Population density balance

Modern industrial crystallization theory dates essentially from the work of Randolph and Larson (1962), who developed the concept of population density insofar as it applies to mixed suspension, mixed product removal crystallization equipment. A detailed treatment of the development of these concepts is given in Chapter 4 of this volume. The population density concept is useful because it allows the user to take the data developed from a crystal screen analysis along with knowledge of the operating parameters of the crystallizer, such as the retention (drawdown) time, slurry density. 

By definition, therefore, the nucleation rate, which is the birth of crystals at size = 0 is shown in Eq. (5.3). As will be shown later, EP is also a function of other properties of the system and the crystallizer. 

The total number of crystals, the total area of the crystals, and the mass of the crystals in the sample are shown in equations below. 

Since nucleation is also a function of supersaturation, it is frequently expressed as a power law model similar to that shown in Eq. (5.7) when the crystallizer design effects on the nucleation rate can be neglected. 

This power law model contains an expression for slurry density and, as will be shown later, there are other considerations that could have been added that have a demonstrated influence on nucleation rate. In order to satisfy any given condition, the mass of crystals per unit volume of the crystallizer, as shown in Eq. (5.7), must be consistent insofar as the nuclei population density and growth rate are concerned with that shown in Eq. (5.2). 

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Mote quantitative relationships of the CSD obtained from batch operations can be developed through formulation of a population balance. Using a population density defined in terms of the total crystallizer volume rather than on a specific basis (n = nU), the general population balance given by equation 42 can be modified in recognition of there being no feed or product streams ... 

Employing two co-ordinates of overall particle size, L, and degree of agglomeration, S (which is, of course, proportional to the mean primary particle size) to define the population density, n S, L, t), the population balance during precipitation with agglomeration is described as ... 

In general, it can be stated that the population density of an animal depends on the balance between the rate of recruitment and the rate of mortality. In the context of ecotoxicology, the influence of pollutants upon either of these factors is of fundamental interest and importance. When a population is at or near its carrying capacity, these two factors are in balance, and the critical question about the effects of pollutants is whether they can adversely affect this balance and bring a population decline. 

Growth and nucleation interact in a crystalliser in which both contribute to the final crystal size distribution (CSD) of the product. The importance of the population balance(37) is widely acknowledged. This is most easily appreciated by reference to the simple, idealised case of a mixed-suspension, mixed-product removal (MSMPR) crystalliser operated continuously in the steady state, where no crystals are present in the feed stream, all crystals are of the same shape, no crystals break down by attrition, and crystal growth rate is independent of crystal size. The crystal size distribution for steady state operation in terms of crystal size d and population density // (number of crystals per unit size per unit volume of the system), derived directly from the population balance over the system(37) is ... 

Discretizing the population balance in K+1 grid points results in K ordinary differential equations as the population density n(LQ,t) is determined by the algebraic relation Equation 4. The equation for dn(LQ,t)/dt is therefore not required. In the overal model of the crys lllzer, the population density must be integrated in the calculation of c(t) and In the calculation of m (t) in the nucleatlon rate. As the population balance is discretized these integrals have to be replaced by numerical integration schemes. 

As shown by Eq. (54), growth rate G can be obtained from the slope of a plot of the log of population density against crystal size nucleation rate B° can be obtained from the same data by using the relationship given by Eq. (57), with n° being the intercept of the population density plot. Nucleation rates obtained by these procedures should be checked by comparison with values obtained from a mass balance (see the later discussion of Eq. (66)). 

The rate of cooling, or evaporation, or addition of diluent required to maintain specified conditions in a batch crystallizer often can be determined from a population-balance model. Moments of the population density function are used in the development of equations relating the control variable to time. As defined earlier, the moments are... 

Within 50 years different land use systems have - partly successively, partly simultaneously - dominated the agricultural practice. Until the 1960s the traditional system was driven from within, the relation of the production, the maintenance of the production-basis and the population density were balanced. This system was mainly based on self-sufficiency. There was hardly any external trade, innovation was derived from within and on-farm research was carried out. All the social functions were locally divided within the family/extended family or the community -production, food security, conservation of resources, production of medicines, advising, and trade (see Table 1, Table 2, and Table 3). In their fulfilment of the daily duties, people were dependent on each other. There were barely any innovations, neither negative nor positive, introduced from outside the system. 

This expression for the rate of change in the population density of nuclei will be used later in the population balance model as an initial condition. Katz has shown that classical nucleation theory predicts well the dependence of the supersatvuntion ratio [10] on the nucleation rate. 

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. 

It will be shown in a later section that the solution of a differential population balance requires a knowledge of tbe relationship between growth rate and size of the growing ciystals Moreover, this relationship can often be deduced from the form of population density deta. A special condition, which simplifies such belances, results when all crystals in the magma grow at (he sama constant rate. Crystal-solvent systems that show this behavior ate said to follow tha AL Law proposed by McCabe.1 while systetes that do not are said to exhibit anomalous growth. 

It is important to note that either clear liquor volume or slurry volume can he taken os a besis for the definition of n, For example, if clear liquor volume is taken as the besis, VT becomes the volume of dear liquor in the control volume. With this definition of the population density, it can be shown thet the population balance on a well-mixed continuous crystallizer is given by... 

Using the same approach as is used to write material and energy balances, a population balance can also be constructed. Consider an arbitrary size within the crystallizer (Figure 4.5), for the size range L to Z.2 with population densities ti and 2, respectively, the balance is... 

Analysis of the PSD Data. From the population balance for a CMSMPR crystallizer operated under steady-state condition, the population density n for size-independent crystal growth is given by Equation (1), where n , G, 6 and 1 are nuclei density, growth rate, residence time of reactants and particle size, respectively. 

The relationship between crystal size, L, and population density, n (number of crystals per unit size per unit volume of the system), derived directly from the population balance (Randolph and Larson, 1988) (section 9.1.1) is... 

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