If you've ever wondered what it means to say one electrical appliance works better than another one does, there is actually a way that these devices are measured for their efficiency and effectiveness. The **coefficient of performance** formula explains the usage of the word "better" when talking about how appliances and other devices perform.

## Coefficient of Performance Formula

You can calculate the coefficient of performance by dividing how much energy a system produces by the amount of energy you input into the system. This coefficient of performance formula applies across fields. This formula is very similar to the **formula of efficiency**, which is work a system outputs divided by the work put into the system, letting you easily compare coefficient of performance vs. efficiency.

Because work is the transfer of energy from one place and form to another, if you can represent the change in energy of a system using work, the two formula are equivalent.

Coefficient of performance example problems demonstrate how useful it can be. If you were using four tons of water to heat a closed ground loop of a geothermal heat pump that produces 35,600 Btu/hr (British thermal units per hour) while consuming 2,700 watts of power, you can calculate the coefficient of performance.

Converting the Btu/hr units to watts, a measure of power, you can follow the manual for a geothermal heat pump or find the conversion online. One Btu/hr is equivalent to 0.293 watts.

This means that 35,900 Btu/hr is roughly 10,518 watts. Though power represents energy divided by time, you can assume the time to input the energy and output it are the same for this problem. Dividing 10,518 by 2,700 as shown by the coefficient of performance formula, you get 3.89. For each watt of power or joule of energy input into the system, the pump produces 3.89 watts of power or joules of energy.

Through examples such as this one, you can compare the coefficient of performance across systems and even across fields. This lets engineers compare the efficiency of different systems such as the comparisons between hybrid cars and regular or electric cars.

## Coefficient of Performance Refrigeration Example

The coefficient of performance can take many forms that are unique to or inherently based on the principles of specific disciplines. The effectiveness of refrigerators or air conditioners represents one way of comparing coefficient of performance as *Q _{C}/W_{in}* for

*Q*the heat the refrigerator gives off

_{C}*Q*and the work input to the system

_{C}*W*. This gives you a method of comparing refrigerators when you want to save money or energy for specific purposes.

_{in}Scientists and engineers study the chemical substances used in refrigerators for cooling, known as refrigerants, to figure out how to make the most energy-efficient appliances they can. Using a refrigerator and heat pump, you can figure out as a refrigerant's coefficient of performance.

You can use calculations that measure the heat given off by the parts of a refrigerator like the evaporator (which serves as a cold reservoir of water) and the condenser (a hot reservoir). It also involves the pressure given off by the heat exchange in which ammonia is compressed as it changes from gas to liquid.

Dividing heat extracted from the evaporator by the work done by the compressor gives you the coefficient of performance for the refrigerator. You can also divide the heat transferred from the condenser by the work done by the compressor to get the heat pump's coefficient of performance.

The specific formula for refrigerators also relates to the **Carnot coefficient of performance**, which should equal the maximum coefficient of performance for a refrigerator. It's given by *T _{C}/(T_{H}-T_{C})* for the

*T*temperature of the cold reservoir, the evaporator, and

_{C}*T*as the measure of the hot one, the condenser.

_{H}References

- Hyperphysics: Coefficient of Performance
- Nordic Geothermal Heat Pump: How to Calculate Coefficient of Performance
- Energy Education: Coefficient of Performance
- AccountingTools: The formula for calculating efficiency
- Lumen: Kinetic Energy and Work-Energy Theorem
- ScienceDirect: Coefficient of Performance

About the Author

S. Hussain Ather is a Master's student in Science Communications the University of California, Santa Cruz. After studying physics and philosophy as an undergraduate at Indiana University-Bloomington, he worked as a scientist at the National Institutes of Health for two years. He primarily performs research in and write about neuroscience and philosophy, however, his interests span ethics, policy, and other areas relevant to science.