Coefficient of Performance Calculator
Solve the refrigeration COP relation COP = Qc / W for any of its three unknowns — cooling capacity, work input, or COP itself — with step-by-step math, a live refrigeration cycle schematic, and companion thermodynamic metrics.
COP is the efficiency of heat pumping
COP measures how much cooling (or heating) you get per unit of work input. A COP of 3.5 means every kilowatt of electrical power moves 3.5 kW of heat out of the cold space. Unlike thermal efficiency, COP is not limited to 1 — it can be 3, 5, or even 10, because the device is not converting heat to work but moving heat from a cold place to a hot place.
Carnot COP sets the upper bound
The maximum possible COP for any refrigerator operating between cold temperature T_c and hot temperature T_h is COP_Carnot = T_c / (T_h − T_c). For T_c = 273 K and T_h = 318 K, that is 6.07. Real devices achieve 30–60% of the Carnot value, limited by compressor inefficiency, finite heat-exchanger area, and pressure drops. The Carnot COP decreases as the temperature lift (T_h − T_c) increases, so cooling to a lower temperature costs more.
Heat rejected is Q_c + W
By the first law of thermodynamics, the heat rejected to the hot reservoir is Q_h = Q_c + W. The COP for a heat pump (which is what you want for heating) is CO_HP = Q_h / W = COP + 1. So a refrigerator with COP = 3.5 has a heat-pump COP of 4.5 — it delivers 4.5 kW of heat for every 1 kW of electrical input, making heat pumps far more efficient than resistive heaters (COP = 1).
Higher COP means lower operating cost
Because electricity is billed by the kilowatt-hour, a higher COP directly reduces running cost. Raising COP from 2.5 to 4.0 cuts the electricity bill by 37.5% for the same cooling delivered. This is why modern inverter-driven heat pumps with COP above 4 are replacing older fixed-speed systems with COP around 2.5. The trade-off is higher capital cost, which is recovered over the equipment's lifetime through lower energy bills.
How to Use This Calculator
The COP relation links three quantities: COP, Qc, and W. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick COP, Qc, or W from the toggle. The calculator rearranges COP = Qc / W automatically for whichever you choose.
2. Use realistic presets
Presets cover typical values: COP from 1.5 (old fridge) to 7.0 (large chiller), cooling capacity from 0.2 kW (fridge) to 1,000 kW (industrial), and work input from 0.1 kW to 200 kW.
3. Match units carefully
Use kW for both Qc and W — the ratio is then dimensionless. If you have BTU/h or tons of refrigeration, convert first — or use the unit converter further down the page. COP is always dimensionless.
4. Read the result and verdict
The result card shows the solved value plus equivalent units. A colour-coded verdict classifies the COP from poor (below 2) through good (3–5) to excellent (above 6).
5. Check the schematic and charts
The schematic shows the evaporator, compressor, condenser, and the heat and work flows. The chart plots COP against cooling capacity for three work values.
6. Switch to Advanced for depth
Advanced mode adds a chart of COP vs. W, a sensitivity check on a +1% input change, and companion metrics including Carnot COP, second-law efficiency, and heat rejection.
What This Tool Offers
Built around the refrigeration COP relation COP = Qc / W, the fundamental link between cooling capacity, work input, and thermodynamic performance.
Three-way solving
Solve for COP, cooling capacity, or work input. The calculator rearranges the formula for whichever variable you choose.
Clean cycle schematic
An evaporator-compressor-condenser diagram with the cooling Qc, work W, and rejected heat Qh arrows, scaled with magnitude.
Performance-class verdict
Each result is classified from poor (below 2) through good (3–5) to excellent (above 6), with a coloured badge.
Multi-unit hints
Each field shows the value converted to related units — tons of refrigeration and BTU/h for cooling; hp and W for work; Carnot fraction for COP.
Dual comparison charts
COP vs. Qc for three work values, plus an advanced COP vs. W chart for three cooling capacities.
Reference tables
Typical COP by application, Carnot limits by temperature, and a full variable reference table.
Reference Tables & Conversions
A refrigeration power unit converter, a COP lookup grid, and typical values.
Refrigeration power unit converter
Carnot COP Limit by Temperature
Maximum theoretical COP = T_c / (T_h − T_c), with temperatures in kelvin.
| T_c (K / °C) | T_h = 308 K (35°C) | T_h = 318 K (45°C) | T_h = 328 K (55°C) | T_h = 343 K (70°C) |
|---|
Typical COP by Application
Representative values for real equipment at rated conditions.
| Application | Typical COP | Carnot COP | % of Carnot | Notes |
|---|---|---|---|---|
| Domestic refrigerator | 1.5 – 3.0 | 4.5 – 6.0 | 30 – 50% | Small compressor, frost-free losses |
| Window air conditioner | 2.5 – 3.2 | 6.0 – 7.0 | 40 – 50% | Fixed-speed, low-cost |
| Split-system AC (inverter) | 3.5 – 5.0 | 6.5 – 8.0 | 50 – 65% | Variable-speed compressor |
| Air-source heat pump | 3.5 – 5.5 | 7.0 – 9.0 | 50 – 65% | Heating mode, mild climate |
| Ground-source heat pump | 4.5 – 6.5 | 8.0 – 11.0 | 55 – 70% | Stable ground temperature |
| Commercial chiller (water-cooled) | 5.0 – 7.0 | 9.0 – 12.0 | 55 – 65% | Large, efficient compressors |
| Industrial ammonia plant | 4.0 – 6.0 | 8.0 – 10.0 | 50 – 60% | Low evaporating temperature |
| Absorption chiller (heat-driven) | 0.6 – 1.2 | — | — | COP defined per unit heat input |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| COP | Coefficient of performance (cooling) | 1 – 8 | — |
| Qc | Cooling capacity (heat removed from cold space) | 0.1 – 10⁴ | kW |
| W | Work input (compressor power) | 0.05 – 10³ | kW |
| Qh | Heat rejected to hot reservoir | — | kW (Qc + W) |
| Tc | Cold reservoir temperature | 233 – 300 | K |
| Th | Hot reservoir temperature | 300 – 350 | K |
| COPCarnot | Maximum theoretical COP | — | Tc / (Th − Tc) |
What Does COP = Qc / W Measure?
The coefficient of performance measures how effectively a refrigeration or heat-pump system moves heat using a given amount of work. It is the ratio of the cooling delivered (heat removed from the cold space) to the mechanical or electrical work supplied to the compressor. Unlike thermal efficiency, COP can be greater than 1 — and often is — because the device is moving heat rather than converting it.
COP Formula
The defining relationship for a refrigerator or heat pump in cooling mode is:
Rearranged to solve for the other two unknowns:
W = Qc / COP
The heat rejected to the hot reservoir follows from the first law:
For a heat pump in heating mode, the COP is defined with respect to the heat delivered:
The maximum theoretical COP is set by the Carnot limit:
and the second-law efficiency of a real device is:
COP Variables
| Symbol | Meaning | Notes |
|---|---|---|
| COP | Coefficient of performance | Cooling delivered per unit work; dimensionless |
| Qc | Cooling capacity | Heat removed from the cold space per unit time |
| W | Work input | Mechanical or electrical power to the compressor |
| Qh | Heat rejected | Qh = Qc + W by the first law |
| Tc | Cold reservoir temperature | Absolute temperature of the refrigerated space |
| Th | Hot reservoir temperature | Absolute temperature of the rejection environment |
| COPCarnot | Carnot COP | Maximum possible COP; Tc/(Th−Tc) |
How Each Variable Affects COP
COP is directly proportional to cooling capacity and inversely proportional to work input. For a given compressor power, a system that removes more heat has a higher COP; for a given cooling load, a more efficient compressor that needs less work has a higher COP. In practice, both are determined by the cycle design: the refrigerant, the evaporating and condensing temperatures, the compressor efficiency, and the heat-exchanger sizes.
The single most important factor is the temperature lift Th − Tc. Because the Carnot COP is Tc/(Th − Tc), a larger lift rapidly reduces the theoretical maximum. This is why air conditioners work better on mild days, why heat pumps lose capacity in extreme cold, and why industrial refrigeration to −40°C is far less efficient than air conditioning to +20°C. Real systems achieve 30–65% of the Carnot COP, with the fraction depending on compressor quality, heat exchanger area, and refrigerant choice.
COP Calculation Examples
COP = 10.5 / 3.0 = 3.50
Qc = 3.5 × 3.0 = 10.5 kW
W = 10.5 / 3.5 = 3.0 kW
COPCarnot = 273.15 / (318.15 − 273.15) = 273.15 / 45 = 6.07
ηII = 3.5 / 6.07 = 57.7%
Frequently Asked Questions
Because a refrigerator or heat pump does not convert work into heat — it moves heat from one place to another. The work input is only used to lift the heat "uphill" from the cold reservoir to the hot reservoir. The heat itself is supplied by the cold space, not by the work. A COP of 3.5 means that for every 1 kW of electrical power, 3.5 kW of heat is moved out of the cold space. This is entirely consistent with the first law, because the total heat rejected to the hot reservoir is Qc + W = 4.5 kW.
The Carnot COP is the maximum theoretical COP for any device operating between two temperatures: COPCarnot = Tc / (Th − Tc). It is set by the second law of thermodynamics and cannot be exceeded, no matter how perfect the engineering. Real devices achieve 30–65% of this limit. The Carnot COP is used as a benchmark: the ratio COP / COPCarnot is the second-law efficiency, which measures how close a real device is to the theoretical ideal. It tells you how much room there is for improvement.
The two are related by COPHP = COP + 1. A refrigerator is rated on the heat it removes from the cold space (Qc), so its COP is Qc/W. A heat pump is rated on the heat it delivers to the hot space (Qh), so its COP is Qh/W = (Qc + W)/W = COP + 1. The same physical device can be both: an air conditioner running in reverse becomes a heat pump, and its rated COP changes from 3.5 (cooling) to 4.5 (heating) at the same operating point. This is why heat pumps are so attractive for heating — they deliver more heat than the electrical energy they consume.
Several irreversible losses pull a real COP below the Carnot limit: (1) compressor inefficiency — real compressors have isentropic efficiencies of 60–85%, so more work is needed than the ideal cycle requires; (2) finite heat-exchanger area — the refrigerant must be colder than the cold space and hotter than the hot space to transfer heat, increasing the effective temperature lift; (3) pressure drops in pipes and valves; (4) heat gains through insulation; (5) defrost cycles, fans, and controls that consume power without contributing to cooling; (6) refrigerant properties that limit the achievable cycle. Improving any of these raises COP toward the Carnot limit.
As the outdoor temperature rises, the temperature lift (Th − Tc) increases, so the Carnot COP falls and the real COP falls with it. A typical split-system AC rated at COP 4.0 at 35°C outdoor might drop to COP 3.0 at 45°C and rise to COP 5.0 at 25°C. This is why air conditioners work harder and cost more to run on the hottest days. Inverter-driven units mitigate this by slowing the compressor at part load, but the fundamental temperature-lift penalty remains. The same effect makes heat pumps less efficient in cold weather.
A ton of refrigeration (RT) is a traditional unit of cooling capacity equal to 12,000 BTU/h, or about 3.517 kW. It originated from the rate at which one ton of ice at 0°C melts over 24 hours, absorbing heat. In modern engineering, RT is still used in the USA for chillers and large HVAC systems; most of the rest of the world uses kW. To convert: 1 RT = 3.517 kW = 12,000 BTU/h = 3,024 kcal/h. The unit converter card above performs all these conversions in one step.
Yes. Qc can be measured from the refrigerant mass flow rate and enthalpy change across the evaporator (Qc = ṁ · Δh), or from the water/air side (Qc = ṁ · cp · ΔT). W can be measured with a power meter on the compressor and any auxiliary drives. COP is then computed directly. In the field, technicians often use the compressor's power draw and the measured temperature difference across the evaporator to estimate COP without breaking into the refrigerant circuit. Standards such as AHRI 210/240 and EN 14511 define the test conditions for rated COP.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| COP | Coefficient of performance (dimensionless) | |
| Qc | Heat removed from the cold space | kJ |
| W | Work input to the cycle | kJ |
What it means
The coefficient of performance is a measure of the efficiency of a refrigeration cycle or heat pump. For a refrigerator, it is defined as the ratio of the heat extracted from the cold reservoir (Q_c) to the work input (W): COP_R = Q_c / W. This is essentially the "benefit" (cooling) divided by the "cost" (work). For a heat pump, COP_HP = Q_h / W, where Q_h is heat delivered to the hot reservoir. The COP is dimensionless and typically greater than 1 for useful devices. The Carnot COP gives the maximum possible value: COP_R, Carnot = T_c / (T_h − T_c). Real refrigerators have lower COP due to irreversibilities. The COP is an important parameter for designing and selecting refrigeration systems, air conditioners, and heat pumps. Higher COP reduces operating costs and environmental impact. The COP depends on the temperature difference and the cycle efficiency.
Worked example
COP Refrigeration – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Q_c | 300 kJ |
| W | 100 kJ |
| Parameter | Value |
|---|---|
| Q_c | 500 kJ |
| W | 150 kJ |
Common mistakes
- Qc vs. Qh: COP for refrigeration uses heat removed from the cold space (Qc), not heat rejected to the hot space.
- Work input W: This is the net work supplied to the compressor (or total work input).
- Units: Qc and W must be in the same units (Joules, kJ, etc.).
- COP > 1? Refrigerators typically have COP > 1; if you get < 1, check your inputs.
- Heat pump COP: For a heat pump, COP = Qh/W – don’t confuse the two.
Applications
The coefficient of performance (COP) for a refrigerator is the ratio of heat removed from the cold space to the work input. It is a measure of the efficiency of refrigeration and heat pump systems. A higher COP indicates better energy utilisation, which is crucial for reducing electricity consumption in residential, commercial, and industrial cooling. Engineers use COP to compare different refrigeration cycles, select components, and design systems that minimise operating costs. The COP is also applied in the design of air conditioning systems and in the evaluation of heat pump performance for both heating and cooling. By improving COP through better compressors, heat exchangers, and refrigerants, significant energy savings can be achieved.
- Refrigerator and freezer design
- Air conditioning system performance evaluation
- Heat pump design for heating and cooling
- Energy efficiency optimisation in HVAC
- Selection of refrigerants and cycle configurations
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