Semilogarithmic population density

Figure 4-24 Summary of semilogarithmic population density plots and potential causes.

Figure 4.6 Semilogarithmic population density versus size plot predicted by application of the MSMPR crystallizer formalism.

Upon constructing the semilogarithmic population density versus size plot. Figure 4.7 results. The growth rate is calculated... 

Figure 4.7 Semilogarithmic population density versus size plot obtained by MSMPR analysis of data in Example 4.3. (Data from Belter et al. 1988.)...

Unfortunately, in many cases the semilogarithmic population density size plot shown in Figure 4.6 is not obtained. A common occurrence when analyzing data at smaller sizes, e.g., possible reasons and approaches to analyzing such data. 

Constant Crystal Growth Model. In this instance, crystals have an inherent constant growth rate, but the rate from crystal-to-crystal varies. The modeling of this phenomenon must be accomplished by use of probability transform techniques due to the presence of a growth rate distribution. The complete solution for the population density yields a semilogarithmic population density plot that is concave upwards similar to size-dependent growth (Berglund and Larson 1984). Since it is relatively difficult to handle, a moment approximation was developed for an MSMPR crystallizer (Larson et al. 1985). 

Figure 4.11 Semilogarithmic population density size plot depicting use of linear extrapolation to determine effective zero-size population density. Data are for the sucrose-water system. (Liang and Hartel 1991.)...

Figure 4.12 Semilogarithmic population density size plot for the sucrose-water system depicting the quality of fit when using a sum of two exponentials for the data. (Berglund and de Jong 1990.)...

A typical semilogarithmic population density size plot obtained from a 1400-1 pilot scale sucrose crystallizer is shown in Figure 4.12. It was found that a single growth rate distribution is insufficient to the data. Surprisingly, a combination of two growth rates was all that was necessary, i.e., the sum of two exponentials reproduced the experimental data. This is particularly useful since it now permits the use of effective rates that can be determined unambiguously. 

Figure 4.13 Curvature in a semilogarithmic population density size plot caused by agglomeration. [From Mechanisms and Kinetic Modeling of Calcium Oxalate Crystal Aggregation in a Urinelike Liquor, R.W. Hartel, B.E. Gottung, A.D. Randolph, and G.W. Drach (1986), AIChEJ. 32(7), pp. 1176-1185. Reproduced by permission of the American Institute of Chemical Engineers. 1986 AIChE.]...

An example of aggregation and its modeling can be seen in Figure 4.13, taken from Hartel and Randolph (1986). Clearly, aggregation can result in similar semilogarithmic population density size plots to those found in size-dependent growth and growth rate dispersion. 

Population Density Distribution. Integration of Equation 1 yields a semilogarithmic relationship. 

Therefore, a semilogarithmic plot of the population density versus size yields n° from the intercept and G from the slope (Figure 4.6). 

Effective Nucleation Rates. A simple approach to handling the problem of upward curvature of the semilogarithmic size plot at small size is to use linear extrapolation. Figure 4.11 shows an example of such data for the sucrose-water system. Previously, we saw data for sucrose crystallization in Figure 4.7. These data obeyed the MSMPR formalism since small-size population densities were not measured. The data shown in Figure 4.11, also for the sucrose-water system, were measured using a Coulter counter, which allowed the smaller sizes to be determined. Since the linear portion of the plot represents the product sized crystals, a linear extrapolation to zero size yields an effective zero-size population density. Therefore, an effective nucleation rate can also be determined. This approach does not require knowledge of the cause of the curvature. 

Figure 6.4.2. MSMPR crystallizer, (a) Scherrmtic of a well-mixed crystalUzer (b) semilogarithmic plot of population density function n(rp) vs. crystal size tp.

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