The behaviour of a granular material is closely related to the nature of the interactions between grains. In this chapter, we focus on these forces at the grain level. We first discuss solid contact, which is dominant in the case of dry granular media made of macroscopic particles (Section 2.1). The basics of Hertz elastic contact, solid friction and the rules of inelastic collisions between solid particles are given. We then discuss other kinds of interaction between grains such as electrostatic and adhesive forces, capillary cohesion and solid bridges (Section 2.2). The last part of the chapter gives a brief overview of the hydrodynamic forces produced on a particle immersed in a fluid (Section 2.3). Our aim in this chapter is to provide some background in contact physics and hydrodynamics that will be useful for our study of granular media. More detailed treatments can be found in the classical books given in the text.

Solid contact forces

The contact force between two dry grains is usually split into a normal force and a tangential force. The physical origin of these forces at the microscopic level involves many phenomena, such as surface roughness, local mechanical properties (elasticity, plasticity, viscoelasticity) and physical and chemical properties (the presence of electrical charge, oxidation, temperature, the presence of lubricant film). In the following, we will not consider these microscopic features in detail but rather focus on the macroscopic laws of solid contact. At the macroscopic level, these laws are dominated by elastic repulsion (Hertz contact) and solid friction (Coulomb’s law).

Sand, gravel, rice, sugar … Granular matter is familiar and abounds around us. However, the physics of granular media is still poorly understood and continues to fascinate scientists and other people, more than three centuries after the work of Coulomb on slope stability. A pile of grains actually exhibits a great variety of behaviours with unique properties. Strong enough to support the weight of a building, grains can also easily flow like water in an hourglass or be transported by wind to sculpt dunes and deserts. For a long time, the study of granular materials has remained the preserve of engineers and geologists. Therefore, important concepts arose from the need to build structures on solid ground, store grains in a silo or predict the history of a sediment. More recently, the study of granular media has entered the field of physics, at the crossroads of statistical physics, mechanics and soft-matter physics. The combination of results from laboratory experiments on model materials, discrete numerical simulations and theoretical approaches from other fields has enriched and renewed our understanding of granular materials.

This book has been written in this context. Our goal is to provide an introduction to the physics of granular media that takes into account recent advances in this field, while describing the basic concepts and tools useful in many industrial and geophysical applications. This book is intended primarily for students, researchers and engineers willing to become familiar with the fundamental properties of granular matter.

The previous chapters show that a granular medium can behave as a solid. In the opposite limit, when grains are strongly shaken in a box, particles are agitated and interact mainly by binary collisions. The medium is then more similar to a gas. This chapter is devoted to this ‘gaseous’ regime of granular matter. The analogy between agitated grains and molecules in a gas was the basis for the development of kinetic theories of granular media that provide constitutive equations for rapid and diluted granular flows. In this chapter, we first introduce the notion of granular temperature and briefly discuss the analogies and differences between a granular gas and a classical molecular gas (Section 5.1). We then present a first approach to the kinetic theory, which gives insight into the physical origin of the transport coefficients (Section 5.2). A more formal presentation of the kinetic theory that is based on the Boltzmann equation for inelastic gases is given in the next section (Section 5.3). We then apply the hydrodynamic equations of the kinetic theory to various situations that highlight the role of inelastic collisions in the behaviour of granular gases (Section 5.4). Finally, some limits of the kinetic theory are discussed, in particular concerning dense media (Section 5.5).

Analogies and differences with a molecular gas

Figure 5.1 gives two examples of granular materials in a ‘gaseous’ state. The first one is obtained by shaking vertically a box containing beads. The second example shows steel beads flowing down a steeply inclined plane under the action of gravity. In both cases, the medium looks like a gas. Particles are strongly agitated and move independently, except when collisions occur.

Most granular flows encountered in nature and industry lie between the quasistatic and gaseous regimes seen in the previous chapters. In this intermediate ‘liquid’ regime, particles remain closely packed and interact both by collision and through long-lived contacts. Understanding and modelling the flow of dense granular media is challenging and many questions remain to be answered, despite important advances having been made during the last decade. In this chapter, we first present the basic features of dense granular flows (Section 6.1), before focusing on the rheology of this peculiar liquid (Section 6.2). A phenomenological constitutive law that is based on dimensional analysis is presented, in which the medium is described as a viscoplastic fluid with a frictional behaviour. The success and limitations of this approach are then discussed, in particular close to the solid–liquid transition where complex collective behaviours are observed. The second part of the chapter presents a hydrodynamic description of dense flows that is valid for a shallow layer flowing under gravity (the Saint-Venant equations) (Section 6.3). This depth-averaged approach enables one to gather the complex rheology into a single basal friction term and is commonly used in geophysics to describe rock avalanches and landslides. We close the chapter with a presentation of the phenomenon of size segregation, which occurs when the medium is composed of particles of different sizes. The consequences of segregation for polydisperse granular flows in various configurations are presented (Section 6.4).

In this chapter, we study erosion and sediment transport from the point of view of the physics of granular media. Situations involving erosion, transport and deposition of particles subjected to fluid flow cover a wide range of applications, from the transport of grains in a pipe to the evolution of landscape on geological scales. We focus in this chapter on the study of erosion and transport of natural sediments under the influence of a water flow (streaming, fluvial erosion, tides, waves and glaciers) or of the wind (dunes, sand invasion, desertification). Besides, we wish to describe sediment transport in the perspective of understanding the geological phenomena that will be discussed in the next chapter. The goal is to propose a description of these phenomena, to model them through basic equations and to explain the dynamical mechanisms at the scale of grains. To do this, we will use the concepts introduced throughout this book.

We begin by briefly outlining the characteristics of the different modes of transport and the most important concepts which allow one to characterize erosion and sediment transport (Section 8.1). We then discuss the nature of the threshold above which a flow may entrain grains into motion (Section 8.2), before presenting the formalism used to describe erosion and transport starting from conservation laws (Section 8.3). Once we have introduced the concepts of saturated transport and saturation transient, we apply them to the different modes of transport: bed load (Section 8.4), aeolian transport (saltation and reptation) (Section 8.5) and turbulent suspension (Section 8.6).

In the previous chapter, we discussed the statics and the elasticity of granular media, when deformations are small and reversible. In this chapter, we address the plasticity of granular media, i.e. irreversible deformations occurring beyond the elastic regime. The two issues associated with plasticity are the following: what is the maximum stress level a granular medium can sustain before being irreversibly deformed and how does the deformation take place beyond the threshold? These questions are covered by soil mechanics, which aims to predict and understand soil stability in nature or during construction in civil engineering. The approaches are mainly based on macroscopic and phenomenological models derived from continuum mechanics. More recently, physicists have been interested in the plasticity of disordered materials, focusing on the microscopic features and trying to understand how rearrangements occur at the grain scale. The link with the continuum models proposed in soil mechanics is still a challenge. In this chapter we will focus on simple macroscopic continuum models, and will only briefly discuss the microscopic properties in a box. The first section (Section 4.1) is dedicated to the phenomenology of plasticity. Several configurations that are used for studying the deformation of a granular medium are described. Section 4.2 is dedicated to the plane shear configuration, for which all the properties of the plasticity of granular media can be introduced using scalar quantities. Tensors, which are necessary to model plasticity, are introduced in Section 4.3. The Mohr–Coulomb model is described and Mohr’s circle used to represent the stress tensor is introduced. In Sections 4.4 and 4.5, we discuss briefly more complex models and unresolved questions. Finally, the plasticity of cohesive materials is presented in Section 4.6.

Definition and examples of granular media

From sand to cereals, from rock avalanches to interplanetary aggregates like Saturn's rings and the asteroid belt (Fig. 1.1), granular media form an extremely vast family, composed of grains with very different shapes and materials, which can span several orders of magnitude in size. However, beyond this great diversity, all these particulate media share fundamental features. They are disordered at the grain level but behave like a solid or a fluid at the macroscopic level, exhibiting phenomena such as arching, avalanches and segregation.

In this book, we shall broadly define a granular medium as a collection of rigid1 macroscopic particles, whose particle size is typically larger than 100 μm (Brown & Richards, 1970; Nedderman, 1992; Guyon & Troadec, 1994; Duran, 1997; Rao & Nott, 2008). As we shall see in Chapter 2, this limitation in size corresponds to a limitation in the type of interaction between the particles (Fig. 1.2). In this book, we will focus on non-Brownian particles that interact mainly by friction and collision. For smaller particles, of diameter between 1 μm and 100 μm, other interactions such as van der Waals forces, humidity and air drag start to play an important role as well. This is the domain of powders. Finally, for even smaller particles, those of diameter below 1 μm, thermal agitation is no longer negligible. The world of colloids then begins (Russel et al., 1989).

This chapter illustrates and discusses some of the most widely used mesoscale models for describing particulate processes. The reader should keep in mind that the chapter is not a comprehensive discussion of all the possible mesoscale models, but is simply a collection of some example models, used in this context to highlight the major issues typically encountered in the simulation of multiphase systems. Although some of the models illustrated in the next sections have general validity, they typically assume slightly different forms when applied to the simulation of solid–liquid systems (e.g. crystallization and precipitation processes), solid–gas systems (e.g. fluidized suspensions, aerosol reactors), liquid–liquid systems (e.g. emulsions), gas–liquid systems (e.g. bubble columns and gas–liquid stirred tanks), and liquid–gas systems (e.g. evaporating and non-evaporating sprays). In what follows we will often refer to the elements of the disperse phase as “particles” to indicate both solid particles (such as crystals, solid amorphous particles, and solid aggregates) and fluid particles (such as droplets and bubbles). The remainder of the chapter is organized as follows. After providing an overview of the philosophy behind the development of mesoscale models in Section 5.1, specific examples of phase-and real-space advection processes (and the consequent diffusion processes) are discussed in Sections 5.2–5.5. Subsequently, phase-space point (discontinuous) processes are presented in Sections 5.6–5.8. For each of these processes, the corresponding functional form of the mesoscale model appearing in the final GPBE is derived and discussed.

Disperse multiphase flows

The majority of the equipment used in the chemical process industry employs multiphase flow. Bubble columns, fluidized beds, flame reactors, and equipment for liquid–liquid extraction, for solid drying, and size enlargement or reduction are common examples. In order to efficiently design, optimize, and scale up industrial systems, computational tools for simulating multiphase flows are very important. Polydisperse multiphase flows are also common in other areas, such as fuel sprays in auto and aircraft engines, brown-out conditions in aerospace vehicles and particulate flows occurring in the environment. Although at first glance the multifarious industrial and environmental multiphase flows appear to be very different from each other, they have a very important common element: it is possible to identify a continuous phase and a disperse phase (usually with a distribution of characteristic “particle sizes”).

Historically the development of the theoretical framework and of computational models for disperse multiphase flows has focused on two different aspects: (i) the evolution of the disperse phase (e.g. breakage and coalescence of bubbles or droplets, particle–particle collisions, etc.) and (ii) multiphase fluid dynamics. The first class of models is mainly concerned with the description of the disperse phase, and is based on the solution of the spatially homogeneous population balance equation (PBE). A PBE is a continuity statement written in terms of a number density function (NDF), which will be described in detail in Chapter 2.

In this chapter, the governing equations needed to describe polydisperse multiphase flows are presented without a rigorous derivation from the microscale model. (See Chapter 4 for a complete derivation.) For clarity, the discussion of the governing equations in this chapter will be limited to particulate systems (e.g. crystallizers, fluidized beds, and aerosol processes). However, the reader familiar with disperse multiphase flow modeling will recognize that our comments hold in a much more general context. Indeed, the extension of the modeling concepts developed in this chapter to many other multiphase systems is straightforward, and will be discussed in later chapters.

The primary purpose of this chapter is to introduce the key concepts and notation needed to develop models for polydisperse multiphase flows. We thus begin with a general discussion of the number-density function (NDF) in its various forms, followed by example transport equations for the NDF with known (PBE) and computed (GPBE) particle velocity. These transport equations are written in terms of “averaged” quantities whose precise definitions will be presented in Chapter 4. We then consider the moment-transport equations that are derived from the NDF transport equation by integration over phase space. Finally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations.

Number-density functions (NDF)

The disperse phase is constituted by discrete elements. One of the main assumptions of our analysis is that the characteristic length scales of the elements are smaller than the characteristic length scale of the variation of properties of interest (i.e. chemical species concentration, temperature, continuous phase velocities).

In this chapter we discuss issues specific to applying moment methods with spatially inhomogeneous systems. In particular, we focus on the spatial transport of moment sets by advection, diffusion, and free transport. In Chapter 7, issues related to transport in phase space are thoroughly treated and, here, we will discuss such terms only inasmuch as they affect spatial transport. In Section 8.1, the principal modeling issues that arise with spatially inhomogeneous systems are briefly reviewed. In the sections that follow, we discuss separately moment methods for (i) the inhomogeneous population-balance equation (PBE) (i.e. where the internal coordinates do not include or affect the velocity) in Section 8.3, (ii) the inhomogeneous kinetic equation (KE) (i.e. where the only internal coordinate is velocity) in Section 8.4, and finally (iii) the full inhomogeneous generalized population-balance equation (GPBE) in Section 8.5. Concrete examples, and the corresponding discretized formulas, are provided for each type of system in order for the reader to understand fully the issues that arise when simulating inhomogeneous systems. An important theme running through the entire chapter is the issue of realizable moment sets, and how realizability is affected by spatial transport. Thus, in order to have explicit examples of the numerical issues, we introduce kinetics-based finite-volume methods (KBFVM) for moment sets in Section 8.2. Nevertheless, the reader should keep in mind that these numerical issues are generic to moment transport and will arise with all spatial-discretization methods.

In this chapter we discuss the basic theory of Gaussian quadrature, which is at the heart of quadrature-based moment methods (QBMM). Proofs for most of the results are not included and, for readers requiring more extensive analytical treatments, references to the literature are made. In addition to a summary of the relevant theory, different algorithms to calculate the abscissas (or nodes) and the weights of the quadrature approximation from a known set of moments are presented, and their advantages and disadvantages are critically discussed. It is important to remind readers that most of the theory for quadrature formulas was developed for mono-dimensional integrals, namely integrals of a single independent variable. Therefore the discussion below starts from univariate distributions, for which the Gaussian quadrature theory applies exactly, and subsequently moves to bivariate and multivariate distributions. Although in the latter cases the quadrature is no longer strictly Gaussian, most of its interesting properties are still valid. In the univariate case, the weights and abscissas are used in the quadrature method of moments (QMOM) to solve moment-transport equations. Thus, we will refer to moment-inversion algorithms that use a full set of moments as QMOM, while other methods that use a reduced set will be referred to differently.