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7/30/2019 C1 Cussler

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C H A P T E R IModelsor Diffusion

If a fw crystalsof a coloredmaterial ike coppersulfateareplacedat the bottomol'a tall bottle illed with water, he color will slowly spreadhrough he bottle. At first thecolor will be concentratedn the bottom of the bottle. Afier a day t will penetrate pr"'arda few centimeters.After several ears he solutionwill appear omogeneous.Theprocessesponsiblebr themovement f thecoloredmaterial s diffusion, he sub.jectof hisbook.Dif l ision iscaused y randommolecularmotion h at eads o completemixing.It ca nbe a slowprocess.n gases, iffusion rogressest a rateof about 0 cm n a minute:in iquids, ts ate s about .0 5cm/min: n solids,ts atemaybeonlyabout .00001 m/min.ln general,t varies esswith temperaturehan do manyotherphenomena.This slow rateof diffusion is responsiblebr its importance. n manycases, iffusion()ccurs equentiallywith otherphenomena.When it is the sloweststep n the sequence,tlrrnits he overall rateof the process.For example,diffusionoften imits theefficiencyof. trntmercial istillationsand he rateof industrial eactions singporouscatalysts. t limitsthe speedwith which acid and base eactand the speedwith which the human ntestine,rbsorbs utrients. t controls he growth of microorganisms roducingpenicillin,the rate'l thecorrosionof steel,and he release f flavor rom food.In gases nd iquids, he ratesof thesediffusionprocessesan ofien be accelerated y.:_jrtation.or example, hecoppersulfate n the all bottlecanbe completelymixed n a few'-- inutesi fthesolut ionisst i r red.hisacceleratedmixingisnotduetodif1isionalone,but' ' thecombinationof diflision and stirring. Diffusion still depends n randommolecular,rttons hat take placeover small moleculardistances.The agitationor stiring is not a- ,lc-cular rocess, ut a macroscopic rocess hat movesportionsof the fluid over much.::rerdistances.After this macroscopicmotion, diffusion mixes newly adjacent ortions' :hc luid. In othercases, uchas he dispersal f pollutants,he agitationof wind or water- 'Juceseffectsqualitativelysimilar to diffusion; theseeffects,called dispersion,will be-:-,redseparately.The descriptionof diffusion involvesa mathematicalmodel basedon a fundamental: ,thesisor "law." lnterestingly, hereare two common choices or such a law. The:: fundamental, ick's law of diffusion,usesa diffusion coefficient.This s the aw that- ,nrmonlycited in descriptions f diffusion. The second,which has no formal name,' r esa mass ransfer oefficient,a type of reversible ateconstant.--noosing etween hese wo models s the subjectof this chapter. Choosing Fick's- .elds to descriptions ommon to physics,physicalchemistry,and biology. These- - .- ::frtlons reexplored ndextendedn Chapters -7. Choosingmass ransfr oefficients- -,..r's conelationsdeveloped xplicitly in chemicalengineering nd used mplicitly in- - : ..aIkinetics nd n medicine. hese orrelationsredescribedn Chapters -14. Both-: : - i jhes areused n Chapters5-19..r: Jircuss he difTerencesetween he two models n SectionLl of this chapter. n:,:- r l.l we show how the choiceof the most appropriate odel s determined. n


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I / Models or Diffusion : (

_ . : . . : :- : .

UftstrJz.9EFztdoz

NT IM E

Fig. 1.1-1. A simple diffusion experiment. Two bulbs initially containing different gasesareconnectedwith a long thin capillary. The changeof concentration n eachbulb is a measureofdifusion and can be analyzed n two different ways.Section1.3we concludewith additionalexamples o illustratehow the choicebetweenhemodels s made.

1.1 The Two Basic ModelsIn this sectionwe want to illustrate he two basicways n which diffusioncanbedescribeil.To do this,we first imagine wo largebulbs connected y a long thin capillary(Fig. 1 - ). The bulbsareat constantemperature ndpressure ndareof equalvolumes.However.onebulb contains arbondioxide,and he other s flled with nitrogen.To indhow ast hese wo gaseswill mix, we measureheconcentrationf carbondioxidein thebulb that nitially containsnitrogen.We make hesemeasurements henonly a traceofcarbondioxidehasbeen ransferred, ndwe f,nd hat heconcentration fcarbondioxidevaries inearlywith time. From this,we know the amount ransferred er unit time,We want o analyzehis amount ransferredo determine hysicalproperties hatwill beapplicablenot only to this experimentbut also n otherexperiments.To do this, we firstdefine he flux:

( 1 . 1 - 1 )In other words, f we double he cross-sectional rea,we expect he amount ransportedto double. Defning he flux in this way is a first step n removing he influences f ourparticularapparatus nd making our resultsmore general.We next assumehat he flux isproportionalo th ega sconcentration:

/ amount sa s emoved(carbon ioxidelux) t - -- i -* I\ t lme (area apl l la ry /

/ carbon ioxide\(carbonioxideluxl k I concentralionI\ difference /( r .1-2)

The proportionality onstantt is calleda mass ransfer oefficient. ts introductionsignalsone of the two basicmodelsof diffusion. Alternatively,we canrecognize hat ncreasingthecapillary's engthwill decreasehe flux, andwe can henassumehat/ carbondioxideconcentration ifference( c a r b o n d i o x i d e f l u x ) : " ( ) t t ' t - 5 )


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,t)rDiffusion

I lses ar g\ .i measureof

be \\'eenhe

ca nb eihtn capil laryVolumes.

' .:: lonl) a trace-:rbtrl l dioxide1l . l1t ha tw i l l bee f i r s t

( 1 . 1 - l )-.: l i ransportedf ou f: i , i t the lux s

( 1 . t - 2 )., lonsignals.i increasing

l ( l . r - 3 )

- Cht,ttsi ttgBefireett he TtvoModels

.' ne i proportionality onstantD i s the diffusioncoefficient. ts ntroduction mplies he'r.3rmrrdel br diffusion. he model often calledFick's aw.Theseassumptions ay seemarbitrary, ut they aresimilar o thosemade n manyother-::r.hes of science.For example, hey are similar o thoseused n developingOhm's aw,-:;h stateshat

( 1 . 1 - 4 )u'. the mass ransfercoefficient r is analogouso the reciprocalof the resistance.An3rnative orm of Ohm's aw is( r,"l$i',11^){*"**)h:ffi

i t y \ / | 1 / n o t e n t i a li : (...'.t'tty,)\ffitr// cunent densI or luxof\ electrons ( 1 . 1 - 5 )- :e diffusioncoefficientD is analogouso the reciprocalof theresistivity.\either the equationusing the mass ransfercoefficient t nor that using the diffusion- ','tlcient D is always successful. This is becauseof the assumptionsmade in their:: r elopment.For example, he lux may notbe proportionalo the concentration ifference. the capillary s very thin or if the two gases eact. In the sameway, Ohm's law is not..*avsvalidatveryhighvoltages.Butthesecasesareexceptions;bothdiffusionequations,,..rrkwell in mostpracticalsituations,ust as Ohm's aw does.Theparallelswith Ohm's aw alsoprovidea clueabouthow thechoicebetween iffusion:lodels s made.The mass ransfer oeffic ientn Eq. 1.1-2an d he resistancen Eq. l.l-4:i simpler,best used or practicalsituationsand rough measurements.The diffusion-,retficientn Eq. 1.1-3and he resistivity n Eq. l.l-5 are more undamental,nvolving:hrsical propertiesike those ound n handbooks.How thesedifferences uide he choiceretween he two models s the subiectof the next section.

1.2 Choosing Between the Two ModelsThe choicebetween he two modelsoutlined n Section1.1 represents compro-misebetweenambition andexperimental esources.Obviously,we would like to express.ur results n themostgeneraland undamentalwayspossible.This suggests orking withJiffusioncoefficients.However. n many cases ur experimentalmeasurements ill dictatel more approximateand phenomenological pproach. Suchapproximations ften implyr.nassransfer oefficients, ut they usuallystill permit us to reachour research oals.This choice and theresultingapproximations re best llustratedby two examples. nthe first, we considerhydrogendiffusion n metals. This diffusion substantiallyeducesa metal'sductility, so much so that partsmade rom the embrittledmetal requently rac-ture. To study his embrittlement,we might expose he metal o hydrogenundera varietyt,f conditions and measure he degreeof embrittlementversus heseconditions. Suchempiricismwould be a reasonableirst approximation, ut it would quickly flood us withuncorrelatednformation hat would be difficult to useeffectively.As an improvement,we can undertake wo setsof experiments.First, we can saturatemetalsampleswith hydrogenanddetermine heir degrees f embrittlement .Thuswe knowmetalproperties ersus ydrogen oncentration.Second,we canmeasure ydrogenuptake

I


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1 . \ 1 , , . j t t t t t r [ ) i l l u s i t r r

tr.-l. ."1:

urtllt "ltlL[n!rI I rrFiD! uri lLJ l r. L", l t :{fllr . -Jtil .'

|l&

Anolyze os moss ronsferFlux= lA(concenlrol ion)* is not consfonivoriotionwith timecorreloted vorioionwilh posil ion gnored

Anolyze os diffusionf nx = - D LAz(concentrolion), is conslontivoriotionwih time on dpositionpredicted

Fig. 1.2-1.Hydrogen diffision into a metal. This processcan be describedwith either a masstransfrcoellci"ni ft o. a diffusion coefficient D. The description with a diftsion coefficientcorrectly predicts he variation ofconcentration with position and time, and so s superior.

versus lme, aSsuggestedn Fig. 1.2-1, ndcorrelate ur measurementss maSSransfercoefficients.Thuswe know average ydrogen oncenfrationersus ime.To our dismay,hemass ransfer oefficientsn thiscasewill bedifficult o nterpret.Theyareanythingbutconstant.At zero ime, heyapproachnfinity; at arge ime, heyapproachzero. At all times, hey vary with the hydrogenconcentrationn the gassurroundinghemetal.Theyarean nconvenient ay to summarize ur results'Moreover, hemass ransfercoefcientsgive only theaverage ydrogen oncentrationn themetal.They gnore he actthat he hydrogenconcentration ery near he metal'ssurfacewill reachsaturation ut theconcentration eepwithin thebar will remainzero.As a result, he metalnear he surfacemay be very brittle but thatwithin may be essentially nchanged'We can nclude hesedetails n the diffusion model describedn the previoussection.This modelassumedhat

/ h y c l r o g e n \ - / h Y d r o g e n I/ hydrogen\ - . . , concentrat iontz :0 / \concentrat iont - / /\ f lux ) - " ( th icknessrz : l ) - ( th icknessatz 0) ( 1 . 2 - )

t 1 l . - 1 y t ' 1 1 ; : t t l ) - ) \/ - 0where he subscript symbolizeshe diffusingspecies. n theseequations,he distanceis that overwhich diffusionoccurs. n theprevioussection,he engthof thecapillarywasappropriatelyhis distance; ut in this case,t seems ncertainwhat he distance houldbe'If we assumehat t is very small,

or , symbolically,j t : D

, t ' Dl$ cr .- :" cl : : ;+t: l :+1 : l : d c r- - D ,o?. (1 .2-3)We can use his relationand the techniques evelopedater in this book to correlateourexperimentswith only one parameter,he diffusioncoefficientD. We thencan correctly


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1.2 ChoosinpBetween he TwoModelsrtr Di.ffusion

r I \F-tr!I l oliJ" ISolid /u-)orus rl!/

Ano lyze os chem ico l eoc l i on/ dc t/ - i - = K ( c 1 ( s o f ) - c t )/ o l/ r< s reocf on roie/ constoni for o/ f i c l i t i ous reoc t i on' / Anolyze os moss tronsfer

S o l u r o t i o n

/ vor ieswi lh s i i r r ingilj. %1,; o"Fig. | .2-2. Rates of drug dissolution. In thi s case,describng he systemwth a mass ransfercoeffcient t is b est because t easily correlates he solution's concentrationversus ime.Describing the systemwith a diffusion coeffcient D gives a similar correlation but introducesan unnecessary arameter, he film thickness . Describing the systemwith a reaction rateconstant also works, but this rate constant s a finction not of chemistry but of physics.

predict he hydrogenuptakeversus ime and the hydrogenconcentrationn the gas. As a,iividend,we get he hydrogenconcentration t all positionsand imeswithin the metal.Thus hemodelbased n hediffision coeffcient ives esults f more undamental aluethan he model basedon mass ransfercoefficients. n mathematicalerms, he diffusionmodel s said o havedistributed arameters,or thedependentariable theconcentration)s.rllowed o varywith all independent ariableslikepositionand ime). In contrast,hemassrransfrmodel s said o have umpedparameterslike theaverage ydrogen oncentrationrn he metal).These esultswould appear o imply that the diffusion model is superior o the mass:ransf'ermodel and so should alwaysbe used. However, n many interestingcases he:nodelsare equivalent.To illustrate his, magine hat we arestudying he dissolutionof a.,rliddrug suspendedn water,as schematicallyuggestedy Fig. 1.2-2.The dissolution'i thisdrug s known to be controlledby thediffusionof the dissolved rug away rom the.,,lid surfaceof the undissolvedmaterial. We measurehe drug concentration ersus ime-: . shown.andwe want to correlate hese esults n termsof as w parameters spossible.One way to correlatehe dissolution esults s to use a mass ransfercoefcient. To do:li\. we write a mass alance n the solution:

/ accumulat ion - \I . . . I / total ateoI \I or drug n I : l , a ir .otut ion\ solut lon /

zoFFzozoOlr

=.-......-------

d c tV - : A j ta t: Aklcr(sat) crl

dC,vZ i = k A ( c t $ o I ) - c t l ,* vor ieswi lh s i r r tng.Note hof kA V =x.

Anolyzeos d i f fus ionr/r. nv = fA (c lbo I l - c1 )

-- lncr mass:r r 'et c ient. . upe r i o r .

r;.\ ransferheylpproach-, ,Lrndingthe:r.\ transf'er

_:ri\rrehe ac t: . r i i r rnut he-, : he surface. ,ru.sect ion.

( 1 . 2 - 1 )

( 1 . 2 - 2 ):r ; distance,.rpi l ary as' - r .hould e.

( 1 . 2 - 3 ), !rfelate OUr,,!n coffectly (1 .2-1)

T I M E


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I / ModelsJ'orDiffusionwhere V is the volumeof solution,A is the total areaof the drug particles,c1(sat)s thedrug concentration t saturation nd at the solid's surface,and c1 s the concentrationnthe bulk solution. ntegrating his equation llowsquantitativelyitting our resultswith oneparameter,he mass ransfrcoeffcient t. This quantity s independent f drug solubility,drugarea, ndsolutionvolume,but t doesvarywith physicalpropertiesike stirring ateandsolutionviscosity.Conelating he effectsof these ropertiesurnsout to be straightforward.The alternative o mass ransfer s diffusion heorv. or which the massbalances

(r.2-s)in which / is an unknownparameter, qual o the average istanceacrosswhich diffusionoccurs.This unknown,called a film or unstirred ayer hickness,s a function not only offlow andviscositybut alsoof the diffusion coefficient tself.Equations1.2-4and | .2-5 areequivalent, nd hey share he samesuccessesnd short-comings. n the ormer,we mustdetermine he mass ransfer oefficient xperimentally;nthe atter,we determine nstead he hickness Thosewho like a scientific eneerprefer omeasure, for it genuflectsowardFick's law of diffusion.Thosewho are morepragmaticpreferexplicitly recognizing he empiricalnatureof themass ransfer oefficient.The choicebetween he mass ransferand diffusion models s thus often a questionoftasterather than precision. The diffusion model is more fundamental and is appropriatewhen concentrations re measured r neededversusboth positionand time. The masstransfermodel s simplerandmore approximate nd s especially sefulwhen only averageconcentrations re nvolved. The additionalexamplesn section1.3shouldhelp us decidewhich model s appropriateor our purposes.Before going on to the next section,we should mention a third way to correlate heresultsother han he two diffusion models.This third way is to assume hat dissolutionsa first-order, eversible hemical eaction.Sucha reactionmight be described y

d c ' / D \V d t : O ( . f J l c r ( s a t )c r l

dcrd t : r r t ( s a t ) - K c l (1 .2-6)

Inthisequation,thequantityKcl(sat)representstherateofdissolution,rcrtandsfortherateof precipitation, nd r is a rateconstantor this process.This equations mathematicallyidentical with Eqs. 1.2-4 and 1.2-5 and so is equally successful. However, he idea oftreatingdissolutionas a chemical eaction s flawed. Becausehe reaction s hypothetical,the rate constant s a compositeof physical actors ather han chemical actors. We dobetter o consider he physicalprocessn termsof a diffusion or mass ransfermodel.1.3 ExamplesIn this section,we giveexamples hat llustrate he choicebetween iffusioncoef-ficientsandmass ransfer oeffcients. his choice s often difficult, a uncturewheremanyhave rouble.I often do. I think my troublecomes rom evolving esearch oals, rom thefact hat as understandhe problembetter, hequestionshat am trying to answer end ochange. notice he sameevolution n my peers,who routinely startwork with onemodelandswitch o the othermodelbefore he end of their research.We shall not solve the following examples. Instead,we want only to discusswhichdiffusion model we would initially use or their solution.The examples ivencertainlydo

E.ramples,.rveral l ypes:l in the ast

rample 1.3-1:-- : b1 react in.- \ \ater o di

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