## Introduction

Pore-scale flow and dispersion in porous media like in fixed beds of spheres has been modeled in different fields of engineering. One example is compact-bed filters that are used for on-site sanitation systems.[1-3] Modeling the flow through such packed beds in laboratory scale is important for aspects such as geometrical scale-up and optimization of operation parameters.[4] In chemical engineering, the modeling of packed-bed reactors, especially those with low-tube-to particle diameter ratio, which are used for highly exothermal reactions, has been studied extensively (see e.g., Nijemeisland and Dixon[5]). Another area of interest is the impregnation of fabrics during composites manufacturing,[6-10] in which the flow is often on several scales and different kinds of particles may be added to the fluid to add functionality to the molded composite. Manufacturing of new biocomposites and papermaking are other examples where dispersion matters.[11-14]

One way to numerically study the flow through fixed beds of spheres is to use traditional Computational Fluid Dynamics (CFD). The flow in fixed-bed reactors has, for example, been simulated with CFD with usage of finite volume discretization of the Navier–Stokes equations (see e.g., Refs. [5, 15], and [16]). Another approach for complex flow systems is the Lattice–Boltzman method,[17, 18] that has been used to derive the detailed flow in porous media and in fixed-bed reactors (see e.g., Refs. [19] and [20]) even for moderate Reynolds numbers. However, it may not be well suited for curved particle interfaces and sharp spacing between the spherical particles. As an alternative approach, accelerated Stokesian dynamics method[21] can be applied, but it requires inverting large matrixes for increased number of particles. Thus, although such methods can be successfully used to simulate the flow and study transport/transfer properties in packed beds of particles, it may be difficult to study large systems due to mesh refinement problems and the requirements of large computational resources. Hence, many of the early simulations were performed on unit cells with periodic boundary conditions.[22, 23] Today, despite the advances of computational power, CFD simulation of a large number of particles often becomes very time consuming and expensive. In addition, there are certain difficulties related to the discretization of small distances near the particle-particle and particle-wall contact points. Therefore, the suitability of the mesh in general and particularly near these contact points should be analyzed carefully and methods should be developed that are stable regarding convergence, computational efficient and that has a geometrical configuration that can be altered in a simple way,[24, 25] for example, due to deformations of porous media. Following these demands, a three-dimensional (3-D) porous-medium model is here built that consists of thousands of spherical particles that are divided into cells using Voronoi diagrams. The aim of building this model is to reproduce mass transfer in a tracer column experiment and the resulting effluent curves. The dispersion is obtained by fitting the continuous model to either effluent curves or spatial distribution of the concentration front inside the column. It requires transient calculations that are less efficient than methods employing B-field[26] or other methods. However, with the approach it is possible to examine how the fitted dispersion develops as the concentration front moves inside the column. Of particular interest is the behavior at the inlet of the column where the discrete character of the particles can change the continuous interpretation of dispersion. On the other hand, it is still not clear whether the dispersion stabilizes if the column length increases depending on the condition at the side walls. Therefore, a study of a prolonged system is carried out.

The 3-D packed-bed model consists of thousands of randomly distributed spheres. Voronoi diagrams are applied to discretize the system into cells that each contains one sphere and Laurent series are applied to find the local flow fields inside the Delaunay tetrahedrons. The whole flow pattern is then obtained by minimization of the dissipation rate of energy[27, 28] for the dual stream function.[29] The obtained stream function provides an excellent tool for flow visualization in porous media. As a result, a discretized 3-D packed-bed model is obtained that is employed to derive the longitudinal (*D _{L}*) and transverse (

*D*) dispersion coefficients. The results are presented as

_{T}*D*and

_{L}/D_{m}*D*vs. Pe

_{T}/D_{m}_{m}(=

*ud*/

*D*), and are further compared to experimental results from the literature. In these expressions,

_{m}*D*is the molecular diffusion coefficient;

_{m}*d*is the inert sphere diameter; and

*u*is the average interstitial liquid velocity, defined as

*u = U/*

**, where**

*ε**U*is the superficial velocity and

*ε*is the porosity. The flow rate in this study is limited to Pe

_{m}< 10 to analyze the dynamics in the tracer column experiment rather than to obtain the dispersion directly. At this range of velocities, the dispersion predominantly depends on the flow variation due to random packing of the porous media rather than variation of flow velocity between single solid particles, so called Taylor dispersion.[30, 31] The Taylor dispersion is essential for higher velocities, that is, Pe

_{m}> 10. As only the low Reynolds number flows are considered, the boundary layer dispersion[30] is neglected. The boundary layer dispersion occurs at Pe

_{m}≫ 1 and results from tracers that come close to no-slip solid boundaries in the medium that could not escape the slow moving region near the boundary without the aid of molecular diffusion.[30]

This work is founded on an earlier published article[32] that used a 2-D packed-bed model to derive the dispersion coefficients. Moreover, similar to the previous 2-D packed-bed model,[32] the model derived herein is generic and can be applied to a number of areas of application such as investigations of the effects of particle-size distribution on the *D _{L}*,[33] drying of iron ore pellets,[34] filtration mechanisms during composites manufacturing,[35] and internal erosion processes.[36]