Some electronic components is cooled by annular fins (heatsinks) which conduct heat away from the component and provide a larger surface area for that heat to dissipate to the surroundings.

The cooling efficiency of an annular fin of width $2w$ and inner and outer radii $r_0$ and $r_1$ may be written in terms of modified Bessel functions of the first and second kinds: $$ \eta = \frac{2r_0}{\beta(r_1^2-r_0^2)}\frac{K_1(u_0)I_1(u_1) - I_1(u_0)K_1(u_1)}{K_0(u_0)I_1(u_1)+I_0(u_0)K_1(u_1)}, $$ where $u_0 = \beta r_0$, $u_1 = \beta r_1$ and $$ \beta = \sqrt{\frac{h_c}{\kappa w}}. $$ $h_c$ is the heat transfer coefficient (which is taken to be constant over the fin's surface) and $\kappa$ is the thermal conductivity of the fin material.

What is the cooling efficiency of an aluminium annular fin with dimensions $r_0 = 5\;\mathrm{mm}$, $r_1 = 10\;\mathrm{mm}$, $w = 0.1\;\mathrm{mm}$? Take $h_c = 10\;\mathrm{W\,m^{-2}\,K^{-1}}$ and $\kappa = 200\;\mathrm{W\,m^{-1}\,K^{-1}}$.

Calculate the heat dissipation, $\dot{Q}$ (the product of the efficiency, the fin area and the temperature difference) for a component temperature of $T_0 = 400\;\mathrm{K}$ and ambient temperature $T_\mathrm{e} = 300\;\mathrm{K}$.