Abdelmalek Hasseine

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.

CFD Modelling of Bubbly Gas Flow using Coupled OPOSPM-Two-Fluid Model

Abstract Bubbly gas flow in vertical tubes received considerable attention due to its applications in chemical, biochemical industries and nuclear reactor design and safety consideration. The use of CFD is called for to avoid the dependence of steady state operation and accident analysis on empirical correlations. This is due to the dependence of these correlations on flow regimes, scaling and geometrical factors. The situation becomes more complicated when the gas bubble-bubble and bubble-continuous phase interactions are taken into account in which bubble growth, breakage and coalescence could not be neglected. In such cases, the application of the two-fluid bubbly flow model becomes limited where only bubble expansion is be taken into account. In this contribution, we bridged the gap between the bubbly flow two-fluid model and the population balances by introducing the OPOSPM as a consistent and reduced population balance model. This adds an extra degree of details by considering the instantaneous bubble breakage and coalescence in the source term of the total bubble number concentration transport equation. The 2D CFD-OPOSPM model was validated against published experimental data of bubbly flow in vertical tubes.

On the Solution of the PBE by Orthogonal Expansion of the Maximum Entropy Functional

Abstract We present a continuous approximation to the population balance equation which has few analytical solutions that are only of academic interest. The proposed solution is a stable and well-conditioned converging sequence of continuous approximations to the number concentration function. Instead of using the moments of the number concentration function as constraints when applying the Maximum Entropy (MaxEnt) method, we require the MaxEnt functional to satisfy pointwise local information sampled from the number concentration function. The solution of this constrained optimization problem results in a continuous Lagrange multiplier which is then expanded using a complete set of orthogonal Chebyshev basis functions. The coefficients of the expansion are then derived in a closed form using local information about the number concentration function. As an application, the present method is validated using an analytical solutions of additive particle aggregation frequency plus first-order particle depletion rate and steady state particle breakage in a continuous homogeneous flow system. The method is found to produce comparable results to that predicted by the Chebyshev-QMOM as the pointwise local information is increased.

CFD modelling of pulsed sieve plate liquid extraction columns using OPOSPM as a reduced population balance model

Abstract The simulation of pulsed sieve plate liquid-liquid extraction columns is performed by coupling a 2D-CFD simulation with a reduced population balance method. In this regard, the One Primary One Secondary Particle Method (OPOSPM) is utilized and implemented in Fluent 17.1 commercial software as a special case of the general Sectional Quadrature Method Of Moments (SQMOM). The droplet-droplet interactions (breakage and coalescence) are taken into account using OPOSPM, while the required information about the velocity field and energy dissipation is estimated by a CFD model. A positive validation of the column hydrodynamics behaviour is with experimental data at different operating conditions.

Measurement and correlation of liquid–liquid equilibria for water + ethanol + mixed solvents (dichloromethane or chloroform + diethyl ether) at T = 293.15 K

Abstract Liquid–liquid equilibrium (LLE) data for the quaternary systems (water + ethanol + dichloromethane (DCM) or chloroform (CHCl3) + diethyl ether (DEE)) were experimentally investigated at 293.15 K. The thermodynamic consistency of the data was performed using the Othmer–Tobias and Hand plots. The experimental tie-line data were correlated using the non-random, two-liquid (NRTL) model. As a result, the comparison of the extracting capabilities of the mixed solvents with respect to the distribution coefficients and separation factors showed that the (50% DCM +50% DEE) system had a higher separation factor for the (water + ethanol + DCM + DEE) system. On the other hand, the (50% CHCl3 +50% DEE) system had a higher separation factor for the (water + ethanol + CHCl3 + DEE) system. The last solvent (50% CHCl3 +50% DEE) was found to be the best solvent, with a positive synergistic effect on DEE, high separation factor, and very low solubility in water.

Population Balance Modelling of Liquid Extraction Columns using the Sectional Quadrature Method of Moments (SQMOM)

Abstract In this work, the Sectional Quadrature Method Of Moments (SQMOM) is extended to solve the nonhomogeneous population balance equation along the spatial domain to model the hydrodynamics and mass transfer behaviour of liquid-liquid extraction columns. The required quadrature nodes and weights are calculated analytically using the Two-Equal Weight Quadrature (TEqWQ) formula derived by Attarakih et al., ( Attarakih, M., Drumm, C., & Bart, H.-J., (2009), Solution of the population balance equation using the Sectional Quadrature Method of Moments (SQMOM). Chem. Eng. Sci., 64, 742-752 ). As a numerical test, the SQMOM was validated using PPBLab software which utilizes the detailed extended fixed pivot as a built-in solver. Moreover, the SQMOM was experimentally validated using the available published steady state experimental data for both chemical test systems: Water-acetone-toluene and water-acetone-butyl acetate chemical test systems for RDC DN80 liquid extraction column.

Variational iteration method and projection method solution of the spatially distributed population balance equation

Abstract In this work, two major hydrodynamic parameters, the holdup of the dispersed phase and the Sauter diameter, are considered. This is done for describing the hydrodynamics of interacting liquid–liquid dispersions using different particle breakage, coalescence and growth models in a particle population balance model. Based on the semi-analytical solution method of the population balance, namely, the variational iteration method (VIM), different process cases have been performed, and it is possible to find the exact solution or a closed approximate solution of a problem. For the simultaneous growth and coalescence terms comparisons between the present method and projection method which include discontinuous Galerkin and collocation techniques are made, respectively. The VIM technique overcomes the difficulties of discretization of the variables, introduces an efficient algorithm that improves the standard discretization method and is able to handle quite successful these process of population balance equations. The results are encouraging and the new method has proven to be suitable to predict holdup and Sauter diameter profiles.