Population balance modelling of particle flocculation with attention to aggregate restructuring and permeability

New insight of biophysical aggregates' geometric distributions from side and bottom views during their flocculation and settling in saline water

Biophysical mud aggregates, consisting of clay minerals and extracellular polymeric substances (EPS), play critical roles in water quality and aquatic ecosystem health. However, the geometric properties from different side views and flocculation dynamic behavior of these aggregates with highly irregular shapes in saline water environments remain poorly understood, particularly under the combined effects of buoyancy and gravity during settling processes. This study introduces a novel dual-view approach, combining an inverted depth-of-field microscope and side camera, to capture high-resolution images of floc geometries from both side and bottom perspectives. A series of laboratory-controlled experiments were conducted to investigate floc formed from pure clay mineral and clay-EPS mixtures, analyzing their size, shape and settlings. The results reveal significant discrepancies between side-view and bottom-view measurements, demonstrating that bottom-view imaging provides a more accurate understanding of floc geometry considering buoyancy and gravity during flocs settling in saline water. Moreover, flocs formed from clay-EPS mixtures exhibited greater size and irregularity compared to those formed solely of clay minerals. EPS-mediated biological cohesion was found to promote the formation of larger, stable macroflocs (≥200 μm) capable of maintaining their structure under turbulent conditions, while pure mineral flocs remained smaller and more compact. Despite increasing floc size, EPS reduced effective density, resulting in overlapping settling velocities between macroflocs and microflocs (<200 μm). This study provides a foundational dataset and innovative methodology for analyzing mud floc geometries, offering new insights into the role of bio-cohesion in natural sediment flocculation dynamics. These findings underscore the need for flocculation models to incorporate organic influences and advocate for multi-perspective observations to improve predictions of fine-grained sediment transport in saline environments.

Growth of Floc Structure and Subsequence Compaction into Smaller Granules through Breakup and Rearrangement of Aluminum Flocs in a Constant Laminar Shear Flow

We have constructed an outer-cylinder-rotating Couette device for high-speed shear flow in laminar flow conditions and visualized the structure formation and subsequent rearrangement of PACl (flocculant made of aluminum hydroxide gel) and kaolinite flocs by visible light imaging. In a previous report, we analyzed the case of relatively low shear rate (G-value = 29 1/s) and confirmed that the flocculation process could be separated into two stages: a floc growth stage and a breakup/rearrangement stage. Once the large bulky flocs that reached the maximum size appeared, they rearranged and densified through structural fracture and rearrangement. In this report, this process was further investigated by conducting experiments under two different high shear rates (58 and 78 1/s) at which breakup and rearrangement became more pronounced, and three different aluminum kaolinite ratios (ALT ratios) that were over and under the optimum dosage (neutralization point by Zeta potential). Visualization results confirmed that, during the growth stage, the flocculation rate could be approximated by a scaling relationship between floc size and elapsed time, which depended on the ALT ratio. After reaching the maximum size, the floc rapidly became compact and dense following adsorption of the gel, incorporating fine fragments from erosion breakup. The over and under dosages created a lot of fragments of erosion breakup, but less so in the optimum dosage. In the optimum ALT ratio, fragments did not remain because they were incorporated into the flocs and densified, and the floc size was thought to be maintained. The floc circularity distribution peaked at around 0.6 and 1, suggesting that the flocs were spherical in shape due to erosion breakup.

On the importance of temporal floc size statistics and yield strength for population balance equation flocculation model

Many aquatic environments contain cohesive sediments that flocculate and create flocs with a wide range of sizes. The Population Balance Equation (PBE) flocculation model is designed to predict the time-dependent floc size distribution and should be more complete than models based on median floc size. However, a PBE flocculation model includes many empirical parameters to represent important physical, chemical, and biological processes. We report a systematic investigation of key model parameters of the open-source PBE-based size class flocculation model FLOCMOD (Verney, Lafite, Claude Brun-Cottan and Le Hir, 2011) using the measured temporal floc size statistics reported by Keyvani and Strom (2014) at a constant turbulent shear rate S. Results show that the median floc size d₅₀, in terms of both the equilibrium floc size and the initial floc growth, is insufficient to constrain the model parameters. A comprehensive error analysis shows that the model is capable of predicting three floc size statistics d₁₆, d₅₀ and d₈₄, which also reveals a clear trend that the best calibrated fragmentation rate (inverse of floc yield strength) is proportional to the floc size statistics considered. Motivated by this finding, the importance of floc yield strength is demonstrated in the predicted temporal evolution of floc size by modeling the floc yield strength as microflocs and macroflocs giving two corresponding fragmentation rates. The model shows a significantly improved agreement in matching the measured floc size statistics.

Persistent reshaping of cohesive sediment towards stable flocs by turbulence

Cohesive sediment forms flocs of various sizes and structures in the natural turbulent environment. Understanding flocculation is critical in accurately predicting sediment transport and biogeochemical cycles. In addition to aggregation and breakup, turbulence also reshapes flocs toward more stable structures. An Eulerian-Lagrangian framework has been implemented to investigate the effect of turbulence on flocculation by capturing the time-evolution of individual flocs. We have identified two floc reshaping mechanisms, namely breakage-regrowth and restructuring by hydrodynamic drag. Surface erosion is found to be the primary breakup mechanism for strong flocs, while fragile flocs tend to split into fragments of similar sizes. Aggregation of flocs of sizes comparable to or greater than the Kolmogorov scale is modulated by turbulence with lower aggregation efficiency. Our findings highlight the limiting effects of turbulence on both floc size and structure.

Multiphase Flow Characteristics in the Classification Process of a Novel Wide Neck Thickener: Experiment and Simulation

A novel thickening equipment known as a wide neck thickener (WNT) was designed to solve the problem of depending only on gravity settlement of the thickener. The computational fluid dynamics method with the Reynolds stress and the volume of fluid models and the particle image velocimetry experimental method were both applied to investigate the pressure and velocity variation and turbulent characteristics of the WNT under different parameter settings. The results indicate that experiments and simulations are consistent. Under four parameter settings, the axial and tangential velocities decrease to the minimum and then increase from the wall to the center. Under different feed velocity, cone angle, and spigot diameter settings, turbulent kinetic energy k and intensity I decrease from the cylinder to the cone and from the wall to the center; the max k and I correspond to the area near the inlet followed by the cylinder, and k and I in the cone are the smallest. When the classification overflow outlet (COO) diameter is 200 mm, k and I increase rapidly, the max k and I are transferred from near the inlet to near the cylinder wall at the COO, and the k and I near the wall decrease significantly.

Aggregation and Gelation in a Tunable Aqueous Colloid-Polymer Bridging System

We report a colloid-polymer model system with tunable bridging interactions for microscopic studies of structure and dynamics using confocal imaging. The interactions between trifluoroethyl methacrylate-co-\emph{tert}-butyl methacrylate (TtMA) copolymer particles and poly(acrylic acid) (PAA) polymers were controllable via polymer concentration and pH. The strength of adsorption of PAA on the particle surface, driven by pH-dependent interactions with polymer brush stabilizers on the particle surfaces, was tuned via solution pH. Particle-polymer suspensions formulated at low pH, where polymers strongly adsorbed to the particles, contained clusters or weak gels at particle volume fractions of $\phi = 0.15$ and $\phi = 0.40$. At high pH, where the PAA only weakly adsorbed to the particle surface, particles largely remained dispersed and the suspensions behaved as a dense fluid. The ability to visualize suspension structure is likely to provide insight into the role of polymer-driven bridging interactions on the behavior of colloidal suspensions.

Theoretical Study on the Micro-Flow Mechanism of Polymer Flooding in a Double Heterogeneous Oil Layer

Critical issues in the development of oil fields include the differences in the layer properties as well as serious interlayer conflicts and disturbances that can lead to the formation of a preferential flow pathway. In order to understand the interlayer disturbance mechanism between the heterogeneous oil layers, mathematical models of the polymer, and oil two-phase micro-flow in porous media are established based on the Navier-Stokes equation. The phase-field method is used to track the two-phase interface during the displacement process. Then, the influences of wettability, injection modes, and permeability contrasts on the front length coefficient and the displacement efficiency are studied. The results showed that when the rock surface is water-wet (oil-wet), the polymer displaced the low (high) permeability layer first, and the interlayer breakthrough is obvious in the early stages of displacement. After the front broke through, the water-wet (oil-wet) rocks began to displace the high (low) permeability layer, and the preferential flow pathway is formed, which slowed the subsequent polymer flooding. When the rock surface is oil-wet, the perforation degree of the inlet had a greater effect on the micro-oil displacement efficiency. The micro-oil displacement efficiency of the full perforation and commingling production model is 26.21% and 37.75% higher than that of the separate-layer injection and commingling production, as well as the partial perforation and commingling production-injection models, respectively. The larger the permeability contrast, the more obvious the interlayer breakthrough. This study reveals the influence of different wettability characteristics, injection modes, and permeability contrasts on the front length coefficient and the displacement efficiency in a micro-heterogeneous model and provides an important theoretical basis for the formulation of enhanced oil recovery schemes for heterogeneous oil layers.

Population balance modelling captures host cell protein dynamics in CHO cell cultures

Monoclonal antibodies (mAbs) have been extensively studied for their wide therapeutic and research applications. Increases in mAb titre has been achieved mainly by cell culture media/feed improvement and cell line engineering to increase cell density and specific mAb productivity. However, this improvement has shifted the bottleneck to downstream purification steps. The higher accumulation of the main cell-derived impurities, host cell proteins (HCPs), in the supernatant can negatively affect product integrity and immunogenicity in addition to increasing the cost of capture and polishing steps. Mathematical modelling of bioprocess dynamics is a valuable tool to improve industrial production at fast rate and low cost. Herein, a single stage volume-based population balance model (PBM) has been built to capture Chinese hamster ovary (CHO) cell behaviour in fed-batch bioreactors. Using cell volume as the internal variable, the model captures the dynamics of mAb and HCP accumulation extracellularly under physiological and mild hypothermic culture conditions. Model-based analysis and orthogonal measurements of lactate dehydrogenase activity and double-stranded DNA concentration in the supernatant show that a significant proportion of HCPs found in the extracellular matrix is secreted by viable cells. The PBM then served as a platform for generating operating strategies that optimise antibody titre and increase cost-efficiency while minimising impurity levels.

Modeling behaviors of permeable non-spherical micro-plastic aggregates by aggregation/sedimentation in turbulent freshwater flow

This study developed and evaluated a behavior model for permeable non-spherical micro-plastic aggregates in a turbulent flow of freshwater based on fractal theory, as conducting experimental and modeling studies. Laboratory-scale experiments evaluated attachment efficiency α to aggregation kinetics in an aquatic environment (pH 6, 20 ℃) of the electrolyte (Al³⁺). The experimental α was dependent on characteristics of plastics (type, size, and density) and ranged from 0.062 to 0.2772 (averaging 0.1) with a high correlation with the modeled α (R² > 0.92). Model validation was conducted under two simulation conditions: one drawn from a previously published study of impermeable spherical aggregates and the other based on fractal theory. The simulations verified the limited primary particle size with the lowest retention rate based on the previous study but it was difficult to determine the specific particle size with the lowest retention rate as a limiting factor. The sum of residual errors for aggregation/sedimentation between the two types of structures showed an overestimation of spherical structures. Such overestimation influenced the aggregate number concentration and distribution pattern. Therefore, the model needs to more detailed express the aggregation mechanism of permeable non-spherical aggregate structures in terms of surface growth.

Numerical Validation of a Population Balance Model Describing Cement Paste Rheology

Rheology control is essential during the period in which cement and concrete pastes are encountered in the fresh state, due to the fact that it directly affects workability, initial placement and the structural performance of the hardened material. Optimizations of clinker formulations and reductions in cement-to-water ratios induced by economic and environmental considerations have a significant effect in rheology, which invokes the need for mechanistic models capable of describing the effect of multiple relevant phenomena on the observed paste flow. In this work, the population balance framework was implemented to develop a model able to relate the transient microstructural evolution of cement pastes under typical experimental conditions with its macroscopic rheological responses. Numerical details and performance are assessed and discussed. It was found that the model is capable of reproducing experimentally observed flow curves by using measured cluster size distribution information. It is also able to predict the complex rheological characteristics typically found in cement pastes. Furthermore, a spatially resolved scheme was proposed to investigate the nature of flow inside a parallel-plates rheometer geometry with the objective of assessing the ability of the model of qualitatively predicting experimentally observed behavior and to gain insight into the effect of possible secondary flows.

Modeling of Flocculation and Sedimentation Using Population Balance Equation

Flocculation is a special phenomenon for fine sediment or silt in reservoirs and estuaries. Flocculation usually results in changes of size, morphology, and settling velocity of sediment particles and finally changes of bed topography of reservoirs and estuaries. The process of flocculation and sedimentation was simulated based on population balance modeling (PBM) and computational fluid dynamics (CFD); the changes of particle or floc size and their settling velocities over time were examined. The results showed that flocculation is a dynamic and nonlinear process containing aggregation, breakage, reaggregation, and rebreakage between particles, microflocs, and macroflocs. Furthermore, the visual process of flocculation and sedimentation was directly created by the simulation results and is in good agreement with the results of the previous experiments.

A review on sediment bioflocculation: Dynamics, influencing factors and modeling

Sediment in a water column provides excellent substratum for microorganism colonization, and biological processes would alter the physical and chemical of sediment, resulting in substantial changes in sediment dynamics. The flocculation of sediment with biological processes are defined as sediment bioflocculation, which has been ubiquitously observed across aquatic ecosystems, activated sludge plants and bioflocculant applications, as a result of various processes involving particle aggregation and breakage under the complex effects of microorganisms and their metabolic products (e.g., extracellular polymeric substances EPS). EPS are complex high-molecular-weight mixtures of polymers, which are the primary components that hold microbial aggregates together by acting as a biological glue. Several mechanistic aggregation theories such as the alginate theory, adsorption bridging theory, divalent cation bridging theory, and Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, and a number of influencing factors (e.g., sediment properties, microbial activity, EPS quantities and components, and external environment conditions) have been proposed to elucidate the role of microorganisms and EPS in sediment aggregation, promoting the investigation of the sediment bioflocculation evolution and kinetics models. However, due to the complex interrelationships of multiple physical, chemical, and biological processes and the incomprehensive knowledge of microorganisms and EPS, considerable research should be further conducted to fully understand their precise roles in the sediment bioflocculation process. In this study, a review of dynamic characterizations, mechanism, influencing factors and models of sediment bioflocculation are given to obtain a more comprehensive understanding of sediment bioflocculation dynamics.

Mechanisms behind overshoots in mean cluster size profiles in aggregation-breakup processes

Aggregation and breakup of small particles in stirred suspensions often shows an overshoot in the time evolution of the mean cluster size: Starting from a suspension of primary particles the mean cluster size first increases before going through a maximum beyond which a slow relaxation sets in. Such behavior was observed in various systems, including polymeric latices, inorganic colloids, asphaltenes, proteins, and, as shown by independent experiments in this work, in the flocculation of microalgae. This work aims at investigating possible mechanism to explain this phenomenon using detailed population balance modeling that incorporates refined rate models for aggregation and breakup of small particles in turbulence. Four mechanisms are considered: (1) restructuring, (2) decay of aggregate strength, (3) deposition of large clusters, and (4) primary particle aggregation where only aggregation events between clusters and primary particles are permitted. We show that all four mechanisms can lead to an overshoot in the mean size profile, while in contrast, aggregation and breakup alone lead to a monotonic, "S"-shaped size evolution profile. In order to distinguish between the different mechanisms simple protocols based on variations of the shear rate during the aggregation-breakup process are proposed.

Significance of Early and Late Stages of Coupled Aggregation and Sedimentation in the Fate of Nanoparticles: Measurement and Modeling

Despite aggregation's crucial role in controlling the environmental fate of nanoparticles (NP), the extent to which current models can describe the progressive stages of NP aggregation/sedimentation is still unclear. In this paper, 24 model combinations of two population-balance models and various collision frequency and settling velocity models are used to analyze spatiotemporal variations in the size and concentration of hydroxyapatite (HAp) NP. The impact of initial conditions and variability in attachment efficiency, α, with aggregate size are investigated. Although permeability models perform well in calculating collision frequencies, they are not appropriate for describing settling velocity because of their negative correlation or insensitivity in respect to fractal dimension. Considering both early and late stages of aggregation, both experimental and model data indicate overall mass removal peaks at an intermediate ionic strength (5 mM CaCl₂) even though the mean aggregate size continued to increase through higher ionic strengths (to 10 mM CaCl₂). This trend was consistent when different approaches to the initial particle size distribution were used and when a variable or constant α was used. These results point to the importance of accurately considering different stages of aggregation in modeling NP fate within various environmental conditions.

Effect of long-range interactions on nanoparticle-induced aggregation

The process of attaching liquid media molecules to dispersed nanoparticles is studied by numerically investigating the time evolution of the size distribution of the emerging aggregates. Within the considered mechanism of aggregation, both the primary particles and the resulting aggregates are assumed to connect freely dispersing molecules, but the particles and aggregates are not allowed to self-link or self-assemble at each evolution stage of the system, due to, e.g., repulsive interactions. The process of random attachment of dispersing molecules to immersed nanoparticles and aggregates is considered to be driven by attractive long-range interactions of the van der Waals type. The molecule binding rate is, in consequence, modeled as being dependent not only on the size and surface morphology of the existing aggregates, but also on the van der Waals forces, whose strength is itself treated as dependent on the aggregate size. It is demonstrated that these forces diminish, in general, the inhomogeneity of aggregate size. Such an effect is shown to be especially distinct when the interaction strength is relatively large but does not increase as aggregates increase in size, i.e., when strongly attracted media molecules functionalize the resultant aggregates to prevent the increase of the interaction strength. This result can be helpful to construct stable complex substances containing aggregates with low size dispersion. Surprisingly, the evolution of aggregating systems toward more significant inhomogeneity takes place when the interaction strength is initially large and increases fast enough with the size of aggregates.