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 $30 \mathrm{ft}^{2}$. The inside board surface is at $175 \mathrm{F}$, the outside surface is at $70 \mathrm{F}$, and the board material has a conductivity of 0.05 Btu/h ft $\mathrm{F.}$ How thick should the board be to limit the heat transfer loss to $720 \mathrm{Btu} / \mathrm{h} ?$

Key Concepts

Fourier's Law of Heat Conduction

This law provides the quantitative basis for conduction, stating that the rate of heat transfer through a material is proportional to the temperature difference across it, the area through which the heat flows, and the material's thermal conductivity, and inversely proportional to the thickness of the material. It is the fundamental relationship used to calculate the required insulation thickness to achieve a desired heat transfer rate.

Thermal Conductivity

Thermal conductivity is a physical property of materials that measures their ability to conduct heat. In scenarios involving insulation, a lower thermal conductivity indicates that the material is a better insulator, as it resists heat flow more effectively. This concept is essential when selecting or designing insulation to control heat losses.

Heat Transfer Rate

The heat transfer rate quantifies the amount of heat energy transferred per unit time. In engineering problems, it is used to ensure that heating or cooling systems perform within specified energy loss or gain limits. Understanding this concept is crucial when setting design targets for insulation systems.

Temperature Gradient

The temperature gradient, which is the difference in temperature between two surfaces, drives the heat flow in conduction problems. A larger temperature gradient results in a higher rate of heat transfer, assuming the properties of the material and other conditions remain constant. This concept is key for determining the necessary insulation thickness to achieve a desired reduction in heat transfer.

Insulation Thickness

Insulation thickness is a critical design parameter that directly affects the thermal resistance of an insulating layer. Greater thickness increases the resistance to heat flow, thereby reducing the rate of energy loss. Determining the proper thickness is essential in applications where controlling temperature differences and maintaining energy efficiency are important.

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