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Abstract A hierarchical approach for modelling and simulation of coupled hydrodynamics and mass transfer in liquid extraction columns using detailed and reduced bivariate population balance models is presented. The hierarchical concept utilizes a one-dimensional CFD model with detailed bivariate population balances. This population balance model is implemented in the PPBLAB software, which is used to optimize the column hydrodynamics. The optimized droplet model parameters (droplet breakage and coalescence) are then used by a two-dimensional CFD reduced population balance model. As a reduced bivariate population balance model, OPOSPM (One Primary and One Secondary Particle Method) is implemented in the commercial FLUENT software to predict the coupled hydrodynamics and mass transfer of an RDC extraction column with 88 compartments. The simulation results show that the coupled two-dimensional-OPOSPM model produces results that are very close to that of the one-dimensional PPBLAB detailed population balance model. The advantages of PPBLAB are the ease of model setup, implementation and the reduced simulation time (order of minutes), when compared to the computational time (order of weeks) and computational resources using FLUENT software. The advantages of the two-dimensional CFD model is the direct estimation of the turbulent energy dissipation using the k-ε model and the local resolution of continuous phase back mixing.

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... The developed model is capable to predict the initial droplet size and velocity distribution in droplet formation region of sprays. In bubble columns, Attarakih et al. (2015a), (2015b), (2015c) used the coupled the Shannon Maximum Entropy Method with the OPOSPM reduced population balance model to predict the inlet and axial bubble size distribution along a pilot scale bubble column where the model predictions show good agreement with the published experimental data in the bubbly flow regime. Sobrino et al. (2015) investigated the diameter and velocity of bubbles in a three-dimensional two-fluid model simulation of a cylindrical fluidized bed. ...

... The accuracy and efficiency of the method is tested against known analytical solutions of the PBE including aggregation, coupled aggregation and growth. As practical case studies, the method is used to extend the numerical capabilities of our previous population balance modelling of liquid-liquid extraction columns as developed by Attarakih et al. (2004, 2008, 2013, Attarakih and Bart, 2014a, 2014bAttarakih et al., 2015aAttarakih et al., , 2015bAttarakih et al., , 2015c for mechanically stirred equipment including the RDC extraction column. On the other hand, our models for non-agitated liquidliquid extraction columns (Jaradat et al., 2011, Attarakih et al., 2012, 2017b are extended to test the present local maximum entropy population balance model. ...

... Also, the oscillation of the expansion coefficients using the direct expansion method is more pronounced and decrease slowly as compared to the LMEM where overdamped oscillation is observed. As shown by Attarakih et al. (2015a), (2015b), (2015c), this overdamped oscillation decreases faster and the overshoot can be minimized or even eliminated if the left boundary condition (f(a,r,t) = 0) is enforced. Note that in the LMEM the right boundary condition (lim f(b,r,t) = 0, as b ? 1) is automatically satisfied thanks the maximum entropy functional. ...

We propose continuous approximations to the population balance equation based on maximization of the Shannon entropy subject to the expected properties of the particle size distribution (PSD). This solution is used to close the source term of the PBE with careful sampling of the PSD at prescribed points as roots of the Nth-degree Legendre polynomial. Being a maximum entropy functional, the solution is unique and converges to the exact solution as the number of sampling points increases with accurate calculation of PSD integral properties. The accuracy and efficiency of the method are demonstrated by trying different analytical case studies (particle aggregation, aggregation and growth, and particle breakage) where we show it is not restricted to prespecified particle kinetics and functional forms. As practical case study, we modelled the coupled hydrodynamics and mass transfer in different liquid extraction columns and compared the calculated results with published steady state experimental data.

... Even though population balance approach is promising from the academic point of view, it is not widely used in the design process due to the rigorous models, abundance of parameters, and scale-up difficulties (Grinbaum, 2006). Computational fluid dynamics (CFD) modelling of liquid-liquid extraction columns takes into consideration the hydrodynamics and mass-transfer simultaneously (Mate et al., 2000;Nabli et al., 1998;Bardin-Monnier et al., 2003;Kolb, 2004;Yadav and Patwardhan, 2009;Xiaojin and Guangsheng, 2011;Attarakih et al., 2015). It involves the solution of the conservation equations of mass, momentum and volume fractions for different fractions, and is regarded as a powerful tool in chemical engineering. ...

... It would be beneficial to use population balance modeling with break up and coalescence predicting the droplet size distribution. This type of extraction column modeling has been done in 2D by Drumm et al. (2009) and Attarakih et al. (2015) using a one-scalar model for the population balance. Generally, the population balance models are not predicting reliably the size distribution without adapting the model to the process to be modeled. ...

... Their measured Kühni column profiles showed a typical "jump" at the continuous phase inlet due to the back mixing, and the absence of a jump at the dispersed phase inlet was onserved. Similar kind of phenomena was noticed by Attarakih et al. (2015) while modeling the upper part of the RDC-column with a 1D detailed population balance model. Thus, the fast mass tranfer rate and the back mixing phenomena taking place at the continuous phase inlet may be the reasons why fitting of the upper part of the column was more challenging. ...

1D axial dispersion and 3D CFD models for the extraction of levulinic acid from dilute aqueous solution by applying 2-methyltetrahydrofuran as a solvent are presented. The models are validated by comparison with the measured levulinic acid concentration profile data obtained in a bench-scale Kühni column. The 1D model contains NRTL parameters for the system levulinic acid-water-2MTHF. Correlations for drop size and hold-up for Kühni columns were taken from literature. The values for overall mass transfer coefficient ranged from 1.4E-5 to 2.2E-5 ms ⁻¹ , and increased as a function of the rotor speed. The fitting of the column performance resulted in a very good prediction of the solute concentration profiles in the extraction column, and the average absolute value of relative error for the 1D model was 23%. CFD model visualized the column performance at the column height of 150.5–160 cm giving valuable information on back mixing, phase velocities, dispersed phase volume fraction, and mass transfer. Dispersed phase volume fraction and mass transfer contours revealed, that the mass transfer rate (app. 0.25 g L ⁻¹ s ⁻¹ )is at its highest just below the rotor, and that there are blind spots in the compartments close to the extractor and just above each down comer. Values for the dispersed phase volume fraction are highest in the same area where the mass transfer reaches the highest values. The highest slip velocity values (app. 0.03 m ⁻¹ )are located in the tip of each compartment partition plates. General correlations, such as hold-up and drop size correlations, can successfully be applied in levulinic acid-water-2MTHF system reported in this work. The 1D axial dispersion model proved to be valuable tool for scale-up purposes, and CFD model, despite the long time needed for each simulation, gave useful information for the design purposes.

... This enables the calculation of the local resolution of the particle (drops, bubbles, solid particles) size distribution. Attarakih et al. (2015) and Zhang et al. (2017) additionally implemented mass transfer between the phases in the PBEs. This method offers highly accurate results since during the calculation of the fluid flow the effects of changes in drop size distribution and mass transfer are included. ...

... However, CFD with PBEs demands high computational effort since the PBEs have to be solved for each grid cell. Attarakih et al. (2015) for example required for a liquidliquid extraction column up to 27 days computing time for a 2D CFD simulation coupled with PBE for coalescence and breakage and one transfer component. In addition, the computing time of the numerical solution of the CFD-PBE further increases, especially when considering not only one dispersed phase, multi component mass-transfer and reactions . ...

... Including PBEs and mass transport into CFD, we expect a significant increase of computing times than calculating only the simplified fluid dynamic. For example Attarakih et al. (2015) simulated a liquid-liquid extraction column in technical scale (DN80, height 2.95 m) with a 2D CFD simulation coupled with PBE and one transfer component and consumed 27 days (648 h) computing time (2 AMD Opteron (4184) processors; each 6 cores). ...

In multiphase devices, fluid dynamics have a high impact on concentration profiles and mass transfer between the phases and therefore influence efficiency. Standard models often assume ideally mixed conditions or plug flow. The application of such models for multiphase devices with complex flow patterns causes inaccuracies, if the flow deviates from ideally mixed or plug flow conditions. Therefore, for a precise model based design and operation parameter determination of devices with complex flow patterns, the local fluid dynamics should be considered. CFD simulations for multiphase systems including mass transfer, population balance equations for coalescence and breakage as well as reactions are still time consuming. Thus, we developed a compartment-model based on prior calculated CFD flow-data. In the CFD simulations, the time consuming population balance equations for coalescence and breakage, mass transfer and reactions are neglected. These phenomena are considered in the compartment-model. Thereby we reduce the overall computing time.
This paper presents the CFD based compartment-model applied on a loop-reactor. First, a three-phase CFD model of the developed multiphase loop-reactor is introduced. Following, the paper presents the compartment-model and the application of a time-driven constant-number Monte-Carlo approach to solve population balances. Finally, the compartment-model is applied to the liquid-liquid extraction part of the loop-reactor calculating the drop size distribution and mass transfer based on previously calculated CFD data.

... As a result of this, a detailed description of the dispersed phase behavior is required. Indeed, the aforementioned back-mixing and dispersion models used in industry ignore the permanent droplet-droplet interactions and hence they predict neither the correct steady state nor the actual dynamic behavior of the dispersed phase (Steiner, 1994;Schmidt et al., 2006;Attarakih et al., 2015a;Alzyod et al., 2016a). In addition to this, they neglect the discreet nature of the dispersed phase by assuming it to behave as a pseudo homogeneous phase and the deviation from the ideal plug flow is taken into account using the axial dispersion coefficient (Thornton, 1992;Steiner, 1994;Mohanty, 2000;Attarakih, 2004;Drumm, 2010). ...

... This behavior is characterized by the coupled interacting hydrodynamic and mass transfer phenomena such as: droplet breakage, coalescence, growth (or shrinking) and the interfacial physical and reactive mass transfer process. Such unit operations include: crystallization (Hulburt and Katz, 1964;Czapla et al., 2009;Liu et al., 2010;Fysikopoulos et al., 2017), bubble columns (Deen et al., 2001;Petitti et al., 2010;Nauha and Alopaeus, 2013;Attarakih et al., 2016;Buffo et al., 2016), and liquid-liquid extraction columns (Weiss and Bart, 1993;Attarakih et al., 2004;Vikhansky and Kraft, 2004;Schmidt et al., 2006;Tiwari et al., 2008;Drumm et al., 2010;Sharma, 2009;Hlawitschka 2013;Wächtler et al., 2014;Attarakih et al., 2013aAttarakih et al., , 2014Attarakih et al., , 2015aAlzyod, 2016aAlzyod, , b, c, 2018. A first attempt to utilize the DPBM in chemical engineering applications was presented by Hulburt and Katz (1964), where they presented a mathematical formulation of the DPBM for crystallization process. ...

... Firstly, the pure column hydrodynamic equation is solved (by m = 0), then the mass transfer equation is solved (by setting m = 1). This assumption is valid for liquid extraction equipment since the breakage and coalescence terms are weak functions of the solute concentration and the operating solute concentration is usually less than 10 percent (Garthe, 2006;Schmidt et al., 2006;Attarakih et al., , 2013aAttarakih et al., , 2015a. In this regard, the required breakage and coalescence kernels, which are used to describe droplets interaction, are geometrical dependent functions. ...

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 applied pulsation intensity characterizes the flow patterns inside the column and increases the available interfacial area, which enhances the column efficiency. The available lumped correlations and mixture models are still insufficient to describe the actual behaviour of the dispersed phase, where they ignore the droplet-droplet and the interphase interactions (Attarakih et al., 2015). These interactions include: breakage, coalescence and interphase mass transfer. ...

... A first attempt to couple OPOPSM and CFD solvers was done by Drumm et al., (2010) to model a lab scale RDC liquid extraction column. In the same direction, Jaradat et al. (2011) simulated a pulsed sieve plate column using the LLECMOD software and Attarakih et al., (2015) implemented OPOSPM to simulate the hydrodynamics and mass transfer behaviour of a pilot plant RDC column. In a recent work, presented a coupled OPOSPM-CFD framework for modelling the hydrodynamics behaviour of a pulsed sieve plate extraction column. ...

... Where, c5 depends on the column geometry, and Hc is the compartment height. The solute mass transfer conservation equation for the i th phase is given by (Attarakih et al., 2015): ...

A reduced coupled 2D-CFD and Population Balance Model (PBM) framework (Alzyod et al., Comput. Aided Chem. Eng., 40, 61-66) is extended to model the mass transfer behaviour of pulsed sieve plate liquid extraction columns. The Euler-Euler approach is used to model the two phase flow inside the column, while the One Primary One Secondary Particle Method (OPOSPM), the simplest form of SQMOM, is utilized as a reduced population balance solver. 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 MOM can be used to solve the PBM with relatively high accuracy and low CPU time (Attarakih et al., 2009a;Buffo et al., 2013;Falola et al., 2013, Attarakih et al., 2015a. Since the bubble size dependency of ( ; , ) has been integrated out, some information about the bubble size distribution will be lost especially its shape . ...

... Since the PBM couples both mass transfer and hydrodynamics to model particle-particle interactions, the resulting model will be a complex system of nonlinear integro-partial differential equations that needs to be solved numerically with low computational cost (Shimizu et al., 2000;Attarakih and Bart, 2014). Among the previously mentioned method of moments, the OPOSPM is considered as the simplest one (Attarakih et al., 2009b;Jildeh et al., 2012;Attarakih et al., 2013;Hlawitschka, 2013, Attarakih et al., 2015a. Despite its simplicity, the OPOSPM can be used to accurately predict the main population properties such as particle number density, dispersed phase holdup, mean bubble diameter and solute concentration (Attarakih et al., 2013;Hlawitschka, 2013. ...

... The OPOSPM framework can be used as a rapid approximation of the continuous PBM, where this model conserves both the total number of particles as well as the volume concentration of the dispersed phase using transport equations for these population parameters (Attarakih et al., 2009b(Attarakih et al., , 2015a. In addition to solute concentration in both phases, these population parameters can be tracked directly by solving coupled OPOSPM-mass transfer models (Attarakih et al., , 2015a. ...

Bubble columns are gas-liquid contactors and reactors which are intensively used in chemical, biochemical and petrochemical industries. Nowadays, they are considered as one of the most important technologies employed to reduce carbon dioxide emissions from industrial gas streams. In bubble columns, gas bubble size is an important design parameter which defines the gas–liquid interfacial area available for mass transfer, determines the bubble rising velocity, residence time and the gas void fraction. Therefore, the bubble behavior plays an important role in designing and scaling-up bubble column reactors. Therefore, in this research thesis, a new mathematical framework for modeling bubble interactions in such equipment is developed. In this regard, the One Primary and One Secondary Particle Method (OPOSPM) developed by Attarakih et al. (Attarakih, M., Abu-Khader, M., Bart, H.-J., 2013, Modelling and Dynamic Analysis of an RDC extraction column using OPOSPM, Chemical Engineering Science, 91, 180-196) was used as a reduced population balance framework to predict the bubble column hydrodynamics and its mass transfer performance. The OPOSPM is coupled with the drift-flux model, bubble breakage, coalescence and growth phenomenological models with necessary derivations and modifications to suit the OPOSPM hierarchical structure.
The coupled OPOSPM-mass transfer model has many advantages over the existing bubble column models. The present model takes into account bubble interactions which occur within the column such as bubble growth, breakage and coalescence which are important to determine the interfacial area concentration. This integral bubble distribution property is essential to predict many transport phenomena such as mass and heat transfer. As a result, accurate prediction of the different hydrodynamics and mass transfer characteristics that affect column design as well as scale-up are obtained using this coupled model compared to the existing models where these interactions are ignored in most cases. This model is also considered as a simple one when compared to the detailed population balance modelling framework, since it consists only of four transport equations for the total number concentration of bubbles, gas void fraction and solute mass balance equations in both gas and liquid phases.

... The MOM can be used to solve the PBM with relatively high accuracy and low CPU time (Attarakih et al., 2009a;Buffo et al., 2013;Falola et al., 2013, Attarakih et al., 2015a. Since the bubble size dependency of ( ; , ) has been integrated out, some information about the bubble size distribution will be lost especially its shape . ...

... Since the PBM couples both mass transfer and hydrodynamics to model particle-particle interactions, the resulting model will be a complex system of nonlinear integro-partial differential equations that needs to be solved numerically with low computational cost (Shimizu et al., 2000;Attarakih and Bart, 2014). Among the previously mentioned method of moments, the OPOSPM is considered as the simplest one (Attarakih et al., 2009b;Jildeh et al., 2012;Attarakih et al., 2013;Hlawitschka, 2013, Attarakih et al., 2015a. Despite its simplicity, the OPOSPM can be used to accurately predict the main population properties such as particle number density, dispersed phase holdup, mean bubble diameter and solute concentration (Attarakih et al., 2013;Hlawitschka, 2013. ...

... The OPOSPM framework can be used as a rapid approximation of the continuous PBM, where this model conserves both the total number of particles as well as the volume concentration of the dispersed phase using transport equations for these population parameters (Attarakih et al., 2009b(Attarakih et al., , 2015a. In addition to solute concentration in both phases, these population parameters can be tracked directly by solving coupled OPOSPM-mass transfer models (Attarakih et al., , 2015a. ...

Bubble columns are gas-liquid contactors and reactors which are intensively used in chemical, biochemical and petrochemical industries. Nowadays, they are considered as one of the most important technologies employed to reduce carbon dioxide emissions from industrial gas streams. In bubble columns, gas bubble size is an important design parameter which defines the gas–liquid interfacial area available for mass transfer, determines the bubble rising velocity, residence time and the gas void fraction. Therefore, the bubble behavior plays an important role in designing and scaling-up bubble column reactors. Therefore, in this research thesis, a new mathematical framework for modeling bubble interactions in such equipment is developed. In this regard, the One Primary and One Secondary Particle Method (OPOSPM) developed by Attarakih et al. (Attarakih, M., Abu-Khader, M., Bart, H.-J., 2013, Modelling and Dynamic Analysis of an RDC extraction column using OPOSPM, Chemical Engineering Science, 91, 180-196) was used as a reduced population balance framework to predict the bubble column hydrodynamics and its mass transfer performance. The OPOSPM is coupled with the drift-flux model, bubble breakage, coalescence and growth phenomenological models with necessary derivations and modifications to suit the OPOSPM hierarchical structure.
The coupled OPOSPM-mass transfer model has many advantages over the existing bubble column models. The present model takes into account bubble interactions which occur within the column such as bubble growth, breakage and coalescence which are important to determine the interfacial area concentration. This integral bubble distribution property is essential to predict many transport phenomena such as mass and heat transfer. As a result, accurate prediction of the different hydrodynamics and mass transfer characteristics that affect column design as well as scale-up are obtained using this coupled model compared to the existing models where these interactions are ignored in most cases. This model is also considered as a simple one when compared to the detailed population balance modelling framework, since it consists only of four transport equations for the total number concentration of bubbles, gas void fraction and solute mass balance equations in both gas and liquid phases.

... PPBLab is a MATLAB based environment for modelling and simulation of discrete flow processes, and in particular liquid-liquid extraction columns, using detailed population balances. The software utilizes the recent PBE solution methods to model and simulate agitated, non-agitated and pulsed liquid extractions columns (Attarakih et al., 2015). These solution methods include: Mixed fixed-pivot and Quadraure Method Of Moments (QMOM), and the One Primary One Secondary Particle Method (OPOSPM). ...

... In addition to this, PPBLab provides many toolboxes with intensive hydrodynamics and mass transfer correlations as well as models to correlate the single droplet velocity and liquid-liquid phase equilibria. PPBLab was validated against a wide range of published experimental data and 3D CFD simulation tools (Attarakih et al., 2015). Accordingly, the simulation results using PPBLab were found competitive with those obtained using 3D CFD tools. ...

... The droplet rising velocity (uy) depends on many factors such as: The droplet terminal velocity (ut), continuous phase velocity (uc), droplet diameter and the column internal geometry which can be taken into account using droplet slowing factor 01 v k . In general, the droplet rising velocity is given by (Attarakih et al., 2015): ...

In this work, we present a new population balance based module for modelling the hydrodynamics and mass transfer processes in pulsed packed bed liquid extraction columns. The new module is fully implemented using PPBLab software, which utilizes recent population balance model solution algorithms. In this regard, the PPBLab detailed and reduced extended fixed pivot solvers are used to discretize the internal coordinates, while the PPBLab built-in space-time solver is used to discretize the physical spatial domain. In addition to this, a user-friendly interface is designed to facilitate the user inputs and outputs and to allow a full access to the CAPE-OPEN thermodynamics package (TEA). As a case study, this PPBLab column module is validated using the published steady state experimental data for water-acetone-toluene chemical system in a DN80 pulsed packed bed liquid extraction column. The predicted column performance is found to agree well with PPBLab software simulation results.

... The DPBM consists of geometrical dependent nonlinear integro-partial differential equations and hence it has no general analytical solution (Attarakih et al., 2009a,b;Ramkrishna, 2000;Vikhansky, 2013;Mohanty, 2000). As a result of this, accurate numerical solution algorithms with low computational cost are required (Attarakih et al., 2015a(Attarakih et al., ,b, 2006aDrumm et al., 2010;Gimbun et al., 2009). During the last decade, several numerical methods have been proposed to achieve these goals. ...

... Compared to the previous mentioned moment based methods, the SQMOM is considered as a comprehensive mathematical framework. This is because it combines the advantages of the method of classes and the QMOM and reduces their drawbacks (Attarakih et al., 2015a(Attarakih et al., ,b, 2009a. In fact, the droplet size distribution can be reconstructed using the primary particles, while the secondary particles are responsible for breakage and coalescence events (Attarakih et al., 2015a(Attarakih et al., ,b, 2009a. ...

... This is because it combines the advantages of the method of classes and the QMOM and reduces their drawbacks (Attarakih et al., 2015a(Attarakih et al., ,b, 2009a. In fact, the droplet size distribution can be reconstructed using the primary particles, while the secondary particles are responsible for breakage and coalescence events (Attarakih et al., 2015a(Attarakih et al., ,b, 2009a. Attarakih et al. (2009a,b) proved that most of the available numerical methods are special cases from the general SQMOM by varying the number of primary and secondary particles. ...

In this work, the Sectional Quadrature Method Of Moments (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 derived by Attarakih et al. (Attarakih et al., 2009attarakih, M., Drumm, C., & Bart, H.-J., (2009), Solution of the population balance equation using the Sectional Quadrature Method of Moments (SQMOM). Chemical Engineering Science, 64, 742-752). By applying the finite volume method to the spatial domain, we end up with a semi-discreet ordinary differential equation system which is solved using the MATLAB standard ODE solvers (ode45). As a case study, the SQMOM is used to investigate the dynamic behavior of a Kühni DN150 liquid-liquid extraction column. As an independent validation step, the SQMOM prediction is compared with the PPBLab software which utilizes the extended fixed pivot technique as a built-in population balance model solver. Furthermore, the SQMOM is validated using the available dynamic experimental data from a Kühni liquid extraction column using water-acetone-toluene chemical test system. The dynamic analyses of the Kühni column show very interesting features concerning the coupled column hydrodynamics and mass transfer and the droplet breakage and coalescence as well.

... Bart et al. (Drumm et al., 2009;Schmidt et al., 2006) coupled the population balance model (PBM) and the CFD to investigate the flow field, hydrodynamics, and mass transfer performance of RDCs. They further developed the one-dimensional simulation tool PPBLAB (Particulate Population Balance LABoratory) to predict the hydrodynamic and mass transfer performance of RDCs (Attarakih et al., 2015). Sen et al. (Sen and Singh, 2016;Sen et al., 2018Sen et al., , 2019 reported the Eulerian two-fluid approach to predict the holdup, axial dispersion, and droplet size distribution in PSPCs. ...

... Nevertheless, CFD modeling of mass transfer performance of extraction columns is rare (Bart et al., 2020). Only few one-dimension numerical results were presented (Attarakih et al., 2015;Jaradat et al., 2011). At present, this predicament deserves to be changed with strong demand for better understanding of mass transfer performance of extraction columns. ...

Liquid–liquid interphase mass transfer was experimentally and numerically studied in a pilot-scale pulsed disc-and-doughnut column (PDDC). The mass transfer coefficients, which were determined by analyzing experimental data through a transient axial dispersion model, increased from 2.27 × 10–6 to 3.06 × 10–6 m/s with an increase in the pulsation intensity from 0.0065 to 0.0105 m/s. The Euler–Euler two-fluid model was applied to simulate the interphase mass transfer in the PDDC. Three key issues, including the formulation of governing equations, selection of the mesh size, and determination of the mass transfer coefficient, which significantly influence the numerical accuracy, were discussed in detail. The simulated concentration profiles accorded well with that predicted by the experimentally verified mass transfer model. Local mass transfer performance was uncovered based on simulation results to elucidate the spatial and temporal variation of the solute concentrations, hold-up, and mass transfer coefficients in the PDDC.

... • One-dimensional simulation tools, which only describe the drop properties along the column height, such as ReDrop or PBELab. [1][2][3][4] • CFD-models (computational fluid dynamics) with PBE, which calculate the fluid dynamics and mass transfer in an either two-or three-dimensional simulation domain. [5][6][7][8][9][10][11][12][13][14] • Compartment-models, which decouple the fluid dynamic simulation from the calculation of the drop-property-distribution of the PBE. ...

... Nevertheless, each independent property, which has to be considered in the PBE, increases the dimensionality of the PBE and therefore the computational effort. 19 For example Attarakih et al. 3 simulated an extraction column type RDC (rotating disc contactor) with a two-dimensional CFD-PBE model and required 4.63 days (without mesh refinement) to 27 days (with mesh refinement) to simulate the steady state operation point. The simulation with a three-dimensional CFD-PBE model would require even more simulation time. ...

For detailed simulation and evaluation of stirred extraction columns a CFD based compartment‐model was developed. Instead of simulating all effects in a computational expensive PBE‐CFD‐model, the velocity field calculation of the continuous phase is decoupled from the calculation of the dispersed phase (one‐way coupling). In CFD only the continuous phase is simulated and the resulting velocity profile is used in the compartment‐model to simulate the drop movement, coalescence, breakage and mass transfer for a representative number of drops (Monte‐Carlo Method). This decoupling has a major impact on the calculated fluid‐dynamics. Thus, the velocity profile of the CFD results is modified in the model to account for phase interaction. The compartment‐model is applied for the simulation of a Kühni extraction column with the system toluene/water/acetone. The simulation results, namely holdup, drop size and concentration profiles over the column height, are in good agreement with experiments for different loads and different stirrer speeds.

... For the drop-based modeling of extraction columns, population balances are applied [3,6,7]. In the present study, a Monte-Carlo method is used to solve the population balance [6,8]. ...

... For example, small amounts of surface-active components can greatly affect the coalescence efficiency [11]. Consequently, one pilot-plant experiment is required to fit the coalescence parameter [7,9]. The population balance model is used to predict the operation domain of a stirred extraction column. ...

For the fast approximation of liquid‐liquid extraction column designs via HETS (height equivalent of a theoretical stage) values and flooding points, a population balance model was developed. The model requires only one pilot‐scale column experiment to fit a coalescence parameter. The comparison of experimental data with simulations shows good agreement for a stirred extraction column. With the model, the operation domain of the column is simulated and visualized in a performance map, showing the holdup, the distance to the flooding point and HETS values. A population‐balance model is applied for the calculation of the operation window of liquid‐liquid extraction columns, showing holdup, flooding point and HETS, which can be used for the estimation of the column design. This model only requires the physico‐chemical properties and one pilot‐plant experiment to determine the coalescence parameter.

... To predict the evolution of DSD, the two-fluid model coupled with a breakage and coalescence model have been adopted in the literature. [6][7][8] The multifractal kernel model was successfully applied to predict the liquid-liquid dispersions in stirred tanks by Li et al. and Gao et al. [9,10] To describe the DSD of dispersion phase in a continuous phase, many semi-empirical phenomenological models, [11][12][13][14] named PBMs, were available in literature to calculate the breakage rate and coalescence rate. However, there is no universal model applicable to all multi-phase systems due to the complex interactions between different phases. ...

... Droplet diameter d in Equation (6) implies that the drag force acting on each droplet is different. In order to simplify the computation, the mean Sauter diameter 32 d is employed to estimate the average drag force acting on a population of droplets. ...

Liquid‐liquid two‐phase flow in a mixer of mixer‐settler has been studied via a computational fluid dynamics (CFD) simulation combined with the population balance model (PBM) and verified with particle image velocimetry (PIV) experiments. The simulation was performed using the multiple reference frame (MRF) approach, the Eulerian‐Eulerian two‐fluid model, and the standard k‐ϵ model. The effects of impeller speed, flow ratio, and impeller type on flow field, droplet diameter, and dispersed phase holdup were investigated. The results showed that CFD simulation combined with PBM could predict droplet size distribution (DSD). The smaller droplets were mainly in the bottom region of the mixer, larger ones were in the top part of the mixer, and the largest droplets appeared in the impeller centre region. The DSD and holdup were more sensitive to impeller speed than to the organic/aqueous flow ratio. A dual‐impeller mixer configuration was designed to enhance the mixing performance. Compared with single‐impeller, the installation of dual‐impellers could effectively avoid the dispersed phase dead zone above the lower impeller. When hc /T = 0.3, the best dispersing effect, such as uniform DSD and high mixing chamber space utilization, could be obtained. This article is protected by copyright. All rights reserved

... Liquid-liquid extraction is considered as one of the most commonly used separation techniques after distillation in pharmaceutical and petroleum industries (Attarakih et al., 2015a(Attarakih et al., , 2015bBart, 2001). In such a technique, the droplet size distribution (DSD) plays the major role in dispersed phase by assuming a pseudo homogeneous phase behavior (Attarakih et al., 2015b;Mohanty, 2000). ...

... For an intensive review concerning these numerical methods, the reader can refer to Attarakih et al. (2004) and Attarakih and Bart (2014). In general, the moments based methods were found efficient in terms of implementation and computational cost (Attarakih et al., 2015a;Wächtler, 2014;Drumm et al., 2010). However, these methods conserve the moments of the distribution but unable to reproduce the distribution itself which is needed in many engineering applications (Buffo and Alopaeus, 2016;Lebaz et al., 2016;Souza et al., 2010;Attarakih and Bart, 2014). ...

The Sectional Quadrature Method Of Moments (SQMOM) is extended to solve the bivariate nonhomogeneous Population Balance Equation (PBE) along the spatial domain. The resulting unclosed integral terms are approximated using Gauss-Christoffel quadrature rule, where the required quadrature nodes and weights are calculated analytically using the Two-Equal Weight Quadrature (TEqWQ) formula (Attarakih, M., Drumm, C., & Bart, H.-J., (2009), Solution of the population balance equation using the Sectional Quadrature Method of Moments (SQMOM). Chemical Engineering Science, 64, 742-752). The resulting set of moment equations is dominant by convection due to the large spatial spontaneous gradients. To take this into account, a finite volume scheme with flux vector splitting technique is designed and coupled with the standard MATLAB ordinary differential equation solvers. To speed up the system approach to steady state and to enforce numerical stability, the present numerical scheme is fifth order accurate in time while it is first order accurate in space. As a numerical test, the model prediction is validated using PPBLab software, where the extended fixed pivot technique with a multi-sectional grid (w.r.t droplet diameter) is used. In addition to this, the SQMOM is experimentally validated using the published data for water-toluene and water-acetone-toluene chemical test systems, where the ability of reconstructing the droplet size distribution is successfully examined. The required number of sections to discretize the internal coordinates (droplet diameter and solute concentration) is found to be 15, while the required number of spatial numerical cells is found to be 50 for accurate steady state pilot plant RDC liquid-liquid extraction column simulation. In this regard, the SQMOM is able to follow the extraction column behavior during column scale-up, where two column diameters are investigated (RDC DN80 and DN150). The SQMOM is found flexible to predict the column hydrodynamics as well as the mass transfer profiles as compared to the published experimental data using only the first four low-order moment equations.

... To date, there have been several investigations into CFD informed mass transfer modelling in liquid-liquid extraction columns, with several studies looking at application within an RDC [13,14] and a PSEC [15]. However, these were done with models unable to distinguish between the flow regimes. ...

In this work, the GEneralised Multifluid Modelling Approach (GEMMA) is applied to the simulation of liquid–liquid extraction in a Rotating Disc Column (RDC) and a Pulsed Sieve-plate Extraction Column (PSEC). A mass transfer modelling methodology is developed, in which the multiphase flows, droplet size distribution and dispersed phase holdup predicted with computational fluid dynamics are coupled to mass transfer correlations to predict the overall mass transfer. The numerical results for the stage-averaged dispersed phase holdup, Sauter mean droplet diameter and axial solute concentration in the RDC and PSEC agree with experimental observations. The proposed modelling method provides an accurate predictive tool for complex multiphase flows, such as those observed in intensified liquid–liquid extraction, and provides an alternative approach to column design using empirical correlations or pilot plant study.

... (10) and (12) are adopted in this study as they are well established in the literature, see e.g. [6,7,14,[59][60][61]. The kernels were derived for a water-oil liquid-liquid system in a stirred tank very similar to the present application in terms of density and viscosity ratio as well as interfacial tension. ...

Liquid-liquid disperse multiphase flow in a baffled tank stirred by a Rushton turbine is investigated by experiments and 3D simulations for different impeller speeds and volume fractions. Droplet size distributions are experimentally measured with an inline shadowgraphic probe and compared with fully transient coupled CFD-PBM simulations using the Eulerian multi-fluid framework, the inhomogeneous Multiple Size Group approach and a statistical turbulence model. The simulation model is assessed by a variation of size class and velocity group numbers, grid resolution and simulation domain extent. The model parameters for the breakup and coalescence kernels are fitted to match the measured droplet size distribution at a selected operation point. A subsequent application to different operation points shows that experimental droplet size trends are captured, albeit aside from the fitting point, deviations to data increase. The resulting droplet size distribution is analyzed by an assessment of the turbulent mixing, breakup and coalescence time scales.

... But their work was limited to small bubble columns < 0.3 m in diameter. Attarakih et al. [31] used a reduced bivariate PBM model to compare the K L a between CFD -Particulate Population Balance Laboratory (PPBLAB) model and CFD -OPOSPM (one primary and one secondary particle method) model. The limitation of their approach was that both their models relied on experimentally correlated K L a for droplet interactions. ...

The diffused aeration process is the most energy-intensive operation of bioreactor treatment, amounting to 45–75 % of the plant energy costs. To improve its efficiency, it is essential to measure the oxygen transfer rate from the aerators to wastewater. In this study, a multiphase mixture computational fluid dynamics (CFD) model is developed using k-ε turbulence closure equations along with a discrete population balance model (PBM) add-on with specific bubble classes, to predict the oxygen mass transfer. The transfer of oxygen species from air to water is modeled using the species transport model. The PBM is used to analyze the formation, growth, breakage, and coalescence of air bubbles. The validated model is then extended for sensitivity analysis for a diffused aeration system in a bench-scale aeration tank. Results show that, the volumetric oxygen mass transfer coefficient increases by 15 %, with a decrease of air bubble size by 10 %. The air bubbles have a wider distribution, with a larger diameter near the bottom of the bioreactor and a narrow distribution with a smaller bubble size at the top. Results show that, in the bioreactor, the dissolved oxygen concentration reaches the equilibrium or saturation value when the height by breadth ratio is 2.5 and does not increase further with increase in height of the water column. Also, the air bubble size of 6 mm was the efficient bubble size for a fixed airflow rate of 1.45 m³ h⁻¹.

... The first step is based on the parameter optimization of existing single droplet literature correlations for velocity, slowing factor and breakage probability using single droplet measurements or CFD simulations. The second step involves the determination of the energy dissipation and axial dispersion correlations using CFD tools [48,[73][74][75], which allow a good determination of these parameters also for industrial-sized columns, where correlations are rare. In the third place, solving an inverse population balance approach together with swarm experimental data obtained from lab-scale col-umns enables a determination of the required coalescence parameters. ...

The digitization of extraction columns requires a profound knowledge of the present hydrodynamics/mass transport interaction as well as appropriate measurement techniques for the detection of relevant input and target values. In this article, the different techniques for droplet size distribution as well as concentration determination are presented and new methods for online evaluation are discussed. In combination with the simulation of droplet size, holdup and solute concentration distribution, an online‐capable process tool for controlling and optimizing extraction columns will be obtained. Different measuring techniques for extraction columns for drop size and concentration determination are described and new methods for online evaluation are presented. In combination with the simulation, an online‐capable tool for process control of extraction columns is introduced.

... The k−ε turbulent model is used to calculate the turbulent viscosity as adopted by numerous researchers. 18,25,32 F ij ⃗ is the interphase interaction force from phase i to phase j, which includes the drag force, lift force, and virtual mass force, and so forth. The drag force is the dominant one in liquid−liquid two-phase flows. ...

In this study, augmented computational fluid dynamic−population
balance model (CFD−PBM) simulations were carried out to predict hydrodynamics in pulsed disc and doughnut columns (PDDC) with wettable internal plates. The droplet breakage and droplet−liquid layer interaction submodels were integrated to construct an augmented PBM, which was embedded in the CFD to calculate droplet-size distributions (DSD). It was found that the augmented PBM-predicted
DSDs are better than the PBM with only the breakage submodel. The influences arising from the wettability of internal plates should be carefully considered. In a pilot-scale PDDC with wettable internal plates, the augmented CFD−PBM simulation results indicated that the shape of DSDs changed little with an increase in the throughput from 80 to 200 L/(dm2·h), but was significantly influenced by the pulsation intensity as it increased from 0.0065 to 0.0125 m/s.

... The modeling and simulation of the extraction devices are necessary for determining the affecting parameters and their impact on the performance of equipment. The model of the process involves a wide set of equations, and solving them simultaneously is difficult and timeconsuming (Gomes et al., 2006;Rode et al., 2013;Attarakih et al., 2015). On the other hand, the numerical methods of artificial neural networks (ANN) have found a wide interest in the modeling of process systems in recent years. ...

In this work, the dispersed phase holdup in a Kühni extraction column is predicted using intelligent methods and a new empirical correlation. Intelligent techniques, including multilayer perceptron and radial basis functions network are used in the prediction of the dispersed phase holdup. To design the network structure and train and test the networks, 174 sets of experimental data are used. The effects of rotor speed and the flow rates of the dispersed and continuous phases on the dispersed phase holdup are experimentally investigated, and then the artificial neural networks are designed. Performance evaluation criteria consisting of R 2 , RMSE, and AARE are used for the models. The RBF method with R 2 , RMSE, and AARE respectively equal to 0.9992, 0.0012, and 0.9795 is the best model. The results show that the RBF method well matches the experimental data with the lowest absolute percentage error (2.1917%). The rotor speed has the most significant effect on the dispersed phase holdup comparing to the flow rates of the continuous and dispersed phases.

... Another effect that is not directly calculated 23 in one-dimensional simulations of extraction columns is the accumulation and coalescence 24 of drops underneath stators, since only moving drops are simulated and the accumulation is 25 neglected. It is usually considered indirectly by fitting the parameters for the coalescence 1 between moving drops to column experiments (Klinger, 2007; requires high computing resources, in the time-scale of several weeks to several months 10 (Attarakih et al., 2015). This is especially the case when calculating the full geometry of the 11 column and when the dimensionality of the population balance increases, for example with 12 multi-component mass transfer, multiple phases or reactions . ...

The simulation of stirred liquid–liquid extraction columns with CFD still requires significant computing resources. Therefore, a CFD-based compartment model was developed for a simulation of a representative amount of drops while keeping the computing effort low. The model is based on the velocity profile of the continuous phase generated by a single-phase CFD simulation. The drop movement, mass transfer, accumulation of the dispersed phase under stators, coalescence, and breakage are modeled according to this flow profile. The simulation of the fluid dynamics is in good agreement with published experiments of a DN80 Kuehni extraction column.

... Accordingly, phenomenological models can be inserted as submodels (kernels) into the CFD model, thus, allowing a space-dependent evolution of the drop size distribution. Examples for the integration of PBM in a CFD environment have recently been published [86,87]. However, these examples do not consider reactions due to high computational demands. ...

Product selectivity and yield in chemical reactions can be limited by side product formation or low conversions due to equilibrium limitations. Extractive reaction systems (ERS) employ an in situ liquid‐liquid extraction that separates the product from the reaction phase to overcome these difficulties. The design of ERS requires a broad knowledge of the discipline of process intensification and extraction and reaction engineering. Furthermore, specific knowledge about the interplay of reaction and extraction phenomena is unique to ERS. This review gives an overview of the design of ERS and enables their application to any suitable reaction. Extractive reaction systems employ an in situ liquid‐liquid extraction that separates the product from the reaction phase to overcome, e.g., equilibrium limitations. This review gives an overview of the unique interplay of reaction and extraction phenomena in order to enable design of extractive reaction systems for any suitable reaction.

... Drumm et al. (2010) presented a coupled OPOPSM-CFD framework to model a lab scale RDC liquid extraction column. In the same direction, Attarakih et al., (2015) implemented OPOSPM to simulate the hydrodynamics and mass transfer behaviour of a pilot plant RDC liquid extraction column. Recently, Attarakih et al., (2016) presented a two dimensional CFD-OPOSPM model to model bubble column reactors. ...

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.

... Many researchers applied the CFD modeling methods across the ejectors to characterize the rate of gas entrainment in gas-liquid systems (Kandakure et al., 2005;Mukherjee et al., 1988). Several researchers have shown that CFD can be used for predicting the phenomena during the investigation of ejector performances (Fernandez, 2001;Giacomelli et al., 2016;Hakkaki-Fard et al., 2015;Hanafi et al., 2015;Lee et al., 2016;Marynowski et al., 2007;Palacz et al., 2017;Rusly et al., 2005;Yuan et al., 2015) and liquid jet in liquid phase systems (Bhattacharjee et al., 2017;Carcasci et al., 2016;Sandhya and Tide, 2017) as well as liquid-liquid extraction systems (Amokrane et al., 2016;Attarakih et al., 2015;Onink et al., 2010;Ye et al., 2016;Zou et al., 2016). In a CFD simulation the local pressure, flow rate and momentum associated with the fluids flow are calculated completely, while they are almost very difficult to be determined experimentally. ...

The hydrodynamic aspect of a novel eductor type contacting device for liquid–liquid extraction
(LLE) system has been evaluated by computational fluid dynamics (CFD) modeling. CFD
results were validated by droplet rise velocity and dispersed phase holdup in water/toluene
system; errors were 20.7% and 15.4%, respectively. The results of CFD simulation have shown
that the existence of venturi above the nozzle-jet improves the mixing due to extension of
mixing region. Educator mixing performance was investigated by the effects of the parameters:
jet velocity, the throat to nozzle area ratio (At/An), the column to nozzle diameter ratio (Dc/Dn),
the projection ratio (Ltn/Dt), and two phases flow ratio (Qc/Qj) on the dispersed phase holdup,
the suction ratio (Rs), the mixing efficiency (ηm) and the mixing energy efficiency (ηe). A new
efficiency named “overall efficiency” was defined as ηo=ηm.ηe to determine the optimum
operation and design condition of the eductor. The values of important parameters were
determined as a) the jet velocity less than 2 m/s; b) At/An around 100; c) the ratio of Dc/Dn<50;
d) 1<Ltn/Dt<2 and the value of Qc/Qj although being flexible, but smaller values are preferred
for achievement of higher mixing efficiency.

... Many methods have been developed to solve the PBM, such as the classes method (CM) [5], the Monte Carlo method (MCM) [6][7][8][9], the Quadrature Method of Moments (QMOM) [10], the Direct Quadrature Method of Moments (DQMOM) [11], the Adaptive Direct Quadrature Method of Moments (ADQMOM) [12], Taylor-series Expansion Method of Moments (TEMOM) [13], and the Extended Quadrature Method of Moment (EQMOM) [14]. These methods have been employed extensively to investigate different industrial processes [15][16][17][18][19][20][21][22][23][24][25]. ...

The accurate prediction of the droplet size distribution (DSD) in liquid–liquid turbulent dispersions is of fundamental importance in many industrial applications and it requires suitable kernels in the population balance model. When a surfactant is included in liquid–liquid dispersions, the droplet breakup behavior will change as an effect of the reduction of the interfacial tension. Moreover, also the dynamic interfacial tension may be different with respect to the static, due to the fact that the surfactant may be easily desorbed from the droplet surface, generating additional disruptive stresses. In this work, the performance of five breakup kernels from the literature is assessed, to investigate their ability to predict the time evolution of the DSD and of the mean Sauter diameter, when different surfactants are employed. Simulations are performed with the quadrature method of moments for the solution of the population balance model coupled with the two-fluid model implemented in the compressibleTwoPhaseEulerFoam solver of the open-source computational fluid dynamics (CFD) code OpenFOAM v. 2.2.x. The time evolution of the mean Sauter diameter predicted by these kernels is validated against experimental data for six test cases referring to a stirred tank with different types of surfactants (Tween 20 and PVA 88%) at different concentrations operating under different stirrer rate. Our results show that for the dispersion containing Tween 20 additional stress is generated, the multifractal breakup kernel properly predicts the DSD evolution, whereas two other kernels predict too fast breakup of droplets covered by adsorbed PVA. Kernels derived originally for bubbles completely fail.

... (2) S B,r,m The transformed breakage source term as given by Eq. (6) S C,r,m The transformed coalescence source term as given by Eq. (7) S M,r,m The transformed mass transfer source term as given by Eq. (8) conserving method of classes the most commonly used methods. This is because they provide fairly accurate solution with a reasonable computational cost and hence they are suitable to be coupled with CFD solvers and flowsheet simulations (Vikhansky, 2013;Lage, 2011;Marchisio and Fox, 2005;Attarakih et al., 2015c;Drumm et al., 2010). However, these methods have the disadvantage of destroying the shape of the droplet size distribution by conserving only its moments (Attarakih and Bart, 2014). ...

We present a new steady state algorithm for modeling the hydrodynamics and mass transfer behavior of liquid
extraction columns based on the SQMOM in a one dimensional domain. The SQMOM is extended to solve
the spatially distributed bivariate population balance equation (w.r.t. droplet diameter and the solute
concentration) at steady state. The integral spatial numerical flux is closed using the Multi Primary one
Secondary Particle Method, while the hydrodynamics integral source terms are closed using the analytical
Two-Equal Weight Quadrature formula. To facilitate the source terms implementation, an analytical solution
based on the algebraic velocity model is derived to calculate the required dispersed phase mean droplet
velocity. In addition to this, the hydrodynamics moment transport equations are coupled with the One Primary
One Secondary Particle Method to close the mass transport equations. The resulting system of ordinary
differential equations is solved using a fifth order numerical scheme in space. As a case study, the model
prediction is validated using published experimental data for DN80 Kühni extraction column. By using
SQMOM, fifteen sections (w.r.t. the droplet diameter as an internal coordinate) are found enough to predict
the droplet volume distribution and the column hydrodynamics as compared to the experimental data. On the
other hand, one section (w.r.t. the droplet solute concentration) along droplet size distribution is found enough
to predict the mass transfer profiles along the column height.

... Among the different methods available, the one most interesting for the investigation of industrial scale liquid-liquid multiphase systems is the two-fluid model (TFM) (Drew, 1982;Drew and Passman, 2006), implicitly assuming that one of the two phases is continuous, whereas the other one is disperse. The TFM is coupled with PBM to predict the evolution of the DSD (Ramkrishna, 2000), as shown in numerous applications (Hu et al., 2015;Attarakih et al., 2015;Favero et al., 2015). Many methods have been developed to solve the PBM, such as the classes method (CM) (Kumar and Ramkrishna, 1996), the Monte Carlo method (MCM) (Lin et al., 2002;Buffo et al., 2013b;Zhang and You, 2015;Hussain et al., 2015) and the method of moments (MOM) (Hulburt and Katz, 1964). ...

... Mass balances for gases in the bed of porosity ε b resulted in the following equation [16]: ...

Activated carbon was prepared from black liquor by steam activation. The BET specific surface area, pore volume and average pore diameter of resultant carbon activated at 900°C can reach 1010m2/g, 0.65m3/g and 3nm respectively. The relative humidity contributes to the adsorption of H2S on carbon, while the increased adsorption temperature and inlet concentration show an unfavorable condition for adsorption. Kinetic model yield a satisfactory result in parameters estimation and prediction for breakthrough time with different inlet H2S concentrations. The surface diffusion is significant to the effective diffusivity during mass transfer under experimental condition. The dynamic experimental results indicate a good performance in H2S removal.

CFD‐PBM numerical simulation is a powerful tool in the research of droplet swarm behavior. In this work, an artificial neural network (ANN) based droplet breakage frequency function is established based on the directly measured data from our previous studies. Then, the weights and biases of ANN are embedded into the CFD‐PBM code in the form of matrices and vectors. For the first time, a CFD‐PBM‐ANN simulation framework is established. Simulation results are in good agreement with the experimental data under different operation conditions. The cumulative droplet size distribution decreases with the increase of interfacial tension and pulse intensity. It is also found by the simulation that the droplet breakage frequency is relatively high at the edge of disc and doughnut plate, which is accordant with the distribution of turbulent energy dissipation and velocity gradient.

The dynamics of drop formation have been investigated in the presence of interfacial mass transfer through controlled flow visualization experiments. The mixtures of n-hexane (solvent) and acetone (solute) were used as a dispersed phase, having different initial compositions varying over a broad range. Drops were formed at the submerged position in the continuous phase (water) at the same operating flow conditions. The unsteady force balance model is developed to analyze the implications of the simultaneously occurring interfacial transfer of the solute on the formation dynamics in real time, and predictions are validated with experimental results. Based on initial compositions, the analysis of the transient drop shape shows a sharp transition in the drop formation regime. At lower initial solute concentrations, i.e., ϕ0 < 0.2, axisymmetric drop formation occurs and the interfacial solute transfer has negligible effects on the formation dynamics. Over an intermediate range of solute concentrations, i.e., 0.2 < ϕ0 < 0.5, Marangoni instability is triggered along the evolving interface, and therefore, the interface deformations and contractions occur during the drop formation. At ϕ0 = 0.5, the drop takes highly nonaxisymmetric shapes and remains away from equilibrium until its detachment from an orifice. For ϕ0 > 0.5, the spontaneous ejection of plumes of the solute results in the rapid generation of multiple droplets of smaller size. This work shows that higher solute concentration gradients not only lead to faster solute transport but also induce strong interfacial instability simultaneously. Thus, the coupled effects of transient change in composition and fluid properties govern the drop size and its formation time in such systems under non-equilibrium.

An original one dimensional population balance model (PBM)-based model of liquid-liquid extraction columns is reported. Compared to existing simulators, ColHySE implements a more realistic description of the flow patterns in the contactor, and predicts its effect on the local droplet-droplet interactions (i.e. breakage and coalescence rates). Proper turbulent properties, extracted from single-phase flow CFD simulations, are used in the source terms of the PBM to evaluate locally the inhomogeneous breakage and coalescence rates, using the averaged Coulaloglou and Tavlarides kernels (Castellano et al., 2018). The sensitivity of the predicted droplets mean diameter, d32, and the holdup, φ, to the parameters of the used empirical and phenomenological models, on the one hand, and to the operating conditions of the column, on the other hand, was studied. Although some model parts must be refined, and an experimental validation remains necessary, the results confirm that the 1D-PBM methodology used in ColHySE is relevant for predicting the interfacial area in the pulsed column as a function of the operating conditions and geometry, hence highlighting its relevance to study the hydrodynamic stability and tendency to flooding. The sensitivity analysis has moreover highlighted the needs for an improved slip velocity model.

In this study, a high-efficient CFD-based framework was developed for the comprehensive hydrodynamic evaluation of extraction columns. The core concepts of pseudo-3D and mean age theory were responsible for the optimal model geometry and steady transformation, respectively. The results indicate that the traditional 2D simplification is easy to cause the problems of geometrical deformation and turbulence distortion. By contrast, the pseudo-3D keeps all of the 3D hydrodynamic characteristics with smaller computational domain, so that it realizes the synergy of high accuracy and low cost. Meanwhile, the mean age theory transfers the unsteady species transport equation into a steady form so that it only takes tens of seconds for calculating the axial mixing. It was determined that the computational cost was reduced by ∼20 times for the complete hydrodynamic estimation of a pulsed sieve-plate column. Hence, it is believed to be a high-efficient framework for the computer-aided process development of extractors.

Detailed measurements and CFD investigations of the hydrodynamics in a bubble column containing internal features causing flow disturbances are presented for both air and helium gases. An optical needle probe has been used to measure profiles of bubble size, bubble velocity, and gas holdup at different locations across the cross‐section of the column. An approach combining CFD with population balances is able to represent observed multiphase flow phenomena such as the effect of the pipes to remix and redistribute the gas as well as the tendency of the gas to channel through a slit in the pipes rather than go around the pipes. The comparison of CFD simulation to experimental measurements reveals that the overall decrease in gas holdup observed when switching from air to helium gas can be explained by swarm effects, whereas the steeper decrease in the gas holdup profile across the column is due to coalescence effects.
This article is protected by copyright. All rights reserved.

The results of calculations of hydraulic resistance of an unirrigated crossflow packing using ANSYS Fluent computational fluid dynamics (CFD) software are reported. A comparative analysis of various designs of crossflow packings is carried out. Graphic relationships and empirical equations are derived for calculating pressure drop in a crossflow packing.

In liquid-liquid extraction processes, the mass transfer coefficient is one of the key design parameters. In the present study, the impact on mass transfer during droplet formation in a quiescent continuous phase is investigated analytically. Therefore, confocal Raman spectroscopy in a novel measurement cell combined with interfacial tension measurements are applied. It could be shown, that Marangoni effects in the system toluene-acetonitrile-water (mass transfer direction from the dispersed to the continuous phase) lead to 50 % of mass transferred in the first 5 s when 90 % is transferred in 1 min.

Extraction of metals in columns is scarce despite the advantage of low solvent inventory and the feasibility of high throughputs. Fast extraction kinetics allow short residence times and thus small apparatus height. A simulation of column concentration profiles via droplet population balances is less sophisticated as with surface active ion exchangers droplet breakage dominates and coalescence is hindered (rigid droplet surface). The extraction of Zn with D2EHPA (di(2-ethylhexyl)phosphoric acid) in a countercurrent Kühni column was investigated and the impact of different breakage kernels analyzed. Experiments in a lab scale test cell based on a DN32 Kühni extraction column geometry gave the basis to validate existing correlations and kernels for droplet breakage probability and daughter droplet size distribution. Since single droplet breakage tests were performed, a breakage dominated regime prevailed in the investigations. This yielded into a new unified correlation for the latter valid for different column geometries and physical and reactive extraction systems. As to this, a more firm basis for the simulation and design of stirred columns in hydrometallurgical application is established.

Many bio-based processes exhibit low yields resulting from inhibitions. This innovative concept for a multiphase-loop reactor can overcome these limitations with a simultaneous aeration and extraction of valuable compounds in one unit operation.
One dispersed phase is used to initiate a loop flow in the reactor while the other phase is dispersed in the downcomer. Therefore, the second dispersed phase rises in the counter-current.
The multiphase flow was simulated using a 2D axis-symmetric Euler-Euler approach in CFD. Besides friction force and gravitation the turbulent dispersion force was considered applying the standard k-ε turbulence model. The simulation of the reactor was validated in pilot-plant experiments. A comparison of the holdup of the dispersed phases and a visual evaluation of the flow field presented very good agreements between the experiments and simulations. Water, Shellsol T and synthetic air were used in the experiments. An improvement of the reactor design was possible based on additional simulation studies.

The operating regimes for a pilot-scale rotating disc contactor (RDC) were investigated by a computational fluid dynamics/population balance model (CFD–PBM) simulation. The model successfully predicted the critical rotor speed, which divided the entire operating range into two regions. In the low-rotor-speed region, the input energy was insufficient to break droplets, resulting in an almost constant droplet diameter. Therefore, the increasing revolution slightly affected the interfacial area, while the axial mixing became severe. In contrast, the interfacial area increased significantly in the high-rotor-speed region because of the increased breakage rate. Moreover, the axial mixing extent increased slightly, because the dispersed-phase accumulation enhanced the advection effect. The results indicate that the CFD–PBM approach can be applied to engineering practice for extractors.

Droplet coalescence and breakage in turbulent liquid-liquid dispersions is simulated by using computational fluid dynamics (CFD) and population balance modeling (PBM). The multifractal (MF) formalism that takes into account internal intermittency was here used for the first time to describe breakage and coalescence in a surfactant-free dispersion. The log-normal Extended Quadrature Method of Moments (EQMOM) was for the first time coupled with a CFD multiphase solver. To assess the accuracy of the model, predictions are compared with experiments and other models (i.e., Coulalogou and Tavlarides kernels and Quadrature Method of Moments, QMOM). EQMOM and QMOM resulted in similar predictions, but EQMOM provides a continuous reconstruction of the droplet size distribution. Transient predictions obtained with the MF kernels result in a better agreement with the experiments. This article is protected by copyright. All rights reserved.

In this work mass transfer in liquid–liquid extraction is investigated with the two-dimensional high-order moment-conserving method of classes (2D-HMMC) (Buffo and Alopaeus, 2016). The solution of a realistic liquid–liquid test case, a counter-current rotating disc column (RDC) composed of three stages where the droplets exchange mass with the continuous phase, is studied. This detailed modelling approach is compared with two other possible approximated models. In the first all the droplets are assumed to have the same size and concentration, and in the second all the droplets are assumed to have the same concentration but different sizes. The results of this comparison show that the information regarding the two-dimensional droplet size-concentration distribution may be needed to properly evaluate the mass transfer rates and therefore the behaviour of the system for all the operating conditions investigated.

Although many investigations have been carried out in liquid-liquid dispersions, new questions still emerge related to the treatment of mathematical simulations for such systems, which would be useful as a complement to experimental scaled-down practices with the aim of predicting the behaviour of real industrial full-scale systems. In order to simulate these processes, three different models characterized by different level of details are analyzed in this work for a stirred tank. They are mainly divided in two types: two zero-dimensional (0D) models, in which spatial homogeneity and perfect mixing of the disperse and continuous phases is assumed, and three-dimensional (3D) models, where the inhomogeneous mixing and spatial distribution of the phases is considered. One of the 0D models considers the spatial distribution of the turbulent dissipation rate (homogeneous model), while the other one employs only the average value of this variable in the tank (lumped model). The 3D model is instead based on the Eulerian-Eulerian two-fluid approach, implemented in computational fluid dynamics codes. The comparison of the results obtained imposing the very same operating conditions between the simplest 0D models (implemented in MATLAB), which are computationally very cheap, and the complex 3D models (implemented in OpenFOAM-2.2.x), which are computationally intensive, highlights their range of validity, allowing to establish a-priori which level of details or approach is needed to simulate a particular system.

In this study, the computational fluid dynamics-population balance model (CFD-PBM) was developed for a five-compartment pilot-scale rotating disc contactor (RDC) using the solving methods of finite volume method (FVM) and quadrature method of moments (QMOM) as implemented in Fluent 14.0. The differences between a simplified two-dimensional (2D) axisymmetric swirl model and a full three-dimensional (3D) model were investigated, and a pseudo-three-dimensional (pseudo-3D) model was developed to combine the benefits of both the 2D and 3D frameworks. The comparison results of the velocity distributions, droplet mean diameter and turbulent dissipation rate indicated that the 3D approach was suitable for quantitative investigations, whereas the 2D axisymmetric swirl model had only qualitative predictive capabilities. The differences are probably attributed to the 2D framework ignoring the cylindrical geometry characteristics of the problem. Based on this knowledge, a pseudo-3D model was proposed by constructing a fan-shaped geometry to overcome this disadvantage, and a parametric study was conducted to determine the angle of the sector. The results showed that the pseudo-3D model had nearly the same computational accuracy as the full 3D simulation, which verified the approach. Moreover, the sector angle had almost no effect on the simulated results, and only two layers of grids in the tangential direction were sufficient. Thus, the pseudo-3D model was found to be a powerful tool with computational costs approximate to those of the 2D framework, but with a calculation accuracy near that of the 3D model.

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 Kühni 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.

This work presents a new windows-based MATLAB program, which is called PPBLAB (Particulate Population Balance Laboratory) for modelling and numerical simulation of particulate systems. As a first step, liquid-liquid extraction columns are modelled using the population balance equation as a mathematical framework. Up to date population balance models and solvers are incorporated. The discretization of the spatial domain is based on the finite volume method with flux vector splitting. A strongly stable semi-implicit first order time integration scheme is used to resolve such a large and stiff ODE system. The MATLAB GUI is used to make PPBLAB a user friendly program, which allows the user to define and simulate liquid extraction columns. A thermodynamics package TEA-COCO, which obtained from CAPE-OPEN, is linked to PPBLAB. Therefore, a special interface is designed for the purpose of data exchange between PPBLAB and CAPE-OPEN TEA tool. The solute distribution coefficient in ternary systems is predicted using the UNIQUAQ model with a special optimization tool to estimate the binary interaction parameters based on infinite conditions. A pilot plant Kühni extraction column is simulated and tested. Full analysis and performance of the PPBLAB are carried out and validated against experimental data.

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.

Mathematical models are reviewed for different types of commonly used extraction columns, viz. pulsed sieve plate column, rotating disc contactor, Kühni column, spray column, Scheibel extractor, packed column, Oldshue-Rushton contactor and reciprocating plate column. In addition, numerical techniques, process simulators and some estimation methods for model parameters have also been included. The review cites over 120 references.

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.

In this paper we present the liquid-liquid two-phase flow simulations of a stirred extraction column with the help of our
own developed meshfree method called the Finite Pointset Method (FPM). The primary (continuous) phase is modeled by the incompressible
Navier-Stokes equations. The motion of the secondary (dispersed) phase is simulated by solving the equation of motion in which
inertia, drag and buoyancy forces are taken into account. The size of the droplets is obtained by solving the droplet population
balance equation (DPBE). The DPBE is solved by the Sectional Quadrature Method of Moments (SQMOM). The coupling between both
phases is performed by considering the momentum transfer from each phase. In this work, some simulations in two and three
dimensional cases with constant breakage and aggregation kernels are presented.

Phenomenological models are proposed to describe drop breakup and coalescence in a turbulently agitated liquid-liquid dispersion. Based on these models, breakage and coalescence rate functions are developed and used to solve the general population balance equation describing drop interactions in a continuous flow vessel. Parameters of the models are evaluated by comparison with experimental data on drop size distributions and mixing frequencies obtained in a continuous flow vessel over a range of operating conditions. The favorable agreement between experimental observation and the model are encouraging that the model is suitable for predicting dispersion properties such as drop size distributions, interfacial areas and mixing frequencies.

The subject of multiphase flows encompasses a vast field, a host of different
technological contexts, a wide spectrum of different scales, a broad range of
engineering disciplines and a multitude of different analytical approaches.
Not surprisingly, the number of books dealing with the subject is voluminous.
For the student or researcher in the field of multiphase flow this broad
spectrum presents a problem for the experimental or analytical methodologies
that might be appropriate for his/her interests can be widely scattered
and difficult to find. The aim of the present text is to try to bring much
of this fundamental understanding together into one book and to present
a unifying approach to the fundamental ideas of multiphase flows. Consequently
the book summarizes those fundamental concepts with relevance to
a broad spectrum of multiphase flows. It does not pretend to present a comprehensive
review of the details of any one multiphase flow or technological
context though reference to books providing such reviews is included where
appropriate. This book is targeted at graduate students and researchers at
the cutting edge of investigations into the fundamental nature of multiphase
flows; it is intended as a reference book for the basic methods used in the
treatment of multiphase flows.

A combined computational fluid dynamics (CFD), population balance modeling (PBM) and mass transfer code is used to simulate a rotating disc contactor pilot plant column. Droplet breakage and coalescence is accounted by a one group model and literature models for the breakage frequency, coalescence propability and efficiency. Mass transfer is described based on the two-film theory. The simulation results as droplet size, hold-up and concentration profile along the column height are compared to experimental data. The simulated droplet size shows a slight increase underneath the first compartment whereas it is in good agreement to experimental data in the active height of the column. Also the solute concentrations profiles could be good represented by the simulations.

An online monitoring and simulation tool (OMST) is used to determine, analyse, simulate and predict the multiphase flow behaviour in an extraction column within the context of model predictive control (MPC). The simulation was done with the droplet population model OPOSPM using an adapted moments method for online simulation. For validation, steady-state and transient experiments are performed, where rotor speed and dispersed volume flow rate are changed using the EFCE system toluene–water. OMST deviations are between 3% (droplet sizes in steady state) and 20% (hold-up in dynamic step-changes) which are highly dependent on the quality of the parameter estimation.

A rotating disc column (RDC) with inner diameter 68 mm and 28 compartments is used in this study. Parameters including Sauter mean diameter, hold-up and mass transfer coefficient are measured experimentally under different operating conditions. The correlations in literature for molecular diffusion and enhancement factor equation including eddy diffusion, circulation and oscillation of drops are evaluated. A new equation for the effective diffusion coefficient as a function of Reynolds number is proposed. The calculated values of mass transfer coefficient and column height from the previous equations and present equation are compared with the experimental data. The results from the present equation are in very good agreement with the experimental results, which may be used in designing RDC columns.

The theory of extraction of a substance, dissolved in droplets of a fluid, when these are falling (or rising) in an other fluid under the influence of gravity, is specialized for the case that the rate of falling (or rising) is very small. A perturbation method, well-known from quantum mechanics, is applied to determine the time of diffusion in second approximation.

Empirical correlations for the prediction of mass transfer coefficients for single drops are presented. Published experimental results for both circulating and oscillating drops are considered. Correlation for the individual continuous-phase mass transfer coefficient, which is based on data from 596 measurements taken from 10 different groups of investigators, reproduces the data with an average absolute error of 14.1%. This is then used to determine a correlating equation for the individual dispersed-phase mass transfer coefficient on the basis of data for overall dispersed-phase mass transfer coefficient taken from 21 sources. The average absolute value of the relative error in the predicted values of overall dispersed-phase mass transfer coefficient from the experimental points by using the correlations for individual mass transfer coefficients is 24.5%. It is further shown that by allowing for the effects of power input per unit mass and dispersed-phase hold-up, the correlations for single drops can be extended to extraction columns. The correction factors required for this purpose have been obtained by using simulated values of overall mass transfer coefficients for pulsed perforated-plate, Karr reciprocating-plate, Kühni, and rotating disc columns.

This article is about single particles, bubbles and drops. It considers their falling (or rising) velocities and mass transfer coefficients. These are presented as graphs of dimensionless groups; the groups are chosen such that the graphs can be recognized as plots of a velocity against a diameter. The result is more intuitive than are plots using the traditional dimensionless groups such as the Reynolds, Sherwood and Schmidt numbers.

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 two-phase flow solver implemented in the open-source OpenFOAM code was extended to a multiphase flow formulation (n dispersed and one continuous phases) and then coupled to the population balance equation (PBE) solution by the Direct Quadrature Method of Moments (DQMOM), originating a polydispersed multiphase flow solver. Although each dispersed phase has its own velocity field, the present implementation considers only the interfacial momentum exchange between the continuous and the dispersed phases. The multiphase flow formulation was described and the details of the PBE-CFD coupling algorithms in OpenFOAM were provided. The implementation of the multiphase flow code was verified and evaluated against the original OpenFOAM two-phase flow solver for flow through a 2D backward facing step, using simplified breakage and aggregation kernels. The computational cost of both codes were compared for serial and parallel simulations.

Modeling and dynamic analysis of liquid extraction columns are essential for the design, control strategies and understanding of column behavior during start up and shutdown. Because of the discrete character of the dispersed phase, the population balance modeling framework is needed. Due to the mathematical complexity of the full population balance model, it is still not feasible for dynamic and online control purposes. In this work, a reduced mathematical model is developed by applying the concept of the primary and secondary particle method (Attarakih et al., 2009b, Solution of the population balance equation using the one primary and one secondary particle method (OPOSPM), Computer Aided Chemical Engineering, vol. 26, pp. 1333–1338). The method is extended to solve the nonhomogenous bivariate population balance equation, which describes the coupled hydrodynamics and mass transfer in an RDC extraction column. The model uses only one primary and one secondary particles, which can be considered as Lagrangian fluid particles carrying information about the distribution as it evolves in space and time. This information includes averaged quantities such as total number, volume and solute concentrations, which are tracked directly through a system of coupled hyperbolic conservation laws with nonlinear source terms. The model describes droplet breakage, coalescence and interphase solute transfer. Rigorous hyperbolic analysis of OPOSPM uncovered the existence of four waves traveling along the column height. Three of these are contact waves, which carry volume and solute concentration information. The dynamic analysis in this paper reveals that the dominant time constant is due to solute concentration in the continuous phase. On the other hand, the response of the dispersed phase mean properties is relatively faster than the solute concentration in the continuous phase. Special shock capturing method based on the upwind scheme with flux vector splitting is used, with explicit wave speeds, as a time–space solver. The model shows a good match between analytical and numerical results for special steady state and dynamic cases as well as the published steady state experimental data.

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.

The prediction of turbulent gas–liquid systems, and in general of multiphase flows, has been historically performed with the implicit assumption of separately considering fluid dynamics issues from the evolution of the dispersed phase (i.e., the gas bubbles). These two aspects can be simultaneously accounted for by means of a multidimensional population balance model, able to describe the interactions between the continuous liquid phase and the gas bubbles, as well as the interactions among different gas bubbles (e.g., coalescence and break-up), both in terms of momentum and mass coupling. The model has to quantify the effects of these interactions on the population of dispersed bubbles and has to estimate the distributions of bubble velocity, size and composition, as well as the state of the continuous phase. A novel approach based on the direct quadrature method of moments is here formulated, tested on several simplified cases and adopted to describe the evolution of the gas bubble in a realistic gas–liquid stirred tank reactor. Results are eventually validated through comparison with experimental data from the literature.

The impact of Marangoni convection on the extraction efficiency during the drop formation stage is investigated in the system toluene/acetone/water for different initial solute concentrations and different drop diameters. Both mass transfer directions of the solute have been considered. Marangoni instabilities are supposed to increase the internal mixing and thus enhance mass transfer coefficients. Experimental results show a strong dependency on the mass transfer direction. The amount of solute extracted is between 19% and 55%. The total transferred mass MA increases with drop diameter and initial concentration. Present models from the literature which predict extraction efficiencies do not take into account interfacial effects like Marangoni convection. A correlation is proposed introducing an effective diffusivity which depends on the initial solute concentration. The diffusivity factor increases linearly with initial solute concentration and is more sensitive in the mass transfer direction c→d.

Laser based measurement techniques were used to investigate the local velocities, the energy dissipation and the phase fraction inside a Kühni miniplant extraction column. The velocity field was determined with particle image velocimetry (PIV). The energy dissipation was obtained based on these measurements. The velocity measurements show that at extreme low dispersed phase flow the flow pattern alters extremely. The local time averaged phase fraction was measured with a laser induced fluorescence (LIF) system. It is shown that the accumulation underneath the stirrer has a high impact on the concentration of the dispersed phase in a single compartment. A higher rotational speed leads to a better distribution of droplets inside the column, whereas at lower stirring speed the droplets by-pass the impeller. Due to the shear force, especially near the stirrer outflow region, drop deformation and breakage occurs.

A new turbulent oscillating drop model for predicting drop-side solution mass transfer in agitated liquid—liquid systems is proposed. The model assumes that drop deformation and oscillation due to the surrounding agitated flow field results in random radial mixing within the drop. An effective diffusivity dependent on power input, drop size and interfacial tension is derived as follows: The above coefficient is substituted for the molecular diffusion coefficient in the spherical drop diffusion equation which is solved in the normal manner to predict drop-side mass transfer rates. The proportionality constant, κ, must be determined from extraction column mass transfer data, e.g. by comparing experimental and column model predicted longitudinal solute concentration profiles.

ReDrop is a simulation tool which can predict the behaviour of pulsed extraction columns on pilot-plant scale. Because it follows a certain number of drops during their entire lifetime in an extraction column, it can be seen as a Monte-Carlo solution of drop-population balances. ReDrop is based on single-drop models which have system-specific parameters. These parameters are determined from lab-scale experiments with single drops. In this study, ReDrop is extended to RDC extraction columns and simulations are carried out for the standard test systems toluene (d)+water (c)+acetone (c→d) and n-butyl acetate (d)+water (c). The calculated hold-up and Sauter mean diameter show good agreement with the experimental data obtained in pilot-plant scale experiments.

A reduced non-equilibrium bivariate population balance model is developed to model simultaneous reversible chemical reaction and extraction of an industrial scale high-pressure oil-splitting reactor with a first-order reversible kinetics. The model includes details about the reacting mixture by considering the discrete (particulate) nature of the dispersed phase. This, however; increases the mathematical complexity of the model and results in a coupled system of partial differential equations (PDE) of hyperbolic type with nonlinear source terms. The characteristics analysis shows that these hyperbolic PDEs have two dominant and distinct characteristic speeds: The mean droplet speed and the mean continuous phase (oil) speed. This generates a set of contact waves moving along the reactor height in opposite directions. Therefore, modern numerical methods based on CFD literature are applicable to provide a numerical solution of the developed model. As a first step in the numerical investigation, a finite volume method with a first-order upwind scheme is developed and implemented. Also, sufficient conditions of stability are derived and implemented. The numerical model is validated against analytical solution of a special case, where numerical convergence in the sense of l2-norm is established. Numerical simulation results showed that the unreacted oil concentration profile decreased exponentially along the reactor height. On the other hand, the glycerine concentration in the oil phase passes through a maximum due to the simultaneous reaction and extraction processes. Three operating parameters: temperature, mean droplet diameter and excess water flow rate were studied to highlight their effect on the reactor performance. The most important parameter was the reactor operating temperature, where 240 °C was found sufficient for complete oil conversion. This agrees with the operating temperature in industrial practice.

In this work, a one-group reduced population balance model based on the one primary and one secondary particle method (OPOSPM) developed recently by Attarakih et al. (In Proceedings of the 19th European Symposium on Computer Aided Process Engineering, ESCAPE-19, Cracow, Poland, June 14−17, 2009; Jezowski, J., Thullie, J., Eds.; Elsevier: New York, 2009; ISBN-13: 978-0-444-53433-0) is implemented in the commercial computational fluid dynamics (CFD) package FLUENT 6.3 for solving the population balance equation in a combined CFD−population balance model (PBM). The one-group reduced population balance conserves the total number (N) and volume (α) concentrations of the population by directly solving two transport equations for N and α and provides a one-quadrature point for closing the unclosed integrals in the population balance equation. Unlike the published two-equation models, the present method offers accuracy improvement and internal consistency (with respect to the continuous population balance equation) by increasing the number of primary particles (sections). The one-group reduced population balance provides the possibility of a one-equation model for the solution of the PBM in CFD based on the mathematically consistent d30 instead of the classical d32 mean droplet diameter. Droplet breakage and coalescence are considered in the PBM, which is coupled to the fluid dynamics in order to describe real droplet behavior in a stirred liquid−liquid extraction column. As a case study, a full pilot-plant extraction column of a rotating disk contactor (RDC) type consisting of 50 compartments was simulated with the new model. The predicted results for the mean droplet diameter and the dispersed-phase volume fraction (holdup) agree well with literature data. The results show that the new CFD−PBM model is very efficient from a computational point of view (a factor of 2 less than the QMOM and a factor of 5 less than the method of classes). This is because the one-group reduced population balance requires the solution of only one equation (the total number concentration) when coupled to the CFD solver. It is therefore suitable for fast and efficient simulations of small-scale devices and even large-scale industrial processes.

Behavior of turbulent flow and particle size distribution in stirred tanks can be predicted by using the concepts of local isotropy.

The experimental data on the holdup of the dispersed phase in a Rushton impeller agitated stirred tank are presented. Experimental measurement is performed utilizing the sample withdrawal method to obtain the local dispersed-phase holdup in a laboratory-scale stirred tank under a variety of operating conditions. Three-dimensional turbulent two-phase liquid−liquid flow in the stirred tank is also numerically simulated by solving the Reynolds-averaged Navier−Stokes equations of two phases formulated by the two-fluid model. The turbulence effect is formulated using a simple two-phase extension of the well-known k−ε turbulence model by adding an extra source term generated from the presence of the dispersed phase in the turbulent kinetic energy transport equation of the continuous phase. A modified “inner−outer” iterative procedure is employed to model the interaction of the rotating impeller with the wall baffles. The model-predicted mean velocity, turbulence characteristics of the continuous phase, and holdup profiles of the dispersed phase are compared against the published experimental data and the present measurements to validate the computational procedure, and good agreement is found up to a rather high overall dispersed-phase holdup case (30 vol %).

Hydrodynamic relations arising from single droplet movement in a RDC geometry are described. These relations can be regarded as the basis for a general estimation strategy for the determination of dispersed-phase flow parameters in connection with droplet population balance modeling. Droplet population balance based modeling of countercurrent liquid-liquid extraction columns offers a considerable advance in describing the real droplet swarm behavior. The main problem encountered is the determination of the characteristic droplet model parameters.

The single and two-phase flow field of a rotating disc contactor (RDC) extraction column is simulated with the help of computational fluid dynamics (CFD). The simulations were validated by particle image velocimetry (PIV) measurements. The single phase setup was used to test different turbulence models, and a 2D and 3D grid approach. For the two-phase simulations, a 2D computational grid and the Euler-Euler model was used. The two-phase PIV measurements are possible when using an iso-optical system, where the refractive indices of both liquid phases are identical.

Mass and heat transfer rates in extraction are studied theoretically and experimentally for the practical range of the variables involved. For the particular but typical case of liquid drops moving through another liquid a simple correlation for the over-all mass transfer coefficient is presented, which holds with a probable error of 20%.
Included are systems in which the rate is limited by either coefficient, as well as systems in which both coefficients are significant. The correlation, valid for both directions of transfer with either phase dispersed, is useful for the extrapolation of performance from system to system in a given piece of equipment. Also, together with correlations for transfer area and effective driving force, it is part of the information needed for design.

A precise model for predicting liquid‐liquid extraction column efficiency based upon assumed hydrodynamic, axial mixing and mass transfer behaviour has been formulated and solved numerically.
The complex nature of the dispersed phase can be better described by drop‐size‐dependent residence time distribution (RTD). Both the variation of axial velocities due to drops of different sizes, i.e. forward mixing, and the axial dispersion for the drops of the same size have been considered in this model.
The computed results reveal that the effects of both varying velocities and dispersion of drops on extraction efficiency are appreciable and cannot be neglected, and the efficiency may be overestimated if only a forward mixing model is adopted. The comparison of the experimental values of N ODP with those predicted shows that the mass transfer data obtained in RDC agree well with the values predicted by the present model for the case of solute transfer in c → d direction, and are slightly higher than the predicted ones for the transfer in d → c direction.

For the design of counter-current liquid–liquid extraction columns, there is a strong industrial demand for more straightforward, faster and money-saving simulation methods. One possibility in this direction that has a great potential is the coupling of computational fluid dynamics (CFD) with population balance models (PBM). Therefore, a combination of CFD and droplet population balance modelling (DPBM) is applied to simulate the drop size distributions and flow fields in a liquid–liquid RDC extractor. The simulations are carried out in the commercial CFD code Fluent. The liquid–liquid flow is modelled using a Reynolds averaged turbulence model in conjunction with the Eulerian two-fluid equations. Models for coalescence and breakup, from Luo and Svendsen [1996. Theoretical model for drop and bubble breakup in turbulent dispersions. A.I.Ch.E. Journal 42, 1225–1233] Coulaloglou and Tavlarides [1977. Description of interaction processes in agitated liquid–liquid dispersions. Chemical Engineering Science 32, 1289–1297] and a mixed model [Martínez-Bazán, C., Montañés, J.L., Lasheras, J.C., 1999. On the breakup of an air bubble injected into a fully developed turbulent flow. Part 1. Breakup frequency. Journal of Fluid Mechanics 401, 157–182; Prince, M.J., Blanch, H.W., 1990. Bubble coalescence and break-up in air-sparged bubble columns. A.I.Ch.E Journal 36, 1485–1499] are implemented in the CFD code as user defined functions. For the solution of the PBM a classes method (CM) [Kumar, S., Ramkrishna, D., 1996. On the solution of population balance equations by discretization—I. A fixed pivot technique. Chemical Engineering Science 51, 1311–1332] and the quadrature method of moments (QMOM) [Marchisio, D.L., Pikturna, J.T., Fox, R.O., Vigil, R.D., Barresi, A.A., 2003a. Quadrature method of moments for population-balance equations. A.I.Ch.E. Journal 49, 1266–1276] are used. Simulated droplet distributions for the systems toluene–water and butyl acetate-water are compared to experimental measurements. The model of Luo and Svendsen was modified to predict the droplet size distribution. The mixed model allows the prediction of the Sauter mean diameter without any adjustable parameter. Pros and cons of the combined model as well as future needs and trends such as multi-fluid CFD–PBM models are discussed. The results show that the link of PBM and CFD is a suitable design tool which can significantly improve the layout of industrial columns.

The extraction of a substance, dissolved in droplets of a fluid, when these are falling (or rising) in an other fluid under
the influence of gravity, is investigated. The circulation currents which are caused in the droplets by the viscous forces
between the two fluids modify the rate of extraction as compared with the case of droplets at rest. Under certain simplifying
assumptions a differential equation governing the combined action of convection and diffusion is derived. The rate of extraction
follows from the solution of an eigenvalue problem.

In the previous part of this work (Chem. Eng. Sci. 54 (1999) 5887), a multiblock simulation model was developed in order to allow the close examination of different regions of a stirred tank for drop size distribution calculations. In this paper, that model is tested in a parameter fitting procedure. The drop breakage and coalescence parameters are fitted against drop size measurements from dense liquid–liquid dispersions, which were assumed fully turbulent. Since the local turbulence and flow values of a stirred tank are used in the present model, the fundamental breakage and coalescence phenomena can be examined more closely. Furthermore, the present model is capable of predicting inhomogeneities occurring in a stirred tank. It is also to be considered as an improved tool for process scale-up, compared to the simple vessel-averaged population balance approach, or use of correlations of dimensionless numbers only. The present model can use two sources of data for fitting parameters in the drop rate functions. One is to use transient data of the measured drop size distribution as the impeller speed is changed. The other is to use time-averaged data measured at different locations of the