Fins extend from a base material into a convective fluid, enhancing heat transfer from the base to the fluid. We may observe an energy balance between the conduction into and out of a section and the convection to the surrounding fluid, as depicted in the following equation. When the cross section and thus perimeter are constant, a simpler for results.
$$ \frac d{dx}(A_c\frac{dT}{dx})-\frac{hP}{k}(T-T\infin)=0~~~~~~\frac{d^2T}{dx^2}-\frac{hP}{kA_c}(T-T\infin)=0 $$
To further simplify, we will let $\theta=T-T_\infin$. Additionally, since the parameter will continue to appear in out analysis of fins, we will define the quantity $m$.
$$ \frac{d^2\theta}{dx^2}-m^2\theta=0,~~~m=\sqrt{\frac{hP}{kA_c}} $$
This differential equation and its solutions should be familiar, and are as follows.
$$ \theta=C_1e^{mx}+C_2e^{-mx} $$
The simplest solution derivation entails assuming that the fin is effectively infinite in length.
$$ \lim_{x\rarr\infin}\theta(x)=0,~~\theta(0)=T_b-T_\infin\Rarr C_1=0,~~C_2=T_b-T_\infin\Rarr\\
$$
$$ \frac{T(x)-T_\infin}{T_b-T_\infin}=e^{-mx} $$
An insulated tip boundary condition is a much safer assumption that the infinite length assumption, and not too much more complicated. Note that $\cosh=(e^x+e^{-x})/2$.
$$ \begin{matrix} \theta'(L)=0\Rarr C_1e^{mL}-C_2e^{-mL}=0\\ \theta(0)=T_b-T_\infin\Rarr C_1+C_2=T_b-T_\infin \end{matrix} \Rarr\\ \begin{bmatrix}C_1\\C_2\end{bmatrix}= \begin{bmatrix} e^{mL}&-e^{-mL}\\1&1 \end{bmatrix}^{-1} \begin{bmatrix}0\\T_b-T_\infin\end{bmatrix}=\frac{T_b-T_\infin}{e^{mL}+e^{-mL}} \begin{bmatrix}e^{-mL}\\e^{mL}\end{bmatrix}\Rarr $$
$$ \frac{T(x)-T_\infin}{T_b-T_\infin}=\frac{\cosh(m(L-x))}{\cos(mL)} $$
Other profiles, such as convection from the tip and prescribed tip temperature, are contained in the formula sheet.
For a fin or series of fins on a base surface, we define efficiency and effectiveness metrics. The fin efficiency is the ratio of (1) the heat transfer through the fins to (2) the heat transfer that would occur on the un-extended base surface. The overall efficiency is the ratio of (1) the heat transfer through the fins and un-extended surface to (2) the total heat transfer that would occur were all the fins surfaces and un-extended surfaces at the base temperature. The fin effectiveness is the ration of (1) the heat transfer through the fin to (2) the heat transfer that would occur through the cross sectional area of the fins were they at the base temperature