Chapter -2-Interphase Transport and Transfer Coefficients

8/13/2019 Chapter -2-Interphase Transport and Transfer Coefficients

1/23

Inlelphase ranspori 3

Chapter two

lnterphase ransportand

Transfer oefficients

In engineering alculations, e are nterested n the determination f the ate of momenum,heat, and mass ransfer rom one phase o another cross he phase nterface, his can beachieved y integrating he lux expression ver he nterfacial rea.Equation l-22) givesthe value of the flux at the nferface as

/ Inrerph.se | ^. "" . . t Cradjent f \I n ,* /-

l ' ' ' n' ' ' \ t )

\quant i ry tvoru .e , l/euanrrrl / Characteri(lic\l\

v"tr* I \ velociry 1,","".""

Note that the delermination of the nterphase lux requires he values of the qua tit /volumeand ts gradient o be known at the nterface. Therefbre, equations f change must be solvedto obtain he distribution f quantity/yolume s a function of position. hese nalytical olutions, owever, re not possible most of the ime, n that case we resort o expedmentaldata and conelate he resulfs by the fansfer coefficients, namely, he friction factor, he heattnnsfer coefficient, nd he mass fansfer oefficielt. he esulting o.relations re hen usedin designing quipment.

This chapter eals with the physical ignificance f these hree ransfer oefficients. naddition, he elationships etween hese ransfer oefficients ill be explained y using di,mensionless umbers ndanalogies.

2.'I FRICTIONFACTOR

Let us consider flat plate of length a and width lu suspended n a uniform stream avrlgan apprcach elocityu- as shown n Figure .1.

't_:L ---

Flgure ,1,Ftow na lal ptate.


2/23

24 Chemical nqineering rocesses

As enginee$, we are nterested n the determination ofthetotl]ldrag force, .e., the com_ponent of the force in the dircction of floq exerted by the flowing s[e;m on the plate. This

force can be calculated by integrating he total momentum lux a^t he wall over re surf.acearca. he otalmomentum lux at he wall, ',lr=0, is

o1-1,=s. , . l r= . r r r, , , l ro (2 )where rr :y:0 s the value of the shear tress t the wall. Since he plate s stationar-y, he luidln contact wiih the plate is also stagnantl and both o, and o, are zero at y : 0. Therefore,Eq. 2- ) reduce.lo

o.'l '=o:.r,lr=o:.=,+lr=.

(2-2')

Note hat he minus sign s omitted n Eq. (2-2) since he value of I, increases s he distarrso) increases. he drag orce, Fr, on one sideof the plate s calculated rom

Evaluation of the integral n Eq. (2-3) requires he value of the velocity qradient at thewall to be known as a function fposition. Obtainjng nalyrical xpressions or the velocitydistribution rom the solution of the equations f change, owever, s almost mDossiblenmost cases. hus, t is customary n engineerjng racrice o rcplace u wjth a dimensiontessEr called he riction fador,, /, such har

t w t LFo=

Jo Jo udtdz

I , "ru = tpual

(2 3)

(2-4)

(2-6)

Substitution fEq. (2-4) into Eq. (2-3) gives

I r w r L / t ^ \t6 - ;p , )_ | | 1ax ,=,w ,Gp, l ) rn (2_r )t ' J u J o ' \ t - /

where /) is the riction actoraveraged ver he area f theplate2, .e.,

r w r LI l ra*a, t te t li f l - r o J o

r i , L- = , -

| | I d x d z| | a , , t , .

w L r a r oJ O JO

Equation 2-5)car be generalized n the orm

Fo: A"nK,*\f)-:-

'Thish known s he o-rlip oundary.onl on.'See sclrionA I iir Apl)cnilrA.

(2-'7)


3/23

Interphase ranspo( 25

in vr'hich he ermsA.r, chamcteristic rea, and -l(i.r, charactedstic inetic erergy, arc defined

by

lwelled surlace rea for f low n conduil .A"-

lPro; . . , .d t"n for f lowaround ubmerged bjecrsr r-d '

t ^K.h=;p,:h

where r,../, s the characteristic velocity.Power, li, is defined as the rate at which work is done. Therefore,

(2-9\

p.*"r:H.=

trc?Djt""") = Gorce)(Velocitv)

til : FDuch (333.4)(2'/ 78) 9262\tt

(2-r0)

(2-l )

Example 2.1 Advefiisements or cars n magazines Eive a complete ist of their features,one of which is the ftiction factor (or drag coeflicient), based on the frcntal area. Sportscars, such as he Toyota Celica, usually have a friction factor of around 0 24. If the car has awidth of 2 m and a height of l 5 m,

a) Determine he power consumed y the car when t is goingat 100km/h.b) Repeat part (a) if the wind blows at a velocity of 30 km/h opposite o the dircction of

the car,

c) Repeat part (a) if the wind blows at a velocity of 30 km/h in the direction of the car.Solution

Physical propertis

Forairat 20 C (293K): p=l.2kg/rf

Assumption

1. Air is at 20'C.

Analysis

a) The characteristic elocity s

+,, r oo'f| 9-'):u.rsm/" \1600/

The drag orce can be calculated rom Eq. 3.1-7) s

, l ^ \ f t , lF p - A , 1 ' l , v l l I ( / ) - ' 2 > 1 . 5 ) I . I t . 2 t ' 2 7 8 f I r 0 . 2 4 ) = 3 3 N\ r ) L z t

The use olEq. (3.1-l ) gives he power onslrmed s


4/23

26 Chemical nsineedng roce$es

b) In this case he charactedstic elocity s

u,r , oor olf+g) :16.r m/(Thereforc, he drag orce and he power consumed rc

| I ^ lFD = Q x t.5) l; (1.2)(36.11)'I (0.24) 563.3L Z J

w : (s63.3)36.rr)20,34t Wc) In this case he characteristic elocity s

, . ,=, roo. o,(f_9 - re.44 /s

Therefore, he drag orce and he power consumed re

f l ^ lF2 e x t.s)l ; (r.2)(19.44),I(0.24) r63.3L z l

ri = (163.3)(19.44)317s w2.1.1 Physical nterpretatton f F ction Factor

Combination fEqs. 2-2)and 2-4) eads oI " 1 , Du , l2" p,k ay "=o

ax , l um

at J1=o 6

b) narred &s

Figure 2.2. The titm modetfof momentum ansier

The friction factor can be determined rom Eq. (2 12) f rhe physical properties of the fluid(vLscosityand densiry). he pproach elocily f the luiO, ni the vetocity radi""i"iif,","llrc nown. tnce he alcularion f rhe elocjty radienteqLrireshe elo;iry isributionnthe luid phase o be known, he actual ase s dJafized sshown n Figure '2.'

-""

Tbe entire resistance o momentum ransport s assumed o be du; to a laminar film ofthickness next o the wa . The velocity radient n the ilm s constant nJ ."0J,"-

(2-12)

(2-3)

'L..


5/23

Inlerphase ransport 7

Substitution f Eq. (2-13) nto Eq. (2 12)and multiplication f the csult ing quation )the characteristic ength, L"r, yield

lt"":t#where he dimensionless erm Re is the Re)nolds number, defined by

Equation 2-14) indicates hat the prcduct of the friction factor with the Reynolds numberis dircctly prcportional o the chamcteristic ength and nversely prcportional o the hickness

of the momentumboundary ayer.

2-2 Heat Transfer Cofficient

2,2,'l Conveclion Heal Transler Coeffieienl

I-et us consider a flat plate suspended n a uniform stream of velocity o@ and emperatue -as shown n Figure 2.3.The emperature t he surface f the plate s kept constant t fu.

As enginee$, we are nterested n the otal rate of heat ransfer rom the plate o the lowingstream. This can be calculated by integrating he total energy lux at the wall over the surfacearea. he orrlenerey fuxal he wall. J r o. .

"" r=oc"

r=o(,od"rr")lr=o (2-16)

where Jlr=o s the molecular or conductive) nergy lox at the wall. As a result f the no-slip boundary condition at the wall, the fluid in contact with the plate is stagnant arld heatis transfered by pure conduction brough he fluid layer mmediately adjacent o the plate.Thercforc, q. (2-16) educes o

""lr=oorlr=nQ,*

f ]"=.,(2-t1)

The rate of heat transfer, 0, ftom one side of the plate to the flowing sffeam s calculatedfrom

q,dx dz (2 18)

(2-14)

(2 15)

. tw tLO : I I

T-

'r+f' 'I+ . -

L )

Figure 2.3. Flow ovor a fLalplare.


6/23

28 Chemical ngineeing rocesses

Evaiuation f the ntegral n Eq. t2 I8l requires heemperatue mdient t the wall to beKnown sa tunc0on t position. o\ ever, he luidmolionma(es he nalyLicaloluljon tthe emperature istribution mpossible o obrajn n most cases. ence, w;'".;lr;;; r"experimentally determined alues of rhe energy lux ar a solid_fluia o""O*'i"

"1.. "rrn"conyection eat transfer coeffcient, h, as

q . : h ( 7 , - T * )

azl61"=o

(2 19)

(2-2t)

(2 23)

which s known as Newton,s a , of cooling.me convection eathansfer oefficient,, hastheuits of w/m2.K. It depends ; the luid lo, rn""r,nni.-, noiJfrop"rti",'ioi".ir""",osrLy,hermal onducri\ty.hear apcciq and no$ Beomelry.Sub.liludon t Eq. 2- Sr nto Eq. 2- 8r gi\es he alfot heal ransferas

Q : Q . (2-20)

where (r) is the heat ransfer coefficient avemged ver the area of the plate and s defined by

- r-,lu* lotoo^o,

- twL)\h)tr - .,6l

r | l rL

l^ l^ h(t,dz r rw tLtnt='o;io;2.:#Jo Jo-oo,o,lo J"o,o,

Equation 2-20) an be generalized n the orm

Q: AH (h)(LT\ch (2 22)where Ar is the heat ransfer area and (AZ)., is the cha$cteristic tempemture iffercnce.

2,2,I.1 Physical terpretation f heat ransfer oefficientCombination f Eqs. (2-17) and 2-19) eads o

T. _ T-

The convection heat tuansfer oefficientcan be determined rom Eq. (2_23) f the thermalconducriviry.of he lujd, the overall emperature ifference, na fr"i"_p".utu* fuA"n, u,me \ arrare Known. ince le calculrtjon f the ernpefalurcrddient l he wall i_quirrsnetemperafure.distdbutionn the luid phase o be known, he aitual case s a"atir"a .

"nornTIie entire resistance o heat hansfer s assumect o be due to a stagnant ilm in the ttuidnext o the wall. The hickness f the ilm, ,,, is such hat t provides h" ,u_" a".irtun"" oheal ansfer as he resistance hat exists or the actual onvection roc"... tt. te_ferut*egradient n the ilm is constant nd s equal o

T*-T,r ld) l )=o

(2-24\


7/23

Interyhase ransport 9

'L_.a) acul casc t) Idulized casc

Frgure .4, I 'e r model or energy ralsler

Substitution fEq. (2-24) nto Eq. (2-23) ives

(2-2s\

Equation 2-25) ndicates hat he hickness f the ilm, dr, determines he value of lr. Forthis eason he erm i is frequently efered o as he th heat ransfer oefrcient.

Example 2,2 Energy genention ate per unit volume as a result of fission within a spherical

reactor of mdius R is given as a function of position asT ' -n-n"t - f l IL

\R , / ]where " s the radial distance measued rcm the center of the sphere. Cooling luid at a tem-pemtue of f- flows over the reactor. f the average eat ransfer coefficient (r) at the sur-face of the reactor s known, detemine the surface empefttuae f the reactor at steady-state.

Solution

System: eactor

Analysis

The nventory ate equation or energy becomes

Rate of energy out : Rate of energy generatron ( r)The rate at which energy eaves he sphere y convection s given by Newton's aw of coollng as

Rate of energy ut (4 t R2) h)(7", T-) (2)

where Zu is the surface emperatue of the sphere. The rate of energy genemtion can bedeiermined y integratirg :)l over he volume of the sphere. he result s

k

t


8/23

30 Chemil Engineeing rccesses

Raterenersyenerari": frt" 1"" "^n"lr-

- 11 r u p lt )

Substitution ofEqs. (2) and (3) inro Eq. (l) gives the surface temperarue as

- _ 2 9ioRr u : , ff 5 0 d

2.2.2 Radiation Heat Trdnster Coefficient

The heat lux due o radiation, 4n, from a small object o the suroundings wall is given as

qR ,"lrl - r )

(i)'l*"^'o,ouor(3)

(4)

where is the emissivity of the small object, d is the Stefan-Boltzmann onstant (5.6.1x10 8 W/m2 K"), alrd Tt and Tz arc the temperatues of the small obiect and the wall indegrees elt n. respectivel).

In engineering ractice, q. 2-26) s wriften n a fashior analogous o Eq. 2-19)as

qR hR r, - rz) (2-27)

Q 26)

where tR

is the ,'adidri on heat ransfer coefrciert. Comparison f Eqs (2 26\ and, 2-2.t)glves

Q28)^ - , a 4 t 4 ,

h- __- :---:--_. =4rolT):T t - 1 2

provided hat (?) >> n - h)/2, \yhere T) : (Tr t T)/2.

2.3 MASSTBANSFER OEFFICIENT

Let us consider a lat plate suspended n a uniform stream f fluid (species B) having a velocity

o- and species "4 concentmtion Ao as shown n Figue 2.5. The surface of the plate s alsocoaled irhspecies 4 wilhconcentrationA,,.

'l_:r -- ---l

Flgure 2.5. Flowover a itat ptaie.


9/23


10/23

32 Chemical ngjreedng rocesses

Equation 2 33) can be generalized n the orm

whereAM s the mass mnsier rea nd 4c,4)., s the characteristic oncentration ifference.

2.3.1 Physical nterpretation l Mass Transfer Coefficient

Combination fEqs. (2-30)and 2-32) eads o

nA: AM \kc)(N: Arch (2.35)

, D,ta fictt : - . r, - . r- a] l r=aThe convection mass hansfer coefficient can be determined rom Eq. (2 36) if the diffusroncoefncient, he overall concentration ifference, and he concentration radient at the wall areknown. Since he calculation of the concentration radient at the wall requires he concentra-tiondi.tribudono be known. he cturl ase s dealized . shown n Figure .6.

The entire esistance o mass ransfer s due o a sragnant ilrn n the luid nexr o the walt.The hickness f the ilm, d., is such hat t provides he same esistance o mass ansferby molecular iffusionas the rcsistance hat exists or the actual onvection rocess. heconcenlradontrcdient n he ilm cconslan( ndequal o

d c t l- ld) l}'=o

Substitution fEq. (2-37) nto Eq. 2-36)gives

(2-37)6.

, Deo' d .

(2-36)

(2-38)

Equation 2 38) ndicates hat he mass ransfer oefficient s directlypropotional o thedifltsion coefficient and nve$ely proportional o the hickness f the conce;tration boundarvlayer

't_.

Figure2.6. The ilmmodettormds rransler


11/23

InleFhase ranspod 3

Figure .7.TransferJ pecies 4 rcm he orid0he tuid hase.

2.3.2 Concentration t the Phase ntedaceConsider he ransfer of species 4 from the solid phase o the fluid phase hrough a ffat inter-face as shown n Figure 2.7. The molar flux of species 4 is expressed y Eq. (2-32). n theapplication of this equation o practical problems of interest, here s no difficulty in definingthe concentration n the bulk fluid phase, ,,l-, since his can be measured xperimentally.However, o estimate he value of c,{,,, one has o make an assumption bout he conditions atthe nterface. t is generally assumed hat he wo phases re n equilibrium with each other atthe solid fluid interface. f Iu represents he inteface temperature, he value of c4. is givenbv

- . _ | P ; " ' /Rr {Assuming idea lgasbehav io r,l u id -ga' " " - l S o t u U i t i q o f s o l i d i n l i q u i d a r l ,f l u i d - l i q u i d

(2-39)

TheAntoine quationis idely used o estimate apor ressues nd i is given n AppendixC.

Example 2.3 0.5 L of ethanol s poued into a cylin&ical tank of 2 L capaciry and he op squickly sealed. The total height of the cylinder s I m. Calculate he mass ransfer coefficientif the ethanol concentration n the air reaches 7, of its saturation value n 5 minutes. Thecylinder emperature s kept constant at 20 oC.

Solution

Physical properties

For ethanol ,4) at 20 "C (293 K):mmHg

Assumption

1. Ideal gas behavior.

Analysis

The mass ransfer oefficient an be calculated tom Eq. 3.3-4), .e.,

m3

lg:ltii

Nl.. : k,(ch - ct*) (r)


12/23

34 Chemiel Engineedng rccesses

The concentmtion difference n Eq. (1) is given as the concenftation of ethanol vapor atthe sudace f the iquid, cAu,minus hat n the bulk solution, 1_. The concentation t heliquid surface s the saturation oncenfation while the concentration n ihe bulk s essentiallvzero at relatively hort imes o hat clu - c/F : c/,. Therefore q. (1) simplifies o

N4' = k'c4' '

The satumtion concenhation ofethanol is

P;* 43.6/7h0,A.:1;T =anr610+21r:2.3e

\ l0-'kmol/m' (3)

Since he ethanol concentration within the cylinder reaches 2% of its saturation value n 5

minutes, he moles f ethanol vaporated uring his pe od aren A : ( 0 . 0 2 ) ( 2 . 39 x 1 0 - 3 ) ( 1 . 5x 1 0 - 3 ) = 7 . 1 ?x t 0 kmol ( 4 )

where 1.5 x 10-3 m3 s the volume of the air space n the tank. Thercfore, he molar flux a5 minutes an be calculated s

^, Number f moles t cpecie.(Area, Time)

7 . 1 7 l 0 .- d - r - t . 2 \ t 0 7 k m o t / m 2 . "

Substitution fEqs. (3) and 5) nto Eq. (2) Bives he mass ransfer oeflicient s, 1 . 2 t l O ' - - ^ _ ,/ " :u6; l i -3 : ) \ tu - m/s

2.4 DIMENSIONLESSNUMBEBS

Rearrangement f Eqs. 2-4), 2 19)and 2-32)gives

Ir. : ; fu"6 A(pt.1) L(pt") = pu- - 0 (2-40\

h ^q. : - - - ; - L(o C eT) L(pCpT):pCpTt-pCpTe e-4t)p v P

Na,, , : k" Lca Lc1=c1.-t :^- (2-Q)

Note that Eqs. (2-40)-(2 42) have he geneml orm

(2\

(6)

/lnterphase\ / Tran.fer \/ DilTerencen \ . . , , .\ flux ,/- \coefficient,/ Quantiry/Volumerf


13/23

Interphase fansport 5

and he emsftt,n/2,hlpep,andk,

all have he same nits, m/s. Thus, he atio of thesequantities ust yield dimensionless umbels.

Heat ransfer tanton umber stg: l- Q-44\P C P r c h

Mass ransibr tanton umber: Stu = & Q 45)uch

Since be erm /2 is dimensionless tself, t is omitted n Eqs. 2-44)and 2-45)Dimensionless umbers an also be obtaited by taking he ratio of the fluxes For example'

when he concentration radienf s expressed n the folm

Gradienr of Quanriry/volumeDifference n guanqty/v:lume

Characteristicength

the expression or the molecular lux,Eq. 1.2 5), becomes

I - p t t ,nLa I " - i - = - t K e

h L,h --k

" " -Nu ' - shDer

Table .1. Transfef oefiicient, iifusivitynd lLrx allobr i,he lanspon l momeniurn

Prccels rraNfer coefficien Drftusivity #;"IlH

Momenrum )no i )t*iPk hL,1

p C P k

^ kcLcnUAB _

(2 46')

(2-41)Molecular lux:{Diffu.i\r) Ditferencen Qutnrity/Volume)

Characteristic ength

Thereforc, he atio of the otal nterphase lux,Eq. (2-43), o the molecular lux,Eq. 2-47)'

Interphaseflux (Transfercoefncient)(Characteristiclength) (2-48)Molecular flux Diffusivity

The quantities n Eq. (2-48) or various ransport rocesses regiven n Table 1.The dimensionless erms epresenting he ratio of the nferphase lux to the molecular lux

in Table .1 are defined n terms f the dimensionless umbers s

(2 4e)

(2,50)

(2-s1)

Energy


14/23

3 Chemrcal ngineeing rocesses

Table .2. Analogous imensionlessumbeE neney

Ener8y

P,= 4

N.: t

s , .=\ : -L' P.ePr pC px.h

" " = u"= tD*nuy : sl = & 4

^ s h k l

where Nu is ihe reat t/dnsfer Nusseh number and NuM is the md.rr transfer Nusselt um-bel. The mass ransfer Nusselt number s generally cdled, te Sherwood ninber, Sh. F-qua-tions 2-49> 2-51) ndicare hat he product /Re/2) is more closely nalogous o theNusselt nd Sherwood umbers han is itself. A summary f the analogous imensron-less numbers or energy and mass ansfer covered so far is given n Table 2.2. The Stantonnumbers or heat and mass ansfer arc designated y StH and StM, espectively.

2-4-1 Dimensionless umbe6 and Time ScalesA characteristic ime is the time over which a given process akes place. Consider, or exam_ple, he ree all of a stone f mass .5 kg ftom the op of a skyscraper. f the height, ,,of thebuilding s 250 m, how long doe-s t take or the stone o reach he ground? Since he accelera_tionof gravity, .e., g : 9.8 m/s2, s responsible or the alling pro-cess,hen he chamctelsttc

time rcpresenting he free fall of a stone s given by

lzso:Vei:5rsFrom physics, he actual ime of fall can be calculated rom the ormula

t ^t= ; e t

(Characteristic ensrh 2|r.h)nol: - -=:a:--: --- -

urllusrvlty

(2-53)

which s different rom 5.1 s. It shodd be kept n mind hat he time scale gives a rcughestimate, or otder-of-magnitude, f the characteristic ime of a given process. As far as theorder-of agnirude. concemed.he alue\ . . and . s are imo.( quivalenr.

Diffusivities rr,c|, DeB) all have he same nirs. m27s. herefore, he charactedstic ime(or time scale) or molecular ransport s given by

s (m/s'?)

(2.s4)


15/23

Inlerphase ransport 7

Tabte .3. Time cales ot ditierentransporlmecnanrsms

Type of

Heat

Mas

+11,

17,,D^"

Lcn

Lch

L. h

Note hat eachprccess xperiences nunsteady-state eriod eforercaching teady tate on-

ditions.Thus, Eq. (2-54) gives an dea of the ime it takes or a given process o reachsteady-state.

Tr;nsfer oefficients f t,7/2. h pe p. and ") allhave he same nits,m/s' The'efore' hecharactedstic ime or ime scale) or convective ranspofl s givenby

Viscous ime scaleDillusive ime scale

Characteristic engthTrirnsferCoe{fcient

(2-ss)

Table 2,3 summarizes he molecular and convective ime scales or the tlansport of momen-

tum. heat. and mass. The tricry issue n the estimation forder of magnitude s how to identify

the characteristic ength. n general, he characteristic eogth Dsed n the molecular ine scale

may be different from that used in the convective time scale.Since he //2 term s dimensionless tself, t is onlitted rom the convective ime scale ol

momentum. Note that he convective ime scale or momentum hansport' a.r/a'./r, s the timeit takes or the fluid to move hrough he system, lso known as he rcstde,ce im?'

'It is possible o redeiine he dimensionless umbers n terms ofthe time scales s ollows:

Conducti\ time scale (2 s6)

(2-5',7)

(2s8)

(2 s9\

(2 60)

Viscous ime scale DtBDiffusive ime scale d

Conductive ime scale DAB

CondDctiveime scale

D t c

2.5 TransportAnalogies

Existing nalogies n vadous ranspofi rocesses epend n the elationship etween he dimensio;less umbers elined yEqs. 2-49) 2-51).InSection2l.lweshowedthat

Convective ime scale or momentum ransport

Diffusive ime scaleConvective ime scale br momentum ransport

it",:? (2-6t)


16/23

38 Chemi@l nsineeing rccesses

On the other hand, ubstitution f Eqs. 2-25)and 2-38) nroEqs. 2-50)and 2-51),respectively, ives

Nu=146r

(2-62)

(2-6t)

(2-64)

(2-66)

and

sh: 143c

Examination f Eqs. 2-61)-(2-63) ndicates hat

Interphaseflux CharacteristiclengthMolecular flux Effective ilm thickness

Comparison fEqs. 2-48)and 2-64) mplies hat

Effecrivellm rhickness " +ry",J_ (2-65)lransrer oerDclentNote that the effectiye lm tficlz?sr is the hichess of a fictitious film that would be requiredto account or the entire resistance f only molecular ransport were nvolved.

UsingEqs. 2-61){2-63), t is possible o express he characterisricength s

1a./,

t /Re6:Nud, Sh 6.-

Substitution f Nu = StHRePr and Sh StMReSc nro Eq. (3.5-6) ives

j /u= ,n".u,r,"."r"

j:r,n:r,"

(2-67)

2.5-1 The Reynolds Analogy

Similarities between the transport of momentum, energy, and mass were fi$t noted byReynolds n 1874. He Foposed that the effective film thicktesses or the transfer of mo-mentum, nergy, nd mass reequal, .e,,

Therefore, Eq. (2-67) becomes

j = Stn ,: stt s"

Reynolds urther assumed hat Pr: Sc l. Under hese ircumstances q. (2-69) educes

(2-68)

(2-6e)

(2-70)

which s known as he Reynous analogJ. Physical propeties in Eq. (2-70) must be evaluatedatT : (Tu +TA/2.

The Reynolds analogy s reasonably alid for gas systems ut should not be considered orliquidsystems.


17/23

Interphase rarsport 9

2.5.2The Chilton-Colburn nalogy

In the Chilton-Colbum analogy he relationships between he effective {ilm thicknesses reexprcssed s

1:p . '7 t 16r d"

(2-7 )

Substitution fEq. (2-71) nto Eq. 2-67) ields

, = s t q P r, t ' = i H

and

(2-72)

where n and rl are he Colburn -factory ot heat and mass hansfer, espectively. hysicalproperties n Eqs. 2-72\ and 2-73')mnstbe evaluated t f : (f, + 7@)/2. Note hatFrls. 2-'72) nd.2-'73) cduce o the Reynolds nalogy, q. 2-70), or fluidswith Pr: Iand Sc 1.

The Chilton Colbumanalogy s valid when0.6 < Pr < 60 and 0.6 < Sc < 3000.However,even fthese cdteria are satisfied, he use of the Chilton-Colburn nalogy s rcstricted y theflow geometry. The validity of the Chilton-Colburn

analogy or flow in differentgeometries

is given n Table .4.Examination of Table 3.4 indicates hat the erm //2 is not equal o the Colburn facto$

in the case of flow around cylinders and spheres. The drag force is the component of thefbrce n the direction f mean low and both viscous nd pressure orces ontdbute o thtsforce4. For flow over a flat plate, he pressure lways acts norinal to the surface of the plateand the component of this force in the direction of mean low is zero. Thus, only viscousforce contributes o the drag force- In the case of curved surfaces, owever, he componentof normal orce to the surface n the direction of mean low is not necessa y zero as shown

Table 2.4. Vahoito 'rhe Ciiron-ColoJr. naogv br va oJsgeometries

L2

: Stv Sc2/J i,r.r (2-7 )

Chilton Colbum Analogy

' : J I I : J M

.^ lNu>>Jr =. /M 1r lsh >>

if Re> 10,000 Smooth ipe)

aThe drag force dising from viscous and presslre orces s called nction lor skin)dng and ottu druE, tc\pe.

t = J E = J 1 [ 4


18/23

40 Chemical nglneering ro@sses

Figur2.8. Psue lorce cting n curved nd lat surfaces.

in Figure 2.8. Therefore, he friction factor for flow over flat plates and for flow inside cir-cular ducts ncludes only friction dlag, whereas he friction factor for flow amund cylinde$,spheres, nd odrer bluff objects ncludes both friction and form drags. As a result, the //2term or flow around cylinders and spheres s greater han he -factors.

Example 2.4 Water evaporates mm a wetted surface frectangular shape when air at I atrnand 35"C is blown over he suface at a velocity of 15 m/s. Heat ransfer measurementsindicate hat or air at I atm and 35'C the average eat ransfer oefficient s givenby thefollowing empirical relation

( f t ) : z tu$where t) is in W/m2.K and u, air velocity, s in m/s. Estimate he mass ransfet oefn-cient and he rate of evaporation f water rom the surface f the atea s I .5 m2.

Solution

Physical properties

For water t 35"C (308K): Pra'= 0.0562 ar

lo :1 .1460ke /n3F o f a i f a r 5 . c (308 ) l

u = lo 47 t0 -6 m2l \

| .p = t.005 r/ksKlPr=6711

Diffusion oefficient f water ,4) n air (B) at 35"C (308 K) is

x l0-5 m2/s,edzos(De B(#)" ' : , r. rr, , ,o, ,(#)t" =r.r,

r 16.47 10 6

The Schmidt umber s

Dte 2.81 < 0-5= 0.586


19/23

InteQhase Grsport 41

Assumption

l. Idealgas behavior

Analysis

The use of the Chilton-Colbum analogy, s : jM, gives

(h) I cr 12/r 2 u$ 7 er 12/li / { , ) = - - = - l - | - - l - l' pi" \ Sc r p do \s.,/

Substitution f the values nto Eq. (t) gives he average ass mnsfer coefficient s

(*,- -9lgl: (9J11"'-0.,0,/,1 . 1 4 6 0 ) '0 0 5 )0 . 5 8 6 /Saturalion oncenradon f water s

P":' 0.0s62' 4 , -? i

=r 8 J r 4 l 0 - r , ( i 5 17 3 , :2

19 l0 - - k m o l / m '

Therefore, he evaporation mte of water rom the surface s

i I = A k,) c , , ca) : (1.5)(0.105)(2.19 10 3 - 0) = 3.45 t0 a k-ot7s

NOTATIONA area, m2As heat ransfer area, m2

\u mass ransfer area, m2Cp heat capacity at constant ressue, kJ/kg.Kcr concentmtion f species , kmol/n3DAs diffusion coefncient or system 4-8, m2/sFp drag orce, N/ friction factorft heat larlsfer coefficient, w/m2 K

js Chilton-Colbum factor for heat ansferju Chilton-Colbum factor for mass hansfer,( kinetic energy per unit volume, J/m3,t thermal conductivity, W/m.K*c mass ransfer coefficient, m/sL length, mM molecular weight, kg/kmolN total molar flux, kmol/m2.s/rr molar low rate ofspecies , kmol/sP pressure, PaO heat ransfer ate, W

o)


20/23

42Chemicalngineedngroesses

q heatfl:dx,W/m2qR heat lux due o radiation,W/m2

7l gas constant, /moI.Ktt energy genemtion mte per unit volume, W/m3f tempemture, C or Kt time, s, velocity, m/sI4l mte of work, W.r rectangolar oordinate, m) rectangular oordinale,z rectangular oordinate, m

d thermal diffusiviry, m2/sA difference, fictitious {ilmthickness or momentum ansfer, md. fictitious film thickness or mass ransfet mdr fictitious film tlickness for heat mnsfer, m3 emissivity11 viscosity, g/m.sl) kinematic viscosity (or momentum diffttsivity), m2/s1r total momentum lux, N/m2p density, kg/m3a stefan-Boltzmannconstant,w/m2.K4rr, flux ofr-momentum n the J-direction, /m2

Bracket

\al average alueofa

Superscript

sat satulauon

Subscripts

,4, B species n binary systemsc,4 characteristici species n multicomponent ystemsr, sudace or walloo Iiee,stream

Dimensionless Numbers

NUE Nusselt number or heat ransferNuM Nusselt number or mass ransfer


21/23


22/23

44 Chemical ngineering rocesses

2.5 In the system hown below, he rate of heat generation s 800 w/ml in Region A,which s pedectly osulated n the eflhand side. Given he conditions ndicated n thefigule,calculate he heat lux and emperatue t the right-hand ide, .e., at r: 100 cm,under teady-state onditions.

(Answer: 20w/m2, 41.3'C)

2.6 Unifom energy eneration ate per unit volume t l = 2.4 x 106 W/m3 s occumngwithin a sphedcal nuclear uel element of 20 cm diametet Under steady conditions hetemperature istribution s given by

I :900 - 10,000r"2

where is n degrees elsius nd is n meters.

a) Detemine he hermal onductivity fthe nuclear uel element.b) wllat is the average eat hansfer coefficient at the suface of the spherc f the ambient

temperature s 35'C?

(Answer: a) 40 W/m K b) 104.6 Wm'z K)

2,7 A plane wall, with a surface reaof 30 m2 and a thickness f 20 cm, separates hotfluid at a temperahrre f 170"C rom a cold luid at 15"C. Under steady tate onditions,the emperature istribution cmss he wall s givenby

Z = 150 600;1 50x2where r is the distance measured tom the hot wall in meters and I is the temperature mdegrees elsius.Ifthe hemal corductivity fthe wa1lis 0 W/m K:

a) Calculate he average eat ransfer oefficients t he hot and cold sudaces.b) Determine he mte of energy generation within the wall.

(Answer: a) (l)r,r = 300 W/rn2 K, \h)"uu

: 477 W lm2 b) 6000W)

2.8 DeriveEq. (2-28).

(Eint: Exprcss and 2 in terms f (1).)

Rare of heat gcneratior = 800 W / nrr


23/23

hteFhase Transport E

2.9 It is also possible o interpret the Nusselt and Sherwoodnumbers as dimensionless. tempemtue and concentration radients, espectively. Show hat the Nusselt and Sherwood

numbers an be erpressed s

-(dT /dy)"=s

and

Qu -r*)/Lch

Documents

Chapter -2-Interphase Transport and Transfer Coefficients