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Design of Electrode Profile In Electrochemical Manufacturing Process
Deslgn of Electrode Proflle In Electrochernlcal Manufacturlng Process D. Zhu’(21, K. Wang’, J. M. Yang’ Research Center for Nontradttlonal Machlnlng College o f Mechanlcal and Electrlcal Englneerlng 1
Nanjlng Unlverstty o f Aeronautlcs and Astronautlcs, Chlna
Abstract Accurate predldlon of electrode shapes 1s most Important In electrochemlcal manuladurlng process whlch Includes eladrochemlcal machlnlng and eledroformlng. In the former process, the electrode deslgn Is focused on the predlctlon of the cathode tool whlch slgnitlcantly mWeds the machlnlng accuraey. In the electrofomlng process, the anode electrode shape detemlnes the current denslty dlstrlbutlon and so the propertles of the deposlted metal. Thls paper proposes a finite element approach to accurately determlne these electrode profiles. The proposed method does not requlre Iterdlve redeslgn process, therefore provldes excellent convergence and emclent computlng. Close agreements between theoretlcal and experlmental resutts have been observed. Key words : Electrode, Eledr&chemlcal machlnlng. Eledrofomlng
1 INTRODUCTION Eledrochemlcal manuladurlng technologles based on Fa raday L m , Inc lud Ing sledrochem leal m a chl nln g (ECM) and electroformlng. plays an Important role In the manufacturlng Industry. E C M Is an eledrochemlcal dlssolutlon process that can remove eledrlcally condudlve materials regardless of thelr hardness and toughness. It produces smooth and burr free surfaces. In the electrofomlng, Inversely, a metal part I s produced by the deposttlon of metal Ions onto a preshaped cathode mould. Thls process has excellent repetttlve accuracy and hlgh fidelity of shape reproductlon from the mould. These advantages of ECM and eledrofomlng are of great Importance t o the manulacturlng of parts wlth complex shape and preclse dlmenslon. ECM and eledrofomlng are currently belng used In alrcrafl and aerospace, eledronlcs, automoblle, molds and dles Industries [I-31. Flgure l a Illustrates the b a s k prlnclple of the ECM process. A low voltage Is applled across the gap between a pre-shaped cathode tool and the anode workplece. The tool Is fed towards wrkplece at a constant rate to keep an Interelectrode gap. The anode materlal dlssolved eledrochemlcally accordlng t o Faradiifs law Is nushad away together wlth generated gas bubbles and Joule heat by noAng eladroiyta. The shape produced In the workplece approxlmately coples the tool shape. The eledrofomlng Illustrated In Flgure I b Is an Inverse process to ECM. The metal Ions deposit on the preshaped cathode mould when low voltage Is applled across the A d e r gap between the anode and the cathode mould. The deposttlon process contlnues untll the requlred thlckness Is reached. Then the deposited metal layer Is mechanlcally separated from the cathode mould. The Inner or outer shape of the electrofomed part Is an e x a d negative mlrror Image o f t h e cathode mould. The electrode deslgn Involves wlth geometry shapes of the cathode tool In ECM and the anode In eledrofomlng (cathode shape In eledrofomlng Is lust
same to the Inner or outer shape o f the part to be produced), respedlvely. Tool deslgn In ECM malnly deals wlth predldlng gap dlstrlbutlon for a glven workplace shape. Accurate deslgn of tool shape 1s declslve to machlnlng accuracy.
Eledrowe Produd
Anode wrkplece
(a) ECM Eledrowe
Cathode mould Produd
@) Eledrofomlng
Flgure 1: Prlnclples of ECM & Eledrofomlng Comparatlvely, the anode shape In eledrofomlng I s not always as Important as the cathode shape In ECM. The purpose of anode deslgn Is to unlfom the dlstrlbutlon of the current denslty on the surface of the cathode mould. A uniform current denstty dlstrlbutlon results In even metal deposttlon. In fad, the uniform metal dlstrlbutlon Is usually obtalned b y post machlnlng (mllllng or tumlng) M e r the eledrofomlng. Therefore the preclse anode deslgn Is not always necessary and slmple anode shapes are usually used, as shown In Flgure I b .
However the mlcrostructures and propertles o f deposited materlals have close relatlonshlp wlth the current denslty. The elTed o f current denslty on the crystal graln s h e has been reported [3]. In alloy eledrofomlng, the content of each element varles wtth the current density. Consequently the dlwerence o f current density leads to the varlatlon of the graln s h e and element content lnslde the deposited materlal and so adversely Mects the propertles of the electroformed produd. Therefore the accurate anode deslgn has to b e conducted to keep unKom current denslty dlstrlbutlon on the surface cathode mould In some Important appllcatlons of eledrofomlng. Mathematlcally. both of cathode deslgn In ECM and anode deslgn In eledrofomlng can b e consldered as an Inverse boundary problem of Laplace equatlon. The solutlon of problem requlres adjustment of some free boundary to satlsfy the Imposed boundary conditlons at electrode surfaces. An accurate analytical solutlon Is usually Imposslble. Emplrlcally driven redeslgns by trlal and error procedures are stlll Adespread In pradlce. The iterative toollng procedure results In a longer leadt h e and costly machlne domrtlme [I]. Researchers have attempted to find emclent and accurate method o f predldlon for deslgnlng E C M electrode slnce the lntroductlon of ECM Into Industry. Also research works have been done on the optlmhatlon of electrode shape or electrode Iocatlon to mlnlmhe the unevenness of the metal dlstrlbutlon In eledrofomlng. It Is posslble to develop more accurate approach due to rapld advancements In computer and numerlcal methods [4-91. Therefore, the present study prlmarlly addresses thls Issue based on the Laplace equatlon o f electrlc neld b y uslng numerlcal methods. A method on the basls o f Rnite element technology Is developed to predlct electrode pronles for both ECM and electroformlng process. 2 MODELLING Accordlng t o thaorlas of eledrlc Reid and eledrochemlstry. Laplace equatlon Is used to descrlbe eledrlc potentlal p dlstrlbutlon wtthln gap domaln n (see Flgure 2):
XR v2q
=
0
For eledrofomlng where the unKom current density dlstrlbutlon Is expected, the second boundary conditlon on cathode electrode Is:
% cons tan t m
-- -
on cathode
Therefore the model descrlblng the electrlc Rind In ECM and eledrofomlng Is In R
(3)
q = U (voltage)
on anode
(4)
q=o
on cathode
(5)
v2q
=
0
*=
Vr . c o s B
m % -= m
on anode only for ECM
K . 7 . W
cons tan t
on cathode only for eledrofomlng
(7)
Among above equatlons. equatlons (3>(5) are common to E C M and eledroformlng, but equatlon (6) Is only for ECM and (7) Is only for eledrofomlng. The problem dlscussed here dttYers from typlcal value boundary problems, whlch baslcally deal wlth the solutlon for the potentlal fundlon uslng only one boundary condltlon for each electrode boundary. In partlcular. the electrode geometry predldlon In electrochemlcal m a n d a d u r l n g process looks for a boundary geometry proRle Instead of potentlal dlstrlbutlon. It Is called Inverse boundary value problem or free boundary problems of Laplace equatlons [I, 91. 3
SOLUTION
Thls study develops a method to apply the two boundary conditlon successively on the anode In ECM or the cathode In eledrofomlng. It Is dlscussed below. Based on the Rnite element method and varlatlon prlnclple. the solutlon of equatlon (El) $9
*m
= -
0
wtthln R
vq.cos e
on anode for ECM
K.7.W
or
% cons tan t m
-=
on cathode for eledrofomlng
Is Identlcal to Rndlng a mlnlmlzes functlon G(9)
potentlal fundlon whlch
(9) rp= U
Flgure 2: Eledrlc potentlal dlstrlbutlon In E C M (left) eledrofomlng (rlght). Besldes the boundary conditlons shown In above Rgure. there are additlonal boundary conditlons for ECM and eledrofomlng, respectively. The addltlonal boundary condltlon for ECM can be derlved from ECM equlllbrlum state conditlon [9]:
where VI - tool feed rate, K - electrolyte condudivlty, B angle b e m e n the feedrate and the normal to anode, 7 - current emclency, w - eledrochemlcal equivalency o f anode metal.
The gap domaln R can be dklded Into a serles of small trlangular elements as shown In Flgure 3. n n-I
n+l
n+l
Flgure 3 : Meshlng and numberlng. A potentlal functlon q(x, y) M l c h varles Ilnearly lnslde each trlangular element Is defined as:
~ x . Y =) Nr .cpr + Ns .cp5 + Nt.cpt
(10)
where N , N, and N, are llnear functlons o f x and y , N ~ ( x , y ) = a a + b a . x + c a . y , h = r , s,t Substitutlon of (10) and (11) mlnlmbatlon of G(9) gives,
Into (9)
(11)
and then
has been found that the dtf7erence of current density causes slgnmcantly nowunKom dlstrlbutlon o f element content, leadlng to the poor materlal performance such as stress even crack. Therefore eWective methods must be developed to mlnlmlze the devlatlon of the current density dlstrlbutlon over the cathode.
P=fL
Length (mm)
where L Is the number of nodal polnts. Assembllng a11 elemental equatlons together (5 l y e r s a s an example) glves
k i i k12 0 k21 k22
k23
kzz 0 0
k43
0 0 0
km 0
0
0
0
0
kw 0 k a k45 ka k s
where 4 Is n x n matrix wHh nonzero determlnants Is column vector potentlal. assembled from N(x, y). SpeclRcally, and- are column vectors of potentlal at the electrode boundaries. pz to p4 are potentlals lnslde gap and bl Is a column vector generated by path lntegratlon In equatlon (9).
Furthermore, the matrlx equatlon (13) Is Identlcal to a recursion formula: cpkl = - ~ ~ ~ l ~ ~ I ~ l ~ c p c1 l> + 1 ~ l , ~ c p I(14) ~
Anode c ont a Ine r
insulation plates
Cathode m w l d
Flgure 4 : Electroformlng of nozzle. Accurate deslgn of the anode shape Is a practlcal method. Usually, the anode pronle can be obtalned In the way o f equakgap opfset. Although equakgap offset can reduce the dlllerence of the current denslty, the anaiysls b y uslng a nnlte element software (ANSYS) Indlcates that thls method stlll leads to a devlatlon o f up to 4096, shown In Flgure 5 In whlch the devlatlon of current denslty ISdenned as (Im - Idn)Amln where I m n : mwlmum current density and Imln: lowesl current density on the cathode. 3UI5 0 % 7
where k i l Is an Inverse matrlx o f Equatlons (14) and (15) glve the potentlal dlstrlbutlon. = U at the anode The second boundary conditlon, ( for ECM) Is applled just by lettlng q1 = U In (15). Then R can be obtalned by (15) and - t o by (14) layer by layer. Consequently an approprlate equakpotentlal c u m can be determlned from the known potentlal dlstrlbutlon. Accordlng to the electrochemlcal manufacturlng theory, the selected curye Is an electrode boundary. Additlonally, above dlscusslon Indlcates that the proposed method does not requlre Iterative process. Therefore, it provldes excellent convergence and computlng efTlclency. 4
RESULTS AND DISCUSSIONS
In order to v e r l v the proposed method, slmulatlons and experlments have been conducted. 4.1 Eleetrofomdng Flgure 4 shows the eledroformlng process of a nozzle used In aerospace Industry. A cathode mould whlch takes the shape of the Inner surface of the nozzle Is rotated lnslde an anode unlt. Electrolyte n o w between the cathode and the anode, both of whlch are In an electrolyte tank. The metal deposltlon usually takes days for the requlrement of several mllllmeters thlckness. A recent development of thls appllcatlon Is alloy eledroformlng, such as nlckekmanganese alloy eledroformlng. As dlscussed earller, the element content Is sensitlve to the varlatlon of current density. It
40%
> m 0
1
0
I
\
200
400
600
Length (mm)
Flgure 5: Devlatlon of current denslty along the length of the nozzle for equal-gap offset. Flgure 6 gives the computlng result of the anode by uslng the proposed method. Anode shape Is descrlbed In the gap varlatlon A t h the length of nozzle. In prlnclple, thls anode shape should resuit In Identlcal current denslty a11 over the cathode mould. Because of computatlon errors, ANSYS anaiysls glves the mwlmum dlllerence O f 4%. 13 2
- 13 0 128 Y
126 L?
124 12 2
0
1 m m m 4 m 5 m m Lmrn (mn)
Flgure 6: Gap varlatlon by the proposed numerlcal method.
In practlce. small places of anode materlal are used Instead of whole anode. They are put In an anode contalner made of Insoluble materlal, as shown In Flgure 4. The contalner takes the shape suggested b y the proposed numerical method. 4.2 ECM ECM experlments were conducted on an Industrlal scale system. The eledrolytlc cell Is shown In Flgure 7. The electrolyte of I50gA NaCI and the anode of low carbon steel were used durlng the vertflcatlon experlment. A semlclrcular tool of stalnless steel wlth a radlus of 20 mm was used. DC voltage of IN was applled. The gap dlstrlbutlon was measured b y uslng a preclslon tool mlcroscope aRer each machlnlng experlment.
Cathpde tool Cathode Electrolyte now Anode Workplece Flgure 7: Electrolyte cell and Interelectrode gap for ECM experlment.
Predlctlon accuracy Is wlthln acceptable range for most of Industrlal appllcatlons. Average relatke devlatlon Is less than 4%. In ECM, the local gap Is a fundlon of not only local current denslty but also electrolyte conductklty and many other parameters. All of these parameters and thelr Interactlons vary durlng the process, leadlng to a complex non-uniform gap dlstrlbutlon. The proposed method provlders the basls for the development of more comprehenske approach. 5 SUMMARY A method for estlmatlng electrode shapes In eledrochemlcal manfladurlng processes has been proposed. The method Is appllcable to both electroformlng and ECM process. It has hlgh computlng efTlclency and good accuracy. In ECM experlment, an average devlatlon of less than 4 % between experlmental and theoretlcal resutts has been observed, well wlthln the accuracy requlrement for most Industrlal appllcatlons. The proposed method makes ECM tool deslgn procedure relatlvely Inexpensive wlth a shorter lead t h e . It also makes contrlbutlon to the Improvement of the quallty of the electroformed products. 6 ACKNOWLEDGMENTS Authors acknowledge Rnanclal support from the Chlna Natural Sclence Foundatlon (50075040). Authors are thankful t o Prof. K. P . Rajurkar (Unlverstty of Nebriaski+ Llncoln, USA) for hls comments and suggestlons.
7 REFERENCES
I 30
40
50
60
70
80
e (degree)
(a) Frontal gap = 0.41mm, voltage = 17V, electrolyte = I50gA NaCI, anode materlal = low carbon steel
30
40
59
60
70
I 80
0 WY=I @) Frontal gap = 0.36mm, voltage = 17V, electrolyte = I50gA NaCI, anode materlal = low carbon steel
Flgure 8 : Theoretlcal and experlmental results. The experimental and theoretlcal resutts are compared In the way of Intereladrode gap versus angle B In Flgure 8. Flgure 8a and Flgure 8b show the resutts for front gaps (gap at B = O ) of 0 . 4 l m m and 0.36mm, respedlvely. More than 12 machlnlng experlments have been conducted to verify the computed resutts. All of these experlmental resutts show close agreements between the theoretlcal and experlmental resutts.
[I] Rajukar K. P., Zhu D., McGeough J. A , , 1999, New Developments of Electrochemleal Machlnlng, Keynote paper, Annals of the CIRP, 4812567-569. [2] McGeough, J. A., Leu, M., Ralurkar, K. P., De Slka, A,, Llu. Q.. 2001, Electroformlng Process and Appllcatlon to MlcrolMacro Manflacturlng, Annals of the CIRP, Keynote paper. 50n:49S514. [3] Zhu, D., Lel, W. N., Qu, N. S.,Xu, H. Y., 2002, Nanocrystalllne Electroformlng Process, Annals of the CIRP, 5111: 17S176. [4] Rajurkar K. P., We1 8. and Kozak J., 1995, Modellng and Monttorlng Interelectrode Gap In Pulse Electrochemleal Machlnlng, Annals of the CIRP,44/1: 177-180. [5] Hardlsty H., Mlleham A. R., and Shlrvarnl, H., 1993, A Flnlte Element Slmulatlon of the Electrochemlcal Machlnlng Process, Annals of the CIRP,42/1: 201 -204. [6] Zhou Y., and Derby J. J., 1995, The Cathode Deslgn Problem In Electrochemleal Machlnlng, Chemlcal Englneerlng Sclence, Vol. 50, No. 17: 2679-2689. Kozak J., 1998, Mdhematlcal Models for Computer Slmulatlon of Electrochemleal Machlnlng Processes, Journal of Materials Processlng Technology, Vol. 76: 17D175. [El Masuku, E. S., Mlleham, A . R., H.Hardlsty, Bramley. A.N., Johal, C., Detassls, P., 2002, A nnlte element slmulatlon of the electroplatlng process, Annals of CIRP, 51/1:16S172. [9] Zhu. D., Rajurkar, K . P., 1999, Modellng and Verlflcatlon of Intereladrode Gap In ECM wlth Passkatlng Electrolyte IMECE-99, ASME, MED V 10: 589-596.
m
Nanjlng Unlverstty o f Aeronautlcs and Astronautlcs, Chlna
Abstract Accurate predldlon of electrode shapes 1s most Important In electrochemlcal manuladurlng process whlch Includes eladrochemlcal machlnlng and eledroformlng. In the former process, the electrode deslgn Is focused on the predlctlon of the cathode tool whlch slgnitlcantly mWeds the machlnlng accuraey. In the electrofomlng process, the anode electrode shape detemlnes the current denslty dlstrlbutlon and so the propertles of the deposlted metal. Thls paper proposes a finite element approach to accurately determlne these electrode profiles. The proposed method does not requlre Iterdlve redeslgn process, therefore provldes excellent convergence and emclent computlng. Close agreements between theoretlcal and experlmental resutts have been observed. Key words : Electrode, Eledr&chemlcal machlnlng. Eledrofomlng
1 INTRODUCTION Eledrochemlcal manuladurlng technologles based on Fa raday L m , Inc lud Ing sledrochem leal m a chl nln g (ECM) and electroformlng. plays an Important role In the manufacturlng Industry. E C M Is an eledrochemlcal dlssolutlon process that can remove eledrlcally condudlve materials regardless of thelr hardness and toughness. It produces smooth and burr free surfaces. In the electrofomlng, Inversely, a metal part I s produced by the deposttlon of metal Ions onto a preshaped cathode mould. Thls process has excellent repetttlve accuracy and hlgh fidelity of shape reproductlon from the mould. These advantages of ECM and eledrofomlng are of great Importance t o the manulacturlng of parts wlth complex shape and preclse dlmenslon. ECM and eledrofomlng are currently belng used In alrcrafl and aerospace, eledronlcs, automoblle, molds and dles Industries [I-31. Flgure l a Illustrates the b a s k prlnclple of the ECM process. A low voltage Is applled across the gap between a pre-shaped cathode tool and the anode workplece. The tool Is fed towards wrkplece at a constant rate to keep an Interelectrode gap. The anode materlal dlssolved eledrochemlcally accordlng t o Faradiifs law Is nushad away together wlth generated gas bubbles and Joule heat by noAng eladroiyta. The shape produced In the workplece approxlmately coples the tool shape. The eledrofomlng Illustrated In Flgure I b Is an Inverse process to ECM. The metal Ions deposit on the preshaped cathode mould when low voltage Is applled across the A d e r gap between the anode and the cathode mould. The deposttlon process contlnues untll the requlred thlckness Is reached. Then the deposited metal layer Is mechanlcally separated from the cathode mould. The Inner or outer shape of the electrofomed part Is an e x a d negative mlrror Image o f t h e cathode mould. The electrode deslgn Involves wlth geometry shapes of the cathode tool In ECM and the anode In eledrofomlng (cathode shape In eledrofomlng Is lust
same to the Inner or outer shape o f the part to be produced), respedlvely. Tool deslgn In ECM malnly deals wlth predldlng gap dlstrlbutlon for a glven workplace shape. Accurate deslgn of tool shape 1s declslve to machlnlng accuracy.
Eledrowe Produd
Anode wrkplece
(a) ECM Eledrowe
Cathode mould Produd
@) Eledrofomlng
Flgure 1: Prlnclples of ECM & Eledrofomlng Comparatlvely, the anode shape In eledrofomlng I s not always as Important as the cathode shape In ECM. The purpose of anode deslgn Is to unlfom the dlstrlbutlon of the current denslty on the surface of the cathode mould. A uniform current denstty dlstrlbutlon results In even metal deposttlon. In fad, the uniform metal dlstrlbutlon Is usually obtalned b y post machlnlng (mllllng or tumlng) M e r the eledrofomlng. Therefore the preclse anode deslgn Is not always necessary and slmple anode shapes are usually used, as shown In Flgure I b .
However the mlcrostructures and propertles o f deposited materlals have close relatlonshlp wlth the current denslty. The elTed o f current denslty on the crystal graln s h e has been reported [3]. In alloy eledrofomlng, the content of each element varles wtth the current density. Consequently the dlwerence o f current density leads to the varlatlon of the graln s h e and element content lnslde the deposited materlal and so adversely Mects the propertles of the electroformed produd. Therefore the accurate anode deslgn has to b e conducted to keep unKom current denslty dlstrlbutlon on the surface cathode mould In some Important appllcatlons of eledrofomlng. Mathematlcally. both of cathode deslgn In ECM and anode deslgn In eledrofomlng can b e consldered as an Inverse boundary problem of Laplace equatlon. The solutlon of problem requlres adjustment of some free boundary to satlsfy the Imposed boundary conditlons at electrode surfaces. An accurate analytical solutlon Is usually Imposslble. Emplrlcally driven redeslgns by trlal and error procedures are stlll Adespread In pradlce. The iterative toollng procedure results In a longer leadt h e and costly machlne domrtlme [I]. Researchers have attempted to find emclent and accurate method o f predldlon for deslgnlng E C M electrode slnce the lntroductlon of ECM Into Industry. Also research works have been done on the optlmhatlon of electrode shape or electrode Iocatlon to mlnlmhe the unevenness of the metal dlstrlbutlon In eledrofomlng. It Is posslble to develop more accurate approach due to rapld advancements In computer and numerlcal methods [4-91. Therefore, the present study prlmarlly addresses thls Issue based on the Laplace equatlon o f electrlc neld b y uslng numerlcal methods. A method on the basls o f Rnite element technology Is developed to predlct electrode pronles for both ECM and electroformlng process. 2 MODELLING Accordlng t o thaorlas of eledrlc Reid and eledrochemlstry. Laplace equatlon Is used to descrlbe eledrlc potentlal p dlstrlbutlon wtthln gap domaln n (see Flgure 2):
XR v2q
=
0
For eledrofomlng where the unKom current density dlstrlbutlon Is expected, the second boundary conditlon on cathode electrode Is:
% cons tan t m
-- -
on cathode
Therefore the model descrlblng the electrlc Rind In ECM and eledrofomlng Is In R
(3)
q = U (voltage)
on anode
(4)
q=o
on cathode
(5)
v2q
=
0
*=
Vr . c o s B
m % -= m
on anode only for ECM
K . 7 . W
cons tan t
on cathode only for eledrofomlng
(7)
Among above equatlons. equatlons (3>(5) are common to E C M and eledroformlng, but equatlon (6) Is only for ECM and (7) Is only for eledrofomlng. The problem dlscussed here dttYers from typlcal value boundary problems, whlch baslcally deal wlth the solutlon for the potentlal fundlon uslng only one boundary condltlon for each electrode boundary. In partlcular. the electrode geometry predldlon In electrochemlcal m a n d a d u r l n g process looks for a boundary geometry proRle Instead of potentlal dlstrlbutlon. It Is called Inverse boundary value problem or free boundary problems of Laplace equatlons [I, 91. 3
SOLUTION
Thls study develops a method to apply the two boundary conditlon successively on the anode In ECM or the cathode In eledrofomlng. It Is dlscussed below. Based on the Rnite element method and varlatlon prlnclple. the solutlon of equatlon (El) $9
*m
= -
0
wtthln R
vq.cos e
on anode for ECM
K.7.W
or
% cons tan t m
-=
on cathode for eledrofomlng
Is Identlcal to Rndlng a mlnlmlzes functlon G(9)
potentlal fundlon whlch
(9) rp= U
Flgure 2: Eledrlc potentlal dlstrlbutlon In E C M (left) eledrofomlng (rlght). Besldes the boundary conditlons shown In above Rgure. there are additlonal boundary conditlons for ECM and eledrofomlng, respectively. The addltlonal boundary condltlon for ECM can be derlved from ECM equlllbrlum state conditlon [9]:
where VI - tool feed rate, K - electrolyte condudivlty, B angle b e m e n the feedrate and the normal to anode, 7 - current emclency, w - eledrochemlcal equivalency o f anode metal.
The gap domaln R can be dklded Into a serles of small trlangular elements as shown In Flgure 3. n n-I
n+l
n+l
Flgure 3 : Meshlng and numberlng. A potentlal functlon q(x, y) M l c h varles Ilnearly lnslde each trlangular element Is defined as:
~ x . Y =) Nr .cpr + Ns .cp5 + Nt.cpt
(10)
where N , N, and N, are llnear functlons o f x and y , N ~ ( x , y ) = a a + b a . x + c a . y , h = r , s,t Substitutlon of (10) and (11) mlnlmbatlon of G(9) gives,
Into (9)
(11)
and then
has been found that the dtf7erence of current density causes slgnmcantly nowunKom dlstrlbutlon o f element content, leadlng to the poor materlal performance such as stress even crack. Therefore eWective methods must be developed to mlnlmlze the devlatlon of the current density dlstrlbutlon over the cathode.
P=fL
Length (mm)
where L Is the number of nodal polnts. Assembllng a11 elemental equatlons together (5 l y e r s a s an example) glves
k i i k12 0 k21 k22
k23
kzz 0 0
k43
0 0 0
km 0
0
0
0
0
kw 0 k a k45 ka k s
where 4 Is n x n matrix wHh nonzero determlnants Is column vector potentlal. assembled from N(x, y). SpeclRcally, and- are column vectors of potentlal at the electrode boundaries. pz to p4 are potentlals lnslde gap and bl Is a column vector generated by path lntegratlon In equatlon (9).
Furthermore, the matrlx equatlon (13) Is Identlcal to a recursion formula: cpkl = - ~ ~ ~ l ~ ~ I ~ l ~ c p c1 l> + 1 ~ l , ~ c p I(14) ~
Anode c ont a Ine r
insulation plates
Cathode m w l d
Flgure 4 : Electroformlng of nozzle. Accurate deslgn of the anode shape Is a practlcal method. Usually, the anode pronle can be obtalned In the way o f equakgap opfset. Although equakgap offset can reduce the dlllerence of the current denslty, the anaiysls b y uslng a nnlte element software (ANSYS) Indlcates that thls method stlll leads to a devlatlon o f up to 4096, shown In Flgure 5 In whlch the devlatlon of current denslty ISdenned as (Im - Idn)Amln where I m n : mwlmum current density and Imln: lowesl current density on the cathode. 3UI5 0 % 7
where k i l Is an Inverse matrlx o f Equatlons (14) and (15) glve the potentlal dlstrlbutlon. = U at the anode The second boundary conditlon, ( for ECM) Is applled just by lettlng q1 = U In (15). Then R can be obtalned by (15) and - t o by (14) layer by layer. Consequently an approprlate equakpotentlal c u m can be determlned from the known potentlal dlstrlbutlon. Accordlng to the electrochemlcal manufacturlng theory, the selected curye Is an electrode boundary. Additlonally, above dlscusslon Indlcates that the proposed method does not requlre Iterative process. Therefore, it provldes excellent convergence and computlng efTlclency. 4
RESULTS AND DISCUSSIONS
In order to v e r l v the proposed method, slmulatlons and experlments have been conducted. 4.1 Eleetrofomdng Flgure 4 shows the eledroformlng process of a nozzle used In aerospace Industry. A cathode mould whlch takes the shape of the Inner surface of the nozzle Is rotated lnslde an anode unlt. Electrolyte n o w between the cathode and the anode, both of whlch are In an electrolyte tank. The metal deposltlon usually takes days for the requlrement of several mllllmeters thlckness. A recent development of thls appllcatlon Is alloy eledroformlng, such as nlckekmanganese alloy eledroformlng. As dlscussed earller, the element content Is sensitlve to the varlatlon of current density. It
40%
> m 0
1
0
I
\
200
400
600
Length (mm)
Flgure 5: Devlatlon of current denslty along the length of the nozzle for equal-gap offset. Flgure 6 gives the computlng result of the anode by uslng the proposed method. Anode shape Is descrlbed In the gap varlatlon A t h the length of nozzle. In prlnclple, thls anode shape should resuit In Identlcal current denslty a11 over the cathode mould. Because of computatlon errors, ANSYS anaiysls glves the mwlmum dlllerence O f 4%. 13 2
- 13 0 128 Y
126 L?
124 12 2
0
1 m m m 4 m 5 m m Lmrn (mn)
Flgure 6: Gap varlatlon by the proposed numerlcal method.
In practlce. small places of anode materlal are used Instead of whole anode. They are put In an anode contalner made of Insoluble materlal, as shown In Flgure 4. The contalner takes the shape suggested b y the proposed numerical method. 4.2 ECM ECM experlments were conducted on an Industrlal scale system. The eledrolytlc cell Is shown In Flgure 7. The electrolyte of I50gA NaCI and the anode of low carbon steel were used durlng the vertflcatlon experlment. A semlclrcular tool of stalnless steel wlth a radlus of 20 mm was used. DC voltage of IN was applled. The gap dlstrlbutlon was measured b y uslng a preclslon tool mlcroscope aRer each machlnlng experlment.
Cathpde tool Cathode Electrolyte now Anode Workplece Flgure 7: Electrolyte cell and Interelectrode gap for ECM experlment.
Predlctlon accuracy Is wlthln acceptable range for most of Industrlal appllcatlons. Average relatke devlatlon Is less than 4%. In ECM, the local gap Is a fundlon of not only local current denslty but also electrolyte conductklty and many other parameters. All of these parameters and thelr Interactlons vary durlng the process, leadlng to a complex non-uniform gap dlstrlbutlon. The proposed method provlders the basls for the development of more comprehenske approach. 5 SUMMARY A method for estlmatlng electrode shapes In eledrochemlcal manfladurlng processes has been proposed. The method Is appllcable to both electroformlng and ECM process. It has hlgh computlng efTlclency and good accuracy. In ECM experlment, an average devlatlon of less than 4 % between experlmental and theoretlcal resutts has been observed, well wlthln the accuracy requlrement for most Industrlal appllcatlons. The proposed method makes ECM tool deslgn procedure relatlvely Inexpensive wlth a shorter lead t h e . It also makes contrlbutlon to the Improvement of the quallty of the electroformed products. 6 ACKNOWLEDGMENTS Authors acknowledge Rnanclal support from the Chlna Natural Sclence Foundatlon (50075040). Authors are thankful t o Prof. K. P . Rajurkar (Unlverstty of Nebriaski+ Llncoln, USA) for hls comments and suggestlons.
7 REFERENCES
I 30
40
50
60
70
80
e (degree)
(a) Frontal gap = 0.41mm, voltage = 17V, electrolyte = I50gA NaCI, anode materlal = low carbon steel
30
40
59
60
70
I 80
0 WY=I @) Frontal gap = 0.36mm, voltage = 17V, electrolyte = I50gA NaCI, anode materlal = low carbon steel
Flgure 8 : Theoretlcal and experlmental results. The experimental and theoretlcal resutts are compared In the way of Intereladrode gap versus angle B In Flgure 8. Flgure 8a and Flgure 8b show the resutts for front gaps (gap at B = O ) of 0 . 4 l m m and 0.36mm, respedlvely. More than 12 machlnlng experlments have been conducted to verify the computed resutts. All of these experlmental resutts show close agreements between the theoretlcal and experlmental resutts.
[I] Rajukar K. P., Zhu D., McGeough J. A , , 1999, New Developments of Electrochemleal Machlnlng, Keynote paper, Annals of the CIRP, 4812567-569. [2] McGeough, J. A., Leu, M., Ralurkar, K. P., De Slka, A,, Llu. Q.. 2001, Electroformlng Process and Appllcatlon to MlcrolMacro Manflacturlng, Annals of the CIRP, Keynote paper. 50n:49S514. [3] Zhu, D., Lel, W. N., Qu, N. S.,Xu, H. Y., 2002, Nanocrystalllne Electroformlng Process, Annals of the CIRP, 5111: 17S176. [4] Rajurkar K. P., We1 8. and Kozak J., 1995, Modellng and Monttorlng Interelectrode Gap In Pulse Electrochemleal Machlnlng, Annals of the CIRP,44/1: 177-180. [5] Hardlsty H., Mlleham A. R., and Shlrvarnl, H., 1993, A Flnlte Element Slmulatlon of the Electrochemlcal Machlnlng Process, Annals of the CIRP,42/1: 201 -204. [6] Zhou Y., and Derby J. J., 1995, The Cathode Deslgn Problem In Electrochemleal Machlnlng, Chemlcal Englneerlng Sclence, Vol. 50, No. 17: 2679-2689. Kozak J., 1998, Mdhematlcal Models for Computer Slmulatlon of Electrochemleal Machlnlng Processes, Journal of Materials Processlng Technology, Vol. 76: 17D175. [El Masuku, E. S., Mlleham, A . R., H.Hardlsty, Bramley. A.N., Johal, C., Detassls, P., 2002, A nnlte element slmulatlon of the electroplatlng process, Annals of CIRP, 51/1:16S172. [9] Zhu. D., Rajurkar, K . P., 1999, Modellng and Verlflcatlon of Intereladrode Gap In ECM wlth Passkatlng Electrolyte IMECE-99, ASME, MED V 10: 589-596.
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