A water heater is covered up with insulation boards over a...

A water heater is covered up with insulation boards over a total surface area of $3 \mathrm{m}^{2}$. The inside board surface is at $75^{\circ} \mathrm{C}$, the outside surface is at $20^{\circ} \mathrm{C}$, and the board material has a conductivity of $0.08 \mathrm{W} / \mathrm{m}$ K. How thick should the board be to limit the heat transfer loss to $200 \mathrm{W} ?$

Key Concepts

Fourier's Law of Heat Conduction

Fourier's Law is the fundamental principle that describes how heat energy is transferred through a material by conduction. It states that the rate of heat transfer through a material is proportional to the negative gradient of temperatures and the area through which the heat flows, and it is inversely proportional to the thickness of the material. This concept is crucial in setting up relationships between the material’s properties, geometry, and the temperature difference driving the process.

Thermal Conductivity

Thermal conductivity is a material property that indicates how well a material can conduct heat. It is defined as the rate at which heat passes through a unit area of a material with a given temperature gradient. High conductivity materials transfer heat quickly, while low conductivity materials, such as insulators, transfer heat slowly. This parameter directly influences the design of insulation systems to control heat loss or gain.

Temperature Difference as a Driving Force

The temperature difference between two regions drives the process of heat conduction. A greater difference results in a higher rate of heat transfer, while a smaller difference reduces the transfer rate. Understanding this concept is essential when designing systems to limit or control heat flow, as it helps determine the necessary material properties and dimensions to achieve a desired thermal performance.

Thermal Resistance

Thermal resistance is a measure of a material's opposition to heat flow and is analogous to electrical resistance in circuits. It is determined by the material’s thickness, area, and thermal conductivity. In insulation design, higher thermal resistance is desirable to reduce heat loss. This concept enables engineers to determine the required thickness or selection of materials to achieve specific thermal performance targets.

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