Felix Gebauer ,Mark W.Hlawitschka ,2,Hans-J?rg Bart,2,*

1 Chair ofSeparation Science and Technology,University of Kaiserslautern,Kaiserslautern 67653,Germany

2 Center for Computational and Mathematical Modelling(CM2),University of Kaiserslautern,Kaisers lautern 67653,Germany

1.Introduction

Liquid–liquid extraction is applied in many chemical,petrochemical,biochemical,hydro metallurgical and nuclear separations processes.Column contactorsare widely used for extraction due to the advantages of high throughput and low footprint[1]compared to alternative equipment options.An essential factor governing extraction-column performance is the competitive relationship between breakage and coalescence of droplets,which determines the interfacialarea available for mass transfer.Characterization and prediction of both components are necessary,for process optimization purposes,in unit-scale modeling.Understanding the complex relationship between the various factors in fluencing coalescence,and increasing the accuracy and reliability of model predictions,have been an ongoing topic of scientific research[2–5].All currently available models depend on the use of adjustable parameters in order to modelspecific testsystems and apparatus geometries.Existing correlations neglect important factors such as electrostatic effects;which depend on the electrolyte ion species,concentration,pH value;as well as parameters characterizing relative droplet inertia,collision angle and the presence of additives.As such,previous studies have given inconsistent results.In order to increase the reproducibility of collision conditions,Eiswirth and Bart[5]and Villwock et al.[6]developed a standardized,automated experimental approach and used it to characterize a water/toluene test system.Generation ofnumerous collision sequences is of crucialimportance in establishing a statistically valid coalescence probability database for the detailed characterization of test systems.The recently developed methodology is used,presently,in the development and validation of new coalescence models.

2.Theory

Coalescence is more complex than breakage,because there are hydrodynamic effects determined by the energy input and geometry of the apparatus,but also by the composition of the phases.Additionally the mass transfer has a decisive effect on the coalescence behavior.An early modeldescribing coalescence of droplets is given by Coulaloglou and Tavlarides[3].Here the first condition for success fulcoalescence is the knowledge of the collision frequency hcoal,depending on the number of droplets in the considered volume element.The determination of the collision frequency depends on swarmexperiments and is notapplicable to single droplet investigation.A description of the coalescence behavior with the collision frequency is not sufficient,since not every droplet interaction results in a coalescence event.It is necessary to introduce the coalescence efficiencyλto account for the observed behavior.Asimple definition of the coalescence efficiency depending on the droplet sizes of the colliding droplets d1and d2is the ratio between coalescence events(Ncoal)and droplet interactions(Nint):

The focus in the following is on different physicaland empiricalapproaches to calculate the coalescence efficiency.The prediction and the modeling of the dropletsize distribution(DSD)with population balance equations(PBE)partially supported by CFD simulations depend mainly on the accuracy of the coalescence kernel.The DSD is in fluenced by physicalproperties,apparatus geometry and process parameters,but severalfactors,for instance the electrochemicaleffects or the impact conditions,are not considered in current kernels.The existing coalescence probability models can be roughly distinguished into two main groups.The basic film drainage model of the first group is given by Coulaloglou[7]and is characterized by the coalescence time needed for coalescence and the contact time between the two droplets:

Coalescence can only occurwhen the force which brings the droplets into contact is acting for a sufficient time(tcontact).The contact time must exceed the time to reach the critical film thickness(tdrainage).Due to the different definition of the characteristic times in various research groups severalmodels were developed.One approach is the implementation ofelectrostatic effect in coalescence kernels by using the DLVO theory[8]giving:

An alternative is based on the consideration of the interfacialenergy and the kinetic energy of the droplets[10]:

Different extensions with the droplet size and the relative velocity,leading to a momentum-based calculation of the kinetic energy are given by Simon[11].A more detailed list of the existing coalescence models is published by Simon[11]and Liao and Lucas[12].

3.Experimental Setup

Due to the sensitivity of the coalescence process to impurities,pH-value,etc.a firm definition of the exper imental conditions is necessary for a precise understanding of the occurring phenomena.Reproducible and statistically reliable single droplet investigations were performed with a specially designed test cell.A detailed description of the experimentalsetup with allfeatures and the basic controlroutine is described by Kamp and Kraume[9].The testcellhas been improved and allchanges are described in detailby Villwock etal.[6].The experiments were recorded with a high speed camera(Photron Fastcam APX RS)equipped with a macro lens and an adjustable bellow to register the coalescence with 30,000 frames per second(fps)and a high magni fication(Fig.1).The physicalproperties of the used EFCE testsystem are given in Table 1.

4.CFD Setup

Fig.1.CAD draft of the single droplet test cellbased on Villwock et al.[6].

Table 1 Test system physicalproperties of the laboratory scale experiments,at 20°C[13]

The volume of fluid(VoF)approach called multiphase Inter Foam with the additional extension of adaptive mesh re finement based on the OpenSource toolbox OpenFOAM?version 2.3.0 was used for the simulations.The numerical framework to specify the experimental single droplet test cell is built with a purely structured hexahedral mesh created with the pre-processing utility blockMesh.The boundary conditions for velocity and phase fraction in the simulation framework are set to zeroGradient with the exception for the velocity at the inlet,which is defined as a fixed value(0 0 0).The initialization of the continuous phase(water)and the droplet regions(toluene)is done with the setFlieds OpenFOAM?utility.The solver needs to adapt the time step to keep the Courant number below 0.5.The re finement conditions and initialsolve options are given in Table 2.Setting a maximum cellnumber is necessary to limit the computational time.The mesh re finement at each time step improves the accuracy of the interface tracking.Simulations were performed on a single core(Xeon E5345,2.33 GHz,with a maximum requested 8 GB RAM)using the ELWE-Cluster from the University of Kaiserslautern.

Table 2 Re finement conditions and solution options

Taking into account the drainage time given in Eq.(2)an additional algorithm is implemented in the CFD code to avoid a meaningless numerical coalescence.A three phase system with two different phases for the droplets and one for the continuous phase is used.The coalescence is initiated by an algorithm that defines the second dispersed phase from then on as the first dispersed phase droplet by using an experimentally derived contact time.The triggering of the algorithm requires a definition for the initial contact of the droplets.The contact is defined by a phase fraction of0.999 of both dispersed phases in a single computational cell and a phase fraction ofmore than 0.2 for each phase.

5.Results and Discussion

To ensure a good comparability with the experimental investigations,a numerical mesh representing the experimental setup was built.The description of the coalescence behavior requires a high resolution of the interfacialarea of the droplets and an adaptive mesh re finementto follow the moving droplet.To narrow the computational time the re finement algorithm generates a coarser grid in the region of the bulk continuous phase surrounding the droplets and the inside of the droplets.An example of the mesh prior to the collision with a pendant and a rising droplet,which is used for the simulation,is given in Fig.2.Due to the dynamic mesh re finementin each time step of the transient simulation the number of grid cells reaches a maximum at approximately 1.6 million.

Fig.2.Re fined adaptive mesh at the droplet interfaces.

A droplet coalescence of two different sized droplets(here:2.2 mm and 3.0 mm)is simulated using the modified CFD code.Fig.3 shows a simulation of droplet coalescence(top)compared to the experimental coalescence investigations(bottom).The sequence of the simulation is given by a 2Dplot as a cut through the center of the domain.Increasing film pressure between the droplets leads to an ellipticaldroplet deformation for both droplets before the actualcontact occurs.Due to the empirically determined film rupture after 19.1 ms a similar droplet shape could be observed in the experimentalstudies as well as in the simulations.The measurement of coalescence time was done with a standardized single droplet investigation,based on the results of more than 150 droplet interactions for the droplet sizes of 2.2 mm(pendant/top)and 3.0 mm(rising/bottom).The test system of analytical grade toluene and reversed osmosis water(conductivity belowκ=0.5 μS·cm-1)without any addition of additives was used.As can be seen from Fig.3,the presented simulation of droplet coalescence with the implemented film rupture time is in good accordance with the experimentalobserved coalescence behavior with a good representation of the occurring droplet shape.

In addition to this,the left part of Fig.4 shows the formation of the entrainment during coalescence recorded with a high speed imaging system at30,000 fps,which is typicalfor the described conditions.The inclusion ofmicro droplets can be attributed to the fact that the critical film thickness or the film rupture is achieved in the edge region of the contact area before the continuous phase can completely flow out of the contact area.A good accordance with the CFD simulation is given in the righthalfofFig.4,which shows an increase in pressure in the contact area by forming micro water droplets inside the toluene droplets.The evaluation of the phase fraction con firms the inclusion of continuous phase in the droplet.

Fig.4.Entrainment of continuous phase after coalescence;high speed image(left);OpenFOAMsimulation of pressure(right).

Asequence of2Dplots of the contactarea given in Fig.5 contains further information about the distribution of the continuous phase between the droplets.The minimum film thickness occurs on the outer circle of the contact surface(Fig.5 frame 1),leading to the critical film thickness and film rupture(Fig.5 frame 4).The continuous phase(coarse structure in the center)cannot drain out completely before the formation of the coalescence bridge is completed(Fig.5 frame 6)which is also noticed experimentally by Eiswirth[14].

CFD simulations enable more perspectives and local in formation beyond the limits of the opticalaccessi bility.Pressure and velocity gradients during the droplet interaction and the film drainage could be visualized and compared with experimental phenomena.Especially the entrainment ofcontinuous phase could be reproduced for the considered conditions.

Fig.3.Image sequence ofcoalescence;CFD(top);experimental(bottom).

Fig.5.Film rupture and coalescence bridge building(legend:water phase fraction).

6.Conclusions

The numericalinvestigation of single droplet interactions opens up many possibilities for the modeldevelopmentto describe the important competing phenomena of breakage and coalescence.The experimental investigation of droplet coalescence is limited by technical and physical accessibilities,which can be expanded by CFD simulations.The OpenFOAM toolbox,with good access to adapt the source code,allows accounting the droplet contact time prior to a coalescence event.Especially localspatial and time resolved information about the hydrodynamics and the film drainage can be found with CFD simulations,which allow a better understanding of the coalescence behavior.Experimentally observed local phenomena at ms time-scale(micro droplets and droplet deformation)could be exactly reproduced by the simulation.The results of the numerical investigations can be used for the development of new and more accurate and predictive breakage and coalescence kernels.Further studies aim at the implementation of surface acting forces to account for the in fluence of ionic ingredients(salts,pH-value)on the coalescence in CFD simulations.

Nomenclature

Acknowledgments

The authors want to thank the DFG for their financial support(BA 1569/55-1)and the RHRK for the computational resources(ELWE-Cluster).

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