(a) A spherical shell has inner and outer radii a and b ,...

(a) A spherical shell has inner and outer radii $a$ and $b$ , respectively, and the temperatures at the inner and outer surfaces are $T_{2}$ and $T_{1}$ . The thermal conductivity of the material of which the shell is made is $k$ . Derive an equation for the total heat current through the shell. (b) Derive an equation for the temperature variation within the shell in part (a); that is, calculate $T$ as a function of $r$ , the distance from the center of the shell. (c) A hollow cylinder has length $L,$ inner radius $a$ , and outer radius $b$ , and the temperatures at the inner and outer surfaces are $T_{2}$ and $T_{1}$ . (The cylinder could represent an insulated hot-water pipe, for example.) The thermal conductivity of the material of which the cylinder is made is $k$ . Derive an equation for the total heat current through the walls of the cylinder. (d) For the cylinder of part (c), derive an equation for the temperature variation inside the cylinder walls. (e) For the spherical shell of part (a) and the hollow cylinder of part (c), show that the equation for the total heat current in each case reduces to Eq. $(17.21)$ for linear heat flow when the shell or cylinder is very thin.

Key Concepts

Thin-Wall Approximation

When the thickness of a shell or cylinder is very small relative to its overall dimensions, the temperature gradient across the wall is nearly constant. This thin-wall approximation simplifies the heat transfer equations to a form that resembles linear heat flow, which is a useful consistency check against the more general expressions derived for thicker geometries.

Boundary Conditions

Applying appropriate boundary conditions, such as specifying the temperatures at the surfaces of the shell or cylinder, is crucial for solving the differential equations governing heat conduction. These conditions ensure that the solutions accurately reflect the physical temperature distribution in the system.

Integration in Heat Conduction Problems

To account for the spatial variations in temperature and cross-sectional area in non-uniform geometries, integration is necessary. By setting up differential equations based on Fourier's law and integrating between the boundaries (inner and outer surfaces), one can derive the expressions for the total heat current as well as the temperature profile within the medium.

Heat Transfer in Spherical Coordinates

For heat conduction through a spherical shell, the analysis is done in spherical coordinates. This involves considering the radial dependence of the surface area, which varies with the square of the radius. The integration of the temperature gradient over a spherical surface allows one to derive the total heat current and the temperature distribution within the shell.

Fourier's Law

Fourier's law forms the foundation of analyzing heat conduction, stating that the heat flux is directly proportional to the negative temperature gradient and the material's thermal conductivity. It serves as the starting point for deriving heat transfer equations in both spherical and cylindrical geometries.

Thermal Conduction

This is the process by which heat is transferred within a material through the molecular interactions, driven by a temperature gradient. The material's thermal properties, such as thermal conductivity, dictate how efficiently heat is conducted from regions of higher temperature to regions of lower temperature.

Heat Transfer in Cylindrical Coordinates

Similarly, heat conduction in hollow cylinders is analyzed using cylindrical coordinates. Here the relevant area through which heat flows depends on the radial distance multiplied by the length of the cylinder. The differential form of Fourier's law is integrated over this geometry, leading to expressions for both the total heat current and the radial temperature distribution.

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