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.

In this chapter we consider models for collisions between smooth (i.e. frictionless) spherical particles with identical (monodisperse) or different (polydisperse) densities and diameters. For simplicity, we consider only collisions during which the particle mass and diameter are conserved, and exclude other processes that might change these properties (e.g. surface condensation or aggregation (Fox et al., 2008)). Likewise, assuming smooth spherical particles means that the particle angular momentum does not change during a collision, and hence only the particle velocity need be accounted for in the kinetic equation. We limit the discussion to hard-sphere collisions, which implies that the particle velocities after a collision can be written as explicit functions of the particle velocities before the collision, but also consider inelastic collisions with a constant coe.cient of restitution and finite-size particle effects. More details on hard-sphere collisions can be found in the books by Cercignani (Cercignani, 1975, 1988, 2000; Cercignani et al., 1994).We will also briefly discuss simpler collision models that are often used to approximate the hard-sphere collision model in the dilute limit. These include the Maxwell model (Maxwell, 1879) and two linearized collision models (i.e. BGK (Bhatnagar et al., 1954) and ES-BGK (Holway, 1966)) for monodisperse particles, and an extension of the ES-BGK model to polydisperse particles. A discussion of kinetic models for collisions that are not of hard-sphere type can be found in Struchtrup (2005).

In this chapter we consider the solution of the generalized population-balance equation (GPBE) with a generic set of internal coordinates ξ = (ξ1, ξ2, …, ξM) under the hypothesis of spatial homogeneity. Under this hypothesis, all spatial gradients are null and the GPBE depends only on time t and on ξ. As discussed earlier, we refer here to the GPBE as a general equation describing the evolution of a number-density function (NDF) in space (neglected in this chapter), in time, and in the phase space generated by the internal coordinates. The extension to inhomogeneous systems is discussed in Chapter 8. It is also worth mentioning that the GPBE is given different names in different fields. It is called the population-balance equation (PBE) in crystallization, precipitation, and, in general, in the literature on particulate processes. In the simulation of aerosols, it is called the general aerosol dynamic equation (Friedlander, 2000), whereas in the simulation of sprays it is generally known as the Williams–Boltzmann equation (Williams, 1985) and, especially in this case, the droplet velocity is included as one of the internal coordinates. When dealing with solid particles, if the particle velocity is included as the only internal coordinate, the GPBE is called the Boltzmann equation. In general, when the particle velocity is the only internal coordinate, the GPBE is also called the kinetic equation (KE). Many of the challenges associated with the KE occur with inhomogeneous systems, which are discussed in Chapter 8.