Finite element approach to the two dimensional analysis of electrochemical machining

V.K. Jain* and P.C. P~ndey +

In this paper, use of the finite element technique for the two dimensional analysis of ecm using simplex triangular elements is described. Good correlation between analytical and experimental results has been observed. This model has been used to predict the anode shape produced by a com- plex shaped tool The paper also describes the details of the computational scheme used

Nomenclatu re

Ac,a,b,c Ce E

Ev F FF J K K,m O

n

T t V

X

g

~v Pe Pm 0

Ae

At AT

Constants Specific heat of the electrolyte, W/kg °C Electrochemical equivalent of the work- material, kg Effective applied voltage, volts Faraday's constant, coulombs Front feed, mm/s Current density, A/mm 2 Electrolyte electrical conductivi ty,/24 mm 4 Stiffness matrix Exponent Normal to the work surface Temperature, °C Time, s Electrolyte flow velocity, m/s Distance in electrolyte flow direction, mm I nterelectrode gap, mm Temperature coefficient of electrolyte conduct iv i ty, /°C Void fraction Electrolyte density, kg/m 3 Work density, kg/m 3 Angle of inclination between feed direction and normal to the tool surface at a node (Radians or degrees) Area of an element 'e', mm 2 Computational cycle time, s Change in temperature, °C

Subscripts

i,j . . . . Node numbers o Condition at the start of machining t Condition at time t

Superscripts

e Element

*Research Scholar and +Professor, Department of Mechanical & Industrial Engineering, University of Roorkee, Roorkee-247 672, India

Electrochemical machining (ecm) can be defined as metal removal by anodic corrosion 1'2. This process offers the advantages of the abil ity to produce intricate shaped workpieces by shaped tools 3 and the high metal removal rate which, unlike conventional machining, is independent of the mechanical properties of the work material such as its hardness and strength, ecru has also been known to pro- duce stress free machined surfaces. Oyer the years this technique has successfully been applied to operations such as ec turning, trepanning, embossing, milling, electrojet machining and ec capillary machining. The principle of electrochemical dissolution has also been successfully applied to produce fragile or thin sectioned components, thin slices of metal crystals, deep slender holes, gun bores, cooling channels in turbine blades and razor blades 3 .

One of the problems encountered in ecm is the formation of an anodic fi lm 2's-12 which in.turn governs the metal cutting in passive and transpassive regions, wild cutting, surface finish and form inaccuracy in the machined surfaces. The mechanics of anodic fi lm formation are not well understood as yet. Inter- granular attack, ecm of composite materials, and the choice of electrolyte suitable for alloys are a few of the problems that have already been considered. An important aspect of research in ecru is the attempt to predict the work surface obtainable from a tool of given shape for J~nown machining conditions. Unfor- tunately this aspect of ecm is still not ful ly under- stood and in practice trial and error methods are quite common. To predict the work shape obtainable in ecm I , models based on the complex variables 13'14'21 approach and numerical analysis ~4'19 have been used. In such cases, however, the agreement between analytical and experimental results has normally been poor even for simple shaped workpieces. The application of the finite difference technique 19 to this problem has not been very successful.

On account of the complex nature of the process 23 of metal removal and interaction of a large number of parameters, the finite elements technique (fet) for the shape and size analysis of the anode in ecru would seem to be a better approach. This method has no constraint on the shape and size of the workpiece 24.

This paper describes in detail the results of work- piece shape analysis during ecm using two dimensional simplex triangular elements. The analytical results have been found to agree well with available experimental data. As an illustration, the surface generated by a complex shaped tool has been predicted analytically.

Finite element analysis (model l e t -22)

A unidimensional heat transfer finite element model fet-11 for ecm has been developed to compute the

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current density, metal removal rate etc. 2° The model has been found to be accurate for simple cases involv- ing plane parallel electrodes, but in the case of complex shaped workpieces the existing analysis needs modif i- cations as the assumption that the electric field remains uniform throughout the gap is not valid. It has been pointed out 2] that the electric field at a point where the curvature of the electrode surface is large and convex is much greater than the field at a plane surface. Short wavelength irregularities also have much greater concentration of field at their crests than those of large wave length.

For a curved interelectrode gap, electric potential distr ibution within the interelectrode gap would be governed by the Laplace Equation 19'2] :

a2~ a2~ ax--~- + ~ = 0 (1)

A knqwledge of the field potential distr ibution within the interelectrode gap would be useful for the determination of current density J at the work surface using Equation (2):

a~ J = K(~-~ ) (2)

The field vector ¢ should be determined in such a manner that it satisfies the boundary conditions as discussed later and also minimises the function /(~b)given by Equation (3):

/(~b) = 1/2 [ (~_)2 + (~_)2 ] dxdy (3)

Assuming that the interelectrode gap is represented by triangular elements (Fig 1 ) which uniquely and continuously represent the field variable ¢(x,y) in the solution domain and vary linearly within the element, then:

~(x,y) = Ni(;b i + Nj~j + Nk¢ k = [N e] {(;b e } (4)

where ~b e is the column vector of nodal potentials for element e whereas the interpolation functions N e are defined as follows:

N B = aB + bBX + CBY where B = i,j,k " 2 A e

aB, b B and c B can be determined 2; as follows:

a i=x jYk - - xkY j , b i = Y j - - Y k andc i = x k - x j , . . .

Electrolyte

Work

y k

Interetectrode gap os two dimensionol continuum

Fig 1 Interelectrode gap as a two- dimensional cont inuum

Similarly, other terms can be evaluated in cyclic permut- ation of the subscripts i, j and k.

To derive element equations, Equation (4) has to be substituted into Equation (3) and differentiat ion leads to

O/e [(~_NN i cqNj~) aN k .~ ~ 0N i ae~i - f f ax ~ ) i + ~ "- j+~-~--~kl~)~---

ohNi ~)Nj~ b aNk cqN i + ( ~ - ~ - ~ i + ~ - j+a--~--~k)~- ~- ] dxdy . . . 15)

Similar equations can be derived for the nodes j and k. These equations of nodes i,j and k of an element combined together can be writ ten in a standard form as follows:

I al e

I d@i

i63/e "1 a~--'-j '= [Km]e {¢}e = {0} (6)

!c3/e L a~k

where K m is the stiffness matr ix with the fol lowing coefficients for the element 1.

fK K Kji Kjj Kjk ~ (~j = { 0 } (7) /

Kki Kkj Kkk J ~)k

Coefficients of the stiffness matrix in Equation (7) could be evaluated from Equation (8).

1 K~ = 4 -~ (bibj + cicj) (8)

where Ae is the area of an element under consideration. Reactions at anode and cathode cause current

density dependent overpotentials 21 . Their presence at the electrodes would alter the boundary conditions as fol lows:

q~ = f(J) at the cathode (9)

~b = E.v-g(J) at the anode

where f(J) and g(J) are arbitrary functions for the cathodic and anodic overpotentials respectively. For the sake of simplici ty it has been assumed that the electrode surfaces are equipotentials which means

= 0 at the cathode

~b = E v at the anode (10)

The boundary condition described by Equation (10) has been used for the computation of results discussed in the next section.

Having obtained the value of current density from Equation (2), Equation (11 ) can be applied to compute the temperature gradient along the electrolyte f low path. Similarly Equation (12) gives the electrolyte conductivity at the nodes.

dT j2 - - AcK (11)

dx K V Pe Ce

K = Ko(1 + e AT)(1 -- ev) n (12)

The interelectrode gap at the instant t for zero feed rate (FF=0) is given by Equation (13) and for f inite feed rate (FF~0) is given by Equation (14).

• 2 ~ ½ Yt = (2ct + yo~ (13)

where, c = E E v K/F Pm

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

1 In(Y°-Ye) ] (14) t = ~ [Yo - Yt + Ye Yt-Ye

and the equilibrium gap Ye can be evaluated from Equation (15).

Ye = C/FF (15)

The computational scheme employed for obtaining the nodal potential, temperature distribution, current density, metal removal etc. is shown in Fig 2.

Results and discussion

Fig 3 shows a comparison between theoretical (fet-22 model) and experimental 27 temperature values for the conditions given in Table 1 and a good correlation between them has been demonstrated. In a few cases, however, fet-22 under- estimates the temperatures compared to the corresponding experimental values. This may be attributed to the inherent defect in the experimental set-up used for temperature measurement (Fig 4) and ineffective filtration of the electro- lyte before its recirculation (contamination has been found to increase electrolyte conductivity which would lead to higher temperatures). During computation, anodic

I Read and print input data I

f I Compute No. of computation cycles, nc I

t Generate K, v and Y for different nodes I

t I Compute feed rate at every node I

I Determine band width, MB I

I Initialise parameters I ..J - l f

I Generate element matrices I

t Assemble element matrices into a global matrix (Sub. AGCSGI)

t Apply boundary conditions (Sub. ABC)

¼ Solve set of simultaneous eqns. to deter

mine values of unknown (Sub. 8ANSOL)

I Compote J. T. K. AM,,. AM,V. TM,, etc [ ¼

Reset Y coordinates of every node I

Compute flow veloca ity v for new value of y I

t I Pri . . . . t th . . . . . its (Sub. PRNTOT) I

Table 1 Experimental i=onditions

C e = 4.18W/g°C F = 96 500.0C Ko = 0 . 1 4 1 2 £ 4 m m 4 x 10 4 To = 30 ° C At = 30 s Total machining time = 300 s Yo = 0.8 mm

Problem No. Variable 19 20 21 22 23 24

FF, mm/s 0.019 0.019 0.019 0.020 0.02 0.C

E V, V 12.0 16.0 20.0 12.0 16.0 20.C

O,mm3/s x 10361.00 55.67 57.67 75.33 72.83 84.E

dissolution efficiency has been assumed to be 100% which is not generally true for actuaJ machining. A more realistic figure would be 90%. A lower interelectrode gap would then be Predicted which would mean a higher temperature 2s Use of actual anodic dissolution or machining efficiency would improve the results as compared to those shown in Fig 3.

Fig 5 shows the current density and electrolyte conductivity variation along the electrolyte f low direction. It is evident from this figure that the current density is high at the corners which agrees with the

I0

15

? to

E

0

O V,m/s 5.81 ,FF, mm/s 0.019

I0

_2t p,

-- / / / ~

X V,m/s 7.34, FFF, mm/s 0.020

_ 2 3

- . / x /

• V, mls 6.17, FFF,mm/s 0.019

19

V,m/s 5.61 , FF,mm/s O.OI9

_ 2O

_ / X /

,,X /

• V,m/s "~60, FF,mrn/s 0.O20

- 2 2

,o

/o.>J I ~ ~ t i

IO 20 IO 20 Distonce, mm Distonce, rnm

R

o V,m/s 8.55, FF,mm/s 0.020

I 2 4 /

Fig 2 Computational scheme used in the fet-22 mode/. AMRR: linear metal removal rate, AMRV: volumetric metal removal rate, TM R R : specific metal removal rate

Fig 3 Comparison of the results from the fet-22 model (heavy line) and experimental results (broken line) for six different problems 27

P R E C I S I O N E N G I N E E R I N G 25

experimental observation (Fig 3) and with that of McGeough 21 . From the theoretical results, however, this is not reflected at the point ot entry of the electrolyte but can be seen clearly towards the exit. This is due to the boundary conditions chosen for obtaining the solution, ie

AT= 0.0 (16)

at the point of entry. Since electrolyte conductivity is temperature dependent it increases continuously along the flow direction. Similar trends have been obtained for other cases shown in Fig 3 (ie problems 20 to 23). The current density was calculated using Equation (2) and the corresponding electrolyte conductivity using Equation (12). These values.were used for consequent computations of temperature.

Fig 6 shows the finite element descritisation scheme used for the interelectrode gap for a complex shaped tool.

a

- - Ball bearing Outside diameter- 42 Inside diameter- 20 Height-12

~ - 8 8 - 1

---~ 25 ¢ .~-~

! O I

I

' d o

L 11'8 F

- E L B bearing Outside diameter-19 Inside diameter-6 Height-6

----------- Thrust bearing o Outside diameter-35 - Inside diameter-20

j Height-IO

- = - - r ~ 1

.[ J

Width- 6

b

_L --P 2.5 i ~ - |

,~--6.5-P I

24.2

t7.5 21.5

ILl III Ji r I

_1 L. 12.4 -

Ill Insulating sleeve

Fig 4 (a) The screw-nut assembly used 2~ to provide vertical feed to the machine table having a leakproof Perspex machining box. The nut has four ball bearings mounted on four sides to help prevent stick slip (all dimensions in mm). (b) Tool shown with four holes (each of 1.2mm diameter) to house copper constantan thermocouples for the measurement of electrolyte temperature at four locations

Table 2 Conditions used for FET-22

Feed rate, F F = 0.0 mm/s Ev = 10.0 V K o = 0.1412 ~Q-i mm-X x Pm = 7.87x 103 kg/m 3 To = 30°C Q = 31.25x 103 mm3/s

10 -1

Node Init ial Inter electrode gap at t ime t, mm No. Inter-

electrode gap, mm t=150 s t=300 s t=450 s t=600 s

1 1 5.0 5.1543 5.3040 5.4496 5.5915 2 4 5.0 5.1546 5.3047 5.4506 5.5926 3 7 5.0 5.1556 5.3065 5.4532 5.5962 4 10 7.25 7.3575 7.6359 7.5681 7.6712

5 12 14.0 14.0557 14.1111 14.1663 14.2214 6 15 7.0 7.1119 7.2221 7.3307 7.4376 7 18 5.0 5.1567 5.3087 5.4565 5.6003 8 20 5.0 5.1567 5.3087 5.4564 5.6001

9 23 5.0 5.1599 5.3153 5.4660 5.6123 10 26 7.5 7.605 7.7092 7.8117 7.9129 11 28 14.0 14.0557 14.111 14.1664 14.2214 12 31 6.5 6.6219 6.7417 6.8593 6.9749

13 34 5.0 5.1591 5.3134 5.4632 5.6089 14 36 5.0 5.1587 5.3126 5.4620 5.6073 15 39 5.0 5.1589 5.3130 5.4625 5.6080

Table 2 shows the interelectrode gap at different nodes located on the work surface for 150, 300, 450 and 600s intervals for the case of zero feed rate. For shapes involv- ing curves which cannot be fitted exactly by triangular elements the model presented here can be modified by using isoparametric elements 2s'26 which also reduces the number of elements to be used. Modifications can also be incorporated by accounting for the heat conducted away to the cathode and anode. Such work is in progress.

Conclusions

This paper has demonstrated the use of the finite element technique in predicting the anode shape obtained, metal removal, current density, temperature etc. in ecru using simple and complex shaped tools. The correlation between limited amount of experimental data and thebretical results is quite good. The model has been applied to the prediction of transient anode shape and it has been concluded that the fet-22 model can help in achieving an accurate tooling design in ecm.

Before analytical justification of the fet, for the analysis of ecm problems, is ful ly established, compre- hensive experimentation must be performed and the oa~a thus obtained tested against the analytical values. In this paper, the limited amount of experimental data aviiilable has been used to test the analysis, but this is not sufficient and currently the authors are engaged in experimentation to test the validity of their models.

References

1. Hoare J.P. Electrochemical Machining. Research Publications GMR-2427(1977) 1-35

2. LeBode M.A . et el. The Importance o f the Electrolyte in Electro- chemical Machining. Collection of Czechoslovak Chemical Communications, 1971, 36, 680- -688

26 PRECISION E N G I N E E R I N G

0.160

'o 0.155 "~

f

o., o w

0.145

51.0 --

50.6

50.2

49.8

/

~,9.4

/ /

/ /

/ /

/ /

i i

I 10

- - Current density

---- Conductivity

t=3OOs

/ /

/ /

/ /

/ /

/ /

/ /

- 55.0

0.165 -

54.6

0.160 --

54.2

- - i

' 0J55 - o

io

~ 0 J 5 0 - ~

5:5.4

0.145. -

53.0

24

/ /

/

/ /

/ /

/ /

/ /

/ /

/ /

/ /

/ /

/ /

/ /

/ /

/ /

/

/

0.140 -- I • 20 oJ4o - o I I I

I0 20 a Distance. mm b Distance, mm

Fig 5 Variation of the current density and electrolyte conductivity along the electrolyte flow direction for two problems 27

3. Bannard J. Electrochemical Machining. J. App. Electrochern. 1977, 7, 1-29

4. Bannard J. Effect of Surface Finish Obtained by Electro- chemical Machining on the Fatigue Life of Some Titanium Alloys. J.App. Electrochem. 1974, 4,229-234

5. Flatt R.K., Wood A.W. and Brook P.A. The Concentration Profiles in Solution at Dissolving Anode Surface - Zinc, Copper and Brass by the Freezing Technique. J. App. Electrochem. 1971, 1,35-39

6. Bannard J. The Use of Surface Active Additives in Electro- chemical Machining Electrolytes. J. App. Electrochem. 1974, 4, 117-124

7. Bannard J. On the Electrochemical Machining of Some Titanium Alloys in Bromide I~lectrolytes. J. App. Electrochem. 1976, 6, 477--483

8. Hoara J.P. i t el. An Investigation of the Differences between NaCI and NaCI0 s as Electrolytes in Electrochemical Machining. J. Electrochem. Soc., 1969, 116 (2), 199-203

9. Hoare J.P., Mira K.W. end Wallace A.J. ECM Batlaviour.of~eel in NaCI04 Electrolytes. Corrosion, 1971, 27 (5), 211-21.~

10.Hoare J.P. i t el. Electrochemical Mach!ning of High Temperature Alloys in NaC1Q s Solutions. J. Electrochem. Sac., 1973, 1 2 0 (SJ, 1071-1077

! ! .Moo K.W., LaBoda M.A. and Hoare J.P. Anodic Film Studies on Steel in Nitrate Based Electrolytes for Electro{:hemica[ Machining. .I. Electrochem. Sac., 1972, 119 (4), 419-427

5.0

IO.O

15.0

20.0

25.0

0 -

Electrolyte

t I 2 5 41 40 39

4 ~ 6 3 8 ~ 3 6

I / /,.~\\-.~.&rr 22 2 5 / j ~ / / " . t : I

12 15 18 20 25 26 28

Work

I I I I I I I I I 0 5.0 I0.0 15 20 25 50 35 40

Fig 6 Finite element descritisation scheme used for the interelectrode gap o f a complex sh.. aped tool

PRECISION E N G I N E E R I N G 27

12 LaBoda M.A. and Hoare J.P. Intricate Pattern ECM on Ferrous Alloys. J. Electrochem. Soc., 1975, 122 (1 I), 1489-1491

13.Collet D.E. et al. A Complex Variable Approach to Electro- chemical Machining Problems. J. Eng. Maths, 1970, 4, 29

14.Tsuei Y.G. et al. Theoretical and Experimental Study of Work- piece Geometry in Electrochemical Machining. ASME Paper No. 76WA/Prod. 1 - 5

15.Tipton H. The Dynamics of Electrochemical Machining. Proc. 5th Int. MTDR, 1964, 509 -522

16.K~nig W. and Hiimb H.J. Mathematical Model for the Calculation of the Contour of the Anode in ECM. Annals CIRP, 1977, 25 ( I ) 83 - 8 7

17. Loutrel S.P. and Cook N.H. A Theoretical Model for High Rate Electrochemical Machining. Trans. ASME Paper No. 73 Prod-2, I - 6

18.Kawafune K. Studies of Electrochl~mical Machining. Bull. JSME, 1968, 11 (45), 554 -469

19.Tipton H. The Determination of the Shape of Tools for Use in Electrochemical Machining. Research Report No. 40, 1971, The Machine Tool Industry Research Association, UK

20.Jain V .K. and Pandey P.C. Design of ECM TooHngs Using Finite

Element Technique. Proc. 8th A IMTDR Conference, 1978, 5 6 6 - 5 7 0

21 .McGeough J.A. Principles of Electrochemical Machining. Chapman and Hall, London, 1974

22.Huebener K.H. The Finite Element Method for Engineers. John Wiley and Sons, New York, 1975

23.Jain V.K. and Pandey P.C. On Some Aspects of Tool Design in ECM. Mech. Eng. Bull., 1977, 8 (3), 6 6 - 7 2

24.Zienkiwicz OoC. and Cheung Y.K. Finite Element in the Solution of Field Problems. The Engineer, September24, 1965

25.Desai C.S. and Abel J.F. Introduction to the Finite Ele- ment Method, Aff i l ia ted East West Press Ltd., New Delhi, 1977

26. Ziankiwicz O.C. The Finite Element Method. McGraw Hi l l Book Co., London, 1977

27.Bhatia S.M. Effect of Electrolyte Conductivity on the ECM Process. M. Tech. Dissertation l i t Bombay, India

28.Jain V.K. and Pandey P.C. Design Analysis of ECM toolings. Precision Engineering, October 1979, 1 (4), 199-206

28 PRECISION ENGINEERING