Consider a one$-$dimensional fin shown in the Figure below. Let the cross$-$section area of

the fin be $A_c.$ Let the perimeter of the fin be $P.$ The outer surface of the fin exchanges heat

with the environment through radiation. Assume the environment to be at $T_{},$ which is

greater than the fin temperature. Assuming that the base of the fin $(x=0)$ and end of the fin

$(x=L)$ are at constant temperatures $T=T_b$ and $T=T_{}$ respectively, Derive a governing steady$-$

state energy conservation equation and accompanying boundary conditions. Comment on

the linearity and homogeneity of the equation.

Hint $1$: Approach this problem by considering a small element of the fin of thickness

$dx,$ and carrying out an energy balance on the element, similar to the energy balance

we carried out in class to derive the general governing energy equation. Be sure to

account for all modes of energy coming in and leaving $($both conduction and

radiation$).$

Hint $2$: A simplified radiation model says that given two bodies at temperatures $T_1$

and $T_2$ exchanging heat with each other radiatively, the heat gained$/$lost by body$1$ is

given by $A_1(T_1^4-T_2^4)$ where $$ is the emissivity of body$1,A_1$ is the area of body $1$ and

$$ is Stefan$-$Boltzmann constant. ok