A Fully Optimized Electrowinning Cell for Achieving a Uniform Current Distribution at Electrodes Utilizing Sampling-Based Sensitivity Approach

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1 J Electr Eng echnol Vol. 10, No.?: 74-?, ISSN(Prnt) ISSN(Onlne) A Fully Optmzed Electrownnng Cell for Achevng a Unform Current Dstrbuton at Electrodes Utlzng Samplng-Based Senstvty Approach Nak-Sun Cho*, Dong-Wook Km*, Jeonghun Cho** and Dong-Hun Km Abstract In ths paper, a znc electrownnng cell s fully optmzed to acheve a unform current dstrbuton at electrode surfaces. o effectvely deal wth an electromagnetcally coupled problem wth mult-dmensonal desgn varables, a samplng-based senstvty approach s combned wth a hghly tuned multphyscs smulaton model. he model nvolves the nterrelaton between electrochemcal reactons and electromagnetc phenomena so as to predct accurate current dstrbutons n the electrownnng cell. In the samplng-based senstvty approach, Krgng-based surrogate models are generated n a local wndow, and accordngly ther senstvty values are extracted. Such unque desgn strategy facltates optmzng very complcated multphyscs and mult-dmensonal desgn problems. Fnally, ten desgn varables decdng the electrolytc cell structure are optmzed, and then the unformty of current dstrbuton n the optmzed cell s examned through the comparson wth exstng cell desgns. Keywords: Electrochemstry, Electromagnetcs, Fnte element analyss, Krgng method, Optmzaton, Senstvty analyss. 1. Introducton Snce the development of the hydrometallurgcal znc producton, electrochemcal deposton of metals s wdely used n varous ndustry felds. Producton of noble metals such as Cu, N, and Zn tradtonally nvolves electrownnng as the last process, where anodc dssoluton and electrodeposton on a cathode occurs. he process s carred out n a cell house typcally consstng of many electrownnng cells, coolng towers and electrolyte storage. It s revealed that the znc cell house s responsble for approxmately 80% of the energy used by an electrolytc znc refnery. hus, ntensve research has been conducted to lower energy consumpton or ncrease current effcency per mass of znc produced [1-6]. Several studes have examned optmal condtons for znc electrownnng [3-5]. hey utlzed a varety of expermental cells and electrodes to test the effect of varables on cell performance. Even though they are relable methods to analyze the electrownnng cell, the number of requred expermental permutatons rapdly ncreases as the number of manpulated varables s gettng larger. herefore, t s dffcult to defne a global optmum for the cell house. Meanwhle, Bouzek et al. n [6] nvestgated the effects of current dstrbuton flowng through an electrolyte between Correspondng Author: Dept. of Electrcal Engneerng, Kyungpook Natonal Unversty, Korea (dh9km@ee.knu.ac.kr). * Dept. of Electrcal Engneerng, Kyungpook Natonal Unversty, Korea. ({nakedsun98, bloodkdw}@nate.com). ** School of Electronc Eng., Kyungpook Natonal Unversty, Korea. (bloodkdw@nate.com). Receved: October 18, 014; Accepted: October, 014 the anode and cathode on znc electrodeposton, mathematcally and expermentally. It s concluded that the quantty and morphology of electrodeposts strongly depends on the current dstrbuton between electrodes. Even though good electrode algnment s mantaned, the nonunformty of the current dstrbuton s caused by varous desgn factors, such as spacng, thckness, and poston of electrodes and ther nsulatng edge strps. Such nonunform current dstrbuton generates local dendrtes on the cathode surface or decrease avalable electrode surface areas for znc deposton. he growng dendrte or effectve electrode area reducton specfcally occurrng at the edges of cathodes can gve an ncreased tendency to short crcuts or loss n current effcency. he objectve of ths paper s to fnd an optmzed cell structure for znc electrownnng, whch can enhance the unformty of electrode current dstrbuton. o accurately predct the current dstrbuton between electrodes, a hghly tuned numercal model s ntroduced. he complcated nterrelatons between electrochemcal and electromagnetc phenomena are taken nto account. An effcent optmzaton method, called samplng-based senstvty approach, s appled to the multphyscs desgn problem of an electrownnng cell. he method bascally generates a Krgngbased surrogate model n a local wndow, and accordngly ts senstvty value s extracted. Such unque desgn strategy facltates ntegratng very complex multphyscs systems nto the optmzaton process. Fnally, an electrolytc cell wth ten desgn varables s optmzed under the assumpton of a steady-state secondary current dstrbuton. hen the unformty of current dstrbuton n the optmzed cell s 74

2 Nak-Sun Cho, Dong-Wook Km, Jeonghun Cho and Dong-Hun Km nvestgated through the comparson wth two conventonal cell desgns (.e. ntal and emprcally obtaned ones).. Mathematcal Model A znc cell house conssts of electrolytc cells, where plate-shaped anodes and cathodes are suspended alternately as shown n Fg. 1. Fresh electrolyte enters the house at one end, passes along the cell n the drecton normal to the electrodes, and the depleted soluton leaves the tank at the opposte end. Electrochemcal reactons nvolve the transfer of electrons across the nterface between an electrode and electrolyte. When electrodes are connected to the power supply, hydrated metal ons receve electrons from the electrode, and the resultng metal s deposted at the cathode surface. Due to the geometrcal symmetry, the area marked wth a red box n Fg. 1 s selected for a two-dmensonal fnte element analyss. he geometry s a unt cell contanng an electrolyte doman, where the anodes and cathodes are modeled as electrode surfaces on the boundares. he ends of the electrodes are solated usng edge strps of solatng materal. o predct the dstrbuton of the galvanc potental f l n the nterelectrode space and the current densty at the electrodes, the Laplace equaton of (1) has to be solved wth several electrochemcal boundary condtons. Ñ = 0 (1) f l A steady-state secondary current dstrbuton at both anodes and cathodes s consdered under the assumpton that the electrolyte has a constant conductvty of 36. S/m. At the cathode, the znc deposton and smultaneous hydrogen occur as follows: - Zn + + e = Zn () H + e = H (3) On the other hand, oxygen s evolved on the anode. 1 H O = H + + O + e - (4) he total current passng through an electrode s the sum of the currents arsng from all reactons occurrng on ts surface. he Butler-Volmer expresson of (5) s used to establsh a quanttatve relatonshp between the electrode reacton knetcs and ts current [1, 6, 7]. æ æ h a h ö af ö æ - cf ö j = j0 ç expç - expç è è R ø è R øø (5) (a) Sde vew where j s the electrode current densty, j 0 s the exchange current densty, a a and a c are the anodc and cathodc charge transfer coeffcents respectvely, F s the Faraday constant, R s the gas constant, s the temperature n Kelvn, and h s the overpotental for the electrode reactons. he overpotental s the prncpal factor determnng the current densty for the three reactons from () to (4), and ts value s gven by h = f -f - E (6) s l eq (b) op vew Fg. 1. A schematc vew of a znc electrownnng cell where f s and E eq denote the potental externally appled to the electrode and the equlbrum potental due to the electrochemcal reactons, respectvely. he equlbrum potental s calculated from the Nernst equaton of (7), whch s the bass of all thermodynamc calculatons for electrochemcal processes. E eq R a = E - ln nf a 0 0 r (7) Fg.. A two-dmensonal fnte element model where E 0 s the equlbrum potental n a normal state, n s the number of electrons, and a 0 s the actvty of the 743

3 A Fully Optmzed Electrownnng Cell for Achevng a Unform Current Dstrbuton at Electrodes Utlzng Samplng-Based Senstvty~ oxdzed speces, and a r s the actvty of the reduced speces. he above equaton mples that the equlbrum potental due to the electron transport s related to the cell temperature and the actvtes of the oxdzed and reduced speces. Accordng to (7), the equlbrum potental values of 1.47 V and V are respectvely obtaned for the two cathodc reactons of () and (3), whle the other value for the anodc reacton of (4) s V. he Laplace equaton of (1) s solved wth the electrolyte-electrode boundary nterface of (5), above electrode reactons and appled potental value f s of V. he other detaled smulaton condtons comply wth [6] and [7]. 3. Samplng-Based Senstvty Approach he optmzaton of the electrownnng cell requres somewhat complcated electrochemcal analyss descrbed n the prevous chapter, and also deals wth the multdmensonal desgn varables, of whch the number usually s more than ten. o obtan an optmum structure of the electrownnng cell, a samplng-based senstvty approach s employed. he method conssts of the local wndow concept and Krgng-based surrogate modelng. In ths secton, the two key components are brefly explaned. 3.1 Hyper-cubc local wndow he global wndow generates samplng ponts over the whole desgn space, whle the local wndow samples a very small regon at the center of a nomnal desgn pont. Consequently, the concept of the local wndow s much more sutable for obtanng the accurate senstvty value of a surrogate model than the global wndow. he local wndow sze, R, s gven by U L R = c( d - d ) = 1,, Lnd (8) where c s the coeffcent whch s usually between ~5%, d s the th desgn varable n a nd-dmensonal space, and the superscrpts, U and L, are the upper and lower bounds, respectvely. After decdng the local wndow, evenly dstrbuted N r ntal samples are generated on the wndow based on the Latn Centrodal Vorono ssellaton [8, 9]. A surrogate model s created based on the responses at samples and unversal Krgng (UKG) method. he accuracy of the model s checked wth a certan crteron. If unsatsfed, addtonal samples are nserted n the wndow. he desgn senstvty at the center of a desgn pont of nterest s extracted from approxmated functons of the surrogate model. Fg. 3 explans how to explore an optmum n a desgn space wth the proposed samplngbased senstvty approach. In the llustraton, only two desgn varables, d 1 and d, are nvolved along wth two constrant condtons, g 1 and g. It s also assumed that an Fg. 3. Illustraton of a hyper-cubc local wndow and ts senstvty-based searchng technque objectve functon monotoncally decreases as the desgn varable values are ncreased. Starng wth an ntal pont, an elaborate surrogate model s produced n the hypercubc local wndow depcted n Fg. 3. Accordngly an accurate desgn senstvty value at the ntal pont s calculated. A next mproved desgn s obtaned wth the ad of a gradent-based searchng technque. he above process s repeated untl an optmum s found out. 3. Surrogate model and ts senstvty In the Krgng method [9-11], the outcomes are consdered as a realzaton of a stochastc process. he goal s to estmate a response y=[y(x 1 ), y(x ),, y(x n )] wth y(x )ÎR 1 based on n samples, x=[x 1, x,, x n ] wth x ÎR m. he response conssts of a summaton of two parts as: y = Fβ + e (9) he frst term of the rght sde of (9), called the mean structure of the response, s ntended to follow the general tendency of the functon to be modeled. It s generally composed of the frst/second-order bass functons F and the regresson coeffcent vector b, whch s obtaned from the generalzed least square method. he second term e s a realzaton of the stochastc process, and ts mean s assumed zero. he covarance of e s defned by ( ( ), ( )) (,, j = s j ) Cov e x e x R θ x x (10) where s s the process varance, q s the correlaton parameter vector estmated by applyng the maxmum lkelhood estmator (MLE). he symbol R denotes the correlaton functon of the stochastc process. he term e makes t possble to follow the fluctuatons around the general tendency. In most engneerng applcatons, the correlaton functon s set to be a Gaussan form expressed as follows j Õ = nr l= 1 (- θ ( x - x ) ) R( θ, x, x ) exp (11) l,l j,l 744

4 Nak-Sun Cho, Dong-Wook Km, Jeonghun Cho and Dong-Hun Km where x,l s the lth component of varable x. Under the decomposton of (9) and the optmal q to maxmze MLE, the nose-free unbased response ŷ at a new pont of nterest denoted by x 0 s wrtten as a lnear predctor 0 ŷ ( x = w y (1) 0 ) where w 0 == [w 1 (x 0 ), w (x 0 ),, w n (x 0 )] means the n 1 weght vector for predcton at the pont. It s obtaned usng the unbased condton E ŷ( x )] = E[y( )] as w 0 [ 0 x0 æ ö = R çr0 + Fλ è σ ø (13) where R s the symmetrc correlaton matrx wth the jth component R,j =R(q, x, x j ), r 0 =[ R(q, x 1, x 0 ),, R(q, x n, x 0 )] s the correlaton vector between x 0 and samples x, and l s the Lagrange multpler. After substtutng (13) nto (1), the predcton of Krgng model whch nterpolates the n sample ponts s expressed as λ ˆ æ F ö -1-1 y ( x0 ) = w0 y = çr0 + R y = f0 β + r0 R ( y - Fβ) (14) è σ ø where s =1/n(y-Fb)R -1 (y-fb) and b=(f R -1 F) -1 (F R -1 y) are obtaned from the generalzed least square regresson. Fnally, the dervatve y ˆ of the predcton model at x 0 s extracted from (14) lke f y ˆ ( x = J β + J r (15) 0 ) f where J and J r denote the Jacoban transformaton of f 0 and r 0, respectvely. r 4. Results he nonunform current dstrbuton on electrode surfaces s lable to cause the locally growng dendrte or effectve electrode area reducton, whch can lead to the short crcut n electrownnng cells or loss n current effcency. o acheve the unformty of cathodc current dstrbuton, a prevous work n [6] performed a parametrc study based on an emprcal way, and obtaned an mproved cell desgn. here, the only effect of the edge strp locatons of cathodes was nvestgated, whereas other geometrc parameters relatng to the whole cell structure were fxed. In ths paper, all geometrc parameters decdng the cell structure n Fg. 4 are selected and optmzed as desgn varables. he desgn goal s to fnd a fully optmzed cell structure, whch can yeld a more unform cathodc current 0 Fg. 4. en desgn varables shapng an electrownnng cell dstrbuton. he objectve functon f s mathematcally defned as follows ò ( ) n mnmze f ( d) = j ( d ) -1 dl (16) where d s the desgn varable vector, L s the ntegral path along the cathode surface, and j n s the normalzed cathodc current densty. he proposed samplng-based senstvty approach was mplemented by means of Matlab functons and applcaton programmng nterface language [1]. he electrochemcal smulatons at samples were executed wth a commercal multphyscs software package, called COMSOL [7]. For comparson wth conventonal cell desgns, the optmzaton started wth the same ntal desgn as the emprcal desgn method used n [7]. After 16 teratve desgns requrng total 673 fnte element smulatons, an optmzed cell desgn was sought out. Desgn varable values between three dfferent desgns of the ntal, emprcally obtaned, and fully optmzed cells are compared wth each other n able 1. In the emprcal method, only two desgn varables of l and l 5 n Fg. 4 were tuned through examnng the mpact of changng the edge strps of cathodes. able 1. Comparson of desgn varables between three dfferent desgns (unt: mm) Lower bounds Intal desgn Emprcal method Proposed method Upper bounds d d d d d l l l l l he galvanc potental and current dstrbutons n the electrolyte soluton for the three dfferent cells are compared n Fg. 5 and 6, respectvely. It s clear that the current unformty around the edge strps of anode as well as cathodes of the optmzed cell s much mproved when compared wth the ntal and the emprcal ones. Moreover, Fg. 6 shows that the current densty value around the L 745

5 A Fully Optmzed Electrownnng Cell for Achevng a Unform Current Dstrbuton at Electrodes Utlzng Samplng-Based Senstvty~ electrode center n the optmzed cell s ncreased by more than 0% of those n two other cells. Fg. 7 presents the normalzed current densty dstrbutons along the cathode surface for the three cells. It s observed that the unformty of cathodc current dstrbuton n the optmzed cell s enhanced specfcally around the edge strp, whch s located at about 90 mm. In Fg. 8, three dfferent cell structures are compared wth each other. (a) Intal cell Fg. 7. Current densty dstrbutons between three dfferent cell desgns (b) Emprcally obtaned cell Fg. 8. hree dfferent cell structures 5. Concluson (c) Optmzed cell Fg. 5. Potental dstrbutons between three dfferent cell desgns he samplng-based senstvty approach was successfully combned wth the multphyscs smulator to optmze the whole cell structure for znc electrownnng. he results show that the optmzed cell provdes a hgher unformty of cathodc current dstrbuton when compared wth the emprcally obtaned desgn. It s nferred that the proposed cell desgn wll be very useful for mprovng the current effcency and preventng the local dendrtes n the electrownnng process. (a) Intal (43 A/m at the electrode center) Acknowledgements hs research was supported by Basc Scence Research Program through the Natonal Research Foundaton of Korea (NRF) funded by the Mnstry of Educaton, Scence and echnology (014R1A1A4A ). (b) Emprcal (43 A/m at the electrode center) (c) Optmzed (51 A/m at the electrode center) Fg. 6. Current densty dstrbutons between three dfferent cell desgns References [1] G. Barton, and A. Scott, A valdated mathematcal model for a znc electrownnng cell, J. Appl. Electrochem., vol., pp , 199. [] A. J. Bard, and L. R. Faulkner, Electrochemcal method; fundamentals and applcatons: John Wley & Sons,

6 Nak-Sun Cho, Dong-Wook Km, Jeonghun Cho and Dong-Hun Km [3] I. Ivanov, Increased current effcency of znc electrownnng n the presence of metal mpurtes by addton of organc nhbtors, J. Hydrometallurgy, vol. 7, pp , 004. [4] P. Gullaume, N. Leclerc, C. Boulanger, J. Lecure, and F. Lapcque, Investgaton of optmal condtons for znc electrownnng from aqueous sulfurc acd electrolytes, J. Appl. Electrochem., vol. 37, pp , 007. [5] K. Bouzek, K. Borve, O. Lorentsen, K. Osmundsen, I. Rousar, and J. honstad, Current dstrbuton ar the electrodes n znc electrownnng cells, Canadan J. Chemcal Eng., vol. 9999, pp. 1-10, 013. [6] M. Mahon, S. Peng, and A. Alfantaz, Applcaton and optmzaton studes of a znc electrownnng process smulaton, J. Electrochem. Soc., vol. 14, pp , [7] COMSOL Users Manual, Electrochemcal method; fundamentals and applcatons: COMSOL Inc., 011. [8] L. Zhao, K. K. Cho, and I. Lee, Metamodelng method usng dynamc Krgng for desgn optmzaton, AIAA J., vol. 49, no. 9, pp , 011. [9] I. Lee, K. Cho and L. Zhao, Samplng-based RBDO usng the stochastc senstvty analyss and dynamc Krgng method, Struct. Multdsc Optm., vol. 44, no. 3, pp , 011. [10] D. Km, G. Jeung, K. K. Cho, H. Km and D, Km, Effcent methodology for relablty assessment of electromagnetc devces utlzng accurate surrogate models based on dynamc Krgng method, J. Magn., vol. 17, no. 4, pp , 01. [11] N. Cho, D. Km, K. K. Cho, and D, Km, Samplngbased senstvty approach to electromagnetc desgns utlzng surrogate models combned wth a local wndow, J. Magn., vol. 18, no. 1, pp , 013. [1] N. Cho, D. Km, G. Jeung, K. K. Cho, and D, Km, Smultaneous desgn approach to transent electromagnetc and thermal problems based on a black-box modelng concept, IEEE rans. Magn., vol. 50, no., pp , 014. Dong-Wook Km was born n Korea n 198. He receved hs B.S. and M.S. degrees n Electrcal Engneerng from Kyungpook Natonal Unversty, Daegu, Korea, n 009 and 011, respectvely. Currently, he s on hs Ph.D. course at Kyungpook Natonal Unversty. Jeonghun Cho receved hs M.S. and Ph.D. degrees n Electrcal Engneerng from Korea Advanced Insttute of Scence and echnology, Daejeon, Korea, n 1998 and 003, respectvely. He was a senor researcher at MCU Applcaton eam of Hynx Semconductor Inc. n Cheongju, Korea, from 003 to 005. He s currently an assocate professor at the School of Electroncs Engneerng n Kyungpook Natonal Unversty, Daegu, Korea. Hs man nterests nclude automotve embedded systems, model-base desgn, and HW/SW codesgn. Dong-Hun Km receved hs M.S. and Ph.D. degrees n Electrcal Engneerng from Seoul Natonal Unversty, Seoul, Korea, n 1994 and 1998, respectvely. He was a senor researcher at the Dgtal Applance Research Center of LG Inc. n Seoul, Korea, from 1998 to 001. He contnued hs research at the Unversty of Southampton n Unted Kngdom as a research fellow for two years (00-003). He s currently an assocate professor at the Department of Electrcal Engneerng n Kyungpook Natonal Unversty, Daegu, Korea. Hs man nterests nclude electromagnetc feld analyss, desgn optmzaton of electrcal applances, and bomedcal applcaton. Nak-Sun Cho was born n Korea n He receved hs M.S. and Ph. D. degrees n Electrcal Engneerng from Kyungpook Natonal Unversty, Daegu, Korea, n 006 and 009, respectvely. He s currently a senor researcher at the Defense Agency for echnology and Qualty. 747