Heat Transfer from Fins(Chapter 3)Zan Wu zan.wu@energy.lth.se 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

QQ

1

1

from the finfrom a similar fin but with λ

QQ

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