Particle number density function

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)... 

Example 2.4.2 Consider a particle number density function fi(rp) having the form n(rp) = n exp(-arp). 

If there is a particle size distribution indicated by n(rp), the particle number density function (equation (2.4.2a)), such that Up,(rp) is the terminal velocity of particles in the size range of tp to tp + Atp, the particle flux Up across a surface area perpendicular to Up,(rp) is... 

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 will now derive an equation of change for the particle number density function n(rp) by developing a particle population balance in a small control volume of dimensions Ax, Ay and Az (Figure 6.2.1) ... 

These source term expressions contain the continuous particle number density function, /(, r, t). However, in order to close the equation the source terms must be expressed entirely in terms of the dependent variable j(r, f). The fixed pivot technique is based on the idea of birth modification. The pivot concentrates the particles in the interval at a single representative point. The number density distribution function/i(, r, t) can be approximated as ... 

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

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 Einstein equations Eqs. (6)-(9) contain two extra variables compared to the more familiar 4 dimensional case, namely P5 and 4>. However, P (n) is a known function of the particle number density and specified by the actual interaction in the matter. Thus (r ) is the only new degree of freedom determined by the extra equation. 

The gas molecules fly about and among each other, at every possible velocity, and bombard both the vessel walls and collide (elastically) with each other. This motion of the gas molecules is described numerically with the assistance of the kinetic theory of gases. A molecule s average number of collisions over a given period of time, the so-called collision index z, and the mean path distance which each gas molecuie covers between two collisions with other molecules, the so-called mean free path length X, are described as shown below as a function of the mean molecule velocity c the molecule diameter 2r and the particle number density molecules n - as a very good approximation ... 

The beam intensity is given by / = ni v, the incident number of molecules 1 per unit area per unit time. The nt, n2, are particle number densities. The total cross section Q12 = Qn(vn) has units of area it is a function of the relative speed, vn, which in this case is given by v. 

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. 

Fig. 14 The plasmon angle shift plotted as a function of particle number density for 30 (open diamond), 35 (open rectangle), 45 (open triangle), and 59 nm (open circle) diameter colloidal Au. Reproduced with permission from [33], Copyright 1999 American Chemical Society...

The half-life is given in the following as a function of initial particle number density, Nq in water at 25°C [27] that result from ty2 = 3tj/... 

The results from a series of flames at different temperatures are analyzed by Equation 5 in Figure 2 where the function kT In (Ue/AT ) is plotted against rie. The intercept (3.6 eV) agrees well with the literature value for uranium dioxide, and the slope gives a value of 3.3 X 10 m for the product n a. The sprayer had been calibrated with cesium, and it was known thus that the total number density of uranium atoms in the flame was 1.6 X 10 m" which corresponds to a relative volume (47rnpfl /3) of 6.5 X 10" of UO2 (density 10.97 g cm ). This leads to separate values for a (the mean particle radius = 6.8 X 10 m) and (particle number density = 4.8 X 10 m ) and to the number of charges per particle (rie/rip = 7.2). 

By taking the mass moments of the distribution function we define the following important multiphase flow parameters. The particle number density ... 

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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