Population balance approach

In general, both nucieation and crystal growth depend on supersaturation and to lesser extent temperature and magma characteristics. Such data must therefore be collected to gain maximum benefit from the population balance approach (Jones and MuIIin, 1974 Jones, 1974). Further simplifications to the describing equations are also possible, however (as follows). 

The development of the differential equations which describe the evolution of particle size and molecular weight properties during the course of the polymerization is based on the so-called "population balance" approach, a quite general model framework which will be described shortly. Symbols which will be used in the subsections to follow are all defined in the nomenclature. 

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. 

A discussion of the population balance approach in modelling particulate systems and the derivation of the general equation are given in Appendix II. 

Particle Size Distribution Determination. To consider the full PSD, a population balance or age distribution analysis on particles must be employed. Table II gives a summary of recent work concerning the determination of PSD s in emulsion systems, using both the "monodispersed" approximation and the population balance approach. More details can be found in the literature sources cited in the Table. 

To our knowledge, this is the first time that an emulsion copolymerization model has been developed based on a population balance approach. The resulting differential equations are more involved and complex than those of the homopolymer case. Lack of experimental literature data for the specific system VCM/VAc made it impossible to directly check the model s predictive powers, however, successful simulation of extreme cases and reasonable trends obtained in the model s predictions are convincing enough about the validity and usefulness of the mathematical model per se. 

In general, different sized particles may have different cycle time distributions and different mass deposition distributions in the spray zone. One approach would be to use small discrete size distribution increments and then to apply Eq. (12) to each size fraction. Inherent in this approach is the assumption that each particle size fraction acts independently. This assumption may not be valid, especially if different particles take different circulation paths within the bed. From the population balance approach, Randolf and Larson (1988) have suggested the use of an effective growth diffusivity coefficient to account for random fluctuations in growth rate. Thus Eq. (6) would be modified to give ... 

The development of the population balance approach has done much to further our understanding of CSD transients and instabilities which are observed in the plant and the laboratory. Advances in computing and control technology which have occurred in the chemical process industry together with the recognition that on-line CSD control might be achievable has led to an increased level of interest and investigation of this area. 

The dynamic model used in predicting the transient behavior of isothermal batch crystallizers is well developed. Randolph and Larson (5) and Hulburt and Katz (6) offer a complete discussion of the theoretical development of the population balance approach. A summary of the set of equations used in this analysis is given below. 

In conclusion to this section, research in the RTD area is always active and the initial concepts of Danckwerts are gradually being completed and extended. The population balance approach provides a theoretical framework for this generalization. However, in spite of the efforts of several authors, simple procedures, easy to use by practitioners, would still be welcome in the field of unsteady state systems (variable volumes and flow rates), multiple inlet/outlet reactors, variable density mixtures, systems in which the mass-flowrate is not conserved, etc... On the other hand, the promising "generalized reaction time distribution" approach could be developed if suitable experimental methods were available for its determination. 

For the description of the particle property distribution, a population balance approach is recommended which is mathematically challenging but which provides valuable insight into the steady-state and dynamic process operating behavior. 

All the pseudo-population balance approaches given to this point have neglected that grains actually are aimihilated in order for other grains to grow. This annihilation can be accounted for in the population balance given in Chapter 3 ... 

The SAXS/TGA approach has been demonstrated to be a useful technique for time-resolution of porosity development in carbons during activation processes. Qualitative interpretation of the data obtained thus far suggests that a population balance approach focusing on the rates of production and consumption of pores as a function of size may be a fruitful approach to the development of quantitative models of activation proces.ses. These then could become useful tools for the optimization of pore size distributions for particular applications by providing descriptions and predictions of how various activating agents and time-temperature histories affect resultant pore size distributions. 

The population balance approach is employed for the description of droplet dynamics in various flow fields. A significant advantage of the method is that a vehicle is provided to include the details of the breakage and coalescence processes in terms of the physical parameters and conditions of operation. A predictive multidimensional particle distribution theory is at hand which, in the case of well-defined droplet processes, can be employed for a priori prediction of the form and the magnitude of the particle size distribution. The physical parameters which affect the form... 

Penlidis, A. Macgregor, J.F. Hamielec, A.E. Mathematical-modeling of emulsion polymerization reactors—a population balance approach and its applications. ACS Symp. Ser. 1986, 313, 219-240. 

The population balance approach to measurement of nucleation and growth rates was presented by Randolph and Larson (1971, 1988). This methodology creates a transform called population density [n(L)], where L is the characteristic size of each particle, by differentiating the cumulative size distribution N versus L. shown in Fig. 4-22, where N is the cumulative number of crystals smaller than L. Per unit volume, the total number of particles, total surface area, and total volume/mass are calculated as the first, second, and third moments of this distribution. 

Prank T, Zwart PJ, Shi J-M, Krepper E, Lucas D, Rohde U (2005) Inhomogeneous MUSIG Model - a Population Balance Approach for Polydispersed Bubbly Flows. Int Conf Nuclear Energy for New Europe 2005, Bled, Slovenia, September 5-8... 

This relation expresses that not all collisions lead to coalescence. The modeling of the coalescence processes thus means to find adequate physical expressions for hc d d, Y) and pc d d, Y). Kamp et al [39], among others, suggested that microscopic closures can be formulated in line with the macroscopic population balance approach, thus we may define ... 

For bubbly flows most of the early papers either adopted a macroscopic population balance approach with an inherent discrete discretization scheme as described earlier, or rather semi-empirical transport equations for the contact area and/or the particle diameter. Actually, very few consistent source term closures exist for the microscopic population balance formulation. The existing models are usually solved using discrete semi-integral techniques, as will be outlined in the next sub-section. 

The population-balance is a powerful tool for modeling foam displacement and flow in porous media because it correctly predicts the evolution of foam microstructure from well documented pore-level events, and because it merges with current reservoir simulation practice. Perhaps the main power of the population-balance approach is its general framework. As understanding of mechanistic detail improves, this information may be incorporated in the modeling effort. 

The population balance approach to analysis of crystallizers was formalized and presented by Randolph and Larson (1971, 1988). The technique parallels other balance approaches such as material and energy balances, which are familiar to process engineers. The population balance is used to account for both the size (an attribute to be described later in this section) and number of particles. Therefore, before a discussion of the development and application of the population balance can be enjoined, it is necessary to consider size and size distributions of particles. 

The population balance approach to analysis of crystallizers has been presented. Table 4.6 summarizes the various outcomes discussed for continuous crystallizers. 

As it will be shown for evaporative crystallization, a population balance approach subject to the same assumptions will yield similar results as far as the mathematical form of the solution is concerned. 

Using the population balance approach is most readily seen by considering several examples. One could rederive all of the age distribution formalism by choosing Ci = (age) in the macroscopic population balance Eq. 12.6-2 ... 

Ouchiyama N, Rough SL, Bridgwater J, 2005, A population balance approach to describing bulk attrition. Chemical Engineering Science, 60,1429-1440... 

A population balance approach can be used to derive a similar expression for a steady-slate CSTR. Equation (8.3) still applies to this reactor but a new relationship for N must be developed. The particle size distribution for Smith-Ewart Case 2 kinetics in a CSTR is given by... 

Since the population balance relationships have already been developed in some detail above (section 9.1.1) most of the following examples are based on the population balance approach and/or nomenclature, for reasons of continuity. It is important to understand, however, that other design methods are available, some based on mass or supersaturation balances rather than on population balances. For details of these other methods reference should be made to original literature sources, e.g., as summarized by Toyokura et al. (1984) and Nyvlt (1978, 1992). 

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