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3D features in the calendering of thermoplastics a computational investigation.

INTRODUCTION

Calendering is a widely used manufacturing process that involves a set of usually four corotating heated calenders (rolls) for the production of thin plastic sheets and films. The thermoplastic melt is fed behind the nip region of the two calenders, the rotational movement of which forces the material to flow in the machine direction, with subsequent detachment from the rollers surface at a specific thickness. Simultaneously, the material flows and spreads in the lateral direction. Two-roll calendering lines are usually used for rubber processing, whereas four-roll calenders are generally used for the production of double-coated products meeting strict surface quality requirements.

Gaskell [1] presented a theoretical analysis of the calendering process based on the assumption that the diameter of the rolls is large compared to the gap between the calenders, and as a result, the flow in the machine direction can be approximated as one-dimensional. He considered Newtonian fluids as well as Bingham plastics and derived the relevant equations to predict the pressure drop in the machine direction, that being of primary interest in design. McKelvey [2] and Middleman [3] extended the study of Gaskell [1] to power-law fluids. A theoretical approach to study the calendering of power-law fluids was also shown by Brazinsky et al. [4]. Alston and Astill [5] presented a unidirectional analysis for the calendering process of non-Newtonian fluids described by a hyperbolic tangent viscosity model, exhibiting Newtonian behavior at high and low shear rates and a shear-thinning behavior at intermediate shear rates. Vlachopoulos and Hrymak [6] studied (theoretically and experimentally) the calendering of rigid PVC (polyvinyl chloride) using a nonisothermal power-law model based on lubrication theory. More recently, numerical studies were carried out by Sofou and Mitsoulis [7, 8] for the calendering of viscoplastic and pseudoplastic materials using the lubrication approximation with no-slip and slip at the calender's surfaces. Kiparissides and Vlachopoulos [9] studied computationally the pressure distribution in the machine direction for symmetric and asymmetric calendering of Newtonian and non-Newtonian fluids using the finite element method. The same authors [10] used a finite-difference nonisothermal model to study the temperature distribution in the calender gap, including the effect of viscous dissipation; they predicted local temperature maxima near the surfaces of the calenders--regions associated with high shear--but only a small rise in the temperature of the calendered product. Agassant and Espy [11] and Agassant [12] studied the flow between the calenders computationally using a 2D finite element method and complemented computational results with calendering experiments using rigid PVC; they observed the formation of a melt bank showing a large recirculation zone, with smaller vortices developing near the entry region. Mitsoulis et al. [13] performed 2D nonisothermal analyses for Newtonian and power-law fluids by means of the finite element method in which they included the melt bank shape determination among other results.

Incidentally, calendering is not confined to thermoplastics. The process of rolling is used extensively in the production of metal sheets and in the forming of ceramic pastes [14-22]. In the context of the production of a metal strip from the molten state, Matsumiya and Flemmings [23] studied the spreading of the material using the lubrication approximation. The authors describe the rheological behavior of the alloy as a power-law fluid and assume that the pressure does not vary in the machine direction; the only non-zero velocity is, therefore, the one in the transverse direction. Johnson [24] studied small-scale spreading of metals undergoing plastic deformation during metal sheet rolling--using an asymptotic formulation with the spreading sides treated as free surfaces--as well as the pressure distribution along the cylinder axis. Sezek et al. [25] studied cold and hot metal plate rolling using a three-dimensional mathematical model considering lateral spreading of the plate based on the minimization of power consumption during the process; a third order polynomial approximation was used to describe the shape of the free side surfaces. Considering two-dimensional flow (in the machine and in the transverse directions) and using gap-averaged quantities, Levine et al. [26] developed a 2D model for the pressure field in the region between the rotating cylinders using the essential tenets of the lubrication approximation. Use of this approach allowed for the determination of the shape of the spreading sheet as well as of the two-dimensional variation of pressure both in the machine and the transverse directions.

Study of three-dimensional effects in the calendering of thermoplastics has been limited. Unkriier [27] carried out experiments using a calender having two rolls of 80 mm diameter and 1600 mm length, using PVC and PS (polystyrene) as calendered materials. His conclusions were that (i) the calendered sheet spreads, (ii) three vortices are formed in and near the melt bank, and (iii) the material in the melt bank conveys to the sides through a spiral motion. He also observed and measured the pressure drop in the transverse direction. A 3D simulation of the calendering process was carried out by Luther and Mewes [28], with their results also predicting the spiral motion of the material in the melt bank. The authors, however, did not include a prediction for the spreading of the calendered sheet or of the existence of a pressure profile in the direction of the calender axes (transverse direction).

Given the state-of-the-art outlined above, the scope and contribution of this work is to solve the fully 3D Stokes equations in geometries realistically relevant to the calendering process and predict the spreading of the sheet, as well as the fully 3D pressure and flow patterns occurring as the material passes through the calenders. The study is performed under isothermal conditions, as explained and justified in subsequent section. We use the generalized Newtonian fluid (GNF) constitutive model for the calendered material, with material and processing parameters relevant to plastics calendering, e.g., calender radius 125 mm, final thickness of the sheet ~1.25 mm and a polymer melt such as PVC. It is also assumed that the material behaves as a purely viscous fluid, thus any elastic effects caused by a second stage calender pair (i.e., pulling tension) are neglected. In our analysis, the spreading sides of the calendered sheet are deformable free surfaces the shape of which is unknown and determined in the course of the computation. This determination is carried out by means of a decoupled numerical procedure, as outlined in subsequent section. The results are validated by comparing the predicted pressure distribution in the machine direction to existing analytical solution [3, 6].

MODEL EQUATIONS

The governing equations for isothermal incompressible steady state flows of polymer melts are the continuity equation

[nabla] x u = 0 (1)

where u is the velocity vector, and the equation for the conservation of momentum in the absence of fluid inertia (Stokes equations)

0 = -[nabla]p + [nabla] x T (2)

where p is the pressure and T is the deviatoric stress tensor. For purely viscous (inelastic) fluids, the rheological constitutive equation that relates the stresses to the velocity gradients is the GNF model

T = [eta]([II.sub.D]) x D (3)

where D is the rate of strain tensor

D = [nabla]u + [nabla][u.sup.T] = [[partial derivative][u.sub.i]/[partial derivative][x.sub.j]] + [[partial derivative][u.sub.j]/[partial derivative][x.sub.i]] (4)

and [II.sub.D] is the second invariant of the rate of strain tensor given by

[II.sub.D] = D : D (5)

For fluids with Newtonian behavior [eta] = [mu], where [mu] is the Newtonian viscosity. The non-Newtonian rheological model used in this work is the power-law in which the viscosity is a function of the second invariant ([II.sub.D]) of the rate of strain tensor D. The power-law viscosity model is given by

[eta]([II.sub.D]) = m[[absolute value of 1/2 [II.sub.D]].sup.n-1/2] for [eta]([II.sub.D]) < m (6)

[eta]([II.sub.D]) = [[eta].sub.T] for [eta]([II.sub.D]) > m (7)

where m the consistency index, n the power-law exponent, and [[eta].sub.T] a truncation viscosity. A schematic of a typical geometry at the mid-plane of a spreading calendered sheet is shown in Fig. 1. The machine, thickness and transverse directions correspond to the x, y, and z axes, respectively, of a Cartesian coordinate system. The size of the gap H(x) between the calenders is found through the following simple equation

H(x) = [H.sub.o] + R - [square root of ([R.sup.2] - [x.sup.2])] (8)

where R is the radius of the calenders and [H.sub.o] is the half gap at the nip region (x = 0) (see Fig. 1). The equation above (Eq. 8), treats the surface of the calenders as cylindrical, in contrast to earlier formulations (e.g., Gaskell [1], McKelvey [2], and Middleman [3]) that assumes parabolic calender surfaces (a reasonable approximation for flow fields extending only a small distance from the nip region). Applying the approximation of Middleman [3] with [H.sub.f]/[H.sub.0] = 1.226 to Eq. 8 and solving with respect to x, yields the detachment point x{ (the point where the sheet leaves off the calenders surface). In the same way, for a given initial thickness 2[H.sub.i], substitution into Eq. 8 yields the biting position [x.sub.i] of the sheet. An actual computational model of the full 3D geometry we consider, including a curved side (free) surface (as determined at the end of the solution procedure) is shown in Fig. 2. The model equations are nondimensionalized using

x' = [x/[square root of 2[RH.sub.o]]], z' = z/[W.sub.i] and p' = [p/m] [([H.sub.o]/[U.sub.r]).sup.n] (9)

where instead of the consistency index (m) the Newtonian viscosity ([mu]) is used when n = 1 and [U.sub.R] is the velocity at the cylinder surfaces. The following boundary conditions for the velocity apply (see Fig. 1):

a. No-slip at both cylinders surface (wall conditions).

b. Fixed value for the volumetric flow rate ([q.sub.in]) at the entrance boundary (x = [x.sub.i]) (inlet condition), given by [q.sub.in] = 2[H.sub.i][U.sub.i][W.sub.i], where [H.sub.i] the half-gap, [U.sub.i] the feeding sheet speed, and [W.sub.i] the sheet width at the entrance boundary.

c. At the exit boundary (x = [x.sub.f]) a Neumann condition is used (fully developed outlet condition), namely [partial derivative]u/[partial derivative]n = 0, where n is the unit vector normal to the outflow boundary.

d. At the side (free) surface z = z(x) boundaries free surface conditions apply, namely u x n = 0 (no flow passes across these surfaces) and T x n = 0 (shear-free condition on the side spreading surfaces, or, equivalently, zero values for the components of the traction vector). The boundary conditions for the Stokes equation (Eq. 2) include also the value of pressure at the boundaries, which was set to a fixed (ambient) value (p = 0) for the entrance (x = [x.sub.i]), exit (x = [x.sub.f]) and side (free) surface (z = z(x)) boundaries.

NUMERICAL PROCEDURE

For the solution of the continuity equation (Eq. 1) and the Stokes equations (Eq. 2), we use the OpenFOAM I (Open Source Field Operation and Manipulation) package, using the Finite Volume Method (FVM). The FVM subdivides the flow domain into a finite number of smaller control volumes (cells) (see Fig. 2 in which the same flow domain is shown from three different view angles). The size of the flow domain in Fig. 2 (8 x [10.sup.5] cells) was determined based on the aspect ratio of the cells filling the domain. In all cases, the maximum allowed cell aspect ratio for the Newtonian fluid was set to 30. For the cases with small spreading (e.g., [W.sub.i]/ 2[H.sub.o] = 200) of the sheet, the maximum aspect ratio of the cell was set to 10. Concerning the cases with higher sheet spreading (e.g., [W.sub.i]/2[H.sub.o] = 80), filling the domain with 8 x [10.sup.5] cells would give a maximum allowed cell aspect ratio of 30. For the non-Newtonian simulations, a more dense mesh arrangement was used with 2.4 x [10.sup.6] cells which gave a maximum cell aspect ratio of 20 for both the lower (e.g., [W.sub.i]/2[H.sub.o] = 200) and higher (e.g., [W.sub.i]/ 2[H.sub.o] = 80) sheet spreading. The continuity and the Stokes equations are then discretized and integrated over each control volume by properly approximating the variation of flow properties between computational nodes. The steady-state incompressible solver uses the SIMPLE algorithm (Semi-Implicit Method for Pressure Linked Equations) introduced by Patankar and Spalding [29].

Validation

The results of the 3D model were validated against the L analytical solution as derived from the lubrication approximation theory (Middleman [3]) for a power-law fluid. In dimensionless form, the analytical solution for the pressure drop in the machine direction is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The material utilized (PVC with m = 46120 Pa [s.sup.n] and n = 0.34) and the geometry for the construction of the 3D computational flow domain for the simulations (R = 125 mm, [H.sub.o] = 0.3 mm, [U.sub.R] = 73.83 mm/s, [H.sub.f] = 0.3765 mm) were those used by Vlachopoulos and Hrymak [6], The validation is based on the comparison of the computed pressure distribution in the machine direction (x-direction) to the result obtained by integrating Eq. 10 using a fourth-order Runge-Kutta algorithm. For this comparison to be relevant, it was assumed that the material is fed from an infinite reservoir located far behind the nip region and that no spreading of the calendered sheet takes place; specifically, the boundary conditions at the vertical surfaces of the sheet sides were a zero pressure gradient and a zero velocity gradient in the transverse direction (z-direction) thus simulating a calender of infinite width. Figure 3 illustrates the calculated pressure distribution in the machine direction for three different mesh densities and for a power-law fluid. It is clear that for 2.4 x [10.sup.6] cells the pressure distribution is in excellent agreement with the analytical solution (Eq. 10). Decreasing mesh density to 2.4 x [10.sup.4] cells a relatively small underestimation of the pressure distribution is observed while further lowering of mesh density leads to increased underestimation of pressure. Comparison of the pressure distributions for a Newtonian fluid, using the same geometry and assuming Newtonian material with [mu] = 4612 Pa s are also shown at Fig. 3 to be in excellent agreement with the analytical solution. The insert in Fig. 3 represents a magnified view of the pressure distribution for a small region around the nip (x' = 0).

Concerning the effect of nonisothennality, Kiparissides and Vlachopoulos [10] presented results of nonisothermal 2D calendering simulations for Newtonian and power-law fluids. They showed that viscous dissipation is small and that under usual operating conditions (e.g., for R = 150 mm, [U.sub.R] = 0.4 m/s, [H.sub.o] = 0.25 mm and a power-law fluid with m = 2.5 x [10.sup.4] Pa [s.sup.n] and n = 0.25) the temperature rise in the produced film is less than 2[degrees]C. This was also investigated in this work for the geometry shown in Fig. 2 (R = 125 mm, C/R = 0.1 m/s, [H.sub.o] = 0.5 mm) and a power-law material (n = 0.34) with temperature dependent consistency index m given by an Arrhenius expression m = [m.sub.o]exp[--b(T -- Tref )] where [m.sub.o] = 5 X [10.sup.4] Pa [s.sub.n] is the consistency index at the reference temperature ([T.sub.ref.] = 180 C) and b = 0.07 [C.sub.-1] is the temperature sensitivity parameter. The material enters the flow domain at the same temperature as the calenders ([T.sub.melt] = [T.sub.wall] = 180[degrees]C) and the spreading sides are regarded as adiabatic. In this simulation, the maximum observed temperature rise was 3[degrees]C and the effect on the sheet spreading and vortex formation, which are the main objectives of this article, were nondiscernible. In the following, we present results of strictly isothermal simulations.

Prediction of Sheet Spreading

In calendering, the rotational movement of the calenders creates a drag flow which, due to the reduced flow gap in the machine direction (x-direction), results in a pressure build-up behind the nip region. This pressure not only affects the melt flow in the machine direction, but, since the sides of the calendered sheet are at ambient pressure, forces the melt to move (spread) in the transverse direction (z-direction) as well. It is this transverse pressure gradient that causes the polymer sheet to spread as it passes through the calenders. In this study, it is assumed that the material is fed as a sheet of uniform thickness and speed (e.g., from a conventional flat coathanger die attached to an extruder located behind the calenders). It is also assumed that the shape of the side (free) surfaces varies only in the x and z directions--no curvature is assumed in the y (thickness) direction. Computationally, the shape is determined based on the physical condition that the points of the free surface belong to a streamline, thus satisfying the following equation at each point on the side surface

dz/dx = [bar.w]/[bar.u] (11)

where dz/dx is the local slope of the free surface (see for example Fig. 1b) and [bar.u], [bar.w] are the gap-averaged velocity components in the x and z directions respectively, at the same position. These gap-averaged velocities are calculated as

[bar.u] = 1/[N.sub.y] [[N.sub.y].summation over (i=1)] [u.sub.i] (12)

[bar.w] = 1/[N.sub.y] [[N.sub.y].summation over (i=1)] [w.sub.i] (13)

where [N.sub.y] is the number of the gap-wise mesh nodes in the y direction. For the Newtonian simulations, we used [N.sub.y] = 11, whereas for the power-law simulations [N.sub.y] = 31. The total number of cells (control volumes) filling the computational flow domain was 8 X [10.sup.5] for the Newtonian and 2.4 X [10.sup.6] for the power-law simulations. Determination of the shape z(x) of the sides of the calendered sheet is achieved through a decoupled iterative procedure, in which the continuity equation (Eq. 1) and the Stokes equations {Eq. 2) are solved (starting with initial (base) configuration of a sheet of uniform width) and in which the shape of the side surface z=z(x) is adjusted, based on solution of Eq. 11. Following solution of Eqs. 1 and 2, the velocities u and w at each location of the side surface in the y direction are known and thus the corresponding gap-averaged values can be calculated from Eqs. 12 and 13 at each nodal position on the side surfaces. A new shape of these side (free) surfaces is then obtained by integrating Eq. 11 utilizing the above-computed gap-averaged velocities. A simple integration formula such as

[z.sup.s+1.sub.j] = [z.sup.s.sub.0] + [[N.sub.x].summation over (j=1)] [[bar.w].sup.s.sub.j]/[[bar.u].sup.s.sub.j] ([x.sup.s.sub.j] - [x.sup.s.sub.j-1] (14)

was found sufficient. In Eq. 14 [z.sub.j] represents the updated z-coordinate at the corresponding point at the 5th iteration, [z.sub.o] is the z-coordinate of the corresponding point at the entrance (x = [x.sub.i]) (equal to the half-width of the sheet at the entrance), [x.sub.j] - [x.sub.j - 1] the distance between two successive surface nodes in the x-direction and [N.sub.x] the number of positions in the x-direction ([N.sub.x] = 201 for typical Newtonian and power-law simulations). After the new z-coordinates are predicted (via Eq. 14) and thus a new shape of the side (free) surface is obtained, a new three-dimensional computational flow domain is constructed, the continuity and Stokes equations {Eqs. 1 and 2) are solved and the procedure is repeated until convergence. The speed of convergence to the final shape of the spreading sheet is improved by using a relaxation procedure

[[??].sup.s+1.sub.j] = (1 - [omega])[z.sup.s+1.sub.j] + [omega][z.sup.s.sub.j] (15)

with relaxation factors [omega] (0.3 < [omega] <0.8) where [[??].sup.s+1] is the z-coordinate at the 5th iteration. Convergence is obtained when the gap-averaged velocities ([bar.u], [bar.w]) and the obtained side (free) surface shape z = z(x), satisfy Eq. 11 so that the sum of the squares of the errors is less than a prescribed tolerance ([epsilon] = [10.sup.-3]), namely:

1/[N.sub.x] [square root of [[N.sub.x].summation over (j=1)] [([z.sub.j]-[z.sub.j-1]/[x.sub.j]-[x.sub.j-1] - [[bar.w].sub.j]/[[bar.u].sub.j]).sup.2]] [less than or equal to] [epsilon] (16)

RESULTS AND DISCUSSION

Prediction of Sheet Spreading

Figure 4 illustrates the shape (geometry) of the side (free) surface for the dimensionless ratio [W.sub.i]/2[H.sub.o] = 80 (Fig. 4a) and [W.sub.i]/2[H.sub.o] = 150 (Fig. 4b) for a Newtonian fluid with [mu] = 6500 Pa s and [rho] = 1000 kg/[m.sup.3]. The feed flow rate is set at [q.sub.in] = 100 kg/hr. Most of the spreading is shown to occur before the material reaches the nip region. For the case with [W.sub.i]/2[H.sub.o] = 80, nineteen iterations where sufficient to reach convergence according to the criterion described by Eq. 16, as shown in Fig. 4a. Even from the fifth iteration the geometry of the free surface assumes a shape which is very similar to the final convergent shape. In the case of a relatively narrow sheet (WJ 2[H.sub.o] = 80) (Fig. 4a), the sheet spreads by ~84%. Increasing the width of the entrance sheet so that [W.sub.i]/2[H.sub.o] = 150 for the same Newtonian material, the amount of spreading is reduced to about 27% as illustrated at Fig. 4b. The side (free) surface shape converges even faster and the tendency of the material to spread is less intense than the case of [W.sub.i]/2[H.sub.o] = 80 (Fig. 4a). At a small region near the entrance, the sheet shows a small (see Fig. 4b), if any, tendency to spread (dz/dx~0) whereas in Fig. 4a the material starts spreading at the biting position (with reference to x = [x.sub.i] of Fig. 1) of the calenders. The fact that the extent of spreading is a function of the width of the fed sheet has also been predicted by Levine et al. [26] using a 2D (x-z) lubrication analysis.

Figure 5 shows the effect of the entrance velocity (for a fixed flow rate at the entrance [q.sub.in] = 100 kg/hr), non-dimensionalized with the velocity of the calenders [U.sub.R], on the spreading ratio [W.sub.f]/[W.sub.i], for the Newtonian fluid. It can be seen that as the ratio [U.sub.i]/[U.sub.R] increases, implying a faster feeding of the melt, the spreading ratio [W.sub.f]/[W.sub.i] increases. At larger values of [U.sub.i]/[U.sub.R], the relationship becomes almost linear (the maximum spread for the studied Newtonian cases is obtained at [W.sub.i]/2[H.sub.o] = 50 with [U.sub.i], being 40% of the calenders' velocity). At the same figure (Fig. 5), the effect of the entrance width of the sheet on the spreading ratio [W.sub.f]/[W.sub.i] is also illustrated. For large values of [W.sub.i], the spreading ratio approaches asymptotically unity--for [W.sub.i]/2[H.sub.o] = 400, [W.sub.f]/[W.sub.i] = 1.05 (i.e., the unidirectional approach as described by the lubrication theory), whereas for smaller sheet width, the corresponding spreading is proportionally larger (e.g., for [W.sub.i]/2[H.sub.o] = 50, [W.sub.f]/[W.sub.i] = 3.17). Inversely, small values of the ratio [U.sub.i]/[U.sub.R] result in small spreading, whereas larger amounts of spreading are associated with increased velocity of the fed sheet. As in all cases, the volumetric flow rate is kept constant, large [U.sub.i] correspond to small [W.sub.i] and vice-versa.

Figure 6 illustrates the effect of the thickness of the fed sheet (for a specific entrance flow rate [q.sub.in] = 100 kg/ hr), expressed as a multiple of [H.sub.o], on the final spreading ratio [W.sub.f]/[W.sub.i] for the case of [W.sub.i]/2[H.sub.o] = 100. An almost linear dependence is observed for [H.sub.i]/[H.sub.o] > 4.

Figure 7 illustrates the spreading of a non-Newtonian fluid for [q.sub.in] = 100 kg/hr, for [W.sub.i]/2[H.sub.o] = 80 (Fig. 7a) and [W.sub.i]/2[H.sub.o] = 150 (Fig. 7b) using the power-law model with m = 50000 Pa [s.sup.n] and n = 0.35 (typical values for PVC). For [W.sub.i]/2[H.sub.o] = 80 (Fig. 7a) and for a small region near the entrance (biting position), the geometry of the side (free) surface implies a negligible spreading tendency--probably affected by the relatively high melt viscosity in this low shear rate region. As the material progresses towards the exit, the prevailing shear rates increase as well as the pressure, and as a result the sheet begins to spread. The maximum extent of spreading achieved close to the nip region is due to the high pressure build up in this area. For [W.sub.i]/2[H.sub.o] = 150 (Fig. 7b), and using the same m and n values, the side surface shape shown at Fig. 7b exhibits a lower spreading tendency than the case with [W.sub.i]/2[H.sub.o] = 80 (Fig. 7a). Moreover, comparison of the final predicted side (free) surface shape for [W.sub.i]/2[H.sub.o] = 150 between the Newtonian (Fig. 4b) and the power-law (Fig. 7b) fluid shows almost the same amount of final spreading. However, since the non-Newtonian material exhibits shear thinning behavior, near the entrance the spreading is less abrupt than the cases with Newtonian material (e.g., [W.sub.i]/2[H.sub.o] = 80 with reference in Fig. 4a and Fig. 7a) due to the correspondingly higher viscosity. Although the volumetric flow rate at the entrance is held constant ([q.sub.in] = 100 kg/hr), the entrance velocity of the sheet changes when the width of the fed sheet changes. Since boundary conditions (d) (in Section Model Equations) ensure that the mass balance is maintained, reducing the entrance width sheet for the given volumetric flow rate, the entrance velocity becomes higher and vice versa resulting in different spreading ratios [W.sub.f]/[W.sub.i].

Pressure Profiles in the Transverse Direction

The predicted pressure profiles in the transverse direction for the case of [W.sub.i]/2[H.sub.o] = 80 for a Newtonian fluid ([mu] = 6500 Pa s) are shown in Fig. 8, at three different positions along the length of the sheet with [x.sup.J] = 0 representing the position of the nip region and [x.sup.J.sub.f] being the detachment point. At the nip region, the pressure distribution appears to have a fairly constant value for a large portion of the sheet width with rapid decrease near the two free surfaces (side edges) where the spreading takes place. At positions behind the nip region ([x.sup.J] = -[x.sup.J.sub.f] and [x.sup.J] = -2[x.sup.J.sub.f]), the pressure profiles exhibit somewhat greater variability with the pressure being maximum at the central region of the sheet, and a sharp pressure gradient near the edges. The dimensionless pressure contours on the symmetry plane x-z (see Fig. 1) for the Newtonian fluid with [W.sub.i]/2[H.sub.o] = 80 is shown in Fig. 9. Pressure profiles at the same [x.sup.J] positions as in the Newtonian fluid are obtained for the power-law fluid (m = 50000 Pa [s.sup.n], n = 0.35) as shown in Fig. 10 for the case with [W.sub.i]/2[H.sub.o] = 80 and in Fig. 11 pressure contours on the x-z symmetry plane are plotted. The pressure contour results presented by Levine et al. [26] show a similar transversal rapid decrease of pressure near the edges. The predicted pressures are also in qualitative agreement with the experimentally measured pressure results reported by Unkruer [27].

The Spiral Motion of the Melt

In the following, we study the motion of fluid particles as they move from the entry plane and through the calender gap to form the calendered sheet. Figure 12 shows the streamlines for a Newtonian fluid ([mu] = 6500 Pa s). The material is entering the flow domain through line AB, which in Fig. 12a is located 5 mm off the x-y symmetry plane, whereas in Fig. 12b line AB is located 50 mm off the x-y symmetry plane (the length of line AB is equal to the entrance thickness of the sheet, 2[H.sub.i]). It is evident that material entering at the top and bottom segments of line AB (that is, nearest to the rotating rollers) is taken up by the drag flow and conveyed directly to the exit. However, material entering the domain from the central portion of line AB (e.g., see Fig. 12a) appears to progress through a spiral motion in the transverse direction, and exit at the sides of the calendered sheet (see Fig. 12a and b). This spiral motion is evidently the result of the combined action of the recirculation forming behind the nip region and of the transverse pressure gradient discussed earlier, and is therefore a fully 3D motion. Similar patterns are observed for material entering at other locations off the x-y symmetry plane such as at 15 mm, 30 mm, and 50 mm from the symmetry plane (the half width of the sheet at the entrance is [W.sub.i]/2 = 75 mm). Similar streamlines patterns are observed for the power-law fluid.

These results appear to be in qualitative agreement with the experimental observations by Unkruer [27] for flow within the melt bank. The implications of the spiraling motion of the material from the entrance to the spreading sides could be significant vis-a-vis the prediction of molecular (or fiber) orientation in the calendered product. However, a more striking implication of the flow patterns revealed by Figs. 12 and 13 is the evident rearrangement of the calendered material. Material at the surface of the fed sheet follows the drag flow near the calender surfaces and ends-up at the corresponding (surface) positions of the produced sheet. This is intuitive and is shown in Fig. 13a for the middle section of the calenders (extending from the symmetry plane at z = 0 to z = [W.sub.f]/6) in which the outer layer of the final calendered sheet is shown to originate from the corresponding outer layer of the fed sheet. Material ending up at the inner layer of that section of the produced sheet (0 < z < [W.sub.f]/6) (streamlines marked as red in Fig. 13) originates from the feed section's inner layer--this material experiences a small lateral displacement due to the transverse pressure developed (Fig. 13a). The analogous streamlines for an intermediate region ([W.sub.f]/ 6 < z < [W.sub.f]/3) are shown in Fig. 13b. Here, the material at the inner layer of the fed sheet experiences a higher lateral displacement than the corresponding material of Fig. 13a, but still ends up forming the inner layers of the corresponding section of the produced sheet--while, as in Fig. 13a, the surface layer still follows the drag flow and ends up forming the surface of the produced sheet. Examination of Fig. 13c, however, shows that the material coming off the calenders at the edges of the calendered sheet has originated to a large extent at the inner region of the entire width-span of the fed sheet. It is also interesting to observe in Fig. 13c that this material (originating at points in the interior of the fed sheet) ends up predominantly at the surface of the calendered sheet. These results point to an as yet unreported rearrangement of the material during the calendering of a sheet and opens up potentially interesting applications, e.g., in the case of the calendering of multilayer extrusions. It should be noted that 2D computational models (e.g., Agassant and Espy [11], Mitsoulis et al. [13]) have demonstrated the existence and extent of the recirculation flow occurring behind the nip region as well as within the melt bank. However, it is only a fully 3D model, including the presence of deforming side surfaces (spreading), that is capable of predicting the full extent of melt motion as it travels through the calenders. Experimental investigation and confirmation of this is pending, as is an investigation of how the presence of a melt bank might influence these flow patterns.

CONCLUSIONS

We have presented a three-dimensional computational analysis of the calendering process for Newtonian and non-Newtonian materials. The usefulness of earlier unidirectional and two-dimensional analyses notwithstanding, there is no denying the fact that the actual process is three-dimensional and that only a limited number of published studies has been concerned with its 3D features. Of these 3D features, we have focused on: (i) the change in the width of the calendered sheet as it passes through the calenders (spreading), (ii) the development of a 3D pressure profile, including a pressure gradient in the transverse direction, and (iii) the existence of a spiraling transverse flow pattern of the material in the melt feeding section, as a result of the combined action of recirculation and of the transverse pressure drop. The study reveals that the spreading is affected by (i) the entrance width of the fed sheet (see Fig. 5)--the lower the entrance width the higher the spreading, whereas for higher entrance width the spreading ratio declines asymptotically to unity, (ii) the feeding speed of the sheet (see Fig. 5)--the higher the feeding speed the higher the spreading and (iii) the thickness of the fed sheet--increasing the thickness the spreading also increases (see Fig. 6). Pressure profiles in the transverse direction (z-direction) and at different locations of the machine direction (x-direction) reveal the existence of rapid pressure decrease near the spreading sides and a mild pressure variation in the central region of the calenders. The aforementioned pressure development in the transverse direction, which is observed for both the Newtonian and power-law material, along with the recirculation occurring behind the nip region, forces the material to flow laterally, following a spiral path. Examination of the flow paths reveals a remarkable material rearrangement from the feeding section to the exit. Material from the skin of the fed sheet (locations close the to the calender surfaces--see Figs. 13a and b) follows the drag flow and exit forming the outer layers of the produced sheet. However, material from the inner part of the fed sheet follows a spiral flow pattern in the transverse direction, ending up at the surface in the region around the edges of the produced sheet.

NOMENCLATURE

x          x-coordinate
Z          z-coordinate
U          velocity vector
n          unit normal vector to a boundary
T          deviatoric stress tensor
D          rate of strain tensor
[II.sub.D] second invariant of the rate of strain tensor
p          Pressure
[mu]       Newtonian viscosity
[eta]      non-Newtonian viscosity
n          power-law index
T          Temperature
b          temperature sensitivity coefficient of viscosity
m          consistency index
R          calender radius
[x.sub.f]  detachment point of the sheet
[x.sub.i]  entry point of the sheet
H(x)       half-gap between the calendars
[W.sub.i]  width of the sheet at the entry point
[W.sub.f]  width of the sheet at the detachment point
[U.sub.R]  velocity of the calendars
[U.sub.i]  feeding speed of the sheet at the entry point
[q.sub.in] volumetric flow rate at the entrance
z(x)       side (free) surface boundary
[bar.u]    gap-averaged velocity in the x-direction
[bar.w]    gap-averaged velocity in the z-direction


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Nickolas D. Polychronopoulos, (1) Ioannis E. Sarris, (2) T.D. Papathanasiou (1)

(1) Department of Mechanical Engineering, University of Thessaly, Volos, Greece

(2) Department of Energy Technology, Technological Educational Institute of Athens, Athens, Greece

Correspondence to: T.D. Papathanasiou; e-mail: athpapathan@uth.gr

DOI 10.1002/pen.23719

Published online in Wiley Online Library (wileyonlinelibrary.com).
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Author:Polychronopoulos, Nickolas D.; Sarris, Ioannis E.; Papathanasiou, T.D.
Publication:Polymer Engineering and Science
Article Type:Report
Geographic Code:4EUGR
Date:Jul 1, 2014
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