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Ind. Eng. Chem. Res. 2004, 43, 2264-2270
Trajectories of Charged Drops in a Liquid-Liquid System: The Effect of Geometrical Scale-Up A. P. Hume,† J. Petera,‡ and L. R. Weatherley*,† Department of Chemical and Process Engineering, University of Canterbury, Christchurch 8004, New Zealand, and Faculty of Environmental and Process Engineering, The Technical University of Lodz, Lodz, Poland
The work in this paper builds on earlier research, which demonstrated the ability to predict charged drop motion in one-dimension and by using a single contactor geometry. Here we have extended this approach to compare the accuracy of the modeling methods in different sizes of contactor. The liquid-liquid system studied here comprised drops of water dispersed in sunflower oil The predicted drop trajectories are compared with those measured using video photography. Comparisons are presented for three different contactor sizes based on similar rectangular geometry. The largest contactor had dimensions of 100 × 100 mm2 cross section and a maximum interelectrode distance of 142 mm, the smallest contactor was of 50 × 50 mm2 cross section with a minimum interelectrode distance of 70 mm. The experimental variables studied included applied voltage, dispersed phase flow rate, interelectrode distance, field polarity, and contactor size. The space charge profiles and the electric field were calculated successfully, and good agreement between the predicted and the experimental trajectories was achieved in all cases. It is concluded that the predictive model can accurately accommodate geometries of different sizes and has the potential as a tool for scale-up design based on small scale experimentation. Background In liquid-liquid contacting operations, drop size and motion relative to the continuous phase are key determinants of the hydraulic capacity of equipment, and ultimately of performance efficiency.1 The diameter and the hold-up of drops directly influence the interfacial area available for mass transfer during the passage of the phases through the contactor. The cyclic coalescence and break-up of drops during their passage through the contactor is also important in reducing drop-side mass transfer resistances. The relative velocity between moving drops and the continuous phase is also important for reduction of film mass transfer resistances which may exist in the vicinity of the liquid-liquid interface. Back-mixing and forward mixing of both liquids are also important phenomena observed in continuous liquidliquid operations and can contribute to reduction of performance due to the distortions in residence time distributions which can accrue from such effects.2-4 In the case of electrostatically charged drops, the motion may be enhanced due to the additional electrical forces acting on the drops. The additional accelerative forces acting on drops will increase velocity and thus may enhance film mass transfer rate around the drop5-7 and produce different patterns of mixing and residence time behavior,8 and thus overall separation efficiency. In this paper we are concerned with unhindered motion of single drops in an electrically charged column contactor. Previous work by the authors 9 focused on the measurement and prediction of the motion of charged water drops through nonconducting liquids, including n-decanol, sunflower oil, and silicone oil. The theoretical approach adopted was based upon the fundamental * Corresponding author. Tel.: +64 3 364 2139. Fax +64 3 364 2063. E-mail:
[email protected]. † University of Canterbury. ‡ The Technical University of Lodz.
relationships governing the electrical field, the potential distribution, and the equations of motion. In that work, an algorithm was successfully developed to calculate the space charge distribution in the continuous phase liquid, after the application of the external electrical field. This was a key development in being able to describe accurately the local electrical field variations at different points in the contactor, and thus enable calculation of the forces acting on each drop locally. There has been extensive experimental study of uncharged drop motion in liquid-liquid systems over many years.10-13 Most of the relationships developed have been either empirical or semiempirical, and it is only comparatively recently that quantitative analysis of the fluid dynamics was used to determine the motion of uncharged drops in a liquid-liquid system. Bozzi et al.14 showed that motion could be accurately determined by solution of the Navier Stokes equations applied to each phase. The steady axisymmetric flow in and around a deformable drop was considered when the drop was assumed to be moving under the influence of gravity along the axis of a vertical tube at intermediate Reynolds number. The equations were solved using the nonlinear free-boundary problem using a Galerkin finite element method. That work considered the motion of deformable drops (albeit uncharged) but only along the axis of symmetry, and such an approach is insufficient for trajectory prediction. More recently9,15 a more rigorous approach was adopted to determine the motion of solid spheres and drops in an electrical field. A drag coefficient relationship was used to simplify the complex hydrodynamics in the vicinity of the interface. The results of this study were interesting since the novel theoretical approach was validated experimentally, also taking into account the more complicated situation of motion in an electrical field. Good agreement between the predicted and experimental motion trajectories was obtained.
10.1021/ie030700n CCC: $27.50 © 2004 American Chemical Society Published on Web 03/30/2004
Ind. Eng. Chem. Res., Vol. 43, No. 9, 2004 2265
Later work16 developed the model to accurately predict mass transfer from a single unhindered falling drop. This was achieved by incorporation of the StefanMaxwell equations into the system of equations. The scope of the work described here focused on charged drop trajectories in a liquid-liquid system in the presence of an externally applied field. The effects of applied field strength, flow rate, and contactor size were determined and the measured trajectories compared with the theoretical predictions. The study of the effect of contactor size was the main focus. Theoretical Approach The key equations and boundary conditions are reproduced in the Appendix at the end of the paper. An outline summary of the theoretical approach is given here, and full details are published elsewhere.9,15 The key features of the theoretical approach are summarized as follows: (1) The electric field and the distribution of the electrical potential in the domain are described by the well-known Poisson differential equation. The boundary conditions comprised the electrical potential at each of the electrodes, and the assumption of zero electrical flux at the domain boundary. (2) The space charge distribution occurring on the right-hand side of the Poisson equation was obtained by using the pure transport equation to simulate the migration of space charge entities in the continuum between the electrodes. It should be noted that space charge migration refers to the migration of undefined ionic impurities, present in very small concentrations. These migrate toward the electrodes in response to the external electrical field and can exert an important influence on the distribution of local electrical field strength, which in turn influences drop motion. (3) The influence of the space charge on the resultant electrical field was therefore required. This was determined by (a) setting an initial, arbitrary, artificial concentration field for the space charge impurities, and (b) determining a value of the mobility parameter (required in the Poisson equation) using a “trial and error” method for solution of the transport equation. The final value of mobility parameter used was that which gave agreement with earlier published experimental observations of space charge transient behavior. (4) Τhe local velocity of migration occurring in the transport equation was determined from the mobility R and the local field strength E. (5) The local field strength is related to the electrical potential, see Petera et al.15 (6) The numerical solution of the equations by numerical discretization of the equations to yield the space charge distribution and the electric field is fully described by Petera and co-workers.15 The Galerkin discretization procedure was used for the calculation of the electrical field, and the Lagrangian numerical framework was used for the calculation of the space charge distribution. The resultant field distribution was obtained by iterative solution of pertinent equations. The overall electrical field was obtained by superposition of the externally applied electrical field and the field due to the space charge. Hence, the overall field was used in the calculation of trajectory. (7) The forces acting on each drop which were used in the final calculation of the trajectory comprised the electrical forces, the drag forces, buoyancy forces, and inertial forces. (8) The calculation of each trajectory involved the setting up and solution of the equations of motion for a single charged drop moving through the continuous phase. The influence of the resultant electrical field had
Table 1. Geometric Dataa column B D E I ID OD S no. geometry (mm) (mm) (mm) (mm) (mm) (mm) (mm) 1 2 3 3 3
expt 1 expt 2 expt 3 expt 4 expt 5
10 10 20 20 20
10 10 10 20 10
4.3 4.3 4.2 4.2 5.02
70 140 142 142 142
2.3 2.3 2.35 2.35 4.48
3.23 3.23 3.3 3.3 6.45
50 50 100 100 100
a B ) distance between the charged electrode and the top of the inverted cone; D ) electrode insertion depth; E ) exposed electrode length in which no insulation is present; I ) interelectrode distance; ID ) earthed electrode inside diameter; OD ) earthed electrode outside diameter; S ) square column side length.
been calculated at each successively found position in a time marching procedure and was thus incorporated into the prediction of the trajectory. Experimental Section The liquid used as the continuous phase was sunflower oil, and the dispersed phase was deionized water, which had been equilibrated with the sunflower oil. The dispersed phase and continuous phase were mixed to form a liquid-liquid dispersion and separated out prior to the experiments. This ensured that the two liquids were in equilibrium with each other and that no mass transfer occurred during the drop trajectory experiments. The drop trajectory behavior in contactors of three sizes was studied, and the details of the key dimensions of each are summarized in Table 1. The columns consisted of a PVC base with glass walls and were equipped with a charged, partially insulated stainless steel electrode near the bottom of the column that protruded horizontally into the center of the column. The columns also had a PVC lid with a protruding earthed, partially insulated stainless steel electrode, which also acted as a nozzle for the injection of the dispersed phase liquid into the continuous phase liquid. An outline diagrammatic arrangement of the contactors is shown in Figure 1. Full details of the column design and the method of operation are given in an earlier paper.9 The dispersed phase was injected into the continuous phase through the earthed nozzle, using a syringe pump (Perfusor VI) to give a constant flow. The electrical field was applied to the contactor electrode by means of a high dc voltage source (Brandenburg, models 2507, 2878N, 828P). Prior to the trajectory measurements, the external field was applied to the contactor in the absence of any flow for a period of 3 h to allow space charge migration to occur and the attainment of space charge equilibrium. Earlier trials17 showed that space charge migration equilibrium was typically achieved in a period of up to 30 min. The drop trajectories were filmed using a standard Sony Handycam video camera with a recording speed of 25 frames per second. Parallax error was minimized by locating the camera 3 m from the column. To avoid the need for two cameras, a high quality optical mirror was located next to the column at angle of 45° to the wall in order that the drop trajectories could be recorded simultaneously from adjacent sides of the column, thus enabling recording in three dimensions. The column was illuminated from the rear and sides using spotlights and a white reflector board. The droplet trajectories were digitized from the recorded images using two approaches. In the case of single drops in unhindered and
2266 Ind. Eng. Chem. Res., Vol. 43, No. 9, 2004 Table 2. Physical Property Data for the Sunflower Oil/Water System property
Figure 1. Layout and essential dimensions of the contactor.
stable motion, it was possible to simply manually transpose the images of each drop directly from a television monitor screen onto a pair of transparent sheets, for the two adjacent image views, respectively. The droplet trajectories were tracked frame by frame by marking a small dot on the sheet corresponding to the position of the drop in successive time frames. The numerical coordinate-time data [(x, y, z)-t] were then determined by overlaying the transparent sheet onto scaled graph paper and reading off the values. The trajectories of each of the drop were averaged to obtain a final trajectory. The drop trajectories were determined relative to the center of the tip of the earthed nozzle, which was taken as the origin, with 3D coordinates (x, y, z) ) (0, 0, 0). The 3D coordinates of each drop were taken from the center of the drop in all cases. In cases where the drop motion was less stable, for example when occasional coalescence was observed, use of image analysis software proved more effective. Size distribution was less uniform, and when droplet coalescence occurred, a second method involved the use of image analysis software (Optimas version 6.5). Here, the video images were imported frame by frame into the computer. The coordinates of every drop in every frame were then found using the “Point Morphometry” function of the package. Full details of the processing of the measurements are given elsewhere.9,18 The relevant physical property data for the system (density, viscosity, and interfacial tension) were measured independently using established techniques which are described in detail elsewhere or from the literature.18 The density of the sunflower oil was measured as a function of temperature using a PAAR digital density
relationship - 14.167] [3343.5 T
viscosity, Pa s (T in K)
µsunflower-oil ) exp
interfacial tension, N m-1 (T n °C) dielectric constant conductivity of sunflower oil, S m-1 mutual solubility
γsunflower-oil ) -1.29502 × 10-5T3 + 1.6147 × 10-2T2 - 0.102139T + 35.87 3.11 1.4 × 10-11
