Size-distributed particle population

If the objective is to obtain a particle-free fluid, then perfect separation means no solid particles in the overflow (equivalent to the heads stream of Section 2.2) and no carrier fluid in the underflow (the tails stream of Section 2.2). If particle classification is the goal, then perfect separation requires all particles above a given size to be in the underflow and all particles smaller than the given size in the overflow. In an imperfect separation, some particles are always present in the overflow (when the goal is to have a particle-free carrier fluid). Similarly, due to imperfections in the separator, some particles coarser than the given size are in the overflow, just as some finer particles are in the underflow from the separator functioning as a classifier. 

The composition of the particle population is usually indicated by the particle size density function f(rp), where tp is the characteristic particle dimension of importance. We denote the values of this function for the feed stream and the product streams by fj rp), fi rp) andfftp), respectively. Since the fraction of particles in the size range rptotp-y dtp is given byV(t ) dtp when the particle size density function is/(rp), such that 

The maximum value of f( p), namely 1, is achieved when the upper limit of integraUon is r ax- 

A variety of particle size distributions are used in practice. The most widely used example is the Normal distribution (Gaussian distribution). 

Other distributions employed are the log-normal distribution, the gamma distribution, the Rosin-Rammler distribution, etc. We will encounter some of them later in the book. 

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Size Distribution. Particle populations are rarely monodispersed. The polydispersity leads the material to depart from the behaviors described by the classical equations. Generally, at a given concentration of particles, the minimum interaction between them occurs for an optimum proportion of the small particles The small particles will either isolate or lubricate the larger ones, but a too high content of small particles can induce a flocculation which can entrap a part of the continuous phase and, consequently, increase the product viscosity by artificially increasing the volume fraction. Refer to Fig. 37. 

The treatment of size-distributed particle populations has so far assumed that the number of particles in the population is quite large and there is almost a continuous distribution in particle sizes in the particle population under consideration. If the number of particles in the population is not large, then we have essentially a discrete distribution in particle sizes. Suppose the number of particles in the size range tp to tp +AVp is ANj. Then the population density function ra,- of particles in this size range is... 

These quantities are thus analogous to those we have already defined for a size distributed particle population. Instead of particle size, we have a distribution variable r, which is intrinsic to a given chemical species. For example, for a flash vaporizer single-entry separator (Figure 2.4.7), the material balance for 1 mole of feed having a molecular weight density function off/(M) is... 

Particle separation devices, aerosol/hydrosol separation in granular media, crystallization and precipitation devices generally involve a size-distributed particle population. 

The notion of a size-distributed particle population was introduced in Section 2.4 via a particle size density function/(rp) the quantity/(rp)drp represents the fraction of particles in the size range of Vp to Vp + dVp in a unit fluid volume. It is also the probability of finding a particle having a size in the size range Vp to Vp + dVp in a unit fluid volume. 

Chapter 5 will be devoted to solid phase synthesis of ceramic powders Chapter 6, to liquid phase synfiiesis and Chapter 7, to gas phase synthesis. Other miscellaneous methods of ceramic powder synthesis are discussed in Chapter 8. All of these ceramic powder synthesis methods have one thing in common, the generation of particles with a particular particle sized distribution. To predict the particle size distribution a population balance is used. The concept of population balances on both the micro and... 

A population of particles is described by a particle size distribution. Particle size distributions may be expressed as frequency distribution curves or cumulative curves. These are illustrated in Figure 1.3. The two are related mathematically in that the cumulative distribution is the integral of the frequency distribution i.e. if the cumulative distribution is denoted as F, then the frequency distribution dF/dx. For simplicity, dF/dx is often written as f x). The distributions can be by number, surface, mass or volume (where particle density does not vary with size, the mass distribution is the same as the volume distribution). Incorporating this information into the notation, /n(x) is the frequency distribution by number, /s(x) is the frequency distribution by surface, Fs is the cumulative distribution by... 

Chapter 2 presents the description of quantities needed to quantify separation in open systems with flow(s) in and out of single-entry and double-entry separators for binary, multi-component and continuous cheimcal mixtmes, as well as a size-distributed particle populatioiL Separation indices useful for describing separation in open systems with or without recycle or reflux are illustrated for steady state operation (Sections 2.2 and 2.3) those for a particle population are provided in Section 2.4. At the end (Section 2.5), indices for description of separation in time-dependent systems, e.g. chromatography, have been introduced. 

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. 

The diametei of average mass and surface area are quantities that involve the size raised to a power, sometimes referred to as the moment, which is descriptive of the fact that the surface area is proportional to the square of the diameter, and the mass or volume of a particle is proportional to the cube of its diameter. These averages represent means as calculated from the different powers of the diameter and mathematically converted back to units of diameter by taking the root of the moment. It is not unusual for a polydispersed particle population to exhibit a diameter of average mass as being one or two orders of magnitude larger than the arithmetic mean of the diameters. In any size distribution, the relation ia equation 4 always holds. 

The mode of distribution is simply the value of the most frequent size present. A distribution exhibiting a single maximum is referred to as a unimodal distribution. When two or more maxima are present, the distribution is caUed bimodal, trimodal, and so on. The mode representing a particle population may have different values depending on whether the measurement is carried out on the basis of particle length, surface area, mass, or volume, or whether the data are represented ia terms of the diameter or log (diameter). 

In industrial practice, the size-distribution cui ve usually is not actually construc ted. Instead, a mean value of the population density for any sieve fraction of interest (in essence, the population density of the particle of average dimension in that fraction) is determined directly as AN/AL, AN being the number of particles retained on the sieve and AL being the difference between the mesh sizes of the retaining sieve and its immediate predecessor. It is common to employ the units of (mm-L)" for n. 

Crystallizers with Fines Removal In Example 3, the product was from a forced-circulation crystallizer of the MSMPR type. In many cases, the product produced by such machines is too small for commercial use therefore, a separation baffle is added within the crystallizer to permit the removal of unwanted fine crystalline material from the magma, thereby controlling the population density in the machine so as to produce a coarser ciystal product. When this is done, the product sample plots on a graph of In n versus L as shown in hne P, Fig. 18-62. The line of steepest ope, line F, represents the particle-size distribution of the fine material, and samples which show this distribution can be taken from the liquid leaving the fines-separation baffle. The product crystals have a slope of lower value, and typically there should be little or no material present smaller than Lj, the size which the baffle is designed to separate. The effective nucleation rate for the product material is the intersection of the extension of line P to zero size. 

Particle-Size Equations It is common practice to plot size-distribution data in such a way that a straight line results, with all the advantages that follow from such a reduction. This can be done if the cui ve fits a standard law such as the normal probability law. According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. In order to maintain a symmetrical beU-shaped cui ve for the frequency distribution it is necessary to plot the population density (e.g., percentage per micron) against size. 

In this ehapter, the transport proeesses relating to partiele eonservation and flow are eonsidered. It starts with a brief introduetion to fluid-particle hydrodynamics that deseribes the motion of erystals suspended in liquors (Chapter 3) and also enables solid-liquid separation equipment to be sized (Chapter 4). This is followed by the momentum and population balances respeetively, whieh deseribe the eomplex flows and mixing within erystallizers and, together with partieulate erystal formation proeesses (Chapters 5 and 6), enable partiele size distributions from erystallizers to be analysed and predieted (Chapters 7 and 8). 

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

The general form of the population balance including aggregation and rupture terms was solved numerically to model the experimental particle size distributions. While excellent agreement was obtained using semi-empirical two-particle aggregation and disruption models (see Figure 6.15), PSD predictions of theoretical models based on laminar and turbulent flow considerations... 

The significance of this novel attempt lies in the inclusion of both the additional particle co-ordinate and in a mechanism of particle disruption by primary particle attrition in the population balance. This formulation permits prediction of secondary particle characteristics, e.g. specific surface area expressed as surface area per unit volume or mass of crystal solid (i.e. m /m or m /kg). It can also account for the formation of bimodal particle size distributions, as are observed in many precipitation processes, for which special forms of size-dependent aggregation kernels have been proposed previously. 

At the crystallization stage, the rates of generation and growth of particles together with their residence times are all important for the formal accounting of particle numbers in each size range. Use of the mass and population balances facilitates calculation of the particle size distribution and its statistics i.e. mean particle size, etc. 

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

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. 

For fluid particles that continuously coalesce and breakup and where the bubble size distributions have local variations, there is still no generally accepted model available and the existing models are contradictory [20]. A population density model is required to describe the changing bubble and drop size. Usually, it is sufficient to simulate a handful of sizes or use some quadrature model, for example, direct quadrature method of moments (DQMOM) to decrease the number of variables. 

The slopes of the calibration curves for the HDC and Fractosil systems are 0.512 and 0.289, respectively. This indicates that the "resolution of the peak separation" for the Fractosil system is superior to that of HDC, since resolution is considered to be inversely proportional to the slope of the log particle diameter - AV calibration curve (26). However when peak spreading is taken into account, the actual relative resolution between particle populations is less for the Fractosil system 2k) a result which indicates that overall, for size distribution resolution, the HDC system is superior. 

At the simplest level, the rate of flow-induced aggregation of compact spherical particles is described by Smoluchowski s theory [Eq. (32)]. Such expressions may then be incorporated into population balance equations to determine the evolution of the agglomerate size distribution with time. However with increase in agglomerate size, complex (fractal) structures may be generated that preclude analysis by simple methods as above. 

Leblanc and Fogler developed a population balance model for the dissolution of polydisperse solids that included both reaction controlled and diffusion-controlled dissolution. This model allows for the handling of continuous particle size distributions. The following population balance was used to develop this model. 

Models for emulsion polymerization reactors vary greatly in their complexity. The level of sophistication needed depends upon the intended use of the model. One could distinguish between two levels of complexity. The first type of model simply involves reactor material and energy balances, and is used to predict the temperature, pressure and monomer concentrations in the reactor. Second level models cannot only predict the above quantities but also polymer properties such as particle size, molecular weight distribution (MWD) and branching frequency. In latex reactor systems, the level one balances are strongly coupled with the particle population balances, thereby making approximate level one models of limited value (1). 

Population Balance Approach. The use of mass and energy balances alone to model polymer reactors is inadequate to describe many cases of interest. Examples are suspension and emulsion polymerizations where drop size or particle distribution may be of interest. In such cases, an accounting for the change in number of droplets or particles of a given size range is often required. This is an example of a population balance. 

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