Fins can be designed to operate via radiative, rather than convective exchange from their outer surfaces. In such instances, we must use the Stefan-Boltzmann radiation law instead of Newton's Law of Cooling in the balance equation.

a. Derive the differential equation describing the temperature profile in a fin that exchanges heat with the environment via radiation. The boundary conditions include a constant base temperature, \(T_{b}\), and an insulated tip.

b. The differential equation is nonlinear and cannot be solved analytically. However, we can define a radiation heat transfer coefficient as follows:

\[q=\sigma \varepsilon A\left(T^{4}-T_{\infty}^{4}\right)=\underbrace{\sigma \varepsilon\left(T^{2}+T_{\infty}^{2}\right)\left(T+T_{\infty}\right)}_{h^{r}} A\left(T-T_{\infty}\right)=h^{r} A\left(T-T_{\infty}\right)\]

\(h^{r}\) is approximated using the average \(T\) for the fin. Solve the differential equation using the radiation heat transfer coefficient.

c. If the temperature of the environment is \(300 \mathrm{~K}\), the base temperature of the fin is \(500 \mathrm{~K}\), the fin material is copper with \(\varepsilon \sim 1\), and each fin has a perimeter of \(0.04 \mathrm{~m}\) and length of \(0.03 \mathrm{~m}\) :

1. Define an average fin temperature and calculate the fin tip temperature.

2. Using the new fin tip temperature, recalculate the radiative heat transfer coefficient and determine a new tip temperature.

3. How many iterations are required to converge to a final solution where the fin tip temperature changes by no more than \(1 \%\) ?