Influence of the adsorbent particle size distribution on the adsorption parameters from stirred tank dynamic experiments

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1 Influence of the adsorbent article size distribution on the adsortion arameters from stirred tank dynamic exeriments Proceedings of Euroean Congress of Chemical Engineering (ECCE-6) Coenhagen, Setember 2007 Influence of the adsorbent article size distribution on the adsortion arameters from stirred tank dynamic exeriments J. A. Hernandez, a J. O. Marroquin b, B. Castro b, G. C. Laredo b, J. A. Ochoa c a Centro de Investigación en Ingeniería y Ciencias Alicadas, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamila, C. P , Cuernavaca, Morelos, México. b Instituto Mexicano del Petróleo, Programa de Tratamiento de Crudo Maya. Eje Central Lázaro Cárdenas 152, México. D. F. C. P México. c Área de Ingeniería Química, Universidad Autónoma Metroolitana, Unidad Iztaalaa, Av. San Rafael Atlixco 186, Col. Vicentina, México, D. F. CP México. Abstract The arameters that affect the searation of n-hetane from a binary mixture conformed of n- hetane and iso-octane, by adsortion in a microorous material called IMP-C12 are resented in this aer. The rocedure involves a mass transfer model that considers a mean article size calculated from the article diameter distribution. Exerimental adsortion curves of n-hetane were determined in a stirred tank by reviously robed that iso-octane is not adsorbed by the IMP-C12. The system of nonlinear differential equations was solved by the finite differences method. The model reroduced with good accuracy (R=0.99) the exerimental adsortion kinetics with a determined mean article size obtaining a Biot number and an intraarticle diffusivity coefficient. In order to redict the n-hetane adsortion kinetics and validate the model, solute transfer simulations were erformed. The results (simulations and exerimental kinetics test) were in good agreement rovided that the rocess uses the same article diameters distribution. The influence of the mean article size in the adsortion kinetics was studied considering different article diameters distribution. The results showed that article size is essential to redict with good accuracy the adsortion curves, therefore it is extremely imortant to have a homogeneous (size and shae) article distribution to determine adsortion kinetics. Keywords: Modelling; mass transfer; adsortion kinetics; article size distribution 1. Introduction Today, adsortion rocess lays a key role in modern industries esecially in the field of environmental rotection engineering, and is alied widely in chemical, biochemical and agro-alimentary industries (Hui et al., 2003, Montesinos et al., 2001, Viard, 1991). In order to redict or evaluate the erformance of an industrial searation rocedure involving

2 J.A. Hernandez et al. adsortion, the mathematical descrition of the dynamic rocess is required. The acked column with orous articles as adsorbent is a conventional oeration in the industries for mixture searation. Normally, in the common oeration, for modeling adsortion kinetics it is necessary to know the model arameters (i.e. adsortion equilibrium arameters and effective intraarticle diffusivity). Exerimental data from tank systems are used to obtain equilibrium arameters and transort arameters for adsorbents. Although, while erforming batch adsortion exeriments, the estimation of the intraarticle diffusivity becomes more difficult when the stirring seed is slow, in which the liquid-solid interface mass transfer resistance is not negligible. In the other hand, the searation of linear and branched alkanes resent in hydrocarbons mixtures is a challenge for the etroleum industry. However, the microorous materials have attracted the attention in the field of environmental rotection engineering (Do and Do 2002, Chuang et al. 2003), as a valuable adsorbent medium for removing numerous unnecessary substances from the gas-liquid hase. Consequently, the microorous materials may be used in the searation of hydrocarbons mixtures. Mathematical modeling of adsortion kinetics has been reorted by a number of researchers who consider the henomenon as a diffusive transort from liquid/gas hase to solid hase into adsorbent article volume (Qinglin et al., 2003, Li et al., 2003, Kaczmarski and Bellot, 2003, Alay et al., 2003, Hui et al., 2003, Lee et al., 2003, Yang et al., 2003, Marroquin et al., 2002, Montesinos et al., 2001, Reid and Thomas, 2001). This henomenon has been modeled with different levels of comlexity considering two different ways for solving the nonlinear equation system: analytical and numerical solution. Usually adsortion model in the batch system considers two couled simultaneous nonlinear differential equations, one for the liquid hase and another for the solid hase. Normally most systems under investigation behave as comlex henomena, where the effect of nonlinearity of the equilibrium isotherm is reresented by assuming that the effective diffusivity coefficient deends on the concentration. The only way to solve such models is by numerical methods as finite differences or finite elements (Alay et al., 2003, Kaczmarski and Bellot, 2003, Lee et al., 2003, Yang et al., 2003, Montesinos et al., 2001). These models are of great imortance in the scientific study of adsortion kinetics mainly to determine non-easy-measured arameters. However, the nonlinear equations system can be simlified and solved analytically and therefore, the most common assumtions are the following: the equilibrium isotherm is linear, there is no article volume change, there is no susension volume variation in the batch system, and the effective intraarticle diffusivity is considered as indeendent of the solute concentration. Marroquin et al., (2002) reorted an analytical solution of the equations that govern the reactant transort in a well stirred tank that contains articles where diffusion and first-order reaction occur. A Lalace transform method was used to solve the reactant transort equations in the fluid and article. They concluded that these equations can be used to determine the effect of mass transfer resistances on transient exeriments and to obtain a maing of situations when these kinds of effects are imortant. Recently, Kaczmarski and Bellot (2003) have evaluated the effect of article size distribution and the variation of article orosity on the mass transfer kinetics. The authors concluded that the error in the rediction of utake curves decreases when the article diameters distribution (obtained aroximated by mean article diameters) is reduced. However, a detailed analysis is necessary to indicate recisely the accetable intervals of the article size distribution. The objective of the resent work was to evaluate the imortance of the articles size distribution on the adsortion arameters (equilibrium arameters and effective intraarticle

3 J.A. Hernandez et al., diffusivity) of n-hetane from binary mixtures (n-hetane/isooctane) into IMP-C12 in a stirred tank. Additionally, an analysis was made to test the efficiency of the mathematic model to redict the adsortion rocess in a batch system. The validation of the model was made with exerimental data obtained by the stirred tank. 2. Materials and methods Theoretical art In Fig. 1 the system is shown. For the molecules it is assumed that the susension is well mixed. The orous articles of different size are considered sherical and susended in the fluid; therefore, only radial transort takes lace through the orous articles. We assumed that the transort and adsortion in all different size articles is reresented by the mean diameter (see Eq. 13). Figure 1. Exerimental stirred tank and fluid samle with orous susended articles. According to Li, et al., (2003), Qinglin, et al., (2003), Marroquin, et al., (2002), Do, (1998), Viard, (1991), and Ruthven, (1984) the mass transort equations that govern the adsortion kinetics in a well mixed batch system that contains sherical orous articles where the interarticle diffusion is considered, the solute concentration variation in the adsorbent orous article is described by the following equation: C 1 C C 2 + κ ( 1 ) = Dr 2 t r r r t 0 r r (1) and to describe the solute concentration change in the bulk fluid when there is change of volume:

4 J.A. Hernandez et al. dcβ dβ Af β + Cβ = k f ( Cβ r= r Cβ ) dt dt V (2) C In Eq. (1), is the article void fraction, and C κ are the solute volume average concentrations in the fluid and adsorbent in the orous articles resectively, reresents the intra-article effective diffusivity and r is the radial coordinated sherical in the articles. β Cβ In Eq. (2), is the fluid volume fraction of the susension, reresent the solute concentration in the well mixed fluid, volume occuied by the susension, A f k f D is the article-fluid interfacial area, V reresent the is the film mass transfer coefficient of the fluid around the article and r is the article radius. In terms of the adsortion caacity, the equilibrium relationshi between fluid hase and adsorbent or adsorbed hase concentrations may well be reresent by linear isotherms (Viard, V., 1991, Ruthven, D. M., 1994), nonlinear isotherms (Li et al., 2003, Yang et al., 2003, Kaczmarski and Bellot, 2003) or irreversible isotherms (Montesinos R. M., 2001). The exerimental equilibrium data reresented by nonlinear exression comlicates the mathematical reresentation of the searation rocess since the article diffusivity is deendent of the intra-article concentration. However, sometimes in the nonlinear henomenon a linear isotherm equation can be assumed in the model in order to redict the adsortion kinetics giving accetable redictions (Viard, V., 1991). Therefore to simlify the solution of the comlex equations, the micro-ore diffusivity has been assumed as indeendent of the intraarticle concentration, which is generally a good aroximation (Ruthven 1984, Viard 1991). The Eq. (1) can be re-written as (Ruthvem 1984): C D 1 C 2 = r 2 t + (1 )K r r r 0 r r (3) where: D D a = + ( 1 )K (4) This Eq. 4 is reorted as the aarent diffusivity by different authors (Ruthven, D. M., 1984, Viard, V. 1991). Eqs. (2) and (3) are couled by the following interfacial boundary conditions: C r = ( ) r D = k f Cβ r= r Cβ at, r (5) r = r at C, β = C (6) C β r = r In Eq. (5), is the solute concentration in the fluid at the surface article. The initial conditions, for equations (2) and (3), are taken to be: for t = 0 Cβ = Cβ0 C = C0 0 r r, and for (7a-b) Eqs. (2-7) were rewritten in terms of the following dimensionless variables: C Cβ Da r U = U f = τ = t ξ = 2 C C 0 β r, 0 r,,, (8a-d)

5 J.A. Hernandez et al., C C β0 0 where and reresent initial solute concentration for the solid and fluid hase, resectively. Thus, the corresonding mass balances Eqs. (2-3) are given by: In the article U 1 U 2 = ξ 2 τ ξ ξ (9) In the fluid 2 du r A K U d f f f f β = ( U ξ= 1 Uf ) dτ βda V dt (10) Those are subjected to the interfacial boundary condition and initial condition given by U = Bi ( U ξ= 1 U f ) at ξ =1, ξ (11) for τ = 0 U, f = U f U 0 = U 0 and for 0 ξ 1 (12a-b) Model solution Eqs. (9-12) reresent the comlete mass transort model for describing the adsortion kinetics in the batch system. This nonlinear equations system of mass transfer was discretized and solved for the well knows method of exlicit finite differences. This exlicit method is very easy to rogram but fails to give a correct solution when Δξ is too small. However, in this solution, we have carefully taken in account the articularity of the method. To describe an adsortion rocess where variations of bed fluid volume caused by the extraction of samles for analysis during the rocess are resented, the mechanism of numerical solution was referred. Exerimental art The IMP-C12 used as adsorbent for this study was roduced in IMP (atent in ste). This material adsorbent is obtained of a modification from the method described by Dacey and Thomas (1954). Adsortion kinetics The IMP-C12 was reviously sieved to obtain a fraction named A, with article sizes within 106 and 180 µm and a second one named B, with articles smaller than 106 µm. One gram of the IMP-C12 (samle A or B) was laced into a stirred tank microreactor (50 ml volume) and 10 ml of a mixture of n-hetane and isooctane (v/v) were added. The mixture was reared with 10% (v/v) of n-hetane. Assays were erformed at constant temerature (30 o C), with mechanical agitation at 300 and 450 rm. A star-shaed magnetic stir bar (2 cm diameter) was used. Samles were taken at different times from the microreactor with a constant volume (200 µl) and the liquid was laced into chromatograhic vials for GC analysis. The number of samles taken from the microreactor was tiically seven for each exeriment which means a reduction of 1.4 ml of the n-hetane and iso-octane mixture. Liquid hase aliquots were withdrawn avoiding the removal of adsorbent. This decrease in the volume of the liquid hase involves a variation of the adsorbent volumetric fraction inside the reactor that must be taken into account in the model to redict the adsortion kinetics.

6 J.A. Hernandez et al. Proerties estimation Porosity (microore volume, ). The orosity of the IMP-C12 was estimated from a nitrogen gas adsortion quantification using Micromeritics ASAP 2405 equiment (Micromeritics Inc., USA). Density. The density of IMP-C12 was quantified by using a calibrated volumetric tube where 1 ml of the adsorbent was laced and weighed afterwards. The material was reviously sieved to obtain articles with diameters within 106 and 180 µm. Particle size distribution. It was estimated by laser ray disersion technique using Horiba LA500 equiment (Horiba, USA). Adsortion Isotherm. Adsortion of n-hetane for the isotherm was quantified once the thermodynamic equilibrium was reached, tyically after 150 minutes. This is when no more changes in n-hetane adsortion were observed as a function of time. These exeriments were conducted at constant temerature 30 o C, mechanical agitation at 300 rm, an adsorbent load of 100 g of material er liter of solution. The adsortion isotherms exerimental data are analyzed and fitted to simle mathematic models. The method of otimization used to fit the equilibrium constant is the Nelder-Mead simlex method of the Otimization Toolbox of Matlab (Demuth and Beale, 2000). 3. Results and discussion A mass transfer model is used to evaluate the imortance of the adsorbent article size distribution on the adsortion arameters. The effective intraarticle diffusivity of n-hetane on IMP-C12 and Biot number were estimated for a stirred microreactor at two different seeds of agitation (300 and 450 rm). The model considers a mean article size calculated from the broad diameter size distribution. In addition, a volume variation of the susension is considered in the mass transfer model due to the extraction of samles for analysis during the rocess. This model was solved using the well known method of exlicit finite differences. However, in order to obtain the solution of the transort model, it was necessary to determine some roerties of the adsorbent. These roerties are dislayed below. Proerties estimated of the adsorbent IMP-C12 Porosity (microore volume, ), Values of microore area ( m2/g) and orosity (0.38 cm3/g) were estimated by the t-lot method reorted by Do (1998). Density, A density of the IMP-C12 of 0.5 g/ml was obtained by weighing the samles in a volumetric tube. Particle size, Figure 2 shows the article diameters distribution obtained for two samles: Samle A, material with article diameters of µm and Samle B, material with article diameters smaller than 106 µm. A unimodal distribution was found in each case with a broad distribution due to an inefficient sieving rocedure. For the A samle ( µm), a article diameter distribution resents a eak at µm (18.6%) with 0.2% of articles smaller than 6.2 µm and 7.1 % of articles bigger than 200 µm. For the B samle (<106 µm), the maxim (19.5%) was found at article diameter of 67.5 µm. The article size distribution ranges from 6.7 to 116 µm. Particles smaller than 34.2 µm accounts for the 11.1% of the oulation. This broad distribution of article diameter has influenced the adsortion as it deends on the article surface area and hence on the article diameter. Therefore to consider

7 J.A. Hernandez et al., this broad article size distribution, we assumed that the equation of average diameter reorted by Morales et al., (2003) could be used: d m = i= N t i= 1 i= N t X d i X d i 3 i 2 i i= 1 (13) where Xi is the size distribution in article number and di reresent the article diameter for i-th article size <106 µm µm Percentage (%) Particle size (µm) Figure 2. Broad article diameters distribution ( and <106μm). Adsortion Equilibrium Isotherm of n-hetane in IMP-C12. This isotherm was quantified after reaching equilibrium (150 min, Figure 3). The exerimental data, were analyzed and fitted to linear and Langmuir tye-equations. The best fit values of the equilibrium constant were determined by linear regression. For the linear model, the ratio of n-hetane adsorbed by the IMP-C12 and n-hetane concentration in the fluid is (R = ) and for the Langmuir model, the constant is (R = ). The two models show satisfactory results to redict the equilibrium isotherm. However, in order to simlify the solution of the equations, the simle linear model was used.

8 J.A. Hernandez et al. Figure 3. Adsortion isotherm data and models fitting of n-hetane. The Table 1 summarizes the results obtained and used in the mathematical model to redict the adsortion kinetics for a stirred tank. TABLE I. Estimated data for the mathematical model of mass transfer. Parameter Value Particle orosity = 0.38 Equilibrium constant K = 0.67 Density of IMP-C12 ρ = 0.5 g/ml Mean article size diameter μm to μm and μm to <106 μm Prediction adsortion kinetics For the IMP-C12 adsortion exeriments in a stirred tank (Fig. 1) a binary mixture of n- hetane and iso-octane was used. Preliminary exerimentation showed that the iso-octane was not adsorbed by the IMP-C12, therefore it was used as a solvent. The nonlinear equations

9 J.A. Hernandez et al., system (Eqs. 9-12) for the n-hetane mass transfer, includes an equation for the variations of the susension due to the aliquots required to exerimentally determine the adsortion kinetics. Figure 4 shows the volume variation of the susension versus time. In order to redict the exerimental data of the volume variation, we roosed an accuracy exonential tye-model (r2 = ) which contains four fitted arameters. This exonential tye-model is resented in Figure 4. The method of otimization used to fit the arameters was the Nelder-Mead simlex method of the Otimization Toolbox of Matlab (Demuth and Beale, 2000). This fit was made for each one of the exeriments. Figure 4. Volume variation against time of the binary mixture (iso-octane/n-hetane, 90/10). To the A samle, the Figure 5 [A] resents the n-hetane adsorbed values (exerimental and simulated) versus time for the binary mixture (iso-octane/n-hetane, 90/10) in IMP-C12 in a stirred tank (450 rm). Figure 5[A] and [B] show that the rediction of adsorbed n-hetane in the article is correct (r2 = 0.995), considering an effective intraarticle diffusivity coefficient 9 D of = 1.76x10 cm2s-1, with a Biot number of 5.3 and a mean article size of m calculated from the article diameters distribution range ( m). The effective intraarticle diffusivity coefficient is comarable to the reorted by Kaczmarski and Bellot, 2003, and Montesinos et al., However, it is interesting to mention that if the mean article diameters is different of , the arameters calculated (effective intraarticle diffusivity coefficient and Biot number) are also different, therefore this confirms that the arameters obtained are deendent of the mean article diameters calculated. Figure 5 [C] gives the rediction of n-hetane concentrations in liquid hase from Eq. (2). This result resents an accetable behavior of n-hetane in the binary solution. It is evident that the concentration of n-hetane in the liquid susension into the stirred tank diminishes when the n-hetane is adsorbed by the IMP-C12.

10 J.A. Hernandez et al. Figure 5. Exerimental and simulated curves for the adsortion kinetics of n-hetane in IMP-C12.

11 J.A. Hernandez et al., Validation of the roosed model Figure 6 deicts the ability of the model to redict adsortion kinetics for the A samle. This Fig. 6 resents the simulated adsortion utake curves and adsortion kinetic exerimental data against time at two different stirring velocities (300 and 450 rm) with the same intraarticle diffusivity coefficient obtained and mean article diameter calculated from the A samle (1.76x10-9 cm2s-1 and m). A very good agreement between calculated and exerimental adsortion curves is observed (a r2 = at 450 rm and a r2 = 0.99 at 300 rm) with Biot numbers of 5.3 and 4.4 resectively. It is evident that the model was successful in redicting exerimental adsortion kinetics because the same broad article size distribution was used. It is clear that the adsortion curve resent a minor adsortion caacity at 300 rm, comared to the values obtained at 450 rm, due to the interfacial resistance in the solid-liquid hase. Consequently, It will be very interesting to study the influence of the article size distribution on the arameters in the adsortion kinetics. Figure 6. Validation of the roosed model to redict adsortion kinetics of n-heane in a stirred tank with 300 and 450 rm. Effect of the adsorbent article size distribution on the develoment of the adsortion kinetics In this section, an analysis of the mean article diameter on the evolution of the adsortion kinetics was made to test the imortance of the article size distribution during the rocess. The arameters calculated from the A samle are used to redict the adsortion kinetics with different article size distribution (B samle). The Biot number calculated in the A samle was emloyed to estimate the Biot number for the B samle in the adsortion kinetics

12 J.A. Hernandez et al. rediction. Figure 7 resent the exerimental and simulated data of adsortion curves for the B Samle. In Fig. 7[A], the discontinuous line reresents the simulated data obtained by the model considering a mean article diameter of m, with an estimated Biot number of 2.5 from the two mean article diameters ( and μm) and the Biot number of 5.3. The discontinuous line does not redict the exerimental adsortion curves robably due to the inefficient values calculated: mean article diameter, effective intraarticle diffusivity coefficient and Biot number obtained; as a result of the broad article size distribution. In order to confirm the imortance of the mean article diameter in the simulation of the adsortion kinetics, Figure 7 [B] shows simulated data at different mean article diameters (73, 90, 120 and 150 μm). This Fig. 7 [B] demonstrates that the mean article diameter is strongly critical since from 73 μm to 150 μm, the adsortion kinetic changed almost 42 %. A good agreement is observed between the redicted and the exerimental adsortion curve (R = 0.89) in the B samle with mean article diameter of 150 μm. It is imortant to remark that in each adsortion kinetics rediction the Biot number was different because the Biot number is function of the article size. However, in Figure 7[C], the mean article diameter is considered constant and only the Biot number is modified (2.5 to 0.4). This Figure 7[C] shows that if the Biot number decreases the adsortion curve also decreases, which is normal since great resistance exist in the interface liquid-solid. Nevertheless, a Biot number of 0.4 does not redict satisfactory the results. Therefore, in order to redict with good accuracy the adsortion kinetics, a mean article diameter is required in all samles. Consequently, a homogeneous article size distribution with same diameter is indisensable to obtained the arameters corrects. Finally, the article size, effective intraarticle diffusivity coefficient and Biot number (when there is small mechanic agitation) are very imortant arameters in the simulation of adsortion kinetics. The numeric solution of the model shows the imortance to redict the adsortion kinetics in a wide range of oerations conditions. 4. Conclusions This study shows that the article size distribution is strongly imortant to redict the adsortion kinetics when the mass transfer tye-model is used. Therefore if the article size diameter is homogeneous and with a same article size, it is ossible to estimate the effective intraarticle diffusivity of the n-hetane into IMP-C12 in batch system and redict with good accuracy the adsortion kinetics. Particle size diameter, intraarticle diffusivity and Biot number (at low stirring rates) are significant factors for the simulation of the adsortion rocess. Volume variation redictions were well reresented by the roosed model. Acknowledgements The authors wish to thank the Instituto Mexicano de Petroleo for the financial suort through D roject.

13 J.A. Hernandez et al., Figure 7. Comarison between exerimental and simulated data of n-hetane adsortion kinetics for the B samle (article diameters distribution <106 μm) 9 considering D = 3.6x10 cm 2 s -1.

14 Nomenclature A f K f r B = D i C, C and C κ β article fluid interfacial area, cm2 Biot number J.A. Hernandez et al. solute concentration in the fluid orous article, in the article and in the fluid, gcm-3 resectively C 0 and Cβ0 initial solute concentration in the fluid of the adsorbent microorous and in the fluid, gcm-3 D a D K K f r r U and U f V Greek letters β τ ξ Subscrits f micro-ore article aarent diffusivity, cm2s-1 micro-ore article diffusivity, cm2s-1 equilibrium constant article liquid film mass transfer coefficient, cms-1 radial distance in the adsorbent article, cm article radius, cm solute dimensionless concentration in article and fluid hase, resectively volume occuied by article and fluid, cm3 article void fraction fraction of volume occuied by the fluid dimensionless time dimensionless radial osition fluid article References Alay, E., Chadwick, D., Kershenbaum, L. S., Barrie, P. J., Sivadinarayana, C. and Gladden L. F. (2003). TEOM analysis of the equilibria and kinetics of n-hexane and n-hetane adsortion on FCC catalyst/silicalite. Chemical Engineering Science, 58, Chuang, C. L., Chiang, P. C. and Chang, E. E. (2003). Column study of benzene adsortion onto activated carbon. Carbon, 41, Dacey, J. R., and Thomas, D. G. (1954). Adsortion on saran charcoal A new tye of molecular sieve. Trans Faraday Soc, 50, Demuth, H., and Beale, M. (2000). Otimization Toolbox for Matlab-Users Guide Version 2.1. Natrick, MA: The MathWorks Inc. Do, D. D. (1998). Adsortion analysis: equilibria and kinetics. Imerial College Press, London.

15 J.A. Hernandez et al., Do, D. D. and Do, H. D. (2002). Effects of adsorbate-adsorbate in the descrition of adsortion isotherm of hydrocarbons in micro-mesoorous carbonaceous materials. Alied Surface Science, 196, Hui, Ch., Chen, B. and Mckay, G. (2003). Pore-surface diffusion model for batch adsortion rocesses. Langmuir, 19, Kaczmarski, K. & Bellot, J. Ch. (2003). Effect of article-size distribution and article orosity changes on mass-transfer kinetics. Acta Chromatograhica, 13, Lee, V. KC, Porter, J. F. and Mckey G. (2003). Fixed bed modeling for acid dye adsortion onto activated carbon. Journal of Chemical Technology and Biotechnology, 78, Li, P., Xiu, G., & Rodrigues, A. E. (2003). Modeling searation of roteins by inert core adsorbent in a batch adsorber. Chemical Engineering Science, 58, Marroquin de la Rosa, J. O., Morones Escobar, R., Viveros Garcia, T., & Ochoa-Taia, J. A. (2002). An analytic solution to the transient diffusion-reaction roblem in articles disersed in a slurry reactor. Chemical Engineering Science, 57, Montesinos, R. M., Guzman, R. & Tejeda-Mansir, A. (2001). Simulation of stirred tank affinity rocesses alied to searation of roteins. International Journal of Bio- Chromatograhy, 6, : Morales E., Lobo-Oehmichen, R., Ochoa-Taia, J. A. (2003). An analytic solution to transient diffusion-reaction roblem with article diameter distribution in a slurry reactor. Submitted in Chemical Engineering Science. Qinglin, H., Farooq, S. & Karimi, I. A. (2003). Binary and ternary adsortion kinetics of gases in carbon molecular sieves. Langmuir, 19(14) Reid, C. R. and Thomas, K. M. (2001). Adsortion kinetics and size exclusion roerties of robe molecules for the selective orosity in a carbon molecular sieve used for air searation. Journal of Physics Chemistry B, 105, Ruthven, D. M. (1984). Princiles of adsortion and adsortion rocesses. John Wiley & Sons, New York. : Viard, Virgine. (1991). Modelisation de la searation chromatograhique du glucose et du fructose en vue de la simulation des rocedes industriels. PhD Thesis, Eécialité Génie des Procedes, Ecole Nationale Suérieure des Industries Agricoles et Alimentaires. Yang, X., Otto, S. R., Al-Duri, B. (2003). Concentration-deendent surface diffusivity model (CDSDM): numerical develoment and alication. Chemical Engineering Journal, 94,