A multi‐QMOM framework to describe multi‐component agglomerates in liquid steel

A variant of the quadrature method of moments (QMOM) for solving multiple population balance equations (PBE) is developed with the objective of application to steel industry processing. During the process of oxygen removal in a steel ladle, a large panel of oxide inclusions may be observed depending on the type of oxygen removal and addition elements. The final quality of the steel can be improved by accurate numerical simulation of the multi‐component precipitation. The model proposed in this article takes into account the interactions between three major aspects of steelmaking modeling, namely fluid dynamics, thermo‐kinetics and population balance. A commercial CFD code is used to predict the liquid steel hydrodynamics, whereas a home‐made thermo‐kinetic code adjusts chemical composition with nucleation and diffusion growth, and finally a set of PBE tracks the evolution of inclusion size with emphasis on particle aggregation. Each PBE is solved by QMOM, the first PBE/QMOM system describing the clusters and each remaining PBE/QMOM system being dedicated to the elementary particles of each inclusion species. It is shown how this coupled model can be used to investigate the cluster size and composition of a particular grade of steel (i.e., Fe‐Al‐Ti‐O). © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Application of the direct quadrature method of moments to polydisperse gas-solid fluidized beds

Most of today's computational fluid dynamics (CFD) calculations for gas-solid flows are carried out assuming that the solid phase is monodispersed, whereas it is well known that in many applications, it is characterized by a particle size distribution (PSD). In order to properly model the evolution of a polydisperse solid phase, the population balance equation (PBE) must be coupled to the continuity and momentum balance equations. In this work, the recently formulated direct quadrature method of moments (DQMOM) is implemented in a multi-fluid CFD code to simulate particle aggregation and breakage in a fluidized-bed (FB) reactor. DQMOM is implemented in the code by representing each node of the quadrature approximation as a distinct solid phase. Since in the multi-fluid model, each solid phase has its own momentum balance, the nodes of the DQMOM approximation are convected with their own velocities. This represents an important improvement with respect to the quadrature method of moments (QMOM) where the moments are tracked using an average solid velocity. Two different aggregation and breakage kernels are tested and the performance of the DQMOM approximation with different numbers of nodes are compared. These results show that the approach is very effective in modeling solid segregation and elutriation and in tracking the evolution of the PSD, even though it requires only a small number of scalars.     

Limitations of quadrature-based moment methods for modeling inhomogeneous polydisperse fluidized powders

Computational fluid dynamics (CFD) is extensively used to investigate the behavior of dense fluidized suspensions. Often modelers assume that these are formed by few solid phases of particles with constant size. But real powders are continuously distributed over the particle size, and their distribution functions change continuously in time and space reflecting the physical and chemical phenomena occurring within the system. To account for this key feature, models have to include a population balance equation (PBE), which needs to be solved in place of or along with the customary fluid dynamic transport equations. The recently developed quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) permit to solve PBEs in commercial CFD codes at relatively low computational cost. These methods, however, still need testing in the context of multiphase flows. Investigating a simple problem, namely the dynamics of two inert polydisperse fluidized suspensions initially segregated, we highlight an important limitation of these methods, which fail to properly model diffusion in real space. We explain where the problem originates and comment on a possible way to overcome it. To conclude the work, we discuss some simulations based on the original and revised formulations of the methods, describing how the code numerics affects the results.     

Solution of population balance equations using the direct quadrature method of moments

The implementation of a population balance equation (PBE) in computational fluid dynamics (CFD) represents a crucial element in the simulation of multiphase flows. Some of the available methods, such as classes methods (CM) and Monte Carlo (MC) methods, are computationally expensive and simulation of real cases of practical interest requires intractable CPU times. On the other hand, other methods such as the method of moments (MOM) are computationally affordable but have proven to be inaccurate for a number of cases. In recent work a new closure, the quadrature method of moments (QMOM), has been introduced, applied and validated. In our earlier work, QMOM was shown to be an efficient and accurate method for tracking the moments of the particle size distribution (PSD) in a CFD simulation. However, QMOM presents two main disadvantages: (i) if applied to multi-variate distributions it loses simplicity and efficiency, and (ii) by tracking only the moments of the PSD, it does not represent realistically polydisperse systems with strong coupling between the internal coordinates and phase velocities. In order to address these issues, in this work the direct quadrature method of moments (DQMOM) is formulated, validated, and tested. DQMOM is based on the idea of tracking directly the variables appearing in the quadrature approximation, rather than tracking the moments of the PSD. Nevertheless, for monovariate cases we show that QMOM and DQMOM yield identical results. In addition, we show how it is possible to extend the DQMOM to multivariate cases and some of relevant theoretical and numerical issues are discussed. These issues are discussed in the present work for homogeneous and one-dimensional flows. References to recent CFD applications of DQMOM to multiphase flows are provided as further proof of the utility of the method.     

Automatic differentiation‐based quadrature method of moments for solving population balance equations

The quadrature method of moments (QMOM) is a promising tool for the solution of population balance equations. QMOM requires solving differential algebraic equations (DAEs) consisting of ordinary differential equations related to the evolution of moments and nonlinear algebraic equations resulting from the quadrature approximation of moments. The available techniques for QMOM are computationally expensive and are able to solve for only a few moments due to numerical robustness deficiencies. In this article, the use of automatic differentiation (AD) is proposed for solution of DAEs arising in QMOM. In the proposed method, the variables of interest are approximated using high‐order Taylor series. The use of AD and Taylor series gives rise to algebraic equations, which can be solved sequentially to obtain high‐fidelity solution of the DAEs. Benchmark examples involving different mechanisms are used to demonstrate the superior accuracy, computational advantage, and robustness of AD‐QMOM over the existing state‐of‐the‐art technique, that is, DAE‐QMOM. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

A novel algorithm for solving population balance equations: the parallel parent and daughter classes. Derivation, analysis and testing

A novel numerical method, the parallel parent and daughter classes (PPDC) technique, for solving population balance equations (PBEs) is presented in this paper. In many practical applications, the PBE of particles under investigation is coupled with the thermo-fluid dynamics of the surrounding fluid. Hence, the PBE needs to be implemented in a computational fluid dynamics (CFD) code, which leads to an additional computational load. The computational cost becomes intractable when techniques such as methods of classes (CM) or Monte Carlo method are used. Quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) are accurate and require a relatively low additional computational cost when applied to CFD. The PPDC is shown to be as accurate as QMOM and DQMOM, and even more accurate in some cases, when the same number of classes is used. In the present work, the PPDC technique has been derived and tested. This technique can be used for solving a wide class of problems involving PBE such as polymerization, aerosol dynamics, bubble columns, etc. Numerical simulations have been carried out on aggregation processes with different kernels and on simultaneous aggregation and breakage processes. The numerical predictions are compared either with analytical solutions, when available, or with the numerical solutions obtained by methods of classes.     

Solution of the population balance equation using the sectional quadrature method of moments (SQMOM)

A discrete framework is introduced for simulating the particulate physical systems governed by population balance equations (PBE) with particle splitting (breakage) and aggregation based on accurately conserving (from theoretical point of view) an unlimited number of moments associated with the particle size distribution. The basic idea is based on the concept of primary and secondary particles, where the former is responsible for distribution reconstruction while the latter is responsible for different particle interactions such as splitting and aggregation. The method is found to track accurately any set of low-order moments with the ability to reconstruct the shape of the distribution. The method is given the name: the sectional quadrature method of moments (SQMOM) and has the advantage of being not tied to the inversion of large sized moment problems as required by the classical quadrature method of moments (QMOM). These methods become ill conditioned when a large number of moments are needed to increase their accuracy. On the contrary, the accuracy of the SQMOM increases by increasing the number of primary particles while using fixed number of secondary particles. Since the positions and local distributions for two secondary particles are found to have an analytical solution, no large moment inversion problems are anymore encountered. The generality of the SQMOM is proved by showing that all the related sectional and quadrature methods appearing in the literature for solving the PBE are merely special cases. The method has already been extended to bivariate PBEs.     

Solution of PBE by MOM in finite size domains

A new approach to solve PBE (Population Balance Equations), FCMOM (Finite size domain Complete set of trial functions Method Of Moments), is presented. The solution of the PBE is sought, instead of the [0,∞] range, in the finite interval between the minimum and maximum particle size; their evolution is tracked imposing moving boundaries conditions. After reformulating the PBE in the standard interval [-1,1], the size distribution function is represented as a series expansion by a complete system of orthonormal functions. Moments evolution equations are developed from the PBE in the interval [-1,1]. The FCMOM is implemented through an efficient algorithm and provides the solution of the PBE both in terms of the moments and in terms of the size distribution function. The FCMOM was validated with applications to particle growth (constant, linear, diffusion-controlled), simultaneous particle growth and nucleation, particle dissolution, particle aggregation (constant, sum, product, Brownian kernels) and simultaneous particle aggregation and growth.     

Implementation and analysis of numerical solution of the population balance equation in CFD packages

Simulation of polydisperse flows must include the effects of particle–particle interaction, as breakage and aggregation, coupling the population balance equation (PBE) with the multiphase modelling. In fact, the implementation of efficient and accurate new numerical techniques to solve the PBE is necessary. The direct quadrature method of moments, known as DQMOM, is a moment-based method that uses an optimal adaptive quadrature closure and came into view as a promising choice for this implementation. In the present work, DQMOM was implemented in two CFD packages: the commercial ANSYS CFX, through FORTRAN subroutines, and the open-source OpenFOAM, by directly coding the PBE solution. Transient zero-dimensional and steady one-dimensional simulations were performed in order to explore the PBE solution accuracy using several interpolation schemes. Simulation cases with dominant breakage, dominant aggregation and invariant solution (equivalent breakage and aggregation) were simulated and validated against an analytical solution. The solution of the population balance equation was then coupled to the two-fluid model, considering that all particles classes share the same velocity field. Momentum exchange terms were evaluated using the local instantaneous Sauter mean diameter of the size distribution function. The two-dimensional tests were performed in a backward facing step geometry where the vortex zones traps the particles and provides high rates of breakage and aggregation.     

Integral formulation of the population balance equation: Application to particulate systems with particle growth

Numerical solution of the population balance equation (PBE) is widely used in many scientific and engineering applications. Available numerical methods, which are based on tracking population moments instead of the distribution, depend on quadrature methods that destroy the distribution itself. The reconstruction of the distribution from these moments is a well-known ill-posed problem and still unresolved question. The present integral formulation of the PBE comes to resolve this problem. As a closure rule, a Cumulative QMOM (CQMOM) is derived in terms of the monotone increasing cumulative moments of the number density function, which allows a complete distribution reconstruction. Numerical analysis of the method show two unique properties: first, the method can be considered as a mesh-free method. Second, the accuracy of the targeted low-order cumulative moments depends only on order of the CQMOM, but not on the discrete grid points used to sample the cumulative moments.     

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.     

Continuous distribution theory for Ostwald ripening: comparison with the LSW approach

A numerical solution for a general population balance equation (PBE) for Ostwald ripening is compared with the usual approach developed by Lifshitz-Slyozov-Wagner (LSW). The PBE incorporates denucleation for unstable particles smaller than the critical nucleus size and reversible growth or dissolution of stable particles. The PBE theory shows how supersaturation decays to equilibrium, and (unlike LSW) how the particle size distribution (PSD) and its moments evolve to a final monodisperse state. The LSW model is known to correctly depict time dependence of particle number concentration and average particle size, but misrepresents the PSD higher moments.     

Chemically resolved aerosol dynamics for internal mixtures by the quadrature method of moments

The quadrature method of moments (QMOM), a promising new tool for aerosol dynamics simulation, is extended to multicomponent, internally mixed particle populations. A new moment closure method, the Jacobian matrix transformation (JMT), is introduced and shown to provide an efficient procedure for evolving quadrature abscissas and weights directly and in closed form. For special growth laws where analytic results are available for comparison, the QMOM is also found to be exact. The JMT implementation of the QMOM is used to explore the asymptotic behavior of coagulating aerosols at long time. Nondimensional reduced moments are constructed, and found to evolve to constant values in excellent agreement with estimates derived from ‘self-preserving’ distributions previously obtained by independent methods. Our findings support the QMOM as a new tool for rapid, accurate simulation of the dynamics of an evolving internally mixed aerosol population, including the approach to asymptotic behavior at long time, in terms of lower-order moments.     

Dimension reduction of bivariate population balances using the quadrature method of moments

Crystallization models with direction-dependent growth rates give rise to multi-dimensional population balance equations (PBE) that require a high computational cost. We propose a model reduction based on the quadrature method of moments (QMOM). Using this method a two-dimensional population balance is reduced to a system of one-dimensional advection equations. Despite the dimension reduction the method keeps important volume dependent information of the crystal size distribution (CSD). It returns the crystal volume distribution as well as other volume dependent moments of the two-dimensional CSD. The method is applied to a model problem with direction-dependent growth of barium sulphate crystals, and shows good performance and convergence in these examples. We also compare it on numerical examples to another model reduction using a normal distribution ansatz approach. We can show that our method still gives satisfactory results where the other approach is not suitable.     

A CFD‐PBM‐PMLM integrated model for the gas–solid flow fields in fluidized bed polymerization reactors

Although the use of computational fluid dynamics (CFD) model coupled with population balance (CFD‐PBM) is becoming a common approach for simulating gas–solid flows in polydisperse fluidized bed polymerization reactors, a number of issues still remain. One major issue is the absence of modeling the growth of a single polymeric particle. In this work a polymeric multilayer model (PMLM) was applied to describe the growth of a single particle under the intraparticle transfer limitations. The PMLM was solved together with a PBM (i.e. PBM‐PMLM) to predict the dynamic evolution of particle size distribution (PSD). In addition, a CFD model based on the Eulerian‐Eulerian two‐fluid model, coupled with PBM‐PMLM (CFD‐PBM‐PMLM), has been implemented to describe the gas–solid flow field in fluidized bed polymerization reactors. The CFD‐PBM‐PMLM model has been validated by comparing simulation results with some classical experimental data. Five cases including fluid dynamics coupled purely continuous PSD, pure particle growth, pure particle aggregation, pure particle breakage, and flow dynamics coupled with all the above factors were carried out to examine the model. The results showed that the CFD‐PBM‐PMLM model describes well the behavior of the gas–solid flow fields in polydisperse fluidized bed polymerization reactors. The results also showed that the intraparticle mass transfer limitation is an important factor in affecting the reactor flow fields. © 2011 American Institute of Chemical Engineers AIChE J, 58: 1717–1732, 2012     

Description of Aerosol Dynamics by the Quadrature Method of Moments

ABSTRACT

The method of moments (MOM) may be used to determine the evolution of the lower-order moments of an unknown aerosol distribution. Previous applications of the method have been limited by the requirement that the equations governing the evolution of the lower-order moments be in closed form. Here a new approach, the quadrature method of moments (QMOM), is described. The dynamical equations for moment evolution are replaced by a quadrature-based approximate set that satisfies closure under a much broader range of conditions without requiring that the size distribution or growth law maintain any special mathematical form. The conventional MOM is recovered as a special case of the QMOM under those conditions, e.g., free-molecular growth, for which conventional closure is satisfied. The QMOM is illustrated for the growth of sulfuric acid-water aerosols and simulations of diffusion-controlled cloud droplet growth are presented.     

Treatment of size-dependent aerosol transport processes using quadrature based moment methods

The use of moment methods for simulation of aerosol settling and diffusion phenomena in which the settling velocity and diffusion coefficient are functions of the size of the particles leads to difficult computational problems, especially if the moment equations need to be closed. In this study, a simple one dimensional problem of aerosol diffusion and gravitational settling is carried out using quadrature method of moments (QMOM) and the direct quadrature method of moments (DQMOM). Analytical solutions can be obtained for the number density function, and issues related to the integration of the solutions to get the moments are discussed. Comparison of the solutions of the moment equations to the moments obtained from the analytical solutions reveals that solutions depend on the initial choice of moments. Results also indicate that the proper choice of moments of the initial number density function may be a significant factor in obtaining more accurate solutions from QMOM or DQMOM.