Chapter -3-Evaluation of Transfer Coefficients Engineering Operations

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Chapter hree

Evaluationof TransferCoefficients

Engineering orrelations

Sincemostengineering roblems o not have heorctical olutions, largeportionof en-gineering nalysiss concerned

ith expeimentalnformation, hich s usually xpressedin lemlsof engineeringonelations. hese onelations.owever.,re imited o a specificgeomeffy,quipmentonfigumtion,oundary onditions,ndsubstance.s a result, heval-uesobfainedromcoffelations renotexact nd t is possibleo oblain wo differcnt nswersfiomtwo different offelationsor thesameproblem.herefore,neshouldkeepn mindthattheuseofa correlationntroducesnerror n lheorderof +25./..

Engineeringorelationsarcgiven n termsof dimensionlesslrmbers. orexample,hecorrelatioassed o dete nine he riction actor, eat ransfer oefficient,ndnasskansfercoefficient regenerally xpressedn the orm

I : I (Re)

Nu: Nu(Re,Pr)Sh: S1t1B"'"'

ln thischaptetsome f theavailableorrelationsor momentum,nergy, ndmassrans,po in different eomefieswi]l bepresented.mphasis ill beplaced nthecalculationsfforce or rateofwork),heat ansfer ate,andmass ransfertte under.steadyonrJitions.

Transfercoeffcents6

REFERENCE EMPERATURENDCONCENTRATION

Theevaluationfthedimensionlessumben hatappearn thecorrelationequireshephysical propefiies

f the luid to be knownor estimated.hese roperties,uchasdensity ndviscosity, epend n tempefaturcnd/or oncentration.emperaturend concenfuation,ntheotherhand, aryas a iunctionof position. wo commonly sed eferenceempemturesandconcenfiationsrc the bulk temperaturer conce ratiofiand heJtlm emperaturer

3.1.1 BulkTemperaturendConcentration

For flow insidepipes, hebulk emperaturer concentrationt a particularocationn thepipe s the averageemperaturcr concenradonf the luid were horoughly ixed, some-times alled hemirrg cup emperaturer concentration.hebrlk temperaturend hebulk



47 ChemicatEngineegprocesses

concentration are denoted by 16 and c6, respectively, and are deflned by

| [ , ,raerb:u#-

lJ^*oo

[["'oond

"= #-

JJ^,"0o(3-l)

(3-2)

\rhrreu, rs hec-omponenl[ velocir)n hedirectlonf mean oq.ror.ne case t flowpast odiesmmersednan nfinite luid, heburk emperaturendburk

:oncentrationecomeheree heamempeftru."no ."""J*'n "oii""i*',ir"

.?#1,"o,

3.1.2 Film Tempe,atureand Concentration

Thefln tenpe ratu re TL andthe il n conrcntrarior, c/, are defined asthe arithmefic averageof the bulk andsurface alues. .e..

(l-3)

fa:r- l

Tf:: - :- :- - and cl=

For flow oversubmerged bjects

n",=3=11s9L . V

where ubscript@ epresentsheconditions t hesualacer thewall.

3.2 FLOWPASTA FLATPLATE

Let usconsidera flat plate suspendedn a uniform streamof velocity r,@andtemperalure?€_as hownn Figue 3.1.Th; Iength f theptaten thedirection f flow s l, and rswidttris Iry.The.localvaluesof rhefriction factor, 'he l**f,""_U"r,-"ra

ifr"'Sln"i*rJi,i".*,aregiven n Table .1 or both aminar nd urbulentlow condirions.i,. ,"r_ il, ]. ,n"Reynoldsumber ased nthedistance, anddefined v

(3-4)

Theexpressionor the friction factorun(canebrainednaryricary;,r,"""i",r"',,'iiiilJIil.lx":Tir;,ft:rr,:;l,fl,.i

Elil*,1

Table3.1.,he otutvatLes . he t.ictionactoi the NLsse.t Jnbe,,ano nesheMoodnLrmberforiowovera iarpta,€

Shr

(D)

(E)

(F)

0.664Re;l /2 (A) o.Otoro.,

0.332Re,1/2prr/3 G) 0.0296R#5pfr/3

0.332Re.1/2scr/3 (q 0.0296R#5scr/3

Rer < 500,000 5 x 1 0 5 < R e r < 1 0 7

0.6 Pr<60 0 . 6 < S c < 3 0 0 0



T.arsferoeffiients48

tho irst oobtain h;ssolution singa mathematicalechniquealled hesimilaritysolunonor themethodof combinationof variables.Note rhatEqs. B) and (C) in Tabte3.1 canbeobtainedlom Eq.(A) by using he ChiltonColbumanalogy. inceanalytical olutions reimpossibleor turbulentlow'Eq. D) n Table .1 s obtainedxperimen;ly.Theuseofrhisequationn theChilton-Colbumnalogy ieldsEqs. E) and F).

The avemgevaluesof the ftiction factor, he Nusseltnumber,andthe Sherwood umbercanbe obtained rom the ocal valuesby theapplicationof the meanvalue heorem.n manvcases,owever,hebansitionrom aminaro turbulentow will occuron theplate. n rh;case,both the laminar 4nd turbulent low regionsmust be taken nto account; calcuiuunstheaveEgevalues.For example, f the transition akesplaceat r., where0 < r" < l,, thentheaveragetiction factor is givenby

Change f variable rom r to Rex reducesEq.(3-5)to

rn *| l/-"'rr,r,,.an",/*"r1r,,,,an",]

r<e14e

Substitutionf Eqs. A) and D) n Table .I inroEq.(3-6)gives

l.328Re:/2 o.o74ReX/5

t f / r . t L 1; l l ( f ^ tu . , t ^ I \ f ) to ,bJx lL L J 1 J t , J

( l-5)

(3-6)

whereRec, heRelnolds numberat thepoint of transition,andRe., the Relnolds numberbased n the engthof the plate,aredefinedby

, ". 0.074(J )= - i ;+

R.;-

TakingRe.= 500,000esultsn

1743

(3-9)

(3-i )

(3-7)

(3-8)

(3 -11)

(3-12)

. -. 0.074

n"l/t Rer,

The average aluesof thefriction factof theNusseltnumber,and the Sherwoodnumbercanbe calculatedn a similarway or a varietyof flow conditions. heresults resiven n

Table .2. n these orrelationsllphysical roperties usrbeevaluaredt the ilm t;mpgra-ture.

Once heavengevaluesof theNusseltandSherwood umbers redetermined,he aveRgevalues f theheatandmassfansfer oefncientsrecalculatedrom

. , . (Nu)k

(sh)D,4



p

9€e

008

: 9 a

€

H

E

399

aG

E6

€

.e

g

;P seE

5

E



TransferoeffcienlsE0

On the other hand, the rate of momentumhansfer, .e., the drag force, the rate of heat

transfetand herate of massansfer of species4 fromone sideof theplateare calculateals

/ t _ \FD= twLt

\ ie i l i ) \ f )

Q: QrL) lh)lrtu- r@

nA = (WL)(kc)lcA_ cA_)

(3-13)

(3-14)

(3-15)

Engineeringproblemsassociatedwith the flow of a fluid over a flat plate areclassifiedasfollows:

. Calculate he ffansferrate;giventhephysicalproperties,he velocity of the fluid, andthedimensionsf theplate.

. Calculate he lengthof theplate n thedirectionof ffow; given he physical properties,the velocityof the fluid, and hetransfer aie.

. Calculaiehe luidvelocity; iven hedimensionsf theplate, he ransferate,and hephysicalpropertiesof the fluid.

Example .1 Waterat 20.C flowsovera 2 m long latplatewith a velocityof 3 m/s. Thewidth of theplate s 1 m. Calculate he drag orceon one sideof theplate.

Solution

Physicalprcperties

fo r wre ra l20 .c (2sJ * , ln :a9oke /mr - - -l r

= 1001 l0 6 kg /m

Assumption

1. Steady-stateonditionsplevail.

Analysis

To determinewhich conelationto use for calculating he averaBeriction factor (/), wemust irst determinehe Reynoldsnumber:

^ Lu*p f2)13|994)

- u l 00 l 10 -6

Thercfore,both aminarand turbulent low rcgionsexiston theplate.TheuseofEq. (D) 1nTable3 2gives he friction factor as

. ". 0.074 1143 0.074 1743

n" ] t Re r 16 lgb ) r 6^ l 0o -- ' "

Thedrag orcecan henbe calcularedromEq. 3-13)as

/ l , \ Tr ^ ' rFD- twLt l ,ouLl r t - r t ,2 ' l (9q9r{3 l l { l t0 i )_27N

\ z / l 2 I



51ChemicalngineedngDcesses

Example3.2 AL at a temperature f 250c ffowsover a 30 cm wide electricresistancelatplateheaterwitha velocityof t3 m/s.Theheater issipatesnergynto h; ;;;;;;;;,","",r?.te f n30 W rr2. How long must theheaterbe in thedirectio*n f flo]' to, tn" .*u""temperatureot to exceed155 C?

Solution

Physical properties

The ilm tempemtures (25+155)/2=90.C.

l u=21 .95x10 -6 .m2 l sForairat90 C (363K) and1atm: ltr=30.58 x l0 3Wm.K

I

lPr = 0.704Assumptions

1 Steady-stateonditionsprevail.2. Both laminaralrd urbulent low regionsexistover he plate.

Analysis

lhe average onvectionheat transfercoefficientcanbe calculated rom Newton,s aw ofcooling s

@t#r*= A+:2rw/m2Todetermine hichcorelation o use,t is necessaryo calculateheReynolds umber.HowevetheReynoldsumberannot edeterminedpriorisincehe enj*, of m. t

"ur.,sunknow0_herefore.rjal-and_errorrocedue,no"tUe""A.Since,eas_.omeJaiiotftrarrunarnd ubulentlow egionsxist verhehearetheuse fEq. E) nTable .2glves

, \lNu)= tj:1: __ 00J7R"j 5

szrler. l

(21 )L I r , r r , , l / 5 |ru . )d lu - -

I 1 r , . , ISimplificationfEq. 2)yields

F(L) = L - 1.99 a/5+ l.l3 =0

(D

(3)

(4)

The lengthof the heatercan be determinedrom Eq. (3) by usingone of the numencalmethodsor roor indinggiven n Section .7.2 n appenaix . mJiterution ."t

"_"giu"o

by Eq. (A.7-25)s expresseds

0.02Lk tF (Lk_)

F(1 .01L* -t t - F t0 .99L r )Assuming al5^:1,, a starting aluecan be estrmateds L,=l.l4l.The iterations regivenn the ablebelow:



Transfercoefiicients 52

Lp0I

23

r.1411.249

1.252|.252

Thus, he engthof theplaie s approximately .25m. Now it is necessaryo check hevalidityof thesecond ssumption:

(1.25)(13)

Example3.3 A waterstorage ank open o the atmosphercs 12 m in length and6 m rnwidth. The water and the surounding air are at a tempemtureof 25.C, and the relativehumidityof the air s 607,. f rhewind blowsara velocity f 2 m/s along he ongsideofthe tank, what s the steady ateof water ossdue o evaporationrom thesurface?

Solution

Physical properti€s

Forairat25 C (298K): r,:15.54 x 10 6m2/s

Diffusioncoefficientof water("4) in air (6) ar 25.C (298K) :

r )aR r t /2 r . roc, l .2rDar r2o8 (DABr r ,1 l- :

I _ 12 .88 tO 5 r { i j| _2 .79 , tO-5m2 /<

\ J t J l

TheSchmidt umbers

=1.4 v 105 =+ Checks21.95 10 6

v 15.54 10-6

Dta 2 .19x10 5

For water t 25'C (298K): Pr.r': 0.03165 ar

Assumptions

l. Steady-stateonditionsrevail.2. Idealgasbehavior

Analysis

Todeterminewhichcorrelation o use,we must irst calculate heReynoldsnumber:

ne1 3e = .-(*111

- - r 's+' o'u t J . J 4 X t ( ' "

Since oth aminar td turbulent onditionsxist, heuseofEq. (F) n Table .2gives

(sh) (0.037 e1/5 871) c'/3 [0.037(1.s4106;a/5 s71]19.56;r/:2996



53 Chemicalngineednsrccesses

Therefote,heaveragemass ransfercoefficients

,0 " , {Sh)Dea_ r2000 l |2 .79t0 - . )t

- - - -t 2

- -465x l0 Jm/s

Thenumberof molesof H2O("4)evaporatedn unit time is

ne: A{k)k'f, _ ca@iil: A(k)(c"ft o.6c,ft):o.4Ak"t ";tThe aturadononcentradonf $aLer,xr. s

. - , -r e _ 001165

'A -Rr

:$314;1onQ5 + 2:7,

= 128 x 10-r kmol/m3

Hence,he ateof water osss

rit: ieMe =0.4A\k) 4tMe

: (0.4)(12 6)(4.65 l0 3)(1.28x 10-3)(18)(3600)l l . l kslh

3.3FLOWPASTA SINGLESPHERE

Consider single pheremmersedn an nfinite luid.Wemayconsiderwo exactly qulva_lentcases: i) thespheres sta$ant, rhe luid flowsoverthesphere, ll *re nolJ i.'stig1lant,thespheremoves

hrcughthefluid.According to Newton's second aw of motion, thebalanceof forcesactingon a single

:ri".'::t1,]:,:-":.1'A:jerDp. atrinsnastasnanttuidwithaconstantenijnal elocity,r. ls exprcsseon the onn

Cravitational orce= Buoyancy+ Dragforce (3-16)

l rD r r \ t "D3 " \ l nD2 , t t | . t\ i lpPg_ ( u Jrsr \ i ) \ i t , , i ) r

B_t l l

where p andp representhedensitiesfthe pafljcleand luid, espectively.n the iteraNre,thefriction facror / is also catled he lrag cifficrznr and s denotedby ir. sirnpiii"",""ofEq. (3-17) ives

t . , , - 4 g D p l p p _ p l' " _ t - -p (3_18)

Equafion3-18) anbe earrangedn dimenstonlessb.mas

^ 4/Rei =

JAr (3-19)



Transiercoefiicients 54

where he Reynoldsnumber,Rep, and he Archimedesnumber,Ar, aredefinedby

D3pgP@r P)

fi

I : - K e p < l

18 .5l= * 2<Rep <500

tleii-

f : 0 .44 500 (Rep<2x105

Engineering roblemsassociated ith the motionof sphericalparticlesn fluids areclassifiedas ollows:

. Calculatehe erminal elocity, r given hev scosiiy f flu d, . and hepa ticle diameter,Dp.

. Calculateheparticle iameter, p; given heviscosity f the luid,p, and he eminalvelocity,,.

. Calculatehe luidviscosity,.;given heparticle iametetDp, and he eminalvelocity' v

The dimculty in theseprcblemsarises rom rhe act that the ricrion factor / in Eq.(3-19) sa complex unction of theReynoldsnumberand he Reynoldsnumbercannotbe detetmined

3.3.1 FrictionFactorCorrelations

For flow of a sphere hrough a stagnant luid, Lapple ard Shepherd 1940)presentedherexperimentalata n the brmof / versus ep.Theirdata anbe approximateds

0.413

(3-20)

(3-21)

(3-22)

(3,23)

(3-24)

( l-25)

Equations 3-22) and (3-24) are generally eferred to as Stokes' aw and Newton's aw,

respectively.In recent ears, fforts ave een irectedo obtain single omprehensivequationor he

friction actor hatcoversheentire ange f Rep.TuftonandLevenspiel1986) roposedhefollowing five-constant quation,whichcofielates he experimeltaldata or Rep ( 2 x 105:

1: 11r 1o.rr :n.orut t )KeP I * 16,300Re roe

3.3,1.1 Solutionso theengineeringroblems Solutionso theengineeringroblems e-scribedabovecannow be sunmatizedas ollows:



55 Chemicalngineenngrocesses

I Calculaterr; giyen p and Dp

SubstitutionfBq.

(3-25)ntoEq.(3-19) ives

Ar= t8(Rep0.173Ref65?)-j 31n4-' I + 16,300Re;loe

(3-26)

SinceEq. (3-26)expresseshe Archimedesnumberas a functionof theReynoldsnumoelcalculationof the terminalvelocity for a givenparticlediameterandfluid viscosityrequiresan iterativesolution.To circumvent his problem, t is necessaryo expresshe Reynoldsnumberas a function of the Archimedesnumber.The following explicit expressionelatingtheArchimedesnumber o the Reynoldsnumber s proposed y Turtonana btart 1Dt; r.

ReP #(l +o.o579Aro4r2)214 (3-27)

Theprocedue o calculatehe erminal elocitys as ollows:

a) CalculateheArchimedesumbertomEq.(3-21),b) SubstituteheArchimedesumber into Eq. (3-27)anddeiermine heReynoldsnum_

bet,

c) OnceheReynolds umbers determined,he erminal elocity anbe calculatedtomme equalon

,rlRep(3-28)

PDeExample 3.4 Calculate he velocitiesat which a drop of water,5 mm in diameter,wouldfall in air at 20'C and hesame izeair bubblewould;se throughwater t 20"C.

Solution


Forwarer r20'c (29JK.J: p-- oqg e/']

l / - l 00 l l0 6 kg /m"

Forairat20'C (293K):

Analysis

ln=1.zulks/nf[ r r= 18 .17 10okg/ms

Water droplet falling in air

To determile the teminal velocity of water, t is necessaryo calculate he Archimedesnumber singEq.(3-21):

A r_u 'pqp tp p t

_(5x l0 ) ' j ( q .8 ) r.2047) tqqg-.2047,

u- €l? ' fo-tr--4 46x to"



Transfermefftcients 56

TheRe),iolds umbers calculatedromEq. 3-27):

A rR e P= 1 " 1 - 0 . 0 5 7 q r 04 1 2) i 2 1 4

4.46x 106:fflr +o.os'ts(4.46t06\0t2l-t ta:3slr

Hence,he erminal elocity s

4 Ree ( l 8 . l 7x l 0 b r ( J581 )i / :

pDp-

( t2o47 l (5x 0 f= rudm/ \

Air bubble rising in water

In thiscase,heArchimedesumbers

, D'ogprpp pt (5 l0-rrr (9.8)gos) .2047 sqg)----n- -ilnoa' F-- Izrq ru

Theminussign ndicates hatthe motion of the bubble s in tie direction opposite o gravity,i.e., t is rising.TheReynolds umber nd he erminal elocity re

ArRec=

rr--r- 0.0579 jo4 l i 12 4

1.219 n6 .. ^ ̂ _-= _ f * :L t +0 .0s?q r.2 ts . , t 0b )0 .4lr 2 r ,_ t825

/ 1Rep t l00 l ^ l0 6) (1825) - -_o =7o,

:rsogrts to lt : o ; m/'s

I CalculateDp; givenprand u;

In thiscase, q.(3-19)mustbe reanangeduch hat heparticlediameters eliminated.fbothsidos f Eq.(3-19)aredivided y Re3p,he esulrs

JRt = Y (3'2e)

where , which s ndependentf Dp, is a dimensionlessumber efined y

y : s@e )p (3-:lo)3 ptri

SubstitutionfEq. (3-25)ntoEq.(3-29) ields

v - j1( t - o . ; : n"oosr ' *04 lJ - t3-Jt )

R"," ' 'Rep 16.300 e;o o



57 Chemicalnqineennsbcesses

SinceEq. (3-31)expresses as a functionof theReynolds umber, alculation f thepaticle diameteror agiven erminalvelocityand luid viscosity equiresan t.rutiu" .olut on.

Tocircumvenrhisproblem.he ollowing xplicir xprelsionelaring to theRe\notdsnumbersproposedy To\un ndAL$ahinl992)as

E(v)(3-32\

t 6 y t 3 / 2 0 y 6 / 1 )) n / n

vlryexp(:s 19?W -'#P) (3-31)

The procedureo calculatehe pafticlediameters as ollows:

a) Calculate fromEq. 3-30),b) Substituie inroEqs. 3-32)and 3-33) ndderermine ep,c) Once he Reynoldsnumber s determined,he particlediametercanbe calculatedtom

theequation

whereV(f) is givenby

& ReP

Example3.5 A gravity settlingchambers oneof the diverse angeof equipmentused oremoveparticulatesolidsfrom gassl€ams. In a settlingchamber,heeniering gasstreamencounters largeandabrupt ncreasen cross_sectionalreaas shown n the figurebeiowAs a resultof the sharpdecreasen the gasvelocity,he solidparticles eftlelown with

gravity.In practice,the gasvelocity through he chambershouldbe kept below 3 m/s toprcvent he re-enftainment fthe settledDarticles.

(l-34)

Sphedcaldustparticleshaving a densityol 2200kg/m3 are to be separatedrom an arstreamat a temperafiue f 25oC.Determine he diameterof the smallestparticle hat canberemovedn a settling hamber m long,2 m wide,and m high.

Solution


l r :1 .184skc/rn3l l l . : 18.41 lO-bkg/m.s

F_r _ -

For airat 25"C (298K) :



Tlansfercoefticients 58

Analysis

For the minimumparticlesize hat can be removedwith 1007.efficiency, he time requiredfor rhisparticte o fall a distance/ must be equal o thetime required o move hisparticle

hodzontally distance , i.e.,

H L . . /H \t -u ' : ( ' ) - - } ' ' - t ' l \Z /

where o) representsheaverageas elocityn thesettling hamber aking (u) 3 m/s,

the settling velocity of thepafiiclescan be calculated s

/ 1 \o1 (3){; }:0.a3 rnls

\ r /

Thevalue f y is calculatedrom Eq. 3-30) s

f^ .- 0.052 0.007 0.0001q'l ^, .= e \pLr 'r '{4J4r ' l i * . i4 t t ,

-t+u31n -

z+)

. . 4 g tpp p tp 4 {q .8 ' t2200 - 1845 , (18 .410 6 ,, - ,t - l

p-, ; ' .- r

' r ' rsas; 'o""-* ' "

Substitutionf thevalue f / intoEq. 3-33) ives

/ o.09 0.007 0.00019, {y )=exp \3 . t5+yt a I yn

_ _1 t r)

Therefore, he Reynoldsnumberand heparticlediameterare

^ v { } ) ZA .3n" " =

Gy . , ' . l 0 y "u , t ,-

l 6 \4 ;2 , t : , 2oA i4 )o , t1 t1 'n -z ) )

^ u Rer ( 18 .41 l0 6 t t2 .55 r^ , . , ^ -o ,_

' prr (1.1845)(0.43)

I Calculate ; given p and u,

In this case, q. (3-19)mustbe reanangedo hat he luid viscosity anbe eliminated.f

bothsides f Eq.(3-19)aredivided y Rezp,he esult s

f : x (3-35)

whereX, which s ndependentf&, is a dimensionlessumber efined y

Y : eDe (PP- -P) (3 -36)"3 pu?

SubstitutionfEq. (3-25)ntoEq.(3-35)gives

x_ j1{ t+o.ptn.o.osr1 0.411 __ \ j_J t )Kep l * l6 .300Re l ' "



59Chemic€lngineeing rocesses

SincoEq.(3-37) xFesses asa function ftheReynoldsumber, alculation fthe fluidviscosity or agiventeminal velocityandparliclediametercquiresan terativesolution.lbcircumventhis problem, he ollowing explicitexpressionelat;g X to the Reynoldsnumberrsproposedy Tosun ndAktanin 1992):

nee 2 j r t + t2oX o l t ti t t t X >0 .5 (3-3 )

The procedureo calculatehefluid viscosity s as ollows:

a) Calculare fromEq. l-16r,

b) Substitute intoEq.(3-39)anddetermineheReynoldsumber,

c) Once he Reynoldsnumbers determined,he luid viscositycanbe calculatedtom lneequatron

(3-3e)p \ P

Exl''.nph3.6 One way of measudngluid viscositys to usea fallingball viscometernwhicha sphericalall of knowndensitys droppedntoa fiuid,filted r;duaredylinder ndthe ime of fall for the ball for a specifieddistances recoraled.

A sphericalball, 5^mm n diameter,hasa alensiry f 1000kg/m3. Ir falls througha liquidof density 10kg/m3 ar 25.C and ravels

distance f l0 cim n f.f rnin.OetJnJne reviscosity f the iquid.

Solution

Theteminal velocityof thespheres

Distance l0 ^ l0 2:9 .26 ' t j - a m/sTime

Thevalueof X is calculatedrom Eq.(3-36) s

(1.8x60)

(9.8)(5 10 3)(1000 910) 4 8 D p \ p p - p )

3 pu? 3 (910)(9.26 l0 a)zSubstitutionf thevalueof X intoEq.(3-38) ivesheReynoldsumber s

R"" ?l + rlox-'o,u)n"t Ll, * ,20(is36)-20/11)4/tt3.2x ro 3

Hence, leviscosity f the luid s

Dp \g (5 l0 r ) (9 .26x t0 4 ) t9 l/ - f t : r - i= - l l2(g/ms



TEnsiercoefficients 60

3.3.1.2 Deviationsfrom idealbehavior It should e noted harEqs. 3_19) nd 3_25)areonly valid for a singlesphericalparticlefalling in an unboundedluid. The Dresence f

container allsandotherparticles swell asanydeviationsrom sphericalhape ffect neterminal elocityof particles. or example, s a resultof rheupflowof disphc;dfluid n asuspensionf uniformparticles,hesettling elocity fparticlesn suspensions slower hanthe erminal elocity fa singleparticlefthesame ize. hemostgeneralmpirical quationrelating hesettling velocity to thevolume ractionof pafticles,a), s givenby

ur(suspension)(3-40)

where he exponent dependson the Reynoldsnumberbasedon the teminal velocityof aparticle n an unboundedluid. In the iterature,valuesof r are cpofiedas

(3-41)

Particleshape s another actoraffecting eminal velocity.Thetenninalvelocityof a non_)phericalrnicle< e\. han hcr fa sphericalne ya facror frr 'r?ri . i ry,. . . ; . ,

u, single phere)

ur(non pherical)

Nu:2 + 0.6Ref;2prr/3

All propetiesn Eq.(3-44)mustbeevaluatedt he ilm temperature.

[4.65 5.00 Rep< 2": lz.soz.es 5oo Rep 2x 105

Sphedcity s definedas the ratio of thesuface areaof a spherehavingthe samevolumeasthe non-spherical article o the actualsurfaceareaof theparticle.

3.3.2 HeatTranslerCorrelations

Whena spherc s immelsedn an nfinite stagnantluid, the analyticalsolution or steadvs[a[econducrionr po.siblelnd he esullserpressedn he orm

u,(spherical)(3-42)

(3-43)

In thecase ffluid motion,hecontributionf theconvective echanism ustbe ncludedin Eq. 3-43).Cor.elationsor including onvectiveeat ransfer reas ollows:

Ranz-Marshall correlation

RanzandMarshall 1952) rcposedhe ollowingcorrelationor constant urfaceempcra_ture:

(3-44)



61Chemicalngneerins rccesses

Whitak€r correlation

Whitaker (1972) consideredheat tansfer fromthe sphere o be a result of two parallelprocesses ccuring simultaneously. e assumedhat the aminarand urbulentcontriburrons

areadditiveandproposedhe ollowing equation:

Nu:2+ (0.4Rev2+ 0 06Re213)pp.10r*l1,)1/4 (3-45)

A11 ropertiesexcept ,.u shouldbe evaluated t fe. Equation 4.3-30) s valid for

3.5 Rep ?.6x lOa 0.71 pr ( 380 t.O< LL6lt .u 3.2

3.3.2.1 Calculntion f theheat ransfer ate Once heaverage ear ansfer oefficientsestimated y usingcorelations, the ate of heat ransferreds calculatedas

A: (1,D2)lhl lr. -r* l i l ' -46)

Example3.7 An instrument s enclosed n a protectivesphericalshell,5 cm in diame@r,and submergedn a river to measureheconcentrationsf pollutants.The temperature ndthevelocityof lhe riverare 10'C and 1.2m/s, respectively.oprevent nydamageo theinsfument as a resultof the ow river temperatue, he surface emperatures keptconstantat 32'C by installing lectrical eatersn theprotectivehell.Calculateheelectrical owerdissipated nder steadyconditions.

Solution

Physicalproperties

Forwater t 10'C (283K):

Forwaterat 32 C (305K). p :7 69 x 10 6kg/m.s

Analysis

System: rotectivehell

Understeady onditions, he electricalpowerdissipaied s equal o the mteof heat oss romthe shellsuface to the river The rate of heat oss s givenby

A: (" Dzp)De, _ r*)

To determiner), it is necessaryo calculateheRe),noldsumber

Dpuap (5 x l0-2)(1.2)(1000)

l,:

r:o+,. o_o

Ir: 1000e/mr-

fr: 1304 l0 oks/ms

It :587 x l0-3w/tn.K

Ik= 9 .32

(1 )

=4 .6x l ja (2 )



TErsier coeffrcients62

TheWhitakeronelation,q. 3-45),Bives

Nu 2+ (0.4Rey, 0 O6Re2/3)p'40L_ p;t /4

Nu:2+[0.4(4.6x l0a)r/2 0.06(4.6"

toa12/311s.32,10.+

/ 1304 10-6\ /a> . t - t :456 (3 )

\769 ,106 /

The verage eat ran\fer oefficients

. . / k \ / 58 / . t0 ' \( f t )=Nu l̂ l=(as6r l - - , l_sJsJw/m.K t4)\ uP / \ ) ^ ru )

Therefore,he rute of heat oss s calculatedrom Eq.(l ) as

a =[16 x 10 r)2](5353)(32- 10):925 w (5)

3,3.3 MassTJansfer orrelations

Whena spheres ilnmelsed n an nfinite stagnantluid, theanalyticalsolution or steadystatediffusionspossible2nd he esult sexpressedn the orm

sh: 2 (3-41)

Inthecase ffluidmotion, hecontdbutionfconvection ustbe akenntoconsidemtion.Corelations for convectivemass ransferare as ollows:

Ranz-Marshall coftelation

For constant urfacecompositionand ow masshansfer ates,Eq. (3-44)may beapplied omasshansferproblemssimply by replacilg Nu andpr with Sh and Sc, respectively,.e

sh:2+0.6Ref;2scr/3 (3-cx)

Equation3-48)s valid or

2<Rep<200 0.6<Sc<2.7

Fmsslingcorrelation

Frossling1938) roposedhe ollowing ofelation:

sh:2 + 0.552Re /2Sc1/3 Q_4s)


2<Rep<800 0.6<Sc<2.7



Steinberger-Tfeybalcorrelation

Thecofielation_originallyroposedy SteinbergerndTrcybal 1960)nciudes corecriontermfor naturalconvection.The lack of experimentaldata,however.makes his ter; v€rydifncult o calculaten mostcases. heeffeciof natualconvectionecomesegligibl;whentheReynoldsnumber s high, and heSteinberger_Treybalorrelation educesJ

"

63 Chemical lrqineeringrocesses

SteinbergerndTreybal 1960)modifiedheFrossling orelationas

which s vaiid or

sh : 2+ 0.552Re0r53cl/3

1500 Rep< 12,000 0.6< Sc< 1.85

Forwater 6) at 25oC(298K):

TheSchmidt umbers

(l-50)

(3-s2)

Sh= 0.347Re962cr/3 (3-5)Equation3-51)s recommendedor iquidswhen

2000<Rep<16,900

3,3.3,1-,Calculationofthe mass ransfer ate Once heaveragemassransfercoefficients

estimatedy using orrelations,hemteof mass f species4 tra-nsferreds catculateJ:r"

,ha = (r D2p)k)lc t. - u)Me

Example3.8 A solidsphere fbenzoic cid (p: 126.7g/m3)wirha diameter f 12mmis droppednto a longcylindricai ank iiledwith purewirer at 25.C. It tfreneignfof tfretank s3 m,detemineheamount fbenzoic ciddissolvedrom hesphere henlt eachesthebottom f the ank.Thesaturationolubility f benzoic cid n waier s 3.412 s/m3.

Solution

Physical properties

Ia= 1000s/m3

I u:892 x 10-6 s/m.s

IDo" : t .z t x lo em2 ls

pD AB

892x 0 o

(1000)(1.2110-e)

Assumptions

1. Initialaccelerarioneriod s negligible nd hesphercinstantaneously.

reachests terminal elocity

2. Diameter f thespherc oes otchange pprcciabty.hus, heReynolds umber ndthe erminal elocity emain onstant.



Tnansfer oeffcients 64

3. Steady-stateonditions revail.4. Physicalpropetiesof waterdo not changeas a result of mass mnsfer.

Analysis

To determine he erminal velocityof the benzoicacid sphere,t is necessaryo calculateheArchimedesumber singEq. 3-21):

, D t rgp rpp p ' r l 2x I 0 r r J {9 .8 ) ( l 000 r i l 2b7 -1000 ] - ^ . ^A- - - -n- =1892 t ; - - r 'odx

ru

TheReynolds umbers calculatedrcm Eq. 3-27):

nep= $1 t . 0 .057qAru4 l2J2 r ' 1Ius AR tn6

- J U o - r uJ l r n n s ? q r s ^ R Y I n o r 0 4 l 2 l l 2 l 4 - 1 n s A

l 8

Hence, he teminal velocify is

, ,_ ,uRep _ '892 , l 0 o ' {40 t -0 ,_O. r_ r , ,'

pDp (1000) (12l0 i )

Since he benzoicacid sphere alls the distance f 3m with a velocity of 0.3 m/s, the alling

time s

Di\rance 3 _/__ =_ : l us

Time U.J

TheSherwood umbers calculatediom the Steinbergerreybal orrelation, q. 3-5-,,as

sh = 0.347Reg62c1/30.34'7(4056)a2('73't)1/3= 54t

Theaverage ass ransfer oef6cients

L .2 l l 0 - 't * . . t s l ( " r '4 I=rsa l r l __t- ) .46xlu 'm/s'\Dr , / \ t2"10- ' /

The mte of transferof benzoicacid(species"4)

to water s calcuiatedby using Eq. (3-52):

tnA= (t D2p)\k,)(cA,, cA-lMA: (n D2p)\k,l@t, pe-)

= [z(12 r0 t)']ts.+e"

to sy1:.+tzoy= s.+: t0-8 gTs

TheamounrfbenToiccid issol\edn l0 s s

MA = th^t = (8.43 10 8)(10): 8.43 l0-7kg



65 Chem'€lEngineeringtoesses

Verification of assumption# 2

The nitial massof the benzoicacid sphere, l4, is

f t r l 2 \ l 0 - l ) l lM^= t 1 f l 267 )_ . t46 , t0 ,kBL6r

Thepercentdedease n the massof t}le sphere s givenby

/ 8.43 l0-7 \r_ 11 . t00 0 .014qa\ 1 .146 l0 7

Therefore,he assumed onstancy f ,p and u, is ustified.

3-4 FLOWNORMAL O A SINGLECYLINDER

3.4.1 FriclionFactorCorrelations

For cross low over an nfinitely long circularcylindet LappleandShephed(1940)presentedtheir experimentaldata n the form of / versusReD, the Reynoldsnumberbasedon thediameterof thecylinder Their datacan be apprcximated s

. 6 .18J = m ReD 2 (3-53)

K€n

f

: t.2 104 Re,<

1.5 105 (3-54)

The friction factor / in Eqs.(3-53)and(3-54) s basedon iheprojectedareaof a cylinder,i.e., diameter imes ength,and Re2 is definedby

"Du*P

xeo=- ; - t3 -55 ,

Tosun ndAk$ahin 1992)proposedhe ollowingsingleequationor rhe riction actorthatcovers he entiremnge of the Reynoldsnumber n theform

6 t R

1: "]|1r +0.:on/e)8/5

Reo 1.5 105 (3-56)KC;

Once he riction factor s determined, hedrag orce s calculatedrom

r ,=rnt( \p, ,*\ r (r_57)

Example3.9 A distillation colurnnhasan outsidediameierof 80 cm and a heightof 10 m.Calculatehedrag orceexerred y air on hecolumnf thewindspeeds 2.5 m/s.



Transieroeff ients 66

Solution

Physicalproperties

Forairar25.c (298K,. l l :1 1845 e/m3

'{rr: 18.41 10 6kg/m.s

Assumption

1. AiI temperatures 25'C.

Anallsis

l .romEq. 3-551lheeynold\umbers

- - Du -p (0 .8(2 .5 t .184518 ,41l0o

r l zex lu -

TheuseofEq. (3-56)giveshe ricrion actoras

6 t R

r - ^"$ lr+o.J6Relq)8"R"l -

6 .18

d##+rtt* 0 36(t sx 105)5/el8/512

Therefore,he drag orce s calculated romEq, (3-57)as

l I . \ T t ^ lF D (DL , l,p t i l f : r 0 .8 l 0 ) l { L 84s (2 .5 ) t rl ( 1 .2- 35 .5

1 L 2 I

3.4.2 HeatTransferCorrelations

As statedn Section .3.2, he analytical olutionor steady-stateonductiontom a sphereto a stagnantmedium givesNu = 2. Therefore, he correlations or heat transfer n spheri_cal geometry equirethat Nu --+2 as Re-->0. In the caseof a single cylinder,howevel nosolutionor thecase f steady-stateonductionxists.Hence,t is rcquired hatNu -->0 asRe-+ 0. The following heat ransfercorelationsareavailablen this case:

Whitaker correlation

Whitaker(1972)proposeda conelation n the orm

Nu: (o.4Rey'?0.06Re 3)pP4@*/;1/a (:-sr)

in whichall propertiesexceprr. arc evaluared t 7t. Equarion 3-58) s valid for

1 .0 (Rep (1 .0x105 0 .67<p r<300 0 .25<pe lpo<5 .2



67 Chemicalngineeing rccesses

Tabte 3.3. Consrantsor Eq. (3_59)or rhe ckcurarcylindern cross low

ReD

l-40 0.75 0.440 1000 0.51 0.5

1x 1032 x i05 0.26 0.62 x ld I x 106 0.076 o.j

Zhukauskascorrelation

Thecorelationproposedy Zhukauskas1972)s givenby

Nu: CRe3pr"(pr_/pr.)r/a (3_59)

10.37 if pr < 10

10.36 i P r l 0

and hevaluesof C and m aregiven n Table4.3.All prcpertiesexceptpru shoulal e evalu,ated t f6 in Eq.(3-59).

Churchill-Bernstein corelation

Churchilland Bemstein

1977)proposed singlecomprehensivequationhat coversheentircmngeof ReD for which data areavailable,as well as for a wide rangeof h. Thjs

equations in the form

0.62Rej.2Prr [. / Re., r5,8l ' /5 ,1_60) | t , 10 .4 /p t ' ,3 l t / 4

I'

\ 282 .000 /I

, .

whereall propefiiesareevaluated t the film temperature. quation(3_60)s rccommendedwhen

ReDpr > 0.2

3,4,2.1 Calculation of the heat transferrate Once he average eat ransfercoefficlent sestimated y using corelations,the rateof heat ransferreds calculatedas

Q = QtDL\ \h|lru - T6 (J -61 )

Example3.10 Assume hat apenon can be approximated sa cylinderof 0.3 m diameterand 1.8m heightwith a surfaceempemturef 30.C. Calculateherateof heat oss romthebodywhile hispersons subjecredo a 4 m/s wind wirha remperatuef _ 10.C.



TEnsfer@efiicients 68

Solution

Physical pmperties

The llm temperatures (30 l0) 2: lO"C

lu - t t . ' t . 106kg /m. .

Fo ra i ra r r0 ' c (263 ) . lu :12 4 I0 6m27s

l k=z l .zZ" t0 r w/m.K

ln: o.zz

Iu -14 .18xt0om2/sFora i ra r0 .Cr280K, :

l l -2a .96^t0 'W/m.K

IPr= 6 714

For irar o.c (30J 1' u- tt '"+ 10-6 gTm

lPr :0 .71

Assumption

1. Steady-stateonditionsrevail.

Analysis

The ate f heatossrom he ody an ecalculaiedromEq. 3_61):

a : Q,DL) \h tQura ( l )

Dererminationf {1,)nEq. 1) equiresheReynoldsumberobeknown. heReynoldsnumberst 7; and7/ are

a t /6 - - l 0oc o '^ -Du ' - r03 r l4 'ReD ---= =i)7;;fr

=e.65 roa

at rJ=to"c ReD:919=: {q3r(11- =8.46x 0a14 . t8^ l 0 6 - - '


TheuseofEq. (3-58)givesheNusselt umber s

Nu = (0.4Rey?+ 0.06Re2/3) f.4Q1- p,\t /4

: [0.4 9.65 104)r/2 0.06(9.65 tg+,vr1,t.tr,or f1.6?rl0-6-)r/a

'\ t 8 .64 , t0 .6 /

, Hence,he average eat ransfercoefficients

^ - . / 23 .28 . l 0 r \(r ' ) Nu{i |, , , ,

- , r ,u , (__J_)=

t6.6w/n. .K



69Chemi€lEngineeringro@sses

Substitutionf thisresult ntoEq.(l) giveshemteof heat ossas

a=(nxo.3 x r .8) (16.6) [30(- 10) ] :1126Zhukauskascorrelation

ForReo= 9.65x 104 ndPr < 10,,? 0.3?,and romTable .3 heconstantsreC = 0.26

andm : 0.6.Hence.heuleof Eq. 3-5s) i les

Nu = 0.26Reg6ProJT r- / Pr@I 4

-0 .2b {q .65 ,00 ,00 ,6 .72 ,01 ' (9? ) 'o

: : : o\v . / . /

Thercfore, heaverage eat tansfercoefficientand he rate of heat oss rom the body are

, l \ t ) ' ^(h) Nu( : ) - (226,(jj i- -) : r7.sw/m':

\ t ) / \ 0 .3 /

Q=Qt x0 .3x 1 .8 ) (L? .5 ) [30 - ( 0 ) ] 1188

Churchill-Bemstein corrclation

TheuseofEq. (3-60)gives

o .b2Re l2 r r / r [ , R .^ r t ' t - l ot

N , , - n 1 _ u - t t t - - - - - - - - - - t I' - " " l l - {0 .4 /Pr '2r l r4L'

\282.000/I

^ . .0 .b2(8.46\ 0arr r0 . t t4r r l r f, 18.a6too\" - lo / ' ,^ ," .+l t+aAlc i t4 t r t r l

^L

t \,sr "ooo I

: ' "

The average eat ransfercoefficientand he mie of heaf oss tom the body are

t l r t / 24 .80 .10 r r _ . . . , . .( h )= Nu{; l - r o r t l - l - l 6 w /m- K

\u . / \ 0 .3 I

a: (r x 0.3 1.8)(16)[30(-r0)] 1086

Comment: The rate of heat ossFedicted by the Zhukauskas onelation s 97ogreater

than hat calculatedusing heChurchill-Bernstein orrelation.t is important o note that no

two conelationswill giveexactly he same esult.

3,4.3 MassTransferCorrelallons

Bedingfield ndDrew(1950) roposedhe ollowing ofielationor cross- ndparallel-flow

of gaseso the cylinder in which mass ransfer o or from the ends of the cylinder is not

considered:

sh = 0.281Rey2 co44Q-62)



Transrer oeffi ienrs 70

Equation 3-62) s valid for

400< ReD< 25,000 0.6< Sc< 2.6

For iquidsthe conelationobrainedby Linton andSherwood1950)maybeused:

sh= 0.281ReB6Scr/3 (3_63)


400< ReD< 25,000 Sc<.3000

3.4.3,1 Calculationof themass ransferrav Once he averagemass ansfer coelficientsestimated y usingcorelations,the rateof massof species4 ffallsferreds calculatecl s

nA: @DL)(kc)lcA. cA_lMA e_64)

whereMe is themolecularweightof species4.

Example3.11 A cylindricat pipeof 5 cm outsidediameters covercdwith a thin layerofethanol. ir at 30.C flowsnomal to the pipewith a velocityof 3 m/s. Derermirc heaveragemass ransfercoefficient.

Solution

Physical propertiesDiffusioncoefncientof edranol ,4) in air (B) at 30.C (303K) is

/ l n 1 \ l / ' / r ^ r \ t / 2

tDea) ro r=rD,qa t r r ' l ( '# l - r t .4s \ l0 5 r ( i : ) - t . 38x l0 5m27s\ r r J / \ J t J l

Forairat30'C 303K): v:16 x 10 6m2ls

The Schmidtnumber s

s":r|=ffiffi=rroAssumptions

l Steady-stateonditionsprevail.2. Isothermalsystem.

Anal)sis

The Reynoldsnumber s

Du* (5> t0 , r ( j )^^__( (D - -=

16 \ l oo-e r l )



where n1 and ? are the mass lorr mte and the specific volume of the ffuid, respectively.Note hat he erm n parenthesesn rhe ight-hand ideof EJ. (3-65) s knownas he rdlIork in thermodynamicsr.oran ncompressibleluid, .e.,y : 1/p: constant, q. 3-65)simplifies o

w:QLP)

71 Chemical ngjneeringrocesses

Theuseof thecorelationproposedy BedingfieldndDreq Eq. 3-62), ives

sh = 0.281Rev2c0.410.281(g3:'rt/20.rcro4:29

Thercfore,he avemgemass ransfercoefficient s

( / , . ) -sh l? ' ) : ,2q, / | 18 tn- :\ D / 1s ' ro : J :8r lo 'm/s

3.5 FLOW N CIRCULAR IPES

Therate of work done,W, to pumpa fluid canbe determiled rom the expression

w=*o:*(l oar) (3-6s)

(3-66)

whereQ is thevolumetriclowrateof the luid.CombinationfEq. (3-66)wirhEq. 3.1 1l)glves

Fp luJ : QJLPI

| l l , \ ' ll tnDL) l ;p \uJ.l / l (u) et^p lL \ z / l

Exprcssinghe average elocity n termsof the volumetric low late

(3-67\

. . a, - '

n D2 4

^ ̂ , 32pLfQ'1

n2Ds

reduces q.(3-68) o

(3-68)

(3-69)

(3-70)

Engineering roblemsassociated ith pipeflow areclassifiedas ollows:

. Determinehepressurerop, APl, or thepumpsize, -iz; iven hevolumetriclow rate,8, thepipediameter,D, and hephysicalpropertiesof the fluid, p and -,.

. Determinehevolumetriclow rate,Q; given hepressurerop, A P , thepipediameter,D, and hephysical ropertiesfthe fluid,p andp.

. Determine hepipe diameter,D; giventhe volumetric flow rate,Q, thepressure rop,A Pl, and le physical ropertiesf the luid,p andp.

Jwork doneon thesystenrs considered ositive-



Transter oefficients 72

3.5.1 FrictionFactorCorelations

3Srl.l. Laminarfow correlatio For aminarlow n acircular ipe, .e.,Re D (ulp/ tr <2100,hesolution f theequatioN f changc rves

(3-71\

The riction actor appearingn Eqs. 3-70)and 3-71)s alsocalled heFanninpnc-tionfactor. lowever,his s not heonlydefinirionor / availablen the jterature.inothercommonlyuseddefinitionfor / is the Darcyt.tion factor, fD, vthich s four times argerthan he Fanning riction factor, .e.,fD : 4 mercfore, for laminar low

(3-72)

3.5.1.2 Turbulentfow correlatio, Since o theoreticalolution xists or turbulentlow,the riction factor s usuallydetermined.om the Moodycharl (Ig44) in which t is exoresseclasa unction f theReynolds umbetRe,and he elative ipewall oughness./D.'Moodypreparedhischartby using heequarionroposedy Colebrook1938)

- 16' Re

a^ - :-,

| . . / r l D l . 2 6 l j \- _ - 4 l o s l - t - l

"J

--\3.706s Re,///

where€ is thesurfacecughness f thepipewall in metels.

3.5.1,3 SoLutionso theengineering roblems

I. Laminar flow

Fof flow in apipe,tle Re),notds umber s definedby

- D luJ 4pQ

subsrirurionfEq. 3-7+ynto q.1:-zr; ila"

r pD

41t D

(3-73)

(3-74\

I Calculat€APl

SubstitutionfEq.

or t7; givenQ and D

(3-75) nroEq.(3-70) ives

128pLQ

P Q(3-7s)

(3-76)



73 Chemical igineering rocesses

Thepumpsize anbe calculatedrom Eq.(3-66) s

I281LLQ2(3-77)

(r-78.)

I CalculateQ; givenIAP and,

RearrangementfEq. (3-76)gives

I CalculateD; givenQand API

Reanangementf Eq.(3-76)gives

^ t D 1 L P-

lz8ttL

^ 1 1281LLQ1t/a

\ raP l(3-7e)

II. Ttrbulent flow

I CalculateAPl or W; givenO and ,

For hegiven alues f Q andD, theReynolds umber anbedetemined singEq. 3-74).

However, hen he values f Re ande/D areknown,determinationf / ftom Eq.(3-73)requiresan terativeproceduesince appears n bothsidesofthe equation.To avoid terativesolutions,effortshavebeendirected o express he friction factor, /, as an explicit functionof theReynolds umbetRe,and he elative ipewall roughness,/D.

Grcgory ndFogarasi1985) omparedhepredictionsf the welve xplicit elations rthEq. 3-73)and ccommendedheuseof theconelation roposedy Chen 1979):

| . . / t lD 5 .0452 \_= 4 tog l_ _ r ^oa r / r .80 )\/ [ \ 3.7065 ne

--- -,/

/ /D \ ll oo8

r 7 .1490 to8o8 lA= l . l + l | ( l -R l r

\25491 \ Re /

Thus, in order to calculate he prcssuredrop using Eq. (3-80), the following procedureshould e ollowedhroughwhichan temtive olutions avoided:

a) CalculateheReynolds umberrom Eq. 3-74),b) Substirute e nto Eq.(3-80)anddetemine ,c) UseEq. (3-70) o find hepressurerop.Finally, hepumpsizecanbedeterminedy

usinBEq. 3-64).



Transfer oefiicients 74

Example 3.12 What s the requiredprcssure ropperunit length n order o pumpwaterat

a \olumetric low rateof 0.03ml/s at 20"C through commercial teel ipeG = 4.0 x10 ) m) 20cmin diameter?

Solution

Physicalprop€rties

nlForw.terat20'C 1293 r I

p: >v'tKc/l-

[ r : l 00 l l 0 6kg /m.s

Anal)sis

TheReynold\umbersdeterminedromEq. t-741 \

^ 4pQ (4 ) tqqq r (0 .03 jnpD n(1001 t0 6 ) (0 .2 )

Substiturionf thisvalue nroEqs. 3-81) nd 3-80)gives

, t E lD \ i l oo8 /7 . t4oo \o8o8 l

\2 .s4o1 \ Re ./

T (4 .o l0 5 /0 .2 )l l l@8 r 7 .1490 08n81

L z.s4e i I' \ rer . ro j /

| . . / t /D s .0452 . \- - 4 roc \ too5 - -n t 'o8o , l

- f (4 .6 l0 5 /0 .2 ) 5 .0452 " I- -4 roc l3J065

-l9 l , l0 r

o8(1 8 l0 ' ) l = ls 14

Hence,he riction actors

f : 436x103

Thus,Eq.(3-70) ives hepressurcropperunirpipe ength s

LP 32p lQ2 i J2 r (asq) r4 .3b0 1 ) (0 .03 )2

r-

7 pr:-

,rr,6215-+urarm

I CalculateQ; givenlAPl and D

In thiscase,earrangementfEq. (3-70) ives

f : ( : ) '\a)whereY is definedby

(3-82)

12D5 lLP l

32pL(3-83)



(3-84)

Thus, heprocedureo calculate he volumetric low mte becomes:

a) Calculate fiorn Eq.(3-83),b) Substitute into Eq. (3-84)and determine he volumetric low mte.

Example 3,13 What is the volumeffic flow rate of water n m3/s at 20oC that can be de-livered through a commercialsteelpipe (s = 4.6 x 10 | m) 20 cm in diameterwhen theprcssureabopperunit lengthof thepipeis 40 Pa/m?

SolutionPhysicalproperties

for warerr20'c {29i , In=o9oks/m]' - ' " ' '[ a

- l 00 l 10 -6kB /ms

Analysis

Substitutionfthegivenvaluesnto Eq.(3-83) ields

. , l n ' o ' l tP l / : r ' r 0 .2 r5 r40 t

\

3zPL

Y

rl2)rgqo)

Hence, q. 3-84)gives hevolumetic low rateas

75 Chemical ngineeringrocesses

SubstitutionfEqs.(3-74)and 3-82)ntoEq. 3-73) ields

a:-4Yb8(1k+ 2)

q=-4Ybc(:1L+Y2)

= (4x1.ee10')r"clq-##S2 (1001 t0 6)(0.2

(999)(1.99 10 3)il:0.03 m3/s

I CalculateD; givenQ and lAPl

Swamee ndJain 1976) ndCheng ndTurton 1990) rcsentedxplicitequationso solveproblems f this ype.These quations,owever,reunnecessarilyomplex. simpler qua-

tioncanbe obtained y us ing heprocedureuggestedy Tosun ndAklahin (1993)as ol-lows.Equation 3-70)can be reaffangedn the olm

whereN is deflnedby

f : (DN)5 (3-85)

. , 1 n l r .e111/5^=\n;td )

(3-86)



Transfer@efficienls 76

For turbulent floq the value of/

variesbetween0.00025and0.01925.Using an averagevalueof 0.01for / givesa relationshipbetweenD and N as

r04

,=

"

(3_87)

SubstitutionfEq.(3-85)nto he efr-handide fEq.(3-73), ndsubsriuionofEqs. 3-74), (3-87), nd = 0.01 nto he ight-handideof Eq. 3-73)give

o .s ta / t I r , r t t r \ - r l 5D:": : I l roelv +s.8?5(--r-_,1 o. l7rI ( ] -88)' " \ t L \P?1v ' l J | /

Theprccedurco calculate hepipediameterbecomes:a) Calculate fromEq.(3-86),b) SubstituteN into Eq. (3-88)and determinehepipediameter.

Example 3,14 Waterat 20"C is to bepumped hrougha commercialsteelpipe (r = 4.6 x10 ) m) at a volumetriclow rateof 0.03mr/s. Dete.minehe diameter f thepipe f theallowable ressureropperunit ength fpipe s 40 Pa/m.

Solution

Physical roperties

( ^^^ .

Forwaler r20'c ,rn., ,. I a-

9q9ke/ml- - -[ r - l 00 l ^ l 0 6kg /ms

Analysis

Equation3-86)gives

, , I n ) Lp l \ t / s I n1 \4q l t t t ^"-\ ir;Ld) :L-t"t""ortl - 'oe

Hence, q. 3-88) iveshepipediameters

o .s t , t t r i a \ ' l r , \' / '

D=;{lrogl,N+s.87s{-+; l o.r7r l" \ l L \ p v , / ) | /

0 5 1 4 t T'

-ff i ( lr"el,+.0ro-,,,r.0s,+'5811,,100rI0o,l-o.,t, l ' ). - . \ r l qqq ) (0 .01 r i 1 .69 ) "

1 )

:0 .2 m

3.5.2 HeatTianslerCorrelations

For heat ransfer n circular pipes,variouscorrelations avebeensuggestedepending n theflow conditions,.e., aminar r turbulent.



77Chemicalng'neeringro@sses

3.5.2.1 Laminar fow coffelation For 7un\nar low heattransfer n a circular tube withconstant all emperalue, ieder ndTate 1936) roposedhe ollowing orelation:

Nu= L86[Repr(r/a))t/t*/,.)o.tn (3-89)

(3.e0)

Nu = 0.027Rea/5 rr/3(p./p,)0 la

in which all propertiesexceptp& are evaluatedat the mean bulk temperature.Equa_tion 3-89)s valid or

13< Re< 2030 0.48< Pr< 16,700 0.0044 p/ u.tu 9.j5

Theanalytical olutiono thisproblems onlypossibleor very ong ubes,.e.,Z/D -+ oo.In thiscaseheNusselt umberemains onstantt 3.66..

3.5,2,2 Turbulentow correlatiors The following correlationsapprcximatehephysicalsituation uitewell for thecases fconstantwall emperaturendconsiant all heat lux:

Dittus'Boelter correlalion

DittusandBoelter 1930) roposedhe ollowing onelarionn whichall physical ropertiesareevaluated t the meanbulk tempenture:

Nu = 0.023Re4/5 l

10.4 torheadng

t 0.3 for coolingThe Dittus-Boeltercorelation is valid when

0.7< Pr< 160 Re 10,000

Sieder-Thtecorrelation

Theconelation roposedy Sieder ndTare 1936)s

L/D > t0

(3-91)

in whichall propertiesexceptpu aieevaluated t the meanbulk temperaturc.Equation 3-91

) is valid or

0.7< Pr< 16,700 Re 10,000 LID >,10


Theequation roposedy Whilaker 1q72)s

Nu = 0.0 15ReO 3Pro42(p/p, )0 14 (3-92\



TEnster oefiicients 8

in which the Pmndtlnumberdependences basedon the work of Fdendand Metzner 1958),

and he functionaldependence f p//ru is from SiederandTare 1936).Al1physicalproper-ties except&o areevaluated t themeanbulk temperature. heWritaker correlation s validfor

2300<Re<l x 105 0 .48<Pr<592 0 .44< tL l pu<25

3,5,2.3 Calculatio of theheat tansfer rute Once heaverage eat ransfer coefficient scalculatedrom corelations y usingEqs. 3-89){3-92), her the rateof energy ans-ferred s calculatedas

a:(rDL)\h)^TLtt, (3-93)

\\,hereLTLM, logarithmicmetlfl emperature iffelence, s deined by

(7.. Tt\,,, (T, - tn\.,,] / t , v_# ( i - q t )

, ^ l t I a - t b t n I'Ltr' - tb"" )

Example 3.15 Steam condensingon the outer surfaceof a thin walled circular tube of65 mm diametermaintains unifom surfaceemperaturef 100'C. Oil flows hough hetube at an average elocity of 1 m/s. Determine he engthof the tube n order to increaseoil temperaturerom 40'C to 60'C. Physical ropertiesf theoil areas ollows:

I t t : | 2 .4" ru " Kg /m

At50 'C : I r - , : 4 .28 105

m2lsI

l P r : 143

At 100'C:pr. 9.3x 10 3kg/m.s.

Solution

Assumptions

l. Steady-stateonditions rcvail.2. Physical rcpertiesemain onstanl.3. Changesnkineticandpotential neryiesrenegligible.

AnalysisSystem: il in thepipe

The nventorymte equation or massbecomes

Rate f massn= Rare f massout:r i :p{r,)(ftr2/4) (1)

On tle otherhand, he nventory ate equation or energy educeso

Rateof energy n : Rateof energyout (2)



79Chemical ngineeringrccesses

The terms n Eq. (2) are expressed y

Rateof energyn: m d p(Tu, - Toi * t DL\hJLhu (3)Rateofenergyout=m ep96.,, -f*9 @)

SincehewaII emperatures constant,heexpressionor AZaM,Eq. 3-94), ecomes

m"r:W (s)tnl

'' -"'" I

\7, - ru ,,/

SubstitutionfEqs.(l), (3), (4)and 5) ntoEq. 2) gives

= \'):'9"^( =') turD 4 (h \ \ t - - t , , , )

Noting hatStH= (r)/((r)pdpl :11u7G.p4, Eq.(6)becomes

9 -1. . t n( 4,) - | RePrrnf' rb i

) a,D 4 SrH r.-r t . , , ) 4 Nu \/ , - r r , , , /

To determineNu (or (l')), firut the Reynoldsnumber must be calculated.The mean bulkiemperatures (40+ 60)/2= 50'C and heReynolds umbers

" .D ( r ' ) ( 65x l0 l , { i r - 1519

, , " Lamina rl owe = - :+ l g * to s

Since he low s aminar, q. 3-89)mustbe used,.e.,

Nu= L86[RePr(r/r )ft/t ,It,,to" (S)

SubstitutionfEq. (8) ntoEq.(7)yields

. . - r , 4 ? " , r ,t l l

2- - -Repr _ l l j ' ' r n l 3

- ', ,

D L r4)fl.861 \T. - k"", ) J

- - f ( t 2 .4 \ t0 r / s . J \ t0 r l 0 ra ,7 t00 40 t1 "2 ^ . ^ ^- rrsrs)rr43)L ioi; i .86;

' " l ro-oo/ l :2602

ThP nrhe lcnorh i ( rhPn

L = (2602)(65 tO-3; = tO9m

Example 3.16 Ab at 20"C entersa circularpipeof 1.5cm niemal diameterwith a velocityof 50 m/s. Steamcondensesn the outsideof thepipeso as o keep he surface emperatureof tie pipeat 150"C.

a) Calculate he ength of thepiperequired o inqease air tempemtwe o 90oC.b) Discuq"heelTecr f surfaceoughne\\ n he ength f rhepipe.



Transfer@effdenls 80

Solution

Physicalproperties

Themean ulk emperatures (20+ 90)/2 55'C

Forair at 20'C (293K): p :1.204 t kglr.ll3

l r ,- t o .S , l 0 -6kg /m.o

Fo ra i ra t 5 'C 328 r : { r - l8 Jq l0 6 m ' / sI

I Pr 0.707

Forair at 150C (423K)'.p = 23.86x 10 6kg/m.s.

Analysisa) System: ir in thepipe

The nventory ate equation or mass cduceso

Rateof mass fair in: Rateof mass f air out =m (l)

Notethat for compressible luids like air both density andaverage elocity dependontemperature ndpressurc.Therefore,using he nlet conditions

_ f - ' o n r< '2 rt i 1 : t r D ' t l L ' \D ) ) , - , " ,i " ' " ;

' '1 , t . 2047 , { s0 , -.06 , l 0

' kg /s

" - L 4 J

In Foblems dealing with the flow of compressibleluids, it is customary o deflnemassvelocity,G, as

c : = p \ t ) (2 )

The advantage f usingG is the act that t remains onstant or steady low of comptess-ible fluids throlgh ductsof unifom cross section. n this case

G : (l .2M'/)(50)= 60.24 g/n2 s

The nventorymte equation or energy s written as

Rateof energy n : Rateof energyout (3)

Equations3)15) of Example .15arealsoapplicableo thisproblem. herefore, eget

a lRePr . /7 . - h ' \_=_ - l n t -| ( 4 )

D 4 Nu \ l - - | a , , )

TheNusselt umbern Eq. 4)canbedetemined nly f theReynolds umbers known.TheReynolds umbers calculateds

^ DG (0.015)(60.24)Re- - - - -45 ,636 + Turbu len ll ow

rl 19.80 t0-o

The valueof, depends n the correlationsas ollows:



8l Chemi@l ngineeing rccesses

Dittus-Boelter conelation

Subsdtutionf Eq. -90r nroEq. 4 give\

t _ Re02pp6 ln / , - rb . "\ _

(45 .03b )02 (0 .707 )0o , "150 - 20 \ _ .0 ,D 0.0s2 \r. - Tb",, 0.092 \ lso _ 90/

-" "

Therefore,he required ength s

a : (58 .3 ) (1 .5 ) :87m

Sieder-Tatecorrelation

Substiru nof Eq. l-al , inLo q. 4 gi\e.

t Reo2P1 'J r / /p . lo ' 4

, t Tu , -7o "D 0. 08 \r. - Tu,,

{45 .636 )02 {0 .70712 ' l4 .80 . 0 " \ -0 ro . / t50 -20 \-oro8 \: : : " - ro "/

'n \rso o6/=t* '

Thercfore, he required ength s

, : ( 49 .9 ) (1 .5 ) :75m


SubstitutionfEq. (3-92)nroEq.(4) gives

L Reo17p ro58 { t / / - / d t -0 ) " ,1 r , - T0 , "

n- 0-06-\h-r^)

r45 .b jo )07r0 .?07)058/ t9 .80 r t0 o \ 0 ra .

/ t50 -20 \

0 .06 \23 .80 t0 b / " ' \ t50 q0 / -" '

Therefore,ie required engths

t: (67)(1.5) 101 ln

b) Note that Eq. (4) is also exprcssedn the orm

t : ''n1

'-ru") " ' 4SLH \7. - rb,,,/

Theuseof theChilton-Colbumnalogy,.e., /2: StnPr2/3,educes q.(5) o

L I p? /3 . I T , -h . " 1 r i 0 .702 ) ) r . / r 50 -20 \ 0 .1068

o=z rt ' \ r , -u,1=, r

'" \"0r7- ,

16)

The riction actor anbecalculatedrom heChen orelation,Eq.(3-80)

| / e/D 5.0452 \

r'T:-o.cl]ro6s

*" ,ocrJ



TEnsreroefficieris2

. / | l D \ t o o 8 / 7 . 1 4 9 0 o 8 o 8 lA _ t _ | f | _ l\ 2.s4e1/ \ Re ./

Forvaious aluesf€/D, thecalculatedaluesf f, LID and arcgiven s ollows:

;7D--j --z/D

0 0.0053 57.9 86.90.001 0.0061 50.3 '75.5

0.002 0.0067 45.8 68.70.003 0.0072 42.6 63.90.0M 0.0077 39.8 59.7

Commenb The increase n surface roughnessncreaseshe friction factor and hencepowerconsumption.On theotherhand, he ncrcasen surface oughnessauses n ncreasein the heathansfercoefficientwith a concomitantdecrcasen pipe length.

3.5.3 MassTransterCorrelaiions

Mass ransfer n cylindrical tubes s encounteredn a vadetyof operations, uch as wettedwallcolumns,everse smosis, ndcross-flow ltmfihation.As in thecase f heat lanster,masshansfercorrelationsdependon whether he low is laminaror turbulent.

3.5.3.1 Laminar low correlation For laminar low mass tansfer n a circular tube with aconstant allconcentration,nexprcssionnalogouso Eq. 3-89)s givenby

sh 1.86[Rec(D/.)] ' /3 (3-e5)


lxescqoryll/32 z

3.5.3.2 TurbulentfowcorrcIations

Gilliland-SherwoodcorrelationGillilandandSherwood1934) onelatedle experimenralesLiltsbrainedrom wettedwalcolumns n the fom

Sh:0 023Reo 3Sco44 (3-96)

which is valid for

2000 Re 35,000 0.6< Sc< 2.5



83 Chemical ngineeringro@sses

Linton-Shenvoodcorrelation

Thecorelation roposedy LintonandSherwood1950)s givenby

Sh= 0.023Reo3Scl/3 (3-97)


2000 Re< 70,000 0.6< Sc< 2500

3.5,3,3 CaLculation f the mass ransferrate Once heaveragemass tansfercoefficient scalculatedromcorelations ivenby Eqs. 3-95)-(3-97),len dre ateofmassofspecies,4 transferreds calculared)

rir=

QrDL)(k")(Lc1) yMa (3-98)where"M,{ is the molecular weightof species"4, and(Lca)'y, logaithmic me.rnconcen-tration difference,s delnedby

(3.ee)

Example ,17 A smooth ubewith an ntemaldiameter f 2.5cm s cast tom solidnaph-thalene.Pureair enters he tube at an average elocity of 9 m/s. If the average ir pressureis I atm and the temperature s 40'C, estimate he tube length required or the averagcconcenhation f naphthalene apor n the air to reach257a

of the saturation alue.Solution

Physical properties

Diffusion oefficient fnaphthalene,{) in ail (6) ar40 C (313K) is

/ l l l \ 1 2 / ? 1 1 \ l / 2rDaa t : r - r : Dq r ) roo (

* )- r0 .b2 ' l 0 5 ) {# ) :6 .61> l0 om2 ls

\ JUUI \JUU/

Forairat40 C (313K): ,:16.95 x 10 6m2/s

TheSchmidt umbers

16 .q5>0 -o ^ -_r r - tB=6;r ' ro=

-z)o

Assumptions

1. Steady tate onditionsrevail.2. Thesystems sothermal.

Analysis

System: ir in thenaphthaleneube



Tlansfer@efficients 84

If naphthalenes designated s species"4, then the rate equation or the conservationof\pecie ,4becomes

Rateof moles f,4 in: Rateofmolesof,4 out (1) :

The ems in Eq.(1)areexpressedy

Rareofmotesof -4 n = nDLlk)(Lc LM e)

Rare f mofes f "4out= e(cA).d: (n D2 4)lu)(cA)Nt (3)

Since heconcentrationt the wall s constant,he expressionor (AcA)LM,Eg. 3_gg),becomes

(Lca,1p=--- -@ L-

rnl tA'I

L,e. - @e).*)

SubstitutionfEqs. 2)-(4) nroEq.(1)gives

;--;#'"1 ?:]= +1_q,"rr-o.xr=ooz((,1))Notethat Eq. (5) canalsobe expressedn the orm

f -oo' : ( . ' - )-o.o?r(R=) (o), \ s rM / \ Sh , /

The valueof a depends n thecofielationsas ollows:

Chilton-Colburn analogy

SubstitutionfEq. (2-73)nroEq.(6)gives

f,=o.onlscrTheReynolds umbers

- D lu ) (2 .5 10 2 ) r9 )Ke : - : . : - - i :_ -13 ,214 _+ Tu lbu lenrl ow

The friction factor can be calculated rom the Chen conelation, Eq. (3_g0).Takingt /D , .0 ,

, I e /D \ r rm8 /7 . t490 \0 .808r 7 . t490 \o .8s8l" - \zs4si) +(

R. /- \ t r *1 - r '16Yro-r

I f 5 .04s2 - l

{ t=-o^eL__ tosf l . t6,0 J, l - } /_0.0072

(4)



E5Chemicalngineeringrocesses

Hence q.(7)

becomesL

D

The requLed ength s then

(0.072)(2\2.sq2/3

0.0072

L = (3'7.4)(2.5) 93.5cm

Linton-Sherwood correlation

SubstirutionfEq. l q7)inLo q. 6)gives

L ^ . ^ ^ . ^

i : 3. Reo s.2/l - J. (1 .274r0t'12.56t2'- 2o.4D

The tube ength s

L: (29.4)(2.s)'73.s cm

3,5.4 Flow n Non-Circular ucts

The conelationsgivenfor the friction factor, heat ransfercoefficient,and mass ansfer co-efficient are only valid for ductsof circular cross section.Thesecorelations can be used orflow in non-circular ducts by introducing be conceptof hJdraulic equivalentdiatneter,Dh,definedby

/ Florr area \

' \WefledperimeFr/

The Reynolds number based on the hydraulic equivalent diameter is

^ Dt\a)p

so hatthefriction factor, basedon the hydmulic equivalentdiameter, s related o Re, in theform

':'(#)

(3-100)

(3 -101)

(3-ro2,

where Q dependson thegeomefiyof rhe system.Sinceg : I only for a circularpipe, fte

useof the hydraulic quivalent iameters not recommendedor laminar low (Birdead1..2002;Fahien, 983). he hydraulic quivalent iameteror vaiiousgeometdess shownnTable3.4.Example 3.18 Water flows at an average elocity of 5 m/s through a duct of equilateraltriangulardoss-sectionwith one side, a, being equal o 2 cm. Elecffic wLes are wrappedaround he oufer surfaceofthe duct to providea constantwall heat lux of 100Wcm2. ffthe nletwater empemtules 25'C and heduct ength s 1.5m, calculate:

a) Thepowerrequircd o pumpwater hrough he duct,b) The exit water emperatue,c) The average eat ransfercoefficient.



TEnsier coefticients 86

Iable3.4. Thehyd€ulic quivalenliameterorvarious eometnes

II

T

I o997ke/nJ

-

l1= ssz lo-o slmI Cp= 4180 /ks.K

-L.{3

Solution

Physi@l properties

For waterat 25

C

(298K) :

Assumptions

l Steady-stateonditionsprevail.2. Changesn kinetic andpotentialenergies rc negligible.3. Variations n p aDdCp with temperature renegligible.

Analysis

System:Water n theduct



E7Chemical ngineedngrccesses

a) Thepower equired s calculaied rom Eq.(2-11)

w:root:lo,tr(lrr,r,)r' lr,r (r)L \ 2 . / J

The friction factor in Eq. (1) can be calculated iom the modifiedform of the Chencorelation,Eq.(3-80)

| / €/D 5.0452 \

v7--o 'os| \ r .zoo5-

*o'o to , t2 l

where

/ F / D \ I l o q s / 7 . l 4 9 o t o 8 q 8 lA: \Ls4s i ) l \R . , /

( 3 )

The hydraulicequivalentdiameterand he Re),nolds umberare

u,: l- 1= t.tss"^/3 \/3

Dh lu )o { 1 .155 l0 - r } r5 r {997 rRe/,-.-=

8tt x t0 o-::-:---- =64.548 -+ Tubulent low

Substitutionof thesevalues n0oEqs.(3) and(2) and raking€/D 0 give

/7 . t490 t08e81 7 .1490 \o8o8 l^=l. .n.,/

: l* . '* / -28xro"

| .. | 5.04s2 . r

" rT--4" tL-60.*8109{2.8. l0 ' ) l+ / - 0.0049

Hence,hepowerrequired s calculated rom Eq.(1) as

W Ir:rrz to- ' ] r1.stfroozrrs, ',o.o*n,,r, rr. ,t Lz l l

{ b) The nventory ateequation or mass s

Rateof massn : Raieof mass ut ,? : p (D)f @) (4)'\ 4 , /

f J3t2 x Io-2 21. a= (997){5) l - | = 0.86Je/sL4l -

The rvertory rateequation or energy educeso

Rateofenergy n: Rateofenergy ut (5)



Transfercoefticients 88

Theterms n Eq. (5) areexpressedy

Rateof energyn = m dp e6,^_ Ta) + e.. (6)

Rateof energyout: r/,epq6*, _ f,4 (j)

wherep, is the rateof heat ansfer to water rom the ateralsurfaces f the duct.Sub_sirul ion f Eqs.6rand 7) ntoEq. 5)give.

b"..rb*--rbt,+:+ 2s riI?11+r9o'- so"c'n( p (0.863)(4t80)

c' Themean ulk emperaLue( t25 50)/2 17.5.C. r his emperature

tr:628 x 10-3rym.K and pt:4.62

Theuseof dreDittus-Boeltercorelation,Eq.(3 9O),gives

Nu = 0.023Re75pro.a 0.023(64,54g)a/s4.62)o.a2g9

Therefore,heaverage eat mnsfercoefficients

/L \ / i 28x l0 l \ ^(h) Nu{ l : tzqsr l\ uh \ rJ5s . lo t l- 16 57 /mzK



89chemical ngineedngbceses

NOTATION

A area,rrta packingsudacearcaperunit volume'1/m

e p heatcapacityatconstantpressrtre, J/kg K

ci concentration f species, kmol/m3

D diameter,m

Dn hydraulicequivalentdiameterm

De particle diameter,m

D,la diffusion coefficient or system 4 B, m2/s

Fp drag orce.N

/ ftiction factor

G massvelocity,kg/m' sI acceleration f gravitY, n/s2

j s Chilton-Colbum-factor for heat ansfer

j a Chilton Colbum -factor fol massfadsfer

k thermalconductivity,W/m K

&. massransfercoefficient,m/s

, length,m

M mass, g

m masslow rate, g/s

M molecularweight,kg/kmol

i molar lowmt€,kmol/s

P pressule,Pa

C heat ransfer ate,W

Q volumetriclowrate,m3/s

4 heat lur, W/m2

? gasconstant, /mol K

7 iemperatue,'Cor K

tr umq s

V volume,m3

, velocity,m/s

o superficialvelocity,m/s

,r terminalvelocitY,m/s

/ wo*, J: width, m

li/ rateof work, w

x rectangular ooldinate,m

A difference

€ porosity

s surfaceoughness f thePiPe,m

p viscosity,kg/m s

l) kinematicviscosity,m2/s

p density,kglm3



Tlansfercoeflicients 90

Overlin€s

Bracket

Superscript

permoleperunitmass

\a) averagealueof.1

sat saturation

Subscripts

A, I speciesn bina.ryystems, bulkc transition rom laminar o turbulentcft characteristic

/ filmi speciesn multicomponent ystemsin inletLM log-mean

out outletpb packedbedu wall or sudaceoo free stream

Dimensionl€ssNumbers

Ar ArchimedesnumberPr PmnddnumberNu NrrsseltumberRe Reynoldsnumber

ReD Reynoldsnumberbasedon thediameterRet Reynoldsnumberbasedonthe hydmulicequivalentdiameterRe, Reynoldsnumberbasedon the engthRe, Reynoldsnumberbasedon the distarce_rSc SchmidtnumberSh Sherwood umberStn Stantonnumber or heat ransferSfu Santon number or masshansfer



91 ChemicalEiqineeg Pbcesses

REFERENC€S

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smoolhnd cu8h ipe aqs. . l lsr .Civ. lEng. . ,J3.Dittus. F.W:and L.M.K. Boelter, 1930,Un;venity of Califomia publicationson Engineering,Vol. 2, p_ L4J.

Berkeley.Dwivedi.PN. and S.N. Upadhyay, 977.panicte-fllid massbansfer n fixed and luidizedbeds. nd. Ens. Chem.

Prce$ De..De\.16.157Ergun, ..1952, luidnow hrcu8h acked olumns,Chem.ng. tog.48,89.Fahien, .W, I 983,Fundanentalsf Transporthenomena,Mccraw,Hilt,Newyork.Frlend,WL andA.B. Metzner, I958, Tufbulenrheat mDsfernside ubesand treanatogyamongheatjmassjod

momentumrdsfer, AIChE Joornal4. 393.Frossling. .. 1938,Beitr ceophy .52,170.Gilliland.E.R.andT.K. Sherwood.934, iffusion f vaporsntoairstreans,nd_ ne.Chem. 6,516.GregoryC.A-mdM. Fogarasi.985,Altenrareo stddard ricrion actor quarion. il Gas .83. 120.I apple..F.and .B Shepherd.qao.cr l .utJr .onotpxni .elrajccro. ie\ .j , l Eng.chem 2. 05.Lirton, WH. a.d T.K. SheNood, 1950,Ma$ r.esfer from sotid shapeso water n streamlineand urbulenl low.

Chen.Eng.Prog. 6,258.

Moody,L.F.. 19lu, Frjction facton for pipeflow, Truns.ASME 66, 671.Ranz,W:E. d WR. Mdshail, I952,Evaporationromdrops par I Chen.Eng.piog.48,l4l.Sieder, .N.dd c.E. Tate, 936.Heathansfer ndpressureropofljquids n lubes,nd.Eng.Chen.28, 1429.Steinberger, .L. dd R.E. Treybal. 1960,Mass ransfer rom a solid sphere o a flowing liqujd stream.AIChE

Iornal 6,227Swmee,PK. andA.K.Jain.1976. xplicir quationsor pipenow problems,. Hydr.Div ASCE102,657_Touson2007,Modelingn Transporthenomen,2ndd..Etsevier cience Technolos/ ooks.Tosun.L a.d I. Ak$hin. 1992,Explicit exlre$lons for the ricrion facror.Unpubtishcdeport,Middle EastTech

Tosu. L andL Aktahin.1993, alculareiticalpjping damelers,hem. ng. 00 Mdch), r65.Forconectronssee lsoChen.Ens. 100 July).8.

Tunon.R. andN.N.Cldk. 1987,An explicir clarionshipo redict sphericalanicle eminatvetociixpowde.Technolo$ 1. 2l .

Tunon,R. and o. Levenspiel,1986,A sho]1ote on rhe dragcofielarion o. spheres, owderTechnolosy47. 83.Whitaker, .. 1972, orced onvecrionearransferorrelationsor now n pipes, ast larptates.ingte ytindeB,

single sphercs, nd or flow in packedbedsdd tubebundtes.AIChEJournal I 8. 361.Zhukauskas, .. 1972,Adveces in HeatTransfer,Vol_8: Heai Transfer ilJmTubes . CrossFloq Eds.J.p.Hart

nettandT.F. rvine. Ji. AcademicPrcss.New Yor^.

SUGGESTEDEFERENCESORFURTHER TUDY

Brodkey. .S.andH.C.Hcrshey,988,Truspod Phenomena:UnifiedApproach. ccraw HiU,Newyork.Hines,A.L-andR.N.Maddox, 985,MassTransfer-FundmentatsndAppiications.rcnriceHall,Englewood

Clifl'. \etr Jer"e)Incrcpem,F.P md D.P.Dewitt, 2002,FundmentalsofHeat andMassTmnsfer.5th Ed..Witev. Newyofk.



TEnsfercoefficienls 92

Middleman,S., 1998,An IntrodDction o Mass dd Heat T.ansfer_ prirciptes of Analysisand Design,witey,

Skelland,A.H.P.,1974,DitrurionatMassTrmsfer.wiley, Newyork.Wbitakei,S., 1976,ElemerraryHeat TransferAnalysis,pergmon press,Newyork.

PROBLEMS

3.1 A flat plareof length2 m and width 30cm s to beplacedparallel o anair stueam rateinperaruef25oC.Whichsideofrheplare,.e.,lengtb rwidth,should einthedirectionol tlowsoas o minimizehedrag orce f:

a) Thevelociry fair is 7 m/s,b) Thevelocity f air s 30 m/s.

(Ans\yer:a) Lengrh b) Widrh)

3,2 Air at atmosphericressurend200.C lowsatg m/s overa latplate150 m ong nthedircction fflow and70 cm wide.

:lE^slimitele rareol coolingof rheplareso as o keep he surface cmperaruret 30oC.

b) Calculatehe drag orceexertedon theplate

(Answer:a) 1589W b) 0.058N')

3.3.^Water t l5'C flowsat 0.15m/s overa flatplate m Iong n thedirection f tlowand0.3 m wide. If energy s transfered from the top and botto; sudacesof the platetothe flowing streamat a steadymte of 3500 W, determinehetemperature t Ae piate sur_face.

(Answer:35'C)

3.4 Fins areused o incrcaseheareaavailable or heat ransferbetweenmetalwalls andpoorlyconductingluidssuchasgases. simple ectangularin s shown elow



93ChemicalEnginees Processes

If oneassumes,

. I= I (1 ' on l y .

. No heat s Iost rom the end or fiom the edges,

. The avengeheathansfercoefficient, t), is constantand uniformover the entire

surfaceof the fin,. The thermalconductivityofthe fin, k, is constant,

. The emperatwefthe medium uroundinghe in, fo, is unifbrm,

. Thewall temperature,u, is constant,

ihe resulting steady-stateempemture istribution s givenby

r r* '""["( ' - ; ) ]T. -T -

If the rate of heat oss rom the fin is 478 W, determinehe aveEgeheat ransfercoeflicient

for the followingconditions:€ : l'75"C; T. :260"C; k : I05 w/m K; t :4 cm;

W:30cm; -8 :5mm.

(Answer: 400wm2.K)

3.5 Considerhercctangularin Biveon Problem .4. Oneof theproblems f practical

interests the determination f the optimumvaluesof B and , to maximize he heat ransfer

mte rom he in for afixedvolume, , andW.Show hat heoptimum imensionsreglven

by

t ,h \V2 r t t t / ky \ l lI a N L J o l , r ^ | |. r ,_ \ r ,wr ) , , , , , . , ,

3.6 Consider herectangularin given n Problem4.4. If a laminar flow regionexistsover

theplate,show hat the optimumvalue of w for the maximumheat ansfer rate1]om le

fin for a fixedvolume,,

and hickness, , is givenby

N1't'vrl

wherekl is the thermalconductivityof the ffujd

3.7 A thin aluminumln (k :205 W/m K) of length L 20 cm has wo endsattached

to twoparallelwallswith temperatures = 100'c and7, = 90'c asshownn the igure

belovr'' he in loses ealby convectiono theambient ir at 7€ : 30'c with an average

heat ransfercoeflicientof (ft) : 120Wrn2 K through he top andbottom surfacesheat

loss rom heedgesmaybeconsideredegligible).

rzvusn rsl(l)iln

2lhlL2

KB



TGnsfer oetrcients 4

F__ r.=,o ._- _l

Oneof yolu friendsassumeshat there s no ntemal genetationof energywithin the rnanctdetermineshe steadystate emperature istributionwithin the fin as

' r _T

7,,- T-,

in whichN andQ aredefined s

- 2Q sinhNz

and Q=

- tL l T t -T* \' - \ r , -r- l

2stnhN L

a) Show hat here s ndeed o nremal enemtionf energywithin he in.b) Determine he ocationand he valueof theminimLrmJmperaturewithin the fin.

(Answeri :0.1 cm,7:30.14.C)

3.8 ReworkExample4.8 by using he RanzMarshallcorrelarion, q. (4.333), .heFrossling orrelarion, q. (a.3-34), nd hemodifiedFrossling oneJation, q. (4.335).

Why do the resultingSherwood umbers iffer significantly rom 541?

3,9 In an experiment arried ur at 20 C,^aglasssphere f density2620kg/ml falls

rh rou8hca rbon re l rach lo r i de tp= l5q0kg /mrandg -q .58 \ l 0akg /m. \Ju i tha re rm i_nalvelocity f65 cm/s.Detelminehediameter fthesphere.

(Answer:21mm)

3.10.

A CO2_bubbles dsing n a glass f beer20cm rall.Estimatehe ime equiredor abubble5 mm in diameter o reach he top if theprope iesof CO2and beercan e taker asequal o thoseof air andwater, espectively.

(Answer:0.54s)

2lhl

KB



95 Chemicalnglneedngrccesses

3.11 Show hat he useof theDittus-Boelterotelation,Eq (4 5 26)' together ith the

Chilton-Colbumnalogy, q.3.5-12)' ields

/ :0.046 Re-o

which is a good power-lawapprcxination for the friction factor in smoothcircularpipes

Calculate for ne : 105,106and107using hisapgoximate quation ndcompalehe

values ith hose btained y lsing theChen orrelation' q (4 5-16)

3.12 For aminariow of an ncompressibleewtonianluid n a circular ipe'Eq (45_12)

indicateshatthepressure rop spr;portional to thevolumetric lowmte For fully tu$[lent

nor .fro* tttut ttt" pres.uredrop n apipe sproportionalo thesquare f thevolumetic flow

rate.

3.13 Determinehepower o pumpa fluid at a volumetriclowmte of 1 1x l0 3m3/s

throueh : cm aiameter orizontalmoothpipe 10m- ong Thephysical ropertiesf rie

fluid re girenasp = ol5 kg/mrand / = l q2- l0 i kg/m<

(Ans\ter: 10.4W)

3.14 Thepurpose f bloodpressuren thehuman ody s to pushblood o the issues f

the organis; s; that theycanperform their functions Each ime theheart beats' l pumps

out btloodnto thearteries. he bloodpressue eachests maximumvalue' e systolic

Dressure. hen heheall contractso pump the blood ln betweenbeats' heheart s at rest

i"J rft" Uto"a pressure alls to a mi;imum value,diastolicpressureAn averag€healthy

Dersonassystolic nddiastolic ressuresf 120and80 nrmHg' espectivelyhehuman

iody hasabout5.6 L of blood. i it takes 0 s for blood o circulatehroughouthebody'

estimatehepower outputof theheart

(Answer:3.73w)

3,15 Waters in isothermalurbulentlowat 20'C through horizontal ipeof circular

cross-sectionwithl0cminside.liametel,Thefollowingexpefimentalvaluesofvelocityalemeasured( a function l rcdialdictance':

Thevelocitydistributionsproposedn the olm

where -ax is themaximum elocityandR is themdiusof thepipe Calculatehepressure

dropper unit length of thePiPe

(Answer:12.3Pa/m)

":^*('-1)"'

, '(cm) I 0.5 2.5 3.5

u . rm/s)0 .394



. l

Transfercoeffi ienis 96

3,16 In Example4.15, he ength o diametermtio is expresseds

I - ' t ^1t '' ro'1

D 4SrH \T" _ 16,,,,l

UseheChilton-Colburnnalogy,.e.,

andevaluatehevalueof L/D. ls it a realisricalue? fiy/why nor?

3.17 Waterat l0'C enten a circularpipeof intemaldiameter .5 cm with an aveugevelocityof 1.2m/s. Steam ondensesn the outside f thepipesoas o keep he surtacetemperature f thepipeat 82"C. If the engrhof thepipejs 5 m, determinehe outlettgm_peratureof water.

(Answen5l 'C)

3.18 Dry air at I atmpressueand50oC enterca circular pipeof 12 cm intemaldiameterwith an average elocity of l0 cm/s. The inner surfaceof the pipe is coatedwith a rtunabso$entmaterialsoakedwith waterat 20.C. If the ength of thepipeis 6 m, calculateheamountof watervaporcaried out of thepipeperhour.

(Answer:0.067 g/h)

z= StnPl"

Chapter -3-Evaluation of Transfer Coefficients Engineering Operations

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Transcript of Chapter -3-Evaluation of Transfer Coefficients Engineering Operations