Academic Journal
Mathematical modeling of a pH swing precipitation process and optimal model design.
| Τίτλος: | Mathematical modeling of a pH swing precipitation process and optimal model design. |
|---|---|
| Συγγραφείς: | Hiremath, Sandesh Athni, Hegde, Chinmay, Voigt, Andreas |
| Πηγή: | Applied Mathematics for Modern Challenges; Jun2026, Vol. 8, p1-53, 53p |
| Θεματικοί όροι: | Precipitation (Chemistry), Calcium carbonate, Stochastic partial differential equations, Stochastic differential equations, Particle size distribution, Artificial neural networks, Chemical kinetics, Mathematical models |
| Περίληψη: | In this work we consider the semi-batch process of precipitation of calcium carbonate solids from a solution containing calcium ions by adjusting the pH of the solution. The change in pH is induced either by the addition of an alkaline solution such as sodium hydroxide (NaOH) or by the addition of carbon dioxide gas (CO2) to the given ionic solution. Under this setup we propose a system of degenerate stochastic partial differential equations that is able to explain the dynamical behavior of the key components of precipitation process. In particular, we propose a semi-linear advection equation for the dynamics of particle size distribution (PSD) of the precipitated particles. This is in turn coupled with a system of stochastic differential equations (SDEs) that is able to explain the chemical kinetics between calcium ions (Ca2+), calcium carbonate (CaCO3) in aqueous state, and pH of the solution. The resulting coupled system is first mathematically studied, in particular conditions for the existence of a mild-solution is established and also the long time behavior of the system is established. Following this we consider the validation of the model using experimentally obtained lab-scale data. To this end, we propose three methods to fit the model with the data which also validates the suitability of the proposed model. The three methods include manual intuitive tuning, the classical forward backward SDE (FBSDE) method, and finally a DNN based method. The FBSDE method is based on the stochastic optimal control formulation for which we provide the necessary and sufficient condition for the existence of an optimal solution. Lastly, we compare the three methods and show that DNN method is the best in terms of the lowest error and as the most economical in terms of computational resources necessary during online use. [ABSTRACT FROM AUTHOR] |
| Copyright of Applied Mathematics for Modern Challenges is the property of American Institute of Mathematical Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Βάση Δεδομένων: | Complementary Index |
| FullText | Text: Availability: 0 |
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| Header | DbId: edb DbLabel: Complementary Index An: 194997441 RelevancyScore: 1082 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 1082.4189453125 |
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| Items | – Name: Title Label: Title Group: Ti Data: Mathematical modeling of a pH swing precipitation process and optimal model design. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Hiremath%2C+Sandesh+Athni%22">Hiremath, Sandesh Athni</searchLink><br /><searchLink fieldCode="AR" term="%22Hegde%2C+Chinmay%22">Hegde, Chinmay</searchLink><br /><searchLink fieldCode="AR" term="%22Voigt%2C+Andreas%22">Voigt, Andreas</searchLink> – Name: TitleSource Label: Source Group: Src Data: Applied Mathematics for Modern Challenges; Jun2026, Vol. 8, p1-53, 53p – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Precipitation+%28Chemistry%29%22">Precipitation (Chemistry)</searchLink><br /><searchLink fieldCode="DE" term="%22Calcium+carbonate%22">Calcium carbonate</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+partial+differential+equations%22">Stochastic partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+differential+equations%22">Stochastic differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Particle+size+distribution%22">Particle size distribution</searchLink><br /><searchLink fieldCode="DE" term="%22Artificial+neural+networks%22">Artificial neural networks</searchLink><br /><searchLink fieldCode="DE" term="%22Chemical+kinetics%22">Chemical kinetics</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+models%22">Mathematical models</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In this work we consider the semi-batch process of precipitation of calcium carbonate solids from a solution containing calcium ions by adjusting the pH of the solution. The change in pH is induced either by the addition of an alkaline solution such as sodium hydroxide (NaOH) or by the addition of carbon dioxide gas (CO2) to the given ionic solution. Under this setup we propose a system of degenerate stochastic partial differential equations that is able to explain the dynamical behavior of the key components of precipitation process. In particular, we propose a semi-linear advection equation for the dynamics of particle size distribution (PSD) of the precipitated particles. This is in turn coupled with a system of stochastic differential equations (SDEs) that is able to explain the chemical kinetics between calcium ions (Ca2+), calcium carbonate (CaCO3) in aqueous state, and pH of the solution. The resulting coupled system is first mathematically studied, in particular conditions for the existence of a mild-solution is established and also the long time behavior of the system is established. Following this we consider the validation of the model using experimentally obtained lab-scale data. To this end, we propose three methods to fit the model with the data which also validates the suitability of the proposed model. The three methods include manual intuitive tuning, the classical forward backward SDE (FBSDE) method, and finally a DNN based method. The FBSDE method is based on the stochastic optimal control formulation for which we provide the necessary and sufficient condition for the existence of an optimal solution. Lastly, we compare the three methods and show that DNN method is the best in terms of the lowest error and as the most economical in terms of computational resources necessary during online use. [ABSTRACT FROM AUTHOR] – Name: Abstract Label: Group: Ab Data: <i>Copyright of Applied Mathematics for Modern Challenges is the property of American Institute of Mathematical Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.3934/ammc.2026003 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 53 StartPage: 1 Subjects: – SubjectFull: Precipitation (Chemistry) Type: general – SubjectFull: Calcium carbonate Type: general – SubjectFull: Stochastic partial differential equations Type: general – SubjectFull: Stochastic differential equations Type: general – SubjectFull: Particle size distribution Type: general – SubjectFull: Artificial neural networks Type: general – SubjectFull: Chemical kinetics Type: general – SubjectFull: Mathematical models Type: general Titles: – TitleFull: Mathematical modeling of a pH swing precipitation process and optimal model design. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Hiremath, Sandesh Athni – PersonEntity: Name: NameFull: Hegde, Chinmay – PersonEntity: Name: NameFull: Voigt, Andreas IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 29947669 Numbering: – Type: volume Value: 8 Titles: – TitleFull: Applied Mathematics for Modern Challenges Type: main |
| ResultId | 1 |