Q$3$: $(15+5$ pts$)$ Heat Transfer in a Thin Rod

Figure $1$: Schematic of heat transfer in a thin rod with round cross$-$section.

In class $($videos$),$ a model of heat transfer in a thin rod $($Figure $1)$ was presented. This might represent the cooling effect of a heatsink in a computer, or the cooling of a rod in a nuclear reactor. In the case shown, the ODE for temperature, $\backslash ($ T $\backslash ),$ with respect to location on the rod, $\backslash ($ x $\backslash ),$ is:

$\backslash [$

$\backslash$frac$\{$d$^\{2\}$ T$\}\{$d x$^\{2\}\}=\backslash$frac$\{$h P$\}\{\backslash$kappa A$\}\backslash$left$($T$-$T$\_\{\backslash$infty$\}\backslash$right$)$

$\backslash ]$

where the parameters are defined as follows:

$-\backslash ($ h $\backslash )$ is the heat transfer coefficient $($in W$/$m K$)$

$-\backslash (\backslash$kappa $\backslash )$ is the thermal conductivity of the rod material $($in W$/$m $\backslash (\backslash$cdot $\backslash$mathrm$\{$K$\}\backslash ))$

$-\backslash ($ P $\backslash )$ is the perimeter of the cross$-$sectional area of the rod $($in m $)$

$-\backslash ($ A $\backslash )$ is the cross$-$sectional area of the rod $($in $\backslash (\backslash$mathrm$\{$m$\}^\{2\}\backslash ))$

$-\backslash ($ T$\_\{\backslash$infty$\}\backslash )$ is the ambient temperature $($in K $)$

Finite Difference Method

The finite difference method $($Section $27\mathrm{.}1\mathrm{.}2$ of the Textbook$)$ involves replacing derivatives in a differential equation with finite difference approximations. For this assignment, you will use this method AND one of your linear system methods $($e$.$g$.$ Gauss elimination, Gauss$-$Seidel$)$ to solve for temperatures in a cooling rod.

Parameter Definitions

For all exercises, use the following parameters:

$-\backslash ($ D $\backslash )$ is the diameter of the round rod $($in m $)$

$-\backslash ($ L $\backslash )$ is the length of the round rod $($in m $)$