# The effect of surface tension on extrudate swell from square and rectangular channels.

INTRODUCTION

Flat extrusion dies are commonly used to extrude films, sheets and coatings from "coathanger" style dies with rectangular exit slots. Such dies are typically designed to produce a uniform flow rate across the exit of the die assuming that this will result in an extruded product downstream with a uniform thickness (1, 2). However, there is a difference between the size and shape of the final product and that of the exit slot. (The change in size or shape experienced by the extrudate after it exits the die is commonly referred to as extrudate swell or simply "die swell.") Average thickness of the extrudate will differ. Edges of the extrudate will be rounded or will taper causing thickness variations near these edges. The average thickness differences are usually not a concern since target thicknesses can be achieved by adjusting line speed or the tension applied by a winder, chill roll or similar device downstream. However, edge thickness variations can not be easily eliminated; some amount of edge trim is usually necessary. The extent of this edge effect is undoubtedly influenced by polymer rheology, polymer flow rate, non-isothermal effects (from nonuniform heating or cooling), surface tension, die exit geometry and downstream winder tension. The effect of surface tension is considered here.

Although the effect of surface tension on extrudate swell from simple one-dimensional circular and planar channels has been examined (3-8), its effect on swell from more complicated channels appears to have not. Investigating extrusion from a one-dimensional channel (e.g., a tube) only requires a two-dimensional free surface flow analysis. The extrusion from a two-dimensional channel (i.e., square or rectangular channel) or a three-dimensional die (i.e., coathanger, fish tail, T-die, etc.) requires a fully three-dimensional analysis.

Fully three-dimensional formulations have been used to analyze extrudate swell from two-dimensional channels (9-21) and three-dimensional dies (21) and to solve related free surface flow problems (22-28). However, in all of those studies, the effect of surface tension was neglected. A fully three-dimension analysis of a free surface flow problem including the effect of surface tension appears to be absent from literature. Such is the subject of the present investigation, which examines swell of a Newtonian fluid from square and rectangular channels. Surface tension is included but the effects of inertia and gravity are not.

THEORY

An analysis of extrusion from such channels requires three-dimensional flow modeling of the fluid between a plane, [Delta][[Omega].sub.1], positioned in the channel and a second plane, [Delta][[Omega].sub.2], in the extrudate (20). Three components of velocity and pressure are determined between [Delta][[Omega].sub.1] and [Delta][[Omega].sub.2] along with the shape of the extrudate between the exit of the channel and [Delta][[Omega].sub.2].

The steady state equations of motion for a Newtonian fluid is described by the Navier-Stokes equation, which in vector form is (20)

[Mathematical Expression Omitted] (1a)

where

[Mathematical Expression Omitted]. (1b)

In addition, continuity requires

[Mathematical Expression Omitted]. (2)

Here [Rho] is the fluid density, [Mu] is its viscosity, P is the pressure and

[Mathematical Expression Omitted]

where U, V and W are the components of velocity in the X, Y and Z directions, respectively.

Equations 1 and 2 are solved in the domain between [Delta][[Omega].sub.1] and [Delta][[Omega].sub.2] subject to boundary conditions.

Boundary Conditions

The upstream plane, [Delta][[Omega].sub.1], should be placed far enough upstream so that the effect of extrusion is not yet felt. A fully developed flow is then assumed there, i.e.

V = W = 0 and [Delta]U/[Delta]X = 0 (3)

where the coordinate X is taken in the extrusion direction.

At the surfaces of the channel, [Delta][[Omega].sub.3], no slip is assumed, i.e.

[Mathematical Expression Omitted] (4)

On the surface, [Delta][[Omega].sub.4], drag from the surrounding air is neglected such that

[Mathematical Expression Omitted] (5)

where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are unit vectors normal and tangent to the free surface, respectively. Also on [Delta][[Omega].sub.4],

[Mathematical Expression Omitted] (6)

where [P.sub.A] is the pressure in the surrounding air (which is taken to be zero here), [Sigma] is the surface tension of the fluid and H is the mean curvature of the free surface given by (29)

[Mathematical Expression Omitted].

No flow across the free surface demands that the kinematic condition,

[Mathematical Expression Omitted] (7)

be satisfied there as well.

Finally, at the downstream plane, [Delta][[Omega].sub.2], the flow is assumed to have become fully developed. The boundary conditions applied in this case are

[Mathematical Expression Omitted] (8)

where [T.sub.2] is the total normal stress at [Delta][[Omega].sub.2].

In these simulations the mass flow rate, [Mathematical Expression Omitted], entering the channel is specified. Therefore, the additional condition

[Mathematical Expression Omitted] (9)

must also be satisfied.

Solution Procedure

In all previous three-dimensional extrudate swell analyses (9-21), surface tension, [Sigma], was assumed to be zero. For the present investigation, the algorithm presented by Gifford (20, 21) was extended to include the effects of non-zero values of [Sigma], That addition requires some discussion.

Evaluating the Mean Curvature of the Free Surface

The addition of surface tension introduces the need W evaluate the mean curvature, H, of the free surface (Eq 6). As an example, consider a case where the location of the free surface, [Z.sub.s] can be described in rectangular Cartesian coordinates by [Z.sub.s] = h([x.sub.y]). The mean curvature is then given by (29)

[Mathematical Expression Omitted] (10)

The above equation shows that a calculation of the curvature requires determining second derivatives of coordinates of the free surface with respect to two spacial coordinates. This means that at least a quadratic representation of the free surface in terms of the spacial coordinates is necessary to obtain a non-zero value of the mean curvature. Here, the above equations are solved using a Galerkin finite element algorithm with quadratic isoparametric elements. (Details of that algorithm which has been used to solve a wide variety of 3-D flow problems are discussed elsewhere (1, 2, 20, 21, 27, 28).) Of course, with quadratic elements, a quadratic representation of the surface of all elements already exists including those at the free surface. The position of the free surface is determined in terms of a sum of quadratic basis functions. The curvature of the free surface will then be a function of the spacial derivatives of these same basis functions. No separate coordinate representation for the free surface is required in a finite element procedure employing basis functions that are quadratic or higher. A finite element procedure using linear basis functions may be easier to implement when surface tension is neglected but will require a separate representation for the free surface when it is not.

Determining the Downstream Boundary

In addition, the locations of the upstream plane, [Delta][[Omega].sub.1], and the downstream one, [Delta][[Omega].sub.2], must be chosen. Satisfactory positions of both planes are usually unknown a priori and should be found as part of the solution. Two-dimensional die swell studies, where the effect of surface tension and/or inertial effects were included, indicate that the placement of the upstream plane, [Delta][[Omega].sub.1], is not crucial (3-8). The same was found in a three-dimensional die swell case when surface tension was neglected (20, 21). Both cases indicate that placing [Delta][[Omega].sub.1] upstream of the die exit at a distance equal to the die opening is sufficient.

However, those same analyses also indicate that the placement of the downstream plane, [Delta][[Omega].sub.2], is important. In some cases, the plane may be fairly close to the exit of the die but in other cases it may not be, By analyzing the "smoothness" of the exudate profile and the transition of the axial velocity to fully developed flow downstream, Gifford (20) was able to determine satisfactory locations for [Delta][[Omega].sub.2] for 3-D die swell computations. A similar analyses was used for the present investigation.

RESULTS

Extrudate Swell From a Circular Channel

Any new algorithm should first be tested on a problem where the results are known before it can b relied upon to solve ones where they are not. As mentioned earlier, this author is unaware of any prior solutions to a fully 3-D free surface flow problem that includes surface tension. Therefore, a known 2-D problem was considered but posed in such a way that the algorithm treated it as a 3-D one. The problem examined was that of determining the effect of surface tension on die swell from a tube with radius [R.sub.0]. In this case, if no tension is applied by a winding device downstream, then [T.sub.2] in Eq 8 becomes (5, 8)

[T.sub.2] = -[Sigma]/[R.sub.2]

where [R.sub.2] is the radius of the final extrudate.

Here the downstream plane, [Delta][[Omega].sub.2] was channel radii downstream, which is more than adequate for this problem (5, 6, 8). One quadrant of the domain was tesselated into 825 elements containing 4452 nodes. A typical finite element mesh is shown in Fig. 1. Equations 1-9 were then solved using the procedure described by Gifford (20). Results showing the final extrudate profile for various values of capillary number, Ca, are presented in Fig. 2. (Actually, the inverse of the capillary number is depicted here where [Ca.sup.-1] = [Sigma]/([Mu][U.sub.av]) with [Mathematical Expression Omitted]. These results agree extremely well with those from 2-D analyses (5, 6, 8). [The results of (5), (6)and (8) differ by about 2% from one another. The results here agree within 0.5% of those from (8), which is probably the most accurate of the three.] The effect of the capillary number on the final die swell attained by the extrudate is depicted in Fig. 3. [Values shown for [Ca.sup.-1] greater than one are those from Omodei's 2-D analysis (5).]

Die Swell From a square Channel

Now let us consider the effect of surface tension on extrudate swell from a square channel shown schematically in Fig. 4. The dimension of each side of the square is taken to be 2H. Figure 5 depicts one quadrant of the domain tesselated into 825 elements. Analogous to the previous example, the downstream plane, [Delta][[Omega].sub.2] was placed at [X.sub.2] = 6H. The value of [T.sub.2] in Eq 8 was taken to be zero in this case.

Figure 6 shows the effect of the capillary number on the final extrudate profile (i.e., that at [Delta][[Omega].sub.2]) from the square channel. Here [Ca.sup.-1] = [Sigma]/([Mu][U.sub.av]) where [U.sub.av] = [Mathematical Expression Omitted]). As with the circle, extrudate cross sections decrease with increasing surface tension. The swell is minimum at the "corners" of the extrudate and maximum at the center of the sides. Notice that surface tension has a greater effect on the swell near the corner than it does on the sides. As [Ca.sup.-1] increases, the shape appears to approach a circle.

The influence of the capillary number on the maximum value of the swell is plotted in Fig. 7 and compared to the circular case discussed above. The swell in both cases approaches an asymptotic value as surface tension increases. In the circular case, that limit is a circle of radius of one. It can also be shown that the limit for the square is also a circle but with a radius of 2/[-square root of [Pi]]. Extrudate swell from the square channel approaches its limit much more rapidly with increasing surface tension than does that from the circular channel.

Figure 8 shows the maximum die swell as a function of downstream position for the cases shown in Fig. 6. Such plots give an indication of whether or not the downstream plane, [Delta][[Omega].sub.2], has been placed too close to the exit of the channel (20). Since slopes reach zero well before [X.sub.2]/H = 6, it appears that the position of [Delta]/[[Omega].sub.2] is adequate for the cases considered here.

The kinematic condition (Eq 7) was used for the free surface iteration in this study (20). Although this method gave rapid convergence for low values of surface tension, slower convergence was experienced as surface tension increased. For the [Ca.sup.-1] = 1 case above, a significant amount of underrelaxation was required. Similar convergence behavior has also been observed in the 2-D die swell case (3-5). For a 2-D free surface problem, Silliman and Scriven (30) showed that the kinematic condition gave rapid convergence for low values of surface tension ([Ca.sup.-1] [less than] 1); however a free surface iteration scheme employing the normal stress boundary condition (i.e., Eq 6) gave much faster convergence for high values of surface tension ([Ca.sup.-1] [greater than] 1). An analogous result undoubtedly occurs in 3-D. However, for the square, the extrudate profile at [Delta][[Omega].sub.2] for the [Ca.sup.-1] = 1 case is very close to the asymptotic circular limit (shown by the dashed curve in [ILLUSTRATION FOR FIGURE 6 OMITTED]). Therefore, it was unnecessary to alter free surface iteration schemes for higher values of surface tension. Such may not be the true for all three-dimensional problems, however (as the remaining examples will indicate).

In practice, few extrusion dies are designed with square exit channels. However, most fiat extrusion dies have rectangular channels in the exit lip region. The effect of surface tension on extrudate swell from rectangle channels is examined next.

Die Swell From a Rectangular Channels

Let us consider extrusion from a rectangular channel with an aspect ratio of 4 (i.e. a channel with dimensions 2H x 0.5H) shown schematically in Fig. 9. The final extrudate shapes (i.e. that at [Delta][[Omega].sub.2] where [X.sub.2] = 6H) obtained by solving the same equations is shown in Fig. 10 for several values of capillary number. [Once again, [Ca.sup.-1] = [Sigma][U.sub.av]) but [Mathematical Expression Omitted] in this case.] Again, cross sections of the extrudate decrease with increasing surface tension. Notice here that surface tension has the greatest influence on the swell of the longer side of the extrudate while that of the shorter side is virtually unaffected. Also significant here is that the sensitivity of the extrudate shape to surface tension is more pronounced for the rectangle than the square. The maximum difference in the dimensions of the extrudate from the square channel between the [Ca.sup.-1] = 0 and [Ca.sup.-1] = 0.02 cases is only about 0.4% (cf. [ILLUSTRATION FOR FIGURE 7 OMITTED]). For the rectangle, that difference is about 3.8%. A similar comparison between the [Ca.sup.-1] = 0 and [Ca.sup.-1] = 0.05 gives about 1% for the square and 7.5% for the rectangle.

Figure 11 shows similar results for a rectangular channel with an aspect of 10 (i.e. a channel with dimensions 2H x 0.2H). ([Ca.sup.-1] = [Sigma]/([Mu][U.sub.av]) with [Mathematical Expression Omitted] in this case.) Once again, cross sections decrease with increasing surface tension with the greatest change in swell occurring along the longer side. Here the maximum swell is decreased by 5.5% by decreasing [Ca.sup.-1] from 0 to 0.02. A comparison of this with the previous cases, indicates that surface tension becomes increasingly important as the aspect ratio of the channel increases. As the aspect ratio grows, so must the curvature at the edges and consequently the importance of surface tension increases (cf. Eq 6).

Another difference between the results of the square and rectangular channels is in convergence. As mentioned earlier, results for an inverse capillary number of one converged for the square channel. However, as the aspect ratio of the channel increases beyond one (i.e., the square case), the rate of convergence decreases. Convergence for an inverse capillary number greater than 0.05 for the 4 x 1 channel and 0.02 for the 10 x 1 one became extremely slow. A free surface iteration scheme employing the normal stress boundary condition would likely produce faster convergence at higher inverse capillary numbers.

As mentioned, most fiat extrusion dies have rectangular channels at the exit lip. Inverse capillary numbers in fiat sheet and film processes normally range from about [10.sup.-3] to [10.sup.-1]. Profile dies are typically extremely complex In shape but often contain long narrow regions with large aspect ratios. In profile extrusion, inverse capillary numbers typically lie roughly between [10.sup.-4] and [10.sup.-2]. The above results Indicate that the effect of inverse capillary numbers in these ranges can be significant even when aspect ratios are moderate.

However, in the extrusion of flat sheets, surface tension may actually help reduce the edge effect and the amount of edge trim necessary (cf. [ILLUSTRATION FOR FIGURE 11 OMITTED]). Surface tension is likely to be more significant in profile extrusion processes where "edges" cannot be trimmed, especially when aspect ratios are high.

CONCLUSIONS

A fully three-dimensional analysis of a free surface problem with surface tension has been considered, apparently for the first time. The effect of surface tension on the die swell of a Newtonian fluid from square and rectangular die is presented. Surface tension decreases extrudate cross sections in all cases examined. The influence on the shapes of final extrudate profiles is found to be greater for channels with higher aspect ratios. Surface tension has the greatest influence on the swell of the longer side of the extrudate while that of the shorter side is virtually unaffected. The effect of the capillary number can be significant and in some cases should be considered in the design of extrusion dies.

NOMENCLATURE

Ca = Capillary number, [Mu][U.sub.av]/[Sigma].

H = Half of the longer side of the rectangle.

H = Mean curvature of the free surface.

[Mathematical Expression Omitted] = Unit vector in the X direction.

[Mathematical Expression Omitted] = Unit tensor, [Mathematical Expression Omitted].

[Mathematical Expression Omitted] = Unit vector in the Y direction.

[Mathematical Expression Omitted] = Unit vector in the Z direction.

[Mathematical Expression Omitted] = Mass flow rate through the channel.

[Mathematical Expression Omitted] = Unit vector normal to the surface.

P = Pressure.

[P.sub.A] = Pressure in the surrounding air.

[R.sub.0] = Radius of the circular channel.

[R.sub.2] = Final radius of the circular extrudate.

[Mathematical Expression Omitted] = Unit vector tangent to the surface.

[Mathematical Expression Omitted] = Stress tensor.

[T.sub.2] = Total normal stress at the downstream plane.

U = X component of velocity.

[U.sub.av] = Average velocity in the channel.

V = Y component of velocity.

[Mathematical Expression Omitted] = Velocity vector, [Mathematical Expression Omitted].

W = Z component of velocity.

X = Coordinate in the extrusion direction.

Y = Coordinate in the horizontal direction.

Z = Coordinate in the vertical direction.

[Z.sub.s] = Z coordinate of the free surface.

[Mu] = Fluid viscosity.

[Rho] = Fluid density.

[Sigma] = Surface tension.

[Delta][Omega] = Surface of the flow domain.

[nabla] = Gradient operator.

[[nabla].sub.[Pi]] = Surface gradient operator.

REFERENCES

1. W. A. Gifford, J. Reinf. Plast. Compos., 16, 661 (1997).

2. W. A. Gifford, SPE ANTEC Tech. Papers, 42, 253 (1996).

3. K. R. Reddy and R. I. Tanner, Comput. Fluids, 6, 83 (1978).

4. B. J. Omodei, Comput. Fluids, 7, 79 (1979).

5. B.J. Omodei, Comput. Fluids, 8, 275 (1980).

6. A. Dutta and M. E. Ryan, AIChE J., 28, 220 (1982).

7. R. L. Gear, M. Keentok, J. F. Milthorpe, and R. I. Tanner, Phys. Fluids, 26(1), 7 (1983).

8. G. C. Georgiou, T. C. Papanastasiou, and J. O. Wilkes, AIChE J., 34, 1559 (1988).

9. M. B. Bush and N. Phan-Thien, J. Non-Newt. Fluid. Mech., 18, 211 (1985).

10. T. Tran-Cong and N. Phan-Thien, J. Non-Newt. Fluid. Mech., 30, 37 (1988).

11. T. Tran-Cong and N. Phan-Thien, Rheol. Acta, (1988).

12. T. Tran-Cong and N. Phan-Thien, Rheol. Acta, 27, 639 (1988).

13. A. Karagiannis, A. N. Hrymak, and J. Vlachopoulos, AIChE J., 34, 2088 (1988).

14. A. Karagiannis, A. N. Hrymak, and J. Vlachopoulos, Rheol. Acta, 28, 121 (1989).

15. C. R. Beverly and R. I. Tanner, Rheol. Acta, 30, 341 (1991).

16. V. Legat and J. M. Marchal, Int. J. Num. Meth. 14, 609 (1992).

17. K. R. J. Ellwood, T. C. Papanastasiou, and J. O. Wilkes, Int. J. Num. Meth. Fluids, 14, 13 (1992).

18. O. Wambersie and M. J. Crochet, Int. J. Num. Meth. Fluids, 14, 343 (1992).

19. V. Legat and J. M. Marchal, Int. J. Num. Meth. Fluids, 16, 29 (1993).

20. W. A. Gifford, Can. J. Chem. Eng., 71, 161 (1993).

21. W. A. Gifford, Handbook of Applied Polymer Processing Technology, Marcel Dekker, New York (1996).

22. A. Karagiannis, A. N. Hrymak, and J. Vlachopoulos, Rheol. Acta, 29, 71 (1990).

23. J. M. Marchal and V. Legat, SPE ANTEC Tech. Papers, 37, 819 (1991).

24. A. Torres, A. N. Hrymak, J. Vlachopoulos, J. Dooley, and B. T. Hilton, Rheol. Acta, 32, 513 (1993).

25. A. Torres, A. N. Hrymak, J. Vlachopoulos, J. Dooley, and B. T. Hilton, Rheol. Acta, 33, 241 (1994).

26. K. Sakaki, R. Katsumoto, T. Kajiwara, and K. Funatsu, Polym. Eng. Sci., 36, 1821 (1996).

27. W. A. Gifford, Polym. Eng. Sci., 37, 315 (1997).

28. W. A. Gifford, SPE ANTEC Tech. Papers, 43, 356 (1997).

29. R. A. Brown, J. Comp. Phys., 33, 217 (1979).

30. W. J. Silliman and L. E. Scriven, J. Comp. Phys., 34, 287 (1980).

Flat extrusion dies are commonly used to extrude films, sheets and coatings from "coathanger" style dies with rectangular exit slots. Such dies are typically designed to produce a uniform flow rate across the exit of the die assuming that this will result in an extruded product downstream with a uniform thickness (1, 2). However, there is a difference between the size and shape of the final product and that of the exit slot. (The change in size or shape experienced by the extrudate after it exits the die is commonly referred to as extrudate swell or simply "die swell.") Average thickness of the extrudate will differ. Edges of the extrudate will be rounded or will taper causing thickness variations near these edges. The average thickness differences are usually not a concern since target thicknesses can be achieved by adjusting line speed or the tension applied by a winder, chill roll or similar device downstream. However, edge thickness variations can not be easily eliminated; some amount of edge trim is usually necessary. The extent of this edge effect is undoubtedly influenced by polymer rheology, polymer flow rate, non-isothermal effects (from nonuniform heating or cooling), surface tension, die exit geometry and downstream winder tension. The effect of surface tension is considered here.

Although the effect of surface tension on extrudate swell from simple one-dimensional circular and planar channels has been examined (3-8), its effect on swell from more complicated channels appears to have not. Investigating extrusion from a one-dimensional channel (e.g., a tube) only requires a two-dimensional free surface flow analysis. The extrusion from a two-dimensional channel (i.e., square or rectangular channel) or a three-dimensional die (i.e., coathanger, fish tail, T-die, etc.) requires a fully three-dimensional analysis.

Fully three-dimensional formulations have been used to analyze extrudate swell from two-dimensional channels (9-21) and three-dimensional dies (21) and to solve related free surface flow problems (22-28). However, in all of those studies, the effect of surface tension was neglected. A fully three-dimension analysis of a free surface flow problem including the effect of surface tension appears to be absent from literature. Such is the subject of the present investigation, which examines swell of a Newtonian fluid from square and rectangular channels. Surface tension is included but the effects of inertia and gravity are not.

THEORY

An analysis of extrusion from such channels requires three-dimensional flow modeling of the fluid between a plane, [Delta][[Omega].sub.1], positioned in the channel and a second plane, [Delta][[Omega].sub.2], in the extrudate (20). Three components of velocity and pressure are determined between [Delta][[Omega].sub.1] and [Delta][[Omega].sub.2] along with the shape of the extrudate between the exit of the channel and [Delta][[Omega].sub.2].

The steady state equations of motion for a Newtonian fluid is described by the Navier-Stokes equation, which in vector form is (20)

[Mathematical Expression Omitted] (1a)

where

[Mathematical Expression Omitted]. (1b)

In addition, continuity requires

[Mathematical Expression Omitted]. (2)

Here [Rho] is the fluid density, [Mu] is its viscosity, P is the pressure and

[Mathematical Expression Omitted]

where U, V and W are the components of velocity in the X, Y and Z directions, respectively.

Equations 1 and 2 are solved in the domain between [Delta][[Omega].sub.1] and [Delta][[Omega].sub.2] subject to boundary conditions.

Boundary Conditions

The upstream plane, [Delta][[Omega].sub.1], should be placed far enough upstream so that the effect of extrusion is not yet felt. A fully developed flow is then assumed there, i.e.

V = W = 0 and [Delta]U/[Delta]X = 0 (3)

where the coordinate X is taken in the extrusion direction.

At the surfaces of the channel, [Delta][[Omega].sub.3], no slip is assumed, i.e.

[Mathematical Expression Omitted] (4)

On the surface, [Delta][[Omega].sub.4], drag from the surrounding air is neglected such that

[Mathematical Expression Omitted] (5)

where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are unit vectors normal and tangent to the free surface, respectively. Also on [Delta][[Omega].sub.4],

[Mathematical Expression Omitted] (6)

where [P.sub.A] is the pressure in the surrounding air (which is taken to be zero here), [Sigma] is the surface tension of the fluid and H is the mean curvature of the free surface given by (29)

[Mathematical Expression Omitted].

No flow across the free surface demands that the kinematic condition,

[Mathematical Expression Omitted] (7)

be satisfied there as well.

Finally, at the downstream plane, [Delta][[Omega].sub.2], the flow is assumed to have become fully developed. The boundary conditions applied in this case are

[Mathematical Expression Omitted] (8)

where [T.sub.2] is the total normal stress at [Delta][[Omega].sub.2].

In these simulations the mass flow rate, [Mathematical Expression Omitted], entering the channel is specified. Therefore, the additional condition

[Mathematical Expression Omitted] (9)

must also be satisfied.

Solution Procedure

In all previous three-dimensional extrudate swell analyses (9-21), surface tension, [Sigma], was assumed to be zero. For the present investigation, the algorithm presented by Gifford (20, 21) was extended to include the effects of non-zero values of [Sigma], That addition requires some discussion.

Evaluating the Mean Curvature of the Free Surface

The addition of surface tension introduces the need W evaluate the mean curvature, H, of the free surface (Eq 6). As an example, consider a case where the location of the free surface, [Z.sub.s] can be described in rectangular Cartesian coordinates by [Z.sub.s] = h([x.sub.y]). The mean curvature is then given by (29)

[Mathematical Expression Omitted] (10)

The above equation shows that a calculation of the curvature requires determining second derivatives of coordinates of the free surface with respect to two spacial coordinates. This means that at least a quadratic representation of the free surface in terms of the spacial coordinates is necessary to obtain a non-zero value of the mean curvature. Here, the above equations are solved using a Galerkin finite element algorithm with quadratic isoparametric elements. (Details of that algorithm which has been used to solve a wide variety of 3-D flow problems are discussed elsewhere (1, 2, 20, 21, 27, 28).) Of course, with quadratic elements, a quadratic representation of the surface of all elements already exists including those at the free surface. The position of the free surface is determined in terms of a sum of quadratic basis functions. The curvature of the free surface will then be a function of the spacial derivatives of these same basis functions. No separate coordinate representation for the free surface is required in a finite element procedure employing basis functions that are quadratic or higher. A finite element procedure using linear basis functions may be easier to implement when surface tension is neglected but will require a separate representation for the free surface when it is not.

Determining the Downstream Boundary

In addition, the locations of the upstream plane, [Delta][[Omega].sub.1], and the downstream one, [Delta][[Omega].sub.2], must be chosen. Satisfactory positions of both planes are usually unknown a priori and should be found as part of the solution. Two-dimensional die swell studies, where the effect of surface tension and/or inertial effects were included, indicate that the placement of the upstream plane, [Delta][[Omega].sub.1], is not crucial (3-8). The same was found in a three-dimensional die swell case when surface tension was neglected (20, 21). Both cases indicate that placing [Delta][[Omega].sub.1] upstream of the die exit at a distance equal to the die opening is sufficient.

However, those same analyses also indicate that the placement of the downstream plane, [Delta][[Omega].sub.2], is important. In some cases, the plane may be fairly close to the exit of the die but in other cases it may not be, By analyzing the "smoothness" of the exudate profile and the transition of the axial velocity to fully developed flow downstream, Gifford (20) was able to determine satisfactory locations for [Delta][[Omega].sub.2] for 3-D die swell computations. A similar analyses was used for the present investigation.

RESULTS

Extrudate Swell From a Circular Channel

Any new algorithm should first be tested on a problem where the results are known before it can b relied upon to solve ones where they are not. As mentioned earlier, this author is unaware of any prior solutions to a fully 3-D free surface flow problem that includes surface tension. Therefore, a known 2-D problem was considered but posed in such a way that the algorithm treated it as a 3-D one. The problem examined was that of determining the effect of surface tension on die swell from a tube with radius [R.sub.0]. In this case, if no tension is applied by a winding device downstream, then [T.sub.2] in Eq 8 becomes (5, 8)

[T.sub.2] = -[Sigma]/[R.sub.2]

where [R.sub.2] is the radius of the final extrudate.

Here the downstream plane, [Delta][[Omega].sub.2] was channel radii downstream, which is more than adequate for this problem (5, 6, 8). One quadrant of the domain was tesselated into 825 elements containing 4452 nodes. A typical finite element mesh is shown in Fig. 1. Equations 1-9 were then solved using the procedure described by Gifford (20). Results showing the final extrudate profile for various values of capillary number, Ca, are presented in Fig. 2. (Actually, the inverse of the capillary number is depicted here where [Ca.sup.-1] = [Sigma]/([Mu][U.sub.av]) with [Mathematical Expression Omitted]. These results agree extremely well with those from 2-D analyses (5, 6, 8). [The results of (5), (6)and (8) differ by about 2% from one another. The results here agree within 0.5% of those from (8), which is probably the most accurate of the three.] The effect of the capillary number on the final die swell attained by the extrudate is depicted in Fig. 3. [Values shown for [Ca.sup.-1] greater than one are those from Omodei's 2-D analysis (5).]

Die Swell From a square Channel

Now let us consider the effect of surface tension on extrudate swell from a square channel shown schematically in Fig. 4. The dimension of each side of the square is taken to be 2H. Figure 5 depicts one quadrant of the domain tesselated into 825 elements. Analogous to the previous example, the downstream plane, [Delta][[Omega].sub.2] was placed at [X.sub.2] = 6H. The value of [T.sub.2] in Eq 8 was taken to be zero in this case.

Figure 6 shows the effect of the capillary number on the final extrudate profile (i.e., that at [Delta][[Omega].sub.2]) from the square channel. Here [Ca.sup.-1] = [Sigma]/([Mu][U.sub.av]) where [U.sub.av] = [Mathematical Expression Omitted]). As with the circle, extrudate cross sections decrease with increasing surface tension. The swell is minimum at the "corners" of the extrudate and maximum at the center of the sides. Notice that surface tension has a greater effect on the swell near the corner than it does on the sides. As [Ca.sup.-1] increases, the shape appears to approach a circle.

The influence of the capillary number on the maximum value of the swell is plotted in Fig. 7 and compared to the circular case discussed above. The swell in both cases approaches an asymptotic value as surface tension increases. In the circular case, that limit is a circle of radius of one. It can also be shown that the limit for the square is also a circle but with a radius of 2/[-square root of [Pi]]. Extrudate swell from the square channel approaches its limit much more rapidly with increasing surface tension than does that from the circular channel.

Figure 8 shows the maximum die swell as a function of downstream position for the cases shown in Fig. 6. Such plots give an indication of whether or not the downstream plane, [Delta][[Omega].sub.2], has been placed too close to the exit of the channel (20). Since slopes reach zero well before [X.sub.2]/H = 6, it appears that the position of [Delta]/[[Omega].sub.2] is adequate for the cases considered here.

The kinematic condition (Eq 7) was used for the free surface iteration in this study (20). Although this method gave rapid convergence for low values of surface tension, slower convergence was experienced as surface tension increased. For the [Ca.sup.-1] = 1 case above, a significant amount of underrelaxation was required. Similar convergence behavior has also been observed in the 2-D die swell case (3-5). For a 2-D free surface problem, Silliman and Scriven (30) showed that the kinematic condition gave rapid convergence for low values of surface tension ([Ca.sup.-1] [less than] 1); however a free surface iteration scheme employing the normal stress boundary condition (i.e., Eq 6) gave much faster convergence for high values of surface tension ([Ca.sup.-1] [greater than] 1). An analogous result undoubtedly occurs in 3-D. However, for the square, the extrudate profile at [Delta][[Omega].sub.2] for the [Ca.sup.-1] = 1 case is very close to the asymptotic circular limit (shown by the dashed curve in [ILLUSTRATION FOR FIGURE 6 OMITTED]). Therefore, it was unnecessary to alter free surface iteration schemes for higher values of surface tension. Such may not be the true for all three-dimensional problems, however (as the remaining examples will indicate).

In practice, few extrusion dies are designed with square exit channels. However, most fiat extrusion dies have rectangular channels in the exit lip region. The effect of surface tension on extrudate swell from rectangle channels is examined next.

Die Swell From a Rectangular Channels

Let us consider extrusion from a rectangular channel with an aspect ratio of 4 (i.e. a channel with dimensions 2H x 0.5H) shown schematically in Fig. 9. The final extrudate shapes (i.e. that at [Delta][[Omega].sub.2] where [X.sub.2] = 6H) obtained by solving the same equations is shown in Fig. 10 for several values of capillary number. [Once again, [Ca.sup.-1] = [Sigma][U.sub.av]) but [Mathematical Expression Omitted] in this case.] Again, cross sections of the extrudate decrease with increasing surface tension. Notice here that surface tension has the greatest influence on the swell of the longer side of the extrudate while that of the shorter side is virtually unaffected. Also significant here is that the sensitivity of the extrudate shape to surface tension is more pronounced for the rectangle than the square. The maximum difference in the dimensions of the extrudate from the square channel between the [Ca.sup.-1] = 0 and [Ca.sup.-1] = 0.02 cases is only about 0.4% (cf. [ILLUSTRATION FOR FIGURE 7 OMITTED]). For the rectangle, that difference is about 3.8%. A similar comparison between the [Ca.sup.-1] = 0 and [Ca.sup.-1] = 0.05 gives about 1% for the square and 7.5% for the rectangle.

Figure 11 shows similar results for a rectangular channel with an aspect of 10 (i.e. a channel with dimensions 2H x 0.2H). ([Ca.sup.-1] = [Sigma]/([Mu][U.sub.av]) with [Mathematical Expression Omitted] in this case.) Once again, cross sections decrease with increasing surface tension with the greatest change in swell occurring along the longer side. Here the maximum swell is decreased by 5.5% by decreasing [Ca.sup.-1] from 0 to 0.02. A comparison of this with the previous cases, indicates that surface tension becomes increasingly important as the aspect ratio of the channel increases. As the aspect ratio grows, so must the curvature at the edges and consequently the importance of surface tension increases (cf. Eq 6).

Another difference between the results of the square and rectangular channels is in convergence. As mentioned earlier, results for an inverse capillary number of one converged for the square channel. However, as the aspect ratio of the channel increases beyond one (i.e., the square case), the rate of convergence decreases. Convergence for an inverse capillary number greater than 0.05 for the 4 x 1 channel and 0.02 for the 10 x 1 one became extremely slow. A free surface iteration scheme employing the normal stress boundary condition would likely produce faster convergence at higher inverse capillary numbers.

As mentioned, most fiat extrusion dies have rectangular channels at the exit lip. Inverse capillary numbers in fiat sheet and film processes normally range from about [10.sup.-3] to [10.sup.-1]. Profile dies are typically extremely complex In shape but often contain long narrow regions with large aspect ratios. In profile extrusion, inverse capillary numbers typically lie roughly between [10.sup.-4] and [10.sup.-2]. The above results Indicate that the effect of inverse capillary numbers in these ranges can be significant even when aspect ratios are moderate.

However, in the extrusion of flat sheets, surface tension may actually help reduce the edge effect and the amount of edge trim necessary (cf. [ILLUSTRATION FOR FIGURE 11 OMITTED]). Surface tension is likely to be more significant in profile extrusion processes where "edges" cannot be trimmed, especially when aspect ratios are high.

CONCLUSIONS

A fully three-dimensional analysis of a free surface problem with surface tension has been considered, apparently for the first time. The effect of surface tension on the die swell of a Newtonian fluid from square and rectangular die is presented. Surface tension decreases extrudate cross sections in all cases examined. The influence on the shapes of final extrudate profiles is found to be greater for channels with higher aspect ratios. Surface tension has the greatest influence on the swell of the longer side of the extrudate while that of the shorter side is virtually unaffected. The effect of the capillary number can be significant and in some cases should be considered in the design of extrusion dies.

NOMENCLATURE

Ca = Capillary number, [Mu][U.sub.av]/[Sigma].

H = Half of the longer side of the rectangle.

H = Mean curvature of the free surface.

[Mathematical Expression Omitted] = Unit vector in the X direction.

[Mathematical Expression Omitted] = Unit tensor, [Mathematical Expression Omitted].

[Mathematical Expression Omitted] = Unit vector in the Y direction.

[Mathematical Expression Omitted] = Unit vector in the Z direction.

[Mathematical Expression Omitted] = Mass flow rate through the channel.

[Mathematical Expression Omitted] = Unit vector normal to the surface.

P = Pressure.

[P.sub.A] = Pressure in the surrounding air.

[R.sub.0] = Radius of the circular channel.

[R.sub.2] = Final radius of the circular extrudate.

[Mathematical Expression Omitted] = Unit vector tangent to the surface.

[Mathematical Expression Omitted] = Stress tensor.

[T.sub.2] = Total normal stress at the downstream plane.

U = X component of velocity.

[U.sub.av] = Average velocity in the channel.

V = Y component of velocity.

[Mathematical Expression Omitted] = Velocity vector, [Mathematical Expression Omitted].

W = Z component of velocity.

X = Coordinate in the extrusion direction.

Y = Coordinate in the horizontal direction.

Z = Coordinate in the vertical direction.

[Z.sub.s] = Z coordinate of the free surface.

[Mu] = Fluid viscosity.

[Rho] = Fluid density.

[Sigma] = Surface tension.

[Delta][Omega] = Surface of the flow domain.

[nabla] = Gradient operator.

[[nabla].sub.[Pi]] = Surface gradient operator.

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Author: | Gifford, W.A. |
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Publication: | Polymer Engineering and Science |

Date: | Jul 1, 1998 |

Words: | 3608 |

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