{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T12:02:52Z","timestamp":1768478572942,"version":"3.49.0"},"reference-count":61,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,11,4]],"date-time":"2018-11-04T00:00:00Z","timestamp":1541289600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51509004"],"award-info":[{"award-number":["51509004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The flocculation of cohesive sediment plays an important role in affecting morphological changes to coastal areas, to dredging operations in navigational canals, to sediment siltation in reservoirs and lakes, and to the variation of water quality in estuarine waters. Many studies have been conducted recently to formulate a turbulence-induced flocculation model (described by a characteristic floc size with respect to flocculation time) of cohesive sediment by virtue of theoretical analysis, numerical modeling, and\/or experimental observation. However, a probability study to formulate the flocculation model is still lacking in the literature. The present study, therefore, aims to derive an explicit expression for the flocculation of cohesive sediment in a turbulent fluid environment based on two common entropy theories: Shannon entropy and Tsallis entropy. This study derives an explicit expression for the characteristic floc size, assumed to be a random variable, as a function of flocculation time by maximizing the entropy function subject to the constraint equation using a hypothesis regarding the cumulative distribution function of floc size. It was found that both the Shannon entropy and the Tsallis entropy theories lead to the same expression. Furthermore, the derived expression was tested with experimental data from the literature and the results were compared with those of existing deterministic models, showing that it has good agreement with the experimental data and that it has a better prediction accuracy for the logarithmic growth pattern of data in comparison to the other models, whereas, for the sigmoid growth pattern of experimental data, the model of Keyvani and Strom or Son and Hsu model could be the better choice for floc size prediction. Finally, the maximum capacity of floc size growth, a key parameter incorporated into this expression, was found to exhibit an empirical power relationship with the flow shear rate.<\/jats:p>","DOI":"10.3390\/e20110845","type":"journal-article","created":{"date-parts":[[2018,11,5]],"date-time":"2018-11-05T10:43:45Z","timestamp":1541414625000},"page":"845","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A Simple Explicit Expression for the Flocculation Dynamics Modeling of Cohesive Sediment Based on Entropy Considerations"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0579-608X","authenticated-orcid":false,"given":"Zhongfan","family":"Zhu","sequence":"first","affiliation":[{"name":"College of Water Sciences, Beijing Normal University, Xinjiekouwai Street 19, Beijing 100875, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s10652-007-9050-7","article-title":"Flocculation model of cohesive sediment using variable fractal dimension","volume":"8","author":"Son","year":"2008","journal-title":"Environ. Fluid Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.ecss.2009.09.028","article-title":"Factors controlling the field settling velocity of cohesive sediment in estuaries","volume":"87","author":"Pejrup","year":"2010","journal-title":"Estuar. Coast Shelf Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1080\/00221689809498621","article-title":"A simple model for turbulence induced flocculation of cohesive sediment","volume":"36","author":"Winterwerp","year":"1998","journal-title":"J. Hydraul. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2668","DOI":"10.1016\/j.csr.2008.09.001","article-title":"Modeling flocculation processes of fine-grained particles using a size-resolved method: Comparison with published laboratory experiments","volume":"28","author":"Xu","year":"2008","journal-title":"Cont. Shelf Res."},{"key":"ref_5","first-page":"14327","article-title":"Sediment processes in estuaries: Future research requirements","volume":"94","author":"Dyer","year":"1989","journal-title":"J. Geophys. Res. Oceans 1978\u20132012"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4385","DOI":"10.3390\/w7084385","article-title":"Fractal dimension of cohesive sediment flocs at steady state under seven shear flow conditions","volume":"7","author":"Zhu","year":"2015","journal-title":"Water"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2118","DOI":"10.1002\/jgrc.20086","article-title":"The settling velocity of mineral, biomineral, and biological particles and aggregates in water","volume":"118","author":"Maggi","year":"2013","journal-title":"J. Geophys. Res. Ocean"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.margeo.2017.10.001","article-title":"Investigation of flocculation dynamics under changing hydrodynamic forcing on an intertidal mudflat","volume":"395","author":"Guo","year":"2018","journal-title":"Mar. Geol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.jhydrol.2009.07.040","article-title":"Biological flocculation of suspended particles in nutrient-rich aqueous ecosystems","volume":"376","author":"Maggi","year":"2009","journal-title":"J. Hydrol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7536","DOI":"10.1002\/2017WR020628","article-title":"Modeling sediment transport with an integrated view of the biofilm effects","volume":"53","author":"Fang","year":"2017","journal-title":"Water Resour. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1061\/JSEDAI.0001389","article-title":"Floc breakup in turbulent flocculation processes","volume":"98","author":"Parker","year":"1972","journal-title":"J. Sanit. Eng. Div."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1006\/jcis.1996.4710","article-title":"Aggregation and breakup of particles in a shear flow","volume":"187","author":"Serra","year":"1997","journal-title":"J. Colloid Interface Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/S0001-8686(98)00062-1","article-title":"Kinetics of fine particle aggregation in turbulence","volume":"78","author":"Lu","year":"1998","journal-title":"Adv. Colloid Inter. Sci."},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s10236-011-0384-9","article-title":"Macroflocs, fine-grained sediment transport, and their longitudinal variations","volume":"61","year":"2011","journal-title":"Ocean Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.margeo.2014.11.014","article-title":"Modeling floc size distribution of suspended cohesive sediments using quadrature method of moments","volume":"359","author":"Shen","year":"2015","journal-title":"Mar. Geol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1016\/j.scitotenv.2017.07.194","article-title":"Fractal dimensions of large aggregates under different flocculation conditions","volume":"609","author":"Moruzzi","year":"2017","journal-title":"Sci. Total Environ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.ecss.2006.02.003","article-title":"A heuristic formula for turbulence-induced flocculation of cohesive sediment","volume":"68","author":"Winterwerp","year":"2006","journal-title":"Estuar. Coast Shelf Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1314","DOI":"10.1016\/j.csr.2010.04.014","article-title":"An idealized model study of flocculation on sediment trapping in an estuarine turbidity maximum","volume":"30","author":"Xu","year":"2010","journal-title":"Cont. Shelf Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"S64","DOI":"10.1016\/j.csr.2010.02.005","article-title":"Behaviour of a floc population during a tidal cycle: Laboratory experiments and numerical modelling","volume":"31","author":"Verney","year":"2011","journal-title":"Cont. Shelf Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2142","DOI":"10.1002\/2015JC011169","article-title":"Flocculation in a decaying shear field and its implications for mud removal in near-field river mouth discharges","volume":"121","author":"Strom","year":"2016","journal-title":"J. Geophys. Res. Oceans"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1849","DOI":"10.5194\/gmd-11-1849-2018","article-title":"Cohesive and mixed sediment in the regional ocean modeling system (ROMS v3.6) implemented in the coupled ocean atmosphere wave sediment-transport modeling system (COAWST r1179)","volume":"11","author":"Sherwood","year":"2018","journal-title":"Geosci. Model Develop."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6736","DOI":"10.1029\/2018JC014154","article-title":"A shear-limited flocculation model for dynamically predicting average floc size","volume":"123","author":"Kuprenas","year":"2018","journal-title":"J. Geophys. Res. Oceans"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/0021-9797(92)90149-G","article-title":"Shear-induced aggregation and breakup of polystyrene latex particles","volume":"154","author":"Oles","year":"1992","journal-title":"J. Colloid Interface Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1016\/j.powtec.2012.11.014","article-title":"Effect of shear rate on aggregate size and structure in the process of aggregation and at steady state","volume":"235","author":"Bubakova","year":"2013","journal-title":"Powder Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/S0021-9797(03)00446-6","article-title":"Characterizing flocculation under heterogeneous turbulence","volume":"264","author":"Hopkins","year":"2003","journal-title":"J. Colloid Interface Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/S0025-3227(99)00013-4","article-title":"A laboratory examination of floc characteristics with regard to turbulent shearing","volume":"160","author":"Manning","year":"1999","journal-title":"Mar. Geol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1579","DOI":"10.1016\/S0043-1354(98)00392-3","article-title":"Flocculation modelling: a review","volume":"33","author":"Thomas","year":"1999","journal-title":"Water Res."},{"key":"ref_29","first-page":"24","article-title":"Theory on orthokinetic flocculation of cohesive sediment: a review","volume":"2","author":"Zhu","year":"2014","journal-title":"J. Geosci. Environ. Prot."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2739","DOI":"10.1016\/S0043-1354(03)00082-4","article-title":"Floc morphology and size distributions of cohesive sediment in steady-state flow","volume":"37","author":"Stone","year":"2003","journal-title":"Water Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1006\/jcis.1998.5714","article-title":"Structure of the aggregates during the process of aggregation and breakup under a shear flow","volume":"206","author":"Serra","year":"1998","journal-title":"J. Colloid Interface Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.jhydrol.2007.05.035","article-title":"Effect of variable fractal dimension on the floc size distribution of suspended cohesive sediment","volume":"343","author":"Maggi","year":"2007","journal-title":"J. Hydrol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.margeo.2014.04.010","article-title":"Influence of cycles of high and low turbulent shear on the growth rate and equilibrium size of mud flocs","volume":"354","author":"Keyvani","year":"2014","journal-title":"Mar. Geol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3582","DOI":"10.1016\/j.watres.2009.05.016","article-title":"The effect of variable yield strength and variable fractal dimension on flocculation of cohesive sediment","volume":"43","author":"Son","year":"2009","journal-title":"Water Res."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"218","DOI":"10.2112\/JCOASTRES-D-15-00110.1","article-title":"A dynamic model for coastal mud flocs with distributed fractal dimension","volume":"33","author":"Xu","year":"2017","journal-title":"J. Coast. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.jcis.2016.12.042","article-title":"Dynamics of aggregate size and shape properties under sequenced flocculation in a turbulent Taylor-Couette reactor","volume":"491","author":"Guerin","year":"2017","journal-title":"J. Colloid Interface Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"641","DOI":"10.3390\/e19120641","article-title":"Tsallis entropy theory for modelling in water engineering: A review","volume":"19","author":"Singh","year":"2017","journal-title":"Entropy"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1061\/(ASCE)0733-9429(1995)121:1(26)","article-title":"Maximum and mean velocities and entropy in open-channel flow","volume":"121","author":"Chiu","year":"1995","journal-title":"J. Hydraul. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1061\/(ASCE)HE.1943-5584.0000793","article-title":"One dimensional velocity distribution in open channels using Tsallis entropy","volume":"19","author":"Cui","year":"2014","journal-title":"J. Hydrol. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1061\/(ASCE)HE.1943-5584.0000319","article-title":"Entropy theory for two-dimensional velocity distribution","volume":"16","author":"Luo","year":"2011","journal-title":"J. Hydrol. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1007\/s00477-016-1221-y","article-title":"One-dimensional velocity distribution in open channels using Renyi entropy","volume":"31","author":"Kumbhakar","year":"2017","journal-title":"Stochastic. Environ. Res. Risk Assess."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1061\/(ASCE)0733-9429(2000)126:1(16)","article-title":"Mathematical models of distribution of sediment concentration","volume":"1","author":"Chiu","year":"2000","journal-title":"J. Hydraul. Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"966","DOI":"10.1061\/(ASCE)HE.1943-5584.0000865","article-title":"Suspended sediment concentration in open channels using Tsallis entropy","volume":"19","author":"Cui","year":"2013","journal-title":"J. Hydrol. Eng."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"494","DOI":"10.1016\/j.physa.2016.08.068","article-title":"Derivation of Rouse equation for sediment concentration using Shannon entropy","volume":"465","author":"Kumbhakar","year":"2017","journal-title":"Physics A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/s00477-002-0088-2","article-title":"An attempt at using the entropy approach to predict the transverse distribution of boundary shear stress in open channel flow","volume":"16","author":"Sterling","year":"2002","journal-title":"Stochastic. Environ. Res. Risk Assess."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00477-014-0959-3","article-title":"Comparison between Shannon and Tsallis entropies for prediction of shear stress distribution in open channels","volume":"29","author":"Bonakdari","year":"2015","journal-title":"Stochastic. Environ. Res. Risk Assess."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communications, I and II","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_48","first-page":"168","article-title":"Stochastic flocculation of cohesive sediment: analysis of floc mobility within the floc size spectrum","volume":"440","author":"Maggi","year":"2008","journal-title":"Water Resour. Res."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"2527","DOI":"10.3390\/w7052527","article-title":"Stochastic flocculation model for cohesive sediment suspended in water","volume":"47","author":"Shin","year":"2015","journal-title":"Water"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics I","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1103\/PhysRev.108.171","article-title":"Information theory and statistical mechanics II","volume":"108","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1109\/PROC.1982.12425","article-title":"On the rationale of maximum entropy methods","volume":"70","author":"Jaynes","year":"1982","journal-title":"Proc. IEEE"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1061\/(ASCE)0733-9429(1987)113:5(583)","article-title":"Entropy and probability concepts in hydraulics","volume":"113","author":"Chiu","year":"1987","journal-title":"J. Hydraul. Eng."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.physa.2017.08.023","article-title":"Formulating the shear stress distribution in circular open channels based on the Renyi entropy","volume":"490","author":"Khozani","year":"2018","journal-title":"Physics A"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"123","DOI":"10.13031\/2013.36266","article-title":"A Shannon entropy-based general derivation of infiltration equations","volume":"54","author":"Singh","year":"2011","journal-title":"Trans. ASABE"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"447","DOI":"10.13031\/2013.29585","article-title":"Tsallis entropy theory for derivation of infiltration equations","volume":"53","author":"Singh","year":"2010","journal-title":"Trans. ASABE"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"8323","DOI":"10.1029\/JC094iC06p08323","article-title":"The flocculation of fine\u2014Grained sediments in estuarine waters","volume":"94","author":"Burban","year":"1989","journal-title":"J. Geophys. Res. Ocean"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1048","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_60","doi-asserted-by":"crossref","first-page":"1974","DOI":"10.1021\/la010702h","article-title":"Aggregation mechanisms of rates of different particle sizes in a controlled shear environment","volume":"18","author":"Selomulya","year":"2002","journal-title":"Langmuir"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"2994","DOI":"10.1016\/j.watres.2005.04.076","article-title":"Experimental analysis of coagulation of particles under low-shear flow","volume":"39","author":"Colomer","year":"2005","journal-title":"Water Res."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/11\/845\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:27:54Z","timestamp":1760196474000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/11\/845"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,4]]},"references-count":61,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2018,11]]}},"alternative-id":["e20110845"],"URL":"https:\/\/doi.org\/10.3390\/e20110845","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,4]]}}}