Figure 19-43 shows (in cross section) a wall that consists...

Figure $19-43$ shows (in cross section) a wall that consists of four layers. The thermal conductivities are $k_{1}=0.060$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, k_{3}=0.040 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $k_{4}=0.12 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\left(k_{2}\right.$ is not known). The layer thicknesses are $L_{1}=1.5 \mathrm{~cm}, L_{3}=2.8 \mathrm{~cm}$, and $L_{4}$ $=3.5 \mathrm{~cm}\left(L_{2}\right.$ is not known). Energy transfer through the wall is steady. What is the temperature of the interface indicated?

Figure $19-43$ shows (in cross section) a wall that consists of four layers. The thermal conductivities are $k_{1}=0.060$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, k_{3}=0.040 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $k_{4}=0.12 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\left(k_{2}\right.$ is not
known). The layer thicknesses are $L_{1}=1.5 \mathrm{~cm}, L_{3}=2.8 \mathrm{~cm}$, and $L_{4}$ $=3.5 \mathrm{~cm}\left(L_{2}\right.$ is not known). Energy transfer through the wall is steady. What is the temperature of the interface indicated?

Key Concepts

Interface Temperature under Steady-State Conditions

At the junction between layers in a composite wall, the temperature is determined by the continuity of heat flux and the relative thermal resistances of the adjoining layers. In steady-state conditions, the constant heat flow ensures that temperature gradients adjust to satisfy the energy balance across each interface.

Series Combination of Thermal Resistances

In multi-layer wall systems, individual thermal resistances of each layer add in series, meaning that the total resistance is the sum of the resistances of all layers. This concept is crucial for determining overall heat transfer and for calculating temperature drops across each layer in steady-state conduction.

Fourier’s Law of Heat Conduction

This principle states that the rate of heat transfer through a material is proportional to the negative of the temperature gradient and the material's thermal conductivity. It forms the basis for analyzing heat flow in solids and is essential for calculating how temperature differences drive energy transfer in layered systems.

Thermal Resistance

Thermal resistance quantifies a material’s opposition to heat flow and is analogous to electrical resistance in circuits. It is calculated as the layer thickness divided by the product of thermal conductivity and cross-sectional area, and it allows for a simplified approach to analyzing conduction by summing resistances in complex systems.

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