Assessing the accuracy of volume averaging effective diffusivity estimates with Brownian dynamics simulations C.G. Aguilar-Madera, F.J. Valdes-Parada, L. Dagdug, J. Alvarez-Ramirez n Divisio ´n de Ciencias Ba ´sicas e Ingenierı ´a, Autonoma MetropolitanaUniversidad Auto ´noma Metropolitana, Apartado Postal 55-534, Me ´xico D.F. 09340, Me ´xico article info Article history: Received 19 November 2011 Received in revised form 28 March 2012 Accepted 5 April 2012 Available online 12 April 2012 Keywords: Transport processes Diffusion Mass transfer Dynamic simulation Volume averaging method Brownian dynamics simulation abstract The volume averaging method (VAM) is widely used for estimating the effective diffusivity for solute transport in constrained geometries (e.g., porous media). Comparisons with experimental results for different geometric configurations have indicated that the VAM can provide accurate results over a wide range of porous media configurations. This work uses Brownian dynamics simulations (BDS) to estimate effective diffusivities for constrained geometries and to assess the accuracy of estimations from the VAM. For simple microscale geometric configurations, both isotropic and non-isotropic, it was found that the results predictions of effective diffusivity from VAM agree well with results from the BDS for high porosity values. However, some discrepancies are found for low porosity values, which were attributed to a smoothing effect of the VAM closure problem in the vicinity of very small apertures. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction The problem of diffusion in geometrically constrained systems arises in different contexts. Examples include transport of parti- cles in biological cells and membranes (Hille, 2001; Alberts et al., 2007) and in zeolites (Karger and Ruthven, 1992), catalytic reactions occurring on templates or in porous media (Somorjai, 2010), transport in polymer blends, foams and ceramic mixtures (Christensen, 1979), separation techniques of size disperse parti- cles on microscales and nanoscales (Buonasera et al., 2009), controlled particle release (Slowing et al., 2008) and many more. A thorough understanding of the phenomena underlying trans- port in constrained geometries should allow accurate manipula- tion and design of structures oriented for controlling diffusion in, e.g., drug release and migration of contaminants in porous media. A key parameter for modeling transport in constrained geo- metries is the effective diffusion coefficient, which is intended to be used with macroscopic equations representing the average behavior of the system. Although for specific geometries experi- mental runs can be performed for estimating the transport properties, the experimental approach may be expensive and lead to inaccurate results due to unavoidable measurement errors. In recent decades, computer simulations have been employed to estimate effective transport properties in porous media. The volume averaging method (VAM) has been widely used in the chemical engineering field to obtain averaged equa- tions that include the effective diffusion parameter (Whitaker, 1999). The approach consists of averaging the local diffusion equations within a control volume to obtain a macroscopic description of the problem while generating at the same time a boundary-value problem corresponding to the effective diffusiv- ity tensor. Reported results in the literature are extensive and show the flexibility of the VAM to accommodate for different conditions. For instance, the success of the VAM to predict directional differences in the effective diffusivity for anisotropic porous media has been demonstrated (Kim et al., 1987). The effective diffusivity of cellular systems (tissues and biofilms) was also studied with the VAM, showing that, under many practical circumstances, reasonable estimates can be obtained (Ochoa- Tapia et al., 1986; Wood et al., 2002). The effect of homogeneous chemical reaction on the effective diffusivity was also studied by showing that chemical reaction can increase the diffusion trans- port (Valdes-Parada and Alvarez-Ramirez, 2010; Valdes-Parada et al., 2011). Modifications of the VAM for addressing the effects of boundary conditions for the closure problem (Lux, 2010) and the porosity asymmetry of the porous media (Valdes-Parada and Alvarez-Ramirez, 2011) have been reported recently. The VAM has been shown to be applicable to a large diversity of geometric conditions, from simple representative cells to complex porous media. However, the evaluation of the accuracy of the effective diffusivity estimates is hampered by the limited availability of reliable experimental data. This work proposes to Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ces Chemical Engineering Science 0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.04.008 n Corresponding author. Tel.: þ52 5558044650; fax: þ52 5558044900. E-mail address: jjar@xanum.uam.mx (J. Alvarez-Ramirez). Chemical Engineering Science 75 (2012) 418–423