A new back-off methodology is presented to address mixed integer dynamic optimization (MIDO) formulations that arise from modeling the integration of scheduling and control of flow-shop batch plants under stochastic parametric uncertainty. The core idea of the methodology is to generate optimal scheduling and control decisions combined with optimal unit operation times that offer dynamic feasibility in the presence of stochastic parametric uncertainty using a back-off strategy. Back-off terms are introduced in the model constraints to represent the variability of the system caused by the stochastic uncertainty. The proposed framework decomposes the MIDO problem into a parametric sensitivity analysis, a scheduling problem, a dynamic optimization problem, and a set of dynamic feasibility problems (dynamic feasibility test) that are solved in an iterative fashion until convergence. A key novelty in this work is the addition of a parametric sensitivity analysis that generates correlations included in the scheduling problem to account for process dynamics. A case study featuring a flow-shop batch plant consisting of two dynamic reaction processes and two steady-state separation processes is used to test the performance of the proposed back-off methodology. The results show that unit operation times chosen from optimization are better suited to accommodate stochastic parametric uncertainty, while the control actions enforce process operational and product quality constraints at reasonable economic costs. Hence, the method proposed in this work has the potential of addressing optimal scheduling and control problems under stochastic realizations in flow-shop batch plants.

Crystallization of α-glycine by addition of an anti-solvent (ethanol) assisted by ultrasound is studied. The experiments of crystallization are conducted at 303.15 K in a solution of 150 ml with continuous agitation by a magnetic rod. Ultrasound is then applied at powers ranging from 8 to 41 W thanks to an ultrasonic horn at 20 kHz. The supersaturation ratio (S) is followed throughout all the experiment. At the end of the experiment, the suspension is filtered, the solid is washed with ethanol and dried at 333.15 K. The resulting crystals are characterized by their final size distributions measured by laser granulometry, their morphologies observed by scanning electronic microscope (SEM) and their crystalline structures by differential scanning calorimetry (DSC). The influence of ultrasonic power (continuous 13, 28 and 40 W or pulsed modes), measured by calorimetry method, is studied for different addition rates (0.05 to 0.36 g of ethanol/min). Ultrasound permits to reduce the metastable zone width and to decrease the size of crystals due to an increase of the nucleation rate. The rate of de-supersaturation is higher in presence of ultrasound, inducing a higher nucleation rate, a higher growth rate or both. At 40 W, the decrease of supersaturation is faster, and the crystallization is finished in 40 min instead of 80 min (at 13 and 28 W) or 120 min without ultrasound. The use of pulsed ultrasound (50 on/50 off) is interesting from an economic point of view because similar results are obtained: comparable size distributions and resembling concentration profiles.

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