{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,17]],"date-time":"2026-09-17T22:37:28Z","timestamp":1789684648927,"version":"4.0.1"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,18]],"date-time":"2024-01-18T00:00:00Z","timestamp":1705536000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study presents a mathematical description of the solid fraction aggregation process in the presence of a flocculant and its result. The basis is a population balance equation. The model is realized in Python language. Verification was carried out using red mud from the investigated enterprise; Flomin AL P 99 VHM was used as a flocculant. The mean square deviation for the parameter \u201cmean aggregate diameter\u201d is equal to 19.88 \u03bcm. The time required for the model calculation is about 3 min. The time spent on modeling depends on the number of calculation channels. In this study, 40 channels (20 with PSD source data, and 20 with empty values required for the calculation) were used for the calculation. The time spent on the model calculation is much shorter than the inertia via each of the communication channels for the studied symmetric radial type thickener. A user interface is developed, where the input parameters are the initial pulp particle size distribution, viscosity and density of pulp in the thickener, particle surface area, concentration and flow rate of flocculant, concentration of solid particles, inner diameter and height of the feed well, and simulation time. The result of the simulation is particle size distribution in the feed well of the washer and the mean flocculus diameter.<\/jats:p>","DOI":"10.3390\/sym16010114","type":"journal-article","created":{"date-parts":[[2024,1,18]],"date-time":"2024-01-18T06:41:22Z","timestamp":1705560082000},"page":"114","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Modeling of Particle Size Distribution in the Presence of Flocculant"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0459-9754","authenticated-orcid":false,"given":"Elmira","family":"Fedorova","sequence":"first","affiliation":[{"name":"Department of Mineral Processing, Automation of Technological Processes and Production, St. Petersburg Mining University of Empress Catherine II, 199106 St. Petersburg, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elena","family":"Pupysheva","sequence":"additional","affiliation":[{"name":"Department of Mineral Processing, Automation of Technological Processes and Production, St. Petersburg Mining University of Empress Catherine II, 199106 St. Petersburg, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vladimir","family":"Morgunov","sequence":"additional","affiliation":[{"name":"Department of Mineral Processing, Automation of Technological Processes and Production, St. Petersburg Mining University of Empress Catherine II, 199106 St. Petersburg, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Concha, A.F. (2014). Solid\u2013Liquid Separation in the Mining Industry, Springer. Fluid Mechanics and Its Applications.","DOI":"10.1007\/978-3-319-02484-4"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1039\/tf9524800166","article-title":"A theory of sedimentation","volume":"48","author":"Kynch","year":"1952","journal-title":"Trans. Faraday Soc."},{"key":"ref_3","first-page":"41","article-title":"Study of the effects of steel and polymer pipe roughness on the pressure loss in tailings slurry hydrotransport","volume":"4","author":"Serzhan","year":"2023","journal-title":"Ore Process. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"686","DOI":"10.31897\/PMI.2022.91","article-title":"Monitoring of grinding condition in drum mills based on resulting shaft torque","volume":"256","author":"Zhukovskiy","year":"2022","journal-title":"J. Min. Inst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1080\/08827508.2017.1415207","article-title":"Effect of grinding on the fractional composition of polymineral laminated bituminous shales","volume":"39","author":"Nikolaeva","year":"2018","journal-title":"Miner. Process. Extr. Met. Rev."},{"key":"ref_6","first-page":"261","article-title":"Study of the kinetics of the process of producing pellets from red mud in a hydrogen flow","volume":"254","author":"Khalifa","year":"2022","journal-title":"J. Min. Inst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"14","DOI":"10.17580\/tsm.2023.04.02","article-title":"Augmented reality as a means of metallurgical equipment servicing","volume":"4","author":"Koteleva","year":"2023","journal-title":"Tsvetnye Met."},{"key":"ref_8","unstructured":"Laros, T., Slottee, S., and Baczek, F. (2002). Mineral Processing Plant Design, Practice, and Control: Proceedings Volume 1, Society for Mining, Metallurgy, and Exploration."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Fawell, P., Farrow, J.B., Heath, A., Nguyen, T., Owen, A.T., Rudman, M., Scales, P., Simic, K., Stephens, D., and Swift, J.D. (2009, January 21\u201324). 20 years of AMIRA P266 \u2018Improving Thickener Technology\u2019\u2014How has it changed the understanding of thickener performance?. Proceedings of the 12th International Seminar on Paste and Thickened Tailings, Vi\u0148a del Mar, Chile.","DOI":"10.36487\/ACG_repo\/963_7"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/S0301-7516(01)00038-2","article-title":"Settling velocities of particulate systems: 12","volume":"63","author":"Concha","year":"2001","journal-title":"Int. J. Miner. Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/S0301-7516(03)00073-5","article-title":"A mathematical model for batch and continuous thickening of flocculated suspensions in vessels with varying cross-section","volume":"73","author":"Damasceno","year":"2004","journal-title":"Int. J. Miner. Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"400","DOI":"10.4028\/www.scientific.net\/DDF.410.400","article-title":"To the Issue of Mathematical Modeling of the Red Mud Thickening Process","volume":"410","author":"Salamatov","year":"2021","journal-title":"Defect Diffus. Forum"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"108364","DOI":"10.1016\/j.mineng.2023.108364","article-title":"Structural evolution of bed drainage channels under the shear effect of the whole process of tailings thickening","volume":"203","author":"Chen","year":"2023","journal-title":"Miner. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.powtec.2021.07.008","article-title":"Effect of interparticle interactions on the yield stress of thickened flocculated copper mineral tailings slurry","volume":"392","author":"Ruan","year":"2021","journal-title":"Powder Technol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.ijmst.2013.01.010","article-title":"Simulation of a semi-industrial pilot plant thickener using CFD approach","volume":"23","author":"Gheshlaghi","year":"2013","journal-title":"Int. J. Min. Sci. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.watres.2017.01.049","article-title":"Batch settling curve registration via image data modeling","volume":"114","author":"Derlon","year":"2017","journal-title":"Water Res."},{"key":"ref_17","unstructured":"Ardila Labiosa, A. (2010). Dynamic Simulation of Red Mud Washers Used in Aluminum Industries. [Master\u2019s Thesis, RMIT University]."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"121280","DOI":"10.1016\/j.ijpharm.2021.121280","article-title":"In-line particle size measurement based on image analysis in a fully continuous granule manufacturing line for rapid process understanding and development","volume":"612","author":"Csorba","year":"2022","journal-title":"Int. J. Pharm."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"61","DOI":"10.17580\/tsm.2023.04.08","article-title":"Modern equipment by Soyuztsvetmetavtomatika for detecting aerosols and spills of harmful pollutants","volume":"4","author":"Oksengoyt","year":"2023","journal-title":"Tsvetnye Met."},{"key":"ref_20","unstructured":"Golberg, G.Y. (2019). Development of the Theory of Formation and Destruction of Flocculation Structures in the Processes of Separation of Suspensions of Fine Coal Enrichment Products. [Ph.D. Thesis, Russian Academy of Sciences]."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"41","DOI":"10.2166\/wst.2002.0092","article-title":"Modelling the activated sludge flocculation process combining laser light diffraction particle sizing and population balance modelling (PBM)","volume":"45","author":"Nopens","year":"2002","journal-title":"Water Sci. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.seppur.2017.03.048","article-title":"Flocculation properties and kinetic investigation of polyacrylamide with different cationic monomer content for high turbid water purification","volume":"182","author":"Ma","year":"2017","journal-title":"Sep. Purif. Technol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"108304","DOI":"10.1016\/j.mineng.2023.108304","article-title":"Review of influence of steric effect on aggregation behavior of fine particles","volume":"203","author":"Xu","year":"2023","journal-title":"Miner. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Quezada, G.R., Ayala, L., Leiva, W.H., Toro, N., Toledo, P.G., Robles, P., and Jeldres, R.I. (2020). Describing Mining Tailing Flocculation in Seawater by Population Balance Models: Effect of Mixing Intensity. Metals, 10.","DOI":"10.3390\/met10020240"},{"key":"ref_25","unstructured":"Health, A.R., and Koh, P.T.L. (2003, January 10\u201312). Combined population balance and CFD modeling of particle aggregation by polymeric flocculant. Proceedings of the Third International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, VIC, Australia."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2893","DOI":"10.1016\/S0009-2509(03)00133-7","article-title":"Particle size distributions and viscosity of suspensions undergoing shear-induced coagulation and fragmentation","volume":"58","author":"Barthelmes","year":"2003","journal-title":"Chem. Eng. Sci."},{"key":"ref_27","unstructured":"Ahrens, R. (2020). Efficient Numerical Treatment of Aggregation Integrals in Multivariate Population Balance Equations. [Ph.D. Thesis, Technischen Universit\u00e4t Hamburg]."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2542","DOI":"10.1016\/S0043-1354(99)00431-5","article-title":"Activated sludge flocculation: On-line determination of floc size and the effect of shear","volume":"34","author":"Biggs","year":"2000","journal-title":"Water Res."},{"key":"ref_29","first-page":"129","article-title":"Versuch einer mathematischen Theorie der Koagulations kinetik kolloider L\u00f6sungen","volume":"129","year":"1917","journal-title":"Z. Phys. Chem."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1641","DOI":"10.1002\/aic.10749","article-title":"Polymer flocculation of calcite: Population balance model","volume":"52","author":"Heath","year":"2006","journal-title":"AIChE J."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1821","DOI":"10.1002\/aic.690341108","article-title":"A discretized population balance for nucleation, growth, and aggregation","volume":"34","author":"Hounslow","year":"1988","journal-title":"AIChE J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1016\/0043-1354(95)00253-7","article-title":"Shear-induced flocculation: The evolution of floc structure and the shape of the size distribution at steady state","volume":"30","author":"Spicer","year":"1996","journal-title":"Water Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1006\/jcis.1996.0005","article-title":"Hydrodynamics of fractal aggregates with radially varying permeability","volume":"177","author":"Veerapaneni","year":"1996","journal-title":"J. Colloid Interface Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1017\/S0022112056000020","article-title":"On the collision of drops in turbulent clouds","volume":"1","author":"Saffman","year":"1956","journal-title":"J. Fluid Mech."},{"key":"ref_35","unstructured":"Gandurina, L.V. (2023, October 20). Organicheskie flokulyanty` v Texnologii Ochistki Prirodny`x i Promy`Shlenny`x Stochny`x vod i Obrabotki Osadka. Inzhenernoe Obespechenie ob``Ektov Stroitel`stva: Obzornaya Informacii\/VNIINTPI,-M. Available online: https:\/\/meganorm.ru\/Data2\/1\/4293841\/4293841858.htm."},{"key":"ref_36","first-page":"404","article-title":"O vliyanii flokulyantov na kinetiku processov obezvozhivaniya i promy`vki krasny`x shlamov iz nizkokremnisty`x boksitov pri proizvodstve glinozema","volume":"23","author":"Salamatov","year":"2019","journal-title":"Vestn. Irkutsk. Gos. Texnicheskogo Univ."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1605","DOI":"10.5829\/IJE.2023.36.09C.03","article-title":"Standards for Selection of Surfactant Compositions used in Completion and Stimulation Fluids","volume":"36","author":"Petrakov","year":"2023","journal-title":"Int. J. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"677","DOI":"10.31897\/PMI.2022.77","article-title":"Adaptive approach formation using machine vision technology to determine the parameters of enrichment products deposition","volume":"256","author":"Romashev","year":"2022","journal-title":"J. Min. Inst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"35","DOI":"10.17580\/or.2023.03.06","article-title":"Development of composite materials based on red mud","volume":"3","author":"Piirainen","year":"2023","journal-title":"Obogashchenie Rud"},{"key":"ref_40","first-page":"1","article-title":"Mathematical modeling of the electric field of an in-line diagnostic probe of a cathode-polarized pipeline","volume":"2023","author":"Krizsky","year":"2023","journal-title":"J. Min. Inst."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Ilyushin, Y.V., and Kapostey, E.I. (2023). Developing a Comprehensive Mathematical Model for Aluminium Production in a Soderberg Electrolyser. Energies, 16.","DOI":"10.3390\/en16176313"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"32","DOI":"10.17580\/or.2022.06.06","article-title":"Specific features of the use of pulverized coal fuel in combined chemical processing","volume":"2022","author":"Dubovikov","year":"2022","journal-title":"Obogashchenie Rud"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"53","DOI":"10.17580\/tsm.2023.04.07","article-title":"Automatic burden balance monitoring and control in the production of metallurgical silicon","volume":"4","author":"Bazhin","year":"2023","journal-title":"Tsvetnye Met."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/114\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:49:39Z","timestamp":1760104179000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/114"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,18]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["sym16010114"],"URL":"https:\/\/doi.org\/10.3390\/sym16010114","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,18]]}}}