Coupled CFD-DEM Modeling: Formulation, Implementation and Application to Multiphase Flows

By Hamid Reza Norouzi, Reza Zarghami, Rahmat Sotudeh-Gharebagh and Navid Mostoufi

Discusses the CFD-DEM method of modeling which combines both the Discrete Element Method and  Computational Fluid Dynamics to simulate fluid-particle interactions.

• Deals with both theoretical and practical concepts of CFD-DEM, its numerical implementation accompanied by a hands-on numerical code in FORTRAN
• Gives examples of industrial applications

• Language: English
• Publisher: Wiley
• Release date: Oct 21, 2016
• ISBN: 9781119005292

Book preview

Preface

This book provides an up-to-date description of the formulation, implementation, and applications of combined (CFD-DEM) modeling. It is an integrated text that deals with theoretical and practical concepts of CFD-DEM, its numerical implementation accompanied by a numerical code and industrial applications. In the DEM part, different contact force models for spherical and non-spherical particles, as well as free-shape bodies, are discussed, along with their applicability and limitations. In the second part, couplings between solid and fluid equations for momentum, energy, and mass for particles and fluid are described and implementation of external forces on particles in multiphase flows is presented.

Over the years, many excellent books have been published dealing with various aspects of CFD. The level of sophistication of these books varies from academic to complex industrial systems and this is the main reason why we started this book with thorough treatment of the DEM. The distinctive feature of this book is its emphasis on coupled CFD-DEM (momentum, energy, and mass) as compared with books written on CFD or DEM alone. In addition, hands-on numerical codes are also delivered with the book in order to be used by readers as is, or modified as desired.

CFD-DEM has found wide range of applications in nearly all systems dealing with solids in various fields of science, engineering and technology such as chemical, food, pharmaceutical, biochemical, mechanical, energy, material, and mineral engineering. However, the prime concern of this book is to provide a more comprehensive treatment of DEM and CFD-DEM in chemical and process engineering with applications in granular and multiphase flow systems. In these systems, the DEM is commonly used for analysis of granular flow, including solid mixers, hoppers and silos, and CFD-DEM for fluid-solid flows, such as fluidized beds and conveyers, spouted beds, coal combustors, and solid incinerators.

Experimentations on multiphase flow systems are of vital importance in research and engineering. Nevertheless, they are lengthy, cost intensive, tedious, and challenging. We are unable to conduct experiments on the micro- and meso-scales in many cases and this is why most industrial processes fail at the early stages of development, design and operation. With the constant evolution of efficient computational tools, we now can analyze these issues and provide solutions. This book also helps the reader to acquire a better insight into these complex systems. With the diffusion of computational skills in industry and academia, we see the future of computation in process engineering rather promising. This would allow the better utilization of existing computational knowledge along with limited experimentation efforts.

The content of this book has gelled over the last 10 years through the collaborative research efforts of the authors on the subject. The book is primarily intended to serve students, scientists, and practitioners in process, chemical, mechanical, and metallurgical engineering. However, other engineers, consultants, and scientists concerned with various aspects of multiphase flow systems may also find it useful. Scientists and graduate students who want to learn and excel in DEM and CFD-DEM would find this book helpful. The content of this book can be used in a graduate course on advanced modeling and simulation in chemical engineering or as a complementary book to other engineering areas.

The authors acknowledge the contribution of many colleagues, former, and current students from the University of Tehran. Special thanks is extended to Dr. Zahra Mansourpour, Sedigheh Karimi, Bahram Haddani-Sisakht, and Shahab Golshan who have greatly contributed to some of important results presented in this book. We also express our gratitude to Mohammad Amin Hassani, Yasaman Norouzi, Mohammad Foroughi-Dahr, Mahsa Okhovat, Maryam Karimi, Maryam Sanaie-Moghadam, Hanieh Sotudeh-Gharebagh, Dr. Jaber Shabanian, Dr. Ebrahim Alizadeh, Dr. Rouzbeh Jafari, and Mr. Christian Jordan for their help extended to us during the completion of the book. (INSF) is acknowledged for supporting our research efforts in the multiphase flow laboratory, and process design and simulation research center where experimentations on and simulations of multiphase flow processes are the main concern.

Finally, we should emphasize that much remains to be done in this area and the utilization of CFD-DEM is expected to be increased rather than diminished. Adapting CFD-DEM to new areas will undoubtedly keep scientists and engineers busy for a long time. We can only hope that we have provided a useful base from which to start. The authors hope that this book would serve the industry and academia in the coming years. No human attempt is flawless, including this book. With your help, shortcomings and mistakes can be remedied and corrected. You are kindly requested to send your comments and corrections to [e-mail].

Hamid Reza Norouzi

Reza Zarghami

Rahmat Sotudeh-Gharebagh

Navid Mostoufi

June 2016, University of Tehran, Tehran, Iran

1

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.

Schematic illustrating momentum, energy, and mass transport between solid and fluid-phases.

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:

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).

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.

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.

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.

Meso-micro approach (fluid–meso, particle–micro): In this approach, referred to as CFD-DEM and shown in Figure 1.2, the flow field is divided into cells with a size larger than the particle size but still less than the flow field. Effect of motion of particles on the flow of fluid is considered by the volume fraction of each phase and momentum exchange through the drag force. This approach is the focus of this book and is explained in detail in the following sections.

Schematic diagram illustrating modeling scales in fluid–particulate systems.

Figure 1.2 Modeling scales in fluid–particulate systems

Let’s consider an example for illustrating the abovementioned approaches for modeling. Various modeling approaches including TPM, TFM, CFD-DEM, and DNS-DEM for a gas-fluidized bed are shown in Figure 1.3. The macro approach TPM, which is a one-dimensional model, is mostly used in industrial applications for long term simulations. For multi-dimensional modeling, the TFM can be used to predict the characteristics of fluidized beds at the meso-scale. The increased accuracy of the model is obtained at the expense of more computational costs and simulations are restricted to shorter periods. Handling the solid particles at a micro scale and fluid at meso scales is carried out in the CFD-DEM approach. While this increases the computational cost, it provides results with a higher resolution when compared with the TFM. If a higher resolution is needed for the fluid phase, DNS-DEM is the choice of modeling considering the fact that it needs higher computational effort. It should be mentioned that applying CFD-DEM, and especially DNS-DEM, is mostly limited to lab scale units.

Schematic illustrating different modeling approaches of TPM (left), TFM (top right), CFD-DEM (middle right), and DNS-DEM (bottom right) for a gas-fluidized bed.

Figure 1.3 Different modeling approaches for fluid–solid systems

In a gas-solid fluidized bed, various structures (micro, meso, and macro) coexist with different scales, as shown in Figure 1.4. Single particles and individual particles in clusters are typical examples of micro scale phenomena, while small bubbles and clusters are considered meso structures. Large bubbles, as well as the whole reactor, are at macro scale. However, it is important to mention that we are not limited to use the scales of these phenomenological structures in the modeling. Nevertheless, modeling with a finer scale would provide characteristics of that structure and larger ones while coarser scale modeling would provide only an averaged description of finer structures. For example, if using the CFD-DEM model, characteristics of particles, clusters, and bubbles can be captured, while in the TFM only characteristics of clusters and bubbles can be obtained and individual particles cannot be observed.

Schematic illustrating various structures, such as micro, meso, and macro, coexist with different scales in a gas–solid fluidized bed.

Figure 1.4 Different scales in a fluid–solid system

1.3 Modeling with DEM

DEM is a type of modeling tool through which the dynamics of a system comprising of a large number of distinct bodies with arbitrary shapes are studied. In granular flows, these distinct bodies are solid particles. Particles may interact with each other through their contact area or interparticle effects. Particles are assumed to be either rigid or deformable, leading to two different formulations of their collision, that is, hard-sphere/event-driven and soft-sphere/time-driven. In the soft-sphere formulation, which is the main focus of this book, particles are allowed to overlap and their contact lasts for a certain period. This allows a particle to be in contact with more than one particle at a time. This formulation is suitable for the motion of particles in both dense and dilute phases at quasi-static and dynamic conditions. Translational and rotational motions of a particle are tracked by integrating Newton’s and Euler’s second laws of motions, respectively. In addition to contact forces between particles, inclusion of other forces, like interparticle and fluid–particle interaction (if the fluid effect is significant), can be performed by introducing proper terms in the equation of motion.

The contact force is calculated according to normal and tangential overlaps of particles (or particle and wall) using a set of force-displacement expressions combined with friction laws. Many force-displacement models, such as linear and non-linear visco-elastic, elasto-plastic, and visco-plastic models have been developed for calculating the contact force between particles depending on the material properties and operating conditions. Particles are surrounded by walls as system boundaries. Among different methods, walls can be introduced to the model by decomposing the actual geometry into triangular or quadrilateral elements. This seems to be the flexible and rather general method to deal with simple to complex geometries in DEM simulations.

All motion and force equations associated with the DEM should be solved using proper integration methods. The explicit integration of these equations, in contrast to the implicit integration, increases the flexibility of the DEM simulation, even though it requires adapting a small time step for integration. A small time step and existence of a large number of interacting particles in the DEM simulation demands a huge computational resource. Without implementing efficient numerical algorithms and parallelization, this modeling approach is restricted to short-time simulations with the number of particles not exceeding 10⁵. Nowadays, the DEM is a powerful technique, allowing scientists and engineers to analyze rather complex systems for which analysis and understanding are not possible by current experimental techniques. In Chapters 2–5 of this book, essential components related to formulation, implementation, and sample application of DEM can be found in detail.

1.4 CFD-DEM Modeling

CFD coupled with DEM is a computational approach used to model fluid–particle systems. In the CFD-DEM, the fluid phase is assumed as a continuum and its meso-scale motion is described by the volume averaged Navier–Stokes equation, while the micro-scale motion of the solid phase is described by Newton’s and Euler’s second laws. Forces acting on a particle are gravity, contact between colliding particles, fluid–particle interaction, and interparticle forces. Normally, there are thousands or millions of distinct bodies in the system for which the equation of motion should be simultaneously solved along with fluid equations over fluid cells. These equations are usually presented for phenomena occurring with different length scales in the system. The flow field of the CFD-DEM is shown in Figure 1.5, which demonstrates the relation between micro- and meso-scales for modeling of a gas-solid system. The coupling between fluid and particulate phases is performed through the local porosity and the mutual fluid–particle interaction forces. Comprehensive reviews of CFD-DEM technique and its applications have been published [5–7].

Schematic illustration of a typical bubbling gas–solid fluidized bed with inset on the right.

Figure 1.5 A typical bubbling gas–solid fluidized bed. Averaged fluid fields (pressure, velocity, porosity, etc.) are calculated in a fluid cell at the meso-scale. The cells are small enough to capture all properties of the flow (e.g., bubbles) and large enough to contain a number of particles to ensures proper averaging. Vectors show the fluid velocity interpolated on the center of fluid cells

In the CFD-DEM, fluid–particle interaction for the fluid phase is based on total interactions per unit volume of the fluid cell (data exchange from the micro- to meso-scale), while the fluid–particle force for the solid phase is based on the force acting on each individual particle (data exchange from meso- to micro-scale). When equations of these phases are coupled together, at each time step, the DEM calculates the position and velocity data for each individual particle and the CFD provides the fluid flow profiles over fluid cells for the next time step. The resulting profiles allow calculation of the fluid–particle interaction acting on particles and in fluid cells. A fully explicit scheme is assumed for coupling, meaning that the coupling terms are calculated with the information at the current time step. Then, equations of motion of both phases are solved to advance the calculations one step at a time.

Outputs of the CFD-DEM are velocity, pressure, temperature, and species concentration fields of the fluid phase as well as contact time, position, velocity, temperature, and interparticle forces of particles at any moment of the simulation. Obviously, these variables are not easily accessible via experimental measurement. Unique capabilities of the CFD-DEM enable one to deeply investigate overall and local quantities of the whole system based on information generated for individual phases. These quantities include mixing and segregation rates, formation, growth, coalescence, and breakage of bubbles, pressure drop, solid flow rate, agglomeration and aggregation of particles, residence time distribution, and concentration and temperature profiles.

The CFD-DEM simulation is computationally expensive and has two major limitations; time step of integration and the size of problem. The first limitation is related to the integration time step for equations of motion of both phases. For the particle phase, integration is performed using the explicit scheme. This is essential for a general purpose DEM code. This integration scheme, however, imposes a stability condition on the particle time step in a system with physical contacts. As a rough estimate, the particle time step should be a fraction (at least 1/10th) of the natural frequency of the linear elastic mass-spring system or a fraction of critical time step determined by Rayleigh analysis (see Chapter 3). Based on this condition, the particle time step generally falls in the range of 10−7 to 10−4 s. The stability condition for the fluid phase is expressed by the Courant number. The Courant number should be kept to less than 0.5 to ensure a converged solution of coupled pressure-velocity fluid phase equations, even if the solver is implicit [8]. In practice, the fluid time step falls in the range of 10−5 to 10−2s based on this condition.

These conditions are applied when dealing with the numerical solution of equations of fluid and particulate phases separately. When solving equations of motion of the coupled fluid–particle flow, the interphase momentum exchange terms in these equations impose new conditions on particle and fluid time steps. The condition for the particle time step is imposed by the particle response time defined as the time required for acceleration of a particle to the fluid velocity due to the drag force [9]. Also, the condition for the fluid time step is imposed by the fluid response time defined as the time required for the acceleration of fluid to the particle velocity due to the drag force in the fluid cell [9]. Ranges of particle and fluid response times are 10−6–10−1 and 10−8–10−1 s, respectively. To ensure stability of solution and prevent lagged integration of equations of motion of phases, particle and fluid time steps should be less than these response times. In practice, the fluid time step is chosen equal or larger than the particle time step in typical fluid-solid systems. This difference leads to developing different coupling strategies (treatment of momentum exchange between phases) that have different stability behaviors as detailed in [10].

The second limitation in the CFD-DEM simulation is size of the problem. The size refers to the number of fluid cells and number of particles in a simulation. Basically, a CFD-DEM numerical code is developed to be executed in either sequential or parallel mode. With the sequential mode, and based on the current computational capacities, a few seconds of real-time simulation of a fluid–particle flow with less than 10⁵ particles and cells is feasible while long simulations are not feasible in this mode. Therefore, implementation of efficient numerical algorithms and parallelization is a must for any CFD-DEM code.

Parallelization falls into two categories: coarse-grained distributed-memory parallelization and fine-grained shared-memory or multi-threaded parallelization. The distributed-memory parallelization gives an almost unlimited number of processing units and memory space while it suffers highly from communication and inter-node data transfer latencies, which make it suitable for coarse-grained computations like CFD simulations. However, the parallelization and load balancing of the code for a dynamic DEM is a crucial task, which still needs more sophisticated algorithms to be implemented. The shared-memory parallelization suits DEM-based codes, since it benefits from loop level load distribution among processors. However, the number of processing units and memory space provided by the shared-memory machines are limited, which restrict the size of problem in some cases.

Accordingly, a unique and robust parallelization model cannot be suggested for a CFD-DEM code and mixed solutions have been used so far [11–13] to benefit from advantages of both parallelization models. This would lead to various configurations for software packages. Current open source and commercial CFD-DEM software packages are listed in Table 1.1. They use different parallelization models, ranging from distributed memory in Message Passing Interface (MPI) to shared memory in OpenMP®. Some packages employ advanced platforms such as compute unified device architecture (CUDA), which requires graphical processing units (GPUs) for computation (can be categorized as the shared-memory parallelization).

Table 1.1 Open source and commercial software packages available for CFD-DEM simulation (accessed on November 2015)

Nowadays, software companies have been developing ready-to-use tools for users with a basic knowledge of CFD-DEM. They regularly update their packages with new models and integrate them with mathematical tools and allow users to develop their own computational models within the software. However, troubleshooting, customizing, and extending of these packages need one to properly understand the basis of CFD-DEM as provided in this book (or in similar books coming in the future).

1.5 Applications

CFD-DEM modeling and simulation can be applied to many multiphase flows in scientific and engineering applications. This can be included in initial stages of process and product design. While these flows have been the subject of numerous experimental investigations, their CFD-DEM simulation can provide more detailed knowledge in different time and length scales to capture all features of such flows. The CFD-DEM has been widely used for analyzing the behavior of particles in fluid systems such as gas fluidization with cohesive and non-cohesive particles, dense phase and pneumatic conveying, drying, granulation, coating, and blending, segregation, agglomeration and clumping, caking, tribo-charging, and particle dispersion in solid and fluid. Industrial processes also can be successfully analyzed by this technique. Examples of such processes are gas cyclones, filtration, solid-liquid mixing, spray coating, drying, cooling, heating, and handling of powders. Moreover, complex processes such as flow in a circulating fluidized bed with application to coal combustion and incineration, fluid catalytic cracking (FCC) units for cracking high-molecular-weight hydrocarbon fractions of crude oil, metallurgical processes (e.g., blast furnaces), chemical reactors, off-gas scrubbing, industrial dust recycling, sedimentation and erosion, trickle beds, and so on, can be properly handled with this numerical technique.

Among the many applications, we chose to investigate chemical engineering examples such as packing behavior of particles, discharge of particles from a hopper, and hydrodynamics of fluidized beds (i.e., bubble behavior, gas and solid flows, and particle mixing). In addition, heat transfer, mass transfer, and chemical reactions are common phenomena in chemical engineering processes such as combustors, incinerators, fluidized bed reactors, dryers, granulators, and coaters. With the wide applications mentioned here, we can see that the future of CFD-DEM modeling and simulation is promising. CFD-DEM modeling would mimic chemical engineering applications as close to reality as possible and expand its application to new challenging problems while reducing the cost of experimentation to a great extent. Research institutions and companies are now integrating CFD-DEM into the early stages of their research and product development. This can largely decrease the development costs, increase the productivity, and shorten the marketing time. With powerful computers and knowledgeable users, modeling and simulation of more complex geometries will become believable in the near future.

1.6 Scope and Overall Plan

The focus of this book is to deal with granular and multiphase flows. The book is organized into two interconnected parts. Part I deals with the flow of solid in the absence of fluid in four chapters and Part II treats the flow of solid in the presence of fluid within three chapters. Although there are excellent books on CFD and DEM alone, limited number of books can be found on CFD-DEM coupling basics. The book chapters contain the most recent theoretical developments in this advanced modeling approach as well as the corresponding numerical implementation. It is worth mentioning that, due to the availability of more books on CFD, this book starts with DEM with more focus on the coupling of CFD-DEM models and corresponding applications.

A large number of references are cited and discussed, covering the essence of most published literature on CFD-DEM. A DEM code in FORTRAN standard accompanies the book to help graduate students and researchers in solving the formulations addressed, since it takes a long time to develop such a long code. This code is mature enough to be directly used (with no modification) in most related applications. However, it also can be partially modified to suit other applications not discussed in this book. Methods of setting up codes for different applications are also presented. It is worth mentioning that the elements of CFD-DEM model described in this book have been assessed by different investigators in terms of overall quantity. It is not our intention to verify the quality and the correctness of their approaches and we leave this to readers. In this book, there are eight chapters in two parts on various relevant topics with the following general descriptions:

Chapter 1 gives a brief introduction to topics related to multi-scale modeling, DEM, and CFD-DEM. A few industrial applications, among many applications that can be handled by this efficient technique, are described in brief. Organization of different chapters of the book and online content forms the last section of this chapter.

Part I

Chapter 2 begins with an overview of DEM for granular flow and is followed by the basic formulation of hard-sphere and soft-sphere frameworks. Since the focus of this book is the soft-sphere framework, the rest of the chapter is devoted to common models for evaluating contact forces and torques among spherical particles in detail. Boundary (walls) and initial conditions are also covered in this chapter. In addition, the calculation procedures and sequences are detailed wherever deemed necessary.

Chapter 3 provides an in-depth treatment of the numerical implementation and algorithms of DEM. Contact search algorithms, integration methods, as well as wall treatment are addressed. Dealing with a large number of equations needs choosing efficient numerical algorithms and parallelization techniques as addressed in this chapter. In order to help the reader to understand the implementation properly, the codes are partially given in this chapter, which eases understanding of the techniques presented.

Chapter 4 describes the implementation of non-spherical particles in DEM. The kinematics and dynamics of non-spherical rigid body are presented first. Superellipsoid and multi-sphere methods, as two main approaches for shape representation of non-spherical particles in a DEM simulation, are also discussed in detail.

Chapter 5 provides applications of the DEM code accompanied with this book to granular flow as formulated and implemented in previous chapters. Packing of particles as a common practice in the powder industry, flow pattern in hoppers, mixing operation and design, screw conveying, and film coating are among the application presented in this chapter with details.

Part II

Chapter 6 covers the CFD-DEM formulation and coupling of momentum, energy, and mass equations. Different coupling strategies and interaction parameters among phases and fluid flow field are also discussed in this chapter with details. This is followed by detailing the solid phase flow, interphase coupling, coupling framework, fluid volume fraction, and mapping from the Eulerian to Lagrangian domains. The heat and mass transfer formulations with and without chemical reactions for both solid and fluid phases as well as their coupling strategies are also presented.

Chapter 7 summarizes the most practical applications related to fluidization, spouting, and pneumatic conveying from literature with some figures and facts generated by our in-house CFD-DEM code developed with formulations and implement approach presented in this book. In subsequent sections, non-isothermal and reactive flows are presented in brief in order to cover the essence of these flows. The chapter ends with miscellaneous applications related to fluid–solid interaction systems practiced in chemical industries.

Chapter 8 briefly introduces interparticle forces and external fields in particulate systems. The governing equations of the system as well as interparticle forces such as van der Waals, liquid bridge, and electrostatic and effect of external fields such as electric, magnetic, vibration, and acoustic are discussed due to their importance in the CFD-DEM modeling. The chapter ends with some relevant applications.

1.7 Online Content

The supplementary online content associated with this book contains the numerical code, animations, and simulation setups for case studies and applications addressed in the book. The content was prepared as the assembly of (i) numerical FORTRAN code that accompanies this book, (ii) animations related to problems solved in Chapters 5 and 7, and (iii) the code files and guides to setup most of the simulations in Chapter 5 using the numerical code in order to support the reader’s hands-on simulation experience. The online materials are available from the Wiley website at: www.wiley.com/go/norouzi/CFD-DEM.

References

[1] Anderson, T.B. and Jackson, R. (1967) Fluid mechanical description of fluidized beds. Equations of motion. Industrial and Engineering Chemistry Fundamentals, 6(4), 527–539.

[2] Tsuji, Y. (2007) Multi-scale modeling of dense phase gas–particle flow. Chemical Engineering Science, 62(13), 3410–3418.

[3] Van der Hoef, M., Ye, M., van Sint Annaland, M., Andrews, A., Sundaresan, S., and Kuipers, J. (2006) Multiscale modeling of gas-fluidized beds. Advances in Chemical Engineering, 31, 65–149.

[4] Klinzing, G.E., Rizk, F., Marcus, R., and Leung, L. (2011) Pneumatic Conveying of Solids: A Theoretical and Practical Approach, Vol. 8, Springer Science & Business Media.

[5] Deen, N., Annaland, M.V.S., Van der Hoef, M., and Kuipers, J. (2007) Review of discrete particle modeling of fluidized beds. Chemical Engineering Science, 62(1), 28–44.

[6] Zhu, H., Zhou, Z., Yang, R., and Yu, A. (2008) Discrete particle simulation of particulate systems: a review of major applications and findings. Chemical Engineering Science, 63(23), 5728–5770.

[7] Zhu, H., Zhou, Z., Yang, R., and Yu, A. (2007) Discrete particle simulation of particulate systems: theoretical developments. Chemical Engineering Science, 62(13), 3378–3396.

[8] Jasak, H. (1996) Error analysis and estimation for the finite volume method with applications to fluid flows. PhD thesis. Imperial College of Science, Technology and Medicine.

[9] Crowe, C.T. (2005) Multiphase Flow Handbook, CRC Press.

[10] Radl, S., Capa Gonzales, B., Goniva, C., and Pirker S. (2014) State of the art in mapping schemes for dilute and dense euler-lagrange simulations. 10th International Conference on CFD in Oil & Gas, Metallurgical and Process Industries, Trondheim, Norway.

[11] Yakubov, S., Cankurt, B., Abdel-Maksoud, M., and Rung, T. (2013) Hybrid MPI/OpenMP parallelization of an Euler–Lagrange approach to cavitation modelling. Computers & Fluids, 80, 365–371.

[12] Amritkar, A., Deb, S., and Tafti, D. (2014) Efficient parallel CFD-DEM simulations using OpenMP. Journal of Computational Physics, 256, 501–519.

[13] Liu, H., Tafti, D.K., and Li, T. (2014) Hybrid parallelism in MFIX CFD-DEM using OpenMP. Powder Technology, 259, 22–29.

Part I

DEM

2

DEM Formulation

The discrete element method (DEM) refers to a type of modeling approach in which the dynamic/time transient behavior of a system comprising of a large number of distinct bodies with arbitrary shapes is studied. These bodies may be considered as either deformable or rigid, which continuously come into contact with each other and rebound. The translational and rotational motions of each body are formulated in the model and are highly dependent on type and strength of interactions of the bodies with the surrounding medium through contact and interparticle effects. The bodies may interact with each other through their contact points/areas. Depending on the operating conditions of the system, interparticle interactions, such as van der Waals, liquid bridge, and electrostatic forces, may exist between distinct bodies even if they are not in contact. In multiphase flows, the fluid interacts with the distinct bodies in the system. The interaction between each body and the fluid is mutual. During the motion of bodies, they interact with stationary or moving walls. Moving walls increases the kinetic energy of bodies leading to convective movement.

In the DEM, the bodies are considered either deformable or rigid. The choice of deformable or rigid formulation depends on operating conditions and available computational resources. These two are mostly referred to as soft-sphere and hard-sphere formulations, although these formulations are not specific to spheres. In general, the soft-sphere formulation covers more practical granular and multiphase flows than the hard-sphere formulation. For a dilute system, the hard-sphere formulation is computationally effective, although soft-sphere formulation is also applicable. This is due to the fact that the contact time of each body is much shorter than the mean time between successive collisions and hence each contact can be considered instantaneous and pairwise. In a dense system, for which long contact times and multiple contacts exist between particles, the soft-sphere formulation is more appropriate.

Discrete element formulations detailed here are mainly based on the soft-sphere, considering the following justifications. First, it covers most of granular and multiphase flows with and without interparticle forces. Second, modeling time-progressive phenomena, like solid-bridge formation, liquid-bridge progression, agglomeration, caking, and particle breakage, are straightforward in the soft-sphere formulation. Third, the soft-sphere approach can be modified and extended to suit the hard-sphere approach. Fourth, the rapid development in computational resources and numerical algorithms make it possible to simulate a system containing hundreds of thousands of distinct bodies in a reasonable time.

In this chapter, both hard- and soft-sphere formulations are covered for spherical particles. These are followed by introducing common models for evaluating contact forces and torques. In addition, the ways we can treat different boundary conditions/walls in the model formulation are also covered here. The rest of the chapter is devoted to boundary (walls) and initial conditions. In addition, the calculation procedures and sequences are detailed wherever deemed necessary. The DEM formulation for non-spherical particles is covered in Chapter 4.

2.1 Hard-Sphere

The soft-sphere formulation is based on the collision forces that are established during a contact between bodies. The collision force is calculated based on the overlap between colliding particles. This overlap, and hence the collision force between bodies, changes in the course of a collision. Then, the Newton’s second law of motion is applied to balance the acceleration and the forces acting on it (see Section 2.2). Integration of the Newton’s second law gives new velocity and position of each body in each time step. This is known as the force-based/soft-sphere/time-driven formulation in the DEM. As we will discuss in Section 2.2, the time step for numerical integration and tracking bodies should be so small that each contact is processed in several time intervals. Moreover, the procedure of finding new collisions and relaxing old collisions should be applied in each time step.

Let us consider a situation in which the collisions between bodies are not very long and frequent. In this system, each body travels freely (based on external forces and translational velocities) until the next collision happens, similar to what we have in the dynamics of granular gases. Thus, new states (position and velocity) of bodies are not evaluated until the next collision/event happens. The post collision velocities of colliding bodies are calculated based on pre-collision velocities via establishing the momentum conservation law. This is the collision-based/hard-sphere/event-driven formulation in the DEM. From the computational point of view, this formulation, in some cases, is more efficient than the force-based formulation since the velocity of bodies is updated whenever a collision/event happens.

This formulation implies that the mean time in which a body freely travels is much more than the duration of the contact. In this case, we can confidently assume that contacts between bodies are pair-wise and there are no multiple contacts. Moreover, we further assume that each collision between bodies is established with no overlap and lasts for an extremely short time ( ). The hard-sphere formulation works well for describing the dynamics of dilute systems where the number density of particles is low and the previously mentioned assumptions are held. The speed gain of the calculation is mostly due to elimination of redundant calculation of the system state (velocity and position of bodies) between successive events. Although its robustness has been proven for dilute systems, it has been successfully implemented also for dense systems. This approach was first introduced by Campbell and Brennen [1] to simulate a granular system and then was used to simulate particles flow in different systems like rotating drums [2, 3]. The hard-sphere DEM was combined by computational fluid dynamics (abbreviated as CFD-DEM) to study bubble and slug formation in a gas-solid fluidized bed [4]. The hard-sphere CFD-DEM model was used to study size segregation in fluidized beds [5–7], fluidization characteristics at high pressure [8, 9], fluidization regimes in spout-fluid beds [10], and granulation in spout-fluid beds [11].

As we discussed before, most theories and implementations in this book are based on the soft-sphere formulation. We bring the hard-sphere formulation here mainly for the sake of comparison between different aspects of these modeling approaches. In addition, this gives the reader a better perspective to enable choosing between these two types of formulations for each application. Thus, we just present the headlines of the hard-sphere formulation to give a general overview of it. The interested reader is referred to other references [12–14] for more detail on theoretical aspects as well as numerical implementation of this approach.

2.1.1 Equation of Motion

The velocity change of each particle with mass mi during its free motion (between successive collisions) in the system is individually tracked by integrating the Newton’s second law of motion.

(2.1)

The first integration of Equation 2.1 yields the new velocity of the particle and the second integration yields its new position. The velocity of particle changes due to external forces, like gravitational force and total fluid–particle interaction force ( ). We considered the fluid-particle interaction force in this equation in order to extend the basic equation of motion to multiphase flows where fluid–particle interactions are important. There are different types of fluid–particle interactions in a multiphase flow such as, buoyancy, drag, and lift forces. We will discuss them in detail in Chapter 6. The equation of motion for the hard-sphere formulation is very similar to that for the soft-sphere formulation. However, there are some differences that will be explained in Section 2.2 after we present the soft-sphere formulation. Note that this equation gives the velocity change of particle i during the free motion of the particle. At the moment of collision between particle i and another particle j, which is assumed to last for an infinitely short time, the hard-sphere collisional model is used to calculate the post-collision velocities (both translational and rotational) based on the pre-collision velocities. Therefore, Equation 2.1 is not valid for the moment of collision between particles. After the collision, the post-collision translational velocity will be used as the initial condition for Equation 2.1 until particle i comes into contact with another particle. In the following section, we present the main framework of the hard-sphere collision model.

2.1.2 Collision Model

Consider two particles with radii Ri and Rj, position vectors and , and masses mi and mj. They are traveling with translational velocities and , and rotating with angular velocities and . We consider that the collision is impulsive and any external force, like attractive, drag and so on, are zero during the collision. This is true since the basic assumption of the hard-sphere formulation is the instantaneous collision. The relations between pre-collision and post-collision translational and rotational velocities are established by impulse equations as follows:

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

where superscripts 0 and 1 represent particle properties before and after the collision, respectively; is the unit vector pointing from particle i to particle j and is perpendicular to the contact plane; is the tangential unit vector at the contact point that lies on the contact plane and Ii is the moment of inertia of the particle.

(2.7)

(2.8)

(2.9)

There are three parameters that are used to determine the impulse force in these Equations 2.2–2.9. These parameters are coefficients of restitution in normal and tangential directions and dynamic friction coefficient. Once the impulse force is known, the post-collision translational and angular velocities can be calculated. The impulse force represents the mutual repulsion of colliding particles due to their elasticity (see Figure 2.1).

Schematic illustrating successive steps in an instantaneous collision between two spherical particles in the hard-sphere formulation, displaying (left–right) before collision, collision, and after collision.

Figure 2.1 Successive steps in an instantaneous collision between two spherical particles in the hard-sphere formulation. and are translational velocities, and are rotational velocities, and are normal and tangential impulse forces. Superscripts 0 and 1 refer to pre- and post-collision velocities, respectively

The relative velocity between colliding particles at the contact point is calculated as follows:

(2.10)

(2.11)

(2.12)

The coefficient of restitution is a measure to show that how much kinetic energy is lost during an inelastic contact between two particles. It is simply defined as the ratio of post- and pre-collision velocities. The coefficients of restitution in normal direction, en, and tangential direction, et, are defined by the corresponding velocities in the contact plane.

(2.13)

(2.14)

The normal component of the impulse force is related to the normal coefficient of restitution and pre-collision relative velocity by:

(2.15)

(2.16)

Calculation of the tangential component of impulse force needs more attention since there are some circumstances that lead to sliding or sticking collisions. When the tangential component of the relative velocity is high in comparison with the normal velocity, sliding occurs throughout the collision. In this situation, the tangential component of impulse force is obtained by Coulomb’s friction law:

(2.17)

In other circumstances, when the normal relative velocity is high in comparison to the tangential relative velocity, sticking collision occurs. Therefore, the tangential component of impulse force is obtained by:

(2.18)

The normal coefficient of restitution varies between 0 and 1. It was experimentally observed that it decreases as the impact velocity increases. We can assume two following extreme cases. When the impact velocity approaches zero, the normal coefficient of restitution becomes unity, meaning no loss of the kinetic energy during the contact. On the other hand, when the impact velocity approaches infinity, the coefficient of restitution approaches zero, meaning that all the kinetic energy is lost during the contact. However, the restitution coefficient is not only a function of impact velocity but also a function of particle properties and the type of particle deformation (elastic or plastic). The same is true for the tangential coefficient of restitution. It is a function of impact velocity, material properties, and contact conditions.

2.1.3 Interparticle Forces

Interparticle forces are mostly referred to as non-contact interactions between bodies that are separated by a distance of d. These forces may be attractive or repulsive. Generally, the attractive or repulsive interparticle force is a continuous function of separation distance of bodies. For example, in a simplified situation, the attractive van der Waals force between two bodies is inversely related to square of the separation distance [15]:

(2.19)

Interparticle forces exist between particles during their movements and each particle interacts with all other surrounding particles. The physics of interparticle forces does not allow us to directly incorporate them into the hard-sphere formulation since this formulation is based on instantaneous and binary events. However, Weber et al. [16] and Weber and Hrenya [17] introduced the square-well method to handle granular flows with interparticle cohesive forces in the hard-sphere formulation. We introduce this methodology here to give a good picture to the reader of how this formulation can be further extended to a similar system in which interparticle forces play a key role. For example, dilute phase electrification of gas-particle flow in pneumatic conveying of particles [18] can be modeled by event-based/hard-sphere DEM. Charge transfer and induction occur in particle–particle and particle-wall collisions/events [19, 20]. At the same time, charged particles experience electrostatic force from neighboring charged particles that can be represented by the square-well method. The