{"id":66945,"date":"2026-10-13T10:37:28","date_gmt":"2026-10-13T07:37:28","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66945"},"modified":"2026-10-13T10:37:28","modified_gmt":"2026-10-13T07:37:28","slug":"carnot-efficiency-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/physics\/carnot-efficiency-calculator\/","title":{"rendered":"Carnot Efficiency Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Find the Maximum Efficiency of a Heat Engine<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66945\" data-favorite-title=\"Carnot Efficiency\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/physics\/carnot-efficiency-calculator\/\" data-favorite-excerpt=\"Enter the hot-reservoir and cold-reservoir temperatures to find the maximum theoretical (Carnot) efficiency of a heat engine, plus the temperature ratio and COP values for a refrigerator or heat pump.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-physic\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/physics\/carnot-efficiency-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66945\" data-post=\"66945\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Hot-reservoir temperature (Th)<\/label><div class=\"term\">Th<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"number\" name=\"hotreservoir_temperature_th\" id=\"field-1\" aria-describedby=\"calc-info-1\" placeholder=\"e.g. 500\" min=\"-273.15\" step=\"any\" value=\"500\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"unit\" aria-label=\"Unit\"><option value=\"k\" selected>K<\/option><option value=\"c\">\u00b0C<\/option><option value=\"f\">\u00b0F<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-1\">Temperature of the heat source. Must be higher than the cold-reservoir temperature.<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-3\">Cold-reservoir temperature (Tc)<\/label><div class=\"term\">Tc<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"number\" name=\"coldreservoir_temperature_tc\" id=\"field-3\" aria-describedby=\"calc-info-3\" placeholder=\"e.g. 300\" min=\"-273.15\" step=\"any\" value=\"300\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"unit\" aria-label=\"Unit\"><option value=\"k\" selected>K<\/option><option value=\"c\">\u00b0C<\/option><option value=\"f\">\u00b0F<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-3\">Temperature of the surroundings or cooling medium (often ~300 K \/ 27 \u00b0C for ambient).<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66945\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Carnot Efficiency Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-2\"><span class=\"field-title\"><span>Details \u2014 absolute temperatures, work ratio and heat-pump performance<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"66945\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Carnot Efficiency<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\eta_{max} = 1 - \\frac{T_c}{T_h} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(\\eta_{max}\\) \u2014 maximum possible (Carnot) efficiency of a heat engine, as a decimal fraction<\/li>\n<li>\\(T_h\\) \u2014 absolute temperature of the hot reservoir (K)<\/li>\n<li>\\(T_c\\) \u2014 absolute temperature of the cold reservoir (K)<\/li>\n<\/ul>\n<p>The formula gives the upper limit on the fraction of absorbed heat that any engine operating between two thermal reservoirs can convert into work, as required by the second law of thermodynamics. Both temperatures must be on an absolute scale (kelvin); if your temperatures are in \u00b0C or \u00b0F, the calculator converts them automatically.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Carnot Efficiency \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a heat engine works between a hot reservoir at 500 K and a cold reservoir (the surroundings) at 300 K.<\/p>\n<p>\\[ \\eta_{max} = 1 - \\frac{T_c}{T_h} = 1 - \\frac{300}{500} = 1 - 0.6 = 0.4 \\]<\/p>\n<p><strong>Result:<\/strong> the maximum efficiency is 0.40, i.e. <strong>40 %<\/strong>. Even a perfectly reversible engine must reject the remaining 60 % of the absorbed heat to the cold reservoir. Running the same cycle backwards gives a Carnot coefficient of performance of \\(300\/(500-300) = 1.5\\) as a refrigerator and \\(500\/(500-300) = 2.5\\) as a heat pump.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Enter the <strong>Hot-reservoir temperature (Th)<\/strong> \u2014 the temperature of the heat source, e.g. combustion gas, steam or a geothermal field.<\/li>\n<li>Enter the <strong>Cold-reservoir temperature (Tc)<\/strong> \u2014 the temperature of the cooling medium, typically ambient air or cooling water (about 300 K \/ 27 \u00b0C).<\/li>\n<li>Select the unit (K, \u00b0C or \u00b0F) for each temperature from the dropdown next to the input field. The calculator converts everything to kelvin internally.<\/li>\n<li>Read the result: the maximum efficiency in percent and as a decimal, the temperature ratio, and the Carnot COP values if the cycle were run as a refrigerator or a heat pump.<\/li>\n<\/ol>\n<h2>What is Carnot efficiency?<\/h2>\n<p>The Carnot efficiency is the theoretical maximum efficiency that any heat engine can achieve when operating between two thermal reservoirs at temperatures \\(T_h\\) and \\(T_c\\). It is named after Sadi Carnot, who showed in 1824 that no engine \u2014 however cleverly designed \u2014 can exceed this limit. The reason is the second law of thermodynamics: an engine must reject part of the heat it absorbs to a colder body, and the amount it must reject is fixed by the ratio \\(T_c\/T_h\\).<\/p>\n<p>Because the formula uses a ratio of temperatures, the temperatures must be absolute (kelvin or Rankine). Using Celsius or Fahrenheit directly gives wrong answers: 100 \u00b0C and 0 \u00b0C are not a 100 % efficient pair, but 373.15 K and 273.15 K give a Carnot efficiency of only about 27 %.<\/p>\n<p>Real engines never reach the Carnot limit because friction, turbulence, heat loss and finite-speed heat transfer make them irreversible. Practical heat engines typically achieve 30\u201370 % of the Carnot value. For example, a modern coal-fired power plant with steam at ~810 K exhausting to ~300 K has a Carnot limit of about 63 %, while its actual efficiency is around 40 %. Engineers use the Carnot value as a benchmark: it tells you whether a design is worth improving further or whether it is already close to the physical ceiling.<\/p>\n<p>The result also shows how much heat is inevitably wasted: the fraction \\(T_c\/T_h\\) of every joule absorbed must be dumped to the cold reservoir. This is why power plants need cooling towers and rivers, and why the efficiency of any engine rises when you raise the source temperature or lower the sink temperature.<\/p>\n<h2>Assumptions, limitations and tips<\/h2>\n<ul>\n<li><strong>Absolute temperatures only.<\/strong> The formula is valid only with temperatures on an absolute scale. If you enter \u00b0C or \u00b0F, the calculator converts them to kelvin before computing the ratio.<\/li>\n<li><strong>Two reservoirs at fixed temperatures.<\/strong> The Carnot model assumes the engine exchanges heat only with reservoirs that stay at constant temperature. Real sources (burning fuel, expanding steam) change temperature during the process, which lowers the achievable efficiency.<\/li>\n<li><strong>Reversible operation.<\/strong> The Carnot engine is infinitely slow and frictionless. Every real process is irreversible, so treat the result as an upper bound, not a design target.<\/li>\n<li><strong>Efficiency cannot reach 100 %.<\/strong> A finite cold reservoir above absolute zero always forces some heat rejection. 100 % would require \\(T_c = 0\\) K, which is unattainable (third law of thermodynamics).<\/li>\n<li><strong>COP values.<\/strong> The refrigerator and heat-pump coefficients of performance shown in the details refer to the same two reservoirs run in reverse. Note that a heat pump's COP is always 1 greater than the refrigerator's COP for the same temperatures.<\/li>\n<li>To compare how well a real machine converts input work into useful output (rather than comparing it with a thermodynamic limit), see the <a href=\"https:\/\/wpcalc.com\/en\/physics\/mechanical-efficiency-calculator\/\">Mechanical Efficiency calculator<\/a>.<\/li>\n<\/ul>\n<h2>Typical Carnot efficiency values<\/h2>\n<table>\n<thead>\n<tr>\n<th>Application<\/th>\n<th>Th (approx.)<\/th>\n<th>Tc (approx.)<\/th>\n<th>Carnot efficiency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Car engine (gasoline)<\/td>\n<td>800 K<\/td>\n<td>300 K<\/td>\n<td>62.5 %<\/td>\n<\/tr>\n<tr>\n<td>Steam power plant<\/td>\n<td>810 K<\/td>\n<td>300 K<\/td>\n<td>63 %<\/td>\n<\/tr>\n<tr>\n<td>Combined-cycle gas plant<\/td>\n<td>1600 K<\/td>\n<td>300 K<\/td>\n<td>81 %<\/td>\n<\/tr>\n<tr>\n<td>Geothermal (low enthalpy)<\/td>\n<td>360 K<\/td>\n<td>300 K<\/td>\n<td>17 %<\/td>\n<\/tr>\n<tr>\n<td>Ocean thermal energy (OTEC)<\/td>\n<td>300 K<\/td>\n<td>278 K<\/td>\n<td>7 %<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These are theoretical ceilings; actual efficiencies are always lower. For other thermodynamic and physics quantities, you may also find our <a href=\"https:\/\/wpcalc.com\/en\/physics\/pressure-calculator\/\">Pressure<\/a> and <a href=\"https:\/\/wpcalc.com\/en\/physics\/density-calculator\/\">Density<\/a> calculators useful.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter the hot-reservoir and cold-reservoir temperatures to find the maximum theoretical (Carnot) efficiency of a heat engine, plus the temperature ratio and COP values for a refrigerator or heat pump.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[62],"tags":[91],"class_list":["post-66945","post","type-post","status-publish","format-standard","hentry","category-physics","tag-energy-work"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66945","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=66945"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66945\/revisions"}],"predecessor-version":[{"id":67022,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66945\/revisions\/67022"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66945"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66945"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66945"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}