Modeling of complex parison formation in extrusion blow molding: effect of medium to large die heads and fuel tank geometry.

INTRODUCTION

Extrusion blow molding is a low-cost and efficient manufacturing technique for complex hollow parts (1), including automotive fuel tanks. This process consists of three main phases, namely, parison formation, parison inflation, and part cooling and solidification (2). The parison formation is the most critical phase as the parison dimensions strongly affect the final part thickness distribution and its mechanical performance.

The complexity of part geometry could dictate the use of advanced manufacturing technologies to meet the target part thickness requirements. In standard extrusion blow molding, the parison axial thickness distribution is controlled using a vertical wall distribution system (VWDS), also known as parison programming. This is achieved by manipulating the die gap opening at a given number of programming points. However, this technique may not be sufficient to produce the desired outcome in the manufacturing of highly complex fuel tank shapes driven by design constraints of available irregular design space and the ever increasing demand of increasing fuel capacity. Thus, the manufacturing of these complex part shapes may require the use of advanced die shaping technologies such as partial wall distribution system (PWDS) and die slide motion (DSM). The PWDS technology allows efficient circumferential parison wall thickness control according to a predetermined profile for an optimal part thickness distribution and the lightest part weight. As the parison thickness has a highly nonlinear relationship with die gap opening because of the extrudate swell phenomenon, a considerable amount of time, cost, and skill is required to get the desired parison dimensions upon any variations in the die geometry and operating conditions. Furthermore, the PWDS technology comes at a substantial increase in capital cost, which adds to the cost of manufacturing. Hence the accurate assessment of tooling requirement using predictive techniques is imperative to minimize tooling change cost and customer delivery time.

Parison swell and sag are factors that significantly influence the parison dimensions, and are strongly affected by die geometry, resin characteristics, and operating conditions. Parison swell is a result of molecular orientation generated during the flow in the die (3). A combination of elongational and shear stresses are applied to the molten polymer as it travels through the die. Once the melt exits the die in the form of a parison, the stress is relieved and the swell occurs. The parison sag is a result of parison stretching under gravitational forces. The degree of swell and sag is controlled by die design, resin characteristics (e.g., extensional viscosity and damping coefficients), and processing parameters, such as melt temperature, suspension time, and total parison length (4).

Parison swell is defined as an increase in the cross sectional area of the extruded parison, and is represented in terms of the diameter swell ([B.sub.1]) and the thickness swell ([B.sub.2]) defined as:

[B.sub.1] = [D.sub.parison]/[D.sub.die] (1)

[B.sub.2] = [h.sub.parison]/[h.sub.die] (2)

where [D.sub.parison] and [D.sub.die] are the parison and die diameters, [h.sub.parison] is the parison thickness, and [h.sub.die] is the die gap opening. The final part thickness distribution is directly related to both diameter swell and thickness swell (5). Therefore, these two parameters can be combined to define the are a swell as follows:

[B.sub.area] = [B.sub.1] x [B.sub.2]. (3)

A pinch-off mold, originally proposed by Sheptak and Beyer (6) can be used to estimate the diameter and thickness swell in Eqs. 1, 2. For simple blow-molded parts, it is shown in a previous study that the weight swell from the dissected final part can efficiently represent the combined effect of the diameter and thickness swell, in the same manner as the area swell (7). Therefore, the weight swell can be estimated as follows when the pinch-off mold measurement is not available:

[B.sub.w] = w/[w.sub.0] (4)

In this equation, w is the weight of the dissected section after swell and sag, whereas [w.sub.0] is the corresponding weight of a section without swell and sag, featuring the same length and assuming a cylindrical geometry.

The effect of die geometry on diameter and thickness swell has been studied in the literature. Orbey and Dealy (4) studied the effect of die design on the swelling of different HDPE melts using a variety of tapered annular dies. They found out that for all resins examined, the diameter swell was always highest for the converging, followed by the straight and then the diverging dies. Garcia-Rejon and coworkers (8) reported that the shape of parison profile and the diameter swell are influenced by hoop stresses that are associated with geometrical configuration of the die assembly. Moreover, they attributed the thickness and diameter swell to the die inclination angle, die land contraction ratio, and die gap opening.

The main contribution to the extrudate swell comes from the memory effect. This is primarily the elastic response to the elongational stresses prevailing at the entrance of the die and, to some extent, to the shear stresses imposed on the melt in the flow channel. Koopmans (9-11) has reported that the maximum swell and the time to reach this value are very sensitive to the molecular weight distribution of the resin. Moreover, according to his findings, small variations in the die geometry are usually much more effective in altering the swelling characteristics of an HDPE resin than most of the operating parameters. Given the nonlinear relationship between parison swell and die gap opening observed in previous studies (7), (12-14), the experimental data collected at a wide range of die geometries, resin characteristics, and operating conditions can be the most efficient technique to validate the finite element simulations, and to verify the practical validity of the viscoelastic constitutive equations and the mathematical swell models (11).

The main goal of this work is to investigate the effect of die geometry and die gap manipulation on swell and sag for complex parisons produced by a combination of VWDS and PWDS techniques. The collected experimental data are compared with the numerical prediction of the parison formation, performed using BlowParison[C] software developed at IMI. Since it is well established that small differences in die geometry can lead to significant swell differences (7), (11), it is very important to study the performance of BlowParison[C] software in predicting the parison dimensions upon any variation in die geometry and die gap profile. This software has been successfully used in the past to model the swell and sag, combined with the nonisothermal effects, for several industrial parts including fuel tanks (15), (16). The software couples a fluid mechanics approach to represent the die flow, with a solid mechanics approach to represent the parison behavior outside the die, and a mathematical swell model to account for the pronounced elongational and shear stresses at high Weissenberg numbers. The fundamentals of this modeling tool can be found elsewhere (7), (17).

THEORETICAL BACKGROUND

Manipulation of Die Gap Profile

The final part thickness distribution is directly related to both the diameter swell and thickness swell. For equivalent parison thicknesses, a section with a smaller diameter will result in a section with a thinner final part thickness (18). To satisfy the design constrains for complex parts, e.g., fuel tanks, parison thickness distribution should be controlled using different techniques, so that the final wall thickness of the blown part is as uniform as possible after inflation (19). Although the die gap programming (VWDS) allows a desired distribution along the extrusion axis, the irregular shape of the final part requires circumferential parison wall thickness control. To this end, the static flexible deformable ring (SFDR), PWDS, and DSM techniques are used in the industry. The die gap manipulation techniques are briefly discussed in the following section.

Vertical Wall Distribution System. The desired vertical variation in parison thickness can be attained by VWDS, as schematically illustrated in Fig. la. The conical core part (mandrel) is axially shifted relative to the die ring (bushing) during parison formation, leading to the desired die gap opening at a given moment during extrusion. The parameters used in the industry to control the die gap opening are shown in Fig. 1b. The die gap for a given programming point ([h.sub.die]) is calculated based on the stroke parameters defined in Fig. 1b as follows:

[h.sub.die] = [h.sub.min] + tan ([[empty set].sub.m])[S.sub.total] (5)

[S.sub.total] = [S.sub.max][([P.sub.basic]/100) + ([P.sub.prog]/100) ([P.sub.select]/100)] (6)

where:

[P.sub.basic] = 100 x ([S.sub.basic]/[S.sub.max]) (7)

[P.sub.prog] = 100 x ([S.sub.prog]/[S.sub.max]) (8)

[P.sub.select] = 100 x ([[S.sub.max] - [S.sub.limit]]/[S.sub.max]); [P.sub.select] [less than or equal to] 100 - [P.sub.basic]. (9)

[FIGURE 1 OMITTED]

In these equations, [h.sub.min] is the minimum gap at the machine setting where the mandrel and bushing are in flush configuration, and [[empty set].sub.m] is the mandrel angle from vertical. As it can be seen from Fig. lb, the VWDS stroke parameters [S.sub.total], [S.sub.basic], [S.sub.prog], and [S.sub.limit] are in millimeters, whereas [P.sub.basic], [P.sub.prog], and [P.sub.select] are in percentage. The definition of these parameters is consistent with machine parameters used in the industry. Figure 1c and d schematically show the position of the mandrel for 0% and 100% die gap programming, respectively. It should be mentioned that a [P.sub.basic] = 0% corresponds to a machine setting where the mandrel and bushing are in flush configuration at 0% die gap programming ([h.sub.die] = [h.sub.min]).

Static Flexible Deformable Ring. This technique is used in the industry to offer a circumferential variation in parison wall thickness without permanent shaping or profiling of the tooling. In particular, for producing nonsymmetrical parts or square bottles, the thin corners in the final part can be avoided by SFDR. However, it only creates a fixed pre-set circumferential wall thickness distribution throughout the whole parison length (19), as schematically shown in Fig. 2. For complex parts, the uniform thickness requirements can be better satisfied once this technique is combined with the PWDS technology.

[FIGURE 2 OMITTED]

Partial Wall Distribution System. The need for creating a circumferential variation in the thickness at a specific section along the parison length is the driver of this technology. The pulling and/or pushing actions (strokes) can be achieved at every desired point, circumferentially and axially (19), as illustrated in Fig. 3a-c. Figure 3d shows the typical variation in the die gap, corresponding to the PWDS action in Fig. 3c. The thin areas in the corners and in the deep-draw sections of the final part can be avoided by using this technique.

[FIGURE 3 OMITTED]

Die Slide Motion. Similar to the PWDS technique, the DSM creates a circumferential variation in the thickness at a specific section along the parison length. However, the pulling and/or pushing actions do not affect the zones outside the DSM stroke action, as illustrated in Fig. 4a and b. Figure 4c shows the typical variation in the parison thickness, corresponding to the DSM action in Fig. 4b.

[FIGURE 4 OMITTED]

Modeling of Parison Formation

Because of the complex microstructure of molten polymers, modeling of annular flow in blow molding operations is governed neither by the current state of deformation nor by the current state of motion; instead, the overall stress field depends on the whole history of the deformation (20). Many existing models today are based either on molecular considerations or on modifications of established theories such as linear or nonlinear viscoelasticity (21-23). In an extensive review. Crochet and Walters (24) presented the numerical simulation of the flow of highly elastic liquids in complex geometries. Brasseur et al. (25) solved the time-dependent compressible Newtonian extrudate-swell problem with slip at the wall, in an attempt to simulate the stick-slip extrusion instability. Tanoue and Iemoto (26) calculated the steady-state annular extrudate swell of polymer melts with die gap programming (VWDS) using the Giesekus constitutive model. The need for experimental trials to validate the simulations was emphasized. A neural network-based model approach was presented by Huang and Liao (27), in which the effects of the die temperature and flow rate on the diameter and thickness swell for HDPE were investigated. A good agreement was reported between experimentally determined parison swell and the predictions, at low flow rates, using the trained neural network model. In another work (28), they presented a combination of finite element modeling using K-BKZ model (29), (30) and artificial neural network to predict the parison formation in extrusion blow molding. Their results showed that the die gap had a smaller effect on the diameter swell but a greater effect on the thickness swell. An increase in both diameter and thickness swell was reported by an increase in die angle. Huang and Huang (31) used a hybrid method consisting of finite element method, artificial neural network, and genetic algorithm to find the optimal parison thickness distribution for a blow molded part with required thickness distribution. The die design results of Orbey and Dealy (4) with tapered dies were simulated for Newtonian fluids by Mitsoulis and Heng (32). They showed that Newtonian fluids gave the opposite results from the experiments. Luo and Mitsoulis (33) predicted the flow of HDPE resins through dies of different geometry using the K-BKZ model. Their numerical results compared well with experiments and showed the ability of the K-BKZ model to capture important memory phenomena in parison extrusion based on die design. The importance of using the Papanastasiou damping function (34) was emphasized for HDPE melts by Luo and Mitsoulis (35). Recently, Mitsoulis (36) derived numerical solutions for the extrudate swell and exit correction in annular flow of pseudoplastic and viscoplastic fluids. An extensive analysis of the existing approaches for the numerical simulation of non-Newtonian and viscoelastic fluids can be found elsewhere (37), (38).

Governing Equations

To calculate the flow stresses in BlowParison [C] software, the flow kinematics is derived using a generalized Newtonian model, and it is subsequently used as an input for the viscoelastic equation to calculate flow stresses (39). Although the flow kinematics in the die was predicted based on the Hele-Shaw model, assuming Carreau viscosity model type behavior for the melt (40), the particle tracking and the deformation prediction was preformed using the K-BKZ viscoelastic model (28-30). Compared to the direct approach where the viscoelastic behavior of the material is taken into account from the beginning, this approach is more cost effective (41). That is, the calculation time is significantly reduced upon using the generalized Newtonian model as it does not require the calculation of the whole history of the deformation (20).

Flow in the Die. The viscosity takes the following form for a fluid obeying the Carreau model (40):

[eta] = [[eta].sub.0][a.sub.T][[1 + [([lambda][a.sub.T][gamma]).sub.2]].sup.[[n - 1]/2]] (10)

In this equation, [eta] is the viscosity of the polymer melt at the processing temperature, [[eta].sub.0] is the zero-shear Newtonian viscosity at a reference temperature, [gamma] is the shear rate, n is the power-law index for the shear-thinning zone, [lambda] is a characteristic time representing the transition between the Newtonian and shear-thinning zones, and [a.sub.T] is the temperature shift factor according to the WLF shift function (42), (43).

Parison Formation After Exiting the Die. The K-BKZ model relates the stress to the strain history as follows (29), (30):

[sigma](t) = - q[delta] + [1/[1 - [theta]]][1.[integral] -[infinity]][N.summation over (k = 1)][[m.sub.k] (t - [tau])[h.sub.k]([I.sub.1], [I.sub.2])] x {[c.sup.-1] ([tau], t) + [theta]C([tau], t)}d[tau] (11)

where q is the hydrostatic pressure, [delta] is the identity tensor, t is the time, [tau] is the relaxation time, m is the memory function given by the Maxwell relaxation spectrum, k is the index of the linear-viscoelastic parameter sets for the memory function, N is the total number of the modes, c is the Cauchy deformation tensor, [c.sup.-1] is the Finger deformation tensor, h is the damping function based on the invariants of the Finger tensor ([I.sub.1] and [I.sub.2]), and [theta] is a parameter that refers to the second normal stress difference in the deformation (biaxial effect) and influences the extensional properties of the material (44). That is, an increase in -[theta] is associated with a decrease of the uniaxial extensional viscosity (45).

The Papanastasiou damping function was used to represent the strain dependency under nonlinear viscoelastic deformation (34):

[h.sub.k]([I.sub.1], [I.sub.2]) = [[alpha]/[[alpha] - 3 + [[beta].sub.k][I.sub.1] (t, [tau]) + {1 - [[beta].sub.k]} [I.sub.2]([t, [tau]]]] (12)

In this equation, [alpha] and [beta] are the damping coefficients. Although an increase of [alpha] reduces the level of nonlinear viscoelasticity at high shear strains, an increase of [beta] decreases the importance of the extensional viscosity (46). In this work, a total of six parameter sets were used for the constitutive model (N = 6). The same value of [alpha] and [beta] were used for all modes. The thermal dependence of the K-BKZ model was accounted for with the WLF temperature shift function.

The flow kinematics from the Carreau model was used as input to the K-BKZ model so as to calculate the flow stresses developed in the die. The overall stress components evaluated for each element were considered as the initial boundary conditions, knowing that these stresses were being removed at the moment the element emerged from the die. Therefore, the stress relaxation of the semisolid extrudate was predicted based on the solid-mechanics principles. The distance in the die over which the particle tracking was performed had a pronounced effect on the swell prediction. Upon a wide range of experimental trials, this length was defined in a dimensionless form by the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

In this equation, [a.sub.1] and [a.sub.2] are constants, [L.sub.die] is the die length, [S.sub.1] is a resin-dependent swell model parameter, DE* is a modified Deborah number related to the flow time in the die ([t.sub.f]) and terminal relaxation time of the resin ([[tau].sub.t]) as follows:

DE* = ([a.sub.T] [[tau].sub.t][a.sub.3]/[t.sub.f]) (14)

where (3):

[[tau].sub.t] = [[eta].sub.0][J.sub.s.sup.0] = [[eta].sub.0] [N.summation over (k = 1)] [G.sub.k][[tau].sub.k.sup.2]/[([N.summation over (k = 1)][G.sub.k][[tau].sub.k]).sup.2] (15)

and [D.sub.f] is a parameter related to the die-head size, approaching unity for small die heads, according to the following correlation:

[D.sub.f] = 1 - ([a.sub.4]/WE*); 0.1 [less than or equal to] [D.sub.f] [less than or equal to] 1 (16)

In the same manner as DE*, the modified WE* in Eq. 16 is defined as:

WE* = Q([a.sub.T][[tau].sub.t][a.sub.3])/[A.sub.die][h.sub.die] (17)

where Q is the volumetric flow rate and [A.sub.die] is the flow channel area at the die exit. In these equations, [a.sub.3] and [a.sub.4] are constants, [a.sub.T] is the temperature shift factor, [J.sub.s.sup.0]is the steady state compliance, and [G.sub.k] and [tau].sub.k] are the linear viscoelastic parameter sets for the memory function (relaxation moduli and relaxation times, respectively) given by the generalized Maxwell relaxation spectrum (N = 6).

It is reported that the steady state compliance in Eq. 15 is independent of the average molecular weight, while it is strongly affected by the molecular weight distribution (3), and could be potentially related to various moments of the molecular weight distribution (3), (47). Thus, in the absence of long-chain branching, it is hypothesized that by incorporating the terminal relaxation time as a parameter in the Hybrid approach, the polydispersity of HDPE grades could be accounted for in the swell prediction.

Mathematical Swell Model. The presented numerical formulation does not account for the elongational forces developed at the entrance to the die and in the die land area. Moreover, the Hybrid fluid mechanics-solid mechanics approach presented in this work is based on the membrane elements for both the flow in the die and parison formation phase, which disregards the three-dimensional nature of the shear flow in the die, particularly at high shear rates. These simplifications may not significantly affect the prediction of shear-dominated thickness swell at low flow rates (7), (17), typical in fuel tank manufacturing operations. However, this could have a negative impact on the predicted diameter swell. To compensate for these shortcomings, the Hybrid approach is coupled with a mathematical swell model. This model makes use of dimensionless quantities ([D.sub.die]/[L.sub.die]) aspect ratio and DE* number) to compensate for the elongational forces ignored by the Hybrid approach. The swell model is developed based on a wide range of experimental trials conducted using different die geometries and operating conditions. This model identifies a hoop factor (HF) for each parison programming point, so as to compensate for the underestimated diameter swell. Therefore, the overall parison swell predicted for each element at each time-step is a combination of the numerically-predicted swell and the model-based hoop factor, defined as follows:

HF = [C.sub.H] + [C.sub.R][[D.sub.die]/[L.sub.die]](1 + DE*) (18)

In this equation, [C.sub.R] and [C.sub.H] are dimensionless die-geometry dependent and resin-dependent parameters, respectively, taking the following forms:

[C.sub.R] = [a.sub.5] + (1 - [a.sub.5])([[[empty set].sub.m] - [[empty set].sub.b]]/45) (19)

[C.sub.H] = ln([tau]/[[tau].sub.ref] (20)

where [a.sub.5] and [[tau].sub.ref] are constants, [[empty set].sub.m] and [[empty set].sub.b] are mandrel and bushing angles, respectively, and [tau] is the Maxwell relaxation time at the processing temperature. It should be mentioned that the mathematical swell model in Blow-Parison[C] software has additional features to accommodate high flow rates, which could lead to pronounced thickness swell for the extruded parison. This software has been validated at high Weissenberg numbers (7), and is capable of predicting parison formation at high production rates (flow rates > 2000 g/s). The details of the Hybrid approach dealing with high Weissenberg numbers can be found elsewhere (7), (17).

Modeling of Heat Transfer. Since a temperature gradient exists within a parison due to exposure to ambient air temperature, the heat is transported from the zones of higher temperature to that of lower temperature. If an energy balance is set up for a parison element, as the thickness is much lower than the other two dimensions, one obtains the differential equation for the temperature field as follows (48), (49):

[rho][C.sub.P][[partial derivative]T/[partial derivative]t] = k[[[partial derivative].sup.2]T/[partial derivative][x.sup.2]] (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

where [rho], [C.sub.p], and k are the density, specific heat and thermal conductivity of the resin, respectively, T is the parison temperature, x is the coordinate of the parison in the thickness direction, t is the elapsed time, h is the heat transfer coefficient at the boundary [GAMMA], and [T.sub.[infinity]] is the ambient air temperature.

EXPERIMENTAL

Material

A commercial linear HDPE blow molding resin, Basell Lupolen 4261 A, was employed in this study. The resin was rheologically characterized to obtain the material parameters for the Carreau model and K-BKZ constitutive equation, and for the mathematical swell model.

Rheological Characterizations

Dynamic frequency sweep measurements were conducted on the molten resins on a Rheometrics ARES rheometer with a parallel plate setup. The measurements were conducted at three different melt temperatures on 25-mm diameter compression-molded disks. For each sample, frequency sweeps were made from [10.sup.-1] to [10.sup.2] rad/s at a strain of 10% (linear regime). The storage and loss moduli (G' and G") and complex viscosity (([eta]*) curves were obtained as function of frequency (Fig. 5a). The relaxation times and moduli for the Maxwell relaxation spectrum (N = 6) were estimated using the UMAT[C] software developed at IMI. The normalized damping curves were also constructed by dividing the G"/G' values at each frequency to the respective value at the highest frequency ([10.sup.2] rad/s). Figure 5b gives the normalized damping curve, which is used in this study to estimate the swell model parameter [S.sub.1] for the HDPE resin. The zero-shear Newtonian viscosity ([[eta].sub.0]), Maxwell relaxation time ([tau]), and terminal relaxation time at the processing temperature ([[alpha].sub.T][[tau].sub.t]) are listed in Table 1.

[FIGURE 5 OMITTED]

TABLE 1. Zero-shear Newtonian viscosity ([[eta].sub.0]), damping
coefficients ([alpha] and [beta]), Maxwell and terminal relaxation
times ([tau] and [a.sub.T] [[tau].sub.t]), swell model parameter
([S.sub.1]), thermal properties (Cp and k), melt density ([rho]), and
K-BKZ fits for Basell HDPE Lupolen 4261 resin.

Material properties (a)                K-BKZ model fits (b)

[[eta].sub.0] (MPa.s)           0.21       [G.sub.k] (MPa) and
                                           [[tau]. sub.k] (s)
[alpha]                        16.0
[beta]                          0.1
[theta]                        -0.11
[tau] (s)                       1.0         0.300     0.006
[a.sub.T] [tau] (s)            32.0         0.122     0.035
[S.sub.1]                       0.46        0.065     0.207
[C.sub.p] (J/kg/[degrees]C)  2700           0.027     1.220
k (W/m/[degrees] C)             0.23        0.009     7.170
[rho] (g/[cm.sup.3])            0.746       0.005     42.15

(a) Estimated at the processing temperature: [T.sub.melt] =
205[degrees]C.
(b) Estimated at the reference temperature: [T.sub.ref] =
178[degrees]C.

To determine the strain-dependent part of the K-BKZ model in shear, stress relaxation experiments after a step strain were performed on a true-shear sliding-plate rheometer (Interlaken). The resin was characterized using this technique at 190[degrees]C by imposing shear strains ranging from 0.05 to 10. The Papanastasiou damping coefficient [alpha] was calculated using the following equation (34):

h([I.sub.1], [I.sub.2]) = ([alpha]/[[alpha] + [[gamma].sup.2]]) (23)

where [gamma] is the imposed shear strain. As the elongational forces in the Hybrid approach are accounted for by the swell model, the effect of the damping coefficient [beta] was found to be negligible through a sensitivity analysis (results not shown). Hence, a default value of [beta] = 0.1 was used in this study. For a similar reason, a default value of [theta] = -0.11 was used in the simulations (45).

The swell parameter [S.sub.1] (Eq. 13) was estimated based on a correlation developed at IMI (17). Knowing that a significant decay of the normalized damping curve as a function of frequency represents a more dominant loss modulus at high temperatures, this correlation relates the swell parameter [S.sub.1] to the slope of its normalized damping curve (Fig. 5b) as follows:

[S.sub.1] = [p.sub.1] ln([n.sub.d]) + [p.sub.2] (24)

where [p.sub.1] and [p.sub.2] are the correlation parameters, and [n.sub.d] is the absolute value of the slope calculated for the normalized damping curve (Fig. 5b). The higher the slope is, a more pronounced sagging and a less significant swelling is expected for the resin, leading to a reduced value for the swell parameter [S.sub.1]. All the estimated material parameters are listed in Table 1.

Parison Formation Trials

The resin was extruded on a continuous industrial-scale extruder at Kautex using four diverging die geometries and different sets of operating conditions. Figure 6 gives the die geometries considered in these studies, denoted as I to IV. The die length and bushing diameter, as well as the minimum die gap, are displayed in this Figure. All die heads had a bushing angle of 10[degrees] and a mandrel angle of 30[degrees], featuring the same die land length (350 mm). The target parison length was achieved by adjusting the extrusion time for the given number of VWDS and PWDS programming points (NPPt = 64). Table 2 gives the operating conditions for these trials, denoted as (a) to (d).

TABLE 2. Operating conditions used for the parison formation trials.

Trial  Die  [T.sub.melt]  (a) [T.sub.[infinity]]  w(g/s)  [t.sub.ext]
            ([degrees]C)   ([degrees]C)                     (S)

(a)      I      205            20                  268         51
(b)     II      205            20                  173        100
(c)    III      205            20                  158         95
(d)     IV      205            20                  166        120

                                Parison length    Weight swell
                                     (mm)          ([B.sub.w])

Trial    Die       Die gap       Exp.    Sim.     Exp.    Sim.
                manipulation

(a)       I      VWDS-PWDS      1440    1508     2.03    1.94
(b)      II      VWDS-PWDS      1670    1664     1.80    1.80
(c)     III      VWDS-PWDS      1570    1507     1.08    1.13
(d)      IV      VWDS-PWDS      1530    1602     1.01    0.96

The experimental and predicted parison length and weight swell results
are also compared.
(a) h = 5 W/[m.sup.2]/[degrees]C.

[FIGURE 6 OMITTED]

For comparing the experimental and predicted weight swell profiles, the definition of parison weight without swell ([w.sub.0]) in Eq. 4 was identified to be too simplistic, due to the complex shape of the extruded parisons created by VWDS/PWDS die gap manipulations. Therefore, the following equation was used to estimate the overall weight without swell for the extruded parison:

[w.sub.0] = [NPPt.summation over (ippt = 1)][([[A.sub.ippt] + [A.sub.ippt - 1]]/2])[L.sub.ippt]([L.sub.parison]/[L.sub.0])[rho]] (25)

and the overall weight with swell and sag for a continuous extrusion was defined as:

w = [NPPt.summation over (ippt = 1)]w([t.sub.ippt] - [t.sub.ippt - 1]) = w[t.sub.ext] (26)

In these equations, w is the experimental mass flow rate, [t.sub.ext] is the total extrusion time, [t.sub.ippt-1] and [t.sub.ippt] and the extrusion times at two consecutive VWDS/PWDS programming points, [rho] is the melt density, and [L.sub.parison] is the final parison length with well and sag. In Eq. 25, [A.sub.ippt-1] and [A.sub.ippt] are the flow channel areas at two consecutive programming points after VWDS/PWDS actions, without swell and sag, and [L.sub.ippt] is the extruded parison segment length between [t.sub.ippt-1] and [t.sub.ippt], without swell and sag, calculated by IMI's parison mesh generation software (ParMesh[C]: See Numerical Simulations section). It should be mentioned that the term ([L.sub.parison]/[L.sub.0]) in Eq. 25 is introduced in an effort to compare the weight for the same parison length before and after swell (see Eq. 4), where [L.sub.0] is the total parison length without swell and sag calculated by ParMesh[C] software. Because of the mass balance conservation, it can be seen that by incorporating the Eqs. 25, 26 in Eq. 4, the weight swell can be simply reduced to:

[B.sub.W] = [[L.sub.0]/[L.sub.parison]] (27)

For the trial (c), the final blown part was also axially dissected into 10 sections with equal lengths. The sections were weighed and the values were compared to the predicted section weights coming from the simulations.

NUMERICAL SIMULATIONS

IMI's ParMesh[C] software was used to create the finite element mesh of the die and the initial parison mesh, without swell and sag, for given sets of VWDS/PWDS profiles and input die geometry information (Eqs. 5--9). The die mesh contained 9600 "3-node" membrane elements in all casess (see Fig. 6). Depending on the parison length, 18000 to 25000 membrane elements were used to create the finite element mesh of the parison. The parison formation, accounting for swell and sag as well as the nonisothermal effects, was subsequently predicted using BlowParison [C] software. The simulation times on a single-processor PC varied between 43 to 65 min for the specified number of elements. The Newton-Raphson iterative scheme was used to solve the set of equation, being the most time-consuming operation in the simulations.

RESULTS AND DISCISSION

Initial Parison Without Swell and Sag

Figure 7a shows the initial parison mesh generated using ParMesh[C] software for the four trials (both front and back views). Although the axial variation in the thickness profile is created by VWDS programming, the circumferential thickness variation is produced by PWDS manipulations. The front and back views of the final blown parts are shown in Fig. 7b. The irregular shape of the fuel tanks in these pictures clearly explains the need for the combined VWDS/PWDS actions. For the four trials, the flow channel area as a function of VWDS/PWDS programming points, calculated by ParMesh[C] software, is given in Fig. 8a. In this figure, ippt = 1 corresponds to the parison bottom section while ippt = 64 represents the uppermost section at the die exit. The increase in the flow channel area for the particular sections of the fuel tanks is meant to compensate for the higher blow-up ratio in these zones. The corresponding parison length evolution, without swell and sag ([L.sub.0]), as a function of extrusion time is presented in Fig. 8b. Despite much shorter extrusion time in the trial (a), the initial parison is much longer compared to the other cases because of the lower VWDS/PWDS actions used for this small die, and also because of its lower minimum gap ([h.sub.min] in Fig. 6). To compare the circumferential thickness distribution of the initial parisons, the highest PWDS pull and/or push actions used for the four trials as function of circumferential die angle are presented in Fig. 8c and d. These ultimate values correspond to the middle axial zone of the parts. It can be seen that up to 3.5 mm of PWDS pulling action, leading to the same amount of circumferential increase in the die gap, is required so as to compensate for the irregular part shape in the trial (d).

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Mathematical Swell Model

For the four trials, Fig. 9 gives the swell model variables [L.sub.pt] and HF as a function of DE* number. With the exception of the die I, the dimensionless distance used in the particle tracking ([L.sub.pt]) increases with decreasing die head size, primarily because the flow time in the die reduces for smaller die heads (see Eqs. 13, 14 and Fig. 6). The reduced [L.sub.pt] value for the die I is mainly attributed to its lower [D.sub.die]/[L.sub.die] aspect ratio, according to Eq. 13. In this Figure, the HF increases monotonically with decreasing die head size. Given the lower minimum gap for the smaller die heads (see Fig. 6), the flow time in the die reduces while the elongational forces in the die land area tend to increase as a result of a higher contraction in the die land zone. This leads to a more pronounced diameter swell for the dies I and II. The increased diameter swell for the die I is also attributed to the higher mass flow rate in the die. All these factors result in a higher HF, via enhancing DE* in Eq. 18.

[FIGURE 9 OMITTED]

Comparison Between Experimental Results and Numerical Predictions

Figure 10 shows the simulated parisons using Blow-Parisor[C] software for the four trials. It can be seen that the final parison lengths in all cases are very similar regardless of the pronounced differences in their respective initial parison lengths, without swell and sag (see Figs. 7a and 8b). This implies a more significant swelling in trials (a) and (b). As mentioned before, this is primarily attributed to the higher contraction in the die land zone for the dies I and II, and to some extent, to the more drastic changes in the die diameter along the flow channel. Hence, because of the resin memory, the parison tends to regain the diameter it originally had at the die entrance, leading to a pronounced sudden diameter swelling in both cases (hoop stresses). Moreover, the smaller [h.sub.min] also leads to more pronounced shear stresses in the die land zone for these dies. It has been shown that this kind of contracting configuration could cause a significant thickness swell for the extruded parison (7), (17). Therefore, the resulting weight swell is higher for these dies, as shown in Table 2. A comparison between the predicted and experimental length and weight swell data in Table 2 shows less than 5% error in all cases, demonstrating that the Hybrid approach used here is capable of discriminating between the effect of die geometry and die gap profile on the final parison swell. Future work will focus on the potential of comparing the predicted parison dimension with the experimental data coming from a pinch off mold for these parts. This will also provide more insight into the validity of some assumptions taken here to relate the geometrical and processing parameters to the variables of the mathematical swell model.

[FIGURE 10 OMITTED]

Figure 11 compares the experimental and simulated weight profiles for the dissected final part (case c). A very good agreement can be seen between the two, showing the validity of the predictions for part design purposes. The current practice for designing new parts could be quite time consuming due to the complex nature of the swell and sag characteristics for the part produced by die gap manipulation techniques. A modeling tool that could predict the parison dimensions upon any variations in the die geometry, die gap programming, resin characteristics, and operating conditions could significantly reduce the part design times and costs (17). More validation data on the successful use of the presented modeling tool for a multitude of complex industrial parts can be found elsewhere (15).

[FIGURE 11 OMITTED]

Figure 12 shows the predicted temperature distribution at the outer surface of the parison for the four trials ([T.sub.[infinity]] = 20[degrees]C and h = 5 W/[m.sup.2]/[degrees]C). It can be seen that the temperature drop at the bottom is proportional to the extrusion time ([t.sub.ext]: see Table 2). Since the memory function in the K-BKZ model is temperature dependent (Eq. 11), the higher the extrusion time, the higher the parison swelling at the bottom due to a more pronounced temperature drop. In light of this, a higher swelling at the bottom is quite expected because of both time-dependent swelling and nonisothermal effects, as it can be seen in Fig. 10.

[FIGURE 12 OMITTED]

CONCLUSION

This work presented the strong dependence of parison dimensions on die geometry and die gap manipulation (VWDS/PWDS). IMI's ParMesh[C] software was used to create the finite element mesh of the initial parison, without swell and sag, for given sets of VWDS/PWDS profiles and input die geometry information. IMI's BlowParison[C] software was subsequently used to predict the parison formation, taking into account the swell and sag, as well as the nonisothermal effects. The comparison between the predicted parison dimensions and experimental data demonstrated the capability of this modeling tool in predicting the parison length and weight profiles. In light of this, the parison dimensions from simulation could be readily used as the starting point for process design, reducing the part design times and costs and customer delivery time. This work also demonstrated that a modeling approach capable of creating the link between the die gap manipulation and parison swell and sag could be the key to a better control over the thickness distribution for the final part. Future work will focus on the potential of comparing the predicted parison dimensions with the experimental data coming from a pinch-off mold. This will also provide more insight into the validity of some assumptions taken here to relate the geometrical and processing parameters to the variables of the mathematical swell model.

ACKNOWLEDGMENTS

The authors thank Stephanie Chang and Marc-Andre Rainville for their technical contribution. They also thank the members of the IMI's SigBlow consortium for their continuing support and Kautex for providing experimental data for validation of the IMI's modeling software.

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Azizeh-Mitra Yousefi, (1) Haile Atsbha (2)

(1) Industrial Materials Institute (IMI), National Research Council of Canada, Boucherville, Quebec, J4B 6Y4, Canada

(2) Kautex Textron, Windsor, Ontario, N8W 5B1, Canada

Correspondence to: Azizeh-Mitra Yousefi; e-mail: [e-mail]

DOI 10.1002/pen.21243

Published online in Wiley InterScience (www.interscience.wiley.com).

[C] 2008 Government of Canada. Exclusive worldwide publication rights in the article have been transferred to Wiley Periodicals, Inc.