Number density function definition

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... 

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. 

We are interested here in the actual number of particles dA/(rp) in the size range of Vp to Vp + dVp per unit volume of the fluid. This quantity is related to a particle number density function n(rp) (the population density junction, see Figure 2.4.3(b) and definition (2.4.2a)) by... 

We begin with the definition of the grand potential f] as a functional of the number density of a fluid [49], p(r)... 

Besides the already mentioned Fukui function, there are a couple of other commonly used concepts which can be connected with Density Functional Theory (Chapter 6). The electronic chemical potential p is given as the first derivative of the energy with respect to the number of electrons, which in a finite difference version is given as half the sum of the ionization potential and the electron affinity. Except for a difference in sign, this is exactly the Mulliken definition of electronegativity. ... 

A demand density is by definition a continuous density function defined over the set of all nonnegative numbers, thus demands are always positive or zero. 

More insight into these processes is obtained by studying the particle number dependent properties of density functionals. This of course requires a suitable definition of these density functionals for fractional particle number. The most natural one is to consider an ensemble of states with different particle number (such an ensemble is for instance obtained by taking a zero temperature limit of temperature dependent density functional theory [84]). We consider a system of N + co electrons where N is an integer and 0 < m < 1. For the corresponding electron density we then have... 

Enlarging the domain of definition of Eqn (1) to all positive n, one could assume the minimum of the density functional E lp], i.e. the ground-state energy Eo n) for a given external potential u(r), to be a continuous and even a differentiable function of the number of electrons n. From the Lagrange multiplier theory, it would further follow that... 

The HSAB principle can be considered as a condensed statement of a very large amount of experimental information, but cannot be labelled a law, since a quantitative definition of the intuitive concepts of chemical hardness (T ) and softness (S) was lacking. This problem was solved when the hardness found an exact, and also an operational, definition in the framework of the Density Functional Theory (DFT) by Parr and co-workers [2], In this context, the hardness is defined as the second order derivative of energy with respect to the number of electrons and has the meaning of resistance to change in the number of electrons. The softness is the inverse of the hardness [3]. Moreover, these quantities are defined in their local version [4, 5] as response functions [6] and have found a wide application in the chemical reactivity theory [7],... 

In astrophysics, particle sizes in protoplanetary disks are usually described in the context of a distribution, dn = f(a)da, where a is the particle radius and (in is the number density of particles with radii between a and a + da. Note that in cosmochemistry, grain size usually refers to the largest diameter of a monomer. The description is often associated with two assumptions. First, it is often assumed that the distribution function, /(a), is close to being a power law in the ISM, except at very large or very small sizes (Weingartner Draine 2001), and power laws are therefore often used in protoplanetary disk studies. Second, the definition of a particle radius assumes that the particles are spherical - a convenience for converting the size distribution to an opacity law. 

The reversibility of molecular behavior gives rise to a kind of symmetry in which the transport processes are coupled to each other. Although a thermodynamic system as a whole may not be in equilibrium, the local states may be in local thermodynamic equilibrium all intensive thermodynamic variables become functions of position and time. The definition of energy and entropy in nonequilibrium systems can be expressed in terms of energy and entropy densities u(T,Nk) and s(T,Nk), which are the functions of the temperature field T(x) and the mole number density Y(x) these densities can be measured. The total energy and entropy of the system is obtained by the following integrations... 

Nevertheless, it is a great advantage in science to have quantitative definitions so that one can measure what one is speaking about, and express it in numbers. Fortunately this is what has happened to chemical hardness. The means by which this has come about lies in density functional theory. This will be the topic of the next chapter. 

Here, pa is the average number density of molecules of species a, i.e., pa = Na)/V, with V the volume of the system. We also recall the definition of the spatial pair correlation function... 

The second approach to defining the absolute hardness rj has a companion parameter taken from density functional theory, called the electronic chemical potential ji. The value of pis essentially the same as the negative of %, as defined in Equation 3.2, and the value of rj is essentially the same as in the more approximate definition in Equation 3.3 but both are defined differently. If the total electronic energy of an atom or molecule is plotted as a function of the total number of electrons N, the graph takes the form of Fig. 3.2 in which the only experimental points are at integral values of but between them it is convenient to... 

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