Calculation of the Optimal Macropore Size in Nanoporous Catalysts

May 6, 2008 - Macropores act as broad highways for molecules to move in and out .... Illustration of the hierarchical pore structure (left) and its su...
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Ind. Eng. Chem. Res. 2008, 47, 3847–3855

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Calculation of the Optimal Macropore Size in Nanoporous Catalysts and Its Application to DeNOx Catalysis Gang Wang† and Marc-Olivier Coppens*,†,‡ Howard P. Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, and Physical Chemistry and Molecular Thermodynamics, DelftChemTech, Delft UniVersity of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands

Macropores act as broad highways for molecules to move in and out of a nanoporous catalyst. The macropore “distributor” network in such a hierarchically structured porous catalyst, containing both nanopores and macropores, is optimized with the aim to find the optimal effectiveness factor, ηopt, of a single reaction with general kinetics in the catalyst. Molecular diffusion is assumed to dominate transport in macropores. It is found that the ηopt-Φ0 relation qualitatively recovers the universal η-Φ relation when the generalized distributor Thiele modulus, Φ0, is defined in a way analogous to the generalized Thiele modulus, Φ, but using the molecular diffusivity in the macropores rather than the effective diffusivity in the nanopores. This is because the concentration gradient inside the optimal hierarchically structured, porous catalyst exists only in one principle direction (e.g., the radial direction in a spherical catalyst particle), and molecular diffusion in the macropores dominates the transport process in this principle direction. The universal ηopt-Φ0 relation is used to design a catalyst for power plant NOx emission control. Overall catalytic activity in a mesoporous catalyst with a median pore size of 32.5 nm could be increased by a factor of 1.8-2.8 simply by introducing macropores (occupying 20-40% of the total volume of the catalyst) with a width of 2-22 µm into the mesoporous catalytic material, so that the remaining mesoporous macropore walls are 5-33 µm thick. In practice, this would correspond to a deNOx catalyst consisting of mesoporous particles with a diameter of 5-33 µm and macropores in between them with a size of around 2-22 µm. Information like this is readily applicable to practical catalyst synthesis. 1. Introduction Zeolites and other nanoporous catalysts have an extremely large internal surface area, which contributes to their high intrinsic catalytic activity. However, their small pore size limits the accessibility to the active sites. This has triggered significant interest in the synthesis of nanoporous catalysts with a bimodal, hierarchically structured pore size distribution, so as to facilitate molecular transport into the catalysts via macropores and large mesopores, and, consequently, improve overall catalyst performance. Recent progress in the synthesis of porous materials with independently controlled pore sizes at multiple length scales should allow us to optimize their pore size distribution.1–6 For example, mesopores and/or macropores of a desired size could be introduced in a zeolite in a well-controlled way, rather than as a side product of a poorly controlled steaming or leaching process. Despite the advances in materials science, the question, however, remains what this optimal pore size distribution is? On the theoretical side, modeling of diffusion and reaction in a hierarchically structured porous catalyst, containing both macropores and nanopores, has been studied for decades.7–12 Dogu12 gave an excellent overview. Earlier theoretical studies aimed at assessing the extent of diffusion limitations in a class of catalyst pellets commonly used in industry and made by compressing nanoporous particles. These pellets contain intraand interparticle pores and, therefore, have a bimodal pore size distribution. A popular approach to study diffusion and reaction in these pellets is the pellet-particle model.12 This approach employs a one-dimensional (1D) continuum model, in which two continuity equations are used to account for diffusion and * Correspondence should be addressed to [email protected]. † Rensselaer Polytechnic Institute. ‡ Delft University of Technology.

reaction in intra- and interparticle pores, respectively.7–10 Loewenberg11 used a pore network model to simulate diffusion and reaction in a hierarchically structured porous catalyst. Sahimi et al.13 gave a detailed review on continuum and pore network models, the former employing effective, volumeaveraged diffusivities and the latter considering the pores explicitly. To maximize the effectiveness factor of a hierarchically structured porous catalyst, structural optimizations have been carried out based on both continuum and pore network models.14–22 Hegedus and co-workers14–16 optimized a catalytic converter under deactivation conditions and a deNOx catalyst for power plant emission control based on a 1D continuum model. They developed a new type of deNOx catalyst guided by the optimizations. Fifty percent improvement in activity was observed in the experiments, in agreement with theoretical predictions. Keil and Rieckmann17 used a 3D random pore network model and optimized the nanopore radius and the nanoporosity of a deactivating hydrodemetallization catalyst. Prachayawarakorn and Mann18 used a pore network model to calculate the effectiveness factor of a first-order reaction in four hierarchically structured porous catalysts, which share the same porosity and numbers of macropores and nanopores but have different pore structures. They showed that the minimum shielding structure (where macropores are placed in the exterior part of the catalyst particle and nanopores are placed in the interior part) is the best of all and gives an 800% higher catalytic yield than its random counterpart (where nanopores and macropores are randomly placed in the catalyst particle). Morbidelli et al.23 optimized the catalytic material distribution in a catalyst pellet. Coppens and co-workers19–22 optimized macropore networks in nanoporous catalysts. Wang et al.21 optimized distributions

10.1021/ie071550+ CCC: $40.75  2008 American Chemical Society Published on Web 05/06/2008

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Figure 2. Illustration of the hierarchical pore structure (left) and its subunit (right). The pore structure is formed by repeating the subunit in the y-direction. The nanoporous, catalytically active material is indicated in white, and the macropores are indicated in black.

Figure 1. Effectiveness factor of a porous catalyst, η, as a function of the generalized Thiele modulus, Φ, for a single reaction with different chemical kinetics and in catalyst particles of different shapes.

of macroporosity and macropore diameters in a square-shaped catalyst particle, based on a 2D continuum model. The square geometry was used to simplify the simulations. They found that the optimal yield is essentially the same, regardless of whether broad distributions of macroporosity and macropore diameters are allowed in the optimizations. They also found that, in the optimal structures, the rate-determining step is the diffusion in the macropores and the dominant transport pathway involves two diffusion processes in series, namely, diffusion in the macropores over a characteristic length scale on the order of the catalyst particle size and local diffusion in the nanoporous macropore walls. Johannessen et al.22 optimized macroporosity and macropore size of a hierarchically structured porous catalyst using a 2D pore network model. They found that concentration gradients vanish between positions located at the same distance from the external surface of the catalyst, when the macropore size and the number of macropores are optimized. The optimal, hierarchically structured porous catalyst is much more active than the nanoporous catalyst. They also developed a 1D effective model, which could predict the performance of the optimal, hierarchically structured porous catalyst with almost equal accuracy to the 2D pore network model. It should be noted that Coppens and Froment19 and Gheorghiu and Coppens20 found that the optimal macropore network should have a broad distribution in macropore diameters, which is in contradiction with the more recent conclusions of Wang et al.,21 who found that that all the macropores have the same size in the optimum. This is because Coppens and Froment19 assumed that diffusion limitations exist only in the nanopores, and Gheorghiu and Coppens20 performed structural optimizations only for numbers of macropores that were fixed a priori. We will address this issue further in a separate paper. The generalized Thiele modulus, Φ, can be used to characterize the extent of diffusion limitations on a single reaction in a porous catalyst, regardless of the chemical kinetics, the reversibility of the reaction, the shape of the catalyst particle, and the number of reactants and products involved: Φ)

V r(c0) S √2

[∫

c0

cc

-1⁄ 2

]

Deffr(c) dc

(1)

where V is the volume of the catalyst particle; S is the external surface area of the catalyst particle; r is the reaction rate; Deff is the effective diffusivity in the porous catalyst; c is the concentration; c0 is the concentration on the external surface of the catalyst, and cc is typically assumed to be zero for an irreversible reaction or the equilibrium concentration for a

reversible reaction.24 Note that this common assumption is only correct for fast enough reactions, which reach their equilibrium inside the pellet. As shown in Figure 1, the η-Φ curves converge to the same values for a single reaction with general kinetics and in catalyst particles of different shapes when Φ < 0.1 and Φ >10, while they follow qualitatively the same trend when Φ is in between.24,25 On the basis of these curves, the effectiveness factor, η, of a single reaction in a porous catalyst could reasonably be estimated solely on the basis of a given value of Φ. Since molecular diffusion dominates in the optimal hierarchical pore structure,21,22 the universal η-Φ relation for the optimal hierarchically structured porous catalyst should be recovered by defining a generalized distributor Thiele modulus, Φ0, in a manner analogous to the generalized Thiele modulus, Φ, but using the molecular diffusivity, Dm, in the macropores rather than the effective diffusivity in the nanopores, as follows: Φ0 )

V r(c0) S √2

[∫

c0

cc

-1⁄ 2

]

Dmr(c) dc

(2)

Recovery of such a universal relation is crucial from a practical point of view, because this universal relation could yield a simple and general approach, i.e., irrespective of chemical kinetics or catalyst particle shape, to estimate the optimal effectiveness factor, ηopt. This approach would complement current, mainly empirical experimental efforts in the synthesis of hierarchically structured porous catalysts. Identifying the universal ηopt-Φ0 relation could also improve our understanding on other important questions on the rational design of the large-pore architecture in a nanoporous catalyst, e.g., what the general features of the optimal hierarchical pore structure are and how sensitive the catalyst performance is to the structure of the large-pore network. Until now, to the best of our knowledge, such a universal ηopt-Φ0 relation has not been mentioned in the context of the optimal design of the macropore architecture in a nanoporous catalyst. To this end, the aim of this paper is to show that the optimal effectiveness factor ηopt of different pore structures studied by Gheorghiu and Coppens,20 Wang et al.,21 and Johannessen et al.22 is governed by Φ0 in a universal way. We also employed the ηopt-Φ0 relation to design a catalyst for power plant NOx emission control. 2. Optimization of the Macropore Size 2.1. Hierarchical Catalyst Slabs. Johannessen et al.22 optimized the hierarchical pore structure based on a 2D pore network model. As shown in Figure 2, the studied structure consists of macropores and nanoporous catalytically active material. In this structure, all the macropores share the same size and shape, as

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is the case for all the macropore walls. Thus, the entire structure could be constructed by periodically repeating its subunit (as shown in the right side in Figure 2). Johannessen et al.22 developed a 1D effective model based on the fact that concentrations are equal along the y-direction for the structure shown in Figure 2 when the structure is optimized. This 1D effective model could be extended to hierarchically structured porous catalysts with two other geometries: cylinders and spheres. A spherical, hierarchically structured porous catalyst could be constructed by introducing macropores, e.g., as cones connecting the external surface of the sphere, straight in a radial direction to the center of the sphere, with equal spacing between two neighboring macropores. An infinitely long, cylindrical, hierarchically structured porous catalyst could be constructed in a similar way as the structure shown in the top right in Figure 12. Dogu12 did a survey on the ratio Dm/De (where De is the effective diffusivity in the nanopores) for hierarchically structured porous catalysts among published works and found that this ratio is typically far greater than 100. Dm is a fundamental property of fluids. Its value is on the order of 10-5 m2/s for gases and 10-9 m2/s for liquids.26 De is a lumped parameter, since it accounts for all the factors (e.g., fluid/pore-wall interactions) that affect diffusion of a fluid in a nanopore network. In zeolites, for example, its value is on the order of diffusivities in liquids or even orders of magnitude lower: diffusivities on the order of 10-12-10-10 m2/s are not uncommon.27 Therefore, we could further simplify the 1D effective model by neglecting De. All these considerations yield dc d εDmxm-1 - xm-1(1 - ε)r(c) ) 0 dx dx with the boundary conditions

(

)

dc )0 dx x ) L : c ) c0

x)0:

(3)

(4a) (4b)

where ε is the macroporosity (macropore volume fraction) and m ) 1 for slabs, 2 for cylinders, and 3 for spheres. Johannessen et al.’s22 aim was to maximize the effectiveness factor of the hierarchically structured porous catalyst ηhs, ηhs )

∫∫Ωr(c) dx dy r(c0)A

(5)

where Ω is the area (2D) or volume (3D) occupied by nanoporous material and A is the area (2D) or volume (3D) occupied by both nanoporous materials and macropores. The effectiveness factor ηhs is a scaled, area-aVeraged (2D) or Volume-aVeraged (3D) yield. Its definition is the same as the one for the scaled yield by Wang et al.21 but differs from the definition by Gheorghiu and Coppens20 and Dogu12 by a factor of 1 - ε. For the 1D effective model (eq 3), ηhs is reduced to ηhs )

m

∫

L

0

r(c)(1 - ε) xm-1 dx r(c0)Lm

Figure 3. (a) Effectiveness factor ηopt and (b) macroporosity εopt of the optimal, hierarchically structured porous catalyst as a function of the generalized distributor Thiele modulus Φ0 for a single reaction with different chemical kinetics and in the catalyst particle of different shapes. The optimization was performed based on the 1D effective model, eq 3.

(6)

In this study, we used an optimization procedure, similar to the one by Johannessen et al.,22 to calculate the optimal effectiveness factor. Both the 1D effective model (eq 3) and the 2D pore network model by Johannessen et al.22 were employed in this calculation. The two models were discretized using a finite-volume method.28 A (2D) 257 × 65 grid was found to be sufficient to obtain grid-independent results for the 2D pore network model, and a (1D) 150 grid was sufficient for

the 1D effective model. Discretized equations were solved using the multigrid method “MGD9V” for the 2D pore network model and were solved using a hybrid iteration method for the 1D effective model.29,30 Because there exist two degrees of freedom (i.e., w and d, as labeled in Figure 2), optimizations based on the 2D pore network model were performed by inspection of the response surfaces. Please note that the two optimization variables are reduced to one variable, namely, ε, for optimizations based on the 1D effective model, since the information that diffusion limitations vanish in the macropore wall was already used in the development of the 1D effective model. Figure 3a shows the optimal effectiveness factor of the hierarchically structured porous catalyst, ηopt, as a function of Φ0. As expected, the ηopt-Φ0 relation closely resembles the universal η-Φ relation as shown in Figure 1. The observed similarities between the ηopt-Φ0 relation and the η-Φ relation are the following. (1) The ηopt-Φ0 curves are essentially the same, regardless of chemical kinetics and the shapes of catalyst particles; in particular, the ηopt-Φ0 curves are virtually independent of chemical kinetics and the shapes of catalyst particles when Φ0 > 10. (2) The optimal effectiveness factor ηopt approaches 1 when Φ0 < 0.1 and is 10. (3) An asymptote, ηopt ≈ 1/ Φ0, is approached when Φ0 > 10. The scaling of the asymptote is not obvious in Figure 3a but clearly appears after some manipulations. As mentioned earlier, the ηopt-Φ0 curve is independent of chemical kinetics and the shapes of catalyst particles when Φ0 > 10. Therefore, this part of the ηopt-Φ0 curve could be obtained by optimizing a hierarchically structured porous catalyst with arbitrary chemical kinetics and shape. For ease of calculation, we chose the case with a first-order reaction in a catalyst slab. In this case, ηopt is

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Figure 5. Square, hierarchical pore structure (containing 2N × 2N macropores). Here, N ) 3 for illustration only. The nanoporous catalytically active islands are indicated in white, and the macropores are indicated in black.

extend insights obtained from the cases without concentration jumps to those with them. In principle, concentrations across an interface could be correlated via an isotherm, which could be cast, either implicitly or explicitly, into the expression cnano ) H(cmacro) Figure 4. (a) Effectiveness factor ηopt and (b) macroporosity εopt of the optimal, hierarchically structured porous catalyst as a function of the generalized distributor Thiele modulus Φ0. The optimization was performed based on the 1D effective model (eq 3) and the 2D pore network model by Johannessen et al.22 Results are presented for a first-order reaction in the hierarchically structured porous catalyst shown in Figure 2.

ηopt ) (1 - εopt)

tanh(Φ0√(1 - εopt) ⁄ εopt) Φ0√(1 - εopt) ⁄ εopt

(7)

As shown in Figure 3b, when Φ0 > 10, εopt approaches 0.5 (see also Johannessen et al.22). Therefore, ηopt )

tanh(Φ0) 1 ≈ 2Φ0 2Φ0

(9)

where the subscripts “macro” and “nano” refer to the macropore side and the nanoporous-material side of the interface, respectively. H is the functional relation between cnano and cmacro. To ease the discussion, we considered the structure shown in Figure 2. In this case, concentration gradients in the y-direction vanish in both the nanoporous material and the macropore in the optimal structures, but the concentration in the nanoporous material is different from that in the macropore. In analogy to eq 3, the 1D effective model is d dc εDmxm-1 - xm-1(1 - ε)r[H(c)] ) 0 dx dx

(

)

(10)

with the corresponding effectiveness factor (8)

As shown in Figure 3b, the εopt-Φ0 curves are essentially the same, regardless of chemical kinetics and the shapes of catalyst particles; when Φ0 > 10, εopt approaches 0.5, in agreement with the result by Johannessen et al.22 The fact that Φ0 dictates ηopt and εopt implies that De and, consequently, Dm/De have no effect on ηopt and εopt because the definition of Φ0 does not contain De. To verify this point, we performed optimizations based on the 2D pore network model (by Johannessen et al.22) for different values of Dm/De. As shown in Figure 4, ηopt and εopt are functions of Dm/De when Dm/De is small (e.g., Dm/De ) 10) but converge to certain values when Dm/De > 30. Optimizations based on the 1D effective model (eq 3) predict these asymptotic values. Because Dm/De is typically greater than 100, as pointed out by Dogu,12 calculations based on the 1D effective model (eq 3) are sufficient for engineering design. All the above treatments are based on the assumed continuity of concentrations across the macropore/nanoporous-material interface. However, a concentration jump might occur in the case of strong confinement effects in the nanoporous material, e.g., adsorption in zeolites. We show here that it is trivial to

ηhs )

m

∫

L

0

r[H(c)](1 - ε) xm-1 dx r[H(c0)]Lm

(11)

Consequently, a problem with concentration jumps could be treated just like the one without concentration jumps, but with modified intrinsic kinetics. Because general kinetics were considered, problems with concentration jumps could be handled the same way as discussed above. It should be mentioned that Dogu,12 and work cited by Dogu,12 used the pellet-particle model to examine the effectiveness factors of hierarchically structured porous catalysts when diffusion limitations vanish in nanoporous catalysts. They also found that the distributor Thiele modulus (which was named differently in their papers) governs the effectiveness factors in this limiting case. However, to the best of our knowledge, it was not pointed out earlier that the optimal, hierarchically structured porous catalyst belongs to this limiting case. 2.2. Hierarchical Structures with a Macropore Network. To see whether the ηopt-Φ0 curves are pore-structure dependent, we optimized a square, hierarchically structured porous catalyst. As shown in Figure 5, the square hierarchical pore structure is a nanoporous catalytically active material, in which 2N × 2N

Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008 3851

Figure 6. Optimal effectiveness factor ηopt of the square, hierarchically structured porous catalyst (containing 2N × 2N macropores) as a function of N.

macropores are introduced. N is an arbitrary positive integer, typically 1-100. It was assumed that the macropores are straight, perpendicular to the square sides, and have a constant diameter as they go through the entire square. Eight-fold symmetry was imposed. The macropores are uniformly distributed and share the same diameter. Gheorghiu and Coppens20 optimized the square, hierarchical pore structure based on a 2D pore network model, and Wang et al.21 based on a 2D continuum model. Here, we optimized the square, hierarchical pore structure using both approaches. We used the same approach as Wang et al.21 for the continuum model based optimization, while, for the pore network model based optimization, our approach differs in one significant way from the one used by Gheorghiu and Coppens.20 The difference is that the macropore number is a manipulation variable in our optimizations, while it was fixed a priori by Gheorghiu and Coppens.20 As shown in Figure 6, the optimal effectiveness factor increases with an increasing number of macropores, N, until a plateau is reached, explaining why the macropore number is left free in our optimizations. The underlying reason for the phenomenon in Figure 6 is that, at a fixed macroporosity, the volume-average diffusivity remains constant with an increasing number of macropores, while the volume-averaged rate constant increases until the catalytically active nanoporous islands are sufficiently small so that diffusion limitations inside these islands disappear. Similarly, optimizations were carried out by inspection of the response surface, which was generated by numerically solving a finite-volume discretization of a 2D continuity equation using the multigrid method.28,30 Figure 7 presents a typical example where the optimum is easily found. As shown in Figure 8, the ηopt-Φ0 relationship is essentially independent of values of Dm/ De, while the ηopt-Φ relationship presents three distinctiVe curves for three values of Dm/De. Furthermore, interestingly, but not surprisingly, the ηopt-Φ0 relationship also recovers the universal η-Φ one for the square, hierarchical pore structure, in agreement with the aforementioned universality of the ηopt-Φ0 relation. 2.3. Hierarchically Structured Eggshell Catalysts. It is clear that the effectiveness factor is governed by Φ for nanoporous catalysts and by Φ0 for the optimal, hierarchically structured porous catalyst. As shown in Figure 9, introduction of the macropores could remove diffusion limitations completely when Φ0 < 0.1 and partly when Φ0 ≈ 1, while significant diffusion limitations still exist in the optimal, hierarchically structured porous catalyst when Φ0 > 10. Therefore, when Φ0 > 10, it would be desirable to deposit catalytic materials only

Figure 7. Optimization of the square, hierarchically structured porous catalyst (containing 2N × 2N macropores) by inspection of the response surface.

Figure 8. Optimal effectiveness factor of the square, hierarchically structured porous catalyst ηopt as a function of (a) the generalized Thiele modulus Φ and (b) the generalized distributor Thiele modulus Φ0 at different values of Dm/De. The solid line represents the results from the optimizations based on the pore network model,20 and the dashed line represents the results from the optimizations based on the continuum model.21 Results are shown for a first-order reaction.

in the surface layer or “skin” of the catalyst particle, as shown in Figure 10. The skin thickness, Ls, is a crucial design parameter. The reason is that the catalytic activity per unit volume is low when the skin is too thin, while catalytic material is wasted when the skin is too thick. To calculate the desired skin thickness, we computed the ratio, χs, of the optimal production in a skin with a certain thickness to that in the entire hierarchically structured porous catalyst. For a single reaction with general kinetics in the catalyst slab,

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Figure 9. Concentration profiles in the optimal, hierarchically structured porous catalyst at different values of the generalized distributor Thiele modulus Φ0, for a first-order reaction.

Figure 11. Optimal scaled production in the skin of a catalyst slab, χs, as a function of the generalized distributor Thiele modulus for the skin, Φ0,S. Results for different reaction kinetics are shown.

Figure 10. Illustration of part of a hierarchically structured eggshell catalyst. The nanoporous catalytically active material is indicated in white, the macropores are indicated in black, and the inert solid is shaded.

χs )

r(c0)Lsηopt,s Lsηopt,s ) r(c0)Lηopt Lηopt

(12)

where ηopt is the optimal effectiveness factor for the skin. When Φ0 > 10, combining eqs 8 and 12 leads to χs ) 2Φ0,sηopt,s

(13)

where the generalized distributor Thiele modulus for the skin Φ0,S is defined as Φ0,s ) Ls

r(c0)

(∫

c0

Figure 12. End view (top) and side view (bottom) of a single channel in a honeycomb monolith reactor. The top left shows a nanoporous washcoat (without macropores), and the top right shows a hierarchically structured washcoat. Void space is in black.

-1⁄ 2

)

Dmr(c) dc (14) √2 cc ηopt,s is dictated by Φ0,S in the same way as ηopt is dictated by Φ0. According to eq 13, χs is a function of Φ0,S only and, therefore, is independent of L. As shown in Figure 11, for a first-order reaction, the production in a skin with a scaled thickness Φ0,S ) 1, 2, and 3 accounts for 83%, 97%, and 99.5% of the optimal production of the entire catalyst pellet, respectively. Johannessen et al.22 recommended using Φ0,S ) 5 for the estimation of the skin thickness, which agrees well with our results. For typical gasphase reactions with Dm ) 10-5 m2/s and k ) 10-1000 s-1, and liquid-phase reactions with Dm ) 10-9 m2/s and k ) 0.001-1 s-1, Φ0,S ) 3 means somewhere from 100 µm to 3 mm. Figure 11 also shows the curves for different kinetics; these curves are similar to the one for a first-order reaction. The results for cases in a catalyst slab can be extended to other geometries, e.g., a cylinder or a sphere, when skins are very thin. In the case of thick skins, curvature would play a role, but this is purely a geometric effect, and Φ0,S is still the governing factor. 3. Case Study: deNOx In this section, we apply the theoretical insights (e.g., the ηopt-Φ0 relationship) obtained in the previous section to design

a catalyst for power plant NOx emission control. NOx is a precursor of acid rain and a contributor to ozone formation near the ground, which is a hazard to human health. Many countries have regulated NOx emissions from power plants. SCR (selective catalytic reduction) using ammonia could be used to abate NOx emissions. The overall SCR reaction is given below:16 4NO + 4NH3 + O2 f 4N2 + 6H2O The SCR is usually carried out in a honeycomb monolith reactor, which consists of a bundle of parallel channels. Figure 12 shows a single channel in a monolith reactor. Porous catalyst is coated on the internal surface of the channels, forming what is called a “washcoat”. When flue gas flows through the channels, NOx molecules diffuse into the washcoat and react on its internal surface. V2O5/TiO2 catalysts are usually used. The reaction temperature is 300-400 °C. A typical monolith reactor is 70-100 cm long with a cross-sectional area of 15 × 15 cm2. The square-shaped opening of the channels has a side length of 0.3-1.0 cm. The washcoat is 0.05-0.15 cm thick.16 Although performing SCR in a monolith reactor is a proven strategy for NOx emission control, further efforts have to be made to improve the performance of the monolith reactor, in order to meet imminent, increasingly strict regulations on environmental protection. For instance, the Clear Skies InitiatiVe

Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008 3853 Table 1. Parameters Needed to Design deNOx Catalysts symbol

La

kb

Dmc

Deb

value unit

0.5-1.5 mm

22.4 s-1

2.2 × 10-5 m2 s-1

6.15 × 10-7 m2 s-1

a The range of the washcoat thickness L, as labeled in Figure 12, is from Beeckman and Hegedus.16b The rate constant k and the effective diffusivity in the nanoporous material De were obtained from Buzanowski and Yang.32c The molecular diffusivity Dm is calculated using Fuller et al.’s method, discussed by Reid et al.26

Figure 13. Generalized distributor Thiele modulus, Φ0, and generalized Thiele modulus, Φ, as a function of the washcoat thickness L.

set a goal to reduce NOx emissions from power plants in the United States by 67%, from the emissions of 5 million tons in 2002 to a cap of 2.1 million tons in 2008, and to 1.7 million tons in 2018.31 One way to address this issue is to mitigate diffusion limitations in the washcoat by introducing macropores into the washcoat, as shown in the top right in Figure 12. It should be noted that, though channels with a square-shaped opening are often used in practice, channels with a circular opening were treated in this study to facilitate the discussions and illustrations. However, similar effects were found for both types of channels. Because the NOx in flue gases from power plants is present in trace amounts only, e.g.,