Comparison of objective functions for batch crystallization using a simple process model and Pontryagin’s minimum principle

Tseng, Yu-Ti; Ward, Jeffrey D.

doi:10.1016/j.compchemeng.2017.01.017

Cited by 14 publications

(19 citation statements)

References 28 publications

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“…Among them, in particular, is Pontryagin's maximum principle, which makes it possible to solve a wide class of problems for objects described by systems of linear or nonlinear differential equations, to search for optimal control of processes with distributed parameters and discrete processes. This is confirmed by studies devoted to the development of the maximum principle for a number of applications: the search for optimal control of technological processes [12][13][14], transport in the fields of its production and operation [15][16][17][18], the design of structures in power engineering [19,20], and the field of economics [21].…”

Section: Introductionmentioning

confidence: 86%

Synthesis of optimal control of technological processes based on a multialternative parametric description of the final state

Demin

2017

EEJET

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Запропоновано метод пошуку оптимального управ-ління технологічними процесами, заснований на аналізі рішення системи диференціальних рівнянь (СДР), яка є математичною моделлю керованого процесу. Показано, що отримані при цьому рішення узгоджуються з прин-ципом максимуму Понтрягіна для задачі про швидкодію, але відкриваються додаткові можливості в управлінні кінцевим станом. Запропоновано спосіб мультиальтер-нативного параметричного опису кінцевого стану Ключові слова: оптимальне керування технологічни-ми процесами, принцип максимуму Понтрягіна, швид-кодія, лінія кінцевого стану, мультіальтернатівний опис кінцевого стану, рідж-аналіз Предложен метод поиска оптимального управления технологическими процессами, основанный на анализе решения системы дифференциальных уравнений (СДУ), являющейся математической моделью управляемого процесса. Показано, что получаемые при этом решения согласуются с принципом максимума Понтрягина для задачи о быстродействии, но открываются дополни-тельные возможности в управлении конечным состоя-нием. Предложен способ мультиальтернативного пара-метрического описания конечного состояния Ключевые слова: оптимальное управление техно-логическими процессами, принцип максимума Понтря-гина, быстродействие, линия конечного состояния, мультиальтернативное описание конечного состояния, ридж-анализ UDC 681.5:519.24

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Section: Introductionmentioning

confidence: 86%

Synthesis of optimal control of technological processes based on a multialternative parametric description of the final state

Demin

2017

EEJET

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“…Section introduces the basic method of moment model. , Section presents a dimensionless framework that is a modified version of a framework proposed previously by Ward et al Sections – introduce a coordinate transformation and method of solution using Pontryagin’s minimum principle proposed and extended by Hofmann and coworkers − ,,, and subsequently also used by Ward and coworkers. ,, …”

Section: Modeling and Theorymentioning

confidence: 99%

“…Previous research has shown that a strong trade-off exists between objective functions based on lower moments and higher moments. , Therefore, μ 0, n , f ′ (nucleated number) and μ 3, n , f ′ (proportional to nucleated volume) are set to be two single objective functions discussed in this work. PFs of multi-objective optimization problems with μ 0, n , f ′ and μ 3, n , f ′ as single objectives can be constructed using the weighted sum method (WSM), which expresses the objective function as where β 1 and β 2 are the scaling factors, which account for the fact that the magnitude of μ 0, n , f ′ and μ 3, n , f ′ are often quite different.…”

Section: Modeling and Theorymentioning

confidence: 99%

“…On the other hand, OCT provides the necessary condition of optimality (NCO) of the optimal solution so that nearly optimal solutions can be found. For systems with relatively simple kinetics (e.g., univariate system with growth and nucleation only), boundary-value problems (BVPs) related to the NCO can be integrated nearly analytically so that the optimization problem can be solved with low computational cost. − The methodology had also been extended to multivariate systems − in which the BVPs were solved numerically by a shooting method. As for systems with more complex phenomena such as external fine removal loop, OCT can also provide analytical expressions for the gradient based on the NCO, which assisted in solving the DOPs by a gradient descent method …”

Section: Introductionmentioning

confidence: 99%

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Dimensionless Framework for Seed Recipe Design and Optimal Control of Batch Crystallization

Pan

Ward

2021

Ind. Eng. Chem. Res.

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In order to study and propose guidelines for the design and control of batch crystallization systems with different growth and nucleation parameters, optimal control theory and a dimensionless batch crystallization model are applied to solve multi-objective optimization problems in a nearly analytical and computationally efficient way. Optimal growth rate trajectories and the corresponding critical seed recipes (seed loading ratio and seed mean size that make nucleated mass negligible) representing a suitable trade-off between two objective functions (number of nuclei and nucleated mass) are derived for 32 chemical systems. The results are also compared with results from applying supersaturation control, a strategy that can be implemented without a kinetic model. The analysis suggests that the benefit from applying an optimal growth rate trajectory is significant only when the sensitivity of nucleation rate to supersaturation is significantly greater than that of the growth rate. Furthermore, the width of the crystal size distribution has little effect on the critical seed recipe. The effect of the batch time on the critical seed loading ratio is also studied for individual systems. It is found that increasing batch time helps to reduce the seed loading requirement for systems in which the nucleation rate is highly sensitive to supersaturation.

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“…Ref. 27,51,52) Furthermore, to the knowledge of the authors, MPC has not been examined for racemic compound forming systems in the literature, so the results of this work are also novel in showcasing the possible benefits of MPC compared to traditional cooling strategies in the literature for racemic compound forming systems such as linear cooling.…”

Section: Inclusion Of Fines Dissolution Loopmentioning

confidence: 99%

Optimal operation of batch enantiomer crystallization: From ternary diagrams to predictive control

et al. 2017

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In this work, the modeling and control of batch crystallization for racemic compound forming systems is addressed in a systematic fashion. Specifically, a batch crystallization process is considered for which the initial solution has been preenriched in the desired enantiomer to enable crystallization of only the preferred enantiomer. A method for determining desired operating conditions (composition of the initial pre-enriched solution and temperature to which the mixture must be cooled for maximum yield) for the batch crystallizer based on a ternary diagram for the enantiomer mixture in a solvent is described. Subsequently, it is shown that the information obtained from the ternary diagram, such as the maximum yield attainable from the process due to thermodynamics, can be used to formulate constraints for an optimization-based control method to achieve desired product characteristics such as a desired yield. The proposed method is demonstrated for the batch crystallization of mandelic acid in a crystallizer with a fines trap that is seeded with crystals of the desired enantiomer. The process is controlled with an optimization-based controller to minimize the ratio of the mass of crystals obtained from nuclei to the mass obtained from seeds while maintaining the desired enantioseparation.

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Comparison of objective functions for batch crystallization using a simple process model and Pontryagin’s minimum principle

Cited by 14 publications

References 28 publications

Synthesis of optimal control of technological processes based on a multialternative parametric description of the final state

Synthesis of optimal control of technological processes based on a multialternative parametric description of the final state

Dimensionless Framework for Seed Recipe Design and Optimal Control of Batch Crystallization

Optimal operation of batch enantiomer crystallization: From ternary diagrams to predictive control

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