A plastic sheet is made by extruding heated plastic through a rectangular die and then allowing the sheet to cool in air prior to being wound on a roller as shown in the schematic diagram below. The roller rotates in such a way that the velocity of the sheet in the z$-$direction is constant and equal to $v_0.$ The thickness of the sheet is $2B,$ which is far smaller than the length $(L)$ and width $(W)$ of the sheet. The sheet leaves the heated die at a temperature of $T_w$ and the air temperature is $T_a.$ The temperature of the sheet reaches the air temperature at the roller. You may assume that the temperature of the sheet is only a function of $z,$ the distance from the heated die. Assume that the density $(),$ heat capacity $(C_p),$ and the thermal conductivity $(k)$ of the plastic sheet are constant. and the heat transfer coefficient between the sheet and the air $(h)$ is also constant. Further assuming that this process has reached steady state, do the following:

$($a$)$ List all appropriate assumptions and reduce the equation of energy to obtain differential equation describing the temperature of the sheet, $T$ as a function of $z$ and solve the general solution for temperature profile.

$($b$)$ Simplify the differential equation in $($a$)$ for the case when $v_0=0$ and solve the differential equation with the appropriate boundary conditions to obtain the temperature distribution along the sheet.

$($c$)$ If the thickness of the sheet in $($b$)$ is not much smaller than the length, which means that there is a temperature gradient in the thickness direction as well. Apply differential equation of change approach to derive the governing differentiation equation.

$($d$)$ Write down the appropriate boundary conditions that are necessary to solve the differential equation in $($c$).$