This chapter also elucidates the issue of particle-particle contact in a multiphase system. The focus is, however, different than in the previous chapters. Those chapters considered the modelling of collision processes where the objective was to explore the collision dynamics (e.g., deformation and velocity). In this chapter, however, we study heat conduction between particles. This indicates that the particles have different temperatures as they collide. First, we examine a mathematical model for heat conduction if the collision is elastic. Later, it is shown how the same strategy can be used if there is a permanent (plastic) deformation during a collision. Finally, consider dissipative forces.

This chapter introduces the reader to the modelling of particle-particle collisions. We assume that two spherical particles collide along the normal axis to plane of contact – that is, we only examine a head-on impact. In the beginning, attention is paid to the contact mechanics. The objective is therefore to prepare the reader for fundamental analysis. First, we investigate a simple case of a single force acting on a surface. This problem gradually extends to a similar contact between two spherical bodies (Hertz theory). Next, these bodies are allowed to move towards each other, and we observe their deformation – i.e., a head-on collision. The collision is also elastic, so there is no mechanical energy loss upon impact. Later, this issue is expanded upon by introducing dissipative forces during the contact in addition to the elastic forces discussed above. These dissipative forces are of different types: both linear and non-linear. Finally, another topic is introduced, which is plastic deformation. Here, the colliding bodies are allowed to deform permanently.

Interactions between particles in multiphase flow may also involve adhesion – i.e., an attraction between the particles. This issue is the main topic of this chapter. The first sections of the chapter, however, focus on a primary case: forces acting between two solid surfaces close to each other. A typical example is an interaction between two spherical bodies, which mimic two particles in a multiphase flow. This situation is later extended to a more complex case: the bodies change their shape due to these adhesive interactions. For this, two theories were developed in the literature (JKR and DMT), and they are fully described in the chapter. Later, it is shown how these theories can be adopted to investigate particle-particle collisions in a multiphase flow. In other words, this topic constitutes an extension of the previous chapter, where the focus was on purely “mechanical” interactions without considering any adhesive forces. Finally, the last section of the chapter describes rough surfaces. There is a brief description of how this real-life issue influences the adhesion between two bodies in contact.

One of the parameters that describe particle-particle collision is a coefficient of restitution. This can be simply defined as a ratio of the post-collisional and pre-collisional relative velocity. This chapter is devoted to this topic. As it is straightforward to measure this parameter experimentally, different practical techniques have been used by the researchers, and they are depicted here. Factors such as material properties and pre-collisional conditions are discussed, and it is shown how they influence the value of the coefficient of restitution. It is worth noting that the coefficient of restitution can also be found theoretically by exploiting the relationships previously discussed in the book, especially in Chapter 3. This is described in detail in this chapter. The chapter therefore returns to the previously considered mathematical models. Finally, the chapter concludes with two additional sections focusing on special cases: collisions of granules and nanoparticles., respectively. These particular types of particles have unique features that greatly influence the collision process and restitution coefficient.

This chapter summarises the topics previously discussed in Chapters 2-8. The objective is to illustrate how to create a computer code that simulates a flow of solid particles in a fluid. First, a model is shown that accounts for the motion of particles due to various particle-fluid forces introduced in Chapter 2. Later, it is emphasised that the particles may collide, and this can be described using the techniques mentioned in Chapters 3-8. Finally, a new problem is introduced (not considered in the previous chapters) – collision detection. This issue is crucial for deciding which particles flowing in a system could potentially collide during a time step. The chapter also unveils an algorithm in which the collision detection mode is implemented.

This chapter explains the hard-sphere model of particle-particle collision. This model exploits impulse equations that directly relate the pre-collisional and post-collisional velocities of the particles. Thus, this model does not track the deformation history that was done in the prior chapters. As a result, we obtain ready analytical solutions so that the computational time is short. First, the chapter shows a standard hard-sphere model for a “mechanical” collision of two bodies. Different strategies are presented, such as the so-called two- and three-parameter hard-sphere model. Later, an extension of these models is shown that also accounts for adhesive interactions. Although, due to its simplicity, the hard-sphere model may not account for various physical phenomena between colliding particles, it may still be used in many applications. In this chapter, the reader is again provided with a computer code.

The book’s final chapter pays attention to various issues that can be encountered when investigating multiphase flows. This chapter can be read independently, although on a few occasions it refers to some selected problems from the prior topics. First, this chapter treats a multiphase flow as a system of spherical particles with some given concentration and with some average distance between the particles. Later, the chapter looks into the particle reaction as immersed in a fluid (discussion so-called response times), and it is shown how the presence of the particles influences the fluid flow by discussing the concept of phase coupling and suspension viscosity. Next, we consider the issue of the dispersion of particles as they are subject to turbulent flows, and how the particles may gather in some selected flow zones (preferential concentration). The fact that the particles may be of different sizes is later analysed by investigating the particle size distribution. The final sections of the chapter are dedicated to collision frequency and a particular case of a flow through a particle bed.

The first chapter describes the main structure of the book, but also reveals an algorithm that the book is built on. The ultimate goal is the creation of a strategy that can be used for modelling fluid flows laden with particles. Therefore, this chapter depicts the main steps: first, modelling the flow with a single particle, then introducing two particles that may interact, and finally, modelling of the whole set of particles. The details are provided in the subsequent chapters.

Whilst the previous chapter focused solely on head-on collisions, this chapter also considers tangential contact. The objective is to extend the previous analysis to include oblique collisions. The strategy also resembles the prior chapter. First, we pay attention to contact mechanics by analysing tangential forces acting on a surface. The analysis is later enhanced to a contact of two spherical bodies. This knowledge is exploited in the subsequent sections of the chapter, where we consider a full oblique collision of two bodies. The collision process is described in detail by following a study case, which is solved using a computer code provided for readers.

This chapter is dedicated to the elementary problem, which concerns interactions between a single particle and the surrounding fluid. First, we explore the drag force, which is the most common interaction. It is shown how this force is derived and applied in practice. This topic is further expanded upon by introducing Basset and added mass force – both are crucial for unsteady cases such as accelerating particles. Next, lift forces (Magnus and Saffman) are shown that may result in the particle’s motion in the lateral direction. To some extent, this is associated with the next issue explained in the chapter: the torque acting on a particle. The following sections pay attention to other interactions: Brownian motion, rarefied gases and the thermophoretic force. These interactions play a role for tiny particles, perhaps of nano-size. Ultimately, we deliberate heat effects when the particle and fluid have different temperatures. Thus, this last section scrutinise convective and radiative heat transfer.