Other Melting Models

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. 

Dekker [69] made observations of the melting behavior of polypropylene. He did not detect a clear melt pool at any side of the channel, but the solid bed was more or less suspended in a melt film. Lindt [72,73] developed a mathematical model describing this melting behavior, which is shown in Fig. 7.45. 

The solid bed Is completely surrounded by a melt film and melting occurs at the entire circumference of the solid bed. 

The melt film Is constant In the cross-channel direction. 

The assumptions are more realistic and broader than most other analyses of melting. This can improve the accuracy of the model, but, unfortunately, it also increases the complexity of the computations quite substantially. 

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The solid product, BaO, was apparently amorphous and porous. Decomposition rate measurements were made between the phase transformation at 1422 K and 1550 K (the salt melts at 1620 K). The enthalpy and entropy of activation at 1500 K (575 13 kJ mole-1 and 200 8 J K"1 mole-1) are very similar to the standard enthalpy and entropy of decomposition at the same temperature (588 7 kJ and 257 5 J K-1, respectively, referred to 1 mole of BaS04). The simplest mechanistic explanation of the observations is that all steps in the reaction are in equilibrium except for desorption of the gaseous products, S02 and 02. Desorption occurs over an area equivalent to about 1.4% of the total exposed crystal surface. Other possible models are discussed. 

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. 

A complication arising from the extension of the theory to flexible macromolecules is that in general, the intermolecular and intramolecular radial distribution functions depend on each other.In modeling the bulk of a one-phase polymer melt, however, the situation resolves itself because the excluded volume effect is insignificant under these conditions the polymer chains assume unperturbed dimensions (see also the section on Monte Carlo simulations by Corradini, as described originally in Ref. 99). One may therefore calculate the structure of the unperturbed single chain and employ the result as input to the PRISM theory to calculate the intermolecular correlation functions in the melt. 

One possibility that would lead to larger inferred porosities for the U-series was introduced by Qin (1992, 1993), who proposed that the retained melt was only in complete equilibrium with the surface of minerals and that solid-state diffusion limited the re-equilibration of the retained melt with the solid. In other respects, this model is identical to the ACM model of Williams and Gill (1989). Qin introduced a specific microscopic melting/diffusion model for spherical grains and coupled it to the larger-scale dynamic melting models. The net affect of this... 

Disequilibrium single porosity models. In addition to complete equilibrium transport, several other variations to the basic model have been proposed. The first relaxes the assumption that moving melt remains in chemical equilibrium with the solid at all times (Spiegelman and Elliott, 1993), although instantaneous melts are assumed to be in chemical equilibrium with the mantle that produced them. For stable elements, this disequilibrium transport produces a residue that reflects perfect fractional melting and the melts have compositions identical to accumulated fractional melts. These models are similar to the dynamic melting models in the limit that but... 

Other melting rules are formulated in terms of properties of the saturated liquid phase, focusing in particular on some feature of the structure as quantified by the radial distribution function g r). Ideally, these rules could be applied to predict the freezing transition of a molecular model using a correlation function given by (say) an integral equation theory for g r). 

Different types of liquid-like behavior can be related to polymeric liquids. In some cases, polymers are just dispersed in the bulk of another material, for instance, in polymeric suspensions such as some types of liquid paints. In some other cases, the polymer forms the bulk, as in the case of polymer melts. Models describing the rheology of the latter type of liquid polymers are needed to perform numerical analysis of processing operations and... 

Isomer effects cannot be predicted. This means the same activity coefficients are obtained, for example, for o-/m-/p-xylene or phenanthrene/anthracene with the different solvents. But at least in the case of VLE or SLE calculation this is not a great problem, since the required standard fugacities, that is, vapor pressure, melting point, and heat of fusion are of much greater importance than small differences of the activity coefficients. Similar problems are also observed for other predictive models, for example, the quantum chemical approach. 

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