Coefficient of Performance Formula, Definition, Solved Examples

Coefficient of Performance Formula: The purchase of any appliance is influenced by its efficiency, and people tend to choose products based on this factor. The coefficient of performance (COP) plays a pivotal role in determining the efficiency of appliances like air conditioners, refrigerators, and heat pumps. When selecting these items at a store, the initial step is often to examine the COP value. Now, let's discuss the formula for the Coefficient of Performance (COP).

Coefficient of Performance Formula

The Coefficient of Performance (COP) is determined by the ratio of heat dissipated to electrical power consumed. This parameter serves as a crucial metric for evaluating the performance of an air conditioning unit. The COP essentially illustrates the effectiveness of heating and cooling systems, with the following points outlining its characteristics:

The Coefficient of Performance (COP) is defined as the ratio of heat dissipation to electrical power intake, serving as a critical parameter for evaluating the performance of air conditioning units.

It reflects the efficiency of heating and cooling systems, and specific characteristics include:

• The COP is the ratio of compressor power input to refrigeration output.
• An increase in COP indicates a proportional rise in cooling output relative to the power input.
• Higher COP values equate to greater efficiency in the system.
• As COP rises, efficiency increases, and conversely, as COP falls, efficiency diminishes.
• It specifically refers to the ratio of evaporator cooling output to the work input of the air conditioner or refrigerator.
• Higher COP values are associated with higher power consumption and running costs.

The formula for calculating the Coefficient of Performance (COP) is expressed as follows

Coefficient of Performance Formula COP= Output ​ / Input

For heating, the Coefficient of Performance is expressed as:

K= Q _H ​ / W _in

Where:  K = Coefficient of Performance, Q _H = Heat pump's output,

Win = Work required by the considered system.

For cooling, the Coefficient of Performance is given by:

K= Q _C ​/W _in

Where:  K = Coefficient of Performance, Q _C = Heat dissipated by the refrigerator, W _in = Work required by the considered system.

If the work input is zero, the Coefficient of Performance becomes infinite. The Coefficient of Performance for a refrigerator is given by:

K _R ​ = Q ₂ ​ ​ / (Q ₁ ​ –Q ₂ ​ )

For a reversible refrigerator, where ​ Q ₁ ​/ Q ₂ ​ = T ₁ / T ₂ ​ ​ ​ , the formula simplifies to:

Coefficient of Performance Formula K _R ​ = T ₂ ​ ​/ (T ₁ ​ –T ₂ ​ )

The Coefficient of Performance for a heat pump is expressed as:

K _H ​ = Q ₁ ​/ (Q ₁ ​ –Q ₂ ​ ) ​

In the case of a reversible heat pump,

Where ​ Q ₁ /Q ₂ ​ ​ = ​ T ₁ /T _2​ ​ , the formula can be written as:

K _H ​ =  T ₁ ​/ (T ₁ ​ –T ₂ ​ ) ​

Coefficient of Performance Formula Solved Examples

Example 1: An air conditioning system operates between a high temperature of 300 K and a low temperature of 270 K. Calculate the Coefficient of Performance (COP) for the air conditioner.

Solution: Given: T ₁ ​ =300K (high temperature), T ₂ ​ =270K (low temperature)

Using the Coefficient of Performance Formula K _R ​ = T ₂ / (T ₁ ​ −T ₂ ​ ) ​ ​ ,

K _R ​ = 270 ​/ (300−270)

K _R ​ = 270 ​ / 30

K _R ​ =9 Therefore, the Coefficient of Performance (COP) for the air conditioner is 9.

Example 2: A heat exchanger functions between a high-temperature reservoir at 450 K and a low-temperature reservoir at 300 K. With 150 kJ of heat absorbed from the hot reservoir and 80 kJ rejected to the cold reservoir, calculate the Coefficient of Performance (COP) for this heat exchanger.

Solution: Given: T ₁ ​ =450K (hot reservoir), T ₂ ​ =300K (cold reservoir), Q ₁ ​ =150kJ (heat absorbed), Q 2 ​ =80kJ (heat rejected) Using the Coefficient of Performance Formula K _H ​ = Q 1/ (Q ₁ ​ −Q ₂ ​ ) ​ ​ ,

K _H ​ = 150 ​ / (150−80)

K _H ​ = 150 / 70 ​

K _H ​ =2.14

Therefore, the Coefficient of Performance (COP) for the heat exchanger is 2.14.

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