Question

In order to size the compressor of a new refrigerator, it is desired to determine the rate of heat transfer from the kitchen air into the refrigerated space through the walls, door, and the top and bottom section of the refrigerator. In your analysis, would you treat this as a transient or steady- state heat transfer problem? Also, would you consider the heat transfer to be one-dimensional or multidimensional? Explain.

Short answer

And should it be treated as one-dimensional or multidimensional? Answer: The heat transfer should be considered as a steady-state heat transfer problem and should be treated as multidimensional.

Step-by-step solution

1. Transient vs Steady-State Heat Transfer Problem

A transient heat transfer problem occurs when the temperature distribution in an object changes over time, whereas a steady-state heat transfer problem is when the temperature distribution does not change with time. In the case of the refrigerator, once its temperature reaches the desired level, it maintains a constant temperature inside. The refrigerator runs periodically to maintain this temperature, and the temperature changes very slowly. Therefore, it is more appropriate to consider this as a steady-state heat transfer problem.

2. One-dimensional vs Multidimensional Heat Transfer

A one-dimensional heat transfer problem assumes that the heat transfer occurs predominantly in one direction, while a multidimensional heat transfer problem considers heat transfer in more than one direction. In the case of the refrigerator, heat transfer occurs through the walls, door, top, and bottom section of the refrigerator. The heat transfer is predominantly one-dimensional through the walls, namely in the direction from the outer surface to the inner surface; however, since heat transfer happens from multiple surfaces (door, top, and bottom sections), the overall heat transfer should be considered as multidimensional. In conclusion, for sizing the compressor of the refrigerator, it is useful to treat this problem as a steady-state, multidimensional heat transfer problem. The analysis should focus on determining the overall rate of heat transfer through each section (walls, door, top, and bottom) to estimate the cooling demand and thus the size of the compressor.

Key concepts

Steady-State Heat Transfer

When engineering a refrigerator's compressor, understanding how it interacts with the surrounding environment is crucial. Steady-state heat transfer comes into play when the temperature within the refrigerated space remains constant over time. Once a refrigerator reaches its set operating temperature, its primary job is to maintain this temperature.

This is achieved by the compressor periodically activating to counteract any heat that has entered the space. In the context of the exercise, treating the situation as a steady-state problem simplifies the calculation as it assumes the internal temperature of the refrigerator does not change over time, despite the intermittent operation of the compressor.

The steadiness allows for a fixed rate of heat transfer to be considered when sizing the compressor, making it a more predictable and manageable calculation for engineers.

Transient Heat Transfer

Contrary to steady-state, transient heat transfer occurs when temperatures within a system change over time. It's the phase a new refrigerator experiences when it's first turned on and starts cooling down from room temperature to its set point.

The reason why the steady-state assumption is more appropriate for sizing the compressor in this case is that the transient phase is temporary, whereas the refrigerator will spend most of its operational life maintaining a constant internal temperature, a scenario best described by steady-state conditions.

However, understanding transient heat transfer is still important, especially during the initial design and testing phase of the refrigerator, where the appliance's ability to reach the desired temperature from a non-operational state is critical for performance assessment.

Multidimensional Heat Transfer

In real-world scenarios, heat transfer often involves more than one dimension. For a refrigerator, heat can penetrate its insulated walls from all sides, including the door, and top and bottom sections—each contributing to the total heat entering the refrigerated space.

While it might be tempting to simplify calculations by considering heat transfer to be one-dimensional, doing so could lead to an underestimation of the heat load and consequently, an undersized compressor. Multidimensional thermal analysis ensures a comprehensive understanding of how heat moves through varied paths and angles, considering conduction, convection, and even radiation that can occur in different materials used in the refrigerator's construction.

Addressing multidimensional heat transfer head-on ensures that all factors contributing to the thermal load are accounted for appropriately, leading to a properly sized compressor that can efficiently maintain the desired temperatures within the refrigerator.

Thermal Analysis

Thermal analysis involves examining how heat transfer affects an object—such as a refrigerator—under different operating conditions. This encompasses both steady-state and transient phenomena, and in both one-dimensional and multidimensional contexts.

Performing a thermal analysis for a refrigerator, one would need to assess all the routes through which heat can enter the system. This means evaluating the insulating properties of the walls, door, and other sections, as well as the potential hot spots where seals might not be as effective.

Through such a rigorous analysis, engineers can predict how external temperatures and conditions will affect the internal environment of the refrigerator, which is invaluable for determining the required performance and size of the compressor. By simulating various scenarios and conditions, a thermal analysis helps in designing a refrigerator that operates efficiently under the expected range of environmental temperatures.

Compressor Sizing

Sizing the compressor is the final step in ensuring a refrigerator meets its cooling needs. It involves taking all the information gained from steady-state and multidimensional heat transfer analyses and translating it into the physical requirements for the compressor.

The capacity of the compressor must be sized to offset the total calculated heat load entering the refrigerated space. This ensures that the interior temperature remains steady at the set value. An optimal compressor size is both energy-efficient — not working harder or more often than necessary — and effective, preventing temperature fluctuations that could compromise food preservation.

Engineers must also consider other factors such as the operational duty cycle, the expected ambient temperature range, and even the opening and closing frequency of the refrigerator door. Compressor sizing is a balance between robustness and efficiency, aiming to provide the necessary cooling power without incurring excessive energy costs.