <i>T</i> since following this curve means that the gas-to-piston force decreases (<i>P</i> decreases) as the volume expands (the piston moves outward). If this assistance is so strong that the volume expansion is significant, the system may deviate from <a href="/wiki/Thermal_equilibrium" title="Thermal equilibrium">thermal equilibrium</a>, and the process fail to be reversible (and thus not a Carnot cycle).</li></ul> <p>Such "infinitesimal" requirements as these (and others) cause the Carnot cycle to take an <i>infinite amount of time</i>, rendering the production of work impossible.<sup id="cite_ref-hang_9-1" class="reference"><a href="#cite_note-hang-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Other practical requirements that make the Carnot cycle impractical to realize include fine control of the gas, and perfect thermal contact with the surroundings (including high and low temperature reservoirs).<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. (March 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot_heat_engine&action=edit&section=9" title="Edit section: Notes"><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 mw-references-columns"><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">Figure 1 in Carnot (1824, p. 17) and Carnot (1890, p. 63). In the diagram, the diameter of the vessel is large enough to bridge the space between the two bodies, but in the model, the vessel is never in contact with both bodies simultaneously. Also, the diagram shows an unlabeled axial rod attached to the outside of the piston.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">In French, Carnot uses <i>machine à feu</i>, which Thurston translates as <i>heat-engine</i> or <i>steam-engine</i>. In a footnote, Carnot distinguishes the steam-engine (<i>machine à vapeur</i>) from the heat-engine in general. (Carnot, 1824, p. 5 and Carnot, 1890, p. 43)</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"><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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.e-education.psu.edu/egee102/node/1942">"The Carnot Efficiency | EGEE 102: Energy Conservation and Environmental Protection"</a>. <i>www.e-education.psu.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-01-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.e-education.psu.edu&rft.atitle=The+Carnot+Efficiency+%7C+EGEE+102%3A+Energy+Conservation+and+Environmental+Protection&rft_id=https%3A%2F%2Fwww.e-education.psu.edu%2Fegee102%2Fnode%2F1942&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.worldcat.org/search?q=ti%3AReflections+on+the+motive+power+of+heat+au%3Acarnot">"Sometimes translated as <i>Reflections on the Motive Power of Heat</i>"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Sometimes+translated+as+Reflections+on+the+Motive+Power+of+Heat&rft_id=http%3A%2F%2Fwww.worldcat.org%2Fsearch%3Fq%3Dti%253AReflections%2Bon%2Bthe%2Bmotive%2Bpower%2Bof%2Bheat%2Bau%253Acarnot&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">English translation by Thurston (Carnot, 1890, p. 51-52).</span> </li> <li id="cite_note-PlanckBook-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-PlanckBook_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlanck1945" class="citation book cs1">Planck, M. (1945). <i>Treatise on Thermodynamics</i>. Dover Publications. p. 90. <q>§90, eqs.(39) & (40)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Treatise+on+Thermodynamics&rft.pages=90&rft.pub=Dover+Publications&rft.date=1945&rft.aulast=Planck&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span></span> </li> <li id="cite_note-FermiBook-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FermiBook_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFermi1956" class="citation book cs1">Fermi, E. (1956). <i>Thermodynamics</i>. Dover Publications (still in print). p. 47. <q>below eq.(63)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics&rft.pages=47&rft.pub=Dover+Publications+%28still+in+print%29&rft.date=1956&rft.aulast=Fermi&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBryant1969" class="citation journal cs1">Bryant, Lynwood (August 1969). "Rudolf Diesel and His Rational Engine". <i>Scientific American</i>. <b>221</b> (2): 108–117. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1969SciAm.221b.108B">1969SciAm.221b.108B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0869-108">10.1038/scientificamerican0869-108</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24926442">24926442</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Rudolf+Diesel+and+His+Rational+Engine&rft.volume=221&rft.issue=2&rft.pages=108-117&rft.date=1969-08&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24926442%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0869-108&rft_id=info%3Abibcode%2F1969SciAm.221b.108B&rft.aulast=Bryant&rft.aufirst=Lynwood&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span></span> </li> <li id="cite_note-hang-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-hang_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-hang_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiuMeng2017" class="citation journal cs1">Liu, Hang; Meng, Xin-He (2017-08-18). <a rel="nofollow" class="external text" href="https://doi.org/10.1140/epjc/s10052-017-5134-9">"Effects of dark energy on the efficiency of charged AdS black holes as heat engines"</a>. <i>The European Physical Journal C</i>. <b>77</b> (8): 556. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1704.04363">1704.04363</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1140%2Fepjc%2Fs10052-017-5134-9">10.1140/epjc/s10052-017-5134-9</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1434-6052">1434-6052</a>. <q>...since the Carnot heat engine, setting an upper bound on the efficiency of a heat engine is an ideal, reversible engine of which a single cycle must be performed in infinite time which is impractical and so the Carnot engine has zero power.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+European+Physical+Journal+C&rft.atitle=Effects+of+dark+energy+on+the+efficiency+of+charged+AdS+black+holes+as+heat+engines&rft.volume=77&rft.issue=8&rft.pages=556&rft.date=2017-08-18&rft_id=info%3Aarxiv%2F1704.04363&rft.issn=1434-6052&rft_id=info%3Adoi%2F10.1140%2Fepjc%2Fs10052-017-5134-9&rft.aulast=Liu&rft.aufirst=Hang&rft.au=Meng%2C+Xin-He&rft_id=https%3A%2F%2Fdoi.org%2F10.1140%2Fepjc%2Fs10052-017-5134-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenentiCasatiWang2020" class="citation journal cs1">Benenti, Giuliano; Casati, Giulio; Wang, Jiao (2020). <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/2007.08573.pdf">"Power, efficiency, and fluctuations in steady-state heat engines"</a> <span class="cs1-format">(PDF)</span>. <i>Physical Review E</i>. <b>102</b> (4). <q>However, fluctuations [in reservoir temperature] make impractical such engines.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+E&rft.atitle=Power%2C+efficiency%2C+and+fluctuations+in+steady-state+heat+engines&rft.volume=102&rft.issue=4&rft.date=2020&rft.aulast=Benenti&rft.aufirst=Giuliano&rft.au=Casati%2C+Giulio&rft.au=Wang%2C+Jiao&rft_id=https%3A%2F%2Farxiv.org%2Fpdf%2F2007.08573.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD2020" class="citation web cs1">D, Bob (2020-01-15). <a rel="nofollow" class="external text" href="https://physics.stackexchange.com/q/525217">"In the isothermal expansion phase of a Carnot cycle, why does the gas expand?"</a>. <i>StackExchange</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-01-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=StackExchange&rft.atitle=In+the+isothermal+expansion+phase+of+a+Carnot+cycle%2C+why+does+the+gas+expand%3F&rft.date=2020-01-15&rft.aulast=D&rft.aufirst=Bob&rft_id=https%3A%2F%2Fphysics.stackexchange.com%2Fq%2F525217&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot_heat_engine&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation audio-visual cs1"><a rel="nofollow" class="external text" href="https://m.youtube.com/watch?v=d6eJ8mccvu0&t=939"><i>Episode 46. Engine of Nature: The Carnot engine, part one, beginning with simple steam engines</i></a>. <a href="/wiki/The_Mechanical_Universe" title="The Mechanical Universe">The Mechanical Universe</a>. <a href="/wiki/Caltech" class="mw-redirect" title="Caltech">Caltech</a> – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Episode+46.+Engine+of+Nature%3A+The+Carnot+engine%2C+part+one%2C+beginning+with+simple+steam+engines&rft.series=The+Mechanical+Universe&rft.pub=Caltech&rft_id=https%3A%2F%2Fm.youtube.com%2Fwatch%3Fv%3Dd6eJ8mccvu0%26t%3D939&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span> </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=Carnot_heat_engine&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBryant1969" class="citation journal cs1">Bryant, Lynwood (August 1969). "Rudolf Diesel and His Rational Engine". <i>Scientific American</i>. <b>221</b> (2): 108–117. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1969SciAm.221b.108B">1969SciAm.221b.108B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0869-108">10.1038/scientificamerican0869-108</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24926442">24926442</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Rudolf+Diesel+and+His+Rational+Engine&rft.volume=221&rft.issue=2&rft.pages=108-117&rft.date=1969-08&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24926442%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0869-108&rft_id=info%3Abibcode%2F1969SciAm.221b.108B&rft.aulast=Bryant&rft.aufirst=Lynwood&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarnot1824" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot, Sadi</a> (1824). <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_QX9iIWF3yOMC"><i>Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance</i></a> (in French). Paris: Bachelier.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=R%C3%A9flexions+sur+la+puissance+motrice+du+feu+et+sur+les+machines+propres+%C3%A0+d%C3%A9velopper+cette+puissance&rft.place=Paris&rft.pub=Bachelier&rft.date=1824&rft.aulast=Carnot&rft.aufirst=Sadi&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbub_gb_QX9iIWF3yOMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span> (<a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_QX9iIWF3yOMC">First Edition 1824</a>) and (<a rel="nofollow" class="external text" href="https://books.google.com/books?id=YcY9AAAAMAAJ">Reissued Edition of 1878</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarnot1890" class="citation book cs1">Carnot, Sadi (1890). <a href="/wiki/Robert_Henry_Thurston" title="Robert Henry Thurston">Thurston, Robert Henry</a> (ed.). <a rel="nofollow" class="external text" href="https://archive.org/stream/reflectionsonmot00carnrich"><i>Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power</i></a>. New York: J. Wiley & Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reflections+on+the+Motive+Power+of+Heat+and+on+Machines+Fitted+to+Develop+That+Power&rft.place=New+York&rft.pub=J.+Wiley+%26+Sons&rft.date=1890&rft.aulast=Carnot&rft.aufirst=Sadi&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Freflectionsonmot00carnrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot+heat+engine" class="Z3988"></span> (<a rel="nofollow" class="external text" href="https://books.google.com/books?id=tgdJAAAAIAAJ">full text of 1897 ed.</a>) (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20120204034518/http://www.history.rochester.edu/steam/carnot/1943/Section2.htm">Archived HTML version</a>)</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist 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href="/wiki/Jet_engine" title="Jet engine">Jet</a></li> <li><a href="/wiki/Minto_wheel" title="Minto wheel">Minto wheel</a></li> <li><a href="/wiki/Photo-Carnot_engine" title="Photo-Carnot engine">Photo-Carnot engine</a></li> <li><a href="/wiki/Reciprocating_engine" title="Reciprocating engine">Piston</a></li> <li><a href="/wiki/Pistonless_rotary_engine" title="Pistonless rotary engine">Pistonless (Rotary)</a></li> <li><a href="/wiki/Rijke_tube" title="Rijke tube">Rijke tube</a></li> <li><a href="/wiki/Rocket_engine" title="Rocket engine">Rocket</a></li> <li><a href="/wiki/Split-single_engine" title="Split-single engine">Split-single</a></li> <li><a href="/wiki/Steam_engine" title="Steam engine">Steam (reciprocating)</a></li> <li><a href="/wiki/Steam_turbine" title="Steam turbine">Steam turbine</a> <ul><li><a href="/wiki/Aeolipile" title="Aeolipile">Aeolipile</a></li></ul></li> <li><a href="/wiki/Stirling_engine" title="Stirling engine">Stirling</a></li> <li><a href="/wiki/Thermoacoustic_heat_engine" title="Thermoacoustic heat engine">Thermoacoustic</a></li> <li><a href="/wiki/Manson_engine" title="Manson engine">Manson engine</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beale_number" title="Beale number">Beale number</a></li> <li><a href="/wiki/West_number" title="West number">West number</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Timeline of heat engine technology</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="background:#F0DC82;"><div><a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Thermodynamic cycle</a></div></td></tr></tbody></table></div>    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