Consider an aluminum pipe which is wrapped with an electric heater at its outer surface with voltage V and current I (electric heaters is insulated), and cooled with a coolant fluid at its inner surface at temperature $T_{\infty}$ and convection coefficient of h. The pipe has a length of L, density of $\rho$, specific heat capacity of c, thermal conductivity of k, and it is initially at temperature $T_i$. You are going to use an implicit finite difference numerical method to solve for the instantaneous temperature distribution of 30 nodes. Find the finite difference equations for both the inside and outside boundaries. Please use * superscript for the end of a time step and no superscripts for the beginning of a time step, along with $r_i$ and $r_o$ for inner and outer radius of each cell. Also, draw a volume element with appropriate parameters you used in the nodal equations.
