Crystallizers population density balance

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 ... 

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 ... 

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... 

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... 

The total solids content, Mj, and hence the production rate of an MSMPR crystallizer are both controlled by the feedstock and operating conditions. Equation 9.26 is, in effect, a growth rate constraint because for a given population density of nuclei only one value of G will satisfy the mass balance. The mass of crystals dM in a given size range dL is... 

Base approach on mass and energy balances, population or number balances. Follow the population density of number versus size. Must know the type of crystals and the mode of operation. 

The population balance provides the mathematical framework incorporating expressions for the various crystal formation, aggregation and disruption mechanisms to predict the final particle size distribution. Note, however, that while particles are commonly characterized by a linear dimension the aggregation and particle disruption terms also require conservation of particle volume. It was shown in Chapter 2 that the population balance accounts for the number of particles at each size in a continuous distribution. The quantity conserved is thus the number (population) density and may be thought of as an extension of the more familiar mass balance. The population balance is given by (Randolph and Larson, 1988)... 

Jager etal. (1992) used a dilution unit in conjunction with laser diffraction measurement equipment. The combination could only determine, however, CSD by volume while the controller required absolute values of population density. For this purpose the CSD measurements were used along with mass flow meter. They were found to be very accurate when used to calculate higher moments of CSD. For the zeroth moment, however, the calculations resulted in standard deviations of up to 20 per cent. This was anticipated because small particles amounted for less then 1 per cent of volume distribution. Physical models for process dynamics were simplified by assuming isothermal operation and class II crystallizer behaviour. The latter implies a fast growing system in which solute concentration remains constant with time and approaches saturation concentration. An isothermal operation constraint enabled the simplification of mass and energy balances into a single constraint on product flowrate. 

Consider a continuous crystallizer of volume V, as shown in Figure 6.4.2(a). A feed stream having a particle (crystal) number density function ra/(rp) (which is also the population density function), volumetric flow rate Qf and species i mass concentration enters the crystallizer continuously. Product stream 1, having a particle (crystal) number density function n (tp), volumetric flow rate Qi and species i mass concentration Pf, leaves the crystallizer continuously. The particle (crystal) number density function n rp) in the well-mixed crystallizer is the same throughout the crystallizer. The macroscopic population balance equation for a stirred tank separator may be written using equations (6.2.60) and (6.2.61) as follows ... 

The population balance is used as a method of accounting for particles as they go throng a process, such as grinding, dassiiication, crystallization, a [regation, or grain growth. This chapter is devoted to the development of population balances, because it is of fundamental importance to several other of the diapters in this book. The chapter draws heavily on the excellent text Theory of Particulate Processes by Randolph and Larson [1]. The number density of particles N(L) (with units of number of particles per unit volume) is equal to the integral of the population 7)q(L) from size L to L + AL and is defined as... 

One method is to solve the population balance equation (Equation 64.6) and to take into account the empirical expression for the nucleation rate (Equation 64.10), which is modified in such a way that the expression includes the impeller tip speed raised to an experimental power. In addition, the experimental value, pertinent to each ch ical, is required for the power of the crystal growth rate in the nncleation rate. Besides, the effect of snspension density on the nucleation rate needs to be known. Fnrthermore, an indnstrial suspension crystallizer does not operate in the fully mixed state, so a simplified model, such as Equation 64.6, reqnires still another experimental coefficient that modifies the CSD and depends on the mixing conditions and the eqnipment type. If the necessary experimental data are available, the method enables the prediction of CSD and the prodnction rate as dependent on the dimensions of the tank and on the operating conditions. One such method is that developed by Toyokura [23] and discussed and modified by Palosaari et al. [24]. However, this method deals with the CTystaUization tank in average and does not distinguish what happens at various locations in the tank. The more fundamental and potentially far more accurate simulation of the process can be obtained by the application of the computational fluid dynamics (CFD). It will be discussed in the following section. 

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