Chapier lPRIHCIPLES OF HEAT

TRAIISFER

1.1 INTROOUCTION

Whenever a temperatrr.anrr..r.a-rroln't'J:::-1lff:rence exists in the universe. energ/ will bcremperarure.A.'.".o;'r.lJ.,?:H"ffi

:.*f ::,:::,':.,[:::-""".U,Jffitransferred as a resurt "r "

rr*fl.r*..ffi.;:. is cared rear. Arthouehtire laws of rhermodvnamia aeat ;,h;;;nsfer, they can only tre_etsysreins rhar are in equitibrium. n.;;;;:.1r1.., u. used to predict rheamounr of ener_*y required ,o "1.uis,.;;r;#;;". one cquilibrium stare.to.anorher' but rhey cannor predic,i"* i* ii.* changes wi'occur. The,scrcnce of heat rransfer rupjr.rn.no ,h.;;,';il second laws of crassicalthermodynamics by providinc ;;il;;l;;il., rhar can be used ropredrct.rates of energy transfei. -- -- v' *'.ry

ro tltustrate the difft1'."11iy'-".il;;:T::i:::,[j]::Hii"1, ji:,","i:,T;,i:1ff :;:ff':'l#triiff'l#::-lt" *"*' ri#'or"?mic ta*s can be used rJ'equiribrium.;;il;'i,,11;:?.J$**H..il,n'"AHT'*j|;il::i:r;I,.*:; o"t ,n* "i,l*i."i',T*r",rhc tarc or hcatro ne. i r *iii t;; #;ilT::i;:;,H#,f.:f n;ruI:fi f;on the orher hand' can predict ,r'r.."i"Ji'i.",?in.t, from rhe warer ro

f,

2 hnerls or Heer Trlr.rsrm,thc stccl r'rd and from rhis informadoa cc caa rhcn carcurate lhc tcmpcra-H:." rh; rcd as wcll as ttc r.lnpcn;;il wrrcr.s r function of

For a comprctc heat-''ansfcr. anarysis it is ncccssary to dcat with thrccdifferenr mcchanisms: conduction, ;ilfi;d radiatisn The designand analysis of hcar-cxchaage and ;r;;;;;on sysrens requiresfamiliarity with cach of these-mech^ri;;;;;transfcr, as wclr as theirlll.T"Uorl In this.chapter we will consider rti UaSc principles of heattransfcr and. somc simple applicarions In *Ldu.ot cbapten each heat_transfcr modc will bc trcared in dcrail

1.2 CONDUCTION HEAT TRANSFER

conduction is thc only hear-rransfer mode in opaque soiid media- when atempe*rurc.eradienr exisrs in such a body. h; *i' U. "r*f.,,Jldthe highcr- to the lower-temperarure roa Trrc iate dt rhich hcat istransfcrred by conducrion, q1, is proponi;Ja dr. renpcranrrc gradicnldT/ d", times the area rhrougi *hi"lrhe"t ;, **i*.C,f fFig t-l(a), or

u^*4*'here

f _ tem;rcraturcx = direcdon of rrrarrcrr,

The actual rare ol heat flow depcnds on rbc thcrmal conductivity, /r, aphysical .propeny of the mediu;.

'n. ;Gdon caa therefore bequan dtativel-v expressed as

.dTq^: -ME

Dircction ot hcst tlos.

re+lFqg11c l-t {r) S*ctch illustrar;lg sign oo,avcltba for oonducrbl hcar flow.

{r-r)

(t-2)

I

I

.. ,.r.: j :t-i..

colquctrol l{err Te'rrartr 3

The minus sip is a consequcnce of thc sc^con-d. t"* of tlttttsdynamic$which requires that hcat

-tt no'" in the direction of lower tcrnpcraTlt:'n" tt"aL;i, as sho*n in Fig l-l(b)'

-wiil b.c ncgative if thc tempcraturc

decrcascs wirb increasing ""r"L or *. rr

vye designate that hcat transferred

ilTlfiri,i". dircction*is to bc a poitiui-quanrity' the negativcsign mustbe inscned in thc right-hand side of Eq' l-2'

*:=JJ# -niqetiaa-of hcat condrnicn'

Eqg:rion 1-lisfo!rt?r''t !at' of t*at cfu*tiott'and scrves o ac!11yrt.rirat co"a,rctivity ft. lf the aiea is in sguarc melers' thc tcmellurc 'ndegrees Kelvia'

"

irr' *t'*"oiJ't't-ttt "f-!-"it

transfer in natts t harrkunits of wattstrermcter Per degrcc Kclvin (w/n'{)'Plane Wall

A simple illustration of Fourier's law is the case-of heat transfer through

. ;i;;[ ri,.*" i" Fis:l-i' when both zurfaces of thc wall are atunifornr, bur different, ,.rlip.r","r.r iear ,rill flow only in one dire.;doruperpendicular to trre *arr s,iriiJ-s.-if

-the thermal conductivity is unifonrL

integration of Eq. l-2 gives

dT7?

T

o,:-fG,-r,)= le,-r) (r-3)where' L:thickness of wall $e: 'I Tr: temperature et left surface (x:0)

fl=:cmPeraturc at right surface (x: t). Friiiricr sas r Frerch methcrnatiqlan ( l ?6E- r

g30) wiro madc importert conrribudoas rotib

rnilytic rcatmqrt o( conduci'on (sce Rcf' l)'

;_:: I nnr

cio"'ucnoN HE^TTI !$FEr 5

Lin "'"1

cases intcgration of Eq' f-2

in more dctail in ChaPtcrAs showngivcs

n, = $\tr,- r) + +(r "_

r'1] (r-5)

or .r,=bl{rr-T)

(t-5)

u\cre k- is dre '"lue of k at rhc nlu""'" tempraturc

(TiT)/7'

plane Walts tn scd )od thermal conlacq-as

nmff***gg$**mi.mt*'** =(f ),t', -",=t?) !'- n-\Y)'tt'-

t;l (1-7)

Etiminating the intermediate tempcratures I' and T' in Eq'

l-?' the ratc of

. q,

';fr$J#"iil'(*)" (*), (*).

,.""n**H*l?;tl':ffi';rbroughJ-l

. q,

5 .hnrcrr.es or HrlrTr^ilscnhcat llow caa bc rnittca in thc iorm

(l-8)

(l -e)

is tbc surfacc

Tr-?,'t (L/IcAra+{Lltat)r+(L/kt)6

For a iaultilaycrcd slab of IV layas io pcrfcct thcrmat contact, thc ratcof beat flow is

n,:TfI+= ,r-:,-r'*'(L/ktr), ,{ e/*),

-

,-l

where T, is the surfacc tcmpcratur of layer I and 7rr*,temperature of laycr N.

Example.l-l A furnacc wall (sec Fig- l-a) consists of a l.2-cm-thickstainlcss stcel inner laycr covcrcd by a lcrn-thick outcr laycr of asbcstosboard insularion. Thc tcmpcratnrc of the insidc surface ef &g slainlesssteel is 800 K and the outsidc surfacc.of the asbestos is 350 K. Determfurcthc hcat-transfci i?tc through tb furnacc wall per lnit arca and theternperature of the interfacc bctqrcea lhe stainlcss stecl a:rd the asbestosThc thcrnral conducrivitics for tlrc steel and thc asbestos arc, respectively:

&r-19W/m'Kkz-OJ W/m'K

ft 'tOO X fr-350N

-q

T1 T' T2@

Iigtu l-l Furnacc *all for cpnf.lc l-2

L2i;;Lrffi

.', :. ::

CoNDUcrroNHeltTielarer, 7

Solution: Thc hcat-uansfer.rate is

^ - Tt-TzYt Lt/k/+12/kl

The heat-transfcr rate per unit arca is

o! : =,T'-T-' r. - = ='=8,9-3=1,= = =6245 wlmzA Lt/ k* Lzllk2 0-012/19+0.05/0.7

The interface lempcrarurg ?,, b determined from the equation4*

_

Tr- T,A Lt/ kl

Solving for I gives,. -,, -*(+) :'*-u'o'( lF ) -7e6 K

The lempcrature drop across the stainlcss steei is thcreforc only about4tK;the temperature drop across the asbestos is 446'K.

Eleclrlc AnaloJj lotConducllonThis is a convmient staning point to introduce a diflcrent vicwppinf for

thc analysb of trcat transfer *hich can be appliett to morc corrplexproblenis and will be followed up in later chaptcrs. The new approachmakes usc of conccpts developed in electric-circuit theory and is oftcncallcd thc analogt bmryeen the low ol heat and electricity.Il thc leat-trans-fer rate is considercd to be analogous to the flow of clectricitn thecombination L/kA is viewed as a resistance. and the tempcrature dif-ference as analogous to a potential differcnce, Eq. l-2 can bc irnitten in aform similar to Ohm's /ar+ in elEctric-circuit theory:

ATqr:T (t-ro)whcre

' A T=Tr- 12, a thermal potentialrRr = I rthermal resistancekA.

The reciprocal of thc thermal resistance is referred to as the therrnalconductance, and k/ L, the thennal conductance per unit arear is called theunit thermal conductance for conduction hcat flow. Similarln Eq. l-8 canbe extcnded to heat flow through thrre scctions in series, ns shonn in Figl-4. in the form

ATt:-rt R.. + Rs + Rc. (r-r r)

6r

whcrcLT-Tt-Tt

^^-lhl;,^,_(h\,*:(#),

Thc clectric'analog approach can also be uscd to solvc morc complex

oroblcms. For cxamplc' tt"';;il;;s conductlon occurs in matcrials

il#; ;:'*:l i rully.X.:;:mf.m:',X !j!i:! Izrea A, and 12 rtr Paralrhe right of thc physical fi;-';;;it this.lroblcm

note that for a givcn

rcmperature dilrcttnce i*;'tilscuaycr of lhc compositc can bc

^r,"irzcd scParatcly' #id"d t;;.condiT1s,:cccssarv for onc

dimcn'

,ior,.l .ondu'tion ri"ou]' J""-t'it * lt:::::t;':*il t:t?tt:IT:";,lr"r.-aifi"..ntc between thc adjacenr matcnals

;"","ffi ;t-";i. ,i,;,#I*g*::ililil1,':$',::"i'liand thc Problcm maY tloss of accurrcY'

Sincc ireat can flow through thtj::iT:als along seParatc paths' the

*li-...f *tt I!o* is lhc sun dtltceo:

4*- 4t* th

-ffi,.ffi-l{*d)rr'-r; (r'r2)^'' #

l'--*--1Rgvc f'5 Hcat colduction through

r rrall with two sections ia parallcl'

Coootrctror HerrTrer.rsen 9

^Lt' li,tt R.= !t' ko-.|o

and the ratq of heat flow is

R.- RtRc' Rr+-Rc

e-:*3sncXn.

ly': number of tayers in scries !& - thermal resistance of nth layer

A ?"o*.,, = tempcrature differcnce acrriss two outer surfaces

Figure l-6 Scries/parallet thermar circuir. rectangurar coordinarcs.

observe that rhe totar heat'transfer area is trrc sum of the two iodividualT.T q 6"-, {: reciprocal "i tfr" ot"f ,.rirln-* equals rhe sum of therecrprocals of all the individual ,.rirt n".r--a, ,fro*r, in Fig l-5, thethermal circuit for this problem i,

" ;;;L';*"ngemenr of rwo resis-tances. Rr and R..

...

A more "ompl.x application of the therrial network

. approach isruustrated in Fig. l-6, where heat is *;;; through a compositestrucrure involving thermal resistance ir.i.l*'*a in paralrel. For thissystem rhc resistance of the niddle,"y*, X;;;;t{, becomes

n.=3'4c..lc

(l-13)

(l-r4)

where

il hnriinrs a llrrrfhArsrrrTbc prcoedia cirotit anab'sis assumes that thc hcat flor is one-dimco-

tiun{ If thc resistenccs R, and X. arc sigdlicantly diffcrcot, n*,o-diroco-siond cffccts caa beeomc imporunr Such twodimcnsiooal conduaionproblcns sill bc discusscd in Cbaptcr 2

Contacl ReslstanccWhco diffcrcnt conducting surfaccs are placed i:o contacr as shown in

Fig la, a thermal rcsistancc is oftcn prcscnt at the interfacc of the solids.Tbc intcrfacc resistancc, frcquently izlled the contact rai.stance, is dcvcl-opcd whcn two materiah will oor fit rigbtly rogether and a thh layer ofnuid is trapped bcwecn rhem. Examinad6o of an cnlarged view of thecontact betveen thc two surfaces shows that lhc solids toucl only at pcaksin thcsurfacg and that.tbc valleys in the mating surfaccs arc occupied by afluid, possibly air, a liquid or a vacuum.

The iatcrfacc resistance is primarily a function of surfacc roughness, theprassu* holding tbe two surfaces in contacl, thc inrerfacc flui4 and theintcrfacc temperaturc. At the interfacc. tle mechanism of heat transfcr is-*-drx- 4eaducrionl*ce pl*e $*ea3h *-*oracr g**s of rirc eoli4ttrih tcut is transferred Ly convection and radiation across thc rrappedinrerlacc fluid.

If lhc hcat flux through rwo solid suilecee in contact k q/A andthetcmperature differcncc across the fluid gap scpararing tlre trvo solids is Af,,the intcrface rcsistancc R, is defined by

LT,R,:fr (t-r5)Ulhen two surfaces are said to be in perJect thermal contact, the intcrfaccrcsistancc approaches zcro and therc is no tcmpemturc differcnce acrossthc interfacc. For imperJect thermal conract. a tempcraturc differcnccocqrrs al thc intcrfacc.

Mct of thc problems at thc end of the chaprcr do not consider intcrfacercsistaacc, orcn tbough it cxists whencvcr solids arc rincchanically joincd.Rcgardlcss of this fact, we should always be aware of lhe existencc of lheinrcrfacc rcsisrancc and thc rcsuhing tempcraturc diffcrcnce across $einrerfacs Panicularly with rough surfaces and low bonding pressurcs,tbc tcmpcraturc drop across thc interface can bc significant and should notbc ignorcd-

The subjcct arca of intcrfacc resistancc is a compler oDci aDd no singleSoory or ea of empirial dara rccrrrarcly dcccribcsthc incrfacc rsictaaefor surfaccs of cngineedng importancc. .Rcfcrcnces 2 and 3 should bcconsultcd for a morc dctailed discussion of this subjcct

F=='--=. ,+_-. .\

CoxouclroN HertTnrrsrer' ll

. Exemplc l-3- The wail of a house consists of a laycr of common brick(1,:0.tOC 6. ft=0.?0 W/m'K) and a la-rcr of gyPsum plastcr (Io:0'038rr''k:O.rt8 W/m.K). Compare $e rate of heat transfcr througb this wallwith anothcr which has bctween thc brick and Spsum an interfaccresistance of 0.1 K/w.

Solutioo: Thc ntc of heat transfer through the idealized wall pcr squarcmeter area apd per dcgrec kelvin temperature difference is

-J!-=A(Tt- Td Lt/ ki L2/ k2 :4.50 W/m2-K0. r@/0.70 + 0.038/0.48The intcrfacr will add a third resistance in series. and thc rate of heattransfer will be reduce4,to

,4r , :===] -:3.11w/m2'KA(Tt- To) Rr+ R.+ Rhi., 0.222+O-l

Thermal ConducllvltY'I'lre thennat conductir;ity is a material property defined by Eq' l-2'

Ercept for qases at low temperatures it is not possible to predict thisproperty analytically. Available inJormation about the thermal conductiv-ity of materials is therefore largely based on experimental measurcmcntsIn general the thermal conductivity of a material varics with temPeratrrfe,but in many practical situations I constant value based on the averagetemperaturc of the systcm will give salisfactory results Tablc l'l lists

Teble l-l Tbermal Conductirities of Sorc Met'l\Noioehllic SoUds. Uquids. and Gascs

Mertnrer Trcr.vrr Coxoucrrvrrret 300 K (W/ra'K)

CoppcrAluminumCarbon stcclGlassPlasticsWaterEthylcne glycolEnginc oilFreon (liquid)HydrogcnAir

386.20l..

54.0.?50.2-0.3

.0.60.250.t50.070.r80.026

12 hnergorHarTrnrsetqpicai valucs of thc tbcrrril condrrcrivitic fm lomc mcta\ noanctallicsolidt liquids, and gascs to illusrraa Oc ordcr of magnitudc to bc cxpcctcdin practice. Addirional informatiol is prcscutcd in Appcadixcs E GoughH.

Thc mcchaaism, of thcrmal conduction in gascs caa bc cxplaincd quaFtativcly by thc Linaic rhcory. All molcctlcs in a g;as arc in random motionand cxchrngc cnergr and momcntum wbcn thcy collidc with onc another.Howcvcr, sincc higher temperatucs arc associated with motcculcs posscss-ing morc kinctic crcrgr, whcn a molcculc from a high-tempcrature rcgionmovcs into a region of loyer tempcf,aturg it transports kinetic energr oa amolccular scale to the lowcrtemperaturc rcgion. Upon impact *iO

"molcculc of lowcr kinetic encrgr, aD cnergt transfcr occun which is sccnas a transfcr of hcat from a macroscopic viewpoint The physicalmcchanics of conducrion in liquids is qualitarively similar, but sincc

Bcm

v

o ,50 .s0 t50 6J0 7t0 tJolopnrmtll

Figru! l-7 Veriation of tbcrmal coaductivity rirh. EmPcflalut" for rrarious gascs ald liguids"

L_., _

CoNDrrcnoN HrerTnelrsrsn 13

molecules in liquids arc morc closcly spaced and thcir forcc ticlds play asignificant role in the cnergr rransfer during collision, rhe picturc is evenmore complex dran in gascs.

fi$rc tt shows how the lbcrmal conductivity oI rcmc gascs varies withtcmpcrature Thc thermal conductivity of gascs is almosiindepcndcnt ofpressure, excepr near thc critical point. According to a simplificd analysisbascd on a kinctic exchange modcl the thcrmal conductiviry of gascs willincreasc as the squarc root of the abdutc tmperature.

Figure l-7 also shows rhc rhermal conductiviry of some liquids as afunction oI temperaturc. It can be scen that except for warer, thc thermatconductivity of liguids dccrcascs wirh incrcasiag rempcraurcs, bur the

F

:s0 150 4J0 550 650 .!SO 850TffiFErurc(l(l

Figurc l-E Variation of thcrmal conductivity wirh tempcraurc for somc mctals.

la hnorr-cs gf HEAT Tr.rNsFEl

$*g. is rc small that in most practical situations the thcrmal conducthrity nay bc assumcd constant at some average tcmperature; tbere is aoapprcchblc dcpcndcncc on prc$ruc in liquids-

figurc l-E shows thc thcrmal conductivitics of some metals and non-ngrrllis solids In solids, thermal cncrgy is transporrcd by frce elccronsand by.vibrations ia thc latdcc stnrcturi In gcncrat thc movcmcnt of frceclet'ons is thc morc important modc, and since in good clccrrical conduc-tors a largc number of frec elcctons move within rhe lanice structurc,good clcctrical conductors are alrc good hcat conducton (c.g. coppcr,silvo, ald aluminum). on the o&cr iand good,elcctricar insutarions'arcabo good thcrmal insularor: (e.g; glass and plastics). Thc bcst rypcs ofthermal insulators, howevcr, rely for thcir insutating effcctivcness ontrapping e 3ns within a porous slrucrurc. In those matc;ah rhe transfcr ofhc11may occ'r by several modes: conduction through a fibrous or poroussolid strucnrrc; conduction and,,for convcction ttrrough air rrapped in thcvoid spaccs; and radiation between portions of tlre solid srru.tut , which iscspecially important at high tempeiatures or in cvacuatd enclosurcrSpccial tlpes of superinsularion marerials have been developcd for cryo-gcrfc appacatlons at verJ torv tcmpcrerures, donn to aboui 25 K. Thisckinds of supcrinsulators oonsist of scvcral raycrs of highty reflecdve mareri-als,-scparared by cvacuated spaccs lo minimizc conduirion and convecriorlaad caa achievc effective conducriviries as lou,as 0.02 W/m.K. Morc.complerc inJormation on superinsuladon is givcn in References 4 and 5.

T3 CONVESTION HEAT TMNSFER\llhca a fluid comes in contact with a solid surface ar a diffcrent tempcra-turc' lbc rcsulting thermal-energr-exchange process is called conrxctionhut transler. This process is a common experience, but a derailed dcscription of thi mcchanism is complicatcd In this introductory scction *,. tittnot ancnpt to cover analyical procedurcs, but rathcr conccntrate onprcscatiag an overview of the mccbanism and pr.cscni thc basic cquarionsthar e' bc used to calculate thc rate of convection hcat transfcr in thou.su\rstcms which are importatt parts of complerc headng and coolingqlstcms

Thcre arc two kinds of convcction processes: natural or./ree comxctianaad trccd com;ection. In the first tlpe rhe morivc force comes from thcdansity difference in rhe llui4 whictr results from irs contact with a surfaccat .a differcnt rempcrNtwc and gi'es risc to boo!"afl forccs" Tpicalcramplcs of such frcc csnvcction arc rhc heat rransfcr bcqvecn oe vialt orfbc roof of a housc on a calm day, rhc conrrccdon in a rnk in which ahcating coil is immersed, or thc bcat transfer from thc surfacc of a solarcollector whcn there is ho,..ind blowing"

{i

CoNrrEctroNHEAITn^NSFER t5

Forced convection occurs when en outside motive forcc mov6 a fluidpast a surfacc at a higher or lower tcrnperature than thc fluid- Sincc theIluid velocity in forced convectiou is larger than in frec convcctiorl moreheat can bc transferred at a given temperature differerrce Thc pricc to bepaid for this increase in the rate of heat transfcr is thc work requircd tomove the fluid past the surface. But regardless of whether lhe coavcction isfree or forccd, th rate of heat transfer, q., can bc writtcn in the form ofN*tton's lat, ol cooling:

q,; E,A(T,-T|) (I-16)whcre

;1- = unit thcrmal convective conductance, or aver:rgeconvection heat-transfer cocfficient, at fluid-to-solid interface. W/m:. K

rl -

surface area in conract with fldid, m24 :surfacc temperature. K

|.-: temperature of undisturbed fluid farauraytromheat-transfer surface, K

Equation t-16 scrves only as a definition of 1.. The numerical valuc of li.must be determined anal-vtically or experimentally. The SI units for lr. arewatts per square meter per degree Kelvin and Table l-2 lists some.approximatc values of convection-heat-transfer coefficients, including boil-ing and condensation, usually considered to be a part of the area ofconveclion.

Table l-2 Appmxirnate Veluesof Coovection-HeacTraasfer Coefliciens

Co$vEcnoN Mooe euo Fr-rro 4 $/mrK)Free convection, airFrcc convcction, waterForccd convcction. airForccd convcction, wa(crBoiling watcrCondensinq water vapor

5-2520-100tG20050-10,000

3,000-t00"0@5,000-t00.000

. Example l-4. Water at 300 K flows over one sidc of a platc of t x2 m inarea, maintained at 400 K. lf the convection-hcat-traasfcrcscff,icbnr is 200W/ml-f" calculate thc nrc of hear transfrr'by conuection frorn dre ptareto the watcr..

Solutipn: Using Eq. l-t6. the rate of heat transfer isq.

- i

".'t (7,

- 4. -

) - 200 x z x (4oo : ?cn) : ul'o,ooo w

L--/

f6 hnrcu-es or Hr.rr Trensrtr,Fhrid flog

?::T:'r l,- I,4.

Iigute l-9 Vclocity and tcmpcrature disrributionsfor forccd convcctioa ovcr r bcetcd plarc-

Merhods for calculating a hcat-transfcr cocfficient arc ukca up inChaptrs 4 and 5. Herc ure shall mctcly cxanine tlrc p'rocess and qualita'rivcly rclatc tbc coavcction of hcat to thi flow of the fluid- Figurc l-9shows a hcated flat platc coolcd by a srrcam of air flowing ovcr it Also'rtmrar*crctttity'rnfi{tia?rnp.tfiuefistrflrrtioniftrt=fu$ poinr ronotc is that tbc velocity, z(y). dccreases in rtrc dircctioa toward rlcsurfaccas a rcsuh of viscous forccs. Sinct the vclqgity of rhe tluitl layer adjaccnt tothe n'all is zcro, thc hcat traasftr pct unit arcat betwecn lhc surface andthis Ruid laycr must bc by conduaioa alonc:

(r-17)

Atbougb this viewpoint sugg6ts thar thc proccss can be viewed asconduction, lhe temperature gradicnt at tbc surface' (?I/dy)lr-" is de-rermiacd by the ratc at which thc fluid farrher from thc wall can transpontlrc cncrgr into the mainsuearn- Thus thc tcmPcraturc gradicnt at thc walldcpcods on the flow nd4 s'ith highcr vdocitics ablc to produce largcrtcmpcrrurc gradicnts and bight rares of lrcat transfcr. At rhe samc timghovcrrcr, the thcrmal conduaivity of thc fluid plap a role. For cxamplc,tbc valuc of /9 for the watcr is an ordcr ol magrritudc larger tban that ofair; tbug as iho*n in Tablc l-1 lbc convection-hcat-traasfcr coefficientfor rratcr G largcr than thc cocfficicnr for air.

Tbc situation is guite similar in free convecdorl as shown in Fig l-10.The principal difference is that in forccd convcction thc vclociry approacbes the frec-strcam value impoccd \t an cxtcrnd forcc, rhcrcar iairce oavcction lhc vclocity at frrst incrcascs witl incrcasing distane frorn

.t! rhb r.xl r prioc npcrsig'. indicercs r qmtity pa uit lcagtb, r doublc primc is rbrqursiiry p.r urit rra, ud r u^ph Firc dgilics the quearity PGt urit volur

* - o: = - \#l ,-=E

(r'- T.-)

F-lgure l-10 Vclocity aad tcmpcraturc disaibutioos for frce @Evecu:onovcr a lreatcd platc inclincd oo *t" p-i; thc horizontal.!

the plate because rhe action of viscosity diminishes rather rapidry whire thedensit_v difference dec1";t;;,i;;il"il:::.:*T:i,itx?;,i"ilffi H;J:H:I;"':;thesurroundingfl uid;this"dussse;:4];;bi;re.rirrnainnrmandapjroach zcro lar auzv frcm dr ,."r.d

"rrh;;: ilc rempcrature fields infrec and forced convicti"" ;;;-;i"J;:;"p"r, and in bsth cases theheat-transfer mechanism at rhc fluidr/solid;r*; is couduction.Thc.preceding discussion ioa;"ai", trr"t--ttJ-"on"."tion-heat-transfercoefficient will depend on thc density,dr*i,y,'""i velociry of the fluid aswell as on its thermar nroperties (thermar *io*,iu,y and specific heat).w''creas in forced

.onu."rion trt. i.ro.itrL *ily 'rpor.a

on the svstemby a pump or a fan and can be directly spccifid l"?;;;;;;H;:velocity will depend on the-tempcraturc difference between the surface andthe fluid, the coefficient of th.'-rd;;p;ri*; * fluid (it determines:11.,*fit change per unit

".p.-i"[Iin .*""i and the force field,wnrch rn systems located on curth is simply ,f," gru"io,ional forcelAlso, convecrion hcat traosfer oo U" irJ"rA fiti" ,t. framework of alh.*^1. resisrance nerwork. Fr"c, ;;--i:iltiltr,""r"t resistance inconvection heat transfer is given by -'

Cor.rrccnoxHrmTn*rsrrn 17

(t-18)

(r-le)

&-+hoA

and this resistance at a surfac.e to fluid interface can easity be incorporatedInto a network. For example, the heat *rii* irr'o thc interior of a room'at f' through a walr to atmosphcre oursia. "i{-;"rrtr*t, in Fig. r-r l. Therate of heat transfer.is gu.n Ly

o=1-T.: T,-r,'

't& - RFT'l+&t-t

It tinrcu.rs orHr,{tTrensa'

nr"M'r,-=l n..! n..J-

' hc-t{ *..1 ' hrn.lF4ure l-ll Tbcrmal nctwork for bcat rratsfcr through

r planc wall with convcction ovcr bolh rutfacca

shcre

n,- |' F,.,A

Rr-*Rr:i

h..oA

F,xemple !.i A 0.1 rn-thick brick wall (/t-0.7 W/m'K) is cxPoscd m acold wind at 270 K through a convcction-heat-ransfer cocfficicnt of 40W/m2-IC On thc otber side is calm air at 330 K with a frecconvection'bcar-tassfcr coefficient of l0 W/m2'K. Calculatc thc ratc of bcar rransferpcr.unit arca (i-e. the hcat flux).

Solutioo: From Eq. t-19 thc thrcc rcsistanccs are

^,-*-"h=0.o25 K/wn,=h: #=0.t43 K/w\**:fth:oro,:

-:

Riou.norrIIeetTrAr.rsfER t9.

i{- RADIATION HEAT TRANSFERWhereas conduction and convecdln heat transfcr can ouly takc placcthrough a material mcdiunr, radiation can transport heat cnen tbrough aperfect vacuum. In the radiatiae mde o! heat tmrcfer thc eacrgt istransportcd in thc form of electromagnctic wavcs which tnvcl at thc spccdof light-. Thcre are many different electromagnctic radiatioa phcaoocoa(e.9. x rays), bur here we will consider only thermal radiation thattransports cnerSl as heat.

Thc quantity of energy leaving a surfacc as radiant hcat depca'ts rpoothc absolutc temperarure and the narurc of thc surfaie- A perfcct radiator,or blackbd.v,' cmits radiant energy from is surface at a rate q. given by

q..= aATa (r-20)The ratc of heat flow by radiarion, a, will ti in r.atts if the sr.sface arcalis in square meters (m:), rhe surface remperature f in K and rhe dimen-sio- ^ I 66s51rr,4-{,.4alld the S*{a'el&olr:.rleria torrom, * t&:a !r iB 6{value of 5.6?x l0-r w/m:.Kr.

An inspcction of Eq. l-20 shows that any blact surface radiatcs iratc proportional to the fourth power of dre abcolute terapcrranrathc ratc of cmission is independent of the conditions of the surroun

20 hnorrs cHr.rrTrlrsrnwbcte c1 t ihc ..mitrancc of tf,: gray surfacc and is cqual to thc ratio ofcmission frm rbc gray surfacc to the cmission from I pcrfec ndiator atthc sanc trycratura

Ex.ilrylc l-6. Calculatc tbe ratc of beat loss into spacc by radiation fromthc uppcr surfacc of a horizonral squarc flat platq 2x2 m in arca, at atcmpcraturc of 500 K with aa cmittancc of 0.6.

Solution: From Eq. l-20 thc ratc at which radiation is cmitted from ablackbody at 500 K is .

c, -5.6?x tO-rxn x1500)':14,180 W

Howevcr, since rhc ernittancc is 0.6 for the surfacc, lhe acrual heat loss willbc 0.6 x 14,180*8508 W.

If ncithcr of two bodics is a pcrfet tad'iatort and if rhc rwo bodicsposscss a givcri geomctrical relarionship to cact other, lhc Dct hat transfcrby radiation bctwcen thcm is givcn by

{.-4'{rq.q(jli'-4'J {=23)wherc $-1 ise :nodulus whjch rnd.ifics.tbc cquarioa for pcrfccr radiarorsto accotrnt for thc emittances and relative Sqomctries ol {he actual bodics.

In many cnginccring problcms, radiation is combincd with othcr modcsof beat ransfer. Thc solution of such problems can oftcn bc simplificd byusing a thermal rcsistancc, rR. for radiation- The dcfinition of .R, is similarto that of &e tbcrmal rcsisancc for convcction and conduction. I[ thc heattr:nsfcr by raciiation is writtcn

^:Tt-Tia' R,the resistancc, by comparison with Eq. l-23, is given by

.- I a5r-r(rl-rr')RJ, Tr-Ti

(t-24)

( r -25)&- Tr- Tiote rSr-r(ff - fr1)

Also a unit thcrmal conductancc can be dcfincd for radiation r1], by

(r-26)

whcrc I is any convenicnt refcrcncc tcmPcraturg whosc choice is oficndidarcd by lbc convection cguation.

Exrmph l-?. Calculatc the radiation unit thcrrnal oonductancc for esmall spberical thermocouple junction locarcd in a large black pipe carry-ing air- Thc pipe temperaturc is 300 K tbc thermocouplc tcrnpcraturc is500 K and thc i^nittancc of rhc thcrmocouplc surface is 03.

Cou{rmo Heet-TnHrsrrn lr{Eo{^M$,rs 2lSolutioa: From Eqs. l-22 arrd l-26, ,rsumirt that thc refereacc tcmpcra-

ture is thc pipc tcmpcraturc Q wc getF _ er (r._ ra\,r- 1,_2rr\.t '21

=d(r(rrr+ rr2)(rr+rS= 5.67 x ro- t x oi(5od + rod;1tm1:463 W/rn2-K

1.5 COMBIIIED HEAT.TRANSFER MECHANISMS

ln practice, heat is &ually transferred in steps through a number ofdifrerent series-connecred scctions and heat transfer frequently occurs by1qp6 ps6henisms ia garallcl. The trrnsfer of heat from the producs ofcombustion in the chambcr of a rocker ms161 thlerrgh a thin u/all to alrralu:{goudpSjalarpa4{$ovcrlhc ourside of the wall will illustratesuchaosc{Fig. t-12).

Producrs of cornburtios conrain gases, such as CO, COr, and HrO,ryhich cmit and absorb radiation. In the first section of this systcrg hcat istherefore transferred. fiom thc hot gas to the inner surface of thc wall ofthc rocket motor by thc mcchanisms of convection and radiadon acting inparallel. Thc total rare of heat flow 4 to the surface of thc wall some

Phvsiel Sl srem

Coolanrr..n.

Thcrml Groir

6)

'la - al.+ tt4 I1

Itgurc l-f2 Hcat lraasfef in a rockct motor.

Qc

9t

; 22 hlrcrrrs q llrrr Trrxsrs,-distane from thc aorzlc is

4- 4.* 4,- it(rr- Trl+ F"t(Tr- T",l

olo -(F,,t + i4)(rr- rr)

T -',r

.

-T 0-27)wberc

Tr= tcmPeraturc of bot gas

L - temperature al inner surfacc of e'all

-

xr= #. f?:T:#L'fd resistancc

In the steady statg hcat is conducrcd through thc shcll, thc sccond scctionof rhc sysrcra, at thc sa.src ratc as lo thc surfacc and

e-4t-lQ,-r*\.T'

_T: 'o

=

'* (l-2S)rR2

whereI-:surface temPcraturc ar wall on coolant sidc.Rr: thermal resistancc of sccond scction

Afrer passing through the wall, the heat llows through thc third scction ofthc sysrem by convection to thc ooolanL Thc ratc of hcat flow in the last

. stcp is

q= qr-irA(T*-'7")=T.JT, (l-29)i,

wherc

4 : tcmPcraturc of coolanl-

thcrmal resistancc ino'- third scrrion of rysrcmNt sbortd bc notcd that the symbol f] srands for conrycction-unir-surfaccconductancc il gcncral but thc numcrical valtcs of thc convcction ccf-ficicnts in thc irst and third sccrions of thc systcm dcpcnd on many

Covatxto He^r-TrlNsrER MEfi^tq$.s Zt

factors and will in general be diffcrent- ,rlso, the arcas of thc thrcgheat-flow sections arc not cqual- But sincc the wall is vcry thin" thc ctrangcin thc heat-flow area is so small thar it can be neglected in this systcm-

In practicg oftcn only the tempcratures of the hot gas atrd trc coolarrtarb known. If intermediate tempratures are eliminated by algcbraic addi-tion of Eqs. l-27,1-28, and l-29, the rate of heat flow is

whcrc the thermal resistances of the three series-connected scctions rheat-flow steps in thc system are defined in Egs- l-27, lA8, and l-29.;

In Eq. l-30 the ratc of heat llow is expressed only in terms of an overalltemprature potential and the heat-transfer'characteristics of individualsections in the heat-flow path. From these relations it is possible toevaluatc quantitativelyJhe importance of each individual rhcrmal resis-tance in the path- An inspection of the order of mapitudcs of theindividual tcrms in the dcnominator often indicates thc means of simplify-ing a problem. When one or the other term dominates quandtativcly' it ises+acrins gcr**imiSh *o {Egh r**'ecs{-'4s*ea.,+iury ir r$'techniqucs of dctermining individual rhermal rcsisranctt and unduc-tanccs, thcrc will bc numcrous occasions wlrcre such approximations wiilbe illustrated- There are, however, certain typs of problcnrs. notably in ?hcdesign of heat exchangers, where it is convenient to simplify the writing ofEq. l-30 by combining the indiviciual resistances or conducances of thethermal s)rstem into one quantity, called the overall unit conductance, theoverall transmittance, or the overall coefficient of heat transfer, U. Tiic usco[ an overall coefficienr is a convcnience in notation' and it is importantnot to lose sight of the significance of the individual iacto6 that detcrminethe numerical value of U.

Writing Eq. l-30 in tcrms of ap overall coefficient givcs

Tr- T, a?,_nrt R,+ R2+ Rr Rr +R2+ Rl

q= UAJT-',

ll,rA--

Rr+R,+Rt Roor

(r-30)

(l-: r)

(l-32)wherc

Thc overall cocfficient (J may be bascd on any choscn arl To avoidmisundcrstandings, thc area basis of an ovcrall coefficicnt should thaeforcalways bc stated. Additional informatioh about the ovcrall hcat transfercocfficicnt U will be presentcd in larcr chaptcrs.

Thc ovcrall [at-tf,ansfr cocfficicnt witl bc fmrad rreful prinnrily inproblems involving thermal systcms consisting of serrcral scries-connectedsections. Thc enalysis of hcat flow at boundafes of complicatcd Scometryand in unstcady-statc conduction problems can bc simplificd by using a

6

2f Prnrcrrr.rs dr Hser lluxrrnrcombincd unit'thcrmal'surfacc coodudancc l-' Thc combincd unit'tt

"i"t-srrrtac. colductalcc' or wit'nrlu ondrctancc lor sbort' corn-

;;;;"ff*ts of hcat now ty aootrcri'on and iadiatioa bctwecn asurfacc and a lluid antl's &Iincilby

F-fr.+E ( r -33)

The unir-surface conductancc spccifice &c averagc total ralc of hcat flow

ocr unit arca between a surface "na "

noia pcr unit lemPeraturc diffcrcncc'

irc unis arc w/m2'K.

,E.xrqpls ls. A O5-mdiarncrcr ernc {e-O'l).carrying stcam ,has a,";;i;;;rature of soo rc' rnt pl-[ is tocatca in a room at 300 K andthe convcction-heat-traRsfcr cocflicient bctwcen the pipc surfacc and-rhc

"ii it o. room is 20 wr/mz'K Calcularc rhc

^combincd unit surfacc

;;"d"";;;;,1 ,h" -t" of hcat loas pcr mtitcr of pipe lcngth'

solufion: This-problcm may te id..ti.4 as a small objea (the pipe)iJJt; iargc utact cnclosurc-1rhc room! Ihus rhe radiation-hcat'transfercoeffiocaris

i : a4T : + T :}(7, +r; - 13'e w/trf ' KThe combined unit surfacc conduoanccis

i: i* i,-20+ t3-9=33.9 w/mr'Kand rhe rate of heat lsss pcr mettris

;DLi('T'*-r'i')=ax05x I x33'9x200- 10'650 w

Example l-9. In &c dcsign of a hcar exchangcr for aircraft application(Fig. l-13), the maximum *"ull t"tptra'oo is not to exceed 8S K' For thel"t?ii*t'Lbulat"d bclw' dacrminc tlrc maximum pcrmissible unitthergral rcsistancc Per squarc-mctcr area of thc mctal wall betwcen

hot gas

on the onc sidc and cold gas on lhc otbcr'

Hor-gas.temPcraturc- 4- 13fl) K

Unit'surface conductane on hor sida ft-' -200 w/m2'K

Unit-surfacc conductancc on cold sidc' il:am W/maK

Ccolanr tcrnPeranr- ?i-3d) K

Solution: In thc stcady state urc can *ritc(q / A) trom gas to bor sidc of walt

-(q / tl) from hot sidc of wal! t'\rough wall to cold gas

DBcNsioNsAt{D Uxns 5

?lryrioi Systm

Ir----1llcril nll

Hot trr

(lfor suafrcct-Tr

fCold surt'aet

k-r+!D.rrilcd Thcml G.ait

r,.= #SimpliliclCiroitT, fr. Tn Tco 1nn- o 1n/v o \n'v c

lLlR, -;;; n: " rlr Rr = ..rI.

Figurc l-13 Physical system and thermal circuit for example l-9'

or

4 =Tg|\s = T"-T' = t3fl;9w: ,, , ,1,3s.-3 .{r-r'==,7--Tt- -E+&+& - t/2ao (l/200)+Rr+(l/4o0)

whcrc 1,, is the hot surfacc rcmperature. substiturlng oumcfical valucs forthc uniiihJrnral rcsistances and tcmPeratures yields i

1300-800 1300-300-0fo5 -I;55O

Solving for R. givesfir-0.0025 mxrc/w

.A unit thcrmal resisrance larger than 0.0025 nf K/W would raisc theinncr wall tcmperaturc above 800 K.

n-6 DilJrENSlOfis AllD ul{lTsA dinrnsionis a namc givcn to any mcasurablc quantity' For cxanplc' thcspacc occupicd by an objcct is qualified by tbc dineasion callcd thcvolumc. The distancc bet*een two points is qualificd by thc dimension

6

n,.'-.1=..1h.

26 hnorrs or HrrrTtrxsrrlcallcd thc lcngdr crnrnon dimensions t,.ced h r hcat:raDsfcr coursc arclcng&, &!g masq forcg heat. and tcmperaturc.

Bcfore oumcrical calculatioiu caa bc madg cach dirnersion mrst bcquantificd *ith a dcfincd, rcproducible nrl. units arc the arbitrary namcrthat specify the magnitudc of cach di'cnsion- For cxamptg thc mctcr is aunit for the dimension of length. othcr unirs of rcngth havc also bccn usedto quantify the dimcnsion of lcngth. Some of thcic arc foot

-var4 milc,millimcter, centimctcr, and kilomctcr._

sevcral differcnt unit systems arc prcscntly in usc throughout the world-In iadusrry and rescarch and devclopm.nt,-th. SI systcm qslsrime Inter-narional d'unit6s) is fast becoming rhe mosr widely uscd slitem of unitsThe SI slrsrem has becn adopred by the Inrernarional Oigaaization torStaadardization and is recommenaea by a large numbcr of nationalsrardard oryanizations- For these rcasons wc n'ill-use sl uniu throuehour\yh" used in rhe SI s-vsrem are dcscribed in epp.nai* L ecomplcrc lisr of conversion factors bcrwcrn tlrc sl sysrem

"ni-rh. cnginecr-ing systcm of unirs that_is freguently rrscC ;a r,trc lairca Srercc @appcars in Tablc

'rt-6. For convc11i6n6g. a cqrdensalion of this rablcappears on tlre inside cover of the rext.

The unirs assigned to thc SI sysrcm and other commonly uscd s-vsrcmsarc sumnarizcd in Table l-3-

Tablc t-3 .Brse Units rnd Derivcd Units tor Several SnsqSvsrsv

Dlt'tgxsrox Excn=m-_cLcngthTimeForccLl"ssTcmpcraturcHcat

ft5ectbtlb..FBtu

mmssNNkg kgK.CJ kcal

cm5

d-rrcs'ccal

I.7 DI}IENSIONAL ANALYSTSTlre iacoavcnicncc of changing from onc sa of unis to anotbcr c.n ofrcnbc evoidcd by using dimensionless pa&tnctcrs rvhosc valucs are tbc sameia e1I sct of unirr The proccss of dcterniaing approprialc dirncnskrnt$snuubcrs is called dittinsional aulysis. Tnir mcooa-not only conrbincsscvcral variables into dimcnsionlcss groups which arc indcpcodcat of rhcsystcrn of unirs, but it facilirates iotcrprcudon of cxpcrimenral dara

Tbc most scrious l;niration of dimensionat anarysis is rbat ir givcs noinformation abour thc naturc of a phenomcnon. ti tacq to apply dimcn

DncxstorerArrrr:rs /7sionar anarysis it is necessary to know bcforchand what nariabrcs inruencethe phenomenon, and rhc sucse oi e.t * o" *i;;;;ff.. ,*.proper serection of these variabres- rt is thcrefore important to have at reasta preliminary theory or a-thorough physiJ unaerstanding of a phenome_non before a dimensionar

"n"tyrit il * *n".rned- Howcver, oncc theperrinenr variabres are known, ii-.nsioJ Inuryri, can be appried to mosrproblems by a routinc procedurc.whictr is o,ritinea bdow..Prlmary Dlmenslons and Dlmenslonal &rqalns

The first step is to serect a slarem of primary dimensions- Thc choice ofthe primary dimensions ir "ruit.".y, i,Ii,'iJii,,,.nrionar formutas of a'pertinent variabres nnust be .*pro*ur. r" ,".r*

"r thcrn In thc sI systcmthe primary dimensions of length l- ri;;:';p._turc li aad mass

.,t./arc used.The dimcnsionat formura of a. physical quantity fotows from definitionsor physicar laws. For ins-tance, rhe dimensional fogckJaf.d&.&4r5ef asaf +{drtv drfinirionJ'rh.;;;;;;; efliidaaticrc is equal toa disrancc divided bv rhe.lime ;,;;",

"{.;to raycrsc ir- The dimen_sional formula of velociry.ir 1!*.* iiZTi or 1tl_r1(-c- a distance orlengrh divided by a time). The unir'oi "lfo"iry could be cxpressed inmerers per second' feet per second. or m'es p.. irou., rir* o.y-al"i.'"lengrh divided bv a time.

'n. Jil;;;"iril"r* and the symbols of

iJ+:;ii quantitiis occurring r..qr.ii1r.i"i.oi_,run.r., problems are gi'en

Bucklngham z TheoremTo determine the numbcr of indepcndent dimensionlcss groups recpiredto obtain a reration describing

"

prti"iJpi"""*enon, thc Buckingham :r(pi) thcoren mav be usea.r e-ccora;;i; ih#i" rhc rcquired number ofindcpendenr dimensionless go"p, ;;;; t"-L.,o.a by combining tbe.physical v-ariabres peninent to a probre' is equal to thc totar number ofthese phyfical quantities z (e.g., dcnsij.;;"ril hcat-trarsfcr coefficient)'Thc dgcbraic thcory of dimcarioot aarl.vsir ritt aot be dcrrcIopcd Lcra For r rigorous aad:T:1ffi:::rtucnr of the methcuaticat b""I;;i-i;otcnr ! rad 4 of Rcfcrcacc 7i}ffi.o."t* ( | dccorc drr rbc quatiry rrr tic dicuiod,cod. nelc.! rl.6ir lbe,#f":t:l;#iffi bv :n Tist (Rcr' 6! nows rhrt tte r ocoren hor& er3jT'-il+-"*ff;f.H j:,1f r*ffi il1trffi*Lilffioac or Eorc of rlc orhet corudols (te. u u.

"q,,1,i.-li il-rr, a.p"oa_r[ rhc nurabcrf.H"T1"ft group3 i'' "qu"tiJ ". .ii ili*.-"iJ.T'"t; . ;; ;. ;;;;;;tl

/' 2S prnrorr.rsont&rrtlerrsrq

Trbrc l-a sone ptysicd auliddc. r*r r*.n.a sy.bob rod DiocasioosrrlttEl$toNs rN

er^rrry snoor {KLcng& %Timc I- z LMass t eForcc rt MTcmpcrarur i F"tt'HcatVclocity 9, ML2/c2Acccteiatior : , L/ewpr& a't L/0,2hcssurc w ML2/e2Dcnsiry P M/o:Llnrcraal cncrgr n, u t ttEnthalpy h L1/0'spccificlcr : L:/o:euJ"L-"i**i.y : i;iiTKinematic viscosiry r_ r/o Lr/e331:gioi1'"'r * u{44.'

-Tbsrmat afifusiviq/ - e/r- -Thcrmalr6ist.:rc! e Tgr1trt1tC-ocfficirtfroforpaasion , , l/TSurfacc rcnsioa

shcar pcr unit area ", utt'

H::,#:$0,*'* I r,ir:;minus thc number of orimary dimcmions m requircd to express thedimcnsional formutas of'thc albysi"n O*",i,i.r.'If wc call rhcse groupsttv 7?2'

"" the cquarion cxpressing-oc raaooorrip-"rnong tbc variabrcs hasa solution of tbc formF{rry*rar-.)-Q (l-34)

:tL_"^O*r.T involviag fivc p\rsical quantitics and rhrce prirrury dimen-srons, ,r -

n is equal to 2 aad thc sotution .itUoia, rlrc formFlrr,z)-6 ( I -35)

, or the form

r

\-t(s) (l-36)F-xpcrincntal date for such 1 .casc caa bc p,rcscnad comcdcatly byot:tPt l! Tarnsl ar- Tbcrcsurrint ..piri; J"J. r*."r, rhc functional*,*y- bcrwccn s, and z2.,ije ;;;dljuc"a 6ep rrirncnsional

-f:j".fh:.enon that can bedcscn-bed t" ;;:::.:;*groups (i.e- if.n-ar=3), Eq- l-34 fr", ,r,"L._ "'"F(curr,ir)=g (t-37)but caa also be written aszr-f('r,zr) (l-3s)For such a casc, cxpcrimental

.data can be corrrrr, for vario''s

""r-ul olor- sometirncs it i, po.lliill:tJf;nrfr"ffff:n's in somc manncr and o plot tti, p*"..1.. againsr the remaining:r ona singlc curve, :rs shown in Cfr"pt i;. -.---

Dstarmlnallon ol Dlmenslonless GroupsA simplc method for derermining dimensioniess groups will now betrT,:",{ by applying it to

" "ooJu'",1""-ii",-,r"rrfer problem and to aproblem ia lluid flow_

Example l-l* Determine dimensioalcss paramerers to datc tte maxi-i",:-;ir;.H:,k'"il1*,?:1ff :T::#;I*.,H*:'J:T:surfacc is Baintaincdsulated- at temperature I, while the otheruurf""" i, i*

Solution: We Ue8rn_ly wriring z as a product of the variables, eachraiscd to an unknown power

and then substitute "..#"?l;t",Irro,n r,ure r

30 nnrcu-sc lte rTr^NsEr.

rathcr &aa four. Wc can rbcfcforc choosc vrlucs for two of ttc exponcnrsin cach of rhc dimensionlcss groupc. Tbc oaly reiriction on the choicc oftbe cxponcnts is that cach of thc sclccred srporsnc bc indcpcndcnt of tbcothcrs. Aa cxpotrcot is indcpcndcat if tbe dctcrminant formcd *ith thccocfficicats of tle rcmaining terns does not vanish (i.c. is not cqual tozcro).

Sincc 7i is tbc variable we cventu.lly s'ant to cvaluate, sct its cxponenl,a, equal ro l- .Sincc wc want q'i to be the indcpendent variablg we do not*'an! to combine it with I and therefore set its cxponcnr c:O This givcstlrc cquations

I +D-c:0c +0:0

c+/-0:0Solving the equarions simuhancously we get ,-

- t. c-O 4-Q and the

first dimcnsionlcss group is T^

",_Tthc prio of rbc marinun rcngc&irruc 4e:hc-sgdecc rcppcrarnrg

For r, wc let a be zero, so 1- ritl mtapptr again. and "-1, no harrcrhc indepcodcar rariable appear ro rhc lirst powcr in rr. Simulancous

solution of thc cquations with thcsc choiccs liclds r- -1, b-*1, d-Z

thus

L2qT,r=T

Tbis problem can be expresscd in terms of Eq. l-36 as

T^ .( r'c';\

rr:J\ k'T JDmcnsionlcss analysis cannot rdeal lhe narurc of th,e funcrional relationbctwccn a, and tr, but in Chapter 2 it will bc slrown tlat

r^ . .

L2cTTt '' 2krl

Exanple t-ll. Find dimensionless paramcrcrs ro correlare dara for thcpressune drop Ap in a pipe of diamcrcr D and length ,l. whcn a fluid ofviscosity p and dcnsiry p is llowing throu;! ihe pipc ar ln avcrage vctociryv.

Solution: Thse arc six variables in this problem, bur only rhrcc dimcn-sions lcading to thrcc indcpcndcnt cquadons- Tbus, thrcc variablcs must bc

dDrMENsroN& ANAI.ySts llsclected in cach cvatuation.of a dimcnsionle,s r paf,amctcr. The variablesof rhc problcrn, their dimcnsions, and rheir cxponcns arc tabuleted bclow-

Vr,nu.au Dncxsrox - F-polrnnabcdcI

LAppDtt,,

trtlM / rtzlTM / L'I

trl\M/ulw/01

Next wc rrrritcr u1'[M1'r,vlM1'[L]to- [ r]"1' ' L4J IFJt') Lu ]Lo )

For thc fint dimcnsionlcss group wc let a: l, 6:0, and c -.() ancl obrainthe Tollowing cquatiocs :

L: l+d*e+f:QM: e:0

4z -**y':

T1lus d: -l aid

For thc sccond dimcnsionless group we let.a=0,6-1, arr

I:

.E

.Y,. .r"\lj.,il*irir.rf r. :r. | :,t-i..

figutc l-14 Relarion bctwccn lricrion facror,/, ead rrynolds n,'nbcr, Rcr, forsmooth and anificially rougtcncd pipcs [og1q(rra2) vcrsus log,o(l/rr)]. Fron J.Nikuradse, 'GescraassigJreitcn der lurbulcatca Stroaung il glanen-Rolreo,-VDI ForschungsheJt, volr 356 19321 -Stromungsgcsctz. iD rauhsa Roiycn"- ZD.I

Fondangs$t vol. 351, 1933.

But when l/4)2300, thc functional rclarionship berween J and r,changcs becausc thc flow undergoes a ihanse from'larninar- lo ..turbu-lent- flow. Dimensional analysis cannol, of coursc, prcdict this physicalphcnomcnon. Morcover, il turbuicnt flon'not only ar. but also the surfaceroughncss, affecrs thc valuc of_/. This is clcarly illustrared in Fig. l-14,whcre expcrimcnkl results Ior rr/;, tersus a3-l are ptotted in thelaminar- and turbulent-flow regimcs. In turbulent flo*. rhc &ta show aproDounccd depcndcncc on tlc rario of tbc avcrage hcight of surfaccroughncss elcmen6, rt, to thc pipc diaracrer, D. For smooth pipcs, how-cver, thc cmpirical rclation

n 2/ d, * J - 0.046rru -O.ffi ,/ P"ro:correlatcs thc experimenral data in thc.rurbulcnt-flow rcgime ovcr a widcraagc of a, @cf. 8)-

REFEREI{(TSl- J- B. Fouricr. Thi*ic aulytiquc & Ia Mar.Pariq t822; A FrccEaD. traDs..

Povcr Publicationg lac- Na'York, t955.2-.W- lv{. Robcscaow aad I P. Hartnctr, ds. Itandbok ol Hcat TransIcr, Scc.3

(by P. J. Schncider), McGraw-Hill Book C.o. Ncw York N.y_ 1973.

? -'r 't. t !^-'rF lD l:6 |Turbulcnt flo*

ir sm6rh pip.

I

5..1 5.6 5.E 6.0

R.ernnancrs 33

3. T. N. Veiziroprl 'Corrctation of Thcroal CCnact Conductaacc ExpednorrlRcsults." Pro& Astron- Acro- 20, Academ_lc prcss, tnc- Ncw yorlq i967.

4. R W. Vancc and W- M. Duke, cds_ Ar:ttied Ctyogenic Engineering,John Wiley& Sons, Ncrr York, 1962"

5. R. Barron, Cryogenic Sysrcns, McGraw-Hilt Book Co., Ncw yorlc, 1967.6. E R. Van Drigst 'Oo Dirncasional Ana.lysis and thc prcscaration of Data in

Fluid Flow Problcor".f.tppl. Mech.. vol- 13, 1940-7. H- L L:agbaar, Dirercioaal Anabsis and Tleory of Modcts, Joha Wilcy &

Sons, Inc. Ncr Yodc, 1951.8. J. Nikuradse, -Gcscanissigkeitca dcr turbulcnten Str6ntr.g ia laneaRohrcq,- VDI Fonchungshy'r,

'iol 356, 1932; -Str.6mungsg6crzc in r.uh"oRohrcq" YDI ForschuqslvJt, vol. 361, 1933.

R

PROBLEMS

Thc problcrns in this chaptcr arc or-qanized in thc manncr shown in thc tablc

r-1 ro l-1!1-19 ro t-25

1-3't b r-34l-35 to 1-37t-3aftn{t

1-2

1-3

't-a

Conduclion healtranstet

Convection hentransfer

Fadiationiat!ranstet

Combined nbcF;Dirnensional

analysis

r-5't-?

t-l Derczninc lhc hcat-$ansfer ratc per unir surfacc area rhrougit a brick wall(A'-03 w/m'K; whcn onc surfac of o..ur"r. i, "i'li.c "nd tbc oilrcr surfacc isar - l0'C. Thc rhickness of rhc ral is t0 crn.

l-2 A furnace wall is cons_rruaed of silica brick, /

113'i*p,,#/

.,.

. n:ain6ined. at a tcn PnoslEvs 35

,u.r""" "nJ ;.;'ffiil::,1.'Xe,*:ffi: FL-[.*'",u,. or rhc othcr

l{ Thc thcrmal,T:.#i:*|l*,;i;:;m"!i::Jffi ::"ffi f l"'":S:;,1.:1,#,:,rfii,*ii.tffiJreguenttl used to describc rhe resistancc to hear f.low of

*.here L is rhc rhich a'*=

* - .rr,i.tn.rr..-oi't"'i;il:: tt thc.insularion. Calcutate thc .R factlr for lGcmu.t.u,.-,i.;:'#i""T,li"*-T*":Ti*:"'l-i..'orra*f i,i::#.'

,- t4,

.considcring heat-transfer principres. " r.aff:--,

-

- . ,m making cookinq urcnsit.r w*L r^lt.^ll' : ).46s stccl a 8d matcrial to use

:::,*-qF;,;:,lff J'j#H"'1.,,:;:#F"iTi;.:ff#::y;1""*scra{ ?t5riq v q -a-'r"'!"\ rucr :rs cootwarc' lbrt is pr*chrscd by tie

l-9 Several rods I cm in diamercr and l0 cm lonp ar" in

\ _.

36 hNcDlts or HsrrTtr,trn

rralt r,hco.4-200'C Ir-500'CL-ljcn *o-15Vn'K,- to-.K-r C- l0-rK-2

l-14 A ocrallic rcfrigcraror *'all is to bc covcrcd wirh e rigid foam insulationrbat has e ibcnnal cmductivity of 0.03 W/n'K- The iatcrior of thc rcfrigerator is'o O.

-"iot'ined rt -20'C Tbe cooling capacity of tbc rcfrigcrarot is 2 kW andtbc surfacc arca of tlc rcfrigcrator wall is 100 #. Dctcrminc lbc minimuminsulatioq thickncss iccdcd so that condcnsation will not occur oD thc cxtcriorsurfacc of tbc isulatiol, assuming that the dcw-poi:rt tempcratutc of rhc airoutsidc tbc rcfrigcrator is l5"C

l-15 Arsuoc oncdimcnsiooa.l hcat transfcr througb tbc compositc wall showain thc figurc:

r.500'C I. 100'C

F- La'-*Ls ' Lc;f- Lo---)(a) Draw thc tbcrmal circuit for thc wall, labcliag all rcsistaaces and

known potcnrials with appropriarc qrmbols.(b) Dctccnrinc trc heat-transfcr ratc through thc wall(c) Dcarminc thc tcmpcraturc of tlrc lcft-baad facc of matcrial D.h-75W/m'K Lt-20sn&a-60 w/m-K Lr= 4-25

"

&c-58 W/m'K Lo-4Ocm&a-20Wm'K 1^-lo-];6,2

At-Acl-16 At $cadJ Etatc lbc tcmpcraturc profilc in a larninatcd sysrem is as shown.

Which natcrial has tbc highcr thcrmal conduoivity? Sbos sullicicnt nork andrcasoning rojustify your answcf,.

'-fl-*

$ii

1_.

I

cDA

lrlercriel I

Pnofig{s 37l-l? consi&r rhc composirc valt shorpn in thc figurc. Thc lcft rurface of thc.

waii is submcrgcd in watcr that has en ambicnt tcmpcraturc of ?0.C and theconvcctivc-heaa-transfcr cocfficicnt on thar sugfacc is 60 w/n2.tc Dctcrnine thcvaluc for kr.

a - :0o w/m.K I . J0 w/m.K

:- J0 (m--;

*,

-15 cm-i :l5ern

r - iO"C

4 = to w'n:'xf-'?0'C

r. o"C

t-t8 At stcady statc rhe Emperature profilc in a tfuce-material compositc bodyis shown in the figure. Derermine thc correct conclusion:(a) k.,>kr>k6.(b) k. >&c >&r.

. (c) k" >k. >&..(d) .*c >*i >{tr.(e) /

3E hnrqnsorHeer?h^}rsrnt-23 A t0GW ctccrric]*-ttfiffi :H;T:t .f_',*T.fi

"TnT* j::i.J#l*

.

l-Z Air rt 20.C O OIT orrcr &c lop surrlcc of r- IG.cm_6ick purc ironaonzontal plata Tbc corvccrivc_hcar-,-".i*-ilrF"l;trffi j *.;: ffi 1 *. */r"{;;tilfi:: J ;:,K[,H.,t:o."."rctrLiltrJ.,T*tT. 2-00 {m: ro rhc surroundnss Draw rncil; ;; ;";il*toa aad cajcula tc th c sreadv-sra rc

"rp.iirl'oiu"Til:2S ODc nrfacc of ! flal

l1T'g *. #;;;,fr'#f.jm"?: lS;1 : :*ienr rcmpcraturcnsuledon ltrar bas r tbcrm:*].*:T##Hi;1qp##r#;ffi affi :S.r#r":: " +. ;ffiT:,tr'tr ffil#T*r*ifi""J:it:jffi#f rn,T,ffi#a affr;'.T.il-*".,* nr rhsurb rhg-j#.ffin}1'* m r],e risuc is coqprirad or so dincrtmro air wirh aa a-abicnr r.r"ff:lto#r20'c vhi.lc,r,.'tr-

"rrr"Jr;;:#

,-. =-i \l,m:.]ir-, = r io'('

(e) Dnr.thc tbcrrnal circrit for this problcru(b) Celcutae.tt t.rr,ri Grrl"il(c) Dacrminc o.r,-r-*J.l_ir.*alt grc rer-tr&DStcr ratc through thc uzall pc, uDit depo of

_ ..,:) H*" rhc surfacc rcnpcrirr$c of rlc wall rhar is crposcd ro rbel-l-t, .l* rougl mcut rurf.cca rrc Drcscdncss ol l^l o radrHffitrf,ffil* T"r,*fr"Hntr #"k lffi i#rr'.fl,**

+O(jctrt+

l: -

lh4'\ --.-ltr R---'

h,oauris 393i &:::"m#",-1:l--'::^"13'" p'r unit surrace area.(d)o",..*".'ii j"#flil",ii"Till':F,nntcioatccresisraace.

rcsistance. rr arca assuEling no interfaccl-29 A small tiansisrc

;tx*ru::l#{,'{:llirf:ilJ::.::ffETii."ffH:#cocrhcienr

""-;:;;:: -T:, rT: is 25"c and

r,om aruminu-,.'l;ffi: ;Lt'.otH **#kT.;,::ilHT:H::,':;';lt"j;;;1';:.*,n,u.r""..D"ffi ;;:"*_".]flf ls"ffi **

t l-29 In problem l_28 r

**;irtttlr:**l*n*,ri:*t:a+.rut,sqrop across $c L,lterface ,ari.r.".].' us rt:ursISto!- Dcterminc drc tcmpcratur.resisrancc. -"-'J rcsrstance- suggest waJT ro .u-i*,.-ul*il,.ri".l;;'gp:fliji;hip'aggltgi#i*x.ffi r*"t

C = -to"c4c=:0 w/m:.(

I-' :s"clr.' t0 Wm:.Kl.3l A btackbody rvith a surfac. ,-- ^, ^ , _,ji,T:l1e-.hcat-rransf crla,.1;'.il;,:j-ol:t.iq"5P:cordrurround;ng.,?l:ffi Ati;;:a:i: ;f,Hi"l?'"i" *l. AtH f ,: #..HHftof.(a)17'c G) siz;C'",i i;iait.*{fth"fr,i;l*iil1iili H'm l;iFkqdr in'io a vacu"'n

r t, catcurarc,r,. l.i**#:r"*".t ol;ffi,J.t **t,n.t,li"t ffHo,l#l S:-"-:tl "r 19.".' has. an cmittancc * ni. oo.,,,,,oorarl cncrry if irs rcapcreture i fmO l{-

zD

.fl:,{r'{S Prstcnrs or llElrTratsFrr,

l-31 A gnybody wi& arca ! mr ald coinancc of 0J b oain:ebcd rt ?00'C iae lerg. blact room rihocc rcmpcnturc t t00'C. Dacmiac tbc oa rarc of bcatraasfcr bctwccs tbc gray body and room by ibc radiatitn rnode of hcat trandcr.

f-35 Cicuht thc overalt bcat-tralsfa cocflicient U fc hoblco l-l5..Base thcvaluc of U oa thc lo A-

l-3,6 Calodatc thc ovcrall bcat-rransfcr cocflicicat U for hoHcn l-26. Basc thcvaluc of U on thc torat cross-scctional area of rhc crall.

l-37 Calculate-tbe overall heat-uansfer cocfficicnt U for hoblcm t-3O Basc thcvaluc of U on a qpical cross-scctional arca of I mi.

;...

l-36 Thc physcal p:ramcrcrs rhat govcrn thc local rcnpcnturc io e fia arcknown to bc:fil- convcttivc-hcat-rransfd coefficicnr for fluid surrouadiag fioA-- thcrrnal coaducrivity of fin srarcrialI,

- charactcrisric dimension of fia

I- -

ambicnt tcmpcralurc of fluid surrounding fiafr- tcmpcrarure of basc of fin.r

- location measurcd from basc of fia

Using thc Buckiogham pi thcorcm. show rbat rbc dimcosionless groups that can bcused ro dcscribe thc rempcraturc disrrib'trioa in thc fin arc

dimensionlcss rcmpcrarurc: Tl-rt'

' dimcnsionlcss thcrmal rcs;"*r.., $dimcnsionlcss locarion: f

l-39 By subsritudag dimensions for cach ph-vsical qr.ranrity givcn in Tablc L-1,show rhat cach group is dimcrrsionlcss.

l-40 In ransicnr conduction, it iq knonra 'hat thc par.Dcurs 6er gowra rhc

Iocal tcmpcraturc distribution in a solid arc:p- dcasity of solid9- spccific bcat of solidL- cbaradcrbtic dimcnsion of rc[d*- tbernal conducrivity of solidt r rihcr- locatioa wirhin solid

Usc tbc Buctiogban pi Orcorco rc tbow- 6af ibc riac'rdrotit &opcfrlurcdistributioa caa bc otprcssed in tcros.ot r diwnlbnlcrc poddo!

x7

Ps'- Pnoar-rr': I

. aad a dimcasioaless group callcd *t t*;'' mmber'

Et ljl When e hcatcd surfece is placed in a stGam of crrolit fluid, thc rurfmc will

losc hcat by forced convection- For this casc thc paramctc6 tleat gOvcm rhchcat-transfcr PFoccss are:il

- convcctinc-hcat-traasfcr socfficicnt

l- charactcristic dimcnsion of surfaccP- dcmity of lluid/- frcc.sream velocitY of fluidI -

thcrmal conductivitY of fluidc,- sPccific hcat o[ tluidtr: viscosity of lluidl.lsc thc results of thc Buckingfuam pi theorcm to show that &c threc dimcasionlcsgroups that govcrn thc hcar-transfcr pro6s arc:

- hrL

' Nussclt numbcr: a-Rcynolds nuor6o, 4)L?nndUaud:tn f

z-..-. ' ^ A