Extended SurfacesExtended Surfaces

Chapter ThreeChapter ThreeSection 3.6Section 3.6

3.6 Heat Transfer from Extended Surfaces (Fins) An extended surface (also know as a combined conduction-convection system or a

fin) is a solid within which heat transfer by conduction is assumed to be one dimensional, while heat is also transferred by convection (and/or radiation) from the surface in a direction transverse to that of conduction.

Why is heat transfer by conduction in the x-direction not, in fact, one-dimensional?

If heat is transferred from the surface to the fluid by convection, what surface condition is dictated by the conservation of energy requirement?

What is the actual functional dependence of the temperature distribution in the solid?

If the temperature distribution is assumed to be one-dimensional, that is, T = T(x) , how should the value of T be interpreted for any xlocation?

How does vary with x ?,cond xq

When may the assumption of one-dimensional conduction be viewed as an excellent approximation? The thin-fin approximation.

Extended surfaces may exist in many situations but are commonly used as fins to enhance heat transfer by increasing the surface areaavailable for convection (and/or radiation). They are particularly beneficial when is small, as for a gas and natural convection.

Some typical fin configurations:

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

, ,

( ) ( )

( ) (if k = a constant)

( )

=

cond x cond x x conv

x c

xx dx x c c

c c

conv s

c c

q q dqdTq kAdx

dq dT d dTq q dx kA kA dxdx dx dx dx

dT d dTkA k A dxdx dx dx

dq h dA T T

dT dT dkA kA kdx dx

+

+

= +

=

= + =

=

=

2

2

( ) + ( )

( ) ( / ) ( )

1 1 ( ) ( )( / ) ( ) =

= 0

or 0

c s

s

s

c

c

c c

dTA dx

dA dAd T dT h k T Tdx A dx d

h dA T Tdx dx

dAd dTor A h k T Tdx dx

x dx

d

A

x

+

A General Conduction Analysis

The Fin Equation Assuming one-dimensional, steady-state conduction in an extended

surface of constant conductivity and uniform cross-sectionalarea , with negligible generation and radiation , the fin equation is of the form (dAc/dx = 0, dAs/dx = P):

( )k( )cA ( 0)q = ( )0radq =

( )2

2 0 =c

d T hPT T

kAdx(3.62)

or, with and the reduced temperature ,( )2 / cm hP kA T T2

22 0

d mdx

= (3.64)

-1 2

1 2

The general solution, (x) = C C!

mx mxe eC C

+

Solutions (Table 3.4):

Base (x = 0) condition

( )0 b bT T =

Tip ( x = L) conditions

( )A. : Conv ect /i n |o x Lkd dx h L = =B. : / |Adiabati 0c x Ld dx = =

( )Fixed temperC. : atu re LL =

( )Infinite fin D. ( > 2.65) 0 : mL L =

Fin Heat Rate: ( )0 | f

f c x sA

dq kA h x dAdx == =

Fin Performance Parameters Fin Efficiency():

,max f ff

f f b

q qq hA

= (3.86)

Expressions for are provided in Table 3.5 for common geometries.f

( )1/ 2222 / 2fA w L t = + ( )/ 2pA t L=

( )( )

1

0

212f

I mLmL I mL

=

Consider a triangular fin:

Fin Resistance: ,1b

t ff f f

Rq hA

= (3.92)

Consider a straight fin with insulated tip:

,max

tanh( ) tanh( )

tanh( )corrected fin length: ( / 2)

f ff

f f b b

cc f

c

q q M mL mLq hA hpL mL

mLL L tmL

= = =

= + =

Fin Effectiveness ():

,

ff

c b b

qhA

with , and /f ch k A P

(3.81)

Circular FinCircular Fin

2

2

0 1 2 0

b

1 2 ( ) 0

The solution is composed of zero-order Bessel functions of the first and secondkinds.

( ) ( ) ( )( )

b

dT dT h T Tdr r dr kt

I mr K mr K mrT r TT T

+ =

+ =

1 2

0 1 1 2 0 1 1 2

( )( ) ( ) ( ) ( )

I mrI mr K mr K mr I mr+

Fin Arrays Representative arrays of

(a) rectangular and(b) annular fins.

Total surface area:

t f bA NA A= +

(3.99)Number of fins Area of exposed base (prime surface)

Total heat rate:

, ( ) bt f f b b b o t b

t oq N hA hA hA

R

= + = (3.100)

Overall surface efficiency and resistance:

,1b

t ot o t

Rq hA

= = (3.103)

( )1 1fo ft

NAA

= (3.102)

Equivalent Thermal Circuit :

Effect of Surface Contact Resistance:

( )( ),

bt t bo c

t o c

q hAR

= =

( )1

1 1f fo ct

NAA C

=

( )1 , ,1 /f f t c c bC hA R A = + (3.105b)

(3.104)

(3.105a)

Effect of Surface Contact Resistance:

,

,, ,

,,

, ,

; ( )

( ) ( )( )

[1/( )]

( )

[ /( )]

t f N b b b b b b

c bf N f f c b

f f

b c bf N

t c c b

q q q q hA T T hA

T Tq N hA T T

N hA

T Tq

R NA

= + = =

= =

=

Tc,b

, ,

, , , ,

, , , ,

,, ,

1 (1) ( ) [ ] ( )

(2) ( ) [ /( )]

1( ) [ ] [ /( )]( )

[1/( )] [ /(

c b f Nf f

b c b f N t c c b

b b f N f N t c c bf f

bf N

f f t c c b

T T qN hA

T T q R NA

T T q q R NAN hA

qN hA R NA

=

=

= +

=+

,, ,

, ,

, ,

)]

= [1/( )] [ /( )]

(1 ) ( ) [ ]1 ( )( / )

= 1 ( )[1 ]1 ( )( / )

bt b f N b b

f f t c c b

f f fto

t b t t f f t c c b

f f

t f f t c c b

q q q hAN hA R NA

NA NAqhA A A hA R A

NAA hA R A

= + ++

= ++

+

3.7 3.7 PennesPennes BioheatBioheat EquationEquation2

2 0 where metabolic heat source

perfusion heat source

=

( )

m p

a

m

p b bq c

q q

T

d T qdx

Tk

=

++ = =

3 blood perfusion rate (m /s) blood density blood specific heat

b

bc

==

=

2

2

22

2

blood temperature tissue temperature

( )

0

m, 0

a

m b b a

TT

q c T Td Tdx k

converti g dx

nd

=

=

+

=

=

+

2

where

( fin equat

/ ; m =

ion

/

)

a m b b b bT T q c c k

Problem 3.132: Determination of maximum allowable power for a 20mm x 20mm electronic chip whose temperature is not to exceedwhen the chip is attached to an air-cooled heat sink with N=11 fins of prescribed dimensions.

cq85 C,cT =

Schematic:

Assumptions: (1) Steady-state, (2) One-dimensional heat transfer, (3) Isothermal chip, (4)Negligible heat transfer from top surface of chip, (5) Negligible temperature rise for air flow, (6) Uniform convection coefficient associated with air flow through channels and over outer surface of heat sink, (7) Negligible radiation.

Analysis: (a) From the thermal circuit,

c cc

tot t,c t,b t,o

T T T Tq

R R R R = =

+ +

( )2 6 2 2t,c t,cR R / W 2 10 m K / W / 0.02m 0.005 K / W= = =

( )2t,b bR L / k W= ( )W / m K 20.003m /180 0.02m 0.042 K / W= =From Eqs. (3.103), (3.102), and (3.99)

( )ft,o o f t f bo t t

N A1R , 1 1 , A N A Ah A A

= = = +

Af = 2WLf = 2 0.02m 0.015m = 6 10-4 m2 Ab = W2 N(tW) = (0.02m)2 11(0.182 10-3 m 0.02m) = 3.6 10-4 m2 At = 6.96 10-3 m2

Comments: The heat sink significantly increases the allowable heat dissipation. If it were not used and heat was simply transferred by convection from the surface of the chip withfrom Part (a) would be replaced by

2tot100 W/m K, 2.05 K/W= =h R

21/ hW 25 K/W, yielding 2.60 W.cnv cR q= = =

( )( )c

85 20 Cq 31.8 W

0.005 0.042 2.00 K / W

= =+ +

Rt,o = 2.00 K/W, and

o = 0.719,

ff

f

tanh mL 0.824 0.704mL 1.17

= = =

With mLf = (2h/kt)1/2 Lf = (200 W/m2K/180 W/mK 0.182 10-3m)1/2 (0.015m) = 1.17, tanh mLf = 0.824 and Eq. (3.87) yields