Population, Density, Growth and Nucleation Rate

Example 3 Population, Density, Growth and Nucleation Rate. 18-43... 

Calculate the population density, growth, and nucleation rates for a crystal sample of urea for which there is the following information. These data are from Bennett and Van Biiren [Chem. Eng. Frvg. Symp. Ser., 65(95), 44 (1969)]. Slurry density = 450 g/L Crystal density = 1.335 g/cm ... 

Example 4 Population Density, Growth, and Nucleation Rate... 

Calculate the population density, growth, and nucleation rates for a crystal sample of urea for which there is the following information. These data are from... 

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 ... 

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. 

Analysis of equation 48 shows that a single sample taken either from inside the crystallizer or from the product stream will allow evaluation of nucleation and growth rates at the system conditions. Figure 12 shows a plot of typical population density data obtained from a crystallizer meeting the stated assumptions. The slope of the plot of such data maybe used to obtain the growth rate, and the product of the intercept and growth rate gives the nucleation rate. 

This removal function gives rise to a discontinuity in the population density at the cutsize of the fines. The nucleation parameters are given in equation 19 In Figure 3 the responses are shown of the population density at 120 pm and of the growth rate after a step in the heat input to the crystallizer from 120 to I70 kW for three simulation edgorithms. The cut-size of the fines was 100 pm, a size dependent growth rate was used as described by Equation 4 with a= -250 and the number of grid points was kOO. When the simulation was performed with the method of lines, severe oscillations are present in the response of the population density at 120 pm, which dampen out rather slowly. Also the response of the Lax-Wendroff method shows these oscillations to a lesser extend. 

Marsh (1988), Cashman and Marsh (1988), and Cashman and Ferry (1988) investigated the application of crystal size distribution (CSD) theory (Randolph and Larson, 1971) to extract crystal growth rate and nucleation density. The following summary is based on the work of Marsh (1988). In the CSD method, the crystal population density, n(L), is defined as the number of crystals of a given size L per unit volume of rock. The cumulative distribution function N(L) is defined as... 

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. 

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). 

The magma density Mr (mass of ciystais per unit volume of slurry or liquor) may be obtained from the third moment of the population density function. As shown above, this quantity is related to nucleation and growth rates by Eq. (II. 2-38). Although magma density is a function of the Idnetic parameters n° and C, it can be measured independently of crystal size distribution and, where possible, it should be used, as indicated above, as a constraint in evaluating nucleation and growth rates from measured crystal size distributions. 

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