Effect of wall slip on the uniformity of flow from sheeting extrusion dies

A theoretical study for analyzing the uniformity of flow from sheeting extrusion dies is presented. In this study it is assume that a slip condition exists at the wall of the die, the magnitude of slip velocity is proportional to the shear stress at the wall, the flow is isothermal and steady state, and a power law model is valid for viscosity. Two extrusion dies, T-dies and coat-hanger dies, are examined. The flow uniformity at the exit of the die is calculated and compared with that for a nonslip analysis. The discrepancies between the slip and nonslip models imply that the wall slip condition induces a significant nonuniform flow distribution. Traditional design criticism based on the nonslip model are invalid for flow with the wall slip condition, and it is necessary to increase the length of the die land to even the flow distribution at the exit of the die.     

Inertia and gravitational effects in extrusion dies for non-Newtonian fluids

An analysis is presented which allows the sheet or film die designer to estimate when inertial and gravitational effects are important. General theoretical equations are developed for end fed dies with arbitrary variation of the cavity cross sectional shape, cavity taper, slot length, and gap over the width. The method assumes viscous flow and a two dimensional approximation for the cavity flow. For fluid flow properties, it is assumed only that the apparent viscosity is a single valued function of the shear rate. In the important special case of constant die geometry and power law fluids, three dimensionless numbers plus the power law index are the parameters controlling the uniformity of flow from the die. Results are presented that illustrate when die orientations with respect to gravity and when fluid inertia are important. When they are not, simple expressions for die inlet pressure and uniformity index are given.     

Optimization‐based design of polymer sheeting dies using generalized Newtonian fluid models

A polymer sheeting die design methodology is presented, which integrates finite element flow simulations, numerical optimization, and design sensitivity analyses to compute die cavity geometries capable of giving a near‐uniform exit velocity. This work extends earlier die design methods to include generalized Newtonian fluid (GNF) models that represent the shear‐thinning behavior of polymer melt. Melt flow computations and design sensitivity analyses are provided using the generalized Hele‐Shaw flow approximation with isothermal power‐law, Carreau‐Yasuda, Cross, Ellis, and Bingham fluid models. The nonlinear equations for die cavity pressure are solved using the Newton‐Raphson iteration method and design sensitivities are derived with the adjoint variable method. The die design method is applied to an industrial coat hanger die, in which a design parameterization is defined that allows for an arbitrary gap height distribution in the manifold of the die. In addition, die performance is assessed and compared for power‐law and Carreau‐Yasuda fluid flow over a range of die operating conditions. Pareto optimal die designs are also considered in this study. POLYM. ENG. SCI., 45:953–965, 2005. © 2005 Society of Plastics Engineers     

The influence of power‐law rheology on flow distribution in coathanger manifolds

Coathanger dies are effective in delivering uniform flow if a polymer melt; however, when the fluid flow index varies from the design values, the flow is not uniform. Although mechanisms such as die lip adjustments have been effective tools for adjusting flow profiles, the issue of a variable flow index has not been fully addressed at the design stage. An analytical solution, based on the assumptions present in the 1‐D design equation, has been developed for the flow distribution in a coathanger manifold. This solution determines the flow distribution for a power‐law fluid with a flow index n* in a manifold designed for a separate flow index n*. From this solution, a uniformity index and a critical design angle are defined. The critical design angle is the angle at which the local derivative of the uniformity index with respect to n* approaches a maximum (for n* < n) or a minimum (for n* > n) as a function of the design angle. The critical design angle is independent of n and is presented as a function of the manifold aspect ratio.     

Theoretical Analysis on Flow of Polymer Melts in Screw Die

Abstract

Flow of polymer melt in screw dies is theoretically analyzed by the broken section method with the uniformity of the extrudates. An isothermal, laminar and steady state power law fluid is assumed. The analysis is discussed in two parts, i.e., screw flow and die slit flow. A way of computer calculation by means of a method of iteration is presented by considering volume balance between screw flow and slit flow.

An ideal screw die is one in which pressure distribution is constant along the screw axis, i.e., the shape of the die slit is constant along the axis and the screw is such that the depth of screw channel decreases almost linearly.     

Flow behavior of polypropylenes with different molecular structures in a coat‐hanger die

An experimental investigation of the flow behavior of three polypropylene melts with different molecular structures during extrusion through a coat‐hanger die is presented. Two linear and one long‐chain branched material, rheologically characterized in shear and elongation, were investigated. Using laser–Doppler velocimeter measurements of the velocity profiles across the gap height were performed at five various locations along the die. The uniformity of the velocity distribution along the die has been assessed using the maximum velocities v₀ of the corresponding velocity profiles across the gap. The velocity distribution along the die changes with throughput and temperature. Regarding the rheological properties, it was found that the power‐law index of the viscosity as a function of shear rate has a decisive influence on the uniformity of flow but that the pronounced strain hardening in elongation typical of the long‐chain branched polypropylene is not reflected by the velocity distribution along the die. POLYM. ENG. SCI., 2012. © 2012 Society of Plastics Engineers     

Analysis of the fluid–structure interaction in the optimization‐based design of polymer sheeting dies

A polymer‐sheeting‐die‐design methodology is presented that integrates a simulation of the polymer melt flow and die‐cavity deformation with numerical optimization to compute a die‐cavity geometry capable of giving a nearly uniform exit flow rate. Both the polymer melt flow and sheeting‐die deformation are analyzed with a general‐purpose finite‐element program. The approach includes a user‐defined element that is used to evaluate the purely viscous non‐Newtonian flow in a flat die. The flow analysis, which is simplified with the Hele–Shaw approximation, is coupled to a three‐dimensional finite‐element simulation for die deformation. In addition, shape optimization of a polymer sheeting die is performed by the incorporation of the coupled analyses in our constrained optimization algorithm. A sample problem is discussed to illustrate the die‐design methodology. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 103: 3994–4004, 2007     

Model for the outer cavity of a dual‐cavity die with parameters determined by two‐dimensional finite‐element analysis

A coating die forms liquid layers of uniform thickness for application to a substrate. In a dual‐cavity coating die an outer cavity and slot improves flow distribution from an inner cavity and slot. A model for axial flow in the outer cavity must consider the ever‐present cross flow. A 1‐D equation for the pressure gradient for a power‐law liquid is obtained as a small departure from a uniform flow distribution and no axial flow. The equation contains a shape factor dependent on cavity shape, Reynolds number, and power‐law index. The shape factor for five triangular cavity shapes is obtained by finite‐element analysis and correlated for application to die design up to the onset of flow recirculation which arises at the junction of the cavity and outer slot. The performance of the combined cavity and slot is considered and the most effective design determined. © 2017 American Institute of Chemical Engineers AIChE J, 64: 708–716, 2018     

A systematic approach for the design of a spiral mandrel die

An effective design strategy is presented for the determination of optimal flow channel geometry of a spiral mandrel die, which can produce annular, thin, and uniform products of polymer melt. This strategy includes two steps: First, a two-dimensional flow simulation associated with the Taguchi approach is applied to search several sets of die geometric parameters, from which process the flow uniformity is assessed. Second, the optimum one parameter set selected by further considering total pressure drop, mixing degree and residence time distribution (RTD). The strategy is illustrated through the design of a spiral mandrel die with four spirals traveling a full 360°, and the viscosity of polymer melt is assumed to be power law model. The results indicated that the design criteria such as high flow uniformity, low pressure drop, good mixing degree and narrow RTD are difficult to attain simultaneously. For a particular polymer melt, there exists a design window for the geometric parameters; within the window the geometric parameter set may satisfy most of the design criteria.     

Three-dimensional finite element analysis of polymeric fluid flow in an extrusion die. Part I: Entrance effect

The Galerkin finite element method has been applied to study the three-dimensional flow field of power-law fluids inside an extrusion die. Two inlet designs, i.e., center-fed and end-fed, have been considered. The effects of inertial force as represented by the Reynolds number Re, inlet geometry, and the power-law index n on lateral flow uniformity and vortex formation in the entrance region have been examined. A flow visualization technique has been carried out to experimentally verify the theoretical prediction of the three-dimensional flow field inside a die. It has been found that increasing Re or decreasing n will deteriorate flow uniformity. Depending on the direction of the inlet jet stream, the inertial force may create a flow peak in the central region of a center-fed die, or the maximum flow rate will appear close to the end of the die for an end-fed die. For highly shear-thinning fluids, lower flow rates are always observed close to the end of the dies. It is concluded that creating a plug flow in the inlet tube of the extrusion die is advantageous for both center-fed and end-fed designs.     

Elongation viscosity estimates of linear low-density polyethylene/ low-density polyethylene blends

The elongational viscosity (EV) of two series of linear low-density polyethylene/low-density polyethylene blends was estimated using an entry flow analysis. The difference, t ? n, between the power law index t of the elongational viscosity and the power law index n of the viscosity, is proportional to the LDPE content for both series of blends investigated. Comparison of the EV of the LLDPE/LDPE blend estimated from the analysis of the flow into an orifice die to the EV value estimated from the analysis of the flow into a capillary die with a flat entry, showed that the difference in geometry had little effect on the EV estimates.     

Analysis on Composite Flow of Polymer Melts in Sandwich Sheeting Die

Abstract

A broken section method for analyzing the two-phase flow of polymer melts in a sandwich sheeting die under a pressure gradient, is presented. The method assumes that the laminar flows are isothermal and the incompressive materials are power-law fluids. Two polymer melts having different flow behaviors are extruded from two extruders respectively into a coaxial manifold and simultaneously, flow into slits perpendicular to the manifold axis, and then contact with each other in the final slit of the sheeting die for laminating the sandwich sheet. The immiscible interface of these adjacent flows in this final slit is determined by a method of variation. Simultaneous equations in many broken sections are analyzed by computerized successive approximation. Uniformity is defined for both core and skin layers.     

Geometry design of a coat-hanger die with uniform flow rate and residence time across the die width

This article provides an approximate but sufficiently practical method of geometry design of a coat-hanger die having both uniform flow rate and residence time across the full width of the die. Such a die is often needed when a heat-sensitive resin such as poly(vinyl chloride) is sheeted, because different residence times across the die width are apt to cause a sheet defect in the transverse direction of the sheet. Although some assumptions were made facilitating mathematical analysis, an application of this method to a sheeting die 1 m wide gave good uniformity of flow rate and residence time.     

Experimental studies on extrudate swell behavior of PS and LLDPE melts in single and dual capillary dies

Extrudate swell behavior of polystyrene (PS) and linear low‐density polyethylene (LLDPE) melts was investigated using a constant shear rate capillary rheometer. Two capillary dies with different design configurations were used, one being a single flow channel and the other being a dual flow channel. A number of extrudate swell related parameters were examined, and used to explain the discrepancies in the extrudate swell results obtained from the single and dual flow channel dies, the parameters including output rate and output rate ratio, power law index, wall shear rate, wall shear stress, melt residence time, pressure drop induced temperature rise, flow channel position relative to the barrel centerline, and the flow patterns. It was found in this work that the power law index (n value) was the main parameter to determine the output rate ratio and the extrudate swell between the large and small holes for the dual flow channel die: the greater the n value the lower the output rate ratio and thus decreased extrudate swell ratio. The differences in the extrudate swell ratio and flow properties for PS and LLDPE melts resulted from the output rate ratio and the molecular chain structure, respectively. The extrudate swell was observed to increase with wall shear rate. The discrepancies in the extrudate swell results from single and dual dies for a given shear rate were caused by differences in the flow patterns in the barrel and die, and the change in the melt velocities flowing from the barrel and in the die to the die exit. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 87: 1713–1722, 2003     

Steady Flow of Power Law Fluids across a Circular Cylinder

The momentum equations describing the steady cross‐flow of power law fluids past an unconfined circular cylinder have been solved numerically using a semi‐implicit finite volume method. The numerical results highlighting the roles of Reynolds number and power law index on the global and detailed flow characteristics have been presented over wide ranges of conditions as 5 ≤ Re ≤ 40 and 0.6 ≤ n ≤ 2. The shear‐thinning behaviour (n < 1) of the fluid decreases the size of recirculation zone and also delays the separation; on the other hand, the shear‐thickening fluids (n > 1) show the opposite behaviour. Furthermore, while the wake size shows non‐monotonous variation with the power law index, but it does not seem to influence the values of drag coefficient. The stagnation pressure coefficient and drag coefficient also show a complex dependence on the power law index and Reynolds number. In addition, the pressure coefficient, vorticity and viscosity distributions on the surface of the cylinder have also been presented to gain further physical insights into the detailed flow kinematics.     

Computer-aided geometric design of the preform dies for flat film extrusion through 3-D finite element flow analysis

A rational computer-aided design procedure is presented for determining the optimum flow channel geometry of a flat film die, which yields a minimum pressure drop and produces a uniform transverse flow rate distribution. The three-dimensional die surface is generated by analytic expressions that represent a dumbbell-like contour. The die surface may be specified by several geometric parameters. The length of the transition zone turns out to be the controlling parameter. Because of the complicated geometric boundary, it is not possible to optimize the flow channel geometry explicitly; instead, a computer trial procedure is employed. The numerical computation is based on an isothermal three-dimensional flow model, which assumes power law behavior for the polymer melts. The calculated results indicate that for a particular polymer and a particular aspect ratio of slit, there may exist an optimum transition length from which the flow channel geometry of a flat film die may be uniquely defined.     

Flow distribution in spiral mandrel dies

An analysis of flow of a power law fluid in a spiral mandrel die is presented. The analysis is applied to study the effect of various die design parameters on the flow distribution at the end of the spiral mandrel section. Three variables that have a very strong effect on the flow distribution are the number of grooves, the initial clearance, and the groove helix angle. The distribution is improved by increasing the number of grooves, by using a non-zero initial clearance, and a relatively small helix angle. Two more variables that have a significant effect on the flow distribution are the taper angle and the initial groove depth. The optimum taper angle was found to be between 1 and 3 degrees. The distribution uniformity improves with the initial groove depth, while the pressure drop reduces at the same time.     

发射药药料离模膨胀和流动均匀性模拟及在模具设计中的应用

为了分析挤出成型过程中模具结构参数对七孔硝基胍发射药离模膨胀率及流动均匀性的影响规律,采用计算流体力学方法,对挤出成型过程进行模拟计算,讨论了模具各结构参数重要性的主次关系;对七孔发射药制备模具进行了结构优化,并进行了实验验证。结果表明,模具收缩角对膨胀率和药料出口速度均匀性的影响最大,压缩段高度次之,成型段长度的影响最小。模具优化后流道出口端速度分布均匀性提高36.53%,表明该模拟计算的可靠性与实用性。     

New slit die rheometer: some results with a butadiene-styrene block copolymer

The slit die rheometer described, is a modification of the extrusion part of a Zwick rheometer (piston apparatus with constant applied pressure). The new slit has a variable depth, and pressure-temperature transducers are flush mounted at the die wall. The duct is 85 mm long and has a rectangular cross-section, 10 mm wide; four depths are available (0.50, 0.75, 1.00 and 1.50 mm). The hydraulic system gives pressures ranging from ～80 to 430 kg/cm², and the volumetric flow rate is determined by collecting extrudate samples. Some results are presented on Solprene 415, a butadiene-styrene block copolymer. Parabolic pressure profiles are accepted along the longitudinal distance of the die. A method is proposed to linearize the pressure gradient, in order to calculate the wall shear stress. A flow curve is obtained which is not strictly described by a power law relation.     

基于灵敏度分析的挤出平缝口模优化设计

给出了以直接微分法计算稳态非线性系统设计灵敏度的方法。在基于挤出平缝口模熔体流动机理分析的基础上，建立了口模内熔体流动数学模型。并把设计灵敏度分析法和数值成型模拟技术相结合，运用于口模流道优化设计中，降低挤出平缝口模的出口速率变化率，从而改善了产品质量。通过实例验证，提出的口模优化设计方法非常有效。