A metallic sphere having inner and outer radii a and b...

A metallic sphere having inner and outer radii $a$ and $b$ respectively has thermal con- ductivity $$ K=\frac{K_{C}}{r} \quad(a<r<b) $$ The thermal resistance between inner surface and outer surface is: (a) $\frac{(b-a)}{4 \pi K_{0}}$ (b) $\frac{\left(b^{2}-a^{2}\right)}{4 \pi K_{0} a b}$ (c) $\frac{4 \pi K_{0}}{(b-a)}$ (d) none of these

A metallic sphere having inner and outer radii $a$ and $b$ respectively has thermal con- ductivity
$$
K=\frac{K_{C}}{r} \quad(a<r<b)
$$
The thermal resistance between inner surface and outer surface is:
(a) $\frac{(b-a)}{4 \pi K_{0}}$
(b) $\frac{\left(b^{2}-a^{2}\right)}{4 \pi K_{0} a b}$
(c) $\frac{4 \pi K_{0}}{(b-a)}$
(d) none of these

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Key Concepts

Thermal Resistance

Thermal resistance is a measure of a material's opposition to the flow of heat. In conduction problems, it quantifies how difficult it is for heat to travel through a medium and is often computed by integrating the reciprocal of the product of thermal conductivity and cross?sectional area over the distance through which the heat flows.

Conduction in Spherical Coordinates

When dealing with spherical geometries, the area through which heat is conducted changes with the radius. This requires special consideration because the cross-sectional area for heat flow is a function of the distance from the center, typically involving a term like 4?r². This variation leads to integrations that often result in logarithmic terms when the conductivity is uniform or modified forms when conductivity varies with radius.

Variable Thermal Conductivity

In many real-world scenarios, a material’s thermal conductivity is not constant but varies with position. This variation must be accounted for in the conduction analysis by including the spatially-dependent thermal conductivity in the integral that defines thermal resistance. Understanding how to incorporate a function like K(r) into the integration process is key to accurately determining the heat transfer characteristics in non-homogeneous materials.

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