A wall consists of a layer of wood outside and a layer of...

A wall consists of a layer of wood outside and a layer of insulation inside. The temperatures inside and outside the wall are $+22^{\circ} \mathrm{C}$ and $-18^{\circ} \mathrm{C}$; the temperature at the wood/insulation boundary is $-8.0^{\circ} \mathrm{C} .$ By what factor would the heat loss through the wall increase if the insulation were not present?

Key Concepts

Multi-layer Composite Walls

When a wall consists of several materials, each with different thermal properties, the overall heat transfer is determined by the combined thermal resistances of these layers in series. Evaluating the temperature drops across each layer allows for the determination of the heat loss through the composite wall, and highlights the role of each material in reducing or facilitating heat flow.

Insulation Efficiency

Insulation is designed to reduce heat transfer by providing a material with high thermal resistance. Its inclusion in a multi-layer wall significantly lowers the overall heat loss by reducing the effective heat current. Removing insulation decreases the total thermal resistance, allowing more heat to pass through, which is quantified by comparing the temperature drops across the remaining material.

Thermal Resistance

Thermal resistance represents a material's opposition to heat flow. In multi-layer systems, each layer contributes its own thermal resistance, and the overall resistance is the sum of these individual resistances. A higher thermal resistance means less heat is conducted across the layer, and adjusting the composition or removing a layer affects the total heat transfer by changing the effective resistance of the wall.

Fourier's Law of Heat Conduction

Fourier's Law states that the rate of heat transfer through a material is proportional to the temperature gradient and the area through which the heat flows, and inversely proportional to the length of the material. This principle underlies the understanding of how different materials, like wood and insulation, affect the overall heat transfer when arranged in series as layers in a system.

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