Number-density functions

We can perform spatially resolved Carr-Purcell-Meiboom-Gill (CPMG) experiments, and then, for each voxel, use magnetization intensities at the echo times to estimate the corresponding number density function, P(t), which represents the amount of fluid associated with the characteristic relaxation time t. The corresponding intrinsic magnetization for the voxel, M0, is calculated by... 

We represent the NMR relaxation distribution by the continuous number density function, P( t), of characteristic relaxation time t. Our measurements correspond to a series of CPMG echoes, represented by... 

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). 

A number density function, fa(b), is defined so that fn(b) d b represents the particle number fraction in a size range from b to b + db. Thus,... 

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... 

Ho et al. (1990) have presented an approach to the description of independent kinetics that makes use of the method of coordinate transformation (Chou and Ho 1988), and which appears to overcome the paradox discussed in the previous paragraph. An alternate way of disposing of the difficulties associated with independent kinetics is intrinsic in the two-label formalism introduced by Aris (1989, 1991b), which has some more than purely formal basis (Prasad et al, 1986). The method of coordinate transformation can (perhaps in general) be reduced to the double-label formalism (Aris and Astarita, 1989a). Finally, the coordinate transformation approach is related to the concept of a number density function s(x), which is discussed in Section IV,B.5. 

The Ho and White (1994) analysis is based on the idea of a constant spacing 8 1/8 then represents the (very large) number of individual components that are in reality present in the mixture over a unit segment of the label axis. In a strict interpretation of the continuous description, 6 = 0 (this is, essentially, De Donder s 1931 approach) but of course reality is a different matter. One can define a number density function i(x) such that the number of individual compounds in any interval [xi,X2 is... 

Figure 2. Inner dynamics number density function qc vs. fluorescence intensity for specific time points post infection, (dots experimental results, solid line simulation results, MOI = 3.0, tg t = 4.5 h)...

The inner dynamics are determined by the cell distribution over the fluorescence changing with time. For comparability the cell concentrations have to be converted into number density functions, which are obtained by normalization with the overall cell concentration at the specific time point and division by the specific class width in logarithmic scale. All cells (uninfected, infected and dead) contribute to the distribution as they all show fluorescence. Figure 2 shows the comparison between simulation results and the flow cytometric data reported by Schulze-Horsel et al. [3] for MOI = 3.0. The simulation peak lags behind in the beginning and catches up for later time points, but the overall tendency of increasing mean values can be reproduced quite well. However, the present model has a drawback for an unknown biological reason the experimental distributions fall back to smaller fluorescence intensities at later time points (data not shown). So far, this effect cannot be simulated with the presented model formulation adequately. 

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 ... 

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)... 

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... 

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. 

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