Suppose that heat conduction occurs at a constant rate d Q /...

Suppose that heat conduction occurs at a constant rate $d Q / d t$ in a hollow sphere with an inner radius $r_{1}$ at temperature $T_{1}$ and an outer radius $r_{2}$ at temperature $T_{2}$. Show that for constant thermal conductivity $K$, the temperature difference between the two surfaces is given by $$ T_{1}-T_{2} \frac{d Q / d t}{4 \pi K}\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right) $$

Suppose that heat conduction occurs at a constant rate $d Q / d t$ in a hollow sphere with an inner radius $r_{1}$ at temperature $T_{1}$ and an outer radius $r_{2}$ at temperature $T_{2}$. Show that for constant thermal conductivity $K$, the temperature difference between the two surfaces is given by
$$
T_{1}-T_{2} \frac{d Q / d t}{4 \pi K}\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)
$$

Key Concepts

Fourier's Law of Heat Conduction

Fourier's law states that the rate of heat transfer through a material is proportional to the negative gradient of the temperature and the area through which the heat flows. In this context, it provides the fundamental relationship between the heat flux, the thermal conductivity, and the temperature gradient in the spherical shell.

Steady-State Heat Conduction

In a steady-state condition, the temperature distribution does not change with time, meaning that the heat entering any region equals the heat leaving it. This principle allows us to apply conservation of energy and equate the heat transfer rate across any spherical surface within the shell, leading to an expression for the temperature difference.

Heat Transfer in Spherical Coordinates

When dealing with spherical geometries, the area through which heat flows is a function of the radius (given by 4?r²). This variation with radius must be accounted for when deriving the temperature gradient, as it affects how the constant heat transfer rate is distributed over the surface area of the sphere.

Thermal Resistance in Spherical Shells

The concept of thermal resistance simplifies complex conduction problems by analogizing them to electrical resistance. In spherical shells, the resistance is not constant but varies with radius. By integrating the differential thermal resistance of thin spherical layers, one obtains the overall temperature difference in relation to the heat transfer rate and the geometry of the sphere.

Integration Techniques in Conduction Problems

The derivation involves setting up an integral that relates the differential temperature drop across thin layers of the sphere to the heat flux. This integration takes into account the radial variation in area, thereby linking the macroscopic temperature difference between the inner and outer surfaces to the constant rate of heat conduction.

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