Compensating for die swell in the design of profile dies.
Designing a profile extrusion die to produce a part with a desired (and often very complex) shape usually involves an arduous "cut and try" procedure. Typically, two steps are involved. A profile die is usually composed of three regions: inlet region, a transition region and a "land" region. The shape of the inlet to the die is normally specified and not varied in the design. The designer usually begins by making the exit or land region of the die the same shape (or "profile") as the part to be produced. If the effects of die swell are small, the design procedure usually involves merely changing the geometry of the transition region or the local land length until the die is "balanced," i.e., a design is found that gives a near uniform flow at the exit of the die along the entire profile (1).
However, even when a die is balanced, if the effects of die swell are appreciable, the shape of the extrudate produced downstream will be substantially different from that at the die exit. Some designers try to compensate for this by making changes to the profile at the exit of the die. But in most cases, this problem is overcome by adding a sizing or calibration system downstream of the die, which holds and cools the extrudate to the desired shape with the aid of a vacuum, water or air.
Here our goal is to solve what has become known as the "inverse" problem. That is, we wish to determine what the shape of the exit opening of the die should be to produce a profile with a specified shape down stream. There have been a few fully 3-D analyses of the inverse problem, but most have been limited to a few shapes of academic interest (2, 3). In any case, no efficient tools are available that can help the profile die designer compensate for the effects of die swell downstream of the die. This paper demonstrates how the 3-D CFD finite element program DIEFLOW is being used in a practical way for that purpose.
Here we must solve for the flow downstream in the extrudate along with that in the die. To do this, a 3-D finite element mesh of the domain must be created. Unfortunately, creating a 3-D mesh containing thousands of nodes and elements for an arbitrary geometry can take many hours, if not days. Although Gifford has streamlined the process of quickly creating meshes for flat dies (4) and some profile dies (1), fast and efficient mesh generators for profile dies with arbitrary internal flow channels are not yet available.
However, if the cross section of the flow channel of the die is uniform over its entire length or if the die has a very long land with a constant cross section so that only the land need be modeled, then mesh generation becomes simpler. Such dies are considered here. A mesh generator was written that reads a 2-D profile from either an ASCII file or a CAD (i.e., DXF) file and creates a 3-D finite element mesh with a prescribed length within seconds. Once the mesh is created, the flow equations governing the flow are solved everywhere in the die and in the downstream extrudate.
In 3-D, the steady momentum and continuity equations are, respectively:
[rho]V * [nabla] V = [nabla] T (1)
T = - PI + [mu]([gamma]) ([nabla]V + [nabla][V.sup.T]) [nabla] * V = 0. (2)
Here [rho] is the fluid density, P is the pressure and the velocity is
V = Ui + Vj + Wk (2a)
where U, V, and W are the components of velocity in the X, Y and Z directions, respectively. The function, [mu]([gamma]) is the shear-dependent viscosity of the polymer where the local shear rate, [gamma] is given by
[gamma] = [([gamma] : [[gamma]/2]).sup.[1/2]] with [gamma] = [nabla]V + [nabla][V.sup.T] (2b)
Equations 1 and 2 are solved from the inlet of the die at X = 0 to a position, [X.sub.f] downstream in the extrudate where a fully developed shape is effectively attained (5).
Fully developed flow is assumed, i.e.
[??]U/[??]X = 0 and V = W = 0 (3)
at both the X = 0 and X = [X.sub.f] planes.
No slip is assumed on all die surfaces, i.e.
V = 0 (4)
The mass flow rate, M, entering the die is usually specified. Therefore, the additional constraint
M = [integral][A.sub.0] [rho] U dA (5)
must also be satisfied where [A.sub.0] is the cross-sectional area of the channel entering the die.
On the free surface of the extrudate, drag from the surrounding air is neglected such that
T : n t = 0 (6)
where n and t are unit vectors normal and tangent at the surface of the extrudate, respectively.
Similarly, neglecting surface tension, a balance of normal stresses at the surface gives
T : n n = 0 (7)
Finally, no flow across the surface demands that the kinematic condition be satisfied, i.e.,
n * V = 0 (8a)
Equation 8a also requires that the flow at every point on the extrudate surface be in a direction tangent to it. Particles on the surface will remain there. Therefore, the free surface will be described by the path of surface particles traveling downstream from the exit of the die at X = [X.sub.0]. Their paths are determined by
dX/dS = V or dS = dX/U = dY/V = dZ/W
Y = x[integral][x.sub.o] V/U dX (8b) and Z = x[integral][x.sub.o] W/U dX. (8c)
Equations 1-8 are solved for the three components of velocity and pressure at node points. The 3-D CFD program DIEFLOW was used. The program has been used to solve a wide range of Newtonian and non-Newtonian free surface problems. Details for solving such problems are presented elsewhere (1, 5-8). In all the simulations presented here, Newtonian flow has been assumed and inertia neglected. The lengths of the channel and extrudate are 8 and 10, respectively, for all cases examined here.
In the first case, our goal is to design a die used to extrude the profile shown in Fig. 1. Calculations were begun using a die with a flow channel having the same shape. This particular die has four-fold symmetry. Each quadrant of the domain contains 1,008 quadratic 20-noded "brick" elements containing 5,245 nodes on which 17,160 values (i.e., the U, V, W and P at node points) are determined. An example of the finite element mesh used here is displayed in Fig. 2.
The profile of the final extrudate from the die along with that of the die channel is shown in Fig. 3. Because of the effects of die swell, the two profiles are significantly different. The question here, however, is what should the shape of the die channel be so the target shape shown in Fig. 1 is the final shape of the extrudate?
[FIGURE 1 OMITTED]
In order to determine this, we track surface particles (i.e., points at die exit surface node locations) to see where they go. The procedure is summarized in Fig. 4. In the previous simulation, we saw that a particle starting at the surface point [X.sub.0] at the exit of the die traveled to the point [X.sub.f1] in the final extrudate profile. For the next assumed shape of the die channel, we move this point in the exact opposite direction to point X[c.sub.2].
We do this with all surface node points, i.e.,
X[c.sub.i+1] = X[c.sub.i] - [OMEGA] ([X.sub.ft] - [X.sub.0]) (9)
where [OMEGA] is a relaxation parameter (7) between 0.5 and 1. We then re-create the mesh in the extrudate and repeat the process until no significant change in the extrudate shape occurs. Typically, only 4 or 5 such iterations are required. The final result showing the shape of the die channel along with the final extrudate profile produced is shown in Fig. 5.
[FIGURE 2 OMITTED]
In this example, the target profile to be extruded has two planes of symmetry. In such cases, a particle at the center point of the exit channel will travel to the center point in the final extrudate with no vertical or horizontal movement. In other words, although die swell may significantly affect the profile shape, it will not move it horizontally or vertically. This is NOT the case with asymmetric shapes.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
PROFILE WITH ONE PLANE OF SYMMETRY
Consider the profile shown in Fig. 6. This profile has only one plane of symmetry (at Z = 0). Similar simulations as above were also run for this profile shape. The result showing the profile extruded from a die channel of this shape is depicted in Fig. 7. Notice that in addition to die swell causing the change in shape downstream of the die, it has also caused it to move horizontally along the plane of symmetry.
To determine the shape of the die to produce the target profile in this case requires a similar procedure as above but also one that accounts for the translation of the extrudate along the plane of symmetry. Figure 8 shows the iterations involved here. The final result showing the shape of the die channel required to extrude the target profile is shown in Fig. 9.
Finally, let us consider the profile outlined by the solid line in Fig. 10. This profile has neither horizontal nor vertical symmetry. In this case, the domain is composed of 1,080 elements consisting of 5,657 nodes on which 18,510 values are determined. The result showing the profile extruded from a die channel of this shape is depicted in Fig. 11. In this case, there is a "twisting" of the extrudate as it exits the die.
To determine the shape of the die to produce the target profile in this case requires a similar procedure as before but this time accounting for the twisting of the extrudate. The final results of the target profile and the shape of the die channel necessary to extrude it are shown in Fig. 12. The dashed line in Fig. 10 also depicts this channel shape.
[FIGURE 6 OMITTED]
Although the horizontal or vertical translation of the extrudate from dies with one axis of symmetry has been reported once before (9), the twisting of a completely asymmetric profile appears to have not. When observed in practice, the twisting of asymmetric profiles is usually attributed to non-Newtonian or temperature effects. These simulations show that twisting will occur, even in the isothermal Newtonian case.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
This analysis shows that the effects of die swell can be compensated for in profile die design for both asymmetric and symmetric profiles. For profiles with one plane of symmetry, this includes compensating for the sideways translation of the extrudate as well as the change in shape that the extrudate experiences. Completely asymmetric profiles undergo a "twisting" downstream of the die. This twisting, which appears not to have been reported before in the literature (at least for the isothermal case), is also compensated for here along with the change in shape that the extrudate undergoes. With the use of a fully three-dimensional finite element flow algorithm along with quick mesh generating capabilities, the usual cut and try involved in the design of many profile dies can be greatly reduced if not eliminated.
The translation or twisting of profiles downstream of a die is often attributed to non-Newtonian or non-isothermal effects. The effects of a non-Newtonian viscosity of a polymer on the amount of die swell can be much larger than the Newtonian examples presented here. Also the effects of temperature variations can be significant. The examples presented here are all isothermal Newtonian cases. These results clearly show that asymmetry of the profile will result in a translation and twisting of the extrudate even in the isothermal Newtonian case.
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
[A.sub.0] = Area of the channel entering the die.
i = Unit vector in the X direction.
I = Unit tensor. ii + jj + kk
j = Unit vector in the Y direction.
k = Unit vector in the Z direction.
M = Mass flow rate through the die.
n = Unit vector normal to the surface.
P = Pressure.
t = Unit vector tangent to the surface.
T = Stress tensor.
U = X component of velocity.
V = Y component of velocity.
V = Velocity vector, Ui + Vj + Wk
W = Z component of velocity.
X = Coordinate in the extrusion direction.
[X.sub.c] = Position vector of a surface point at the exit of the channel at X = [X.sub.0].
[X.sub.0] = X coordinate of die exit.
[X.sub.f] = Position vector of a surface point at X = [X.sub.f].
Y = Coordinate in the horizontal direction.
Z = Coordinate in the vertical direction.
[gamma] = Shear rate.
[gamma] = Shear rate tensor. [gradient]V + [gradient][V.sup.T].
[micro] = Fluid viscosity.
[rho] = Fluid density.
[ohm] = Relaxation parameter.
[gradient] = Gradient operator.
1. W. A. Gifford, SPE ANTEC Tech. Papers, 48, 142 (2002).
2. V. Legat and J. M. Marchal, Int. J. Numer, Meth. Fluids, 16, 29 (1993).
3. Y. D. Rubin and T. M. Marchal, SPE ANTEC Tech. Papers, 44, 299 (1998).
4. W. A. Gifford, SPE ANTEC Tech. Papers, 42, 253 (1996).
5. W. A. Gifford, Polym. Eng. Sci., 38, 1167 (1998).
6. W. A. Gifford, Polym. Eng. Sci., 37, 315 (1998).
7. W. A. Gifford, Polym. Eng. Sci., 40, 2095 (2000).
8. W. A. Gifford, SPE ANTEC Tech. Papers, 47, 49 (2001).
9. A. Kargiannis, A. N. Hrymak, and J. Vlachopoulos, Rheol, Acta, 28, 121 (1989).
W. A. GIFFORD
18852 64th Ave., Chippewa Falls, WI 54729-6422
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|Author:||Gifford, W. A.|
|Publication:||Polymer Engineering and Science|
|Date:||Oct 1, 2003|
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