挤出平缝口模优化设计

挤出平缝口模通常用于加工膜和片材，对产品厚度的一致性有很高要求。本文给出了结合聚合物成型模拟技术、设计灵敏度分析和数值优化技术的口模形状优化设计方法。以最小压力降为目标函数，口模出口处任意点的速率与出口已知速率相对误差的平方和小于容许误差为约束函数，口模形状参数为优化设计变量，给出了目标函数的表达式，推导了日标函数、约束函数对优化设计变量的灵敏度公式。利用灵敏度分析和基于梯度的优化算法即序列二次规划算法(SQP法)求解最优设计参数。通过算例表明，应用该法进行口模优化设计可以减小压力降和口模出口速率变化率。     

Simulation of coating flows with slip effects

This work is concerned with the numerical prediction of wire coating flows. Both annular tube‐tooling and pressure‐tooling type extrusion–drag flows are investigated for viscous fluids. The effects of slip at die walls are analysed and free surfaces are computed. Flow conditions around the die exit are considered, contrasting imposition of no‐slip and various instances of slip models for die wall conditions. Numerical solutions are computed by means of a time marching Taylor–Galerkin/pressure–correction finite element scheme, that demonstrates how slip conditions on die walls mitigate stress singularities at the die exit. For pressure‐tooling and with appropriate handling of slip, reduction in shear rate at the die exit may be achieved. Maximum shear rates for tube‐tooling are about one quarter of those encountered in pressure‐tooling. Equivalently, extension rates peak at land entry, and tube‐tooling values are one third of those observed for pressure‐tooling. With slip and tube‐tooling, peak shear values at die exit may be almost completely eliminated. Nevertheless, in contrast to the pressure‐tooling scenario, this produces larger peak shear rates upstream within the land region than would otherwise be the case for no‐slip. Copyright © 2000 John Wiley & Sons, Ltd.     

Upper bound analysis of thick wall tubes extrusion process through rotating curved dies

In this study, extrusion process of thick wall tubes through rotating curved dies is investigated by the method of upper bound. Total deformation region is divided into four deformation zones and a velocity field is developed for each deformation zone. The twist moments generated on container and mandrel surfaces are calculated and by equating them with the twist moment exerted by rotating die, the twisting length of tube inside the container is determined. Then, the internal powers, the powers dissipated on frictional and velocity discontinuity surfaces for a rigid-perfectly plastic material are evaluated and they are used in upper bound model. By optimizing the total power with respect to the slippage parameter between die and the tube material, the required relative extrusion pressure for a given process conditions and die angular velocity is determined. The results of finite element simulations are also presented and satisfactory agreement between the calculated and FEM results are demonstrated.     

Imaging and analysis of wave type interfacial instability in the coextrusion of low-density polyethylene melts

This report covers experimental studies and numerical modelling of interfacial instability in the bi-layer coextrusion flow of two low-density polyethylene melts. Melt streams are converged at an angle of 30° to a common die land. Melt stream confluence was observed in two coextrusion die arrangements. In one die design, which we term ‘bifurcated’ the melt stream is split by a divider plate in the die after being delivered from a single extruder. In the other design melt streams are delivered to a die from two separate extruders. In each die design melt flow in the confluent region and die land to the die exit was observed through side windows of a visualization cell. Velocity ratios of the two melt streams were varied and layer thickness ratios producing wave type interfacial instability determined for each melt for a variety of flow conditions. Stress and velocity fields in the coextrusion arrangements were quantified using stress birefringence and particle image velocimetry techniques.     

Multi-scale phenomena induced by fluoropolymer processing aids during the extrusion of linear-low density polyethylene

Using polymer processing aids (PPA) based on fluoropolymers not only reduces die pressure, but also postpones extrusion defects for linear polyolefins. The purpose of this work is to investigate the mechanisms explaining PPA efficiency. An experimental set-up was developed to study the PPA actions at different scales, based on a slit die with transparent walls and removable steel inserts, in line with a laboratory extruder. Original results are presented, correlating the evolution of die pressure, surface defects, velocity profiles measured by Laser Doppler Velocimetry and PPA die coating analyzed by Scanning Electronic Microscopy. During extrusion, PPA droplets and stretched aggregates appear first randomly on the whole die surface. Then, coating morphology progressively changes, leading to a network of interconnected and heterogeneous PPA streaks. Quantification of surface deposition shows that the PPA concentration grows towards the die exit, in concordance with spatially evolutionary wall slip. Surface defects elimination is correlated to the presence of discontinuous deposits at the die exit.     

Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method

Accurate prediction of extrudate (die) swell in polymer melt extrusion is important as this helps in appropriate die design for profile extrusion applications. Extrudate swell prediction has shown significant difficulties due to two key reasons. The first is the appropriate representation of the constitutive behavior of the polymer melt. The second is regarding the simulation of the free surface, which requires special techniques in the traditionally used Eulerian framework. In this paper we propose a method for simulation of extrudate swell using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional-step ALE technique is employed in which the Lagrangian phase of material motion and convection arising out of mesh motion are decoupled. In the first step, the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation besides allowing for natural evolution of the free surface with the flow. In the second step, mesh is moved in ALE mode and the associated convection of the variables due to relative motion of the mesh is performed using a Godunov type scheme. While the mesh is fixed in space in the die region, the nodal points of the mesh on the extrudate free surface are allowed to move normal to flow direction with special rules to facilitate the simulation of swell. A differential exponential Phan Thien Tanner (PTT) model is used to represent the constitutive behavior of the melt. Using this method we simulate extrudate swell in planar and axisymmetric extrusion with abrupt contraction ahead of the die exit. This geometry allows the extrudate to have significant memory for shorter die lengths and acts as a good test for swell predictions. We demonstrate that our predictions of extrudate swell match well with reported experimental and numerical simulations.     

Sharkskin instabilities and the effect of slip from gas-assisted extrusion

This paper is concerned with a polymer extrusion instability and the effect of introducing slip by means of a thin lubricating gas layer between the extrusion die wall and the flowing polymer melt. Gas-assisted extrusion (GAE) experiments were carried out using high-density polyethylene (HDPE) and linear low-density polyethylene (LLDPE) for a number of different gas injection die geometries. The stress distributions within the polymer melt were monitored during extrusion using flow birefringence. Polyflow numerical simulations were used to calculate the local stress concentrations in the melt at the die exit, as these were believed to be related to the occurrence of sharkskin. Simulations were also used to observe the effect of a full slip boundary condition as imparted by GAE. A key finding of the paper is that GAE in the parallel section of the die wall simply moved the local exit stress concentration upstream to the point of gas injection, and therefore did not reduce sharkskin. Simulations indicated that for correctly designed dies, the local surface stress concentration would be reduced. However, it was found experimentally that it was not possible to obtain a stable gas layer for this die design with upstream gas injection. A numerical investigation, involving simulations of varying levels of partial slip along the die wall, indicated an optimum level of slip where the stress concentrations were reduced. It is speculated that this is the reason that coatings such as PTFE, which impart a partial slip, can reduce sharkskin while GAE does not. The findings show that a controlled level of partial slip lowers the overall stress concentrations.     

挤压凹模强度的光弹性与有限元优化综合分析

本文介绍用光弹性试验测定挤压凹模工作时承受的内压力分布规律,在此基础上作了凹模的有限元优化设计,最后进行了光弹性实验验证.这种综合分析方法为提高模具的承载能力、节省模具材料和改进模具结构设计提供了有益的经验.     

The effect of die exit curvature,die surface roughness and a fluoropolymer additive on sharkskin extrusion instabilities in polyethylene processing

This paper reports experimental observations and numerical simulations relating to sharkskin extrusion instabilities for two different types of polyethylene, a metallocene high-density polyethylene (HDPE) and a linear low-density polyethylene (LLDPE). Experimental results are presented for both the effect of die exit curvature and die surface roughness for slit die geometry. Matching polyflow numerical simulations are also reported and are shown to be qualitatively consistent with experimental observations. The onset of the sharkskin instability is correlated with the magnitude of the stress concentration at the die exit, and is found to be sensitive to both the melt/wall separation point for a curved exit die, and the level of partial slip at the die wall. Additional observations on the effect of a fluoropolymer additive also support the sensitivity of the sharkskin instability to partial slip at the wall.     

Delayed die swell

The experiments reported here establish that there is a general critical condition associated with die swell which we call delayed die swell. This condition is defined by a critical speed which is the area-averaged velocity, the extrusion velocity, at the exit of the pipe when the swell is first delayed. The delayed swell ratio and delay distance first increase for larger, post-critical values of the extrusion velocity; then the increases are terminated either by instabilities or by smoothing. The maximum post-critical velocity at the pipe exit was always greater than the shear wave speed measured on the shear-wave-speed meter. The post critical area averaged velocity at the position of maximum swell before termination was always less than the shear wave speed. There were always points in the region of swelling where the ratio of the local velocity to the shear wave speed, the viscoelastic Mach number, was unity. The swelling of the jet is a nonlinear phenomenon which we suggest is finally terminated either by instability or when the variations of the velocity, vorticity and stress field are reduced to zero by the inward propagation of shear waves from the free surface of the jet. This propagation is generated by discontinuous “initial” data along χ in which the prescribed values of velocity at the boundary change from no-slip in the pipe to no-shear in the jet. The measurements raise the possibility that the delay may be associated with a change of type from supercritical to subcritical flow.     

A new analytical definition of the dead material zone for forward extrusion of shaped sections

This article aims to provide a new formulation for the analysis of the extrusion process for non-axisymmetric sections. The upper bound theorem has been used to obtain a generalized kinematically admissible velocity field. The geometry of the deforming region has been formulated considering variation of the dead zone size at different angular positions and three-dimensional curved surfaces have been employed to define the entry and exit surfaces of the deformation zone. Using this analytical method, extrusion of square, rectangular and L-shaped sections were analyzed and the effect of shape complexity on material flow and dead material zone (DMZ) formation under different conditions has been investigated. Physical modelling experiments and finite element analysis were carried out to reveal the capability of the proposed theoretical method.     

Constant velocity motion of a strip die on the boundary of a viscoelastic base

We consider a contact problem on the interaction of a rigid strip die with the boundary of a viscoelastic base. We assume that the die moves at a constant velocity on this boundary and is indented into it by a constant normal force. Friction in the die—surface contact region is neglected. The die base is corrugated in the direction perpendicular to the direction of motion. At the first stage, we determine the displacement of the base boundary due to the normal load applied to it. Then, at the second stage, we derive the integral equation of the contact problem for determining the contact pressure. At the third stage, we construct an approximate solution of this integral equation by using the modified Multhopp—Kalandiya method.     

Stability of plane poiseuille flow of a highly elastic liquid

The stability of fully developed pressure driven plane laminar flow of a Maxwell fluid has been studied using linear hydrodynamic stability theory. Elasticity is destabilizing in the inertial regime, but the flow is found to be stable to infinitesimal disturbances at low Reynolds numbers. This result contradicts previous calculations, which predicted a low Reynolds number flow instability at a critical recoverable shear of order unity. The previous calculations were carried out using less accurate numerical methods; the eigenvalue problem which must be solved is a delicate one, requiring sophisticated umerical techniques in order to avoid the calculation of spurious unstable modes.This work has direct bearing on the question of the mechanism of a low Reynolds number extrusion instability known as “melt fracture”. It is observed that the intensity of melt fracture increases with increasing die length for high density polyethylene, and it is therfore believed by some experimentalists that fully-developed die flow is unstable for this polymer above a critical recoverable shear. The analysis appears to be at variance with this interpretation of the experimental results.     

Effect of pressure-dependent slip on flow curve multiplicity

Various microstructural pictures for slip at polymer/solid interfaces lead to relations which have a region where multiple values of slip velocity are predicted for the same shear stress. This leads to the expectation of multivalued flow curves, which has been verified in specific cases by numerous researchers. We study the effect of pressure dependence on flow curve multiplicity using a simple multivalued slip relation to model the phenomena of hysteresis and spurt flow in polymer extrusion. A continuation technique is used to trace out the boundaries of the region of flow curve multiplicity as pressure drop and die length to diameter (L/D) ratio are changed. Results for Newtonian, shear thinning and viscoelastic constitutive equations show that, despite the multivalued nature of the slip model, multiplicity (and thus hysteresis) is absent at high L/D.  For the sake of completeness, we also carry out time-dependent simulations at constant piston speed taking fluid compressibility into account. These simulations show that oscillations in the pressure drop and exit volumetric flow rate result only if the system is operated in the multiplicity region of the steady state flow curve, in agreement with the results of similar simulations by researchers using various multivalued slip models without pressure dependence. The results demonstrate that a multivalued slip model does not guarantee multiplicity in the flow curve for the constant pressure drop operation, nor oscillations for constant piston speed operation. Received: 18 August 1997 Accepted: 30 March 1998     

Zylindrische Gleitlager mit nicht-Newtonschen Schmiermitteln

übersicht Um den Einflu? von nicht-Newtonschen Schmierstoffen auf die Vorg?nge im Zapfenlager zu untersuchen, wird die Schmiermitteltheorie auf Potenzgesetz-Fluide erweitert und auf zylindrische Gleitlager angewendet. Mit Hilfe einer numerischen Integration werden die Druckverteilung und die Geschwindigkeits-verteilung im Schmierspalt berechnet. Es zeigt sich, da? bei strukturviskosen ?len ein Druckabbau gegenüber Newtonschen Schmierstoffen stattfindet. In beiden F?llen wird gleiche Tragf?higkeit im Lager vorausgesetzt.

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Summary In order to study the influence of non-Newtonian lubricants as far as the flow field in journal bearings is concerned, the hydrodynamic lubrication theory is extended to power-law fluids and it is applied to cylindrical bearings. After having carried out a numerical integration procedure the pressure distribution and the velocity distribution are calculated for the lubricating film. It can be shown that for pseudoplastic lubricants the pressure peak will be decreased in comparison to Newtonian fluids. In both cases the same load of the bearing is assumed.

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Diese Arbeit wurde mit Unterstützung der Deutschen Forschungsgemeinschaft durchgeführt. Herrn Dipl.-Ing. A. Waterstraat, der mich bei der numerischen Auswertung tatkr?ftig unterstützt hat, sei hier gedankt     

Stick-slip flow of high density polyethylene in a transparent slit die investigated by laser Doppler velocimetry

The oscillating flow instability of a molten linear high-density polyethylene is carefully studied using a single screw extruder equipped with a transparent slit die. Experiments are performed using laser Doppler velocimetry in order to obtain the local velocities field across the entire die width. At low flow rate, the extrusion is stable and steady state velocity profiles are obtained. During the instability, the velocity oscillates between two steady state limits, suggesting a periodic stick-slip transition mechanism. At high flow rate, the flow is mainly characterized by a pronounced wall slip. We show that wall slip occurs all along the die land. An investigation of the slip flow conditions shows that wall slip is not homogeneous in a cross section of the slit die, and that pure plug flow occurs only for very high flow rates. A numerical computation of the profile assuming wall slip boundary conditions is done to obtain the true local wall slip velocity. It confirms that slip velocities are of the same order of magnitude as those measured with a capillary rheometer.     

Visual observation of development of sharkskin melt fracture in polybutadiene extrusion

We studied the sharkskin melt fracture phenomena of polybutadiene (PBD), which exhibits similar flow properties and instabilities to linear low density polyethylene (LLDPE). The advantages of using PBD are that it exhibits larger distortions and slower development due to its high viscosity. By using a video camera with close-up lenses focused at the die exit, we observed the development of sharkskin in profile. It was shown that the sharkskin melt fracture develops with the cohesive failure at the die exit due to a peeling of surface layers as described by Howell and Benbow (1962). Oddly enough, this mechanism is similar to that proposed by Ovaici et al. (1998) for the extrusion of chocolate. A soap solution coating around the die exit to induce a slippery interface eliminated the sharkskin fracture, similar in the effect of fluoro-elastomer coating with LLDPE. Based on our visual observations, we qualitatively modeled the sharkskin fracture by modifying Cogswell‘s (1977) idea on exit stretching at the die exit and the force balance of Ovaici et al. on the ring formation of the chocolate extrusion. Received: 22 May 1998 Accepted: 31 August 1998     

含油轴承表面供油行为与自润滑机理分析

含油轴承是一类重要的减摩自润滑零件,轴承基体中油液的渗流特性对表面润滑性能有重要影响. 建立含油轴承孔隙渗流与表面油膜润滑的耦合力学模型,分析含油轴承系统中油液的渗流特性,探讨轴承表面油液的供油行为与自润滑机理. 结果表明:在含油轴承的收敛区内同时存在周向旋转流、径向伸缩流和法向渗析流,油液在各方向上的流动状况由该向流体压力梯度决定;受油膜压力影响,油液在接触区向多孔基体渗入,在接触区入口部位向多孔表面析出,由此构成了油液渗入和析出的闭环速度流线,增强摩擦界面间的泵吸效应. 油液在法向上的渗析速度随中心膜厚增加而减小,随转速升高而增大,渗析速度越大,对泵吸效应的增强作用越显著,接触区入口的油液也更易进入摩擦界面,保障含油轴承的良好自润滑效果. 研究结果对揭示含油轴承的供油自润滑机理具有重要意义.      

随动耦合变阶梯径向滑动轴承动力特征及稳定性的研究

用有限差分法循环迭代求解了随动耦合变阶梯结构径向滑动轴承油膜压力的雷诺方程和两阶段油膜耦合变阶梯结构的流量控制方程。在分析轴承油膜压力形成机量及静力特性的基础上,采用对位移和速度的小扰动法计算了轴承的动力特性系数,考察了运转参数对这种轴承承载特性、动力特性系数、等效刚度、界限涡动比以界限失稳转速的影响,结果发现合理地选择设计参数可以使这种轴承具有较好的静力特性、动力特性和稳定性。     

The extrusion of a model yield stress fluid imaged by MRI velocimetry

Extrusion tests were performed by forcing a well-characterized model yield stress fluid from a cylindrical cartridge through various cylindrical extrusion dies using a variety of different piston velocities. In this study the Bingham number within the die ranged from 0.1 to 10. MRI techniques allowed for the non-invasive determination of the local velocity within the extruded material in the range [0.015; 20 mm s^?1]. The velocity profile within a very long die was determined by MRI and agreed very well with the analytical results for the flow of a Herschel–Bulkley fluid within a conduit using parameters determined from independent rheometrical tests, validating both the rheological approach and the accuracy of the MRI techniques. Although the velocity was determined by MRI in the upper and lower zones separately, the intersection of these zones showed great agreement, providing an entire view of the extrusion process. In the range of Bingham number studied, the velocity field for a given contraction ratio appeared similar when scaled by the piston velocity, with a dimpled acceleration zone above the die and lateral dead zones varying negligibly with the piston velocity. For a further analysis the experimental results were compared with the results of numerical simulations. Finite element simulations using an elastic solids model were performed to provide this comparison. It was found that this model did well in representing the characteristics of extrusion flow seen in the experiments; an aspect that was not present in the biviscous simulations. The MRI results show that for the range of values studied, both the piston velocity and the contraction ratio have little effect on the characteristics of the flow, including the size and location of the apparent dead zones. It was found that with an appropriate scaling the central, longitudinal velocity follows a master curve. A decreasing contraction ratio, on the other hand, appears to increase the size of the weak velocity region, in contrast with the simulation results.