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Fourier's Law of thermal conduction is a fundamental principle that describes how heat energy transfers through a material due to temperature gradients. In the context of a spherical shell, this law can be expressed mathematically as:
\[\begin{equation}\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 k \frac{\partial T(r)}{\partial r}\right) = 0\end{equation}\]
where \(r\) is the radial position within the sphere, \(k\) is the thermal conductivity of the material, and \(T(r)\) is the temperature at any radius \(r\). Fourier's law in spherical coordinates takes into account that the area through which heat is conducted increases with the square of the radius, hence the term \(r^2\) in the equation.

Heat conduction in spherical systems, such as our spherical shell, differs from planar conduction due to geometry. When dealing with spheres, the heat transfer is radially outwards or inwards, and the amount of material through which heat must conduct gets larger as we move away from the center. This geometry is accounted for in the governing heat conduction equation, which, for a sphere with constant thermal conductivity, simplifies to:
\[\begin{equation}\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial T(r)}{\partial r}\right) = 0\end{equation}\]
Integrating this equation gives us a general form of the temperature profile within the spherical shell. Understanding how to solve this differential equation is essential for predicting temperature variation in such systems, which is especially crucial in applications involving thermal insulation or energy storage.

Boundary conditions play a significant role in solving heat transfer problems as they define the behavior of a system at its limits. For our spherical shell, we have two key boundary conditions:

• The inner surface has a uniform heat flux, \(\dot{q}_{1}\).
• The outer surface experiences convection with the ambient, characterized by a convection coefficient \(h\) and an ambient temperature \(T_{c \infty}\).

These conditions allow us to determine the unknown constants from the general solution of the temperature profile. They provide the necessary information to predict how the spherical shell will interact with its environment. Specifically, the first condition helps us find the value of constant \(C_{1}\), while the second condition is used to determine \(C_{2}\) and thereby complete our temperature expression.

The temperature profile in a material describes how temperature varies with location. In the spherical shell scenario, this profile is determined by integrating the heat conduction equation, considering the respective boundary conditions for the system. Once we find the constants \(C_{1}\) and \(C_{2}\), we can write the temperature profile as:
\[\begin{equation}T(r) = -\frac{C_{1}}{r} + C_{2}\end{equation}\]
In the final step of our solution, the profile shows how temperature changes from the inner radius to the outer radius of the spherical shell. By applying the boundary conditions properly, we can accurately predict and express the outer surface temperature of the shell, which is crucial for designing thermal systems and understanding how they will perform in real-world scenarios.