Chapter 2

STEADYSTATEcoilDucTroil

2.1 INTRODUCTION

Condtction is rhat modc of hear uansfer in .which hcat travels from ar-egion of high temperaturc ro a region of lower lempcrature because ofdirecr contact berwcen the morecures of the medium. n" ..r^,i*rirrpbelween thr heat-transfcr rare by conduction and the temperrrure ;#;;:rion in the medium is the Fouricr law.

Conduction can occur in solids, tiquids. and gascs. However, in liquidsand gases that arc arowcd ro circurate, ir ii usualry combined withconvcction- Thercfore, purc conduction occurs primariljin opuqu. souas,whcrc morion of thc me16ii1 is rcsrricrc4 In rhis chaprc.u. *ili

"o*ia.irbe conduaing medium lo \ a rclid, bur rhc principics O"u"fop.a *i U.oppli* ro liquids and gascs in which convcctivc rnotion is rcstrictcd.A discussion of hear conducrion can bc broken dowa inro tiri.. ,";o,subjcct arcas- The first invorvcs sready conductiou in which *. ,r*;;:turc is a funcrion of only onc coordinare dircction G* a;;;;; iJthrough 2{). The second arca conccrns stcady conduction in wirich ttretcmpcrarurc is a f'sctbn of two or thrcc coordinarc oiredio,ns iscc s*d*2-[ Thc rlird arca is transicnt or unsrcady conducrion- Thi, ,G;";iibe dcalt wirh in Cbaprer 3.

Digital computcrs and com?utcr prognmrning are imponant aspccb ofhcat lransf6, as-wc frcquentry cncounier prob[rns rco difficurt or time-consuping to solve by han4 In such c-s6 we often tunr lo computcrs,42

f..f:;.{

Coxoug.nox EQrrArroN 43

cirher digiral or prograrnmable hand calculators, to providc solutions. Toillustrate the t)pes of programs that can be uscd to solvc conductionproblems, four computer programs are included here- Thc progranE arewrirren in a general form so thar they can be applied to a bioad'range ofproblems.

The gcneral conduction equation is derivcd in the next scction- Much ofthc material that follows starts with a solurion of &is cquation- It providesthe tcmperature distribution in the material and oncc thc tempcraturcdistribution is known, the heat-transfer rate by thc conduction nrode canbe cvaluated by applying rhe Fourier law.

2.2 CONDUCTION EQUATION

The conducrion equarion is a mathematical expression of thc conservationof ener,ey in a solid subsrance. It is derived b1r performing an enerrybalance on an elemental volume of material in-which hJar is beii!transferrcd by conduction. Heat transfer by convecrion and radiation iiassumed to be negligible within rhe solid. Hear-rransfer rates by csnduc-tion are related to the temperature disrribution in thc soiid by the Fourierlaw (Eq. l-2).

The cnergy balance accounts for the fact rhat energlr can be generatedinside the material. Tlpical examples of inrernal .n.rgy generation in asolid would be heat generared by chemical reacrions, hiaigenerared as aresult of clectric currents passing throu_eh a resisrance, and heat generatedbv nuclear reactions.

The leneral form of the conduction equa.rion accounts for storage ofenergy in the material. we know from thermodynarnic considlradonithat,thc intcrnal energy of a material will increase if the temperature of the'matcrial increases. A solid material can therefore expericnci 3. net increasein stored enerry when its temperature increases with time and a netdccreasc in stored energy when its temperature decreascs wirh time If thernaterial tcmperature remains consrant, no cnerry can bc stored and steadyconCitions are said to exist.

Hcat-transfcr problems arc classilied in broad carcgories accordiag tothe variables that the temperarure depcnds orr. If...the temperaturc isindependcnt of time, the problem is callcd a steady ot stea4v-state problem.,lf lrc tmpcntwc is a function

4l Stlrur_c:ere Oor.rouqrox

r- n+} rhe probrem i, "t"..in.o as onc4imensiorur od uansienr- lIT-T{t,c) in cylindrical

"*rt;;;;"-;;;i". is ctaaiticd es ,,"*.d-ttletrsiaul atd teady.Fectangular Coordlnates

To sirnplify tre dcrivation of the conductiou cquation, wc wiII considcr aoae-dimcnsional rcctangutar "."rai"r. qyri;L *,o'.n in Fig 2-l aadassume rhat the rcmocraturc in ,n. rn"rcl"ii u tun"tioo of only thc .rcoordinarc and drnc..or r: (r,r). w;;i;,r".*rumc lbar rhc conduc-tiriry'E densitv o. and specific il"t

" "io-.i,ia "r. at constanl T'ccffecr of variabre conduciviry ,.ilr u ir**r.o r" section 2_4.A srakncnt of the corof Fig; 2-l is as follows:

scrvation of energr applied to tt c "oit ot volumc

I rarc of cpcrg I I rutc of cnerw ]I conducled inro f + | generated l#.a. Il conrot volumc J Icontrol uolume J

f rarc oI enerw 'l -

-

| conducted *our I I rate of energy II of control 1+f stored inside Ilvolume J l*n,tot volume J

(2_l)Using thc Fourier law ro express the rwo conduction tcrms and dcfininsthc symbol q'i as the,ur. oi .n..g, g.nlrir.Ji"rio. rbc conrrol ,rohmf

T - Tlt. ,tGcDcntiotrqc-AAxConruntpropcrlks

k- e-c

__: 'tt -IlI={Figure 2-t Conrrol votumc in.recangutar coordiD.tca-

ffil-

CoxoucnoxEquenox 45.per unit volumg Eq. 2-l may be expresse{ in thc form

- u$ft)+ q,;,

;?.j+ q.:,;.*il, t

-, li

'ir:

46 Srrroy-St-^rz Aor-.rcTrT

Wc.rrc rcldorn rcquircd ro dctcrminc lhc tcarpcraturc distribution in asolid matcrial by rclving thc conduaion cquation in the form of Eq- 2-6.A solution to Eq- 2-6 would involve solving a partial diffcre.ntial cquation-Ia most problems wc can make simplilying assumptions that will climinatctcrms from thc conducti

i:*,i. :*:

and a dimensionless time as

CoNDUcnoN Eormor {7

(2-r3)

(2-r4)

(2-15)

It,

Thc symbols T,, L;, aad I, rcprescn! a reference. temperaturg lengtla and,i*";

".rp"",i".ry.

rn" choice of refercnce quanriries is arbitraqr, altboughr*tii"f values should bc selected once the problcm b complctcly;.d. A dirnensionless ratio of temperature differences is oftea prefcr-.Lt. ,o

" ratio of temperaturc; and the choice of dimensionlcss groups

*"y *ty from problem to problen- The f9-rm of.thc dimcnsionlcss grouP6

"i" oft.n sclected so that thqy- linit the dimensionless variables bctwcen

.onu"ni.n, extremes such as zero and l' The value for t is usuallyiclected as rhe maximum x dimension of thc system for which thetmperature distribution is being detcrmined'

When the definirions of the dimensionless temperature, coordinate' andtime are substiruted into Eq- 24' the conduction cqua{on wrinen in anondimensional form becomes

azo qT L,2 L,2 aodt'* kr; :

"', a'

The quantity ot,/ C is a dimensionless group called the Fourier number'which is designated by the svmbol Fo:

at-Fo: ---L,'

The choice of reference time and length used in the Fourier numbcr mayvarv from problem to problenq butthe basic form remair$ unchanged.The Fourier number will always bi: a thermal diffusivity multiplied by timedivided by the square of a charactcristic lcngth.

Thc Fouricr number is the rate of heat transfer by conduction dividedby the ratc of cnergv stored in a rnaterial- The Fourier number is animportanr dimension'iiss group used in transient conduction problems andit will appcar frequently in thc work that follows.

-

The other dimensionless group appearing in Eq' 2-14 is a tcrm involvingheat gencration. Wc will ut" tt i ty-Uot {o to represent the dimc'nsioalcssgeneration:

_ c:i,L:4a=a7 (2-16)

This term is a ratio of internal heat generated pcr unit time to heatconducred through the volumc pcr unit timc- '

2ll

43 STaTDYSI TE C-otcDuc'nox

Thc oncdimcnsional form of thc conducdoadimensionlcss form now bccomcs

a20 -

la,p+1e-fr6

cquation cxPrcsscd in

(2-17)

Exemple ?1. Dqcrminc thc simplificd fgrm.of thc gcncral colduction

.q;;;b.t applics to stcady' oncldiy*i""tt-T,lduction in a rcctangu-lar solid with consrant p'op"ruo and no gcncration' Solvc thc rcsultingconduction cquadon f"t d;';;;"rato" ?ito;ludm and hcat-transferratc in thc sotid in terms of conslanls of inteEntiott'

Sotution: Thc general form of the conducdon cquarion is

-(#.#-#) +qz-r*Since thc problem is srcady and one-dimcnsioral thc rcmpcraturc

is

#;;;';tirrc antl it titnvir'-ctq" dontfr*efftntffidircctions. If we assumc t}iai fit only a funnioa of thc x coordnatc'

or-o 4:o - {-o0t 01" d:'Also, since no generation is prcscnt'

s';:0The simplified form of the conduction equation bccomes

lir:odxl

Inrcgrating this sccond-ordcr diffcrendal equarign twicc yiclds the tcmpcr-

aturc distriburion in ""t

oi i*o -ntono of inrcgrarioa' C' and C':

T:Cf+Czvalucs for ihe rwo oonstants of integrarion would bc dctsoincd in

a

pardcllar problcm uy tp"liyitg and-a:nd1ng rwo tbun&rlr conditions'Thc beat-transr", ra,"

"otia"iJ thru"S rt'c otid in ttrc r dircction is

givcn bY thc Fouricr law:

s*-U*-'ktC1Noticc rlrat lhc bcat-transfcr ratc is uniform ia egr*rncnt

with -Examplcs

l-l and l-2 and i,r"'"Jnito"i "fr

tot"tio* ia tbc solid under stcadycondidons-

CoxouqnoxEeurrror 49,

Cyllndrlcal Coordlnales

The conduction equation s.ritten in thc iorn of Eq. 2.6 applics only to arcctangular coordinate system. Thc generation and ericrgr stoage tcrrrxlarc indepcndcnt of coordinatc system, but the net conduction termsdepend on geometry and therefore on the coordinate q6rem. The dcpen-dcnce on lhc coordinate system uscd to formulate thc problem can bcremoved from the analysis by replacing the net conduction terms with theLaplacian operator. The form of the Laplacian is differcnt for erchcoordinatc system. Thc Laplacian operation in rcctangular, c,vlindricaland cphcrical coordinate qrs(ems is gi'rcn in Appedix A. Tlrc esnductioncquation written in terms of the Laplaciar is

u:r+$:*#For a gcacral lransient three-dimensional problem in cylindrical coordi.

nates, I: T(r,$,:,tl,. Thc coordinale symbols are shbwn in Fig- 2-2(a). Ifthe Laplacian is substituted into Eq.2-18, the general form of the conduc-,tica+$a*ies js@drC+orair* ttms

la/a71 .taT.d2r.q7 tdr;?ul'a,)-7 a6r* ur* e -;a (2-le)If the transient temperature in a cylindrical shape is one-dimensional so

that I- I(r,r), the special case of the conduction equation bccomcs

+ *tE).+:*# eza)Furthermorq il the temperature is steady and a function of only the radial

(2-r8)

7*

50 Srrrsr-srrrrObnoucrrox

666rlinale' the conduaion "q*ri- can bc rducc'l

to

L d (.{\*&-o (2-21)rdr\'?rl k '

Noticc ia Eq. 2-21 that thc tcmPeraturc is now a hmction of only a singlc

variablc r end &c "q*ioo "i" Ut writtco

as aa ordinary diffcrential

equation.-'Wh"o

no internal cncrSi gencration is prcscnt and thc tcmPeraturc rs a

function of thc radius ottfi' 6t iroay forur of thc conduction equation forcylindrical coordinates is

*(,+)-' (2-22)Example 2-2. Determine thc stcady PTqcraturc distribudon

and hcat-

ransfer rate in a cytindcr *ith lc;gth l'in^tcrrns of two constants ofiri.gtiti.tt The lcmpcratu'c is a fuiction of thc radius r onl1" and noinre"tnat gcneradon is prescnt in thc cylindct'

Soludon:The appropriaefwnr tf lhectnduction cquadon is Eq' 2-22:d {,{\=odr[ dt'l

Integraring once with respect to t;f,* tlddt'-F - t,

OI

A second inte-eration givesT= CJnr+' Ct

Thc constants of inregration caa bc detcrnriacd oce two boundary condi-

tions arc specified- Thc heat'rransfcr ratc across a cj'lindrical surfacc witharbitrary radius r is

c : - u # : - k(znrt)g - -ZtHCtNodcr rhat thc heat-transfcr ralc actoss any c1'lindrical surfacc is constantfor stcadY condirions.

Spherlcal Coordlnales

For a sphcrical shapc where the tcmperaturc is a funaion of the thrcc

"*Ji*,L and timc, o, 7-T(r,0'4'lj' thc gencral form of the conduc-

t,.s&,=.

dl Lrdrr

wSiEr.Dy, Oxe-ourre.rsroxrt Cor,oucnop Wmrour Gei..Enlno$ 5l

(2-23)

Figure 2-2 @) Spherical coordinare sy.srem_

tion equarion which inchies generation oi energ-v is

;*('#).*7#("",#), I A2T.q;', laTfsin:daa: k a?t

The spherical coordinate system is illusrrared in Fig- 2-2ft).Special cases for one-dimensionaf transient conducrion and onedirnen-

sional, steady conduction for spherical coordinates can bc simplificd fromEq.2-23. The simplified forms are lefr as an exercise.

2-3 STEADY, ONE.DTMENSTONAL CONOUCTToNWITHOUT GENERANON

We will now appl.v the conduction cquation to problems in which thetemperaturc is a function of a single coordinate onll In the rectangularcoordinate system the temperature will be a function of the x coordinatconly, and in bgrh thc cylindrical and spherical coordinare slatems thetemperature will be a function of thc radial coordinate only. The thcrmalconductivity is assumed constant and no generation is considered.-.

The general procedure wc wirl use consists basicalry of two steps. Thefirst involves. determining thc tcmpcratule distribution by solving thcappropriate simplilied form of thc conduction cquation. Thi, prLo,":*il5 of solving an ordinary, second-ordcr diffc*eatial cquation. Oncethc diffcrential cquatioa it rolvcd, troo bounda,ry ondirions are imposcd rodetcrminc the two consrants of integration. -The second step involvcssolving for the rate of heat transfer through the solid by applying therouncr law.

iL

52 Srtery-Srrtt @rroucrrox

Fectangular CoordtnalesThc srcady, oncdimcnsional tcmpcrature distribution in a rcctaryguhr

planc wall *'ith no en"'ry gcner:rtion is govcrned by thc simplifica roin ofthe conduaion cquation (Eq. 2-10),

o'T -odxJ

solving this diffrential cquation in tcrms of two constans of integratiorlC, and Co rcsuks in -

T(xl-Crx+C2Sec Example 2-1.

Thc- consranb of inregration can bc dercrmined once lwo boundaDrcondidons arc spccificd. Lcr us assumc rhar the rwo boundary conditioriare dctermined by spccifying the temperaturts at trre two crrr?n* surfactrof thc *.all as sho*n in Fig. 2-3:

Appl-nng thesc trrvo boundarylempcralure in the wall of

r(0)- r!t(D-:t:

conditioas rcsulrs in the dimensionless

tE_ L (2-24)The t:mperature distribution is therefore linear *'ith x. The heal-rransferrate through thc wall as determjned by the Fourier law is

r{x\- r,Tr-?],

q=-k'4#-L4L*The hcal-ransfer ratc per unit arca throu-eh the wall is thc hcar flux and is

TaOt- Tt

T. ltxtI . dmrtJln

,tt;- . o

-{ .tftlt. T.

Frgure 2-3. Rccrangutar gcomctry and boundary condidons.

(2-2s)

ffi.ffr:'

SrE^Dv, Orc-ouvexsloxr,r- Conoucnqx Wmrotn Garmrnor j3denotcd by 4'. The double-prime superscript O".oto that thc quantity q iscvaluated pcr unit area. For a plane wall

. q k(Tr-T)c:7= L

When Eq. 2-25 is written in the form of Ohm's law,AT

tR,

the thermal resistance for a plane wall asgoted in Chapter I becomes ,'

6+l (2'27)t/The tlow ol lteat b1r conducri\_@celugh a plane wall is a result of atemperature difference acrofs the wall and is inhibited by &ermai resis-ance. which is proportional ro rhe wall thickness and inversell, propor-tional to the thermal conductivity of the wall and in cross-scctionai area.

Ifheat flows by conducrion through several plane surfaces, the tempera-ture distribution and heat-ransfer rale taa bc 4"t.'u,t^rd&y ssumirgfrrat rhe teat llows tb^rolghaa.eqgivalcnt thcrrnz{ cin:rit in *hich ttrc heatllorvsconsecutivcly through a series olrcsistances,cach corcsponding to a

' separate wall material,As an cxample of a series circuir, consider a plane wall, denoted by the

subscript l, covered with two different t)pes of insulating materiais,denoted by subscripts 2 and 3. The geometry is shown in Fig. 2{. Thesame amount. oi heat fiows consecutively through each resistance, and

irrti- ^rlJt ""r-:'n,-fr

^,-# ^,.#Figurc 2-.1 Scrics thermal circuit, rectangular coordinatcs.

ATLlkA (2-26)

zt

!latcrial l

5| Srs^Dy-Sr^TE Gorqorrcrro :thcrefore thc rhermal circrrrit is a scrics circdl If the thrcc r:ratcriatpropcrtics are known, thc gconctry is givcn aod thc two cxtrcme surfacetcmperaturcs arc spccifid the hcat-rransfcr nlc can thcn bc dererminedfrom an cxprcssion similar to Ohm's law, as gircn by Eq. l-14:

,:f4f) =Ar-.* T,-7,' \ 4/,",,, i A .R,+.R2+.R3,- |

oncc the hcat-rransfir rate through the.compositc *all is known, interfacetemperatures among thc three matcrials can bc dctermincd by applyingOhm's law to only one wall. ylerial, For examplg n" rcrnp.ru,uri'{ "ilhc inrcrfacc berween marcrials I and 2 can Uc Octirmincd fiom

o= Tr- T, -

Tr- T,' 'Rr Lt/kfi

Frequentll', plane walls arc cornposed of compositc materials subdividedso that the hear must flow rhrough scvcrar materials simuhancously rarhei&an,cqrsntively- rilrhen rhis ii thc cqsg ttc (hcrmal circuit becomes apa::lie1 circuir- A rypical cxample of a parallcl circuit is shown in nig. f _0.The heat flow is derermined by

(2-28)

(2-2e)

(2-30)n:(+:),", = T'- T'^,*(ffi)*^.

Individual resislances arc determincd by the cguation

R,= klL, (,:r,z.3,4)Intermediare remperarures such as \ ^ y bc dercrmined by Eq- 2-29.The parallcl circuir assumcs that thc hcat flow is onc-dimcnsionar. and ifthe resistafices R, and lR, arc significantll, diffcrcnt, two-dimensionalcffects can bccomc imponanr (scc S-ction 2-|Cyllndrlcal Coordlnates J

The mos-t common condnction-problem involving a cylindrical geometryis one in which bcar is conducred radially through-a Irig" i,"h;"yii;L',as-shown in Fig 2-5. The rcmpcraturc olthe imioc surficc of rh.

"iil;;;: t"T to bc 2!. and thc rcmperaturc of thc oursidc surfacc t f;"*" abt.!' F stcaty-srarc tempc'rarure disrriburion in rhc consrant DroDcnv;]".:.1:-:: ll,:ll, gcncradon is prcscnr is givcn ty,h;;;;';iEiz-zl. suDJcctcd to thc two boundary conditions

T(r)-7,T(r.)=\

;;l'ii[r{'

Srreoy, Oxe-omgrcroxet Corvoucnow WrrHouT Gexenrnox 55

Figure 2-5 C.vlindrical geometry and .boundary condirions.

The solution for the local temperarure. l.(r), isr(r)=r,+(r,-r)WA (2_3r)

Sec Etample 2-2. Equarion 2-31 written in dimensionless form is

(2-32'

therefore logrithmic withiradiallylaw for

(2-34)

T(r)-r _h(r/r,)

To- T, ln(r"/ r,)Thc temperarurc distribution in the cylhder isradius.

-

Once thc tcmperature distribution is knorvn, the heat flowthrough the cylinder may bc dctcrmincd by using the Fouricrclli ndrical coordinarcs,

' q--u{4#-..k(?nn# " {2-33)*L.t I is rhc length of rhc q{inder--.,Piftu*$",ing the tcmperaturc distribution giVcn in Eq. 2-31 and substrtutrng thc rcsult into Eq. 2-33 yiclds

o= 4-7.' k(r"/r,)/2*l

zq

T = T(t\li = constrnt

tu- =o

56 Srrrol-Srrrt Cd{Dttc?toN

Equation 2-3rl is *rittcn in thc form-of Ohm's.law and the denominatort"ptl*" thc tbermal rcsistancc of a hollow cylinder:

-

ln(r./ rt)E--

2nHThe principlcs of a scrics and parallcl circuit dcvelopcd for-a planc-wall

,.*irlar Lordio"t. systcm can also bc applied to a hollow-cylindcr;;;; F";;ple, supPosc that a Jluid flows. through a tubc which isI""*.J uy

-

insuLting rnatcrial as shown in Fig 2-6' Tbc avcrage fluid;;;;; L"*o ,J be r, an

-

thc ouside surfacc tcmpcrature of thc;;J;,t* is T2- Thc tubc marcrial is designared tv sutscrlp.l Td lt;;;;fi;; i, no*u.r 2- Thc convcctivc resistancc of thc fluid is givcn $t;;. l-ig.-]]r. fluid rcsisrance is connecrcd in series wirh rhc rwo conduc-iirl ,.risrunc.s of thc r$o solid matcrials becausc the hcat musr flowconsccutively through cach rnetcrid'

Thc hcar-flow rate for the problcm is givcn by

(2-35)

(2-36)/ Ar\e-{ 4 r--Tr-Tz

- 1 tnirr/rJ,frt(ttlrt)i,z-f - lrkl 2zkJ

Tr T, rt --TP\,\,t44r1 f'4Al'4R'#;,\^,'#

tnl ,.i', IR' "-fillr

ngttrc 2{ Scrics rhermal circuit cylindricat ScomcrDr'

ffi-r

STEADY, Oxe'orxtrlstoxir 6*ot'tt'on' Wmrorrr Gersr'rtrot 57

The rhcrmal resis,nce uscd in Eq. 2-36 nust h the total resisuncc

bctwecn the two t"o*" lffi"i*l]'litrt" tw:'known tcmpcraturcs hadili"'c -o r,,,': $"iffmru,"1*tr":1i$n1i:

;:li*::J:i:f;H;: ';

;#;;;;"i-i**r"' '*e

is knownis

T,- T, (2:37)c:@,r,l',.t1 ''t' r'

Exarnple 2-3' An aluminum pip" &"i"'-$eam at ltO"C Thc PiPe

"l&-185 w/m'K) h* ;;;;^i'[i"t"trctGa'l of l0 cm and an outsidc

- iir."t., (o.d-) of tt ;: TiJ;; iiToottain 1'*- wbcrc thc ambient

$itr:?lffi lrf fl $fr;"flk:,:Hl?'ffi::: .xiys: Fudt length of pipe if thc pipe ts urunsurarcq

'Ai"a with a S-cro-&ickZil-;;"d;cc ,ir. t.u, toss from e:_.prfl t, I_:"."'":"*"-":-'r:;f*;;";laver of insularion t*-dlO'W7t'I(i' Dercrmine the heat-ransfer rate

per

unit length from the inJuittta'pipt' i"u*t that the convectivc rlistance

#;;;il*;rc-elieible' i' ' ' ir {'Solution: For thc uninsulated pipe the -only

siirificant reiistances 'tohcat llow are lhe

"o"j"'ti"t"l ttti"i"tt :t^.:l: pipe and the convecuve

resisancq of Ult 'oom

oii' Sint" the convective resistance of thc steam n

negligible, the inside su'rf;;;;;t"1;tt ot the pipe is thc same as the

steam tempcr"tu"' '{t't"at-transfer

rate Per 'init tength

in terms of

symbols shbwn in the figure tt -

-,.-.:q ='r - 1- hQ2/ rr)f 2rkr+l/2trrrh*

(r.57x l0-1)+o-177For thc insulated otot' lit tttt:LnJc ot tt'c insulation must bc

added and

th. ."pr.rrion for the heat loss bccomes

q'= 1=T,-Tn

Itttv al . l,- Zrk, Ztrk, 2rrthn

ffi-m#'m

ilo-30@7;V.rrTt7'z"xo'o6x 15

80 :452w lm

l l0-30

- 138 W/m

[i3x tot'1 +0-482 +o'oe6EO

q

5E Srteor-Srrre @loucnox

Airf,o'L

AitinuT-

Thc prescncc of the insulation has reduced .the heat loss from tbe steam by?0 pcrcent Noticc that in both cases the resistancc of thc aluminum pipecan be neglected wi&out loss in accurac;* for thc calculation of heat-trans-fer rate.

Spherlcal Goordinates

Thc tempcrature distribution and. heat-transfcr ratc through a hollowspherc arc dcrcrmined in a manner similar to that'outlincd for the planewall and hollow cylinder- Thc stcady, oncdimcnsional kmPctaturc dislrib-ution with no gJcrarion pr*cnt is detcrrnincd by solving thc simplifiedform of thc cr:nduction cqnation writteo in spherical coordinates. llhisequation is

Assuming that rhc boundary conditiom gpccify that thc inner and outcrsurfacc 6f thc spherc arc at knolvn temPetaturcs as shonm in Fig'2'7'

I d (*{\_t d2(rT):std,\'&l t dl

T(r'l-7,T(r"l-7|.

--_

ffi^,W,i'v,l

t,/

t/v

Srrrov, oxr-orrnsrcitrer Corrot'crrox wmlour Gei'anrnor 19

Ftgute L7 Sphcrical gcomctlv and boundary conditions'

The ternpcrarurc dierributioe ie rlr lollow sphcrc i' thenrlnl: % /r_1) Qtitt7=T -'. 1\' r I

The temperature in a hollow sphcre thercforc varies hyperbolicnlly vttlt

radius.The heat-transfer rate in thc sphere is dctermincd by applvirrll

ll''Fourier law to Eq. 2-38. The result is

T,-To (/ t',tc: Wat+"W

The thcrmal resistance for a hollow sphere is therefore I

D-3'i4;kror,

(t. 'l(, )

Overall Heat-Transfer CoefllcientAs shown in Chapter I' when a hear-ransfcr.problcm tt::1"::,::";'11

thcrmal rcsistances in scrics' parallcl or combinations ot tne ,i*-l.i:..ii,,'.convcnicnt to dcfinc ea overell tcat'tranrfcr cocfficicnt or ovcrrrr

u'r'""

tance. Thc symbol for thc ovirefl heat'transtcr cocfticicnt is U' rn

eo Sre^Dx-SrerE*ryThc unirs of both U and tr arc W/ml'tC \iihcn f-q' 2-41 is comparcd

to

'=t{}-' {242)wc sec lhat U can bc writtcf, in tcrms of tbe total thcrmal rcsistancc

of tlccircuit:

u/-'G;i;As an cramplc oI thc ovcrall bcat-transfcr coefficienL considcr

thc tbree

;#;ffi; ,i"*" i"Fls e-+n. vatuc oI Ulor this exanrplc is_ffitffi

ln rhb cxarnptc. thc croes'cctionrl arcer of rll rhrcc matcrials arc cqual' sothere can oe liulc *"r*ioilJtt-l'hat area should bc uscd in Eq' 243'Howevcr, when rhc

"r*

-io cach rcsisunc: Ittn varics' wc mun bc

#.*t**."*u"t"g 2t arca to bc uscd in Eq' 243'*n pt"tft*At.*tn ttlr"lattt rnrl'b thc onc involving a comP(Fne

cvlinds in ryhich ,b. t dt;;.tare cormecttd in ecrics' Thc valuc for Ultir thc circuit ibowa in Fig 2-5 b

q: UA{ATl,oor=-7--'n (r,/ r,l ln(rt/ rr)l.r*rJr= - z'*J 2ek;

(243\

T-1i:,./r,t ln(h/rz)ffi,*-frt'zqNoticc that the product ol llA is a consunt. but thc valuc

of U varics

dcpcnding upon the "h;;t-;;

corrcsponding'arca' For cxamplq

suppose that we *o*t tIi insidc pipc zre4 A;' as our rqferencc ^te4whcrc

Ai:2zrJThcn the {/ value bascd on '{, would beU;n@

i.lf U is bascd on thc outsidc pipc arca '{o' wherc

Ao-2mtl

UA=

then

"Srzeoy. Olre-onreNstoxAr- CoNDucnoN Wmrour Gglantror 6l

uo: \ rrln(rr/ 11) . rln(rt/ rr)-;i- 0,, ---,

'

Evcn though the values for U, and U. arc differcnt' the Ul product isalways consunt:

U,A,= UoAu

Example 21. A plastic (k=0.5 W/n'Q,tlP:.carries a fluid such thattJcooicAvc heai-transfei cocfficient is 300 W/'2'IC The a'reragc fluidlip"ra** is 100"c. The pipc has an i'd' of-].T and an o'd- of;Lsrr. Ifii"'t"".,.*fer rate through the pipe pei unit length is 500--W/n'ioluf",. tbc externa.l pipa-timperature. Also calculate the overall heat'transfer cocfficient based on thc pipe outside surface aree'

Solution: A sketch of the pipe system is shown in the accompanfingfigure- The heat-transfer rate is given b-v

Tr-TzI ln(rr/ rr)+-

i,(2rrrl)' 2rkrl

1 ln(1/r1)mtrrt -F

or thc heat-transfer rate per unit lcngth is

-\- c':+^--=--Tr-7,

| . ln(rr/ rr)-f-hr2ar, ztkr

6l Snrory-SrrcCororrcnot

UoAo-

Jop- loo-r2'I .

l^(z/rs\ffid:oit-.ffir:-365"C

Thc overall bcat-rransfcr ocfficicnt based on ,{o is

ffiiI

1 .

ln(rr/ rr)@rtt-Trk-kJ

,, -

|vo r, .

rrln(rr/ rr);F,* k,

I2

.0.02x1n(2/1.5)tJ>(300 - ---63-

-62.69 W/m2-K

As a check on thc valuc for O, ve can calculate fhelheat-ranSer rarcbascd on the calcutatcd ualuc of IJ.:

q': U,A,(Tt- T)-62.69x2r x 0.02(100 -

36.53) -

5gg 1v7t

Grltlcal lnsulallon Thlckness tor a CyllnderAn intcresring situarion ariscs when a c-vlinder with low thermal rcsis-

rancc is coverid by an insularion layer and thc insulation is sunounded bya fluid- Tbc gcomctry is shown in Fig. 2-8. Assume thar thc i.nncr surfaceof the iosularion has a known constant valuc of temperature cqual to 1..Suppose that we wisb to dctcrmine thc cffect additional insulation willhavc on thc bcat-transfcr ratc frorn the cylindcr. lt is nor obvious wbc.tficr

Frgurc Lt Crirical redius of pipc insulation-

w:'{i

Sraov. Ore-oryaxstoxl @xoucrrox Wmrour Geterenox 63

addirional insulation will incrcasc or decreasc thc heat-transfer ratc. Thehcat transfer at srcady state through rhc grlinder and insulation should bccqual to lhc convection ralc from &c surfacc:

q: h-,.t.(T.- T-lwhere thc symbols are defined in Fig 2-8.

As insulation is adde4 lo increeses but I decrcases. To dcterminewhich cffcct dominates, we can writc thc hcat-transfer ratc as

q= T'*T* -

ln(r"/ r,)/2rk,l+ l/ h"2rr"lTo determine thc effect of variable insulation thickness on dre heat-trans-fer rate, wc can takc the derivative of q with respect to r. and cquatc theresult to zero to deterrnlge oprimum condilions. The resulting condition foroprimum heat flow is the condition

Ttr q*n*iry.l.

..

6{ Sraov-srrrr6lotrnorExemplc2.5.Ai-mmdiamcrcrclcctricalwircis-covcredwitha2.mm.

thict laycr of plastic insulario (t-OS W/n'K} The wirc is surroundediy.i,

",itf, an ambient EmPeralurc of 23'C and i]-10 W/m2'K' Thc

Jr. r"-p"oturc is 100"CDctermincthcratcolbcatdissipatcdfromthcwircpcrunitlengthwith

"ra *ti,ou, thc insulatiou Assumc that lbc wire tempcraturc is not

affcctcd by prescncc of thc insularion"

Solution: First we calcularc rhc Biot gumbcr:.

Bi -

& -

lo(2 +o'5] x lo-t -0.0,k, 0-5

Sincc the Biot number is less than t, the presence of insulationincrcasc rhc bear ransfcr from the *'irt. The heat-traasfer rate perlength with thc insularion oo *rc nirc is

t.- 1t*J%-' ln{r./r,) . I7-2ak, 2zrni,

willunit

100- 25-

10.90W/mln(2-510-51

.++?rX0.5 2ex(25x l0-3) x t0Without insulition thc heat-traasfcr rate is

4: E,!G"-7-): l0x2'rx(o'5x l0-3x lo0-25)-z'36 *,'Thc addition of the insulation incrcascs lhe rate of heat transfer from thewirc by a factor of 4.6.

24 EFFECT OF YARIABLE THERMAL CONDUCTilITY

The rhermal conductiviry of srost matcrials is not constant but varies with;;;";. so far wc bavc assumcd rhat rhc thcrmal conducrivity was**ont ln this sccrion, howcrrcr. wc will dcrcrminc the cffect a variablerhcrmal conduaivity would harrc on the heat florvand tcmpcraturc distrib-ution in a plaac *ail, a hollow c1'liadcr, and a hollow sphcrc'

lf rlre conctuctioo equarioa for thc rccrangular coordinatc qtstem derivcdin Section 2-2had bccn dcrivcd asgrrning thc thcrmal conductiviry was avariable tbe form of Eq.2-5 *ould bccomc

*(-#).*(-f).*(*#).';-*# Q41)

Errrcr or Verren-s Trcnxrr. @xoucrrvrrr 65

if thc tempcrature distribution in ihe rect4ngular solid is steadn onc'di'mensional, and does not involve internal cnerry gencration, Eq.247 anbc simplified to

*lo"#l:' Q4stBcforc wc can solvc &is cquation, we must know how $c rhcrmalconductivity varies with lemPcrature, lc(I), over thc rangc of tcnperatucscncountcrcd in &e solid. For many materials, very little accrrraqr b lost byassuming thc conductivity-temperature variation is lincac :

where p is a constant. *k(r):'to(l + Br) (249'

Integrating Eq. 2a8 once with resPect to x gives

kl)#:q (2-50)The heat flow through the wall will be a consrant for steady-statc condi-tions. The Fourier law applied to the wall is

q':-k(n# (2-5r)By comparing Eq. 2-50 with Eq. 2-5 l, we see that the heat flux is

4": - crSubstituting Eq.2 9 into Eq- 2-50 followed by integration with rcspect tox yields

r,(r*o$\-c,x*c,The values for the two constants of inlegration can bc deternined dyspccifying two boundary condirions. Assuming that thc boundaries of thesolid are at known tempcraturcs such aq those shown in FiC. 2J' thcboundary conditions arc

f(0)= 1,T(L)=7,

'Wt can dctcrmine the values of the constaqts of iotcgration C, a-nd C2,which arc

9t

6 STI DYSTATEOonoucrrox11g .limcnsionlcss tcmpcratrlrc distnlburioa ia the wall is

lhcD

W-;*{[rr,+r;,;^

r,2- r(x)2 IT,-Tt IThe tcmpcraturc profile in a plaoe wall yth.:"t)n"n thcnnal conductiv'

itv is not linear, but *'. ""ol"l it" eq'2-52

will "aul to thc lincar

result

;iL:;;;*;'Gi ,l"ttJ conductivitv is oonstan! or *hca p-0'' ihc b.at flux through thc wall is

Q'- -Cr= - *lo,-r,)+ +(r:-r,')lwhich can bc rcwrittca in the form

c":ro(r.ury)+The quandry in parerthcscs is rtre ttrtrtnal cosduaivity

s'duatod at thc

mean or avcragc temPeraturc, ?-' of thc *'all' rvhcrc

r^_!,lrr'The thcrmal conductivity cvaluatcd al 1- is

r-:ro(r-ury)The heat flux in terms of k- becomes simply

_ k^(Tt- T2\

s" =--T-

Q-52)

(2-53)

Equation 2-53 is written h a particularly convenicat form' It shoss thatrhe hcat flux througtr

"

*"ll';i ;ermal conductivity'that rraries [rybili;;;* io uc Jcutatca bv tsig. thc. form of thc heat-flux.qiri*'a*i"p.a fo' **^t

"ointtuctit;ty if the conductiviry is

o.atuatca at the avcrage of thc two wall-surface tempcraturcsIf a hollow cylindcr * ioUo*

"pltt" 3nsiss of a matcrial lor which thc

thcrnal coriductivity *ti; li"early with tempcrature a similar proccdurcvill allow us to dctcrminltlt i#p"o*"-Rmlilc and bcat flux throrrghtn * r"*i"rt Thc dctails ir" tcttior pout!11.at tbe end ol thc cbaper'

Tbe hcat flux througb i lout* cythicr with lincarly varying coniluaiv'ity ald tlown surfacc tcmPcranrrcs b

T,-Toq-w,

f"

' ErFEcr oF vArt^rLE THERII L Cot'rDucrrvfrl 67

and for a hollow sphere the hcat flux is

--

\-T" -q (ro- r,l/4tk^ror,

where the values for &- arc dctermincd by

*^-u(t+r,ff)we can now sce that the previous expressions for heat flux through a

olane wall, a hollow cylindei, and a hollow sphcre with constant thcrmalio"au*i"i' may etill-&c {'"d ro dcrcnninc rh. heat transfcr by simplyr.pl"cing tie constant thermal conductivity with the thcrmal conductivitycvaluated al lhe avcrage did lempcraturc-

when thc thermal conductivity docs not rzry linearly, the heat llux ca1bc shown ro be surnmarized bi the Fourier law written in the form ofOhm's law:

s:+A-where fl_ Iepressnts tlle mtn ihcrma'l resistance of rhc solid tffitfia{-Regardtess oi geometry, the mean thermal resistance h based upon themean thermal conducriviry of r'he soiid defined by

r-- -]

- f"*g1/f-l1- llJTr

where the temperatures T1 and T2are .ihe extreme temPeratures across thcsurface, or Ai:I,-?"1 'Then ti..."tt thermal rcsisance for a plancwali is

O:Lk^A

For a hollow cylinder R- is-

ln(r.lr,)R^=_llFTand for a hollow sphere

p : fo-fi"n 4rk^r"r,,

' Exearplc 2-6. .t largc planc wall is 0J5 m thick One surfacc b main-taincd ai a tcmperature uf lS'C and thc other surface fo at ll5'C' Onlytwo valucs of rhermal conductivity are a.vailable for the wall matcrid. At0"C k:26W/m'K and at 100'C k=32 Vru'IC Detcnninc thc heat fluxrhrough ttre walt assuming the thcrmal conducti"ity varics lincarly withtemperature.

'lq

6t SrE^Dy-Sr^TE@xoucnoNSolutiou The mcan tcnpcrature of rhc wall is

T^-Tr!rTr-ry -4s.c,

Thc mean thcrmal conductivity 6q f3 qfuined by linearly interpolatingbctwccn tbc two givcn ccnductiviry valucs

32-k^* 100-2532-26 100-0

or

t-=30JW/m.KThe hcat flux through rhe *,all becomcs

a,:!_- 3r il5-35- 7 - qC: d-lsz:os -6e70 w /m2

2.5 STEADY, ONE-DIMENSIONAL CONDUCTION WITH GENERATION

umil now wc havc not considcrcd a conduction probrem that invorvcsgeneradon of heat insidc tbe marerial. The.pp.o..h to a probrem invotu-ing interna.l generation is idenrical to ,i't^, *.a in the previous sections.Fint" the appropriate form of rhe encrg, equation is sorved ror rhctempcraturc distnlution ia dre marcrial. The solution will result in twoconsran$ of integration tbat musr bc deternined by two boundary condi-tionr Ncxt' thc Fourier law is used ro derermine the hear nux rhroug ttrcsolid-

Hcar cao bc generared inrernally in a number of wavs. Chemicalrcactions. both endothermic or cxorhermic. can occur in a solid marcrial.An exothcrmic rcaction will gcnerarc hcaq whercas an cndolhcrmic rcac-tion will absorb heat from the marerial causing a negatitx source or a heatsrn&. Elcctric c_.rrenr passing through a resislnce !.n.rut., h*t ; th;conductor- Hcat gencration arso ocq's in fissionablc matcrials *

" ,.Ji

of lhe nuclcar reaction tfiat rakes place within the material.

Eeclangular Coordlnates

{s an clmp]c of a problem involving inteinal hear generadon, "or,rid",a plane wall with a consrant gcncrarion disrnbutcd urrltororty ,h.o"gh;;;

9:li*,ytyc.. fhc-gcorrrion raic pcr unir votumc i, ai",.J Ll-*r.symbol ge, dic[ in this cxrmple.is a consranr value. Supposc r# ;;planc wall hac onc surfacc maintained at a known temperature I, and thatthc othcr surfacc is insulatcd- Thc geomcrry and boundary ;J;;;givca in thc problem arc illustrared in'Fig. 2-9.

ffi'#rtai. l

SrElDy. Oxg-onrersroxlr Coroucnou wmr Gri,rtrenox

lnt0htcd $rrli(

figur! 2-9 Conduction in a plane wall with uniforrn gcncrrtio!-

The appropriate formbT the conduction equation is

* * #:o (2-54)dx- Asince the problem assumes a stearil; onc-dimeosioa.d rem{sslrelisrribgr-tion. Aftcr integrating Eq. 2-54 t*'ice. the ternpcratfic disrributicrnis

I("):- #"t*C1xlC" (2-55)wherc C' and C1 are constants of integration that rvill bc satisfied by thcboundary conditions.

The fint bouadary condition for thjs problern is simplyr(0): r, (2-56)

The second boundary condition must specify that the surface rt r:.L is aninsulated or adiabatic boundary. Since the heat transferred to thisboundary is conducted to the surface, the condition of an adiabaticstrrfaccwould, be. q"l,_,_. rffl,_"_oor

+l :o (2-s7)dx l,- LAn insulatcd boundary in a solid material,is one for which thc tcmpcraturegiadient is zero at thc boundary. Substituting borh boundary orditilrns,Eqs.2-56 and 2-57,,into Eq. 2-55 results in thc ternperaturc distribution foithc solid:

o

V

(2-58)

Thc temperature distribution is parabolic with x and its maximum valuc

T(x)-f, qtxLt. r \T=/.r,t'-7L)3g

46- . constrntt = coostrnt

79 Srsry$,r^ts C.orsoucnoxoccurs at the insulated surfacc, r-r. Thc ooditioa for a maximurntempcr.tun, dT/ dx*0, uras satisfied by the insularcd boundary

".iAit.iat r-L Thc maxim'- tcmpcraturc iatbc ualt is fhcrcforcr@)-r^-rr* otlrf

The equatioa abovc can be recast in terms of dimcosionlessparametcrs, asshown in Example l-10:

T -...t2rmr

. ,lCL

4__t_frwe should also noticc that all the encrgr gcncratcd insidc the urall musr bcconducred from thc surface ar x:0. NL lrcar may bc transferrcd ,fighthe righr-hand surface becausc it is insuraled and no cncrry may bc storedin thc marerial because stcady conditions havc bccn assumcd rhcrefor,gan encrs/ balance on t}e wall at the surfacc r-0rcquircs that

4lr,-o= - C;Vof

-u#1,_,---cTALDifferenfiarion of Eq. 2-5g will show thar rhis condition is auromaricalysatisfied.

.

Problems involving nonunirorm hear gencredoDr or on.. with differentboundary conditions. are approached in i manncr simirar ro tlu pro..Ju.iillustrared above.

Cyllndrlcal Coordlnales

.

A common problcm in c-vlindrical coordinares involving cncrs/ gener&tion is the case of a solid wirc carrying an elecrric "u,,tt *ii i. ,i,i,showa in Fig 2-10. The currenr is l and -thc ctecrrical rcsistancc

"ith. ;;;;is .R. The cxtcrnal surface remperature of the wire is a knona d;.t Th;energr gcnaated per unit volume within thc wirc is

l2R4c=TIf thc currcnt and elecrrical resistance arc co'stanls rhc internar heargcncr:.tior is also a consanl

Thc steady, onedimensional form oJ rhc conductioa equadon in cylin-drical coordinatcs which inctudcs consrant gencrarion is Eq. 2-21:

+*(#).$-o (2-se)

SnADY' Orrt-oruexsroxet. Coxoucnox wmr Gorenraox Zt

figurc2-t tonilucfionin a cylinilcr wirh uniform generadon

lntegrating Eq.2-59 twice yields rhe rcmperature distriburion in the wire interms of the two constants of integration, C, and C,:

r(r)=Crln.- S*Q (2-60)To determine values for C, and C, we musr have tu,o boundarv

conditions. At first glance it apoears that we have only one boundarycondition, which is

T(r"): T" ,

But we also know that all locations in the wire must havc a finirelempcrature- If we try to determine the cenrertine temperature of thc wireby-cvaluating Eq. 2-60 at r-0, we wourd arrivc ar

"o infioit" rcmpcrature

as long as the Inr term remains. To prevcnt the unrealistic tempcrature atthe centerline of the wirc we must sct C,

-e.-

Alother way of visuatizilg the sccond boundary condition is to realizelhat tbe ccnterline of the wire as an insulated location:

#l*,=oThe ccatcrline must bc insulated becausc of the fact that it is a linc oftJ^T.try. This boundary condition provides thc samc result rs before,Cr-0'

7tt

dfl&4^l1'.t

.i72 Sreeoy-Srerr C.onoucnox

Whcn thc two bouadary conditions arc rsed to dctcrminc valucs for C,and C2, thc tcmpcrarurc disributioa in thc wire bccomcs

ry-#l-(;)'l (2{r)

(2{3)

(2-u)

Thc maxinum tcmpcraturc in thc wirc occurs at thc ccalcr and is cqualto

r@1: r*= qX?2 +r,

Example 2'7- Determine thc marimum clurent that a r-mmdiamcrcrbarc aluminum (&-2o4 w/m.K) wirc caa cargr withour cxcccding atcmpcrarurc of 200"c. The wirc is suspendcd in air with ao ambilnttcmperaturc of 25'C and the convectivc heal-transfer cocfficicnr bclwccnthc wirc p4airis lOW/m2.K. Thc clectrical rcsisrancc of rhis wirc frunir fcqgth of conductor is 0-937 g/a-

. Tlori*, This example is a slighr variarion from rhe siruarion used roderire Eq. 2-61. In this problem ihe ,mUient air remperarure is knownrather than the surface remperarure of the wire. The apiropriar. uounau,_ucondition is therefore one for which rhe heat conduoed ro rhe .r,.;o,surface ofrhe uirc is equar to the hear convected-into the air. Mathemati-can),. thts boundar],condition is expressed as

F,irl1")- r-l= - r#1,-.. (:52)

As bcforc, the sccond boun&ry condirion is{l =otlf l._o

This boun&ry condidon imprics that rhe maximum *ire remperarure wilroccur at thc ccntcr of the wirc.

Tbc appropriarc form of the conducrion cquation is Eq. 2-59 and rlcyludg for consrant gcncration is given in eq. Z-60. SuUstiruring;;;;boundary condidons 2-62 and 2-63*rcsurrs in thc tcmpcrarurc distributionin tbc wirc of

w=ffi{,.*-#)T}c maximum wirc tcmpcraturc is thercfore

r(0)=?.*.- ,**ff(r-*)

Srr.rbv, Oxt-onrrxslos,rt- Coxoucnox wrnr Ge.rrnr'nor ?3

The gcncration term expressed in terms o! the current and rcsistancc perunit length is

o;:+:++:#+so

r*u=r-+*+(,-+)2nt

2',(lo-3/2) x l0oo3zlr.$$aI172200:25+Solving for the current yields

! =12.2 A

At this point in our development of the princlples of conductiol' weshould again recognize the occurrencc of several dirn:nsironless ros?6which rccur throughout the chapter. Equation 2-64 is writtcn ia dimeasion-less form. Thcreforc, the grouPs

qi;'r" and L+i,r*

arc also dimensionless. Thc first term is actually a dinemionless geaera-tion and the product of the two is the dimensiontress generation flrstidentificd in Eq. 2-16. The second dimensionless group is thc Biot aumbcr,which appears in problems involving the combined coaductio.n/convectionmodes of heat transfer.

tn "aaitioo

to ,..ogriring lhe existence of the Biot number' we shouldalso bc asarc of its cffcct on thc hcat-transfer process. Tbe Biot numbcr isthc ratio of conductive rcsistance in thc solid to convectilrc resistancc inthe fluid. Therefore, the physical limits on thc Biot nurnber are

Bi-+O when R*oa-+Oor when k-+co, and

Bi-rco whcn .R*,-t0or r,hcn A]--e.

. S/hcn &c Biot uunrbcr .approachcs zcrq the solid is practically bothcr'mal and thc tcmpcraturc varics nost in ihc fluid- As the Biot numberapproachcs infinity, thc oppositc is true. Thc resistancc in thc solid is muchlar3er thaa that in thc flui4 thc lluid is nearly isothcrnal' and thetcmpcraturc differences occur predominantly. in thc solid.

3t

71 Srgruv$rrrt@xouclox

2.5 TIEAT TRANSFER FROI$ FNTS

Hcat conductcd through a solid substance is oftco rcmovcd fr'ono thc rclidpurcly by thc convectioa modc- Siacc thc contrcction ratc is proportionalro tbc surfacc area, the heat dissiparcd at thc surfacc can bc iacrcascd bymercly extending the surface- The cxrcndcd surlacc is called ain.

A simplc straight fin with constant cross-scctional ale.a A is shown inFig- 2-l t. Thc heat is conducred &tot{gb$c.rolid rutmialof $cfiaaadiris rcmoved from thc surfacc to the surrounding fluid by convcaion. Thclempcraturc oJ the ambicnt lluid is I- anil ihe combincd-lreal-transfercocfficicnt is 1.. borh ol which are assurned consunL

To dctcrmine the tempcrature distribution in tbc fin, and a,cnrually theheat-transfer rate from the surface. r must firsf perform an cncrsrbalancc on a differential volunre of fir marsial We cdrnot use theconduction cquation developcd in Section 2-2 becausc it accounrs only forrlc sdrsion so&ad

#-Hgrx)-r-l:s (2-65)Equation 2-65 may be nondimensionalized by defining a dimensionlesstemperature and coordinate as

,,v>-ffi

ture distribution,

HeerTnrxsizn rnou Fws 75

and

wherc I is the base (x-0) temperature of the fin. In terms of the newvariablcs, Eq. 2-65 becomes

^xl:-L,L

aze _4Pt2 n_odEz kA (2-66)

(2-68)

The dimcnsionless group (i"PLz/kA) can be simplified to a form resem-bling the Biot number.The perimeter tirnes the length of the fin is equal tothe total surface area A, of the fin:

A,: PLThen

4=+ e-67)AAwherc ,{ is thc cross-sectional area of the fin. Equi.tion 2-67 has dimcn-sions sf hngth and it can therefore be considered to be thc characteristidlengih of thc fin l:

PLZ7-l

The dimensionless group in Eq. 2-66 can now be expressed as

Fort _n,tkAk

vhich is similar to the Biot numbcr used in previous problems involvingcombined co{id&1iril and onvcctira. Thc Bist nunber is then

it FptzBi-f -t Q-6e)We should havc expccted somc form of the Biot numbcr to appcar in a finproblem nhich combines the conductivc and conveclivc modes of hcattransfcr.

+d

76 Srrroy-Strn @rrnrrnq.r

Thc diurcnsionlcss form of thc fin-cnergr cquationyrtt o in tcrms of thc Biot nurnbcr:

f;':*'lt=oThc solution of ft.2-70 is

,{9 -

Cic -(Bl)'/:r.r 6'r"Gi)rntThe valucs of the-rwo consranB of intcgration caa bc dctcrmincd once

two boundary conditions are spccified. The most frcqucntly known tcm-pcrarure along thc lcngrh of thc fin is thc base tcmperatufc,-4; $ritren inthc form of a boundary condir.ioo,

T(0-T'

Case l: A vcry long fin such rhar thcambient tempcraturc of the fluid:

T(L-el:7*or

0(r;:sCase IL A fin with an insulated tip ar -t: l:

(2-66) can now be

(2:70)

(2-7r)

(2-72)

tip tcmpcraturc rcaches tlre

(2-73)

(2-74)

(2-7s)

This equation will sen'c as the fint boundary condition. The secondboundary condition ma)' t"kc onc of sevcrat dificrcnt forms. Threc of rhcmost commbnb/ used boundary condidons arc considcrcd in thc followingthree cases-

{t :oel,4l =Qdt lrr

Casc trI: '{ fin with a convccrivc hcat loss from the tip surfacc arca Thisboundary condition bccomes

- k#1,-El^L)-r-f

or

-#1."-#^'Thc boundary condition 2-72, along wirh onc of cacb of thc thrcc

f#

Heet Ti.er.rsmr mox Fncs 7?

boundary condirions 2-73,2-74, or 2-75, will provide rhree differcnt formsfor rhc tcmpcrature distribution in a fin of consant cross-seciional arca-

once thc tempcraturc distribution in the fin is knowu, &c icer dissipated from thc fin can bc determined. The casiesr method of cnaluaa*gthc hcat-transfcr rate from the fin involvcs determining thc amount of hearconducted through thc base of thc fin:

t --*#1,-o-- #rr,-r-r#lr_" (2.76)we can noqr deterrrine th temperature distribution and heat-transfer ratesfrom rhc fins rhat satisfy thc three given sets of boundaryconditions-

case I: For an infinitely long firu thc dimensionless tempcraturc distrib-ution is a

Tl t\ _ 'r'd(0- j#+-:.-"-VETtrt b- I

-

Bur rhe length of the fin is indeterminate, so ii .is more convenic.nt ro

llcp*ess rhc *aperar+cdisrci&lrrion in +qrm of *:T(

-\ _.rA(r): j::1--1-:e :.-Y i,e"1t,rTo- T-The heat-transfer rate is

(?i -

r_) -

VEi l Go- r-)4t: irr. j' rrr

(2-77)

(2-78)

(2-7eh

case II: For a fin wirh an insulated tip, the dimensionless temperaturedistribution is

,g),,G. _

cosrr[(ei)'/'(r _0]oG): ir- r- cosh(Bi)r/l

and the heat-transfer rate from the fin is

t1=gil/2$(ra- rJtanh(Bi)t/2 e-80)Case III: For a fin wirh convection from its tip, the tempcrature

distribution is

g(E)=#*-

oosh [ (Bi)'^( I - {).] + (Bi),/r(l / pr) sinh [ (BD r/r( I - ) ]cosh(Bi) I /2 + (Bi)t /z(A / p L)sinh(Bi;'/z

(2-81)

9

\ii:{'

7a,srrruv-Srrrr Coorrnaand thc bcar-rrans?cr ralc lr

q,-(Btnfl(r:-al{ sinh(Bi)tn +(BD'2(z /z) cosl(ni)trr JJ

(2.E2)

Exemph 2{. A stainless sreel (,t=20 W/n.K) fia has a circularcross-scctional arca with a diamcrcr of 2 crn and a lcngth of l0 cm- Thc finis attachcd to a wall that has a tempcrature of 300'C. Thc lluid surround-ing thc fin has an ambient rempcraturc of 50'C and the hear-traasfcrcoclficicot is t0 VrcF- K.,fhe end of the fin is insulated- Dercrminc:a. Thc rate of hcat dissipAred from tbc fia.b. Thc tcmperalure at rhe end of thc fin-c Tbc ratc of beat rransfcr from thc walt arca covercd by thc fin if thc fis

b nor ued-d.Thc hcat-transfer ratc from thc samc fin gcomcrry if thc suinless srccl. fin is replaced by a ficticious fin with infinirc thermal conducriviry.; .,i.

Solutim: T?rc remperature disrribution and heat-rransfcr rarc from thcfia arc givcn b-v Eqr 2-79 and 2-8Q rcspcrtivcl,v. First, wc will calcularc ticfin paramctcrs:

A : z R2 -

d(0.01;r -

o * t0-' m2i,y

_

roxr(o.o)Xo.r)2 _,.0-

"t- - rn 2ox; x ro-.&l

-

2o.oxi_x lo-' =0.06283 w/K

a Jhe heat-rransfer rate is

ct : @i)' /'z +( [ - i'-) unh(Bi)'/2-

(1.0X0.06283)(300- 50) tanh(1.0) = I 1.96 wb-Tbc fi*'tip tcmpcaturc is thc tcmperaturc at {: l:

8(t): cosho. .- : -L - I

cosh(Bi)r/l cosh(l'0) ls4t -0'6+8T( L,

- f- + 0.648( fD

- I-)

- 50+ 0.648(3m

- 50)

-212'Cc If rrc assume that tbe heat-transfcr cocfficient ovct Lhe surface of thc

wall is the same as that over lhe surface of the fin, the hcat-transfer ratc

okHeArTRATsFER rnolr Frxs ?9

from thc srall without a lin attachcd iso= i,s$r- 2'-)= tOxnx t0-.(300-50)-0.785 W

llc prcscnce of tirc fin has increascd tie heat dissipation from thc- surfacc area covercd by the fin by a factor of I 1.96/0.7g2:152,'d. If rhe fin thermal conductivity approaches infinii, the Bior i,rmb.r

'l would epproach zero. The hcat flow by conduction through thc fin' r-iddtenal would have no rcsistancc and the cntire length of thc fin would

becomc isothermal at thc basc temperature. The heat-transfer rate from,/,: i.

gro -

h,A,(T6-T-)= l0z'(0.02X0.1X300-50)= l5.Zl w ,'.,. t il

The ideal heat-transfer rarc is the maximum possible ur(olunt of heatthat can be transferred hm a fin of equal size. The stainless steel findissipates

, ,,

this ideal fin would thcn bccome

ts-7!:_t_t.96 _24Vo

less hear than the ideal fin.

Fln EfllclencyTbc previous analysis used to determine the temperature disrribution

aad heat-transfer rate from fins only applies to fini that have consrantcross-sectional areas. whcn the fin is tapered, the cross-sectional areavarics resulting in a more complex equation for the tempcrature distribu_tion. The temperaturc distriburioa and heat-transfer rarcs from taperedfins arc cxpresscd in tcrms of Bessel functions. A complete t *t rr.rrt onthe subjcct of tapered fins can bc found in References t and 2..

A convenicnt conccpt that can bc used to provide a value for thiheat'transfer rate from fins is the fin cfficiency. fre fin elficiency is aciirrc

f) SreroY-lrec C-or'tDucrloxThe hcar-transfcr ratc from thc aaud lia will thco bc

q^--niJ,(Tr'T-|Wc are alrcady in a position to dcrcrsrir crprcssions for the fin cfficicncy'no, ir"-pte -Oc fin cfficienry for a fia with consrant cross-scctional areaand an insulatcd riP would bc

grcur -

fs')t"(u/txrt- r't=t--arL(Tb-T-\r:5fur"*,tai)'z'

(2-84)

(2-85)

A plot of Eq. 2-85 is sho*n in Fig; 2-ll Thc figurc shows that rhe"ft;.i.n.y drops rapidly

as thc Biot numbcr incrcascs' A fin *'ith a large

""fu. oiniot'numbcr-dissipates less hcar than onc with a smaller Biot

aumber. If rhc cfficicnry diops ro a vcq'low valuc, it is possible for thcsurface of the ryall witbout tbc lia grcsant rc rransfcr morc beat lhan tromrhe wall wirh fin in piacc. wc could hrre anticipated this situation. ThcBiot numb.r opro.ns thc rario of conductivc to convcctivc rcsistances.For

"

largc valuc of Bi thc conducrivc rcsistancc is large compared.^ro rhcconvccrivi resisancc, and rhc tcmpcrrturc drop in the fin is significant-

Figufc Ll2 Efficicncy for consrant-cross-scctione}$Ga fiD with innlated tip.

HrerTnexsrrn rnou Fnc tt.

Whcn thc Biot number is largc, thc poorly conducting lin occuSrics al at^:t,i"i"", af*,ively transfer hcat by convccrion, and thc prcence of the fini.au".t the heat dissipation lrom thc wall'"?in

,*taob should bc selcaed rhat havc high values 9{ 6:Td.oniu.,i"i,yr that is, metallic fins are superior to fins madc of insulating;;,;;;ir. in siruations when thc convcctive-hcat-transfer cocfficient isi"... ,fr. Biot numbcr incrcascs and the advantage of adding fins for the#;;;i t*reasing thc hcat transfer rate is diminished. If rhc fluidiil;;; ptros. uy cith-er boiling or co1{gnsr1q, rhe hcat rransfcr cocfficicntt".oio'quir t-g*

^ sloi ia Tablc l-2' Thereforc' when thc fluid

;;;; piasc' it-is possiblc for the rrn to actually reduce thc hcatdissioated from the plane surface'-'r'r'g"t.

z:iz -ei.'es ihe fiq-efficicncy fora fin with constant

cross'sectional

"r.""ii,t. riritras an inilatcd tp. rnc curvc musr bc modified il it is

;;;.."d ," ;pply ro a fin wh:ch loscs heat from is cnd surfacr arca' Thei.'r, ,runrt., iiom thc tnd surfacc may bc compeirsated for b'v addin-e an;;;;i;;t length ro the fin. Thc added length is such that the additionai*ni." arca *ifl re+{ *ca*ss {ras:bc.lipaea ofrlrc actual fin and the errd sufiacc of ttre cnendcdfirmill te ireulatcd.

Jakob (Rei. 3) recommends that thc addcd lcngrh be cqual lo thc {alioof the tin'cross-sectional area to perimacr. Thc corrtctcd lengrh of tbe fin.

Figu!. 2,l3 Efficicnca for fia rith'triangnlar profilc' 1

tn

'H

':

82 Srrrov-Srrrt oxoucnox

ri''i l \"'Figurc 2-14 Efficicncy ,.,

""rrljlr,'" wirh rccrangutar profile.1., necessarl' ro sarisfl, rhc ins,.rlared dp boundary condition is tlen

L,- L* +The crror involvcd in thc approximarion of adding to the fin lenst' rocompcnsarc for hear loss from tbc tip is less rhan ld dien

Ir;wbcrc l is rhc tia rhickness. Vt

Fin cfficicncies for scverat other t)pcs of fins are sbown in Figs. 2-13and 2-14' Additioaal fin'cfficicngr "uilo -" avaitaule in Refcrcncc 4.

Exery. 19' Dcremnine the hcar-tnnsfer ratc from thc rccrangurar fiasfown in rhc figurc. The dp of rbc fin i, oor iosJat"d "oJ-,hJ;;;$cT{-:onducliviry of 150 W/r:o.lC Thc base i.rnp"ra,*. b 100.C andthc lluid is at 20"c rhc bear-ransfcr cocfficicnt bctwccn rbc ria and fluidis 30 W/maIC

.i

Lr- L + tl2

Ar- Lrr II

-r,2r' Lr, ,, I

i.

Ur,.,hf"

Heet Trrxsrrn rnov Frus E3

Solution: To account for heat toss from the tip area we determine thccorrccred lcngth of the fin: *

L: L+ $ -to*ff :ron, ",n

The Biot number based on the correcred length is then

. B ,= ^"!1,' - 3ox 9:84 lr9?9e5F

-o.ez2" kA 150x0.008and the surface area of the fin with length l. is

A,: L, p : e.2095 x 0.g4 :0.176 rn2The efficicncrv from.Fig. 2-12 is

\-0'775Thc heat-transfer ratc is then

q = qF"A "(7, - T *) = 0-77 5 x 30 x 0. t76( 100 - 20i

*327 WExanple 2-10. Ar aluminum {k:200w/m.K) annular fin is placcd on

a copper tube that carries a fluid. fie tube is g cm o.d. Tnc nuia is at250'C. Thc fin is 0.5 cm thick and ld cm o.d. The surrounding fluid is at70"C and the convcctive-hcat-transfer coefficient i, 60-W7ilt.K" D*termine the heat-transfcr rate from the fin.

I

ttlpI

-+l,F-

ttII&l SrrroYSrert Corownox '

'ntr-} -

Solufcn: Thc corcclcd t.o$h ,fo*o in Fig 2-14 *.a rofli'.lunt t*hcat loss frcm tbe tip is

L,-L+j -{r-o)* T-l:s "n

Thc profilc arca isA,

- L,t

- 4.25 x 0.5

- 2-125 cm2

200x2.125x l0-' )'":O-33

r* =

rtl L, _ r* +_ r+

af5 -z.oe

. rt 7l rl

The cfficicnc;- from Fig. t ,ott ,_0.r,

and thc bcat-rransfer rate from thc fia irc - ni, A,\7 b :T J- "ril?t7r 2,2 - t tyf; t -%)

= 0.89 x 60 x2 x r(D-O825r -D.Of 1pSO -ml:31{ W

Thc basc temperalurc of the fin is assumcd to bc thc sarnc as rhc fluidtempcnalure inside thc tubc bccausc thc rapcrarurc drop across ticcoppcr rube *ill be small.

2-7 STEADY, TWO, AND THREE-DIMENSIONAL CONDUCTION

Wc havc assumed so far that the tempcrature disrribution in the solid wasa funaion of onl;" a singlc coordinatc; that is, thc situarion involvcd onlyone-dimensional conduction. However, we now nccd to develop techniquesthat caa bc uscd to detcrnrine thc heat-transfer'rate and lemperaturcdistribution yhen tbc tempenturc is a function of two or threc coordinarcvariablc" Thc two- and drecdimensional solution urill bc morc involvcd,and so wc vill have to usc approximatc and indircct or analog merhods toprovidc a solution.

Thc compledty and length of solutions lo lwo- and rhreedirnensionalproblcne suggcst that mlutin dth a .ligiral conrputer will be desirableThcrdorc, two compuler piognms arc includcd in this scction. Tleprograe language is FORTRAN IV. Thc tlgc of cxamplc programssclect d art relativcly ci*plc. m that tbc readcr can follow thc programdcvclopmcnt without unusual cffort More complex programs arc sug-gested in thc problcms at thc cnd of thc chapter.

'

;' :(&)' " : ( 0'0425 )r/': ( 60

Sneov. Twq' exo Trntr-oncrsrone! CoilDtETtoil 85

Analytlcal Melhodt

The most obvious approach to iietermining the tempcrature distributionin a solitl for which thc temperature is a function of two or threecoordinatcs would bc' to attcmpt an cxact solution of the governingequation. For the case of steadv conduction in a rclid with constantthcrmal conductivity and no internal gencration, the govcrning cquation isrhc conduction cquation derived in Scction 2-2:

912':0

Th,iscquatioa ie Laplaoc's cquarion- Tlclocnof laptgr,;cl "TElioa in thcdiffcrent coordinatc systems is given in Appcndir B.

Laplacc's cguation is a linear partial differential cquation. Scvcral stan-dard techniques for sq!'ing it are available. One metho{ separation ofvariables, is pardcularly usefrrl in heat-transfcr worlc Although thismerhod is not coverEd here, the interested reader b referred to Rcfcrences5, 6. and 7 for comptete details on this and oths mcthods of soh'ingLaplace's equation.?nct the tcmperaturc fistrrturion is dacrmine{ te-rarAess tf ttfu-

ftc hcar flux js dctcrnincd by the Fourier law. ln two- and thrccdirncn-sional systems this law is most conveniently exprcssed in vector form as

4":-kvr (2-86)qherc V l" ir thc gndient of thc scalar tempcrature. Thc form of 'thcgradient in rcctangular, cylindrical and sphcrical coordinates is girrcn inAppendix B.

The gradient of a scalar quanrir.v such as the temperatur-e rcsults in avector quantity which, according ro the vecror form of the Fouricr law (Eq.2-86), is the heat flux, g'. Usuall;-, we do not consider the heat flux to be avcctor quantity since it has dimensions of energy per unit area, ncither ofwhich are vcctor quantities. Howevcr, it is convcnicnt to imaginc hci.t tobe "flowing" in a certain direction; thereforc, {'ir often rcferrcd to as rheheat-fux oector.

An imponant geometric property of the gradient is the fact that the'heat-flux yector is directed pcrpendicular to an isotherm. a line of constanttempcraturc, at each point in the solid. As an illustration of ttris properry,Fig. 2-15 shows several isotherms and representativc heat-flux vcctor atpoints l, B, and C in a twedimensional rcctangular rclid- The lcngth ofcach of thc three heat-flux vectors is proponional to thc local temperaturegradient. That is, wherc the isothcrms are closc\r spaccd, the gndient islarge and the heat flux is also large. Where dre isothcrnrs arc *idclyspaced, thc heat flux is proportionally smaller. In Fig 2-15 the heat flux atpoint ,{ ir grcater than at point I, wherc- the tcmpcraturc gndicnt issmaller.

trt

t6 SrrrotrSrereC.onoucrroxs

{lI

f. r cmLrt

Flgure 2':5 Hear-flux vccror and its g-comctric rcrarionship to isorhcrms.

since we can visualizc thc hcat flux as a vcctor. it shourd have propertieslike any other vecror quantiry. wc should thereforc bc able ro resofvc thchcat-flux vecror'inro i'' componenr in the directions of the coordinatcaxes. Expressions for thc vcctorcomponcnts can bc determin.o uy."p"nJ-i"g t. form of the gradient For a rccrangular coordinar. ,jrr.rn, rt.heat-flux vecror is

a"--r({i*#;-84Thcrcforc, thc heat flux in the r dircction would bc

_ .dr4,- - KESimilar. cxprcs.sions can be written for thc @mponcnts of the heat-fluxvrctor in thcy and z directions

Thc hcat-transfer ratc in thc -r direcrion across a ptanc area p which ricsin theyr plane is then

',--ol^,(E)l,* (2-87)

,t

t&BF

Snenv; Two_ ero TtncE-DnancroN^L CoirDtcnou &fThc subscripr p indicares that rhe derivative of thc tempcraturc m.st becvaluatcd ai cach point on the plane beforc integration over rhc a-rea of thePlane.

Graphlcal MethodsExact' anarytricar sorutions to the conduction equation for tc,o and threcdimcnsions are often impossiblc to achic,ve- For cases in which;"ly,i;

solutions are difficurt to obtain, approximate merhds "r"

r**td rilgraphical method is a simpre.techniquc that can provide answenr for thehcat-transfcr ratc wirh surprising "."u.u"y.The graphical method is based on the geometricar requirement of thevccror form of the Fouricr raw, which srtcifies that the isotherms and,constan! heat-flux rines are ajways perpeniicurar at points whcre the twolines intersect. We can rherfrre,["t"i, tt" isotherms and consront fluxIines and conrinue ro revise them until they sarisfy rhe perpcadicurar

condition-Thc accuracy of the sketched temperature distribution wit be dirccdyrelared to lhc carc raken in ttre consttnction-of ttre lines- T,rittr a littlecxperience we can obtain reasonauty accurate resurts in a shorr aaount of

.time.The sreps used in the graphical method can bc outlined as follows:Step l: Draw an accurare scare model of the materiar in which thetemperarure distribution and heat_transf.. rui. a-re desired.Step 2: Sketch the heat-flux lines and isotherms on the modet. The llu.rlines and isotherms form.curvirine"r rq;r;;.-;t au interse*ioas of theheat flux and isotherms, the tangenrs ;o- ,h";;;.t wiu bc pcrpcndic.tar.The diagonals of curvili-rrear squares bisect each other and are pcrpcndicu-lar' Rcmember ihat adiacent irotr,.*rl unJi""un* fines caanot *, tcach other' Isotherms are pcrpcndicurar to adiabatic boundarics sincc anadiabatic boundary is aline oi

"o*t oi n,r-, th"t is, g":Q. Atrso, lincs ofsyrnmerry arc adiabadc boundaries.Step 3: Continue to redraw the isothcrms and flux lines by adjustingthcir locarion undl thcy mcct the

"onaitions sp"Cncd in Stcp Z-

.h:: ygu Tc satisficd with :h: "".c,roq of your drawing the teopera-turc distribution is known and thc t."t nu* is acterminc4 as usual, byepplying the Fo'ricr o*: T: ilil;; thir;'fo;or", oroo:d., {h##lem of dctcrmining rhe heat-trdnsf., ,"tc Urriugii .t u"uof I .bcam uscdm a furnace wall Thc bcam is ,u..ouna.Jri-"in., siae ty-hr"il;;placcd in rhe wa' of the furnace,;;;;i,t ;ti'r-lu."), The nrfa* of

SS Srrrry-Srere Coxorrcnol

ln3uLliq

tigurc 2-f5 Graphical rncrhod applied ro ea t bcam: (a) p}ysical modcl;O) scale dnwiag of bcam and skcrc$ of llru lioclrend isothcrms'

tbc bcam on lbc insidc of thc furnacc has a tcmpcraturc of fr and thcsudacc of tbc bcam ncar thc .xtcrior wdl b ar Ir Tbc thcrmal conductiv'ity oI Scbcarn b*.

The ccatcrlinc of the bc.rr b e linc of ry@{ry md it is thacforc anadiabatic surfaca Constant-flux lincs aad isothcrms skctcbcd on a scalcdrawing of trc bcam arc shown in fig. 2-16(b|

The bcat is rcsuicrcd to llos' in four chanacls limited oa cach sidc byflux liaes- Thc'rctal hcat transfcr rate through onchalf of thc beam is

Flur lincs

I

Adbb4i$surfre

ffiSrerov, Jwo er.o Trneonararixrer Colorlcrron S9

therclore

c*,- i *Thc inset in the figure shows a rypical curvilincar square through which thehcat-rransfer rate is 4r. Thc Fouricr law applicd to thc single curvilinearsquarc per unit dcpth of thc figurc is

,:r=rorL# (2-s8)II car.h curviliacar squarc is skcrchcd such thar it mcets the condition

Ax= 6y, cach tempcraturc subdivision is cqual The tempcraturc diffetenceberwcen two adjacent isotherms can then bc cxprcsscd in terms of thcoverall temperaturc diffeppce across the cntire surface aed, M, the aum-ber of equal temperaturc subdivisions ia thc figure:

(aT)*, Tt-TTi*r-Ti:ffI{ rle flsi liacs ha* t'etn dfuided irno t/ qtat subdivisiong tre heat

transfcrlhrough cach of the channcls formcd by adjaccnt heat-flux lincs iscqual. and the total heat transfer through the bcam is

{..r-lf4 (2-eo)substituting Eqs. 2-88 and 2-89 into 2-90 givcs an cxpression for the totalheat-transfer rate of

(2-er)when the grid is squarq that is, whcn Ax:rl1.

The heat-transfer rate can thercforc be determincd by drawing a scriesof curvilinear squares and then counting the number of equal tempcraturFsubdivisions, M, and the numbcr of equal hcat-flux subdivisionl iy.

Exa.mple 2-lI. Detcrrnine thc hcat-transfcrratc through thc beam shownin Fig. 2-t6 if Ir-500"C, Ir:200oC and k:?0 \trr/m.K.

Solunr'on: For &is cxampleM-13 jy-4 (AI)*I-300"C

Then the heat-transfer ratc through rbc bcarn pcr unit depth is givca byEq.2-91:

(-z[Sttarl-"J: rem v,oThc factor of 2 is uscd bccausc the tzluc of rVras shoyn for only onc-halfof the bcam. ' ..

,.#.s.3

'{,i90 Srerpf-Srerribrwcnor

cl:+

+.6 slh rFa+nvsgrg

v-tdt\O .O l$l 6T ; ll=l- Fil- E Rl E---

ci I

rSI

4.tIs.

I

go

?

tt)s

U

dIclFI

.5=ncE+. r!: e

.Es* 3;- E5r !x Ert=-! sg t =

E 3

-t=o

;-eCoAc.C;'

=iotro

*-=- -l

Sttrot Two- exo Txnrs-or*rapsrorrr Cclxirraon gl

\-8Al

ldFl>tlv

-lts

3A

l^t-

_,1; ,asli tlSl3 '19ls l:

-qlA

l-J l*tsl> Fh-Fl! dlvlv lv

lc lc

:I

*y'.--\ i

I -

\./

ca.=

-oc-? E-t 5i.E*:E

E F5=

ooq

.oo

A

g'oc

o 9xt3E=E:x3

{u660o= c'c 63-'g G': gatr

9a Srrror-Srere Ooxoucnor.

IEE{li t-albb'tl$tqI ttgIi.(lg3{ls E*ls !.Fl:* flr;qlr itllirt:;lI 72-'l.s;ilexEIE?E[=F-:l-

-

.l3r!I ?;ili-l rl:a e; :-J*r IF;: **;= i.E E?Si

-

Eiia:3{F-a oEi-i>

-.41>

t ee--.E .E.r 5%ts tso

tsF'

t.-rl]dldlvlc

-ONrA

!c

o-=e'-

=

=-^eC

-q.: '=

o=

E.E ?-i =Eei t3

4Ee

o

-J.9:;

co.:N

tr

r; . .[

SrEADy, -fwo- rso Trnte-onct.rsrorer Corroulnoy fSiThe facror N / M in Eq- 2-9 t is callcd the conduction slnpe faaor, Si

s:-#,so that rhc hca( lransfer written in t:rms of .S is

y'- kS(A?n)"_.[ e-g?l)Thc conduction shape factor for severar shapes are cataroged in Tabrc 2-1.

.

Exgressions for the conduction shapc factor are known for sveralsimple geometries. For cxamplg when iq. 2-26 is arranged in the form ofEq.2-92, rhc conduction shape factor for "

piun" wall is A/L. For. ahollow cylinder of lcngrh I, rhe conductioo ,nifi ra"t* o Zri1tnp.lr,l.Notice that the shape factor has dimensions of I'.rg*,.Ana?og i,lc,lhods

L

."1i.^:t_..:t-:flricaf potenrial f in a material with constant resistivityano no tnrernal sources of potendal is governed by Laplace,s cquation, the:amc equation rhat governs rhe tempcraturc distribution in a coastantpropedy solid wirh no inrcrnal cnergy gencration- The ,t .quudc; io;transport ol hear (Fourier's law) and- ttte transport of charge tbrt^" r"*lare also similar, as shown in the table. Owing to ttre siiritarity of th.cquations rhat govcru thc two phenomen4 the" transport of chargc andhea! arc said to bc analogous.

lglgous Equations for Thermet and Elecrrical SpremsErrcrnrclr-

SvsrevTsrr-VerSrsrev

Conscrvation cquationRatc cquation

v2E-0-AE'::n--

grf-0:1f

'&,,I:-l]i:lionlcss potential in the clectrical systm is anaiogous ro theormensronless tempcrature in a thermal system. It is to our

"d';;;" ;use this analogy bccause voltages are easii,o n,."rur. than tempcratures.By measuring the rocation or ti.,. consrni;,.;,i"r rines with a vortmeter,ye can determine rhe locations of thc isotherms.The analogy is carried out as forlows- e-*"r" model of thc thermarScometry is cut from a comrncrciary avairablc irecrrically ilil;

l1f:,,t"d e bancry.b conncccd o ttc p.p.r !o previde the overallcnung porcnriar across the modcl Thc crcerricar uiu"a..y;Jil;;lTOg::O on thc paper must bc

"n"fogou, . ,fr. io.rponding boundary:_":1r,1*t in rhe rhermat problcrn. noi UOermat UounAaricr the conduct-ng papcr must have a boundary of constant potentiaL Thir _n ."rity t

a''4t

X Sreroy.,Srrrt C-onoucnorachi*cd by- coating rhc oo-undary with a hie*rry conducting paint andco-nnccring ir ro a bauery. Tbc thermally insuiarcd bouoa-ary ;n b.sia-utarcal in the clccrrical qnrcm by an cicctricaily insularcJ 6;;;:which is simply rhe cdgc of thc papcr.

Qrcc:thc oonsrant potcntiar rincs arc rocated with a probc conaecrcd toa voltmercr, rhc boundary conditions may bc swirched and the o.ti,og*"ilinel or thc lincs of constanr currcnt, may bc located. 'llhcsc lincs clrrc-spond ro lines of consrant hcar flur- By using this proccdurc tn. *naccuratcly gcncrate thc compretc sct of curvilinlr squ"r., in the moder.and a value for thc conduction shapc facror

""n t dercrmined with

ercatcr accuracy than by the graphical rcchniquc.Thc analog mcthod has thc advanragc rhar l rocarcs rhc isorherms andflur lincs without rhc rriar-and-crror proccdure of rhc graphicat ;.,h"d,but ir has rhc disadvanragc thar it rcguires rhc purchasJoi ,p.A^i.qJi

mcnr T:hc -sraphical method requircs only pcncil. papcr. and p"rj;;;.Botb merbods. howcver. arc pracricatl'rimiri to rwodimcnsionui g.o-.-

rrics and simplc boundary conditions such as isothermar aad adlabaticboun&rics. A morc 6lar-il+'cl ,l:.,rroioaf *hcanelog,nethod can be foundia X,cfcrcnccs 8 and 9.

Numerlcal lrletbodsNumerical solutions are po*'erfur and versatire rechnigues when applicdto slca.dy conducrion probrems. Numericar merhods ."n * *...Jr',uty,

applicd 1o prgblems rhar cannot be sorved convenienrty uy ort.ii."i'-liCuo.For examplc, numerical merhods can be used ro J,u. proUt..r,,invoh'ing radiadve boundary condirions or inrcrnar .n.rg' gJn.iui;".Graphical and analog rechniques cannor be conveniently ,-,iOlo pr;;i;;solutions to thesc rwo t)?es of problems.

Tlrc finitcdiffcrcnce numericar merhod invorves dividing thc sorid inro anumbcr oI da- An cncrgl' barance is appried to cach nolc o,hich r.suhsin en- algcbraic cquation for the tcrrp.ratur. of cach node. A ,.p;;r;cqution b dcrived for cach nodc tooted on thc,boundary

"i ,h;-J;Thc rcsult of thc finitediffcrencc rechnigue is z argebrai. .qL,io^ r* ,hia nodcs in thc rclid Thc, algcbraic equations ,ipt".. *,1 sirgl. p;Jdiflcrcorid cquation and thc applicablc Lundary conditionsIf tbe numbcr of nodes in thc sorid is reraivery smarl wc can uscstandard mathcmaticar tcchnigucs ro sotvc the rcsurting algcuraic

"qu.-tions As tbc numbcr of nodcs incrcascs, thc rimc rcquired to achicvc ancxact solutbn becomcs unrcasonabrc. Approximate solurions bccomcadrraotagcors ir thesc cascs Wc will firsr considcr

-

";p;;;;;;mctho4 a,llcd re I axa t io n.. .

A p.-o*ber of equations grows large, the application of programma-blc calculators and digitar contpurcrs biomes i.por,.nt. Two computcr

iSreeov, Two- ero Tnrer-onrrr.rgorget Coxoucnox 95programs are includcd later in this section to illustratc thc typ6 0fsodimensional conduction probrems that can bc b.*t rctvcc by digitarcomputers coupled with numerical techniques.

Thc finitedifference rechnique wiil be iirustrared by considering a two-dimensional conduction problem. Firsr. we dividc thqsolid into aiumbcrof equal-size squares- The sorid within each subdivision is imagincd to beconcentralcd at the ccnter of the square and thc conccatratcd. mass is thcnodc- The_interior rcgion of a typical two{imcnsional sotd is *own inFig-2'17- Each subdivided square has a rength in the x dircction ofAr anda length in they direction- of Ay. Thc node designatcd by thc subscdpt zcrois shown surrounded by the four adjacent noaes. Each noa. ir imag:ncd tobc connected to adjacent nodes by a smalr conducting rod- Heaican bcconducted only along rfu: imaginary rods. That is, inducrion bctweennode 0 and node l, which actually occurs across an interface ofheight ay,in the continuous material is imagined to rake plage through thc inrlginaryrod connecting nodes 0 and l.

For steady conditions an energy balance applied to node 0 whcn there isno encrgy generatiorr gives

9i-o=0

Figurc 2-17 Arrangemcnt of nodcs for an interiorscciioaof r twodimcnsionel solid-

4

: Q-e3)

'40

96 Sreroy-SrenC-orcDncnox

Ncx! wc apply the foi." law to cacL of tlese temu to "xpr.r.cquatioo in terms of aodal tcEpcraturies Thc frnt tcrm would bc,

cxanplg

o,-o--.#=k$ra1.J-:l3-whcrc drc tempcraturc gradient is cvaluatcd at thc midplanc bctwecn thetwo nodcs and d is the depth of thc twodimcnsional gcomclry measuredinto thc planc of thc figurc. Similar cxprcssions caa bc c/ritten for thc threeremaining terms:

T -',rq.-o-klxt fr:

nr-*=x4'affT

-',re,.'-kh,xdfrfIf rbc suMjvisiors arc drawn so rbribg/:rc.aI squarg At:Ay and

cach heat-flow cguation becornes indcpandanlof the gpornetry. Horrcvcr,thc accurary of replacing the temperaturc gredicnr by lbc frnitc dillcrcnccof two tcmpcraturcs is dcpendcnl upoo lbd ciu- of cach squarc. As cachsquarc is madc smaller, tbe approximation for thc tempcraturc gradientbecomes more accurate.

By substiruting the four finitediffcrence rquations into Eq- 2-93 we seethat thc energr balance for nodc 0 is simply dcpendent upon the rempera-ture of node Q and the temperature of thc four adjacent nodes when thegrid is square and the thermal conductivity is constant:

Tt+ T2+ Tr* Tr-4To-O (2-e4)Aa cquation similar to Eq. 2-94 wiI apply to all interior nodes: thar is, ir

applies to all nodcs that arc not locarcd on thc boundagr of the solid andarc surrounded by an cqually spaccd squarc grid-

A scparate cnergl balancc must bc applicd to cach nodc lhat is locarcdon thc boundary of the solid- Consider, for cxamplg a nodc dcnored bytbc subscript 0 located on thc boundary of e did rhat is in conract wilh afluid- Thc ambicnt fluid tcmperarurc is I and rhc convccrive-hcat-rrans-fer coefficicnt betwcen'the solid tnd rhc flull b i.. Thc geometDr is shownin Fig- 2-18. Each nodc is located at the ccnta of iu rcspecrivc suMivision.Nodcc rhar cach boundary no,ilc rcprcscnrs only onc{alf of thi massrcprcscntcd by cech intcrior nodg

Nodc 0 on thc boundagr cao cxcbange hcat by oonduction wit} threeadjaccnt nodes ia thc soli4 and il can also rransfcr hcat by convcc{ionwith thc fluid- Thc cnergr balancc applicd b node 0 is &crcforc

4r-o* 4z-o* 1ra* g---o:0

,fl'1U

sfiij:l

tlrefor

Sraoy, Two ervo Trnre-onreNsrorer. (l*ougron 9I

r-rt'E 2-rs .[i,frffif:l#f; :T"1j:*imcnsionar

Thc first threc tcrms represcut. conduction in the solid and thc rast termrepresents lhe convection ratc to node 0 from the ambient fluid dcsignatJby the symbol o. Subsrituting finitcdifference. approximations for theFourier law for the first three terms and Newton'i iaw for the last termyields

kA.vd!#+*faIfi+*f a!3+ i,d,ya1r_- ro11:g (2-95)

Once.again Eq.2-95 can bc simplified if we choose "

rq*d grid. orAx:A.v. Equarion 2-95 may bc rewriuen in thc formi(r,+rr*r,+(+)'--['.(+)]n-, (2-,ou)

-

Thc tempcrarures.at the boundary are functions of the tempcraturrs o[thc n-eighboring nodes and the parl.mctcr i,i;77. we should rccogrizethis-dimensionless group as the iiot number.The finir+dilference method is illuaretcd in.thc,following cxamplc.Exaopte 2-12- Detcrmine thc stcady tcmperature distribution aad hcat-transfer raks from all four surfaccs oi tt" woOi*.*i.rii*fii,h;;

thc figurc' Two of the boundaries "*

i,*,rt"*ia,'" ,rri.a is insurated, andthe fourth transfers heat by convcction- ---' -

l

#";{t'j

98 SriE^D"SrlrtCoxnJcnox

Surfrcr 8Ia - l0O"C

t - I lvtn.li4a"=o

./ r Dcprh

L--:o(m----JSurfrcr DIn' 100'C

t-j r-----i1t.:l,irl--r-----r-

Ui

--i--_-_1,___tt'l6to

Solurion: Tbe solid is fint subdivided inro a square g,d as sho*,n in theaccompan-ving figurc. The nodes are numbered from I t.o 9. The grid issquare with A.x:A/= l0 cm. Thc only nodes with unJcnoqn t.-p.rutur.sare nodes 4 5. and 6. Node 5 is an inrcrior nde. so Eq. 2-94 applies.

T.* T2+ T6+ Tr-4Tr:{1Node 4 is on a boundary rhat rransfcrs hear by convecrion, so Eq- 2-96applier

l(Tt + T) + rr +(Bi) r_ - (2 + Bi) 7.: 0where

Bi h.A,x j0x0.t0,*-T- *-T:,Nodc 5 is on aa insulared boundary, so thc appropriatc enerry balance is

qr-..| qr_"* qs-=Oor

Sorfa: Ii.: 50 t in::l(

t_ - -io"C

Ii

l0 cmII

I

IL

Surfecc Cinqht..l

I

J.:

;

lr

Jr

!'

k+dfi++k^yd#+*faW.-s

ffitis

Sneov. Two- l.tD Trnre.Do,lelsror.A.t. C;olroucrrox- I .

or

i(4+rr)+?'j -ZTr=sThc remaining six boun&ry nodes are mainrained at knosn tcmPcratures'so energy balances are not nceded at thesc nodes- The six boundarylemPeratures are

Tt=Tz:4-200"CT7:Ir: l'r:100"C

Substituting these tempcraturcs into thc encrry-balancc cquatioru for nodes4,5, and 6 Yields

M+Tt-7Tq:0 (node 4)300+ i"r+ T^-4Ti=O (node 5)

150+ fi-2f6=0 (node 6)Values for the temperatures ?i, fr, and Io may be daer:nined tt sfuirutta-neously solving these three equations- il'hc solutions arc:

T+-75'5"C" T':128'?"C

Ie= t39.4'C

. To determinc thc heat-transfer rates pcr unit depth at each surfacq wewill use the finite-differcnce form of the Fourier law when heat is uans'ferred by conduction and Ncwton's law when hcat is traasferred by \convcction.

At surfac I the heat-transfet rate per unit depth into the did is l. il:fl-a+4*-t+4*'t

- lT

-T. T--Trfil:lUl\r+Q--r')+ -T )-t: _621.5w/m

That is, 627.5 W/m is convcctd away frorn thc solid al th surfacc l' Thenegativc sign indicatcs that thc.heat is rcrnoved from thc solid-.

At surface I the hcat-transfer tate per'unit dcpth iato thc solid is. qL= q\a+ qlt-s+ q!a+ qi*

=*6r(l rt, t *"1"+f rr;r'1-^-^\tTt-- aY '2 AY I+f'.$1n- r-l

:538.8 W/m

5n

'#tq+ilI$n

:

fOo Srrrov-Srrtt CorcDncrroN

Surfacc C is insulatcd, rc

At surfacc D(c-0

I I Ts-Tc _

Tr-7, _

t D- ?: IqL* k Axl- 1f

^y *T-i

^y )+ F,!e,_r)

-88.8 W/mAs an ovcrall chcck on our heal-raasfer rateg wc k:row rhat for srcady_sratc conditions, thc nct hcat-uanslcr ratc inro the solid must bc ,tro:

q'"^: q) + q's + q'c + q'D= -

627-5 +538.8 +0+ 88.8:0.1 vm

Tbc sizc of .the net heat flow inro rhc solid givcs an indicariol of thcaccuracy of the finitcdilfcrencc mctbod for this panicular.problcm-

Selarallon TechnlguesIn Example 2-12 thc solid r'-rcrial r.as rubdividcd into a gid ia *'irictr

thrcc nodes had unknowa remperarures. Thc solution rcsulted in thrcealgcbraic cquations for the tlree unkno*n tcmperatures. If rrc had wantedto incrcase thc accuracy of rlre solurior by dccrcasing thc grid spacing wtwould havc had more nodes with unkaown tcmpsratures and addirionalequations to solve. In general cach node with aa unknonm temperatureresuls in an algebraic equation that must be solved simultaneously withthc other nodal equations.

Wlrcn the number of nodes is relativety small. as ir was in Framplc 2-12,we havc lirdc roublc solving tlrc qmcm of simulraneous cqrurions. \\henthc n'mber of cquations becomcs largg borlever, it bccomcs ncccssary tousc an approximatc mcthod to solve tlre eguarions- A tcchniguc occasircn-ally uscd in hcat transfcr is thc rclaxarion srethod- whilc tbe relaxationmcthod has limited applications ro practical heat traasfcr problcms, it is apcdagogcd tool that can illustratc lrow numcrical tcchnlues can beapplld to simplc problcms Tbc conccprual Echcmc of tbc relaxationnctbod will also belp us undcrstand rhe morc praaical nurnerical mcthodswhich follow latcr in thc chaprcr.

Tbc prposc of thc relaxation method is lo cstimatc thc tcmpcrarurEs ofcach oodc ia the solid such tbat thc cncrgfba,lelce eqrntlrnr rrt rpproximately satisficd. lnstead of scning all cnergr-balance cquations suchas Eqp 2-94 and 246 cqual ro zcro, c/c coutd cquare thcm toi tcrn callcda rai&nl- The remperatures arc rhen qnrematicary changcd uatil rhcrcsidual is rcduccd to a smalt valuc- Thc size of tlrc residrut wiu indicarc

ti

Srtrov. Two- rrrD Tt{r.EDncrrfloNlr. Cor.rorrcrror l0lthc accuracy to which the tcmperatures of all the nodcs have beencstimated- If residuals for ill nodat cquadons arc reduccd cxacdy to zero,rhcn rhc tcmpcraturcs arc cxact solutioasio the cncrry balancc cquations.

To illustrate the rclaxation metho4 we c:ln apply it to thc thrce cqua-tions used in Example 2-12. The three nodal enerry balanccs in thecxample werc

4ffi+Tr-7Tr: R,300+ L+ T"-4Tr: p,

150+ ?i-2Ir3 R.The right-haud side of cach cquation [as bcen replaced by a rcsidua! .(,wherc thc subscript indicates the respcctivc node. Our job now is todetermine values of Tr, Tr, and. ?, so that the residuals are reduced toreasonably small values.The magnitude of the residuals will dctcrminc &eaccuracy of thc approximation of the tcmperaturc. We notice, for examplgtbat an crror in thc tcmpcraturc of nodc 4 of otre degree will produce arcsidual of seven degrces. The dimensions of the residuals arc tcmperature.Oncc r&c rodd cacrgyta.bace cqsarioas levc bcca derjvcd, &c rcla;rationtectmiqw according ro tfic foltwing four steps-

Step l: Thc first step in the relaxation method is to assumc values for allunknown nodal temperatures- We should use our knowledge oi heattransfer to $ress tempcratures as closc to the actual values as possibte. InExample 2-12 we must guess values for Ta, T', and f6. The cxtremetcmperaturc limia in the problcm are the 50"C fluid and thc 200.Ctempcrature on thc upper surface of the solid. Thereforg thc guesses forthe unknown steady temperatures musr lie between thesc limi.ts We wouldcxpect 2"o to bc thc lowest of thc three tcmperatures ana f6 to be thehighcst because it is on an.insulated bou.ndary. Suppose we assume thatthe thrce initial values for the temperatures are t

Ir:80"CIs:100'C'Ic = l50oC

Step 2: The next step involves substituting thc initial tcmpcraturegucsses into the residual equations and calculating each rcsidual The

'tesiduals for this example "t"

*.= -60-c

frs- 130'CRe

- -50"CSinct thc rcsiduals ar non,rcno, we must @ntinuc to change the. ternpera-tures until each rcsidual is rcduced toqard zero-

102 SmrmSrrrt*:Srep 3: To reduce thc residuals we changc the tempcratuic corrcspond-

ing to thc abaolutc valuc'of rhe largest residual until that residual isreduced to zcro. Thc convcrgcncc to the correct set of tempcratures isoften hclpcd by changing thc nodal tcmpcraturc so that thc residual is norrcduced cxactly to zcro but is changed to a small value with a sigr oppositcthe sign of the rcsidual prior to change in temperaturc. This process iscalled o*nelaxation.

In our cxamplc thc lar-eest residual conesponds to nodc 5. The residualequation for nodc 5 shows that if wc increase I by 35'C rhe new valucfor thc rcsidual R, will be reduced by 140'C. thereby changin_r ir ro a lowvaluc with an opposirc sign. Norice that a change in I, will also affecr rhevalucs for R, and R.. A summary of the new values for the threelemperatulcs and corrcsponding rcsiduals are

?"r:80'C R.: -lJog

. l":: I35'C Rs: - IQ"CIe- 150'C Ro:

- l5"C

Step 4: Tbe next srep in tbe rclaxarion process is to repeat the prcvioussrcp until the desircd degree of accurac]' is achicvcd. The largest residual isno*' R.. so we chan-qe 7. by an amount to change .R. to a small positivevalue. Assume that we decrease I. b1, 4"C. The new temperature valuesaad corresponding residuals are

_

Tt:76"C Rr:3"Cl"s= 135"C Rr:

- l4"C

Io:150"C Re: -

lS"CRepeating this step twice, first changing Io. then 1'5, results in the follo*.ing values:

Dccrcasc I by l0'C:Tf 76"C Rr=3oCl"s=135"C Rs=

-24'C?.e = I40"C Ro:5"C

Decrcasc 7j by ?"C:Tr-76"C Ra-

-4oCIt= 128"C fis:4"CIo:140"C fie:-2"C

Irt four rclaxation stepc Orc tcmpcraturcs arc all within I oC of thc cxactvalucs dcrcrmind in Examplc}-l2.The preceding steps arc besr organizcdin a rablc similar to the one shown in Table 2-2. By organizing thcrclaxation srcps and rccording rbe &ta in tabular forrn, thc amount ofripeatcd work is rninimircd.

H;{q:i

m,*'ffi".i

Srreot Two- rxo Tnnee-os{rNsror,r& e.oNDuclloN

Tzbie 2-2 Summery of Tempentures rod Residuab for Exrnpk 2.12SrrP T1 R. Tr-Iniial gucssInitial rcsidualstncrcasc i"5 bY 35New residualsDccrcasc 7. bY 4Ncw rcsidualsDecrcasc fr bY l0Ncw rcsidualsDccrcasc 2'r bY 7Ncrr residuals

R4,T.R,

5

_t

150

r50

150

l4tt

t40

100

135

r35

t35

128

3

3

-4

EO

80

16

76

76

-60

-25

t30

-10

-14

-244

-50

-t5-t5

The finite-difference apfioach using a relaxation methd can be ex-lended to cylindrical coordinates. and the resulting differencc cquationsarc described in Reference 5.

lf internal generation is present in the solid, the relaxation technique canbe used without any complicarioa- Suppose rlat ar a particular rrode theenergy generation ratc per unir volume is a'f,. Thc cncrry balancrc for aninterior node 0 in a two-dimensional systcm with four neighboring nodesas shown in Fig- 2-17 is

Qr-a* 1za* 4t-* qr-u+ qo:0Replacing each heat-flow term by the finite-difference fortn of the Fourierlaw gives

k^yd!J:&-+k^xdf+*Uaff+ r*aff +q,f, ax6yd=o (2-e?

If the grid is squarq Eq- 2'97 becomes

(2-e8)

Whencver a nodq is located on the boundary of a rcli4 the rcsidualcquation depcnds upon thc type of boundary condition at thc surface. Forcxampte, t-he residual equation for a surface node on a flat surface incontact rvith fluid is givcn by Eq. 2-!16. Residual cquations fior otherboundary conditions arc $unrnrarizcd in Tabh 2-3. In each casc theeacrgy-balance cquation is witten for the node dcnotcd ty trc mbccript O

To this point wc have considcrtd problems iin which the temPerature is afunction of two coordinates only. Ho*cvcq thc tcchniqucs we haw devel-oped for two-dimensional problems can easily be extended to thrcedimen-sional problems. For cxample, if wc considcr a typical node 0 in a cctnstantProPerty solid {,ith no gencration rurroundcd by six nodcs as shown in

fr + f2+ Tt+ T4- 4To+ O'; @X\' :O

4.

lol Srrrov{rrru Oolrpucnorl .Trtfc 2-3 Residud Eqrnriotrs rr Bo.ndery Nodes rn Tno-Dhrnsio;ret

Sfsteos, &t!$c Gri& (&-Af,lConorriox Gtoranny Noorr.Eeuenon

Fbt surttcqisotbcraalbouodary

Fler surfacc,iasularcdboun&4r

Flar nrlaccia conuctvirh floid

Encrbr corncr,both surfaccsilsuhrcd

,Errcri:r orncr,borh surfaccsin contaowitb r lluid

7-lr

o'E *ti- ro-&(7i- 12- 13, beatinput at surfacrpcr unit rrca

- g')

i(rr+ :rJ)+4-2To- 4

i(r:+ rr)+ I,+(Di)I--f2+Bilfo- Ro(Bi-itx/k)

i(r'+ rJ- r"- &

i(r,+ rr+{BDr--(l +Bi)ro