$1)$ A $0\mathrm{.}4$ m thick rectangular plane wall with a thermal conductivity of $\backslash (20\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}\backslash$cdot $\backslash$mathrm$\{$K$\}\backslash )$ and thermal diffusivity of $\backslash (5\backslash$cdot $10^\{-6\}\backslash$mathrm$\{$~m$\}^\{2\}/\backslash$mathrm$\{$s$\}\backslash )$ is initially at a uniform temperature of $\backslash (150^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ One side of the wall is insulated. The other side is suddenly exposed to an environment at $\backslash (5^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The convection coefficient associated with the exposed surface is $\backslash (100\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$cdot $\backslash$mathrm$\{$~K$\}\backslash ).$

In case A$,$ the wall is considered as it is$.$ In case B$,$ the exposed surface of the wall is covered with a $10$ cm thick layer of low$-$density foam with a thermal conductivity of $\backslash (0\mathrm{.}03\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}\backslash$cdot $\backslash$mathrm$\{$K$\}\backslash ).$ For both cases, the times, $\backslash ($ t$\_\{1\}\backslash )$ and $\backslash ($ t$\_\{2\}\backslash ),$ at which the midpoint in the wall $(0\mathrm{.}2$ m away from the exposed surface$)$ reaches a temperature of $\backslash (145^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ and $\backslash (30^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ respectively, are to be determined.

a$)$ Formulate a solution for time $($do not solve$)$ with the following modeling approaches:

i$.$ the lumped capacitance model,

ii$.1-$term series solution for spatial effects,

iii. $4-$term series solution for spatial effects,

iv$.$ semi$-$infinite solid.

State the common assumptions for all approaches as well as the specific assumptions for each approach.

b$)$ Determine $\backslash ($ t$\_\{1\}\backslash )$ and $\backslash ($ t$\_\{2\}\backslash )$ for both cases A and B $.$ For each case and time point, select two modeling approaches that you consider most appropriate from the options above and compare the results. Make sure to explain your reasoning for selecting a particular modeling approach over others.