Nonisothermal Newtonian Model

In the following analysis we first present the Newtonian isothermal model, which leads to an analytical solution. Then we discuss the Newtonian nonisothermal model, which gives insight into the complexities of the coupled heat and momentum transfer equations. PET, Nylon, and polysiloxanes are three typical polymers which are almost Newtonian at spinning conditions. Finally, we introduce the non-Newtonian isothermal model together with its associated difficulties. High-density polyethylene (HDPE), LDPE, polypropylene (PP), and polystyrene (PS) are all pseudoplastic and viscoelastic and fall into the latter category. 

Still, sophisticated, exact, numerical, non-Newtonian and nonisothermal models are essential in order to reach the goal of accurately predicting final product properties from the total thermomechanical and deformation history of each fluid element passing through the extruder. A great deal more research remains to be done in order to accomplish this goal. 

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. 

Newtonian model, isothermal Power-law model, nonisothermal Newtonian, nonisothermal, gravity effects included Maxwell and Leonov models, nonisothermal... 

Analyses of the Newtonian isothermal and nonisothermal models can be even further complicated by the introduction of the viscoelastic nature of the polymers in the melt state. The viscoelasticity of the polymer is important in cases where the relaxation time. A., is of the same order of magnitude or slower than the characteristic time constant of the process, which might be taken to be equal to vo/L. The ratio of these two time constants is called the Deborah number (Eq. 3.90), and it is equal to... 

The Newtonian and Isothermal Model as a Special Case of the Newtonian and Nonisothermal Model. Prove that the solution of the Newtonian isothermal model, Eq. 9.26, can be deduced from Eqs. 9.66 to 9.68 with the appropriate simplifications. 

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. 

Fig. 10.48 Numerical simulation results of nonisothermal flow of HDPE, Melt Flow Index MFI = 0.1 melt obeying the Carreau-Yagoda model for a typical FCM model wedge of e/h — 3 and =15. (a) Velocity (b) shear rate and (c) temperature profiles [Reprinted by permission from E. L. Canedo and L. N. Valsamis, Non Newtonian and Non-isothermal Flow between Non-parallel Plate - Applications to Mixer Design, SPE ANTEC Tech. Papers, 36, 164 (1990).]...

Once the continuum hypothesis has been adopted, the usual macroscopic laws of classical continuum physics are invoked to provide a mathematical description of fluid motion and/or heat transfer in nonisothermal systems - namely, conservation of mass, conservation of linear and angular momentum (the basic principles of Newtonian mechanics), and conservation of energy (the first law of thermodynamics). Although the second law of thermodynamics does not contribute directly to the derivation of the governing equations, we shall see that it does provide constraints on the allowable forms for the so-called constitutive models that relate the velocity gradients in the fluid to the short-range forces that act across surfaces within the fluid. 

Newtonian model, nonisothermal including the effect of polymer crystallization... 

By the 1960s, there were efforts to simulate flow in screw extruders including both non-Newtonian and nonisothermal character. This is found notably in the work of Griffith [G17] and Zamodits and Pearson [Z2]. They basically formulate in Cartesian coordinates a lubrication-based model with equations of motion of the form... 

Numerical simulation of the fiber-spinning process began with the early work of Matovich and Pearson [92], who analyzed the spinning of a Newtonian liquid and arrived at an analytical solution. Attempts were then made to analyze the process with differential constitutive models. Early work by Denn et al. [93] considered the upper converted Maxwell model, including nonisothermal effects [94]. Later, Gagon and Denn [95] used the PTT model and included nonisothermal effects to simulate... 

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. 

The fully melted polymer now enters the third zone of the extmder where it is pressurized. The buildup of pressure is required in order to pump the melt through the die at the end of the extruder. The pressurization of the melt is based on a viscous drag mechanism. We first illustrate how viscous drag can lead to a pressurization of the melt. This is followed by the development of a nonisothermal non-Newtonian model of the metering section. Because numerical methods are required to solve the equations generated in this model, we end the section by presenting the isothermal Newtonian case where an analytical solution is possible. 

The isothermal Newtonian model is a useful model, because it reveals most of the characteristics of the tubular film blowing process. Nevertheless, it suffers from two disadvantages the actual film blowing process is basically a nonisothermal process, and the polymer melt is non-Newtonian in character. In this section we address the nonisothermal case, and in the next section the matter of the non-Newtonian character of the polymer melt. 

A purely viscous non-Newtonian approach was followed by Han and Park (1975b). They used the power-law model and the energy equation, assuming that the effects of crystallization were insignificant. The agreement of this model with experimental data in terms of the bubble radius and thickness as a function of the axial distance for LDPE and HDPE was reported to be reasonable. In terms of viscoelastic models, Luo and Tanner (1985) considered the Leonov model, and Cain and Denn (1988) considered the upper convected Maxwell and Marrucci models in nonisothermal cases of film blowing. In some of the cases analyzed, multiple steady-state solutions were present (see also Problem 9C.2). 

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