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Introduction

Industry demand for efficient and faster computational tools has facilitated the development of Computational Fluid Dynamics (CFD). This has allowed the utilization of CFD as a specialized tool to solve mass, momentum, energy, and species conservation equations. Advances in computer technology have now changed the entire frame of CFD modeling, allowing it to be a tool for engineers and scientist to carry out design, simulation, and optimization of various processes. Today, the application of CFD not only covers the conventional engineering fields, such as chemical and mechanical engineering, but is also widely extended to multidisciplinary areas, such as environment and healthcare.

With CFD, fluid and solid particulate phases can be modeled by the "Eulerian-Eulerian" approach, which is a way of looking at the motion of fluid and particles from a continuum point of view. This hypothesis may be true for fluids but it may bring less accurate results when considering solid particles as a continuum. In order to properly model particle motion, the Discrete Element Method (DEM) has been developed, in which the motion of individual particles is tracked in space and time using the Lagrangian approach. This approach is complementary to the Eulerian approach for modeling multiphase flows and is referred to as the " Eulerian-Lagrangian " approach, detailed in the following sections.

Multiphase flows exist in many industrial applications such as gas or liquid fluidized bed reactors, fluidized bed dryers, spotted beds, three-phase gas-liquid-solid fluidized beds, pneumatic conveying of solids, and so on. A detailed knowledge of these flows is crucial for design, scale-up, optimization, and troubleshooting of such processes. Although this may be achieved by experimental techniques, modeling can be considered as an alternative tool for exploring different aspects of multiphase flows. Modeling enables us to understand different phenomena occurring in these processes, to perform sensitivity analysis on different input parameters and to test different configurations and operational conditions at lower expense compared with experimental methods. In the following, we discuss the overall view of the modeling of granular and multiphase flows.

1.1 Multiphase Coupling

Phase coupling, in terms of momentum, energy, and mass, is a basic concept in the description of any multiphase flow. The coupling can occur through exchange of momentum, energy, and mass among phases as shown in Figure 1.1. In principle, fluid-particulate properties can be described by position, velocity, size, temperature, and species concentration of fluid and/or particle. While the phenomenological description of multiphase flow can be applied to classify flow characteristics, it also can be used to determine appropriate numerical formulations. In various modes of coupling, depending on the contribution of phases and phenomena, different coupling schemes can be adapted. This may allow independent treatment of phases or simultaneous integration of momentum, heat, and mass exchanges between phases. In general, modeling complexity increases as more effects associated with time and length scales are included in the simulation.

Figure 1.1 Momentum, energy, and mass transport between solid and fluid phases

1.2 Modeling Approaches

Real systems are rather complex in nature and modeling allows analysis and simulation of these systems to be conducted more accurately. Depending on the length scales considered for fluid and particle systems, various combinations of modeling scales can be suggested. These are classified as micro-, meso-, and macro-scale models. In a micro-scale model, trajectories of individual particles are calculated through the equation of particle motion and the fluid length scale is the same as the particle size or even smaller. At the same time, instantaneous flow field around individual particles is calculated. In the meso-scale model, both solid and fluid phases are considered as interpenetrating continua. The conservation equations are solved over a mesh of cells. The size of the cells is small enough to capture main features of the flow, like bubble motions and clusters, and large enough (essentially larger than the size of individual particles) to allow averaging of properties (porosity, interactions, etc.) over the cells. Anderson and Jackson [1] first presented this formulation for fluid-particulate systems. In the macro-scale model, the fluid length scale is in the order of the flow field. This means that motions of the fluid and the assemblage of particles are treated in one dimension based on overall quantities [2]. It is also possible to develop some intermediate models in which the length scales of fluid and solid phases are different. For example, the length scale of solid phase can be kept at the micro-scale while changing the length scale of fluid phase to meso or macro. Under these conditions, the affective interactions in the larger scale can be calculated by averaging the information in the smaller scale.

In multi-scale modeling, the smaller scale model takes into account various interactions (i.e., fluid-particle, particle-particle) in detail. These interaction details can be used with some assumptions and averaging to develop closure laws for calculating the effective interactions (e.g., drag force) in the larger scale model [3]. This allows capture of the essential information needed on the larger scale. Alternatively, calculation of effective interactions can be performed through the local experimental data, if available. Combination of fluid/particle motion with different modeling scales can provide different modeling approaches, as sketched in Figure 1.2 and detailed here:

1. Micro approach (fluid-micro, particle-micro): In this approach, the fluid flow around particles is estimated by the Navier-Stokes equation. Since the forces acting on particles are calculated by integrating stresses on the surface of the particle, the empirical correlation for drag and lift forces are not required. This approach is used in cases where particle inertial force is relatively small (e.g., liquid-particle flow) or the fluid lubricating effect on particles is rather significant (e.g., dense-phase liquid-particle flow). A typical example of such an approach, shown in Figure 1.2, is the direct numerical simulation-discrete element method (DNS-DEM).
2. Meso approach (fluid-meso, particle-meso): In this approach, which is shown in Figure 1.2 and is referred to as the two-fluid model (TFM), in addition to the real fluid, the assemblage of particles is also considered to be the second continuum phase. The flow field is divided into a number of small cells to capture motions of both phases, provided that the cell size is larger than the particle size. The two continuous phases are modeled by applying laws of momentum and mass conservations in each fluid cell, leading to averaged Navier-Stokes and continuity equations. Capability of the TFM in capturing the solid phase motion greatly depends on the closure laws used for this phase. These closure laws always involve some simplifications or are obtained by semi-empirical correlations. While this approach is preferred in commercial packages for its computational simplicity, its effectiveness depends on the constitutive equations and is not easily applicable to all flow conditions. The TFM has been successfully utilized to obtain the flow behavior of various non-reacting and reacting multiphase flows in laboratory, pilot, and industrial scales.
3. Macro approach (fluid-macro, particle-macro): This approach provides a one-dimensional (1D) description of gas-particle flows [4]. The main output of such a model is the pressure drop, which is considered as the sum of pressure drops due to flow of fluid and particles. Usually, a formula for the single phase flow, such as Darcy-Weisbach equation, is used for the fluid pressure drop and that of particles is balanced with the fluid drag formula from the momentum balance. This approach would also allow the calculation of averaged flow properties by empirical correlations that are essential in design and analysis of industrial processes. A typical example of such approach, shown in Figure 1.2, is the two-phase model (TPM) in fluidization. In this model, conservation equations are written for bubbles and emulsion, both having the length scale of the system in a fluidized bed.
4. Macro-micro approach (fluid-macro, particle-micro): In this approach, shown in Figure 1.2 by 1D-DEM, the fluid forces acting on particles are calculated from empirical correlations (e.g., drag and lift) while translational and rotational motions of particles are described based on Newton's and Euler's second laws. At very low concentration of particles, effect of particles on the fluid motion can be neglected. However, at higher concentrations, closure laws should be modified to account for the closeness of surrounding particles. Generally, in this approach the flow field, which is considered to change in one dimension, is not divided into cells and additional pressure drop is taken into account to reflect the effect of particles on the fluid motion.
5. Meso-micro approach (fluid-meso, particle-micro): In this approach, referred to as CFD-DEM and shown in Figure...