$5-38$ A plane wall with surface temperature of $\backslash (350^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ is attached with straight rectangular fins $\backslash (($k$=235\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}\backslash$cdot $\backslash$mathrm$\{$K$\})\backslash ).$ The fins are exposed to an ambient air condition of $\backslash (25^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and the convection heat transfer coefficient is $\backslash (154\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$cdot $\backslash$mathrm$\{$~K$\}\backslash ).$ Each fin has a length of $50$ mm $,$ a base of $5$ mm thick, and a width of $100$ mm $.$ For a single fin, using a uniform nodal spacing of $10$ mm $,$ determine $($a$)$ the finite difference equations, $($b$)$ the nodal temperatures by solving the finite difference equations, and $($c$)$ the heat transfer rate and compare the result with analytical solution.

FIGURE P$5-38$