The walls of a refrigerator are typically constructed by...

The walls of a refrigerator are typically constructed by sandwiching a layer of insulation between sheet metal panels. Consider a wall made from fiberglass insulation of thermal conductivity $k_{i}=0.046 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and thickness $L_{i}=50 \mathrm{~mm}$ and steel panels, each of thermal conductivity $k_{p}=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and thickness $L_{p}=3 \mathrm{~mm}$. If the wall separates refrigerated air at $T_{\infty, i}=4^{\circ} \mathrm{C}$ from ambient air at $T_{\infty, o}=25^{\circ} \mathrm{C}$, what is the heat gain per unit surface area? Coefficients associated with natural convection at the inner and outer surfaces may be approximated as $h_{i}=h_{o}=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.

Key Concepts

Composite Wall Analysis

Composite wall analysis deals with the evaluation of heat transfer through structures composed of multiple layers with different thermal properties. By considering each layer's contribution to the overall thermal resistance, including both conduction through the materials and convection at the surfaces, this approach allows for determining the net heat transfer rate across the entire wall.

Thermal Circuit Analysis

Thermal circuit analysis models the heat transfer process as a network of resistances, analogous to electrical circuits. Each layer’s resistance, whether due to conduction or convection, is considered in series or parallel, enabling a systematic computation of the overall temperature difference and heat flow through the composite system.

Convection Heat Transfer

Convection involves the transfer of heat between a solid surface and a moving fluid, such as air, where heat is carried away by the fluid motion. It is characterized by a convective heat transfer coefficient, which, together with the surface temperature difference, determines the rate at which heat is exchanged between the solid surfaces and the surrounding fluid.

Thermal Resistance

Thermal resistance quantifies how strongly a material opposes the flow of heat. It depends on the material’s thickness and thermal conductivity, typically calculated by the relation R = L/k for conduction. This concept is crucial when analyzing the heat transfer through combined layers, as it allows the addition of resistances from different materials to determine the overall heat transfer rate.

Conduction Heat Transfer

Conduction is the process by which heat energy is transmitted through a material due to a temperature gradient. Governed by Fourier’s law, the rate of heat transfer depends on the material’s thermal conductivity, the temperature difference over its length, and its cross-sectional area, making it a key mechanism when considering heat transfer in solid layers.

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