Rheological models, types

The analysis indicates that presently quite adequate phenomenological models are available for description of the straining of commercial (polydisperse) polymers in the liquid state. A comparatively clear understanding of the mechanics of the processes of manufacturing of sleeve-type and flat films of molten thermoplastics also has been developed. So far, physical approaches have provided rheological models only for monodisperse polymers (the properties of which differ significantly from those of the ones used in industry). 

The results of the latest research into helical flow of viscoplastic fluids (media characterized by ultimate stress or yield point ) have been systematized and reported most comprehensively in a recent preprint by Z. P. Schulman, V. N. Zad-vornyh, A. I. Litvinov 15). The authors have obtained a closed system of equations independent of a specific type of rheological model of the viscoplastic medium. The equations are represented in a criterion form and permit the calculation of the required characteristics of the helical flow of a specific fluid. For example, calculations have been performed with respect to generalized Schulman s model16) which represents adequately the behavior of various paint compoditions, drilling fluids, pulps, food masses, cement and clay suspensions and a number of other non-Newtonian media characterized by both pseudoplastic and dilatant properties. 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

Table 6.1 gives the most widespread rheological models of nonlinearly viscous fluids. Most of these models do not describe all aspects of the actual behavior of nonlinear viscous fluids in the entire range of the shear rate. Instead, they explain only some specific characteristic features of the flow. Table 6.1 contains quasi-Newtonian relations of two types, namely,... 

Predictions for the values of the moduli of idealised fully oriented crystalline polymers will not be discussed, since this topic is dealt with extensively in Chapter 7. Neither shall we consider one-dimensional rheological models of the spring and dashpot type, reviews of which have been given by Kennedy and Reiner Sobotka has generalised such models in two dimensions. 

Most polymer processes are dominated by the shear strain rate. Consequently, the viscosity used to characterize the fluid is based on shear deformation measurement devices. The rheological models that are used for these types of flows are usually termed Generalized Newtonian Fluids (GNF). In a GNF model, the stress in a fluid is dependent on the second invariant of the stain rate tensor, which is approximated by the shear rate in most shear dominated flows. The temperature dependence of GNF fluids is generally included in the coefficients of the viscosity model. Various models are currently being used to represent the temperature and strain rate dependence of the viscosity. 

These types of phenomena can t be described in terms of simple rheological models with constant parameters. Systems that reveal the dependence of the viscosity on the flow rate are referred to as anomalous or non-Newtonian. In dilute suspensions, changes in the viscosity associated with the orientation and deformafion of the particles in the absence of particle-partide interactions are typically not too large. 

Table 6.8 Rheological Parameters of the General Rheological Model [Eq. (6.4)] for Different Generic-Type Thermoplastics...

It is very important, from one hand, to accept a hypothesis about the material fracture properties before physical model building because general view of TF is going to change depending on mechanical model (brittle, elasto-plastic, visco-elasto-plastic, ete.) of the material. From the other hand, it is necessary to keep in mind that the material response to loads or actions is different depending on the accepted mechanical model because rheological properties of the material determine type of response in time. The most remarkable difference can be observed between brittle materials and materials with explicit plastic properties. 

The polymer rheology is modeled by extending the usual power-law equation to include second-order shear-rate effects and temperature dependence assuming Arrhenius type relationship. 

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

There are basically three types of approaches to define the solid stress tensor, or more specifically the solid viscosity. In the early hydrodynamic models— developed by Jackson and his co-workers (Anderson and Jackson, 1967 Anderson et al., 1995), Kuipers et al., (1992), and Tsuo and Gidaspow (1990)—the viscosity is defined as an empirical constant, and also the dependence of the solid phase pressure on the solid volume fraction is determined from experiments. The advantage of this model is its simplicity, the drawback is that it does not take into account the underlying characteristics of the solid phase rheology. 

Figure 4. Rheological properties of the modeling unfilled rubber mixes on the basis of isoprene rubber with a various type and contents of sulfur vulcanization activators...

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