Number-density functions NDF

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. 

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). If this hypothesis holds, the particulate system can be described by a continuum or mean-field theory. Each element of the disperse phase is generally identified by a number of properties known as coordinates. Two elements are identical if they have identical values for their coordinates, otherwise elements are indistinguishable. Usually coordinates are classified as internal and external. External coordinates are spatial coordinates in fact, the position of the elements in physical space is not an internal property of the elements. Internal coordinates refer to more intimate properties of the elements such as their momenta (or velocities), their enthalpy 

Let us consider a population of disperse entities such as solid particles or liquid droplets inside an infinitesimal control volume located at the physical point x = (xi,X2,X3) and of measure dx = dxi dX2 dxs. Let . m) be the internal-coordinate vector, 

It is straightforward that the quantity ( )d represents the number density of disperse entities contained in the phase-space volume d centered at per unit of physical volume. If we integrate the NDF over all possible values of the internal-coordinate vector we obtain the total number concentration N(t, x)  

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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 remaining chapters in this book are organized as follows. Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. There, the mathematical definition of a number-density function (NDF) formulated in terms of different choices for the internal coordinates is described, followed by an introduction to population-balance equations (PBE) in their various forms. Chapter 2 concludes with a short discussion on the differences between the moment-transport equations associated with the PBE and those arising due to ensemble averaging in turbulence theory. This difference is very important, and the reader should keep in mind that at the mesoscale level the microscale turbulence appears in the form of correlations for fluid drag, mass transfer, etc., and thus the mesoscale models can have non-turbulent solutions even when the microscale flow is turbulent (i.e. turbulent wakes behind individual particles). Thus, when dealing with turbulence models for mesoscale flows, a separate ensemble-averaging procedure must be applied to the moment-transport equations of the PBE (or to the PBE itself). In this book, we are primarily... 

Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. In this chapter the many possible number-density functions (NDF), formulated with different choices for the internal coordinates, are presented, followed by an introduction to the PBE in their various forms. The chapter concludes with a short discussion on the differences between the moment-transport equations associated with the PBE, and those arising due to ensemble averaging in turbulence theory. 

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