Question

A refrigerator is removing heat from a cold medium at \(3^{\circ} \mathrm{C}\) at a rate of \(7200 \mathrm{kJ} / \mathrm{h}\) and rejecting the waste heat to a medium at \(30^{\circ} \mathrm{C}\). If the coefficient of performance of the refrigerator is \(2,\) the power consumed by the refrigerator is \((a) 0.1 \mathrm{kW}\) (b) \(0.5 \mathrm{kW}\) \((c) 1.0 \mathrm{kW}\) \((d) 2.0 \mathrm{kW}\) \((e) 5.0 \mathrm{kW}\)

Short answer

Answer: The power consumed by the refrigerator is 1.0 kW.

Step-by-step solution

Write down the definition of the coefficient of performance (COP)

The COP is a measure of the efficiency of a refrigerator, defined as the ratio of the heat removed from the cold medium (Q_C) to the work input (W) required to remove that heat. Mathematically, it's written as: COP = \frac{Q_C}{W}

Use the given COP to rearrange the equation for the work input

We're given a COP of 2. Let's manipulate the equation to find the work input W: W = \frac{Q_C}{\text{COP}}

Convert the heat transfer rate to the appropriate units

We know that the heat is removed at a rate of 7200 kJ/h. First, we should convert the rate to kJ/s. There are 3600 seconds in an hour, so Q_C = \frac{7200 \text{ kJ}}{3600 \text{ s}} = 2 \text{ kJ/s}

Calculate the power consumed by the refrigerator

Now, we can substitute the values of Q_C and the COP into the equation from step 2 to find the work input (power consumed): W = \frac{2 \text{ kJ/s}}{2} = 1 \text{ kJ/s} Since 1 kJ/s is equivalent to 1 kW, the power consumed by the refrigerator is 1.0 kW, which corresponds to answer choice (c).

Key concepts

Refrigeration Cycle

Understanding the refrigeration cycle is fundamental when analyzing the workings of refrigerators, air conditioners, and any appliance that is used to cool a space or substance. The refrigeration cycle is a series of steps that involve manipulating a refrigerant to absorb heat from a cooled environment and then release it to the surroundings.

The process consists primarily of four stages: evaporation, compression, condensation, and expansion. During the evaporation stage, the refrigerant absorbs heat from the inside of a refrigerator and evaporates at a low pressure. This results in cooling the inside area. Then, the vapor is compressed, which increases its temperature and pressure. Following compression, the high-pressure hot vapor moves through a condenser where it releases heat to the outside environment and condenses into a liquid. Finally, the liquid passes through an expansion device that reduces its pressure and temperature, and the cycle begins again.

These mechanisms rely on the principles of heat transfer and thermodynamics, and the refrigeration cycle is designed to maximize energy efficiency through careful control of the refrigerant's physical states and thermodynamic properties.

Energy Efficiency

Energy efficiency in the context of refrigeration is a measure of how effectively a refrigerator uses electricity to remove heat from its interior. A more energy-efficient refrigerator consumes less power to perform the same amount of cooling compared to a less efficient one.

One way to quantify energy efficiency is through the coefficient of performance (COP). The COP is the ratio of the heat removed from a refrigerator's interior (useful cooling provided) to the electrical energy (work) it consumes to drive the cycle. A higher COP value indicates a more energy-efficient system. Energy efficiency is not just beneficial for reducing operational costs; it is also critical for reducing the environmental impact by lowering greenhouse gas emissions associated with electricity production.

Factors affecting the energy efficiency of a refrigeration cycle include the type and quality of insulation, the efficiency of the compressor, the properties of the refrigerant, and the design of the heat exchangers. Optimizing these factors can lead to significant energy savings.

Heat Transfer

Heat transfer is the process of thermal energy moving from a hotter object to a colder one. In the context of refrigeration, heat transfer occurs at both the internal and external interfaces of the refrigerator. Internally, heat is absorbed from the food compartments and transferred to the refrigerant circulating within the refrigerator. Externally, heat is released from the refrigerant to the environment when it passes through the condenser coils.

There are three fundamental modes of heat transfer: conduction, convection, and radiation. In a standard refrigerator, conduction occurs within the walls and shelves, convection occurs as the air inside circulates over these surfaces, and radiation can happen with the heat exchange between the coils and the surrounding air.

Improving heat transfer efficiency can also improve the overall energy efficiency of a refrigeration system. This can be achieved through the use of better heat-conducting materials, increasing surface area for heat exchange, and ensuring proper airflow around the condenser coils to disperse heat more effectively.

Thermodynamic Principles

Thermodynamics is a branch of physics concerned with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. The thermodynamic principles govern the operation of refrigeration cycles and determine their performance and efficiency.

The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. The energy going into a refrigerator (electrical work) must equal the sum of the energy removed (cooling effect) and the energy losses like waste heat. The second law of thermodynamics introduces the concept of entropy and states that in all energy exchanges, if no energy enters or leaves the system, the potential energy of the state will always be less than that of the initial state. This principle dictates that some energy will be lost as waste heat in any real system, making 100% efficiency impossible.

Understanding these principles helps in identifying the optimal conditions for the refrigeration cycle to obtain maximum efficiency. It explains why a refrigerator is more efficient if the difference between the temperature of the refrigerated space and the outside environment is minimized, and why refrigerators must release waste heat to function properly.