$2)$ Consider a plane wall with thickness of $\backslash ($ L$=0\mathrm{.}2\backslash$mathrm$\{$~m$\}\backslash ),$ thermal conductivity of $\backslash ($ k$=65\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$mK$\}\backslash )$ and thermal diffusivity of $\backslash (\backslash$mathrm$\{$a$\}=2\mathrm{.}5\backslash$times $10^\{-6\}\backslash$mathrm$\{$~m$\}^\{2\}/\backslash$mathrm$\{$s$\}\backslash ).$ Heat is generated uniformly at a constant rate of $\backslash (\backslash$mathrm$\{$q$\}=1\mathrm{.}5\backslash$mathrm$\{$x$\}\backslash )\backslash (10^\{4\}\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )$ within the wall. On the left side of the wall there is a constant wall temperature boundary condition of $\backslash (0^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ while the right side of the wall is exposed to fluid flowing at $\backslash (60^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ Heat transfer coefficient associated to the fluid exposed surface, i$.$e$.$ right surface, is $\backslash (\backslash$mathrm$\{$h$\}=50\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}2\backslash$cdot $\backslash$mathrm$\{$~K$\}\backslash ).$ Consider $5$ equally spaced nodes in the medium, $2$ at the boundaries and $3$ in between.

a$)$ Find the right$-$side surface exact temperature at steady state.

b$)$ By using Finite Difference Method, estimate the right$-$side surface temperature at steady state. Show all your work!

c$)$ The wall has an initial uniform temperature of $\backslash (150^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ and is exposed to the specified conditions starting at $\backslash (\backslash$mathrm$\{$t$\}=0\backslash$mathrm$\{$~s$\}\backslash ).$

i$.$ Determine the maximum time step, $\backslash (\backslash$Delta $\backslash$mathrm$\{$t$\}\backslash ),$ possible for a stable explicit solution to obtain the numerical solution for the right wall.

ii$.$ Using that time step, estimate the temperature of the right wall exposed to the fluid, ensuring that the result differs by no more than $\backslash (10^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ from the exact temperature found in part $($a$).$ Specify the time required to reach that temperature.

iii. Plot the temperatures of nodes $3($midplane$)$ and $5($Right Wall$)$ with time.

iv$.$ Plot the temperature profiles across the wall for the time point found in part $($c$.$ii$).$

d$)$ Repeat part $($c$)-$ ii$.$ iii. and iv$.$ with the implicit method using $\backslash (\backslash$Delta $\backslash$mathrm$\{$t$\}=20\backslash$mathrm$\{$~s$\}\backslash ).$

For parts $($c$)$ and $($d$),$ you may use your preferred software, providing detailed sample calculations. Include your code in the appendix and upload your solution file $($MATLAB$,$ Excel etc.$)$ separately.