(Solved): Let us consider a concentric cylinder system as shown by the adjacent figure wherein the central cy ...

Let us consider a concentric cylinder system as shown by the adjacent figure wherein the central cylinder uniformly receives electrical power, Pin. a) Develop an expression for the excess temperature distribution, $?(r)=T(r)?T_a$ assuming 1-D conduction in the radial direction and convection air cooling from both top and bottom surfaces as a function of the given parameters shown by the figure. For simplicity, let's assume that the outer cylinder's diameter $\mathrm{D}?$ d, i.e. the outer cylinder is effectively infinitely long in the radial direction. Let's also assume that the central cylinder's excess length beyond the bottom surface is negligible. $[10]$ b) Plot the excess temperature profile, $?(r)$ for $0?r?20\mathrm{mm}$ and the following data: $t=3mm,P_{\text{in }}=50W,h=75\frac{W}{m^{2}K},T_a=25^{?}\mathrm{C},k=388\frac{W}{mK}$ for both $\mathrm{d}=6\mathrm{mm}$ and $\mathrm{d}=1\mathrm{mm}$ on the same graph. c) Calculate the maximum temperature (in ${}^{?}\mathrm{C}$ ) for the following source diameters; $\mathrm{d}=6\mathrm{mm}$, $\mathrm{d}=1\mathrm{mm}$ and $\mathrm{d}=60\mathrm{nm}$. Comment! [5]