Chapter 2, Problem 2.37P

The steady-state temperature distribution in a semi-transparent material of thermal conductivity k and thickness L exposed to laser irradiation is of the form
$T\left(x\right)=-\frac{A}{k{a}^2}{e}^{-ax}+Bx+C$
where A, a, B, and C are known constants. For this situation, radiation absorption in the material is manifested by a distributed heat generation term, $\stackrel{.}{q}\left(x\right).$

1. Obtain expressions for the conduction heat fluxes at the front and rear surfaces.
2. Derive an expression for $\stackrel{.}{q}\left(x\right).$
3. Derive an expression for the rate at which radiation is absorbed in the entire material. per unit surface area. Express your result in terms of the known constants for the temperature distribution, the thermal conductivity of the material, and its thickness.