CHAPTER8
Fundamentalsof Mass Transfer
Diffusion is the processby which molecules,ions, or other small particlesspon- taneouslymix, moving from regionsof relativelyhigh concentrationinto regionsof lower concentration.This processcan be analyzedin two ways. First, it can be describedwith Fick's law and a diffusion coeffìcient,a fundamentaland scientificdescription used in the fìrst two partsof this book. Second,it can be explainedin terms of a masstransfer coef- ficient, an approximateengineering idea that often gives a simpler description. It is this simpleridea that is emphasizedin this part of this book. Analyzing diffusion with masstransfer coefficients requires assuming that changesin concentrationare limited to that small part of the system'svolume nearits boundaries.For example,in the absorptionof one gasinto a liquid, we assumethat all gasesand liquids are well mixed, exceptnear the gas-liquid interface.In the leachingof metal by pouring acid overore, we assumethat the acid is homogeneous,except in a thin layernext to the solidore particles.In studiesof digestion,we assumethat the contentsof the smallintestine are well mixed, exceptnear the villi at the intestine'swall. Such an analysisis sometimescalled a "lumped-parametermodel" to distinguishit from the "distributed-parametermodel" using diffusion coefficients.Both modelsare much simplerfor dilute solutions. If you are beginning a study of diffusion, you may have troubìe deciding whetherto organizeyour resultsas masstransfer coefficients or as diffusion coefficients.I havethis trouble too. The cliché is that you should use the masstransfer coeffìcient approach if the diffusion occursacross an interface,but this cliché has many exceptions.Instead of dependingon the cliché, I believeyou shouldalways try both approachesto seewhich is betterfor your own needs.In my own work, I havefound that I often switch from one to the other as the work proceedsand my objectivesevolve. This chapterdiscusses mass transfer coefficients for dilute solutions;extensions to con- centratedsolutions are deferredto Section13.5. In Section8.1, we give a basicdefinition for a masstransfer coefficient and show how this coefficientcan be usedexperimentally. In Section8.2, we presentother common definitionsthat representa thicket of prickly al- ternativesrivaled only by standardstates for chemicalpotentials. These various definitions are why masstransfer often has a reputationwith studentsof being a difficult subject. In Section8.3, we list existingcorrelations of masstransfer coefficients; and in Section8.4, we explainhow thesecorrelations can be developedwith dimensionalanalysis. Finally, in Section8.5, we discussprocesses involving diffusion acrossinterfaces, a topic that leadsto overallmass transfer coeffìcients found as averagesof more local processes.This last idea is commonly called masstransfer resistances in series.
8.1 A Definition of Mass Tlansfer Coefficients
The defìnitionof masstransfer is basedon empirical argumentslike thoseused in developingFick's law in Chapter2. Imagine we are interestedin the transferof mass
211 lrl 8 / Fundamentalsof Mass Transfer e.1/ A DeÍir
irom someinterface into a well-mixed solution. We expectthat the amounttransferred is (a) proportionalto the concentrationdifference and the interfacialarea:
/ rateol mass\ _ ,. ( inrcrfacial\ / concenrrarion\ (8.1-I) \transferred/-"\ area i \ diflerence) where the proportionality is summarizedby À, called a masstransfer coeffìcient. If we (c divide both sidesof this equationby the area,we can write the equationin more familiar symbols:
Nr:k(cri-ct) (8.r-2)
where N1 is the ffux at the interfaceand c1; and cl are the concentrationsat the interface Fig and in the bulk solution,respectively. The flux N1 includesboth diffusion and convection: mas it is like the total flux nt exceptthat it is located at the interface.The concentrationc1; is air- at the interface but in the samefluid as the bulk concentrationc1. It is often in equilibrium exc( with the concentrationacross the interfacein a second,adjacent fluid phase;we will defer is tt discussionof transportacross this interfaceuntil sectiong.5. The flux equationin Eq.8.1-2 makespractical sense. It saysthat if the concentration zero. Thus, differenceis doubled,the flux will double. It also suggeststhat if the areais doubled,the total amountof masstransferred will doublebut the flux per areawill not change.In other 4. words, this definition suggestsan easy way of organizing our thinking around a simple constant,the masstransfer coefîcient ft. k Unfbrtunately,this simple schemeconceals a variety of approximationsand ambigui- This valueii ties. Before introducingthese complexities, we shall go over someeasy examples. These examples The time are important. Study them carefully beforeyou go on to the hardermaterial that follows. (r
Example d 8.1-l: Humidification Imaginethat water is evaporatinginto initially dry air in ; theclosed vessel shown schematically in Fig. 8.1- I (a). The vesselis isothermalat 25"C,so the water'svapor pressure is 23.8 mm Hg. This vesselhas 0.8 liter of water with 150cm2 of surfacearea in a total volume of 19.2 liters. After 3 minutes,the air is five percent The air is in saturated.What is the mass transfercoeffìcient? How long will it take to reach ninety percentsaturation? r- Solution The flux at 3 minutescan be found directly from the valuessiven: We usethì. vapor ( I / air \ concentration volume I Nr: \ / \ / Reanangins
^ ^- /23.8\ / | mol \ /273 \ U.U)t" tt lt ,60 _)ttg.4tirersl - r / \ 22.4tirers / \ 298/
:4.4.10-E mol/cm2-sec
The concentrationdifference is that at the water'ssurface minus that in the bulk solution. It takes trr:: That at the water'ssurfàce is the valueat saturation;that in bulk at shorttimes is essentially 'ler 8.1 / A Defnition of Mass Transfer Coe.fficients 213
:r'J is (a) Humidilication (b) Packod Bed t-E''ut .lt) | I r fiàldìf'l rie i: (c) (d) A '-.rliar LiquidDrops Gas Bubble rg1 o"o.l , | ?"o i t) l' à,arI
::: JÙC Fig. 8.1-1.Foureasy examples. We analyzeeach of thephysical situations shown in termsof -,iolll masstransfer coefficients. In (a),we assumethat the air is at constanthumidity, except near the IS air-waterinterface. In (b), we assumethat water flowing through the packed bed is well mixed, --Ltnl exceptvery closeto the solidspheres. In (c) and(d), we assumethat the liquid solution,which i.'Ier is thecontinuous phase, is at constantcomposition, except near the dropletor bubblesurfaces. l-:iIilll zero. Thus, from Eq. 8.1-2, we have i. !he (2:'8 t t:' . il c'f 4.4. l0-" mol/cm2-sec: k . ?l -o\ 22.4.l0' cm' 298 ''--nle \160 / k :3.4. l0-2cm/sec :::ui- -;-- - This value is lower than that commonly found for transferin gases. . - \c The time requiredfor ninety percentsaturation can be found from a massbalance: ,:i.tt / accumulation\ / evaporation\ I l:l I \ In gaspnase / \ rate / a -:.1ltl ;Vc1 : l1Y, dt -'..0 : Ak[ct(sat) - c1] - tÌll --:'llI The air is initially dry, so
/ :0, cr:0
We usethis conditionto integratethe massbalance: Ll _1_"_(kAlv)t-( - I c1(sat) Rearrangingthe equationand insertingthe valuesgiven, we find '' t:-Lmú- \ kA \ cr (sat)/ 18.4.101 cm3 Inl| - 0.9) (3.4. l0-2 cm/sec). (150cm2)
: 8.3. 103sec :2.3hr
It takesover 2 hoursto saturatethe air this much I+ 8 / Fundamentalsof Mass Transfer 8.1/ADt
Example 8.1-2: Mass transfer in a packed bed Imagine that 0.2 centimeter-diameter This valuc spheresof benzoicacid arepacked into a bed like that shownschematically in Fig. 8.1- 1 (b). definition The sphereshave 23 cm2 surfaceper 1 cm3 of bed. Pure water flowing at a superficial A tang, velocity of 5 cm/secinto the bed is sixty-two percentsaturated with benzoicacid after it dissolvine haspassed through 100centimeters of bed. What is the masstransfer coefficient? in problen Solution The answerto this problem dependson the concentrationdifference that it is rh usedin the definition of the masstransfer coefficient. In every definition,we choosethis so muchl differenceas the value at the sphere'ssurface minus that in the solution. However,we Benzoi can definedifferent mass transfer coefficients by choosingthe concentrationdifference at is relatirel variouspositions in the bed. For example,we can choosethe concentrationdifference at base.b1 L the bed's entranceand so obtain carbonor by one mr Nr:klcr(sat)-01 dissolutitrr 0.62c t sat){5 cm/sec)A Third.and 1 : (ct(sat) (23.rr/.*.x loo .,,-')A intertace. aremuch whereA is the bed's crosssection. Thus intellectu " k:1.3. 10 3cm/sec can be a..: chemical. This definitionfor the masstransfer coeffìcient is infrequentlyused. Alternatively.we can chooseas our concentrationdifference that at a position e in the bed and write a massbalance on a differentialvolume AA,zat this position: Erample \\ater.3ì . amountof (accumurarior,: ( -, / \ -ìminut3. "r,"lr"'fJ;"r, ) \dissolution/ 5 o : A(ctrnl-r,rol ) 11a1.;olrt \ l.- l:,1--l where a is the spheresurface area per bed volume. Substitutingfor N1 from Eq.8.l-2, dividing by AAz, and taking the limit as Az goesto zero,we find dc, ka -c1l ' : ^lct(sat) 47. This is subjectto the initial conditionthat
z :0, ct:0 Integrating,we obtainan exponentialof the sameform as in the fìrst example: c1 -_r | -(--\kaluo)z c't(sat) given, Rearrangingthe equationand insertingthe values we find 5. ::l::i l *:('n)',l'-'' I \a:/ \ r'1(sat)/
5 cm/sec ln(l - 0.62) (23 cmzlcm3)(100cm)
: 2.1' 10-3cm/sec 8.I / A Definition rf Mass Transfer Coefficients 215
.:'lC f This value is typical of those found in liquids. This type of mass transfer coefficient . bt. definitionis preferableto that usedfirst, a point exploredfurther in Section8.2. ': , lll A tangentialpoint worth discussingis the specific chemical system of benzoic acid ::': lt dissolvingin water. This systemis academicallyubiquitous, showing up againand again in problemsof masstransfer. Indeed, if you readthe literature,you can get the impression :-:ì.'e that it is the only systemwhere mass transfer is important,which is not true. Why is it used :irir so much? . ,'\e Benzoic acid is studiedthoroughly for three distinct reasons. First, its concentration ,: .lt is relatively easily measured,for the amountpresent can be determinedby titration with -:' ;.it base,by UV spectrophotometryof the benzenering, or by radioactivelytagging either the carbon or the hydrogen. Second,the dissolutionof benzoic acid is accuratelydescribed by one mass transfercoefficient. This is not true of all dissolutions. For example,the dissolutionof aspirin is essentiallyindependent of eventsin solution (see Section 15.3). Third, andmost subtle, benzoic acid is solid,so masstransfer takes place across a solid-fluid interface.Such interfaces are an exceptionin masstransfer problems; fluid-fluid interfaces are much more common. However,solid-fluid interfacesare the rule for heattransfer, the intellectualprecursor of masstransfer. Experiments with benzoicacid dissolvingin water can be compareddirectly with heat transferexperiments. These three reasonsmake this chemicalsystem popular.
Example 8.1-3: Mass transfer in an emulsion Bromine is being rapidly dissolvedin water,as shownschematically in Fig. 8.1-1(c). Its concentrationis abouthalf saturatedin 3 minutes.What is the masstransfer coefficient? Solution Again, we begin with a massbalance:
d - --Vct ANr: Aklct(sat) - crl dt
dct ,:kafct(sat)-crl at
where a(: AIV) is the surfaceíìrea of the bromine dropletsdivided by the volume of aqueoussolution. If the waterinitially containsno bromine,
f :0, cr:0
Using this in our integration,we find
ct -1*r*kat c1(sat)
Rearranging,
ko:-!n(r _-1) / \ c1$at)/ I --* ln(l-0.5) -t mtn : 3.9. 10-3sec I 216 8 / Fundamentalst{ Mass TransJer 8.2/ OtherD
This is as far as we can go; we cannotfind the masstransfèr coefficient, only its product T. with a. Such a product occursoften and is a fixture of many masstransfer conelations. The Effect quantity fta is very similar to the rate constantof a flrst-orderreversible reaction with an equilibriumconstant equal to unity. This particularproblem is similar to the calculationof Masstranstèr a half-life for radioactivedecay. Such a parallelis worth thinking throughand will become a very usefulconcept in Chapter15. Diffusion
Example 8.1-4: Mass transfer from an oxygen bubble A bubbleof oxygen originally 0.1 centimeterin diameteris injectedinto excessstirred water, as shown schematicallyin Fig. 8. 1-1(d). After 7 minutes,the bubble is 0.054centimeter in diameter.What is the mass Dispersion transfercoefficient? Solution This time. we write a massbalance not on the surroundinssolution but on the bubbleitself:
Homogenetru. d/ 4 .\ -lcr-Ír-l - ANr chemical dr\'3 / reaction -4trr2klcr - : (sat) 0l Hetero_9ene.," chemicll This equationis tricky; cl refersto the oxygenconcentration in the bubble,1 mol/22.4liters reacti(r n at standardconditions. but t 1(sat)rel'ers to theoxygen concentration al saturationin water. about1.5 . l0-3 molesoer liter undersimilar conditions. Thus
dr , c1(sat) dj cl E.: -0.034k : \\: i:Iìnitrrrn,: This is subiectto the condition i:nnitrtrn.i: 'r\- rnrni'. - ' ""t"'- t :0, r : 0.05cm il) rt;*, ...., In ir;ì. :.-. - .. L - : sointegration gives ..\ t'.:\-, -l -'31 .1.:L.l3l ,
-i-: . -_- r : 0.0-5cm - 0.034kr .:.' .a._
Insertingthe numerical values given, we fìnd ...---.-
0.021 cm: 0.05cm - 0.034ft(420sec)
k: 1.6.l0 I cm/sec
Rememberthat this coefficientis definedin termsof theconcentration in theliquid; it would be numericallydifferent if it were definedin termsof the gas-phaseconcentration. t:iler 8.2 / Other Defnitions of Mass Transfer Cofficients 211 rJuct Table 8.2- | . Mass transJèrcofficient compared with other raîe coe.lf(ienîs
The Basic Effect equation Rate Force Coefficient :h an .ìn0f Masstransfer Nr : ftAcr Flux per area Diffèrence of The masstransfèr coetfi- - rrllle relative to concentration cientft([:]l/r) is a an interface function of flow Diffusion -ii : DVcr Flux per area Gradientof The diffusion coefficient relative to concentratlon D (I:lL2 lt) is a the volume physical property iaÌ11 average independentof flow i\ lll velocity ll.ìss Dispersion -c'rv't : PY7, Flux per area Gradientof The dispersioncoefficient relative to time average Lll:lL' /f ) depends r but the mass concentratron on the flow average velocity Homogeneous rt: Ktcl Rate per Concentration The rate constant chemical volume rr ([:] 1/r) is reaction a physical property independentof flow Heterogeneousrt: Ktcl Flux per Concentration The rate constant chemical interfacial rc1(:lL lt) is a surface feaction area property often defined : ..i!-rs in termsof a bulk '',rtùf. concentration
8.2 Other Definitions of Mass Thansfer Coefficients
We now want to return to some of the problemswe glossedover in the simple definition of a masstransfer coefficient given in the previoussection. We introducedthis defìnitionwith the implicationthat it providesa simpleway of analyzingcomplex problems. We implied that the mass transfercoefficient will be like the density or the viscosity,a physicalquantity that is well defìnedfor a specifìcsituation. In fact, the masstransfer coefficient is often an ambiguousconcept, reflecting nuances of its basicdefinition. To begin our discussionof thesenuances, we first comparethe mass transfercoefficient with the other rate constantsgiven in Tàble 8.2-1. The masstransfer coefficientseems a curious contrast,a combinationof diffusion and dispersion.Because it involves a concentrationdifference, it has different dimensionsthan the diffusion and dispersioncoefficients. It is a rateconstant for an interfacialphysical reaction, most similar to the rate constantof an interfàcialchemical reaction. Unfortunately,the definitionof the masstransfer coeffìcient in Table8.2- I is not so well acceptedthat the coefficient'sdimensions are always the same. This is not true for the other processesin this table. For example,the dimensionsof the diffusion coefficientare alwaystaken as L2lt. Iî theconcentration is expressedin termsof mole fraction or partial pressure,then appropriateunit conversionsare made to ensurethat the diffusion coefficient llú keepsthe samedimensions. This is not the casefbr mass transfercoefficients, where a variety of definitions are accepted.Four of the more common of theseare shown in Table 8.2-2. This variety is 218 8 / Fundamental.sof Mass Transfer 8.2 / Other Dr
Air w Table 8.2-2. Common definitions of mass transfer cofficientso 'iroc€ omm Basicequation Typical units of /
Nr : tAcr cm/sec Common in the older literature:used herebecause of its simplephysical signifìcance(Treybal, 1980) Nt : kpLpt mol/cm2-sec-atm Common for a gasabsorption; equivalent fbrms occur in biological problems Wcî€' (McCabeSmith, and Harriot, 1985; sotL .: wllt Sherwood,Pigford, and Wilke, 1975) om- Nr : fr.,4-rr mol/cm2-sec Preferred for practical calculations, especiallyin gases(Bennett and Myers, 1974) Fr,s : Nr :ÀAcr f cruo cm/sec Used in an effort to includediffusion- mirt- inducedconvection (cf. fr in Eq. 13.5-2 denr.: et seq.) (Bird, Stewart,and Lightfoot, 1960) diffe:, Irqu:: oIn Notes: thistable, Nt is defìnedas moles per L2t, andct as molesper L3. Paralleldefinitions whereNr is in termsof M lL2t andcr is M f L3t areeasily developed. Definitions mixing moles and massare infrequently used. mixture \\ ith bFor a gasof constantmolar concentration c, k : RTkp : k,lc. For a diluteliquid solution concentratlon k : 6421ùk,, whereúz is ttremolecular weight of thesolvent, antl p is thesolution density. The proponi.' betr"'eenintei- \\'hich ,- largely an experimentalartifact, arising becausethe concentrationcan be measuredin so \ alue many differentunits, including partialpressure, mole and massfractions, and molarity. In this bt'o nrrrìrion in t h. In this book, we will mostfrequently use the first definitionin Table8.2-2, implying that masstransfer coefficients have dimensionsof length per time. If the flux is expressedin ruirh it fron:. '.\ (llten molesper areaper time we will usuallyexpress the concentrationin molesper volume. If L. nlui: the flux is expressedin massper areaper time, we will give the concentrationin massper .'t]lOr€ flC;r.' volume. This choiceis the simplestfor correlationsof masstransfer coefficients reviewed in ::nnrtion $;. '* thischapter and for predictionsof thesecoeffìcients given in ChaptersI 3- I 4. Expressingthe ., .t: Ured ::: masstransfer coeffìcient in dimensionsof velocity is also simplestin the cases .ìn(rther : of chemical -.:'ìnterll-t;t.: reactionand simultaneousheat and masstransfer described in Chapters15, 16, and20. - However,in someother cases, alternative forms of the masstransfer coefficients lead to --:i.1!e iifa; :: simplerfinal equations.This is especiallytrue in thedesign of equipmentfor gasadsorption, .-ir'.:Ì' UfìkL, , j, distillation,and extractiondescribed in Chaptersg-l l. There,we will frequentlyuse k' a- rlr't.tlt. - the third form in Table8.2.2, which expressesconcentrations in mole fractions. -r:J::Ìllnini:: In some - :_ .-3nl l:\. casesof gasadsorption, we will find it convenientto respectseventy years of tradition and --,.... use kp, with concentrationsexpressed as partial pressures. In the membrane separations - _.:-.\..._j in Chapter17, we will mentionforms like k, but will calry out our discussionin terms of forms equivalentto k. The masstransfer coefîcients defined in Table8.2-2 arealso complicated by the choice of a concentrationdifference, by the interfacial areafor masstransfer, and by the treatment of convection.The basic definitionsgiven in Eq. 8.1-2 or Table 8.2-l areambiguous, for the concentrationdiffèrence involved is incompletelydefined. To explore the ambiguity more carefully,consider the packedtower shown schematicallyin Fig. 8.2-1. This tower is basically a piece of pipe standingon its end and filled with crushedinert materiallike broken gìass. Air containingammonia flows upward throughthe column. Water trickles down throughthe column and absorbsthe ammonia: Ammonia is scrubbedout of the gas 8.2 / Other Definitions of Mass Transfer Cofficients 2t9
Air wilh only Pure <_ w0rer
The concentrolion difference of ommonio here...
...moy be very ferent thon lhol here îîî
Fig.8.2- l . Ammoniascrubbing. In thisexample, ammonia is separatedby washinga gas mixturewith water. As explainedin thetext, the example illustrates ambiguities in the definitionof masstransfer coefficients. The ambiguities occur because the concentration differencecausing the mass transfèr changes and because the interfacial area between gas and liquidis unknown. mixture with water. The flux of ammoniainto the water is proportionalto the ammonia concentrationat the air-water interfaceminus the ammoniaconcentration in thebulk water. The proportionalityconstant is the masstransfer coefficient. The concentrationdifference betweeninterface and bulk is not constantbut can vary along the height of the column. Which valueof concentrationdifference should we use? In this book, we alwayschoose to usethe local concentrationdifference at a particular positionin the column. Sucha choiceimplies a "local masstransfer coefficient" to distin- guish it from an "averagemass transfer coefficient." Use of a local coefficientmeans that we often must makea few extramathematical calculations. However. the local coefficient is more nearly constant,a smooth function of changesin other processvariables. This definitionwas implicitly usedin Examples8.1-1, 8.1-3, and 8.1-4in theprevious section. It was usedin parallelwith a type of averagecoefficient in Example 8.| -2. Anotherpotential source of ambiguityin the definitionof the masstransfer coefficient is the interfacialarea. As an example,we againconsider the packedtower in Fig. 8.2-1. The surfacearea between water and gasis usuallyexperimentally unknown, so that the flux per areais unknown as well. Thus the masstransfer coefficient cannot be easily found. This problemis dodgedby lumping the areainto themass transfer coefficient and experimentally determiningthe product of the two. We just measurethe flux per column volume. This may seemlike cheating,but it works like a charm. Finally, masstransfer coefficients can be complicatedby diffusion-inducedconvection normalto theinterface. This complicationdoes not existin dilute solution,just asit doesnot exist for the dilute diffusion describedin Chapter2. For concentratedsolutions, there may be a larger convectiveflux normal to the interfacethat disruptsthe concentrationprofiles near the interface. The consequenceof this flux, which is like the concentrateddiffusion problemsin Section3.3, is that the flux will not doublewhen the concentrationdifference is doubled.This diffusion-inducedconvection is responsiblefor the lastdefinition in Table 8.2-2,where the interfacialvelocity is explicitly included. Fortunately,most solutionsare dilute,so we can successfullydefer discussing this problem until Section13.5. I fìnd thesepoints difficult, hard to understandwithout careful thought. To spur this thought,try solving the examplesthat follow. 220 8 / Fundamentalsof Mass Transfer t ) / Other
Sr Example 8.2-1: The mass transfer coefficientin a blood oxygenator Blood oxygena- correspond tors areused to replacethe humanlungs during open-heart surgery. To improveoxygenator design,you arestudying mass transfer of oxygeninto waterin onespecific blood oxygenator. From publishedcorrelations of masstransfer coefficients, you expectthat the masstransfer coefficientbased on the oxygenconcentration difference in the water is 3.3 ' 10-3 centime- ters per second. You want to use this coefficientin an equationgiven by the oxygenator manufacturer
N1:kp(po,-pó,) wherep0, is the actualoxygen partial pressure in the gas,and pfi. is the hypotheticaloxygen For the gas partial pressure(the "oxygen tension")that would be in equilibrium with water under the experimentalconditions. The manufacturerexpressed both pressuresin millimeters of k 02. You also know the Henry's law constantof oxygen in water at your experimental conditions:
Po' : 44,000 atmxo, where xe, is the mole fraction of the total oxygen in the water. Find the masstransfer coefficient ko. Thesecon, Solution Becausethe correlationsare based on the concentrationsin the liquid. the flux equationmust be Example I Nr :3.1 ' l0 I cm/sec(t'e,, - t'6r) poresconl placedtog whereca., andcs, referto concentrationsat the interfaceand the bulk solution,respectively. We can convertthese concentrations to the oxvsen tensionsas follows: -\
( o no.\ By compa: LUì - L^Ur - \iln1 wherec is thetotal concentration in theliquid water, p is theliquid's density, and M is its averagemolecular weight. Because the solution is dilute, Notethat r R'e u ar 1g/cmr I atm t0' : | I po' expenmen 118g/r.,-rot 4A. 104atn760.rn Hcl \ Combiningwith theearlier definition, we see * here .\' .cm I1.67.l0-emoll , Nr : 3.3 l0-'- | - no,) s secL .-: o,'nHg )(n0., mol 12 - : 5.5'10 \Pot, Poz) cm2 secmm Hg
The new coefficientko equals5.5 ' l0-12 in the unitsgiven. Thus
Example 8.2-2: Converting units of ammonia mass transfer coefficients A packed '.'. tower is being used to study ammonia scrubbingwith 25"C water. The mass transfer hich i. : coefficientsreported for this towerare I .18lb-mol NHrftrr-ft2for the liquid and 1.09lb-mol ::lil3rrth3 NHr/hr-ft2-atmfor the gas. What arethese coefficients in centimetersper second? '..,;;ter 8.2 / Other Definitions of Mass Transfer Cofficients 221
Solution FromTable 8.2-2, we seethat the units of theliquid-phase coefficient , .:ena- correspondto k,. Thus - :-l.ltOf i :-iì,ttOf. ,, úr,, t:-^, l:,rll \lèf p :--:lme- / 18lb/mol \ / t. I8 lb-molNHr \ / 30.5cm \ / hr \ : : l.ltof :t______1tt_ lt lt _l \ 62.4 rb/fr' / \ frr-hr / \ fr / \ 3.ó00sec /
:2.9 .10-3cmrsec '\ \ gen Forthe gasphase, we seefrom Table8.2-2that the coefficienthas the units of kr. Thus : rrr-the : . ii:\ Of k: RTkp -:::ntal _t_lt-_|_lt_[298"K)/l.3l4arm-ftr\ /l.09lb-mol\ /30.5cm\ / hr \ \ lb-mol-K/\hr-ft'-atm/\ ft /\3.600sec/ : 3.6 cm/sec
Theseconversions take time and thousht.but arenot difficult. .1uid.
Example 8.2-3:Averaging a masstransfer coefficient Imaginetwo poroussolids whose pores contain different concentrationsof a particulardilute solution. If thesesolids are placedtogether, the flux N1 from one to the otherwill be (seeSection 2.3) .:: elY " N,: JDlntLr,
By comparisonwith Eq. 8.1-2,we seethat the local masstransfer coefficient is
k: tfDlrt U t. its Note that this coefficientis initially infinite. We want to correlateour resultsnot in terms of this local value but in terms of a total experimentaltime /e. This implies an averagecoefficient. [. definedby
Nt : [at'
where N1 is the total solutetransferred per areadivided by 16.How is I relatedto ft? Solution From the problem statement,we seethat
.. '!^t-'[,""N1,tr fi"JD/n nrlat Nl' : :2V DltrtoLct .[,1,"d, rs Thus * : zvTl"n :.,;ked ::n Slèr which is twice the value of k evaluatedat /e. Note that "local" refershere to a particular .-ntol time ratherthan a particularposition. Mass Transfer 8.3/ Correi, 222 8 / Fundamentalsof
againthe packedbed of and Example 8.2-42Logmean masstransfer coefficients Consider 8.I -2' Masstransfer benzoicacid spheres shown in Fig. 8.1-1(b) that was basic to Example a log meandriving force: À,i.. coeffìcientsin a bed like this are sometimesreported in tetms of The coeffìci Many ari N1 usedrnostl\ closelyrelal criticsasser by thetotal surface Why bother For this specificcase, N1 is the total benzoicacid leavingthe beddivided concentrationat the This ar-eì" areain the bed. The bed is fed with pure water,and the benzoicacid approxlmall spheresurfaces is at saturation;that is, it equalsc1(sat)' Thus ily accomplr - - - cr(out)] . lcr(sat) 0] [c1(sat) thisneed. \ /V1:/<'ut@
\t1(sat)-cr(out)/ Showhowklu*isrelatedtothelocalcoefficientftusedintheearlierproblem. bed,we showed 8.3 Soluiion By integratinga massbalance on a differentiallength of in Example8.1-2 that for a bedof lengthL, Inl cientsand h cr(out) :1_ n-(kaluo)L a method ftr cr (sat) convenlent Reananging,we fìnd Experiment Howerer c1(sat)- r,(o9 _ o_(ka lu\r do not uant - , (t"t) 0 This avoida Takingthe logarithmof both sidesand rearranging' coefficient: someoneel, kaL , I (r(sat)-0 \ ln|-----__-_.I (sat) - (out) 8'-j \ c1 c 1 / Multiplyingboth sidesby c1(out)' In sible. The. - - dimensionìe r(sat)- 0l [cr(sat) cr(oul : engineersu, cr(out)uo kaL / r'r(sat) - 0 \ lnl-'- l so scientiîìù \ r'r(sat) - cl(out) / Thechar correlationi By definition, transfercoe cr(out)uuA of diffusion Nr:_-' a(AL) Damkohler usedfor dif the where A is A ke1'p, physicalsy, will be the dissolvings of thirty pe 8.3 / Correlations of Mass Transfer Cofficients /-z-)
-:.ì : Of ano .: -::l\ fCf ftros: ft
The coefficientsare identical. Many arguethat the log mean masstransfer coefficient is superiorto the local value usedmostly in this book. Their reasonsare that the coefficientsare the sameor (at worst) closely relatedand that ft1o*is macroscopicand henceeasier to measure.After all, these critics assert,you implicitly repeatthis derivationevery time you make a massbalance. :Ì ece Why bother?Why not useft1o* and be done with it? :t the This argumenthas merit, but it makesme uneasy.I fìnd that I needto think throughthe approximationsof masstransfercoefficients every time I usethem and that this review is eas- iÌy accomplishedby making a massbalance and integrating.I fìnd that most studentsshare this need.My adviceis to avoid log meancoefficients until your calculationsare routine.
',red 8.3 Correlations of Mass Tlansfer Coefficients
In the previoustwo sectionswe havepresented definitions of masstransfer coeffi- cientsand have shown how thesecoefficients can be found from experiment.Thus we have a methodfor analyzingthe resultsof masstransfèr experiments. This methodcan be more convenientthan diffusion when the experimentsinvolve mass transferacross interfaces. Experimentsof this sort includeliquid-liquid extraction,gas absorption, and distillation. However,we often want to predicthow one of thesecomplex situations will behave.We do not want to correlateexperiments; we wanf to avoid experimentsas much as possible. This avoidanceis like that in our studiesof diffusion, where we often looked up diffusion coefficientsso that we could calculatea flux or a concentrationprofile. We wantedto use someoneelse's measurements rather than painfullv makeour own.
8.3.I Dimensionless Numbers
In the same way, we want to look up masstransfer coefficients whenever pos- sible. Thesecoefficients are rarely reportedas individual values,but as correlationsof dimensionlessnumbers. These numbers are often named,and they aremajor weaponsthat engineersuse to confusescientists. These weapons are effective because the namessound so scientific,like closerelatives of nineteenth-centuryorganic chemists. The characteristicsof thecommon dimensionless groups frequently used in masstransfer conelationsare given in Table 8.3-1. Sherwoodand Stantonnumbers involve the mass transfercoefficient itself. The Schmidt,Lewis, andPrandtl numbers involve different kinds of diffusion, and the Reynolds,Grashof, and Pecletnumbers describe flow. The second Damkohler number, which certainly is the most imposing name, is one of many groups usedfor diffusion with chemicalreaction. A key point about each of thesegroups is that its exact defìnition implies a specific physical system. For example,the characteristiclength / in the Sherwoodnumber kllD will be the membranethickness for membranetransport, but the spherediameter for a dissolvingsphere. A good analogyis the dimensionlessgroup "efficiency." An efficiency of thirty percent has very different implications for a turbine and for a running deer. In 11A 8 / Fundamentalsof Mass Transfer 8.3/ Corr
Table 8.3-1. Significanr:eof r:ommondimensionLess groups Some i The accur Group' Physicalmeaning Used in elTors are KI masstransfèr velocity &Iì) SOft tr Sherwoodnumber - Usual dependentvariable D diffusion velocity physicalp k masstransfèr velocity forced con Stantonnumber ^ Occasional dependentvariable I t)u flow velocity fbr gase, be reliable diffusivity of momentum Schmidtnumbe. I Correlationsofgas or liquid data plants.the. D diffusivity of mass checkson a diffusivity of energy Lewis number Simultaneousheat and masstransfer Manr r D diffusivity of mass involvea S diffusivity of momentum Heat transfer;included here for Prandtl number.I interest.Tl a diffusivity of energy compìeteness The \ ariali luo inertial forces diffèrenrpi Reynoldsnumber - -;-- ^ or Forcedconvection v vlscouslorces Frrre\anlp- flow velocity r. pumpeJ: "momentum velocity" :hroughri,. lr trther,. buoyancy forces Grashófnumb rr'1#1t Freeconvection -,:Llre\J-:: viscousforces : intem"._ uol flow velocity :,:lutinl.. Péclet number Correlations of gas or liquid data I diffusion velocity The i :: reactionvelocity Correlationsinvolving reactions -.. SecondDamkóhler number .. diffusion velocity (seeChapters 15-16) -. :::;:. ^ rc12 or (Thielemodulus)'- I -... D 'r ,, Note: u The symbolsand their dimensionsare as fbllows: - I: D diffusioncoefficient (L2lt) g accelerationdue to gravity(L/rr) k masstransfer coefficient (l, /r) / characteristiclength (L) u" fluid velociry (Llt) o thermaldiffusivity (L2ll) r first-orderreaction rate constant (t r) u kinematicviscosity (L2/r) Ap/p fractionaldensity change the same way, a Sherwood number of 2 means different things for a membrane and for a dissoh'ing sphere. This flexibility is central to the correlations that follow.
8.3.2 Frequently Used Correlations
Correlationsof masstransfer coefficients are convenientlydivided into thosefor fluid-fluid interfacesand thosefor fluid-solid interlaces.The correlationsfor fluid-fluid interfacesare by far the more important,for they arebasic to gasadsorption, liquid-liquid extraction.and nonidealdistillation. Correlationsof thesemass transfer coefficients are also importantfor aerationand water cooling. Thesecorrelations usually haveno known parallelcorrelations in heattransfer, where fluid-fluid interfacesare not common. ..,..\ler 8.3 / Correlations of Mass Transfer Coefficients 225
Someof the more usefulcorrelations for fluid-fluid interfacesare given in Table8.3-2. The accuracyof thesecorrelations is typically of the order of thirty percent,but larger elTorsare not uncommon. Raw datacan look more like the result of a shotgunblast than any sort of coherentexperiment because the data include wide rangesof chemical and physical properties. For example,the Reynoldsnumber, that characteristicparameter of forced convection,can vary 10,000times. The Schmidtnumber, the ratio (vlD), is about I for gasesbut about 1000for liquids. Over a more moderaterange, experimental data can be reliable' Still, while the correlationsare usefulfor the preliminarydesign of small pilot l:ir plants,they shouldnot be usedfor the designof full-scaleequipment without experimental checkson the specificchemical systems involved. Many :: i ll \tef of the correlationsin Table 8.3-2 have the samegeneral form. They typically involve a Sherwoodnumber, which containsthe masstransfer coefficient, the quantity of interest.This Sherwoodnumber varies with Schmidtnumber, a characteristicof diffusion. The variationof Sherwoodnumber with flow is more complex becausethe flow has two differentphysical origins. In mostcases, the flow is causedby extemalstirring or pumping. For example,the liquids usedin extractionare rapidly stirred;the gas in ammoniascrubbing is pumpedthrough the packed tower; the blood in the artificialkidney is pumpedby theheart throughthe dialysisunit. This type ofexternally driven flow is called "forced convection." In other cases,the fluid velocity is a result of the masstransfer itself. The masstransfer causesdensity gradients in the surroundingsolution; thesein turn causeflow. This type of internally generatedflow is called "free convection." For example,the dispersalof pollutantsand the dissolutionofdrugs areoften acceleratedby free convection. The dimensionlessform of the correlationsfor fluid-fluid interfacesmay disguisethe very real quantitativesimilarities between them. To explorethese similarities, we consider the variationsof the masstransfer coeffìcient with fluid velocity and with diffusion coeffì- cient. Thesevariations are surprisinglyuniform. The masstransfer coefficient varies with the 0.7 powerof the fluid velocity in four of the five correlationsfor packedtowers in Thble 8'3-2. It varieswith the diffusion coefficientto the 0.5 to 0.7 power in every one of the correlations.Thus any theorythat we derivefor masstransfer across fluid-fluid interfaces shouldimpÌy variationswith velocity and diffusion coeffìcientlike thoseshown here. Somefrequently quoted correlations for fluid-solid interfacesare given in Table 8.3-3. Thesecorrelations are rarely important in common separationprocesses like absorption and extractions.They are importantin leaching,in membraneseparations, and in electro- chemistry.However, the realreason that thesecorrelations are quoted in undergraduateand graduatecourses is that they are closeanalogues to heat transfer.Heat transferis an older subject,with a strongtheoretical basis and more familiar nuances.This analogygets lazy lecturersmerely mumble, "Mass transferis just like heattransfer" and quickly comparethe correlationsin Thble8.3-2 with the heattransfer parallels. The correlationsfor solid-fluid interfacesin Table8.3-3 are much like their heattransfer equivalents.More significantly,these less important, fluid-solid correlationsare analogous but more accuratethan the importantfluid-fluid correlationsin Table8.3-2. Accuraciesfor solid-fluid interfacesare typically averageIl07o; for somecorrelations like laminarflow in ..-.ì\e for a singletube, accuracies can be *l%o. Suchprecision, which is truly rare for masstransfèr - :-fluid measurements,reflects the simplergeometry and stableflows in thesecases. Laminar flow :-liquid of one fluid in a tube is much betterunderstood than turbulentflow of gas and liquid in a a--,tr Of€ packedtower. r ÌlOWfl The correlationsfor fluid-solid interfacesoften showmathematical forms like thosefor fluid-fluid interfaces. The masstransfer coefficient is most often written as a Sherwood oo
la E
i-!ar=..i à! .=y L ì - ", ==_-É!3Éi=-+ 6-v .2 e = E i = E Í FX r XF ÈEE: ir È?. É : í r ; E oà Zi, Í€ Èp i t I É = i ,OF E;î4Éa€ e e E ? z: FZ Èiagaa È E : € É È € t: liTZÉi', É E É È s € 7il.î=É-,;É;; a ; z^o\ &ÈòÈ{zùÀÈÈEÉEEE=€ ;^ E V È Z i () 'tt; h i* .= àoh \ *i. € € ; e .c \ :, a .. 9, , I g 5E;è' à s g; ì t g; ì B- Y YE Eg L v? ! 3 Ei ia Es uPpuP; *e oo =u) I! É E íT Èl Eí EÉÉEÉEH3 li OO i=- t I i: =': És r=':€31É€ iA lÈ è = lÈ E; Q6 !E E = .t,E .;a += +a:+aÈ .qí L ;e -: 5 1F az Zi =Íi=i= Ft aZ c èC L ll ll ll tt ll ll ri rì i" ;[ i^rr il^il \ a \q \< \< \-È ,?-è '':. =
o c:i s \ 2 >,- ttr ú,9
Q.) " 9c Òì ts >6(, cl .Í3Èo oo :\- !4-E O ..'!'o,o
>d>= < tr\9 :iì: 7 E\ = :.jli;c, tsoi6
't à =-, ='- -e \ k L:'->9+ ? € 6 X\ u r'6{ t:Ò-- 'azò0d - 3lk - E iz -ó I h,h E .= .=- 'h r= l.= È-.-! h.= =Ò îO >..y . tr € o o= !ù Ya o50)E oF c É +7= t ==+= - c ='t ! z! _ói =.: =.: c F S'= Y ='r, = ? 6h ÉE e.r EZ .ZE :, .: i: .s; òoE ú7. ,.3 -i ;i .E Y ' ! - p ? ú.= 7 = = !UtrF = I e^ = J O È 7tsà 4.9
--l !.0 u n: ,E; E È ; È EE .F É o\ :o ú g 9Òc E : =C E i ;È t.èÉ : ta- qY -é E Z :5 E a É 2 sÈ ;î E:i c\ l* ó i :} ts E É EE È '"Y iÉ :í ql ;;-= - ,=z;- i Èi ? ,i,ii==tEiÉ :: i; = ic = É =*E É= t,, É 9- -= ;í E í€ E ,+ =;e :,2 ít otr E !É,+=rEt1 Ég€zÉ+=,Éí'.; É EIE .2ó E i !\ Ò , w Er.E= E;- : ;?E I:; l? i; oA;ttze,tAiS'o È F'- È!
-: .- -;g:E1'.=:Y.;i 'a c Eà' '= 6àà J. 3 .c .E _c rg E! ,3 = E r= i2- i tE ;€úo",? a i I Z= É; Eà o_ c > 7 - h U: E^É!: 1É ÈF Éé EF O I É:"e ÉE :;.: v u q d E te %Z 3".i.El% <É aà !à trx> 9i: 7. 1 ;: rxo :u1 È s+ 9> ;È, rQ a îd, x= .:: à#3 É+ ? .2.2 -'Ya ftdN f?d.= Aàfi ú. > O> à ! à : ;! * tào o.;; .o! *r lù 'rl irT rrrr r l Lr ll b:€ a. i ;r f,-r it' ítt eFìo-É :t ! è -_: -\ \*t è \ è \ e \ - 6 Ur€ b
o -O -lA "\ É - cPJoo lo .9a= E à l,^\- -- ^t^ !l- _tA Éa:aí) z . \_/ -l\ E€ = q P FI v tr-\È i: Oà'^,^O ù
: -,Y: €- ! a.= L! U rlw\u =--^ =OF=- - o É ,z a-É 2 =-";99 ú u r 13 qaÈ: 6 z;f '{ h I { = = + Fo 3 = í: 7 = = ? - À -= : d ! :'-ù : == = = l-- l= .=,J -a Fetpt o ri > 9 - a F à i e u ó îP'd,E = =.=7y - E b e7 c= Z= É= É; E.Y F C.; : ,t È,=É =i =? == ;+ î. 7=* EE#. Z '-a-É- _ ?;i,h- *X é== = =-, Ò2 : !-o EÉ EÉ E; ;1 aÉ := ,= : !- >.= t I ! -E E; ,=ia= à; ài Zî p2 i i =- C^ =iÉ-iùù,-,.o.q È z5i: - a'= X
- rch < -.F a 228 8 / Fundamenîalsof Mass Transfer E.-ì/ Cttrr, number,though occasionallyas a Stantonnumber. The effect of diffusion coefficientis most often expressedas a Schmidtnumber. The effect of flow is most often expressedas a Reynoldsnumber for forcedconvection, and as a Grashófnumber for free convection. : Thesefluid-solid dimensionlesscorrelations can concealhow the masstransfer coef- ficient varies with fluid flow u and diffusion coeffìcient D, just as those for fluid-fluid interfacesobscured these variations. Basically,ft often varieswith the squareroot of u. The variation is lower for some laminar flows and higher for turbulent flows. It usualÌy varieswith D2l3,though this variationis rarely checkedcarefully by thosewho developthe correlations.Variation of k with D2l3 doeshave sometheoretical basis, a point explored further in Chapter13.
Example 8.3-1: Dissolution rate of a spinning disc A soÌid disc of benzoic acid 2.5 centimetersin diameter is spinning at 20 rpm and 25'C. How fast will it dissolve in a largevolume of water? How fast will it dissolvein a largevolume of air? The diffusion coefficientsare 1.00. l0-s cm2/secin waterand 0.233cm2Tsec in air. The solubilityof benzoicacid in wateris 0.003g/cmr: its equilibriumvapor pressure is 0.30mm Hg. Solution Before startingthis problem, try to guessthe answer. Will the mass transferbe higherin water or in air? In eachcase, the dissolutionrate is
Nr : ftcr(sat)
wherec1(sat) is the concentrationat equilibrium. We can find k from Table8.3-2:
ttutl 2 r U rl I k:0.62D1-l f- ì \r,,/ \D/
For water.the masstransfer coefficient is ! r:mplr t12 2 20160)(2r 0.01cm lsec --' :0.62(1.00. 5 lsec l0 cm27sec, r) (, I al (10'01.tr'/r* .00'l0* sec)"'
: 0.90. l0-3 cm/sec
Thus the flux is
N1 : (0.90. l0 I cmlsec)(0.003g/cm3)
:2.7 .10 6 g/cm2-sec
For air, the valuesare very different:
t,t 20160)(2r/sec 0.l5cm2/sec :0.62(0.233cm2lsec) r) ( ; (10.l-5.t"? sec 02-)-) ;"1/*. )"'
: 0.11cm/sec 8.3 / Correlations of Mass Transfer Coffit:ients 229
' - :3Ì]tts Almoslpure ..r'J asa oir : :1. î Pure woler l:: rO€f- - i-fluid .:0fu. : ..ìtliìlly . .. .rpthe :\:lofed
Air & woter- solublevopor
| ì. acid Gos dissolved in wofer - r..olve lt:lusion Fig. 8.3- l. Gas scrubbing in a wetted-wall column. A water-solublegas is being dissolve6 in a falìing fi lm --:Lit\ of of water. The problem is to calculate the length of the c'olumn necessaryro reach a liquid concentrationequal to ten percenl saturatlon.
r' ll ldòù
which is much largerthan before. However,the flux is
mmHs Imot N1: (0.47cmlsec)f (o': \ ( \ (213\ /122s\l L\760mmHe/ \22.4.t}t.r,/ \2e8l \ not )] :0.9. l0-óg/cm2sec
The flux in air is aboutone-third of that in water,even though the masstransf-er coefficient in air is about500 times largerthan that in water. Did you suessthis?
Example 8.3-2: Gas scrubbing with a wetted-wall column Air containing a warer- soluble vapor is flowing up and water is flowing down in the experimentalcolumn shown in Fig' '.- 8.3-l ' The water flow in the 0.07-centimeter-thickfilm is 3 centimetersper second, )'" thecolumn diameteris l0 centimeters,and the air is essentiallywell mixedright up to the interface' The diffusion coefficientin water of the absorbedvapor is 1.g . 10"5 cà27sec. How long a column is neededto reach a gas concentrationin water that is ten percentof saturation? Solution The first step is to write a massbalance on the water in a differential columnheight A: :
(accumulation): (flow in minus flow out) * (absorption) g:[ndluucfl, _frdlu0ci ]:+a:* tdLzklct (sat)_c1l in which d is the column diameter,/ is the film thickness,u is the flow, and c1 is the vapor concentrationin the water. This balanceleads to
,1... 0 : -/r0",' -.1klcr(saU -r.,l dz 230 8 / Fundamentalsof Mass Transfer \.-t / C,,rr.
From Table 8.3-2, we have rrhere r/ rr flt-r* . Ctrn:i I Prrtl't12 k:0.69( _ I \z/ We also know that the enteringwater is pure; that is, when
z:0, cl :0
Combining theseresults and integrating,we find (r -_ | -(--l.3g(Dz/I2uo,tt/2 c1(sat)
Insertingthe numericalvalues given,
/ l2u \l- / c, \.l2 __t_|lnf l_ tl \1.90DlL \ .,(sat)/l
/ 0.01 cm)2(3-='''.^"r"',_-.'cm/sec) \ : I'"'"' I ltntl_0.t)12 \ t t .oorl.8 . lo-5 cm2lsec/ " : 4.8cm
This approximatecalculation has been improved elaborately, even though its practicalvalue is small.
Example 8.3-3: Measuring stomach flow Imaginewe want to estimatethe averageflow in the stomachby measuringthe dissolutionrate of a nonabsorbingsolute present as a large sphericalpill. From in vitro experiments,we know that this pill's dissolutionis accurately describedwith a masstransfer coefficient. How can we do this? Solution We first calculatethe concentrationc1 of the dissolving solutein the stomachand then show how this is relatedto the flow. From a massbalance. dc, V --; : rd'k[c1 (sat)- c1] clt where V is the stomach'svolume, nd2 is the pill's area,k is the masstransfer coefficient, andc1(sat) is thesolute's solubility. Because no soluteis initiallypresent,
cr :0 when / :0
Inte-rrating,
,V/r'1(sat)\ A:-lnl-l - ndtt \c'1(sat) r'1/
If rl e assume that stomach flow is essentially forced convection, we find, from Table 8.3-3,
kl /du\t"rr.,',, -:2+0.6{-l l-ì D \u/ \D/ !,,itts.fer 8.3 / Correlations of Mass Transfer Cofficients zJl
where d is the pill diameter,D is the diffusion coefficient,and u is the unknown stomach flow. Combiningand rearranging. t': 25/vt:p:'t\f V / cr(sar)\ -'l12 e( a / L"r*'"('.n".l-;) which is the desiredresult. Note the assumptionsin this problem: The flow is dueto forced convection,the pill diameteris constant,and flow in and out of the stomachis negligible.
Example 8.3-4: Gtucoseuptake by red blood cells The uptakeof glucoseacross the red blood cell membranehas a maximum rateranging from 0. I to 5 p*ov.rr-hr. Apparently, these differencesresult from differencesin experimentalconditions. Using the correlation for liquid dropsin Table8.3-2, estimate the effectof masstransfer in thebulk to seewhen it could haveaffected these uptake rates. To make the estimationmore quantitative,assume that a typicalexperiment is madein a beakercontaining 100 cm3 of redbl,ood cells suspended in 1 liter of plasma.The beakeris stirredwith a l/50-hp motor. The cellsoriginally contain little glucose. At time zero,radioactively tagged glucose is addedand its uptakemeasured. The diffusion coefficientof glucoseis about6 . 10-6 cm21sec, and the plasmaviscosity is approximatelythat of water. Solution We are interestedin the casein which glucoseuptake is dominated by mass transfer. In this case, glucosewill diffuse to the membraneand then almost -.,1value instantaneouslybe takeninto the cell. Thus
Nt : kct
where cl is ::ie flOW the bulk concentration.If we can calculatek, then we can estimateN1, the -:. ,1large desiredquantity. we seefrom Tàble8.3-2 that for a suspensionof liquid drops, ,,urately
ft : 0.13 p-t/1v-s/t2 D213 -:r' in the (i)"' tlsohp . t/a :0.r3( 1.45 l}e g-cm2\ 1 1* y-tn \ l. 000cmr hp-secr / \.r, /
-5lr2 , t,2/3 3:nClent, /0.0t cm2\ | . ,^-r-cm'\ to.lu \*./ \ sec/ 3 : 5.7. l0 cm/sec: 2I cmlhr
The flux is
Nt : (21 cmlhr)c1
Whetheror not diffusionoutside the cells is significantdepends -.!1_1 on c1,the amount of glucose used. The flux equals5 pmol/cmz-hrwhen c1 is about 0.3 mmol/liter. If c1 far exceeds 0.3 mmol/liter, then the flux due to diffusion will be much fasterthan that due to the cell membrane.The measurementswill then truly representmembrane properties. However, if the glucose concentrationis less than 0.3 mmol/liter, then the measurementswill be z)z 8 / Fundamentalsof Mass Transfer 8.1/ Drn;
functions both of the membraneand of mass transferin plasma. Such restrictionscan compromisemeasurements in biological svstems.
8.4 DimensionalAnalysis: The routeto Correlations
The correlationsin the previous sectionprovide a useful and compact way of presentingexperimental information. Use of thesecorrelations quickly gives reasonable estimatesof masstransfer coefficients. However, when we find the correlationsinadequate, we will be forced to make our own experimentsand developour own correlations.How can we do this? The basicform of masstranstèr correlations is easily developedusing a methodcalled dimensionalanalysis (Bridgeman, 1922;Becker, l9j6). This methodis easily learnedvia the two specifìcexamples that follow. Before embarkingon this description,I want to emphasizethat most people go through three mental statesconcerning this method. At fìrst they believeit is a route to all knowledge,a simple techniqueby which any set of experimentaldata can be greatly simplified. Next they becomedisillusioned when they havedifficulties in the useof the technique.These difficulties commonly result from efforts to be too complete.Finally, they learnto usethe methodwith skill and caution,benefiting both from their pastsuccesses and from their frequentfailures. I mentionthese three stages '!È. j \. Jl: becauseI am afraidmany may give up at thesecond stage and miss the real benefits involved. We now turn to the examples.
8.4.1 Aeration
Aerationis a commonindustrial process and yet onein which thereis often serious disagreementabout correlations.This is especiallytrue for deep-bedfermentors and for sewagetreatment, where the rising bubblescan be the chief meansof stirring. We want to study this processusing the equipmentshown schematicallyin Fig. g.4-I . we plan to inject oxygeninto a varietyof aqueoussolutions and measurethe oxygen concentrationin the bulk with oxygen selectiveelectrodes. We expectto vary the averagebubble velocity u, the solution'sdensity p and viscosityÉr, the enteringbubble diameter d, and the depth of thehed L. We measurethe steady-stateoxygen concentrationas a function of position in the bed. Thesedata can be summarizedas a masstransfer coefficient in the following wav. From a massbalance. we seethat
- -ùEdc, 0 : r kulcl(sat) - r'1| (8.4-l) u'herezz is the total bubblearea per column volume. This equation,a closeparallel to the nlanyrîass balances in Section8.1, is subiectto the initialcondition : :0, cr:0 (8.4-2) Thus
u/ c1(sat) t.., _ -ln{ (8.4-3) .:\-: \ cr (sat)- cr (z) 1,.;rt.sler 8.4 / DimensionalAnalysis: The route to Correlations 233
, :l\ Can
: ..Ar Of :.' rn&ble : J r't.|U&t9,
.l iìlled r:iad via . .\ Jnt to Fig. 8.4-I . An experimentalapparatus for the studyof aeration.Oxygen bubbles from the :-.,d. At spargerat the bottom of the tower partially dissolvein the aqueoussolution. The concentration .: ' .et Of in this solutionis measuredwith electrodesthat are specific for dissolvedoxygen. The concentrationsfound in this way tlle interpretedin terrnsofmass transfercoefîcients; this .:3n they interpretationassumes that the solutionis well mixed,except very nearthe bubble walls. ::r 3lforts -.--:rrting'.',.'...È Ideally, we would like to measure ft and a independently, separating the effects of mass 1':' .tà89S transfer and geometry. This would be difficult here, so we report only the product ftd. -r rrlved. Our experimental results now consist of the following:
ka : ka(u,O, p, d, z') (8.4-4) We assumethat this function hasthe form
: :':ì .ÈflOUS /s4 [constant)uqpfl Ut 4t re (8.4-5) -. for .,nd whereboth the constantin the squarebrackets and the exponentsare dimensionless.Now \\ l rvant the dimensionsor units on the left-handside of this equationmust equalthe dimensionsor ,\: planto units on the right-handside. We cannothave centimeters per secondon the left-handside -,:.1tionin equaÌto gramson the right. Becauseka has dimensionsof the reciprocalof time (1 lt), u : .r'loctty hasdimensions of length/ttme(L lt), p hasdimensions of massper lengthcubed (M I Lr, : .:ì.'depth and so forth, we fìnd
- 1 /L\" /M\fl :hebed. -r:r (8.4-6) .. Froma (;i (z.j (#)'(Ltò(Lt' The only way this equationcan be dimensionallyconsistent is if the exponenton time on the left-handside ofthe equationequals the sumofthe exponentson time on theright-hand side: S.,1-1) -l:-d-y (8.4-1) .- :l to the Similar equationshold for the mass: 0:f_ty (8.4-8) r8.;1-2) andfor thelength: 0:a-3fl*y*6*e (8.4-e) s.-r-3) Equations8.4-1 to 8.4-9give three equations for thefive unknown exponents. l-'i+ 8 / Fundamentalsof Mass Transfer 8.1 / Dittttr
We can solve these equationsin terms of the two key exponentsand thus simplify Eq. 8.4-5. We choosethe two key exponentsarbitrarily. For example,if we choosethe exponenton the viscosity y andthat on column heighte , we obtain a:l-y (8.4-r0) F:-y (8.4-ll) (8.4-12) A-_", -e-1 (8.4-13) (8.4-14)
Insertingthese results into Eq. 8.4-5 and rearranging,we find
/ kad\ l;/ : [constant](+)'(;)' (8.4-15) The left-handside of this equationis a type of Stantonnumber. The first term in parentheses on the right-handside is the Reynoldsnumber, and the secondsuch term is a measureof the tank's depth. This analysissuggests how we should plan our experiments. We expect to plot our measurementsof Stantonnumber versus two independentvariables: Reynolds number and zld. We want to cover the widest possiblerange of independentvariables. Our resulting correlationwill be a convenientand compactway of presentingour results,and everyone will live happily ever after. Unfortunately,it is not alwaysthat simple for a varietyof reasons.First, we hadto assume that the bulk liquid was well mixed, andit may not be. If it is not, we shallbe averagingour valuesin someunknown fashion, and we may find that our correlationextrapolates unreli- ably.Second,wemayfindthatourdatadonotfitanexponential formlikeEq.8.4-5. Thiscan happenif theoxygen transferred is consumedin somesort of chemicalreaction, which is true in aeration.Third, we do not know which independentvariables are important. We might suspectthat ka varieswith tank diameter,or spargershape, or surfacetension, or thephases of the moon. Suchvariations can be includedin our analysis,but they make it complex. Still, this strategyhas produced a simplemethod of correlatingourresults. The foregoing objectionsare important only if theyare shown to be soby experiment.Until then,we should usethis easystrategy.
8.4.2 The Artificial Kidney
The secondexample to be discussedin this sectionis the masstn.,rsfer out of the tube shownschematically in Fig. 8.4-2. Suchtubes are basic to the artificialkidney. There, blood flowing in a tubularmembrane is dialyzedagainst well-stirred saline solution. Toxins in the blood diffuse acrossthe membraneinto the saline,thus purifying the blood. This dialysisis often slow; it cantake more than 40 hoursper week. Increasingthe masstransfer in this systemwould greatlyimprove its clinicalvalue. The first stepin increasingthis rateis to stir the surroundingsaline rapidly. This mixing increasesthe rateof masstransfer on the salineside of the membrane,so that only a small part of the concentrationdifference is there,as shown in Fig. 8.4-2. In otherwords, we have decreasedthe resistanceto masstransfer on the salineside. The secondstep in increasing 1,.;,t.s.ler 8.4 / DimensionalAnalysis: The route to Correlations 235
:nplify Blood fiow ,.. the
. -1-l0) . l-l l) r J,l?l r +-13) .1-1rl)
. _l_ l5) Fig. 8.4-2. Mass transf'erin an artificial kidney. Arterial blood flows through a dialysis tube that is immersed in saline. Toxins in the blood diffuse acrossthe tube wall and into the saline. If the ':'--lheses sallne is well stirred and if the tube wall is thin, then the rate of toxin removal dependson the concentrationgradient in the blood. Experiments'in this situation are easily correlated using ::'tlf€ Of dimensionalunalysis.
: ot our the rateis to makethe membraneas thin aspossible. Although too thin a membranewould --r.r and rupture,existing membranesare alreadyso thin that the membranethickness has only a :i.ulting minor effect. The resultis that the concentrationdifference across the membraneis not the :-. afvone largestpart of the overallconcentration difference, again as shownin Fig. 8.4-2. The rate of toxin removalnow dependsonly on what happensin the blood. We want to ,rt:UlÎe correlateour measurementsof toxin removalas a function of blood flow. tube size.and so :ìiO! OUf forth. To do this, we find the flux for eachcase: :-.unleli- amounttransferred Thiscan flux N1 - (8.4-r6) .-:,1\tfUe (area)(time) '*: night By definition, .-:'phases :ì.llex. Nt:k(ct-cn)
: :egoing - ^L I (8.4-17) .,.;'.hould Becausewe know N1 and c1,we carìfind the masstransfer coefficient ft. As befbre, we recognizethat the masstransfer coefficient of a particulartoxin varies with the system'sproperties:
k : k(u,p, I,r,D, d) (8.4-18) -:i of the where u, p, and p are the velocity,density, and viscosityof the blood, D is the diffusion There, :. coeffìcientof the toxin in blood. and d is the diameterof the tube. We assumethat this - Toxins relationhas the form .r This - -. :riìnsfer ft [constant)u"pfr uv PdO' (8.4-l9) wherethe constantis dimensionless.The dimensionsor units on the left-handside of this : . nllxlng equationmust equalthe dimensionsor units on the right-handside; so ' .L\mall , '.J have (8.4-20) " -:rrsing i,'(i)"(#)'(X)'(+)'" 236 8 / Fundamentalsof Mass Transfer 8.5/ Mass T',
This equationwill be dimensionallyconsistent only if the exponenton the length on the The reascr left-handside ofthe equationequals the sum ofthe exponentson the right-handside: beginning.in the densitr 1. l:a-3F-V*23+e (R 4-) l\ kd Similar equationsmust hold for mass: D 0:f*y (8.4-22) which is the and fbr time: haveno rea: and make int -l : _o _ y - 3 (9,L_)\\ sharpens.
We solvethese equations in termsof the exponentsa and 6: 8.5 A:A (8.4-24) B:ql3-l (8.4-25) Y:7*a-3 (8.4-26) 6:ó (8.4-21) €:a-l (8.4-28)
We combinethese results with Eq. 8.4-19and collect tems: ó kdo /duo\"/ u \ lconstantì{ } lr^l (8.4-2e) tt \u/ \pD/
Byconvention,wemultiplybothsidesofthisequationbythedimensionlessquarftityttlpD to obtain
kct /ctuo\"/ u \'o r:n- : lconstant](' (8.4-30) , s- ) \A ) This equationis the desiredcorrelation. As in the first example,a key stepin the analysis is the arbitrary choice of the two exponentscv and 6. Any other pair of exponentscould have been chosenand would have given a completely equivalentcorrelation. However, the particular manipulationshere are made so that the dimensionlessgroups found are consistentwith traditionalpatterns. Such traditional patterns sometimes reflect experience and sometimesmerely mirror convention.Multiplying both sidesof Eq. 8.4-29plpD involvesthese factors. The trouble with this analysisis that it is not the whole story. From experimentsat low flow, we would f,nd that the masstransfer coeffìcient k doesvary with the cuberoot of the velocity u and with the two thirds power of the diffusion coefficientD. This suggeststhat Eo. 8.4-30can be rewritten
kd /,1u\r'l -:lconstantl-\D/ l-l (8.4-31) D thatis. thata: (l - ó) : l13. However,if wethenmeasuredftas afunctionof thetube diameterd.wewouldfinditproportionaltod-t/3,notd-2/3 as suggestedbyEq.8.4-3 1. Afier a good start,our dimensionalanalysis is failing. ;t1.\rer 8.5 / Mass Transfer AcrossIntetfaces 231
ìn the The reasonfbr this failure is that we did not chooseall the relevantvariables at the very beginning,in Eq.8.4- l8. We shouldhave included the tube length L; we couldhave omitted the densityp and viscositytr. Had we done so, we would haveobtained: r_ll) kd ( r12u\ttl Iconstant] (8.,1-32) D \DLl
'l \ J- which is the result quotedin Table 8.3-3. However,from dimensionalanalysis alone, w'e have no reasonto be critical of our original result in Eq. 8.4-30. We can only be critical and make improvementsas our experimentalexperience grows and as our physicalinsight ' _r-t3) sharpens.
8.5 Mass T[ansfer Across Interfaces r J- )-ll In the previous sections,we used mass transfercoeffìcients as an easy way of \ J-l5l describingdiffusion occurringfrom an interfaceinto a relatively homogeneoussolution. . J_r6l Thesecoefficients involved many approximationsand sparkedthe explosionof defìnitions ' -+-17) exemplifiedby Table8.2-2. Still, they area very easyway to correlateexperimental results or to makeestimates using the publishedrelations summarized in Tables8.3-2 and 8.3-3. ) +-28) In this section.we extend thesedefinitions to transferacross an interfàce.from one well-mixed bulk phaseinto anotherdifferent one. This caseoccurs much more frequently than doestransfèr from an interfaceinto onebulk phase;indeed, I had troubledreaming up examplesearlier in this chapter.Transfer across an interfaceagain sparks potentially major . l_29) problemsof unit conversion,but theseproblems are often simplifìedin specialcases.
'.-.; ,,-' pD 8.5.1 The Basic Flux Equation
Presumably,we candescribe mass transfer across an interf'acein termsof the same \ +-30) type of flux equationas before: (8.s- - .rrìalysis NI : KAcr 1) ' j'-:. could ..,,here "overall masstransfer '-i Nr is the soluteflux relativeto the interface,K is called an ..\ -r\eî qrr - : -: ":.-iJ Jcl is some appropriateconcentrafton difference. Rut what is Act ? ..-: CTr, - -:nxppropriate value of Ac1 tums out to be difficult. To illustratethis, consider " LÙr',-' ,.:ijirrlsshown in Fig. 8.5-1. In the f,rstterm, hot benzeneis placedon top of -- fl r r : - i-,' benzenecools and the water warrnsuntil they reachthe sametemperature. itf '-,,:i.:irre is the criterion for equilibrium, and the amount of energytransferred - ìtl lD rr- - i to the temperaturedifference between the liquids. Everything -.'L)i ;rroportional 'r Dl llli rcCIllS SOCUf9. -' .'.'\ (.t'r/1Ar As a secondexample, shown in Fig.8.5- 1 (b), imagine that a benzenesolution of bromine is placedon top of water containingthe sameconcentration of bromine. After a while, we find that the initially equalconcentrations have changed, that the bromineconcentration in \ -1-31) the benzeneis much higherthan that in water.This is becausethe bromineis more soluble in benzene,so that its concentrationin the final solutionis higher. : :hetube This resultsuggests which concentrationdifference we can usein Eq. 8.5-I . Vy'eshould r -. r.J-31 . not usethe concentrationin benzeneminus the concentrationin water;that is initially zero, and yet thereis a flux. Instead,we can usethe concentrationactually in benzeneminus the 238 8 / FundamentalsoJ'Mass Transfer 8.5/ Mass Tr,,
(o) Heot tronsfer ww.",",wlì-l-,.n,.n"trF;l Iniliol Finol (b) Bromineextroction t---l _ f---r p,a Benzene .l- aon" lEquol - , I 7777-2 V7-7V lEquot// ) woter----+ lDitute) |,/,/,/) /,/ ,/) lnllioI Finol (c) Bromine voporizotion ['';]-^' F-.;-l F;1F*,* Wú Frg:: Finol diîìu.:: changr' Fig. 8.5-1.Driving fbrcesacross interfaces. In heattransfer, îhe amount ofheat transfèrred coetl.: dependson thetemperature difference between the two l'iquids,as shown in (a). In mass îe\t. transfèr,the amount of solutethat ditfuses depends on thesolute's "solubility" or, moreexactly, on its chemicalpotential. Two casesare shown. In (b), brominediffuses from waterinto benzenebecause it is muchmore soluble in benzene;in (c),bromine evaporates until its :he lett is bein-l chemicalpotentials in thesolutions are equal. This behaviorcompìicates analysis of mass transtèr. .\-. :
.,.here ,('-. ì: rhe concentration that would be in benzene that was in equilibrium with the actual concentration ::r-bulk pre\\u: in water. Svmbolicallv. .reflur acro... Thu.. Nr : Klcr(in benzene)- Hc(inwater)] (8.5-2) where rY is a partition coefficient,the ratio at equilibrium of the concentrationin benzene r..rrr! r-r.- th:(rrr Ii., - to that in water.Note that this doespredict a zero flux at equilibrium. "gu.r -::J r and r A better understandingof this phenomenonmay come from the third case,shown in \\'e norr nee_ Fig. 8.5-1(c)' Here, bromine is vaporizedfrom water into air. Initially, the bromine's .:.rlnto.t ajl... concentrationin water is higher than that in air; afterward,it is lower. Of course,this reversalof the concentrationin the liquid might be expressedin molesper liter and that in |11. -)- gasas a partial pressurein atmospheres,so it is not surprisingthat strangethings happen. .,.:Cre H rr .rlr: As you think about this more carefilly, you will realizethat the units of pressureor i.:. S.-r--ìthrr* concentrationcloud a deepertruth: Mass transfercan be describedin termsof more funda- mentalchemical potentials. If this weredone, the peculiarconcentration differences would ( disappear.However, chemical potentials turn out to be very difficult to usein practice,and so the concentrationdifferences for masstransfer acrossinterfaces will remaincomplicated -i theflur by units.
8.5.2 The Overall Mass Transfer Cofficient 'We want to include thesequalitative observations in more exactequations. To do this, we considerthe exampleof the gas-liquid interfacein Fig. 8.5-2. In this case,gas on l' .,,t.rter 8.5 / Mass TransferAcross Interfaces 239
Fig. 8.5-2. Mass transfer across a gas-liquid interface. In this example, a solute vapor is diffusing from the gas on the left into the liquid on the right. Becausethe solute concentration changesboth in the gas and in the liquid, the solute's flux must depend on a mass transfer -.'J - coefficient in each phase. These coefficients are combined into an overall flux equation in the text. 'r 3\ilctly, the left is being transferredinto the liquid on the right. The flux in the gasis Nt:kp(prc-pn) (8.5-3)
whereftr, is the gas-phasemass transfer coefficient (typically in mol/cmr-sec-atm),p1e is the bulkpressure,and p11is pressure. - r llratlOn the interfacial Becausethe interfacialregion is thin, the flux acrossit will be in steadystate, and the flux in the gaswill equalthat in the liquid. Thus.
Nr : kr (cri - cro) (8.5-4) - -inzene where the liquid-phasemass transfer coefficient k1 is typically in centimetersper second and c1;and c10are the interfacialand bulk concentrations,respectively. . :1,r\\ n ln We now needto eliminatethe unknowninterfacial concentrations from theseequations. -:.rnline's In almostall cases,equilibrium existsacross the interface: .-:.r-.this : .-J thatin Pti Hcti (8.5-5) . ::ppen. where 11is a type of Henry's law or partition constant(here in cmr-atm/mol). Combining l:'-.LlfC Of Eqs.8.5-3 through8.5-5, we can find the interfacialconcentrations -: iunda- krPn * ktcrc \\ Pri - r. OUld " (8.s-6) :,..-t-. OIìd H knHlkl :'riicated and the flux I *t :,It, - Hcrc) (8.s-7) * ul,rr(prc You shouldcheck the derivationsofthese results. To do Before proceedingfurther, we make a quick analogy. This result is often comparedto an electriccircuit containinq !its on two resistancesin series.The flux correspondsto the current. 240 8 / Fundamentalsof Mass Transfer
and the concentrationdifference prc - Hcrc conespondsto the voltage.The resistanceis then I f k,, + H lkL, which is roughly a sum of two resistancesin series.This is a good way of thinking about theseeffects. You must remember,however, that the resistancesl/kn and 1I kl arenot directly added,but alwaysweighted by partition coefficientslike t1. We now want to write Eq. 8.5-7 in the form of Eq. 8.5-1. We can do this in two wavs. First. we can write
Nr :Kz(r:ì-cro) (8.s-8)
where
Kr: (8.5-e) l lkt -l I lkpH
and
. Dtrr 'H (8.5-r0)
K1 is called an "overall liquid-side masstransfer coefficient," and ci is the hypotheticat liquid concentrationthat would be in equilibrium with the bulk gas. Alternatively,
Ny: Kp(pn - pi) (8.5-l r)
where I K.- _ (8.s-l2) '' llkr+ H/kL
and
Pî : Hcut (8.s-l3)
K2 is an "overall gas-sidemass transfer coefficient," and pi is the hypotheticalgas-phase concentrationthat would be in equilibrium with the bulk liquid. We now turn to a varietyof examplesillustrating mass transfer across an interface.These exampleshave the annoyingcharacteristic that they areinitially diffìcult to do, but they are trivial after you understandthem. Rememberthat most of the difficulty comesfrom that ancientbut common curse: unit conversion.
Example 8.5-1: Oxygen mass transfer Estimatethe overall liquid-sidemass transfer coeffìcientat 25'C for oxygen from water into air. In this estimate,assume that each individual masstransfer coefficient is D k: 0.01cm Thisrelation is justified in Section13.1. Solution For oxygenin air, the diffusion coefficientis 0.23 cm2lsec;for oxygen in water.the diffusioncoefficient is 2.1 . l0-5 cm2/sec.The Henry'slaw constantin this caseis 4.,1. 101atmospheres. We needonly calculateft1- and k, andplug thesevalues into T,,irts.fe r 8.5 / Mass TransferAcross Interfaces 241
iJnce lS Eq. 8.5-9.Finding ft1 is easy ,,,,ìway D, 2.1.10-5 cm2/sec :, I lkp " H. 0.01cm 0.01cm ,r r\ &YS. :2.1 .10 3cm/sec
Finding kn and H is harderbecause of unit conversions.From Eq. 8.5-3 and Table8.2-2 s.5-8) k6 D6 t_ Kp: RT (0.01cm)(RZ)
0.23cm2 s.s-9) : lsec (0.01cmX82 cm3-atm/g-mol-'K)(298"K)
: 9.4 . l0-a g-mol/cm2-sec-atm r 5-10) From the units of the Henry's law constant,we seethat the valuegiven implies ,thetical Ptr : H'rti By comparisonwith Eq. 8.5-5, r.5-I l) p1;:Hc1; :(Hc)x1;
Thus s.5-12) / H'\ 4.4' lOaatm H : | - I - -, :J.9. l0' atm-cm'/g-mol \, / lg-mol/l8cmr Insertingthese results into Eq. 8.-5-9,we find s.-5-l3) I Kr-_ " llkt+llkpH ;,r.-phase I ,:. These tl - .: theyare tl lol.rn/*. e|1 : r-rìlnthat : 2.1. 10-3cm/sec
The masstransfer is dominatedby the liquid-sideresistance. This would alsobe true if we calculatedthe overall gas-side mass transfer coeffìcient, a consequenceof the slow diffusion ',:.- transfer in the liquid state.It is the usualcase fbr problemsof this sort. : :hat each
Example 8.5-2: Perfume extraction Jasmone(CllHl60) is a valuablematerial in the perfume industry, used in many soapsand cosmetics. Supposewe are recoveringthis materialfrom a water suspensionof jasmine flowers by an extractionwith benzene.The aqueousphase is continuouslthe mass transfer coefficient in thebenzene drops is 3.0' l0 + - r oxygen centimetersper second;the masstransfèr coefficient in the aqueousphase is 2.4. l0 l -.1.,:rtin this centimetersper second.Jasmone is about 170 times more solublein benzenethan in the - .,,luesinto suspension.What is the overall masstransfer coeffìcient? ''t ^"t 8 / Fundamentalsof Mass Transfer 8.6/ Cor
Solution For convenience,we designateall concentrationsin the benzenephase Whenthe with a prime and all thosein the water without a prime. The flux is
Nr : k(cro- cr,) : k' (c'r,- c1o) are in equilibrium: The interfacialconcentrations ln passin' c'li : Hcli \\-e c;:
Eliminating theseinterfacial concentrations, we find ttl N,:l l(Hc,n-t:,^l" Lllk'+Hlkj The quantityin squarebrackets is the overall coefficientK' that we seek. This coeffìcient Tre c.ì. is basedon a driving force in benzene.Inserting the values,
K,: +' lz0 3.0. 10-1cm/sec 2.4. 10-3cm/sec
: 1.3.10 'cm/sec
Similar resultsfor the overallcoefficient based on a driving force in waterare easily found. Two points about this problem deservemention. First, the result is a completeparallel to Eq. 8.5-12, but for a liquid-liquid interfaceinstead of a gas-liquid interface. Second, masstransfer in the water dominatesthe processeven though the masstransfer coefficient in water is largerbecause jasmone is so much more solublein benzene.
'We Example 8.5-3: Overall mass transfer coefficients in a packed tower are studying gas absorptioninto water at 2.2 atmospherestotal pressurein a packed tower contain- ing Berl saddles.From earlier experimentswith ammoniaand methane,we believethat for both gasesthe mass transfercoefficient times the packing areaper tower volume is l8 lb-mol/hr-ft3for the gas side and 530 lb-molftrr-ftr for the liquid side. The valuesfor thesetwo gasesmay be simiìar becausemethane, and ammonia have similar molecular weights. However,their Henry's law constantsare different: 9.6 atmospheresfor ammonia and41,000 atmospheres fbr methane.What is the overallgas-side mass transfer coefficient for eachgas? Solution This is essentiallya problem in unit conversion. Although you can extractthe appropriateequations from the text, I alwaysfeel more confidentif I repeatparts of the derivation. The quantitywe seek,the overall gas-sidetransfer coefficient K., is definedby
N1a: K,.rut1 10 - r'i) : kra(.yro- yr,) : k,a(xti - r'n) wherer'1 and11 are the gasand liquid mole fractions. The interfacialconcentrations are related by Henry's Ìaw: - - /)1, pJ t, Hxti 8.6 / Conclusirns 243
When theseinterfacial concentrations are eliminated,we find that 1 1 Hlo ----L Kro- kru' kra
In passing,we recognizethat yi mustequal Hxrclp. We can now fìnd the overall coefficientfbr eachsas. For ammonia,
9.6 atmlZ.2 atm -l .T ["d | 6 lb-mot/hr-It- 530 1b-mol/hr-ft3
K,.a - l6 1b-mol/hr-ft3
The gas-sideresistance controls the rate. For methane,
I --]_ 1 41.000 atntl2.2 atm Kr" l8 1b-mol/hr-ft' 530Ib-molftr-ft3
K,'a : 0.03 lb-mol/hr-ft3
The coefficientfor methaneis smallerand is dominatedby the liquid-side masstransf.er : ,ìnd. coefficient. '.:-r 1lel : - "nd. --'lr'fll 8.6 Conclusions
This chapterpresents an alternativemodel for diffusion, one using masstransfer coefficientsrather than diffusion coefficients. The modelis mostuseful for diffusionacross - j\ lng -.i.ìin- phaseboundaries. It assumesthat largechanges in the concentrationoccur only very near theseboundaries and that the solutionsfar from the boundariesare well mixed. Such a : : thàt descriptionis called a lumped-parametermodel. -::lL- 1S resultscan be convertedinto mass -:" for In this chapter,we have shown how experimental can be efficiently orga- :-;u lar transfercoefficients. We have also shown how thesecoeffìcients nizedas dimensionlesscorrelations, and we havecataloged published correlations that are : ì'rflnia commonly useful. Thesecorrelations are compromisedby problemswith units that come ::-Jìent out of a plethoraof closelyrelated definitions. provide usefuldescriptions of diffusion in complex . Jcan Masstransfer coeffìcients especially niultiphasesystems. They are basic to the analysisand design of industrial processes l.:i f iìrtS ìike absorption,extraction, and distillation. They should fìnd major applicationsin the .tudy of physiologic processeslike membranediffusion. blood perfusion.and digestion; physiologistsand physicians do not often usethese models but would benefitfrom doing so. Mass transfercoeffìcients are not useful in chemistry when the focus is on chemical kinetics or chemical change. They are not useful in studiesof the solid state, where ;oncentrationsvary with both positionand time, and lumped-parametermodels do not help ntuch. However,mass transfer coeffìcients are used in analyzingetching processes.like thoseused in makingsilicon chips. All in all, the materialin this chapteris a solid alternativefor analyzingdiffusion near rnterfaces.It is basicstuff for chemicalengineers, but it is an unexploredmethod for many ,rthers.It reDavscareful studv.