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.     

Numerical solution of the bivariate population balance equation for the interacting hydrodynamics and mass transfer in liquid-liquid extraction columns

A comprehensive model for predicting the interacting hydrodynamics and mass transfer is formulated on the basis of a spatially distributed population balance equation in terms of the bivariate number density function with respect to droplet diameter and solute concentration. The two macro- (droplet breakage and coalescence) and micro- (interphase mass transfer) droplet phenomena are allowed to interact through the dispersion interfacial tension. The resulting model equations are composed of a system of partial and algebraic equations that are dominated by convection, and hence it calls for a specialized discretization approach. The model equations are applied to a laboratory segment of an RDC column using an experimentally validated droplet transport and interaction functions. Aside from the model spatial discretization, two methods for the discretization of the droplet diameter are extended to include the droplet solute concentration. These methods are the generalized fixed-pivot technique (GFP) and the quadrature method of moments (QMOM). The numerical results obtained from the two extended methods are almost identical, and the CPU time of both methods is found acceptable so that the two methods are being extended to simulate a full-scale liquid-liquid extraction column.     

Optimal moving and fixed grids for the solution of discretized population balances in batch and continuous systems: droplet breakage

The numerical solution of droplet population balance equations (PBEs) by discretization is known to suffer from inherent finite domain errors (FDE). Tow approaches that minimize the total FDE during the solution of discrete droplet PBEs using an approximate optimal moving (for batch) and fixed (for continuous systems) grids are introduced. The optimal grids are found based on the minimization of the total FDE, where analytical expressions are derived for the latter. It is found that the optimal moving grid is very effective for tracking out steeply moving population density with a reasonable number of size intervals. This moving grid exploits all the advantages of its fixed counterpart by preserving any two pre-chosen integral properties of the evolving population. The moving pivot technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996b) 1333) is extended for unsteady-state continuous flow systems, where it is shown that the equations of the pivots are reduced to that of the batch system for sufficiently fine discretization. It is also shown that for a sufficiently fine grid, the differential equations of the pivots could be decoupled from that of the discrete number density allowing a sequential solution in time. An optimal fixed grid is also developed for continuous systems based on minimizing the time-averaged total FDE. The two grids are tested using several cases, where analytical solutions are available, for batch and continuous droplet breakage in stirred vessels. Significant improvements are achieved in predicting the number densities, zero and first moments of the population.     

CFD based extraction column design——Chances and challenges

This paper shows that one-dimensional (1-D) [and three-dimensional (3-D) computational fluid dynamics (CFD)] simulations can replace the state-of-the-art usage of pseudo-homogeneous dispersion or back mixing models. This is based on standardized lab-scale cel experiments for the determination of droplet rise, breakage, coalescence and mass transfer parameters in addition to a limited number of additional mini-plant experiments with original fluids. Alternatively, the hydrodynamic parameters can also be derived using more sophisticated 3-D CFD simulations. Computational 1-D modeling served as a basis to replace pilot-plant experiments in any column geometry. The combination of 3-D CFD simulations with droplet population balance models (DPBM) increased the accuracy of the hydrodynamic simulations and gave information about the local droplet size. The high computational costs can be reduced by open source CFD codes when using a flexible mesh generation. First combined simulations using a three way coupled CFD/DPBM/mass-transfer solver pave the way for a safer design of industrial-sized columns, where no correlations are available.     

Droplet coalescence model optimization using a detailed population balance model for RDC extraction column

Single droplet experiments in a small lab scale Rotating Disk Contactor (RDC) for two different liquid–liquid systems were used to evaluate the coalescence parameters necessary for column simulations. Five different coalescence models are studied; the models parameters were obtained by an inverse solution of the population balance model using the extended fixed-pivot technique for the discretization of the droplet internal coordinate. The estimated coalescence parameters by solving the inverse problem were found dependent on the chemical test system. The Coulaloglou and Tavlarides model was found to be the best model to predict the experimental data for both test systems. These parameters were used to study the hydrodynamics and mass transfer behavior of pilot plant RDC extraction column using the simulation tool LLECMOD. This is performed for two different liquid–liquid systems as recommended by the European Federation of Chemical Engineering (EFCE) (butylacetate–acetone–water (b–a–w) and toluene–acetone–water (t–a–w)). The simulated Sauter mean droplet diameter, hold-up values and concentration profiles for organic and aqueous phase were found to be well predicted compared to the experimental data.     

A Hybrid Scheme for the Solution of the Bivariate Spatially Distributed Population Balance Equation

The advantages of the generalized fixed pivot technique as extended to mass transfer and the quadrature method of moments are hybridized to reduce the bivariate spatially distributed population balance equation describing the coupled hydrodynamics and mass transfer in liquid‐liquid extraction columns. The key idea in the hybridization technique is to use the available moments furnished by the generalized fixed pivot technique to find the abscissa and weights for the Gaussian‐quadrature based approach, in an attempt to evaluate the integrals over unknown droplet densities. To implement the quadrature method of moments efficiently, an explicit form for the abscissas and weights is derived based on the product‐difference algorithm as described by McGraw [1]. The proposed technique is found to reduce the discrete system of partial differential equations from 2 M_x + 1 to M_x + 2, where M_x is the number of pivots or classes. The spatial variable is discretized in a conservative form using a couple of recently published central difference schemes. The numerical predictions of the detailed and reduced models are found to be almost identical, accompanied by a substantial reduction of the CPU time as a characteristic of the hybrid model.     

Tropfenschwarmanalytik mittels Bildverarbeitung zur Simulation von Extraktionskolonnen mit Populationsbilanzen

Modern methods based on population balance methods permit a fast and accurate calculation of important quantities within extraction processes. The approaches show a good correlation between experiments and simulations with a deviation of the Sauter diameter d₃₂ smaller as 5 % (stirred vessel 450 mm) and 7 – 12 % (column 450 mm). With the help of an optical measuring technique droplet swarms are examined transiently. The current limit of the transmitted light technique is at 16 – 20 vol.‐% without further optimization. The limit of incident light technique is above 30 vol.‐%. Distance transformation and watershed segmentation algorithms enable the analysis of the droplet images.     

Numerical solution of the spatially distributed population balance equation describing the hydrodynamics of interacting liquid-liquid dispersions

In liquid-liquid contacting equipment such as completely mixed and differential contactors, droplet population balance based modeling is now being used to describe the complex hydrodynamic behavior of the dispersed phase. For the hydrodynamics of these interacting dispersions this model accounts for droplet breakage, droplet coalescence, axial dispersion, exit and entry events. The resulting population balance equations are integro-partial differential equations (IPDE) that rarely have an analytical solution, especially when they show spatial dependency, and hence numerical solutions are sought in general. To do this, these IPDEs are projected onto a system of convective dominant partial differential equations by discretizing the droplet diameter (internal coordinate). This is accomplished by generalizing the fixed-pivot (GFP) technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996a) 1311) handling any two integral properties of the population number density for continuous flow systems by treating the inlet feed distribution as a source term. Moreover, the GFD technique has the advantage of being free of repeated or double integral evaluation resulting from the weighted residual approaches such as the Galerkin's method. This allows the time-dependent breakage and coalescence functions to be easily handled without appreciable increase in the computational time. The resulting system of PDEs is spatially discretized in conservative form using a simplified first order upwind scheme as well as first- and second-order non-oscillatory central differencing schemes. This spatial discretization avoids the characteristic decomposition of the convective flux based on the approximate Riemann solvers and the operator splitting technique required by classical upwind schemes. The time variable is discretized using an implicit strongly stable approach that is formulated by careful lagging of the non-linear parts of the convective and source terms. The algorithm is tested against analytical solutions of the simplified population balance equation for a differential liquid-liquid extraction column through four case studies. In all these case studies the discrete models converge successfully to the available analytical solutions and to solutions on relatively fine grids when the analytical solution is not available. Realization of the algorithm is accomplished by comparing its predictions to experimental steady-state hydrodynamic data of a laboratory scale rotating disc contactor of diameter. Practically, the combined algorithm is found fast enough for the computation of the transient and steady-state hydrodynamic behavior of the continuously and spatially distributed interacting liquid-liquid dispersions.     

Two Analytical Approaches for Solution of Population Balance Equations: Particle Breakage Process

Various particulate systems were modeled by the population balance equation (PBE). However, only few cases of analytical solutions for the breakage process do exist, with most solutions being valid for the batch stirred vessel. The analytical solutions of the PBE for particulate processes under the influence of particle breakage in batch and continuous processes were investigated. Such solutions are obtained from the integro‐differential PBE governing the particle size distribution density function by two analytical approaches: the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM). ADM generates an infinite series which converges uniformly to the exact solution of the problem, while HPM transforms a difficult problem into a simple one which can be easily handled. The results indicate that the two methods can avoid numerical stability problems which often characterize general numerical techniques in this area.