WENO scheme with static grid adaptation for tracking steep moving fronts

The numerical solution of a hyperbolic or a convection-dominated parabolic partial differential equation is challenging due to the large local gradients that are present in the solution. A possible method to track the sharp fronts that are associated with large gradients is to adapt the grid and this can be done dynamically or statically, i.e. at discrete points of time during the simulation. In this paper, a novel approach that is based on combining the high-order WENO scheme with a static moving grid method is presented. The proposed algorithm is tested on the viscid Burgers’ equation, the linear advection equation and the population balance equation that describes particle growth in emulsion polymerization. Enhancements in the performance are observed in all case studies when compared with the conventional WENO scheme on a uniform grid making it a promising alternative when dealing with similar problems.     

Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), extended quadrature method of moments (EQMOM), conditional quadrature method of moments (CQMOM), extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.     

Dynamic evolution of PSD in continuous flow processes: A comparative study of fixed and moving grid numerical techniques

The present study provides a comprehensive investigation on the numerical solution of the dynamic population balance equation (PBE) in continuous flow processes. Specifically, continuous particulate processes undergoing particle aggregation and/or growth are examined. The dynamic PBE is numerically solved in both the continuous and its equivalent discrete form using the Galerkin on finite elements method (GFEM) and the moving grid technique (MGT) of Kumar and Ramkrishna [1997. Chemical Engineering Science 52, 4659-4679], respectively. Numerical simulations are carried out over a wide range of variation of particle aggregation and growth rates till the dynamic solution has reached its final steady-state value. The performance of the two numerical methods is assessed by a direct comparison of the calculated particle size distributions and/or their moments to available steady-state analytical solutions.     

Self-preserving theory of particulate systems

The similarity solution of the population balance equation for pure kinetic coagulation (coalescence) has proved to be a useful means of representing the evolution of size distributions of homogeneous particulate systems. The resulting solution is termed self-preserving since neither particle size nor time appears explicitly in the solution. In the present work similarity solutions are developed for a wide class of particulate processes that are governed by a general population balance equation for the inhomogeneous size distribution density function. The general theory has application to the analysis of size spectra of atmosphere aerosols undergoing simultaneous coagulation, turbulent diffusion, and growth by gas-to-particle condensation, to the analysis of size spectra during grinding processes, and to several other physical systems of interest.     

Solution of population balances with growth and nucleation by high order moment-conserving method of classes

The high order method of classes, developed in our earlier work [Alopaeus, V., Laakkonen, M., Aittamaa J., 2006a. Solution of population balances with breakage and agglomeration by high order moment-conserving method of classes. Chemical Engineering Science 61, 6732-6752] for solution of population balances (PBs), is extended to problems with growth and primary nucleation. The growth problem leads to a hyperbolic partial differential equation with fundamentally different numerical characteristics than the PB with breakage and agglomeration only. However, we show that the principle of moment conservation in the numerical solution can also be applied to this advection-type problem, leading to extremely accurate numerical solutions. The method is tested for two numerical cases. The first one is mass transfer induced particle growth, and the second one is primary nucleation with constant growth (similar to the Riemann advection problem). For mass transfer induced growth, we first analyze functional form of the growth rate from mass transfer correlation viewpoint, and derive a general analytical solution for the power-law growth. The numerical results from the moment conserving method are also compared to one well established high resolution numerical method for advection problems, namely the Lax-Wendroff method with van Leer flux limiter. It was shown that the present method is far superior by predicting the distribution moments with several order of magnitudes lower numerical error. For the Riemann problem with constant growth rate, the present method predicts the shock front location exactly without any numerical diffusion.     

A new method for solving population balance equations using a radial basis function network

Abstract

This study presents a new method for solving the population balance equation (PBE) for particle coagulation. The method introduces a radial basis function network to approximate the number density function. The approximate solution should satisfy the PBE at the collocation points in a continuous manner. The coagulation terms in the PBE are directly handled using the Gaussian quadrature techniques. The final solution is a bivariant analytical function of particle volume and time, which can be easily used in any subsequent calculation. Then the method is validated by comparing with analytical solutions and the sectional method for four numerical test problems. These problems are selected to vary in coagulation kernels and initial conditions. The comparative results show that the present method can accurately predict the time evolution of the number density function. Moreover, the computational efficiency of the present method is quite acceptable considering the high accuracy.

Copyright © 2020 American Association for Aerosol Research     

PDF method for population balance in turbulent reactive flow

Turbulent reactive flows with particle formation, such as soot formation and precipitation, are characterized by complex interactions between turbulence, scalar transport, particle formation and particle transport and inter-particle events such as coagulation. The effect of formation, growth and coagulation on the particle size distribution (PSD) must be modelled by the population balance equation (PBE). While the PBE has been studied extensively in homogeneous systems and, recently, in simple flows, its coupling with turbulent reactive flows poses a wealth of new questions. Processes such as nucleation, growth and coagulation are described by kinetic laws that link them to the local concentrations of the reactive scalars, which are random in a turbulent flow. This accounts for additional mechanisms that induce randomness and fluctuations to the particle concentration and PSD. Furthermore, conventional RANS closure of the coagulation term PDE (which describes the evolution of the PSD) leads to unknown correlations. In this work a new pdf approach is developed, based on the transport of the joint pdf of reactive scalars and particle number densities at different sizes, which overcomes the additional closure problems. It is also shown how the pdf method can be solved numerically via Monte-Carlo methods, and this is demonstrated via two applications in a partially stirred reactor: precipitation via nucleation-growth and coagulation. In each case the pdf method is compared with models that neglect correlations at various levels, and it is demonstrated that the interaction of turbulence with particle formation mechanisms accounts for significant deviations in the PSD.     

Solving population balance equations for two-component aggregation by a finite volume scheme

A conservative finite volume approach, originally proposed by Filbet and Laurençot [2004a. Numerical simulation of the Smoluchowski coagulation equation. SIAM Journal on Scientific Computing 25(6), 2004-2048] for the one-dimensional aggregation, is extended to simulate two-component aggregation. In order to apply the finite volume scheme, we reformulate the original integro-ordinary differential population balance equation for two-component aggregation problems into a partial differential equation of hyperbolic-type. Instead of using a fully discrete finite volume scheme and equidistant discretization of internal properties variables, we propose a semidiscrete upwind formulation and a geometric grid discretization of the internal variables. The resultant ordinary differential equations (ODEs) are then solved by using a standard adaptive ODEs-solver. Several numerical test cases for the one and two-components aggregation process are considered here. The numerical results are validated against available analytical solutions.     

Analytical–Numerical Solution of the Multicomponent Aerosol General Dynamic Equation—with Coagulation

ABSTRACT

A numerical procedure based on an analytical solution is presented for solution of the full multicomponent aerosol general dynamic equation. The analytical solution for the equation, accounting for growth, removal, and particle sources, is employed in an iterative procedure to account for coagulation. The iterative process is shown to be rapidly convergent, and its performance is validated by comparison with the exact solution for pure coagulation of a single-component aerosol. A simulation is presented of the evolution of a multicomponent coagulating aerosol, where each component grows-evaporates at a different rate.     

Part III: Dynamic evolution of the particle size distribution in batch and continuous particulate processes: A Galerkin on finite elements approach

The present study provides a comprehensive investigation on the numerical solution of the dynamic population balance equation (PBE) in batch and continuous flow particulate processes. The general PBE was numerically solved using the Galerkin on finite elements method (GFEM) for particulate processes undergoing simultaneous particle aggregation, growth and nucleation. The performance of the GFEM in terms of accuracy and stability was assessed by a direct comparison of the calculated particle size distributions (PSD) and/or their corresponding moments to available analytical solutions. Numerical simulations were carried out over a wide range of variation of particle aggregation and growth rates. Simulations of batch particulate processes revealed that the GFEM was generally more accurate than the standard fixed-grid sectional methods and more robust than the orthogonal collocation on finite elements method. In the case of continuous flow processes, numerical simulations were carried out till the dynamic solution reached its final steady-state. It was found that the calculated PSDs converged exactly to available analytical steady-state solutions of the PBE. The transient PSDs were often characterized by steep moving fronts that required careful discretization of the particle volume domain and the inclusion of an artificial diffusion term in order to eliminate numerical oscillations in the solution. Finally, for continuous particulate processes, the effect of various size-dependent particle growth models on the calculated steady-state PSDs was investigated.     

A new discretization of space for the solution of multi-dimensional population balance equations

In this work, a novel radial grid is combined with the framework of minimal internal consistency of discretized equations of Chakraborty and Kumar [2007. A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62, 4112-4125] to solve n-dimensional population balance equations (PBEs) with preservation of (n+1) instead of 2ⁿ properties required in direct extension of the 1-d fixed pivot technique of Kumar and Ramkrishna [1996a. On the solutions of population balance equation by discretization-I. A fixed pivot technique. Chemical Engineering Science 51, 1311-1332]. The radial grids for the solution of 2-d PBEs are obtained by intersecting arbitrarily spaced radial lines with arcs of arbitrarily increasing radii. The quadrilaterals obtained thus are divided into triangles to represent a non-pivot particle in 2-d space through three surrounding pivots by preserving three properties, the number and the two masses of the species that constitute the newly formed particle. Such a grid combines the ease of generating and handling a structured grid with the effectiveness of the framework of minimal internal consistency. A new quantitative measure to supplement visual comparison of two solutions is also introduced. The comparison of numerical and analytical solutions of 2-d PBEs for a number of uniform and selectively refined radial grids shows that the quality of solution obtained with radial grids is substantially better than that obtained with the direct extension of the 1-d fixed pivot technique to higher dimensions for both size independent and size dependent aggregation kernels. The framework of Chakraborty and Kumar combined with the proposed 2-d radial grid, which offers flexibility and achieves both reduced numerical dispersion and the ease of implementation, appears as an effective extension of the widely used 1-d fixed pivot technique to solve 2-d PBEs.     

Simulation of Condensation Aerosol Growth by Condensation and Evaporation

A new numerical method is reported for the solution of the condensation—evaporation equation, a first-order hyperbolic equation. The solution and properties of the nonlinear integrodifferential equation arising when the mass of the condensing vapor is conserved are discussed. For aerosol evolution in the conserved case it is shown that there develops an asymptotic regime analogous to the asymptotic behavior found for the coagulation process.     

A Bimodal Integral Solution of the Dynamic Equation for an Aerosol Undergoing Simultaneous Particle Inception and Coagulation

A model is developed from the general aerosol dynamic equation, using a bimodal integral formulation that includes particle formation and growth by coagulation in the free molecular regime. The particle inception mode accounts for the introduction of newly formed particles which, through coagulative collisions with one another, constitute the source of the particles in the growth mode. A numerical solution for the system of the first three moments of the particle volume distribution function is discussed, under the assumption of a logarithmic-normal behavior of the two modes of the size distribution function. The bimodal integral solution is subject to a detailed comparison with the MAEROS sectional model for the case of an aerosol that undergoes free molecular coagulation occurring simultaneously with particle formation by a Gaussian source pulse, under flamelike conditions.     

Part IV: Dynamic evolution of the particle size distribution in particulate processes. A comparative study between Monte Carlo and the generalized method of moments

The present work provides a comparative study on the numerical solution of the dynamic population balance equation (PBE) for batch particulate processes undergoing simultaneous particle aggregation, growth and nucleation. The general PBE was numerically solved using three different techniques namely, the Galerkin on finite elements method (GFEM), the generalized method of moments (GMOM) and the stochastic Monte Carlo (MC) method. Numerical simulations were carried out over a wide range of variation of particle aggregation and growth rate models. The performance of the selected techniques was assessed in terms of their numerical accuracy and computational requirements. The numerical results revealed that, in general, the GFEM provides more accurate predictions of the particle size distribution (PSD) than the other two methods, however, at the expense of more computational effort and time. On the other hand, the GMOM yields very accurate predictions of selected moments of the distribution and has minimal computational requirements. However, its main disadvantage is related to its inherent difficulty in reconstructing the original distribution using a finite set of calculated moments. Finally, stochastic MC simulations can provide very accurate predictions of both PSD and its corresponding moments while the MC computational requirements are, in general, lower than those required for the GFEM.     

Modelling study of emulsion latex coagulation processes in coagulators

This paper presents a numerical study of emulsion latex coagulation processes in continuous coagulators based on the full computational fluid dynamics approach. The RANS approach together with the k‐ε turbulence model was used to describe the detailed flow field in the coagulators. The coagulant mixing process was modelled by the convection‐diffusion equation and the emulsion latex coagulation process was formulated by the population balance equation of the particle size with a coagulation kernel including a perikinetic and orthokinetic combined mechanism. The flow and coagulation models were independently validated by means of comparing simulated results to the relevant experimental data from the literature. A series of simulations were carried out to study the effects of coagulator bottom shape, salt solution feeding location, residence time and agitation speed, as well as the influence of four typical scale‐up criteria on the latex particle coagulation process. The presented results would be helpful for the relevant process design, development, and scale‐up of continuous latex coagulators.     

Improved orthokinetic coagulation model for fractal colloids: Aggregation and breakup

An improved discretized population balance equation (PBE) is proposed in this study. This improved discretized PBE has new probability distribution functions for aggregates produced in non-uniform discrete coagulation modeling. The authors extended an improved particle coagulation model previously developed to an adjustable geometric size interval (q), where q is a volume ratio of class k+1 particles to class k particles (υ_k+1/υ_k=q). This model was compared with exact numerical solutions of continuous (uniform discretized) PBEs and applied to simulate the particle aggregation and breakup with fractal dimensions lower than 3. Further, comparisons were made using the fractal aggregate collision mechanisms of orthokinetic coagulation with the inclusion of flow induced breakup.In the course of the investigation the new algorithm was found to be a substantial improvement in terms of numerical accuracy, stability, and computational efficiency over the continuous model. This new algorithm makes it possible to solve fractal particle aggregation and breakup problems with high accuracy, perfect mass conservation, and exceptional computational efficiency, thus the new model can be used to develop predictive simulation techniques for the coupled coagulation using computational fluid dynamics (CFD) and chemical reaction modeling.     

A control oriented study on the numerical solution of the population balance equation for crystallization processes

The population balance equation provides a well established mathematical framework for dynamic modeling of numerous particulate processes. Numerical solution of the population balance equation is often complicated due to the occurrence of steep moving fronts and/or sharp discontinuities. This study aims to give a comprehensive analysis of the most widely used population balance solution methods, namely the method of characteristics, the finite volume methods and the finite element methods, in terms of the performance requirements essential for on-line control applications. The numerical techniques are used to solve the dynamic population balance equation of various test problems as well as industrial crystallization processes undergoing simultaneous nucleation and growth. The time-varying supersaturation profiles in the latter real-life case studies provide more realistic scenarios to identify the advantages and pitfalls of a particular numerical technique.The simulation results demonstrate that the method of characteristics gives the most accurate numerical predictions, whereas high computational burden limits its use for complex real crystallization processes. It is shown that the high order finite volume methods in combination with flux limiting functions are well capable of capturing sharp discontinuities and steep moving fronts at a reasonable computational cost, which facilitates their use for on-line control applications. The finite element methods, namely the orthogonal collocation and the Galerkin's techniques, on the other hand may severely suffer from numerical problems. This shortcoming, in addition to their complex implementation and low computational efficiency, makes the finite element methods less appealing for the intended application.     

Part V: Dynamic evolution of the multivariate particle size distribution undergoing combined particle growth and aggregation

The present paper presents a study on the application of the Galerkin finite element method (FEM) for the solution of the dynamic multivariate population balance equation (PBE) in batch particulate systems undergoing aggregation as well as combined aggregation and growth. The performance of the Galerkin FEM in terms of accuracy and stability was assessed by a direct comparison of the calculated particle size distributions and/or their corresponding moments to available analytical solutions as well as by comparison to univariate numerical solutions. Numerical simulations were carried out for a variety of particle aggregation and growth mechanisms including constant, Brownian, and modified Brownian aggregation as well as constant and linear growth rate functions and for a wide range of values for the aggregation and growth rate coefficients. The simulation results revealed that the proposed Galerkin FEM produces very accurate numerical solutions but at significant computational cost.     

Computationally efficient solution of population balance models incorporating nucleation, growth and coagulation: application to emulsion polymerization

A computationally efficient solution technique is presented for population balance models accounting for nucleation, growth and coagulation (aggregation) (with extensions for breakage). In contrast to earlier techniques, this technique is not based on approximating the population balance equation, but is based on employing individual rates of nucleation, growth and coagulation to update the PSD in a hierarchical framework. The method is comprised of two steps. The first step is the calculation of the rates of nucleation, growth and coagulation by solving an appropriate system of equations. This information is then used in the second step to update the PSD. The method effectively decomposes the fast and the slow kinetics, thereby eliminating the stiffness in the solution. In solving the coagulation kernel, a semi-analytical solution strategy is adapted, which substantially reduces the computational requirement, but also ensures the consistency of properties such as the number and mass of particles.