In all the batch scenarios, the results are shown in 4 plots. The middle plot is the supersaturation (output), the bottom plot is the corresponding MPC input profile, and the two graphs at the top of the figures show the evolution of the number-weighted crystal mean- size (L10) and the phase diagram. In the first batch scenario results are shown in Figure 4-25.
This unconstrained scenario resulting in the MPC successfully tracking the supersaturation trajectory for 142 minutes, at which point the end-point of the batch was triggered because the system temperature dropped to 295 K. Then the setpoint was changed to 0 g/g supersaturation to end the batch, and the batch crystallization process terminated at 160 minutes. The system was seeded with 10 µm seed crystals with a loading of 0.5 g/L. In the phase diagram, the operating profile of the batch is shown to be very close to the solubility curve throughout production and there was little secondary nucleation throughout the process as a result. The crystals in the system did grow from the 10 µm seed size to 32 µm at the end of the batch. This may not appear to be significant growth, but the small seed size coupled with the seed loading results in many small seed crystals in the system, which results in a high surface area per mass of crystals when compared to a larger seed size for the same loading. Therefore, the growth to 33 µm is a reasonable outcome. Increasing the final seed size would be achievable through a smaller seed loading for the same size, or a larger seed size for the same loading.
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Figure 4-25. – Batch Scenario 1 - 10µm seed size and 0.5 g L-1 seed loading unconstrained
The second scenario shown in Figure 4-26 complements the results from the first scenario as it is the same crystallization system but with SFL-Plant constraints. The main differences seen here are that the plant input minimum temperature constraint is reached at 135 minutes, so no further cooling of the system is possible. The supersaturation profile coincidentally begins to diverge from the setpoint and the supersaturation is reduced as the dissolved concentration of paracetamol in the system continues to be consumed by growth. The system temperature does reach the 295 K end-point at 150 minutes, at which point the batch is ended. In this system, the crystal mean size also reached 33 µm from the initial 10 µm seed size, and a similar operating profile as scenario 1 is observed in the phase diagram plot, but a noticeable different is the mean-size trajectory from 135 to 150 minutes in scenario 2 appears to plateau smoothly and gradually, as compared to the almost instantaneous change seen in scenario 1.
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Figure 4-26 – Batch Scenario 2 Results - 10µm seed size and 0.5 g L-1 seed loading with SFL-Plant constraints
Scenario 3 shows a variation of scenario 1 where the seed loading is increased. Insights that can be gained from the model equations and the PBE would suggest then that for an increased mass of crystals in the same batch system, there would be more crystals and therefore a larger surface area and the resulting mean-size at the end of the batch should be smaller. Furthermore, the time to complete a batch should also be shorter because the growth rate is independent of size and dependent on supersaturation, thus dependent on concentration and the 2nd moment (related to surface area). This is clearly seen in the results for scenario 3
in Figure 4-27, with a shorter batch time of 119 minutes and smaller mean-size of 29.5 µm. Comparing these results to scenario 4 in Figure 4-28 with the SFL-Plant constraints active, there is again a noticeable different in the operation of the batch. As the 2nd moment is larger
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the mass balance. So, when the low limit on the input temperature is reached at 108 minutes, the supersaturation diverges from the setpoint at a faster rate than in scenario 2. This batch also ends later than the unconstrained system, at 130 minutes, because the system temperature reaches 295 K at a later time than scenario 3. However, the mean-size is similar in scenario 3 and 4 so it appears there is little effect on the overall size of crystals at the end of the batch.
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Figure 4-28. – Batch Scenario 4 Results - 10µm seed size and 0.75 g L-1 seed loading with SFL-Plant constraints
Scenario 5 is also similar to scenario 1 but the seed size is doubled from 10 µm to 20 µm with the same seed loading. The resulting seed conditions in the model have a smaller zeroth moment and 2nd moment, because the same mass of larger crystals results in fewer crystals
in the system and a smaller surface area. Therefore, it is expected that the batch will require a longer time to complete to the same end-point as previous cases due to a relatively lower consumption of dissolved paracetamol in the system. Also, it is expected that the final mean size of crystals will be larger than the previous cases too. This is seen in the results in Figure 4-29 where the batch time is 277 minutes and the final crystal size reached 65 µm.
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Figure 4-29. – Batch Scenario 5 Results - 20µm seed size and 0.5 g L-1 seed loading unconstrained
In the final batch scenario shown in Figure 4-30, a slightly longer batch time is observed from scenario 5 but because the consumption of supersaturation for growth is slower in this system, the SFL-Plant constraints do not have a significant effect on the overall batch time or supersaturation trajectory. It was possible for the SFL-MPC to maintain the supersaturation target and only reached the low limit for the input at 275 minutes, the batch ended at 278 minutes. The input profile is certainly less aggressive towards the end of the batch in scenario 6 than scenario 5. The mean size was also similar, but slightly larger again in the SFL-Plant constraints scenario. It is suspected that the increased batch time which results from reaching the MPC input limits results in more time for crystal growth and therefore marginally larger crystals at the end of the batch.
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Figure 4-30. . – Batch Scenario 6 Results - 20µm seed size and 0.5 g L-1 seed loading with SFL-Plant constraints
Finally, the CQAs for the six scenarios have been summarised in Table 4-16. The main conclusions from these KPIs are that the mean-size for the SFL-Plant constraints scenarios were marginally larger than their counterpart unconstrainted scenarios, and the COVs are the same, although in practice it would likely not be possible to distinguish the differences seen between scenarios 1 and 2 or 3 and 4 through in-line process measurements.
Scenario Mean Size
(µm) COV Recovery (%) Yield (%)
1 33.15 0.022 47.30 100.00 2 33.45 0.022 48.61 100.00 3 29.29 0.024 48.32 100.00 4 29.34 0.024 48.60 100.00 5 65.82 0.013 46.22 100.00 6 66.90 0.013 48.59 100.00
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The yield from all scenarios was 100 % because every batch ended at the solubility curve, but the recovery was between 46 % and 49 %. Therefore, a large quantity of paracetamol remains dissolved in the system at the end of the batch, but this is a good outcome for this batch crystallization system because the solubility curve that was used (Nagy 𝑒𝑡 𝑎𝑙., 2008a) is only feasible from system temperature of 320 K to 290 K, and the batch end-point was close to this limit.