Heat conduction in solid medium
We are interested of how heat spreads. Basic Newton's cooling law states that heat energy moves from hot parts to cold parts relative to the temperature difference. More accurately, heat flux (conduction rate) is proportional to the temperature gradient
\[
\Phi=-k \nabla T
\]
Where \( \Phi \) is heat flux \( \left[\mathrm{W} / \mathrm{m}^{2}\right], k \) is thermal conductivity \( [\mathrm{W} / \mathrm{mK}] \) and \( T \) is temperature \( [\mathrm{K}] \).
We assume that our medium is solid (i.e. we have only conduction, no convection or radiation). The part we are interested of is a cylindrical part with known height \( h \) and radius \( r \), You will be given which material the part is.
After a couple of intermediate steps (divergence, etc.) we find that
\[
\rho c_{p} \frac{\partial T}{\partial t}=k \nabla^{2} T, \nabla^{2} T=\frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2} T}{\partial y^{2}}+\frac{\partial^{2} T}{\partial z^{2}}
\]
Where \( \rho \) is the density of the material and \( c_{p} \) is the specific heat capacity of the material. You can find these values from handbooks and/or other references. Solving for the partial derivative of heat by time, we find the thermal diffusivity \( \alpha \) of the material which appear in homework set 9.
Your task is to find out what is the steady-state of the temperature in the object after one end of the area is being heated. For simplicity's sake, we assume that the material extends beyond the area we are interested of. Somehow, one circular end of the block is heated to temperature \( T_{1} \) while the rest of the material stays in ambient temperature \( T_{0} \).
The minimum requirement is to setup the differential equations related to temperature on the block related to incoming and outcoming heat flux (the entire block as one object) and solve the resulting system using a solver more advanced than Euler's method.
In the intermediate result requires you to calculate the temperature gradient inside the volume of interest using some numerical method, such as dividing the volume into smalter elements.
You may consider staying in Cartesian coordinates (then the boundaries become slightly more complex) or taking advantage of the volume of interest being cylindricat and applying symmetry (in which case vou need to note that coordinates stretch in cylindrical coordinate svstem).
