Nonisothermal viscosity model

There have been numerous studies on the film-blowing process. Since the initial thin-shell approximation proposed by Pearson and Petrie [125, 126] with the Newtonian model assumed for deformation, various rheological models have been incorporated in simulations, such as the power-law model [127,128], a crystallization model [129], the Maxwell model [130-133], the Leonov model [133], a viscoplasti-c-elastic model [134], the K-BKZ/PSM model [135-137], and a nonisothermal viscosity model [138]. A complete set of experimental data was reported by Gupta [139] for the Styron 666 polystyrene and by Tas [140] for three different grades of LDPE. 

In building mathematical models of product formation in a mold it is possible to treat a polymeric material as motionless (or quasi-solid), because the viscosity grows very rapidly with the formation of a linear or network polymer thus, hydrodynamic phenomena can be neglected. In this situation, the polymerization process itself becomes the most important factor, and it is worth noting that the process occurs in nonisothermal conditions. 

The classic extrusion model gives insight into the screw extrusion mechanism and first-order estimates. For more accurate design equations, it is necessary to eliminate a long series of simplifying assumptions. These, in the order of significance are (a) the shear rate-dependent non-Newtonian viscosity (b) nonisothermal effects from both conduction and viscous dissipation and (c) geometrical factors such as curvature effects. Each of these... 

Distributed Parameter Models Both non-Newtonian and shear-thinning properties of polymeric melts in particular, as well as the nonisothermal nature of the flow, significantly affect the melt extmsion process. Moreover, the non-Newtonian and nonisothermal effects interact and reinforce each other. We analyzed the non-Newtonian effect in the simple case of unidirectional parallel plate flow in Example 3.6 where Fig.E 3.6c plots flow rate versus the pressure gradient, illustrating the effect of the shear-dependent viscosity on flow rate using a Power Law model fluid. These curves are equivalent to screw characteristic curves with the cross-channel flow neglected. The Newtonian straight lines are replaced with S-shaped curves. 

The first milestone in modeling the process is credited to Pearson and Petrie (42—44). who laid the mathematical foundation of the thin-film, steady-state, isothermal Newtonian analysis presented below. Petrie (45) simulated the process using either a Newtonian fluid model or an elastic solid model in the Newtonian case, he inserted the temperature profile obtained experimentally by Ast (46), who was the first to deal with nonisothermal effects and solve the energy equation to account for the temperature-dependent viscosity. Petrie (47) and Pearson (48) provide reviews of these early stages of mathematical foundation for the analysis of film blowing. 

The temperature dependence of viscosity can often be as important as its shear rate dependence for nonisothermal processing problems (e.g.. Tanner, 1985). For all liquids, viscosity decreases with increasing temperature and decreasing pressure. A useful empirical model for both effects on the limiting low shear rate viscosity is... 

The great virtue of this model is that unlike other models, it explicitly incorporates the dependence of viscosity on temperature as well as shear rate through the temperature dependence of the zero-shear viscosity (to be discussed shortly). Equation 14.8 is written to emphasize that point. This makes it particularly well suited for nonisothermal flow calculations. 

Here D/Dt is the material time derivative or the time derivative following the fluid motion. These equations have been written in a form which demonstrates the temperature dependence of the viscosity and thermal conductivity. Furthermore, the equations have been written with the assumption that the rheological properties are described by the GNF model. We now use the nonisothermal equations of change to resolve the examples in Section 5.2.1. 

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