$1$D conduction with generation

In Problem $1,$ we assumed that the glass envelope was trans$-$

parent to solar radiation. In fact, the glass envelope absorbs light

throughout the volume of the glass. In this problem, we consider

only the glass envelope and model the impact of light absorption

as a form of energy generation.

Though the glass is a cylinder, it is relatively large diameter and

thin$-$walled with thickness $th,$ so we can approximate it as a

plane wall with no curvature. Assume that light is absorbed

evenly throughout the glass at a volumetric generation rate of

$g^{}{?}^{\text{'}\text{'}\text{'}}($Units are $\frac{W}{m^3}).$ The inner wall $(x=0)$ is subject to a con$-$

stant flux $q_{\text{l}\text{o}\text{s}\text{s}}^{}{?}^{\text{'}\text{'}},$ and the outer wall is exposed to convection with a

convective heat transfer coefficient $\frac{?}{\text{b}\text{a}\text{r}}(h{)}_{}$ and ambient temperature

$T_{}.$ The geometry is shown in Figure $3.$

a$.)$ Derive a governing ODE in terms of temperature and its derivatives. It will help you to start by drawing the

differential control volume.

b$.)$ Solve the ODE to obtain a solution for $T(x).$ You should have two constants of integration, $C_1$ and $C_2$ in

the solution.

c$.)$ Identify the boundary conditions that will be imposed, then use the boundary conditions to solve for the

two constants of integration. Clearly state the full solution for temperature.