The Effect of Operating Conditions on the Dispersion State of

As an important step in the preparation of supported metal catalysts, the drying of impregnated porous medium is studied by a mathematical model inclu...
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The Effect of Operating Conditions on the Dispersion State of Supported Metal Catalysts: A Model Study† Qiang Qin and Doraiswam Ramkrishna* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907

As an important step in the preparation of supported metal catalysts, the drying of impregnated porous medium is studied by a mathematical model including catalyst-scale mass and heat balances and particle-scale population balance. Both the loading profile and the particle size distribution for the precursors are considered. Calculations show that the combined effect of two migration mechanisms, intrapellet diffusion in the centripetal direction and convection in the centrifugal direction, determine the catalyst structure. Both physical and operational parameters are studied: the former characterize the physical preference toward one migration mechanism, and the latter give rise to the driving forces. Four structure regions are found in the operational space. For some systems with strong preferences, certain regions cannot be accessed in practice. The particle size distributions and hence the surface area for each region are also studied. Studies show that a uniform loading profile could be obtained under very low and very high temperatures, but the particle size resulting from high temperatures is much smaller, implying higher activity and lower thermal stability. In the case of high Thiele numbers, use of egg-shell catalyst may reduce thermal deactivation by having less active material in the inner part of the catalyst pellet where the temperature is much higher. This improvement is compromised by smaller particles, which are always formed in the center during the preparation. Introduction Supported metal catalysts are widely used in chemical industries, such as ammonia synthesis,1 organic synthesis,2 polymerization processes,3 and fuel cells.4 In such catalysts, particles, typically in nanometer scale, of active metal phase are dispersed on the internal surface of inactive porous medium to achieve high surface area and stability.5,6 Our previous study7 has shown that the dispersion state of the metal phase inside the porous medium may play an important role on the catalyst performance including the activity, stability, and product selectivity. Therefore, it will be ideal to use catalysts with dispersion structure specially designed for the process. This work is an attempt to study the effects of preparation conditions on the dispersion state of the supported catalyst so that the preparation process can be controlled properly to obtain the desired catalyst structure. In examining the dispersion state, we mainly focus on two distributions in catalyst scale (of millimeters) and particle scale (of nanometers), respectively: (1) the distribution of metal precursor phase inside of porous medium (metal loading profile); (2) the size distribution of precursor particles. Both factors have significant impacts on the catalyst performance, depending on the characteristics of the reaction (kinetics, selectivity, diffusion, etc.) and catalyst (thermal stability, metalsupport interaction (MSI), etc).8,9 Experimental as well as theoretical studies have shown the effects of metal loading profile on catalyst activity and product selectivity. With different types of metal loading profiles, the † This paper is dedicated to Professor M. Dudukovic, a chemical reaction engineer of distinguished accomplishments. * To whom correspondence should be addressed. Tel: (765) 494-4066. Fax: (765) 494-0805. E-mail: ramkrish@ ecn.purdue.edu.

supported catalysts can be roughly divided into four categories: uniform, egg-shell, egg-white, and eggyolk.10 It is well-known that for nonnegative ordered reactions, egg-shell catalyst has advantages in the case of strong diffusion limitation (high Thiele modulus)11 because in this case the active phase is concentrated in the outer part of the catalyst where the reactant concentrations are highest. In our previous study, the diffusion limitation is found to cause deactivation and change the product selectivity in Fischer-Tropsch synthesis (FTS).7 The effect of metal loading profile on the reaction kinetics is also reported when diffusion is not limiting.12,13 In fluidized bed operation, egg-shell catalyst leads to shorter catalyst lifetime due to the attrition of the outer part of the catalyst pellet.14,15 The effect of particle size distribution on catalyst performance is, however, more complicated. For structureinsensitive reactions, the particle size affects the activity and stability in two opposite directions: smaller particles have larger surface area and thus higher activity, while they are more readily redispersed (sintering) and thermally less stable.7 An optimal particle size should be aimed at by considering both effects. For example, Babier et al.16 suggested an optimum particle size around 9 nm for alumina-supported cobalt catalyst for Fischer-Tropsch synthesis. The situation becomes even more complicated for structure-sensitive reactions in which the kinetics depends strongly on the local crystallographic structure and MSI.17 In this case, in addition to the size dependence of the total surface area, the local kinetics is also size-dependent. Generally, changing particle size in the range from 1 to 5 nm causes significant variation of crystal characteristic (for example, average coordination number), while little effect is expected when the size is above 5 nm.17 Therefore, the local kinetics must depend on particle size in this range due to structure sensitivity. Additionally, for

10.1021/ie048954b CCC: $30.25 © 2005 American Chemical Society Published on Web 04/15/2005

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particles up to tens of nanometers, deviation of catalytic properties of metal particles from bulk is also found as a result of MSI. More deviation is expected for smaller particles, and this also raises the size dependence of local catalyst activity, product selectivity, or both.18 Therefore, for both structure-sensitive and structureinsensitive catalysts, the particle size effects should be considered for the choice of catalyst. Due to the complexity of both effects in the commercial catalytic processes, theoretical optimization of catalyst structure is difficult. In industrial practice, the choice of catalyst structure is usually based on experience.15 In our previous study,7 a methodology for simulating the catalyst structure has been developed. Based on this method, optimization of particle size has been performed. Given a desired catalyst structure, it is therefore of interest to determine how we can prepare it. Wetness impregnation is the most frequently used method in the industry.19 Generally three steps are involved in this method: (1) impregnation of aqueous precursor into a porous medium, typically alumina or silica; (2) drying of the impregnated porous medium and deposition of the precursor on the support surface; (3) conversion of the precursor into active form by calcination and reduction.15,19 Our interest is in the second step, which is drying and deposition. The final size of active metal particles is readily estimated from the size of precursors resulting from this step. When high temperatures are involved, redispersion of the metal phase may occur during the third step, causing changes of the final catalyst structure.20 However, only in cases of excessive sintering do we need to consider the redispersion. The model of the sintering process can be readily formulated using the same methodology as in this study. The temperature during the drying is typically between 50 and 250 °C.21 The migration of the solution is due to both convection and diffusion.15,21,22 In a spherical catalyst pellet, the convection, raised by capillary flow22 and gaseous pressure flow,23 is centrifugal (because the liquid saturation is higher in the center), while the diffusion is centripetal (because the temperature and therefore the solubility are lower in the center). It is these two mechanisms that determine the final metal loading profile. When convection dominates, an egg-shell catalyst is obtained, and an egg-yolk structure is expected for the opposite situation. The drying processes have been broadly distinguished between slow drying and fast drying.22,24 Slow drying occurs when the convection is much faster than the vapor removal and results in an egg-shell, while fast drying leads to an eggyolk or uniform structure. Model studies have been done on the slow-drying case.21,25 Lekhal et al. published a more versatile model, which can deal with both cases.15 However, all past models have considered only the metal loading profile and neglected the particle size distribution, which, as pointed out before, is also an important factor affecting both catalyst activity and stability. The deposition of the precursor on the support surface is a complex process as a result of physical or chemical interactions6,10 between the precursor ions and the support. The process is usually described by an adsorption isotherm,10,15,21,26 which is simple enough to be determined by two parameters. In this study, to calculate the particle size distribution, the deposition is considered as heterogeneous crystallization including two steps: nucleation and growth. Plenty of studies

have been done on the classic nucleation theory (CNT), and a good review has been given by Binsbergen.27 The present study views the interactions between precursor and support (both physical and chemical) as parametrized by the contact angle and surface energies. For the particle growth, Mohan’s model28 is used. Given the nucleation and growth kinetics, a population balance model (PBM)29 can be formulated to calculate the particle size distribution. Some formulations in Lekhal’s model15 are borrowed for the migration of solution inside the porous medium as part of this model, but there are two main differences in this study: (1) this model uses PBM as the deposition model so that the particles size distribution can be described; (2) this study is focused on the effect of preparation conditions on the catalyst structure. Therefore, the modulus in the dimensional analysis is divided into operation parameters (flow conditions and temperature of the drying medium) and physical parameters (diffusion, convection, and thermal conduction properties of the support and surface properties of the precursor particles). More parameters are considered in this study, and new results are revealed. Model Description As mentioned in previous section, two surface properties are of interest in this study: the metal loading profile and the particle size distribution (the term “metal loading” in this paper means the loading of metal precursors instead of active metal which will be obtained after reduction processes). Both of them should be able to be described by the model. This is realized by introducing a number density function for the particle population: n(x,r,t). This function describes the number of particles per surface area of support with size x, around position r in the porous medium, and at time t. In this study, a spherical catalyst pellet is considered so that the position vector r can be replaced with a scalar r, the radial position in the pellet: n(x,r,t). As will be shown, the model can be easily generalized to any geometry. It is clear that the normalized n is the particle size distribution:

f(x,r) )

n(x,r,t)tf)

∫0∞n dx

and the metal loading profile can be obtained by integrating n over x:

lm(r) )

∫0∞R4πx2n(x,r,t)tf)As dx

(1)

where R is the shape factor of the particle, tf is the time when drying is finished, and As is the specific internal surface area of the porous medium. Equation 1 defines the metal loading: surface area of precursor particles per unit catalyst volume. Therefore, the overall metal loading is

Lm )

∫0Rlm(r)4πr2 dr 4 3 πR 3

∫01lm(Rξ)ξ2 dξ

)3

(2)

During the crystallization process, the number density function should satisfy a population balance equation (PBE). With the information of particle number density

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and the size growth rate, one can immediately calculate the kinetics of deposition (the liquid to solid transfer rate) assuming the nuclei size is zero:

Hls(r,t) )

∫0∞AsRFpn(x,r,t)4πx2rg dx

(3)

where Fp is the density of the precursor particle. This deposition kinetics enables us to calculate the catalystscale mass and heat transfer (the migration of solution and vapor) and therefore the catalyst-scale temperature/ concentration profiles, which are external variables of the PBE. Therefore in this model, we have particle-scale PBE and catalyst-scale mass/energy balances coupled to each other. In the formulation of this model, the following assumptions are made: (1) All three phases are considered as continuum, and volume-average variables are used to describe their properties. (2) There is no gas-liquid interfacial mass and heat transfer resistance; that is, at each point inside the porous medium, gas and liquid are in phase equilibrium; Raoult’s law is used to calculate the vapor pressure. (3) Convection (driven by capillary pressure and gaseous pressure gradient) and diffusion happen simultaneously, governed by Darcy’s law and Fick’s law, respectively. (4) The solubility of the precursors depends on temperature and satisfies the Gibbs-Helmholtz relation. (5) CNT is used to describe the heterogeneous nucleation. The nuclei have zero size. The local geometry of the support surface does not affect the nucleation progress because typically the pore size is much higher than the nuclei size. (6) The particles do not move, and there is no aggregation. Particle Size PBE. The population of the precursors during the crystallization is described by the following population balance equation:

∂ ∂ n(x,r,t) + {n(x,r,t)rg[Y(r,t)]} ) 0 ∂t ∂x

rG ) kg(ln S)

n ) 0 at t ) 0 rgn ) I at x ) 0 where I is the nucleation rate from the CNT

(

)

-∆G*  k BT a

(6)

where a is the fraction of area of support surface that is not covered by particles. By assuming that the area covered by one particle is the projection of the particle onto the support surface, one can immediately derive

a ) 1 -

∫0 n(x,r,t)πx ∞

2

dx

E (∆µ)2

where E is a lumped parameter. It depends on the surface properties of the crystal and the interaction with the support. ∆µ is the chemical potential difference between phases, which depends on the oversaturation. Catalyst-Scale Mass and Heat Balances. Mass balances for drying medium, air in this study (eq 7), solvent in both gas and liquid phases (eq 8), and precursor ions in liquid phase (eq 9) are listed below.

∂ ( C ) ) -∇‚(Ng,a) ∂t g g,a

(7)

∂ ( C + lCl,s) ) -∇‚(Nl,s + Ng,s) ∂t g g,s ∂ ( C ) ) -∇‚(Nl,() - R(Hls ∂t l l,(

(8) (9)

where l and g are volume fractions of liquid and gas phases, Cg,a and Cg,s are concentrations of air and solvent in gas phase, Cl( means the concentration of positive or negative ion in liquid phase and R( means the stoichoimetric number for positive or negative ion. N’s are flux terms with the subscript of the same meaning. The flux terms consist of convection and diffusion:

N ) Nc + Nd

(10)

The convection, driven by the gaseous pressure gradient (gaseous pressure flow) and for liquid also the capillary pressure between gas and liquid phases (capillary flow),15

Pc ) 1.364 × 105γ

(

)

lCl,sMl,s Fs

-0.63

is governed by the Darcy’s law:

K Nc ) -C ∇P µ

(5)

Y is a vector of external variables including temperature and concentration. The right-hand side of eq 4 is zero due to the assumption that no aggregation and breakage occurs. The boundary and initial conditions for the PDE are

I ) kn exp

∆G* )

(4)

where rg is the particle growth kinetics. According to Mohan,28 it is expressed as a function of the supersaturation, S: 2

∆G* is the critical free energy, it can be written as

(11)

The diffusion terms are easy to calculate using Fick’s law

Nd ) - CD∇y

(12)

To simplify the model, effective diffusion coefficients are used for diffusion in the gas phase instead of a detailed dusty gas model as Lekhal did; and in the liquid phase, the concentration of the solvent is constant and only binary diffusion between the cation and anion is considered. Therefore, inserting eqs 11 and 12 gives us the expressions for all the flux terms as shown in eq 13:

{

KKg,eff Ng,a ) - gCgDe,a∇ya - Cgyag ∇Pg µg KKg,eff Ng,s ) -gCgDe,s∇ys - Cgysg ∇Pg µg KKl,eff Nl,s ) -Cl,sl ∇Pl µl KKl,eff F Nl,( ) -lD+,-∇Cl,( - Cl,(l ∇Pl - Cl,(z(D+,- ∇φ µl RT (13)

Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005 6469 Table 1. Nondimensional Moduli and Their Physical Meanings modulus ψ de,a, de,s γ hls δ βg, βl χ Sha, Shs θ0 Kls Bi

values or orders of magnitude

physical meaning

1.44

nondimensional heat of evaporation 103-104 nondimensional diffusion coefficient for air and solvent vapor 1-10 nondimensional heat capacities 10-1 nondimensional entropy of dissolution -1 10 -10 nondimensional activation energy for nucleation 10-1-102 (convection inside porous medium)/diffusion 10-1-102 (heat transfer due to conductivity)/(heat transfer due to diffusion) determined by Sherwood numbers for Reynolds’ number mass transfer 0.8-1.3 nondimensional drying temperature 1-10 (liquid to solid mass transfer)/diffusion determined by Biot number Reynolds’ number

where Cg is total concentration of gas phase, which can be calculated using the equation of state for gas, De,a and De,s are the respective effective diffusion coefficients of air and solvent vapor in the gas phase; D+,- is the binary diffusion coefficient between anion and cation. The y’s are molar fractions in gas phase; P is pressure; K is the intrinsic permeability; Kg,eff and Kl,eff are relative permeabilities of gas and liquid phase, respectively. The last term in the fourth equation counts for the electrostatic potential gradient as a result of diffusions of ions. The energy balance is

∂ ( c h + lclhl + Fshs) ) -∇‚(Ng,ahg,a + Ng,vhg,v + ∂t g g g Nl,shl - λ∇T) (14) where h is enthalpy and λ is thermal conductivity. The boundary conditions are symmetric conditions at the center and Neumann conditions at the external surface. Dimensional Analysis. Since the interest of this study is the effect of operating conditions on the catalyst structure, the dimensionless moduli are chosen differently than in Lekhal’s work so that all the operation parameters and physical parameters can be grouped separately. The detailed mathematical derivations are shown in the Appendix. The final nondimensional form for the catalyst scale mass and heat transfers are shown by eqs 22-25. The model parameters and their physical meanings are listed in Table 1. Results and Discussions Parameters of Interest. All the model parameters are shown in Table 1. They are divided into two groups, physical parameters and operating parameters. The mass and heat transfer at the outer surface of the catalyst pellet, described by the Sherwood number and Biot number, as well as the temperature of drying medium (air) outside the catalyst pellet (θ0), are operational parameters. However, the mass and heat transfer coefficients are not independent of each other; they are

Figure 1. Metal loading profile for system with βl ) 5 and χ ) 5 and under conditions with Re ) 5000 and temperatures of 40, 120, and 240 °C.

both related to the flow condition. Indeed, the Sherwood and Biot numbers are readily expressed by the Reynolds’ number using correlations.23 Therefore, the number of operational parameters is reduced to two, Re (Reynolds’ number) for the flow condition of drying medium and θ0 for the temperature. This reduction of parameters makes the analysis much easier. In contrast to the relatively few operational parameters, there are 10 physical parameters. However, among all the components of the system, the physical properties of air (drying medium) and water (solvent) are known, and therefore, most of the physical parameters are of known magnitude. We are particularly interested in three parameters: (1) βl because it contains the two important mechanisms of the migration of the solution, which determines the metal loading profile; (2) χ because it combines the most important heat transfer mechanisms, which determines the temperature profile and therefore the evaporation rate and concentration profile (due to the temperature dependence of solubility); (3) Kls because it describes the rate of deposition of precursors onto the support. Overall, the effects of the two operational parameters on both the metal loading profile and particle size distribution of different systems characterized by the above three physical parameters will be of interest in our discussion in this section. The initial condition for the drying process is the end of impregnation. In this study, the liquid is assumed to be uniformly impregnated into the porous medium; the phase saturation is 0.95:

l(r,t)0) ) 0.95  There is no precursor deposited on the support:

n(x,r,t)0) ) 0 The solution is saturated, and the liquid and gas are in equilibrium. Metal Loading Profile. The metal loading profile is defined by eq 1. However, as the nondimensional form, eq 15 is used in this study.

˜lm(ζ) )

∫0∞Rξ2f(ξ,ζ,τf) dξ

(15)

where ζ is the nondimensional radial position in the catalyst pellet. Figure 1 shows the result for a system with βl ) 5 and χ ) 5 and under conditions with Re )

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Figure 2. Profiles of gas volume fraction inside the porous pellet at different times. The same drying conditions as those in Figure 1 were used with a temperature of 120 °C.

5000 and temperatures of 40, 120, and 240 °C, respectively. This figure shows that both low temperature and very high temperature result in rather uniform distributions, while the intermediate temperature results in an egg-shell-like distribution. When the temperature is very low, the evaporation at the catalyst outer surface is slow and the driving force (the gradient of liquid fraction) for the intrapellet convection is low. Therefore, the precursors are uniformly deposited onto the inner surface of the support. As the temperature gets higher, the evaporation rate keeps increasing and so does the driving force for the convection. As a result, the metal loading profile is more and more egg-shell-like. However, as the temperature continuously increases, the evaporation rate increases progressively until it is much faster than the convection. At this point, the process enters the fast-drying mode under which the liquidphase frontier will shrink down toward the core of the pellet so that the precursors are uniformly deposited. But as the liquid frontier retreats, the vapor removal is determined by the intrapellet diffusion and convection instead of the flow conditions outside the pellet. Therefore the vapor removal rate slows and so does the liquid frontier retrieval. Consequently, the profile appears less uniform toward the center of the pellet. However, the metal loading is more nonuniform over the entire pellet at 120 °C. As shown in Figures 2 and 3, the drying pattern is quite different. At 240 °C, there is a rather clear boundary between wet and dry areas in the pellet at the beginning, and the outer part is dried out quickly before significant convection occurs. However, when temperature is 120 °C, the clear boundary disappears and the migration of precursors due to convection lasts through out the drying process. Figure 4 shows the results for a different system with βl ) 1 and χ ) 1. From the figure, it should be clear that the precursor is less concentrated in the outer part of the pellet due to the stronger diffusion effect in the centripetal direction. At 240 °C, the profile appears as an egg-yolk, suggesting that in this situation, the diffusion effect starts to dominate. This is because of the high concentration gradient raised by the temperature dependence of the solubility and the high temperature gradient. Our interpretations of the events at high drying temperatures are suspect as the model does not account for the possibility of the formation of bubbles at the pore

Figure 3. Profiles of gas volume fraction inside the porous pellet at different times. The same drying conditions as those in Figure 1 were used with a temperature of 240 °C.

Figure 4. Metal loading profile for the system with βl ) 1 and χ ) 1 and under conditions with Re ) 5000 and temperatures of 40, 120, and 240 °C.

surface. Bubble formation can occur when the temperature difference between the liquid and the pore surface is sufficiently high. However, such a phenomenon can be admitted into the model only when there is a mechanism for distinct temperature profiles for the solid and the fluid within the pores. Since the averaging approach used here does not accommodate such disparity between the solid and fluid temperature profiles, the high temperature scenario presented by the model becomes less realistic as the temperature is progressively increased. The above figures show clearly the effect of the combination of convection and diffusion: the former results in egg-shell, while the latter results in egg-yolk. To explore this further, a quantified definition for the terms egg-shell, egg-yolk, and uniform is needed. As shown in Figure 5, for those with 10% higher metal loading at the outer part of the catalyst pellet than the average, we consider them egg-shell. Similarly, egg-yolk catalyst will have 10% higher metal loading in the center part. Therefore, a catalyst is regarded as (admittedly, with some arbitrariness) egg-shell if

∫0.71˜lm(ζ)4πζ2 dζ/∫0.714πζ2 dζ > 1.1 ∫01˜lm(ζ)4πζ2 dζ/∫014πζ2 dζ

(16)

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Figure 5. The definition of egg-shell and egg-yolk. A catalyst is egg-shell if the metal loading in the outer part is 10% higher than the average and egg-yolk if it is 10% higher in the inner part.

Figure 7. Catalyst structure in a Re-θ0 map for system with βl ) 0.1 and χ ) 50, where the solid line is for Kls ) 5 and the dotted line is for Kls ) 1.

Figure 6. Catalyst structure in a Re-θ0 map for a system with βl ) 0.1 and χ ) 0.5, where the solid line is for Kls ) 5 and the dotted line is for Kls ) 1.

and egg-yolk if

∫00.3˜lm(ζ)4πζ2 dζ/∫00.34πζ2 dζ > 1.1 ∫01˜lm(ζ)4πζ2 dζ/∫014πζ2 dζ

(17)

Definitions 16 and 17 allow us to examine the structure that we can get in a Re-θ0 space for a system characterized by βl, χ, and Kls. Figure 6 is a structure map for systems with βl ) 0.1 and χ ) 0.5. In this case, the diffusion occurs much more easily than convection, and a low thermal conductivity raises the possibility of hightemperature gradient, which also implies high concentration gradient. Therefore, at high temperature and low Reynolds’ number, where the former increases the diffusion driving force and the latter decreases the convection driving force, the diffusion dominates and an egg-yolk catalyst is obtained. On the other hand, the uniform structure appears at low temperatures and high Reynolds’ numbers. Due to the low value of βl, convection is not strong enough to produce an egg-shell catalyst despite the value of the Reynolds’ numbers. A path marked in Figure 6 shows that we can change from uniform region to egg-yolk region at point A and change back to uniform at point B by increasing temperature only. This changing-back behavior is produced by the fact that the temperature gradient increases first and then decreases when the drying temperature is continuously increased until it brings all the liquid inside the catalyst to boiling point when a uniform temperature profile is expected. It is also shown that changing the value of Kls from 5 to 1 broadens the area of the egg-

yolk region. This is because the decrease of deposition rate makes the solution more easily oversaturated with a consequent increase in concentration gradient. From the foregoing discussion, one can see that the diffusion/convection behavior is determined by both the physical and operational parameters, the two types playing different roles: physical parameters determine the relative importance of convection and diffusion, while the operational parameters control the driving forces. It is the combination of the two types of parameters that finally determines whether diffusion or convection dominates or whether they are both important. As we increase the thermal conductivity, the temperature gradient decreases until it is almost isothermal in the catalyst; therefore, the concentration gradient results from the oversaturation gradient only, and one can expect that the diffusion plays a less important role and metal loading will move from center to the surface of the catalyst. Figure 7 shows the structure map for βl ) 0.1 and χ ) 50. As shown in this figure, the egg-yolk region disappears due to high thermal conduction. When the convection driving force is low (low Reynolds’ number), uniform structure is obtained, while at higher Reynolds’ numbers, increased convection favors an eggshell structure. For the same reason as mentioned before, the area of egg-shell region is compressed when the value of Kls is decreased. A similar changing-back behavior as in Figure 6 is also observed in this figure: along the marked path, we change from uniform to eggshell at point A and then change back at point B. This changing-back is because the fast drying region is reached at point B as the earlier discussion. Figures 8 and 9 show the structure map for βl ) 10. The high value of βl results in only egg-shell and uniform, regardless of the values of χ. The parameter Kls has little effect in these two cases. Note that the changing-back behavior is lost in Figure 8. This is probably because the low thermal conductivity stops the liquid frontier from continuously retreating to the core as the intrapellet diffusion becomes the bottleneck for vapor removal. Figure 10 shows the case of βl ) 1 and χ ) 5 in which the system does not have a strong preference toward either convection or diffusion. As a result, all structures

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Figure 8. Catalyst structure in a Re-θ0 map for system with βl ) 10 and χ ) 0.5.

Figure 9. Catalyst structure in a Re-θ0 map for system with βl ) 10 and χ ) 50.

Figure 10. Catalyst structure in a Re-θ0 map for system with βl ) 1 and χ ) 5. All three structures appear in this map.

can be obtained for this system by just changing operating conditions. The map can be divided into four regions for different structures. Note that although regions I and III are both for uniform structure, the mechanism is not the same: region I is for slow drying under low temperature and Reynolds’ number (therefore, low convection drying force), while region III is for fast drying where the vapor removal rate dominates. More importantly, as will be shown in the following

Figure 11. The diagram of structure inaccessibility, where the solid line is for Kls ) 5 and the dotted line is for Kls ) 1.

sections, the structure is not the same for these two regions as we consider the particle size distribution. Structure “Inaccessibility”. As discussed in the previous section, physical parameters characterize the physical preference (toward convection or diffusion) of the system, while operating parameters give rise to the driving forces. The combination of the two effects finalizes the structure of the catalyst. One can easily predict the situation if both effects are in the same direction. However, when they are in opposite directions (for example, the system favors diffusion, while the concentration gradient is low), it is more complicated. In extreme cases, in which the physical nature of the system has a strong preference toward one direction, the operating conditions may not be able to compensate this tendency any more. In this case, change of operational parameters cannot reach the corresponding structure region. Therefore, for some systems, some structures may not be accessible. For example, as we discussed in previous section, in Figure 6, the egg-shell structure is absent, while in some other cases, egg-yolk is absent. Theoretically, one may expect to overcome this “inaccessibility” by using extreme operating conditions. But in practice, these extreme operating conditions are hard to achieve for different reasons, such as safety issues and energy consumption. In this study, a temperature between 20 and 250 °C and a Reynolds’ number between 100 and 10 000 are considered to be practical. For a given system, if a certain structure cannot be obtained in this operation range, we consider that structure to be inaccessible. Figure 11 plots all of the structure inaccessibility in a χ-βl space. As we can see, uniform is accessible to all systems because we can always get a uniform structure by simply using low temperature. Egg-shell is not accessible to systems with low thermal conductivity and permeability, and egg-yolk is not accessible to systems with high thermal conductivity and permeability. Decrease of the deposition rate may produce broader areas of egg-yolk accessibility, and this effect occurs mainly in systems with high thermal conductivity. This is easy to understand because only under this situation does the oversaturation become important to the concentration gradient. Particle Size Distribution. As discussed in previous sections, normalizing of the number density function will give the particle size distribution. However, it is more convenient to use the number density function because it contains not only the particle size distribution

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Figure 15. Nondimensional number density function along the catalyst radius for region IV.

Figure 12. Nondimensional number density function along the catalyst radius for region I.

Figure 13. Nondimensional number density function along the catalyst radius for region II.

Figure 14. Nondimensional number density function along the catalyst radius for region III.

information but also the metal loading information. The nondimensional number density functions for the four different regions in Figure 10 are shown in Figures 1215. The figures show that not only the metal loading profiles but also the particle size distributions are quite different for the four regions. The reason for dividing the uniform region into I and III is more clear here: not only the mechanism is different, the structures are also different despite that they are all uniform. The particle

size is much smaller in region III than I, and therefore one can expect that catalyst in region III has higher surface area and less stability. The distributions for I and II look very similar except for the metal loading. In both cases, higher average particle size at the outer surface than in the center is observed. That is because in both situations, the vaporization process occurs mainly at the outer surface leading to a more favorable condition for the particle size growth there. Bigger particle size at the surface occurs also in region IV for the same reason. The difference between particle sizes at the surface and in the center may affect the stability. Generally, in the case of strong exothermal reaction and high Thiele moduli, the temperature is much higher in the center than at the surface; therefore it is easier for the redispersion to occur in the inner part of the catalyst. The smaller particle size in the center aggravates the redispersion. This effect weakens the stability improvement by using egg-shell catalyst, because as we try to avoid the bad redispersion in the center by moving most of the active phase to the outer part, we make the inner part more vulnerable. In addition to the redispersion, kinetic effects may also rise from this pattern of particle size distribution. As from our previous studies with FTS, the reoxidation of the supported cobalt particle as a deactivation mechanism is more serious in the inner part of the catalyst pellet due to the higher water level resulting from the diffusion limitation. Since the reoxidation is favored for smaller particles, similar to the effect of temperature, the use of egg-shell catalyst will make the inner part easier to deactivate. However, region IV is different from the other regions where the particle size is higher in the center. This is because in fast drying, since the liquid-phase shrinks into the core, particles in the center have more time to grow than those outside. The overall average particle size for the whole catalyst is in this sequence: I > II > IV > III, and therefore, the surface area is in opposite direction. Generally, higher temperature and Reynolds’ number leads to finer particles because a higher nucleation rate is expected in this situation due to the high-temperature sensitivity of the nucleation process. Conclusions This study formulated a general model for the drying process in the wetness impregnation method for the synthesis of supported catalyst. Two featured characteristics are studied: the metal loading and particle size distribution. Results show the following: (1) The combination of convection and diffusion finalizes the metal loading profile. Both physical and operational param-

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Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005

eters can affect the nature of convection/diffusion in the system but with different effects: physical parameters characterize physical preference for convection or diffusion, while the operational parameters affect the driving forces. Both changes of these two types of parameters may lead to different catalyst structure. Four regions are defined for different structures. (2) Structure inaccessibility for given systems is studied. For some systems, some structure may not be accessible in practice. A structure accessibility map is given in this study. (3) The particle size distribution changes dramatically in different regions. Higher temperature and Reynolds’ number results in finer particles. The particle size changes along the radius of a catalyst pellet. In most cases, bigger particles are expected at the outer surface, but for region III, we have opposite situation.

(

)

Cs ∂ gpg(yaγg,a + ys) + Flcl,sγl,sθ - Flcl,sψ + γθ ) ∂τ Cg,0 s ∇ ˜ b,(pg∇y˜ a(de,aγg,a - 1) + βggpg∇p˜ g(yaγg,a + ys) + βlFcl,sl∇p˜ lγl,sθ - βlFcl,slψ - Fχ∇θ˜ ) (25) With boundary conditions

(

)

pg pg pg 1 -de,a ∇y˜ a - βg yag∇p˜ g ) Sha - ya n θ θ θ0 θ pg pg pg 1 βlFcl,sl∇p˜ l ) -Shs ysn ∇y˜ s - βg ysg∇p˜ g θ θ de,s θ d+,-∇c˜ l,( + βlcl,(∇p˜ l ) 0

Appendix: Dimensional Reduction

pg∇y˜ a(de,aγg,a - 1) + βggpg∇p˜ g(yaγg,a + ys) + βlFcl,sl∇p˜ lγl,sθ - βlFcl,slψ - χ∇θ˜ ) χBi(θ0 - θ) (26)

To nondimensionalize eqs 7-9 and 4, it is convenient to use the radius of the pellet as characteristic length and the drying pressure as characteristic pressure:

where

P p) Pair

1 ∇ ˜ ) ∇, L

(18)

To separate the operational parameters and physical parameters, the boiling point of solvent is used as the characteristic temperature instead of using drying temperature, and for the same reason, the concentration of pure water is used to reduce the concentrations.

c)

C , Cl,0

θ)

T Tb

(19)

With all the above selections, we found it convenient to use the following nondimensional time:

τ)

t L /D+,-

(20)

2

where the characteristic time is the time for the diffusion of unit of precursor through a cross-section under unit of the concentration. In eq 13, the concentration of gas phase can be calculated using the equation of state:

Cg )

Pg RT

(21)

Inserting eqs 18-21 into the model equation will lead to the nondimensional form:

(

)

(

)

pg pg ∂ pg  y ) de,a∇ ˜ b, ∇y˜ a + βgyag ∇p˜ g ∂τ g θ a θ θ

(

)

(

(22)

pg ∂ pg g ys + lcl,sF ) ∇ ˜ b, Fβlcl,sl∇p˜ l + de,s ∇y˜ s + ∂τ θ θ pg de,aβg ysg∇p˜ g (23) θ

)

L2 ∂ (lcl,() ) ∇ ˜ b,(∇c˜ l,( + cl,(lβl∇p˜ l) - R( H ∂τ D+,-Cl,0 ls (24)

de,a )

De,a Cl,0 Cp ∆h , F) , γ) , ψ) , D+,Cg,0 Cpg,s Cpg,sTdry KKg,r P µg air βg ) De,a

KKl,r P µl air kaL λ βl ) , χ) , Sha ) , D+,D+,-Cl,0Cpg,s De,a ks L hL , Bi ) Shs ) De,a λ Note that in eq 24 the deposition rate is still dimensional. The reduction of the PBE is needed to reduce it. In reduction of eq 4, the characteristic time is same as in eq 20:

t0 ) L2/D+,-

(27)

The characteristic size then takes the following form:

x0 ) kgt0

(28)

Equation 4 is easy to solve analytically given that the external variables are fixed. The solution is

{

I for xkgt

(29)

Equation 29 gives us a very convenient way to reduce the number density function:

f)

n kn/kg

(30)

Equations 27-30 enable us to reduce the PBE to the following form:

∂ ∂ f + {f(ln S)2} ) 0 ∂τ ∂ξ

(31)

Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005 6475

with boundary condition

(

(ln S)2f ) exp -

)

δ θ3(ln S)2

at ξ ) 0

(32)

Note that a in eq 6 is assumed to be 1 due to the low coverage of the support surface. With the nondimensional form of the PBE, eq 24 can be reduced further:

∂ ˜ b,(∇c˜ l,( + cl,(lβl∇p˜ l) - R(Klshls ( c ) ) ∇ ∂τ l l,(

(33)

∫0∞fξ2 dξ

Fp Kls ) 4πR(kgt0)3knt0As Cl,0 The supersaturation, S, is a temperature-dependent term because the solubility is temperature-dependent:

[ (

Cs Hs 1 1 ) exp Cs,r R T Tr

)]

(34)

where Cs is the temperature-dependent solubility, Hs is entropy of the dissolution, and Cs,r is a reference solubility under reference temperature Tr. The supersaturation is therefore expressed by the nondimensional term as

S)

[(

c 1 1 C ) exp hs Cs cs,r θ θr

)]

(35)

where

hs )

Greek Symbols , g, l ) porosity and volume fractions of liquid and gas phase µl, µg ) liquid and gas viscosities (Pa s) φ ) electric potential (V) λ ) thermal conductivity (J/(ms K)) θ ) nondimensional temperature τ ) nondimensional time

where

hls ) (ln S)2

Pl, Pg ) liquid and gas pressures (Pa) R ) pellet size r, r ) position vector and radial position in pellet (m) Re ) Reynolds’ number rg, kg ) particle growth rate and rate constant (m/s) T ) temperature (K) t ) time (s) x ) particle size (m) Y ) vector of external variables ya, ys ) air and solvent vapor molar fractions in gas phase

Hs RT

Notations c ) nondimensional concentration Cl,(, Cl,s ) ion and solvent concentrations in liquid phase (mol/m3) Cg,a, Cg,s ) air and solvent concentrations in gas phase (mol/ m3) De,a, De,s ) effective diffusion coefficients of air and solvent in gas phase (m2/s) D( ) binary diffusion coefficient of ions in liquid phase (m2/ s) F ) Faraday constant (96 500 C/mol) f ) particle size distribution (no./m) ∆G* ) Gibbs function difference (J/mol) hl, hg, hs ) enthalpies of liquid, gas, and solid phases (J/ mol) Hls ) rate of liquid to support deposition (m3/(m3 s)) I, kn ) nucleation rate and rate constant (no./(m2 s)) K, Kl,eff, Kg,eff ) permeability (m2) and relative permeability for each phase lm, Lm ) position-dependent and overall metal precursor loading (m2/m3) n ) number density function for precursor particles (no./ (m2 m)) Ng,a, Ng,s ) air and solvent fluxes in gas phase (mol/(m2 s)) Nl,(, Nl,s ) ions and solvent fluxes in liquid phase (mol/ m2s) Pair ) air pressure (Pa) Pc ) capillary pressure (Pa)

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(22) Neimark, A. V.; Kheifez, L. I.; Fenelonov, V. B. Theory of Preparation of Supported Catalysts. Ind. Eng. Chem. Prod. Res. Dev. 1981, 20, 439. (23) Hager, J.; Hermansson, M.; Wimmerstedt, R. Modelling Steam Drying of a Single Porous Ceramic Sphere: Experiments and Simulations. Chem. Eng. Sci. 1996, 52, 1253-1264. (24) Yiotis, A. G.; Stubos, A. K.; Boudouvis, A. G.; Yortsos, Y. C. A 2-D Pore-Network Model of Drying of Single Component Liquids in Porous Media. Adv. Water Resour. 2001, 24, 439-460. (25) Uemura, Y.; Hatate, Y.; Ikari, A. Formation of Nickel Concentration Profile in Nickel/Alumnia Catalyst During PostImpregnation Drying. J. Chem. Eng. Jpn. 1987, 20. (26) Melo, F.; Cervello, J.; Hermana, E. Impregnation of Porous Supports: I. Theoretical Study of the Impregnation of One or Two Species. Chem. Eng. Sci. 1980, 35, 2165-2174.

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Received for review October 27, 2004 Revised manuscript received February 22, 2005 Accepted March 1, 2005 IE048954B