Mitigating Deactivation Effects through Rational Design of

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Ind. Eng. Chem. Res. 2010, 49, 11087–11097

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Mitigating Deactivation Effects through Rational Design of Hierarchically Structured Catalysts: Application to Hydrodemetalation Sanjeev M. Rao and Marc-Olivier Coppens* Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York, 12180

The broad pore network of a hierarchically structured hydrodemetalation catalyst, containing both nano- and macropores, is mathematically optimized to maximize the conversion of nickel metalloporphyrins in crude oil residue. A random spheres model (RSM) describing the nanoporous catalyst at the mesoscale is combined with a two-dimensional continuum approach to model the entire catalyst at the macroscale. Catalysts with a spatially uniform as well as a nonuniform macroporosity distribution are optimized. The macroporosity profiles of the optimal nonuniform catalysts fluctuate about the optimal uniform value, while the spatially and temporally integrated reaction rates from both types of optimized catalysts are almost the same. Moreover, the integrated reaction rate of the optimal hierarchically structured catalysts are almost 8 times higher than the yield obtained from a purely nanoporous catalyst. For a time on stream of 1 year, approximately 21% less catalytic material is required in the optimal hierarchically structured catalyst, compared to the purely nanoporous one. The mathematical optimization tools employed here can be extended to other industrially important reactions affected by deactivation. 1. Introduction Industrial catalysts are plagued by intraparticle diffusion limitations, which are compounded by the effects of deactivation. Deactivation by deposition of solid products of reactions on the catalyst surface reduces the intrinsic activity of the catalyst by blocking active sites and also limits access to pores located deep within the catalyst. Thus, the rate of reaction over the catalyst decreases with increased time on stream. Below a certain activity level, the catalyst has to be regenerated, which increases production costs and also lowers the lifetime of the catalyst. In this paper, we report results on the optimization of the broad pore network in a hierarchically structured catalyst undergoing deactivation through loss of active surface area and pore plugging. The hierarchically structured catalyst comprises both nanopores (micropores or narrow mesopores) and a network of broad pore channels (mesopores or macropores), which act as highways for reactants to reach the active sites in the nanoporous region of the catalyst. Hierarchically structured catalysts with an optimized broad pore network have been shown to produce significant improvement over purely nanoporous catalysts through an increased reaction yield, for diffusion limited reactions not affected by deactivation.1-4 To the best of our knowledge, the broad pore network for catalysts that undergo deactivation has not been mathematically optimized. Considerable experimental efforts have been directed toward the synthesis and testing of mesoporous and microporous catalysts, especially zeolites, to reduce intraparticle diffusion limitations5,6 and resist deactivation,7,8 but the desired properties of the optimal broad pore network are not well understood. This study aims at providing insight into the nature of the optimized broad pore network. The methodology can be applied to a wide range of industrially important reactions, such as alkylations, isomerizations, reforming, and cracking. Advances in the ability to synthesize well-controlled hierarchical materials is making it increasingly possible to synthesize theoretically optimized structures.5,6,9,10 * To whom correspondence should be addressed. Tel.: 518-276-2671. Fax: 518-276-4030. E-mail: [email protected].

The deactivation of hydrodemetalation catalysts is discussed as a case study. Hydrodemetalation is a problem representative of a large class of industrially relevant processes that undergo deactivation by loss of active surface area and pore plugging. Hydrodemetalation involves the removal of organometallic compounds of nickel and vanadium (metalloporphyrins) from crude oil residue as metal sulfides, using hydrogen in a liquid phase reaction, catalyzed by CoO-MoO3/Al2O3 catalyst.11 The deposited metal sulfides, in the form of crystallites,12 block the active sites for reaction and also plug the pores. The metal sulfides themselves possess some activity for hydrodemetalation; however, it is a fraction of the intrinsic activity of the fresh catalyst. The kinetics of the hydrodemetalation reaction are quite well understood. Either a series reaction scheme11 or a lumped single reaction step mechanism13 is commonly used to describe the conversion of the metalloporphyrins. The hydrodemetalation reaction is also intraparticle diffusion limited,12,14-16 which means that pore mouth catalysis can occur, blocking the pores and limiting reactant access to the fresh catalyst surface. A broad pore network, properly introduced into such a nanoporous catalyst, can be effective in allowing faster transport to the inner regions of the catalyst. Therefore, hydrodemetalation is an interesting case for optimization studies. Researchers from both academia and industry have mathematically modeled hydrodemetalation catalysts in deactivation studies. These models include a conventional cylindrical pore model,17 and the random spheres model (RSM)14,15,18 to describe the purely nanoporous catalyst. The RSM is employed in this study as well and is described in detail further on. Bimodal hydrodemetalation catalysts have been mathematically modeled using cylindrical pore models to account for metal sulfide deposition in both the nanoporous catalyst and the macropores.13 Percolation theory has been used to determine the role of macropore transport in hydrodemetalation catalysts operating below the percolation threshold.19 However, in both models of the bimodal catalyst listed above, the macropores are assumed to be randomly distributed in the catalyst pellet. Therefore, they may not be as effective as an optimized broad pore network in reducing intraparticle transport limitations and providing better

10.1021/ie1009487  2010 American Chemical Society Published on Web 06/28/2010

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Figure 1. (a) Hierarchically structured catalyst composed of nanoporous region (gray spheres) separated by a broad pore network. (b) Purely nanoporous catalyst visualized using the RSM. The nanoporous region of the hierarchically structured catalyst in part a retains its intrinsic character and can be modeled using the RSM.

access to inner regions of the catalyst. Keil and Rieckmann20 optimized the nanopore radius and nanopore volume per unit mass of a bimodal hydrodemetalation catalyst, using the model of Pereira and Beeckman,13 to maximize the average conversion of metalloporphyrins over time. However, they did not optimize the macropore network. An important assumption made in this study is that deactivation does not affect the broad pore network. Our goal is to optimize the properties of the broad pore network. The RSM was employed to model the nanoporous region of the hierarchically structured catalyst, and we used the properties of the nanoporous catalyst, such as the local effective diffusivity and the local effectiveness factor, to calculate the effective properties of the entire catalyst pellet. A two-dimensional continuum approach was used to describe the diffusion-reaction process in the entire catalyst pellet. The objective function for maximization is defined as the spatially and temporally integrated reaction rate, over a given time on stream, for the hierarchically structured catalyst, scaled with respect to the spatially and temporally integrated reaction rate, over the same period of time, for a purely nanoporous catalyst with the same geometry. We discuss how the optimal broad pore network functions in reducing the effects of deactivation. We compare the performance of optimized uniform (single macroporosity) and nonuniform (distribution in macroporosities) hydrodemetalation catalysts and show why they perform equally well. An interesting and unique outcome of this work is that the approach of minimizing intraparticle diffusion limitations can also provide vital clues to better resist deactivation. 2. Mathematical Modeling and Problem Formulation 2.1. Geometry. The hierarchically structured fresh hydrodemetalation catalyst is schematically illustrated in Figure 1. The term “hierarchically structured” is used to describe the hierarchy that exists within the catalyst pellet, wherein the large pores act as transport highways for molecules to and from the nanopores located in the nanoporous catalyst. The catalyst is not ordered within the nanoporous catalyst, and the pore space of the nanoporous catalyst is visualized by means of the random spheres model (RSM). Weissberg21 developed the RSM in his study of diffusion coefficients in porous media. Haller22 applied the RSM to study nucleation and growth of network structures of microphases in the separation of immiscible liquids. The RSM is preferred over the cylindrical pore description because experimental results on aged hydrodemetalation catalysts showed metal sulfide deposits that were larger than the average pore diameter,12 which is theoretically impossible in the cylindrical pore model. A detailed development of the RSM applied to the hydrodemetalation problem can be found in the work of Wei and co-workers.14,15 In the next subsection, we

briefly summarize the important equations of the RSM, as developed in the references mentioned above, and apply it to the problem at hand. 2.2. Random Spheres Model for Nanoporous Catalyst. The RSM describes the pore space of a porous material as the void region that exists between randomly intersecting spheres (either monodisperse or with a size distribution) that constitute the solid fraction. The randomly intersecting spheres can be mathematically described by a Poisson process. The parameters of the RSM for a fresh catalyst are the radius of the spheres, a1, and their number density, n1. In this work, the catalyst spheres are assumed monodisperse. The porosity, εn, and specific surface area, σ, of the fresh nanoporous catalyst are used to calculate a1 and n1 in eqs 1 and 2: 4 εn ) exp - πn1a13 3

(

)

(1)

σ ) 4εnπn1a12

(2)

In the case of deactivation, the metal sulfide crystallites grow from fixed nucleation sites on the CoO-MoO3/Al2O3 hydrodemetalation catalyst.12 Modeling the metal sulfide crystallites as another set of random spheres leads to modifications in eqs 1 and 2, as shown in eqs 3 and 4: 4 εn ) exp - π(n1a13 + n2a23) 3

[

σ ) 4πεn(n1a12 + n2a22)

]

(3) (4)

The parameters n2 and a2, representing the number density of metal sulfide crystallites and the radius of a single crystallite sphere, respectively, are unknown. Equation 3 mathematically describes the decrease in porosity of the nanoporous catalyst, as the metal sulfide spheres grow on the catalyst surface and occupy the pore volume. Equation 4 accounts for the change in surface area due to the growing metal sulfide spheres, including the effect of overlap between spheres. Figure 2 illustrates the deactivation process affecting the nanoporous catalyst. The catalytically active surface area per unit volume of the nanoporous catalyst, which accounts for the low hydrodemetalation activity that the deposits possess through the activity ratio Γ, is defined by σrxn ) 4πεn(n1a12 + Γn2a22)

(5)

Equations 3-5 are all defined with respect to the volume of the nanoporous catalyst. The active surface area expressed with respect to the mass of nanoporous catalyst is given by

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S)

σrxn Fcat

(6)

The value for n2 is assumed constant and set equal to n1/10, based on experimental results reported by Wei and co-workers.12,15 To determine a2, a PDE describing the deposition of metal sulfides, M (kg metals/kg fresh catalyst) as a function of time is solved: ∂M ) Skc ∂t

(7)

From the solution of M, the value of a2 can be easily calculated by converting to equivalent volume terms: a2 )

 3

MFcat 4 F πn 3 deposit 2

(8)

In eq 7, t is the variable for time, k is the intrinsic rate constant for hydrodemetalation, which follows first-order kinetics,13,14,20 and c is the concentration of the metalloporphyrin in the nanoporous catalyst. The assumption of first-order kinetics is justified for Boscan vacuum feed as established experimentally by Pereira and Beeckman.13 In eq 8, Fcat is the density of the fresh nanoporous catalyst and Fdeposit is the density of the metal sulfide crystallites. The updated nanoporosity from eq 3 is used to update the effective diffusivity in the nanoporous catalyst using a correlation for configurational diffusion.23 First, the updated nanoporosity and specific surface area are combined to define an equivalent cylindrical pore radius, rp, which, in turn, is used in the equation to calculate the effective diffusivity in the nanoporous catalyst: rp )

2εn σ

εn De ) Dbulk [1 - (rmetal /rp)]4 τn

(9)

(10)

Equation 10 describes the effect of the molecular size of the organometallic reactant (rmetal) relative to the pore size on the diffusivity of the reactant molecule. Dbulk is the bulk diffusivity of the reactant in the liquid phase. The tortuosity in the nanoporous catalyst, τn, is set to 4.14 Hence, all the information for the nanoporous catalyst is contained in the local Thiele modulus:

Figure 2. Deactivation results from the deposition of metal sulfide crystallites, represented as a set of random spheres (in black), with radii a2 and number density of crystallites n2 growing on the catalyst surface (in gray).

φlocal )

Lm 2



kSFcat DeFoil

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(11)

from which the local effectiveness factor can be determined: ηlocal )

tanh φlocal φlocal

(12)

In eq 11, Lm is the size of the nanoporous regions separated by the broad pores (see section on Simulation) and Foil is the density of the crude oil residue feed. 2.3. Continuum Model. Since the nature of the optimum broad pore network is unknown a priori, we studied and optimized two types of hierarchically structured catalysts: uniform and nonuniform. By uniform catalysts, we mean that the macroporosity has a unique value, whereas in a nonuniform catalyst, there exists a distribution in the macroporosities. In this regard, parallels can be drawn between uniform and nonuniform structures and bidisperse and bimodal structures. Bidisperse catalysts contain nanoporous regions of a unique size separated by broad pores of another unique size. Bimodal catalysts contain a distribution in the sizes of the broad pores and nanoporous regions. The diameter of the broad pores in the hierarchically structured catalyst is not and does not need to be optimized in this work. It is kept constant, because, as is shown further on, the optimal structures are insensitive to the size of the broad pores, as long as there are no diffusion limitations in the nanoporous region. Hence, the nonuniform catalyst is, strictly speaking, not necessarily a bimodal structure, even though there is a distribution in macroporosities and the size of the nanoporous regions. This distinction between bimodal and nonuniform structures does not apply to bidisperse and uniform structures. One way to interpret the optimal nonuniform structures is to consider the macroporosity distribution and a broad pore size distribution, keeping the size of the nanoporous regions constant. Keeping the distinction between bimodal and nonuniform catalysts in mind, we can treat a nonuniform catalyst as being made up of a very large number of uniform substructures (USS), akin to the approach used to represent bimodal catalysts.1 Figure 3 is a schematic representation of a uniform and a nonuniform catalyst for the square geometry, which is employed in this study. The RSM description of the nanoporous catalyst is combined with a two-dimensional continuum model of the hierarchically structured catalyst for the purpose of simulation and optimization. By employing the continuum approach, the hierarchically structured catalyst pellet can be treated as a continuum with effective properties. In this problem, the effective rate constant (kE) and the effective diffusivity (DE) for the hierarchically structured catalyst are defined as functions of the macroporosity, ε. The relevant equations are kE ) kηlocal(1 - ε)

(13)

ε DE ) Dbulk [1 - (rmetal /rbroad)]4 τ

(14)

where rbroad is the radius of the broad pores. The local effectiveness factor ηlocal will be used to establish the link between deactivation in the nanoporous catalyst and the continuum model for the hierarchically structured catalyst. The diffusion-reaction equation does not need to be explicitly solved in the nanoporous catalyst because the hydrodemetalation kinetics obey first-order kinetics. Equation 12 is not a rigorous solution to the problem of diffusion and reaction in a square

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σHS ) σ(1 - ε) SHS )

(18)

σHS σ ) )S Fcat,HS Fcat

(19)

The solution of the diffusion-reaction equation is needed to solve the relevant optimization problem and, also, to provide the concentration profile throughout the catalyst, so that eq 7 can be solved. In the dimensionless form of eq 15, a Thiele modulus for the hierarchically structured catalyst can be defined, choosing half the side of the square catalyst slab (L/2) as the characteristic length: Figure 3. Uniform (a) and nonuniform (b) catalysts. The nonuniform catalyst is represented as a catalyst consisting of N × N uniform substructures (USS). In this illustration, N ) 3.

Figure 4. The uniform catalyst is treated as a continuum with properties which depend on the macroporosity (ε), resulting from the broad pore network.

exposed to different concentrations on all four edges; it is employed here as an approximation. One way to overcome this approximation would be to numerically solve the problem and plot a table of the local effectiveness factor versus the Thiele modulus. However, it is well-known that eqs 11 and 12 can be used to calculate the effectiveness factor with high accuracy.24 In this work, the tortuosity of the large channels τ has not been considered. As pointed out elsewhere,1 the tortuosity of the broad pore network does not affect the qualitative outcome of the optimization problem for diffusion limited reactions without deactivation. In the present case, deactivation does not affect the large channels; therefore, the argument stated above can be applied here as well, but the tortuosity of the large channels can affect the diffusivity.25 Figure 4 shows the application of the two-dimensional continuum approach to a uniform catalyst. The equation for diffusion and reaction in the entire catalyst pellet in two dimensions, with the appropriate boundary conditions, is given by

φHS )

L 2



kηlocal(1 - ε)SFcat FoilεDbulk[1 - (rmetal /rbroad)]4

Finally, the link between the deactivation modeling in the nanoporous region and the continuum model is established by the usage of the effective diffusivity in the nanoporous catalyst to calculate the local effectiveness factor for the effective rate constant (see eqs 11-13). Equations 15 and 16 are applied to both the uniform and nonuniform catalyst structures. For the uniform structure, the diffusion-reaction equation applies over the entire catalyst volume (area in 2D); for the nonuniform structure, the diffusionreaction equation is applied to each uniform substructure with appropriate continuity of flux boundary conditions applied at the interfaces. The reaction yield is integrated over the entire volume for both types of structures. In the next section, we discuss the formulation of the objective function and the simulation of the mathematical model developed in this section. 2.4. Objective Function Formulation. The superiority of the optimal hierarchical structures over the purely nanoporous catalyst is evaluated. The performance of a catalyst is evaluated by spatially integrating the reaction rate over the volume at every time step and then integrating over the total time on stream, giving a reaction yield: reaction yield )

∫ ∫ ∫ Sk cF t

(15)

c(x, 0) ) c(0, y) ) c(x, L) ) c(L, y) ) c0

(16)

(

)

(

)

The terms used in eq 15 require some clarification, since the equation applies to the entire catalyst particle volume, V (large pore network volume, εV, plus nanoporous catalyst volume,(1-ε) V). In eqs 1-12, the basis is the nanoporous catalyst volume. Hence, to obtain quantities based on the total catalyst particle volume, the corresponding pellet parameters have to be modified accordingly. Thus, we have the following parameters for the hierarchically structured catalyst: Fcat,HS ) Fcat(1 - ε)

(17)

x

y

E

cat

dx dy dt

(21)

The reaction yield can be used to define a scaled yield, by dividing eq 21 with the reaction yield from the purely nanoporous catalyst:

scaled yield )

∫ ∫ ∫ Sk cF ∫ ∫ ∫ SkcF t

x

t

Fcat ∂ ∂c ∂c ∂ -DE + -DE ) -kESHSc ∂x ∂x ∂y ∂y Foil

(20)

x

y

y

E

cat

cat

dx dy dt

(22) dx dy dt

The objective function for maximization is the scaled yield. It should be noted that the surface area terms do not cancel in eq 22 because they depend on the rate of deactivation, which differs between the hierarchically structured and the purely nanoporous catalysts. Hydrodemetalation is a liquid phase reaction; hence, the mean free path of the reactant molecules is much smaller than the size of the broad pores, so that Knudsen diffusion does not play a role in this problem. With these considerations, the macroporosity (ε) is the only optimization variable in this study. For a given macroporosity and size of the broad pores, the size of a nanoporous region can be computed as1 d Lm ) (1 - ε + √1 - ε) ε

(23)

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14

Table 1. Simulation Parameters (From Mace´ and Wei ) Properties of Purely Nanoporous Catalyst initial nanoporosity (εinit) initial surface area per unit mass (Sinit) initial surface area per unit volume (σinit) size of catalyst slab (L) density of nanoporous catalyst (Fcat) density of nickel sulfide deposits (Fdeposit)

0.64 1.76 × 105 m2/kg cat 2.62 × 108 m2/m3 cat 1.6 × 10-3 m 1490 kg cat/m3 5820 kg sulfides/m3 Random Sphere Model Parameters 3.27 × 10-9 m 3.024 × 1024 m-3

radius of random spheres (a1) number density of random spheres (n1) Properties of Feed

900 kg oil/m3 5 × 10-4 kg species/kg oil (500 ppm) 1.5 × 10-9 m

density of residue oil (Foil) concentration of nickel metalloporphyrin (c0) radius of metalloporphyrin molecule (rmetal) Diffusion-Reaction Parameters intrinsic rate constant (k) ratio of catalytic activities of deposits to fresh catalyst (Γ) initial effective diffusivity of metalloporphyrin in the nanoporous catalyst (De,init) bulk diffusivity of metalloporphyrin (Dbulk) time on stream

One might argue that, since Lm depends on both d and ε, the broad pore diameter should also be optimized, since Lm affects the local Thiele modulus and, thus, the local effectiveness factor in the nanoporous region of the hierarchically structured catalyst. However, if for a given value of d and ε, the size of the nanoporous region is small enough that there are no diffusion limitations, then it is not necessary to optimize d, also because Knudsen diffusion does not play a role here, and hence the diffusivity in the broad pores is independent of d. For gas phase reactions, the mean free paths of the reactants could be larger than the size of the pores, especially in the mesopore range, and Knudsen diffusion could influence transport in the broad pores. Since Knudsen diffusion depends on the pore size, the diameter of the broad pores would also have to be optimized with ε for gas phase reactions. For this study, the broad pore size, d, was kept constant at 50 nm to minimize the effect of configurational diffusion on the diffusivity. Using the approach developed by Wang,26 an upper bound on the size of the nanoporous region without diffusion limitations can be determined for this problem. We show further on that the size of the nanoporous regions in the optimal structures is much lower than the calculated upper bound; hence, it is justified not to optimize the broad pore size, d. The same constraint imposed by Wang et al.1 to ensure a minimum of five broad pores in either direction in the uniform substructures of the nonuniform catalyst was used in this work. This allows for the use of a continuum model in solving diffusion and reaction in the uniform substructures. This constraint, along with bounds on the macroporosity (0 e ε e 1), were the only constraints applied to this study. 3. Simulation The mathematical model was solved using COMSOL Multiphysics 3.5a on an Intel Quad Core 2.66 GHz workstation using 4GB of RAM. COMSOL Multiphysics is a finite element based simulation package. The optimization required the coupling of MATLAB’s fmincon (constrained minimizer) function with the mathematical model. The default optimization settings of fmincon were initially used to reach the neighborhood of the global optimum. As explained further on, given the shallow nature of the global optimum and the existence of a

5 × 10-8 kg oil/m2 cat s 0.1 5 × 10-11 m2/s 1.355 × 10-9 m2/s 1 year

very large number of solutions that gave very similar integrated reaction rates, the objective function tolerance had to be decreased from the default value of 10-6 to 10-10 to obtain the global optimum. Since this optimizer guarantees only local optima, we performed the simulations many times with the initial macroporosities obtained from a pseudo random number generating function. It was found that the optimized scaled yields did not differ very much from each other. The properties of the purely nanoporous catalyst were obtained from the simulation studies of nickel deposition on hydrodemetalation catalysts conducted by Mace´ and Wei14 and are listed in Table 1. Note that the intrinsic rate constant for nickel metalloporphyrin hydrodemetalation is 2 orders of magnitude higher than that reported in the original paper. The basis for this reported value is as follows: Mace´ and Wei14 report that the intrinsic rate constant in their paper gave Thiele moduli between 1.7 and 3, which is less than the typical industrial Thiele modulus values for diffusion limited hydrodemetalation. Pereira and Beeckman13 have reported an intrinsic hydrodemetalation rate for vanadium removal from Boscan vacuum feed of 5.6 × 10-11 m/s. Since the unit of m/s is not typical for a heterogeneous rate constant, we believe that the units are actually (m3oil/m2cat s). Multiplying this intrinsic rate constant with the crude oil density (Foil) gives a value of approximately 5 × 10-8 kg oil /m2 s, which is the value we use here. Moreover, Smith and Wei27 report that the reactivity for vanadyl etioporphyrin removal is 5 times lower than that for nickel etioporphyrin removal. Finally, Smith and Wei12 report that the average size of the Ni and V crystallites and the number density of the crystallites on aged hydrodemetalation catalysts are quite similar. Therefore, we hypothesize that the intrinsic reaction rates for nickel and vanadium removal are also close to each other in value or at least of the same order of magnitude. The bulk diffusivity of the metalloporphyrin, Dbulk, can be estimated by substituting the initial values from Table 1 for the various parameters in eqs 9 and 10. The ratio of activities of the metal sulfide deposits to the fresh catalyst, Γ, introduced in eq 5 is an important parameter that varies with the type of catalyst employed (e.g., Ni-V/SiO2 catalysts will give different activity ratios compared to CoO-MoO3/Al2O3). The value of the activity ratio is taken to be 0.1 from Mace´ and Wei14 but can be as high as 0.5-0.8.13 It will be explained

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Figure 5. Optimal macroporosity profiles for hierarchically structured catalysts consisting of (a) 1 × 1, (b) 4 × 4, (c) 5 × 5, and (d) 6 × 6 USS. The color bar on the right-hand side represents the range of macroporosities seen in the optimal structures.

further on in the paper that changing this parameter affects the optimization only quantitatively. 4. Results and Discussion 4.1. Optimal Macroporosity. The optimal macroporosities for the optimal uniform (1 × 1) and nonuniform structures (4 × 4, 5 × 5, and 6 × 6 USS) are shown in Figure 5. The optimal macroporosities vary from 10% to approximately 29% for the uniform substructures within the nonuniform catalyst; however, the optimal macroporosities fluctuate about the optimal uniform macroporosity, which is about 21%. Thus, the optimal macroporosities follow the same trend as seen in the work of Wang et al.1 This result is not obvious, since, in the work of Wang et al.,1 the macroporosities were optimized for steady state operation, whereas in the hydrodemetalation problem (and other reactions affected by deactivation) the reaction rate decreases over the time on stream. The explanation for the trends seen in Figure 5 is as follows: in this study, we have limited the effect of deactivation to the nanoporous regions of the hierarchically structured catalyst, with the broad pores acting only as molecular transport highways. At every instant of time, the hierarchically structured hydrodemetalation catalyst can be treated as equivalent to the hierarchically structured catalyst of Wang et al.1 with time dependent nanoporous catalyst parameters. For such a problem, Wang et al.1 have concluded that the optimal bidisperse and bimodal catalysts have very similar performance, since, in the optimal case, the intraparticle diffusion limitations vanish in the nanoporous catalyst and the only diffusion limitations exist in the broad pores because the molecules travel over longer distances (the broad pores span the entire length of the catalyst pellet). During the optimization, the solution converged to the neighborhood of the global optimum very quickly; however, the global optimum itself is very shallow and a very large number of solutions with somewhat asymmetric macroporosity distributions about the diagonal axes gave reaction yields very close to the reaction yields obtained from the symmetric distributions reported in this paper. This is also reflected in a sensitivity analysis, which is addressed further on in this section. We suspect that both the dependence on time of the optimal solution and the choice of the optimizer may be responsible for the many local optima obtained in the neighborhood of the global optimum. One way to check the influence of the time on stream is to carry out the optimization for various time windows within the total time

frame and determine which time window affects the macroporosity distribution the most. For example, it could be that the initial 1 month or so is the period of highest hydrodemetalation activity and that it is this window that determines the optimal macroporosities for the entire time on stream. 4.2. Calculation of Bounds on the Size of the Nanoporous Region. Since the diameter of the broad pores, d, is not optimized in this study, it is necessary to ensure that there are no intraparticle diffusion limitations in the nanoporous region of the hierarchically structured catalyst. Wang26 poses an upper limit on the local, generalized Thiele modulus for a reaction with kinetics r(c) and determines the largest size of the nanoporous regions without diffusion limitations as φlocal )

w r(c0) 2 √2

(∫

c0

ceq

-1/2

)

De(c)r(c) dc

) 0.1

(24)

where w is the nanoporous wall thickness in Wang’s work. For the hydrodemetalation problem, eq 24 reduces to φlocal )

Lm 2



kSFcat ) 0.1 DeFoil

(25)

One would expect the highest diffusion limitations encountered by the catalyst running on stream to be at time t ) 0. For this case, all the parameters in eq 25 take on initial values, which, when substituted, give a value for Lm of 11.7 µm. Now, for the optimal uniform catalyst, the value of Lm computed from eq 23 is 0.4 µm, which is much lower than the calculated upper bound. While the diffusion limitations for the hierarchically structured catalyst are highest at the start of the run, the diffusion limitations in the nanoporous regions are highest at the end of the run. The effective diffusivity in the nanoporous regions decreases due to pore plugging by the metal sulfides, and therefore, the local Thiele modulus, given by eq 11, increases with increasing time on stream, thereby increasing the diffusion limitations at the nanoporous catalyst level. Hence, a lower bound on the size of the nanoporous regions, Lm, can be derived similar to the procedure described above, by substituting the nanoporous catalyst parameters at the end of the time on stream in eq 25 and deriving the corresponding broad pore diameter, for a fixed macroporosity from eq 23. For the simulation results presented in this work, the optimal catalyst running 1 year on stream violated the constraint imposed in eq 25 marginally, only

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Figure 6. Spatially integrated reaction rate plotted against time for the (purely nanoporous) catalyst, uniform catalyst, and 6 × 6 USS. The total time on stream is 1 year. The spatially integrated reaction rate is obtained from eq 21.

toward the end of the run. The maximum value of the local Thiele modulus in the nanoporous regions at the end of 1 year was 0.154, resulting in a local effectiveness factor of 0.99, which implies the absence of diffusion limitations. To satisfy eq 25 throughout the run, the size of the broad pores cannot exceed 32 nm for an optimal macroporosity of approximately 21%. However, reducing the broad pore size will also decrease the effective diffusivity of the hierarchically structured catalyst through eq 14 and increase the Thiele modulus through eq 20. Moreover, in the optimal hierarchically structured catalysts, the spatially integrated reaction rate toward the end of 1 year on stream is very low (see Figure 6); hence, the violation of eq 25 does not lead to significant deviations in the reaction yield. We hypothesize that decreasing the effective diffusivity will significantly reduce the reaction yield of the optimal hierarchically structured catalyst to a greater extent than the decrease in reaction yield brought about by the marginal violation of eq 25 toward the end of the time on stream. The uniform structures optimized for 3 and 6 months on stream (see section 4.5) did not violate eq 25 at any point in time. Therefore, there are indeed no diffusion limitations in the nanoporous walls of the macropores (ηlocal = 1) and there is no effect on d in this problem. 4.3. Spatially Integrated Yield versus Time. Figure 6 compares the performance of the optimal uniform and 6 × 6 USS catalysts against the purely nanoporous catalyst over the entire time on stream. We observe that the performance of the uniform and 6 × 6 USS structures are almost identical and clearly much superior to the performance of the purely nanoporous catalyst. Furthermore, from Figure 7, it is easily seen that the uniform catalyst performs as well as the 4 × 4, 5 × 5, and 6 × 6 optimal USS catalysts, which confirms the findings reported earlier in this section. Moreover, the yield from the optimized hierarchically structured catalysts is almost 8 times higher than that of the purely nanoporous catalyst. The uniform and 6 × 6 USS catalysts perform almost identically well, because in the optimal case, the volume (area in 2D) averaged diffusivities in the broad pores are almost equal in both structures and the diffusion limitations only exist in the broad pores. Equation 14 implies that the diffusivity in the broad pores

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Figure 7. Scaled yield plotted against the number of uniform substructures (NUSS) along either the x or y direction. The scaled yield is obtained directly from eq 22.

depends primarily on the macroporosity, for a constant tortuosity. Because of the assumption of no deactivation by plugging in the broad pores, the macroporosity remains constant over the entire time on stream. The radius of the broad pores (rbroad) is chosen large enough to minimize the effect of configurational diffusion in eq 14. It has also been established that the size of the broad pores is unimportant from the point of view of Knudsen diffusion and intraparticle diffusion limitations. Therefore, it can be concluded that the details of the broad pores are not crucial to this problem. On the basis of the results presented in Figures 5-7, it is apparent that similar results can be quantitatively obtained for different kinetics as well, because the optimization minimizes the diffusion limitations in the hierarchically structured catalyst. For example, if a Ni-V/SiO2 catalyst is employed in place of CoO-MoO3/Al2O3, then the ratio Γ in eq 5 will change, in turn affecting the surface area for reaction through eqs 5 and 6. This change in surface area will affect only the kinetic term in eq 20, i.e., the numerator in the definition of the Thiele modulus for the hierarchically structured catalyst. Γ has no effect at the scale of the purely nanoporous catalyst because the intraparticle diffusion limitations in the nanoporous catalyst vanish in the optimal structures, as explained in section 4.1. Thus, a new optimization problem presents itself for which the results obtained similar to Figures 5-7 will reflect only a quantitative change but will qualitatively remain the same. 4.4. Sensitivity Analysis. The results discussed above indicate that the optimal uniform catalyst performs nearly as well as the optimal nonuniform catalyst due to the reasons elaborated upon in the previous section. Hence, uniform catalysts are preferred owing to the ease of synthesis, compared to optimized nonuniform catalysts. However, in the synthesis process, it is quite possible for the macroporosity to fluctuate about the optimal value, especially if the optimal uniform catalyst is synthesized by assembling together nanoporous grains, followed by pelletization to produce the final catalyst. In this subsection, we tackle the issue of sensitivity of the catalyst to deviations in the optimal macroporosity. Considering a 6 × 6 USS catalyst, the optimal macroporosity is randomly varied between 50 and 200% of the optimal uniform macroporosity in each of the uniform substructures that make up the nonuniform catalyst. These

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Figure 8. Deviation in the reaction yield from the 6 × 6 USS catalysts with a macroporosity distribution randomly fluctuating within 50-200% of the optimal uniform macroporosity, plotted against the number of pore structure realizations.

Figure 9. Optimal macroporosities for uniform catalysts running 3, 6, and 12 months on stream.

structures are then simulated for a very large number of realizations (in this study 200 realizations were used), and the deviation in the reaction yield is evaluated as follows: deviationi(%) ) [(reaction yield)optimal uniform - (reaction yield)i] × 100 (reaction yield)optimal uniform

(26)

In eq 26, the subscript i refers to the pore network realization number. The deviation for each pore structure realization is plotted versus the corresponding pore network realization number and is shown in Figure 8. Figure 8 shows us that over a very large number of pore network realizations, the deviation in the reaction yields varies from approximately -2.5% to about 1.25% and appears to fluctuate around -1%. The negative sign indicates a case where the pore structure realized by randomly varying the macroporosity about the optimal uniform value in the 6 × 6 USS performs better than the optimal uniform structure. The exact opposite reasoning applies to the positive sign in the deviation values. Since the uniform structure is a subset of the nonuniform structure, the optimal nonuniform structure will always have a higher reaction yield compared to the optimal uniform one. Hence, there are likely to be a higher number of negative deviations over a very large number of pore structure realizations, and this is indeed observed in Figure 8. On the basis of the range of the deviations observed from Figure 8, it can be concluded that the optimal structures are insensitive to random fluctuations in the macroporosity of the optimal uniform structure. 4.5. Effect of Time on Stream. In the last part of this section, the effect of the time on stream on the performance of the optimal uniform catalysts is investigated, since we have already established that the optimal uniform and nonuniform catalysts give very similar reaction yields for a given time on stream. The simulation and optimization was performed for 3 and 6 months on stream in addition to the 1 year on stream simulation. The optimal macroporosities versus the time on stream are plotted in Figure 9. We can see that the optimal macroporosities of the uniform catalyst are higher for lower times on stream, which is expected because the intrinsic reaction rates are higher at lower times on stream (see Figure 6) and, therefore, the diffusion limitations in the catalyst pellet will also be higher.

Figure 10. Spatially integrated reaction rate versus time on stream for the optimal uniform catalyst running 3, 6, and 12 months on stream.

Figure 10 compares the spatially integrated reaction rate for the three different cases. Remarkably, the spatially integrated reaction rates appear to be very similar, even though the optimal macroporosities are very different. We explain this trend as follows: From eq 21, consider only the spatial integral, which is the time dependent yield: yield ) )

∫ ∫ Sk cF x

y

E

∫ ∫ Skη

cat

x

y

(27)

dx dy

local(1

- ε)cFcat dx dy

(28) ) kηlocal(1 - ε)Fcat

∫ ∫ Sc dx dy x

y

(29) Thus, if we consider the ratio of the spatially integrated reaction rate for the optimized catalyst for 3 months on stream and for 12 months on stream,

Ind. Eng. Chem. Res., Vol. 49, No. 21, 2010

( (

kηlocal(1 - ε)Fcat (yield)3 ) (yield)12 kηlocal(1 - ε)Fcat

(1 - ε12

x

y

3

x

y

12

(30)

( ∫ ∫ Sc dx dy) )( ∫ ∫ Sc dx dy)

(1 - ε3)

)

∫ ∫ Sc dx dy) ∫ ∫ Sc dx dy)

x

y

3

x

y

12

(31)

Equations 30 and 31 contain the subscripts 3 and 12, which denote the catalyst running for 3 months and 12 months on stream, respectively. Two extremities for the reaction can be considered here: strong diffusion limitations and kinetic control. The strongest diffusion limitations in this problem exist at the start of the run, i.e., at t ) 0. At this point, the specific surface area, S, is equal to the initial specific surface area, Sinit, for both catalysts, and thus the ratio simplifies to

( ∫ ∫ Sc dx dy) )( ∫ ∫ Sc dx dy)

(1 - ε3) (1 - ε12

x

y

3

x

y

12

( ∫ ∫ c dx dy) )( ∫ ∫ c dx dy)

(1 - ε3)

)

(1 - ε12

x

y

3

x

y

12

(32) )

)

[ [

] ]

(1 - ε3) (ηHSc0)3 (1 - ε12) (ηHSc0)12

(33)

(1 - ε3) ηHS,3 (1 - ε12) ηHS,12

(34)

∫ ∫ k Sc dx dy x

y E

2

)

∫ ∫ c dx dy x

y

c0L2

kESc0L

(35)

In the limit of strong diffusion limitation, the effectiveness factor asymptotically approaches a value of 1/φHS as explained by Froment and Bischoff24 and is therefore only a function of the Thiele modulus of the hierarchically structured catalyst, as defined in eq 20. Combining eqs 30 and 34,

[

]

(yield)3 (1 - ε3) φHS,12 ) (yield)12 (1 - ε12) φHS,3

(36)

The ratio of the Thiele modulus for the hierarchically structured catalysts can be simplified as φHS,12 ) φHS,3



ε3(1 - ε12) ε12(1 - ε3)

(37)

Equation 37 results by substitution from eq 20, noting that all the terms independent of the macroporosity cancel out. Finally, combining eqs 36 and 37 gives (yield)3 ) (yield)12



ε3(1 - ε3) ε12(1 - ε12)

It is observed from Figure 10 that the spatially integrated rates for both the 3 and 6 months on stream catalyst start out higher than the 12 months on stream catalyst, which has already been attributed to the extent of diffusion limitations in the three cases. However, at some point in time, the spatially integrated rates for the 3 and 6 months catalysts start to fall below the corresponding values of the 12 month catalyst, indicating a transition from strong diffusion limitations to kinetic control. This transition is more apparent for the 6 month catalyst. Analytically, the trend seen in the kinetically limited regime can be explained using eq 31. The data from the 3 month catalyst did not indicate a complete transition to kinetic control at the end of the run. Hence, the spatially integrated rates from the 6 and 12 month catalyst were compared at the end of the 6 month run. Under kinetic control, the area integral terms in eq 31 cancel, simplifying the ratio of yields to (yield)6 (1 - ε6) ) (yield)12 (1 - ε12)

(39)

Therefore, under kinetic control, the ratio of the spatially integrated reaction rates is simply the ratio of volumes of the nanoporous catalyst regions in the hierarchically structured catalysts. Substituting the values for the optimal macroporosities in eq 39 gives a value of 0.93, while the comparison of simulation data gives a ratio of 0.90, which indicates good agreement. 5. Conclusions

where the definition of the effectiveness factor, for the hierarchically structured catalyst at t ) 0 is ηHS )

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(38)

Thus, the ratio of the spatially integrated reaction rates at t ) 0 can be computed by substituting the uniform optimal macroporosities. In this case, the ratio is approximately 1.14. From comparison with the simulation data, the same ratio is 1.1, which is in good agreement with the analytically obtained value.

The methodology developed in this paper demonstrates quantitatively for the first time the opportunities offered by the optimization of the broad pore network for diffusion-limited reactions undergoing deactivation in the nanoporous region of a hierarchically structured catalyst. The possibility to increase the reaction yield almost 8-fold over purely nanoporous catalysts is very promising for the hydrodemetalation reaction and for other important chemical processes as well. An optimal macroporosity of approximately 21% for the uniform catalyst means that about 21% less catalytic material is needed to achieve this approximately 8-fold increase in the reaction yield over a year on stream. This simple modification in catalyst structure could result in substantial savings for the petroleum industry and have a positive environmental impact by reducing the required resources and increasing efficiency. The effect of the time on stream on the optimal macroporosity of the hierarchical catalyst indicates a possibility to combine higher reaction yields over time while simultaneously reducing the required amount of catalytic material. However, there is clearly a trade-off, because higher macroporosities also result in shorter optimal times on stream. Hence, the question of the cost and time involved in catalyst regeneration or replacement becomes critical. For example, on the basis of the results presented above, two possible scenarios can be considered: one where the catalyst is run for 1 year on stream, thereby requiring 21% less catalytic material than the purely nanoporous catalyst and giving a lower reaction yield overall. The other case would be to operate the catalyst for 3 months, with a macroporosity of 31% and regenerating or replacing this catalyst three times within the total time window of 1 year on stream. Thus, over a 1 year period, running catalysts for 3 months, multiple times will result in a higher reaction yield compared to the catalyst that has been optimized for 1 year on stream. An advantage of this approach is that the deactivation of the optimal hierarchical catalysts can also be reduced in this way. This is an issue that clearly warrants

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further investigation, especially from a process design point of view. The results and the sensitivity analysis indicate that uniformly macroporous catalysts perform as well as catalysts in which the macroporosity is not entirely uniform, as long as it fluctuates about the optimal value. The presented results provide a rational basis for the synthesis of stable and active hierarchically structured catalysts. At this point, we can comment on the synthesis and experimental validation of these theoretically optimized hierarchical catalysts. While synthesis of specifically designed structures is never trivial, it should be feasible in the present case, owing to availability of methods to synthesize and test hierarchically structured catalysts for hydrocarbon processing, with controlled large pore volume and pore size distribution, as reported in the patent literature.28,29 Experimental validation must also include studies on the mechanical strength of the hierarchical catalysts because the introduction of macropores might weaken the mechanical stability of these catalysts under typical operating conditions (temperatures above 700 K and pressures above 10 MPa for hydrodemetalation). For example, U.S. Patent 7,186,32930 discloses a bimodal/trimodal Ni-Mo/Al2O3 hydroprocessing catalyst for converting heavy feedstock containing 10-30% of the total pore volume as macropores that are at least 200 nm in size. The inventors clearly state that exceeding this percentage range can have a substantial detrimental effect on the mechanical strength of the catalyst.

εn ) porosity of nanoporous catalyst described using RSM Γ ) factor accounting for residual activity of metal sulfide deposits η ) effectiveness factor φ ) Thiele modulus for the nanoporous catalyst φHS ) Thiele modulus for the hierarchically structured catalyst based on the broad pore network Fcat ) density of nanoporous catalyst region (kg fresh catalyst/m3 nanoporous catalyst volume) Fdeposit ) density of metal sulfide deposits (kg metals/m3 metals) Foil ) density of oil (kg oil/m3) σ ) specific surface area (m2 catalyst/m3 nanoporous catalyst volume) σrxn ) specific reaction surface area (m2 catalyst/m3 nanoporous catalyst volume) τ ) large pore network tortuosity τn ) nanoporous catalyst tortuosity

Acknowledgment

Literature Cited

The authors gratefully acknowledge financial support for this project from Synfuels, China, and start-up funds for M.-O.C. from Rensselaer Polytechnic Institute. S.M.R. is grateful to the Department of Chemical and Biological Engineering at Rensselaer Polytechnic Institute for the Howard P. Isermann fellowship. S.M.R. also thanks Robert Healey and Vasudevan Venkateshwaran for assistance with the computing resources.

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List of Symbols a1 ) radius of catalytic microspheres (m) a2 ) radius of metal sulfide crystallites (m) c ) concentration of reacting organometallic species (kg metal/kg oil) d ) diameter of broad pores (m) De ) effective diffusivity in the nanoporous square islands (m2/s) DE ) effective diffusivity in the hierarchically structured catalyst (m2/s) Dbulk ) bulk diffusivity of organometallic reactant (m2/s) k ) intrinsic rate constant (kg oil/m2 cat s) kE ) effective rate constant for the continuum model (kg oil/m2 s) L ) length of side of catalyst square slab (m) Lm ) size of the nanoporous square islands (m) M ) weight of metal sulfide crystallites deposited (kg metal/kg fresh catalyst) n1 ) number density of catalytic microspheres (m-3 nanoporous catalyst volume) n2 ) number density of metal sulfide crystallites (m-3 nanoporous catalyst volume) r ) radius (m) S ) surface area available for reaction (m2 cat/kg fresh catalyst) t ) time (s) x ) coordinate along the x direction (m) y ) coordinate along the y direction (m) Greek Symbols ε ) porosity of broad pore network

Subscripts broad ) broad pore network deposit ) metal sulfide deposits HS ) hierarchically structured catalyst init ) initial value local ) local values metal ) metalloporphyrin p ) pore rxn ) reaction

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(25) Coppens, M.-O. Structuring Catalyst Nanoporosity. In Structured Catalysts and Reactors, 2nd ed.; Cybulski, A. J., Moulijn, J. A., Eds.; CRC Press: Boca Raton, FL, 2006, Vol. 22, p 779. (26) Wang, G. Broad Pore Channels as Molecular Highways in Nanoporous Catalysts: Multiscale Modeling, Optimization and Applications. Ph.D. Dissertation, Rensselaer Polytechnic Institute, Troy, NY, 2008. (27) Smith, B. J.; Wei, J. Deactivation in Catalytic Hydrodemetalation. I. Model Compound Kinetic Studies. J. Catal. 1991, 132, 1. (28) Chen, H. C. Hydroprocessing with a Catalyst having Bimodal Pore Distribution. U.S. Patent 4,435,278, March 6, 1984. (29) Johnson, D. R. Hydrodemetalation and Hydrodesulfurization using a Catalyst of Specified Macroporosity. U.S. Patent 4,976,848, December 11, 1990. (30) Abe, S.; Hino, A.; Shimowake, M.; Fujta, K. High-Macropore Hydroprocessing Catalyst and its Use. U.S. Patent 7,186,392, March 6, 2007.

ReceiVed for reView April 22, 2010 ReVised manuscript receiVed June 9, 2010 Accepted June 10, 2010 IE1009487