The walls of a food storage facility are made of a 2...

The walls of a food storage facility are made of a 2 -cm-thick layer of wood $(k=0.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ in contact with a 5 -cm-thick layer of polyurethane foam $(k=0.03 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. If the temperature of the surface of the wood is $-10^{\circ} \mathrm{C}$ and the temperature of the surface of the polyurethane foam is $20^{\circ} \mathrm{C}$, the temperature of the surface where the two layers are in contact is (a) $-7^{\circ} \mathrm{C}$ (b) $-2^{\circ} \mathrm{C}$ (c) $3^{\circ} \mathrm{C}$ (d) $8^{\circ} \mathrm{C}$ (e) $11^{\circ} \mathrm{C}$

The walls of a food storage facility are made of a 2 -cm-thick layer of wood $(k=0.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ in contact with a 5 -cm-thick layer of polyurethane foam $(k=0.03 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. If the temperature of the surface of the wood is $-10^{\circ} \mathrm{C}$ and the temperature of the surface of the polyurethane foam is $20^{\circ} \mathrm{C}$, the temperature of the surface where the two layers are in contact is
(a) $-7^{\circ} \mathrm{C}$ (b) $-2^{\circ} \mathrm{C}$
(c) $3^{\circ} \mathrm{C}$
(d) $8^{\circ} \mathrm{C}$
(e) $11^{\circ} \mathrm{C}$

Key Concepts

Fourier's Law of Heat Conduction

Fourier's law states that the heat flux through a material is proportional to the negative of the temperature gradient and the material's thermal conductivity. It forms the fundamental basis for analyzing heat transfer in materials and is essential for understanding how temperature differences drive heat flow in solids.

Thermal Conductivity

Thermal conductivity is a material property that measures the ability to conduct heat. It indicates how quickly heat energy is transmitted through a substance. Different materials have different thermal conductivities, and these differences play a crucial role in determining how temperature gradients develop across composite structures.

Thermal Resistance

Thermal resistance is a concept used to quantify the difficulty that heat encounters as it flows through a material. It is analogous to electrical resistance in circuits, with higher thermal resistance indicating lower heat flow for a given temperature difference. For a uniform material layer, thermal resistance is calculated by dividing the thickness of the layer by the product of its thermal conductivity and cross-sectional area.

Composite Thermal Insulation

In composite thermal insulation systems, multiple layers of different materials are arranged in series, each contributing its own thermal resistance. The overall thermal resistance of the system is the sum of the individual resistances, and the steady-state heat transfer is determined by the total temperature difference divided by this total thermal resistance. The interface temperature between layers is found by equating the heat flux through each layer.

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