11
Numerical Simulations of Annular Extrudate Swell of Polymer Melts YASUHIKO OTSUKI,* TOSHIHISA KAJIWAFU,** and KAZUMORI FUNATSU*** *Processing Technology Section Polymer Applications Development Laborato y Zdemitsu Petrochemical Co., Ltd. 1-1, Anesaki-Kaigan, Zchihara 299-01, Japan **Department of Applied Chemistry Kyushu Institute of Technology 1 -1 Sensuicho, Tobata-ku,Kitakyusyu 804, Japan ***Department of Chemical Engineering Kyushu University 6-1 0-1 Hakozaki, Higashi-ku,Fukuoka 81 2, Japan Numerical viscoelastic simulations were carried out using a K-BKZ type of sep- arable integral constitutive equation. Both reversible and irreversible models were tried for several types of damping functions to calculate the annular extrudate behavior of high-density polyethylene (HDPE). There are two aims in this study; first, to clarify the properties of these dumping functions, and second, to investigate the influence of rheological characteristics on annular extrudate swell. In these numerical simulations, relaxation spectrum and shear viscosity were fixed, and the other characteristics were varied. The reversional response of the damping function mainly has an effect on the magnitude of the area swell even if the die is straight. The irreversible model expresses the experimental results of annular extrudate swell better than the reversible model. The accurate fitting of N1 by the damping model is important for predicting it. The magnitude of N1 predicted from the Wagner exponential model is lower than that of the PSM model, and the area swell shows the same tendency as N 1. A modified PSM model that allows the N 1 curve to shift can fit the magnitude of area swell. The relationship between the diameter and thickness of the extrudate depends on N2 /N 1, and it was estimated by simple linear elasticity of solids. The time-dependent viscosity varies with the type of damping function, and it influences the time-dependent swell. INTRODUCTION nnular extrusion is widely applied to polymer pro- A cessing, blow molding in particular. It is impor- tant to understand the swell behavior of polymer melts because the parison shape, that is, the thick- ness and the diameter, changes dramatically as a re- sult of swelling, and the swell behavior seriously de- pends on material properties or die dimensions. Some empirical studies of the problem have been conducted (1, 2). However, it is difficult to investigate the influ- ence of rheological characteristics on extrudate swell by experiments because we cannot prepare the poly- mers to change their rheological parameters system- atically. It is known that the first normal stress differ- ence in the die mainly affects the capillary extrudate swell. This is fully discussed in a recent report (3). This paper investigates the effects of rheological character- istics of the melt, not only the first normal stress difference, but also the second normal stress differ- ence, elongational viscosities, and response of revers- ing strain, and clarifies which flow characteristics should be considered in order to control the annular die swell. Numerical approaches to the investigation of the annular extrudate swell, using both integral and dif- ferential constitutive equations, have been carried out. The Giesekus model with a single relaxation mode was used as a differential type, and the results at high Weissenberg numbers were obtained (4, 5). On the other hand, a separable integral constitutive model of K-BKZ type (6, 7) was used by several researchers. POLYMER ENGINEERING AND SCIENCE, JULY 1997, Vol. 37, No. 7 1171

1997-Otsuki-Numerical Simulations of Annular Extrudate Swell of Polymer Melts

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  • Numerical Simulations of Annular Extrudate Swell of Polymer Melts

    YASUHIKO OTSUKI,* TOSHIHISA KAJIWAFU,** and KAZUMORI FUNATSU***

    *Processing Technology Section Polymer Applications Development Laborato y

    Zdemitsu Petrochemical Co., Ltd. 1-1, Anesaki-Kaigan, Zchihara 299-01, Japan

    **Department of Applied Chemistry Kyushu Institute of Technology

    1 -1 Sensuicho, Tobata-ku, Kitakyusyu 804, Japan

    ***Department of Chemical Engineering Kyushu University

    6-1 0-1 Hakozaki, Higashi-ku, Fukuoka 8 1 2, Japan

    Numerical viscoelastic simulations were carried out using a K-BKZ type of sep- arable integral constitutive equation. Both reversible and irreversible models were tried for several types of damping functions to calculate the annular extrudate behavior of high-density polyethylene (HDPE). There are two aims in this study; first, to clarify the properties of these dumping functions, and second, to investigate the influence of rheological characteristics on annular extrudate swell. In these numerical simulations, relaxation spectrum and shear viscosity were fixed, and the other characteristics were varied. The reversional response of the damping function mainly has an effect on the magnitude of the area swell even if the die is straight. The irreversible model expresses the experimental results of annular extrudate swell better than the reversible model. The accurate fitting of N1 by the damping model is important for predicting it. The magnitude of N1 predicted from the Wagner exponential model is lower than that of the PSM model, and the area swell shows the same tendency as N 1. A modified PSM model that allows the N 1 curve to shift can fit the magnitude of area swell. The relationship between the diameter and thickness of the extrudate depends on N 2 /N 1, and it was estimated by simple linear elasticity of solids. The time-dependent viscosity varies with the type of damping function, and it influences the time-dependent swell.

    INTRODUCTION

    nnular extrusion is widely applied to polymer pro- A cessing, blow molding in particular. It is impor- tant to understand the swell behavior of polymer melts because the parison shape, that is, the thick- ness and the diameter, changes dramatically as a re- sult of swelling, and the swell behavior seriously de- pends on material properties or die dimensions. Some empirical studies of the problem have been conducted (1, 2). However, it is difficult to investigate the influ- ence of rheological characteristics on extrudate swell by experiments because we cannot prepare the poly- mers to change their rheological parameters system- atically. It is known that the first normal stress differ- ence in the die mainly affects the capillary extrudate

    swell. This is fully discussed in a recent report (3). This paper investigates the effects of rheological character- istics of the melt, not only the first normal stress difference, but also the second normal stress differ- ence, elongational viscosities, and response of revers- ing strain, and clarifies which flow characteristics should be considered in order to control the annular die swell.

    Numerical approaches to the investigation of the annular extrudate swell, using both integral and dif- ferential constitutive equations, have been carried out. The Giesekus model with a single relaxation mode was used as a differential type, and the results a t high Weissenberg numbers were obtained (4, 5). On the other hand, a separable integral constitutive model of K-BKZ type (6, 7) was used by several researchers.

    POLYMER ENGINEERING AND SCIENCE, JULY 1997, Vol. 37, No. 7 1171

  • Yasuhiko Otsuki, Toshihisa Kajiwara, and Kazumori finatsu

    Luo and Mitsoulis have carried out simulations of HDPE melt emerging from various types of annular extrudate, straight, diverging, or converging, and have compared their simulations with experimental data (8). Garcia-Rejon et al. have employed a numerical and empirical study for a more practical die geometry (9). Recently, the time-dependent annular extrusion simulation of K-BKZ fluid was accomplished by the use of the ALE method (10). The time-dependent prop- erty of die swell is a n important problem. However, very few numerical works have focused on this point. We have evaluated the parison swell as a function of time after emerging from the die.

    The merits of using a separable integral constitutive model of the K-BKZ type are as follows. First, this model can explain simple flow Characteristics of poly- mer melts well. Second, several researchers have had success in getting numerical results of viscoelastic annular exit flow with high shear rates, considering the relaxation spectrum that is indispensable for studying the rheological characteristics of polymer melts. We employed the K-BKZ model for this reason. Some researchers have developed numerical methods to simulate the viscoelastic flow of integral constitu- tive models (1 1). We have employed the stream line element method, considering its economic advantage (12).

    It is important to determine what type of damping function to use in the simulation using the K-BKZ model. In recent years, a considerable number of damping functions of the K-BKZ model have been proposed. However, there are few works concerning the relationship between the characteristics of damp- ing function and die swell behavior. Goublomme, Draily, and Crochet (13) employed the PSM model (14) and the Wagner exponential model (15) for the simu- lation of capillary extrudate flow, and found that the PSM model leads to a larger swelling ratio than the Wagner model. They also pointed out that it is impor- tant to take account of Wagners irreversible theory (16) for the problem of the converging region in the upper stream section.

    In this paper, we have made the annular extrudate swell simulation using the K-BKZ model including both reversible and irreversible types, with not only some typical damping functions, but also the modified PSM model to express various types of shear damping features of polymer melts. The streamline element method with integral constitutive models was em- ployed to simulate the flow emerging from the annular die. And we have analyzed the sensitivity of annular die swelling to flow characteristics, varying the types of damping function, its response under reversing strain, and material coefficients. High-density poly- ethylene (HDPE), commercial blow molding grade, was chosen a s the sample material. Experiments of rheo- logical characteristics and annular extrusion have been done to examine the validity of the numerical results.

    CONSTITUTIVE EQUATION

    Wagner introduced a separable K-BKZ type integral constitutive equation (15). Luo and Tanner modified the equation in the following form in order to take account of the second normal stress difference (17).

    1 m(t - t)h(Ic-l , Z,)(C- + OC) dt (1 1 1 - 8 T = __ 1.

    where 7 is the extra stress tensor. C and C- are the relative Cauchy strain tensor and Finger strain ten- sor, h is the damping function depending upon their first invariant, Ic and Ic-. O is a material coefficient. m(t - t) is the time-dependent linear viscoelastic memory function, as follows:

    Ai and Gi are the relaxation time and the relaxation modulus coefficient, respectively. The question now arises, what is the appropriate damping function for the present problem? First we examined the two typ- ical damping functions that Goublomme and Crochet employed for capillary flow (13). One is Wagners ex- ponential model (15) and the other is the PSM model introduced by Papanastasiou et al. (14). These are represented as Equations 3 and 4, respectively;

    h = exp( -a -1 (3)

    ff h = a + z - 3

    where (Y is the material constant. I is the generalized strain invariant defined as follows (18),

    I = P I c - , + ( 1 - p ) ~ ~ (51 p is the material constant. a and p make a contribu- tion to the response when the deformation is large. We examined another damping function because these models have limits in varying some rheological char- acteristics. It is represented a s follows:

    ff h =

    (Y + (I - 3) (61

    where n is the material constant. This model can be regarded as the modified PSM model. Details of this model are explained in the next section.

    These damping functions are employed to investi- gate the effect of flow characteristics on the swelling behavior of parisons.

    DETERMINATION OF MATERIAL CONSTANTS

    The sample material is an HDPE called 530B (Idemitsu), a commercial grade for blow molding. The relaxation spectrum obtained from the storage and the loss moduli, G and G . is in Table 1. Shear flow characteristics are shown in Fig. 1 . The Cox-Merz rule

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  • Numerical Simulations of Annular Extrudate Swell

    Table 1. Relaxation Spectrum (190C).

    Relaxation Modulus Relaxation Time Is) [Pal

    0.001 0.01 0.1 1

    10 100

    2.30 x 105

    4.15 x lo4 1.34 x 1 o4 3.45 x lo3

    9.56 X lo4

    9.67 X 10

    complex viscosity . capillary 104 1 Ni

    (s-) or w (rad/s) Rg, 1. Steady shear viscosity andflrst normal stress duer- ence as a function of shear rate for HDPE at 190C.

    n a

    PSM - 7.0 - - - - - - - modfled-PSM 1.1 9.0 - - - _. . . . modifiid-PSM 1.2 12.0 - - - - - - . Wagner - 0.25

    holds well for this material. Figure 2 shows the uniax- ial elongational viscosity at 150C measured by a Meissner-type elongational rheometer (19). In this study, the relaxation spectrum and the shear viscosity have been fixed as well as possible for all calculations because these experimental data are rather reliable, and the other characteristics have been varied.

    Simple shear flow characteristics calculated with the Wagner model and the PSM model are shown in Fig. 1 . The Wagner model gives rise to the first normal stress difference lower than the PSM model at high shear rates when their shear viscosities coincide with each other. Table 2 summarizes the dependence of material coefficients on the flow characteristics. For both the Wagner model and the PSM model, the shear damping function is affected only by parameter a. Osaki demonstrated that the feature of the shear damping function could be classified into a few types according to molecular structure (201. It seems to be insufficient to fit these various features by only one parameter. One reasonable way to express the shear damping nature is to use a more general form with a multiple nonlinear parameter, at, corresponding to each relaxation mode. However, adopting this

    7

    104 i 0.122 o.066 i 0.310 3 1 , , I , , , , , , , , , ,j K8,,,j lo- 100 10 1 o2

    t (9 m. 2. Unsteady uniaxial elongational viscosity, HDPE at 150C.

    method, a considerable number of nonlinear parame- ters must be taken. For the purpose of studying the effect of material parameters, this method may be inappropriate. We have taken the constant value of a for all modes. The other way, the damping function that Feigl and Ottinger have employed for the simula- tion of contraction flow, can be cited for the purpose (21).

    1 h = 1 + a(I,-1 - 3)(1, - 3i;; (7)

    n l , n2, and a are material constants. It takes a gen- eralized form of the damping function proposed by Wagner and Demarmels (22). With this model, shear flow characteristics can be controlled by two parame- ters. Namely, one can vary the relationship between the first normal stress difference and shear viscosity. However, it must be noted that uniaxial elongational viscosity is likely to diverge under large deformation, by Feigl and Ottingers model, as shown in Fig. 3. Therefore, we tried the new damping function as Equation 6, the shear flow characteristic of which cor- responds with Feigl and Ottingers model, and elon- gational viscosity does not diverge. Unsteady elonga- tional viscosities calculated with various damping functions are presented in Fig. 3. It indicates that there is a difference between the nonlinear properties in the Wagner model and the PSM model; the Wagner

    Table 2. Relationship Between Parameters and Flow Characteristics.

    Shear Viscosity Elongational and N1 Viscosity N2

    a 0 0 X X 0 X X 0 0

    P 8

    n (extended PSM) 0 0 X

    0. dependent, x: independent.

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  • Yasuhiko Otsuki, Toshihisa Kajiwara, and Kazumori Funatsu

    - v)

    d

    +w t

    c- - - -___- -5% --------I %

    t (s) Q. 3. Unsteady elongational viscosity, HDPE at 150C. d = 0.31 s-.

    PSM model a = 7, p = 0.1 Wagner model a = 0.25, p = 0.1 Feigl atinger model a = 0.0833, nl = 0.12, n2 = 1.08 modifkd-PSM model a = 12, p = 0.05, n = 1.2

    model causes remarkable overshoot. The nature of the extended PSM model could be varied by parameter n.

    One other thing is important for the characteristics of the damping function: the response of deformation histories involving a reversal of flow. The K-BKZ type equation cannot describe the response satisfactorily (23). The irreversible network rupture model intro- duced by Wagner (161, which never causes the in- crease of the damping function under the reversing strain, corrects the defect (24). In this study we have examined the irreversible model for the annular die swell simulation.

    NUMERICAL METHOD

    We assume steady-state creeping flow of an incom- pressible fluid under isothermal conditions. The con- servation of mass and momentum balance are repre- sented a s follows,

    v . v = o (8) - v p + v T = o (9)

    where v is the velocity vector and p is the isotropic pressure. In this study, the numerical scheme has been constructed on the basis of the streamline finite element method (SFEM). Calculating with this method, the nodes for velocity and pressure consist of several streamlines, and the amount of stress ealcu- lation is reduced considerably. See Luo and Tanner for a full account of SFEM (1 7). The method of calculating the velocity gradient is referenced by Luo and Mitoulis (25).

    To check the validity of the developed research code, we compared our predictions of capillary extrudate swell with the results of Goublomme et al. (13). They

    1174

    got 1.700 and 2.214 as the predicted values of the PSM model for their runs No. 1 and No. 6. Our pre- dicted values for corresponding conditions are 1.700 and 2.163, respectively.

    EVALUATIONS OF ANNULAR EXTRUDATE SWELL

    Area swell xa, diameter swell xd, and thickness swell xc are defined a s follows:

    c(d - c ) c(d - C) X a =

    C x c = 7

    (10)

    (111

    (12)

    where d and d are the diameters of die and extrudate, respectively, and c is die gap and c is the thickness of extrudate.

    Experiment

    Some experiments were done by using a continuous extrusion blow molding machine. The experimental data of diameter, thickness, and area swells were taken as a function of time after emerging from die. The die with a converging section shown in Flg. 4 was used here. The outer diameter was 30 mm and the die gap was 2 mm. The die land that is a straight section was 20 mm. The die temperature was 190C. As it is difficult to measure the radial position of the inner free surface, a pinch-off mold was used to measure the weight distribution of the parison; this is a classic

    Fig. 4. Die dimension.

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  • Numerical Simulations of Annular Extrudate Swell

    method (26). We used a video camera for measure- ment of the outer diameter. The experimental proce- dure was as follows: The parison was continuously extruded in the atmosphere. In the beginning, it was cut a t the die lip and the timer started. The new pari- son emerging from the die was monitored by the video camera. After an optimal time, the molds were closed and molded piece was removed. The swell ratio near the end of the parison was evaluated as a function of the progress time after exiting the die to eliminate the effect of sagging as well as possible, because the sim- ulation was made under no-gravity flow, as described in the next section. The area swell ratio was evaluated from weight per unit length and melt density consid- ering mass balance. The thickness was determined from the outer diameter and the area. The influence of

    small effect of that may be involved in the data when the progress time is large.

    Numerical Simulation

    R0FI\ numerical simulation evaluate point at tl m,

    cooling by air is not eliminated here. Therefore, a t=o t=t1 experiment

    Fig. 5 . The dflerence between numerical simulation and ex- periment.

    0 experimental data

    Numerical simulations were carried out assuming steady-state, isothermal, and no-gravity flow. In this study, two different dies were employed. One was a long straight die model that ignores the converging flow section of the upper stream. The other was the converged die used a s an experiment concerning the upper stream. They had 348 meshes with 1521 nodes and 492 meshes and 2145 nodes, respectively. We took the extrudate length 200 times as large as the die gap to sufficiently evaluate the time-dependence.

    x

    Comparison Between Experiment and Numerical Simulation

    For comparing experiment and numerical simula- tion, the locational swelling ratio was estimated as a function of time after the fluid of the location leaves the die. We tracked the movement of fluid on the steady-state numerical result to evaluate the time- dependent property of die swell, and compared this with the experimental result. Figure 5 shows the dif- ference between the experimental result and the nu- merical simulation.

    RESULTS AND DISCUSSION

    The Response of Reversing Strain

    Figure 6 shows the effect of the upper stream region on the numerical result of the area swelling ratio as a function of time, when both the reversible model and the irreversible model are used. The PSM model is used here. Experimental data involving the effect of the upper stream is also plotted in Fig. 6. The converg- ing region causes a remarkable increase of area swell, as Goublomme et al. have indicated for capillary ex- trudate (13). While the reversible model makes a much larger area swell than the experimental data, reasonable results are computed using the irrevers- ible model.

    Rg. 6. Effect of upper stream on area swelling ratio, HDPE at 190"C.ya=32.4s- ' ,PSMmodel ,a= 7 ,p=0 .12 . 0 = - 0 . 1 1 .

    Figure 7 shows the features of parison characterized by the PSM model. It is obvious that the elongational reversing strain appears through the flow from con- verging to diverging, and the swell behavior changes dramatically by the response of reversing strain. On the other hand, it might be expected that the numer- ical result of swell is not affected by the characteristics for the reversing strain when a long straight die with- out a converging flow section at the upper stream is used, since the flow for straight die is thought to be not influenced by any reversing strain caused by elon- gational flow. However, the magnitude of die swell is rather affected by the reversionary characteristic of the constitutive model even if the long straight die is used as shown in Rg. 6.

    For the moment, let us look at the feature of the damping function. Figure 8 shows the transition of the damping function along a streamline located at 0.25 mm from die wall a t the entrance. A standard position

    POLYMER ENGINEERING AND SCIENCE, JULY 1997, Vol. 37, No. 7 1Y75

  • Yasuhiko Otsuki, Toshihisa Kajiwara, and Kazurnori Funatsu

    1 1 upper stream n

    Fg. 7. Final streamltne finiteelement grids for HDPE at 190"C, r, = 32.4 s-*, irreversible PSM model, a = 7, = 0.12, 0 = -0.11.

    reversible -

    irreversible

    __-___--

    -

    I 1 I 1 -4 -2 0 2 4

    Z/C

    Fg. 8. Transition of damping function along a streamline HDPE, 19O"C, = 32.4 s - I , upper stream ignored, PSM model, a = 7.0, = 0.1, 0 = -0.25.

    is taken at the entrance ( z / c = -41 to calculate strain. z and c represent the position of the axis direction and the die gap, respectively. It is recognized that the re- versible model leads to an immediate sharp increase of the damping function just after passing the die lip, while the irreversible model sets the damping function a s the minimum value near the die lip after emerging. It is found that a complex reversing deformation is generated near the die lip while the flow turns from shear dominant to elongation dominant. Clearly, the magnitude of stress after the die lip increases with the reversible model. As the result, die swell increases more than in the irreversible model. These results lead to the conclusion that the constitutive model should explain the response of reversing strain well even if the flow is simple shear in the die. In the following sec-

    1176

    tions of this paper, we describe only the results of the irreversible model, because we have found that the reversionary response mainly affects the magnitude of area swell and the results are qualitatively alike in the two models.

    The Effect of First NormaI Stress Difference

    Figure 9 shows the numerical results of area swell- ing ratio when various types of damping functions are used. The Wagner model causes lower area die swell than the PSM model. Let u s now consider the differ- ences in flow characteristics between the two models. We find that the Wagner model leads to a lower first normal stress difference than the PSM model. The PSM model leads to -50% larger area swell than the Wagner model. The difference quantitatively corre- sponds to the difference of the first normal stress difference between the two models. For the present, it may be useful to look more closely at the point by using the extended PSM model. We can vary the first normal stress difference by this model, as shown in Fig. 1 , while the shear viscosity curves are fixed. Cor- responding numerical results to the flow curves are shown in Fig. 9. It is recognized that the magnitude of area swell changes sensitively by the normal stress difference. When the normal stress difference is cor- responded to the Wagner model by the extended PSM model with n = 1.2, the magnitude of area swell is almost the same as the Wagner model. These results lead to the conclusion that the first normal stress difference must be fitted accurately to experimental data to simulate die swelling quantitatively by the K-BKZ model.

    3 , I

    Experimental data 1 1 1 0 1 0 0

    t (s) Fig. 9. Efect offirst normal stress dgerence on area swelling ratio for HDPE at 190C, -ja = 32.4 s-', upper stream consid- ered.

    n a p 0

    PSM - 7.0 0.12 -0.11 - - _ - - _ modifiid-PSM 1.1 9.0 0.05 -0.11 - . . - . . . . modijled-PSM 1.2 12.0 0.05 -0.11 - - - - . - . Wagner - 0.25 0.13 -0.11

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  • Numerical Simulations of Annular Extrudate Swell

    3

    * 2

    1 Experimental data

    1 1-0 1 0 0 t ($1

    Fig. 10. EBect of parameter 6 on area swelting ratio for HDPE at 190C. ya = 32.4 s-I.

    irreversible PSM model a = 7.0 ___ p=o.13 0 = 0 N2/Nl = 0

    - - - - . p = 0.1 0 = -0.25 N2/N1 = -0.2 p = 0.12 0 = -0.1 1 N2/N1 = -0.1 _ _ _ _ _ _

    The Effect of Second Normal Stress Difference The constancy ratio of the second normal stress

    difference to the first normal stress difference is con- trolled by the parameter 0 in Equation I .

    Figure 10 indicates the effect of parameter 0 on the area swell of the parison. This diagram tells us that the area swell decreases a little with increasing -N2/ N 1, as Goublomme et al. have mentioned for capillary extrudate swell (27). However, the influence is too small compared with the influence of reversibility or the first normal stress difference. What is more impor- tant here is the influence of the relationship between

    the thickness and the diameter of parison. It is desir- able to describe the estimation of the relationship be- fore moving on to the results. The area, thickness, and diameter of the parison are not independent of one another. Henze et al. related the diameter swell and weight swell by a power law (28). This relation is the result of the next expression:

    XUJ = X d x c (141

    Some researchers have considered the power law re- lationship for empirical study (29-3 1). Weight swell x, corresponds to area swell xa when the melt density is constant. The equation is rewritten as a rational form,

    (15)

    where d,,, is the center diameter indicated in Fig. 1 1 , namely

    d , = d - c (16)

    xdm is its swell ratio. Power law relations are repre- sented as

    Xdm = xz (17) xc = x;'-"' (18)

    we evaluated the thickness parameter b.

    b = l - a (191

    Figure 1 1 represents the meaning of the parameter. Now we evaluate the thickness parameter b in our numerical results. Figure 12 shows the area swell x, and b calculated with the PSM model when the flow rates are vaned. Jus t after emerging from the die, thickness swell is dominant, and moving away from near the die lip, b is going to be a nearly constant value. Figure 13 shows the constant value b as a function of -N2/N1 when the long straight die is used. It is clear that an increase of -N2/N1 produces an

    b=O b = 0.5 b = l x,= 1 x d m = x c X d m = 1

    planar elongation biaxial elongation planar elongation Fig. 11. The meaning of b as an evaluation of the relationship between diameter and thickness.

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  • Yasuhiko Otsuki, Toshihisa Kajiwara, and Kazumori Funatsu

    0 PSM model a=7 Wagner model a=0.25

    0 modified-PSM model a =9 A modified-PSM model a=12 0 0.7 -0 PSM model a =7 (upper stream considered) -

    3 1 I I I

    I S -- 1-N2/N1 2+N2M1

    106L . ' ' ' " ' a l ' ' " " " ' ' ' " ' ' I

    I I I ' 0 0.01 0.1 1 10 100

    t (s) Fig. 12. xa and b as a function of t h e , 190C, irreversible PSM model, a = 7.0, p = 0.1, 0 = -0.25.

    -N2/N1 Fig. 13. Effect of -N2/N1 on relationship between diameter and thickness swell, upper stream ignored, irreversible model, 1', = 32.4 s-' .

    increase of the ratio of thickness swell. We must point out that the parameter 0 influences not only -N2/N1 but also the elongational viscosities as shown in Flg. 14. However, a simple estimation with a linear elastic model shows the large influence of N 2 / N l on the re- lationship in the swell ratios. For a Hookean elastic material, the constitutive equation is represented as

    1 E e = - {( 1 + v ) a - Y t r (u )S) (201

    where e is the infinitesimal strain tensor, u is the stress tensor, E is Young's modulus, and v is Poisson's ratio.

    Considering the incompressibility, the diagonal components of Equation 20 become

    1 2E e , , = - (ZN, + N,) (21)

    Y I*

    Fig. 14. Unsteady elongational viscosity at 15O"C, d = 0.31 s-', PSM model, a = 7.0.

    1 2E e2, = - (-Nl + N,) (22)

    (23)

    Now the swell ratio is regarded as the mean strain. The mean strains can be represented in logarithmic form.

    e l l = -ln(x,) (24) Z,, = ln(x,) (25)

    We assume that swell appears as the result of elastic recovery from the normal stress difference in the die. The relationship between the mean strain and the mean stress suffered in the straight die is expressed by Equations 25 and 26. To take the volumetric aver- age for the mean stress because of the plug-flow as- sumption at lower stream region, we have

    -ln(x,) = EQ jR; (2N1 + N,)ur dr (26) ln(X,) = - (-PIl + N,)ur d r (27) EQ = jR;

    where r indicates the radii coordinate location. R, and R, are the outer and inner radius of the straight die, respectively, u is the axis velocity component, and Q is volume flow rate. For the simple shear flow expressed by Equation 1 , N2/N1 is constant. Therefore parame- ter b is estimated as follows:

    In(,y,) JE: (-Nl + Ndur d r 1 - (N , /N~) - - b=- - ln(x,) - -J:; (2N, + N21ur dr 2 + (N,/Nl)

    (28)

    This is the result from the assumption of infinitesimal deformation. However, it is guessed that the relation-

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  • Numerical Simulations of Annular Extrudate Swell

    _ _ _ _ _ _ _ _ _ _ _ _ _

    t (s) Fig. 15. Unsteady elongational viscosity, 150C. 6 = 0.31 s-', PSM model, a = 7.0. 0 = -0.1 1.

    1 1 1 0 1 0 0

    t (s) Fig. 17. Area swelling ratio as a function of time for HDPE at 190C. upper stream considered, irreversible modified-PSM model, a = 8.5, p = 0.05, 0 = -0.1 1, n = 1.08.

    upper stream /-- ,

  • Yasuhiko Otsuki, Toshihisa Kajiwara, and Kazurnori Funatsu

    Table 3. The Influences of Rheological Characteristics on Annular Extrudate Swell Behavior.

    Magnitude of Relationship Between Time-Dependent Characteristics Area Swell Diameter and Thickness Swell of Swell

    Response of reversing strain 0 First normal stress difference 0 Second normal stress difference 0 Unsteady viscosity 0 Nonlinear property of uniaxial elongational viscosity

    other die type, but at least the nonlinear uniaxial property alone little affects deformation out of the die.

    While the uniaxial elongational property under large deformation is not so important here, time-de- pendent viscosity under smaller deformation seems to be more important. The increasing slope of area swell differs from the Wagner model to the extended PSM model with n = 1.2 as shown in F'ig. 9, although the steady shear characteristics are almost the same and their magnitudes of area swell are on an equivalent level on the whole. These results are due to their damping nature because the relaxation spectrum is identified here. For the moment, let us look closely at the unsteady flow characteristics in Fig. 4 to Fig. 6. The features correspond to their swell behavior, that is to say, the Wagner model produces the lower defor- mation resistance and accordingly a larger swell ratio at early time, but as further deformation is added, the viscosity is going to be higher and the swell ratio is going to be lower than in the extended PSM model. This result suggests that the characteristic of the damping function affects the time-dependent swell growth as a result of time-dependent viscosity. Espe- cially, the characteristic of time-dependent biaxial elongation is supposed to be very important consider- ing the deforming feature of annular die swell.

    Propriety of Damping Function

    The Wagner model causes a lower area swelling ratio and the PSM model causes a slightly higher one than experimental results for the sample material as shown in Fig. 15, whereas the extended PSM model can fit the magnitude of area swelling ratio by control- ling the first normal stress difference. Figure 1 7 shows the numerical result when the curve of the first nor- mal stress difference is appropriate for the extended PSM model. Figure 18 involves the results of diameter and area swelling ratios. It is very effective to add the parameter n for the PSM model to predict extrudate swell behavior quantitatively.

    CONCLUSION

    The influence of flow characteristics on the annular die swell behavior of HDPE is examined, employing a numerical simulation with a single separable integral constitutive equation. The conclusions are as follows:

    1 ) The complex reversing deformation exists near the die lip. Therefore, the characteristic for reversing strain changes the magnitude of swell even if the die dimension is very simple.

    2) The area swell is greatly influenced by N 1 when the shear viscosity is fixed.

    3 ) N2/N1 mainly affects the relationship between diameter and thickness of extrudate. The effect is es- timated assuming simple linear elastic deformation.

    4) The nonlinear property of uniaxial elongational viscosity has little effect on the annular die swell for the present die. 5) The time-dependent property of swell varies with

    the type of damping function. It is supposed that the difference is caused by the time-dependent property of viscosity.

    Table 3 summarizes the main effects of annular die swell behavior. We can fit the experimental swell be- havior with the irreversible extended PSM model. However, it is not clear how much the model can characterize these flow properties. Especially, it is questionable whether the model can express the re- sponse of a rapid complex reversing deformation near the die lip. It is reported that the separable model is insufficient for the behavior just after the large step strain (32) and the irreversible model cannot explain the response of double step strain when the interval of two steps is short (33) . We suggest two points as fu- ture themes: first, the empirical study of the above flow characteristics and construction of a constitutive equation of polymer melts; second, the extension of material type or die type to examine the generality of the present simulating procedure.

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