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A Mathematical Model for Non-catalytic

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Pergamon Computers chem. Engng Vol. 22, No. 3, pp. 459---474, 1998 Copyright © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain PlI: S0098-1354(96)00372-9 0o98-1354/98 $19.00 + 0.00 A mathematical model for non-catalytic liquid-solid reversible reactions E I. C. C. Pais a* and A. A. T. G. Portugal b a Deicam International pic, Talbot Way, Small Heath Business Park, Birmingham B 10 0HJ, UK b Universidade de Coimbra, Departamento de Engenharia Qufmica, Largo MarquSs de Pom
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  Pergamon Computers chem. Engng Vol. 22, No. 3, pp. 459---474, 1998Copyright © 1998 Elsevier Science Ltd. All rights reservedPrinted in Great BritainPlI: S0098-1354(96)00372-9 0o98-1354/98 $19.00 + 0.00 A mathematical model for non-catalyticliquid-solid reversible reactions E I. C. C. Pais a* and A. A. T. G. Portugal b a Deicam International pic, Talbot Way, Small Heath Business Park, Birmingham B 10 0HJ, UKb Universidade de Coimbra, Departamento de Engenharia Qufmica, Largo MarquSs de Pombal,3000-Coimbra, Portugal(Received 21 September 1993; revised 2 December 1996) Abstract A sharp interface model has been developed for non-catalytic liquid-solid reversible reactions. It is shown that asharp reaction interface can occur, in certain circumstances, even if the reaction is reversible. Because the pseudo-steady state approximation may not be valid for liquid-solid reactions, this assumption was not made in the model.This led to the need for numerical resolution and the model was solved using orthogonal collocation both in theunreacted core and in the ash layer. Owing to the equilibrium relationship, which holds at the reaction interface, theconcentrations of the liquid species are coupled at this location, resulting in a possibly non-linear set of algebraicequations that must be solved in order to evaluate concentrations at the boundaries. An analysis is made of theinfluence of the parameters that characterize the model. The effect of reversibility is a decrease in the driving forcefor diffusion. © 1998 Elsevier Science Ltd. All rights reserved Keywords: Non-catalytic; Reversibility; Diffusion Notation C=concentration (mol/m 3 liquid) except C ~ (mol/m 3 solid)D~=effective diffusion coefficient (m 2/s)KE=equilibrium constant ((mol/m 3) W(mol/m 3) ~A)KL=mass transfer coefficient (m/s)ntl =total number of collocation points in innerzonen,2=total number of collocation points in outerzonePM=molecular weightr----radial spatial coordinate (m)re=location of the reaction front (m)R=outer radius of the particle (m)t=time (s) v~(j,i,k)=weight of point k of zone j in the evaluation,using orthogonal collocation, of the firstderivative in point i of zonej v2(j,i,k)=weight of point k of zone j in the evaluation,using orthogonal collocation, of the secondderivative in point i of zonejx=volumetric conversion of particle* Author to whom correspondenceshouldbe addressed. E-mail:fpg@delcam.eom; Fax: +44 (0)121 766 5511.459y=adimensional concentration (C/C °)Greek letters e=porosity of the particlerh =adimensional spatial coordinate for inner zonerh=adimensional spatial coordinate for inner zone~--stoichiometric coefficient in reactionp--density of solid species (kg/m 3) Superscripts in =referred to initial concentrationO=referred to bulk concentration Subscripts A=referred to species AB =referred to species Bc--referred to the reaction frontR--referred to species RS =referred to species S1=referred to inner zone (unreacted core of par-ticle)  460E I. C. C. PAXSand A. A. T. G. PORTUGAL2=referred to outer zone (ash layer of particle) 1. Introduction Heterogeneous fluid-solid reactions can be dividedinto catalytic and non-catalytic. Both types are usuallyinfluenced to a high degree by heat and mass transferprocesses (Doraiswamy and Sharma, 1984). A generalheterogeneous reaction, where reactants and productscan be either in the solid or fluid phases, can berepresented byr/AA(f) + r/RR(S)~r/BB(f)+ r/sS(s). (1)Examples of non-catalytic fluid-solid reactions are thereduction, roasting and chlorination of ores, decomposi-tion reactions, carbonyl formation, gasification reactions(Doraiswamy and Sharma, 1984) and the causticizingreaction (Angevine, 1983; Blackwell, 1987; Don'is andAllen, 1985, 1986). Two basic types of model have beenconsidered so far: the sharp interface model (SIM) andthe homogeneous model (or volume reaction model).The SIM was srcinally proposed by Yagi and Kunii(1955) as a representation of the conversion of solidparticles by reaction with a gas. This model assumes thatreaction occurs at a sharp interface between a growinglayer of solid product and a shrinking unreacted solid core. The rate controlling step can be either internaldiffusion, gas film diffusion or chemical reaction(Levenspiel, 1972). Extensions of this model are thecrackling core model (Park and Levenspiel, 1975, 1977)and the grain model (Szekely and Evans, 1970, 1971;Sohn and Szekely, 1972) with or without varyingstructural properties (Ranade and Harrison, 1979; Geor-gakis et al., 1979). The homogeneous model holds whenthe solid is porous and the rate of diffusion of thereactant fluid is high. In this case the fluid will penetratedeeply into the solid and the reaction will take placethroughout the particle. Comparison between these twomodels has been made by several authors (Ishida andWen, 1968; Doraiswamy and Sharma, 1984). An exten-sive model review has been published by Ramachandranand Doraiswamy (1982).For the sharp interface model to be applicable it is notnecessary that the unreacted core be totally non-porous.If the reaction is irreversible, it is sufficient that the masstransfer rate be significantly lower than the rate ofreaction, and therefore whatever reactant is available viamass transfer can be assumed to react instantly. In thiscase the progress of the reaction front will depend onlocal concentration gradients rather than on parameterssuch as the surface reaction rate constant. No studieshave been found in the literature concerning thepossibility of a sharp interface model when the reactionis reversible. For the SIM to be valid in thesecircumstances, it must be ensured that no reaction occursin the ash layer or in the shrinking core. The mainpurpose of this study is therefore to determine theapplicability of such a model to reversible reactions. Ananalysis of the phenomena occurring in each zone hasalready been made by Portugal and Pais (1989) and willbe repeated here for the sake of clarity. It is shown that,under certain circumstances, a sharp interface may indeed exist for reversible reactions.On the other hand, nearly all the analytical modelresolutions found in the literature for shrinking coremodels rely on the pseudo-steady state assumption. Itsphysical significance is that the interface can beconsidered stationary at any time, while a steady statediffusion flux is evaluated to find the concentrationprofile (Doraiswamy and Sharma, 1984). This assump-tion is valid when the concentration of the reacting fluidphase species in the bulk is much less than the molardensity of the solid, that is cO pM -,,----R < 10 -3 (2) PRwhere C ° is the concentration of the fluid reactant in thebulk, PM R is the molecular weight of solid species R,and PR is the density of solid species R. It may not holdfor liquid-solid reactions when C ° has a value compara-PRble to P--~R Because the model developed in this studyis to be applied to a liquid-solid reaction, the pseudo-steady state assumption was not made so that nolimitations exist in the concentrations of the liquid phasespecies. Analytical solutions found in the literature alsoassume, for mass transfer-limited models, that theconcentration of A at the reaction interface is nil, whichis valid only if the reaction is irreversible.In this work, a sharp interface model has beendeveloped for reactions of the type described by (1) forcases (such as most liquid-solid reactions) where thepseudo-steady state approximation is not valid. Moreo-ver, reactions have been considered to be reversiblerather than irreversible. The model relies on theassumptions that mass transfer (either through ash layeror fluid film) is the limiting step, reversible reaction isinstantaneous, and spherical particle is isothermal. Ifreaction is reversible, the concentrations of the liquidspecies are coupled at the reaction interface by theequilibrium relationship. They are therefore not known apriori and must be determined during the solutionprocess. Together with the fact that the pseudo-steadystate assumption was not made, this led to the need fornumerical resolution. Solution is non-trivial since thereis a discontinuity in the concentration gradients at thereaction interface and a possibly non-linear system ofalgebraic equations must be solved in order to determinethe concentrations in all the boundaries. 2. Validity of the sharp interface model forreversible reactions In this section, the validity of the SIM model forreversible reactions is analysed. Several authors refer tothe causticizing reaction as following a moving-bound-  D A Model for non catalytic liquid-solid reversible reactions k Fluid0 C B Fig. 1. Schematic representation of the particle undergoing aSIM reaction.ary pattern (Angevine, 1983; Blackwell, 1987; Dorrisand Allen, 1985). However, no model has been found inthe literature that could represent this situation for areversible reaction such as the causticizing reaction. Thiswas therefore one of the first issues approached in thisstudy. Could a sharp interface model exist for areversible reaction? Based on a study of the fundamentalphysical and chemical mechanisms involved, it can beconcluded that such a situation is possible.Figure 1 represents graphically the classical SIMpattern, where zone 1 corresponds to the inner shrinkingcore and Zone 2 to the ash layer:A very important issue, fundamental to the validity ofthe model proposed, is, since the reaction is reversible,whether it can be guaranteed that the ash layer does notre-react (reverse reaction) to form solid reactant R again(in which case a sharp interface model would not bevalid). Let us examine in greater detail what happens inthis zone.As the particle is immersed in the bulk liquor andstarts to react, a layer of infinitesimal width of R willreact with A to produce S. At this surface, both R and Sare available and both A and B are also available. If thechemical reaction is instantaneous--a commonly usedassumption--chemical equilibrium (which determinesthe ultimate concentrations achievable) is attainedinstantly, i.e. the concentrations of A and B at this layerare such that they satisfy the equilibrium relationship. Ina simplified case, let us suppose that the equilibriumrelationship depends on the concentrations of the liquidspecies only. If, for equilibrium concentrations, KE= C~/ CAA, where r/A and r/Bare the stoichiometric coefficientsof A and B in the reversible reaction, respectively, thenthe concentration of the bulk liquid is such that there isan excess of A and a deficiency of B, i.e. C o ~/C°A A<KE. This condition must be met since the purpose of thereaction is to consume A and produce B. If, in the bulkliquid, the relationship of the concentrations were suchthat C o ~/C ° ~^>--KE, no observable reaction would takeplace.As soon as the forward reaction starts, the solidproduct S is formed and is therefore available forreaction. At the reaction interface, A is being consumed,461producing B. Both A and B were present in the bulkliquid and are present in the whole particle, due todiffusion. Because, in the bulk liquid, there is an excessof A and a deficiency of B with respect to equilibrium,and equilibrium concentrations exist at the reaction front(initially the surface of the particle), A will be trans-ferred from the bulk liquid to the particle, and B will betransferred from the particle to the bulk liquid. As Aarrives at the reaction surface, and B leaves it, theconcentrations at this point are momentarily disturbed bythe diffusion process and will tend to return (instantly) toequilibrium, consuming the excess of A by reaction withthe solid reactant R, available at the inner core. Thiscauses the movement of the reaction front towards thecentre of the particle.The relationship of the liquid concentrations (C~I C~A) ranges from an excess of A in the bulk liquid toequilibrium at the reaction interface. Although S, A andB are present in the ash layer, there is an excess of Acompared to equilibrium. The reaction is reversible andequilibria are dynamic, so both the forward and reversereactions take place in this zone simultaneously. How-ever, if an infinitesimal amount of B reacts with S, in theash layer, to form R again, the large amount of A presentwill immediately react with this amount of R. This is dueto the fact that excess A, with respect to equilibrium,means in practical terms that the rate of the forwardreaction is higher than the rate of the reverse reaction. Asa consequence, although S is present, the reversereaction is not observed, in global terms, because thatwould draw the concentrations even further fromequilibrium, increasing the excess of A with respect to B.In other words, the excess of A inhibits the reversereaction in the ash layer. Therefore, in overall terms(forward-reverse reaction), no observable reactionoccurs in this zone since any R formed is immediatelyconsumed and the forward reaction cannot proceed anyfurther since there is no solid reactant R available; alayer of S alone will form. Please note that thisphenomenon does not depend on the value of theequilibrium constant.If the reaction is irreversible or the inner core is non-porous, then A cannot diffuse into the inner core.However, in this study, the reaction is taken to bereversible and the inner core is porous. Since equilib-rium is assumed at the reaction interface and thereforethe concentration of A at this location is not zero, A isalso present in the inner core. An equally important issueis whether R does not react with A in this zone,producing S and therefore invalidating the SIM patternagain. For the set of parameters used in this study, theforward reaction does not take place in this zone, inobservable terms, because there is an excess of B anddeficiency of A, compared with equilibrium, that inhibitthe progress of the forward reaction. This is representedquantitatively later, in Section 5 (Fig. 4). While it can beensured that the concentrations in the ash layer inhibitthe reverse reaction from taking place, the same cannotbe guaranteed for the inner core with respect to the directreaction. It does happen for this specific set of parame-  462ters, such that the effective diffusion coefficient of B istwice as large as for A (estimate justified later in thepaper) and therefore it is easier for B to diffuse to thecentre of the particle. This leads to an excess of B, in theinner core, compared with equilibrium. For the values ofthe effective diffusion coefficients used in this study, thevalue of the equilibrium constant, i.e. the degree ofreversibility, does not seem to influence this phenome-non. This was checked for trial runs of cases with lowervalues of Ke (namely, all the cases represented in Fig. 8also resulted in an excess of B in the inner core). A trialrun was made for KE=500mol/m 3, for which theconcentration of A at the reaction interface is sig-nificantly higher than the concentration of B at the samelocation (reaction is highly reversible), which alsoresulted in an excess of B in the inner zone. However, ifthe effective diffusion coefficient of A in zone 1 isgreater than the effective diffusion coefficient of B in thesame zone, then A diffuses more easily into the centre ofthe particle than B and a deficiency of B may exist in thiszone, leading to the occurrence of forward reaction in allor part of the shrinking core. This can be observed if thevalues of the effective diffusion coefficients for A and Bare swapped, i.e. value for A twice as large as for B (testruns have been made for this case). For this combinationof parameters, the SIM pattern is not valid, although theconcentration of A at the reaction interface is muchlower than the concentration of B, since there is anexcess of A in the inner core. Because the concentrationsof A and B in the inner core are determined by diffusionalone, their relationship depends on the effective diffu-E I. C. C. PAre and A. A. T. G. PORTUGAL sion coefficients arid on the driving force for diffusion.In this study, the driving force is determined by theconcentrations at the reaction interface, which in turndepend on the equilibrium constant and stoichiometriccoefficients. Since the model was solved numerically,the conditions for a sharp interface will not be gener-alized in quantitative terms. However, its existencedepends on the value of the effective diffusion coeffi-cients of A and B in the inner core, and possibly on thevalue of the equilibrium constant and stoichiometriccoefficients. During simulation, the existence of anexcess of B in the inner core must therefore be verifiedfor the model to be valid.Summarizing, in the ash layer, the existing solid is S,and A and B are present. However, the reverse reactionis inhibited by the excess of A with respect toequilibrium. In the inner core, for the set of parametersused, the existing solid is R, and A and B are alsoTable I. Parameters used for the simulation of the base caseC ° 1150 C o 600C~ 25,700e~ 0.15 e2 0.05 DeAl 1X 10 -ii Dea I 2X 10 -ii DeA2 3.33X 10 -12 DuB1 6.67X 10 -12 KLA 5 X 10 -4 KLa 1 X l0 -3KE 50X 103 R 5>(10 -5~/A= r/R 1 r/a 2 PM R 74.09 PM s 100.09 PR 2240 Ps 2710 1.0 o t = 2s0.8 u t = 1200 s A t = 2240 s= t = 2880 s o 00.6 0.4 0.2'r /R O 0.0~ 0.0 0.2 0.4 0.6 0.8 1.0 r/R Fig. 2. Calculated adimensionalized COl- profiles for various values of time t. For clarity no interpolated profiles have beencalculated and only collocation points are represented and connected by straight lines.
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