Carnot Efficiency & Heat Pump COP Calculator
Find the most any heat engine, fridge or heat pump can do between two temperatures. Get the Carnot efficiency or COP, compare it with a real machine’s figure, and see where every watt of heat and work goes.
How to Use
- Pick the machine: a Heat engine, a Refrigerator or a Heat pump. Use Temperature needed to find how hot or cold one side must be for a target efficiency or COP.
- Enter the hot-side and cold-side temperatures in °C, °F or kelvin. The calculator works in kelvin, so you never have to convert.
- Optionally enter the machine’s real efficiency or COP to see what share of the Carnot limit it reaches (its second-law efficiency).
- Enter the heat flow (heat in for an engine, heat removed for a fridge, heat delivered for a heat pump) in W, kW, MW, hp or BTU/h to get the work and the other heat flow.
- Read the energy-flow diagram, where each arrow is as wide as the energy it carries, and check Show Work for each step.
Worked Example
A steam power plant. Steam at 565 °C is 838.15 K and the cooling water at 30 °C is 303.15 K. The Carnot limit is η = 1 − 303.15 ÷ 838.15 = 63.83%. If the plant actually turns 40% of its 1,000 MW of heat into electricity, it makes 400 MW and sends 600 MW to the cooling water, reaching 40 ÷ 63.83 = 62.67% of what the second law allows.
An air-source heat pump. Taking heat from 7 °C air (280.15 K) and delivering it to 35 °C heating water (308.15 K), the best possible COP is 308.15 ÷ (308.15 − 280.15) = 308.15 ÷ 28 = 11.01. A unit rated at COP 4 delivering 8 kW uses 8 ÷ 4 = 2 kW of electricity and moves the other 6 kW in from outside.
The common mistake: using °C instead of kelvin. For the plant above, 1 − 30 ÷ 565 = 94.69%, which looks like an extraordinary engine but is simply wrong; the true limit is 63.83%. A fridge at 4 °C in a 22 °C kitchen fares even worse: 4 ÷ (22 − 4) gives a COP of 0.2222 instead of 277.15 ÷ 18 = 15.40. Differences can stay in °C (18 °C is 18 K), but each temperature on its own must be in kelvin.
Show Work
Formulas
Carnot’s Ideal Engine
In 1824 the French engineer Sadi Carnot published Réflexions sur la puissance motrice du feu (Reflections on the Motive Power of Fire), asking how much work a steam engine could possibly get from its fuel. He imagined an ideal engine running in a reversible cycle between a hot and a cold body, and showed that its output depends only on the two temperatures, not on the working fluid. Carnot died of cholera in 1832, aged 36, and the book went largely unread until Émile Clapeyron restated it with pressure–volume diagrams in 1834.
William Thomson (later Lord Kelvin) used Carnot’s result to define an absolute temperature scale in 1848, the scale behind the kelvin and the reason the formula needs it. Rudolf Clausius reconciled Carnot’s idea with the conservation of energy in 1850 and in 1865 named the quantity that a reversible cycle leaves unchanged entropy; the Show Work section checks it, since a real machine always produces some.
Running the cycle backwards gives a refrigerator or heat pump. Thomson proposed using one to heat buildings in 1852, pointing out that it could deliver more heat than the work put in. Today’s heat pumps and fridges are rated by their coefficient of performance, and the Carnot COP sets the ceiling every one of them works under.
About This Tool
This calculator works out the Carnot limit for a heat engine, a refrigerator or a heat pump from two temperatures in any scale, and compares it with a real machine’s efficiency or COP to give the second-law efficiency. With a heat flow it splits the energy into work and heat both ways round, checks the entropy produced, and draws the flows as arrows whose width matches the energy they carry. It can also run backwards to find the temperature a target efficiency or COP needs.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Heat Conduction Calculator, Gibbs Free Energy Calculator, and Nuclear Binding Energy Calculator.
Frequently Asked Questions
What is the Carnot efficiency?
It is the most work any heat engine can get from heat flowing between two temperatures: η = 1 − T_c ÷ T_h, with both in kelvin. Steam at 565 °C (838.15 K) with cooling water at 30 °C (303.15 K) gives 1 − 303.15 ÷ 838.15 = 63.83%. No real engine between those temperatures can beat it.
Why do the temperatures have to be in kelvin?
The formula compares temperatures as ratios, which only works on an absolute scale that starts at zero energy. Using Celsius for the 565 °C / 30 °C plant gives 1 − 30 ÷ 565 = 94.69%, far above the true limit of 63.83%. For a fridge at 4 °C in a 22 °C kitchen, Celsius gives a COP of 0.2222 instead of 15.40.
What is the COP of a heat pump?
The coefficient of performance is the heat delivered divided by the electricity used. The Carnot limit is COP = T_h ÷ (T_h − T_c): from 7 °C outdoor air to 35 °C heating water it is 308.15 ÷ 28 = 11.01. A real unit with a COP of 4 turns 2 kW of electricity into 8 kW of heat, drawing the other 6 kW from the outside air.
Why do heat pumps work less well in cold weather?
The bigger the temperature lift, the lower the limit. Heating 35 °C water from −7 °C air gives a Carnot COP of 308.15 ÷ 42 = 7.337, against 11.01 at 7 °C. If a heat pump reaches the same share of the limit in both cases, a COP of 4 at 7 °C falls to 2.667 at −7 °C, which is still 2.667 times better than an electric heater.
Can a heat engine ever be 100% efficient?
Only with its cold side at absolute zero, which cannot be reached. With a 30 °C cold side, 90% would need a hot side of 303.15 ÷ (1 − 0.9) = 3,031.5 K (2,758.35 °C), and 99% would need 30,315 K. That is why power stations raise the steam temperature and use the coldest cooling water they can.
How do I use the Carnot Efficiency & Heat Pump COP Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Power stations
A plant with 565 °C steam and 30 °C cooling water is capped at 63.83%. If it reaches 40%, 1,000 MW of heat gives 400 MW of electricity and 600 MW goes to the cooling water: 62.67% of the limit.
Heat pump sizing
An air-source heat pump with a COP of 4 delivering 8 kW of heat draws 2 kW of electricity and takes 6 kW from the outside air; the Carnot minimum at 7 → 35 °C would be 0.7269 kW.
Fridges and freezers
Holding 4 °C in a 22 °C kitchen has a Carnot COP of 15.40; a freezer at −18 °C manages only 6.379, so each watt of cooling costs more than twice as much work.
Low-grade heat
Ocean thermal energy between 25 °C surface water and 5 °C deep water is limited to 6.708%; a 150 °C geothermal source cooled at 30 °C reaches 28.36%.
Thermodynamics homework
Find how hot the steam must be for a 50% Carnot limit with a 30 °C condenser: T_h = 303.15 ÷ 0.5 = 606.3 K, or 333.15 °C.
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