Polymer processing flows

The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers the Peclet number is defined as pcUUk which represents the ratio of convective to conductive energy transport). As emphasized before, numerical simulation of convection-dominated transport phenomena by the standard Galerkin method in a fixed (i.e. Eulerian) framework gives unstable and oscillatory results and cannot be used. 

Families of finite elements and their corresponding shape functions, schemes for derivation of the elemental stiffness equations (i.e. the working equations) and updating of non-linear physical parameters in polymer processing flow simulations have been discussed in previous chapters. However, except for a brief explanation in the worked examples in Chapter 2, any detailed discussion of the numerical solution of the global set of algebraic equations has, so far, been avoided. We now turn our attention to this important topic. 

Polymer processing flows are always laminar and generally creeping type flows. A creeping flow is one in which viscous forces predominate over forces of inertia and acceleration. Classic examples of such flows include those treated by the hydrodynamic theory of lubrication. For these types of flows, the second term on the left-hand side of Eq. 2.5-18 vanishes, and the Equation of motion reduces to ... 

Capillary Flow Rheometry Next we examine the experimentally obtained results with the capillary flow rheometer shown in Fig. 3.1, which are directly relevant to polymer processing flows, since the attainable shear rate values are in the range encountered in polymer processing. The required pressure drop AP does not increase linearly with increases in the volumetric flow rate Q, as is the case with Newtonian fluids. Rather, increasingly smaller increments of AP are needed for the same increases in Q. The Newtonian Poiseuille equation, relating flow rate to pressure drop in a tube, is linear and given by... 

This section describes two common experimental methods for evaluating i], Fj, and IG as functions of shear rate. The experiments involved are the steady capillary and the cone-and-plate viscometric flows. As noted in the previous section, in the former, only the steady shear viscosity function can be determined for shear rates greater than unity, while in the latter, all three viscometric functions can be determined, but only at very low shear rates. Capillary shear viscosity measurements are much better developed and understood, and certainly much more widely used for the analysis of polymer processing flows, than normal stress difference measurements. It must be emphasized that the results obtained by both viscometric experiments are independent of any constitutive equation. In fact, one reason to conduct viscometric experiments is to test the validity of any given constitutive equation, and clearly the same constitutive equation parameters have to fit the experimental results obtained with all viscometric flows. 

The generalized Newtonian fluid models (GNF), which are widely used in polymer processing flow analysis, since they are capable of describing well the very strong shear rate dependence of melts. 

Fig. 12.16 Entrance flow patterns in molten polymers, (a) Schematic representation of the wine glass and entrance vortex regions with the entrance angle. [Reprinted by permission from J. L. White, Critique on Flow Patterns in Polymer Fluids at the Entrance of a Die and Instabilities Leading to Extrudate Distortion, App/. Polym. Symp., No. 20, 155 (1973).] (b) Birefringence entrance flow pattern for a PS melt. [Reprinted by permission from J. F. Agassant, et al., The Matching of Experimental Polymer Processing Flows to Viscoelastic Numerical Simulation, Int. Polym. Process., 17, 3 (2002).]...

Polymeric fluids are the most studied of all complex fluids. Their rich rheological behavior is deservedly the topic of numerous books and is much too vast a subject to be covered in detail here. We must therefore limit ourselves to an overview. The interested reader can obtain more thorough presentations in the following references a book by Ferry (1980), which concentrates on the linear viscoelasticity of polymeric fluids, a pair of books by Bird et al. (1987a,b), which cover polymer constitutive equations, molecular models, and elementary fluid mechanics, books by Tanner (1985), by Dealy and Wissbrun (1990), and by Baird and Dimitris (1995), which emphasize kinematics and polymer processing flows, a book by Macosko (1994) focusing on measurement methods and a book by Larson (1988) on polymer constitutive equations. Parts of this present chapter are condensed versions of material from Larson (1988). The static properties of flexible polymer molecules are discussed in Section 2.2.3 their chemistry is described in Flory (1953). 

The equations describing polymer processing operations are usually coupled, nonlinear partial-differential or partial-differential-integral equations in which two or three spatial directions and perhaps time appear as independent variables. Fully three-dimensional problems can usually be solved only for purely viscous liquids, and substantial simplification is usually required even for two-dimensional problems of viscoelastic liquids because of limitations of computer speed and memory. In some situations the geometry provides simplifications that lead to closed-form analytical solutions, but these are rare, and numerical methods are usually required in order to obtain process information. Numerical methods relevant to polymer processing flows are discussed in (3,4,16,17). The following three broad classes of numerical methods are in common use ... 

Other applications to fluid mechanics and polymer processing problems can be found on a regular basis in pubhcations such as the Journal of Fluid Mechanics, the Journal of Non-Newtonian Fluid Mechanics, International Polymer Processing, and so forth. We address polymer processing flows in which polymer viscoelasticity is important in Chapter 10. 

These models fit the shear rate dependence of viscosity very well and are very usefid to engineers. They form the backbone of polymer processing flow analyses. If the problem is to predict pressure drop versus steady flow rate in channels of relatively constant cross section, or torque versus steady rotation rate, the general viscous fluid gives excellent results. We need to be sure that we pick a model that describes our particular material over the rates and stresses of concern, however. With numerical methods, the multiple parameter models are readily solved. 

---