A heating element at $\backslash (400^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ is subjected to average convection heat transfer coefficient of $\backslash (25\backslash$frac$\{\backslash$mathrm$\{$~W$\}\}\{\backslash$mathrm$\{$~m$\}\}\backslash )$ when the surrounding air at $\backslash (25^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The heating element is made from iron and has a cylindrical geometry with $2\mathrm{.}5$ mm in diameter and $20$ mm in length.

$($a$)$ Determine the thermal resistance and overall heat transfer coefficient for the heat transfer system if the heating element is in thermodynamic equilibrium

$($b$)$ Determine the power required to maintain the heating element at $\backslash (400^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$

$($c$)$ Provide three engineering assumptions and engineering model for solving $($a$)$ and $($b$)$