Title Natural Convection in CuSO4-H2SO4 S JOURNAL OF THE ELECTROCHEMICAL SOCI (2009), 156(9): F99-F108.

Transcription

1 Ttle Numercal Smulaton of Ionc Mass- Natural Convecton n CuSO-H2SO S Author(s) Kawa, S.; Fukunaka, Y.; Kda, S. Ctaton JOURNAL OF THE ELECTROCHEMICAL SOCI (2009), 156(9): F99-F108 Issue Date 2009 URL Rght 2009 The Electrochemcal Socety Type Journal Artcle Textverson publsher Kyoto Unversty

2 /2009/1569/F99/10/$25.00 The Electrochemcal Socety Numercal Smulaton of Ionc Mass-Transfer Rates wth Natural Convecton n CuSO H 2 SO Soluton I. Numercal Study on the Developments of Secondary Flow and Electrolyte Stratfcaton Phenomena S. Kawa, a Y. Fukunaka, b, *,z and S. Kda a a Department of Mechancal Engneerng and Scence, Kyoto Unversty, Kyoto , Japan b Nano Technology Research Center, Waseda Unversty, Tokyo , Japan F99 The onc mass-transfer rates accompanyng natural convectve electrolyte flow n a CuSO aqueous electrolyte soluton acdfed wth an excess amount of H 2 SO are numercally analyzed. The effects of a supportng electrolyte and an nteracton behavor between both cathodc upward and anodc downward natural convectons are examned. Both anodc and cathodc current densty dstrbutons along the vertcal heght are also calculated. A mathematcal model s extended by ncorporatng an addtonal boundary condton at the lmtng current. A measure of onc mgraton effect, a rato of lmtng current to lmtng dffuson current, nvolvng the transference number of a dschargng metallc on s ntroduced for ths purpose. The present calculaton predcts the oscllaton behavors n transent varatons n both electrode surface concentraton and maxmal natural convectve velocty, whch are deeply related to the perodc fluctuatng electrolyte flow patterns dstorted by secondary flow. The addton of H 2 SO mantans an value around 1 and prevents the further development to the transton or turbulent natural convecton The Electrochemcal Socety. DOI: / All rghts reserved. Manuscrpt submtted September 23, 2008; revsed manuscrpt receved May 27, Publshed July 9, Natural convectve electrolyte flow plays an essental role n several electrochemcal reactors. Copper refnng electrolyss s the most mportant ndustral process n producng hgh purty copper n a large scale, where 1 m hgh vertcal-plane electrode surfaces are placed face to face, separated by a 3 5 cm thck electrolyte layer. A supportng electrolyte of H 2 SO s frequently added to ncrease the conductvty of the bath and to mprove the polarzaton parameter for electrodeposton. Durng the electrolyss, four concentraton boundary layers CBLs for Cu 2+,H +,HSO, and SO 2 ons are formed near both electrode surfaces. They are not completely ndependent of each other because the electroneutralty condton and acd base equlbrum must be satsfed. The nvestgaton applyng the physcochemcal hydrodynamcs to verfy the mportance of natural convectve electrolyte flow was reported by Wagner. 1 Snce then, numerous papers have been publshed on the onc mass-transfer rate accompanyng natural convecton developed along the vertcal-plane electrodes They are mostly based on the boundary-layer theory n sem-nfnte meda. The nteracton behavor between cathodc upward and anodc downward natural convectons has not been frequently dscussed. In the prevous study, the effect of the nteracton behavor between anodc downward and cathodc upward natural convectons developng durng the galvanostatc electrolyss n a tny electrolytc cell was examned both numercally and expermentally. 12 The transent varatons n the onc mass-transfer phenomena assocated wth modulatng an electrolytc current were also nvestgated. 13 The appled current denstes employed n these studes were far below the lmtng one, and copper electrolyss was carred out only n a bnary CuSO aqueous electrolyte soluton. Besdes, the unformty of both anodc and cathodc current-densty dstrbutons along the vertcal heght was assumed. Many theoretcal and numercal studes on the current-densty dstrbuton were reported as well as on the onc mass-transfer phenomena n varous electrochemcal systems such as the rotatng-dsk electrode system, 1-18 and parallel or tubular electrode systems However, only a few studes have been carred out on the currentdensty dstrbuton n the vertcal-plane electrode system. 6,8,25,26 It s thus ndspensable to nvestgate the transent onc mass-transfer phenomena accompanyng natural convectons developng along both electrode surfaces durng the galvanostatc copper electrolyss at an apprecable fracton of the lmtng current densty. * Electrochemcal Socety Actve Member. z E-mal: fukunaka@energy.kyoto-u.ac.jp The goal of ths research s to predct the transent behavor of the onc mass-transfer rates and the current-densty dstrbutons along vertcal-plane electrodes resultng from an nteracton between upward and downward natural convectons. The predctons of both lmtng current densty and crteron of unform current-densty dstrbuton are also necessary to mprove the ndustral productvty of the electrolytc process wth a hgher current densty. Moreover, t s well known that the onc mass-transfer rate consderably couples wth the mcrostructural or morphologcal varatons n the electrodeposted flm at a relatvely hgh cathodc overpotental. The predcton of the electrode surface overpotental wthout masstransfer contrbuton may provde a key to control the physcal and electrcal propertes of electrodeposted copper flms. Numercal smulatons may gve varous nsghts nto the couplng phenomena between the mcrostructural or morphologcal varatons and the onc mass-transfer rate. 27 A mathematcal model s developed for a CuSO aqueous electrolyte soluton acdfed wth an excess amount of H 2 SO under the presumptons of electroneutralty, acd base equlbrum, and constant physcal propertes. The effects of supportng electrolyte and nteracton between both natural convectons are now focused. Calculated results are compared to the optcal measurements. 10,11 The present paper addresses only the numercal study part. The comparsons between present numercal calculatons and optcal measurements are separately carred out n Part II. Expermental The expermental setup employed n the present study was the same as prevous studes. 10,11 Fgure 1 shows a schematc dagram of an electrolytc cell. Both the anode and cathode were vertcally nstalled face to face n a cell, separated by a.8 cm thck electrolyte layer. The electrolyte soluton was poured nto the cell untl ts surface was postoned at a heght of 16 cm from the bottom. Two electrolyte compostons, 0.05 M CuSO and 0.05 M CuSO 1.85 M H 2 SO aqueous electrolyte solutons, were employed. A constant electrolytc current was appled to the cathode. The prelmnary experments showed that the cathodc lmtng current denstes of both electrolyte solutons were about =.6 and 2.0 ma/cm 2, respectvely. 10 The electrolytc condtons employed n ths study are summarzed n Table I. Mathematcal Model A mathematcal model and numercal calculaton procedure for a bnary CuSO aqueous electrolyte soluton have been reported prevously. 12,13 The present study descrbes a mathematcal model

3 F100 Fgure 1. Schematc dagram of the electrolytc cell. a Sde vew and b top vew. for a CuSO aqueous electrolyte soluton acdfed wth an excess amount of H 2 SO. A two-dmensonal 2D mathematcal model s numercally analyzed based on the marker-and-cell MAC method, usng the fnte dfference method and determnstc relaxaton technques. Fgure 2a shows a schematc dagram of the 2D electrolytc cell flled wth an aqueous electrolyte. We use a grd of cells wth an rregular nterval mesh, as demonstrated n Fg. 2b, where the mnmal mesh sze n the vcnty of both electrode surfaces s set to 1 m whch s far larger than the double-layer thckness. The common rato of less than 1.15 s employed to defne the uneven mesh sze represented by a geometrcal progresson. In a CuSO H 2 SO electrolyte, both CuSO and H 2 SO dssocate nto four on speces, Cu 2+,H +,HSO, and SO 2, whose concentratons are denoted as C 1, C 2, C 3, and C, respectvely. When the concentratons of CuSO and H 2 SO are gven as m A and m B, respectvely, and the degree of dssocaton of the HSO on s also assgned as a, the followng expressons are derved accordng to the acd base equlbrum C 1 = m A C 2 = 1 +am B C 3 = 1 am B C = m A + am B 1 These on concentratons satsfy the prncple of electroneutralty z k C k =0 2 k=1 Here, z k s the valency of an on speces k. Table I. Expermental condtons. Composton of electrolyte 0.05 M CuSO Average current densty ave ma/cm , 0.73, 0.96, 1.2, 1 96, 3.65,.6 Composton of electrolyte 0.05 M CuSO 1.85 M H 2 SO Average current densty ave ma/cm , 0.73, 0.96, 1.2, 1.96 Electrode spacng L cm.8 Temperature T K 293 Fgure 2. a Schematc dagram of the 2D electrolytc cell and b unevenly dvded grd used n numercal calculaton. The mathematcal model ncludes the unsteady convecton dffuson equatons for electroneutral compounds CuSO and H 2 SO, the Posson equaton for the electrc potental, and the Naver Stokes and contnuty equatons for the ncompressble vscous flud. The followng equatons are derved from the mass and momentum conservaton laws under the assumptons of the electroneutralty and constant physcal propertes m A t m B t + v m A = D A 2 m A 3 + v m B = D B 2 m B 2 =0 v t + v v = 1 0 P + 2 v + f 0 v =0 Here, D k expresses the dffuson coeffcent, the electrc potental, v the flud velocty vector, P the pressure, the knematc vscosty, 0 the reference flud densty, and t the tme coordnate. f s an external force vector actng over the flud. The buoyancy nduced by flud densty dfference s consdered n the present study f = 0 g Here, s the flud densty and g s the gravtatonal acceleraton vector. Accordng to the electroneutralty and Boussnesq approxmaton, the densty dfference wth respect to the concentraton of each on can be descrbed as follows = 0 = 0 1 A + 0 m Am 1 0 m Bm B

4 = A m A + B m B 9 0 Here, A and B express the densfcaton coeffcents of CuSO and H 2 SO, respectvely. The boundary condton on the electrode surface must be formulated. The no-slp boundary condton s appled. The net mass flux of each on speces k on the electrode surface j k can be expressed accordng to the Nernst Planck equaton for the concentraton of each on governed by dffuson and onc mgraton as follows C k j k = D k kc k 10 Here, k s the moblty of each on. At the ntal condton, the concentraton of each on s set equal to the bulk electrolyte concentraton, and the concentraton gradent of each on s absent. Then, the current densty can be wrtten as follows = F z k j k = F k z k C k=1 k=1 k,0 = 11 Here, s the electrc conductvty of an electrolyte and F s the Faraday constant. The present study assumes that only the Cu 2+ on reacts at the electrode/electrolyte nterface, and no other electrochemcal reacton such as the gas evoluton takes place. The boundary condton for each on on the electrode surface can then be expressed as follows C 1 D 1 = z 1 F 1.0 t 1 12 C k D k = z k F t k k = 2,3, 13 t k = kz k C k,0 1 l=1 l z l C l,0 Here, t k s the transference number of an on speces k, whch represents the onc mgraton effect at the electrode surface. Durng the electrolyss, the CBLs develop due to the electrochemcal deposton and dssoluton of copper at the electrode/ electrolyte nterface. The boundary condton for each on can be wrtten as follows C 1 j 1 = D 1 1C 1 = 15 z 1 F where C 2 = C am B = a for m A m B 21 2m A am B a In the above dervaton, the followng Ensten relaton s also appled RT D = 22 z F Here, R s the unversal gas constant and T s the temperature. Smlarly, the followng relatons are derved C 3 = C 1 23 where = C = C am B 1.0 a 2m A am B a 2 for m A m B 25 2m A + am B = 2a for m A m B 26 2m A am B a In a 0.05 M CuSO 1.85 M H 2 SO electrolyte soluton, the degree of dssocaton of the HSO on s ,10,, and can then be approxmated as 0.81, 0.3, and 0.38, respectvely. These values of,, and can be estmated as , 0.12, and 0.217, respectvely, under the consderaton of m A /m B = at the ntal bulk electrolyte condton. Therefore, ths approxmaton s less accurate partcularly n. Both the accuracy mprovement and assocatng consderaton of the dependences of physcal propertes on the electrolyte composton and concentraton are left for future work. The followng boundary condtons for the concentraton of CuSO and the electrc potental are derved under the presumpton of constant physcal propertes D A m A = 1.0 t 1 z 1 F z 1 F RT + D Am A = 1.0 t 1 z 1 F F C k j k = D k kc k =0 k = 2,3, 16 Accordng to the electroneutralty, the followng equaton can be obtaned = C k z k k=1 =0 z 1 F z 1 /D 1 k=1 k z k C k /D k Then, Eq. 15 and 16 can be rewrtten as follows C 1 = C k = z 1 FD 11.0 z k FD 1 kz k C k /D k l=1 1 z 1 C 1 /D 1 k=1 k z k C k /D k l z l C l /D l k = 2,3, 19 Substtuton of Eq. 1 nto Eq. 18 and 19 yelds the followng relaton D A = D D t 1 = Here, D A and t 1 can be consdered as the dffuson coeffcent of CuSO and the transference number of the Cu 2+ on n a CuSO aqueous electrolyte soluton contanng an excess amount of H 2 SO. Clearly the dffuson coeffcent of CuSO can be expressed by combnng on dffusvtes D wth mobltes. Therefore, D A can be consdered as the bnary electrolyte dffuson coeffcent, extended to the case of a bnary electrolyte soluton contanng an excess amount of the supportng electrolyte. Then, the followng relaton between concentraton gradents of CuSO and H 2 SO toward the electrode surface can be obtaned m B = a m A = 1.0 t ad A z 1 F 29 Assumng that the boundary condton for H 2 SO can be smlarly expressed as Eq. 13, the dffuson coeffcent of H 2 SO n a

5 F102 CuSO H 2 SO electrolyte soluton can be estmated as follows m B D B = t az 2 F 1 z 1 D B = D A z t 1 The current-densty dstrbuton along the vertcal-plane electrode s also calculated. The electrc-potental gradent toward the effectve electrode surface s corrected from the constrant that the ntegraton of local current densty over the whole effectve electrode surface area must be equal to the appled average current densty ave C z = A z = t 2 z 2 1 F t 1 RT + D z Am A z z 2 1 F t 1 RT + D z Am A z C da = A da = ave A Here, A denotes the total effectve electrode surface area. The cathode surface Cu 2+ -on concentraton nevtably starts to consderably vary along the electrode heght. When a hgh level constant electrolytc current s mposed, the lmtng current condton may be partally establshed where the cathode surface concentraton of the Cu 2+ on approaches zero. Because the present study assumes that only the electrochemcal deposton and dssoluton of copper are taken nto account at the electrode/electrolyte nterface and no other reacton such as gas evoluton s consdered, the electrc-potental gradent toward the cathode surface approaches nfnty near the lmtng current densty, as easly magned n Eq. 32 at m A z 0. Then, the above-mentoned calculaton procedure becomes nevtably unstable. In such a stuaton, the cathodc current densty s expressed exclusvely wth the concentraton gradent If m A z 0.05 m A,0 z 1 F C z = 1.0 t 1 A D m Az If m A z 0.05 m A,0 z 2 1 F 2 C z = 1.0 t 1 RT + D z Am A z 3 C da = ave A 35 The present study conventonally assumes that the lmtng current condton s substantally establshed when the concentraton of the Cu 2+ on at the cathode surface drops below 5% of bulk concentraton whch corresponds to M. When the whole effectve cathode surface reaches the lmtng current condton, a correcton factor nvolvng the transference number of the Cu 2+ on s ntroduced. It s not based on the quanttatve techncal arguments, but a technque to overcome the computatonal dffculty. Because the Cu 2+ on s no longer a domnant onc speces near the cathode surface at the lmtng current condton, the conventonal transference number treatment may no longer be accepted. The rato = I L /I D of the lmtng current to the lmtng dffuson current s a convenent measure of the onc mgraton effect, whch depends on the electrolyte composton 29 D,C z = z 1 FD A m Az 36 I D = D,C da I L = ave A C z = D,C z = I L D,C z 37 I D The boundary condtons at both nsulated wall and electrolyte free surfaces are descrbed. Because no electrochemcal reacton occurs at these nterfaces, the boundary condtons on both nsulated wall and electrolyte free surfaces are gven as follows m A n = m B n = n =0 38 Here n ndcates the normal drecton to these boundares. The boundary condtons for flud velocty and pressure felds on both effectve electrode and nsulated wall surfaces are wrtten as follows, accordng to the no-slp boundary condton ux S,z S =0 wx S,z S =0 39 Pz S = 0 x=x 2 ux S,z S 0 S The boundary condtons on the bottom wall surfaces are also expressed as follows ux WS,z WS =0 wx WS,z WS =0 1 Px WS = 0 z z=z 2 wx WS,z WS 0 g A m A + B m B WS 2 The boundary condtons on an electrolyte free surface are wrtten as follows ux LS =0 wx LS,z LS =0 3 z z=z LS Px LS = 0 z z=z 2 wx LS,z LS 0 g A m A + B m B LS An electrolyte soluton s ntally homogeneous and at rest m A = m A,0 = 0.05 m B = m B,0 = 1.85 v = u,w = 0,0 for t =0 5 The ntal condton for the electrc potental can be determned accordng to Eq. 11 z = ave effectve electrode surface regon z =0 nsulated electrode surface regon 6 In the dscretzaton of governng equatons, the second-order central dfference scheme s appled for space dscretzaton except for the convectve terms, for whch the thrd-order upwnd Kawamura dfference scheme s adopted. Besdes, the frst-order backward dfference scheme s appled for tme dscretzaton. The bref flow chart of numercal calculaton at each tme step s gven as follows. 1. The electrc-potental feld s numercally calculated by solvng Eq. 5 wth boundary condtons, where the teratve method s employed.

6 F103 Table II. Physcal constants. Composton of electrolyte 0.05 M CuSO m 2 /s D A m 2 /s A m 3 /mol t kg/m S/m 0.5 Composton of electrolyte 0.05 M CuSO 1.85 M H 2 SO m 2 /s D A m 2 /s A m 3 /mol B m 3 /mol t t kg/m S/m 51 a The concentraton felds of both Cu 2+ and H + ons are numercally calculated accordng to Eq. 1, by solvng Eq. 3 and wth boundary condtons based on the teratve algorthm. 3. Both anodc and cathodc current-densty dstrbutons along the vertcal heght are then calculated by correctng the electrcpotental gradent toward the electrode surface to satsfy the constrant that the ntegraton of local current densty over the whole effectve electrode surface area must be equal to the appled average current densty ave. In ths process, the teratve method s employed. The teraton s fnshed when the average current densty estmated from the modfed electrc-potental gradent s converged wthn the error of 0.01% of the exact value.. Both velocty and pressure felds are calculated by numercally solvng Eq. 6 and 7 wth boundary condtons, accordng to the MAC method, where the teratve calculaton method s used. The convergence of the present calculaton was examned wth varous knds of tme ncrement and spatal grd dmensons. There s almost no dfference n the accuracy between the computatonal grds of and cells. In ths study, the grd of cells s adopted to examne the current-densty dstrbuton along the electrode heght at an acceptable computng tme. The tme step t s set to 100 s n the present calculaton. The crtera of both CuSO and H 2 SO concentraton calculatons at each tme step are set to 2 ave 1.0 t 1 t/z1 F DA and 2 ave t 2 t/z2 F 1/ DB, respectvely, whch are the measures of concentraton change due to both dffuson and onc mgraton effects at the ntal tme step. Besdes, 0.1 m/s s employed as the crteron of velocty calculaton. The crteron of electrc-potental calculaton s 1 mv for the copper electrolyss n a 0.05 M CuSO below =1 ma/cm 2 and n a 0.05 M CuSO 1.85 M H 2 SO below =2 ma/cm 2. For the copper electrolyss n a 0.05 M CuSO above =1 ma/cm 2, 10 mv s adopted as the crteron. The physcal propertes employed n the present study are lsted n Table II, whch are referred to the prevous studes. 9,10 Fgure 3. Stream functon profle at 600 s after startng the electrolyss 0.05 M CuSO soluton, t = 600 s. a = 0.96, b = 1.96, and c =.6 ma/cm 2. Results and Dscusson Fgures 3 and show the stream functon profles at 600 s after startng the galvanostatc electrolyss at varous appled current denstes n both electrolyte compostons. The cathode s placed on the left sde and the anode on the rght sde. The smlar stream functon profles and ther transent phenomena have already been dscussed prevously. 13 The perodc fluctuatng electrolyte flow phenomena are more remarkable n a 0.05 M CuSO electrolyte soluton, as seen n Fg. 3. At the hgher current densty, small vortces developng near both upper and lower edges of the effectve electrodes sgnfcantly dstort the flow pattern n the bulk electrolyte sandwched by two vertcal-plane electrodes. The perodcally fluctuatng electrolyte flow pattern then appears. Vortces sze seems to ncrease at the hgher current densty. The confnement effects, such as the collson between both cathodc upward and anodc downward natural convectons and the effect caused by the nsulated lateral wall and electrolyte free surfaces, may nfluence both concentraton and current-densty dstrbutons along the electrode heght. The confnement effects are less sgnfcant n a CuSO H 2 SO electrolyte, as seen n Fg.. The addton of H 2 SO damps natural convecton and prevents the further development to the transton or turbulent natural convecton. The transent varatons n the calculated electrc-potental feld are also shown n Fg. 5, where the galvanostatc copper electrolyss was carred out at = 0.96 ma/cm 2 n a 0.05 M CuSO aqueous electrolyte. The transent varatons n the calculated electrcpotental profle at the mddle of the effectve electrode heght are also descrbed n Fg. 6. The ntally symmetrc potental feld s Fgure. Stream functon profle at 600 s after startng the electrolyss 0.05 M CuSO 1.85 M H 2 SO soluton, t = 600 s. a = 0.236, b = 0.96, and c = 1.96 ma/cm 2.

7 F10 Fgure 7. Transent varatons n the cathode surface Cu 2+ -on concentraton at mdheght 0.05 M CuSO soluton, z = cm. Fgure 5. Transent varatons n electrc-potental feld 0.05 M CuSO soluton, = 0.96 ma/cm 2. a t = 30, b t = 60, and c t = 300 s. gradually dstorted wth tme. It s because the electrc-potental gradent toward the cathode surface s steeper due to the depleton of the Cu 2+ -on concentraton, whle the electrc-potental gradent toward the anode surface becomes more relaxed due to the enrchment of the Cu 2+ -on concentraton. As the tme progresses, the potental feld starts to dstort even along the vertcal drecton because both anode and cathode surface Cu 2+ -on concentratons vary wth electrode heght due to the development of natural convecton. C 1,C t = C 1, t 1 ave z 1 t 7 F D A Because an electrolyte soluton s ntally homogeneous and at rest, the onc mass-transfer phenomena due to both dffuson and onc mgraton effects are predomnant untl natural convecton starts to develop. The slopes estmated from the present calculaton agree well wth the analytcal ones and the errors are wthn 1%. Transent varatons n onc mass-transfer rate wth natural convecton. Fgure 8 compares the transent varatons n the calcu- The accuracy of present numercal calculaton. Fgure 7 compares the transent varatons n the calculated cathode surface Cu 2+ -on concentraton wth the analytcal solutons at varous current denstes n a 0.05 M CuSO aqueous electrolyte to check the accuracy of the present numercal calculaton. The sold lnes correspond to the analytcal solutons, and the symbols correspond to the calculated results. Calculated results are plotted at 1 s ntervals. The analytcal soluton for the transent varaton n the electrode surface concentraton s based on the unsteady one-dmensonal dffuson equaton n sem-nfnte meda, whch can be expressed as follows Fgure 6. Transent varatons n the electrc-potental profle at mdheght 0.05 M CuSO soluton, = 0.96 ma/cm 2, z = cm. Fgure 8. Transent varatons n the calculated cathode surface on concentraton at mdheght z = cm. a Cu 2+ -on concentraton and b H + -on concentraton.

8 F105 were also seen n the prevous study n a tny electrolytc cell. 12 Compared to the prevous study, the followng two thngs are noted. 1. The magntude of the overshoot s smaller n the present electrolytc cell. 2. It takes much more tme to approach a substantal steady-state value n the present study than n the prevous study. Fgure 9. Transent varatons n maxmal natural convectve velocty at mdheght z = cm. lated cathode surface on concentraton at the mddle of the heght between a CuSO aqueous electrolyte wth and wthout H 2 SO. The sold lnes correspond to the calculated results n a 0.05 M CuSO, whle the dashed lnes to those n a 0.05 M CuSO 1.85 M H 2 SO. Calculated results are plotted at 6 s ntervals. Clearly the Cu 2+ -on concentraton decreases due to the electrodeposton of copper at the cathode surface, whle the H + -on concentraton ncreases due to the onc mgraton effect. These concentraton changes are ntally proportonal to the square root of duraton tme, whch mples that the onc mass-transfer phenomena due to both dffuson and onc mgraton effects are domnant just after statng the electrolyss. As tme progresses, natural convecton develops along the vertcal-plane cathode. The cathode surface on concentraton gradually approaches a substantally steady-state value. It s almost smlar to the prevous study 12 where the galvanostatc copper electrolyss was carred out n a smaller electrolytc cell wth a heght of 10 mm and an electrode spacng of 1 2 mm. The most dfferent pont from the prevous study s that t takes more tme to attan a substantal steady state n the present study. It s ascrbed to the cell dmensons such as the nterelectrode dstance and electrode heght, whch are deeply related to the magntude of the developed natural convecton, the collson and nteracton between anodc downward and cathodc upward natural convectons. The electrode surface concentraton gradually converges to an almost steady-state value wth fluctuaton, and ts ampltude of oscllaton becomes larger at the hgher current densty n an unsupported CuSO. The nteracton behavor between cathodc upward and anodc downward natural convectons nduces such a fluctuaton. The perodc fluctuatng electrolyte flow phenomena resultng from such an nteracton lasts a much longer tme because they are related to the momentum dsspaton by vscous stress. The smlar fluctuatng behavors are also seen n Fg. 9, whch descrbes the transent varatons n the maxmal natural convectve velocty w max at the mddle of the heght n two dfferent electrolyte solutons at varous current denstes. The sold lnes correspond to the calculated results n a 0.05 M CuSO, whle the dashed lnes to those n a 0.05 M CuSO 1.85 M H 2 SO. Calculated results are plotted at 6 s ntervals. The maxmum value of the natural convectve velocty n the hydrodynamc boundary layer s located at m away from the electrode surface. Because the buoyancy of H 2 SO nduced by the onc mgraton effect acts n an opposte drecton to that of CuSO, the magntude of the developed natural convecton s depressed n a 0.05 M CuSO 1.85 M H 2 SO. The overshoots followed by fluctuatng phenomena are clearly seen n both electrolyte solutons. The smlar results As already mentoned n oscllaton behavors of the concentraton, the overshoot followed by fluctuatng behavors n velocty s also determned by the magntude of the appled current densty, cell dmensons, and electrode confguraton. The nterestng result s that the calculated maxmal velocty starts to decrease almost monotonously at 20 s after statng the electrolyss at =.6 ma/cm 2 n a 0.05 M CuSO aqueous electrolyte. Such a phenomenon could not be detected n prevous studes. 12,13 Ths apparent deceleraton n the natural convectve velocty s ascrbed to the electrolyte stratfcaton near the upper cathode, where the lghter and lower concentraton electrolyte s accumulated. It s consdered as very mportant n the electrolyte crculaton system desgn n the actual ndustral electrorefnng tankhouse because the conventonal electrorefnng operaton lasts for several weeks. The transent varatons n the maxmal natural convectve velocty take much more tme to approach a substantal steady state compared to the transent varatons n the electrode surface concentraton. Because the CBL thckness s very thn compared to the hydrodynamc boundary-layer thckness because of the hgh Schmdt number, the effect of secondary flow velocty changes on the convectve mass transfer s far less sgnfcant. Therefore, the concentraton fluctuaton n the CBL s damped out more rapdly than the velocty fluctuaton n the hydrodynamc boundary layer. Current-densty dstrbuton along the vertcal cathode. Fgure 10 shows the calculated cathodc current-densty dstrbuton along the vertcal heght at 600 s after startng the electrolyss n both electrolyte compostons at varous current denstes. Here, z s the vertcal dstance from the lower edge of the effectve cathode. The present calculaton predcts that the unform current-densty dstrbuton s mantaned wthn 600 s except for = 3.65 and.6 ma/ cm 2 n a 0.05 M CuSO and = 1.96 ma/cm 2 n a 0.05 M CuSO 1.85 M H 2 SO.At = 3.65 ma/cm 2 n a 0.05 M CuSO, the partal lmtng current condton s establshed, whle the entre lmtng current condtons are establshed at =.6 ma/cm 2 n a 0.05 M CuSO and at = 1.96 ma/cm 2 n a 0.05 M CuSO 1.85 M H 2 SO. Typcal transent varatons n the cathodc current densty and correspondng cathode surface Cu 2+ -on concentraton dstrbutons along the vertcal heght are also seen n Fg. 11. The cathode surface Cu 2+ -on concentraton decreases wth heght as the natural convecton develops along the vertcal cathode wth tme. Thus, the partal lmtng current condton s establshed at the upper cathode when a hgh level constant current s mposed. A relatvely localzed peak appears due to mass-transfer lmtaton at the upper cathode, and ths peak descends from the upper end to the lower end wth tme. As long as the cathode surface s flat and natural convecton s steady lamnar flow n sem-nfnte meda, the boundary-layer theory gves the followng expresson under the presumpton of unform dstrbuton of the zero concentraton of the Cu 2+ on along the cathode heght detal dervaton has been descrbed n Ref. 1 and 3 C z z Fgure 12 demonstrates the relatonshps between log z/h and log C z/ ave at 600 s after startng the electrolyss n both electrolyte compostons at the expermentally measured lmtng current denstes. Here, h s the heght of the effectve electrode. The boundary-layer theory qualtatvely agrees wth the calculated slope n both electrolyte solutons. The small dfferences are attrbuted to

9 F106 Fgure 11. Transent varatons n the calculated cathodc current densty and Cu 2+ -on concentraton dstrbutons along the vertcal heght 0.05 M CuSO soluton, = 3.65 ma/cm 2. a Cathodc current-densty dstrbuton and b cathode surface Cu 2+ -on concentraton dstrbuton. Fgure 10. Calculated current-densty dstrbuton along the vertcal cathode heght at 600 s after startng the electrolyss at varous current denstes t =600 s. a 0.05 M CuSO soluton and b 0.05 M CuSO 1.85 M H 2 SO soluton. fnte electrode heght and nterelectrode dstance as well as the breakdown of the steady lamnar electrolyte flow, as seen n Fg. 3 and. Correcton factor. The dffculty to numercally predct the lmtng current densty n an unsupported CuSO electrolyte soluton accompanes how to express the onc mgraton effect n the zero concentraton electrolyte soluton. Because the conventonal transference number treatment no longer becomes avalable, t s hard to accurately estmate the contrbuton of onc mgraton effect to the onc mass flux. Fgure 13 shows the transent varatons n calculated correcton factor n both electrolyte solutons at the expermentally measured lmtng current denstes. Before establshng the entre lmtng current condton, s mantaned at constant values of 1.56 n the CuSO electrolyte and n the CuSO H 2 SO electrolyte, respectvely. It s because s expressed n terms of the transference number of the Cu 2+ on as follows = I L 1.0 = I D t 1 After establshng the entre lmtng current condton, starts to ncrease. At the expermentally measured lmtng current densty, approaches 1.7 n the CuSO electrolyte and 1.03 n the CuSO H 2 SO electrolyte, respectvely. The calculated value of gradually decreases wth oscllaton at =.6 ma/cm 2 n the CuSO electrolyte. There exsts no numercal study on the transent varatons n the correcton factor n the electrolytc process wth vertcal-plane electrode confguraton. Newman examned the dependence of on the addtve amount of H 2 SO wth the rotatngdsk electrode n a CuSO H 2 SO aqueous electrolyte soluton, 29 where was estmated around 1.9 n an unsupported CuSO. Ths value agrees well wth the present numercal calculaton, as far as the transent varaton n s concerned. However, further numercal study s ndspensable to examne the valdty of the present numercal estmaton of onc mgraton effect under the lmtng current condton because only two cases are numercally calculated n ths study. Concluson The mathematcal model for the onc mass-transfer rates assocated wth natural convecton nduced along the vertcal-plane electrodes n an aqueous electrolyte soluton s developed. The effects of a supportng electrolyte and an nteracton between both cathodc and anodc natural convectons on the onc mass-transfer rate are examned.

10 F107 The present calculaton predcts that the unform current-densty dstrbuton can be mantaned wthn 600 s after startng the galvanostatc electrolyss below one-half of the lmtng current densty n both electrolyte compostons. Both partal and entre lmtng current condtons along the vertcal cathode are calculated. Under the entre lmtng current condton, the classcal natural convecton correlaton for the vertcal cathode qualtatvely agrees wth the present numercal calculaton. The deceleraton n the natural convectve velocty s notced at the hgh level constant current densty n a 0.05 M CuSO, whch s ascrbed to the electrolyte stratfcaton. Lst of Symbols Fgure 12. Relatonshps between log z/h and log C / ave at t = 600 s. -: 0.05 M CuSO soluton, =.6 ma/cm : 0.05 M CuSO 1.85 M H 2 SO soluton, = 1.96 ma/cm 2. Fgure 13. Transent varatons n the calculated correcton factor n two electrolyte solutons. -: 0.05 M CuSO soluton, =.6 ma/cm : 0.05 M CuSO 1.85 M H 2 SO soluton, = 1.96 ma/cm 2. Greek Subscrpts a degree of dssocaton of the HSO on A total effectve electrode surface area, m 2 C concentraton of an on speces, mol m 3 C,0 concentraton of an on speces n the bulk electrolyte, mol m 3 C,C concentraton of an on speces at the cathode surface, mol m 3 D dffuson coeffcent of an on speces or electroneutral compound, m 2 s 1 F Faraday constant, 96,500 C/equv g gravtatonal acceleraton, 9.8 m s 2 g gravtatonal acceleraton vector, m s 2 h heght of the effectve electrode, m H heght of an electrolyte free surface from the bottom of a cell, m current densty, A m 2 ave average current densty, A m 2 A anodc current densty, A m 2 C cathodc current densty, A m 2 D,C cathodc dffuson current densty, A m 2 I D lmtng dffuson current over the whole effectve electrode area, A I L lmtng current over the whole effectve electrode area, A j k mass flux of on speces k, mol m 2 s 1 L nterelectrode spacng, m m concentraton of an electroneutral compound, mol m 3 m,0 concentraton of an electroneutral compound n the bulk electrolyte, mol m 3 n normal drecton to the boundary P pressure, N m 2 R unversal gas constant, 8.31 J mol 1 K 1 t tme, s t transference number of an on speces T temperature, K u horzontal component of flud velocty of an electrolyte, m s 1 v flud velocty vector, m s 1 w vertcal component of flud velocty of an electrolyte, m s 1 w max maxmum value of natural convectve velocty, m s 1 x horzontal dstance from the cathode surface, m horzontal dstance from the anode surface, m z vertcal dstance from the bottom of a cell, m z valency of an on speces, g equv mol 1 z vertcal dstance from the lower edge of the effectve electrode, m densfcaton coeffcent of an electroneutral compound, m 3 mol 1 m concentraton dfference n an electroneutral compound from the bulk electrolyte, mol m 3 flud densty dfference from bulk electrolyte, kg m 3 correcton factor rato of concentraton gradent of the SO 2 ontothecu 2+ on at the electrode surface rato of concentraton gradent of the HSO ontothecu 2+ on at the electrode surface moblty of an on speces, m 2 V 1 s 1 knematc vscosty of an electrolyte, m 2 s 1 rato of concentraton gradent of the H + ontothecu 2+ onatthe electrode surface flud densty of an electrolyte, kg m 3 0 flud densty of bulk electrolyte, kg m 3 electrc conductvty of an electrolyte, S m 1 electrc potental, V 0 reference value 1 Cu 2+ on 2 H + on 3 HSO on SO 2 on A CuSO B H 2 SO LS at the electrolyte free surface S at the electrode or lateral wall surfaces WS at the bottom wall surface

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