Wire drawing and coating using a combined geometry hydrodynamic unit: Theory and experiment

Journal of Materials Processing Technology 178 (2006) 98–110

Wire drawing and coating using a combined geometry hydrodynamic unit: Theory and experiment S. Akter ∗ , M.S.J. Hashmi School of Mechanical & Manufacturing Engineering, Dublin City University, Dublin 9, Ireland Received in revised form 13 February 2006; accepted 14 February 2006

Abstract In conventional drawing, the diameter of the wire is reduced by pulling it through a reduction die. By plasto-hydrodynamic technique, the wire diameter can be reduced by a limited amount without using any conventional reduction die. In this process, the wire is pulled through a tubular orifice of certain bore which is filled with a viscous fluid (polymer). The smallest bore size of the orifice is always greater than the diameter of the undeformed wire and hence it is possible to avoid metal-to-metal contact. During the past few years, plasto-hydrodynamic wire drawing has been investigated by different researchers using either stepped parallel or tapered bore pressure unit. In this paper, wire drawing and coating using a combined parallel and tapered bore hydrodynamic unit has been modelled considering that the polymer is a non-Newtonian fluid and results are obtained for different wire speeds in terms of the changes in viscosity, shear stress, percentage reduction in area, pressure distribution and coating thickness. © 2006 Published by Elsevier B.V. Keywords: Hydrodynamic; Combined geometry; Drawing; Coating

1. Introduction Hydrodynamic lubrication for wire drawing was first investigated by Christopherson and Naylor [1]. They used oil as a lubrication medium. Following their findings, Wistrich [2] carried out experimental work on the forced lubrication based on a pressure tube system. Dry soap was used for lubrication in a short nozzle as it is a very good boundary lubricant. Orlov et al. [3] developed a double die arrangement using externally pressurised oil. The lubricant was transported into the chamber formed by the exit cone of the pressure die and the entry part of the drawing die, where the pressurised lubricant provides the hydrodynamic lubrication during drawing. Though better result was claimed for reduced die wear and reduced power consumption, there was a lack of substantial evidence to support this claim. Later, Thompson and Symmons [4] experimented with polymer melt as a means of lubrication. Stevens [5] found that the polymer is very effective as lubricant and it also leaves a protective coating on the wire. Crampton et al. [6] further investigated plasto-hydrodynamic polymer lubrication of wire. He showed in his analysis that the reduction and coating thickness on the ∗

Corresponding author. E-mail address: [email protected] (S. Akter).

0924-0136/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.jmatprotec.2006.02.010

wire depends on the wire speed, polymer melt viscosity and the die chamber configuration. This was further investigated by Parvinmehr et al. [7] who used a die-less unit and the smallest bore of which was slightly greater than the diameter of the wire. For both of these works, the tests were carried out with wires of 1.6 mm diameter. They also carried out a non-Newtonian analysis for wire drawing using a stepped parallel bore unit. Later, Hashmi et al. reported, in reference [8], a Newtonian analysis for wire drawing in a conical tubular orifice. Also two different theoretical models have been presented in references [9,10] for plasto-hydrodynamic pressure through a combined parallel and tapered bore unit for wire coating process only without any reduction in wire diameter. In this paper, a non-Newtonian plasto-hydrodynamic wire drawing model has been presented, the solution of which gives the prediction of the coating thickness, percentage reduction in diameter of the wire and the pressure distribution within the unit. 1.1. Non-Newtonian pressure analysis for coating process only Fig. 1 shows a schematic diagram of the plasto-hydrodynamic wire coating and drawing system used in this study. The con-

S. Akter, M.S.J. Hashmi / Journal of Materials Processing Technology 178 (2006) 98–110

99

Nomenclature B dP dX G h k K0 K l N P Q T V x1 Y0 Y

slope for linear deformation change in hydrodynamic pressure (N/m2 ) increment in length (m) temperature–pressure coefficient (◦ C/N m−2 ) radial gap in the unit and the wire (m) coefficient for viscosity (N s/m2 ) strain hardening constant (N/m2 ) non-Newtonian factor (m4 /N2 ) length of section (m) strain rate sensitivity constant pressure gradient (N/m3 ) flow of polymer (m2 /s) temperature (◦ C) wire velocity (m/s) yielding point (m) initial yield stress (N/m2 ) yield stress of the wire material (N/m2 )

Greek symbols α ln(µ/µ0 )/(T − T0 ) (◦ C−1 ) γ shear rate (s−1 ) µ viscosity (N s/m2 ) τ shear stress (N/m2 ) τc wall shear stress (N/m2 ) Subscripts 0 denotes reference point 1 parallel section of the unit 2 tapered section of the unit i, j at the increment of length Unit Pressure, shear stress 1 Bar = 105 N/m2 = 0.1 M Pa Viscosity 1 N s/m2 = 10 Poise

tinuum (wire, strip, tube, rope, etc.) enters the leakage control unit attached to the melt chamber. It then passes through the melt chamber and enters the plasto-hydrodynamic pressure unit where the hydrodynamic pressure assists drawing and coating the continuum. The reduced and coated continuum is then wound on the bull block which is driven by a variable speed motor. Fig. 2 shows the combined hydrodynamic pressure unit where h1 , h2 and h3 , respectively, are the gaps between the wire and the unit at the entrance at, the step and at the exit of the unit. L1 and L2 are the lengths of the parallel part and the tapered part of the unit, respectively. The following analysis is based on the geometrical configuration shown in Fig. 2. To simplify the analysis, the following assumptions were made: (a) The flow of the polymer melt is laminar. This seems to be a reasonable assumption since the viscosity of the polymer is high and the gaps of the pressure unit are low.

Fig. 1. Schematic diagram of plasto-hydrodynamic wire drawing and coating process.

(b) The polymer layer thickness is small compared to the dimensions of the unit. This enabled the analysis to be conducted in cartesian co-ordinate system. (c) The flow is in steady state. (d) The pressure is constant across the film thickness. (e) The continuum is concentrically located. (f) The pressure gradient dP/dx is independent of y. 1.2. Analysis for the parallel part of the unit The relationship between the pressure and shear stress gradient is given by


  dp dτxy dτ (1) = = dx 1 dy 1 dy 1 Two different equations are generally used to express the stress and shear rate relation for a polymer melt. The first is power law equation given by
 dU n τ1 = k (2) dy This equation is applicable for any type of fluid. Here, n is the power law index which equals to 1 for a Newtonian fluid, greater than 1 for a dilatant fluid and less than 1 for a pseudoplastic fluid. In this equation, τ is the shear stress, k the viscosity coefficient and (dU/dy) is the shear rate.

Fig. 2. Schematic diagram of a combined parallel and tapered bore unit.

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The second is an empirical equation relating the shear stress and the shear rate of polymer melts suggested by Rabinowitsch [11] in the form
 dU 3 τ1 + Kτ1 = µ µγ (3) dy Here, K is the non-Newtonian factor, µ the viscosity of the polymer melt and τ and (dU/dy) are the shear stress and shear rate, respectively. In this paper, Eq. (3) has been used for the hydrodynamic analysis within the combined unit. Integration of Eq. (1) gives τ1 = P1 y + τc1 where τ c1 is the shear stress at y = 0 and P 1 = (dP/dx)1 is assumed to be independent of y. Substituting for τ 1 in Eq. (3) gives
 dU µ = P1 y + τc1 dy 1 3 2  + K(P  1 y3 + τc1 + 3P  1 y2 τc1 + 3τc1 P1 ) 3

2

which after integration becomes U1 =

P1 y2 τc1 y + 2µ µ +

K P13 y4 3 2 3 ( + τc1 y + P12 y3 τc1 + τc1 P1 y2 ) + C µ 4 2

(4)

where C is the constant of integration. The boundary conditions are: (a) at y = 0, U1 = V; (b) at y = h1 , U1 = 0. Applying condition (a) in Eq. (3), C = V and hence U1 =

P1 y2 τc1 y + 2µ µ 3 2 K P 3 y4 3 y + P12 y3 τc1 + τc1 P1 y2 ) + V + τc1 + ( 1 µ 4 2

(5)

Also substituting condition (b) in Eq. (4) and after rearrangement it gives 3 3 2 + (P1 h1 )τc1 τc1 2   1 µV P1 h1 P13 h31 2 2 + ( + P1 h1 )τc1 + + + =0 (6) k Kh1 2K 4 The real root of this equation is: ⎛ ⎞ 
3 1/2 1/3 2V 2 µV µ 1 1 1 ⎠ + + τc1 = ⎝− + P 2 h2 2Kh1 27 K 4 1 1 4K2 h21 ⎛

⎞ 
3 1/2 1/3 2V 2 µV µ 1 1 1 ⎠ + ⎝− − + + P 2 h21 2Kh1 4K2 h21 27 K 4 1 1 − P1 h1 2

The flow of the polymer melt in the first part of the unit is given by h1 Q1 =

U1 dy 0

Substituting for U1 from Eq. (5) in the above equation and integrating gives Q1 =

P1 h31 τc1 h21 + 6µ 2µ   3 h2 2 P  h3 P12 h41 τc1 τc1 τc1 K P13 h51 1 1 1 + + + + + Vh1 µ 20 2 4 2 (7)

For continuous flow operation, Q1 = Q2

(8)

Hence, establishment of flow equation in the tapered part is necessary. The pressure gradient in the second part under steady state condition gives

 dp dτ = (9) dx 2 dy 2 The boundary conditions are: (c) at y = 0, U2 = V; (d) at y = h, U2 = 0. The shear stress and shear rate is related by
 dU τ2 + Kτ23 = µ = µγ (10) dy Differentiating this equation with respect to y gives  
 2 2 dτ2 dτ d U 2 + 3Kτ22 =µ dy dy dy2 Substituting (dτ 2 /dy) for (dP2 /dx) from Eq. (9) and integrating it twice it gives 

  2
 dP2 y2 y τ23 dP2 2 dP2 + 3K τ2 + y dx 2 dx 2 3 dx

   τ24 dp2 y 3 dP2 + + τ2 + C2 y + C3 = µU 12 dx dx 3 and then setting the boundary conditions (c) and (d), the velocity profile becomes

y  KP2 y4

y  P2  y U2 = V 1 − − 1− − (h − y) h 4µ h 2µ  y

 3KP2 τ22 y(h − y) τ24 h − 1 − − µ 2 12

(11)

S. Akter, M.S.J. Hashmi / Journal of Materials Processing Technology 178 (2006) 98–110

The flow of the polymer melt in the tapered part of the unit is, h Q2 =

U2 dy

101

In hydrodynamic drawing and coating process, the temperature of the pressure unit is kept constant by heater band. Therefore, in this analysis, it is assumed that there is no change in temperature.

0

Substituting for U2 from Eq. (11) the flow becomes   P2  h3 K τ22 P2 h3 Vh Q2 = − − + 12µ µ 4 2

1.4. Introduction of shear rate and viscosity change (12)

Noting that dQ/dx = 0, Eq. (12) gives   P2 h3 K τ22 P2 h3 + = Vh + C4 6µ µ 2 and then integrating this equation it gives   6µV C5 V   + C6 + P2 = 2h2 M 1 + 3Kτ22 h

⎛ −⎝

−µj−1 γj + 2k

Qi = Q1 =

and

  6µV C5 V C6 = − + 2 M[1 + 3Kτ22 ] h3 2h3 Therefore, the pressure distribution becomes,     3µ 6µV 1 1 1 1 P2 = + − 2 − M[1 + 3Kτ22 ] h h3 M[1 + 3Kτ22 ] h2 h3 (14)

1.3. Solution procedure 1.3.1. Introduction of viscosity and shear stress in pressure distribution Any increase in the hydrodynamic pressure has an effect on viscosity which is equivalent to the reduction in temperature and vice versa [12]. The simplified form of Arrhenius equation to relate the viscosity and temperature can be expressed as (15)

which in finite difference term can be written as µi = µi−1 e(−α(Ti −B.DP−Ti−1 ))

1 3k

3 +

µ

1/2 ⎞1/3 ⎠ (17)

(16)

 h3 P1i τc1 h21 K 1 + + 6µi−1 2µi−1 µi−1   3 h2 2 P  h3 τc1 τc1 P  31i h51 P  21i h41 τc1 1 1i 1 × + + + 20 2 4 2

+ Vh1

(18)

This equation is a cubic equation, where each term except P i is known (viscosity can be determined from Eq. (15)). Solution of this equation gives this term. So that, P1 (i) = P1 (i − 1) + P 1 dX also dP = P1 (i) − P1 (i − 1). The initial values for the parallel part are, hydrodynamic pressure P1 (1) = 0, T(1) = polymer melt temperature and µ(1) = viscosity at T(1). Similarly, the hydrodynamic pressure at the tapered part can be obtained by differentiating Eq. (14) with respect to x and then writing in finite difference form  P2(j) = P2(j−1) +

  6µj−1 V 6µj−1 1 + 2 2 2 ] M[1 + 3Kτ2j ] h(j) M[1 + 3Kτ2j

⎡
 ⎤⎤  2 ] P1 (L1)M[1+3Kτ2j 1 1   2 +V h3 − h2 6µj−1 ⎥⎥ 1 ⎢ ⎥⎥ dX ⎢
 × ⎦⎦ 3 ⎣ h(j) 1 1 − 2 2 h h 3

where α can be replaced by b and B by c/b.

 j−1 γj 2 2k

where γ(j) = V/h(j) and h(j) = h(j − 1) − M dX. Eqs. (6), (7), (12) and (17) may be solved simultaneously. Numerical values of Pm (maximum pressure at the step) and hence P1 and P2 may be substituted in above equations, using an iteration technique, until the condition Q1 = Q2 is satisfied. Thus, shear stresses on the wire and volumetric flow rate can be determined. To determine the hydrodynamic pressure developed at each small increment of length of the first unit, Eq. (7) can be written in the following way

2

µ = µ0 e(−bdT +cdP)




(13)

where M can be obtained from the geometry of the unit, M = (h2 − h3 )/L2 and h = h2 − Mx. The boundary conditions are at x = L1 , h = h2 , P2 = P1 (L1 ) and x = L1 + L2 , h = h3 , P2 = 0. Substituting the conditions in Eq. (13) it gives
  P1 (L1 )M[1+3Kτ22 ] 1 1 + V h3 − h2 2 6µ
 C5 = 1 1 − h2 h2 3

Solution of Eq. (10) gives the shear stress where viscosity and shear rate change can be incorporated ⎛  ⎞1/3 
 3

µ γ 2 1/2 γ µ 1 j−1 j j−1 j ⎠ + + τ2j = ⎝ 2k 3k 2k

2

(19)

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Fig. 3. Geometry used for wire drawing analysis in the combined unit.

The viscosity can be determined from Eq. (16) by rearranging as µj = µj−1 e(−α(Tj −B.DP−Tj−1 ))

(20)

where dP = P2 (j) − P2 (j − 1) and h(j) = h(j − 1) − M dx. The initial values for the tapered part are,P2 (1) = P1 (L1 ), T(1) = T(L1 ), µ(1) = viscosity at T(L1 ), h(1) = h2 and P2 (L2 ) = 0, at h(L1 + L2 ) = h3 .

(22)

Equilibrium of forces on the wire in x direction gives
 1 2 πD1 = (πD1 x1 )τC1 σx1 4

σx1 =

As the hydrodynamic pressure starts increasing in the first section from zero, it is considered that the reduction also starts in the first section. Consider a distance x1 from the entry point of the unit where yielding of wire has just commenced (see Fig. 3a). The principal stresses acting on the wire are: σ 1 = σ x1 and σ 2 = σ 3 = −P1 , where σ x1 is the axial stress and P1 is the radial pressure at point x1 . The stress–strain characteristics of the wire are assumed to be of the form Y = Y0 + K0 ε

P1 + σx = Y

hence

1.5. Wire drawing analysis for the combined parallel and tapered unit—the yielding position of the wire inside the die-less unit

n

Therefore, yielding commences when the condition Y = Y0 is satisfied. Using the Tresca or von-Mises theory of yielding gives

(21)

where Y is the flow stress of the wire material and Y0 is the yield stress.

4x1 τC1 D1

(23)

The pressure at x1 is given by P1 =

Pm x1 L1

(24)

as the pressure in the section builds up linearly along its length. Here, Pm is the pressure at the junction of the two parts of the unit. Substituting Eqs. (23) and (24) in Eq. (22) and rearranging, the position of yield in the wire x1 may be expressed as: x1 =

Y0 4τC1 D1

+

Pm L1

(25)

This equation enables the prediction of the position where the wire starts to yield plastically within the first part of the unit.

S. Akter, M.S.J. Hashmi / Journal of Materials Processing Technology 178 (2006) 98–110

As soon as yielding occurs, plastic deformation will continue as long as Eq. (22) is satisfied.

This is the governing differential equation in the deformation zone for the axial stress in the wire. Rewriting this equation in finite-difference form gives

1.6. Deformation zone σxi = Consider a section of the unit within which the wire is plastically deformed as shown in Fig. 3b. Since the deformation of the wire is not pre-defined as a function of x, the equations containing this variable cannot be solved analytically. Hence, a finite-difference technique is adopted to solve these equations governing the deformation zone assuming that between any two points of small distance dx apart on the deforming wire, the deformation takes place linearly, therefore,

2(Di−1 − Di )Yi 4τCi dx + + σxi−1 Di Di

(29)

This equation is a function of shear stress on the wire and it must be determined independently. The equation for the wall shear stress in the first part of the unit may be written as (from Eq. (5)), ⎛ ⎞ 
3 1/2 1/3 2 2 µi−1 Vi µi−1 Vi 1 1 1 2 ⎠ τci = ⎝− + + + P  i h2i 4K2 h2i

2Khi

⎛

dD = constant = B 2dx



+ ⎝−

Expressing this equation in finite-difference form yields, Di = Di−1 − 2Bi ∆x

103

µi−1 Vi − 2Khi

µi−1 Vi2 4K2 h2i 2

27

+

K

1 27




4

1 1 2 + P  i h2i K 4

3 1/2

⎞1/3 ⎠

1 − Pi hi 2

(26)

(30)

and similarly the variation in the gap, dh is given by dh =

This equation contains the pressure gradient and variation of the wire velocity in the deformation zone and they must be determined separately. The flow of the polymer melt in this region may be expressed as (from Eq. (6)),

dD 2

hence hi = hi−1 + Bi ∆x

(27)

Qi = Q1 =

where Bi is the slope of the deformation profile within distance x. Considering a small section of the wire, the radial equilibrium of forces gives

  πDdx πDdx σr (πDdx) = −P cos α + τC sin αn cos α cos α hence



σr = −P 1 −



τC tan α P

(28)

The value of τ c /P has been shown to be very small and since α is also very small, the term τ c tan α/P can be ignored. Equilibrium of forces in x direction gives


π  1 2 −σx πD + (σx + dσx ) (D + dD)2 4 4

  πDdx πDdx +P sin α + τC cos α = 0 cos α cos α Rearranging and ignoring powers of dD 2dDσx + Ddσx + 4Pdx tan α + 4dxτC = 0 but tan α = dD/2dx hence 2dD(P + σx ) + Ddσx + 4dxτC = 0 Substituting for (P + σ X = Y) and rearranging gives dσx = −

2dDY 4τC dx − D D

 ×

Pi h3i τci h21 K + + 6µi−1 2µi−1 µi−1

P  3i h5i τ 3 h2 P  2 h4 τci τ 2 P  h3 + ci i + i i + ci i i 20 2 4 2

 + Vhi (31)

Eqs. (30) and (31) may be solved simultaneously in order to determine P i and τ ci by iterating P i at point i in the deformation zone. Therefore, P1(i) = P1(i−1) + Pi ∆x

(32)

The variation of speed of the wire in the deformation zone may also be included in the analysis. The continuity of flow of metal through the element (Fig. 3c) gives π π (V + dV ) (D + dD)2 = VD2 4 4 Ignoring power of dD gives 2dD dV =− V (D + 2dD) Rewriting this equation in finite difference form gives Vi =

1−

Vi−1 2(Di−1 −Di ) 2Di−1 −Di

(33)

The true strain–stress relationship of the wire in the deformation zone may be shown as: Yi = Y0 + K0 εni

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where εi = 2 ln(D1 /Di ). Therefore,

n D1 Yi = Y0 + K0 2 ln Di

respectively, as (34)

Substituting for Yi in Eq. (29) gives 



n  Di−1 − Di D1 Y0 + K0 2 ln σxi = 2 Di Di +

4τCi dx + σxi−1 Di

(35)

The new shear stress within the zone where deformation of wire has been observed is then given by τi = Pi hi + τci

(36)

At any point i in the deformation zone, Di , hi and Vi may be calculated from Eqs. (26), (27) and (33), respectively, for an arbitrary value of Bi . By substituting the values of Di and Vi in Eq. (34), the yield stress Yi may be calculated. Also by substituting for Vi and hi in Eqs. (30) and (31) gives P i and τ ci and simultaneously iterating for P i until Eq. (31) is satisfied. Hence, Eq. (32) gives P1 (i). Similarly, τ ci and Di may be substituted in Eq. (35) to evaluate σ xi . This P i , τ ci and hi gives the shear stress within the deformation zone. Other variables in the above equations are known physical properties. Having calculated σ xi , Pi and Yi , the values of Bi may be iterated in the above equations until equation σ xi + Pi = Yi is satisfied. 1.7. After the step (in the tapered unit) Referring to Fig. 3c, which shows an element of the wire within a straight conical profile, the pressure equation in finite difference term is (from Eq. (19)),    6µj−1 V 1 P2(j) = P2(j−1) + 2 ] h2 M[1 + 3Kτ2j j   6µj−1 1 + (37) 2 ] h3 M[1 + 3Kτ2j j The change in shear stress where viscosity and shear rate change can be incorporated is ⎛  ⎞1/3 
 3

µ γ 2 1/2 µ 1 γ j−1 j j−1 j ⎠ τ2j = ⎝ + + 2k 3k 2k ⎛ −⎝

−µj−1 γj + 2k




1/2 ⎞1/3 3

 2 1 µj−1 γj ⎠ + 3k 2k

where γ(j) = V/hj and τcj = τ2j − Pj hj

(38)

The change in gap between the wire and the unit, velocity and diameter within the deformation zone can be expressed,

hj = hj−1 − (M − kj )∆x
 Dj−1 2 Vj = Vj−1 Dj Dj = Dj−1 − 2kj ∆x

(39)

where tan α = −kj . The true strain–stress relationship of the wire in the deformation zone may be shown as:

n 
D1 (40) Yj = Y0 + K0 2 ln Dj The increment in axial stress may be expressed as, 


n 
D1 Dj−1 − Dj σxj = 2 Y0 + K0 2 ln Dj Dj +

4τCj dx + σxj−1 Dj

(41)

The Tresca yield criterion for plastic yielding in finite difference form becomes; P2(j) + σxj ≥ Yj

(42)

The plastic deformation is calculated on the basis of Eq. (42) combined with Eqs. (37)–(41) for small increment of x. The initial conditions for the deformation of wire in the tapered unit are, P2 (1) = P1 (step), htaper (1) = hparallel (step), Ytaper (1) = Yparallel (step), Vtaper (1) = Vparallel (step), σ xtaper (1) = σ parallel (step) and Dtaper (1) = Dparallel (step). For both coating and drawing, the drag force may be given by x=L

Fd =

πDτx dx

(43)

x=0

The drawing stress is given by σx =

4xτx D

(44)

where D is the equivalent diameter. 2. Results and discussions Theoretical results have been calculated on the basis of the equations derived in the theoretical analysis. Results were obtained in terms of pressure distribution, viscosity, shear stress, percentage reduction in diameter and coating thickness for different combination of polymer melt, wire material and geometry of the pressure units. Case 1. The polymer was Borosiloxane and the wire materials were copper and stainless steel. The properties for Borosiloxane and the wire material and the geometry of the pressure units were taken from references [10,12–14].

S. Akter, M.S.J. Hashmi / Journal of Materials Processing Technology 178 (2006) 98–110

G = 4.2 × 10−1 ◦ C/(N mm−2 ), α = 0.016 ◦ C−1 , dX = 1 mm, h2 = 0.25 mm, h3 = 0.05 mm, h1 = 0.5 mm, K = 5.6 × 10−11 m4 /N2 , L1 = 145 mm and L2 = 35 mm. Initial polymer viscosity at polymer melt temperature 270 ◦ C at different wire velocities and shear rates are: from 0.1 to 0.3 m/s, the shear rate is 200 to 600 s−1 and the polymer melt viscosity is 50 N s/m2 . At 0.5 m/s, the shear rate is 1000 s−1 and the polymer melt viscosity is 47 N s/m2 . At 1 m/s, the shear rate is 2000 s−1 and the polymer melt viscosity is 45 N s/m2 . The properties for stainless steel wire were: Y0 = 180 MN m−2 , K0 = 1050 MN m−2 , n = 0.98 and D1 = 2.0 mm. The properties for copper wire were: Y0 = 130 MN m−2 , K0 = 281 MN m−2 , n = 0.76 and D1 = 2.0 mm The combined pressure unit was first used in reference [14] for hydrodynamic wire drawing and coating process. Due to limitations of the apparatus it was not possible to maintain wire velocity more than 0.16 m/s in case of copper wire and 0.1 m/s for stainless steel wire. In this current paper, theoretical results have been calculated and compared with the experimentally measured results based on reference [14]. In Fig. 4a, theoretical pressure distributions have been shown within the unit using copper wire for different wire velocities. In the parallel part of the unit, the hydrodynamic pressure increases linearly from zero to a certain pressure and in the tapered part of the unit it increases gradually upto a maximum pressure and then decreases. Also, with the increase in velocity, the pressure also increases. The increase is not proportional to the increase in velocity where viscosity has a shear thinning effect. In Fig. 4b, theoretical and experimental pressure distributions are presented for copper wire at wire velocity 0.16 m/s. The theoretical pressure is less than the experimental pressure. Fig. 5a shows the theoretical pressure distributions using stainless steel wire at different wire velocities. Fig. 5b shows the comparison of experimental and theoretical pressure distribution at wire velocity of 0.1 m/s. Both Fig. 5a and b show the similar pressure profiles as in Fig. 4a and b. The copper wire is much softer than the stainless steel wire. As such, elastic as well as plastic deformation (if any) of the copper wire would be greater than that for the stainless steel. In reference [14], it was found that the area of the copper wire was reduced only marginally 0.37% at wire velocity of 0.16 m/s and in the case of stainless steel wire at 0.1 m/s it was only 0.02%. However, theoretical solutions are given here for predicted reduction in area and coating thickness at velocity range of 0–1.0 m/s. Fig. 6 shows the theoretical percentage reduction in area for copper and stainless steel wire at different velocities. At velocity of 0.16 m/s for copper wire, the predicted reduction in area is about 0.2%. The theoretical reduction increases with the increase in wire velocities. At wire velocity of 1.0 m/s, this percentage reduction in area is 7.3%. There is no theoretical reduction for stainless steel wire at a velocity of 0.1 m/s while the experimental reduction was 0.02%. At a wire velocity of 1.0 m/s, the predicted reduction in area is about 1.1%.

105

Fig. 4. (a) Theoretical pressure distribution within the unit for copper wire for different wire velocities (polymer—Borosiloxane) and (b) pressure distribution within the unit for copper wire for wire velocity 0.16 m/s (polymer—Borosiloxane).

Fig. 7 shows the predicted coating thickness for different wire velocities for both types of wire. With the increase in the wire velocity, the percentage reduction in area as well as the coating thickness increases. However, the coating thickness is higher for copper wire than the stainless steel. At the velocity of 1.0 m/s, for copper wire this thickness is about 0.086 mm whereas for the stainless steel wire it is about 0.054 mm. Figs. 8 and 9 show the theoretical change in the viscosity of polymer from the entrance of the plasto-hydrodynamic unit

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Fig. 6. Percentage reduction in diameter of wire for different velocities (polymer—Borosiloxane).

Figs. 10 and 11 show the theoretically predicted change in the shear stress for copper and stainless steel wire at different velocities due to the change in the shear rate and viscosity. In the parallel part, the shear stress increases linearly upto the step. At the step, there is a sudden drop in the shear stress. In the

Fig. 5. (a) Theoretical pressure distribution within the unit for stainless steel wire for different wire velocities (polymer—Borosiloxane) and (b) pressure distribution within the unit for stainless steel wire for wire velocity 0.1 m/s (polymer—Borosiloxane).

for both copper and stainless steel at different velocities. In the parallel part, the viscosity of the polymer increases in a linear manner. In the tapered part, the viscosity increases non-linearly upto a certain point and then decreases gradually. It may be noted that in Figs. 4 and 5, in the parallel part of the unit, the hydrodynamic pressure increases from zero to a certain pressure and in the tapered part of the unit, it increases gradually upto a maximum pressure and then decreases. Therefore, the dependence of viscosity on pressure has been established.

Fig. 7. Coating thickness for different wire velocities (polymer—Borosiloxane).

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Fig. 8. Viscosity distribution along the unit during copper wire drawing (polymer—Borosiloxane).

tapered part, it increases gradually upto the maximum. After that, the shear stress decreases near the end of the tapered part. For each type of wire, the shear stress increases with the increase in velocity. Figs. 12 and 13 show the comparison of the theoretical and the experimental drawing force for the copper wire and the stainless steel wire, respectively. The drawing force generally increases with the increase in the wire velocity. The theoretical drawing force agrees well with the experimental force in the case of stainless steel wire. For the copper wire, the experimental value

Fig. 9. Viscosity distribution along the unit during stainless steel wire drawing (polymer—Borosiloxane).

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Fig. 10. Magnitude of change in shear stress during copper wire drawing (polymer—Borosiloxane).

is considerably lower than the theoretically predicted value. The reason for such discrepancy is not clear. However, some problem with the experimental apparatus cannot be ruled out. Case 2. The polymer was Durethan B31F (Nylon-6). Properties of polymer Nylon 6 and the geometry of the pressure unit were taken from references [15,16].

Fig. 11. Magnitude of change in shear stress during stainless steel wire drawing (polymer—Borosiloxane).

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Fig. 12. Drawing load for copper wire at different wire drawing velocities (polymer—Borosiloxane).

Initial polymer viscosity at polymer melt temperature 270 ◦ C at different wire velocities and shear rates are: - at 4 m/s, the shear rate is 5714 s−1 and the polymer melt viscosity is 100 N s/m2 . - at 8 m/s, the shear rate is 11428 s−1 and the polymer melt viscosity is 65 N s/m2 . - at 12 m/s, the shear rate is 17142 s−1 and the polymer melt viscosity is 50 N s/m2 .

Fig. 13. Drawing load for stainless steel wire at different wire drawing velocities (polymer—Borosiloxane).

Fig. 14. Pressure distribution within the unit for different velocities (polymer—Nylon 6).

The other factors are: G = 0.32 ◦ C/(N mm−2 ), α = 0.0114 ◦ C−1 and K = 10.1 × 10−11 m4 /N2 ; L1 = 17 mm, L2 = 5 mm, h1 = 0.7 mm, h2 = 0.25 mm and h3 = 0.051 mm. In reference [16], wire coating process was carried out with Nylon 6 polymer melt on galvanized mild steel wire. Experimental results were shown for coating and drawing force at different velocities, melt temperatures and argon back pressures. In this case, the objective was only to coat the wire, no reduction of wire was intended. The geometrical parameters and other variables were maintained at such values so that there could not be any deformation of the wire. In the present paper, theoretical results have been derived to show the hydrodynamic pressure, drawing force and coating. The theoretical drawing force and coating have been compared with the experimental results based on reference [16]. Fig. 14 shows the theoretical hydrodynamic pressure within the unit for polymer melt temperature of 270 ◦ C and wire velocities of 4, 8 and 12 m/s. The trend of the pressure profiles is similar to those in Figs. 4 and 5. The pressure increases with the increase in the wire velocity. However, the increase is not proportional to the increase in velocity because of the shear thinning effect of viscosity. Fig. 15 shows the drawing force at a temperature of 270 ◦ C and 5 bar argon back pressure. The theoretical drawing force has also been given for argon back pressure of 5 bar. The drawing force generally increases with the increase in the wire velocity.

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Fig. 16. Coating thickness for different wire velocities (polymer—Nylon 6).

Fig. 15. Drawing load for different wire drawing velocities (polymer—Nylon 6).

The theoretical drawing force matches well with the experimental force though it is slightly lower in magnitude. From Fig. 14, it can be seen that the maximum hydrodynamic pressure at the wire velocity of 12 m/s and polymer melt temperature of 270 ◦ C is about 165 bar. According to von Mises theory of yielding, the deformation of the wire starts when the combined effect of the hydrodynamic pressure and axial stress equals or exceeds the elastic limit of the wire material. In this case, the drawing force at polymer melt temperature of 270 ◦ C, back pressure 5 bar and wire velocity of 12 m/s, is about 24 N (Fig. 13). The axial stress due to this load is 621 bar. The combined effect of the hydrodynamic pressure and axial stress is therefore 786 bar (78.6 MPa) which is less than the elastic limit of mild steel which is 280 MPa. As such, the combined effect of the pressure and the drawing load is not sufficient to cause any plastic deformation in the wire. Experimentally [16], it was observed that there was no plastic deformation of the wire at any velocity. If there were no deformation in the wire, then the coating thickness would be equal to the gap between the wire and the exit pressure unit. Fig. 16 presents the theoretical and experimental coating thickness on wire deposited using a combined parallel and tapered bore unit. The variables were drawing speed (upto 12 m/s), argon back pressure (5 and 10 bar) and polymer melt temperature of 230 ◦ C. The coating was found to be continuous and concentric. The experimental thickness was almost equal to the theoretical thickness (exit gap between the wire and the unit, i.e. 0.051 mm). It is evident that the prospect for the application of concentric and continuous coating on wire is very good. For drawing application, where the final size does not have to be substantially reduced and the presence of the coating is desirable, this process would be ideal for simultaneous drawing and coating of a continuum of various shaped cross-sections.

3. Conclusion A non-Newtonian plasto-hydrodynamic model for wire drawing using a combined parallel and tapered bore unit has been presented. The theoretical pressure distribution, percentage reduction in area and coating thickness for different wire velocities have been shown. The theoretical viscosity and change in shear stress also have been predicted. Comparisons with experimental results show reasonable correspondence in trend. References [1] D.C. Christopherson, P.B. Naylor, Promotion of fluid lubrication in wire drawing, Proc. Inst. Mech. Eng. (1955) 643. [2] J.G. Wistrich, Lubrication in wire drawing, Wear (1957) 505–511. [3] S.I. Orlov, V.L. Kolmogorov, V.I. Uralskll, V.T. Stukalov, Integrated development and introduction of new high speed mills and hydrodynamic lubrication system for drawing wires, in: Steel in the USSR, vol. 10, 1974, pp. 953–956. [4] P.J. Thompson, G.R. Symmons, A plasto-hydrodynamic analysis of the lubrication and coating of wire using polymer melt during drawing, Proc. Inst. Mech. Eng. 191 (13) (1977) 115. [5] A.J. Stevens, A plasto-hydrodynamic investigation of the lubrication and coating of wire using a polymer melt during drawing process, M.Phil. Thesis, Sheffield City Polytechnic, 1979. [6] R. Crampton, G.R. Symmons, M.S.J. Hashmi, A non-Newtonian plastohydrodynamic analysis of the lubrication and coating of wire using a polymer melt during drawing, in: Proceedings of the International Symposium on Metal Working Lubrication, San Fransisco, USA, 1980. [7] H. Parvinmehr, G.R. Symmons, M.S.J. Hashmi, A non-Newtonian plasto-hydrodynamic analysis of die-less wire drawing process using a stepped bore unit, Int. J. Mech. Sci. 29 (4) (1987) 239–257. [8] M.S.J. Hashmi, G.R. Symmons, H. Parvinmehr, A novel technique of wire drawing, J. Mech. Eng. Sci. Inst. Mech. Eng. 24 (1982). [9] M.S.J. Hashmi, G.R. Symmons, A numerical solution for the plastohydrodynamic drawing of a rigid non-linear strain hardening continuum through a conical orifice, J. Math. Modell. 8 (1985) 457–462. [10] M.A. Nwir, M.S.J. Hashmi, Plasto-hydrodynamic pressure distribution in a complex geometry pressure unit: experimental results using Borosiloxanes as pressure medium, in: Proceedings of the IMC-11 Conference, Belfast, Northern Ireland, 1994. [11] B. Rabinowitsch, Uber Die Viskostat und Elastizitat Von Solen, Z. Phys. Chem. A145 (1929) 141.

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[12] F.N. Cogswell, The influence of pressure on the viscosity of polymer melts, Plast. Polym. 41 (1973) 39. [13] J.B. Hull, A.R. Jones, A.R.W. Heppel, A.J. Fletcher, S. Trengove, The effects of temperature rise on the rheology of carrier media used in abrasive flow machining, Surf. Eng. Eng. Appl. 2 (1993) 240. [14] M.A. Nwir, M.S.J. Hashmi, Hydrodynamic pressure distribution in tapered, stepped, parallel bore and complex geometry pressure units:

experimental results, in: Proceedings of the International Conference of Advances in Materials and Processing Technologies (AMPT’95), Dublin City University, Dublin, Ireland, 1995, pp. 1431–1437. [15] Bayer, Application technology information, ATI 189e, 1988. [16] S. Akter, M.S.J. Hashmi, Experimental results of high speed wire coating using a combined hydrodynamic unit, in: Proceedings of the International Conference of Advances in Materials and Processing Technologies (AMPT’97), Malaysia, 1997, pp. 378–386.