Modeling of Wire Electrochemical Machining

Transcription

1 A publiation of 91 CHEMICAL ENGINEERING TRANSACTIONS VOL. 41, 214 Guest Editors: Simonetta Palmas, Mihele Masia, Annalisa Vaa Copyright 214, AIDIC Servizi S.r.l., ISBN ; ISSN The Italian Assoiation of Chemial Engineering DOI: 1.333/CET Modeling of Wire Eletrohemial Mahining Vladimir M. Volgin a,b, Van Dong Do a, Alexey D. Davydov b a Tula State University, pr. Lenina 92, Tula 312, Russia b Frumkin Institute of Physial Chemistry and Eletrohemistry, Russian Aademy of Sienes, Leninskii pr. 31, Mosow 11971, Russia volgin@tsu.tula.ru The eletrohemial mahining with a wire tool eletrode was studied theoretially. The Laplae equation for the eletri field potential and the equation of workpiee surfae evolution were used as the mathematial model of the proess. A sheme of omputer simulation of mahining was developed. The sheme involved the numerial solution of the boundary integral equation, whih is a onsequene of the Laplae equation, by using the method of boundary elements; the determination of a new position of workpiee surfae with regard for possible topologial hanges; and the motion of wire tool eletrode along a presribed path. The mahining of typial features (straight slits, slits with orners, openings with square and triangular ross-setions) is analyzed. It is shown that this sheme of simulation an be used for various mahining regimes, inluding the ases of the topologial hanges of the workpiee surfae. The results of simulation agree well with the literature data on the wire eletrohemial mahining. As a result of simulation, the dependenes of the front and side intereletrode gaps on the mahining parameters were obtained for various shemes of mahining. They an be used for determining the path of wire tool eletrode in order to obtain the presribed shape and sizes of workpiee surfae. 1. Introdution For utting the omplex-shaped parts made of diffiult-to work materials, various non-onventional methods of mahining (laser, eletro-erosion, eletrohemial mahining) are widely used (Rajurkar et al., 26). In the ases of laser and eletro-erosion utting, a zone of thermal effet forms on the workpiee surfae; in the eletrohemial mahining (ECM), suh zone is not observed (Davydov et al., 24). Though the shemes of wire eletrohemial utting have been known rather long (Metzger, 1958), insuffiient auray of the mahining limited its appliation (Proklova, 1976). However, in reent years, the wire eletrohemial utting has attrated inreasing interest, espeially for treatment of miroworkpiees (Chung et al., 211). This is aused by several reasons: the absene of onsiderable mehanial ation on the wire tool-eletrode and the workpiee (Lee et al., 211) (as a result, low-stiff workpiees an be treated to a high auray); the absene of thermal ation of the workpiee and tool (Speiser and Ivanov, 213); the absene of tool-eletrode wear (Rajurkar et al., 213); the use of ultrashort voltage pulses (of several nanoseonds) and ultra-small intereletrode gaps (of the order of several mirometers) (Shin et al., 28); the use of axial eletrolyte flushing (Qu et al., 213); the use the small amplitude vibration of tool-eletrode (Wang et al., 211), et. By now, the regularities of eletrohemial shaping for the omplex-shaped workpiees have not been adequately investigated, beause the majority of the works, whih are devoted to the wire eletrohemial utting are experimental. In the ase of linear motion of wire eletrode with a onstant rate, the parameters of utting an be determined by using the known analytial solutions (Zhitnikov et al., 24); in more omplex ases, the numerial methods should be used (Hinduja et al., 213). In this ase, the boundary element method (Volgin and Davydov, 24) is more suitable than the finite element method (Rodrigues et al., 211), beause, it is easier to perform the remeshing of the omputational domain. The aim of this work is to develop the methods of numerial simulation of wire eletrohemial mahining with regard for possible topologial hanges in the workpiee surfae and to study various shemes of Please ite this artile as: Volgin V., Do D., Davydov A., 214, Modeling of wire eletrohemial mahining, Chemial Engineering Transations, 41, DOI: 1.333/CET144116

2 92 formation of typial features (slits and openings of various shapes). 2. Statement of problem, basi equations The model of the wire eletrohemial mahining (Figure 1) involves the Laplae equation for the eletri field potential, equation of workpiee surfae evolution, and equation of the trajetory of wire tooleletrode. For onveniene, the mathematial model is written in the dimensionless form: ( grad ) div = dx a Φ = A X Φ (1) dya Φ = A Y, (2) dx dy = V ( τ ) os[ α ( τ )], = V ( τ ) sin[ α ( τ )] (3) where Φ is the dimensionless potential; X, Y are the dimensionless oordinates; τ is the dimensionless time; V, α are the dimensionless feed rate of wire tool-eletrode and the angle between the diretion of feed rate and the absissa axis, respetively; A is dimensionless parameter, whih haraterizes the mahining onditions; a is the subsript, whih haraterizes the point on the workpiee surfae; с is the subsript, whih haraterizes the loation of the enter of the wire tool-eletrode. Figure 1: Sheme of wire eletrohemial mahining: (a) before mahining, (b) and () during mahining; (1) workpiee, (2) and (3) wire tool-eletrode before and during mahining, (4) trajetory of wire tooleletrode and (5) intereletrode gap filled with eletrolyte solution; S ss is side intereletrode gap, W is utting width When passing to the dimensionless variables, the diameter of wire tool-eletrode ( ) was taken as a unit length, the applied voltage (U) was taken as a unit eletri potential, and the harateristi wire eletrode feed rate ( v ) was taken as the unit rate: d X x y ϕ d v v =, Y =, Φ =, I = i, V =, τ = d d U χu v d t (4) where x, y are the dimensional oordinates; t is the time; ϕ is the potential in the solution; and χ is the ondutivity of eletrolyte solution. The mathematial model (Eq. 1-3) ignores the variations in the solution onentration (Davydov et al., 24). This is aeptable in the ases of suffiiently intense pumping or stirring of eletrolyte solution. Eq. 2 involves the dimensionless parameter A, whih haraterizes the mahining onditions. It an be alulated by the following equation:

3 93 ηε U A V χ = d (5) v where η is the urrent effiieny and ε V is the volumetri eletrohemial equivalent. To solve system of Eqs. 1 3, the boundary and initial onditions should be presribed. For the sheme of wire eletrohemial mahining (Figure 1), in the ase that the polarization of eletrodes an be ignored, the boundary onditions for the dimensionless potential should be as follows: Φ = 1, on the anode, on the athode (workpiee ) (wire tool - eletrode) The initial onditions are presribed under the assumptions that, at the initial instant of time, the enter of wire tool-eletrode is loated at the origin of the oordinates, and the workpiee surfae and the tooleletrode surfae are separated by a dimensionless initial intereletrode gap S (Figure 1 a). Similarly to the eletrohemial mahining with a plane tool-eletrode, whih moves towards the workpiee with a onstant rate, in the ase under onsideration, a steady-state frontal intereletrode gap an be introdued: s ηε V χu v = (7) This enables us to present parameter A in the following form: s A = = S (8) d i.e. parameter A is equal to the dimensionless intereletrode gap for the plane tool-eletrode, all other onditions of mahining being the same. In the wire eletrohemial mahining, the steady-state frontal intereletrode gap ( S ) will be smaller than in the mahining with a plane eletrode ( S ). The relation between these frontal gaps is as follows: S K = (9) S where K is the oeffiient that monotonially dereases and approahes unity with dereasing steady-state intereletrode gap for the plane tool-eletrode. When passing to the dimensional variables, the variation of oeffiient K from unity an be taken into aount by omparing the presribed and alulated values of steady-state frontal intereletrode gap. Using the dimensionless steady-state frontal intereletrode gap, equations (9) an be written as follows: dx a = S Φ, X dya = S Φ Y The boundary value problem for equations (1) (3), (6) is a problem with moving boundary. Then, the equations, whih desribe the transport proesses and the motion of omputational region boundary, should be alulated simultaneously. 3. Method of numerial solution The numerial solution is frequently simplified by using the quasi-steady state approximation. Within this approximation, the entire time of mahining is divided into a number of time steps. For eah step: 1) first, the distribution of eletri potential is alulated (at the eletrode geometry orresponding to the beginning of the step); 2) then, a new shape of the workpiee surfae is determined (at the distribution of urrent density orresponding to the beginning of the step); (6) (1)

4 94 3) then, a new position of the enter of the wire tool-eletrode is determined. At eah time step, the boundary-value problem for Eq. 1 with the boundary onditions in Eq. 6 was solved numerially by the method of boundary elements. The system of differene equations was solved numerially by using the diret and iteration methods. As a result, the distribution of the urrent density over the workpiee surfae was determined. The equation of workpiee surfae evolution (2) was solved numerially using the Level Set method, whih provides the aount for possible topologial hanges of the workpiee surfae. To maintain the ompromise between the auray and amount of omputation, the remeshing of the workpiee surfae was realized after eah time step. 4. Results of modelling and disussion The following values of parameters were taken for the modelling: the dimensionless steady-state intereletrode gap for plane eletrode ( S ) was taken to be.5.5; the dimensionless time step was hosen so that the numerial solution was stable and aurate (ommonly, it was.2.2). From 2 to 1 boundary elements were presribed on the initial surfaes of workpiee and wire tool-eletrode. In the ourse of simulation, a distane between the nodes of boundary element mesh varied: it inreased on the onvex regions of workpiee surfae and dereased on the onave regions. To provide an adequate auray of numerial solution and redue the omputational ost, the boundary element mesh was adapted in the ourse of simulation. The remeshing was realized by the following simple, but suffiiently effetive method. When the length of the boundary element inreased by more than 1.5 times (as ompared with its initial length), the element was divided into two elements of the same length. When the length of boundary element dereased by more than 2 times (as ompared with its initial length), the element was exluded; in this ase, two boundary elements were replaed by one element. At the first stage, the modelling of the wire eletrohemial mahining with a onstant feed rate for the straight slits was performed. As a result, the utting width and the side and frontal intereletrode gaps were determined. The alulated results agree well with the literature data (Zhitnikov et al., 24). In partiular, the frontal gap is by approximately 1 % smaller than that for a plane eletrode. Then, the mahining of slits with orners was simulated for two values of steady-state intereletrode gap ( S =.1,.5). From the alulated results, it follows that the length of transient zone, where the utting geometry differs from that for the straight-line motion of wire tool-eletrode, depends on the steady-state intereletrode gap. With inreasing gap, the length of transient zone inreases, and the utting width near the orner inreases proportionally to the orner value. Then, the effet of the mahining onditions and the tool-eletrode path on the auray of mahining of the openings with square (Figure 2) and triangle (Figure 3) ross-setions was studied. The initial (round) and mahined openings are shown with bold lines. The tool-eletrode trajetory is shown dashed. The tool eletrode just before mahining is shown as gray irle. Two shemes of shaping were onsidered. In the first sheme, prior to the mahining, the tool-eletrode is loated in the enter of the opening; then, it moves normally to the orresponding (square or triangle) ontour. When the enter of tool-eletrode reahes the ontour, the eletrode starts to move along the ontour lokwise and, then, returns to the starting point (Figures 2a, 2, 3a, 3). In the seond sheme, prior to the mahining, the wire tool-eletrode is loated in a orner of the ontour and, then, moves along the ontour with a onstant rate up to returning to the starting point (Figures 2b, 2d, 3b, 3d). In ontrast to the formation of straight and orner slits, in the mahining of more omplex elements, suh as openings with square and triangle ross-setions, the topologial hanges of workpiee surfae an take plae (Figures 2 and 3). In the above ases of mahining of openings, the hanges an be aused by the fat that not total amount of material of the workpiee inside the opening is anodially dissolved in the ourse of mahining. It was assumed that, when a fration of material, whih is loated in the entral part of the opening, is separated from the workpiee due to self-intersetion of workpiee surfae, it is removed from the mahining zone. In addition, sharp edges an form on the workpiee surfae in the ourse of mahining. Within the aepted parametri desription of workpiee surfae geometry, the disruption of workpiee surfae smoothness an lead to the formation of intersetions. These intersetions have no physial meaning and should be eliminated in the simulation. The intersetions of boundary elements, whih desribe the workpiee surfae, were eliminated by using the following algorithm: - the intersetion point of the lines, whih pass through the boundary elements, was alulated for all possible pairs of boundary elements; - the position of the intersetion point was determined and the ase that the intersetion point is loated

5 inside both boundary elements, i.e. the intersetion of ontour takes plae, was reognized; - in the presene of ontour intersetion, three groups of boundary elements were formed: the elements before the first interseting element; the elements between the interseting elements; the elements after the seond interseting element; - the groups of boundary elements, whih really desribe the workpiee surfae geometry, were determined; other groups of boundary elements were exluded from further alulations; 95 Figure 2: The results of simulation of wire eletrohemial mahining of square opening at (a, b).1, (, d) S =.5 for different tool-eletrode trajetories S = Figure 3: The results of simulation of wire eletrohemial mahining of triangular opening at (a, b).1, (, d) S =.5 for different tool-eletrode trajetories S =

6 96 The results of modelling show that the proposed sheme of numerial simulation of wire eletrohemial mahining is suffiiently effetive and an be used to predit the dimensions and shape of workpiee surfae and to improve the trajetory of tool-eletrode. 5. Conlusions A sheme of numerial simulation of eletrohemial mahining with a wire tool-eletrode, whih enables one to predit the shape and dimensions of workpiee surfae, is developed. The mahining of typial elements (the straight and orner slits, the openings with square and triangle ross-setions) is analyzed. It is shown that this method of simulation an be applied to various shemes of mahining, inluding the ases of topologial hanges of workpiee surfae. The results of simulation agree well with the literature data. In the subsequent study, the apabilities of modelling will be extended by taking into aount the dependene of urrent effiieny on the urrent density, and by taking into aount the eletrode polarization. In addition, the pulsed regimes of mahining will be onsidered and the effets of hargedisharge of eletrial double layer will be taken into aount. Aknowledgements This work was supported by the Russian Foundation for Basi Researh, projet no and the Ministry of Eduation and Siene of the Russian Federation, projet no. 196 of the Basi Part of the State Program. Referenes Chung D.K., Shin H.S., Park M.S., Kim B.H., Chu C.N., 211, Reent researhes in miro eletrial mahining, International Journal of Preision Engineering and Manufaturing 12(2), Davydov A.D., Volgin V.M., Lyubimov V.V., 24, Eletrohemial mahining of metals: fundamentals of eletrohemial shaping, Russian Journal of Eletrohemistry 4, Hinduja S., Kunieda M., 213, Modelling of ECM and EDM proesses, CIRP Annals - Manufaturing Tehnology 62, Lee Ch.-L., Kanda Y., Ikeda Sh., Matsumura M., 211, Eletrohemial method for sliing Si bloks into wafers using platinum wire eletrodes, Solar Energy Materials & Solar Cells 95, Metzger M., 1958, Eletrolyti saw, Review of Sientifi Instruments 29, Proklova V.D., 1976, Eletrohemial mahining with non-profiled tool-eletrode, (in Russian), Mashinostroenie, Mosow, Russia. Qu N., Fang X., Li W., Zeng Y., Zhu D., 213, Wire eletrohemial mahining with axial eletrolyte flushing for titanium alloy, Chinese Journal of Aeronautis 26(1), Rajurkar K.P., Levy G., Malshe A., Sundaram M.M., MGeough J., Hu X., Resnik R., DeSilva A., 26, Miro and nano mahining by eletro-physial and hemial proesses, CIRP Annals-Manufaturing Tehnology 55(2), Rajurkar K.P., Sundaram M.M., Malshe A.P., 213, Review of eletrohemial and eletrodisharge mahining, Proedia CIRP 6, Rodrigues J., Coimbra M.D.C., Rodrigues A., 211, Moving finite elements method for investigating Stefan problems, Chemial Engineering Transations, 24, Shin H.Sh., Kim B.H., Chu Ch.N., 28, Analysis of the side gap resulting from miro eletrohemial mahining with a tungsten wire and ultrashort voltage pulses, Journal of Miromehanis and Miroengineering 18(759), 1-6. Spieser A., Ivanov A., 213, Reent developments and researh hallenges in eletrohemial miromahining (μecm), International Journal of Advaned Manufaturing Tehnology 69, Volgin V.M., Davydov A.D., 24, Modeling of multistage eletrohemial shaping, Journal of Materials Proessing Tehnology 149(1-3), Wang S., Zhu D., Zeng Y., Liu Y., 211, Miro wire eletrode eletrohemial utting with low frequeny and small amplitude tool vibration, International Journal of Advaned Manufaturing Tehnology 53(5-8), Zhitnikov V.P., Fedorova G.I., Zinatullina O.V., Kamashev A.V., 24, Simulation of non-stationary proesses of eletrohemial mahining, Journal of Materials Proessing Tehnology 149,