1. Numerical Computation of Neck-in and Edge Beading in Film Casting Tyler Birkel and Jessica Eckles, University of Wisconsin-Eau Claire Faculty Advisors: Robert Langer and Mohamed Elgindi, Department of Mathematics Funded by: UW System Applied Research Grant 2004 and Extrusion Dies Industries, LLC, Chippewa Falls, WI We model the edge bead width by where and α and β are parameters to be determined experimentally. According to [2], the edge bead has an elliptical shape, and the film thickness and edge bead height are related by Conservation of volume yields. The boundary value problem is difficult or impossible to solve analytically. To overcome this we use the Runge-Kutta Method to find an approximate solution of the initial value problem. The solution of the initial value problem depends on the value of E. Then we adjust the value of E until the boundary condition v(1 ) = DR is satisfied. The algorithm is: We implemented this algorithm in Maple. Some representative solutions are shown in the table below. Introduction Thin plastic films are produced by a process called film casting. The melted polymer is extruded through a uniform narrow slit onto chilled roller which stretches the film to a desired thickness. Research Objective To predict accurately the width of the film and the size of the edge bead at roller. x = distance from the die to the roller v = velocity w = width A = ratio of the distance between the die and the roller to the width of the die E = measure of the pulling force in the film DR = “draw ratio” = v(1 )/ v(0 ) = v(1 ) (Recall: v(0 ) = 1.) Variables Choose initial guess for E While | The velocity at the roller minus the Draw Ratio | > tolerance do Compute approximate solution of Initial Value Problem Adjust E End Mathematical Model for Neck-in Numerical Solution Neck-in Final Product To model the edge bead we introduce additional variables As the film is stretched two undesirable phenomena occur – neck-in and edge beading. Neck-in is a narrowing of the width of the film as illustrated below. The excess material created from the neck-in builds up on the edges, forming edge beads which can be several centimeters wide and up to five times as thick as the film at its center. The edge beads must be trimmed from the film and are often wasted. Neck-in can be observed in this photograph of honey being poured through a narrow slit. Modeling Edge bead The principal cause of edge beading and neck-in is the Edge Stress Effect, due to the difference in the ways the edge and center are stretched. Other causes are surface tension and die swell. We consider only the Edge Stress Effect in this project. w e = edge bead width h e = edge bead height h f = film thickness WE = Weissenberg number Combining these two equations and solving for gives Distance to Roller Draw Ratio 10.1 w(1)=.93584, v(1)=10 h(1)=.10665.5 w(1)=.71917 v(1)=10 h(1)=. 13904 w(1)=.93738, v(1)=30 h(1)=. 03556 w(1)=.71465 v(1)=10 h(1)=. 04664 30 where a ( x ) is determined by the quadratic equation Boundary Value Problem w(0) = 1, v(0) = 1, v(1) = DR, and E must be determined along with w(x ) and v(x ) so that v(1 ) = DR is satisfied [1]. The film width and velocity are modeled by the following boundary value problem where We incorporated this into the Maple program for neck-in. The result is illustrated here. (Note: Draw Ratio = 10, and Distance to Roller = 0.5) References 1.Beaulne, M. and E. Mitsoulis, Numerical Simulation of the Film Casting Process, International Polymer Processing XIV, (1999) 3 2. Dobroth, T. and Lewis Erwin, Causes of Edge Beads in Cast Films, Polymer Science and Engineering, Vol. 6, No. 7, 1986 Future Research Question Is it possible to modify the die geometry to reduce or eliminate neck-in and edge beading? Contact Information: Tyler Birkel, birkeltj@uwec.edu; Jessica Eckles, ecklesjj@uwec.edubirkeltj@uwec.eduecklesjj@uwec.edu