Population of particles distribution

In this section the population balance modeling approach established by Randolph [95], Randolph and Larson [96], Himmelblau and Bischoff [35], and Ramkrishna [93, 94] is outlined. The population balance model is considered a concept for describing the evolution of populations of countable entities like bubble, drops and particles. In particular, in multiphase reactive flow the dispersed phase is treated as a population of particles distributed not only in physical space (i.e., in the ambient continuous phase) but also in an abstract property space [37, 95]. In the terminology of Hulburt and Katz [37], one refers to the spatial coordinates as external coordinates and the property coordinates as internal coordinates. The joint space of internal and external coordinates is referred to as the particle phase space. In this case the quantity of basic interest is a density function like the average number of particles per unit volume of the particle state space. The population balance may thus be considered an equation for the number density and regarded as a number balance for particles of a particular state. 

Consider again a population of particles distributed according to a scalar state variable x, which we shall take to vary over the entire real, line and let /i(x, t) be the number density. The scalar state x is presumed to vary in... 

Distribution Averages. The most commonly used quantities for describing the average diameter of a particle population are the mean, mode, median, and geometric mean. The mean diameter, d, is statistically calculated and in one form or another represents the size of a particle population. It is usefiil for comparing various populations of particles. 

If the secondary stream contains emulsifier it can function in three ways. When the emulsion feed is started quickly the added emulsifier can serve to lengthen the particle formation period and hence to broaden the particle size distribution. When the emulsion feed is started later and added in such a manner that the emulsifier is promptly adsorbed on existing particles, one can obtain quite narrow size distributions. If the emulsion feed is started later but added rapidly enough to generate free emulsifier in the reaction mixture a second population of particles can be formed, again yielding a broad size distribution. 

Strawbridge, K.B., Ray, E, Hallett, F.R., Tosh, S.M., Dalgleish, D.G. 1995. Measurement of particle size distributions in milk homogenized by a microfluidizer estimation of populations of particles with radii less than 100 nm.. /. Coll. Interface Sci. 171, 392-398. 

These two equations are similar but have very different interpretations. Equation (2.7) describes an n-particle system in 3-dimensional space. Equation (2.8) can be considered to describe the diffusion of a collection of particles (called walkers here to avoid confusion) in 3n-dimensional space with a source or sink. The function T gives the distribution of the walkers in this space. Each walker is a point in 3n-dimensional space. Equation (2.8) can be solved by permitting each of the walkers to make a 3n-dimensional random walk, and for the population of particles to grow or shrink according to the potential term. 

With a reactor in which single-point nucleation, giving a population with particles all the same size, is followed by growth, the population of particles has the following form [33], plotted in Figure 7.15 as cumulative distribution ... 

The size of a particle may be expressed by a single dimension using one of the diameters defined in Table 2.1. The differences between these dimensions increase as the particle diverges in shape from a sphere. For a population of particles whose shape is not size dependent, distributions obtained using different methods of analysis may be homologous. Multiplying the sizes of one distribution by a constant (shape) factor will therefore generate the other distribution. 

The development here follows that of Lichti et af. (1980). By analogy with the MWD calculation for bulk and solution polymerizations presented earlier, the MWD formalism for monodisperse emulsion systems requires the evaluation of certain types of free-radical growth time distributions. Because of the variable nature of the reaction loci (depending on the state i), a separate growth time distribution is required for the population of particles in each state i. It is therefore convenient to define the distribution of singly distinguished latex particles in state i. denoted as the... 

Equation (2.14) must be coupled with initial conditions given for the starting time and with boundaries conditions in physical space O and in phase space O. Analytical solutions to Eq. (2.14) are available for a few special cases and only under conditions specified by some very simple hypotheses. However, numerical methods can be used to solve this equation and will be presented in Chapters 7 and 8. The numerical solution of Eq. (2.14) provides knowledge of the NDE for each time instant and at every physical point in the computational domain, as well as at every point in phase space. As has already been mentioned, sometimes the population of particles is described by just one internal coordinate, for example particle length (i.e. f = L), and the PBE is said to be univariate. When two internal coordinates are needed, for example particle volume and surface area (i.e. = (v, a)), the PBE is said to be bivariate. More generally, higher-dimensional cases are referred to as multivariate PBEs. Another important case occurs when part of the internal-coordinate vector is equal to the particle-velocity vector (i.e. when the particles are characterized not by a unique velocity field but by their own velocity distribution). In that case, the PBE becomes the GPBE, as described next. 

If a population of particles is to be represented by a single number, there are many different measures of central tendency or mean sizes. Those include the median, the mode and many different means arithmetic, geometric, quadratic, cubic, bi-quadratic, harmonic (ref. 1) to name just a few. As to which is to be chosen to represent the population, once again this depends on what property is of importance the real system is in effect to be represented by an artificial mono-sized system of particle size equal to the mean. Thus, for example, in precipitation of fine particles due to turbulence or in total recovery predictions in gas cleaning, a simple analysis may be used to show that the most relevant mean size is the arithmetic mean of the mass distribution (this is the same as the bi-quadratic mean of the number distribution). In flow through packed beds (relevant to powder aeration or de-aeration), it is the arithmetic mean of the surface distribution, which is identical to the harmonic mean of the mass distribution. 

It has been shown (see Frisch et al., 1987) that Eq. [8] admits equilibrium solutions that lead to a Fermi-Dirac distribution for the mean population of particles Nr This distribution is expressed by... 

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