Chemical Engineering Science 56 (2001) 659}666               Multilevel modelling of heterogeneous catalytic reactors     ...
660                                    B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666                     ...
B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666                                         661volume-averaging...
662                                    B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666                     ...
B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666                           663Macropores * second ( ) level:...
664                                 B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666order N and M, dened as ...
B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666                           665                              ...
666                                      B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666Greek letters      ...
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  1. 1. Chemical Engineering Science 56 (2001) 659}666 Multilevel modelling of heterogeneous catalytic reactors B. G. Lakatos* & & Department of Process Engineering, University of Veszprem, H-8200 Veszprem, HungaryAbstract The paper presents the fundamental elements of the multilevel mathematical models of heterogeneous catalytic reactors. Thesemodels are derived by means of the volume-averaging method, appropriately modi"ed for the dispersed systems, and providea theoretical basis for modelling heterogeneous systems across scales. The properties of the models are shown and analysed.A three-level model of a well-stirred catalytic reactor with bidispersed catalysts is presented. The model equations are solved by meansof multilevel collocation. The transient behaviour of the reactor is investigated by means of computer simulation. 2001 ElsevierScience Ltd. All rights reserved.Keywords: Multilevel model; Volume averaging; Well-stirred catalytic reactor; Bidispersed catalyst; Multilevel collocation1. Introduction hierarchy of the system we can reformulate and general- ise these models about the so-called multilevel dynamical In reaction engineering, pseudo-homogeneous models models, providing in this way a suitable theoretical basisare often used for modelling heterogeneous catalytic re- for modelling and computing the heterogeneous systemsactors, taking into consideration the e!ects of intrapellet across scales.processes only by means of e!ectiveness factors. When, In the paper, some fundamental aspects of the multi-however, the time scales of processes inside the catalyst level dynamical models are presented. The hierarchy ofpellets and those of the continuous #uid phase are of the modelling levels is derived and analysed. A three-levelsimilar order of magnitude then this approach can lead model is derived for well-stirred catalytic reactors withto signi"cant errors, especially under dynamic condi- bidispersed catalysts and some transient phenomena aretions, as was revealed by Ramkrishna and Arce (1989). studied by simulation.The two-phase heterogeneous models represent a morerational basis for modelling heterogeneous systems sincethese models handle all processes occurring simulta- 2. Multilevel dynamical modelsneously in the #uid phase and inside the catalysts. Thesemodels were used mainly for kinetic studies (Dogu 2.1. Volume averaging for dispersed phase& Smith, 1975; Datar, Kulkarni & Doraiswamy, 1987;Burghardt, Rogut & Gotkowska, 1988; Keil, 1996; Mol- K Consider the two-phase heterogeneous systems shownler & OConnor, 1996) and parameters estimation schematically in Fig. 1 in which the interfacial surfaces(Imison & Rice, 1975; Do & Rice, 1982; Kim & Chang, between the phases are complex and may vary in time.1994), while Arce and Ramkrishna (1991), Trinh and Let be a scalar quantity which in the - and -phases isRamkrishna (1994) and recently Vernikovskaya, Zagor- denoted, respectively, by and . Then the variation of ? @nikov, Chumakova and Noskov (1999) applied them for inside the phases is described by the balance equationsstudying dynamic phenomena. In all cases, the hetero- *geneous models were determined on phenomenological G # ( j )" , i" , , (1) G *t G Gbasis. However, taking into consideration the discretenature of the catalyst phase and the characteristic where j is the #ow density and is the volumetric source G G density of quantity . The transport across the - G interface is described by the boundary conditions * Correspondence address: Department of Process Engineering, Uni-versity of Veszprem, P.O. Box 158, H-8201, Veszprem, Hungary. n (j ! w)#n ( j ! w)" , (2) ? ? ? ? @ @ @ @ ?@0009-2509/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 2 7 3 - 6
  2. 2. 660 B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666 volume < for the -phase that ? < "constant ¸ (5) ? ? for which the conditions ;¸ ; , (6) ? ? ? J (7) @ ? are valid then the single-level mathematical model can be converted into a two-level mathematical model by means of the modied volume averaging technique. In this case, the phase average of quantity in the -phase is dened in the usual way (Whitaker, 1967; Slattery, 1967; Gray, 1975) 1 1 1 2 d d, (8) ? ? ? Fig. 1. Two-phase systems with di!erent -phase scales. 4 4? where # and is the characteristic function ? @ ? of the set of points of the -phase, but the phase average of quantity in the -phase takes the formwhere w is the velocity of the interface, while denotes ?@ 4@the surface source density of quantity . 1 2 1 2 n d , (9) G @ @ . @ @ The set of equations (1) and boundary conditions (2) form a microscopic mathematical model of the two- where 1.2 denotes the average of over a particle:phase systems under investigation. In fact, this is . a single-level model since all variables are represented on 1 1 2 d. (10)the same, microscopic, level. However, such mathemat- @ . @ @ 4@ical description is very complicated and impracticable. In Eq. (9) n is the population density function of theOne procedure for working around this problem is the elements of the dispersed -phase, termed next particles,use of local averaging method (Whitaker, 1967; Slattery, dened as n : R;R;RPR by means of which1967; Gray, 1975) by means of which Eqs. (1) are aver- n(x, t, ) d expresses the number of particles havingaged over a representative local volume to obtain @ @ volume ( , #d ) at moment of time t in the aver-a simplied, well-structured, but usually not closed set of @ @ @ aging volume associated with coordinate x.averaged equations. Applying some closure scheme re- A second average that is important in this method issults in an averaged formulation of the dynamic behav- the intrinsic phase averageiour of multiphase systems. This formulation is alsoa single-level mathematical model since it is expressed 1 1 2@ 1 2, (11)purely in terms of the averaged, macroscopic, quantities. @ @ @ This procedure, however, leads to meaningful results where is the volume fraction of the -phaseonly if (1) the characteristic lengths and of both @ phases are of similar order of magnitude, where is the 4@K n d (12)distance over which varies signicantly and is the @ @ @characteristic length of the phases; (2) there exists such averaging volume while the partial volume of the -phase is given as constant ¸ (3) 4@K n d . (13) @ @ @that the three characteristic lengths satisfy the inequali- ties On the basis of Eqs. (9)}(13), the volume-averaging theorem and the general transport theorem can also be ;¸; . (4) written in terms of the population density function of theThese conditions may be satised for systems in Fig. 1a -phase.and b, but not for system shown in Fig. 1c in which the -phase is, in essence, a dispersed phase. Accordingly, 2.2. Two-level model for heterogeneous systemsnext the -phase will be treated as a continuous one.Here, the characteristic lengths of the phases may di!er Applying the volume-averaging operator (8) tosignicantly, but if it is possible to dene such averaging Eqs. (1), making use of the appropriate forms of the
  3. 3. B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666 661volume-averaging theorem and of the general transporttheorem, and considering the boundary conditions (2),for the system in Fig. 1c, the following set of equationsare obtained. For the continuous -phase we obtain one equation: * 1 2?1 2? ? ? #1 2? ( 1 2?1 2?) ? *t ? ? ? ? ! ( 1D 2?1 2?)! 1 2? ? ? ? ? ? 4@L d ! 1 2 @ n d @ . dt @ 4@L Fig. 2. Splitting a single modelling level into two levels of a two-phase # n j n dA d @ @ @ system. @ 4@ 4@L ! n n dA d . (14) ?@ @ @ The set of equations (14)}(17) is termed two-level @ 4@ model, since the dependent variables and system para-The left-hand-side terms of Eq. (14) describe the motion meters that are involved in it can be divided into twoof quantity in the continuous phase, while the right- equivalent classes, termed here modelling levels. Vari-hand-side terms describe the changes of due to the ables 1 2? and n, for instance, are dened as variables ofvariation of volume of the particles, and to the transfer of ? the macroscopic level, while remained a variable of through the surfaces of particles, and production of @ the microscopic level. The former is termed higher mod- due to some surface source, respectively. On the left- elling level, whilst the latter is the lower modelling level.hand side, the conductive term was closed by means of Therefore, a single-level model was converted intothe usual dispersion-type closure model (Carbonell a two-level one as it is shown in Fig. 2. The particles on Whitaker, 1983; Quintard Whitaker, 1993). the higher modelling level are represented as point sinks For the dispersed -phase we obtain two equations. immersed and moving in the continuous phase, while onThe population balance equation the lower level are described as material bodies of nite*n * d volume which have their own inner world, often with # (1 2 n)# @ n 1 2 n (15) complex processes.*t @ . * dt @ . @ In the two-level model (14)}(17), a number of processesdescribes the behaviour of particles immersed and mov- occurring on two di!erent scales is collected and related.ing in the continuous phase, while the equation For instance, it describes three-dimensional motions in the physical space on two scales, particles are assumed to * @ # ( #q ) (16) be of general form, or both surface and volumetric source @ *t @ @ @ @ @ terms are present. This model, being in fact cognition- oriented one, provides a framework for modelling hetero-describes the motion of quantity inside the particles. geneous systems across scales, as well as for their multi-Here, q denotes some non-convective component of @ level computations. A further advantage of this model isthe #ow density that may be of complex nature, depend- that during the course of developing it reliable estimatesing on the structure of particles. Eqs. (14)}(16) are com- can be obtained for the model errors, so that purpose-pleted with the constraint for the volume fractions of oriented models, useful for engineering computations,phases can be derived by its successive reduction allowing at the 4@L same time exact error analysis. # n d 1 (17)? @ @ 2.3. Generalisation: multilevel dynamical modelsas well with the appropriate boundary and initial condi-tions. The boundary conditions for Eq. (14) describe the Generalisation of the two-level models leads to theconnection of the system with the environment while the concept of multilevel dynamical models. In fact, when theboundary conditions for Eq. (16) describe the connection particles are of complex nature, composed, from instance,between the interior of a particle and its continuous from a great number of particles of much smaller size,phase environment. The details of the derivation of Eqs. then the volume averaging process on this second, small-(14)}(17), as well as of the derivation of equations of er, scale may also lead to meaningful and much simplermotion of the system will be published elsewhere. models. In this way a three-level model can be dened.
  4. 4. 662 B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666 We note here that it is often useful to describe even the continuous -phase by means of a two-level model, vis- ualising it as consisting of a large number of #uid ele- ments that simultaneously interact with each other and with the particles of the dispersed -phase. Such model was derived and presented recently by Lakatos, Ulbert and Blickle (1999) for describing micromixing in continu- ous crystallizers. 3. Three-level models of catalytic reactors with bidisper- sed catalysts Consider a continuous catalytic reactor in which an APProduct-type reaction occurs and both the #uid andFig. 3. Coupling the modelling levels in a three-level mathematical solid phases can be considered perfectly mixed. Let therepresentation. catalyst (macro-) particles be agglomerates of small por- ous (micro-) particles what means that the catalyst is of biporous nature. Let us assume that there exists someHere, the modelling levels are coupled with each other as resistance to the mass transfer at the #uid phase}macro-it is shown schematically in Fig. 3. particles interface, adsorption of the reagent A occurs at Based on the extension of a two-level model into the the surface of the microparticles with Freundlich-iso-three-level representation, a denition of the n-level mod- therm, and the rst-order reaction occurs only in theels can be formulated as follows: microparticles. If the variables x , x ,2 and equations F , F ,2 Considering the system in terms of the multilevel mod- with the corresponding boundary conditions B , B ,2 els, here the rst modelling level is formed by the com- of a distributed parameter model can be divided into pletely stirred #uid-phase. The second-level modeln equivalent classes that can be grouped into a hierarchy describes the di!usion-type motion in the macropores,in such a way that and the third modelling level is in fact the mathematicalM :F (x , f [x ]), B (u), description of the di!usion and reaction in the micro- pores. Denoting the third phase as -phase, the rst ( )M :F (x , f [x ]), B (x ), level is described by the averaged equations (14) and (15), M :F (x , f [x ]), B (x ), the second ( ) level is also described by similar averaged equations obtained by replacing the indices by and $ by , and the third level is described by Eq. (16)M :F (x ), B (x ), (18) replacing the index by . Under such conditions, the L L L L L purpose-oriented model of the system, obtained throughthen the model is called n-level (dynamical) model with a number of simplifying steps, and with the furtherthe notion assumptions:x Yx Yx Y2Yx , E the reactor is isotherm; L E the macroparticles can be modelled as spheres, cylin-M YM YM Y2YM , (19) L ders or plates, while the microparticles can bewhere Y denotes the relation higher level } lower level, modelled as spheres;F are the equations of the ith level, f are functionals of E both the macro and microparticles are monodispersed G Gtheir arguments transmitting the in#uence of the (i!1)th with constant characteristic sizes R and R , respec- @ Alevel to the ith level, while B are the boundary conditions tively; Gtransmitting the in#uence of the ith level to the (i!1)th E the catalytic activity is constant,level. consists of the following equations and boundary The multilevel dynamical models are characterised by conditions:a number of special properties because of the character-istic hierarchy of the levels expressed by Eqs. (18). For Fluid-phase * rst ( ) level:instance, they exhibit some memory features and a char- dc m(1! ) D *cacteristic structure of eigenvalues due to the e!ects the F: ? q(c !c )! ? @ @ @ . ? ? dt ?GL ? R *r @ @lower levels, often leading to complex dynamic phe- @ @ P 0nomena. (20)
  5. 5. B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666 663Macropores * second ( ) level: and dimensionless parameters *c D * *c D m(1! ) @ @ @ rK @ p ? @, p ? @p , @ *t rK *r @ *r qR @ @ @ @ ?F : (21) @ 3(1! ) D R c 3(1! ) D *c p @ A $ A @, p @K p , ! @ A A A , c D R 3(1! ) R *r A A @K @ A @ $ A A P 0 k R kRsubject to the boundary conditions p P @ , p R @ , p cI . D D @K A @ A @ *c *c kB : @ 0, @ R [c !c @ @ ]. Eqs. (20)}(25) are written in dimensionless form as @ *r @ @ *r @ @ D ? @P 0 @ P 0 @ P 0 d * (22) p ? ! !p @ , (28) d ?GL ? *Micropores * third ( ) level: E * 1 * * * *c D * *c @ @ !p A , (29)F: A A A r A !k c (23) * * * * A A *t r *r A *r P A K A A A * p *subject to the boundary conditions A A !p (30) * * A *cB: A 0, c A A a cI . (24) A *r A H A P 0 $ @ subject to the boundary conditions H P 0 In general, the initial conditions are * * @ 0, @ p [ ! ], (31) * * ? @ Ec (0)c , c (r ,0)c (r ), E E ? ? @ @ @ @ * *c (r , r ,0)c (r , r ). A A @ A A @ (25) A 0, A p I (32) * * @ Naturally, the rst-level boundary conditions are in- K Kcluded into the equation F since it describes a com- and to the initial conditions ?pletely stirred #uid}solid suspension. Equations F and ? (0) , ( ,0) ( ), ( , ,0) ( , ). (33)F have been simplied signicantly because of the @ ? ? @ @ A Amonodisperse nature of particles, since in this case the The problem (28)}(33) was solved numerically by de-population density functions have the delta function veloping a two-level orthogonal collocation scheme withforms n N (r !R ), i , . By the same reasons, G G G G the aid of which the partial di!erential equations mightthe functionals f and f transmitting the e!ects of the ? @ be converted into a system of ordinary di!erential ones. and levels to the and levels, respectively, have According to the method of orthogonal collocation (Vil-taken the forms ladsen Stewart, 1967), approximate solutions were *c (t) constructed for and in terms of orthogonal poly- @ c (r , t) (r !R ) dr @ A *r @ @ @ @ @ @ @ (26) nomials of degree N and M in and , respectively, @ P 0 written in the formsand , I ( , ) d ( ) L *c (r , t) @ L (34) A @ c (r , r , t) (r !R ) dr . (27) *r A @ A A A A L A PA 0A and + ,4. Solution to the model equations I ( , , ) g ( ) L K, (35) A KL K L Before solving numerically the problem (20)}(25) we where the coe$cients d , n1, 2,2, N#1 andrewrite it in dimensionless form by introducing the di- L g , m1, 2,2, M#1, n1, 2,2, N#1 are un-mensionless variables KL known functions of the dimensionless time . Note that r r D polynomials (32) and (33) satisfy the boundary conditions @ , A , t @ , at the values 0 and 0 due to their appropriate R R R @ A @ forms. c The N and M interior collocation points were selected A , c c A ?K K to be the roots of the shifted Legendre polynomials of $
  6. 6. 664 B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666order N and M, dened as r¸ Q (r)¸ Q (r) dr , n, m0,1, 2,2, (36) L K LK where is the Kronecker delta. The (N#1)-th and LK(M#1)-th collocation points were selected 1 ,and 1. + Substituting functions (34) and (35) into Eqs. (29) and(30), respectively, and requiring those to be satised at allcollocation points, as well as taking into considerationthe boundary conditions (31) and (32), together with theEq. (28), a set of (N#1) (M#1) ordinary di!erentialequations was obtained. The resulting set of ODE was solved by means of thesemi-implicit Runge}Kutta method (Villadsen Michel-sen, 1978). The program was written in C/C# imple-#, Fig. 4. E!ects of the shape of macroparticles and of the rate of themented on PC, and was used for investigating di!erent #uid-particle mass transfer on the transient responses of the reactor.transient phenomena of the catalytic reactors with bidis-persed catalysts by computer simulation. In the simula-tions, the following base-case values of parameters wereused: p 1.65, p 0.2 m, p 2, p 1.2, p 15, p R, p 1, k0.5. 5. Simulation results and discussion First we consider the system without reaction, i.e. thevessel is assumed to be operated as an adsorber withbidispersed adsorbents. Here the transient responses ofthe system to the step changes of the adsorbate werestudied. The numerical experiences have shown that in theinterval 10(¹ /¹ D R/D R(10 of the ratio @ A A @ @ A Fig. 5. Transient processes in the #uid, in the macroparticles and in theof the time scales of macro- and microparticles, the tran- microparticles.sient behaviour of the adsorber was a!ected either by themacro- and micropore mass transport. At larger values ofthis ratio the micropore transport, while for smaller tration proles in the macro- and microparticles arevalues the macropore transport was the dominant factor shown for two moments of dimensionless time .a!ecting the transient responses. Consider now the system with chemical reaction, i.e. Fig. 4 shows the in#uence of the shape of macropar- the vessel is assumed to be operated as a catalytic reactorticles on the relative uptake given as with bidispersed catalysts, but with di!erent boundary O /F@ EQ ds conditions. Namely, we assume that the reagent is #ow-u( ) /E E , (37) ing through the vessel without any catalyst under sta- /F@ EQ ds /E E tionary conditions, when suddenly a given amount offor two di!erent values of parameter p expressing the empty catalyst is introduced into the vessel. If we can in#uence of the rate of mass transfer between the #uid assume that the disturbance of the hydrodynamics of thephase and macroparticles. In the case of negligible mass vessel is negligible, then the initial conditions of thetransfer resistance, only small in#uence of the shape of problem (28)}(32) can be written in the formmacroparticles is observed while for small rate of mass (0) , ( ,0)0, ( , ,0)0. (38)transfer the macroparticle shape becomes an important ? ? @ Afactor of the transients. Numerical experiences have shown that minima might be The whole transient process occurring simultaneously detected in the #uid-phase concentration of the reactor inat the three modelling levels of the system, computed for a wide range of parameters, similarly to that observed forthe empty vessel initial conditions and step change of the monoporous catalysts (Imison Rice, 1975; Do Rice,adsorbate at the inlet, is presented in Fig. 5. The concen- 1982). The place of minima on the time axis and their
  7. 7. B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666 665 6. Conclusions The multilevel mathematical models, presented in the paper in general terms, were derived by means of the volume-averaging method, appropriately modied for the dispersed systems. It might be applied for de- scribing the heterogeneous catalytic reactors because of the discrete nature of the catalyst phase and of the corresponding hierarchy of the system. The multilevel dynamical models, in their original cognition-oriented form provide a suitable theoretical basis for modelling and computing the heterogeneous systems across scales. A three-level dynamical model of a well-stirred cata-Fig. 6. The in#uence of the voidage of the reactor on the minima of#uid concentration. lytic reactor with bidispersed catalysts was presented and analysed in terms of the multilevel models. The model equations were solved by means of a two-level ortho- gonal collocation scheme. This method has proved suit- able for computing the concentration distributions in both the macropores and micropores in transient states of the reactor. By means of computer simulation several transient phenomena have been considered, showing the in#uence of the shape of macroparticles, of the mass transfer resistance at the #uid}macroparticle interface, and of the voidage of the reactor on the transient behav- iour of the reactor. Notation a parameter of the Freundlich isotherm $Fig. 7. Macropore and micropore concentration distributions at di- A surface, mmensionless time 10. C concentration, kg m d coe$cients of the polynomial (Eq. (34)) L D di!usion coe$cient, m svalues are strongly a!ected by the relative amount of the D matrix of the di!usion coe$cientscatalyst as it is seen in Fig. 6. This gure shows the g coe$cients of the polynomial (Eq. (35)) KLhistory of the #uid concentration for di!erent values of j #ow density, quantity m sthe reactor voidage . The capacity of catalyst increases k exponent of Freundlich-isotherm ?with decreasing voidage and the minima arise at smaller k reaction rate coe$cient, s Pmoments of time. Fig. 7 presents the concentration distri- k mass transfer rate coe$cient, s Rbutions of the macropores and micropores for two di!er- ¸Q shifted Legendre-polynomial of order m Kent ratios of the time scales of macro- and microparticles ¸ characteristic length scale of averaging, mat the dimensionless time 10. This gure illustrates n population density function, no mwell that for ¹ /¹ +10 the micropores are lled with n normal unit vector @ Areactant continuously as compared to the macropore p dimensionless parameter Hconcentration distribution, while in the case of q volumetric #ow rate, m s¹ /¹ +10, this concentration is practically zero q #ow density, quantity m s @ Asince the reactant di!used into the micropores is com- r radial coordinate, mpletely consumed by the reaction. R radius, m The minima presented in the reactor responses arise t time, sbecause of the memory e!ects of the lower levels repre- u relative uptakesented by the macro- and microparticles. The di!erences ¹ time scale of di!usion, sin the #uid concentration seem to be characteristic for v velocity, m sapplying it into some preliminary estimation of the sys- volume, averaging volume, mtem parameters. w velocity of the -interface, m s
  8. 8. 666 B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666Greek letters Datar, A., Kulkarni, B. D., Doraiswamy, L. K. (1987). E!ectiveness factors in bidispersed catalysts: The e!ect of di!usivity variations. volume fraction Chemical Engineering Science, 42, 1233}1238. Do, D. D., Rice, R. G. (1982). The transient response of CSTR dimensionless macropore radius containing porous catalyst pellets. Chemical Engineering Science, 37, dimensionless concentration 1471}1475. characteristic microscopic length scale, m Dogu, G., Smith, J. M. (1975). A dynamic method for catalyst characteristic macroscopic length scale, m di!usivities. AIChE Journal, 21, 58}69. dimensionless micropore radius Gray, W. G. (1975). A derivation of the equations for multiphase transport. Chemical Engineering Science, 30, 229}233. volume source density Imison, B. W., Rice, R. G. (1975). The transient response of CSTR to density, kg m a spherical particle with mass transfer resistance. Chemical Engin- surface source density, quantity m s eering Science, 30, 1421}1423. intensive (scalar) quantity Keil, F. J. (1996). Modelling of phenomena within catalyst particles. characteristic function of a set Chemical Engineering Science, 51, 1543}1568. Kim, D. H., Chang, K. S. (1994). Time-domain solutions of un- steady-state mass transfer models for biporous catalyst in a continu-Indices ous-#ow gradientless reactor. Chemical Engineering Science, 49, 2173}2185.m maximal value Lakatos, B. G., Ulbert, Zs., Blickle, T. (1999). Population bal- -phase, #uid phase ance model for micromixing in continuous crystallizers. Pro- ceedings of the 14th Symposium on Industrial Crystallization, -phase, macropore Cambridge. -phase, micropore Moller, K. P., OConnor, C. T. (1996). The measurement of di!usion K -interface in porous catalysts using a CSTR. Chemical Engineering Science, 51,1.2 volume-averaging operator 3403}3408. scalar product Quintard, M., Whitaker, S. (1993). Transport in ordered and dis- ordered porous media: volume-averaged equations, closure prob-J of the same order of magnitude lems, and comparison with experiments. Chemical Engineering Science, 48, 2537}2564. Ramkrishna, D., Arce, P. (1989). Can pseudo-homogeneousAcknowledgements reactor models be valid?. Chemical Engineering Science, 44, 1949}1966. Slattery, J. C. (1967). Flow of viscoelastic #uids through porous media. The author would like to thank the Hungarian Re- AIChE Journal, 13, 1066}1071.search Foundation and the Ministry of Education of Trinh, S., Ramkrishna, D. (1994). Feasibility of pattern formationHungary for supporting this work under Grants in catalytic reactors. Chemical Engineering Science, 49,T 023183 and FKFP 0520/1999, respectively. 1585}1599. Vernikovskaya, N. V., Zagornikov, A. N., Chumakova, N. A., Noskov, A. S. (1999). Mathematical modeling of unsteady-state operation taking into account adsorption and chemisorption pro-References cesses on the catalyst pellets. Chemical Engineering Science, 54, 4639}4643.Arce, P., Ramkrishna, D. (1991). Pattern formation in catalytic Villadsen, J., Stewart, W. E. (1967). Solution of boundary value reactors: the role of #uid mixing. AIChE Journal, 37, 98}110. problems by orthogonal collocation. Chemical Engineering Science,Burghardt, A., Rogut, J., Gotkowska, J. (1988). Di!usion coe$cients 22, 1483}1501. in bidisperse porous structures. Chemical Engineering Science, 43, Villadsen, J., Michelsen, M. L. (1978). Solution of diwerential equation 2463. models by polynomial approximation. Englewood Cli!s, NJ: Pren-Carbonell, R. G., Whitaker, S. (1983). Dispersion in pulsed systems- tice-Hall. II. Theoretical developments for passive dispersion in porous Whitaker, S. (1967). Di!usion and dispersion in porous media. AIChE media. Chemical Engineering Science, 38, 1795}1802. Journal, 13, 420}427.

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