Mathematical Problems in Engineering

Research Article

Thermal Characterizations of Exponential Fin Systems

Table 1

Efficiencies of exponential fins compared to efficiencies of common fins.


Fin type	Cross-sectional area	$m L_{\infty}$	$η_{f}$
Fin type	Perimeter	$m L_{\infty}$	$η_{f}$

Rectangular straight or pin fins with insulated ${tips}^{(i)}$ [20]	$\begin{matrix} 2 H t \\ 4 H \end{matrix},$ $\begin{matrix} π r_{b}^{2} \\ 2 π r_{b} \end{matrix}$	2.65	$\frac{tanh (m L)}{(m L)}$

Triangular straight ${fin}^{(i)}$ [20]	$\begin{matrix} H 0.5 t_{o} {1 - x / L} \\ 2 H \end{matrix}$	—	$\frac{1}{(m L)} \frac{I_{1} (2 m L)}{I_{0} (2 m L)}$

Triangular pin ${fin}^{(ii)}$ [20]	$\begin{matrix} π r_{b}^{2} {1 - (\frac{x}{L})}^{2} \\ 2 π r_{b} {1 - (\frac{x}{L})} \end{matrix}$	—	$\frac{2}{(m L)} \frac{I_{2} (2 m L)}{I_{1} (2 m L)}$

Parabolic straight ${fin}^{(i)}$ [20]	$\begin{matrix} H 0.5 t_{o} {1 - {(x / L)}^{2}} \\ 2 H \end{matrix}$	—	$\frac{2}{{(4 {(m L)}^{2} + 1)}^{0.5} + 1}$ f

Parabolic pin ${fin}^{(ii)}$ [20]	$\begin{matrix} π r_{b}^{2} {1 - {(\frac{x}{L})}^{2}}^{2} \\ 2 π r_{b} {1 - {(\frac{x}{L})}^{2}} \end{matrix}$	—	$\frac{2}{{([4 / 9] {(m L)}^{2} + 1)}^{0.5} + 1}$

Exponential straight ${fin}^{(i) (1)}$	$\begin{matrix} 2 H_{b} e^{- b x} t \\ 4 H_{b} e^{- b x} \end{matrix}$	$\begin{matrix} (\frac{X}{\sqrt{4 X^{2} + 1}}) (5.293 - \ln [1 + (\frac{1}{\sqrt{4 X^{2} + 1}})]); b > 0, \\ (\frac{- X}{\sqrt{4 X^{2} + 1}}) (5.293 - \ln [1 - (\frac{1}{\sqrt{4 X^{2} + 1}})]); b < 0 \end{matrix}$	$\begin{matrix} \frac{0.99 {- 1 + \sqrt{4 X^{2} + 1}}}{2 X^{2} [1 - Exp (- b {(L_{\infty})}_{1})]}, b > 0 \\ \frac{0.99 {1 + \sqrt{4 X^{2} + 1}}}{2 X^{2} [Exp (- b {(L_{\infty})}_{2}) - 1]}, b < 0 \end{matrix}$
Exponential pin ${fin}^{(ii) (1)}$	$\begin{matrix} π r_{b}^{2} e^{- 2 b x} \\ 2 π r_{b} e^{- b x} \end{matrix}$	$\begin{matrix} 0.8233 (\frac{X^{0.8804} - 0.0047}{0.2945 X^{0.8934} + 0.2428}); b > 0, \\ 0.7237 (\frac{X^{0.6906} + 3.3301 X^{1.4019} + 0.3939 X^{0.6902} - 0.0302}{0.9547 X^{1.3923} + e^{- 0.6311 X^{1.9309}} - 0.7425}); b < 0 \end{matrix}$	$\begin{matrix} \frac{0.99}{(1 - e^{- m L_{\infty} / X})} \\ \times \frac{1}{X^{2}} (\frac{X}{2}) {[\frac{K_{1} (2 X)}{K_{2} (2 X)} \\ + (\frac{K_{3} (2 X)}{K_{2} (2 X)})]} - 1, \\ b > 0 \\ \frac{0.99}{(e^{- m L_{\infty} / X} - 1)} \\ \times \frac{1}{X^{2}} {\frac{- X}{2} [\frac{I_{1} (- 2 X)}{I_{2} (- 2 X)} \\ + \frac{I_{3} (- 2 X)}{I_{2} (- 2 X)}] + 1}, b < 0 \end{matrix}$

Exponential decaying temperature-pin ${fin}^{(ii) (2)}$	$\begin{matrix} π r_{b}^{2} {X^{2} + [1 - X^{2}] \\ {\times e^{- 0.5 a x_{o} x}}}^{2} \\ 2 π r_{b} {X^{2} + [1 - X^{2}] \\ \times e^{- 0.5 a x_{o} x}} \end{matrix}$	$\begin{matrix} (2 X) \ln [\frac{X^{2}}{(X^{2} - 1)}]; X > 1.0 \\ 4.605 X; X \leq 1.0 \end{matrix}$	$\begin{matrix} \frac{1}{2 X^{2}} {X^{2} \ln [\frac{X^{2}}{(X^{2} - 1)}] - 1}^{- 1}, X > 1.0 \\ \frac{1}{2 X^{2}} (9.0 - 6.697 X^{2}), X \leq 1.0 \end{matrix}$

{}^{(i)}m = \sqrt{2 h / (k t)};

{}^{(ii)}m = \sqrt{2 h / (k r_{b})};

{}^{(1)}X = m / b;

{}^{(2)}X = m / a