Tadmor melting model

Figure 6.11 Schematics of the solid bed just prior to complete melting (a) the solid bed is pushed to the trailing flight with the Tadmor melting model and barrel rotation physics, and (b) the solid bed is a thin plate and positioned as in the diagram (screw rotation and observation). The cream color represents molten resin...

The new melting model presented in this section qualitatively fits the experimental data observed by many previous researchers. Like the Tadmor and Klein model [8], this model is based on simplistic assumptions and linear mathematics for the melt films. The new model, however, does not require the reorganization of the solid bed like the Tadmor and Klein model. Furthermore, the new model allows viscous dissipation and melting in all four melt films, and does not restrict all melting to the Zone C film. Melting in the Zone D melt film becomes highly important when the boundary conditions are switched from barrel rotation to the actual conditions of screw rotation. 

Modihcations of this model, including a nonlinear temperature prohle in the melt him, channel curvature effects, and an approximate method to account for the flight clearance effect, are presented by Tadmor and Klein (1), together with expressions for power calculations. Numerous other improvements of the melting model have been suggested in the literature (33,41 -6). A detailed discussion of these, however, is beyond the scope of this text. 

S. D. Lipshitz, R. Lavie, and Z. Tadmor, A Melting Model for Reciprocating Screw Injection Molding Machines, Polym. Eng. Set, 14, 553 (1974). 

The generally accepted melting model for SSEs is the one described by Tadmor and Gogos [6] wherein the passive solid bed moves axially downstream being gradually compressed in the transition sections such that a melt film forms at the barrel wall by a combination of direct heat transfer through this wall and friction of the unmelted... 

The dominant melting model (Fig. 5.23 ) was initially developed by Tadmor. At the beginning of the transition zone, a layer of molten... 

After Tadmor s first publication on melting, many others started to study melting in single screw extruders. As a result, many publications appeared in the technical literature in the 1970s and beyond. This section will review the various melting models proposed and analyze their advantages and disadvantages. 

Results from computer simulations for a HDPE polymer show that the Maddock/ Tadmor melting mechanism occurs in the early stages of melting. At the beginning of the compression section, the solid bed tends to become totally encapsulated, while towards the end of the melting, the solid bed breaks apart into several pieces. At the time of publication of the paper by Viriyayuthakorn and Kassahun, no direct comparison was available between theoretical predictions and experimental results. Therefore, no statement can be made about the accuracy of the predictions. However, regardless of the accuracy of the predictions, this model provides new capabilities that no doubt will prove very useful in future work on the analysis of plasticating extrusion. 

There are lots of different models for the description of solid conveying and melting in standard extmders. The most important melting models are for sure the one by Tadmor [3] and the further developed version by Potente [4], whieh allow the description pseudoplastic flow behaviour also. 

The new concepts presented here remove the literature assumption that the solid bed reorganizes, and it allows melting at all solid bed interfaces. The Tadmor model allows melting at only a single interface, that is, as specified by the melting velocity Filni C- The model presented here predicts melting at all four interfaces two in the y direction and two in the x direction. 

It is now useful to examine the melting energy at the four solid interfaces for this new model and for the historical Tadmor model [8]. The dissipation data from the simulations are summarized in Table 6.1 for a PE resin with a viscosity of 880 Pa-s. Examination of the table points out that the vectorial velocities (V)) for Zones C and D are very different for the assumption of barrel and screw rotation, as presented previously and as shown in Table 6.1. Eor the historic model, all energy is dissipated in the Zone C melt film, and the cumulative energy for melting was calculated... 

Assuming one-dimensional heat transfer is the mode of the solid bed heating due to the heating of the film by conduction and dissipation, the temperature will only change in the y direction. The same assumption that was made by Tadmor and Klein will be made here that the heat transfer model is a semi-infinite slab moving at a velocity Vsy c (melting velocity) with the boundary conditions T(0) = and j(-oo) = 7 , This assumption is not strictly correct because it will also be proposed that the other four surfaces are melting. The major error will occur at the corners of the solid bed. is the velocity of the solid bed surface adjacent to Film C as it moves toward the center of the solid bed in the y direction. 

The Tadmor model assumes Newtonian fluids and shallow channels. The channel cross section and that of the solid bed are assumed to be rectangular. The width of the solid bed profile is denoted by X(z), which is the the main objective that we are seeking with the model. The solid bed that develops at steady state conditions is the focal interest here. Furthermore, Tadmor assumed that melting only occurs at the barrel surface and the solid bed is homogeneous, continuous and deformable. 

Inherent Errors in Using the Power Law Model in Pressure Flows The shear rate during pressure flow between parallel plates varies from zero at the center to maximum shear rate at the wall, yw. Most polymer melts show Newtonian behavior at low shear rates, hence using the Power Law model for calculating flow rate introduces a certain error. How would you estimate the error introduced as a function of C, where C is the position below which the fluid is Newtonian [See Z. Tadmor, Polym. Eng. Sci., 6, 202 (1966).]... 

The drag-removal melting mechanism was discovered and mathematically modeled by Tadmor (27) in connection to melting in SSEs (see Section 9.3). It was further rehned, experimentally, verihed, and formulated as a self-contained computer package by Tadmor et al. (28-31). Later Vermeulen et al. (32), and Sundstrom and Lo (26) and Sundstrom and Young (33) analyzed the problem both experimentally and theoretically Mount (34) measured experimental rates of melting, and Pearson (35) analyzed the theoretical problem mathematically in detail, as shown in Fig. 5.12. In this section we follow Pearson s discussion. 

Z. Tadmor, Fundamentals of Plasticating Extrusion. I. A Theoretical Model for Melting, Polym. Eng. Sci., 6, 185-190 (1966). First presented at the Society of Plastics Engineers Annual Technical Conference, Montreal Canada, April 1966. 

As mentioned earlier, the melting mechanism in screw extruders was first formulated by Tadmor (29) on the basis of the previously described visual observations pioneered by Bruce Maddock. The channel cross section and that of the solid bed are assumed to be rectangular, as in Fig. 9.26. The prediction of the solid bed width profile (SBP), that is the width of the solid bed X as a function of down-channel distance z, is the primary objective of the model, which can be experimentally verified by direct observation via the cooling experiment of the kind shown in Figs. 9.20-9.25. As shown by Zhu and Chen (40), the solid bed can also be measured dynamically during operation by equipping the extruder with a glass barrel. 

Fig. 10.47 The effect of the Power Law index in the Carreau model, and the melt-pool size for a characteristic model wedge with e/h — 3 and ot= 15° on the non-Newtonian qp/qd parameters. [Reprinted hy permission from L. N. Valsamis and E. L. Canedo, Mixing in the Farrel Continuous Mixer in Mixing and Compounding of Polymers, I. Manas-Zloczower and Z. Tadmor, Eds., Hanser, Munich, 1994.]...

Tadmor, Z., Fundamentals of Plasticity Extrusion—I. A Theoretical Model for Melting, Polymer Engineering and Science, vol. 6, 1966, p. 185. 

Tadmor Z. Fundamentals of plasticating extrusion—I. A theoretical model for melting. Polym Eng Sci 1966 6 185. 

---