Dimensional change resulting from swell and drawdown of extruded polymers.
In previous work on shape change during drawdown, Griffith and Tsai (ref. 1) concluded that drawdown from the equilibrated (relaxed) swell state for rectangular and t-shaped profiles was generally symmetric. However, small systematic deviations from symmetric shape change in their data, e.g., figures 6 and 7, correspond to the shape change during drawdown reported here.
Earlier work by Stevenson (ref. 2) proposed size and shape change analysis, which is further developed here and linked with predictive models fit to data.
A systematic way is proposed to analyze and model changes in individual extrudate dimensions in terms of size (symmetric) and shape (non-symmetric) changes along the extrusion line. Shape change data are fit to a two-constant power law model, and a three-constant model which accurately predicts the asymptotic dimensional change at large draw ratios. This analysis will be applied to a rubber compound and a rigid polyvinyl chloride (PVC) compound.
The utility of this understanding will be illustrated by modeling dimensional changes when the draw ratio is in error and when the compound swell properties deviate from specification (different from those for which the die was designed). The pattern of the percentage deviations of end-of-line (cold) dimensions from specification will be used to distinguish between drawdown and swell as specific sources of variation.
The following two data sets are presented in this article:
* A rubber compound extruded on a Troester 90 mm pin barrel extruder, and
* A rigid PVC compound, Geon 85857, extruded on a 45 mm extruder as reported by Huneault, LeFleur and Carreau (ref. 3).
The rectangular dies (all without relief along the land) used in these data sets are summarized in table 1.
In the experiment with rubber, the equilibrated (relaxed) swell sample of the rubber compound was collected just outside the die and allowed to swell unconstrained in hot water at a temperature comparable to the stock temperature. Three other samples were collected at increasing draw ratios; at the highest draw ratio, the extrudate was lifted off the conveyor. To sample these highly stretched extrusions, a 0.7 meter section of the extrudate was clamped in a fixture, cut from the line, secured on a board, and allowed to relax over several days. The extrudate profiles were obtained using a profilometer which had a rotatable stylus opposite a reference wheel. Profiles obtained with this device had a flat bottom surface. Figure 1 shows a scaled view (thickness dimension magnified 20x) of the profiles in the equilibrated state, and at low, intermediate and high drawdowns.
The rigid PVC drawdown data were provided by Huneault (refs. 4 and 5) in his master's thesis at the Ecole Polytechnique in Montreal. The thesis and published paper (refs. 3 and 4) also contain data for drawdown of high density polyethylene. Similar to the experiments with rubber, the extrudate was pulled through a cooling tank by a takeaway device (ref. 4). Once the cut sample was cool, the width was measured directly and the average thickness was obtained from the linear weight and density of the sample.
For the purposes of this analysis, consider an extrudate emerging from a die, location 1, and being drawn down along the extrusion line to end-of-line (cold) dimensions at location 3 (figure 2). (Location 2 is an intermediate location along the line not used in this work.) The equilibrated swell sample E is illustrated at location E in the figure. The profiles in figure 2 are idealized (no rounding of comers).
The change in dimensions caused by swell and drawdown between two locations along the extrusion line is analyzed in terms of a symmetric size change factor (Sz) (ratio of cross-sectional areas) and shape change factors (Sh), which apply to individual dimensions, e.g., thickness or width. The shape change is determined by the ratio of extrudate dimensions, e.g., the maximum thickness at two locations along the extrusion line divided by the square root of the cross-sectional area ratio at these two locations. Shape change factors account for the non-symmetric changes in extrudate dimensions between these locations; the symmetric size change having been removed by dividing by the square root of the area change ratio. For dimensions that change symmetrically between the two locations, the shape change factor (Sh) is 1.
A schematic drawing illustrating size and shape changes in nine configurations is shown in figure 3. Size and shape changes are illustrated by the changing extrudate dimensions (red rectangles) superposed on the fixed die dimensions (yellow rectangles). The open symbols represent data from figure 1, as explained following equation 11.
Typically, dimensions are analyzed between the die (location 1) and the end of the line (location 3) so the size (Sz) change and shape (Sh) change factors for thickness (H) and width (W) are defined:
[Sz.sub.13] = [A.sub.3]/[A.sub.1] = D[R.sup.-1] (1)
[Sh.sup.H1.sub.3] = ([H.sub.3]/[H.sub.1])/[([A.sub.3]/[A.sub.1]).sup.1/2] (2)
[Sh.sup.W.sub.13] = ([W.sub.3]/[W.sub.1])/[([A.sub.3]/[A.sub.1]).sup.1/2] (3)
Rearranging equation 2 gives the thickness dimension ratio at the end of the line:
[H.sub.3]/H, = [Sh.sup.H.sub.13] D[R.sup.-1/2] (4)
To predict dimensions, such as [H.sub.3], we need to model how the shape change factor (Sh) for each dimension depends, usually in different ways, on draw ratio.
The profiles in figure 1 can be used to determine the average thickness of the extrudate ([H.sub.a]), the maximum thickness ([H.sub.m]) and the width (W) for the equilibrated swell and during drawdown. These measurements, relative to the die dimensions, are plotted versus the draw ratio (DR) in figure 4. The draw ratio at a particular location is calculated as the ratio of the die open area divided by the cross-sectional area of the extrudate:
DR = [A.sub.1]/A = [H.sub.1][W.sub.1]/([H.sub.a]W) (5)
Note that the plots are almost linear on a log-log graph with the [H.sub.a]/[H.sub.1] and W/[W.sub.1], plots straddling the dashed line representing symmetric drawdown. Note that with [H.sub.m], the maximum thickness has a higher dependence on DR.
The simplest realistic model for dependence of the shape change factor (Sh) on draw ratio (DR) is the power law, which has the following form for a thickness (H):
[Sh.sup.H] = (H/[H.sub.1])/[(A/[A.sub.1]).sup.1/2] = C D[R.sup.-m] (6)
where C and m are constants determined by a curve fit to the thickness data, which could be for the average (a), maximum (m) or some other thickness.
When equation 6 is evaluated for the average thickness ([H.sub.a]/ [H.sub.1]), the corresponding equation for width has the form:
[Sh.sup.W] = (W/[W.sub.1])/[(A/[A.sub.1]).sup.1/2] = [(C D[R.sup.-m]).sup.-1] (7)
and the constants (C) and (m) are the same for predictions of [H.sub.a]/[H.sub.1] and W/[W.sub.1]. Multiplying equation 6 applied to [H.sub.a] times equation 7 gives equation 5.
The power law constants in figure 4 were evaluated by two linear least square fits: (a) to equations 6 and 7 in logarithmic form with the [H.sub.a]/[H.sub.1] and WAV, data, and (b) to equation 6, evaluated for [H.sub.m]/[H.sub.1] with the corresponding maximum thickness data. The constants are given in figure 4.
The power law curves in figure 4 show good fits to the data, but at DRs of 2 or more, the predictions cross over.
The curve labeled "slope" in figure 4 is defined as the rate of change of [H.sub.a]/[H.sub.1] with respect to DR divided by the analogous rate for W/[W.sub.1] as shown in equation 8. The application section of this article shows that the slope value can be useful in detecting deviations from the specified draw ratio.
The "slope" for the power law model in figure 4 is given by:
(m+1/2) C2 D[R.sup.-2m]/(1/2-m) = 256D[R.sup.-0.296] (8) (l/2-m)
A model for drawdown is needed that does not predict crossover of the thickness and width predictions at high DR. The three-constant KCm model shown below gives predictions that run parallel to the symmetric line at high draw ratios. This is accomplished by adding a constant (K) to the power law model, which makes the shape change factor ([Sh.sup.H]) for thickness approach a constant value (K) at high draw ratios:
[Sh.sup.H] = (H/[H.sub.1])/[(A/[A.sub.1]).sup.1/2] = [[K + C D[R.sup.-m]].sup.+1] (9)
As was the case with the power law model, if equation 9 is evaluated for [H.sub.a], then the corresponding equation for width (W) has the form:
[Sh.sup.W] = (W/[W.sub.1])/[(A/[A.sub.1]).sup.1/2] = [[K + C D[R.sup.-m]].sup.-1] (10)
and the constants (K), (C) and (m) are the same for both [Sh.sup.Ha] and [Sh.sup.W] predictions.
The KCm model predictions for data obtained from figure 1 are shown as shape change factors in figure 5. Note the KCm model predictions flatten out at high draw ratios. For [H.sub.a]/[H.sub.1] and W/[W.sub.1], the KCm model predictions cross the line of symmetry at extremely high draw ratios since the curve fit value of K happens to be less than 1.
The "slope" in figure 5 for the KCm model is given by:
Slope = d([H.sub.a]/[H.sub.1])/dDR/d(W/[W.sub.1])/dDR =
[([Sh.sup.Ha]).sup.2] [0.5 [Sh.sup.Ha] + mCD[R.sup.-m]]/[0.5 [Sh.sup.Ha] + mCD[R.sup.-m]] (11)
where [Sh.sup.Ha] is given by equation 9 applied to average thickness [H.sub.a]. Note that at typical draw ratios, the "slope" is about 2, a value commonly observed in practice.
The trajectory of the shape change factor [Sh.sup.Ha] versus Sz as predicted by the KCm model is shown by lines in figure 3. The swell line extends from the die face to the equilibrated swell. The drawdown line shows how [Sh.sup.Ha] changes with Sz during drawdown from equilibrated swell to very high drawdowns. The drawdown line traces a path higher on the figure than the swell line and never crosses the die coordinates. Swell cannot be directly counteracted by drawdown.
The data for rigid PVC of Huneault et al. (ref. 3) are plotted as shape change factors for thickness and width in figure 6. The data cover a wide range of draw ratios, 0.5 to 5.0, and show little dependence on output rate (varied by a factor of 8), except possibly at the highest flow rate and lower draw ratios. Data for rubber (Os) show somewhat higher swell and equilibrated swell at lower draw ratios.
Other data from Huneault et al. (ref. 3) show the ratio of thickness to width ([Sh.sub.13]H/[Sh.sub.13.sup.W]) plotted versus draw ratio for flow through two dies with L,/HI ratios of 10 (figure 6) and 40 (data not shown here). These data show the [Sh.sup.H]/[Sh.sup.W] ratio, same as the (H/[H.sub.1])(W/[W.sub.1]) ratio, is typically 7.8% higher for the shorter [L.sub.1]/[H.sub.1] 10 die compared to the [L.sub.1]/[H.sub.1] 40 die.
For capillary dies, equilibrated swell ratios show a pronounced decrease with increasing L/D up to about an L/D of 10, and then a much shallower decrease at higher L/Ds (ref. 7). (For higher L/D dies, the polymer is said to "forget" its unoriented state prior to entering the die.) The relatively small 7.8% increase in swell for [L.sub.1]/[H.sub.1] = 10 rectangular die compared to [L.sub.1]/ [D.sub.1] = 40 die is consistent with this result.
For the available data, the KCm model constants are rather different for rubber and for two different PVC materials, only one of which is reported here. The C value is higher for rubber (-0.12 versus -0.04 for PVC), and the m value is higher for PVC (-1.5 versus -1.0 for rubber). If additional drawdown data show the KCm model constants for a type of material, e.g., natural rubber, fall within a narrow range, it may be possible to use generic drawdown models for these materials.
Dimensional control of a rectangular profile, which could represent a portion of a tire tread profile, is considered in examples where the dimensional variation is caused by (a) an error in the draw ratio and (b) the swell of the compound deviating from the specified swell (swell for which the die was cut). Only cold dimensions at the end of the line are used in these examples. The effect on dimensions for these two causes is very different, as will be demonstrated graphically. Data in table 2 illustrate how to distinguish between draw ratio errors and swell deviations.
First, consider a compound with the specified swell properties (KCm model parameters 1, 0.1 and 1), but a draw ratio of 0.4, well below the specified draw ratio of 0.6. A schematic of the die and extruded profile is shown in figure 7 (right side), along with a drawdown diagram for the extrudate dimensions (left side). The yellow arrows show that by increasing the draw ratio, the dimensions on the drawdown diagram naturally follow the drawdown curves ([H.sub.3]/[H.sub.1] and [W.sub.3]/[W.sub.1] versus DR), and come into specification at the specified draw ratio of 0.6. The dimensions of the die/profile schematic (right side) also come into specification by contracting along the yellow arrows.
The situation where the swell properties deviate from their specified values is shown in figure 8, where drawdown curves are shown for two different predictions for the KCm model. For swell larger than specified (KCm model parameters 1,0.18 and I), the drawdown curve is given by the dashed lines, which show with higher thickness dimensions and lower width dimensions. At the specified draw ratio of 0.6, the thickness is too high and the width too low. If one wants to bring the too low width dimension into specification, the draw ratio must be reduced to 0.37, as shown by the yellow arrows in the figure. But at this low draw ratio, the extrudate thickness and cross-sectional area (dashed lines in the die profile schematic) are much higher than specification. Conversely, if the thickness dimension was brought into specification, the draw ratio would be increased to bring down the thickness dimension to specification (DR at downward arrow in figure). At this draw ratio, higher than 0.6, the width would be smaller and even further from the width specification, and the cross-sectional area of the profile would be too small.
These examples show it is much easier to deal with the situation where the draw ratio is too high or low, rather than when the swell is out of specification. When swell is higher or lower than expected, it is not possible to bring both the thickness and width into specification, so some sort of compromise is necessary, or a specialized shape control method is needed to allow independent control of thickness and width. Seven methods of shape control were described in a recent publication (ref. 8), and are presented in detail in the Rubber Extrusion Technology short course notes (ref. 6).
Ways to use end-of-line rectangular profile data to identify incorrect draw ratio or deviations in swell properties are illustrated in figure 9 and table 2. In the figure, the specifications for width and thickness ratios are represented by the solid diamonds (DR = 0.5, KCm =1,0.115,1). [H.sub.3]/[H.sub.1] and [W.sub.3]/[W.sub.1] ratios for draw ratio deviations of [+ or -] 0.05 ([+ or -] 10%) with the swell on specification are shown by the open diamonds.
[H.sub.3]/[H.sub.1] and [W.sub.3]/[W.sub.1] ratios for deviations in the swell properties are shown in figure 9 by the open circles for the high swell predictions (KCm 1, 0.145, 1) and the open triangles for the low swell predictions (KCm 1, 0.090, 1).
More detailed insights can be obtained by examining the percentage deviations of the dimensions and cross-sectional areas relative to specification, as illustrated in table 2.
Cases 2 and 5 in the table (boldface) are for both swell and draw ratios in specification. Cases 1 and 3 in the table are for deviations in swell properties with the correct draw ratio. The %[H.sub.3sp] and %[W.sub.3sp] table entries are the percentage deviations for these dimensions from their specified values at the end of the line (predicted dimension-specified dimension)/specified dimension, expressed as a percentage). For a given profile with the DR in specification, cases 1 and 3, the %[H.sub.3sp] and %[W.sub.3sp] values are always of the opposite sign. If there is excess thickness swell, then there must be a deficiency in width swell, since the cross-sectional area (draw ratio) is at specification. The percentage changes in %[H.sub.3sp] and %[W.sub.3sp] are comparable (-4.1% and 4.2% in case 1), so the ratio %[H.sub.3sp]/%[W.sub.3sp] is about -1.
Cases 4 and 6 in table 2 represent the expected swell with [+ or -] 10% deviations in the draw ratio. Cases 4 and 6 correspond to the open diamonds in figure 9 on either side of the set point (filled diamond). With draw down as the source of variation, the ratio of the percentage change %[H.sub.3sp]/%[W.sub.3sp] is positive and between about 2 and 2.5, as shown in cases 4 and 6. The ratio of percentage change corresponds to the "slope" prediction in figure 5. The percentage deviation in the profile cross-sectional area at the end of the line relative to the specified cross-section is given by %[(HW).sub.3sp]. This value is the percentage correction that brings the draw ratio DR into specification: For example, in case 4, 1.111 x 0.45 = 0.5.
The utility of these observations is limited to cases where dimensional variation is due almost exclusively to a deviation in draw ratio or to a deviation in swell properties. More work is needed to identify dimensional variation caused by simultaneous deviations in draw ratio and swell. There are many other causes of variation in extrusion, including batch changes, unsymmetrical dies, material property drift during a run, output rate variation of adjacent materials, surging, loss of feed, startup, cycling of controllers, to name some of the more common causes. Ways of dealing with these sources of variation on rubber extrusion lines are discussed in other publications (refs. 6 and 8).
Most extrusions are more complex than rectangles, and may involve two or more components. Laser gauging can provide accurate and detailed rendering of the profile on line-of-sight surfaces. It may be necessary to examine several thickness and width measurements to see which combinations can represent a rectangular profile for a particular compound. The tread profile in figure 10 shows some candidate thickness and width dimensions.
Dimensional changes (thickness and width) in rectangular profiles caused by drawdown from the equilibrated (relaxed) swell state have been measured and analyzed for rubber and PVC over a range of draw ratios, flow rates and die geometries. Width and thickness dimensions do not change symmetrically; generally, thickness dimensions change at more than twice the rate of width dimensions with drawdown.
These dimensional changes were measured and analyzed in terms of size change factors (symmetric) and shape change factors (quantifies asymmetric change). The shape change factors were shown to be dependent on the draw ratio, and were modeled by a two-constant power law model and a three-constant (KCm) model which accurately described the limiting behavior at high draw ratios. The KCm model was used to illustrate the relative ease of correcting for drawdown errors and the difficulty of dealing with deviations from the specified swell. This model also provided useful criteria for identifying deviations of draw rate or swell from specified values by analyzing the deviation from specification of rectangular profile dimensions at the end of the line (cold location).
by James F. Stevenson, Stevenson PolyTech LLC
(1.) R.M. Griffith and J.T. Tsai, "Shape changes during drawing of non-circular extruded profiles, " Polym. Eng. Sci., 20, 1,181 (1980).
(2.) J.F. Stevenson. ".Analysis of extrudate dimensions: Die design, swell and drawdown, " Plast. Rubber Process Appl. 5, 325 (1985).
(3.) MA. Huneault, P.G. LeFleur and P.J. Carreau, "Extrudate swell and drawdown effects on extrudedprofile dimensions and shape," Polym. Eng. Sci. 30, 1,544 (1990).
(4.) M. Huneault, "Le gonflement postextrusion dans le procede d'extrusion de profiles in materiaux plastiques," M.Sc.A. Thesis, Ecole Polytechnique, Universite de Montreal, April 1989.
(5.) M.A. Huneault, personal communication.
(6.) J.F. Stevenson, "Rubber extrusion technology short course notes, " available from Rubber World, rubberworld.com/bookstore (2018).
(7.) R. Kannabiran, "Application of flow behavior to design of rubber extrusion dies," Rubber Chem. Technol. 59, 142(1986).
(8.) J.F. Stevenson, "Rubber extrusion in an Industry 4.0 environment," Rubber & Plastics News, 18, June 25, 2018.
Caption: Figure 1--profilometer traces of rubber extrudates
Caption: Figure 2--schematic showing dimensional change along an extrusion line
Caption: Figure 3--size and shape change for an extrudate (red) superposed on a die (yellow)
Caption: Figure 4--dimension ratios from figure 1 and power law predictions versus DR
Caption: Figure 5--shape change factors from the data in figure 1 and KCm model predictions versus DR
Caption: Figure 6--shape change data for rigid PVC versus draw ratio from Huneault et al. (ref. 3)
Caption: Figure 7--dimensions are brought into specification by increasing the draw ratio from 0.4 to 0.6
Caption: Figure 8--dimensions for compound with swell deviation cannot be brought to specification with draw
Caption: Figure 9--dimensions for deviations from specified swell and draw ratios
Caption: Figure 10--tread profile dimensions that could approximate a rectangular extrusion
Table 1--dimensions for rubber and PVC dies Compound Thickness Width Land UH W/H (mm) (mm) (mm) Rubber 6.4 229 12.7 2.0 35.8 PVC 1.0 25 10 10 25 PVC 1.0 25 40 40 25 Table 2--predictions by KCm model for deviations in swell properties and draw ratios Die dimensions: [H.sub.1] (mm): 10, [L.sub.1] (mm): 200, [A.sub.1] = [H.sub.1][W.sub.1] ([mm.sup.2]): 2,000 Model constants Case K C m DR 1 1 0.090 1 0.50 2 1 0.115 1 0.50 3 1 0.140 1 0.50 4 1 0.115 1 0.45 5 1 0.115 1 0.50 6 1 0.115 1 0.55 Dimension ratios Case [H.sub.3]/ [W.sub.3]/ %[H.sub.3sp] [H.sub.1] [W.sub.1] 1 1.67 1.20 -4.1% 2 1.74 1.15 0.0% 3 1.81 1.10 4.1% 4 1.87 1.19 7.6% 5 1.74 1.15 0.0% 6 1.63 1.12 -6.3% Percent deviation from specification Case %[W.sub.3sp] %[H.sub.3sp]/ %[H.sub.3sp] %[W.sub.3sp] 1 4.2% -0.96 0.0% 2 0.0% 0/0 0.0% 3 -3.9% -1.04 0.0% 4 3.3% 2.33 11.1% 5 0.0% 0/0 0.0% 6 -3.0% 2.09 -9.1%
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|Author:||Stevenson, James F.|
|Date:||May 1, 2019|
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