Heat Pump & Refrigerator COP Calculator

Calculate coefficient of performance for heat pumps and refrigerators from reservoir temperatures.

Room temp: 293K (20°C)
Outside: 253K (-20°C)
Please check your inputs and try again.

COP Measures Heat Moved per Unit Work Input

A heat pump or refrigerator is judged by how much heat it transfers compared with the work supplied, not by a conventional efficiency limited to 100%. For a refrigerator, COPR=QL/W. For a heat pump, COPHP=QH/W=COPR+1 because energy conservation gives QH=QL+W.

The maximum ideal values are set by a reversible Carnot cycle: COPR,C=TL/(TH−TL) and COPHP,C=TH/(TH−TL). Absolute Kelvin temperatures are essential. Real systems have lower COP because of compressor inefficiency, finite temperature differences, pressure drops, and other losses.

COPR=QL/W,   COPHP=QH/W=COPR+1
SymbolMeaningWhy it appears / units
QLHeat removed from cold reservoirJ or W over a steady interval.
QHHeat delivered to hot reservoirSame energy or power unit.
WWork inputJ or W.
TL,THReservoir temperaturesK for Carnot COP formulas.

A COP of 4 for a heat pump means four units of heat are delivered for each unit of work input; the extra heat comes from the colder environment. As the temperature lift TH−TL grows, ideal and real COP generally decrease.

COP is not a percentage efficiency. For the same device, energy conservation gives COPH=COPC+1. A heat pump operating between closer reservoir temperatures should have a higher ideal Carnot COP than one lifting heat across a larger temperature difference.

Worked Examples

Example 1: Heat pump: T_H=293K (20°C), T_C=253K (-20°C)
COP=293/(293-253)=293/40
Result: 7.3 max COP — 7.3× more efficient than electric heater
Real heat pumps achieve COP 2.5-4.5
Example 2: Refrigerator: T_H=300K, T_C=255K
COP_R=255/(300-255)=255/45
Result: 5.67 max Carnot COP
Real fridges: COP 1.5-3.5
Example 3: Heat pump energy balance
QH=12kW, W=3kW
Result: COPHP=4, QL=9kW
The delivered heat exceeds electrical work because environmental heat is also transferred.
Example 4: Carnot refrigerator
TL=270K, TH=300K
Result: COPR,C=9
This is an upper bound for reversible operation between those reservoir temperatures.

Common Mistakes

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Treating COP above 1 as impossible

COP is heat transferred divided by work, not energy output created from nothing. Heat pumps move existing thermal energy.

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Using Celsius in Carnot COP

Absolute temperature is required; using Celsius can produce nonsensical or negative values.

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Comparing heat-pump and refrigerator COP without definitions

COPHP uses heat delivered to the hot side, while COPR uses heat removed from the cold side, so they differ by exactly 1 for the same cycle.

Frequently Asked Questions

Why is COP > 1?
Heat pumps don't create heat — they move it. The COP represents heat moved per unit of work input. Even at COP=3, you're moving 3 kW of heat for 1 kW of electricity by extracting 2 kW from cold air/ground.
Ground source vs air source heat pumps?
Ground source (geothermal) uses soil at ~10-15°C year-round — more stable T_C gives higher COP. Air source performance degrades in cold weather (T_C drops, COP decreases). Ground source: COP 3-5; Air source: COP 2-4 in mild weather.
Why can a heat pump have COP greater than 100%?
COP is not the same as thermal efficiency. A heat pump uses work to move heat from outside to inside, so delivered heat equals extracted environmental heat plus work input.
Why does cold weather reduce heat-pump COP?
A larger temperature difference between the outdoor heat source and indoor delivery temperature requires more compressor work per unit of heat moved.
What is the relationship between refrigerator and heat-pump COP?
For the same device and operating condition, energy conservation gives COPHP=COPR+1 because QH=QL+W.
Is Carnot COP achievable in practice?
No real device reaches it because reversible operation would require idealized infinitesimal temperature differences and no friction, pressure drop, electrical loss, or other irreversibility.
Why can a heat-pump COP be greater than 1?
A heat pump does not convert work entirely into heat; it uses work to move heat from a colder region and delivers both that transferred heat and the work input to the warm side. Therefore COPH=QH/W can exceed 1 without violating energy conservation.

Formula Explorer connections

Interpretation: This relationship connects pressure, velocity, density, viscosity, geometry or transport in a fluid system. Assumption: Check whether flow is steady, incompressible, laminar, fully developed or one-dimensional. Reynolds and Mach regimes determine whether simplified formulas are valid.

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