Chapter -1-Moleculaar Nd ConvectiveT Ransport

8/13/2019 Chapter -1-Moleculaar Nd ConvectiveT Ransport

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ChapteroneMolecular nd Convective ransport

The otal lux of anyquantitys the sumof themolecular ndconvectiveluxes.The luxesarising rompotentialgradients r driving forcesarc calledmolecularwes. Molecular luxesarc expressedn the brm of cor?.rt#r/tiveot phenomenologicdl)4rdlionr br nomentum,energy, Lnd ass ransport.Momentunlenergy, ndmass analsobe transportedy bulkfluid motionor bulk low,and he esultingluxis called onvectivelry. This chapter ealswith the ormulation f molecular ndconvectiveluxes n nomentum, nergy, ndmasstransport,

1.MOLECULARRANSPORTSubstancesaybehave ifferentlywhensubjectedo thesame radients. onstitutitiequa-rionr dentify he characteristicsf a particular ubstance.or example,f thegradientsmomentum,hen heviscositys deiined y theconstitutivequation alledNev,ton'saw ofrir.ori4J. fthe gmdients energy,hen he hermal onductivitys defiDedby ourier's awof heatconduction.f thegradients concentmtion,hen hediffusion oefficients definedby Fick's rst law of tlffi,sio . Viscosity, hermalconductivity,and diffusion coefficientarecalled ct1 o t prope i es.1,1 Nelrton'sLawof ViscosityConsider fluid contained etweenwo largeparailel lates f areaA, separatedy a verysmalldistance - The systems initially at restbut at tine l:0 the owerplate s set nmoiion n ther-directionat a constantelocityy by applying force in ther-directionwhile heupper lateskeptstatjonary.he esulting elocity rofiles reshownn Figure .1for variousimes.At I :0, the velocity s zeroeverywherexcept t the owerplate,whichhasa velocityy. Then hevelocity istributiontartso develop sa unction ftime.Finally,at steady-state,inear elocity istributions obtained.Experimentalesults how hat he brce equiredo maintainhemotionof the owerplateperunitarea ormomentumlux) s proportionalo thevelociiygradient,.e.,

FVA 2 1

Mom* 'um Dro; l1v kb . | t r' ' g rJdren l

( t -1 )


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2 Chemical ngineeringrocesses

Figure.t.1.Vetocityrcjite evetopmentn /owbetweenaratptares.

and heproportionalityonstant, , is thev6coriD,. quation l -l) is a macroscoDicoua_tion.Themicrcscopicormof thisequations givenbyda.tt^ : -U -,: : - t tV",ay (1 -2 )

which s knownasNewton'saw of t)iscorjryndany luid obeyingEq. (1 2) is caleo aNewtoniannui.d.The rerm /q is called ate of strainl o, ,ot" oj diyor*oiior'ot sn"a, ,or".rne termbf, ls calledJhear fieJs.t containswo subscipts: r epresentshealirectionfforce, .e.,Fl, andy representshe dircction f rhenormal o thesurfac", "., e", "" *fr.f,the force s xcing. Therefore.,tr is simply he orceper unit area, .e_ f,7ii. it-r, .t*po$rbteo nte|preL.Jr s he luxoff,_momenlLtmnrher_dirrl ion," l::^": 1",':1.-"1,,gadienr \ negari\e..e..ur decrea\e\ irh ncrealing . a negarivergnIs nuoduceon he ighlhand ide f Eq. | 2) o lharheslressn en.ionsporit ire.. In5l unrls.hearue\s \ expre\\edn N/mrrpaland elocityradjenln {;/sj /m. Thus,theexaminadonf Eq. I. - | ind;cale\halheunir.of r scori iyln I units re

N /m2 ^ Ns ( kg .m/ . ) t . s kai 1 = - - - - - . = P a \ 1 = _ . . , : _ lr m / S l / m t n z m lMostviscosity atan thecgssystem reusuallyeportedn g/(cm.s), nownasapoise p),or n centipoise1 cP 0.01 P),where

I pa .s t0p_ to rcpViscosity arieswith empelature.hile iquidviscosity ecreasesith ncreasingeurper_

:::"":,C..i:-::.:..1y ll"t:ases ith increasingemperarurc.oncenffariontso ffects-viscosuyrorsorulronsr suspensions.jscosify xlues f various ubstancesregiven n TableD.1 nADDendD-Example1.1 A Newtonianluid with a viscosity f l0 cp is placedbetweeDlwoar.gelarallel plares. he disrance-erweenheplatess 4 mm.The owerptate s pulld in theposrrrver-drrecrjonwith a forceof 0.5 N, while he upperplate s pulled n thenegative

,,rSt,ainsdefired sdefonltion ler unit engih. orexample,fa spring f origjnatength,o k stietchedo alength . rhenhesrajn s (t _ Io)/2".


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MotecutarTransportr directionwith a force of 2 N. Eachplatehasan areaof 2.5 m2. f the velocityof the owerplate s 0.1 m/s, calculate:a) The steady-state omentum fux,b) The velocity of the upper plare.Solution

I = - 2 N . _ - -

a)

b)

Themomentum lux (or forceperunit area) sF 05+)rw= A: : - : - 1Pa

Let V2be hevelocityof theupper late.FromEq.(1.1-2)

lL - r -- r = 0 . 5 N

: : + v 2 = v1 -rY tvz,vJoar: -u Jr, ,' ( t )Substitutionf thevaluesnto Eq. 1) gives

( l ) (4 10 r )Vu0.1 - : _0.3m/s10x10 3 (2)Themjnus sign indicates hat theupperplatemoves n the negative-direction. Notethatthevelocity radients dh/dy: -100 s I'1.2Fourier's awof HealConductionConsider slabof solidmaterial f areaA betweenwo largeparallelplates f a disrancey apan. nitially thesolid material s at temperature throughout.Then the lower plate ssuddenlybrought o a slightly highertemperature, i, and maintainedat that temperature.Thesecondaw of tlermodynamicstateshathearfows pontaneouslyrom hehigherempemtureZt to the ower temperature . As timeproceeds,he temperature rofile in the slabchanges,ndultimately inear teady-stateemperatures attainedsshownn Figure1.3.Experimentalmeasurementsadeatsteadystate ndicate hat he rateof heat low perunitarea sproporlion.tlo he emperarue radienl..e..

O T t T oA \ 2 Y\zr Tmnspon ,rEnergy prop; Temperaruredux Cradient

(13)


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4 Chemical nqneeinqPrcesses

'l_Direcrion fEle'sy Flux

Flgure .3. Temperatwefolile evelopmenin asolidslabbelweenwo plales.

Theproportionalityconstant, , between heenergy lux and he emperaturemdientscalledthemalconductiyitt.l^SI units,0 is n w(J/s) A ]n n2, dT dx in K/m, and in W/m.K.The themal conductivity of a material s, in general,a function of temperature.However,in many engineeringapplications he variation s sufficiendy small to be neglected.Thermalconductivityvalues or varioussubstancesrcgiven n TableD.2 in Appendix D.Themicroscopicormof Eq.(1.1-3)s knownasFowier's aw of heatconduction nd sgivenby(l 4)

in which the subscript) indicates he direction of the eneryy flux. The negativesign inEq. l-4) indicateshatheat lows n thedirection f decreasingempemture.Example 1.2 One side of a copperslab cceivesanet heat nputat a rate of 5000W due ondiation. Theother ace s held at a temperatueof 35 C. If steady-stateonditionsprevail,calculate he surface emperature f the side rcceivingradiantenergy.The surfaceareaofeachace s 0.05m', and heslab hicknesssa cm.Solution

Physical PropeltiesForcopper: : 398W/m.K


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6 chemical nsineerinsrccesses

as Fick's irst law of difusion and s givenb,daAI A , : u A B p d y ( l -6)

(r '7)

where A" and rt representhe molecularmass lux of species4 in the )-directionandmass raition of species"4, respectively.f the total density,p, is constant, hen the telmp(da^/.1 ) canbe replacedby pA/dy alc'dEq.1-6)becomes

' a J

To measure tr6 experimentally,t is necessaryo design n experimentlike heonegivenabove)n which heconvective asslux s almost ero.In mass ransfercalculations,t is sometimesmore convenient o express oncenffationsin molarunits ather han n mass nits, n tems of molarconcentaation.ick's irst aw ofdiffusions writtenas

where q. and.rA epresenthemolecularmolar lux of species'{ n the)-directionand hemole raciionof species4, respectively.fthe totalmolarconcenfation, , is constant,henthe er \c(dxA/dy> canbe replaced y dcAldt, andEq.(l-8) becomes

r e . = - u A B d y

t j , :-t>etcff

'r3/2

(1 -8 )

(r r0)

( l -9)

Thediffusion oemcienthashedimensionsfm2/s n SI units.Typical alues fDAa aregiven n ,{ppe di)i Ll.Examination f these aluesndicateshat hediffusion oefdcient fgasesasanorderof magnitudef l0 ' m'/s under tmosphericonditions. ssumingdealgasbehavior,bepressurend emperatureependencef thediff$ion coefficient f gasesmavbeestimatediom the elation

Diffusion oefficientsor liquidsarc usuallyn theorderof l0 9 m2/s.On the otherhand,D/B valuesor solids ary rom 10 I0 to 10 la m2/s.Exampl 1.3 Air at alnosphericpressureand 95oC flows at 20 m/s over a-flat plate ofnaphthalene0 cm long n the dircction f flow and60 cm wide.Expedmenral easure-mentseport hemolarco0centrationf naphthalenen theair,cA,asa unct ion f distancer ftom theplateas ollows:


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'\ cA(cm) (mol/m3)0 0 .117l0 0.09320 0.0'7630 0.06340 0.05150 0.043

Determinehemolar luxof naphthalee rom heplatesurface nder readyonditions.SolutionPhysicalpropertiesDiffusion oemcient f naphthalene.4) n atu 6) at 95"C (368K) is

r l t ' x r l / 2 - / l r i R \ J / 2tDa6 t36s=1Dl6 r3* {l l i ) - , 0 .0 : t0 -5 r { *1 l l) - 0 .84 r l0 -5m2 ls\ JUU/ \JUUIAssumptions

1. The otalmolarconcentration,, s constant.2. Naphthalene late s also at a temperaturef 95"C.Analysisfhe molar flux of naphthaleneransfered from tle platesurface o the flowing stream sdeterminedrom

t i . l ,=":-o^,(*) t 'r\ qa , / f=0

It is possibleo calculate heconcentration radienton the surfaceof theplateby usingoneof the severalmethodsexplained n Section A.5 in AppendixA.GraphicalmelhodTheplot of c,4versus is given n Figurc 2.5.The slopeof the angento the curve at .{ = 0is -0.0023 (mo1/m3)/cm.Curve fitting methodFrcm semi-logplot of cl versus , shown n Figure2.6, t appea$ hata straight jlt reple-sents hedata airly well. The equationof this ine canbe determined y the methodof least

' squaresn the orm y = mx b (2)


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E Chemical ngineeringrocsses

f.-. \

r (cn)Figure 1.5. Conconliion oi species as a lunclionof posiiion.

s

r(m)Figure 1.6. Concenrrarion fspcies 4 as a funclionof posnion.

wherc) : Iogc,{ (3)

To detemine he values f m and, from Eqs. A.6-10)and ,{.6-11) n Appendix A, therequiredvaluesarecalculatedas ollows:

Yi --20932-1.032- t .1 9-1.201-1.292-1.36'7

0l0203040

0-10.32-22.38-36.03-51 .68-68.35

0100400-90016002500Lyi = -6.943 lri : 150 tx') i = -188.76 Iri = 5500


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The valuesof m and arc(61(-88.76) rsox-6.943)(6)(5500)(rso)z(-6.943)(5s00)1s0)(-188.?6)(6)(5s00)(r50), = _0.94

Thercfore,Eq. (2) takes he form1ogc,4: 00871 0.94 + cA=0.115e-002r (4)

Differentiationof Eq. (4) gives he concentxationradienton the surfaceof theplateas{a | - - (0 . l l 5 r ' 0 .02 i 0 .0023mo l /m '17cm: ' . 21mol /mo\ 4r ,/,=o

Substitutionf thenumeical valuesntoEq. (l) gives he molar lux of naphthaleneromthe suifaceas"r;,1,_o (0.84 10 5)(0.23):19.32 10-7mol/m2.s

DIMENSIONLESS UMBERSNewton'slaw" of viscosity, ouder's law" of heat onduction,ndFick's irst "law" of dif-fusion,n rcality, renot awsbutdefining quationsor viscosity, , thermal onductivity..anddiffusion oefficient, ,aB. he luxes r},, 4), j.{r) and hegradientsdu,/dy, dT dy,lpA/d)) mustbeknownor measurableor theexperimertaleterminationf p, k, andDAB.Newron'saw of viscosity, 4. (1-2),Fouder'saw of heat onduction, q.(1-4),andFick's irst awofdiffusion,Eqs. 1-7)and 1-9), anbegeneralizeds

/l ,totecut:u\ /transpon) / cradient t \\ f l u x / - \p rooe r t r / \ d r i v i ng fo rce / (1 -11 )

1 . 4

Althoughheconstitutivequationsre imilar,heyarenotcompletelynalogoosecausehetranspofi ropefiiesp, &, D13) havedifferenr nits.These quationsanalsobeexprcssedin the following foms:t L dtB = ----\.prr)

K a a ^ .q r : = - - - \ p L p 1 )P L P A Y. ^ d q tJA,= -uAB---' a J

pL,. momentun/volume1-12)pcp Z : energy/volume (1 ,13)

\zp,a= mass f "A/volume (1-14)pdp : constant

The termp/p in Eq. (I . 12) s calledmomentam iffusirit or kinematicvrrcosiry,and heterm k pC p in Eq. (l 13) s called hermal diffusiviry.Momentumand hermaldiffusivities


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10Chemlcal ngineeringrocessesTable ,1 Analogousermsnconstilulivoquationsormomenlum,ne.gy, ndmass ormole)lknsfer n ons-dimenslon

Momentum Energy Mass MoleMoleculdnux ty q, jA, .{ ,T r a n s p o n p r o p e r t y p k D t s D t t

Ju, dl dtA '|c,turao,cnr r dnlne rorce dr n d, ,1,Diffirsivity Dea D,qtQuci l )n olume p, t ' I pT fc .{

o \ t ' j ) d . t c p I I J r e J , , ruraolenrorvudny/vorume d, , \ Jt d,

aredesignatedy v anda, rcspectively.ote hat he ermsu, d, andDA, all have he sameunits,m2/s,andEqs. 1-12)-(l i4) canbeexpressedn thegeneralom as1 Molecular)- rDiffu.ivi / cradienl f \ ' '\ i iu / " ' ( qr"""i'v,^v"r"*")

rl ls)Thequantitieshatappearn Eqs. 1-1i) and l l5) aresunmarizednTablel-1.Since he ems },,d, andD,4sall have hesame nits, he atioof any wo of these iffusivitiesesultsn a dimensionlessumber. or example,he atioof momentum iffusivity othermal iffusivity ive theP undtlnambe, Prl

Prandtlumber=pr 1:914 (1-16)ThePrandtlnumber s a function of temperature ndpressure.However, ts dependence ntemperature, t least for liquids, is much stronger.The order of magnitudeof tle Pranddnumberor gases nd iquidscanbeestimateds

- (103) (10)Pr: :-i 1] :- -:1= 1 for Sases- (103) (10)Pr:_::- _ :10 for iquids

TheSchmidt umbersdefined s le ratioof themomentumomass iffusivities:Schmidt umber. ^ l'L '- sc - D^B- eD^B ( l -17 )

The orderof magnitudeof the Schmidtnumber or gases nd iquids can be estjmated stn ' for Saqes( I l ( 0 t ) -

^ 10,3sc *frr5 = to3 for iquids


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llolecularTranspod1Finally, the ratio of d to D AR gi.r'es he Lewis numbe\ Lel

Lewisnumber Le :

1.5 CONVECTIVETRANSPORTConvectivelux or bulk fux ofa quantitys expresseds

(t"";;:t'") : lquantitv,n/orume)ttff:::llt") ( l le )

(1-20)

Chdacteisnc clocity

kpCpD,sa Sc. (1 - 8 )

Whenair spumpedhrough pipe, t is consideredsinglephasenda single omponentsystem.Inhiscase,here s no ambiguityn defininghecharacte ficvelocify.However,fthe oxygen r the air were reacting, hen the fact that air is composed redominantlyof twospecies,2 andN2,would have o be taken nto account. ence, ir shouldbe considereda single hase, inarycomponentystem. or a single hase ystem omposedf ,?compo-nents,hegeneral efinition f a characteristicelocitys givenby^ , . - \ - p . , .

where i is theweightingactorandui s hevelocity fa constituent.he hreemostcommoncharacteristicelocities re isted n Table1.2.The ermy, in the definition f the volumeaverageelocity epresentshepartialmolarvolumeof a constituent.he molar averagevelocity s equal o the volurne vengevelocitywhen he otal molarconcentration,, isconstant. n heother and,hemass verageelocitysequalo thevolume verageelocitywhen he otalmass ensity, , is constant.Thechoice f a characteristicelocity s arbitrary. or agivenproblem,t is moreconvenient oselect characteristicelocily hatwill make heconvectivelux zeroand hus ieldasimpler roblem.n the itemture,t is common mcticeo use hemolaraverageelocity ordilutegases,.e.,c = constant, nd hemass verageelocity or liquids, .e.,p : constant.It should e noted hat hemolecularmasslux expressionivenby Eq.(1-6) epresentsthenolecularnass lux with respecto themass verageelocity. herefore,n theequationrepresentinghe otalmasslux, hecharacteristicelocityn rheconvective ass lux elmistaken s hemass vemge elocily.On heother and,Eq. l -8) s themolecularmolar fuxwithrespecttohenolaraverageelocity. herefore,hemolaraverageelocitys consideredthecharacteristicelocitv n theconvective olar lux erm.

Table1,2.Common haraclersllcelocilies

voluhe fiacltun ci4)r* = X'r,r,1r :;,r;7;r;


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12Chemicalnginee s Pfocesses

1.6 TOTAL LUXSince he total flux of anyquantity s the sumof its molecularandconvectiveluxes, hen

Convectivelux (Quantity/Volumexcharacteristiceloci9M.l"*l"t fl* : (Dtft.t"tq'Xct"dt""t "f at-rtty/V"1,-")Table .3. analogousems in luxexpressionsorvariousvPes l ransportn one-dimension

/Tolal _ l/ franspon\ / Cradientf \\ nux/ - \ nronenv/ \ drir ng orceMoleculd fllx

Moleuhrnux convecliveluxThequantitieshatappearn Eqs (1-21)and 12-22) regivenn Table1 3Thegeneral lux expressionsor momentum,energy,and mass ransportn different coor-dinate ystemsregiven n,A.ppendirFromEq. (1-22), he ratio of tho convectivelux to themolecular lux is given by

(t 23)

Type ofTransport Total Flux

Momentum nlt

Energy

Mas

-, d- v - - _ -

, .lT,l0e pT)' dv

_p UAB_d,_-^" d"

" A R , l t

t r t p r \a r

"eul

None

Mole


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IMotecutarransportSincethe gradientof a quantityrepresentshe var:iationof thar particular quantitycharacteristicength, he"Gradientof Quantify/Volume" anbe expresseds

Grudjent f Quantity/Volume Diference in Quantity/VolumeCh.racteristicengthTheuseofEq. (l-24) in Eq.(l-23) gives

Convectivelux (Characreristicelocity)(Charactedsticength)Molecularlux Diffusivity

1 3

(1-24')

( r -25)

(1 26)(1-2',7)

The ratio of the convective lux to the molecular lux is known as thepecletnumber-pe.Therefore, eclet umbersor healandmass an\fer\ are

DaaHence,he otal lux of anyquantitys givenby

IMolecular luxTotal flux : .lMolecular lux + ConvectiveluxI Convective lux

1.6.1 Rateol MassEntering nd/orLeavinghe System

. /Mass f \ /averaee\ Flow\'r : \ vot me , \ *ro.i-ty/ \ u'"u/h i : p i \ r ) A : p i Q

Pe (1-28)

(1-30)

The mass low rateof species enterinBand/or eaving he system, i, is expresseds

(1-29)ln general,he massof species may enteranavoreave he systemby two means:

. Entedngand/or eavingconduits,. Exchangeof massbetween he systemand ts surrouldings hrough heboundaries fthesystem,.e., nterphaseansport.Whena massof species enten and./oreaves he systemby a conduit(s), he characteristicvelocity s takenas he average elocity of the lowingstream nd t is usually argeenoughoneglect hemolecular lux comparedo the convectivelux, i.e., PeM>>1. Therefore,Eq.(1-29)simplifieso

[, t*. , / cradienrf \| \ Ditrusivity/ \ Massof i/VolumeL- r*",* #'" ", ** '

(1 -31)


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14 Chemicalnsineeringrcesses

Surnmation f Eq.(1 31) overall specieseads o the otalmass low rate,m, entedngand,/orleaving he systemby a conduit n the folmrh=P(u lA=PQOn a molarbasis, qs. 1-31)and 1-32) ake he olm

(r 32)

i i = ci lulA: c iQ

i : c \ r )A : c Q

(1-33)(r,34)

On the other hand,when a massof species ente$and/or leaves he systemas a resultof interphase ansport, he flux expressiono be used s dictatedby the value of thePecletnumber sshownn Eq.(1-28).Example 1.4 Liquid B is flowing overa vefiical plate as shown n Figue 1.7. The sui:faceof theplate is coatedwith a material, 4, which has a very 1owsolubility in liquid 6 Theconcentratronistibution of species"4 in the iquid is Sivenby Bird et al. (2002)^s

t ' : [ * " , ' a ,cAo f @13)n

L

Suf&e dtcd with species '1Figure1.7.Solid issolulionntoa fallingih.


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MolecularTEnspod5wherec,4" s the solubility of r'4 n 6, q is the dimensionlessarameter ofinedby

/ pe6 \'/ 'n:" \s1"o^"r)and f (4/3) is thegammaunctiondefinedby

/' -rQ) : I p" - te-PB n>oJ OCalculate he rate of transferof species"4 into the lowing liquid.SolutionAssumptions

1 Thetotal molar concentrationn the iquid phases constant.2. In the r-direction, the convective lux is smallcomparedo the molecular lux.AnalysisThe molar rate of transferof species 4 canbe calculated rom the expression

r W r L;o: I I na, l- ,azay (r)Jo Jowhere the total molar fiux of species -4 at the intorface, Nr,l,=0, is given by

Nr, l , -o- J ; , , - r=- r r , ( * ) ,.By the applicationof the chain rule, Eq.(2) takes he formr",l^-r-'^'#(?\ r-"Theerm nlai is

on I pe6 \ t ' )0x \9ttDaaz

(3)

(4)On the other hand, he term dcA/d4 canbe calculatedby the application of the Leibnitzfo.mula,.e.,Eq.(A.4 3) n AppendixA, as

- alcA cAodq | (413)SubsrirurionfEqs. 4)and 5) nroEq. 3)yields

^, oou::.(_E \,,, (6)A. \_o=_ f t4JJ , \opoou , )Finally,the useof Eq. (6) in Eq. (1 gives henolar rate of transferof species"4 as

(5)

,^::ffieP)"'t ,,1,,, (7)


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16 chemical ngineeringrocesses

1.6.2RateoI EnergyEntedngand/orLeavlng he SytemThe ateofenergy ntering nd,/ofeavingheq]slem.E. is exprel.ed s

E

' : (ffi )(i;.':,';):i:: (1 36)EnerBy er unit volume, n theotherhand, s expresseds heproductof energy er unitmass, , andm:rrs erunitvolume..e.. en\iq. uch hatEq ( 1-16l ecomesF_f E*,g)\ l I$ ) 1nu. | .a.)FIo")_E. \ Mass \volume/ \velocrry,/ area (1 -17)

Thermal\/ Gradientof\diffusivity/ \ Energy/Volrne/ . (ffi)(*m:,u*)l(*s)-=""""*t"5--"* -l(1-3s)

As in the caseof mass,energymay enteror leave he systemby two means:. By inlet and/oroutlets0eams,. By exchange f energybetween hesystemand ts suroundings hrcughthe boundariesof thesystem n theform of heatandworkWhenenergyente$ and/or eaveshe systemby a conduit(s), he characte stic velocity istakenas he avemge elocity of the flowing streamand t is usually arye enollgh o neglectthe molecularfux comparedo the convectivelux, .e, PeE>>1. Therefore, q. (1-35)simplifieso

NOTATION

DmE

kti i

atea,m2heatcapacityat constantpressure, J/kg Ktotal concenffation, mol/m3concentrationf species, kmol/m3diffusioncoefficient or system 4-8, m2/srate of energy,Wtotalenergylux,w/m'force,Nmolecularmolar lux,kmol/m2smolecularmass lux,kg/m2sthermal onductivity, /m.Ktotalmasslow rate, g/smass low rateof species, kg/stotalmolar lux, kmol/m2s


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tlotationri totalmolar flow rate, kmol/si, molar fiow rate of species , kmol/sP prcssure,PaC heathansfer ate,WQ volumetriclowrate.m1/.4 heat lux,w/m2Z temperatule,C or Kt time,sy votume,m-V partialmolarvolume f species, m3/kmolu velocity, m/su* molar average elocity, m/sur volumeavemge elocity,m/sW total mass lux, kg/m2 sr rcctangular oordinate,mri mole ractlon of species) rectangular oordinate,md themal diffusivity, m2/s/ mteof sftain, /sp viscosity,g/m.sD kinematicviscosity(or momentumdiffusivity), m2/sr totalmomentum ux,N/m2p total density,kg/m3p; densityof species, kg/m3rt, flux of,y-momentumn they-direction,N/m2rri massraction fspeciesOverlines^ perunit mass- paftialmolarBracket(a) average alueof dSuperscriptsat saturauonSubscriptsA, B speciesn binarysystemsc, characteristicl. i speciesn multicomponent ysems

MolecularTransport7


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18Chemicatngineeing rccesses

DimensionlssNumbersLePeuPeuPISc

LewisnumberPecletnumber or heat ransferPeclet umberor massransferPrandtlnumberSchmidt umberREFERENCESTou:oi_200f.Modetng n Tran.pon l"enomen.2ndEd..Etr\ ier Science Tecnnoto$/ ooksBid,R.B.. wE. Sle$andd r N L,shroor. 002. rcn\oun henomena.2 rdbd..$ iji X* i-*.Kelvin,WT., 18t1,Thesecutarooling f rheeaih,Trans. oy.Soc.Edin.23, 5;.SUGGESTEDEFERENCESORFURTHEB TUDYBrodkev.R S sd H c. Hdshev, r988. Transponphenonena A unified Approach.Mccraw-Hill, New york.cussrer .L..1ee7. itrusionMassra'sferinFrui.rysteD,,"osd-, ;d,idg;;i;.; o*.1, :l.unor".Fahien, .W.,1983, undamentarsf Tranrporphenomena,McGraw_Hilr, ew ork.PROBLEMS1.1 Show hattheforce perunit areacanbe nterpreted s hemomentumlux.1,2 ANewtonian iuidvith a viscosity f 50 cp splaced erweenwo argeparallel latesreparared) a distance f 8 rnm.Each larehasanarea f 2 rnr. f," "oo"r'ofl,. ,""i i,r f l e os r r ] vedr rec t ronwi thave loc i ryo f0 .4m/swh i le lhe lowerp la re i ,kepLs ta r jonary .a) Calcularehe teady orceappliedoLhe rpperlate.b) Theiujd in panta) js replaced irhanother ewtonianluid of viscosity cp If thesteadyorceappliedo heupper lares hesame s hatofpart a),calcui"il,,lr"".r""irvfthe upper late.(Answer:a) 5 N b)4mls)L3,,Threeparal lelf latplatesareseparatedbytwofluidsasshowninthefigurebelow.Whatshouldbethe valueof f2 so as rokeep he plate n themiddlestationaryi '

yr=rmis_r_TrFtuidB (uB = 0-8 P) ,fFluid4(pA = 1 cP)

-(Answer: 2 cm)


19/22

lMolecularTranspoft91.4 The steady ate of heat oss hroughaplaneslab,which hasa suface areaof 3 m2 andis 7 cm thick, is 72 W. Determine he thermal conductivityoi the slab f the temperatuedistdbufion n the slab s givenas

Z : 5.r* 10wherc Z is temperaturen 'C and r is thedistancemeasuredrom onesideof the slab n cm(Answer:0.048Wm.K)1.5 The nnerandoutersudaceemperatues f a 20 cm thick brick wall are30'C and5'C, respectively.hesurface rea f thewall s 25 m2.Detemine hesteadyateof heatloss hroughhewall f the hermal onductivitys 0.?2Wm K.(Answer:3150W)1.6 Energy s generated niforrnly in a 6 cm thick wall. The steady-stateempemtuledistributions

I = 145+ 30002 150022where is temperaturen 'C anda is the distancemeasued rom one side of the wall inmeters.Determine he rate of heatgeneration erunit volome f the themal conductivityofthewall s 15W/m.K.(Answer:45 kw/m3)1.7 The temperature istributionn a one-dimensionalall of thermalconductivity20 w/m K and hickness 0 cm s

I:80 + 10e-o e'sin(r6)wheie Z is temperaturen oC, t is time in houfs, f : z/L is the dimensionless istancewithabeing coordinate easwedromonesideof thewall,and , is thewall hicknessnmeters.Calculatehe total amountof heat ransfenedn half an hour if the surfacearcaofthewall s 15m2.(Answer:15,360 )1.8 Thesieadystate emperature istribution within aplanewall I m thick with a themalconductivity f 8 W/m.K is measuredsa function fpositionas ollows:

wherez is thedistancemeasued rom one sideof the wall. Detemine the miform mte ofenergy enerationerunit volumewithin hewall.(Answn1920w/m3)


20/22


21/22

lMolecularTranspoft1If the sudace area of the slab is A, determine the amount of heat haDsfered into the slab asa function of time./ . ^ 2 k A ( T t T i ^I AJrswer: : _ - Va I\ '/Td /1.12 Ai at 20'C and I atmpressurelowsovera porous late hat s soakedn ethanol.Themolarconcenftationfethanol n theair,cA, s givenby

c A : 4 e - I 5 zwherecA is in kmol/m3andz is the distancemeasurealrom the surface f theplate nmete$. Calculate he molar flux of ethanol tom theplate.(Answen0.283 mol/m2h)1.13 Theformal deflnition of thepartialmolar volume s givenby

Substitutey : Q)

into Eq.(1) and show hat the volume raction s equal o the mole raction or constant otalmolar oncentation,, .e.,

-.''l^[,;)]]

(3)This further mplies that the molaravemge elocity s equal o thevolumeaverage elocitywhen he otalmolarconcentrations constant.1.14 For a gasat constant rcssue,why does heSchmidt umber sually cmain airlyconstantover a arge emperatue ange,while the diffusion coefficientchangesmarkedly?1.15 Gas { dissolvesn liquid6 anddiffusesnto he iquidphase. s t diffuses, pecies.4 undergoes n rreversiblechemical eactionas shown n the figure below.Under steadycorditions,he esulting oncentrationistributionn the iquidphases givenby

- /av\' ' : \u ) , .0. , , , , ( l )

I U A E

in which


22/22

22 Chemicatngineedngrccesses

wherec4 is the surfaceconcentration,+ s the reaction ateconstantandD,aB s thedi11u_sioncoefficient.

a) Determine heraieof molesof,4 elrteing the iquid phasef theffoss-sectionalareaorfte tank s A.b) Detelmine hemolarflux atz = l. W}at is thephysicalsignificanceofthis rcsult?(l,o'",, ur'i,- 4?44341 6,o)' L " ' -J

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