A plane wall is insulated on one side $(x=0),$ and the wall is initially at a uniform temperature of

$T_i.$ At time $t=0,$ the temperature of the surface at $x=L,$ is instantaneously raised to $T_s>T_i.$

a$.$ Build a model $($governing equation and boundary conditions$)$ that may be solved for the time$-$

dependent temperature profile in the wall, $T(x,t).$

b$.$ Sketch the temperature distributions $T(x,0),T(x,t_1)$ where $T(x,t)x{q}^{\text{'}\text{'}}(\frac{L}{2},t)q^{\text{'}\text{'}}(L,t)t_1$ and

$T(x,t)as$ functions $ofx.$

$c.$ Sketch the mid$-$plane heat flux, $q^{\text{'}\text{'}}(\frac{L}{2},t),as$ a function $of$ time.

$d.$ Sketch the heat flux $at$ the temperature specified surface, $q^{\text{'}\text{'}}(L,t),as$ a function $of$ time$\mathrm{.}0$ where $t_1$ and

$T(x,t)as$ functions $ofx.$

$c.$ Sketch the mid$-$plane heat flux, $q^{\text{'}\text{'}}(\frac{L}{2},t),as$ a function $of$ time.

$d.$ Sketch the heat flux $at$ the temperature specified surface, $q^{\text{'}\text{'}}(L,t),as$ a function $of$ time.