Lorentz force influence on momentum and mass transfer in natural convection copper electrolysis Tom Weier ⋆ , J¨ urgen H¨ uller, Gunter Gerbeth, Frank-Peter Weiss Forschungszentrum Rossendorf, P.O. Box 51 01 19, 01314 Dresden, Germany Key words: electrochemistry, fluid mechanics, mass transfer, momentum transfer, magnetic field, convection control preprint, finally published in Chemical Engineering Science 60 (2005), 293–298 1 Introduction The influence of magnetic fields on electrochemical reactions has been the subject of many investigations. For a recent comprehensive review see Fahidy (1999). Besides the controversely discussed magnetic field effects on material properties and electrode reaction kinetics, the influence of the Lorentz force on momentum and mass transfer is a major and well recognized effect. De- spite this, attempts to link direct flow measurements to observed mass transfer modifications are somewhat scarce. As well, scaling laws to estimate possible gains in space–time–yield for real processes under limiting current conditions are not yet available. This short communication does not aim to bridge these gaps completely, but attempts to demonstrate the multitude of flow config- urations attainable by the application of simple permanent magnets to an electrochemical cell. An electromagnetic body force, the Lorentz force, results from the vector prod- uct of current density j and magnetic induction B F = j × B. (1) Under the conditions of common electrochemical processes, the current density is determined solely by the faradaic current to a very good approximation. In Email address: [email protected], Tel.: +49 351 260 2226, Fax: +49 351 260 2007 (Tom Weier ⋆ ).
Transcript
Lorentz force influence on momentum and
mass transfer in natural convection copper
electrolysis
Tom Weier⋆, Jurgen Huller, Gunter Gerbeth, Frank-Peter Weiss
Key words: electrochemistry, fluid mechanics, mass transfer, momentum transfer,magnetic field, convection control
preprint, finally published in Chemical Engineering Science 60 (2005), 293–298
1 Introduction
The influence of magnetic fields on electrochemical reactions has been thesubject of many investigations. For a recent comprehensive review see Fahidy(1999). Besides the controversely discussed magnetic field effects on materialproperties and electrode reaction kinetics, the influence of the Lorentz forceon momentum and mass transfer is a major and well recognized effect. De-spite this, attempts to link direct flow measurements to observed mass transfermodifications are somewhat scarce. As well, scaling laws to estimate possiblegains in space–time–yield for real processes under limiting current conditionsare not yet available. This short communication does not aim to bridge thesegaps completely, but attempts to demonstrate the multitude of flow config-urations attainable by the application of simple permanent magnets to anelectrochemical cell.
An electromagnetic body force, the Lorentz force, results from the vector prod-uct of current density j and magnetic induction B
F = j × B. (1)
Under the conditions of common electrochemical processes, the current densityis determined solely by the faradaic current to a very good approximation. In
with E denoting the electric field and σ the electrical conductivity, containsthe term (u×B) accounting for potential differences caused by the motion in amagnetic field. However, the currents thus induced are very small in magneticfields of moderate strength (∼ 1T) compared to the faradaic current. TheLorentz force acts as a momentum source for the flow in the cell. The rightmostterm in Eq. 2 accounts for the charge transport by diffusion in concentrationgradients ∇c of the electroactive species. D denotes the diffusion coefficient ofthe electroactive species, n its charge number and NF the Faraday constant.Charge transport by diffusion becomes important in case of concentrationgradients which typically evolve at electrodes and may lead to a limitation ofthe current by mass transfer. Details on the derivation are given in Newman(1991) while Olivas et al. (2004) discusses this subject related to Lorentz forceeffects in electroplating.
As the current density is an inherent feature of electrochemical processes, onlya suitable magnetic field has to be added to generate a Lorentz force. In eco-nomic terms, no running costs are added to the process, provided the magneticfield originates from permanent magnets. This is a more comfortable situationthan that faced, e.g., in electromagnetic flow control for naval applicationswhere the electrical currents have to be added (see Weier et al. (2003)).
2 Experimental setup
A small electrolytic cell with inner dimensions as given in Fig. 1 was made fromPMMA. The side walls forming the electrodes consist of 0.5mm thick copperplates. These copper plates are reinforced with PMMA frames which allow atthe same time to fix permanent magnets behind the electrodes. Neodymium–Iron–Boron (NdFeB) permanent magnets 30mm × 10mm wide in x and y–direction and 6mm extension in z, i.e. the magnetization direction, are usedto provide a static magnetic field mainly oriented in z–direction. Its measureddecay with increasing distance from the electrode surface is given in the rightpart of Fig. 1. In all experiments, the lower edge of the magnets coincided withthe lower edge of the cell. From Eq. (1) it follows that a current density in y–direction and a magnetic induction in z–direction will generate a Lorentz forcewith an x–component FL only. The direction of this Lorentz force (upwardsor downwards) depends on the electrical current and the orientation of themagnet as sketched in the left part of Fig. 1.
The chemical reaction under investigation was Cu2+ + 2 e− ⇋ Cu. Three
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different concentrations of CuSO4 (0.1mol/l, 0.2mol/l and 0.375mol/l) in anaqueous 1.5mol/l H2SO4 solution have been used during the experiments. Cop-per electrodeposition at the working electrode was carried out via an EG&GInstruments Princeton Applied Research Potentiostat Model Versastat. Forall concentrations, linear sweep voltammograms have been recorded to deter-mine electrode potentials for limiting current conditions. A working electrodepotential of −400mV versus PtRE has been found sufficient to guarantee limit-ing current conditions for copper deposition under all concentrations and fieldconfigurations used. The experiments were performed at room temperature.Chronoamperometry was applied to compare different Lorentz force configu-rations. A Pt wire quasi reference electrode has been used in order to minimizedisturbances of the flow.
Qualitative information on the concentration distribution in the cell was ob-tained by a shadowgraph technique. It visualises the second spatial derivativeof the density. Velocity fields in the x–y–plane were measured using a simpleDigital Particle Image Velocimetry (DPIV).
3 Results and discussion
Fig. 2 shows the response of the cell current for a 0.1molar CuSO4 solution to apotential step from the rest potential to −400mV versus PtRE at the workingelectrode. The copper deposition caused by the potential change leads to adecrease of the Cu2+ concentration near the working electrode. This diminishesthe density of the near-wall solution and leads to an upwards (positive x–direction) directed natural convection. Correspondingly copper dissolves atthe counter electrode, leading to a denser solution and a downwards directednatural convection there. After a certain time, steady state limiting currentconditions are reached. Without the permanent magnet, this is the case ca. 50 safter the potential step with jlim = 42.4Am−2.
Under action of a downwards directed Lorentz force a steady state is reachedafter approximately the same time, but low frequency current fluctuations arestill present. The limiting current density is increased to jlim ≈ 53.4Am−2
due to the additional convection caused by the momentum input. Turning thepermanent magnet by 180◦ around the x axis leads to an upwards directedLorentz force near the electrode. Under these conditions, the current densitypasses through a local minimum at t ≈ 25 . . . 60 s with jlim ≈ 62.8Am−2 andreaches a steady state only after 130 s with jlim ≈ 69.4Am−2. The minimum inthe current density corresponds to a minimum of the measured x–componentof the mean velocity (u) in the near-electrode region. The values of u wereobtained by averaging over five points in y–direction (y = 2.7 . . . 5.5mm), andare plotted versus the electrode height in Fig. 2 (right). Note that the DPIV
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measurements do not cover the whole cell height. The largest velocities arefound at the beginning of the measurement since the diffusion layer just startsto develop and current densities are much larger than under mass transportlimitation. Therefore the Lorentz force has their highest magnitude for t = 0.With increasing diffusion layer thickness, current densities decrease and sodoes the Lorentz force, thus convection slows down (t = 50 s). However, steadystate current densities are slightly higher inducing again larger velocities (t =150 s).
Fig. 3 shows shadowgraph images for three different CuSO4 concentrationsand a downwards directed Lorentz force at the working electrode 70 s afterthe potential step has been applied. The arrows outside the images show theLorentz force direction. At a CuSO4 concentration of 0.1mol/l, the shadow-graph shows wave like structures in the lower region of the electrode. In themiddle part of the cell plumes rise upwards. The Lorentz force is able to drivethe lighter near-cathode fluid downwards. However, since the magnetic fieldand thereby the Lorentz force decay with the distance from the electrode, thesituation is highly unstable. As soon as the lighter fluid moves away from thecathode, buoyancy seems to be able to counteract the Lorentz force and drivesthe fluid upwards. Since local densities vary strongly, so do local force ratios,resulting in the unsteady structures. The picture changes for the 0.2 molarCuSO4 solution. Due to the higher bulk concentration, the limiting currentdensity is larger (jlim = 71.0Am−2 for the case without magnetic field) com-pared to the 0.1 molar solution. Consequently, a larger Lorentz force will begenerated. On the other hand, buoyancy is increased as well, since the densitydifference between the Cu2+ free solution and the bulk increases. However, theLorentz force obviously dominates the flow. A regular vortex structure with acenter in the lower left part of the cell, where the driving force is maximum,is formed. Recirculation regions in the top left and lower right corner of thecell can be deduced. A similar flow structure evolves for the 0.3 molar CuSO4
solution.
Velocity fields in the cell measured by DPIV are shown in Fig. 4. The subfig-ures a) and b) give the velocity distribution immediately after the potentialstep. In both cases, the relatively high current densities result in pronouncedvortex structures in the whole cell. In case of an upward directed Lorentz force(Fig. 4a), the fluid moves in clockwise direction around a center in the upperleft part of the cell. For the downward directed Lorentz force (Fig. 4b), themotion is counterclockwise and the vortex center is in the lower left part ofthe cell. Despite the different directions of rotation, both velocity fields arerather similar (note that the measured region omits a part of the upper cellregion and does not extend directly to the cell walls). The maximum velocityfound at t = 0 s for a downward Lorentz force is 22mm/s, for the upwardsdirected force it is only slightly higher with 23mm/s. Under steady state con-ditions (t = 150 s) the picture changes. While the upward force (Fig. 4c) still
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maintains a dominant single vortex flow in the whole cell, the vortex due tothe downward directed Lorentz force is much smaller. Likewise, the maximumvelocities differ now considerably, 15.3mm/s for upward versus 8.2mm/s fordownward forcing. The differences in the velocity magnitudes explain the dif-ferences in the limiting current density measurements shown in Fig. 2. Thevelocity field in Fig. 4d) corresponds well to the shadowgraph visualisation inFig. 3.
Fig. 5 shows flow configurations with magnets placed behind the workingelectrode as well as behind the counter electrode. While Lorentz forces actingin the same angular direction amplify the effect of a single side Lorentz force,such configurations do not change the flow field qualitatively. Lorentz forces,however, directed on both electrodes in the same x–direction produce thedouble vortex structures shown in Fig. 5. The shadowgraphs have been takenwith the 0.375 molar CuSO4 solution where the images have higher contrast.Velocity fields were measured with a 0.1 molar CuSO4 solution, since thelower velocities are better resolved by the simple DPIV setup. In both cases,Lorentz forces upwards or downwards at the electrodes, the shadowgraphslook rather symmetric 10 s after the potential step (Figs. 5a and 5d). Twoelongated vortices dominate the flow. Their centers are in equal height in thelower (Figs. 5a) and upper part (Figs. 5d) of the cell, respectively. For thesteady conditions at t = 150 s, the situations have changed (Figs. 5b and 5e).In both cases, the vortex driven by Lorentz force and buoyancy in the samedirection, i.e. the right one in Fig. 5b and the left one in Fig. 5e, extends overa relatively large cell region. In contrast, the vortex where the Lorentz forcecounteracts buoyancy is weaker in both cases. The velocity field measuredwith a 0.1 molar CuSO4 solution at t = 150 s clearly reflects this behaviour,too.
As there is only a practical limitation on the number of permanent magnetsapplicable to the cell, a multitude of different flow fields can be generated.Fig. 6 gives two examples with flows driven by Lorentz forces from four per-manent magnets. The now smaller magnets (16 × 8 × 6mm) are placed behindthe electrodes in such a way that their magnetization direction is turned by180◦. All DPIV measurements have been taken at t = 0 s for the same fieldconfiguration and a 0.1 molar and 0.375 molar CuSO4 solution, respectively.The four vortex structures in Fig. 6a and 6b are symmetric and quite similarfor both concentrations. However, due to the higher current densities for themore concentrated solution, velocity magnitudes are approximately two timeslarger in this case. A double vortex structure turned by 90◦ when comparedto those from Fig. 5 results from the force configuration shown in Fig. 6c andFig. 6d. Again, the structure is symmetric and the velocity magnitudes arehigher for the more concentrated solution.
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4 Conclusions
Momentum and mass transfer during copper electrolysis in a small cell canbe significantly influenced by Lorentz forces. Depending on the magnetic fieldconfigurations, a multitude of flow fields with an accordingly large range ofmass transfer conditions can be arranged. The moderate magnetic field ofsimple permanent magnets placed behind the electrodes, although its actionis limited to the vicinity of the electrodes, is able to promote convection inthe whole cell. The interplay of Lorentz and buoyancy forces is substantial forthe resulting flow structure.
Since faradaic currents are inherently coupled to electrochemical processes,the mass transfer enhancement attainable by Lorentz force induced convectiondoes not increase the running costs. Therefore it seems attractive to performfurther studies in order to determine the underlying scaling laws.
Acknowledgement
We are grateful to Alexander Grahn for allowing us to use his shadowgraphequipment. Support from German Deutsche Forschungsgemeinschaft in frameof SFB 609 is gratefully acknowledged.
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Notation
B magnetic induction, T
Bz magnetic induction in z–direction, T
c concentration of electroactive species, mol/m3
D diffusion coefficient of the electroactive species, m2/s
E electric field strength, V/m
F Lorentz force density, N/m3
FL Lorentz force density, x–component, N/m3
j current density, A/m−2
jlim limiting current density, A/m−2
n charge number, dimensionless
NF Faraday constant, 96485 A s/mol
PtRE Pt quasi reference electrode
u velocity, m/s
u velocity–component in x–direction, m/s
t time, s
x,y,z cartesian coordinates, m
Greek letters
σ electrical conductivity, S/m
References
Fahidy, T., 1999. The effect of magnetic fields on electrochemical processes.In: Conway, B. E. (Ed.), Modern Aspects of Electrochemistry. No. 32.Kluwer/Plenum, New York, pp. 333–354.
Olivas, P., Alemany, A., Bark, F., 2004. Electromagnetic control of electro-plating of a cylinder in forced convection. J. Appl. Electrochem. 34, 19–30.
1 Sketch of the electrolytic cell and the field configurations nearthe electrode (left). Decay of Bz with the distance from theelectrode (right). 9
2 Chronoamperometry for different Lorentz force configurationsat the working electrode (left) and velocity near the workingelectrode (averaged over y = 2.7 . . . 5.5mm) for an upwardsdirected Lorentz force (right) in the 0.1molar CuSO4 solution. 10
3 Shadowgraph images 70 s after potential step for a downwarddirected Lorentz force (arrow) at the working electrode anddifferent CuSO4 concentrations: 0.1mol/l (left), 0.2mol/l(middle) and 0.375mol/l (right). 11
4 Velocity fields for the 0.1 molar CuSO4 solution at t = 0 s (a,b) and t = 150 s (c,d) for upwards (a, c) and downwards (b, d)Lorentz force at the working electrode. 12
5 Shadowgraph visualisation and DPIV measurements forLorentz forces in the same direction at both electrodes. a)0.375mol/l CuSO4, FL downwards, t = 10 s; b) 0.375mol/lCuSO4, FL downwards, t = 150 s; c) 0.1mol/l CuSO4, FL
downwards, t = 150 s; d–f) as before, but FL upwards. Workingelectrode is on the left. 13
6 Velocity fields for 0.1 molar (a,c) and 0.375 molar CuSO4 (b,d)solution at t = 0 s. Lorentz force directions are indicated byarrows outside the diagrams. Working electrode is on the left. 14
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ElektrodeWorking
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Fig. 1. Sketch of the electrolytic cell and the field configurations near the electrode(left). Decay of Bz with the distance from the electrode (right).
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Fig. 2. Chronoamperometry for different Lorentz force configurations at theworking electrode (left) and velocity near the working electrode (averaged overy = 2.7 . . . 5.5 mm) for an upwards directed Lorentz force (right) in the 0.1 molarCuSO4 solution.
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Fig. 3. Shadowgraph images 70 s after potential step for a downward directedLorentz force (arrow) at the working electrode and different CuSO4 concentrations:0.1 mol/l (left), 0.2 mol/l (middle) and 0.375 mol/l (right).
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Fig. 4. Velocity fields for the 0.1 molar CuSO4 solution at t = 0 s (a, b) and t = 150 s(c,d) for upwards (a, c) and downwards (b, d) Lorentz force at the working electrode.
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a) b)
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Fig. 5. Shadowgraph visualisation and DPIV measurements for Lorentz forces in thesame direction at both electrodes. a) 0.375 mol/l CuSO4, FL downwards, t = 10 s; b)0.375 mol/l CuSO4, FL downwards, t = 150 s; c) 0.1 mol/l CuSO4, FL downwards,t = 150 s; d–f) as before, but FL upwards. Working electrode is on the left.
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Fig. 6. Velocity fields for 0.1 molar (a,c) and 0.375 molar CuSO4 (b,d) solutionat t = 0 s. Lorentz force directions are indicated by arrows outside the diagrams.Working electrode is on the left.
