Q$1.$ A pipe subjected to a heating element is used to preheat water going into a continuous petrochemical process. The water being supplied is at $\backslash (50^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ with a heat transfer coefficient of $600\backslash (\backslash$mathrm$\{$W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$mathrm$\{$~K$\}\backslash )$ and is heated using a heating element wrapped around the pipe supplying a heat flux of $\backslash (500\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash ).$ The pipe is fabricated from steel with a thermal conductivity of $\backslash (22\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$mK$\}\backslash ).$ The internal diameter of the pipe is $10$ cm and the thickness is $5$ cm $.$ Ignore any radiation effects and assume the heating element is sufficiently lagged to prevent heat loss to the environment.

$($a$)$ Starting with the one$-$dimensional heat conduction equation, develop the governing differential equation of heat transfer through the container wall for this problem.

$[4$ marks$]$

$($b$)$ State the boundary conditions for this problem.

$[2$ marks$]$

$($c$)$ Solve the differential equation to get the unique solution for $\backslash (\backslash$mathrm$\{$T$\}(\backslash$mathrm$\{$r$\})\backslash )$ and hence or otherwise determine the temperature difference across the wall of the pipe.

$[14$ marks$]$