MASS TRANSFER WITH COMPLEX REVERSIBLE

Mass transfer with complex reversible chemical reactions--Ithe resistancegas phase:againstmass transferis situatedin the2297trations remain bounded. In this way one finds (for thesituation that all stoichiometriccoefficients are equal):( ), , (F) ( ) O.VW2.3. Infinite reaction rate constantsIn case of infinite reaction rate constantsthe boundary condition (9) is no longer valid and the gradients of[B], [C] and [D] are not necessarily zero in order totake into account that these componentsare nonvolatile.To derive the appropriateboundary conditionsonemay return to the mass balances over a small layer Axadjacent to the gas-liquidinterface, depicted in Fig. 1.For the four componentsone finds:For finite reactionrate constantsand finite timederivativesall integrals vanish when Ax approacheszero and eq. (9) is recovered. In the limit situation thatthe reaction rate constantshave infinite values, thereaction rate, R,, becomes indefinite and therefore theintegrals are also indefinite. The way to proceed is firstto eliminate R, by adding and subtractingamong theeqs (10) and then to take the limit that Ax 0, bywhich the accumulationterms vanish under the assumptionthat the time derivativesof all concen-This eliminationprocess gives three independenteqs (1 I), therefore the fourth equation to achieve thecompleteboundaryconditionshas to express thevalue of the quantityR, which just has been eliminated. In case of infinite reaction rate constantsandwith the assumptionthat at the gas-liquidinterfaceequilibrium is maintainedthis fourth equation will bethe equilibriumcondition.In view of applicabilityit is desirable to obtain onegeneral solution for the phenomenonmass transferfollowed by a reversible chemical reaction. Thereforethe set of eqs (3), (4), (5) and (6) will be solved withboundary condition (9). For the limit situation wherethe reaction rate constantsbecome infinite, the descriptionwill be approximatedby extremelyhighvalues of km,n,p,q and k,,,,,“. Afterwards this approximation has to be verified which is possible with the aidof the analytical solution of Olander (1960). In casethat for the descriptionof the phenomenonmasstransfer accompaniedby complex chemical reactionbesides the eqs (3), (4), (5) and (6) also equilibriumequations are used (Cornelisse et al., 1980) boundarycondition (11) must be used instead of condition(9).2.4. Film modelFor the film model the phenomenonmass transferfollowed by a chemical reaction can be representedwith the following set of equations:Damb ax2D a2tc1c ax2-ybR, O y,R, Oa2 CD1 y,,R, ODd--&x2J*and the following(34(34boundary conditions:U2d0AX-xk-;lg.I. Eluxes at the interface.at x 6:[A] [A],,,[Bl [&,[C] [Cl,, CD1 CDlo- (13)

G. F.2298VERSTEEGet al.For infinite reaction rate constantsthe boundaryconditions(11) derived in Section 2.3 should be used.2.5. Enhancement factorThe enhancementfactor, E,, defined as the ratio ofthe mass flux of componentA through the interfacewith chemical reaction and driving force ([Ali[A],,)to the mass flux through the interface without chemical reaction, but with the same driving force, can beobtained for both models from the calculated concentration profiles. The enhancementfactor, E,, is definedby:-penetrationmodel:plane can be manipulatedby defining a x @(.z). Inorder to move the grid points towards the interface thederivative of Q near z 0 should be small comparedwith elsewhere. So an uniform mesh size in z nowautomaticallygenerates a higher concentrationof gridpoints near the interface (see Fig. 2).The same can be done with respect to the timevariable, t. Also here the grid spacing near the origin isrefined, as the steepest time-derivativesoccur for smallt, being induced by the concentrationjump betweengas and liquid at the start of the contact period.An example for the spatial (place) transformationis:x pz (lk,UCKli-IAl,) -film(A) j:-Da(d ) ,dt(14)model:k,Ea(CAli CAl0) --Da(151X 0However,these concentrationprofiles are calculated with boundaryconditions(9a) and (12a) for thepenetrationmodel and the film model respectively andtherefore the value of the numericallycalculated enhancementfactor, as defined above, has only a physical meaning in those situationswhere the gas-phasemass transfer resistancecan be neglected completely(i.e.[Al,. 0 [Al,. iI*J. NUMERICAL TREATMENT3.1. IntroductionIn the film model the main goal of a numericalapproach is to produce the concentrationprofiles assolution of a system of coupled non-linearordinarydifferentialequationswith two point boundaryconditions.From the concentrationprofiles obtainedafter solving this set ofequationsthe mass transfer rate(e.g. the enhancementfactor) can be calculated easily.From the numerical point of view the film model canbe regarded as a special case of the penetrationmodel.Thereforethe numericaltechniqueused for solvingthese differential equations will not be discussed separately.In the penetration(Higbie) model the concentrationprofiles are time-dependent:they develop as solutionof a system of coupled ject to specified initial andtwo point boundary conditions.The approach used tosolve these models is mainly based on the methodpresentedby Cornelisseet al. (1980) however, somenew numericalfeatures are introducedin order toincrease the accuracy and minimise the computationaltime.3.2. DiscretizationThe discretizationitself follows mainly the lines byCornelisseet 2. (1980). Thereforeonly a few additional remarks are presented.The distributionof spatial grid points in the x, r--p)z4(16)with parameter p between 0 and 1, controlling the griddeformation.For p I the identity x z is obtainedand for lower p-values the grid deformationincreases.[It should be avoided that p 0, in order to retain theone-to-onecorrespondencebetween the derivatives(dc/dx) and (dc/dz) given by the formula (dc/dz) (dc/dx) . (dx/dz).lForthe time coordinatea quadratictransformation:t wZ(17)is a suitable choice. A typical grid in the x, t-plane isdepicted in Fig. 3 and from this figure it can be seenthat near the interface (small x) and at the start (short t)the concentrationof grid points is higher than elsewhere. Discretizing (partial) differential equations in atransformedcoordinatesystem is somewhatmorecomplicatedthen in the original variables because ofthe transformationderivatives that occur in the differential operators.For instance, transformingthe second x-derivativein terms of z, z being given by x Q(z), leads to:(18)xFig.2. Effect of transformationon the spatial variable.

2299Mass transfer with complex reversible chemical reactions-IrFig. 3. A typical grid in the x, t-plane.Fig. 4. Discretizationwhere @’ is the derivativeof Q(z), which is knownanalytically. Now in a point zm( mAz), on a grid withuniform mesh size AZ and subscript m indicating theplace level, the discretizationof eq. (18) becomes:CA1 I-CALAZFor the time derivative a three-pointcretizationis used (with superscripttime level) leading to:8 CA1backwardj indicatingdisthe3[A]‘,“-4[A]j,, [A]”at’(20)2AtThe finite difference form of the penetration(Higbie)model thus leads to relations between concentrationsin five grid points, clustered as a “molecule” shown inFig. 4. Only for j O this moleculeis impossible,because grid points with time index - 1 do not exist.Thereforein the first step a two-pointbackwarddiscretization(Euler) is used leading to:The solutionscheme.of eq. (22) is given by:CA1- CAlo erfc[Ali- C4o(24)Concentrationprofiles according to eq. (24) are shownin Fig. 5, illustratingthe well known phenomenonofpenetrationdepth. At any time, say t*, the significantbehaviourof the concentrationprofile is restrictedwithin an interval 0 .x x*, where x* is a function oft*. In view of eq. (24) this function between x* and t* ischosen usually as:x* constant,/4D,t*(25)with a value of the constant depending on what one isinclined to call “significant behaviour”.Now in numerical calculationsit looks worthwhileto restrict the calculationaldomain to the intervalgiven by the penetrationdepth, and to spend no gridpoints on the remainingpart of the x-axis wherealmost nothing occurs. This adaptive grid point distribution is most easily accomplishedif a coordinatetransformationaccording to eq. (26) is employed:(21)at the cost of a lower ordertime step.truncationerrorin this7 3.3. Special transformationIn the present approach a special transformationofthe independentvariables has been introducedtakenfrom the analyticalsolution of the well known onedimensionaldiffusion problem:(22)t Oandx20CA1 CAlo(234t Oandx 0CA1 C4iWb)t Oandx a3CA1 [Alo.(23 )tJS’(26aIWb)In Fig. 6 the effect of the transformationis illustratedstarting with a uniform grid in r, r-space. For shortcontact times the spacing is very fine, leading to a highconcentrationof grid points and so to a high resolution near the spot where the sharpest gradients occur.Besides the above mentionedfeature, several additional advantageouspropertiesconcerningthe numerical treatmentare introducedwith the transformation of the independentvariables as specified byeq. (26). Firstly, the semi-infinite x-domain is mappedin the r-domain on the finite interval [0, 11. Secondly,

G.2300F.VERSTEEG et al.IIA],IA1molem3IAl,,II900I1I1 Ix0II5xiiiFig. 5. Concentrationprofile for the physical absorptionin the r, z-domain the r-derivative of the concentrationeq. (24) at the gas-liquidinterface is bounded when Tapproachesthe value zero whereas the x-derivativeisunboundedfor t O: [Ali.7-zfor several contact times.aC-4- 2r%R,a7(284while the equationsfor the other componentsareaffected with more complicatedexpressions,e.g. forcomponentC:W’b)Finite domains and finite derivatives are highly desirable if not necessaryfor reliable numericalresults.When applying this transformationon the system offour co ledpartial differential equations [eqs (3), (41,(5) and (6)], the maximum diffusion coefficient, D,, ofthe componentsinvolved should be taken in eq. (25) inorder to assure that this componentwill not run out ofthe picture. The equation for this component,say A,becomes:with:t(r) inverseerf {r}(291and& ;.(30)mtt001xrFig. 6. Effectuf transformationaccordingto eq. (26) on the spatial and time variable.

Mass transfer with complex reversible chemical reactions-IThe boundary conditions (8) are now located at I 1:r O, r l:[ 4] [A10,[B] [B],,ccl ccl, c l c%and the boundarycomes:condition(31)(9) at the interface be-2301The initial conditions, however, require special attention because of the topological consequences of thetransformation. In view of eq. (26) the initial line (t 0and x 0) is mapped upon one point r 0, r 1, so theinitial condition (7) in the X, t-plane is moved to thatone point:t O, r l:[A] [A],,[BJ [B&,,Ccl CCL,,- CAL.A Wa)Wb)It is interesting to note that this transformation hasgiven the physical phenomena a chronological order.Under the assumption that R, and all the derivativesof concentrations with respect to z and r are bounded,the terms in eq. (28) can be distinguished by the powersof r.The diffusion operator is independent of 7 and forz O it is the only term present in the differentialequation. Then for T 0 the accumulation term, whichis linear in 7, becomes important because it has to takeinto account the incoming flux generated by theboundary condition (32a), which by itself is coming upalso linearly in t. This is the phenomenon of purelyphysical absorption.Finally, with increasing 7, the reaction term becomes effective, it acts on the concentration profilesbuilt up by the physical absorption and is trying toturn them back to equilibrium values, which existalready at the boundary r 1 [see eq. (31)]. So, in fact,after the physical absorption, coming in from thegas-liquid interface (r O), the reaction term is a kindof response from the liquid bulk. Being quadratic in Tthis effect will predominate in the end, pressing theconcentration profiles back to the left and filling upthe whole interval with the equilibrium values.So for t--r cc this coordinate system {I, 7} maybecome inadequate, but it is advantageous in thebeginning, when the most rapid changes occur, due tothe sudden onset of the interface condition (9a). Especially the Newton-Raphsonlinearization of thenon-linear reaction term, eq. (2), using estimates of aprevious time level may be awkward just because ofthe large changes from one time level to another. Inthis new coordinate system, however, the onset of theinterface conditions is not abrupt but gradual, startingfrom zero and rising linearly in 7, so in the interior allthe differences will be gradual. Accordingly the estimates in the linearization of the reaction term will bemuch more accurate. Besides, the reaction term itself issmall, of order ?.Remarks on the initial conditionThe boundary conditions for the r, r-description areeasily transferred from the original problem, the interface (x 0) coincides with r 0 and the bulk (x to)with r 1.CD1 CDlo- (33)In the meanwhile the origin in the X, r-plane seems tobe stretched upon the whole interval z 0, O rt 1,and it is not obvious beforehand what kind of initialcondition is originating from this point.The answer to this question is a differential problemitself, consisting of ordinary differential equationsalong the r-axis, together with two point boundaryconditions. It is the differential problem that turns upwhen the limit 7 - O is taken in the partial differentialproblem [eq. (28)]. In the differential equations onlythe diffusion operators mentioned earlier are retained[see eq. (28)]:(3kd)andy)] Ox ((c(r))’(34b)and also for the components 5 and D. The boundaryconditions at r 0 follow from eq. (32) for 7 0:t O,r O:( ), , r ), , (F)rC, a(4CD1arr Cl(35)whiie the boundary conditions at r 1 where alreadygiven by eq. (3 I), which again is the limit of eq. (33) forr 0. The solution of the differential problem eqs (33),(34) and (35) is:z O,Otr 1:LA1 II& CSI [No, CC1 Ccl,and [D] [DJe(36)and therefore this is the initial condition in the r,z-plane.In this case the r, z-initial condition eq. (36) is quitesimilar to the x, r-initial-condition eq. (7). However,this section is meant to warn the reader not to copy itfrom eq. (7) but to realise the consequences of thetransformation. In fact the model problem [eqs (22)and (23)] is a non-trivia1 example. In the r, z-plane theformulation is:3.4.(374r Oand r O:[A] [A]i(37b)7 0and r l:[A] [A],,(374

G.2302whilethe initialdifferentialF.VERSTEEGis found from the ordinaryconditionproblem:d2iIAl .ar2et al.this set of equation,he linearized the reaction rateexpressionsaccordingto the method proposedbyHikita and Asai (1963) leading to:[Z?] [Blifor O .x 6r O: [A] [AJr l: [A] [A-jo.The solutionz O, O r of this differential1:[A] [A];-(problemis:[A];-[A&,)*r(37d)This initial condition is clearly quite different from eq.(23a). [In this case the expression(376) satisfies eq.(37a) and so it is identical with the solution of thepartial differential problem (37), given by eq. (24).]4. APPROXIMATESOLUTIONS4.1. DeCoursey approachDeCoursey (1982) derived an approximateanalytical solution for the set of eqs (3), (4), (5) and (6) withinitial and boundaryconditions(7), (8) and (9). Heobtained this solution by changing the instantaneousconcentrationsto time mean concentrationsaccording to the Danckwerts’surface renewal model bytaking “s-multiplied”Laplace transforms.In order toobtain an integrableset linear differential equationswith constant coefficientshe made the following assumptions:--the--thediffusivitiesequilibriumof all species are equal,at any point can be expressedbyCM, cc1 CW(KE l) -[B] is equal to [Rli near the interface accordingto van Krevelen and Hoftijzer (19481,-thereaction rate can be described with R, K,( CA1---theCAlI31 I;(XLreactionrate at x 00 can be expressed by R,and at x 0by R, k2[B],([A]-[A] ) 0 kzCBli(CAI- CAL).Att 0and x cu the concentrationsaccording to:of A, 5, C andD are in equilibriumThe enhancementfactor can be calculated accordmg to DeCoursey’sset of eqs (33), (34) and (35).This approximatesolutionwas checked in thatwork for three limit situations,K ‘a and [A],, O,R, co, and M * 0, and the agreementwith the analytical solutions was extremely good.4.2. Onda approachOnda et al. (1970, 1972) presentedapproximatesolutions for both film model and penetrationmodel.For the film model the set of eqs (3a), f3b), (3 ) and (3d)were solved with boundaryconditions(12) and (13)and for the penetrationmodel the set of eqs (3), (4), (5)and (6) with initial and boundaryconditions(7), (8)and (9). In order to obtain an analytical solution forThe enhancementfactor can be calculated with hiseqs (9), (1 I), (12), (21) and (22) but it should be notedthat for the trial and error method the enhancementfactor, E,, should be taken as iterationparameterinstead ofe,, as was proposed originally by the author,in order to obtain a stable solution method over awide range of conditions.However, the enhancementfactor calculated in this way is not comparablewiththe enhancementfactor defined according to eq. (14)or (15) and the enhancementfactor calculated according to the Onda approximationshould be multipliedwith [A /( [Ali- [A],,). The approximatesolution ofthe penetrationmodel has the additionalrestrictionthat the diffusivity of all species must have the samevalue.It should be noted also that Onda treated thereaction products E and F in a different way in thelinearizationstep which led to his somewhat strangeeq. (19) in which the expressionfor the equilibriumconstant is a function of the reaction orders m and p.Furthermore,the reaction rate expression, R,, shouldbe zero at x 6 indicating that in the liquid bulk theequilibriumcompositionis maintained.For the linearized rate expressionused by Onda, R, can havevalues different from zero at x 6 and only for the veryspecial case that [Alo [E& 0is this conditionalways fulfilled.The approximatesolutions has deen comparedinthat work with numerical solutions; however, this hasbeen restricted to situationswith [A], 0and lowvalues of E,.4.3. Hikita approachHikita et 01. (1982) presented an approximatesolution for the penetrationmodel and for the reaction:A B%C Daccording to the set of eqs (3). (4), (5) and (6) with initialand boundary conditions (7), (8) and (9). The followingassumptionshave been made:--the concentrationsof the species 5, C and Din theliquid near the gas-liquid interface were assumedto be constantand equal to their interfacialconcentrationsand independentof the exposuretime of the liquid to the gas (Hikita and Asai,1963),-thechemical equilibriumhas been established

Mass transferaccordingreversible chemical reactions-Iwith complexreactionto:K k,lk-, (CcloCulo)/(CAloCBlo)-A(I) ZB( P(I)RESULTS(16) and (17):-fix the bulk values on a finite z, far enough to be ofno influence on the results-numberof w-steps (time grid) equal to the numberof z-steps (spatial grid) equal to N-takep O.Ol, eq. (16).(B) transformation(26),(16) and (17):.followedbytrans-formations-take-takeequal numberp O.Ol.of w-stepsand z-stepsThe examples are:(I) physical absorption:A(g) A(1),(2) absorptionand first order irreversiblereaction:( N),chemicalA(g)- A(l)A(I) B(I)(3) absorptionwith R, kl[A]and bimolecularirreversibleTable 1. ComparisonExamplerate constant,chemicalof numericalIwith & L,[A]solutionsExamplewith 99141.85142.58142.80142.82Exact value of1.0000analyticalsolutionI .ooo510003k,:[B].As an illustrationof the performancethe calculatedenhancement factors are presentedin Table 1.From Table 1 it can be concludedthat with theimplementation of an additional transformation according to eq. (26) the accuracy of the numericalsolutions can be

with a reversible chemical reaction which could be regarded as instantaneous with respect to mass trans- fer. Also analytical solutions for both film and pen- etration theory have been presented for first-order reversible and irreversible reactions (