$1.$ Consider a long circular cylinder with energy generation given by $\backslash (\backslash$dot$\{$q$\}=$a$\backslash$left$(1-\backslash$frac$\{$r$\}\{$R$\}\backslash$right$)\backslash ).$ The cylinder is exposed to an environment at $\backslash (\backslash$mathrm$\{$T$\}\_\{\backslash$infty$\}\backslash )$ with the heat transfer coefficient $\backslash ($ h $\backslash ).$

a$.$ Write the governing equation and boundary conditions for transient temperature distribution. $(20$ pts$)$

b$.$ Nondimensionalize the governing equation and boundary conditions $(20$ pts $)$

c$.$ Obtain the Steady State solution analytically $(20$ pts$)$

d$.$ The coefficient a is suddenly doubled from its initial value. Numerically solve for the transient temperature distribution. Create a plot of centerline temperature as a function of time and upload into canvas along with your spreadsheet. $(40$ pts $)$