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Heat Transfer from Fins(Chapter 3)Zan Wu [email protected] Room: 5123
Fins
Fins/Extended surfaces
Why not called as convectors?
Radiators
Fins
Fan cooling is not sufficient for advanced microprocessors
Microfins
Microfin copper tube
Carbon nanotube microfinson a chip surface
Fin analysis
Two basic questions What is the rate of heat dissipated by the fin? What is the variation in the fin temperature from
the fin base to the fin tip?
Rectangular fin
2
2 0 (3 31)d Cdx A
x d x
L
t 1
Q 1
.
t f
bZ
Energy balance on the element from x to x + dx
A: area of a cross section normal to xC: perimeter of this section
)tt( fSteady state1D
Cont’d
Boundary conditions:
Assume a long and thin fin, the heat transferred at the fin tip is negligible
)313(0AC
dxd
2
2
b2
bZZ2
ACm 2
)tt(dxdt:Lx fLx
0dxdt
Lx
f111 tttt:0x
x d x
L
t 1
Q 1
.
t f
bZ
Rectangular fin
Solution:
co sh2
sin h2
m x m x
m x m x
e em x
e em x
1 2
3 4cosh sinh
m x m xC e C e
C m x C m x
Hyperbolic functions
At x = L = 2
1 1
cosh ( ) (3 38)cosh
f
f
t t m L xt t m L
2
1
1c o s h m L
heat transfer from the fin?Q
1 10
sinh ( )coshx
d m LQ A A mdx m L
CmA
1 1 1tanh 2 tanh (3 40)Q C A mL b Z mL
Rectangular fin
Rectangular fin
= 25 W/m2K, b = 2 cm, L = 10 cm
Rectangular fin
If the condition below is used, i.e., to consider heat loss from the fin tip
one has
and
and
LxLxdx
d
)413(mLsinh
mmLcosh
)xL(msinhm
)xL(mcosh
1
)423(mLsinh
mmLcosh
1
1
2
)433(mLtanh
m1
mLtanhmAmQ 11
Fins on Stegosaurus
Those plates absorb radiation from the sun or cool the blood?
Practical considerations
e.g., How to choose a fin material?How to optimize fins?
Criterion for benefit
Fig. 3-13. Arrangement of rectangular fins
1
p referab le if
0dQdL
1 ( )Q function LL
.
Z
t 1
b
1Q
Fin effectiveness, fin efficiency
1
1
from the finfrom the base area w ithout the fin
1
1
from the finfrom a similar fin but with λ
Criterion: maximum heat flow at a given mass
M = b L Z = Z A1 A1 = b L, Z, are given.
Find maximum for constant A1 = bL.
C 2Z , A = bZ
mLtanhACQ 11
b2
ACm 2
1Q
b
Ab
2tanhZb2Q 111
LZb
Optimal rectangular fin
Cont’d
Condition
1 0 gives optimum
after some algebra one obtains
21.419 (3 55)/ 2
dQdb
Lb b
Fin material selection
After some algebra one finds:
)523(b
Ab
2tanhZb2
mLtanhAmQ
11
11
12 1 .4 1 9Aub b
1from the condition / 0dQ db
)a613(4
1utanh
uZ1QA 233
3
1
11
For an optimized rectangular fin
Cont’d
M = b L Z = Z A1 =
/ is the material parameter see Table 3-1.
Aluminum instead of Copper. / Aluminum: 11.8; Copper: 23.0
Why not Magnesium? / Magnesium: 10.2
232
3
1
1
41
utanhu
Z1Q
Straight triangular fin
= t tf
Heat balance
Solution:
K0 as x 0 B = 0 because is finite for x = 0
x = L and = 1
)623(0bL2
x1
dxd
x1
dxd
2
2
bL2
)x2(BKx2AI 00
L2AI 01
LxbZA
L
d x
x
b t 1
t f
1Q
Bessel differential equation
I0 and K0 are the modified Bessel functions of order zero
Triangular fin
)L2(IA
0
1
)653()L2(I)x2(I
0
0
1
Lx1 dx
dtAQ
)663()L2(I)L2(Ib2ZQ
0
111
Table 3.2 for numerical values of Bessel functions
Recap)383(
mLcosh)xL(mcosh
ft1tftt
1
b22m
)403(mLtanh1Zb21Q
mLtanhb
2
mLmLtanh
21.419 (3 55)/ 2L
b b
)653()L2(0I)x2(0I
1
bL2
11 1
0
( 2 )2 (3 -6 6 )
( 2 )I L
Q b ZI L
)L2(0I)L2(1I
b2
L)L2(0I/)L2(1I
21 .309 (3 67 )/ 2L
b b
Optimal fin: Maximum heat transfer at fixed fin mass
mL = 1.419 mL = 1.309
24
Circular or annular fins
Heat conducting area
A = 2r b
Convective perimeter
C = 2 2r = 4r
r 1 r 2
b
Fin efficiency for circular fins
How to use the fin efficiency in engineering calculations
s
flänsarareaoflänsad
QQQ
QQQ
finareaunfinned
( )
( )
fins b f
b b f fins
A t t
Q A t t Q
( ) ( 3 7 1)b f b f in sQ t t A A
Graphene