Simulation of slit dies in operation including the interaction between melt pressure and die deflection.
The primary requirement in the process design of slit extrusion dies is to ensure uniform melt distribution across the width of the die, leading to an extrudate of a sufficiently uniform thickness. Several analytic theories have been proposed for the design of distribution manifolds to achieve this, and numerical methods have also been used to simulate flow in the manifold and slit. Unfortunately, the design performance is often not achieved because of process dependent alterations of the slit dimensions. Extrusion pressures, acting over the large area of the slit, exert forces high enough to bring about significant deflection of the die body, opening up the die, particularly towards the lips and near the center line. This "clam-shelling" can seriously affect melt flow distribution and extrudate thickness, and corrections must then be attempted by on-line adjustment using the choker bar and flexible lips. Deflection of the lips can often be large relative to the required extrudate thickness, and the available adjustment is limited. In fact, incorporation of flexible lips and a choker bar weakens the die, leading to larger deflections in the first place. Attempts to reduce deflection by increasing the thickness of the die body are limited by considerations of die weight and cost. Reduction of the length of the die reduces deflection, by shortening the lever arm of the pressure load about the body bolts, but this can conflict with the need for good, material-independent melt distribution, which is best achieved using the curved "coat-hanger" type distribution manifold, with a large "drop" in the flow direction.
Die body deflection is a well-recognized problem, which complicates the design and operation of slit dies, but, surprisingly, it has received relatively little attention in the literature. In a recent paper (1) we reported experimental measurements of deflection in a 1.2 m wide slit die, designed for high pressure operation [ILLUSTRATION FOR FIGURE 1 OMITTED]. At a pressure drop of 200 bar, for example, the lip gap increased from the design value of 1.9 mm to [approximately]3 mm on the center line. In the same paper, analytic predictions of deflection, all based on beam theory, were found to be highly inaccurate. Three-dimensional linear elastic analyses were then carried out using a number of finite element models of increasing complexity, constructed using a commercial software package (MARC Analysis Research Corporation). Agreement of the results with experimental data was only fair, even for the most elaborate and computationally expensive models. It was concluded that the main reason was the use of loads in the deflection calculations derived from a pressure field in the slit that was not entirely appropriate. This pressure field was obtained from an analysis of flow using the undeflected slit geometry (though the computed overall pressure was adjusted to match the experimental value by adjusting the flowrate). Deflection increases the slit gaps significantly, particularly near the lips, leading to lower pressure gradients in this region. The distribution of the pressure load on the die in operation is therefore different from that assumed in the computations.
It is clear that for a proper simulation of the die in operation it is necessary to couple the analysis of the flow distribution and pressure field in the slit with the analysis of die body deflection; since, by altering the slit geometry, deflection alters the pressure field, which, in turn, determines the deflection. To our knowledge, no such coupled analysis has previously been carried out. All previous calculations of deflection have been based on pressure loads calculated for the undeflected die geometry, often by using approximate, analytic methods. A coupled, numerical analysis of the sort proposed here can be achieved by iteration between pressure field and deflection calculations, but the use of three-dimensional finite element models, such as were mentioned above, is impractical within such a scheme, because of their high computational demands. The analysis of the pressure field in the slit is routinely reduced to two dimensions, by the use of the Hele-Shaw approximation, which simplifies the equations for momentum and mass conservation to a potential equation for pressure in the plane of the slit. Modem thick-plate finite element formulations provide the means to reduce the deflection calculation to two dimensions as well. In this work, we follow this route to develop a fully coupled analysis of melt flow, pressure field and deflection in slit dies, which can conveniently be run on a PC, providing the die designer, for the first time, with the means to simulate the performance of the die in operation. The software is enclosed within a graphical user interface in the MS-Windows operating environment, facilitating problem set up and display of results, and thus provides a highly convenient and powerful computer aided design facility. Further details of the software engineering aspects of the package are provided elsewhere (2).
The outline of the paper is as follows. Previous work on melt flow analysis and process design of slit dies is reviewed, leading to a summary of the Hele-Shaw formulation for melt flow in the slit. A method for including the distribution manifold within the Hele-Shaw treatment is described. The thick-plate formulation for die body deflection is then summarized, and a method for representing the flex lip assembly as an equivalent plate is outlined. The application of these formulations, using finite-element methods, is described and applied to the 1.2 m commercial slit die mentioned previously (1). Details of the iterative scheme coupling the flow and deflection analyses are set out. Finally, results are given and compared with experimental data.
MELT FLOW ANALYSIS AND PROCESS DESIGN OF SLIT DIES - A REVIEW
In analytic methods for slit die design the manifold and slit are generally modeled using formulae for developed isothermal flow in simple slit and tube geometries. These formulae are coupled to provide expressions for pressure drop along all possible flow paths, melt residence time, stresses, shear rates, etc. One or more criteria are then imposed, to define desirable performance characteristics, the most basic being equal pressure drop along all flow paths, leading to uniform melt distribution across the die; uniform residence times, and uniform shear rates have also been used.
The first work seems to have been by Pearson (3), who considered a die with a T-shaped manifold and a linearly tapering slot. The design criterion was equal pressure drop along all flow paths. One-dimensional temperature effects were also considered. Rothemeyer (4) proposed a novel design, without any distribution manifold. The slit height was everywhere constant, and flow path lengths were equalized by making the slit non-flat. The design was therefore a purely geometrical exercise, and independent of melt rheology. However, the necessary increase in the die thickness, and manufacturing difficulties, were a bar to the practical application of this principle. Gormar (5) based his design procedure on the Prandtl-Eyring constitutive equation. In this case the required manifold diameter and slit lengths cannot be calculated explicitly, and Knappe and Schonewald (6), and Schonewald (7), proposed the use of nomograms, with consideration of residence time and wall shear stress. The resulting designs would, however, be specific to given material flow properties. A procedure leading theoretically to material-independent designs was proposed by Wortberg (8), using the so-called representative viscosity, and requiring uniform shear rates throughout the manifold and slit. Matsubara (9) presented an analytic method for design of a coat-hanger die with uniform flow distribution and residence time, taking into consideration, for the first time, the true length of the curving manifold. A year later the same author analyzed the residence time in a T-die, showing that uniform values are not obtainable (10), and went on to investigate the ratio of residence times in the manifold and slot of a coat-hanger die. He showed that slot length could be decreased significantly if an increase in residence time towards the die edges is accepted (11). The resultant shortening of the die is important in reducing die body deflection. Matsubara subsequently published an analysis of residence time in a linearly tapered coat-hanger die (12).
Wortberg and Tempeler (13) proposed a die with exchangeable lips for different materials and operating conditions, claiming that a die designed for the most shear-thinning material could be used successfully for other materials by minor choker bar adjustments. A slit die designed to provide equal melt viscosity in the manifold and slot had a performance independent of the material and operating point, but, on account of its large length, was susceptible to die body deflection. Matsubara (14) concluded his work on slit dies by listing the advantages of curvilinearly tapered coat-hanger manifold dies, which he found to provide the best balance between good performance and fabrication costs. In all the works so far mentioned, the analyses treated the manifold as a circular-section duct.
Liu et al. (15) investigated a linearly tapered manifold coat-hanger die with a non-circular manifold. The flow distribution was investigated as a function of melt rheology, die width, and manifold dimensions; the latter two factors were found to influence strongly the flow uniformity towards the die edges, with flowrate uniformity deteriorating as the power-law index n fell. Winter and Fritz (16) presented a new theory for design of coat hanger dies, with circular- or rectangular-section manifolds. For the latter, provided that the manifold aspect ratio is at least 10, the theory predicts material-independent performance. For wide dies, however, this leads to large lengths making the die susceptible to deflection. A novel feature of this design is that the manifold curves down to the die exit, forming part of the lips at the edge of the die. Numerical simulations of a 1.2 m die designed according to the theory, using the software described in this work, have shown that though melt distribution is excellent across the slot lips (in the absence of deflection), flow rate falls off in the edge/manifold lip region (17).
The analytic design methods reviewed above are restricted by the need to approximate the die by simply shaped geometrical elements, and by the need to use a single, analytic representation of the melt rheology throughout. The restriction to developed flow is not problematical, because Reynolds numbers are generally rather low; neither is the restriction to isothermal flow, since this can be assumed in the majority of practical cases - a detailed study of thermal effects in slit dies has been published previously (18), and gives a criterion for the validity of the isothermal assumption. Numerical methods, however, particularly those using unstructured meshes, such as the finite element method, allow complete geometrical flexibility, and can represent the die geometries accurately. Additionally, they provide complete freedom in the description of melt flow properties. The most convenient approach exploits the particular features of flow in narrow slits between plane parallel, or nearly parallel walls, to reduce the dimensionality of the problem. The Hele-Shaw (lubrication) approximation leads to a potential equation for pressure or stream function on the plane of the slit, which combines momentum and mass conservation. This approach has been widely used, also, for the simulation of cavity filling in injection molding (19). It can be extended to include thermal effects, by discretization across the slit gap, resulting in a three-dimensional problem for temperature. Applications in extrusion are less widespread. Pearson (20) used it for modeling a cross-head die, and Fenner and Nadiri (21) applied it, using the finite element method, to a cable covering die. Nassehi and Pittman (22) also implemented a Hele-Shaw analysis using finite elements, to design the slit in a die extruding multi-layer tubular film. A dual-cavity coat-hanger die was analyzed, using finite elements, by Lee and Liu (23). The foregoing analyses all involved the isothermal flow assumption.
A generalized Hele-Shaw formulation, including temperature effects, has been applied to the analyses of slit dies, using two approaches. In the lumped parameter method the temperature variable is a gapwise averaged value on the plane of the slit [Vergnes, Saillard, and Agassant (24), Vergnes and Agassant (25), and Vlcek et al. (26, 27)]. A three-dimensional temperature formulation, combined with the generalized Hele-Shaw flow model has been used by Arpin, Lafleur, and Vergnes (28), and provides, additionally, temperature profiles across the slit gap. These non-isothermal analyses have been reviewed in more detail elsewhere (18).
A limited number of fully three-dimensional analyses of flow in slit dies have been carried out. Czyborra et al. (29) modeled Newtonian flow in a fish-tail die; Dooley (30) investigated the flow in two coat-hanger dies and compared predicted exit flow distribution with experimental data; transverse flow in a coat-hanger die with a circular-section manifold was studied by Wang for power-law fluids (31), and rubber compounds (32).
In none of the works reviewed above was the analysis of the melt flow and pressure field coupled with die body deflection. The closest approach to this seems to have been by Helmy (33, 34), who used analytic methods, calculating the pressure field and applying this as a distributed load to a beam representing the die body. However, there was no feed back of the modified slit geometry to the pressure field calculation, and this was therefore not a fully coupled analysis.
THE HELE-SHAW FORMULATION
Flow in the Slit
The Hele-Shaw or lubrication approximation has been widely used, as indicated in the previous section, and it is not necessary to repeat the derivation here. The approximation is valid for flow in narrow planar cavities, between parallel or nearly parallel plates, [Delta]H/[Delta]x, [Delta]H/[Delta]y [much less than] 1, see Fig. 2. The derivation of the Hele-Shaw equations, and discussion of the conditions for their applicability has been given by Pearson (19) and Hieber and Shen (20). Additionally, an experimental test of the equations was carried out by Benis (35). The end result of the simplifications is a potential equation for pressure in the plane of the slit, which combines momentum and mass conservation.
[Delta] / [Delta]x [S[Delta]P / [Delta]x] + [Delta] / [Delta]y [S[Delta]P / [Delta]y] = [Q.sub.o] (1)
We write the equation here, unconventionally, with a non-zero right hand side representing a volumetric source term, which will be used to introduce material within the flow domain. S is the flow conductivity, which, for the isothermal flow of a fluid characterized by the power-law model
[Mathematical Expression Omitted]
may be written
[Mathematical Expression Omitted]
A detailed investigation of the applicability of the isothermal flow assumption in slit extrusion has recently been made (18), where it was established as valid in the great majority of practical operating conditions. Gapwise averaged flow velocities may be recovered from the pressure field by
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
and the local flow rates between the plates, per unit width in the coordinate directions, are then
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
It is apparent from Eqs 1 and 3 that the problem for the pressure field is nonlinear, and an iterative solution scheme will be required. This is described later.
The mathematical model is completed by specification of boundary conditions. We require zero, or atmospheric, pressure along the lip exit, and zero flow across the impermeable boundaries in the x-y plane. From Eq 4 this latter is achieved by requiring
[Delta]P / [Delta]n = 0 (6)
where n is the normal at the boundary.
For convenience in the finite element implementation, as will be seen later, all boundaries of the problem domain other than the lip exit are taken to be impermeable, and flow is introduced using a non-zero value of the source term [Q.sub.o] in Eq 10 within a couple of attached, nonphysical elements.
Flow in the Distribution Manifold
The correct representation of the flowrate-pressure drop relationship in the distribution manifold is a key requirement in the simulation of flow distribution in the die. The manifold may have any one of a number of cross sections, the commonest being triangular, rectangular, or tear-drop shaped. In none of these, or other cases, can the flow be treated directly by the Hele-Shaw formulation. It is, however, highly desirable to model flow in the slit and manifold in a unified way. This can be done by replacing the manifold notionally by a Hele-Shaw slit having the same pressure drop-flowrate relationship as the actual manifold channel. The dimension of the equivalent Hele-Shaw slit can be found if developed flows are assumed; this is not strictly valid, as the manifold tapers, and the flow, which is not entirely parallel to the axis, decreases along its length. However, Reynolds numbers in the manifold are low, and a locally developed flow assumption is good. (There is no alternative other than a full three-dimensional analysis.) Pressure drop-flowrate relationships for developed isothermal flow in ducts with a wide variety of cross-sections can then be obtained using the method proposed by Miller (36). Briefly, this proposes that for a material with given flow properties ([Phi] and n in Eq 2), results for ducts with a variety of cross-sections can be represented by a master curve of average wall shear stress [Mathematical Expression Omitted] vs. apparent shear rate [Mathematical Expression Omitted]. These quantities are defined by
[Mathematical Expression Omitted]
where [D.sub.h] is the hydraulic diameter
[D.sub.h] = 4A / p (8)
A being the cross-sectional area, p the wetted perimeter, and [Delta]P/[Delta]L the axial pressure drop per unit distance; and
[Mathematical Expression Omitted]
where [Mathematical Expression Omitted] is the volumetric flowrate and [Lambda] is a shape factor, which depends solely on the duct cross-section shape, and is constant for geometrically similar channels.
Miller claims that this procedure gives pressure drops in a square-section duct accurate within 5% for materials with power law coefficients n in the range 0.4 to 1.0. Liu (37) further verified Miller's method.
The form of the master curve can be obtained using any convenient analytic result; for example, that for developed flow of a power-law fluid in a circular pipe
[Mathematical Expression Omitted]
Converting to the relationship for a duct of arbitrary cross-section, denoted by subscript a,
[Mathematical Expression Omitted]
Comparison with the result for developed isothermal power-law flow in a Hele-Shaw slit of width W, and gap H
[Mathematical Expression Omitted]
yields the required relationship between the geometrical parameters of the arbitrary duct and the equivalent Hele-Shaw slit
[Mathematical Expression Omitted]
The procedure we have adopted here is to retain the actual width, W, of the distribution manifold in the x-y plane, and calculate the gap, [H.sub.e], of the equivalent Hele-Shaw slit using Eq 13, and the appropriate value of [[Lambda].sub.a], drawn from the table provided by Miller. This is automated in the software, requiring only the selection of the manifold cross-section geometry, via the graphical user interface. The analysis of flow in the manifold and slit is then treated in a unified way.
Melt Flow Property Data
Melt shear viscosities are obtained from a data base developed by Schuler (38) linked to the analysis codes, and currently including about 150 industrial polymers. Viscosity as a function of shear rate is represented at each of a number of temperatures by
[Mathematical Expression Omitted]
over a range of shear rates, [Mathematical Expression Omitted], usually between 50 and 500 [s.sup.-1]. Interpolation of viscosity to the required temperature is made assuming an exponential type of temperature dependence. Local values of the power-law parameters, [Phi] and n, for use in Eq 3, 13, etc., are obtained from the tangent of log [Mu] vs. log [Mathematical Expression Omitted] at a representative local shear rate. This is chosen as the value at z = 0.75 h [ILLUSTRATION FOR FIGURE 2 OMITTED].
THICK PLATE FORMULATION FOR DEFLECTIONS
Mindlin Plate Formulation
By integrating through the plate thickness, finite element plate formulations provide a two-dimensional approximation, wherein weighted average transverse displacements w(x, y), and rotations of the normal to the mid-plane, [[Psi].sub.x](x, u) and [[Psi].sub.y](x, y), are obtained [ILLUSTRATION FOR FIGURE 3 OMITTED]. The Mindlin formulation (39) allows for transverse shear effects, providing an alternative to classical Kirchoff thin plate theory, and thereby making possible the analysis of thicker plates such as the die body slabs. The underlying assumptions are that the displacements are small relative to plate thickness, the stress normal to the mid-plane surface is negligible, and that normals to the mid-plane before deformation remain straight, though not necessarily normal to the mid-plane after deformation of the plate. The finite element equations are conventionally obtained by minimization of the functional for total potential energy of the Mindlin plate, which may be written
[Mathematical Expression Omitted]
where [[Epsilon].sub.b] and [[Epsilon].sub.s] are the bending and shear strains, [D.sub.b] and [D.sub.s] are the plate rigidities for bending and shear, q is the normal distributed load per unit area, and [M.sub.n], [M.sub.ns], and [Q.sub.n] are the moments and transverse shear forces per unit length of the portion [[Gamma].sub.[Sigma]] of the plate boundary [Gamma].
From the finite element viewpoint, an important feature of the Mindlin formulation is that only [C.sup.0] inter element continuity is required, compared with the [C.sup.1] requirement of the Kirchoff formulation. This means that, conveniently, the same type of nine-node isoparametric elements can be used for both the Hele-Shaw and Mindlin analyses. However, the most reliable results are obtained for the plate analysis using the Heterosis element, in which the Langrangian shape functions are replaced by 8-node Serendipity functions for the interpolation of the transverse displacements, w. Conventional Langrangian elements are used for the pressure analysis.
An estimate of the probable accuracy of the Mindlin formulation in the present application can be obtained from the work of Gomaa et al. (40). In comparisons with the closed form solution for a simply supported square plate, with uniformly distributed load, the numerical results showed errors of 2.7% at an aspect ratio of 0.2; 5.6% at 0.3; and 14.3% at 1.0. For partially clamped conditions deviations were smaller. The die slabs have aspect ratios between 0.23 (thickness to width) and 0.32 (thickness to length). On this basis, errors between 5% and 10% may be expected. This is very acceptable for practical purposes.
The appropriate boundary conditions for the thick plate analysis, as applied to the present problem, are not immediately obvious. Note, first, that the upper and lower halves of the die will be analyzed separately. It is also important to note the details of the meeting surfaces at the rear of the die halves, see Fig. 4. The bearing surface is along the raised strip, marked S in the Figure. This is the usual arrangement in slit dies; the reduced contact area is designed to increase the sealing pressure produced by the body bolts, and to ease separation of the die halves after use. The body bolts are located immediately behind the sealing surface at the back of the manifold.
To calculate deflections properly it is necessary to model the die slabs completely, from the rear edge to the slit exit, though, because of symmetry, only half of each slab need be considered. After some numerical experimentation, the following boundary conditions were identified as providing the best representation of the physics:
(i) Along the rear edge [[Gamma].sub.3]: zero z-direction displacement, w, and no rotation, [[Psi].sub.y], about the x-axis
(ii) On the center line, [[Gamma].sub.1]: no rotation, [[Psi].sub.y], about the x-axis
(iii) At the body bolt centers: no z-direction displacement, w.
In an earlier version, zero z-direction displacement along the rear of the manifold, on surface S, was used in place of (iii) above. This resulted in an unrealistically strong restraint and too small deflections. An additional advantage of restraining just the body bolt locations is that the analysis then provides loads at these points; these correspond to the real body bolt loads, which are of practical use to the designer.
FINITE ELEMENT IMPLEMENTATION
Hele-Shaw Analysis for Pressure Field and Flow Distribution
A mesh of biquadratic Langrangian elements of the sort used in the Hele-Shaw analysis is shown in Fig. 5. Because of symmetry about the center line, only half the flow area need be analyzed. (The boundary condition [Delta]P/[Delta]n = 0 is applied on the center line). In the mesh shown, the rear two rows of elements represent the distribution manifold. Four nonphysical elements are added adjacent to the manifold at the center line, within which material is introduced into the domain using an appropriate value of the source term [Q.sub.o] in Eq 1. Element boundaries are arranged to coincide with step changes in the slit gap H, at the edge of the region representing the manifold, and between the various regions of the slit, see Figs. 1b and 4. Within each element, values of the slit height H are defined at all nodes; thus, where step changes occur, H can be two- (or multi-) valued at a node. Smoothly changing values of H are interpolated within elements.
Finite element equations, corresponding to the linearized pressure potential equation, Eq 1, are formulated using the familiar Galerkin procedure, leading to a system of equations, which are solved using a symmetrical frontal routine. Linearization is achieved by substituting integration point values of the flow conductivity, S, evaluated using the pressure field found in the previous iteration(s). Convergence is reliably achieved, in most cases, using S values based on the results of the most recent iteration; however, for melts with values of the power law index, n, less than about 0.5, convergence is lost. This difficulty can be overcome by calculating S using a weighted geometric mean of pressure gradients in the previous two iterations. Indeed, convergence can be accelerated for all values of n, if the weighting is adjusted appropriately as a function of the local n value. The theoretical basis of this technique is outlined in an Appendix. Iterations are started using a uniform viscosity corresponding to a typical shear rate within the die.
Thick Plate Analysis for Deflections and Bolt Loads
In the thick plate analysis, information on plate thickness is handled in just the same way as slit height in the Hele-Shaw analysis. Step changes can occur at element boundaries, and continuous changes can be interpolated within elements. Figure 6 shows the main features of the cross sections of the two die halves. The thick plate treatment makes it difficult to include details such as drillings for cartridge heaters, but the effect of these on plate stiffness is very slight. Figure 6 also omits details of the extrusion slit, and the manifold and choker bar gap in the underside of the upper half of the die. While it would be possible to include these in the specification of plate thicknesses, more complicated meshes would be required, and as their effects will be slight, these details have been omitted. Referring to Fig. 6 it can be seen that the upper half of the die requires a minimum of 11 rows of elements in the flow direction to represent the changes in slab thickness, if the recess for the choker bar bolts is approximated using the quadratic curve shown dotted on the Figure. The lower half requires 9 rows, and the resulting "macro-element" mesh is shown in Fig. 7. For the analysis, this type of mesh is subdivided in an automatic mesh generation routine to give a refined mesh, in which certain nodes are located at the body bolt centers. In Fig. 6, details of the flexlip assembly are omitted. The stiffness contributed by this is included in the model by appropriately thickening the plate just upstream of the lip region. A procedure for calculating the required equivalent plate thickness is given in an Appendix.
Coupling the Hele-Shaw and Thick Plate Analyses
The Hele-Shaw and thick plate analyses are coupled in an iterative scheme, which starts with a calculation of the pressure field in the slit, using its undeflected geometry. The resulting pressures are interpolated onto the meshes to be used in the thick plate analyses of the two die halves. The deflections of each half are calculated, corresponding to these loads, and interpolated back onto the Hele-Shaw mesh, to yield modified nodal values of the slit gap H. The pressure field is then recalculated, and so on. Iteration is continued until successive pressure fields agree sufficiently. Recall that the pressure field calculation involves an inner iterative loop, which is converged on each cycle of the outer Hele-Shaw/thick plate iteration. The convergence criterion for pressure on both inner and outer loops is
[absolute value of [P.sub.i,j] - [P.sub.i-1,j]/[P.sub.i,j]] [less than] [10.sup.-4] (16)
where i is the iteration counter. The condition is to be satisfied for all nodes, j, of the Hele-Shaw mesh. Convergence of the outer loop is reliably obtained in 6 to 10 iterations. Figure 8 illustrates the interpolation problem. Meshes for the pressure and deflection analyses necessarily differ. Suppose we wish to interpolate pressure to node N of the thick plate mesh. It is first necessary to identify element e as the host element of the pressure mesh containing node N, and then to find the local coordinates of point N within element e, allowing interpolation of the calculated pressures to this point. This is a non-trivial procedure in isoparametric elements, but sophisticated and efficient algorithms are available. Four interpolations are required: from [TABULAR DATA FOR TABLE 1 OMITTED] the pressure mesh to the two deflection meshes; and from the two deflection meshes back to the pressure mesh. Host elements and local coordinates for each interpolation are identified once, and stored for reuse on each cycle of the iteration. In some cases, it may be possible to assume that deflection in the upper and lower halves of the die are in a fixed ratio. In this case only one deflection analysis is required, together with the definition of this ratio. Provision is made for this in the software. Figure 9 shows the logical structure of the overall calculation, enclosed within the graphical user interface.
Further information, useful in assessing the die design, may be obtained by post-processing the pressure field results from the converged simulation. In the finite element method, pressure gradients, and hence velocities and flowrates (Eqs 4, 5) are best obtained at integration points, and values are discontinuous at element boundaries. Smoothed nodal values are useful, however, and are required for input to the vector plot routine. The smoothing is carried out using a global scheme (41, 42).
Stream function plots are an additional, important aid to visualising flow fields. In the present case, the stream function, [Psi], is conveniently evaluated using the fact that stream lines and isobars are orthogonal.
[Delta][Psi] / [Delta]x [Delta]P / [Delta]x + [Delta][Psi] / [Delta]y [Delta]P / [Delta]y = 0 (17)
Table 2. Comparison of Computed Deflections of the Upper Half of the Die, for the Two Pressure Drops.
EC - die exit, on Center Line, y = 0 mm. EE - Die Exit, at the Edge, y = 600 mm. UC - 82 mm Upstream of the Exit, on the Center Line. UE - 82 mm Upstream of the Exit, at the Edge.
Upper Die Half Lower Die Half
Deflection (mm) Ratio Deflection (mm) Ratio
at at at at Pos. 151.9 bar 244.7 bar 1.61 151.9 bar 244.7 bar 1.61
EC 0.4528 0.6785 1.50 0.1938 0.2971 1.53 EE 0.3868 0.5551 1.44 0.1324 0.1943 1.47 UC 0.1736 0.2687 1.55 0.1214 0.1879 1.55 UE 0.1039 0.1544 1.49 0.0693 0.1028 1.48
Pressure gradients are again obtained at the integration points, and a finite element solution is carried out for nodal values of the stream function, [Psi]. For this hyperbolic equation, we use a version of the Streamline Upwind Petrov Galerkin (SUPG) formulation extended for bi-quadratic elements, (43). Contour plots of the stream function, at equal contour intervals, provide an immediate picture of flow patterns and relative flow velocities.
Information on material residence time in the die is useful in identifying material hold up, assessing self-cleaning properties, and the possibility of thermal degradation. A residence time field, R(x, y), may be obtained, for steady flow without recirculation, by solution of
[Mathematical Expression Omitted]
where gapwise mean velocities are evaluated at integration points, and nodal values of [R.sub.t] are found using the SUPG formulation.
These ideas can be extended further by calculating total material strain within the die, which can be related (qualitatively, at least) to elastic stresses and molecular alignment, or to the alignment of fillers such as flake or fiber. Based on representative local values of the shear rate within the gap, [Mathematical Expression Omitted], a representative total strain, [Mathematical Expression Omitted], is given by
[Mathematical Expression Omitted]
The representative shear rate is chosen as the value at z = 0.75 h [ILLUSTRATION FOR FIGURE 2 OMITTED].
The wall shear stress, [[Tau].sub.w], is also of interest, since it may be correlated with shark skin extrudate defects, which can occur for some polymers when a critical wail shear stress is exceeded: [[Tau].sub.w] is calculated at integration points from
[[Tau].sub.w] = H / 2 [square root of [([Delta]P / [Delta]x).sup.2] + [([Delta]P / [Delta]y).sup.2]] (20)
[TABULAR DATA FOR TABLE 3 OMITTED] [TABULAR DATA FOR TABLE 4 OMITTED] and smoothed to give nodal values, from which contour plots can be generated.
RESULTS AND DISCUSSION
As indicated above, the present techniques have been applied to analyze the performance of a 1.2-m monoextrusion die, as illustrated in Figs. 1a, b, and 4. This has a modified, coat-hanger type manifold, and is designed for high pressure operation, with a short flow-length dimension and thick die body. Detailed experimental measurements of the deflection and pressure drop of this die have been described elsewhere (1), and the results of the present computer simulations will be compared with these data.
Experiments and computations have shown that the deflection varies very nearly linearly with the pressure drop, so it is sufficient to compare results for one typical case. The small departure from linearity is, however, illustrated using a second set of computed results, and the reasons for this behavior are outlined.
In the experimental case, 200 kg/h (7.49 x [10.sup.-5][m.sup.3][s.sup.-1]) of LLDPE (Himont, Italy, batch no. 1001/SF 100923/66) flows through the die, producing an overall pressure drop of 157 bar (152 bar from manifold to exit on the center line). From the material database, the consistency of the melt was given as [Phi] = 2.42 x [10.sup.-5] [Pa.sup.n] [s.sup.-1], and the power law index as n = 0.79. Figure 10 shows contours of the computed deflection of the upper half of the die. Figure 11 shows the sum of the computed deflections of both halves, interpolated onto the mesh representing the flow channel, which is used for the analysis of the pressure field. A maximum combined computed deflection of 0.647 mm occurs at the die exit on the centerline, falling to 0.519 mm at the side of the die.
Computed and experimental deflections of the upper die half are compared in Table 1. In the central region of the die, near the exit, predicted deflections are highly accurate. Nearer the die edge, deflection is over predicted, with discrepancies rising to 0.06 mm. This is believed to be due to a constraint imposed by friction in the end plate assembly, which is not included in the mathematical model. Although it could probably be modeled as a (possibly nonlinear) spring restraining the die edge, its characterization would be difficult, and specific to a particular die. In view of this, and the still rather small resulting discrepancies, the matter is not pursued further here. At the upstream position A, deflections are in general very slightly underpredicted. Experimental results at position B are in all cases very similar to those at position A. This seems unlikely on physical grounds, and some systematic error in the measurements at position B must be suspected. Discounting discrepancies at this position, the agreement between experimental and computed deflections at all points, except the two in the exit-edge region, are remarkably good - mostly within a few thousandths of a millimeter. This level of agreement provides grounds for confidence in the accuracy of the thick-plate modeling of deflection, and for the success of the method used to take account of the stiffness contribution of the flexlip assembly. This latter is essential for the correct prediction of the rapidly increasing deflection of the upper half of the die along the flow direction, towards the exit. These deflection profiles along the flow direction are illustrated in Fig. 12, where computed values for the upper and lower die halves are shown, together with experimental values for the upper half. These plots confirm the features noted above, namely: suspect, low experimental values at point B; highly accurate predictions near the center-line; and slight overprediction near the edge. The computed results also show smaller deflections for the lower half of the die. The upper half is weakened by the choker bar and flexlip arrangements, and the results show clearly the price paid, in terms of increased deflection for the inclusion of these adjustment facilities. Variation of the deflection across the die, close to the exit is displayed in Fig. 13.
Finally, in this section, we examine the dependence of deflection on the die pressure drop. Table 2 compares results for the case discussed above with a second case having a higher pressure drop. It is seen that the ratio of the deflections is somewhat position dependent, and slightly lower than the ratio of the pressure drops. This is due to changes in the form of the pressure field in the slit. At the higher pressure drop, an increased proportion of the pressure loss occurs in the upstream region of the die (as a result of the opening of the lips) where it exerts a smaller moment about the body bolts. The departure from strict proportionality is, however, small, and the figures in Table 2 indicate that deflection results for one pressure drop can be extrapolated linearly to another with fairly small errors (less than [approximately]10% for an approximate doubling of the pressure drop).
Body Bolt Loads
As explained above, the constraint of zero deflection is imposed in the thick plate analysis at nodes positioned to correspond to the body bolt centers, and the analysis then yields the loads at these points, enabling the designer to size the bolts correctly. Table 3 illustrates the results obtained for a case where 200 kg/h LLDPE at 233 [degrees] C flows through the 1.2 m die, producing a pressure drop of 151 bar. Results are shown for three levels of mesh refinement, and also illustrate the convergence of the calculations. The location and identification of the body bolts is shown in Fig. 7; see also Fig. 10. The computed loads show a small negative value for bolt no. 1; it appears that this bolt may be redundant. The value of 2.0 x [10.sup.5] N at bolt no. 2 agrees well with an approximate analytic calculation taking moments about the rear edge of the die. This approximate method, however, is not able to predict accurately the variation of load with position. Loads fall steadily towards the die edge, to about 60% of the center line value, as the flow slit, and hence the lever arm of the pressure load about the bolts, becomes shorter.
Table 4 shows experimental pressure drops for several cases, compared with computed values. Melt flow in the die is effectively isothermal in all cases, and agreement between mean experimental values and computed results from the coupled pressure field-deflection analysis is very close - within 2% in two cases, and 6% in the third. This is within the accuracy of the melt viscosity data. For comparison, computed results based on the undeflected slit dimensions are also shown. These are approximately 25% higher than the predictions from the coupled analysis, illustrating the importance of taking into account the interaction between deflection and pressure fields.
The agreement between computed and experimental pressure drops is very encouraging, and, we believe, indicates that predictions of flow distribution in the die will also be accurate.
A contour plot of a typical pressure field produced by the software is shown in Fig. 14.
Effect of Deflection on Flow Distribution
We now examine flow distribution in the die, showing the effects of deflection, and also the influence of melt pseudoplasticity. First, Fig. 15 shows the results of a simulation based on the undeflected channel geometry of the 1.2-m die, with both the choker bar and flexlips in the null position. Exit velocity and flowrate profiles across the die are of the same form - since the lip gap is constant - with values at the center line 1.2 times greater than those at the die edge. Figure 16 shows results from the coupled simulation including the effects of deflection (but again without use of the choker bar or flexlips) for the same flowrate (150 kg/h) of the same material (LDPE, power law index n = 0.4). Flowrate at the die center has now increased by 13%, and decreased by 16% at the edge; the exit velocity has fallen by 12% at the center, and by as much as 30% at the edge. The ratio of the flowrate at the center to that at the edge is now 1.6:1, and the ratio of the velocities 1.5:1. The overall increase in the lip gap has resulted in lower exit velocities, with the largest decrease occurring at the edge, because more material now flows towards the center. Clearly, in practice, use of the choker bar and flexlips would be essential. The simulation can be used to assess their capabilities in correcting the uneven exit flowrates and velocities, by adjusting the assigned slit heights in the choker bar and lip regions. However, we do not show results of this type here, but next illustrate the influence of the degree of melt pseudoplasticity. Figure 17 displays exit velocity and flowrate profiles for the undeflected geometry, while Figure 18 shows the effects of deflection, from the coupled analysis, for a material with the higher power law index of 0.7. For the undeflected geometry we now see a reversal of the previous case, with more material flowing towards the die sides; the ratio is 0.87:1. Clearly, the performance of this die is not material independent. Interestingly, for this case the effect of deflection is to produce rather uniform distribution and exit velocities. The center to side ratio for the flowrate is 1:1 and for the velocity 0.96:1. This result is, of course, specific to this particular power law index, and die pressure drop; in other cases use of the choker bar and flexlips will generally be necessary.
In this Section we very briefly show some examples of post-processed results produced by the software. Figure 19 shows contours of the stream function for 200 kg/h LLDPE at 235 [degrees] C, giving an immediate picture of flow lines and relative velocities throughout the die; the contours of residence time, in Fig. 20, show that the residence time of melt leaving the die is approximately three times greater at the edge than at the center.
A simulation of slit die operation has been described that couples an isothermal Hele-Shaw pressure field/flow analysis with a thick plate analysis of deflection. Both analyses are two-dimensional, thus making it possible to obtain results in a couple of hours on a PC, and to predict, for the first time, the dynamic operating characteristics of the die, taking into account the interaction of internal pressure, flow distribution, and die deflection. Computed results agree closely with experimental values.
The calculation routines are enclosed within a graphical user interface (GUI) in the MS-Windows operating environment (2). This facilitates problem set up, using customized screens, pop-down menus, and mouse clicks, and incorporates a semi-automatic mesh generation procedure. Post-processing of the primary data on the melt pressure field yields the stream function, melt residence time, melt total strain, and wall shear stress. These quantities, as well as pressure and deflection fields may be displayed as contour plots, selected via the GUI. The package also incorporates a materials property data base, currently including some 150 industrial polymers. Overall, the software provides a powerful and convenient computational aid to the designer and process operator of slit dies.
Because of space limitations, it is not possible to reproduce the Appendices here. They may be obtained by contacting the second-named author.
This work was supported by Sectoral Grant S/BREU-900366 of the Commission of the European Communities.
A = Cross sectional area of duct or manifold, [m.sup.2].
[a.sub.i] = Coefficient in log polynomial representation of viscosity as a function of shear rate, log[(ns/m).sup.2]/log([s.sup.-1]).
[D.sub.h] = Hydraulic diameter, m.
[D.sub.i] = Bending rigidity of Mindlin plate, Nm.
[D.sub.s] = Shear rigidity of Mindlin plate, [Nm.sup.-2].
H = Total extrusion slit height (gapwise dimension), m.
[H.sub.e] = Slit height of a Hele-Shaw slit with the same pressure drop-flow rate relationship as the distribution manifold, m.
[M.sub.n], [M.sub.ns] = Bending moments per unit length of boundary, N.
n = Outward normal to the boundary.
n = Power law index.
P = Pressure, [Nm.sup.-2].
p = Perimeter of duct, m.
q = Normal distributed load per unit area, [Nm.sup.-2].
[Q.sub.n] = Transverse shear force per unit length of boundary, [Nm.sup.-3].
[Q.sub.o] = Volumetric source term, [ms.sup.-1].
[Q.sub.x], [Q.sub.y] = Local flowrates in the slit, in x- and y- directions, [m.sup.2][s.sup.-1].
R = Pipe radius, m.
[R.sub.t] = Residence time, s.
S = Flow conductivity in the Hele-Shaw equation, [m.sup.5][s.sup.-1][N.sup.-1].
[Mathematical Expression Omitted] = Volumetric flowrate in the manifold, [m.sup.3][s.sup.-1].
[Mathematical Expression Omitted], [Mathematical Expression Omitted] = Gapwise averaged flow velocities in the slit in x- and y-directions, [ms.sup.-1].
w(x, y) = Weighted average transverse displacement of the plate, m.
W = Manifold width, perpendicular to flow direction, m.
x, y = Coordinates in the plane of the slit, m.
z = Gapwise coordinate in the slit, m.
[Mathematical Expression Omitted] = Shear rate, [s.sup.-1].
[Mathematical Expression Omitted] = Apparent shear rate at the wall, [s.sup.-1].
[Mathematical Expression Omitted] = Representative total strain.
[Gamma], [[Gamma].sub.[Sigma]] = Boundary, portion of boundary of the plate.
[[Epsilon].sub.b], [[Epsilon].sub.s] = Bending and shear strains in Mindlin plate.
[Lambda] = Shape factor for manifold, or duct cross section.
[Mu] = Viscosity, [Nsm.sup.-2].
[Pi] = Total potential energy for Mindlin plate.
[Tau] = Shear stress, [Nm.sup.-2].
[[Tau].sub.w], [Mathematical Expression Omitted] = shear stress, average shear stress at the wall, [Nm.sup.-2].
[Psi] = Stream function, [m.sup.2][s.sup.-1].
[[Psi].sub.x], [[Psi].sub.y] = Rotations of the normal to the mid-plane of the plate.
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|Author:||Sander, R.; Pittman, J.F.T.|
|Publication:||Polymer Engineering and Science|
|Date:||Aug 15, 1996|
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