Number density function average

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

Z (x, r,t) single number distribution function denoting the number of particles per unit volume of the particle phase space at time t (general) r,t) average single particle number density function using particle diameter as inner coordinate (i— —3)... 

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. 

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. 

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

The task at hand is one of starting from Equation (3.2.14) and deriving Equation (3.2.8) by defining the volume-averaged number density function... 

In view of the previously given definitions of the average number density, mass average velocity, and kinetic energy, the function

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The population balance deals with number of pores rather than the pore volume, in a given size range. However, if the number density function is assume to be equal to the pore volume size distribution divided by the average pore volume in rach pore-size range and both breath and length are independent of time and uncorrelated with the width, then a parameter proportional to the number density hmction can be defined as follows ... 

In a first modeling approach, a macroscopic population balance is formulated directly on the averaging scales in terms of number density functions [80, 102], A corresponding set of macroscopic source term closures are presented as well. Reviews of numerous fluid particle breakage and coalescence kernels on macroseopie scales can be found elsewhere [60,73,74, 122], This modeling framework resembles the mixture model concept. 

Average single particle number density function using particle diameter as inner coordinate... 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

The constants K depend upon the volume of the solvent molecule (assumed to be spherica in slrape) and the number density of the solvent. ai2 is the average of the diameters of solvent molecule and a spherical solute molecule. This equation may be applied to solute of a more general shape by calculating the contribution of each atom and then scaling thi by the fraction of fhat atom s surface that is actually exposed to the solvent. The dispersioi contribution to the solvation free energy can be modelled as a continuous distributioi function that is integrated over the cavity surface [Floris and Tomasi 1989]. 

The description of the atomic distribution in noncrystalline materials employs a distribution function, (r), which corresponds to the probability of finding another atom at a distance r from the origin atom taken as the point r = 0. In a system having an average number density p = N/V, the probability of finding another atom at a distance r from an origin atom corresponds to Pq ( ). Whereas the information given by (r), which is called the pair distribution function, is only one-dimensional, it is quantitative information on the noncrystalline systems and as such is one of the most important pieces of information in the study of noncrystalline materials. The interatomic distances cannot be smaller than the atomic core diameters, so = 0. 

Fig. 6.7. Evolution of the sample averaged (R< ) as a function of MC time. The initial value of e(N) = C = 1.0 was changed to the values indicated after 600 MC steps. The indicated melt value corresponds to a comparable system with explicit chains with repulsive Lennard-Jones interactions and a number density of 0.85 cr-3 (from [45])...

In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function /s(r, v), which can be defined as the number density of the solid particles in the 6D coordinate and velocity space. That is, /s(r, v, t) dv dr is the average number of particles to be found in a 6D volume dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by... 

Nuclides, reaction with monomers, 14 248 NuDat database, 21 314 Nukiyama-Tanasawa function, 23 185 Null-background techniques, in infrared spectroscopy, 23 139-140 Number-average molecular weight, 20 101 of polymers, 11 195, 196 Number density, of droplets, 23 187 Number of gas-phase transfer units (Nq), packed column absorbers, 1 51 Number of overall gas-phase transfer units (Nog), packed column absorbers, 1 52 Number of transfer units (Nt, NTU), 10 761... 

The value of the peaks and troughs in the pair distribution function represent the fluctuation in number density. The peaks represent regions where the concentrations are in excess of the average value while the troughs represent a deficit. As the volume fraction is increased, the peaks and troughs grow, reflecting the increase in order with concentration. We... 

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