Hybrid Modeling of Methane Reformers. 1. A Metamodel for the Effectiveness Factor of a Catalyst Pellet with Complex Geometry Andre´ Luis Alberton, † Marcio Schwaab, † Carlos Eduardo Fontes, ‡ Roberto Carlos Bittencourt, § 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, Brazil, Engineering Simulation and Scientific Software (ESSS), Rua Lauro Mu¨ller, 116, 1404, Rio de Janeiro, RJ, 22290-160, Brazil, and Petro´leo Brasileiro, S.A., Petrobra´s, Ilha do Funda˜o, Cidade UniVersita´ria, Rio de Janeiro, RJ, 21941-598, Brazil In this work, effectiveness factors for methane steam reforming reactions were obtained by solving mass and heat balance equations inside catalytic pellets for different reaction conditions and catalytic pellet geometries with the help of CFD (computational fluid dynamic) techniques. CFD computations were performed for real particle geometries and real kinetic rate expressions as described in the technical literature. A linear correlation was found between the effectiveness factor and the area/volume ratio, which characterizes the methane steam reforming as a diffusion-controlled process. The slopes of the straight lines depend of the external reaction conditions, thermal conductivity, and effective diffusivity. On the basis of the CFD results, empirical metamodels were built to represent effectiveness factors for methane steam reforming reactions at different reaction conditions. The metamodels can be easily inserted into a reactor model for simulation of the full industrial process. 1. Introduction During simulation of industrial catalytic reactors, the calcula- tion of effectiveness factors is of fundamental importance, since catalyst pellets may be subject to important internal mass diffusion and heat transport limitations that must be considered during simulation. The effectiveness factor is generally ex- pressed as a function of the Thiele modulus, although analytical solutions are only available for certain simple catalyst pellet geometries and reaction rate expressions. For this reason, existing analytical expressions can not be used for simulation of most real catalytic industrial processes. Many efforts have been made to model effectiveness factors for more complex reactions. Analytical expressions that correlate modified Thiele modulus and effectiveness factors have been derived for reversible reactions, 1 complex kinetic rate expressions, 2,3 endo/exothermic reactions, 4 catalyst particles presenting mul- timodal pore distributions, 5 and catalyst particles with complex catalytic pellet geometries, 6,7 among others. In some cases, the calculation of the effectiveness factor may require the numerical integration of model equations. 2,3 In spite of the effort to model effectiveness factors for complex processes, no simple and generalized expressions and/ or methodology have been developed so far for the determina- tion of the effectiveness factors of reaction systems where multiple reaction occur simultaneously, involving multiple reaction steps, strong energy effects and carried out in catalyst particles of complex geometries. Nevertheless, it must be pointed out that this is a common scenario in real industrial processes. As a matter of fact, catalytic reactions are often described by complex kinetic rate expressions, such as Langmuir-Hinshelwood- Hougen-Watson (LHHW) equations, as the reaction mechanism may involve a large number of reaction steps over the catalyst surface. Besides, simple catalyst geometries are rarely used in some real industrial processes, such as the methane steam reforming. Instead, in order to maximize the interfacial area and simultaneously keep the mechanical strength of the pellet, it is common to use particles of complex geometries (such as holed cylinders) in industrial sites. Methane steam reforming is a typical industrial process where the proper modeling of the effectiveness factor is of fundamental importance for process simulation and design. This is a nonisothermal heterogeneous process, commonly described by multiple LHHW kinetic rate expressions, where reactions are performed over catalytic pellets of complex geometries. In this case, the solution of the mass and heat balance equations inside the catalyst pellet can only be obtained through numerical integration, which can be very involving when the catalyst particle presents complex geometry, due to the complex definition of boundary conditions. Besides, simulation of real operations requires the simultaneous solution of the balance equations inside the catalytic pellet and along the reactor length. Given the numerical complexity of the resulting model equa- tions, the simulation of a real industrial process constitutes a very complex task, particularly if the balance equations derived for particles of complex geometries are coupled to the reactor flow model. Despite that, Dixon and co-workers 8,9 developed a computational fluid dynamic (CFD) model for a tubular reactor where the steam reforming of methane is performed. However, the proposed CFD model cannot be used for process analysis and catalyst design, because of the extremely large dimension of the proposed numerical problem. A model simplification that is often proposed in the litera- ture 10 consists in approximating the catalyst pellet with unusual geometry by a particle pellet presenting a regular geometry (slab, sphere, cylinder), where balance equations can be solved with the help of classical numerical procedures as orthogonal collocation method. 11 However, such approximations are not necessarily good when very complex pellet geometries are considered. Besides, it is still necessary to solve the balance equations inside the catalytic pellet numerically when more * To whom correspondence should be addressed. E-mail: pinto@ peq.coppe.ufrj.br. Tel.: 55-21-25628337. Fax: 55-21-25628300. † Universidade Federal do Rio de Janeiro. ‡ Engineering Simulation and Scientific Software (ESSS). § Petrobra´s. Ind. Eng. Chem. Res. 2009, 48, 9369–9375 9369 10.1021/ie801830q CCC: $40.75  2009 American Chemical Society Published on Web 04/09/2009