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The growing computational power enables the establishment of the Population Balance Equation (PBE)
to model the steady state and dynamic behavior of multiphase flow unit operations. Accordingly, the twophase
flow
behavior inside liquid-liquid extraction equipment is characterized by different factors. These
factors include: interactions among droplets (breakage and coalescence), different time scales due to the
size distribution of the dispersed phase, and micro time scales of the interphase diffusional mass transfer
process. As a result of this, the general PBE has no well known analytical solution and therefore robust
numerical solution methods with low computational cost are highly admired.
In this work, the Sectional Quadrature Method of Moments (SQMOM) (Attarakih, M. 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) is extended to take into account the continuous flow
systems in spatial domain. In this regard, the SQMOM is extended to solve the spatially distributed
nonhomogeneous bivariate PBE to model the hydrodynamics and physical/reactive mass transfer
behavior of liquid-liquid extraction equipment. Based on the extended SQMOM, two different steady
state and dynamic simulation algorithms for hydrodynamics and mass transfer behavior of liquid-liquid
extraction equipment are developed and efficiently implemented. At the steady state modeling level, a
Spatially-Mixed SQMOM (SM-SQMOM) algorithm is developed and successfully implemented in a onedimensional
physical spatial domain. The integral spatial numerical flux is closed using the mean mass
droplet diameter based on the One Primary and One Secondary Particle Method (OPOSPM which is the
simplest case of the SQMOM). On the other hand the hydrodynamics integral source terms are closed
using the analytical Two-Equal Weight Quadrature (TEqWQ). To avoid the numerical solution of the
droplet rise velocity, an analytical solution based on the algebraic velocity model is derived for the
particular case of unit velocity exponent appearing in the droplet swarm model. In addition to this, the
source term due to mass transport is closed using OPOSPM. The resulting system of ordinary differential
equations with respect to space is solved using the MATLAB adaptive Runge–Kutta method (ODE45). At
the dynamic modeling level, the SQMOM is extended to a one-dimensional physical spatial domain and
resolved using the finite volume method. To close the mathematical model, the required quadrature nodes
and weights are calculated using the analytical solution based on the Two Unequal Weights Quadrature
(TUEWQ) formula. By applying the finite volume method to the spatial domain, a semi-discreet ordinary
differential equation system is obtained and solved. Both steady state and dynamic algorithms are
extensively validated at analytical, numerical, and experimental levels. At the numerical level, the
predictions of both algorithms are validated using the extended fixed pivot technique as implemented in
PPBLab software (Attarakih, M., Alzyod, S., Abu-Khader, M., Bart, H.-J. (2012). PPBLAB: A new
multivariate population balance environment for particulate system modeling and simulation. Procedia
Eng. 42, pp. 144-562). At the experimental validation level, the extended SQMOM is successfully used
to model the steady state hydrodynamics and physical and reactive mass transfer behavior of agitated
liquid-liquid extraction columns under different operating conditions. In this regard, both models are
found efficient and able to follow liquid extraction column behavior during column scale-up, where three
column diameters were investigated (DN32, DN80, and DN150). To shed more light on the local
interactions among the contacted phases, a reduced coupled PBE and CFD framework is used to model
the hydrodynamic behavior of pulsed sieve plate columns. In this regard, OPOSPM is utilized and
implemented in FLUENT 18.2 commercial software as a special case of the SQMOM. The dropletdroplet
interactions
(breakage
and
coalescence)
are
taken
into
account
using
OPOSPM,
while
the
required
information
about
the
velocity
field
and
energy
dissipation
is
calculated
by
the
CFD
model.
In
addition
to
this,
the proposed coupled OPOSPM-CFD framework is extended to include the mass transfer. The
proposed framework is numerically tested and the results are compared with the published experimental
data. The required breakage and coalescence parameters to perform the 2D-CFD simulation are estimated
using PPBLab software, where a 1D-CFD simulation using a multi-sectional gird is performed. A very
good agreement is obtained at the experimental and the numerical validation levels.

The polydispersive nature of the turbulent droplet swarm in agitated liquid-liquid contacting equipment makes its mathematical modelling and the solution methodologies a rather sophisticated process. This polydispersion could be modelled as a population of droplets randomly distributed with respect to some internal properties at a specific location in space using the population balance equation as a mathematical tool. However, the analytical solution of such a mathematical model is hardly to obtain except for particular idealized cases, and hence numerical solutions are resorted to in general. This is due to the inherent nonlinearities in the convective and diffusive terms as well as the appearance of many integrals in the source term. In this work two conservative discretization methodologies for both internal (droplet state) and external (spatial) coordinates are extended and efficiently implemented to solve the population balance equation (PBE) describing the hydrodynamics of liquid-liquid contacting equipment. The internal coordinate conservative discretization techniques of Kumar and Ramkrishna (1996a, b) originally developed for the solution of PBE in simple batch systems are extended to continuous flow systems and validated against analytical solutions as well as published experimental droplet interaction functions and hydrodynamic data. In addition to these methodologies, we presented a conservative discretization approach for droplet breakage in batch and continuous flow systems, where it is found to have identical convergence characteristics when compared to the method of Kumar and Ramkrishna (1996a). Apart from the specific discretization schemes, the numerical solution of droplet population balance equations by discretization is known to suffer from inherent finite domain errors (FDE). Two approaches that minimize the total FDE during the solution of the discrete PBEs using an approximate optimal moving (for batch) and fixed (for continuous systems) grids are introduced (Attarakih, Bart & Faqir, 2003a). As a result, significant improvements are achieved in predicting the number densities, zero and first moments of the population. For spatially distributed populations (such as extraction columns) the resulting system of partial differential equations is spatially discretized in conservative form using a simplified first order upwind scheme as well as first and second order nonoscillatory central differencing schemes (Kurganov & Tadmor, 2000). 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 (Karlsen et al., 2001). The time variable is discretized using an implicit strongly stable approach that is formulated by careful lagging of the nonlinear parts of the convective and source terms. The present algorithms are tested against analytical solutions of the simplified PBE through many case studies. In all these case studies the discrete models converges successfully to the available analytical solutions and to solutions on relatively fine grids when the analytical solution is not available. This is accomplished by deriving five analytical solutions of the PBE in continuous stirred tank and liquid-liquid extraction column for especial cases of breakage and coalescence functions. As an especial case, these algorithms are implemented via a windows computer code called LLECMOD (Liquid-Liquid Extraction Column Module) to simulate the hydrodynamics of general liquid-liquid extraction columns (LLEC). The user input dialog makes the LLECMOD a user-friendly program that enables the user to select grids, column dimensions, flow rates, velocity models, simulation parameters, dispersed and continuous phases chemical components, and droplet phase space-time solvers. The graphical output within the windows environment adds to the program a distinctive feature and makes it very easy to examine and interpret the results very quickly. Moreover, the dynamic model of the dispersed phase is carefully treated to correctly predict the oscillatory behavior of the LLEC hold up. In this context, a continuous velocity model corresponding to the manipulation of the inlet continuous flow rate through the control of the dispersed phase level is derived to get rid of this behavior.