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Chapter 19: Problem 9

Why do refrigerators and heat pumps have different definitions of COP?

Short Answer

Expert verified
The COP definitions of refrigerators and heat pumps differ because their operational goals are different: a refrigerator's goal is to absorb heat from the space to be cooled, while a heat pump's goal is to supply heat to the space to be heated. Thus, the COP of a refrigerator is defined as the ratio of the heat absorbed from the space being cooled to the work done, while the COP of a heat pump is defined as the ratio of the heat supplied to the space being heated to the work done.

Step by step solution

01

Understanding the operation of a refrigerator

A refrigerator operates by removing heat from the space to be cooled (inside the refrigerator) and expelling it to the surroundings (usually the kitchen). Therefore, its efficiency or COP (denoted by \(COP_R\)) is defined as the amount of heat removed from the refrigerator per unit work done, or \(COP_R = Q_{in}/W = T_{low}/(T_{high}-T_{low})\). Here, \(Q_{in}\) is the heat absorbed from the inside of the refrigerator, \(W\) is the work done on the system, and \(T_{high}\) and \(T_{low}\) represent the high (outside) and low (inside) temperatures respectively.
02

Understanding the operation of a heat pump

A heat pump operates by extracting heat from a source and transferring it to a space that is to be heated. Thus, its efficiency or COP (denoted by \(COP_{HP}\)) is defined as the amount of heat expelled to the space to be heated or cooling unit per unit work done, or \(COP_{HP} = Q_{out}/W = T_{high}/(T_{high}-T_{low})\). Here, \(Q_{out}\) is the heat expelled from the heat pump, and the other symbols have the same meanings as in the previous step.
03

Understanding differences between the COP's of a refrigerator and heat pump

The contrasting definitions of COP arise due to the differing objectives of a refrigerator and a heat pump. The goal of a refrigerator is to maintain a space at a cooler temperature than the surroundings, thus its COP is defined in terms of the heat taken in from that space. On the other hand, a heat pump is used to supply heat to a space that is to be maintained at a warmer temperature than its surroundings, thus its COP is defined in terms of the heat supplied to that space. Therefore, even though refrigerators and heat pumps operate on the same principle of heat transfer, their COP definitions differ due to their different operational goals.

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Most popular questions from this chapter

Refrigerators remain among the greatest consumers of electrical energy in most homes, although mandated efficiency standards have decreased their energy consumption by some \(80 \%\) in the past four decades. In the course of a day, one kitchen refrigerator removes \(30 \mathrm{MJ}\) of energy from its contents, in the process consuming \(10 \mathrm{MJ}\) of electrical energy. The electricity comes from a \(40 \%\) efficient coal-fired power plant. The electrical energy a. is used to run the light bulb inside the refrigerator. b. wouldn't be necessary if the refrigerator had enough insulation. c. retains its high-quality status after the refrigerator has used it. d. ends up as waste heat rejected to the kitchen environment.

Why doesn't the evolution of human civilization violate the second law of thermodynamics?

Problem 74 of Chapter 16 provided an approximate expression for the specific heat of copper at low absolute temperatures: \(c=31(T / 343 \mathrm{K})^{3} \mathrm{J} / \mathrm{kg} \cdot \mathrm{K} .\) Use this to find the entropy change when \(40 \mathrm{g}\) of copper are cooled from \(25 \mathrm{K}\) to \(10 \mathrm{K}\). Why is the change negative?

Consider a gas containing an even number \(N\) of molecules, distributed among the two halves of a closed box. Find expressions for (a) the total number of microstates and (b) the number of microstates with half the molecules on each side of the box. (You can either work out a formula, or explore the term "combinations" in a math reference source.) (c) Use these results to find the ratio of the probability that all the molecules will be found on one side of the box to the probability that there will be equal numbers on both sides. (d) Evaluate for \(N=4\) and \(N=100\).

Use appropriate energy-flow diagrams to analyze the situation in Got It? \(19.2 ;\) that is, show that using a refrigerator to cool the lowtemperature reservoir can't increase the overall efficiency of a Carnot engine when the work input to the refrigerator is included.
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