Carnot Efficiency Calculator: Heat Engine, Refrigerator and Heat Pump

The best possible efficiency of a heat engine, and the best possible coefficient of performance of a refrigerator or heat pump, between a hot and a cold temperature.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

No heat engine working between a hot reservoir at Th and a cold reservoir at Tc can be more efficient than a Carnot engine: η = 1 - Tc / Th, with temperatures in kelvin. Real engines are always lower.

A refrigerator or heat pump run backwards has a coefficient of performance limit, COP = Tc / (Th - Tc) for cooling and Th / (Th - Tc) for heating. The smaller the temperature difference, the better the COP.

Formulas used

  • \eta_{Carnot} = 1 - \frac{T_c}{T_h}
  • COP_{cooling} = \frac{T_c}{T_h - T_c}
  • COP_{heating} = \frac{T_h}{T_h - T_c}

Worked example

With the default numbers in the calculator above:

  1. Absolute temperatures: 773 K and 303 K
  2. Carnot efficiency (best possible heat engine): 60.8 %
  3. Maximum work and heat rejected: 608 kJ of work
  4. Best possible heat pump COP: 1.65
  5. Best possible refrigerator COP: 0.645

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

Why kelvin?

The formula uses absolute temperature. Add 273.15 to degrees Celsius.

Is Carnot efficiency achievable?

No. It is a limit. Real engines have friction, heat loss and finite-speed processes.

Why does a heat pump work better on a mild day?

The temperature difference is smaller, so the Carnot limit and the real COP are higher.

Related calculators

Further reading: Cengel and Boles, Thermodynamics: An Engineering Approach; Moran and Shapiro, Fundamentals of Engineering Thermodynamics.

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