Refrigerator COP Calculator: Coefficient of Performance from Heat and Work

The coefficient of performance of a refrigerator from the heat it moves and the work it uses.

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

Choose which two of the three quantities you know: the heat absorbed from the cold space Q_L, the heat rejected to the surroundings Q_H, or the work supplied W. The calculator finds the third from the energy balance Q_H = Q_L + W.

The coefficient of performance (COP) of a refrigerator is the heat taken out of the cold space divided by the work put in, COP = Q_L / W. It is often above 1, because the machine moves heat and does not make it. Used as a heat pump, the same machine has a COP of Q_H / W, which is the refrigerator COP plus 1.

To see how good the machine is, enter the temperature of the cold space and of the room. The ideal (Carnot) refrigerator between those temperatures has COP = T_L / (T_H - T_L) with temperatures in kelvin. No real refrigerator can beat it, and the calculator shows what share of it your numbers reach.

Formulas used

  • Q_H = Q_L + W
  • \mathrm{COP} = \frac{Q_L}{W} = \frac{Q_L}{Q_H - Q_L}
  • \mathrm{COP}_{ideal} = \frac{T_L}{T_H - T_L}
  • T(\mathrm{K}) = t(^\circ\mathrm{C}) + 273.15

Worked example

With the default numbers in the calculator above:

  1. Heat rejected from the heat balance: 400.00 kJ
  2. Coefficient of performance of the refrigerator: 3.00
  3. The same machine used as a heat pump: 4.00
  4. Temperatures in kelvin: 277.15 K, 298.15 K
  5. COP of an ideal (Carnot) refrigerator: 13.20
  6. Share of the ideal COP that this machine reaches: 22.73 %

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

A second worked example

A different case, with these numbers: Which two do you know? Heat absorbed Q_L and heat rejected Q_H; Heat rejected to the surroundings, Q_H 450 kJ.

  1. Work supplied from the heat balance: 150.00 kJ
  2. Coefficient of performance of the refrigerator: 2.00
  3. The same machine used as a heat pump: 3.00
  4. Temperatures in kelvin: 277.15 K, 298.15 K
  5. COP of an ideal (Carnot) refrigerator: 13.20
  6. Share of the ideal COP that this machine reaches: 15.15 %

Common questions

What is the COP of a refrigerator?

The coefficient of performance is the heat removed from the cold space divided by the work supplied. A COP of 3 means 3 kJ of heat is removed for each 1 kJ of work.

Can the COP be more than 1?

Yes. A refrigerator moves heat from cold to hot and does not create it, so the heat moved can be more than the work used. It is not an efficiency, which can never exceed 1.

How is a refrigerator different from a heat engine?

A heat engine takes heat from a hot source and turns some into work. A refrigerator works in reverse: it uses work to take heat from a cold space and reject it to a hotter one.

Why does the COP get better when the temperature difference is smaller?

The ideal COP is T_L / (T_H - T_L), so a smaller difference between the room and the cold space gives a larger ideal COP, and a real machine does less work for the same heat moved.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any physics or thermodynamics text (second law, refrigerators, coefficient of performance); Cengel and Boles, Thermodynamics: An Engineering Approach.

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