Extended Surface Equation Table (3.4)

The following are for extended surfaces of a uniform cross-section.

Case	Tip Condition (x = L)	Temperature Distribution θ/θ_b		Fin Heat Transfer Rate q_f
A	Convection: hθ(L) = −kdθ/dx\|_x=L	$\frac{\cosh m (L - x) + (h / m k) \sinh m (L - x)}{\cosh m L + (h / m k) \sinh m L}$	(3.75)	$M \frac{\sinh m L + (h / m k) \cosh m L}{\cosh m L + (h / m k) \sinh m L}$	(3.77)
B	Adiabatic: dθ/dx\|_x=L = 0	$\frac{\cosh m (L - x)}{\cosh m L}$	(3.80)	$M \tanh m L$	(3.81)
C	Prescribed temperature: θ(L) = θ_L	$\frac{(θ_{L} / θ_{b}) \sinh m x + \sinh m (L - x)}{\sinh m L}$	(3.82)	$M \frac{(\cosh m L - θ_{L} / θ_{b})}{\sinh m L}$	(3.83)
D	Infinite fin (L → ∞): θ(L) = 0	$e^{- m x}$	(3.84)	$M$	(3.85)
$\begin{matrix} θ \equiv T - T_{\infty} & m^{2} \equiv h P / k A_{c} \\ θ_{b} = θ (0) = T_{b} - T_{\infty} & M \equiv \sqrt{h P k A_{c}} θ_{b} \end{matrix}$			A table of hyperbolic functions is given in Appendix B.1.

In the above table referrers to hyperbolic sin of mL or . Same for cosine.

is perimeter of the cross-sectional area.