4. Discussion
4.3. Weight COV and Number COV
In Figure 6, upper panel shows the number based crystal size distribution function for result using minimizing weight coefficient of variation and minimizing number coefficient of variation as objective function.
In this panel we can see that the distribution of result using minimizing number coefficient of variation is more narrow than distribution of result using objective function minimizing weight coefficient of variation.
Lower panel in Figure 6 shows the weighted final crystal size distribution function, both results from using minimizing weight coefficient of variation and number coefficient of variation shows a peak. And the distribution peak of result using weight coefficient of variation is narrower.
Figure 6: Final crystal size distribution for different objectives.
0 0.5 1 1.5 2 2.5 3 3.5 4
0 5 10 15 20 25
L'
f'(L')
0 0.5 1 1.5 2 2.5 3 3.5 4
0 2 4 6
L' L'3 f'(L')
30
Both results from two objective functions show a narrow peak, however it is the nucleated crystals that give a narrow distribution. Figure 7 shows the optimal growth rate using objective function minimizing weight coefficient of variation and number coefficient of variation. When the supersaturation increases, both nucleation rate and growth rate will be enhanced. From Equation 13, we can see that the growth rate directly affect the nucleation rate.
Because of the power in Equation 13 γ is equal to three, when the growth rate increases heavily, the nucleation rate increases much more.
Figure 7: The optimal growth trajectory using objective function, minimizing
weight coefficient of variation and minimizing number coefficient of variation.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 2 4 6 8 10 12
t'
G'
minimizing weight COV minimizing number COV
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Figure 8: The optimal nucleation rate trajectory using objective function,
minimizing weight coefficient of variation and minimizing number coefficient of variation.
Figure 8 shows the nucleation rate trajectory. In this figure we can see that both of the objective functions result in large nucleation peak in the process. The objective function minimizing weight coefficient of variation causes excess nucleation at the beginning of the batch, and the objective function minimizing number coefficient of variation causes excess nucleation at the middle of the batch time. From Figures 6, 7 and 8 we can see that these two objective functions achieve narrow final crystal size distribution by making excess nucleation during the batch. Thus they achieve the minimum of
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 50 100 150 200 250
t'
B'
minimizing weight COV minimizing number COV
32
the objective function, however, the large amount of nucleated crystals and excess nucleation are not desirable.
Figure 9: Weight coefficient of variation and number coefficient of variation
value for different objectives.
Figure 9 shows the resulting weight coefficient of variation and number coefficient of variation when both of these objectives are used as the objective
0.05 0.1 0.15
minimizing Weight COV minimizing Number COV
0.05 0.1 0.15
Weight COV for seed
0.05 0.1 0.15
Number COV for seed
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function. As expected, in panel A, when the objective is to minimize the weight coefficient of variation, the weight coefficient of variation is indeed lower than the number coefficient of variation. In Panel B, again as expected, when the objective is to minimize the number COV, the number COV is indeed less than when the objective is to minimize the weight COV.
However, the situation if the nucleated crystals are filtered out (Panels C and D). In that case, minimizing the weight COV also minimizes the number COV, and using the number COV as the objective results in an inferior performance. Figure 7 shows the number distribution (f(l)) and the weight distribution (L3f(L)) of the product using the saturation concentration trajectory that minimizes the weight COV and the number COV. Minimizing the weight COV results in more growth of the seeds. On the basis of these two results, we conclude that among objectives that aim to minimize a COV, minimizing the weight COV is more suitable than minimizing the number COV.
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Figure 10: The optimal concentration trajectory using objective function,
minimizing weight coefficient of variation and minimizing number coefficient of variation.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t'
C'
35
Figure 11: Different order of moment plot during the batch time for using
objective function minimizing weight coefficient of variation and minimizing number coefficient of variation.
0 0.5 1
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4.4. Weight Mean Size and Number Mean Size
Figure 12 shows the optimal nucleation trajectory using object function minimizing weight mean size and minimizing number mean size. Again we can observer that for trajectory from result using objective function maximizing number mean size causes excess nucleation.
Figure 12: The optimal nucleation rate trajectory using objective function,
minimizing weight mean size and minimizing number mean size.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 10 20 30 40 50 60 70 80
t'
B'
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Figure 13: The optimal growth trajectory using objective function, minimizing
weight mean size and minimizing number mean size.
Figure 13 shows the optimal growth rate trajectory using objective function minimizing weight mean size and minimizing number mean size. We can observe that the batch is actually started when 't =0.16.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t'
G'
38
Figure 14: The optimal concentration trajectory using objective function,
minimizing weight mean size and minimizing number mean size.
Although the optimal trajectory of minimizing number mean size gives an excess early growth and shorten the batch time, it is still reasonable for objective function minimizing number mean size. In Figure 15 we can see that the excess nucleation from trajectory which is optimized using maximizing number mean size causes raising of zeroth moment at the early batch time.
However, at the end of the batch the zeroth moment from the optimal trajectory using maximizing weight means size is larger. Since the number mean size of the crystal is defined as µT,1/µ , the value of the zeroth T,0
moment is very important for number mean size when compare to the weight
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t'
C'
39
mean size. And from Equation 26, the nucleation rate directly affects the value of the zeroth moment. Therefore, it is reasonable that the trajectory using maximizing number mean size as objective function by avoiding high growth rate when the third moment is larger at the end of the batch. And that is the major reason why the program chooses an early growth trajectory and shortens the batch time.
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Figure 15: Different order of moment plot during the batch time for using
objectives function minimizing weight mean size and minimizing number mean size.
We compare objectives involving maximizing a mean size of the final product crystals, either the weight mean size or number mean size. Figure 16 shows the results for maximizing the weight mean size and the number mean size. In Panel A, as expected, if the objective is to maximize the weight mean
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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size, then the product crystals do indeed have a larger weight mean size than if the objective is to maximize the number mean size. Likewise, from Panel B, if the objective is to maximize the number mean size then the resulting product crystals do indeed have a larger number mean size than if the objective is to maximize the weight mean size.
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Figure 16: Weight mean size and number mean size value for different
objectives.
However, as was the case for objectives based on COV, the situation changes if only the seed crystals are considered. (Panels C and D) In this case, maximizing the weight mean size gives a better result for both objectives.
Maximizing WMS Maximizing NMS
0.05 0.1 0.15
Weight Mean Size for seed
0.05 0.1 0.15
Number Mean Size for seed
43
from the saturation concentration trajectories determined using both objective functions. The objective of maximizing the weight mean size results in greater growth of the seed crystals. Therefore, among the two objective functions that involve product mean sizes, we conclude that weight mean size is a superior objective to number mean size.
Figure 17: Final crystal size distribution for different objectives.
0 0.5 1 1.5 2 2.5 3 3.5 4
0 50 100 150 200
L'
f'(L')
0 0.5 1 1.5 2 2.5 3 3.5 4
0 1 2 3 4
L' L'3 f'(L')
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