Question $2$

A thin plastic adhesive is fixed on one end of a metal rod of length $2$ mm $.$ To affix the adhesive to the metal, a laser with a heat flux of $850000\frac{W}{m^2}$ is used for $20$ ms $.$ Both the ends of the metal rod are exposed to convective cooling with air at temperature of $300K$ and convective coefficient of $100\frac{W}{m^2}K.$ The rod has the density of $7850k\frac{g}{m^3},$ specific heat of $435\frac{J}{k}gK,$ and a thermal conductivity of $600\frac{W}{m}K.$ The thickness and the thermal conductivity of the plastic adhesive can be neglected. This affixing process results in unsteady one$-$dimensional conduction heat transfer which is governed by the following energy equation:

$\frac{\text{d}\text{e}\text{l}(c_{p}T)}{\text{d}\text{e}\text{l}\text{t}}=\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}(k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{x}})$

Consider that the metal bar is discretized with $2$ control volumes with the mesh size of $1\mathrm{.}0$ mm as shown in Fig. Q$2.$ Taking the time$-$step as $10$ ms $,$

$($a$)$ Derive the discretized equations for temperatures at nodes $1$ and $2$ in the context of explicit time marching.

$(14$ marks$)$

$($b$)$ Obtain the values of temperatures at nodes $1$ and $2$ at time instants $10,20,30$ and $40$ milliseconds for an initial temperature of $300$K using explicit time marching.

$(12$ marks$)$