heat transfer fins

8/11/2019 heat transfer fins

1/13

UNIT III STEADY STATE CONDUCTION WITH

HEAT GENERATIONIntroduction, I.D. heat conduction with heat sources, Extended surfaces

(fins), Fin effectiveness, 2-D heat conduction, Numericals.

1


2/13

.1 Introduction to conduction with heat !eneration within the

s"stem

#an" $ro%lems encountered in heat transfer re&uire an anal"sis that ta'e into account

!eneration and a%sor$tion of heat within the s"stem. he exam$les are

(i) *"stem in which electrical current flows.(ii) Nucleate reactors

(iii) +om%ustion $rocesses

(iv) +hemical $rocesses(v) Dr"in!

(vi) *ettin! of concrete etc.

.2 lane wall with heat source

et us consider a sla% of co$$er in contact with two fluids. et current I $ass throu!h the sla%and heat !enerated $er unit volume is q . he !eneral non-stead" conduction e&uation inrectan!ular coordinates is

=

+

+

+ t

K

q

z

t

y

t

x

t 12

2

2

2

2

2

For stead" state when tem$erature varies onl" x-direction, a%ove e&uation reduces to

2

2

=

+K

q

xd

td

wo %oundar" conditions are

uttin! two %oundar" conditions in E&. (1) we !et

andClCK

lqts )2(....................................

2 21

2

2 ++

=

2

At x = l , t = ts2At x = -l , t = ts1From a%ove e&uation

K

q

xd

td =

2

2

Inte!ratin!, 1CxK

q

xd

td+

=

Inte!ratin! a!ain,

)1...(..........2

21

2

CxCx

K

qt ++

=


3/13

)(....................................2

21

2

1 ClCK

lqts +

=

/ddin! E&. (2) and E&. (), we !et

K

lqCtt ss

2

212 2

=+

2122

22lK

qttCor ss ++=

uttin! this value of +2in E&. (2), we !et

222

12

2

1

2

2

ss

s

tt

K

lqlC

K

lqt

++

++

=

l

ttC ss

2

12

1

=

*u%stitutin! values of +1and +2 in E&. (1), we o%tain

( )K

lqtt

l

xtt

x

K

qt ss

ss22222

2

12

12

2 +

+++

=

( )222

1212222

ssss ttlxttxl

Klqt +++=

/ll the heat !enerated with the wall sla% must %e convected awa" to the surroundin!s. /lso heat

conducted to each wall surface is dissi$ated to the surroundin!s %" convection. +onse&uentl", the

maximum tem$erature must occur at the center at x 0 or at x 0 , =dx

dt. Now, we will find

tem$erature distri%ution in terms of maximum tem$erature to instead of ts.

hen x 0 , t 0 t, so from E&. ()

stlK

qt +

= 2

2

2

2

lK

qttor

s

=

( )

+

+

=

ss tlK

qtxl

K

qttAlso

222

22

2

2x

K

q=

If ts10 ts20 ts

( ) ).....(..........2

22

stxlK

qt +

=

or ( )222

xlK

qtt s

=

( )222

xlK

q

=


4/13

K

lq

K

xq

tt

ttsos 22

22

=

)3.......(....................

2

=

l

x

tt

ttro

s

If we want to use E&. () or E&. (3), surface tem$erature, ts

, must %e 'nown in terms of surroundin!tem$erature tand convective heat transfer coefficient, h. From ener!" %alance,

otal heat !enerated 0 4eat convected awa" from the faces

)()2()2( = ttAhlAqor s

+

= th

lqtor s

From E&.(), )(2

22xl

K

qtt s

=

uttin! in this e&uation, the a%ove value of ts, we o%tain

l

lqxl

K

qtt

+

= )(

2

22

l

lqxl

K

q += )(2

22

/t x 0 , 0 max, therefore

h

lql

K

q +

= 2

2max

. +"linder with 4eat *ource/ current- carr"in! wire or a fuel element in a nuclear reactor ma" re$resent the s"stem. et us a

consider a c"linder of radius of radius rowith uniforml" distri%uted heat sources. If c"linder is ver"

lon!, the tem$erature ma" %e considered as function of radius onl". he !eneral non-stead" state

conduction e&uation in c"lindrical coordinates is

=

=

+

+

+

+ tt

k

c

k

q

z

tt

rr

t

rr

t 1112

2

2

2

22

2

In stead" state and tem$erature variation onl" in radial direction, a%ove e&uation reduces

to

or

K

q

dr

dt

rdr

d,

12

2

=

++

orK

qr

dr

dt

dr

dr ,

2

2

=

++

K

rq

dr

dtr

dr

d =

5n inte!ratin!, we !et


5/13

C

r

K

q

dr

dtr +

=

2

2

,,,, mesuationbecoandaboveeqCsodr

dtratdiscussedalreadyAs ===

2

2r

K

q

dr

dtr

=

getweegratingon

r

K

q

dr

dtor ,int,

2

=

1

2

C

K

rqt +=

other %oundar" condition is that at r 0 r

, t 0 ts

1

2

, C

K

rqtyieldswhich s +

=

st

K

rqCso +

=

2

2.

1

uttin! this value of +1

in a%ove E&.

Now we shall relate surface tem$erature to the tem$erature of surroundin!s t .

From ener!" %alance

otal heat !enerated 0 4eat convected awa" or

( ) ( )= ttlrhlrq s2

2

+

= th

rqts

2

*o $uttin! value of tsin E&. (1), we !et

( )h

rqrr

K

qtt

2

22

+

=

3


6/13

he tem$erature $rofile is shown in the ad6acent dia!ram

when r is 7ero, %ecomes maximum

h

rqr

K

q

22

max

+

=

3.4. Fins or Extended Surf!es

1. Introduction

4eat transfer %" convection %etween a surface and the fluid surroundin! it ma" %e enhanced

%" attachin! to the surface thin stri$s or $ieces of metals called Fins or extended surfaces. hese

stri$s ma" %e rectan!ular fins, annular fins or trian!ular fins. Fins are !enerall" used on the surfaces

where heat transfer is low. 8" $uttin! fins on a surface, area of heat transfer is enhanced, %ut at the

same time avera!e surface tem$erature decreases. he former effect increases rate of heat transfer

and latter effect decrease rate of heat transfer. Fins are attached to

(i) +ar radiators

(ii) External surfaces of en!ine of a scooter

(iii) 8oiler tu%es

(iv) Electrical transformers and motors

(v) Economisers for steam $ower $lants

(vi) *mall ca$acit" com$ressors

". One Di#ension$ Fins of Unifor# Cross%Se!tion$ Are

9


7/13

4eat conducted in at x 0 x 0 4eat conducted out at x : dx : 4eat convected awa"

5ver width dx

)1...(...........convdxxx QQQ += +

dxx

QQ

dx

dtAKQ xxx

+= ,

dxdx

dtAK

xQQ xdxx

+=+

dxdx

tdAKQx 2

2

=

( ) ( )= ttconvectionforAreahQconv. ( )= ttdxph

uttin! values of ;x

, ;x:dx

and ;conv

in E&.(1), we o%tain

( )+= ttdxphdxdx

tdAKQQ

xx 2

2

( ) 2

2

= dxttphdx

dxtdAK

( ) 2

2

= ttAK

ph

dx

tdor

2

2

2

2

2

,, mAK

ph

dx

td

dx

dttIf ===

)2..(..........2

2

2

=

mdx

d

which is one-dimensional fin e&uation of uniform cross-sectional area. he !eneral solution of E&.

(2) is

).........(21mxmx

eCeC +=


9/13

, ===dx

dor

dx

dAKlxAt

uttin! this %oundar" condition in E&.(), we o%tain,

getweEqfromCvalueofputtingeCeC lmlm ),.(, 221=

( ) lmlmlmlmlm

eeeC

oreCeC

=+=

1

21 )(

lmlm

lm

ee

eC

+=

1

lmlm

lm

lmlm

lm

ee

e

ee

eCC

+=

+==

12

uttin! these values of +1

and +2

in E&.(), we !et

lmlm

xmlm

lmlm

xmlm

ee

e

ee

e

+

+++=

{ }( ) 2?

2?)()(

mlml

xlmxlm

ee

eeor

++

=

ml

xlm

cosh

)(cosh

=

From a%ove e&uation in ex$onential form

( ) lengthiniteoffinofcaseinassamee

ee

ee mxml

xlm

inf2?

)(

=+

+=

=

=

x

fdxdAKQ

cosh

)(sinh

=

+=

xml

xlmmAK

ml

AK

phAK tanh=

mlAKphQf tanh=

(iii) Fin of Finite en!th

/ $h"sicall" more realistic %oundar" condition at the ti$ is

( ) orttAhdx

dAK lxl

lx

==

=

ll

lx

hdx

d

=

=

xmxm eCeCEqfromow += 21),.(,

)()(, 2121lmlm

l

lmlmeCeCheCemCKor

+=

@


10/13

and also +1

0

-

+2

, therefore

( ){ }lmlmllmlm eCeCmK

heCeC += 2222 )(

( ) ( )

+=++

mK

heeemK

heeC

lmlmlmllmlml

12

( ) ( )nowso

eemK

hee

emK

h

Cmlmllmlml

mll

,

1

2 ++

+

=

mxmxeCeC

+= 21

( ) mxmx

eCeC += 22

)(2mxmxmx

eeCe +=

( )

( ) ( )mlmllmlml

mxmxmll

mx

eemK

hee

eeemK

h

e

++

+

+=

1

( ) ( )

( ) ( )mlmllmlml

xlmxlmlxlmxlmlxlmxlm

eemK

hee

eemK

hee

mK

hee

++

++

++++

=

)()()()()()(

1

{ } { }

( ) ( ) 2?2?

2?2? )()()()(

mlmllmlml

xlmxlmlxlmxlm

eemK

hee

eemK

hee

++

++=

mlmK

hml

xlmmK

hxlm

l

l

sinhcosh

)(sinh)(cosh

+

+=

==

xf dx

dAKQ

oxl

l

mlmK

hml

xlmmK

hxlm

mAK=

+

++=

sinhcosh

)(cosh)(sinh

1


11/13

mlmK

hml

mlmK

hml

AKphQl

l

f

sinhcosh

coshsinh

+

+=

3. Usefu$ness of Fins

et us consider a rectan!ular fin as shown in the dia!ram. For a lon! fin

AKphQf =

4eat transfer without finsoAhQ fw ,.. =

bK

Ah

AKph

Ah

Q

Q

f

fw ==

..

For the rectan!ular fin

tbifbtbp =+= 222

K

th

b

bt

K

h

Q

Qso

f

fw

22

.. ==

he ratio should %e much smaller to 6ustif" the cost of fins and la%our involved.

11


12/13

12

K

thso

1. *o convective heat transfer coefficient should %e small. *o it is a $oor $ractice to use fins incondensation and %oilin! where h is ver" hi!h. Fins should %e used in free convection where h has

low value.2. he material of fin should have hi!h conductivit". It will ma'e fins more effective.

. he thic'ness of fins should %e as small as $ossi%le within the constraints of stren!th re&uired.

4. Fin Effi!ien!) It is defined as the ratio of actual heat transfer from a fin to the heat that would %e transferred if

the entire fin surface were maintained at the $rimar" surface tem$erature.

tetemperatursurfaceprimaryatweresurfacefinentireiffinthefromtransferheatIdeal

finthefromtransferheatActualf =

ideal

f

fQ

Q=

(i) For Fin of Infinite en!th

ml

AK

phl

lph

AKph

Q

Q

ideal

f

f

11

)(

====

(ii' For Fin Insu$ted t t(e End

)(tanh

lph

mlAKphQ

Qideal

ff ==

ml

ml

lAK

ph

mlf

tanhtanh==

&iii' Fin of Finite *en+t(

)sinh(cosh)(

)cosh(sinh

mlmK

hmllph

mlmK

hmlAKph

Q

Q

l

l

ideal

f

f

+

+==

mlmK

hml

mlmK

hml

lm l

l

f

sinhcosh

coshsinh1

+

+==

3. Fin Effectiveness

12


13/13

It is ratio of heat transfer from the fin to the heat transfer without fin.

.Ah

QEff

f=

&i' Fin of Infinite *en+t(

.

Ah

AKph

Ah

Q

Eff f

==

Ah

pKEff =.

&ii' Fin Insu$ted t t(e End

tanh.

Ah

mlAKph

Ah

QEff

f ==

mlAh

pKEff tanh. =

&iii' Fin of Finite *en+t(

ml!inhmK

hmlCosh

mlml!inhAKph

AhAh

QEff

l

mK

h

fl

+

+==

cosh1.

ml!inhmK

hmlCosh

mlml!inhAKph

Ah

pKEff

l

mK

hl

+

+=

cosh.

1

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