Fin heat dissipation, efficiency and effectiveness

Fins

A fin is an extended surface that increases the area for convection, so that more heat leaves a hot surface. Heat flow $Q = hA\Delta T$ can be increased by raising $h$, $\Delta T$ or $A$; fins increase $A$ when $h$ is low (gas side). Examples: engine cylinder fins, radiator tubes, heat sinks.

Types: straight (rectangular, triangular, parabolic profile), pin (cylindrical or tapered), annular (circumferential) fins on tubes.

Derivation (uniform fin, insulated tip)

Assumptions: steady one-dimensional conduction along the fin, constant $k$ and $h$, uniform fin temperature across thickness. Let $A_c$ be the cross-section, $P$ the perimeter, $T_b$ the base temperature and $\theta = T - T_\infty$.

Energy balance on a slice $dx$: heat conducted in = heat conducted out + heat convected:

$$-kA_c\frac{dT}{dx} = -kA_c\left(\frac{dT}{dx}+\frac{d^2T}{dx^2}dx\right)+hP\,dx\,(T-T_\infty)$$ $$\frac{d^2\theta}{dx^2}-m^2\theta = 0,\qquad m=\sqrt{\frac{hP}{kA_c}}$$

Solution: $\theta = C_1e^{mx}+C_2e^{-mx}$. Boundary conditions: $\theta = \theta_b$ at $x = 0$; $d\theta/dx = 0$ at $x = L$ (insulated tip). This gives

$$\frac{\theta}{\theta_b} = \frac{\cosh m(L-x)}{\cosh mL}$$

Heat dissipated equals conduction at the base:

$$Q_{fin} = -kA_c\frac{d\theta}{dx}\Big|_{x=0} = \sqrt{hPkA_c}\;\theta_b\tanh(mL)$$

Other tip conditions: infinitely long fin $Q = \sqrt{hPkA_c}\,\theta_b$; convecting tip, use corrected length $L_c = L + A_c/P$ in the above formula.

Efficiency and effectiveness

$$\eta_{fin}=\frac{Q_{fin}}{Q_{max}}=\frac{Q_{fin}}{hPL\,\theta_b}=\frac{\tanh mL}{mL} \qquad \varepsilon_{fin}=\frac{Q_{fin}}{hA_c\theta_b}$$

Efficiency is the actual heat loss over the heat that would be lost if the entire fin were at the base temperature. Effectiveness is the heat with the fin over the heat without the fin.

A fin is worthwhile if $\varepsilon_{fin} > 2$ (preferably much greater). This needs high $k$ (copper, aluminium), thin closely spaced fins, and low $h$ on the fin side (gas, free convection). Fins are useless when $h$ is high, e.g. boiling or condensation.