Heat Transfer Fins

Uses

Used for dissipating heat very effectively by maximizing surface area and using conduction and convection!

Excess temperature is the driving factor for heat transfer

Primarily makes use of…
$Q_{cond}^{\prime }=\frac{k}{L}\left(T_s-T_{\infty }\right)$
$Q_{conv}^{\prime }=\overline{h}\left(T_s-T_{\infty }\right)$
If we want to increase $\dot{Q}$ without changing $\Delta T$ or h, we need to increase the cross sectional area in….
${\dot{Q}}_{conv}=\bar{h}A(T_s-T_{\infty })$

Fin Temperature Distributions

Very long fin

$$\frac{T(x)-T_{\infty }}{T_b-T_{\infty }}=e^{-x\sqrt{\frac{hp}{k{A}_c}}}=e^{-xm}$$

Adiabatic Fin Tip

$$\frac{T(x)-T_{\infty }}{T_b-T_{\infty }}=\frac{\cosh m(L-x)}{\cosh mL}$$

Specified T at Fin Tip

$$\frac{T(x)-T_{\infty }}{T_b-T_{\infty }}=\frac{\left[\frac{T_L-T_{\infty }}{T_b-T_{\infty }}\right]\sinh mx+\sinh m(L-x)}{\sinh mL}$$

Convection From Fin TIp

$$\frac{T(x)-T_{\infty }}{T_b-T_{\infty }}=\frac{\cosh m(L-x)+\left(\frac{h}{mk}\right)\sinh m(L-x)}{\cosh mL+\left(\frac{h}{mk}\right)\sinh mL}$$

Where $m=\sqrt{hp/k{A}_c}$
Note that p is perimeter here!

Temperature distribution

The temperature distribution at a distance can be defined like this $\theta =T\left(x\right)-T_{\infty }$

Boundary Conditions

At the fin tip

1. Very long fin: ${\lim }_{x\to \infty }\theta \left(x\right)=T\left(x\right)-T_{\infty }=0$
2. Adiabatic fin tip: $\frac{d\theta }{dx}=0$
3. Specified T at fin tip: $\theta \left(L\right)=const$
4. Convection from fin tip: $-k\frac{d\theta }{dx}=h{\theta }_L$

Fin Heat Transfer Rates

Very Long Fin

$${\dot{Q}}_{\text{long fin}}=-k{A}_c{\frac{dT}{dx}|}_{x=0}=\sqrt{hpk{A}_c}(T_b-T_{\infty })$$

Adiabatic Fin

$${\dot{Q}}_{\text{adiabatic tip}}=-k{A}_c{\frac{dT}{dx}|}_{x=0}=\sqrt{hpk{A}_c}(T_b-T_{\infty })\tanh mL$$

Specified T at Fin Tip

$${\dot{Q}}_{\text{specified temp.}}=\sqrt{hpk{A}_c}(T_b-T_{\infty })\frac{\cosh mL-\left[\frac{T_L-T_{\infty }}{T_b-T_{\infty }}\right]}{\sinh mL}$$

Convection From Fin Tip

$${\dot{Q}}_{\text{convection}}=\sqrt{hpk{A}_c}(T_b-T_{\infty })\frac{\sinh mL+\left(\frac{h}{mk}\right)\cosh mL}{\cosh mL+\left(\frac{h}{mk}\right)\sinh mL}$$

Conduction and convection both occur as a function of location x so D*O NOT PUT THEM IN SERIES!!!

Example

See lecture notes for an example on how to solve these

Fin Efficiency

${\eta }_{fin}={\frac{Q_{\text{fin}}^{\prime }}{h{A}_f\left(T_B-T_{\infty }\right)}}_{}^{}=\frac{\text{Heat Transfer with fin}}{\text{Heat Transfer of the base without the fin}}$
This describes how well a fin performs compared to the ideal scenario.
$A_f$ in this case is the total fin area

Fin Effectiveness

${ϵ}_{fin}={\frac{Q_{\text{fin}}^{\prime }}{h{A}_B\left(T_B-T_{\infty }\right)}}_{}^{}=\frac{\text{Heat Transfer rate with Fin}}{\text{Heat Transfer rate of a wall with the same cross sectional area}}$

Corrected Fin Length

Taking a cross-sectional area at the tip, $A_c$ and accounting for convection from the area on the side of the fin…
$A_{corr}=A_{conv}+A$
$P{L}_{corr}=PL+A_c$
Then apply a corrected in length… $L_c=L+\frac{A_c}{P}$

Multiple Fins

${ϵ}_{fin,overall}={\frac{Q{\prime }_{\text{with fins, total}}}{Q_{\text{no fins, total}}^{\prime }}}_{}^{}$
${ϵ}_{fin,overall}=\frac{h\left(A_{unfin}+{\eta }_{fin}n{A}_{fin}\right)\left(T_B-T_{\infty }\right)}{h\left(A_{nofins}\right)\left(T_B-T_{\infty }\right)}$
There is a limit to the amount of fins that is optimal to dissipate heat

${ϵ}_{fin,overall}={\frac{A_{unfin}+{ϵ}_{fin}nAc}{A_{nofins}}}_{}$
Where unfin is basically the geometry without the fin where it used to be

Heat Sinks

$Q_{HeatSink}^{\prime }={\frac{T_s-T\infty }{R}}_{}$
This is design based where you select one heat sink from multiple given.