Ellis Model
An analytical solution has been developed for on-line simulation of non-Newtonian and non-isothermal viscous flow in real time polymer processing. The modeling of the non-Newtonian viscous flow utilizes a modified Ellis model that expresses the viscosity as a function of the shear stress; the modeling of the heat transfer utilizes a Bessel series expansion to include effects of heat conduction, heat convection, and internal shear heating. The resulting simulation is suitable for inclusion in real time process controllers requiring sub millisecond response. Numerical verification indicates that the flow rate predictions of the described analysis compare well with the results from a commercial molding simulation. However, empirical validation utilizing a design of experiments for an injection molding process indicates that the described analysis is qualitatively useful but does not possess sufficient accuracy for quantitative process and quality control.
In this work, the feed system in injection molds is modeled as a flow network that consists of cylindrical and annular elements of varying lengths and diameters. For both element types, an Ellis model is utilized to provide reasonable modeling of the viscosity across both the power law and Newtonian regimes:
where t1/2 is the shear stress at which the viscosity is 50% of the Newtonian limit, h0, and a-1 is the slope of the viscosity in the power law regime. For the purpose of modeling the temperature dependence of the rheology, the zero shear rate viscosity is modeled with Arrhenius type dependence as:
The following figure graphs the viscosity behavior of the modified Ellis model and a cross model with WLF temperature dependence for a neat polypropylene resin (Borealis BH340P) with the coefficients listed in Tables 1 and 2. It is observed that the modified Ellis model with four coefficients closely tracks WLF model with six coefficients; the mean absolute percentage error (MAPE) between the two models is 2.95%. The Ellis model does diverge from the WLF model at very low shear rates (too abruptly transitioning to the Newtonian regime), at very high shear rates (underestimating the shear thinning effect of the polymer melt), and for very broad temperature ranges (across which the Arrhenius temperature dependence and constant t1/2 can induce significant errors). For this processing regime (spanning 20 °C and 10,000 1/sec), however, the modified Ellis model is a close approximation to the rheological models used in sophisticated numerical simulations. Furthermore, the modified Ellis model is a vast improvement over Newtonian and power law constitutive models while still retaining an analytical solution for the flow conductance.
Derivation for Rods
Consider the viscous flow in a rod. For a viscosity with the described Ellis model behavior, the relationship between flow rate and pressure gradient can be modeled as:
where Q is the volumetric flow rate, R is the rod radius, DP is the pressure drop, L is the length of the rod, and other coefficients are from the Ellis model fitting.
The shear stress and shear rate are, respectively:
Integrating the shear heating across the radius leads to the viscous dissipation power:
The heat conduction between the polymer melt with an initial temperature Tmelt and the walls of a cylindrical feed system with constant temperature Twall is modeled as a boundary value problem. With a constant mold wall boundary condition instead of a Biot boundary condition, the transient temperature distribution after a time step Dt is solved as:
where a is the thermal
diffusivity of the polymer melt, J0 and J1 are
Bessel functions of the first kind, and bn
is the eigenvalue. The bulk temperature of the polymer melt,
,
can be estimated as:
Neglecting higher order
terms and applying the boundary condition
for cylinders,
from which
The temperature rise due to viscous heating is:
which indicates that the change in the bulk temperature is:
The bulk temperature of the melt considering both viscous heating and heat conduction in a cylinder with Ellis flow is estimated for a pressure drop DP as:
A similar solution can be developed for channels and annuli.
Results
The described analytical solution was implemented in a spreadsheet with the pressures at the inlet and the entrances to the cavities provided by the simulation as boundary conditions. The melt pressure, melt temperature, and melt flow rates predicted by the commercial simulation and the described analytical solution are plotted in the following figure as a function of time. As indicated in Figures (b) and (c), the polymer melt at the inlet remains at a uniform 230 °C and flows near a constant volumetric rate of 22 cc/sec.
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(a) |
(b) |
(c) |
Figure (a) indicates that the melt pressure at the inlet quickly climbs from 0 to 8 MPa as the polymer melt begins to flow, then increases almost linearly to 16 MPa as the melt propagates throughout the cavities. The melt pressures at the entrances to the small and large cavities are very different, however. Since the small cavity is one half the thickness of the large cavity, the melt pressure at the entrance to the small cavity increases more rapidly than the melt pressure at the entrance to the large cavity. Since the small cavity has a shorter flow length, however, the small cavity fills first. As a result, the melt pressure in the small cavity approaches the inlet pressure at 0.6 seconds after the small cavity has filled. By comparison, the melt pressure at the entrance to the large cavity increases linearly to continue the propagation of the melt until the end of filling. The bulk temperatures predicted by the simulation and the analytical solution vary somewhat but demonstrate the importance of non-isothermal modeling.
As indicated in Figure (b), the bulk temperature of the polymer melt being conveyed to the large cavity is increasing by approximately 2 °C. This temperature rise is due to the higher flow rates and internal viscous dissipation of the polymer melt, and has been observed to be even more significant in molding processes utilizing higher flow rates and/or smaller runner systems with greater pressure drops. A smaller rise in the bulk temperature is witnessed for the smaller cavity with lower flow rates and shear rates occurring in the feed system. In general, the analytical solution provides a reasonable estimate of the bulk temperature as a function of the applied pressure boundary conditions. The 0.5 °C difference between the simulation and analytical solution for the large cavity is likely due to either higher order terms that were dropped in the analytic solution, or an under estimation of the melt viscosity at high shear rates. It is unclear why the commercial simulation would indicate a significant decrease in the bulk temperature of the polymer melt entering the small cavity at the end of filling, though it is speculated that the simulation is attempting to model heat conduction from the hot runner to the cooler mold wall.
Figure (c) plots the flow rates estimated by the simulation and analytical solution as a function of time. It is observed that the flow rate to the small cavity is lower than the flow rate to the larger cavity. The flow rate in the small cavity decreases as the melt propagates in the smaller cavity until the cavity is full at 0.6 seconds. Once the small cavity is completely full, the flow rate to the large cavity increases and maintains 22 cc/sec until the end of filling. It is observed that the analytical solution exhibits the same behavior as the simulation: there is a systematic error between the two solutions of approximately 14%. In general, the analytical solution under predicts the flow rate since the Ellis model tends to under predict the shear thinning of the polymer melt at high shear rates. It should be noted that it is possible provide an improved prediction by proportioning the inlet flow rate according to the ratio of the predicted flow rates for the small and large cavities:
When this correction was applied to the analytical solution, the mean average percentage error was reduced from 14% to 0.99%. This result indicates that the presented analytical solution may provide powerful prediction capabilities suitable for integration with real time process controllers, as is next investigated.
Spreadsheet Implementation
This spreadsheet provides the details of the above results. The yellow section is the input pressures, either from a simulation (such as Moldflow) or process data from melt pressure measurements. The green section are the hot runner (HR) parameters and the material constants for the PP. The material constants can be obtained from fitting the Ellis model to material data or the Cross-WLF model. The remainder of the spreadsheet provides the calculations and graphs. Please note that more complex flow geometries require the formation and solution of a flow conductance matrix, which was not implemented in this spreadsheet. Access to this spreadsheet may also be provided by clicking on the spreadsheet figure below.
