Nonisothermal Two-Dimensional Film Casting of a Viscous Polymer.
Film casting is an important process in the polymer industry for the manufacture of such products as food packaging, plastic bags, and magnetic audio and video tape. The process consists of extruding a thin film of molten polymer from a slot die, stretching it through an air gap and then cooling it on a chilled roll, as schematically illustrated in Fig. 1. One goal of film line designers is to maximize the production of film of uniform thickness. This goal is hampered by two problems: neck-in, a reduction in the downstream width of the film; and edge-bead, an increase in the thickness of the film at its edges. Numerical simulation of the film casting process is a useful tool for increasing the understanding of these phenomena. This paper presents an algorithm for simulating steady-state film casting of a viscous fluid. The algorithm, which is developed within a finite element framework, allows for a qualitative study of neck-in and edge-bead under the influence of self-weight, inertia, nonisothermal conditio ns, and nonuniform inlet boundary conditions.
Many of the previous studies on steady-state film casting focus on one-dimensional (1D) isothermal film casting [1-3], which does not include the formation or effects of an edge bead. While some studies allow for an edge bead [4-6] and others accommodate non-isothermal conditions [7-11], the simultaneous treatment of temperature and edge-bead has received only limited attention in the literature . Furthermore, although one study  includes inertia and gravity, these factors have not been included for the case where an edge bead exists. All of the studies cited above consider a uniform thickness and velocity at the die.
The first section presents the governing equations and boundary conditions for the cast film process. Following this, the finite element algorithm is summarized and the results of the simulations are presented and discussed. Results are compared to those presented in previously published studies, and the influence of self-weight, nonisothermal conditions, and nonuniform inlet boundary conditions are investigated. Concluding remarks are given in the final section.
The two-dimensional field equations for steady-state film casting are derived in References ,  and , using the assumptions that the film is thin and that the thickness gradient is small. One should however note that the second assumption does not strictly hold at the edge of the film . For the coordinate system defined in Fig. 1, the equations for the conservation of momentum, mass and thermal energy and the constitutive equation, using dyadic notation , are as follows:
[nabla] [cdotp] (h[sigma]) + [rho]hb = [rho]hu [cdotp] [nabla]u (1)
[nabla] [cdotp] (hu) = 0 (2)
[rho]Chu [cdotp] [nabla]T + 2[alpha](T - [T.sub.air]) - kh[[nabla].sup.2]T = 0 (3)
[sigma] = 2[eta](D + tr(D)1) (4)
where h is the thickness, u is the velocity, T is the temperature and [sigma] and D are the tensors for plane stress and rate of deformation, respectively. The material properties are defined by [eta], [rho], C and k, which, in the order listed, represent viscosity, density, specific heat capacity and thermal conductivity. Self-weight enters the conservation of momentum equation through the acceleration vector b, which for vertical film casting equals [g, 0], with g being the acceleration due to gravity. The sagging of the film, for non-vertical film casting, is not accommodated in the current formulation. If the viscosity of the polymer is high, then the self-weight term can be ignored, as can the inertial term. The source term in the conservation of thermal energy equation uses the symbols [alpha] and [T.sub.air] to represent the one-sided heat transfer coefficient and the surrounding air temperature, respectively. This source term is actually a boundary condition on the film's upper and lower surfaces, whe re Newton's law of cooling is assumed to apply. The coupling between the mechanical and thermal variables comes about through the viscosity, which is considered to be related to the temperature via an Arrhenius relation
[eta](T) = [[eta].sub.o]e[E/R(1/T-1/[T.sub.o])] (5)
in which [[eta].sub.o] and [T.sub.o] are reference values of viscosity and temperature, E is the activation energy, and R is the gas constant (8.314 J [mol.sup.-1] [K.sup.-1]). Equation 5 has the drawback that the viscosity of the material near the solidification temperature may be under predicted when compared to the trends observed in the laboratory. This shortcoming may be eliminated by using an alternative viscosity function, which is introduced in Sidiropoulos et al. (16) for blown film production; that is,
[eta](T) = [[eta].sub.o][e.sup.-a(T-[T.sub.o])+c[1/[(T-[T.sub.s]).sup.d]-1/[([T. sub.o]-[T.sub.s]).sup.d]]] when T [greater than] [T.sub.s]
[eta](T) = [infinity] when T [less than or equal to] [T.sub.s] (6)
where a, c and d are constant parameters and [T.sub.s] is the solidification temperature. Both viscosity-temperature relations are used in the subsequent simulations.
In general, the natural and kinematic boundary conditions for a thermomechanical problem are given by
[sigma][cdotp]n = t on [[gamma].sub.t], u = [u.sub.0] on [[gamma].sub.u],
q[cdotp]n = [q.sub.0] on [[gamma].sub.q], and T = [T.sub.0] on [[gamma].sub.T] (7)
in which [[gamma].sub.t], [[gamma].sub.u], [[gamma].sub.q] and [[gamma].sub.T] are subsets of the problem boundary where the following may be specified, depending on conditions: traction t, velocity [u.sub.0] thermal flux [q.sub.0] and temperature [T.sub.0]. It should be noted that the vector n is a unit outward normal to the bounding surface. The specific boundary conditions for the film casting problem are summarized in Fig. 2, where only half of the film's width is shown, as symmetry is assumed. A zero traction is assumed to exist on the free surface, along with a zero shear stress acting on the line of symmetry. The essential boundary conditions for velocity include specifying [u.sub.die] and [u.sub.roll], the constant velocities at the die and roll. These two values define an important dimensionless number, the draw ratio Dr = [u.sub.roll]/[u.sub.die], that characterizes the degree of stretching. At the die, the out-of-plane velocity is zero; therefore, the thickness at the die [h.sub.die] is also a pres cribed boundary condition. Since the prehistory of the film inside the die is not considered in the present formulation, there is no possibility of modeling die swell. For the thermal boundary conditions, the temperature [T.sub.die] is specified at the die and the thermal flux is set to zero on all other boundaries. The zero thermal flux assumption is reasonable given that the film is thin and that polymers are poor thermal conductors.
To find the free surface, which is defined by w = w ([x.sub.1]), the zero mass flux boundary condition is applied, along with the kinematic relation that the surface normal is the negative reciprocal of the slope of w. Therefore, in terms of w, the zero mass flux condition, u [cdotp] n = 0, can be written as
dw/[dx.sub.1] [u.sub.1] - [u.sub.2] = 0 (8)
FINITE ELEMENT ALGORITHM
Figure 3 shows the algorithm used for solving the governing equations to predict the width, velocity, thickness and temperature of a given cast film problem. Since the equations are nonlinear, it is necessary to gradually increase the draw ratio in steps of [delta]Dr. Following the common finite element notation, the finite element equivalent of Eqs 1 to 4 and 8, along with the appropriate boundary conditions, may be written in compact form as
[[psi].sup.n] [equivalent] [K.sup.n] [a.sup.n] - [F.sup.n] = 0 (9)
where [K.sup.n] is the stiffness matrix and [F.sup.n] is the load vector, which includes both mechanical and thermal effects. Both [K.sup.n] and [F.sup.n] are functions of the degree of freedom vector [a.sup.n], where the superscript n refers to the iteration step. For the formulation presented in this paper, the degrees of freedom are the nodal velocities, thickness, temperature and width. As shown in Zienkiewicz , Eq 9 can be expressed in terms of a truncated Taylor's expansion to provide the following Newton-Raphson recursion algorithm:
[[K.sup.n].sub.T] [delta][a.sup.n] = - [[psi].sup.n] with [[K.sup.n].sub.T] = [(d[psi]/da).sup.n]
and [delta][a.sup.n] = [a.sup.n+1] - [a.sup.n] (10)
The components of the tangential stiffness matrix [K.sub.[T.sup.n]] of Eq 10 are derived in Reference .
As Fig. 3 shows, an initial guess is needed to start the algorithm. For the results presented in this paper, the initial guess consists of a rectangular domain of dimensions [w.sub.die] by L, in which [u.sub.2] = 0, [u.sub.1] and h are determined using the 1D closed-form solutions given in Reference  and T varies linearly in the machine direction. For the mesh update step, the [x.sub.2] coordinates for each column of nodes, as shown in Fig. 2, must be adjusted to agree with the new predicted width. Each node in each column is updated so that its new [x.sub.2] value maintains the same ratio to the new width as it had to the previous width. To determine convergence the following criterion is tested:
Max([parallel][delta][a.sub.u][parallel]/[parallel][a.sub.u][parallel ], [parallel][delta][a.sub.h][parallel]/[parallel][a.sub.h][parallel], [parallel][delta][a.sub.T][parallel]/[parallel][a.sub.T][parallel], [parallel][delta][a.sub.w][parallel]/[parallel][a.sub.w][parallel]) [less than or equal to] Tolerance (11)
in which [a.sub.u], [a.sub.h], [a.sub.T] and [a.sub.w], are the current solutions for the velocity, thickness, temperature and width degrees of freedom, [delta] represents the change in these variables and [parallel] [parallel] denotes the Euclidean norm of the vector.
The results of the simulations presented in this section were obtained using the input parameters summarized in Tables 1 to 3, which make use of References (18) and (19) for the material properties. For the numerical parameters, the 5408 element, 2809 node, 11132 degree of freedom mesh shown in Fig. 2 was used, along with a tolerance of 0.01. So that Fig. 2 is easier to read, the mesh has been stretched in the [x.sub.1] direction. At the die and free surface, the mesh is finer due to the larger gradients in these regions.
Comparison to Published Simulations
In the published film casting research, two studies, namely those of d'Halewyu et al. (4) and Sakaki et al. (6), simulate the same problem, but obtain different thickness and neck-in results. The problem is isothermal with the mechanical variables of Table 1 and a viscosity of [10.sup.4] Pa*s. Figure 4 shows the thickness profile along the chill roll for both studies, and for a simulation using the algorithm developed in this study. This figure shows that the 3D formulation of Sakaki et al. (6) has a greater neck-in than the 2D formulation of d'Halewyu et al. (4). The greater neck-in cannot, however, be accounted for by the difference in dimensionality, as the current study is 2D and it supports the 3D results of Sakaki et al. (6). One possible explanation for the difference is the uncoupling of the width, velocity and thickness in the numerical algorithm of d'Halewyu et al. (4), which, because of the high nonlinearity of the problem, could cause convergence to a solution different from that of the other alg orithms. A consequence of this finding is that the numerical algorithm may play an important role in the solution obtained. As a final point, the findings here indicate that, at least for the simulation in question, there is little need to resort to a 3D formulation over a 2D one, for the prediction of the thickness profile at the chill roll.
Influence of Self-Weight
For low viscosity polymers, the self-weight and inertial terms in the momentum equation cannot be neglected. This is demonstrated by comparing vertical casting with and without self-weight. The problem selected was the casting of polyethylene terephthalate (PET), as described by Barq et al. (9). Tables 1 and 2 summarize the input parameters, and Fig. 5 compares the simulated thickness profiles at the chill roll with and without the influence of self-weight. Figure 5 shows that including self-weight results in less neckin, a larger thickness edge bead, and a more uniform film thickness; therefore, for low viscosity, low elasticity, low temperature-dependent film casting, self weight has a beneficial influence.
Figure 5 also illustrates the insensitivity of this particular setup to nonisothermal effects. Although there was a temperature drop over the air gap of roughly 8[degrees]C, consistent with the results of Barq et al. (9), it had little influence on the thickness profiles at the chill roll. This agrees with the conclusion of Barq et al. (9), who found that a nonisothermal model is unnecessary when simulating the PET in question under typical processing conditions. However, as the next section demonstrates, for some materials thermal effects are important.
Simulations using low density polyethylene (LDPE) were conducted to further investigate the effect of heat transfer on film casting. The input parameters of Tables 1 and 3 were used, with Dr = 4 and 16 and heat transfer coefficients [alpha] of 0, 5, 10, and 15 W/([m.sup.2] K). Figures 6 and 7 show temperature and thickness contour plots for various values of [alpha]. The contour values are normalized by the value of the variable at the die. For both types of plots, the values of the contours decrease in the downstream direction. The difference between contours is 2% of the die value for temperature, and 10% for thickness, or 3.6[degrees]C and [10.sup.-4]m, respectively. With larger heat transfer, the thickness draws down more rapidly and the edge bead becomes more prominent. These sharper gradients are reflected in the more closely spaced contours. The contour plots also show that increases in heat transfer promote a more uniform thickness across the width of the film, as indicated by the overall straightnes s of the contours over most of the film's width. A more uniform thickness along the chill roll, with increasing heat transfer, is shown by the thickness profiles in Fig. 8. As may be observed for both draw ratios, higher heat transfer extends the region of close-to-uniform thickness in the middle of the sheet. A similar trend is found by Debbaut et al. , but in their case an increase in the relaxation time, not the heat transfer coefficient, causes this change in behavior.
The thickness contour predictions appeared to be insensitive to the choice of the viscosity function selected; i.e., the conventional Arrhenius relation was found to be suitable for the analysis. The reason for this is that the special characteristics of Eq 6 only come into play near the solidification temperature, which is not reached in the air gap but when the film contacts the chill roll. A close examination of Fig. 8 reveals that the slope of the film thickness versus the transverse distance decreases and even becomes negative at the outside edge, where the bead forms. A similar observation is reported by Debbaut et al. .
Localized Cooling Effects
In film casting, localized cooling jets are often directed at the edges of the film to reduce the likelihood of tearing. To see the influence of these jets on the thickness field, simulations were performed using several values of Dr and the processing conditions of Tables 1 and 3, with [alpha] = 20 W/([m.sup.2] K) over approximately 7 cm of the film's edge. As may be observed in Figs. 9 and 10, localized cooling jets directed at the edge of the film significantly reduced neck-in and promoted a larger zone of uniform thickness at the chill roll. These findings support experience, which shows that localized cooling jets for LDPE film casting benefit the finished product. It is worth noting that another study of LDPE  shows a trend different than that of Fig. 9. Instead of an increasing draw ratio increasing neck-in, the opposite effect is observed at high draw ratios. A likely explanation of the different behavior is the inclusion of viscoelasticity in the other study .
Nonconstant Thickness at the Die
Although film casting dies are usually designed to extrude a uniform thickness, a nonconstant thickness could be used. This section briefly investigates how a nonuniform thickness at the die effects the thickness field. To do this, a simulation was conducted using the input parameters of Tables 1 and 3, but with Dr = 16 and excluding heat transfer ([alpha] = 0). A die profile was selected to compensate for the tendency of the film, once outside the die, to increase in thickness at the edges. The profile was defined according to
[h.sub.die]([X.sub.2]) = [h.sub.sym] - 0.0002[[x.sup.2].sub.2] (12)
where [h.sub.sym] is the thickness at the line of symmetry. This equation represents an inverse parabolic thickness profile that decreases to half of the value of [h.sub.sym] at the edge of the sheet. In this simulation [h.sub.sym] = 0.001 m, the value previously used across the entire width of the die.
As is clearly observed from the thickness profiles of Fig. 11, the nonconstant thickness at the die compensated for the thickening at the edge. The simulation with nonconstant thickness indicates that the edge bead thickness is reduced when compared to the constant thickness inlet condition, but neck-in is only slightly increased and the region of uniform thickness is almost unchanged. Therefore, less material has been used to produce the same amount of finished product. This one simulation suggests that the die geometry has a dramatic influence on the final film profile, and that it might be possible to adapt the die to not only use less material but also to reduce neck-in or promote a more uniform thickness or even to eliminate edge bead entirely.
A Discussion of the Neck-in and Edge Bead Phenomena
A complex interaction exists between edge-bead and neck-in, which can be better understood by considering a very wide film with constant [h.sub.dle]. For a wide film the streamlines are parallel in the center, while at the edge they are closer at the roll than they are at the die. This means that at the center of the film the behavior is 1D; that is, the variation in the variables only takes place in the machine direction. Moreover, an edge bead must form because the streamlines are growing closer and continuity requires the same mass flux to flow between streamlines. Dobroth and Erwin (20) suggest that the predominant cause of the 1D and edge bead zones is the different stress effects in each zone, where plane strain and uniaxial extension exist, respectively. To obtain plane strain in the film it must be either infinite or constrained in the transverse direction. In the case of cast film the edge beads provide the restraining influence. If the restraining influence of the edge beads increases then the widt h of the plane strain, or 1D, region must increase and neck-in would decrease. Furthermore, for the same mass flux to flow through the now closer streamlines at the edges, the thickness of the edge beads must increase.
The results of the previously presented simulations support the above theory of the interaction between edge bead and neck-in. In the case of self-weight, the natural tendency to form an edge bead provides a restraining influence because of the greater weight of the thicker edge. This same trend occurs in the cases of nonisothermal cooling. The restraining influence of the uniaxial edge is increased as the viscosity is increased. The increase in viscosity is due to a decrease in temperature, as shown in Fig. 6, which is a consequence of the longer duration of cooling at the edge because of the increased length of the streamlines there. Localized cooling is a more dramatic example of the same phenomenon. In this case, the temperature at the edge is explicitly decreased to increase the viscosity and thus amplify the restraining influence of the edge. The interaction of edge bead and neck-in is nicely illustrated by the nonconstant thickness simulation. An effect opposite to that observed in the other simulatio ns is created. By decreasing the edge bead thickness, its restraining influence is reduced and, as Fig. 11 shows, neck-in increases slightly. Given the strong relationship between neck-in and edge bead, some compromise must be struck between the goals of reducing edge bead thickness or reducing neck-in.
Sakaki et al. (6) show that neck-in and edge bead are unaffected by the die width. This is a consequence of the 1D nature of the film at the center of its width. If the film's behavior is 1D at the center, then an increase in die width will only add to the size of the 1D region: it will not effect the shape of the free surface. The simulations of the present study show 1D behavior at the center, as evident by the region of nearly parallel thickness contours in Fig. 7. Therefore, an increase in width would have little effect on the amount of neck-in. However, if the aspect ratio A, which is defined as the ratio of the air gap length to the die width (A = L/wdie), is increased, then neck-in will likely have a dependence on die width. This statement is suggested by the simulation results of Debbaut et al. (5), which do not show a region of ID behavior on the thickness contours for the film casting of a Newtonian fluid, A likely explanation for the absence of a 1D region is their use of A = 1, versus A = 0.2 for Sakaki et al. (6), or A = 0.4 for the current study.
For efficient film casting, neck-in should be limited and the region of uniform thickness should extend over most of the width of the sheet. The simulations of this study suggested that both goals can be promoted by the self-weight of the film (for low viscosity polymers), nonisothermal conditions, localized cooling jets and nonuniform boundary conditions at the die. The simulations of this study also showed that conclusions based on one set of processing conditions and/or material properties do not necessarily apply to all possible film casting problems. For example, the importance of self-weight and nonisothermal conditions vary depending on the problem considered. Further investigation of the influence of various factors on film casting awaits the development of a more comprehensive model, which incorporates the effects of the elasticity of the melt. Another issue of importance to film casting, which is currently being investigated, is the stability of the film casting process.
The financial support provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada is gratefully acknowledged.
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|Author:||SMITH, SPENCER; STOLLE, DIETER|
|Publication:||Polymer Engineering and Science|
|Date:||Aug 1, 2000|
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