Menwer Attarakih

LLECMOD: A Bivariate Population Balance Simulation Tool for Liquid- Liquid Extraction Columns

The population balance equation finds many applications in modelling poly-dispersed systems arising in many engineering applications such as aerosols dynamics, crystallization, precipitation, granulation, liquid-liquid, gas-liquid, combustion processes and microbial systems. The population balance lays down a modern approach for modelling the complex discrete behaviour of such systems. Due to the industrial importance of liquid-liquid extraction columns for the separation of many chemicals that are not amenable for separation by distillation, a Windows based program called LLECMOD is developed. Due to the multivariate nature of the population of droplets in liquid -liquid extraction columns (with respect to size and solute concentration), a spatially distributed population balance equation is developed. The basis of LLECMOD depends on modern numerical algorithms that couples the computational fluid dynamics and population balances. To avoid the solution of the momentum balance equations (for the continuous and discrete phases), experimen- tal correlations are used for the estimation of the turbulent energy dissipation and the slip velocities of the moving droplets along with interaction frequencies of breakage and coalescence. The design of LLECMOD is flexible in such a way that allows the user to define droplet terminal velocity, energy dissipation, axial dispersion, breakage and coalescence frequen- cies and the other internal geometrical details of the column. The user input dialog makes the LLECMOD a user-friendly program that enables the user to select the simulation parameters and functions easily. The program is reinforced by a pa- rameter estimation package for the droplet coalescence models. The scale-up and simulation of agitated extraction col- umns based on the populations balanced model leads to the main application of the simulation tool.

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

Abstract 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.

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

Abstract 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.

The OPOSPM as a Nonlinear Autocorrelation Population Balance Model for Dynamic Simulation of Liquid Extraction Columns

Abstract Dynamic simulation and online control problems in liquid extraction columns are still unresolved issues due to the two-phase flow and the particulate character of the dispersed phase. In this work, the One Primary and One Secondary Particle Model (OPOSPM) with two autocorrelation parameters is used as an alternative to the full population balance model. The model presents the base hierarchy of the SQMOM and consists only of two transport equations for droplet number and volume concentrations. Using the full population balance model or online experimental data, the autocorrelation parameters are identified using a constrained weighted nonlinear least square method. Compared to the experimental data in RDC and Kuhni columns, the autocorrelated OPOSPM predicts accurately the dynamic and steady state mean population properties with a simulation time amounts to only 3% of that required by the detailed model.

Solution of the population balance equation for liquid-liquid extraction columns using a generalized fixed-pivot and central difference schemes

Abstract In this work, the so-called fixed pivot technique is generalized to discretize the full population balance equation describing the hydrodynamics of liquid-liquid extraction columns (LLEC) with respect to droplet diameter. The spatial variable is discretized in a conservative form using a couple of the recently published central difference schemes. These schemes are combined with an implicit time integration method that is essentially noniterative by lagging the nonlinear terms. The combined numerical algorithm is found fast enough for the purpose of simulating the performance of the LLECs.

A CFD-Population Balance Model for the Simulation of Kühni Extraction Column

Abstract In this work, computational fluid dynamics (CFD) calculations coupled with DPBM are compared to LLECMOD (Liquid-Liquid Extraction Column MODule) simulations and to Laser Induced Fluorescence (LIF) measurement of the phase fraction using an iso-optical system of calcium chloride/water and butyl acetate. The results show a good agreement between the simulations and experimental data. The CFD requires a high computational load compared to LLECMOD, but gives local information about the droplet size and the phase fraction and is independent from geometrical constraints.

Integral Formulation of the Smoluchowski Coagulation Equation using the Cumulative Quadrature Method of Moments (CQMOM)

Abstract The integral formulation of the nonlinear continuous Smoluchowski coagulation equation (CSCE) using the CQMOM presents a hierarchical method to couple the QMOM and the physically evolving particle size distribution. This hierarchical nature of the CQMOM can be utilized in process system engineering and at individual unit operation as well. Here, not only the cumulative particle size distribution is reconstructed, but also its low-order cumulative moments. Numerical analysis shows two desirable properties of the CQMOM: First, it can be considered as a mesh-free method, since the solution of each integral equation at the current grid point does not depend on the other ones. Second, the accuracy of the targeted low-order cumulative moments depend only on the nodes and weights of the continuous Gauss-Christoffel quadrature, and not on sampling the continuous low-order cumulative moments. Moreover, at the upper particle size limit, the QMOM is imbedded as a limiting case.

OPOSSIM: A Population Balance-SIMULINK Module for Modelling Coupled Hydrodynamics and Mass Transfer in Liquid Extraction Equipment

Abstract Dynamic behaviour, control and design strategies for liquid extraction equipment are faced by the complex hydrodynamic behavior of the dispersed phase with many droplet interactions (e.g. breakage and coalescence). To take this into account, the population balance modelling framework is used by implementing the bivariate OPOSPM (One Primary and One Secondary Particle Method) with a one-dimensional finite volume method in the physical space. To narrow the gap between the steady state and dynamic design during process synthesis, OPOSPM is implemented in a MATLAB/Simulink flowsheeting environment. As an outcome of this, we present a new OPOSPM-MATLAB/Simulink module which is called OPOSSIM for modeling and simulation the coupled two-phase flow and mass transfer in a Kuhni liquid extraction column.

RDC extraction column simulation using the multi-primary one secondary particle method: Coupled hydrodynamics and mass transfer

Abstract Based on the multivariate population balance equation (PBE) and the primary secondary particle concept a mathematical model is developed for liquid extraction columns. It is extended to include the momentum balance for the dispersed phase. The resulting model is complicated by the integral source term of the PBE. To reduce this complexity, while maintaining most of the information from the continuous PBE, the concept of the primary secondary particle method is used. The effect of the number of primary particles (PP) on the final predicted solution is investigated. Numerical results show that the solution converge fast as the number of PP is increased. The terminal droplet velocity is found to be the most sensitive model parameter to the number of PP. The predicted steady state profiles (droplet diameter, holdup and the concentration profiles) along a pilot RDC extraction column are compared to the experimental data where good agreement is achieved.

A Meshfree Maximum Entropy Method for the Solution of the Population Balance Equation

Abstract In this work, the number density function in the population balance equation (PBE) is approximated in terms of field nodes through a complete set of orthogonal basis functions in a semi-logarithmic space. We proposed the functional values at these field nodes to satisfy the maximum entropy solution. This hybridization of function approximation and information theories based on Shannon Maximum Entropy principle, allowed us to construct a sequence of positive continuous approximations of the PBE. The Lagrange multipliers, which result from the maximization of the Shannon entropy subject to the available average information, was estimated by solving a well-conditioned linear system of algebraic equations. As an application, this meshfree solution of the PBE is validated using an analytical solution of the microbial cell dynamics in a constant abiotic environment with simultaneous cell growth and division for which the analytical solution was derived by using the Adomian method.