Polymer nonisothermal modeling

Example 14.1 A Semiempirical, Simplified, One-Dimensional, Nonisothermal Model [C. D. Han, Rheology in Polymer Processing, Academic Press, New York, 1976, Section 12.3.1.] Assuming steady state and further assuming that there is only one nonvanishing velocity component v(z), which is a function of only ", and that temperature varies only in the z direction, the equation of motion reduces to... 

Properties for a few polymers required in the solids conveying model are presented in Appendix B. Here it is observed that the friction coefficient between the polymer and steel is relatively independent of temperature. As the temperature approaches the melting point the friction coefficients tend to increase, but over a wide range of temperatures /b is nearly constant. Hence, it seems justifiable to use the mass flow rate and pressure rise calculated by means of the isothermal model in the nonisothermal 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. 

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 modern discipline of Materials Science and Engineering can be described as a search for experimental and theoretical relations between a material s processing, its resulting microstructure, and the properties arising from that microstructure. These relations are often complicated, and it is usually difficult to obtain closed-form solutions for them. For that reason, it is often attractive to supplement experimental work in this area with numerical simulations. During the past several years, we have developed a general finite element computer model which is able to capture the essential aspects of a variety of nonisothermal and reactive polymer processing operations. This "flow code" has been Implemented on a number of computer systems of various sizes, and a PC-compatible version is available on request. This paper is intended to outline the fundamentals which underlie this code, and to present some simple but illustrative examples of its use. 

This review has highlighted the important effects that should be modeled. These include two-phase flow of liquid water and gas in the fuel-cell sandwich, a robust membrane model that accounts for the different membrane transport modes, nonisothermal effects, especially in the directions perpendicular to the sandwich, and multidimensional effects such as changing gas composition along the channel, among others. For any model, a balance must be struck between the complexity required to describe the physical reality and the additional costs of such complexity. In other words, while more complex models more accurately describe the physics of the transport processes, they are more computationally costly and may have so many unknown parameters that their results are not as meaningful. Hopefully, this review has shown and broken down for the reader the vast complexities of transport within polymer-electrolyte fuel cells and the various ways they have been and can be modeled. 

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. 

Finite Element Modeling of Nonisothermal Polymer Flows... 

This paper has described an analytical tool which can predict velocities, stresses, and temperatures in nonisothermal flow situations of the sort encountered in many polymer melt processing operations. Such a model cannot be expected to replace the experience and intuition which provide the basis for most process design today, but it is hoped that this inexpensive and easily implemented model can provide a means by which the process designer s intuition might be expanded. Properly used, it can be a valuable additional tool at the process designer s disposal. 

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

The analysis of fast polymerisation reactions has shown that the effects, revealed during the mathematical simulation (diffusion model), are identical to the experimental effects of the cationic polymerisation of isobutylene (as an example). The important consequence of process nonisothermicity is its adverse effect on polymer quality, while the external thermostating is not effective enough in this case [52],... 

Denq, B.L., Chin, W.Y., Lin, K.F. Kinetic model of thermal degradation of polymers from nonisothermal process. J. Appl. Polym. Sd. 66, 1855-1867 (1997)... 

Ozawa extended the Avrami model to quantify polymer crystallization kinetics using noniso-thermal data [289]. It was reasoned that nonisothermal crystallization amounted to infinitesimal short crystallization times at isothermal conditions, given a crystallization temperature T [290]. This analysis led to the following equation ... 

A kinetic analysis based on the Coats-Redfern method applied nonisothermal TGA data to evaluate the stability of the polymer during the degradation experiment. Of the different methods, the Coats-Redfern method has been shown to offer the most precise results because gives a linear fitting for the kinetic model function [97]. This method is the most frequent in the estimation of the kinetic function. It is based on assumptions that only one reaction mechanism operates at a time, that the calculated E value relates specifically to this mechanism and that the rate of degradation, can be expressed as the basic rate equation (Eq. 5.3). This method is an integral method that assumes various... 

Guo, Y., Wang, N., Bradshaw, R.D., Brinson, L.C. Modehng mechanical aging shift factors in glassy polymers during nonisothermal physical aging. I. Experiments and kahr-aK model prediction. J. Polym. Sci. Part B Polym. Phys. 47, 340 (2009)... 

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. 

Hieber has shown that a number of studies of nonisothermal crystallization of isotactic poly(propylene) and poly(ethylene terephthalate) can be treated by the Nakamura model.(88a) This model has also been shown to hold for syndiotactic poly(styrene) and poly(caprolactone). (98,98a) The results for linear polyethylene have not been conclusive. All of the studies have involved unfractionated polymers. In one study curvature was observed in the Ozawa type plot.(89) In another study, with a different sample, the Nakamura model was shown to hold.(88)... 

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