tooling design for ecm

Tooling design for ecm V.K. Jain* and P.C. Pandey**

This paper critically reviews existing analytical models, ie classical ecm theory, the cos 0 method, the analogue method, the finite difference technique, the complex variable technique, a perturbation method and nomo, graphic and empirical approaches to metal removal in ecm. A review of the progress to date made in the fields of application of the finite elements technique to several electrochemical operations, ie ecm, ec drilling, boring, wire cutting etc is also presented. An introduction to the application of three dimensional finite element analysis to ecm problems is included. The paper suggests areas where future work could improve the accuracy of exis'ting models pertaining to ecru tool design

Electrochemical machining (ecm), described 1'2's as the process of accelerated, controlled anodic corrosion, is achieved by a high velocity electrolyte flowing between the anode and cathode (Fig 1). The electrolyte serves multifarious purposes, including removing the heat generated during machining 4, allowing a high rate of metal dissolution, removing the reaction products etc. Rate and direction of anodic dissolution in the process depend on the applied voltage, electrolyte conductivity, current density, anode material, electrolyte f low velocity, presence of anodic film'etc.

Over the years, ecm has found applications in a number of practical machining operations such as turning 6-9'116, trepanning, broaching l°'12s, grinding HA2'I21 fine hole drilling ~3, die sinking ~, cavity sinking ls'~6 , piercing 17'is and deburring tg'sg, and is being used widely in the aeronautics, nuclear 3°, biomedical a and space industries 22. Some typicalexamples of ecru applications are: machining of turbine blades 3z made of high temperature and high strength alloys 21'2a-2s, copying of internal and external surfaces, cutting of curvilinear slots ~4'26, machining of intricate patterns s'2°ns, production of long curved profiles 27, machining of gears 2s'29, production of integrally bladed nozzle-rings for use in diesel locomotives a~ and machining of thin (80/Jm) large diameter diaphragms. The process has several advantages to offer, eg machinability of the work material is independent 32'33'37-40 of its mechanical properties, production of stress and burr free surfaces, comparatively good surface finish 34-36, better corrosion resistance of the parts produced, absence of swarf or chip disposal problems, low overall machining times, and working accuracy independent of the operator's skill 36.

Many research data have been published, but the potential of ecm as a production process has not been explored fully due to practical difficulties faced in the design of tools, the complex nature 71'72 of the process of metal removal and the difficulties encountered in prediction of the anode shape. This paper attempts to highlight some of the problems of tooling design in ecm and the prediction of anode shape. *Mechanical Engineering Department, MNR Engineering College, Allahabad 211004, India and **Professor, Department of Mechanical and Industrial Engineering, University of Roorkee, Roorkee-247 672, India

Tooling design in ecm

Tool design in ecru is mainly concerned with the prediction of anode (work) shape obtainable from a tool while operating under specified conditions of machining. Conversely 64'6s'7°, it also deals with the computation of the tool shape which under specified conditions would produce a work piece having a prescribed shape and accu racy 99'z = s.

Computa t ion o f anode shape

The workshape obtainable from a given tool in ecm under ideal conditions can be computed from the corresponding equilibrium gap. However, these results are influenced by a large number of parameters such as the presence of an anodic film a6'42-4a, electrolyte f low rate 3a, work material microstructure, intergranular attack s°'s~, change in valency 33 of the work material during cutting, type of electrolye s2-e°A26 and the role of additives 4a'49, current density, electrolyte throwing =2s power, electrolyte pressure distribution, stray current attack and passivity s2'sa's9. In some cases, the exact influence of these parameters is not clear. Fig 1 shows the ec hole sinking operation with the interelectode gap (lEG) divided into four different regions. Most of the work 67-s7 available in the literature pertains to the front and side gaps (zones 2 and 4) only, whereas scanty or no information is available about the mode of material removal in the transient 7L~z'l°° and stagnation regions 87'ss (zones 3 and 1 ).

L- Electrolyte

t

/

Transition I--

/

Sol i ; ,

/ aQ ~ ,

/ / / ~" / /~_/ / / ~)~.- ~_~

Stagnation

Fig I Schematic diagram of ecd with outward mode of electrolyte flow. Zone 1 (stagnation), zone 2 ( f ront) zone 3 (transition) and zone 4 (side)

P R E C I S I O N E N G I N EE R ING 0141-6359/80/040195-12 $02.00 © 1980 IPC Business Press 195

For the case of machining with plane parallel electrodes, the local metal removal s9 and the equilibrium gap =° are given by Equations (1) and (2) respectively:

E l c t r n = ( 1 )

F

E v E K ~

Ye F F F Pm cos 0 (2)

Equation (2) is valid in zones 2 and 4 only and when nonpassivating electrolytes are used. For a passivating electrolyte, E v is as follows

E v = A v - A E v (3)

The overpotential 4,/X Ev, in Equation (3) consists of reversible, concentration and chemical overpotentials and can sometimes be high 1° (up to about 5V).

Equation (2) above is based on the assumption that electrolyte conductivity K remains constant during the operation• However, K has been known to be a function of electrolyte temperature, void fraction, distribution of voids within the l EG ~6'63'69'sl , electrolyte concentration, efficiency of electrolyte filtrationS6 etc. For the computation of K, Tipton ~ proposed the following equation which accounts for the effects of electrolyte temperature and hydrogen liberation:

K = K o (1 + a AT)(1 - OZv )n (4)

In Equation (4), Hopenfield and Cole 2° assume a value of n of 1.5 for the case of uniform void distribution; for non- uniform distribution, especially when bubbles are concentrated near the cathode, a value of n of 2.0 has been suggested ~6'63'69'sl. Experimentally, the void fraction (~v can be evaluated 4'20'Ts'Sz'93 from:

C~v- 1 + ~ (5)

where

= Qg/QI and

/ c Rg (273+T+A T) Og - Z F P

The temperature rise 4 z~Tof the electrolyte in ecm is due to chemical reaction in the bulk of the electrolyte, the reversible heat of reaction and the viscous heat generated within the electrolyte• However, the major factor that contributes to temperature rise is the Joule heating:

dT E2v K dx - V Pe C e y2 A c K (6)

which, when solved, yields 1

o~ x) ( 1-O~v)n } -1] (7) A T = ( T - T o ) = - d [Exp {(A c K o

The electrolyte temperature at any point within the l E G depends upon the length of the flow path, x. Equations (2),

Nomenclature A A v

t

Bg,a ,a s ,a s

bb bc,Bl -Bs Ce Cm Ct D m E E v F

F F

I c !

lEG

J

K m I(

K,j k

L

m

m s

mr/"

n n r

n s P P Q

R ~ p f

Cross sectional area T Applied voltage t Interelectrode gaps (See Fig 1) t c Constant V Bare length of the electrode v Constants x Specific heat of the electrolyte W Machine tool-rate Y Cost of drilling tool Z Unitary matrix AE v Electrochemical equivalent Po Effective voltage p Faraday's constant 0 Front feed rate Current Function o~ v I nterelectrode gap v,v' Current density e Stiffness matrix ¢ Electrolyte conductivity X Coefficients of stiffness matrix 77 Wave number Length of electric resistance Metal removal Total number of sparks before the end of tool- c life d Metal removal rate e Exponent F Length of normal to anode surface g Number of sparks per unit length of drilling i , j

Mean gap between electrodes I Electrolyte pressure within l E G m Electrolyte flow rate s Gap resistance t Radius V

Temperature Machining time Time to change the tool Electrolyte flow velocity Volume Distance along electrolyte flow direction Width Interelectrode gap Valency Overpotential Volumetric concentric concentration of hydrogen Density Angle of inclination between feed direction and normal to the tool surface Temperature coefficient of electrolyte conductivity Void fraction Correction factors Amplitude of surface irregularities Electric field potential Wavelength Machining efficiency

S u b s c r i p t s

corner drill electrolyte (C e ), element (H e ), equilibrium front gas node number liquid work material sparks, spike, side condition at time t voltage

196 PRECISION E N G I N E E R I N G

(4) and (7) predict that the lEG would become non- parallel after a time t even for the simplest machining configuration.

Theoretically, the ec reaction would attain an equilibrium only after t approaches inf inity and the corresponding lEG can be computed from Equations (8) and (9) while machining with zero and finite feed rates, respectively.

Y= (Y2 o + 2 CA t) ½ (8)

where

E v E K C=11 Fp m

1 Y - Ye z~t=-~F [Yo-Yt + Ye In ( -o "e) ] (9)

Yt- Ye Equation (9) is implicit and can be solved iteratively. Equation (8) predicts an infinite value of the gap as t approaches infinity. However, in practice, as the lEG increases, the current density decreases so that the mrr (metal removal rate) gradually diminishes and for a large gap the process would come to a standstill. Equations (8) and (9) have been re-derived ]13 with the lEG expressed in terms of the current density, J:

Y= Yo+C' At (10)

Ye = Yo + (C ' - F F) A t (11)

where

C' = T?J/F Pm and z& t is very small

Equations (10) and (11) yield values of Y which are close to those obtained from (8) and (9) and agree well with the experimental 9~']°~ data.

Assuming that the electrolyte behaves like a pure ohmic resistance, the current density J can be computed from Equation (12).

E v K E v - - ( 1 2 ) J y Rgap

Equation (12) is valid within a certain range of experimental conditions only 2° . While machining complicated anode

o

c

3

/ 10 ~-- x = 8 4 m m

L Current 2" 4

[ I I I I f I I I I I I I I I I

shapes, the electric field gets distorted and in such cases Ohm's Law is not obeyed 27 . Under such conditions J can be computed from 37-39 :

J = - ~ (2K o ) (13)

Ippolito 1~7 has also mentioned that the conductivity of the lEG is a function of the machining current and other parameters and hence application of Ohm's Law to ecm is not justified. According to him

R g a p = R e + R a d d ( 1 4 )

where

A ~ 1 V a QbpC Re = ; Radd = ~ K / c ~c (15)

and the values of the constants have been evaluated experimentally nT as a = -0.32, b = -0.12, c = -0.11, ~ = 0.151.

Current density distribution within the lEG has been measured by Kawafune ~s who found it to be lower than the analytically calculated distribution. However, in some cases ~ , experimentally measured current density has been reported to be higher than that obtained analytically (see also Fig 2). The conflicting nature of the results can be explained on the basis of actual anodic dissolution efficiency, r/, achieved and a change in anode material valency with time.

lEG varies along the electrolyte flow direction, with the result that the flow velocity also varies and can be evaluated from the equation of continuity for flow of electrolyte, ie

Vl W Y1 = V2 W Y2 (16)

For a simple shaped lEG, Equation (16) would give satisfactory results; however, for higher accuracy and when a complex shaped lEG is involved, the conformal mapping technique is recommended.

Op t im isa t ion of process variables

During ecm, at certain values of the limiting tool feed rate, the process 16'8° can change from boiling to non-boiling and choking to non-choking. Feed rate greater than the

f x = 57mm Current 2" 3

-- x = 30mm

-- Cur ren t Z 2

x

I I 1 1 1 1 I I I I I I I

IOO 2 0 0 3OO

Machining t ime, s

V = 21.0m/s x = Distance

I I0 I-- x = 2.Smm

L Curren t Z I

5 Oo o 000 °0 0

I I

o IOO 2 0 0 3 o o o

Machining t i m e , s

Zero feed rote Experimental x, c,,o, (3 FET- II - -

Fig 2 Comparison of analytical and experimental results

Machining time

5" o 0 o 0 t = 0 S

~" £~ ~3 • o t = 140S

T o o n 0 t : 320s

-t r t f t i t I I I I I I I

b

o o 9 O t = 4 0 s

o o O o t = 6 0 s

_,', ~, z~ ~ ,', ,,, ~, ~ , ~z~ ~, z~ t : 3 2 0 s

-- Machining accuracy

I I I I I I I I I I I I l l 5 10 14

Distance, x , m x I0 -z

% 2.04 ,c

E

2.02 E

2.00

1.98 ~

PRECISION E N G I N E E R I N G 197

value corresponding to the intersection point m' of choke l imit and boil l imit (Fig 3) implies that the flow in the lEG is either choked or boiling. Maximum mrr is defined to correspond to the point m'. Optimisation analysis carried out by Bhattacharyya 62 with mrr (or feed rate for constant area of machining) as the objective function and the following three constraints

V o i> 1.65 F F (Electrolyte boiling) (17)

V o/> 1.44 F}: (Hydrodynamic instability) (18)

V o ~< 2.66 F F (Passivity) (19)

predicted an optimum feed rate, in ecm, of 1.825 mm/min. This analysis does not include the onset of sparking as a constraint. This is important from a practical viewPoint as beyond a certain frequency of sparking (or feed rate) the tool damage due to sparking could become significant and make the process uneconomical 73'a8. Higher feed rates in excess of the optimum lead to lower tool life. Larsson and coworkers 7a suggest that the feed rate given by Equation (20) would result in minimum machining cost/part.

m s o.5 F F = { b c (t c+~,t/cm) } (20)

For the condition of zero sparking rate, Larsson and co-workers ~a's7 have derived an optimum feed rate of 2.28 mm/min which means the optimum feed rate 62 of 1.825 mm/min takes care of the sparking constraint automatically. However in general, high mrr in ecm can be realised by increasing the feed rate, supply voltage, electrolyte pressure, flow velocity and the electrolyte inlet temperature. To achieve higher feed the quantity J r/should be high, but increase in J leads to a decrease in r/. On the other hand, if Y is decreased beyond a certain value, increase in potential gradient would result in arcing and the process would come to a stop 76 It has also been reported ~27 that, in some cases*, the sludge produced during the process could coat the electrodes resulting in a complete halt of the anodic dissolution and short circuiting of the lEG.

From the discussion above, it is evident that the relationship between the'parameters governing the ecm process is complex and hence diff icult to model analytically. Some progress made in this direction is discussed below.

A n a l y t i c a l models for too l design in ecm

The classical theory of ecm is valid under ideal conditions only. Trial and error methods for tool profile corrections

4 - Theore t i ca l , ,

.c_

_~ 3 g

f n I

I o Run 2 z~ Run I

I I o 0.06 o.12

Feed rote. in/min Fig 3 The ecm characteristic curve'S°.m' corresponds to the point o f maximum metal removal rate

0.18

have been used extensively. An alternative method for tool profile correction is to calculate the working gap and account for it during tool manufacture. With the avail- ability of high speed computers it is now possible to determine the tool shape, that would yield a workpiece of specified geometry, by successive approximations. Since the computing time is only a fraction of that required for carrying out the experiments, applying tool correction analytically has proved to be time Saving and economical.

A number of models for anode shape prediction and tooling design in ecm have been developed. Since the ability to predict the variation in lEG for any given operating conditions is a pre-requisite of proper design of ecm tools, these models are discussed in terms of the equilibrium gap.

Cos 0 m e t h o d

This method is based on the computation of equilibrium gap for the given conditions but excludes the consideration of the mode of e!ectrolyte flow, overpotential, variation in electrolyte conductivity, heat transferred to the environment etc. In this method, equilibrium work shape is computed corresponding to the tool whose profile has to be approximated by a large number of planar sections inclined at different angles.

This method was first proposed by Tipton 6s and is described in detail in reference 92, but its scope is limited by the following

Regions with sharp corners 92 cannot be analysed. Generally applicable only for 0 < 45 ° It is not possible to account for the effects of the mode of electrolyte flow, void fraction, change in electrolyte temperature and conductivity, overpotential, heat conducted away to the tool and work etc. Researchers have also expressed conflicting opinions

about the reference surface for the measurement of the angle 0. Tsuei 7° et al suggest that 0 should be taken as the angle between the tool-feed direction and the normal to the anode surface,and not the cathode surface, as suggested by others 6s . However, in view of the approximations involved, its use is not recommended, especially when complex shaped workpieces are to be analysed.

F in i te d i f fe rence techn ique ( fd t ) :

The cos 0 method is based on the assumption that the lines of electric potential are straight and normal to the electrode surfaces, but this is not true when electrodes with small radii are used. In such cases, it is necessary to know the electric potential distribution in order to determine the current density, mrr, transient and equilibrium anode profile etc. fdt has been employed by several investigators 4'4~'64'7°'83-8s'~19 for ecm tooling design using a nonpassivating electrolyte with constant conductivity and temperature. In such cases, the electric field flow lines are governed by Laplace's Equation (21) and the boundary conditions given by Equation (22)

a2~ a2~ ~2~ . + - - + - - = 0 (21)

~)x 2 ~3y 2 ~)z 2

~b = 0 at the cathode and ~=Ev=Av-AEv at the anode (22)

Equations (21) and (22) would yield a set of simultaneous equations that could be solved for ~ by the backward differences, forward differences, or central differences 9s'96


technique. Corresponding to the potential distributibh obtained, the instantaneous current density s3 at the anode surface can be evaluated from

a ~b ) (23) J = K ( an---;

Fig 4 shows the tool-wo'rk surfaces and lEG drawn in square mesh. The initial potentials at the grid points within the lEG region are set by linear interpolation along the vertical gridlines betweeen the tool and work boundaries. For a point 0 located in a mesh of spacing h (i,j) we can write s3

@ii = (@i+ 1,i + @i-1 ,j+ @i, i+1 + @i, i-1 )/4 (24)

Equation (24) shows that the potential at any point 0 (i,j) on the square mesh is equal to the mean of the potentials at the four nearest adjacent points (ie the points 1,2,3 and 4 in Fig 4).

The procedure of computation in this case is to consider each grid point (i,j) within the field in turn and adjust its potential value to the mean of those at the four points around it and to repeat the process until the potential values attained are correct within the prescribed tolerances. It should be noted that the work and tool boundaries are at fixed known potentials and, therefore, their values are not adjusted during the relaxation process. In some cases, all the points on the tool and work boundary may not lie on grid points (Fig 4) and the regular stars may not be formed; such problems have been attempted by the use of the over-relaxation technique s3. To solve such problems, Nanayakkara and Larsson n9 have suggested the use of irregular grids along with regular grids. In this case, a polynomial Equation (25) was used instead of a linear interpolation Equation (24) to describe the potential distribution in the area around a nodal point including its near neighbours.

0 = B lx 2 + B2Y 2 + B3x + B4y+Bs (25)

After each computational interval of Ats, work and tool boundaries are moved to new locations, according to the cut and feed vector. For simplicity, Tipton sl has only accounted for the vertical component of the cut vector which is proportional to the vertical potential gradient at the work boundary. However, for precision in the results, movenlent of the cut vector in both x and y directions should be considered.

It is thus evident that fdt would yield approximate results. Further, in case of a complex shaped lEG, the tool work boundaries cannot be matched accurately using square mesheswhich introduce still greater approximations. On account of the approximation involved in the use of this technique, the authors recommended 92'u° the use of the fet for tool design in ecm.

Analogue me t i l od

Laplace Equation (21) was first solved by the conducting paper analogue s3'9~'97. In this case, equipotential surfaces representing the anode and cathode, to an approximate scale, have to be drawn on a conducting paper. The work-" boundary is segmented to evaluate local current density and then it is moved to the position they would occupy at the end of the time interval At. This could be obtained by finding point to point movement by the vectorial addition of feed rate vector (F F /k t cos ~) and cut vector

h 1 ~ r l l l ] l l l l l l l l l i l r l l l [ l l l l

i i i J ~ l l ] l i J l i i i i l l / . . . . . . . . I I Ii ~ I I I Ji Ii I[ i i i J '~' j 2 , ; 4

yJ , , , , . . . . . . . . . : : ' ' "

l ~ . . . . . . . j-li_ I - - ~ i i l i+l ~ x

a b

3

0 4 4

I / " z "3 I I

I c d e

Fig 4 Finite difference analysis model s3 of ecru process (a) I E G-tool-work system discretised in square meshes (b) A square mesh of spacing h having four equiplaced adjacent points 1,2,3,4 which form a regular star. (c) Boundary forms irregular star with one short arm-03. (d) Boundary forms irregular star with two short arms 02,03 and (e) Boundary forms irregular star with three short arms 02,03,04

( K i ~ , ) A t). The method is based on the assumption that

feed rate velocity and cut velocities are constant over the time A t. This transforms point A to point B to make a new surface on the work and this process is repeated until all points A are transformed to points B which are on the worksurface.

Before repeating the process, the applied voltage must be adjusted so that the same current values as used earlier are obtained. The process has to be repeated until the workshape does not change appreciably between two successive steps and the final equilibrium shape obtained s3.

The analogue method is approximate and cumber- some, its accuracy depends on the skill of the operator and is not advisable for use when a high degree of precision is desired.

Empir ica l formulae

The exact path of the electric current flow lines, within the lEG, is diff icult to determine analytically. Therefore, normally, the chordal distance between two stations 27 is taken as the length of current flow line and this assumption in the majority of cases is responsible for the discrepancy between the analytical and experimental results. It has also been found that the conformity of the surface radii of the tool and the anode cavity decreases as the angle 0 increases. Therefore, attempts 22,3~,71,1oo have been made to derive empirical equations for the evaluation of the lEG.

Based on experimental data, empirical Equation (26) has been suggested by K6nig and Pah122.

a o = r 0"35 0.35 [(10e*) ye] 0.5 (26) c

for 0.15 ~< Ye ~ 0.6 mm and 0.5 ~<r c ~< 5 mm. e* is Euler's number.


K6nig and Degenhardt 7~ also suggest Equations (27-28) when the bare length of the tool (b b) varies between 1 mm ~< r c ~< 5 mm.

a o = (0.1 + Ye)(0.314 r c + 1.17) for b b = 0 (27)

ao,= 2Ye + 0.1 [6.283(rc-1)] 0"5 for bb/> 1 (28)

Equations (27-28) have been found 91'127 to yield erroneous results at low feed rates (F F <~ 0.006 mm/s) or when the equilibrium gap is large ( Ye ~> 1.0 mm). Based on the regression analysis 127 of experimental data plotted in Fig 5 (mild steel anode, brass cathode, electrolyte 10% w/v NaCI and feed rate 0.0057 mm/s, outward electrolyte f low, coefficient of correlation 0.8) the authors suggest Equation (29) for the computation of a n, mm.

a o = 0.19536 Ye + 0.5779 (29)

The magnitude ofa ' s (Fig 1) can be evaluated 22 from Equation (30), as

t , 2 ,0 5 a s = (2 b b Y e t aol • (30}

!

Analytical evaluation of as, which is affected by stray current, is diff icult. It can, however, be evaluated =z from the empirical equation

0.7 0.123(10e*) Ye] 0.5 a s = [2 b b Ye + rc

+ 0.65 Ye (31)

Test results show that the overcut in ecm is a function of the machining parameters but does not depend on the tool di mensions 22 . The authors' experimental data 9~ , however, point to the contrary. Further, i t should be noted that Equations (26-31) are valid only for the case of non-passivating electrolytes. I ppolito and Fassolio l°° checked the validity of Equation (28) and report that for rc=0 it gives the upper bound whereas Equation (32) gives the lower bound of the side gap; Equation (33) was found to cross the experimental data.

I a s (2 b b Ye +/)2 y2}0.5 = -e" (32)

t a s = (2 b b Ye + 2.9 Ye 2)0"5 (33)

The lateral gap at the end of the non-coated zone of the electrode can be computed from:

a' s = (2Yeb b + (1; Ye*) 2 -c ) 0"5 (34)

where ¢ $ - - = Ye- l .29Ye, c = 0.29 and F F 1.16 mm/min

From the results given in reference 100 it is evident that Equation (34) is valid only for a specified tool-work combination and specified machining conditions and the discrepancy between experimental and analytical results increases with an increase in K Ev/F F. Equation (35) also predicts variation in overcut with the conical taper and it was found l°° to be practically independent of the uncoated length b b.

, 2 B ' Y e , 2x+B' as=as+ ~ larctan ~r - arctan (BI/D')] (35)

where

B'= v'E K E v ; D' = (4F~= a's 2 -B'2 )0.5

FP m v' is a correction factor of conductivity and takes into

account the fact that the lines of electrical potential are not straight and the work-surface wil l be lightly passivated by the presence of the metallic oxides and hydroxides. It is thus evident that for the evaluation of side and front

gaps, Ye can be used as a basic parameter. In some of the cases the side gap has been demonstrated to be independent of the machining time which, however, is not true in practice. It is to be noted that in the majority of cases the effect of electrolyte f low mode (inward, outward etc) has been neglected. Empirical equations are normally valid under specified working conditions only, which limits their use.

Nomographic approach

For the evaluation of equil ibrium anode shape, a nomographic approach has also been used 22:1'1°1. K6nig 22 has prepared a nomogram for the evaluation of side gap for the known equilibrium gap, tool-radius and the bare tool length. Fig 6 shows such a nomogram for bb=0 and bb/>0 for a given tool. Heitmann a4 has used the nomograms for evaluation of electrolyte temperature rise (A T) while working under specified conditions. Use of nomograms has proved to be advantageous in planning an ecm operation.

However, the nomograms are very often based on a number of simplified assumptions 8,9 and hence cannot be recommended for general use; they would be more useful if available in a generalised form.

Complex variables approach

Anode shape prec~iction in ecm has been attempted by employing the complex variables approach for the case of shaped workpieces and using simplified assumptions. Collett et al 1°3 have determined lEG for both completely side insulated and bare and straight sided tools and arrived at Equation (36)

Overcut at corner = 1.159 (36)

Machine gap

However, PERA 118 suggests a value of 1.7 for this ratio. This is because Collett et al 1e3 assumed that the electrolyte conductivity and void fraction remain constant throughout. Hewson-Browne 1°4 extended the work of Collett et al and used the conformal mapping technique for the analysis of two dimensional machining using straight sided tools with a finite land width b b. For an insulated tool i t was assumed that the sides do not participate in metal removal; it was shown for such a case that

0.731 ~ao/Y e ~ 1.159 (37)

where the lower l imit applies to the case of a side insulated tool and the higher l imit is for a bare tool. How- ever, the authors have found experimentally that for the case of machining with bare tools the ratio ao/Y e is a function of machining conditions and tool-work com- binations. Nilson and Tsuei 41'86'94 have solved the problem of anode shape predicted by the inverted method, in which the spatial coordinates were selected as dependent variables on the plane of complex potential and the transformed boundary conditions are known explicit ly on the free boundary. Known asymptotic solutions for the side gap region reduce the size of the field which must be determined.

Use of mathematical models has not so far been generalised because workpieces have complex shapes and there are many parameters to consider.

A per turbat ion method

This analysis has been developed by McGeough ani:l co-workers 77-79 and can be used for the analysis of ec

200 P R E C I S I O N E N G I N E E R I N G

operations like deburring, anode smoothing or shaping when the amplitude of micro-irregularities on the cathode and.anode is small compared to lEG. This analysis has limited applications and is based on the following simplified assumptionsS9 :

Electrolyte conductivity is constant Effects of Joule heating and hydrogen gas bubbles are suppressed by sufficient agitation o.f the electrolyte Machining efficiency is 100% Perturbation boundaries change so slowly that their motion may be ignored at any instant. Consider a case in which cathode and anode shapes

(Fig 7) -are defined mathematically by Equations (38)a-b respectively

Y = 0 (38)a

Y =p + e sin k x where e < < p (38)b

Laplace's Equation (21) can be solved with the boundary condition given below.

~b=0on Y = O a n d ~ = E v o n Y = p + e s i n k x

It has been shown s9 that the potential at any point Y can be obtained from

= E v Y Eve s i n k x s i nhky (39) P p s inhkp

Using Equation (39), a set of equations has been derived which describe separately the behaviours of the anode surface and irregularities respectively:

1 Pe-Po] A t = FFF [P°-P+Pe in (40)

Pe-P

r o

,o E ~o

6

18 --

14 --

A

O S

I I I I I 1

0 . 5 1.0 1.5

x

A

o ~ ~ " --a%~ ~ ~ 0

0 0

I I I 1 l I I I I

2.0 2.5 :3.0 :3.5

~ , m m

4.0

e=eoEXp[_MEvk~ cothks ds] (41} Po M Ev-FFS

dp provided d-~ =/= 0. Here, M = E El(pro F) and s is the coordinate

that specifies the average surface. Equation (41) has been applied s9 to analyse irregu-

larities on the anode shape of the type of short wavelength, long wavelength, arbitrarily shaped and even and arbitrarily shaped.

This analysis is based on the solution of Laplace's Equation (21) and therefore accounts for the field concentration effects. This explains the more rapid smoothing expected with short wavelength irregularities which have a much greater field concentration than larger wavelength irregularities. This analysis uses the Fourier series expansion and has been extended to analyse a more general class of electrode shaping by Fourier Transforms. An equation similar to (40) has also been derived by Tipton 6s in terms of dimensionless quantities for the case of surface smoothing to the required tolerance.

Fini te elements technique (fet)

In view of the shortcomings in the analytical models discussed so far, the authors recommend l°s the use of the fet where the choice of the shape and size of the elements is convenient ~°6, it is easy to incorporate different boundary conditions 1°~ and it is possible to analyse non-homogeneous situations 9s . In this case, the effects of simultaneous variations in different parameters could easily be accounted for. The authors have so far developed one, two and three dimensional models for the anode shape obtainable in ecm.

U n i d i m e n s i o n a l a n a l y s i s ( M o d e l F E T - 1 1 )

For the analysis of the unidimensional electrochemical wire cutting process (ecwcp) l°s'z°9 and the ecru process 92'zl°'ln the lEG has been discretised into a number of elements (Fig 8).

Electrolyte temperature at any nodal point in the continuum, ie lEG is evaluated from (6) which could be written as

51=dT 6Tfj ~j dT d_TT) - ~ - (6 d x = 0 (42}

dx xi xi dx

r , mm :3.03.54.0

2.5

2.0

1.5

1.0

I - - - - I =

0.5

I I I l l [ t 0.7 0.5 410.:3

-

b , . ~ " ~ o.s

0.1 0 0.6 0.4 0.2 0

Og, mrn

~mm ~

I I I I I 0.2 ~1~).4 0.6

y C x Cathode / . z / i / / / / / / / / / / / / / / / / / / / / / / .

" ~ Electrolyte

p(,I I ' - - )' - ' t

Fig 5 (Top left) Relationship between equilibrium gap Ye and overcut, a o

Fig 6 (Left) Nomogram 22 for the determination of the side gap, a s

Fig 7 (Above) Configuration of electrodes s9


On simplification, this reduces to

X = D H e.= K m T e

where

D = ~ i ; He =

and

1

Pe y; Ce V

(43)

x • 1 -1 = f l BTBdx = [ ] for an element (44) Km

--1 1 x i

Assembly of the equations thus obtained for different elements, say three, would result in a symmetric, sparse and banded matrix of the form given below:

IXlt fi 1 !l X2 = 2 - T~

X3 - 1 T3

The set of simultaneous equations can be solved by Gaussian elimination technique 122 for unknown nodal temperatures using a boundary condition of A T = 0 at entry. Thereafter. current density, mrr, conductivity etc can be computed.

The analytical results have been compared with experimental data and a good correlation between them has been shown. As an illustration, a comparison between temperature computed from model FET-11 and obtained experimentally 9° is given in Fig 9. The authors have also conducted a parametric study H2 of the ecru process in attempting to form some guide lines and to evolve a standard procedure for ecm tool designers. This model has also been extended 113 (model SGFET-11) for the anode shape prediction during ec drilling which has been treated as a problem with moving coordinates. It has also been shown that the overcut on the side (a's) is basically governed by the computed value of a o . A comparison between the computed and experimentally measured anode shapes shows 91 a good correlation between the two.

However, the degree of correlation could further be improved if more realistic values of the machining efficiency, T/, anode material valency and heat transfer data were available.

Two dimensional analysis

fet using triangular elements can be used for the analysis of two dimensional machining problems assuming that the temperature rise over an element follows a linear law. But in the case of non-parallel lEG the electric current flow lines would not be normal 84 to the surfaces and the electric field potential in such cases would be governed by Laplace Equation (21). The authors H3'H4 have developed

V 2 X

C

- F % . - ~ - - - - ~ - - - ~ ; ~ I i j ' k Work

a Tool

I I Work ' - -

~, k

X

b

(45)

Fig 8 (a) One, (b) two and (c) three dimensional discretis- ation of interelectrode gap

a computational procedure (Fig 10) for anode shape prediction (model SGFET-22), evaluation of current density and other related parameters by solving the Laplace's Equation (21). Fig 11 shows a comparison between the theoretical and experimental 9° electrolyte temperature distribution within lEG. Further, improvements in FET-22 and SGFET-22 are possible by the use of isoparametric elements 9s'123 instead of triangular elements.

Three dimensional analysis

Equation (21) can be used for the evaluation of three dimensional potential distribution within lEG. The problem is equivalent to finding the 0 distribution that satisfies the boundary condition and minimises the function /

1 [(a0)2 + 09,2 (a0)2] d x d y d z I=fffv 2 ox. (~-~t + o z

or

(46)

I = f f f v 1 { g ~ T [ D m ] { g } dv (47)

09 09 o30] (48) wheregT= [~x Oy Oz

!

/= / (1) +/(2) + / ( 3 ) + . . . + / ( E ' ) = ~ I(e) (49) e=l

where I (e) is the contribution of a single element to I. The minimisation of / occurs when

0 1 0 E' E' 0 I (e) - Z; / e = : S - 0 (50)

63{0] 0 [0 ] e=l e=l 0 [0 ]

To evaluate derivatives 63 / a-~-}' the integral should be wr i t ten

in terms of nodal values, ie

0 (e) = [N re) ] (0]

aN 2 (e) aN¢(eF ax " " " ax

0 N 2 (e) a N¢ (e) ay " ay

63 N2 (e) 0 N e (e) o~z " ' " az

(51)

0 1

92 =[B] (e)[o] (52)

0e

Therefore,

-aN l (o) ax

g(e) = a N z (e) ay

O N t (°) az

Here B contains the information related to derivatives of the shape functions. After simplification 66 it can be shown that

8/ a9 - [Ke ] {0} (53)

where

[K e] = f f fv [Bre) ] T [D(e)] [B(e)] dv (54)

The final system of equations is obtained by substituting (53) into (50)

0/ E [K(e !] - - = ~, [0 ] (55) a0 e=l

Evaluation of stiffness matrix K (e) involves the solution of - - m the integral Equation (54) and its value is a function of the type of element being used. Fig 8 illustrates the use of tetrahedral elements for the analysis.

Assuming linear interpolation functions, field variable O(x,y,z) can be uniquely and continuously represented in the solution domain by Equation (56)

~e (x,y,z) = [N (e) ] {0 e] = NiO i + NjOj

+ NkO k + NI~ I (56)


.= o & E

4 5

40

35

30

Experiment . . . . FET-II

k o = 0.01412 ~-I mm-S To = 30oC Yo : 0.Smm At : 30S

t : 500s

O = 23.33 = I03 mmS/s F" F = O.OI5 mm Is E v : 12.0 v /

45 [ O : 31.67=103mmS/s F F = O.OIOmm/s / E v = 16.0V / /

40 -- /

/ / / / / J 35 - / / ~

, , F

_ Ev=2O.OV - - / /

O : 25,0=103mm3/s

:50 O 8 16 24

Distance, mm Distance, mm Fig 9 Comparison of analytical (FET- 11) and experimental results

Fig 10 (Right) Computational scheme used in mode/SGFET-22. KiN is an index which counts the number of computational cycles. NC is the total number of machining cuts (or number of cycles) for which the computation has been carried out

F" F : O.OIImm/s E v = 12.0V

_ f

- / /

Y" I I ~ 0 8 16 24

Experiment . . . . FET-22 V, m/s FF, mm/s x I0 V, m/s FF, mm/s x I0

x &528 0.0011 x 4.74 0.0015 o 2.:55 0.0015 o 6.02 0.0011 • 2.74 0.0020 • 2.32 0.0020

40

30

E ¢p

: 20

E

I0

/ /

/ e

/ /

/ / /

/ /

0 I0 20 25 O I0

Distance. mm Distance, mm

Fig 11 Comparison of analytical (FET-22) and experimental results

20 25

I Read and print input data I

I Compute no. of computation cycles J

I Generate K,V and Y for different nodes I

I Compute feed rote at every node I

I Determine band width J

Initiolise parameters I

J Generate and assemble J element matrices

Form global matrix J

Apply boundary conditions j

Solve set of eqns.to I determine I

I Compufe I

I Compute T I

I Compute heat transfer to air 1

I Compute decrease in electrolyte temperature

Jco~p~e effective electrolyte temberatureJ

I. Compute K,Y, mrrL, mrrvondmr I

fes

coordinates Outward flow

Yes I Reassign x-y

coordinates ~ 0 ~ Side flow

Yes Reassign x-y coordinates inward flow

I Compute V for new value of Y

NN= number of nodes Electrolyte flow for

IOW<O sidewards TOW= 0 inwards IOW>O outwards

PRECISION ENGINEERING 203

Dif ferent iat ion of funct ional / and s impl i f icat ion 66 leads

to Equation (57)

K['Kji Ki j

i Kj i

K k i Kkj

LKli KIj

K"/[ = .(0}- 1571 Kkk Kkl[ Ck

KIk K I I j ¢1 where Kii, Kjk . . . . . KII are coefficients of stiffness matr ix K (e) that can be evaluated as fol lows:

K(e)_SSSv(a__~N- O~N" ~/V i ~__N/N- ON i ~NIN'~dxdydz (58) ij - \Ox + ov ov +Tz Oz I

Using the def in i t ion of interpolat ion funct ion, Equation (58} can be shown to be reduced as

1 Ki j = 36v2 (bib j +c i c j + did/) (59)

where a,b,c and d are constants for an element and can be computed w~

This analysis wou ld give three dimensional potential d is t r ibut ion w i th in the lEG f rom which current density, mrr, temperature d is t r ibut ion etc could be computed as done in earlier models.

Discussion and conclusions The mechanism of anodic passivation at high current densities in ecru is not well 36'42-47 understood; i t has been found, however, to reduce the anodic dissolution eff iciency. Therefore, attempts have been made to destroy the passivating f i lm during the process by the use of reducing a~jents 39'46 as additives to the electrolyte. Quanti tat ive data regarding the effects of grain size, grain boundaries, density and or ientat ion, on mrr and current density are not available and hence their effect in model l ing cannot be incorporated. Also, governing equations do not exist that can relate the influence of electrolyte mixtures on surface finish °~, dimensional control 43-4s, pH of the electrolyte 124 and mrr, especially w i th comoosite materials. Further, most of the cases do not account for the change in valency 33 wi th t ime. This can often seriously affect the values of machining efficiencies s9 . Presence of lines or str iat ions on the machined surfaces have also been observed exper imental ly but their basic cause has not been resolved 4° However, remedies, set by experience, have been suggested, eg inject ion 32 of gas into the electrolyte and use of mul t iho led electrodes 37-4°.

I t is evident that the problems concerning ecm tool design are highly complex and involve a number of interrelated parameters, such as temperature, electrolyte conduct iv i ty , current density. Further, a number of parameters (effect of microstructure, grain boundary, passivation f i lm etc) whose effects are not well known, make the process more d i f f i cu l t to analyse.

It can thus be concluded that most of the models of the ecru process make use of s impl i f ied assumptions, such as constant conduct iv i ty and neglecting the void fract ion which give rise to discrepancies between the analytical and experimental results. In practice, ecm tool designers often resort to an empirical approach which, however, requires extensive research data, including the effect of passivating electrolytes on metal removal. A nomographic approach can also be useful for tool designers and can be developed to a satisfactory level.

In the opinion of the authors, i t is better to go for numerical methods such as the fd t and fet. The fet is the stronger of the two and should develop into a standard procedure for ecm tool design.

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204 PRECISION ENGINEERING

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PRECISION ENGINEERING 205

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118. Electrochemical Machining. PERA Report No. 145, 1965

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206 PRECISION ENGINEERING

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