Application of the differential maximum entropy method (DMaxEnt) for the solution of population balance equations.

Zacharioudakis, Emmanouil; Ζαχαρουδιάκης, Εμμανουήλ

dc.contributor.author	Zacharioudakis, Emmanouil	en
dc.contributor.author	Ζαχαρουδιάκης, Εμμανουήλ	el
dc.date.accessioned	2025-03-28T12:55:03Z
dc.date.available	2025-03-28T12:55:03Z
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/61530
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.29226
dc.description	Εθνικό Μετσόβιο Πολυτεχνείο--Μεταπτυχιακή Εργασία. Διεπιστημονικό-Διατμηματικό Πρόγραμμα Μεταπτυχιακών Σπουδών (Δ.Π.Μ.Σ.) “Υπολογιστική Μηχανική”	el
dc.rights	Αναφορά Δημιουργού-Μη Εμπορική Χρήση 3.0 Ελλάδα	*
dc.rights.uri	http://creativecommons.org/licenses/by-nc/3.0/gr/	*
dc.subject	Crystallization	en
dc.subject	Maximum entropy	en
dc.subject	Population balance	en
dc.title	Application of the differential maximum entropy method (DMaxEnt) for the solution of population balance equations.	en
heal.type	masterThesis
heal.classification	Computational mechanics	en
heal.language	en
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2024-11-04
heal.abstract	Crystallization is a complex process that includes many independent mechanisms and can be an economical and efficient separation process. Its application in the Pharmaceutical Industry for the separation of chiral Active Pharmaceutical Ingredients is of great importance, since their activity is usually attributed to one of the two enantiomers. There are also examples where one of the two enantiomers has a toxic effect. Finally, the scope and complexity of crystallization applications necessitate detailed mathematical modeling of the process to perform simulations with accurate results. Population Balance Equations (PBE) are widely used to model crystallization processes and predict the dynamic behavior of the distribution of crystalline particles, as a function of system parameters and individual crystallization mechanisms (e.g., growth, breakage, aggregation, racemization). PBEs define a system of partial differential equations, and their numerical solution is usually a demanding task. Among the various numerical techniques, we report the finite element and finite volume method with substantial computational demands. Alternatively, one can resort to the Method of Moments (MoM) or the Differential Maximum Entropy method (DMaxEnt). MoM is used to transform the PBEs into a system of ordinary differential equations, modeling the time evolution of the particle distribution’s statistical moments, thus reducing the problem’s order of magnitude. Similarly to MoM, DMaxEnt is also used to reduce the problem’s order of magnitude, but in this case PBEs are transformed into a system of ordinary differential equations, modeling the time evolution of the particle distribution’s Lagrange multipliers. Those are a series of discrete values used to approach the particle size distribution function. In this thesis, we study the mechanisms of growth, breakage, aggregation, and racemization, and we develop a computational model that calculates the evolution of the statistical moments of crystal distributions. This is achieved in two ways: first, using the Method of Moments combined with the Maximum Entropy Method, which reconstructs the particle distribution from a finite number of moments (the problem of moments), and finally, using the Differential Maximum Entropy (DMaxEnt) method. The aim of this thesis is, initially, to validate DMaxEnt, by solving simple benchmark cases, and comparing the solution with the analytical solution (when available) or the solution resulting from the Finite Element Method (FEM). Then the effect of individual crystallization mechanisms on the particle size distribution and its moments is studied, under isothermal conditions. The results of the model are compared with those of the analytical solution (when available) and those of FEM, obtained by using Comsol Multiphysics software. MoM combined with MaxEnt has a great advantage in terms of computational time of crystallization process simulations, and produces relatively accurate results, compared to FEM. By going one step further and utilizing DMaxEnt, one can achieve results with similar accuracy as MoM, while using a fraction of the total computational time needed from FEM. Finally, we observe that differences bothP a g e 4 \| 80 in accuracy and computational time increases as the complexity of the simulated crystallization process increases.	en
heal.advisorName	Kavousanakis, Michail	en
heal.committeeMemberName	Papathanasiou, Athanasios	en
heal.committeeMemberName	Kokkoris, Georgios	en
heal.academicPublisher	Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Χημικών Μηχανικών	el
heal.academicPublisherID	ntua
heal.numberOfPages	80 σ.	el
heal.fullTextAvailability	false