A commercial grade cubical freezer, 3 m on a side, has a composite wall consisting of an exteri

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A commercial grade cubical freezer, 3 m on a side, has a...

A commercial grade cubical freezer, $3 \mathrm{~m}$ on a side, has a composite wall consisting of an exterior sheet of $6.35-\mathrm{mm}$-thick plain carbon steel, an intermediate layer of $100-\mathrm{mm}$-thick cork insulation, and an inner sheet of $6.35$-mm-thick aluminum alloy (2024). Adhesive interfaces between the insulation and the metallic strips are each characterized by a thermal contact resistance of $R_{t, c}^{\prime \prime}=2.5 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$. What is the steady-state cooling load that must be maintained by the refrigerator under conditions for which the outer and inner surface temperatures are $22^{\circ} \mathrm{C}$ and $-6^{\circ} \mathrm{C}$, respectively?

Key Concepts

Steady-State Heat Transfer

Steady-state heat transfer assumes that the temperature distribution within a system does not change with time, meaning that the heat entering any part of the system equals the heat leaving it. This assumption simplifies the analysis by allowing the use of constant thermal resistances and leads to a reliable calculation of the continuous loading on a cooling system in maintaining a set temperature difference.

Thermal Contact Resistance

Thermal contact resistance arises at the interface between two materials, due to imperfections or gaps in contact, and acts as an additional barrier to heat flow. This resistance must be taken into account in composite systems where different materials are bonded together, as it can significantly affect the total thermal resistance and hence the performance of the insulation.

Cooling Load

The cooling load is the amount of heat energy that must be removed from a space or system to maintain a desired internal temperature. In the context of conduction through composite walls, the cooling load is determined by the overall temperature difference across the wall and the total thermal resistance of the materials and interfaces involved.

Composite Wall Analysis

Composite wall analysis involves studying heat transfer through a multilayered structure. In such an analysis, each layer possesses its own material properties (like thickness and thermal conductivity), and the overall heat transfer is determined by treating the layers as thermal resistances connected in series. This allows for calculation of the temperature drop across each layer and the system as a whole.

Thermal Conduction

Thermal conduction is the mode of heat transfer through a material resulting from a temperature gradient. It is described by Fourier’s law, which relates the heat flux to the material's thermal conductivity and the temperature gradient across its thickness. This concept is fundamental when analyzing how heat moves through solid layers in a steady state.

Thermal Resistance

Thermal resistance quantifies the opposition to heat flow through a material and is calculated as the ratio of the material's thickness to its thermal conductivity. In series thermal systems, these resistances add together, providing a straightforward way to calculate the overall heat transfer rate given a temperature difference, which is crucial for designing and analyzing insulated systems.

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