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Ind. Eng. Chem. Res. 1998, 37, 2158-2171
Diffusion and Reaction in Catalyst Pellets with Bidisperse Pore Size Distribution Timur Dogˇ u† Department of Chemical Engineering, Middle East Technical University, 06531 Ankara, Turkey
Catalysts with bidisperse pore structures are extensively used in chemical industries. In such catalysts, diffusion and reaction in the microporous particles are preceded by diffusion in the macropores. The relative significance of macro- and micropore diffusion resistances on the observed reaction rates with linear and nonlinear rate forms, heat effects, catalyst deactivation and selectivity problems in bidisperse catalysts are investigated in number of published works. In this article, a review of the models proposed in the literature is presented together with a critical analysis of assumptions involved. Criteria are presented to test the relative significance of transport limitations. Also, a review of experimental techniques and experimental values of micro- and macropore effective diffusivities of gases in solids with bidisperse pore structures, such as zeolite pellets and supported porous catalysts, is given. 1. Introduction Many of the industrial catalysts have bidisperse pore size distributions containing micro- and macropores. Supported alumina catalysts, molecular sieve pellets, and acid ion-exchange resin catalysts are some examples of such catalysts. Porous solids with bidisperse pore srructures are also widely used as sorbents and ion-exchange resins. Catalysts with bidisperse pore size distribution are usually recommended for better utilization of the active catalyst surface. In the catalyst pellets that are formed by compacting fine microporous particles, macropores are formed between the agglomerated particles. In such catalyst pellets, most of the active sites usually lie within the microporous particles. Consequently, most of reaction takes place within the particles and macropores contribute to the transport of reactants to these particles. The pore size distribution of such a catalyst pellet is a major factor affecting the relative rates of diffusion in the macroporous and microporous regions. Micropore diffusivity values are usually much smaller than macropore diffusivities. Another factor that determines the relative significance of diffusion resistances in the micro- and macropores is the ratio of microparticle to pellet sizes. As shown in the literature (Ors and Dogu, 1979; Kulkarni et al., 1981), the effectiveness factor of a bidisperse porous catalyst is a function of two parameters; namely, the particle Thiele modulus and a dimensionless parameter R that characterizes the ratio of diffusion times in the macro- and micropore regions:
()
Di R 0 R ) (1 + P)(1 - a) Da r0
2
(1)
where P represents the geometry of the pellet; and P ) 0, 1, and 2 for slab, cylindrical, and spherical geometries, respectively. Some aspects of diffusional effects on the observed rates in bidisperse porous catalysts have been † Fax: (90) 312 rorqual.cc.metu.edu.tr.
2101264.
E-mail:
tdogu@
Figure 1. Schematic representation of the (a) pellet-particle model and (b) branched micro-macropore model.
discussed by Doraiswamy and Sarma (1984) and by Jayaraman and Doraiswamy (1983). One approach in analyzing diffusion and reaction processes in a catalyst with a bimodal pore-size distribution considers the pellet as an agglomeration of microporous particles (Figure 1a). The microporous particles that are agglomerated are generally present in powder or granular form, and their shapes may be approximated as spheres. The ratio of pellet to particle radii is >102 in many cases, and particles are assumed to act as uniformly distributed point sinks. Ruckenstein et al. (1971) used such a pellet-particle model for the analysis of transient sorption by sorbents with bimodal pore-size distributions. In the analysis, competing effects of macropore and micropore diffusion resistances on the sorption rate was illustrated. A second approach of modeling a bidisperse catalyst involves the assumption of having cylindrical macropores and cylindrical micropores that extend from the macropores into the pellet (Figure 1b). This approach can be considered as an extension of Thiele’s single-pore model for the effectiveness factor. Such a branched micromacropore model was used in the early work of Tartarelli et al. (1970) for the analysis of diffusion and reaction in a catalyst containing both micro- and macropores. In a more recent study, a similar model was used by Petersen (1991) for the analysis of adsorption by a sorbent with bidisperse pore structure.
S0888-5885(97)00613-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 03/24/1998
Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2159
In recent years, a number of random and geometrical pore network models have been proposed for the description of transport phenomena in porous solids (Sahimi et al., 1990; Hollewand and Gladden, 1992; Burganos and Sotirchos,1987). Use of such pore network models for the analysis of diffusion, sorption, and reaction in bidisperse catalysts and adsorbents is presented in the works of Petropoulos et al. (1991), Loewenberg (1994), and Andrade et al. (1995). In this review article, an analysis of the models proposed in the literature for diffusion and reaction in bidisperse porous catalysts is presented. Effectiveness factor analysis, including nonlinear rate forms, nonisothermal pellets, diffusivity variations, and catalyst deactivation, are presented. Criteria used to test the significance of macro- and micropore diffusion resistances on the observed rate are generalized. In the last section of the article, experimental values of macro- and micropore diffusion coefficients and predictive models are summarized together with a review of experimental techniques of diffusivity measurements. In all these analyses, the review is focused on bidisperse porous catalysts and sorbents containing both micro- and macropores.
where
( )
)
Di d 2 dCi r - kCni ) 0 dr r2 dr
(
(2)
( )
3(1 - a) Da d dCa dCi R2 Di 2 dR dR r0 dr R
r)r0
)0
(3)
In developing eqs 2 and 3, all the reaction was considered to take place on the walls of micropores (Ors and Dogu, 1979; Kulkarni et al. 1981; Jayaraman et al., 1983). In some recent studies (Ihm et al., 1996; Ferreira and Rodriques, 1993), the contribution of macropore surfaces to the reaction was also considered in macroreticular ion-exchange resin catalysts. Symmetry condition was usually assumed both for the particle and for the pellet in most of the published literature:
dCa dCi )0 |r)r0 ) 0; | dr dR R)R0
(4)
This analysis shows that the effectiveness factor of a bidisperse catalyst is a function of two dimensionless groups, namely, the particle Thiele modulus (φi) and the parameter R. For a first-order reaction, the effectiveness factor expression is (Ors and Dogu, 1979):
)
[( [(
)]
(
9 d(Ca/C0) φ2i R d(R/R0)
)
R/R0)1
1/2 φi -1 tanh φi 9 -1 ) 2 1/2 φi φi R tanh R -1 tanh φi
R
)]
()
φi ) r0
k Di
1/2
(5)
(6)
(7)
and R is defined by eq 1. The effectiveness factor of a bidisperse catalyst pellet can also be expressed in terms of a shape-generalized pellet Thiele modulus (φaG) and the ratio of particle and pellet Thiele modulus values. For an nth order reaction, definitions of these parameters are:
φaG )
() ( ) () ( ) VP Ae
pellet
VP φiG ) Ae
2. Effectiveness Factor Some of the earlier papers on effectiveness of bidisperse porous catalysts are by Mingle and Smith (1961), Wakao and Smith (1964), Carberry (1962a), and Silveston and Hashimoto (1971). Using the pellet-particle model, Ors and Dogu (1979) obtained an analytical solution for effectiveness factor for a first-order reaction taking place in a catalyst with a bimodal pore size distribution. 2.1. Pellet-Particle Model. In this model, a bidisperse catalyst pellet is considered as an agglomeration of microporous spherical particles (Figure 1a). Considering an nth order surface reaction, the pseudohomogeneous mass conservation equations for the microporous particle (eq 2) and for the spherical catalyst pellet (eq 3) can be expressed as follows:
(
η)
k(1 - a)Cn-1 0 Da
particle
( ) φiG φaG
2
kCn-1 0 Di
)G
1/2
(8)
1/2
(9)
(10)
In eq 8, k(1 - a) ) k* is the nth order reaction rate constant based on pellet volume. Parameter G is inversely proportional to the parameter R:
G)
()
3 Da r0 ) R Di R0
2
(1 - a)-1
(Spherical Pellet) (11)
The effectiveness factor expressions obtained for slab, cylindrical, and spherical pellets and the definitions of corresponding dimensionless parameters are given in Table 1. The variation of the effectiveness factor with respect to the Thiele modulus (φaG) for different values of G is given in Figures 2 and 3 for spherical and slab pellet geometries, respectively. With these shapegeneralized pellet Thiele modulus and G definitions, the effectiveness factor values predicted from the expressions given in Table 1 for different geometries are quite close to each other. For a G value of one, the comparison of effectiveness factors for spherical and slab-shaped catalyst pellets is illustrated in Figure 4. The effect of shape on the effectiveness of bidisperse pellets and shape normalization was also discussed in the recent papers of Jayaraman (1993a,1995) and Kucukada and Dogu (1985). Jayaraman (1993a) illustrated that shape normalization of the Thiele modulus brought the curves together for different geometries. In this case, the spread between the curves can be as high as 36%, especially at intermediate values of the Thiele modulus. A modified shape normalization was defined in the recent paper of Jayaraman (1995) by following the approach of Miller and Lee (1983). With this normalization, the spread in the curves for various shapes becomes 2.5 × 10-19 10-10 3.34 × 10-6 1.5 × 10-5 4.8 × 10-5 3.2 × 10-6
ethylene N2 N2 n-butane n-butane ethane butane N2 i-C4H10 N2 He-N2 butane ethane butane cyclopropane isobutane oxygen
alumina activated soda activated soda boehmite NaY zeolite activated carbon activated carbon 5 Å molecular sieve 4 Å molecular sieve 4 Å molecular sieve γ-alumina γ-alumina 5 Å molecular sieve 5 Å molecular sieve 5 Å molecular sieve 13X molecular sieve molecular sieving carbon CaX(Na) zeolite NaX molecular sieve 5 Å molecular sieve 5 Å molecular sieve KY zeolite NaY zeolite
temperature (°C) porosity a method effective micropore diffusivity, Di (m2/s) effective macropore diffusivity, Da (m2/s) diffusing species porous solid
Table 2. Pellet and Microporous Particle Effective Diffusion Coefficients in Some Bidisperse Porous Solids
Dogu and Ercan (1983) Dogu et al. (1987) Dogu et al. (1987) Hashimoto and Smith (1974) Hsu and Haynes (1981) Mayfield and Do (1991) Mayfield and Do (1992) Hashimoto and Smith (1973) Kumar et al. (1982) Kumar et al. (1982) Biswas et al. (1987b) Biswas et al. (1987b) Shah and Ruthven (1977) Shah and Ruthven (1977) Shah and Ruthven (1977) Onyestyak et al. (1995) Nakano et al. (1991)
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sawa and Smith (1973), and Haynes and Sarma (1973) demonstrated that both macro- and micropore diffusion coefficients can be determined from chromatographic experiments. In these dynamic experiments, moments of the response peaks to adsorptive or inert tracers were analyzed for the evaluation of macro- and micropore diffusion coefficients as well as adsorption parameters. In the early work of Hashimoto and Smith (1973), macro- and micropore diffusivities of n-butane in alumina, and nitrogen and n-butane in 5A zeolites were determined. Kumar et al. (1982) used a similar procedure to determine effective macro- and micropore diffusion coefficients of light hydrocarbons in 4A and 5A zeolites. Chromatographic techniques are widely used for the investigation of sorption and diffusion in zeolites. In this review article, primarily the publications that have considered the bidisperse nature of the pellets are reviewed. Some of the macro- and micropore diffusivities reported in the literature are summarized in Table 2. Because micropore diameters of zeolites are usually of the same order of magnitude as the molecular diameters of diffusing species, the diffusion is characterized by a strong molecule-wall interaction. In light of this strong interaction, most investigators did not consider the adsorbed phase and gas phase separately and characterized the adsorption and diffusion process with an intercrystalline diffusivity, Dc (Hsu and Haynes, 1981; Kawazoe et al., 1974; Shah and Ruthven, 1977; Sarma and Haynes, 1974; Haynes, 1975; Chang et al., 1984). In some of these studies, analyses of chromatographic data were carried out in the time domain. In some other studies, adsorbed and gas phase concentrations were considered separately in the micropores. The molecule-zeolite and molecule-molecule interactions and their effects on the order of magnitude and concentration dependence of diffusivity were discussed and a unified diffusion theory was developed in the work of Xiao and Wei (1992). These authors showed that transition from Knudsen diffusion to configurational diffusion depends on the ratio of molecular diameter to pore diameter, molecular length, zeolite structure, and temperature. Limbach and Wei (1990) determined the restricted diffusion coefficient of nickel etioporphyrin in granular γ-alumina and silica-alumina catalyst support materials using a transient uptake technique. They showed that experimental results agreed well with the restricted diffusion theory based on the circular pore model with hydraulic pore radius and Stokes-Einstein radius for the molecular size. One of the first papers that considered macro- and micropore diffusion in bidisperse systems was the transient sorption study of Ruckenstein et al. (1971). The effects of micropore diffusion, macropore diffusion, and adsorption on the transient uptake of bidisperse porous solids were also considered in the studies of Ma and Lee (1976), Lee et al. (1979), Do (1983), Nakano et al. (1991), Lee (1978), and Micke et al. (1994). More recently Liapas and McCoy (1994) investigated the effect of micropore diffusion on column performance in perfusion chromatography. A bidisperse pore diffusion model was also used in the pressure-swing adsorption study of Doong and Yang (1987). In the recent study of Gutsche (1993), concentration dependence of the micropore diffusion coefficient of NH3 in A-type zeolite crystals was analyzed. A differential adsorption bed was used by Mayfield and Do (1991) for
Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2167
the measurement of adsorption rates of ethane, butane, and pentane on activated carbon. Macropore, micropore, and solid diffusion coefficients were experimentally determined from uptake results. A zero-length column chromatography technique was presented by Micke et al. (1994) for the analysis of kinetic data and evaluation of diffusivities of microporous solids. 6.2.2. Single-Pellet Dynamic Technique. The single-pellet moment technique that was originally developed for the investigation of effective diffusivities in porous solids (Dogu and Smith, 1975) was modified by Uyanik (1977) and by Hashimoto et al. (1976) for the investigation of diffusion in bidisperse porous pellets. In this single-pellet procedure, a dynamic version of a Wicke-Kallenbach diffusion cell was used. Diffusivity measurements made using this technique are summarized in the review of Park and Do (1996). In this dynamic procedure, pulses of inert and adsorptive tracers are injected into the stream flowing over one face of the pellet, and the response peak is detected at the outlet of the other chamber. The first moment, which corresponds to the time delay of the response peak, is a function of pellet effective diffusivity, whereas the second moment contains the micropore diffusion coefficient. Dogu and Ercan (1983) used this technique for the evaluation of macro- and micropore diffusivities of ethylene in an R-alumina pellet. In that analysis, the adsorption equilibrium constant was also evaluated. In later studies (Dogu et al., 1987; Burghardt et al., 1988), this technique was applied to evaluate the diffusion coefficient in different bidisperse porous solids. In a recent study, Cabbar et al. (1996) showed that the adsorption in a bidisperse porous solid clay might be characterized either by a micropore diffusivity or with an adsorption rate constant. They suggested the following relation between the adsorption rate constant kads and Di,
kads ) Di
15(1 - a)3 r20
(47)
A similar conclusion was reached by Parker and Valocchi (1986). Biswas et al. (1987a,b) obtained the analytical solution for the problem of transient diffusion with a step input in a Wicke-Kallenbach diffusion cell containing a bidisperse pellet. They presented a time domain analysis. Their results indicated that because of the small diameter of the microporous particle and the large value of diffusivity, the micropore sensitivity of the moment technique became low. The time domain solution of a diffusion process in a tabletted bidisperse catalyst pellet was also given by Jury (1977). Analysis of diffusion and reaction in a single pellet reactor containing a bidisperse catalyst pellet was reported by Kunzru (1984). This work illustrates the use of a single pellet reactor for determining diffusivities, the effectiveness factor, and the intrinsic rate constant for bidisperse catalysts. 6.2.3. Frequency Response and NMR Techniques. Micro- and macropore diffusivities of bidisperse pellets were also measured by frequency response techniques. Recently, Jordi and Do (1993) used the frequency response method for analysis of sorption kinetics in bidisperse sorbents. The limitations of the batch frequency response technique were also discussed. Yasuda and Sugasawa (1984) developed such a tech-
nique to study zeolitic diffusion of gases. In a recent study, Onyestyak et al. (1995) measured the micro- and macropore diffusion coefficients of CO2 and isobutane in different manufactured 5A and 13X zeolite pellets. Macropore diffusion was the controlling step in all these pellets. Bourdin et al. (1996) used another frequency response method for the study of macro- and micropore mass transfer parameters during adsorption. In this thermal frequency response technique, mass and heat transfer kinetics in zeolites were determined by measuring the temperature of the adsorbent sample. The presence of a surface barrier in some adsorption studies of bidisperse solids was indicated by the nuclear magnetic resonance (NMR) pulsed-field gradient and NMR fast tracer desorption techniques (Karger and Ruthven, 1989). 6.3. Some Recent Developments in Diffusion, Adsorption, and Reaction in Bidisperse Solids. In some recent studies, interesting aspects of diffusion and adsorption of bidisperse porous solids were analyzed. Multicomponent sorption and multicomponent micropore diffusion were investigated by Hu and Do (1993) and Van Den Broeke and Krishna (1995). In the publications of Hu et al. (1993), energy distribution of adsorption sites for both equilibrium isotherm and diffusion were included in the sorption kinetics of singleand multicomponent systems. A uniform energy distribution was used in this analysis. Experimental data obtained for ethane and propane sorption in activated carbon was analyzed for binary adsorption, desorption, and displacement dynamics. Analytic solution of cyclic mass transfer in an adsorbent pellet with a bidisperse pore structure was obtained by Carta (1993). Carta and Rodriguez (1993) considered simultaneous diffusion and convection in the macropores and diffusion in the micropores of a bidisperse solid and developed a criterion to determine the conditions under which transport limitations in pores influence the performance of chromatographic processes utilizing permeable particles. The effect of size distribution of microporous particles on the adsorption in bidisperse zeolite pellets was examined in the theoretical work of Tsibranska et al. (1992). Breakthrough adsorption curves were developed for this case. In zeolites, micropore diameters are generally of the same order of magnitude as the diameter of the diffusing species. Diffusion and reaction in pores in which the individual molecules cannot pass each other was described as a single-file system by Ka¨rger et al. (1992) who applied a jump model for the elementary steps of diffusion and simulated adsorption and reaction cases. Soil aggregates also have bidisperse pore structures. In addition to macro- and micropore diffusion resistances, partitioning into the soil organic matter and adsorption on the gas-mineral and gas-water surfaces should be considered in modeling migration of volatile organic contaminants in soil. In the recent work of Arocha et al. (1997), the bidisperse nature of the soil pore structure was taken into consideration in modeling diffusion and sorption, and numerical solutions of the partial differential equations describing this system were presented.
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7. Concluding Remarks This review covers the topic of the impacts of microand macropore diffusion resistances on the performance of catalysts with bidisperse pore size distributions. The review focuses on two aspects of the problem; namely, theoretical studies of the transport limitations and the experimental work on micro- and macropore diffusivities. Sorption by microporous sorbents has also been extensively investigated in the literature. Considering the relevance of diffusion and adsorption processes on the performance of such catalysts, some of the publications on sorption by bidisperse porous sorbents (such as molecular sieve pellets and bidisperse catalyst supports) are also covered in this review. Erroneous predictions may result if the relative significance of macro- and micropore diffusion resistances are not considered and diffusion process is described with a single effective diffusivity in a bidisperse porous catalyst. It was illustrated in the literature that prediction of the effectiveness factor using equations derived for monodisperse catalyst pellets might give overestimated values. The numerical value of a dimensionless parameter R, which is the ratio of diffusion times in the macro- and micropores, demonstrates the relative significance of macro- and micropore diffusion resistances. For very large values of R, diffusion resistance in the micropores lose significance, and performance of the catalyst can be predicted assuming a monodisperse pore structure (macropores only). On the other hand, for very small values of R, micropore diffusion resistance becomes more important and pellet effectiveness approaches particle effectiveness. The criterion reported in this review can be used to test the relative significance of macro- and micropore diffusion resistances on the observed reaction rate. In many practical cases, the pore structure of the catalysts are bidisperse. Catalytic material can be supported on a carrier pellet or formed directly into pellets (Fulton, 1986). In a number of reactions, such as hydrogenation, dehydrogenation, hydrotreating, reforming, etc., supported catalysts with alumina supports are used. The pore size distributions of these catalysts are usually bidisperse. Control of pore size distribution of alumina catalyst supports was discussed in the review paper of Trimm and Stanislaus (1986). Synthetic zeolite catalyst pellets used in number of reactions also have bidisperse pore-size distributions. Two examples in which the bidisperse nature of zeolite catalysts is considered are the isomerization of cyclopropane to propylene (Park and Kim, 1984) and toluene disproportionation (Beltrame et al., 1984). Macroreticular resin catalysts that consist of very small randomly packed gelular microparticles with continuous nongel pores are also treated in a manner similar to that used for bidisperse catalysts. In all these cases, macro- and micropore diffusion resistances should be considered in the predictions of the effectiveness factor, catalyst deactivation, selectivity, and yield calculations. If the diffusion resistances in the micro- and macropores are equally significant (value of the parameter G of the order of magnitude of one) in the diffusion limit observed, then activation energy approaches one-half of the value that would be predicted for a monodisperse porous catalyst. The pellet-particle and branched macro-micropore models predict very similar results for the effectiveness of the bidisperse catalysts. By means of a shape normalization of the Thiele modulus, the spread of the
effectiveness factor curves for various pellet shapes became
