Nucleation population density

If p = 1, the nucleation population density function n is independent of the magma densities, but growth rate G increases as magma density increases. If p > 1, then n increases as Mj increases so does G. Similarly, from (6.4.21), the dominant crystal size Zpd also increases as Mj-increases for constant since G increases. 

Solution We know from our earlier analysis that, for a given Mj if t,es is varied, the growth rate G and the nucleation population density function will vary according to relations (6.4.31b) and (6.4.31c), respectively, for a given p. Therefore a procedure that could be followed is carry out two different experiments for two values of i,es at the same Mj. Obtain, from a plot of n(rp) vs. Cp, the values of Ga, Gt, n and n . Assume different values of p and check whether the same p can describe both of the following relations ... 

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. 

Identification of an initial condition is difficult because of the problem of specifying the size distribution at the instant nucleation occurs. The difficulty is mitigated through the use of seeding which would mean that the initial population density function would correspond to that of the seed crystals ... 

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 ... 

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. 

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 ... 

A plot of the In n versus L yields a straight line where the slope is - 1/G t and the intercept is n (the population density of nuclei). Since the MSMPR is at steady state, the supersaturation is known. This experiment can then be repeated at a number of different supersaturations and fit to growth and nucleation expressions of the form below ... 

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... 

H° = zero side nuclei concentration, also called zero size population density B° = nucleation rate... 

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)). 

System conditions often allow for the measurement of magma density, and in such cases is should be used as a constraint in evaluating nucleation and growth kinetics from measured population densities. This approach is especially useful in instances of uncertainty in the determination of population densities from sieving or other particle sizing techniques. 

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. 

Example 4 Population Density, Growth, and 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. 

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