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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.
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AME 60634 Int. Heat Trans.
D. B. Go 2
Fins: The Fin Equation • Solutions
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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
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AME 60634 Int. Heat Trans.
D. B. Go 4
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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
&
'(
)(
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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
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AME 60634 Int. Heat Trans.
D. B. Go 7
Fins: Efficiency
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AME 60634 Int. Heat Trans.
D. B. Go 8
Fins: Efficiency
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AME 60634 Int. Heat Trans.
D. B. Go 9
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( )
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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 )
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AME 60634 Int. Heat Trans.
D. B. Go 11
sinh Function
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AME 60634 Int. Heat Trans.
D. B. Go 12
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
∫
