Question

Which of the following relations is true, for coefficient of performance (C.O.P)?
A. $\left( {C.O.P} \right){\text{ }}heat{\text{ }}pump{\text{ }}-{\text{ }}\left( {C.O.P} \right){\text{ }}refrigeration{\text{ }} = {\text{ }}1$
B. $\left( {C.O.P} \right){\text{ }}heat{\text{ }}pump{\text{ }}-{\text{ }}\left( {C.O.P} \right){\text{ }}refrigeration{\text{ > }}1$
C. $\left( {C.O.P} \right){\text{ }}heat{\text{ }}pump{\text{ }}-{\text{ }}\left( {C.O.P} \right){\text{ }}refrigeration{\text{ < }}1$
D. $\left( {C.O.P} \right){\text{ }}heat{\text{ }}pump{\text{ }}-{\text{ }}\left( {C.O.P} \right){\text{ }}refrigeration{\text{ }} = {\text{ 0}}$

Answer 1

First of all, we will find the coefficient of performance of the refrigeration. Then we will find the coefficient of performance of the heat pump. After that we will compare these two quantities.

Complete step by step answer:
In terms of the output coefficient (COP), denoted by COPR, the efficiency of a refrigerator is expressed. A refrigerator's purpose is to extract heat (${Q_L}$) from the refrigerated room. It needs a $W\left( {net,\,in} \right)$ , job input to achieve this aim. Then a refrigerator's COP can be expressed as.
$COP{\text{ }}refrigerator = {\text{ }}\left( {desired{\text{ }}output} \right)/{\text{ }}\left( {required{\text{ }}input} \right) = QLW\left( {net,{\text{ }}in} \right)$ …… (1)
So, the conservation of energy principle for a cyclic device requires that
$W\left( {net,\,in} \right) = {Q_H} - {Q_L}\left( {KJ} \right)$ …… (2)
Then the COP relation will be,
COP refrigeration, $= \dfrac{{{Q_L}}}{{\left( {{Q_H} - {Q_L}} \right)}} = \dfrac{1}{{{Q_H}/{Q_L} - 1}}$ …… (3)
We notice that the value of COPR can be greater than unity.

The heat pump is another system that moves heat from a low-temperature medium to a high temperature one. In the same cycle, refrigerators and heat pumps work but vary in their goals. However, the purpose of a heat pump is to maintain a heated room at a high temperature. This is achieved by extracting heat from a source of low temperatures, such as water or cold outside air in winter, and supplying this heat to the medium of high temperatures, such as a home. The coefficient of performance COP heat pump, defined as
$COP{\text{ }}heat{\text{ }}pump{\text{ }} = {\text{ }}\left( {Desired{\text{ }}output} \right)/\left( {Required{\text{ }}input} \right) = QHW\left( {net,{\text{ }}in} \right)$…… (4)

The expression will be,
COP heat pump $= \dfrac{{{Q_H}}}{{\left( {{Q_H} - {Q_L}} \right)}} = \dfrac{1}{{\left( {1 - {Q_L}/{Q_H}} \right)}}$ …… (5)
From equation (3) and (5),
$COP{\text{ }}heat{\text{ }}pump = COP{\text{ }}refrigerator{\text{ }} + {\text{ }}1$
$COP{\text{ }}heat{\text{ }}pump - COP{\text{ }}refrigerator{\text{ }} = {\text{ }}1$
For ${Q_L}$ and ${Q_H}$ fixed values. This relationship means that since COPR is a positive quantity, the coefficient of output of a heat pump is always greater than unity.
Hence, the required answer is$\left( {C.O.P} \right){\text{ }}heat{\text{ }}pump{\text{ }}-{\text{ }}\left( {C.O.P} \right){\text{ }}refrigeration{\text{ }} = {\text{ }}1$

The correct option is A.

Additional information:
Coefficient of performance: A heat pump, refrigerator or air conditioning system's efficiency coefficient or COP (sometimes CP or COP) is a ratio of usable heating or cooling offered for the work required. Higher COPs are equal to lower operating costs. The COP typically exceeds $1$ , especially in heat pumps, since it pumps additional heat from a heat source to where the heat is needed instead of only converting work to heat (which, if $100$ percent efficient, would be a COP of $1$ ). COP calculations for complete systems should involve the energy usage of all auxiliary power consuming devices.

Note: The most fundamental energy-efficiency metric of any heat engine is COP or Output Coefficient, when comparing heat pumps, fridges, and air conditioners, it's very helpful. It is important to remember that the coefficient of performance of the heat pump is always greater than the coefficient of performance of the refrigeration.