Owing to a faulty door contact, the small light bulb (25 W)...

Owing to a faulty door contact, the small light bulb (25 W) inside a refrigerator is kept on and limited insulation lets $50 \mathrm{W}$ of energy from the outside seep into the refrigerated space. How much of a temperature difference to the ambient surroundings at $20^{\circ} \mathrm{C}$ must the refrigerator have in its heat exchanger with an area of $1 \mathrm{m}^{2}$ and an average heat transfer coefficient of $15 \mathrm{W} / \mathrm{m}^{2} \mathrm{K}$ to reject the leaks of energy?

Key Concepts

Heat Transfer Coefficient

The heat transfer coefficient defines how efficiently heat is transferred per unit area per unit temperature difference. It is a key parameter in determining the performance of a heat exchanger, dictating the required temperature gradient to reject a certain amount of heat.

Heat Exchanger Design

In refrigeration systems, the heat exchanger is designed to remove excess heat from the refrigerated space. Its performance is characterized by its surface area and overall heat transfer properties, which together determine the necessary temperature difference between the exchanger and the ambient environment to achieve the desired heat rejection.

Energy Balance in Refrigeration Systems

Refrigeration involves maintaining a lower internal temperature by balancing the heat gained (from internal sources like lights or external leaks) with the heat rejected through the heat exchanger. An accurate energy balance is critical, as it ensures that all incoming heat is effectively removed to maintain the desired temperature.

Newton's Law of Cooling

This law states that the rate of heat transfer from a surface is proportional to the difference between the surface temperature and the ambient temperature. In engineering applications, this relationship is used to calculate how much of a temperature difference is needed to reject a specific amount of heat in a device such as a refrigerator.

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