Problem $1.$ A $1-$cm thick, $60-$cm long cylindrical reactor made of stainless steel $($k $=15$ W$/$m$$C$)$ of $30-$cm diameter is being used for thermochemically processing swine manure slurry. The reactor was loaded with manure slurry $(95\%$ water$)$ and heated in a furnace. During the reaction, the outer and inner walls of the reactor were found to be $550$C and $435$C$,$ respectively. Assuming a one$-$dimensional steady$-$state heat transfer process with no energy generation, $($i$)$ formulate a differential equation to describe the heat transfer process $,($ii$)$ solve the differential equation to determine the temperature profile within the wall $,$ and $($iii$)$ plot the temperature profile across the wall of the cylinder $$and determine the rate of heat transfer through the wall.