The outer surface of a hollow sphere of radius r2 is...

The outer surface of a hollow sphere of radius $r_{2}$ is subjected to a uniform heat flux $q_{2}^{\prime \prime}$. The inner surface at $r_{1}$ is held at a constant temperature $T_{s, 1}$. (a) Develop an expression for the temperature distribution $T(r)$ in the sphere wall in terms of $q_{2}^{\prime \prime}, T_{s, 1}, r_{1}, r_{2}$, and the thermal conductivity of the wall material $k$. (b) If the inner and outer tube radii are $r_{1}=50 \mathrm{~mm}$ and $r_{2}=100 \mathrm{~mm}$, what heat flux $q_{2}^{\prime \prime}$ is required to maintain the outer surface at $T_{s, 2}=50^{\circ} \mathrm{C}$, while the inner surface is at $T_{s, 1}=20^{\circ} \mathrm{C}$ ? The thermal conductivity of the wall material is $k=10 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.

The outer surface of a hollow sphere of radius $r_{2}$ is subjected to a uniform heat flux $q_{2}^{\prime \prime}$. The inner surface at $r_{1}$ is held at a constant temperature $T_{s, 1}$.
(a) Develop an expression for the temperature distribution $T(r)$ in the sphere wall in terms of $q_{2}^{\prime \prime}, T_{s, 1}, r_{1}, r_{2}$, and the thermal conductivity of the wall material $k$.
(b) If the inner and outer tube radii are $r_{1}=50 \mathrm{~mm}$ and $r_{2}=100 \mathrm{~mm}$, what heat flux $q_{2}^{\prime \prime}$ is required to maintain the outer surface at $T_{s, 2}=50^{\circ} \mathrm{C}$, while the inner surface is at $T_{s, 1}=20^{\circ} \mathrm{C}$ ? The thermal conductivity of the wall material is $k=10 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.

Key Concepts

Steady-State Heat Conduction

This concept involves analyzing heat transfer when the temperature distribution does not change with time. In steady-state conditions, all the energy entering a system equals the energy leaving, allowing simplification of the heat conduction equations without transient or time-dependent terms. This is fundamental when deriving temperature distributions in solid materials with constant thermal conductivity.

Fourier’s Law of Conduction in Spherical Coordinates

Fourier's law relates the heat flux to the temperature gradient and the thermal conductivity of the material. In spherical coordinates, the law is modified to account for the geometry, where the heat flow occurs radially and the area perpendicular to the flow changes with the square of the radius. This spatial variation must be considered when deriving expressions for temperature as a function of the radius in spherical shells.

Integration of the Heat Conduction Equation in Spherical Geometry

In problems involving spherical symmetry, the differential form of the conduction equation is integrated over the radial coordinate to obtain the temperature distribution. Integration takes into account the variable cross-sectional area for heat conduction, leading to logarithmic or algebraic expressions that relate temperature differences and radii. This technique is essential for developing analytical solutions in spherical systems.

Thermal Resistance of a Spherical Shell

Thermal resistance quantifies the opposition to heat flow through a material and is a concept analogous to electrical resistance. In a spherical shell, the resistance takes into account the curvature of the surface and the varying area with radius. The expression for thermal resistance is derived by integrating the inverse of the conduction path area, and it plays a critical role in relating heat flux to temperature differences across the shell.

Boundary Conditions in Heat Conduction Problems

Setting appropriate boundary conditions is vital for solving differential equations in heat transfer. Typically, one boundary may have a specified temperature (Dirichlet condition) while another may have a specified heat flux (Neumann condition). These conditions ensure that the solution not only satisfies the differential equation but also aligns with the physical constraints of the problem, enabling the determination of integration constants and complete temperature distribution.

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