High shear breakage of compact polyelectrolyte-bridged flocs: A method for obtaining model-independent breakage rate function data

In this paper, we study a population balance equation (PBE) where flocs are distributed into classes according to their mass. Each class i contains i primary particles with mass m p and size L p. All differently sized flocs can aggregate, binary breakup into two equally sized flocs is used, and the floc’s fractal dimension is d 0 = 2, independently of their size. The collision efficiency is kept constant, and the collision frequency derived by Saffman and Turner (J Fluid Mech 1:16–30, 1956) is used. For the breakup rate, the formulation by Winterwerp (J Hydraul Eng Res 36(3):309–326, 1998), which accounts for the porosity of flocs, is used. We show that the mean floc size computed with the PBE varies with the shear rate as the Kolmogorov microscale, as observed both in laboratory and in situ. Moreover, the equilibrium mean floc size varies linearly with a global parameter P which is proportional to the ratio between the rates of aggregation and breakup. The ratio between the parameters of aggregation and breakup can therefore be estimated analytically from the observed equilibrium floc size. The parameter for aggregation can be calibrated from the temporal evolution of the mean floc size. We calibrate the PBE model using mixing jar flocculation experiments, see Mietta et al. (J Colloid Interface Sci 336(1):134–141, 2009a, Ocean Dyn 59:751–763, 2009b) for details. We show that this model can reproduce the experimental data fairly accurately. The collision efficiency α and the ratio between parameters for aggregation and breakup α and E are shown to decrease linearly with increasing absolute value of the ζ-potential, both for mud and kaolinite suspensions. Suspensions at high pH and different dissolved salt type and concentration have been used. We show that the temporal evolution of the floc size distribution computed with this PBE is very similar to that computed with the PBE developed by Verney et al. (Cont Shelf Res, 2010) where classes are distributed following a geometrical series and mass conservation is statistically ensured. The same terms for aggregation and breakup are used in the two PBEs. Moreover, we argue, using both PBEs, that bimodal distributions become monomodal in a closed system with homogeneous sediment, even when a variable shear rate is applied.

The FCMOM (finite size domain complete set of trial functions method of moments) is an efficient and accurate numerical technique to solve monovariate and bivariate population balance equations. It was previously formulated for homogeneous systems. In this paper, the FCMOM approach is extended to solve monovariate population balance equations for inhomogeneous (spatially not uniform) systems. In the FCMOM, the method of moments is formulated in a finite domain of the internal coordinates and the particle size distribution function is represented as a truncated series expansion by a complete system of orthonormal functions. The FCMOM is extended to inhomogeneous systems assuming that the particle-phase convective velocity is independent of the internal variables (particle size). The method is illustrated by applications to particle diffusion and to particle convection. In the case of particle convection, a gas−solid dilute flow in a pipe was simulated and the FCMOM equations were coupled with the governing equations (mass and momentum balances) of the gas phase.

Floc breakage in agitated suspensions: Effect of agitation rate

Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation—a special case

Neste trabalho foi analisada a influência da inclusão do Modelo de Balanço Populacional (MBP) em simulações por Fluidodinâmica Computacional (CFD) na dispersão do líquido injetado em um Lavador Venturi. Foram comparados os resultados de simulações com e sem a inclusão do MBP, bem como avaliada a inclusão dos efeitos dos modelos de quebra e coalescência de gotas. Os resultados mostraram que a inclusão do MBP apresentou efeito pouco pronunciado sobre a queda de pressão no Lavador Venturi. Nos contornos de fração volumétrica das gotas, foi observado que a inclusão do MBP possibilitou avaliar a variação de diâmetro de gota e a sua dispersão no equipamento. Simulações por CFD utilizando o MBP podem contribuir para projetos de Lavadores Venturi, de modo a verificar a dispersão de líquido injetado que a geometria correspondente proporciona.

Crystallization is an important part of many chemical industries. Efforts are being invested to improve the performance of the crystallization process by designing novel crystallizers. An important aspect in the development of new crystallizers is the ability to describe the behavior of such units in terms of rigorous dynamic mathematical models and solving the resulting models efficiently. The current bottleneck in modeling crystallization systems is the complexities associated with the crystal birth, growth, and death processes using population balance equations. In this article, various crystal birth, death, and growth models are introduced and reviewed. Population balance models as well as solution methods (e.g., analytical, moment methods, discretization (classes/sectional) methods and Monte Carlo methods) are also reviewed, and new advances in solution methods are described. Population balance equations are used in other fields, and developments from other fields that can be extended to crystal population balance equations are included in this review.

Kinetics of fluidised bed melt granulationIV. Selecting the breakage model

Fixed point convergence and acceleration for steady state population balance modelling of precipitation processes: Application to neodymium oxalate

Analysis of the aggregation–fragmentation population balance equation with application to coagulation

This work reviews the semi-analytical technique (SAT) and proposes a unique SAT based on the homotopy analysis method (HAM), called accelerated HAM (AHAM) (recently proposed in Hussain et al . (Hussain S, Arora G, Kumar R. 2024 Semi-analytical methods for solving non-linear differential equations: a review. J. Math. Anal. Appl. 531 , 127821 (doi: 10.1016/j.jmaa.2023.127821 ))) to solve pure aggregation and coupled aggregation–fragmentation population balance equations (PBEs), which are nonlinear integro-partial differential equations. The novelty of this approach is demonstrated through a comparative analysis of numerical results against existing findings from the Adomian decomposition (Singh R, Saha J, Kumar J. 2015 Adomian decomposition method for solving fragmentation and aggregation population balance equations. J. Appl. Math. Comput. 48 , 265–292 (doi: 10.1007/s12190-014-0802-5 )), homotopy analysis (Kaur G, Singh R, Briesen H. 2022 Approximate solutions of aggregation and breakage population balance equations. J. Math. Anal. Appl. 512 , 126166 (doi: 10.1016/j.jmaa.2022.126166 )), homotopy perturbation (Kaur G, Singh R, Singh M, Kumar J, Matsoukas T. 2019 Analytical approach for solving population balances: a homotopy perturbation method. J. Phys. A Math. Theor. 52 , 385201 (doi: 10.1088/1751-8121/ab2cf5 )) and optimized decomposition (Kaushik S, Kumar R. 2023 A novel optimized decomposition method for Smoluchowski’s aggregation equation. J. Comput. Appl. Math. 419 , 114710 (doi: 10.1016/j.cam.2022.114710 )) methods for these models. In addition, the theoretical convergence analysis and error estimation of the proposed method are also investigated. To validate the scheme, we analyse several numerical test cases and the results illustrate that the suggested technique offers the most accurate estimates for the two models under consideration.

A puristic analysis of population balance—II.

Spectral methods are evaluated for the solution of population balance problems. Both a simplified population balance equation (PBE) with an analytical solution available and a rigorous PBE are considered in this numerical analysis. In comparison with the orthogonal collocation, tau, and Galerkin methods, the least-squares method does not produce the same favorable results. On the other hand, the least-squares method with a direct minimization solver is capable of producing equally accurate results of the model equations as the orthogonal collocation, tau, and Galerkin methods. Compared to the conventional least-squares method, the direct minimization formulation is better conditioned but does not produce a symmetric positive-definite system matrix.

Particle size distribution of nanoparticles plays an important role in modelling many scientific and engineering problems. In this article, we proposed a Finite Volume Method (FVM) to model TiO2 nanoparticles formation using population balance equations (PBEs) by incorporating the simultaneous agglomeration and disintegration processes. The superposition of the PBEs for agglomeration and disintegration with different kernels leads to a system of partial-integro differential equations, which are numerically solved by using FVM. The precipitation of TiO2 nanoparticles in the batch reactor is studied experimentally as well as by numerical simulations based on Austin and Diemer disintegration kernels and Shear agglomeration kernel. Finally, the capability of the precipitation model is evaluated and the experimental results on particle sizes are compared with the numerical results.

A numerical procedure has been developed to solve the population balance equation (PBE) for the flocculation of colloidal suspensions. It has the flexibility to cope with different flocculation kinetics and spread in particle volume. The two key steps in this procedure are the adoption of the logarithm of particle volume as the independent variable and the implementation of uniform discretization in the logarithmic domain. Together they allow the PBE to be represented accurately over the entire particle range overcoming the failure of the popular geometric discretization scheme to represent the PBE satisfactorily for large particles. The Method of Lines is used to convert the PBE from an integro‐differential equation into a set of first‐order ordinary differential equations, which is then integrated using commercial scientific computing software. The procedure has been tested for different kinetics and initial particle distributions. Preservation of particulate volume of between 1 and 8% is consistently observed. © 2011 American Institute of Chemical Engineers AIChE J, 58: 3043–3053, 2012

The population balance equations (PBEs) serve as the primary governing equations for simulating the crystallization process. Two-dimensional (2D) PBEs pertain to crystals that exhibit anisotropic growth, which is characterized by changes in two internal coordinates. Because PBEs are the hyperbolic equations, it becomes imperative to establish a high-resolution scheme to reduce numerical diffusion and numerical dispersion, thereby ensuring accurate crystal size distribution. This paper uses Euler’s first-order explicit (EE) method–Peridynamic Differential Operator (PDDO) to solve 2D PBE, namely, the EE method for discretizing the time derivative and the PDDO for discretizing the internal crystal-size derivative. Five examples, including size-independent growth with smooth and non-smooth distributions, size-dependent growth, nucleation, and size-independent/dependent growth for batch crystallization are considered. The results show that the EE–PDDO method is more accurate than the HR method and that it is as good as the fifth-order Weighted Essential Non-Oscillatory (WENO) method in solving 2D PBE. This study extends the EE–PDDO method to the simulation of 2D PBE, and the advantages of the EE-PDDO method in dealing with discontinuous and sharp front problems are demonstrated.