AME  60634    Int.  Heat  Trans.  

D.  B.  Go    1    

Fins: Overview •  Fins

–  extended surfaces that enhance fluid heat transfer to/from a surface in large part by increasing the effective surface area of the body

–  combine conduction through the fin and convection to/from the fin

•  the conduction is assumed to be one-dimensional

•  Applications –  fins are often used to enhance convection when h is

small (a gas as the working fluid) –  fins can also be used to increase the surface area

for radiation –  radiators (cars), heat sinks (PCs), heat exchangers

(power plants), nature (stegosaurus)

Straight fins of (a) uniform and (b) non-uniform cross sections; (c) annular fin, and (d) pin fin of non-uniform cross section.

AME  60634    Int.  Heat  Trans.  

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Fins: The Fin Equation •  Solutions

AME  60634    Int.  Heat  Trans.  

D.  B.  Go    3    

x2 d2ydx2

+ x dydx+ m2x2 −ν 2( ) y = 0

Bessel Equations

with solution x2 d2ydx2

+ x dydx+ m2x2 −ν 2( ) y = 0

Form of Bessel equation of order ν

Jν = Bessel function of first kind of order ν Yν = Bessel function of second kind of order ν

x2 d2ydx2

+ x dydx− m2x2 −ν 2( ) y = 0 with solution y x( ) =C1Iν mx( )+C2Kν mx( )

Form of modified Bessel equation of order ν

Iν = modified Bessel function of first kind of order ν Kν = modified Bessel function of second kind of order ν

AME  60634    Int.  Heat  Trans.  

D.  B.  Go    4    

AME  60634    Int.  Heat  Trans.  

D.  B.  Go    5    

ddx

Wν mx( )!" #$=mWν−1 mx( )−

νxWν mx( ) W = J,Y, I

−mWν−1 mx( )−νxWν mx( ) W = K

&

'((

)((

Bessel Functions – Recurrence Relations

OR

ddx

Wν mx( )!" #$=−mWν+1 mx( )+

νxWν mx( ) W = J,Y,K

mWν+1 mx( )+νxWν mx( ) W = I

&

'((

)((

AND

ddx

xνWν mx( )!" #$=mxνWν−1 mx( ) W = J,Y, I−mxνWν−1 mx( ) W = K

&

'(

)(

AME  60634    Int.  Heat  Trans.  

D.  B.  Go    6    

Fins: Fin Performance Parameters •  Fin Efficiency

–  the ratio of actual amount of heat removed by a fin to the ideal amount of heat removed if the fin was an isothermal body at the base temperature

•  that is, the ratio the actual heat transfer from the fin to ideal heat transfer from the fin if the fin had no conduction resistance

•  Fin Effectiveness –  ratio of the fin heat transfer rate to the heat transfer rate that would exist

without the fin

•  Fin Resistance

–  defined using the temperature difference between the base and fluid as the driving potential

€

η f ≡qfqf ,max

=qf

hAfθb

€

ε f ≡qfqf ,max

=qf

hAc,bθb=Rt,bRt, f

€

Rt, f ≡θbqf

=1

hAfη f

AME  60634    Int.  Heat  Trans.  

D.  B.  Go    7    

Fins: Efficiency

AME  60634    Int.  Heat  Trans.  

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Fins: Efficiency

AME  60634    Int.  Heat  Trans.  

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Fins: Arrays •  Arrays

–  total surface area

–  total heat rate

–  overall surface efficiency

–  overall surface resistance

€

At = NAf + Ab

€

qt = Nη f hAfθb + hAbθb =ηohAtθb =θbRt,o

€

Rt,o =θbqt

=1

hAtηo

€

N ≡ number of finsAb ≡ exposed base surface (prime surface)

€

ηo =1−NAfAt

1−η f( )

AME  60634    Int.  Heat  Trans.  

D.  B.  Go    10    

Fins: Thermal Circuit •  Equivalent Thermal Circuit

•  Effect of Surface Contact Resistance

€

Rt,o =1

hAtηo(c )

€

ηo(c ) =1−NAfAt

1−η fC1

$

% &

'

( )

€

C1 =1−η f hAf$ $ R t,c

Ac,b

%

& '

(

) *

€

qt =ηo(c )hAfθb =θbRt,o(c )

AME  60634    Int.  Heat  Trans.  

D.  B.  Go    11    

sinh Function

AME  60634    Int.  Heat  Trans.  

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Fourier Series Consider a set of eigenfunctions ϕn that are orthogonal, where orthogonality is defined as

φm x( )φn x( )dx = 0a

b

∫ for m ≠ n

An arbitrary function f(x) can be expanded as series of these orthogonal eigenfunctions

f x( ) = A0φ0 x( )+ A1φ1 x( )+ A2φ2 x( )+... or f x( ) = Anφn x( )n=0

∞

∑Due to orthogonality, we thus know

φn x( ) f x( )dxa

b

∫ = φn x( ) Anφn x( )n=0

∞

∑ dxa

b

∫ = φn x( )Anφn x( )dxa

b

∫all other ϕnAmϕm integrate to zero because m ≠ n

Thus, the constants in the Fourier series are

φn x( ) f x( )dxa

b

∫ = An φn2 x( )a

b

∫ An =φn x( ) f x( )dx

a

b

∫

φn2 x( )

a

b

∫