Ind. Eng. Chem. Res. 2009, 48, 10277–10283
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Hybrid Modeling of Methane Reformers. 3. Optimal Geometries of Perforated Catalyst Pellets Andre´ L. Alberton,† Marcio Schwaab,‡ Roberto Carlos Bittencourt,§ Martin Schmal,† and Jose´ Carlos Pinto*,† Programa de Engenharia Quı´mica/COPPE, UniVersidade Federal do Rio de Janeiro, Cidade UniVersita´ria CP, 68502, Rio de Janeiro, RJ, 21941-972, Brasil, Departamento de Engenharia Quı´mica, UniVersidade Federal de Santa Maria, AVenida Roraima, 1000, Cidade UniVersita´ria, Santa Maria, RS, 97105-900, Brasil, and Petro´leo Brasileiro, S.A., Petrobra´s, Ilha do Funda˜o, Cidade UniVersita´ria, Rio de Janeiro, RJ, 21941-598, Brasil
This paper studies the optimization of catalyst pellet geometries for the simultaneous maximization of the mechanical strength and catalyst activity of perforated catalyst pellets used for the reforming of methane. This multiobjective problem can be analyzed with the help of Pareto fronts. A previous work (Part 1: Alberton; et al. Ind. Eng. Chem. Res., in press) demonstrated that the effectiveness factor of catalysts with complex geometry used for the reforming of methane can be expressed as a linear function of the specific area of the pellet; consequently, catalyst activity can be expressed in terms of the geometric parameters of the pellet. It can also be assumed that the mechanical strength of perforated catalyst pellets is related to the minimum wall thickness of the solid piece. Therefore, two optimization problems can be proposed: (i) the simultaneous maximization of the specific area and of the minimum wall thickness of the pellet and (ii) the simultaneous maximization of the overall catalyst activity in the catalyst bed and of the minimum wall thickness of the pellet. These objective functions are evaluated here for several catalyst geometries and analyzed with a Pareto filter. The Pareto fronts obtained for the two analyzed cases are essentially the same, indicating that the maximization of the specific area constitutes a useful criterion for design of perforated catalysts in diffusioncontrolled systems. Optimized catalyst geometries are presented for different catalyst designs and can be used as references for future manufacture of catalyst pellets. 1. Introduction Catalysts can be regarded as the “fundamental intelligence” of most chemical processes, controlling how one or more reacting species are transformed into desired products. Catalyst processes are very important in the chemical field, as around 75% of all chemicals are produced with the aid of catalysts, especially heterogeneous catalysts.2 In heterogeneous processes, the catalyst is normally used as a solid pellet, while reactants are usually gases and/or liquids. In these systems, the overall catalyst performance depends on many variables, including the intrinsic chemical activity of the catalyst and the diffusion limitations of reacting chemical species in the reactor bed and inside the solid pellet. As it is well-known, diffusion limitations are sensitive to modification of geometrical parameters, such as the pellet dimensions and specific area. Therefore, the simultaneous optimization of the chemical nature of the catalyst and of the shape of the pellet is very important for optimization of process performance. A typical and important example is the steam reforming of methane, which is performed in packed bed reactors containing perforated cylindrical pellets of compressed Ni based catalysts. Few works have attempted to optimize the geometric parameters of catalyst pellets, in spite of the practical importance of this problem. In several aspects, the manufacture of industrial catalysts is more an art than a precise science, requiring the accumulated expertise of catalyst manufacturers for achievement of certain desired catalyst properties.2 Therefore, understanding * To whom correspondence should be addressed. Tel.: +55-2125628337. Fax: +55-21-25628300. E-mail:
[email protected]. † Universidade Federal do Rio de Janeiro. ‡ Universidade Federal de Santa Maria. § Petro´leo Brasileiro, S.A.
of how the geometric parameters of catalyst pellets affect the catalyst performance and optimization of pellet shape is a subject of great industrial interest. Besides, the development of guidelines for pellet manufacture based on sound mathematical analysis may be very helpful for those who work in this field. Generally, the two main features of the catalyst pellet are its reaction activity and mechanical strength. An optimum geometry can be defined as the catalyst shape that allows for maximum reactor activity with minimum amount of catalyst, without compromising the mechanical resistance of the pellet. However, this optimization problem is not an easy one, since the main objectives are maximized along opposite directions.3 The mechanical strength can be optimized with bulk and regular geometries, such as solid spheres, while the specific area (and catalyst activity) is maximized for holed pellets with irregular geometries. One of the most common geometries used in industry for maximization of the specific area are perforated cylinders.3 It is commonly accepted that the activity of the catalyst in the catalyst bed can be obtained through maximization of the specific area. However, it must be pointed out that, although holed pellets present larger specific area, the porosity of the catalyst bed also increases, leading to lower amounts of catalyst for the same height of the reactor. Therefore, it is also necessary to evaluate if the maximization of the specific area leads to maximization of the overall activity in the whole catalyst bed. The optimization of the catalyst activity (instead of the specific area) is very difficult, as the effectiveness factor of complex kinetic rates performed in pellets of complex geometry cannot be obtained in a simple way. A typical example of such systems is the methane steam reforming, whose kinetic rates must be represented in terms of Langmuir-HinshelwoodHougen-Watson (LHHW) expressions. Besides, the shape of
10.1021/ie9001662 CCC: $40.75 2009 American Chemical Society Published on Web 10/06/2009
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the catalyst pellets used for steam reforming of methane is very complex. In a previous work,1 the influence of external conditions and geometrical parameters of the pellet on the effectiveness factor of methane steam reforming was investigated. It was shown that the effectiveness factor for methane steam reforming reactions (ηj, j ) 1, 2, and 3), can be written as linear functions of the area/volume ratio, as ηj ) Rj(Deff, k, T, Ci)
A V
(1)
where A is the interfacial area and V is the volume of the pellet; Deff is the effective diffusivity of the reactants in the pellet; k is the thermal conductivity of the solid; and Ci represents the external bulk composition. In eq 1, the indexes 1, 2, and 3 represent the reactions R1, R2, and R3 (as reported by Xu and Froment4), described as follows. CH4 + H2O T CO + 3H2
(R1)
CO + H2O T CO2 + H2
(R2)
CH4 + 2H2O T CO2 + 4H2
(R3)
Hence, it is possible to correlate the local activity of the catalyst with the geometrical parameters of the pellet. Although derived for methane steam reforming, this conclusion can probably be extended for every system with strong diffusion limitations.5 A very important point for optimization of catalyst geometry is the establishment of a useful criterion for evaluation of the mechanical strength of the pellet. The detailed analysis of the tension distributions in the catalyst pellet is not a trivial problem. For instance, tensions are expected to vary along the bed because of the bed weight and flow conditions. However, it seems reasonable to assume that the mechanical strength can be parametrized in terms of the minimum wall thickness of the pellet, as the mechanical strength of the pellet becomes larger when the minimum thickness increases. The proposed problem poses a multiobjective optimization problem, as the perforation of the catalyst increases the specific area and simultaneously reduces the minimum wall thickness. Multiobjective problems are found frequently in many chemical engineering fields. The growing interest and importance of these problems can be illustrated in terms of the increasing number of publications related to multiobjective optimizations.6 Nevertheless, multiobjective optimization techniques have not been used so far to provide the optimal geometrical characteristics of catalyst pellets, which can be related to the overall catalyst activity and to the mechanical strength of the pellets. Solutions for a multiobjective problem are described in terms of the Pareto front, constituted by the set of nondominated points.6,7 According to this criterion, one point i is dominated by another point j when, for all analyzed objective functions (e.g., mechanical strength and activity of the catalyst bed), the point j is better than the point i. The optimum solution for a multiobjective problem must be picked up from the Pareto front. This work analyzes the multiobjective optimization of catalyst pellet shapes, considering two specific cases: (1) the simultaneous maximization of the specific area and of the minimum wall thickness and (2) the simultaneous maximization of the activity of the catalyst bed and of the minimum pellet wall thickness. Mathematical expressions are presented to indicate the influence of the radius and the height of the pellet on the interfacial area. The results presented here can be used as references for understanding whether geometries developed so
far are in fact optimal, establishing guidelines for manufacture of catalyst pellets. 2. Objective Functions and Optimal Solutions Consider the catalyst pellet illustrated in Figure 1A. The interfacial area can be written as A ) 2π(RL + R2 + (nF + C)(rL) - (nF + C2)(r2)) (2) where R is the external radius of the pellet, L is the height of the pellet, nF is the number of external holes, r is the radius of the external holes, and C is the ratio between the radius of the central hole (rC) and the radius of external holes (obviously, the parameter C has no sense for Raschig rings and a full cylinder, as can be seen in Figure 1B): C)
rC r
(3)
For a fixed number of holes and well-defined geometry (see Figure 1), the minimum wall thicknesses (m0, m1, and m2) can be written as m0 ) a - (r + rC)
(4)
m1 ) R - (a + r)
(5)
m2 ) 2a sen(π/nF) - 2r
(6)
The maximization of the minimum wall thickness is obtained when m0 ) m1 or m1 ) m2 (for a fixed geometry, m0, m1, and m2 cannot be obtained independently). Considering the first condition m0 ) m1, the parameter a can be obtained as a)
R + rC 2
(7)
If m2 < m1, then maximization of the minimum wall thickness can be obtained by making m2 ) m1, which is equivalent to locating the external holes closer to the edge of the pellet. In such a case, the parameter a can be obtained as a)
R+r 1 + 2 sin(π/nF)
(8)
In a PFR reactor, the local activity of the catalyst in the bed can be evaluated in the form NR dFi ) (1 - εT)Fcat. ηjνi, jrj dVbed j)1
∑
(9)
where Fi is the molar flow of reactant i, Vbed is the volume of the catalyst bed, εT is the total porosity of the bed, Fcat. is the density of the catalyst, νi,j is the stoichiometric coefficient of reactant i in reaction j, ηj is the effectiveness factor of reaction j, and NR is the total number of reactions. However, according to eq 1, the dependence of the effectiveness factor is linear with respect to the area/volume ratio for internal diffusion-controlled systems. Substituting eq 1 into eq 9 NR dFi A ) (1 - εT)Fcat. Rjνi, jrj dVbed V j)1
∑
(10)
This means that the local activity of the catalyst bed is proportional to
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dFi A ∝ (1 - εT) dVbed V
(11)
The total porosity of the bed is the sum of the porosity induced by the packing of the pellets (εB), which is a function of the shape of the pellets, and the intrinsic porosity of the catalyst pellet (εP), which depends on the geometrical parameters and the number of holes. The total porosity can be written as εT ) εB + εP - εBεP
(12)
The porosity εB can be obtained with correlations available in the literature. In this work, it was calculated as recommended by Pushnov:8 εB )
A1 (D/d)n1
+ B1
(13)
where D is the tubular reactor diameter, d is the pellet characteristic length, and A1, B1, and n1 are functions of the pellet shape (for cylinder shaped pellet A1 ) 0.9198, B1 ) 0.3414, and n1 ) 2). The value of D was taken from industrial methane steam reformers (10 cm). The porosity of the particle can be obtained as follows:
( Rr )
εP ) (nF + C2)
(14)
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To build the Pareto fronts, several geometries have been analyzed, covering a range of meaningful geometrical parameters of the catalyst pellet. The external radius was set equal to 10 mm (R ) 10) and the height to 20 mm (L ) 20). As observed through simulations, the Pareto fronts depend very weakly on the ratio L/R, which means that results presented here are valid for all meaningful ranges of R and L. After definition of the porosity of the particle, of the radius of the external hole, of the number of external holes, and of the parameter C, all the remaining geometrical parameters of the particle can be calculated. These control variables were then allowed to vary within a discretized grid, as presented in Table 1, where the number of elements represents the number of segments used to investigate the analyzed variable in the proposed design range. Powerful algorithms, such as the particle swarm optimization (PSO) technique, are available in the literature to perform multiobjective optimization procedures in more complex problems.9 The proposed grid search procedure usually demands much larger computational effort, when compared to those more elaborated algorithms. Nevertheless, for the purposes of the present paper, grid searching leads to satisfactory numerical results, as objective function evaluations are explicit and very simple and the number of decision variables is small. For this reason, full scanning of the design region can be performed in no more than a couple of minutes. Despite that, a multiobjective optimization algorithm based on the PSO technique was also
Figure 1. Illustrative representation of the basic cylindrical pellet geometry with several holes (A) and Raschig ring (B).
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used to perform the optimizations, leading to exactly the same results presented in the following sections. For all analyzed geometries, the objective functions were obtained and evaluated according to a Pareto filter. A geometry g was eliminated whenever a rival geometry h was able to generate better values for all analyzed objective functions: FOBJ1(g) < FOBJ1(h) FOBJ2(g) < FOBJ2(h)
(15)
Inconsistent geometries, with negative geometrical parameters, were also eliminated. All of the set of noneliminated (or nondominated) geometries form the solution of the problem: the Pareto front. 3. Optimal Solutions 3.1. Maximization of the Interfacial Area and of the Minimum Wall Thickness. In this case, the following objective functions were considered:
FOBJ1 ) m1 FOBJ2 ) A
(16)
The obtained results are presented in Figure 2. The maximization of the interfacial area (or specific area, as L and R are fixed) is obtained for a larger number of holes. However, the maximum allowed number of holes changes with the minimum allowed wall thickness. As clearly shown in Figure 2, when the minimum wall thickness is used as a design criterion, the range of the allowed number of holes is relatively narrow. This means that the pellet strength imposes an optimum geometrical configuration for the pellet (and a certain specified interfacial area). This is a very interesting point that has been neglected in previous publications regarding the pellet design. Raschig rings, illustrated in Figure 1B, were also considered during simulations. It is interesting to observe that Raschig rings and full cylinders constitute the optimum solutions when the pellet is fragile, and the designer should use a large m1 value to avoid the breakage of the pellet. However, it is necessary to point out that the maximum allowed ratio between the internal
Figure 2. Pareto front considering the interfacial area and the minimum wall thickness as objective functions.
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The results presented in this work may be very important for pellet design. They establish guidelines for manufacture of industrial catalyst pellets. The designer of the catalyst pellet can define the minimum allowed wall thickness, as a function of catalyst strength, and then obtain the optimal solution for the specified m value (according the geometries described in detail in the Supporting Information). 4. Influence of the External Radius and the Height of the Pellet on the Interfacial Area
Figure 3. Pareto front of considering the activity and the minimal thickness as objective functions. Table 1. Ranges of Parameters Investigated for Construction of the Pareto Front variable
investigated range
no. of elements
particle porosity (εP) ratio between central and lateral holes (C) no. of lateral holes (nF) central hole
0.02-0.8 0-5
50 50
0, 3-10 0-1
7
A V
(
( )
( ))
1 A 1 r 1 r + + (nF + C) 2 - (nF + C2) )2 VT R L L R R
2
(18)
Differentiating eq 18 with respect to the pellet height (L)
( )
d A 1 ) -2 dL VT L
radius and the external radius is around 0.7. For higher ratios, other geometries lead to better performances. It is interesting to observe that, after the Raschig rings, only five-holed pellets are inside the Pareto front. According to the proposed analysis, commonly used pellet geometries, such as four holed cylinders, are not optimal. 3.2. Maximization of the Bed Activity and of the Minimum Wall Thickness. In this case, two objective functions were considered: FOBJ1 ) m1 FOBJ2 ) (1 - εT)
Now it is interesting to establish guidelines for design of the external radius and the height of the pellet. The previous section demonstrated that the maximization of catalyst activity on the bed can be obtained through the maximization of the interfacial area (assuming a diffusion-controlled process). For a fixed geometry, the specified interfacial area of the pellet can be described as
(17)
Figure 3 presents the obtained results. The geometries presented in Figure 3 are shown in detail in the Supporting Information. The obtained solutions are exactly the same ones presented in section 3.1, indicating that the maximization of the catalyst activity is obtained with the maximization of the interfacial area. This simplifies the optimization problem very significantly, as the catalyst activity, which is a complex function of the compounds and catalyst properties, can be easily parametrized in terms of the interfacial area. It is interesting to observe the behavior of the constant C, the ratio between the radius of the central hole and of the lateral holes. In the optimal solutions, the increase of the number of lateral holes causes an increase of C. This is the same trend observed in commercial pellets. Figure 4 illustrates this trend, also presenting the range of C for a different number of holes in the optimal solution set, as presented in detail in the Supporting Information. Another interesting feature is that the equalities m0 ) m1 ) m2 and r ) rC occur for optimal solutions only when six lateral holes are considered. This geometry is quite similar to some common commercial geometries. In some cases, the lateral holes are shifted toward the external surface in order to maximize the wall thickness in the central region. The equality m0 ) m1 ) m2 can also occur when 10 lateral holes are considered. However, in this case, the central hole is much bigger than the lateral hole.
2
( ) (1 - (n
F
( Rr ) )
+ C2)
2
(19)
Equation 19 may also be written as
( )
1 2 d A ) -2 (1 - εP) dL VT L
()
(20)
Because (1 - εP) and 1/L are positive, the right-hand side of eq 20 is always negative, which means that the increase of the height of the pellet leads to a decrease of the specific interfacial area. Therefore, L should be as small as possible depending on the maximum allowed pressure drop in the reactor and minimum allowed catalyst resistance. Differentiating eq 18 with respect to the external radius of the pellet (R)
( )
(nF + C2)r2 (nF + C)r d A 2 )4 -4 - 2 3 dR VT LR R3 R
(21)
This has an extremum placed at L)
(nF + C2)r2 rnF + rC + R/2
(22)
When the height is lower than the right-hand side of the eq 22, the derivative in eq 21 is positive. In this case, the increase of the external radius leads to an increase of the area per volume ratio. On the other hand, when the height is higher than the value calculated from eq 22, the derivative in eq 21 is negative and the increase of the external radius diminishes the specific interfacial area. From a practical point of view, values of L lower than the value calculated from eq 22 are unfeasible. For example, for a pellet with external radius equal to 5 mm, lateral holes equal to the central hole (C ) 1), with six lateral holes having radii equal to 0.75 mm, the height of the particle should be lower than 0.51 mm to ensure that the derivative in eq 21 is positive. Afandizadeh and Foumeny3 suggest that the ratio between the height and the external radius should lie in the range of 1.5 e L/R e 3 in order to provide the uniform packing of the bed,
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Figure 4. Illustration of the behavior of constant C (C ) rC/r) of optimal solutions.
when cylinders are considered. Therefore, according to the considerations presented above, for practical purposes, both the external radius and the height of the pellet should be as low as possible, in order to maximize the interfacial area, and consequently, the activity of the catalyst bed. Nevertheless, small dimensions of the catalyst pellet would lead to a significant increase of the pressure drop.10 It must be pointed out that the global effect of the catalyst shape on the reactor operation may be influenced by specific features of the reaction system, as shown in studies of methane steam reforming. For example, Kagyrmanova et al.10 recommend the usage of pellets with the minimum allowed radius and the maximum allowed height for improvement of the heattransfer coefficients and increase of the methane conversion. Additionally, the dynamic behavior of the flow and of the heat transfer is influenced by the shape of the pellet, especially near the tube wall,11,12 and the reactor temperature profile can also be a complex function of the pellet shape.1,13,14 Therefore, catalyst design should not disregard the complex thermal and kinetic effects of catalyst shape on the reactor operation.14 A practical aspect related to the good operation of real industrial reactors is the homogeneous loading of catalyst pellets inside the reactor. In the case of methane steam reforming, bad catalyst loadings may result in temperature overshooting,3,15 which must be avoided for safety reasons. Studies related to the optimization of catalyst geometries implicitly assume that catalyst pellets are loaded homogeneously in the bed; otherwise, optimization results may be meaningless, as poor loading can overrule the optimization of the catalyst pellet shape. Another practical issue related to the operation of real industrial reactors is the fact that steam reforming reactions are generally performed with beds composed by catalyst pellets of distinct geometries and compositions. For example, although less active, alkali-promoted catalysts may be used in the beginning of the reactor to enhance the coke formation resistance.15 It must be pointed out that modification of catalyst geometries can exert enormous influence on the dynamic and steady-state temperature and composition profiles along the reactor, especially due to local and global modification of the bed porosity.14 In this case, as shown elsewhere,14 optimization studies should evaluate the influence of the distinct catalyst pellet geometries and compositions on the many process outputs of interest (such as pressure drops, temperature profiles, conversions, and selectivities) with the help of a reliable simulation tool.
In the preset work, new insights about the simultaneous evaluation of mechanical strength and activity of catalyst pellets were presented through the solution of a multiobjective problem, confirming the usually reported commercial trends. The results presented here can be used readily for the optimization of catalyst beds and catalyst geometries in real industrial applications, when more complex diffusive limitations cannot be neglected, with the help of the numerical procedures developed previously by Alberton et al.1,14 5. Conclusions This work analyzed the optimization of perforated catalyst pellets with respect to its mechanical strength and activity in the catalyst bed. The Pareto fronts obtained for the simultaneous maximization of the interfacial area and of the minimum wall thickness and simultaneous maximization of the bed activity and of the minimum wall thickness are very similar, indicating that bed activity can be enhanced through an increase of the interfacial pellet area in diffusion-controlled systems. For practical purposes, the height and the external radius of the catalyst pellet must be kept at the lowest allowed values (depending on maximum allowed pressure drop and pellet resistance) in order to maximize the interfacial area. As shown through simulations, the effect of L and R on the obtained optimum solutions is negligible. The set of optimum catalyst pellets is presented in detail in the Supporting Information for different cylindrical shapes, establishing a guideline for manufacture of industrial catalyst pellets. The designer of the catalyst pellet can define the minimum allowed wall thickness (m1), as a function of catalyst strength, and then obtain the optimal solution for the specified m1 value. As shown, optimum solutions include perforated pellets with a distinct number of holes of different dimensions, including some well-known commercial geometries. Acknowledgment The authors thank CNPqsConselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gicosfor providing scholarships and for supporting part of this research. The authors thank PETROBRAS for providing technical support. Supporting Information Available: Set of optimal solutions considering the maximization of the minimum wall thickness
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and the activity of the catalyst bed. This material is available free of charge via Internet at http://pubs.acs.org.
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Fcat. ) density of the catalyst νi,j ) stoichiometric coefficient of species i in reaction j
Literature Cited Nomenclature a ) distance between the center of the external hole and the center of the pellet A ) interfacial area of the catalyst pellet A1, B1, n1 ) parameters for determination of εB C ) ratio between central and lateral radii Ci ) external bulk composition D ) diameter of the industrial reactor tube d ) characteristic length of pellet Deff ) effective diffusivity F ) molar flow FOBJ1 ) first objective function FOBJ2 ) second objective function k ) thermal conductivity of the pellets L ) height of the catalyst pellet m0 ) minimum distance between the central and a external hole m1 ) minimum distance between the external hole and the edge of the pellet m2 ) minimum distance between two adjacent external holes nF ) number of external holes NR ) total number of reactions R ) external radius of the catalyst pellet r ) radius of external holes rC ) radius of the central hole rj ) rate of the reaction j T ) temperature V ) volume of the catalyst pellet Vbed ) volume of the reactor VT ) volume occupied by a full cylinder with the same R and L of the catalyst pellet Greek Letters εB ) porosity induced by the bed packing εP ) intrinsic porosity of the pellet, defined as the ratio between the volume of holes and the volume of the pellet εT ) total porosity of the catalyst bed ηj ) effectiveness factor of reaction j
(1) Alberton, A. L.; Schwaab, M.; Fontes, C. E.; Bittencourt, R. C.; Pinto, J. C. Hybrid modeling of methane reformers. 1. Empirical modeling of the effectiveness factor of catalyst pellet with complex geometry. Ind. Eng. Chem. Res., in press. (2) Hagen, J. Industrial Catalysts: A Practical Approach, 2nd ed.; WileyVCH: Weinheim, Germany, 2006. (3) Afandizadeh, S.; Foumeny, E. A. Design of packed bed reactors: Guides to catalyst shape, size, and loading. Appl. Therm. Eng. 2001, 21, 669. (4) Xu, J.; Froment, G. F. Methane steam reforming, methanation and water-gas shift: I. Intrinsic kinetics. AIChE J. 1989, 35, 88. (5) Aris, R. On shape factors for irregular particles-I: The steady state problem. Diffusion and reaction. Chem. Eng. Sci. 1957, 6, 262. (6) Rangaiah, G. P. Multi-ObjectiVe Optimization: Techniques and Applications in Chemical Engineering; World Scientific: Singapore, 2009. (7) Rao, S. S. Engineering Optimization: Theory and Practice, 4th ed.; John Wiley & Sons: New York, 2009. (8) Pushnov, A. S. Calculation of bed average porosity. Chem. Pet. Eng. 2006, 42, 14. (9) Debt, K.; Pratap, A.; Agarwall, A.; Meyarivan, T. A Fast and elitist multiobjective genetic algorithm. IEEE Trans. EVolut. Comput. 2002, 6, 182. (10) Kagyrmanova, A. P.; Zolotarskii, I. A.; Smirnov, E. I.; Verikoskaya, N. V. Optimum dimensions of shaped steam reforming catalysts. Chem. Eng. J. 2007, 134, 228. (11) Dixon, A. G.; Taskin, M. E.; Stitt, E. H.; Nijemeisland, M. 3D CFD simulations of steam previous reforming term with resolved intraparticle reaction and gradients. Chem. Eng. Sci. 2007, 62, 4963. (12) Nijemeisland, M.; Dixon, A. G.; Stittb, E. H. Catalyst design by CFD for heat transfer and reaction in steam reforming. Chem. Eng. Sci. 2004, 59, 5185. (13) Mohammadzadeh, J. S. S.; Zamaniyan, A. Catalyst shape as a design parametersOptimum shape for methane steam reforming catalyst. Chem. Eng. Res. Des. 2002, 80, 383. (14) Schwaab, M., Alberton, A. L., Fontes, C. E., Bittencourt, R. C., Pinto, J. C. Hybrid modeling of methane reformers. 2. Modeling of the industrial reactor. Ind. Eng. Chem. Res., in press. (15) Rostrup-Nielsen, J. R. Catalytic Steam Reforming. Catalysis: Science and Technology; Springer-Verlag: New York, 1984.
ReceiVed for reView January 30, 2009 ReVised manuscript receiVed September 5, 2009 Accepted September 14, 2009 IE9001662
