The Number Density Function

We postulate that there exists an average number density function defined on the particle state space, 

If we desire the spatial number density of a selected class of particles belonging to some subset of the space then the integration above must be over the subset A.  

Other densities such as volume or mass density may also be defined for the particle population. Thus, if v(x) is the volume of the particle of internal state X, then the volume density may be defined as v(x)/i(x, r, t). The volume 

The denominator above represents the total volume fraction of all particles. Similarly, mass fractions can also be readily defined. For the case of scalar internal state using only particle size (volume) denoting the number density by /i(v, r, t), the volume fraction density of particles of volume v becomes 

A cumulative volume fraction that represents the fraction of particles with volume at most v, denoted F(v, r, t), is given by 

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A partial differential equation is then developed for the number density of particles in the phase space (analogous to the classical Liouville equation that expresses the conservation of probability in the phase space of a mechanical system) (32>. In other words, if the particle states (i.e. points in the particle phase space) are regarded at any moment as a continuum filling a suitable portion of the phase space, flowing with a velocity field specified by the function u , then one may ask for the density of this fluid streaming through the phase space, i.e. the number density function n(z,t) of particles in the phase space defined as the number of particles in the system at time t with phase coordinates in the range z (dz/2). 

The number density function is usually obtained by using microscopy or other optical means such as Fraunhofer diffraction. The mass density function can be acquired by use of sieving or other methods which can easily weigh the sample of particles within a given size range. 

Given the number density function of Eq. (1.24), the corresponding mass density function becomes... 

The arithmetic mean diameter d is the averaged diameter based on the number density function of the sample d is defined by... 

The distribution function W (t, p) is a function normalised to unity, so that the number density function n(t, q) can be calculated as... 

This modeling approach thus considers the balance principle to the number density function for an arbitrary combined material sub-volume... 

A differential population balance for the number density function is thus achieved ... 

As in all mathematical descriptions of transport phenomena, the theory of polydisperse multiphase flows introduces a set of dimensionless numbers that are pertinent in describing the behavior of the flow. Depending on the complexity of the flow (e.g. variations in physical properties due to chemical reactions, collisions, etc.), the set of dimensionless numbers can be quite large. (Details on the physical models for momentum exchange are given in Chapter 5.) As will be described in detail in Chapter 4, a kinetic equation can be derived for the number-density function (NDF) of the velocity of the disperse phase n t, X, v). Also in this example, for clarity, we will assume that the problem has only one particle velocity component v and is one-dimensional in physical space with coordinate x at time t. Furthermore, we will assume that the NDF has been normalized (by multiplying it by the volume of a particle) such that the first three velocity moments are... 

In the formulation of a mesoscale model, the number-density function (NDF) plays a key role. For this reason, we discuss the properties of the NDF in some detail in Chapter 2. In words, the NDF is the number of particles per unit volume with a given set of values for the mesoscale variables. Since at any time instant a microscale particle will have a unique set of microscale variables, the NDF is also referred to as the one-particle NDF. In general, the one-particle NDF is nonzero only for realizable values of the mesoscale variables. In other words, the realizable mesoscale values are the ones observed in the ensemble of all particles appearing in the microscale simulation. In contrast, sets of mesoscale values that are never observed in the microscale simulations are non-realizable. Realizability constraints may occur for a variety of reasons, e.g. due to conservation of mass, momentum, energy, etc., and are intrinsic properties of the microscale model. It is also important to note that the mesoscale values are usually strongly correlated. By this we mean that the NDF for any two mesoscale variables cannot be reconstructed from knowledge of the separate NDFs for each variable. Thus, by construction, the one-particle NDF contains all of the underlying correlations between the mesoscale variables for only one particle. 

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 NDE transport equation by integration over phase space. Einally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations. 

In this section we will briefly review the collision model for binary hard-sphere collisions using the notation in Fox Vedula (2010). The change in the number-density function due to elastic hard-sphere collisions (Boltzmann, 1872 Cercignani, 1988 Chapman Cowling, 1961 Enksog, 1921) obeys an (unclosed) integral expression of the form ... 

To follow the dynamic evolution of PSD in a particulate process, a population balance approach is commonly employed. The distribution of the droplets/particles is considered to be continuous in the volume domain and is usually described by a number density function, (v, t). Thus, n(v, f)dv represents the number of particles per unit volume in the differential volume size range (v, v + dv). For a dynamic particulate system, undergoing simultaneous particle breakage and coalescence, the rate of change of the number density function with respect to time and volume is given by the following non-linear integro-differential population balance equation (PBE) [36] ... 

The number density function, along with the environmental phase variables, completely determines the evolution of all properties of the dispersed phase system. The population balance framework is thus an indispensable tool for dealing with dispersed phase systems. This book seeks to address the various aspects of the methodology of population balance necessary for its successful application. Thus Chapter 2 develops the mathematical framework leading to the population balance equation. It... 

We now return to the issue of boundary conditions. Basically, this is a question of specifying the component of the particle flux normal to the boundary or (equivalently) the number density at each point on appropriate parts of the boundary. We shall presently see what these appropriate parts are. Note that the population balance equation (2.7.9) features a first-order partial differential operator on the left-hand side. Although the nature of the complete equation is governed by the dependence of the right-hand side on the number density function, the solution to Eq. (2.7.9) may be viewed as evolving along characteristic curves which (are the same... 

The population balance equation in the number density function / (x, t) is given by (2.7.6) with the right-hand side set equal to zero. Thus,... 

FIGURE 2.11.1 Singular behavior of the number density function near the origi for the dissolution process in Section 2.11.1. 

Initially, the cells of total population density are assumed to be all of age zero. The number density function / (t, t) must satisfy the population balance equation... 

In writing the population balance equation for the number density function / (m, x, t), we invoke the general form (2.10.7), remembering that x originates from the spherical coordinate system. 

In this process, if mass is conserved during breakage, then the total mass in the system must remain constant. It is of interest to examine the first moment of the number density function defined by... 

In identifying the steady-state population balance equation for the number density function/ (x, c, t), we appeal to the general form (2.8.3) and drop the time derivative. Also we take note of the fact that drops which appear in the vessel either by entering with the feed or by breakage of larger droplets must necessarily be of age zero so that they are accounted for in the boundary condition at age zero. Thus, the population balance equation becomes... 

It is a functional of the number density function, which requires the specification of the number density over a range of particle states. Such population balance equations are integro-partial differential in nature. Many of the examples from Sections 3.2 and 3.3 yield... 

Not infrequently, practical needs can be fulfilled by calculating the (generally integral) moments of the number density function. The calculation of such moments can occasionally be accomplished by directly taking moments of the population balance equation producing a set of moment equations. 

Thus, not surprisingly, all the moments are analytically accessible for this case. More often, however, the constraints under which moment equations can be obtained directly are violated in examples of applications. Trouble frequently arises due either to the moment equations being unclosed or to the emergence of nonintegral moments. Hulburt and Akiyama (1969) sought to cure this problem by expanding the number density function in terms of a finite number (say M) of Laguerre polynomials, L (x) as... 

Broadly, the self-similar solution identifies invariant domains in the space of the independent variables along which the solution remains the same or contains a part that is the same. Consider, for example, the number density function / (x, t) that may satisfy a population balance equation such as (3.2.8) or (3.3.5). By a self-similar solution of either of these equations we mean one to be of the form... 

We further define the number density function / (t, t) such that... 

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