Die lines in plastics extrusion: film blowing experiments and numerical simulation.
Extrusion is the most important unit operation in the plastics processing industry. It is a convenient way to continuously manufacture lengths of complex cross section to strict dimensional tolerances, mechanical properties and cosmetic features. Extrusion is complicated and sensitive, even to minor changes in the operating parameters. The problems encountered in extrusion range from aesthetic defects to major malfunctions that can shut the line down. Many surface defects appear in extrusion. These can destroy extrudate aesthetics and compromise optical, electrical and mechanical properties. In film, sheet, pipe, tubing, and in wire and cable coating, where surface properties matter, cosmetic problems will challenge profitability. Even if purely cosmetic, surface imperfections can be a costly impediment to sales.
Die lines are longitudinal indentations or protrusions formed on the extrudate surface (1, 2). These striations may extend continuously for hundreds of feet. When die lines severely interfere with extrusion, the processing lines must be shut down to clean the die. If required periodically, this is costly.
Despite the commercial implications of such defects, there is little literature on die lines, either experimental or theoretical studies. We have previously reviewed the mechanisms and affecting factors in die line formation, and ways to suppress these striations (1, 2).
Here we first briefly introduce how to measure and characterize die lines (die line metrology) (2). Because die lines are commonly observed on blown film, we then conduct film blowing experiments to study how die lines depend on die defect size, shape and position and on various operating parameters. The empirical relations between die lines and die defects, material properties and affecting operating conditions are explored using the dimensionless groups. The results establish quantitatively the significance of these factors in die line formation.
To better understand the fluid mechanics of die line formation, we then numerically model this free surface flow of polymer melt leaving the die lips using FIDAP (version 8.52) (3). In particular, we investigate how die defects, material properties and operating conditions affect die lines. The results are compared with the experimental observations.
DIE LINE METROLOGY
Die lines are long surface striations. Like other surface defects, they are visible because they scatter light differently from the rest of the surface. Thus the optical surface metrology can be helpful in developing die line metrology (2).
In our die line metrology, we categorize die lines as either isolated striations on a smooth surface, or distributed ones over the entire surface (2). We use mechanical profilers to measure die lines. Both isolated and distributed die lines can be profiled this way. Measuring die line shapes also helps deepen our understanding of the fluid mechanics of die line formation.
There are three ways to characterize die lines (2):
1. Use full die line profiles.
2. Visually compare die lines with scratch standards. This is a qualitative or semi-quantitative method, focusing on die line cosmetics.
3. Die line statistics.
The first two methods apply to isolated die lines on a smooth surface, while the third applies to distributed die lines. Here, we focus on isolated die lines on blown film, which are characterized using their profiles, and base widths and heights.
Film blowing was performed on a "Yellow Jacket" Blown Film Tower 6536. A Killion extruder was used, with a single 19 mm screw having 19 mm constant pitch and 17.7[degrees] flight angle. The ratio of barrel length to diameter is 24:1. The screw channel depth varies from 3.5 mm at the feed zone to 1.3 mm at the metering zone.
To study how operating conditions affect die lines, screw speed (throughput Q), take-off speed ([V.sub.f]), blowup ratio (BUR) and melt temperature ([T.sub.e]) were varied independently. The temperatures of the three barrel heaters and the die heater can be set individually. Within each extrusion experiment, these temperatures varied by less than 5[degrees]C.
[FIGURE 1 OMITTED]
The blown film die has outer (barrel) and inner (mandrel) pieces (see Fig. 1). Polymer melt is extruded through the annulus. A fixed volume of air is supplied from below, passing through a special channel in the mandrel. This is done at start-up, and occasionally thereafter, to fix the bubble diameter. The bubble's internal air pressure is monitored using a pressure gauge.
Die land and lip imperfections are the main causes of die lines (1, 2). To study how they can trigger die lines, we machined two mandrels having defects of controlled shapes, sizes and positions. Both mandrels were made of low carbon steel 1018. All mandrel defects were deliberately large, leading to die lines with profiles large enough to characterize. Tables 1 and 2 list all die defect dimensions.
Many defects were made on each mandrel to enhance its efficiency. The defects were evenly distributed along the mandrel perimeter to minimize their interactions. Both mandrels had a height of 19.05 mm and a diameter of 21.08 mm. The die barrel had an inner diameter of 26.16 mm, thus forming an annular gap of 2.540 mm. The die barrel had no machined imperfections.
A linear low-density polyethylene (LLDPE) Dowlex[TM] 2045 (Dow Chemical) and a high-density polyethylene (HDPE) Paxon[TM] (Exxon Chemical Company) were used. Table 3 lists their physical properties.
The air must be supplied slowly to obtain a stable bubble having specified BUR. The film bubble is susceptible to ambient temperature and air flow. Hence all experiments were carried out at the same room temperature and with minimal laboratory air currents.
When switching testing materials, the screw channel and die were purged for roughly 30 minutes with the new resin and a screw speed of 25 rpm. Since die lip build-up can trigger die lines, before each extrusion run, the die lips were cleaned with a brass tool to avoid scratching.
In each film blowing run, after the bubble stabilized, film was extruded for one minute, collected and weighed.
To flatten and fasten film for die line scanning, an aluminum sample holder is used (see Fig. 2). The rectangular glass piece in the middle provides a flat film holding area. Two bars then fix the film onto the glass. The glass surface roughness [R.sub.q] is 0.04 [micro]m, low enough to cause negligible errors in die line measurement.
Blown film must be trimmed to fit onto the glass piece in the sample holder. To improve measurement accuracy, double-stick tape was used to adhere the film to the glass piece. The adhesion must be carefully performed to prevent air entrapment under film, which will deform die lines. Excessive force can also distort die lines. Die lines are held perpendicular to the holder's longitudinal direction. Their shapes are contoured by the profiling stylus as it traverses the film.
Die Line Profiling
Die line shapes were measured using a mechanical stylus profiler Surfanalyzer[R] System 4000. It has a stylus of radius 2.54 [micro]m and loading 200 mg. A scanning speed of 0.25 mm/s and normal magnification were used.
[FIGURE 2 OMITTED]
The profiler was calibrated using the two standards of different roughness. The calibration is also required when the profiler beam or diamond stylus is adjusted. However, this is unnecessary when just switching film samples. Before profiling, each film sample was prescanned twice or three times to ensure a proper reading range. The stylus must then be re-positioned until the whole die line can be scanned.
Die line shapes were plotted on grid charts with preset scales and roughness heights were calculated automatically. However, since die lines on the blown film are isolated, not distributed, they were quantified using their base widths and heights, not statistics.
Unfortunately, our stylus profiler is analog. All die line profiles must be digitized manually from the charts, and then characterized and compared using proper height and width scales.
EXPERIMENTAL RESULTS AND DISCUSSION
Figure 3 shows a micrograph of a typical die defect with its die line. Die line measurement using the stylus profiler showed excellent reproducibility (see Fig. 4). The differences were partly caused by the manual digitization.
Die line shapes on our blown film are usually asymmetric. This could be due to 1) slightly misshapen die defects, 2) imperfections in sample preparation, 3) mandrel eccentricity in the film blowing dies, or 4) sample misalignment during stylus profiling.
For our results, the dimensionless groups in our dimensional analysis (2) were employed. We believe that our observations quantitatively describe how various factors affect die line formation. Though specific numerical values can vary with equipment and materials, we believe our results yield correct trends.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
Die line sizes strongly depend on die defect dimensions. Figure 5 depicts the rectangular lip indentation 13 and its die lines on HDPE film. Visual inspection revealed that, without downstream stretching and BUR = 1, the sizes of lip defects and their die lines have the same order of magnitude.
Figure 6 compares die lines on LLDPE film caused by three lip indentations 12, 13, and 14 on mandrel M1. For the indentation I1, its die lines bulge on both blown film surfaces (neither side of the film is flat), rendering stylus profilometry impossible (see Fig. 7). The results for HDPE film are similar.
From Table 1, indentations 12, 13, and 14 have the same heights, but different widths, 12 < 14 < 13. Under identical operating conditions, both the height and base width of die lines increase dramatically with the indentation width (see Fig. 6). Importantly, both the die defect height and width govern the die line size and shape. The dimensionless die line height is roughly 0.02 for the narrowest die defect (12), and 0.1 for 13 and 14. On the other hand, all dimensionless die line widths are roughly between 1/2 and 2. These results hold for both LLDPE and HDPE. Interestingly, the die line dependence (especially the height) on die defect width weakens with die defect width. This is presumably because widening die defects enlarges the die line radii, thus reducing the surface tension and die line dissipation (1, 2). Though the capillary number, based on the die line's base width, is around [10.sup.3], the capillary number based on the die line's peak radius can approach unity.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
Figure 8 shows how die line height and base width depend on die defect sizes and polymer polydispersity (PI). They suggest the following empirical dimensionless relations between the die defect and die line sizes for both polymers.
(h/[h.sub.0]) = 0.05 [PI.sup.0.42] (1 + (3.75[[alpha].sub.0])[.sup.-5])[.sup.-2.3] (1)
(w/[w.sub.0]) = 4.00 [PI.sup.-0.61] (1 + (3.75[[alpha].sub.0])[.sup.-2])[.sup.-0.83] (2)
where [[alpha].sub.0] [equivalent to] [w.sub.0]/[h.sub.0] is the die defect shape factor. Thus, the die line shape factor [[alpha].sub.0] [equivalent to] w/h is
[FIGURE 8 OMITTED]
(w/h) = [[alpha].sub.0] (w/[w.sub.0]) ([h.sub.0]/h) = 80[[alpha].sub.0] [PI.sup.-1.03] (1 + (3.75[[alpha].sub.0])[.sup.-2])[.sup.-0.83] (1 + (3.75[[alpha].sub.0])[.sup.-5])[.sup.2.3] (3)
For engineering purposes, we also plot the 95% confidence interval for the fitted line (4).
Polymer melts remember their recent deformation history. Thus, upstream disturbance to melt flow can affect melt viscoelasticity, causing die lines (1, 2). The proximity of die defects to die lips determines how long the polymer melt has to recover before exiting the die, thus governing the occurrence of these kinds of die lines. To study this upstream generation, several land defects were machined on both mandrels, including asperities (A1-A6 on M2) and indentations (15 and 16 on M1). They have different sizes and proximities to the die lips (see Tables 1 and 2).
Table 4 lists the local Deborah numbers, [De.sub.local] (see Appendix), near the asperity tips under the highest throughput for both melts studied. The Appendix shows how these Deborah numbers are calculated.
Calculating a Deborah number for an indentation is trickier. We assume that the penetration depth of the flow disturbance is arguably equal to the indentation depth (1, 2). Table 5 thus lists the local Deborah numbers near land indentations on M1.
We would have preferred experiments for Deborah numbers approaching unity. For De [much less than] 1, no die lines were caused by any land defects. This result is expected since, in these experiments, the melt fully recovers from the die defect flow disturbance. However, the highest Deborah number in the experiments we could generate approached 1/20. We find that when the Deborah number approaches 1/20 (for A1), still no die lines are observed. For both materials, the melt temperatures approach the melting points, leaving us no way to further increase the Deborah number.
The land defect sizes on our mandrels are large, up to 2/5 of the annular gap, far exceeding anything encountered on a commercial extrusion die. Moreover, not only are our asperities large, but our experiments included cases as close as 3 mm from the die lips. In fact, we find machining asperities any closer to be difficult. Therefore, we find that die land defects matter only when they are remarkably close to the lips, and lip defects are the most crucial in suppressing die lines. This matches the observations of Yapel et al. (1, 2). Hence, we believe that keeping the Deborah number of the tip asperity below unity prevents the upstream generation of die lines.
All lip defects on our mandrels are rectangular. Though having distinct widths (same depths and lengths), they all generate bell-shaped die lines (see Fig. 6). Though no other die defect shapes were tested, die line formation simulations show that semi-circular and triangular lip asperities also lead to bell-shaped die lines. Thus, we believe that all lip imperfections cause bell-shaped die lines, though die line profile details and sizes can depend on die defect shapes.
Extrusion operating conditions are essential in die line formation (1, 2). For instance, increasing throughput can decrease the viscosity of a shear thinning melt, altering the melt flow, thus affecting die lines. We separately altered each of four operating parameters in our film blowing, throughput Q, take-off roll speed [V.sub.f], blowup ratio (BUR) and melt temperature [T.sub.e].
Throughput affects die lines by increasing the upstream melt flow in die channels. We hypothesized that higher throughput worsens extrudate swell, thus enhancing flow disturbance amplification near die lips (1, 2), and enlarging die lines. Figure 9 shows die lines caused by 12, at three throughputs for LLDPE. Other die defects behaved similarly and results for HDPE resembled those seen in Fig. 9. For each lip indentation, die lines heighten and widen with throughput. This supports our hypothesis on extrudate swell.
Extrudate swell can usually be related with upstream Deborah number by (5)
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
[D*.sub.f] = 0.1 + (1 + [1/2] [De.sub.us.sup.2])[.sup.1/6] (4)
[De.sub.us] [equivalent to] [[tau].sub.relax]/[t.sub.us] (5)
where [D*.sub.f] is the extrudate/die radius ratio, [[tau].sub.relax] is the characteristic melt relaxation time and [t.sub.us] is the mean melt residence time on the die land. To relate die line sizes with throughput, Fig. 10 collapses the results at three throughputs for three defects and two materials. The Deborah numbers, [De.sub.us], in Fig. 10 were calculated using the annulus average melt velocity and the average melt relaxation times (see Appendix and Table 6).
Though LLDPE has more short chain branching than HDPE, this short chain branching hardly affects the viscoelasticity of molten plastics (6). The difference between the two polymers' die lines is presumably caused by their differing molecular weights ([M.sub.w]) and polydispersities (PI). Figure 10 suggests two dimensionless empirical relations between die line sizes and throughput for both polymers:
(h/[h.sub.0]) = (1 + (3.75[[alpha].sub.0])[.sup.-5])[.sup.-2.32] (454[[De.sub.us]/[PI.sup.2]] + 0.0379) (6)
(w/[w.sub.0]) = [[PI.sup.-0.670]/[(1 + ([[alpha].sub.0] - 0.510)[.sup.2])]] (2.78 X [10.sup.4] [[De.sub.us]/[PI.sup.2]] + 0.165) (7)
Combining these, we get another for die line shape:
(w/h) = [[[alpha].sub.0][PI.sup.-0.670](1 + (3.75[[alpha].sub.0])[.sup.-5])[.sup.2.32](2.78 X [10.sup.4][[De.sub.us]/[PI.sup.2]] + 0.165)]/[(1 + ([[alpha].sub.0] - 0.510)[.sup.2]) (454[[De.sub.us]/[PI.sup.2]] + 0.0379)] (8)
Take-off Roll Speed
Take-off roll speed governs the downstream stretching in the extrusion direction, which in turn influences melt flow, cooling and, for semicrystalline polymers, crystallization. Thus, all these affect die lines. Increasing take-off speed lessens extrudate swell, and suppresses flow disturbance amplification near die lips, thus flattening die lines. For brevity, we no longer include die line profiles. Details can be found in Ref. 2.
Our experiments showed that for each lip indentation, its die lines flatten with take-off speed for both LLDPE and HDPE. The die lines narrow with [V.sub.f] for LLDPE, but for HDPE, they hardly change. Figure 11 shows the startup extensional viscosities for both materials, computed from linear viscoelasticity given by (6):
[[eta].sub.E.sup.+](t) = 3[[eta].sup.+] (t) = 3 [[integral].sub.0.sup.t] G(s)ds = -3 [N.summation over (i=1)] [G.sub.i][[lambda].sub.i]([e.sup.-t/[[lambda].sub.t]] - 1) (9)
where [[eta].sup.+](t) is the startup shear viscosity, G(s) is the relaxation modulus and [G.sub.i] and [[lambda].sub.i] are the discrete relaxation moduli and relaxation times of the molten plastics. Presumably, HDPE's higher extensional viscosity hinders die line flattening in film blowing.
[FIGURE 11 OMITTED]
Figure 12 collapses the results for both LLDPE and HDPE. It suggests the following empirical dimensionless relations between die line sizes and take-off speed (or drawdown ratio):
[FIGURE 12 OMITTED]
(h/[h.sub.0]) = [PI.sup.0.3] (1 + (3.75[[alpha].sub.0])[.sup.-10])[.sup.2.4] (-3.70 X [10.sup.-3] DDR + 0.0834) (10)
(w/[w.sub.0])[.sub.LLDPE] = (15.7[[alpha].sub.0.sup.2] - 18.2[[alpha].sub.0] + 6.23)[.sup.-1] (-0.0778DDR + 2.73) (11)
(w/[w.sub.0])[.sub.HDPE] = (4.13[[alpha].sub.0.sup.2] - 5.24[[alpha].sub.0] + 2.60)[.sup.-1] (12)
where drawdown ratio DDR = [V.sub.f]/[V.sub.0] and [V.sub.0] is the average melt velocity in die annulus. Combining these yields the dimensionless relations for die line shape:
(w/h)[.sub.LLDPE] = [[[alpha].sub.0][PI.sup.-0.3] (1 + (3.75[[alpha].sub.0])[.sup.-10])[.sup.2.4](-0.0778DDR + 2.73)]/[(15.7[[alpha].sub.0.sup.2] - 18.2[[alpha].sub.0] + 6.23)(-3.70 X [10.sup.-3] DDR + 0.0834)] (13)
(w/h)[.sub.HDPE] = [[[alpha].sub.0][PI.sup.-0.3] (1 + (3.75[[alpha].sub.0])[.sup.-10])[.sup.2.4]]/[(4.13[[alpha].sub.0.sup.2] - 5.24[[alpha].sub.0] + 2.60) (-3.70 X [10.sup.-3] DDR + 0.0834)] (14)
[FIGURE 13 OMITTED]
Blowup ratio (BUR) flattens and widens die lines through stretching in the hoop direction. Our experiments showed that for each lip indentation, its die lines flatten and widen with BUR for both LLDPE and HDPE.
Figure 13 collapses the results for both materials. It suggests the following dimensionless relations between die line sizes and BUR:
(h/[h.sub.0]) = [PI.sup.0.29](1 + (3.75[[alpha].sub.0])[.sup.-10])[.sup.-0.21] (-5.70 X [10.sup.-3] BUR + 0.0649) (15)
(w/[w.sub.0]) = [PI.sup.-0.84] (1 + (3.75[[alpha].sub.0])[.sup.-10])[.sup.-0.9] (2.29 BUR + 2.65) (16)
where BUR = D/[D.sub.0], D and [D.sub.0] are the outer diameters of film bubble and die annulus respectively. Combining these, we have another relation for die line shape:
(w/h) = [[alpha].sub.0][PI.sup.-1.13](1 + (3.75[[alpha].sub.0][.sup.-10])[.sup.-0.69] [[(2.29 BUR + 2.65)]/[(-5.70 X [10.sup.-3] BUR + 0.0649)]] (17)
To study how the hoop stretching deforms die lines. Fig. 14 shows the local strain near die lines and global strain on LLDPE blown film,
[[epsilon].sub.local] = ln (w/[w.sub.0]) (18)
[[epsilon].sub.global] = ln([D.sub.i]/[D.sub.m]) = ln (BU[R.sub.i]) (19)
where w, [w.sub.0] are the die line and die defect widths, [D.sub.m] is the mandrel diameter and [D.sub.i] is the inner film bubble diameter. Importantly, we use [D.sub.i]/[D.sub.m] or BU[R.sub.i] here because die lines occur on the inner film surface. BU[R.sub.i] differs from the quantity BUR in Eqs 15-17. BUR is based on the outer bubble diameter and is commonly used in blown film manufacturing.
The local and global strains are roughly equal near BU[R.sub.i] = 1. Though both enlarge with bubble growth, the local strain increases more slowly, causing less melt deformation near die lines than globally encountered in the surrounding film.
[FIGURE 14 OMITTED]
Alternatively, we can define angles in radians for both die defects and die lines.
[[theta].sub.o] = [w.sub.0]/[R.sub.m] (20)
[theta] = w/[R.sub.i] (21)
where [R.sub.m] is the mandrel radius and [R.sub.i] is the inner film bubble radius. Combining Eqs 18-21, we get,
In([theta]/[[theta].sub.o]) = [[epsilon].sub.local] - [[epsilon].sub.global] (22)
Figure 15 shows how [theta]/[[theta].sub.o] depends on BU[R.sub.i]. The ratio [theta]/[[theta].sub.o] drops with BU[R.sub.i], meaning that the melt is stretched less near die lines than globally. This matches the results from Fig. 14.
Melt temperature [T.sub.e] is crucial in plastics extrusion because melt properties, such as viscosity, strongly depend on it. Also, the frost line height in film blowing increases with [T.sub.e]. This allows more time for die line dissipation and thus presumably helps flatten die lines (1, 2).
Surprisingly, our experiments show that for each lip indentation, the die line shapes and sizes hardly vary significantly with increasing temperatures for both LLDPE and HDPE.
Though such observations are inconsistent with our die line dissipation mechanism, it can be explained with the capillary number (1, 2). In plastics processing, surface tension seldom matters. This is because both [eta] and R are so large. In our experiments, the lowest capillary number (corresponding to 12) is 1090 for LLDPE and 450 for HDPE, far above unity. Thus, the viscous force dominates the melt flow and die lines barely dissipate even at the highest [T.sub.e]. However, defects on commercial extrusion dies can be of O(1) [micro]m, where Ca is O(1). The surface tension can then outweigh the viscous force and help flatten die lines.
[FIGURE 15 OMITTED]
Governing Equations and Boundary Conditions
Here, die line formation is assumed to be isothermal, neglecting viscous dissipation, downstream cooling, and the temperature-dependence of material properties. A power-law viscosity model is employed to represent the shear thinning of molten plastics.
The continuity and momentum equations, in tensor notation, are (7),
[[D[rho]]/Dt] + [rho]([nabla] * v) = 0 (23)
[rho][[Dv]/[Dt]] = -[nabla]p - [[nabla] * [tau]] + [rho]g (24)
To solve these numerically, the flow domain must be discretized into small elements. The Galerkin technique is employed to transform the differential equations into algebraic ones (8).
Figure 16 depicts polymer melt extruded from a slit and the flow domain in our die line formation simulation. The melt is assumed to be incompressible and gravity, negligible. Thus, Eqs 23 and 24 simplify to
([nabla] * v) = 0 (25)
[FIGURE 16 OMITTED]
[rho][[Dv]/[Dt]] = -[nabla]p - [[nabla] * [tau]] (26)
FIDAP provides several viscosity models for generalized Newtonian fluids (9), for which
[tau] = -[eta]([dot.[gamma]])[dot.[gamma]] (27)
where [dot.[gamma]] is the rate of deformation tensor and [dot.[gamma]] is its magnitude. For engineering purposes, we select the truncated power-law model.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)
where m is the consistency index, n is the power-law exponent and [dot.[gamma].sub.0] is the critical (onset) shear rate.
In Fig. 16 (front view), the coordinates are located such that the boundary condition on the right is x-symmetric. Also, in commercial slit dies, the width far exceeds the gap, making edge effects negligible for die lines far from the edges. Thus, we assume [v.sub.x] = 0 on both sides. Furthermore, the flow domain bottom is at the slit midplane, where the flow is z-symmetric, namely [v.sub.z] = 0.
The inlet flow velocity profile is assumed to be uniform. As expected, we found that it quickly becomes parabolic along the die land. The downstream flow (at the outlet of the computational domain) is in the extrusion direction only, so [v.sub.x] = [v.sub.z] = 0. For these calculations, the downstream velocity [v.sub.y] is unspecified, thus it is a dependent variable. The higher the extrudate swell, the lower [v.sub.y]. So for these calculations, the drawdown ratio DDR (defined in Eq 10) is also a dependent variable and always below unity. When extrudate exits the slit, its thickness increases and approaches a downstream value asymptotically. We call these equilibrium swell calculations, and we define their drawdown ratios as DD[R.sub.e] (the subscript denoting equilibrium).
Later, studying the effect of stretching on die lines, we will use DD[R.sub.e] as our no-stretching baseline. This contrasts with many textbook treatments that neglect swell (10-12). Those thus naturally employ DDR = 1 as the no-stretching baseline.
When polymer melt exits the slit, the extrudate surface location is unknown a priori. Two boundary conditions for this interface are kinematic (9):
[[[partial derivative]S(x, t)]/[[partial derivative]t]] + u * [nabla]S = 0 (29)
and stress continuity:
[[sigma].(=)] * n = ([2.sub.[gamma]]H - [p.sub.a])n + [[nabla].sub.s[gamma]] (30)
where S(x, t) = 0 is the free surface, u is the boundary velocity. [[sigma].(=)] is the total stress, [gamma] is the surface tension, H is the mean surface curvature. [p.sub.a] is ambient pressure, n is the surface normal vector, and [[nabla].sub.s] is the surface gradient. For steady state flows, Eq 29 reduces to
u * n = 0 (31)
We first define all dimensionless variables.
x* = [x/d] y* = [y/d] z* = [z/d] (32)
v* = [v/V] p* = [p/[m(V/d)[.sup.n]]] t* = [t/d/V] (33)
[tau]* = [[tau]/[m(V/d)[.sup.n]]] [nabla]* = d[nabla] [D/[Dt*]] = ([d/V])[D/[Dt]] (34)
where d is the slit half gap, V is the slit average melt velocity, and m and n were defined in Eq 28. Inserting these variables into Eqs 25 and 26, and after regrouping, we have,
([nabla]* * v*) = 0 (35)
Re[[Dv*]/[Dt*]] = -[nabla]*p* - [[nabla]* * [tau]*] (36)
where Re [equivalent to] [rho][V.sup.2-n] [d.sup.n]/m is the Reynolds number for power-law liquids. Similarly, the power-law in Eq 28 becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)
where [eta]*([dot.[gamma]*]) = [eta]([dot.[gamma]])/(m * (V/d)[.sup.n-1]), [dot.[gamma]*] = [dot.[gamma]]/(V/d) and [dot.[gamma]*.sub.0] = [dot.[gamma].sub.0]/(V/d).
The boundary conditions at the free surface are adimensionalized by introducing H* = Hd and Eq 30 becomes
[[sigma].(=)]* * n = (2 C[a.sup.-1] H* - [p*.sub.a])n + [[nabla]*.sub.s] C[a.sup.-1] (38)
where Ca [equivalent to] dm(V/d)[.sup.n]/[gamma] is the capillary number for power-law liquids.
Thus, one can vary Re to explore the role of viscosity on die line formation, and Ca to explore surface tension. Table 7 lists typical properties for molten plastics, extrusion die dimensions and values of Re and Ca.
Importantly, the capillary number from Eq 38 is a global value (based on the slit gap d). It differs from the local capillary number near die line (based on die defect size) that was used in the previous experiments. They are related by
C[a.sub.local] = [[R.sub.0]/d]C[a.sub.global] (39)
where [R.sub.0] is the die defect characteristic dimension. For circular die defects, [R.sub.0] is the radius. To solve the dimensionless governing equations, we must specify the global Ca that is commonly used in plastics processing.
A stepwise approach was used to simulate die line formation. We first solved for the velocity profile where the extrudate surface is undeformed, namely, having the same geometry as the die exit. The surface was then freed to compute the full solution using the previous velocity field as the initial condition.
The Newton-Raphson method was used for the fixed surface simulation, and the segregated approach, for the released surface flow. The relaxation technique (12) was employed to ensure convergence in the second step with relaxation factors up to 0.65. The first step usually converged in 4-6 iterations and the second step in 60-150. The average computation time for each full solution was 4 hours on a Sun Microsystem Ultra-10 workstation.
The straight-spine strategy was used to update free surfaces and remesh the flow domain (12). Five layers of internal nodes below the free surface were allowed to move along their spines. Brick elements with 27 nodes were used in all simulations, since solution using 8 node elements converged more slowly.
Though mesh refining increases solution accuracy, it dramatically lengthens computation time, particularly for 3-D problems. Furthermore, too fine a mesh can cause solutions to diverge. We refined the mesh near die lines and near slit lips (common lines), and coarsened it elsewhere (see Fig. 17).
SIMULATION RESULTS AND DISCUSSION
In all simulations, we used die land asperities of triangular cross section, extending upstream from the lip (triangular asperities) and similar ones, of semicircular cross section (semicircular asperities). The former is quantified by its width and height (w X h). and the latter by its radius R. All these values have been adimensionalized by the slit half gap d.
[FIGURE 17 OMITTED]
[FIGURE 18 OMITTED]
Figure 18 is a typical result in simulating die line formation, where polymer exits a slit with a semicircular asperity, using the mesh in Fig. 17. This figure shows the left half of the flow domain only. The uniform inlet velocity is unity. This velocity applies to all simulations. After leaving the slit, the die line becomes shallower and wider, then steadies downstream. This matches our experimental observations.
Shear Thinning and Surface Tension
We used a truncated power-law viscosity model in Eq 28 to explore the shear thinning of molten plastics while neglecting viscoelasticity. In addition, since surface tension can flatten die lines, it must be included. Though in our experiments, the local capillary number always exceeds 400, in commercial die line formation, it can approach unity. Numerical simulation allows us to explore this effect otherwise inaccessible to our experimentation. We next explore how shear thinning and surface tension affect die line formation.
[FIGURE 19 OMITTED]
Figure 19 illustrates the front views of four flow domains for Newtonian fluids, shear thinning fluids with and without surface tension, using the mesh in Fig. 17. Both deformed and undeformed front views are shown. For brevity, we no longer include 3D velocity profiles. For all these simulations, the semicircular asperity has a radius of 0.2. Table 8 lists material properties for each case. Though shear thinning lessens the extrudate swell, it barely affects die line depths and base widths. Unsurprisingly, since Ca [much greater than] 1 in typical polymer extrusion, the viscous force dominates the flow, and the surface tension barely affects the velocity profiles, free surface shapes and die lines.
We used a semicircular and three triangular asperities to probe how die defect shapes affect die lines. All die defects are upstream through the land. Figure 20 shows the front view of four flow domain shapes.
Figure 21 discloses the dependence of die line depth and base width on die defect shape. All four asperities caused bell-shaped die lines, whose detailed profiles depend on die defect shape. This matches the experimental results in film blowing. The die line width magnitudes in both the simulation and the experiments matched, but their depths are an order of magnitude lower in experiments. Presumably, since die line shapes depend on both die defect heights and widths, the narrow, deep indentations on our mandrels (in contrast to the wide and shallow asperities in simulation) triggered only diminutive die lines on the blown film.
[FIGURE 20 OMITTED]
[FIGURE 21 OMITTED]
Though asperities S1 and T1 have the same size, the S1's die line is slightly wider and slightly deeper than its T1 counterpart (see Fig. 21). However, this difference would hardly affect the visibility of small die lines (R = 1 [micro]m). Hence, we believe that these two asperities have the same engineering significance.
Figure 21 further proves that both the die defect height and width govern the die line size and shape. When the asperity width (depth) doubles from T1 to T2 (T3). both the die line depth and its base width change.
We used six semicircular asperities with R = 0.05, 0.06, 0.075, 0.09, 0.1, 0.2 and seven triangular asperities of (w X h) = (0.02 X 0.01), (0.04 X 0.02), (0.07 X 0.035), (0.1 X 0.05), (0.15 X 0.075), (0.2 X 0.1), (0.4 X 0.2). For both die defects, die lines widen and deepen with asperity radius (semicircular) or width (triangular).
Figure 22 shows the dependence of die line depth and base width on semicircular asperity radius. The die line depth increases, but its base width drops with asperity size. Interestingly, the dependence of die lines on die defects weakens with larger die defects. This matches our experimental results. The results for triangular asperities are similar.
FIDAP yielded curious results (see Fig. 23) when we further reduced die defect sizes. Since these results are time steady, there is no numerical instability. We have never seen die lines shaped like these in our experiments.
The polymer density, viscosity and surface tension enter the governing equations through Re (see Eq 36) and Ca (see Eq 38). We next vary these to see how the material properties affect die lines.
In polymer melt flows, Re approaches zero, making the inertial force negligible. Our simulation showed varying Re between [10.sup.-5] and [10.sup.-2] hardly changes extrudate or die line shapes. This confirms that typical variations in the melt density and viscosity in commercial polymer processing hardly affect die line formation. We see no point in increasing Re.
[FIGURE 22 OMITTED]
[FIGURE 23 OMITTED]
The capillary number, which compares the viscous force to the surface tension. is crucial in die line formation (1, 2). We simulated die line formation using Ca = 0.5, 5, 50, 500 and 5000. For all Ca values, we used a semicircular lip asperity of radius 0.05 and Re = [10.sup.-4]. The fluid is also assumed to be shear thinning with n = 0.6.
Surprisingly, these results hardly differ, meaning that the surface tension cannot flatten die lines. This contradicts the literature and our die line dissipation mechanism (1, 2). For Ca < 0.5, the solutions diverged. Thus, we cannot further investigate how Ca can affect die lines.
In the film blowing experiments, die line heights and base widths were found to increase with throughput. In our die line formation simulations, though higher throughputs increase Re and Ca, they hardly affect die lines. This is presumably because the power-law excludes melt viscoelasticity.
Downstream Stretching (Drawdown Ratio)
Our film blowing experiments showed that downstream stretching flattens die lines. We studied downstream stretching by imposing downstream velocity [V.sub.f], exceeding the calculated equilibrium value. Figure 24 shows the side view of the velocity profile with downstream stretching velocity [V.sub.f] = 1, using the mesh in Fig. 17. Without stretching, the uniform equilibrium downstream velocity is [V.sub.e] = 0.936. Unlike flows without stretching, the stretched extrudate first swells, then contracts to accommodate the specified [V.sub.f].
[FIGURE 24 OMITTED]
Figure 25 overlays the front views of the four flow domains for different [V.sub.f]. The extrudate swells less with [V.sub.f], satisfying the mass balance. Die lines flatten with [V.sub.f]. This matches our experimental results. Though die lines slightly widen with [V.sub.f], these tiny increments are negligible in governing die line visibility.
Interestingly, Fig. 25 shows that when stretching, the extrudate bulges near the die lines. So we find that, in principle, stretching can flank an indented striation between two protruding ones. Thus, we believe that stretching can reshape the die lines, neither necessarily worsening nor improving them. In our experiments, we have yet to observe this die line reshaping.
[FIGURE 25 OMITTED]
FIDAP yielded curious, perhaps unphysical, results when [V.sub.f] = 1.025, similar to Fig. 23, so we exclude these calculations from Fig. 25. Importantly, we succeeded in establishing the trend without far exceeding DDR of unity.
In our experiments, both die defects and operating conditions affected die line formation. For die defects, we conclude
1. Die lines heighten and widen with die defect width. Both the heights and widths of rectangular die defects govern the die line sizes and shapes. The die line dependence (especially the height) on die defect width weakens with die defect width.
2. Land defects, indentations or asperities, leave no die lines. We believe that lip defects are more crucial than their land counterparts in causing die lines.
3. All rectangular lip defects create bell-shaped die lines, whose sizes and detailed profiles depend on the die imperfection shapes.
Regarding operating conditions, for indented defects causing protruding die lines, we conclude
4. Die lines heighten and widen with throughput Q.
5. Die lines flatten with take-off speed [V.sub.f]. They narrow with [V.sub.f] for LLDPE, but hardly change for HDPE.
6. Die lines flatten, but widen with BUR. Local strain increases more slowly than the global strain as the film bubble enlarges, causing less melt deformation near die lines than globally (in the surrounding film).
7. Die line shapes and sizes hardly change with melt temperature. This is because our large die defects cause the viscous force to outweigh the surface tension; thus heating the melt hardly flattens die lines.
Our simulations qualitatively match the experiments. From these we conclude that,
1. Lip asperities generate indented die lines.
2. Both semicircular and triangular asperities generate bell-shaped die lines, whose shapes and sizes depend on those of die defects.
3. Both the height and width of non-circular die defects, such as triangular ones, govern die line size and shape.
4. Die lines widen and deepen with the die asperity radius (semicircular) or with its width (triangular). Die line dependence on die defect size weakens as defects enlarge.
5. For the spectrum of melt densities and viscosities (or consistency indexes) encountered in polymer processing, these melt properties hardly affect die lines.
6. The amount of shear thinning (power law exponent) affects neither die line shape nor its size, even though it does affect extrudate swell.
7. Throughput hardly affects die line shape or size for power-law liquids.
8. Die lines flatten with drawdown, but they widen slightly. This slight widening hardly affects die line visibility. Stretching can also flank an indented striation between two protruding ones.
In our film blowing experiments, no land defects created die lines, suggesting that lip defects matter more. However, when land defects are sufficiently large or near enough to the lips, they will cause die lines. Thus, more experiments must be carried out before any conclusion can be drawn on land defects. Our film blowing experiments yielded some interesting relations between die line dimensions and die defect shapes and sizes, material properties and operating conditions. However, we suggest further experiments to examine if these results hold for different extrusion systems and materials. Because defects on our blown film mandrels are deliberately large, we did not capture how increasing melt temperature can suppress die lines. Tiny die defects (R = 1 [micro]m) are recommended for future experiments.
In our die line formation simulations, we neglected melt elasticity, downstream cooling and material property dependence on temperature. These factors will influence die line formation in commercial plastics extrusion. In addition, FIDAP fails for tiny die defects and Ca [much less than] 1; thus we cannot study how surface tension affects die lines. These difficulties must eventually be tackled to better understand die line formation. For instance, the finite element analysis package POLYFLOW might be employed for this, since it claims to incorporate fluid viscoelasticity and to simulate free surface flows (3). Though some simplifications were made, our die line simulation results qualitatively agree with the experiments and some even matched the experimental data.
Our film blowing experiments and die line formation simulations showed that die lines flatten dramatically with downstream stretching. Because shallower die lines are less visible (2), we recommend increasing take-off speed (changing nothing else) to reduce die lines. This requires a larger die gap to maintain the specified extrudate thickness, and the take-off speed must not be so high as to destroy extrudate properties. In addition, the proper take-off speed to suppress die lines depends on extrusion systems and materials.
APPENDIX: CALCULATING LOCAL DEBORAH NUMBERS NEAR ASPERITIES
For a pressure-driven flow of a power-law fluid through an annulus, the velocity profile and volumetric flow rate are (14).
[v.sub.z] ([xi]) = R [[[DELTA] pR]/[2 mL]][.sup.s][[integral].sub.[kappa].sup.[xi]]([[[beta].sup.2]/[xi]'] - [xi]')[.sup.s] d[xi]' [kappa] [less than or equal to] [xi] [less than or equal to] [beta] (A.1)
[v.sub.z] ([xi]) = R [[[DELTA] pR]/[2 mL]][.sup.s][[integral].sub.[xi].sup.1] ([xi]' - [[[beta].sup.2]/[xi]'])[.sup.s] d[xi]' [beta] [less than or equal to] [xi] [less than or equal to] 1 (A.2)
Q = [[[pi] [R.sup.3 + s]]/[3 + s]] ([[[DELTA]p]/[2 mL]])[.sup.s] [(1 - [[beta].sup.2])[.sup.1 + s] - [[kappa].sup.1 - s]([[beta].sup.2] - [[kappa].sup.2])[.sup.1 + s]] (A.3)
where R is the annulus outer radius, [kappa] is the ratio of annulus inner to outer radii, [xi] [equivalent to] r/R, s [equivalent to] l/n. For both mandrels, the annulus outer and inner radii are 13.08 and 10.54 mm, thus [kappa] = 0.8058. The consistency index m and power-law exponent n are obtained from the steady shear viscosities. Location [beta](n, [kappa]) is where the velocity maximizes. Its values are tabulated in Bird et al. (13).
For a given throughout Q, we compute [DELTA] p/2mL from Eq A.3, then insert it into Eq A.1 or A.2 to obtain the velocity, [v.sub.local] at any radial location. The penetration depths of the flow disturbance by asperities on mandrel M2 are the asperity heights. We then calculate [t.sub.flow]
[t.sub.flow] = [D.sub.d]/[v.sub.local] (A.4)
where [D.sub.d] is the die defect proximity to die lips.
The average melt relaxation times are calculated from melt relaxation spectra, using
[bar.[lambda]] [equivalent to] [[N.summation over (i=1)][G.sub.i][[lambda].sub.i]]/[[N.summation over (i=1)][G.sub.i]] (A.5)
Equation A.5 is based on rubber elasticity (14), where the shear modulus is
G = NkT (A.6)
where N is number entanglement density (the number of entanglements per unit volume), k the Boltzmann constant and T the absolute temperature. Thus, Eq A.5 yields the average relaxation time weighted with the mole fraction of entanglements from each discrete relaxation component. Hence, the local Deborah number is
[De.sub.local] [equivalent to] [bar.[lambda]]/[t.sub.flow] (A.7)
Table 6 lists the consistency index m, power-law exponent n and average melt relaxation time [lambda] for the LLDPE and HDPE.
Table 1. Indentation Dimensions on Mandrel M1. Dimension 11 12 13 14 15 16 Width [mm] 1.448 0.4064 1.143 0.7874 1.397 1.778 Height [mm] 1.524 1.524 1.524 1.524 0.5080 1.016 Length [mm] 6.350 6.350 6.350 6.350 3.937 3.937 Proximity to lip [mm] -- -- -- -- 9.525 9.525 Table 2. Asperity Dimensions on Mandrel M2. Dimension A1 A2 A3 A4 A5 A6 Height [mm] 0.7442 0.6833 0.7442 0.6833 0.7442 0.6833 Diameter [mm] 0.7899 1.405 0.7899 1.405 0.7899 1.405 Proximity to lip [mm] 3.048 6.096 9.144 9.144 12.19 15.24 Table 3. Physical Properties of Dowlex[TM] 2045 and Paxon[TM] HDPE. Properties Dowlex[TM] 2045 Paxon[TM] HDPE Density [g/cm[.sup.3]] 0.9200 0.9540 [M.sub.n] [g/mol] 30,000 19,600 [M.sub.w] [g/mol] 118,000 129,500 Polydispersity (PI = [M.sub.w]/[M.sub.n]) 3.93 6.61 Melt Flow Index [g/10 min] 1.0 0.30 Table 4. Local Deborah Numbers Near Asperities on Mandrel M2. Materials A1 A2 A3 LLDPE (Q=36.9 g/min, [T.sub.e]=135[degrees]C) 0.0460 0.0227 0.0153 HDPE (Q=37.6 g/min, [T.sub.e]=190[degrees]C) 0.0414 0.0204 0.0138 Materials A4 A5 A6 LLDPE (Q=36.9 g/min, [T.sub.e]=135[degrees]C) 0.0152 0.0115 0.0091 HDPE (Q=37.6 g/min, [T.sub.e]=190[degrees]C) 0.0136 0.0104 0.0081 Table 5. Local Deborah Numbers Near Land Indentations on Mandrel M1. Materials 15 16 LLDPE (Q=36.9 g/min, [T.sub.e]=135[degrees]C) 0.0135 0.0150 HDPE (Q=37.6 g/min, [T.sub.e]=190[degrees]C) 0.0119 0.0136 Table 6. Material Properties for Calculating Local Deborah Numbers. Materials m [Pa.[s.sup.n]] n LLDPE ([T.sub.e] = 135[degrees]C) 7.70 X [10.sup.4] 0.3262 HDPE ([T.sub.e] = 190[degrees]C) 3.30 X [10.sup.4] 0.3943 Materials [lambda] [sec] LLDPE ([T.sub.e] = 135[degrees]C) 0.062 HDPE ([T.sub.e] = 190[degrees]C) 0.045 Table 7. Typical Melt Properties and Die Dimensions in Plastics Extrusion. Symbol Typical Values Unit Polymer Melt Density [rho] 0.8 X [10.sup.3] kg/[m.sup.3] Consistency Index m [10.sup.4] Pa.[s.sup.n] Power-law Exponent n 0.4 -- Surface Tension [[sigma].sub.s] 0.025 N/m Die Gap d 2 X [10.sup.-3] m Average Melt Velocity V 0.1 m/s Reynolds Number Re 1.27 X [10.sup.-4] -- Capillary Number Ca 2.52 X [10.sup.3] -- Table 8. Material Properties in Studying Shear Thinning and Surface Tension. Re Ca n Newtonian Fluids (I) [10.sup.-4] [infinity] -- Shear Thinning Fluids (II) [10.sup.-4] [infinity] 0.4 Shear Thinning Fluids with Surface Tension (III) [10.sup.-4] 2.5 X [10.sup.3] 0.4 Shear Thinning Fluids with Surface Tension (IV) [10.sup.-4] 2.5 X [10.sup.3] 0.75 [[gamma]*.sub.0] Newtonian Fluids (I) -- Shear Thinning Fluids (II) 0.001 Shear Thinning Fluids with Surface Tension (III) 0.001 Shear Thinning Fluids with Surface Tension (IV) 0.001
The authors gratefully acknowledge the financial support of the Kimberly-Clark Corporation and the 3M Company. We also thank Mr. Walter Wigglesworth from our Physics Department for his assistance in making our blown film mandrels. Further, we appreciate the helpful discussions with and suggestions from Professors Michael Graham. Tim Osswald, David Malkus and Tahn Tran-Cong.
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2. Fan Ding. Ph.D. thesis, Mechanical Engineering, University of Wisconsin-Madison (2002).
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FAN DING and A. JEFFREY GIACOMIN*
Department of Mechanical Engineering and Rheology Research Center University of Wisconsin Madison, WI 53706-1572
* To whom correspondence should be addressed.
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|Author:||Ding, Fan; Giacomin, A. Jeffrey|
|Publication:||Polymer Engineering and Science|
|Date:||Oct 1, 2004|
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