Question

Explain how a household refrigerator can be viewed as a heat pump that heats the kitchen. If you knew the refrigerator's coefficient of performance, could you calculate its coefficient of performance when viewed as a heat pump?

Short answer

Yes, the COP of a refrigerator viewed as a heat pump is given by \( COP_{heat \, pump} = COP_{refrigerator} + 1 \).

Step-by-step solution

Understand the Concept

A household refrigerator can be viewed as a heat pump because it transfers heat from the cooler interior to the warmer kitchen environment. Both devices operate on the same principle of moving heat against its natural direction of flow, using mechanical work.

Define Coefficient of Performance (COP)

The Coefficient of Performance (COP) for a refrigerator is defined as the ratio of the heat removed from the cold space (Q_c) to the work input (W): \[ COP_{refrigerator} = \frac{Q_c}{W} \]

Heat Pump COP

For a heat pump, the COP is defined as the ratio of the heat delivered to the warm space (Q_h) to the work input (W): \[ COP_{heat \, pump} = \frac{Q_h}{W} \]

Relate Q_h and Q_c

In both cases, the work done (W) and the heat transferred have a relationship: \[ Q_h = Q_c + W \]

Express Heat Pump COP in Terms of Refrigerator COP

Substitute the relationship between Q_h and Q_c into the COP equation for the heat pump: \[ COP_{heat \, pump} = \frac{Q_c + W}{W} = \frac{Q_c}{W} + \frac{W}{W} = COP_{refrigerator} + 1 \]

Final Expression

This shows that the COP of the refrigerator, when viewed as a heat pump, can be calculated by adding 1 to the COP of the refrigerator: \[ COP_{heat \, pump} = COP_{refrigerator} + 1 \]

Key concepts

Heat Pump

A heat pump is a device that transfers heat from a colder area to a warmer area. This process goes against the natural direction of heat flow. In a heat pump, mechanical work is used to move heat.
Heat pumps have versatile applications:

• Heating homes during winter by extracting outdoor heat.
• Cooling homes during summer by transferring indoor heat outside.

The efficiency of a heat pump is measured by its Coefficient of Performance (COP).
The COP is calculated as \(\text{COP} = \frac{Q_h}{W}\), where \Q_h\ is the heat delivered to the warmer space and \W\ is the work input. A higher COP means the heat pump is more efficient.

Refrigerator

A refrigerator is essentially a heat pump with a specific focus. It moves heat from the inside of the fridge (cold space) to the room (warm space).
The refrigerator’s efficiency is measured by its Coefficient of Performance (COP) which is calculated as: \(\text{COP}_{refrigerator} = \frac{Q_c}{W}\), where \Q_c\ is the heat extracted from the fridge interior and \W\ is the work input. This means a higher COP indicates a more efficient refrigerator.
Refrigerators keep food and drinks cold by constantly removing heat from the interior, and this heat is then expelled to the kitchen.

Mechanical Work

Mechanical work in the context of a heat pump or refrigerator is the energy required to transfer heat. This work is usually provided by an electric motor powering a compressor.
Here’s how it works:

• The compressor compresses the refrigerant, increasing its pressure and temperature.
• The hot refrigerant gas releases heat to the surrounding area in the condenser.
• The refrigerant then cools down and travels to the evaporator, absorbing heat from the fridge interior.

This cycle repeats, and the work done in compressing and moving the refrigerant is what keeps the interiors cool or warm, depending on the device.

Heat Transfer

Heat transfer is the process of moving heat from one place to another. There are three main methods: conduction, convection, and radiation. In heat pumps and refrigerators, convection and conduction are primarily used.
How heat transfer works in these devices:

• In heat pumps, the refrigerant absorbs heat from the cooler environment during evaporation and releases it into the warmer area during condensation.
• In refrigerators, the refrigerant absorbs heat from inside the fridge during evaporation and expels it to the kitchen during condensation.

The efficiency of heat transfer affects the overall performance of the device. Efficient heat transfer ensures that less work is needed to move a given amount of heat, thus lowering energy consumption.