About

The Coefficient of Performance (COP) quantifies how effectively a thermodynamic cycle transfers heat relative to work consumed. A heat pump with COP = 4.0 delivers 4 kW of thermal energy per 1 kW of electrical input. Miscalculating this ratio leads to oversized equipment, inflated energy bills, and premature compressor failure. This tool computes actual COP from measured energy values, derives the Carnot theoretical maximum from reservoir temperatures, and reports the second-law efficiency ratio η so you can quantify how far your real system deviates from ideal reversible operation.

The calculator handles both heating and cooling modes, converts to EER (BTU/Wh) for North American equipment ratings, and flags physically impossible inputs where COP_actual exceeds COP_Carnot. Approximation assumes steady-state, single-stage vapor-compression cycles. Multi-stage cascade systems or absorption chillers require stage-by-stage analysis not covered here. Pro tip: field-measured COP typically runs 30% - 50% below Carnot due to irreversibilities in expansion valves, heat exchangers, and compressor friction.

Formulas

The actual Coefficient of Performance is the ratio of useful thermal energy to work input. For a cooling system:

COP_c = Q_cW

For a heating system (heat pump):

COP_h = Q_hW

The Carnot COP represents the theoretical maximum for a reversible cycle operating between two thermal reservoirs:

COP_Carnot,c = T_cT_h − T_c

COP_Carnot,h = T_hT_h − T_c

Second-law efficiency compares actual performance to the Carnot limit:

η_II = COP_actualCOP_Carnot × 100%

Energy Efficiency Ratio conversion for North American ratings:

EER = COP × 3.412 BTU/Wh

Where Q_c = heat removed from cold reservoir, Q_h = heat delivered to hot reservoir, W = work (electrical energy) input, T_h = hot reservoir temperature in Kelvin, T_c = cold reservoir temperature in Kelvin. Note: Q_h = Q_c + W by the first law of thermodynamics.

Reference Data

System Type	Typical COP Range	EER Equivalent	Common Application	Source Temperature
Window AC Unit	2.5 - 3.5	8.5 - 12.0	Residential room cooling	35 °C outdoor
Split System AC	3.0 - 5.0	10.2 - 17.1	Residential / light commercial	35 °C outdoor
Central Chiller (Centrifugal)	5.0 - 7.0	17.1 - 23.9	Large commercial buildings	30 °C condenser water
Air-Source Heat Pump (Heating)	2.5 - 4.5	-	Residential space heating	−5 to 10 °C
Ground-Source Heat Pump (Heating)	3.5 - 5.5	-	Residential / commercial heating	8 - 15 °C ground
Domestic Refrigerator	1.5 - 2.5	5.1 - 8.5	Food storage at 4 °C	25 °C kitchen
Commercial Freezer	1.0 - 1.8	3.4 - 6.1	Food storage at −18 °C	30 °C ambient
Industrial Ammonia Chiller	4.0 - 6.0	13.6 - 20.5	Process cooling, cold storage	25 - 35 °C
Absorption Chiller (Single Effect)	0.6 - 0.8	2.0 - 2.7	Waste heat driven cooling	80 - 120 °C heat source
Absorption Chiller (Double Effect)	1.0 - 1.4	3.4 - 4.8	Commercial / district cooling	150 - 180 °C steam
Water-Source Heat Pump	4.0 - 6.0	-	Lake / river water heating	5 - 20 °C water
CO₂ Heat Pump (Water Heating)	3.0 - 5.0	-	Domestic hot water to 90 °C	5 - 20 °C ambient
VRF / VRV System (Cooling)	3.5 - 6.5	11.9 - 22.2	Multi-zone commercial HVAC	35 °C outdoor
Thermoelectric (Peltier) Cooler	0.3 - 0.7	1.0 - 2.4	Electronics cooling, small devices	25 °C ambient
Carnot Ideal (Cooling, 5/35 °C)	9.27	31.6	Theoretical upper limit	5 / 35 °C
Carnot Ideal (Heating, −5/35 °C)	7.70	-	Theoretical upper limit	−5 / 35 °C

Frequently Asked Questions

The Carnot COP is inversely proportional to the temperature difference (T_h − T_c). A larger ΔT forces the compressor to work harder to move heat against a steeper thermal gradient. For example, a heat pump operating between −10 °C and 40 °C (ΔT = 50 K) has a Carnot COP_h of 6.26, while between 5 °C and 35 °C (ΔT = 30 K) it reaches 10.27. This is why ground-source heat pumps outperform air-source units in extreme climates.

No. The second law of thermodynamics prohibits any real cycle from exceeding the Carnot efficiency. If your calculated actual COP exceeds the Carnot value, the measurement data contains errors: inaccurate power metering, heat leakage affecting Q measurements, or incorrect temperature readings. This calculator flags such cases as physically invalid. Real systems typically achieve 30% to 60% of the Carnot limit.

EER (Energy Efficiency Ratio) equals COP multiplied by 3.412 BTU/Wh. The factor converts watts to BTU/h. EER is standard in North American HVAC ratings (AHRI standards). COP is dimensionless and used in international engineering, ISO 13256, and thermodynamic analysis. A COP of 3.5 equals an EER of approximately 11.9. Minimum federal EER standards for residential AC in the US range from 11.0 to 14.0 depending on capacity.

By the first law, Q_h = Q_c + W. Dividing both sides by W gives COP_h = COP_c + 1. The heating COP always exceeds the cooling COP by exactly 1 for the same operating conditions. This is why heat pumps are inherently more efficient for heating than for cooling.

Manufacturer ratings use standardized test conditions (e.g., ISO 13256: entering water at 0 °C for heating, 25 °C for cooling). Field conditions introduce variable ambient temperatures, duct losses (10 - 30%), defrost cycles in air-source units (reducing effective COP by 10 - 15% below 5 °C), refrigerant charge drift, and fouled heat exchangers. Always derate catalog COP by at least 15 - 20% for design load calculations.

The breakeven depends on local electricity and gas prices. The formula is: COP_breakeven = (electricity price per kWh) ÷ (gas price per kWh equivalent) × furnace efficiency. For a 95% efficient furnace with electricity at 0.12 $/kWh and gas at 0.04 $/kWh, the breakeven COP is approximately 2.85. Most modern heat pumps exceed this in mild climates. Below −10 °C, air-source COP can drop to 1.5 - 2.0, making gas cheaper.