AME 60634

Int. Heat Trans. 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.

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AME 60634
Int. Heat Trans. Fins: The Fin Equation
• Solutions

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AME 60634
Int. Heat Trans. Bessel Equations

Form of Bessel equation of order 

d2y dy d2y dy
x 2

dx 2
+ x
dx
+ ( m x
2 2
- n 2
) y=0 with solution x2

dx 2
+ x
dx
+ ( m x
2 2
- n 2
) y=0

J = Bessel function of first kind of order 

Y = Bessel function of second kind of order 

Form of modified Bessel equation of order 

x
d2y
2
+ x
dy
- ( m x
2 2
- n 2
) y=0 with solution y ( x ) = C1In ( mx ) +C2 Kn ( mx )
dx 2
dx

I = modified Bessel function of first kind of order 

K = modified Bessel function of second kind of order 

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AME 60634
Int. Heat Trans.

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AME 60634
Int. Heat Trans. Bessel Functions – Recurrence Relations
ì n
ï mWn -1 ( mx ) - Wn ( mx ) W = J,Y, I
dé ï x
ë n ( )û í
W mx ù =
dx ï -mW ( mx ) - n W ( mx ) W = K
ïî n -1
x
n

OR
ì n
ï -mWn +1 ( mx ) + Wn ( mx ) W = J,Y, K
dé ï x
W
ë n ( mx ) ù
û = í
dx ï n
mWn +1 ( mx ) + Wn ( mx ) W = I
ïî x

AND

ì
dé n ï mx n
Wn -1 ( mx ) W = J,Y, I
ù
ë x Wn ( mx )û = í
ïî -mx Wn -1 ( mx ) W = K
n
dx

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AME 60634
Int. Heat Trans. 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
qf qf
hf º =
q f ,max hA f q b
• Fin Effectiveness
– ratio of the fin heat transfer rate to the heat transfer rate that would exist
without the fin
qf qf Rt,b
ef º = =
q f ,max hAc,bq b Rt, f

• Fin Resistance
– defined using the temperature difference between the base and fluid as
the driving potential
qb 1
Rt, f º =
qf hA f h f
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AME 60634
Int. Heat Trans. Fins: Efficiency

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AME 60634
Int. Heat Trans. Fins: Efficiency

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AME 60634
Int. Heat Trans. Fins: Arrays
• Arrays
– total surface area
N º number of fins
At = NA f + Ab
Ab º exposed base surface (prime surface)

– total heat rate

qb
qt = Nh f hA f qb + hAbq b = ho hAtq b =
Rt,o
– overall surface efficiency
NA f
ho = 1-
At
(1- h f )

– overall surface resistance

qb 1
Rt,o = =
qt hAtho

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AME 60634
Int. Heat Trans. Fins: Thermal Circuit
• Equivalent Thermal Circuit

• Effect of Surface Contact Resistance

q
qt = ho(c )hA f q b = b
Rt,o(c )

NA f æ h f ö
ho(c ) = 1- ç1- ÷
At è C1 ø
æ R¢¢ ö
C1 = 1- h f hA f ç t,c ÷
è Ac,b ø
1
Rt,o =
D. B. Go hAtho(c ) 10
AME 60634
Int. Heat Trans. sinh Function

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AME 60634
Int. Heat Trans. Fourier Series
Consider a set of eigenfunctions ϕn that are orthogonal, where orthogonality is
defined as
b

ò f ( x) f ( x) dx = 0
m n
for m ≠ n
a

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

eigenfunctions
¥
f ( x) = A0f0 ( x) + A1f1 ( x) + A2f2 ( x ) +... or f ( x ) = å Anfn ( x )
n=0
Due to orthogonality, we thus know
b b ¥ b

ò f ( x) f ( x) dx = ò f ( x) å A f ( x) dx = ò f ( x) A f ( x ) dx
n n n n n n n
a a n=0 a
all other ϕnAmϕm integrate
to zero because m ≠ n

Thus, the constants in the Fourier series are b

b b
ò f ( x ) f ( x ) dx
n

ò n ( ) ( ) n ò n ( x)
An = a
f x f x dx = A f 2
b

D. B. Go
a a ò f ( x)
2
n
12
a