Rheological models Carreau

The radial film thickness profiles at different durations of spinning and for the four different rheological models (Newtonian, Power-law, Carreau and Viscoplastic) were obtained by numerical methods (12-13). Figure 3 shows representative film thickness profiles for Newtonian (n = 1) and non-Newtonian (n < 1) liquids. 

Grmela, M., and Carreau, P. J., Conformation tensor rheological models, J. Non-Newtonian Fluid Mech., 23, 271-294 (1987). 

Rheological models have also been developed to describe fluid behavior over the shear rate range which include Newtonian behavior at low and high shear rates. The Carreau Model Pi has been found to fit polymer data satisfactorily. Equation 2 is the Carreau Model A. In Equation 2, i is the Newtonian viscosity in the low shear region, x is the Newtonian viscosity in the high shear regions, and i is the shear rate. The parameter n is the power-law exponent and Tj- is a characteristic time constant. All parameters are determined by fitting experimental data. 

Non-Newtonian characteristics are introduced by expressing the wall shear in the capillary tube as an equivalent shear derived from a rheological model such as the power-law model (Equation 1) or the Carreau Model A (Equation 2). Derivations of polymer flow models based upon power-law and Carreau Model A are found in references 6 and 7. Equation 7... 

The rheological models for unified conq>lex viscosity versus frequency curves can be ea y written based on the modified Cox-Mertz rule discussed in Quarter 5. Thus, using Eq. (5.5), the modified Carreau model for complex viscosity can be written from Eq. (6.1) as... 

The above expression would be valid within the low to medium shear-rate range (i.e., 0 < 7 < 10 s ). Other expressions for il/, could have been obtained by considering the other viscosity functions described in the earlier section. However, this has not been done because the effective upper limit of shear rate of not greater than 10 s" for actual primary normal stress experimental measurements lies in a range consistent with only the Carreau model. In case measurements of primary normal stress at higher shear rates ever become possible in the future due to improved sophistication in experimental techniques, it is recommended that Eq. (2.44) from the General Rheological Model [87] be used... 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

Step 4 - update the value of viscosity (r/) using an appropriate rheological equation (e.g. temperature-dependent form of the Carreau model given as Equation (5.4)). 

Performing numerical simulations of the extrusion process requires that the shear viscosity be available as a function of shear rate and temperature over the operating conditions of the process. Many models have been developed, and the best model for a particular application will depend on the rheological response of the resin and the operating conditions of the process. In other words, the model must provide an acceptable viscosity for the shear rates and temperatures of the process. The simple models presented here include the power law. Cross, and Carreau models. An excellent description of a broad range of models was presented previously by Tadmor and Gogos [4]. 

There are numerous other GNF models, such as the Casson model (used in food rheology), the Ellis, the Powell-Eyring model, and the Reiner-Pillippoff model. These are reviewed in the literature. In Appendix A we list the parameters of the Power Law, the Carreau, and the Cross constitutive equations for common polymers evaluated using oscillatory and capillary flow viscometry. 

Both the Carreau and the Cross models can be modified to include a term due to yield stress. For example, the Carreau model with a yield term given in Equation (2.16) was employed in the study of the rheological behavior of glass-filled polymers (Poslinski et al., 1988) ... 

A number of models have been developed to describe transient viscoelastic behavior and one must have at hand carefully obtained rheological data in order to test the applicability of the models. Another example of the applicability of models to viscoelastic data is the study of Leppard and Christiansen (1975) in which the models proposed by Bogue and Chen, Carreau, and Spriggs were evaluated. In the case of foods, the empirical models have been developed to describe the transient data on stick butter, tub margarine (Mason et al, 1982), canned frosting (Kokini and Dickie, 1981 Dickie and Kokini, 1982), and mayonnaise (Campanella and Peleg, 1987c). 

Kokini, J. L., and Plutchok, G. J. (1987a). Predicting steady and oscillatory shear rheological properties of cmc/guar blends using the Bird-Carreau constitutive model. J. Text Stud. 18, 31-42. 

The basis for the rheological calculation is the description of the viscosity function of the plastic used with the help of the Carreau model, which describes the viscosity in its dependence on shear rate in the three regions ... 

Khalkhal and Carreau (2011) examined the linear viscoelastic properties as well as the evolution of the stmcture in multiwall carbon nanotube-epoxy suspensions at different concentration under the influence of flow history and temperature. Initially, based on the frequency sweep measurements, the critical concentration in which the storage and loss moduli shows a transition from liquid-like to solid-like behavior at low angular frequencies was found to be about 2 wt%. This transition indicates the formation of a percolated carbon nanotube network. Consequently, 2 wt% was considered as the rheological percolation threshold. The appearance of an apparent yield stress, at about 2 wt% and higher concentration in the steady shear measurements performed from the low shear of 0.01 s to high shear of 100 s confirmed the formation of a percolated network (Fig. 7.9). The authors used the Herschel-Bulkley model to estimate the apparent yield stress. As a result they showed that the apparent yield stress scales with concentration as Xy (Khalkhal and Carreau 2011). 

It is well known that the rheological properties of partially hydrolyzed polyacrylamide depend on the stresses associated with a given flow field. In a simple shear flow, the apparent viscosity is constant at low shear rates (Newtonian behavior). At a critical shear rate, the apparent viscosity decreases as the shear rate is increased, i.e., a shear thinning behavior [48]. The viscosity shear-rate data of water soluble-polymers are commonly fitted using the Carreau viscosity model [49]. According to this model, the apparent viscosity, p, is a function of the shear rate, Y, as follows ... 

Two series of PBTA/PI block copolymers were synthesized in this study and solution processed into molecular composite fibers via dry-jet wet-spinning. The unique rheological properties of liquid-crystalline PBTA homopolymers and PBTA/PI block copolymers were studied with a cone-and-plate rheometer. For block copolymers, the critical concentration decreased with an increase in PBTA content. The flow curves of isotropic and anisotropic solutions could be described via the power-law model and Carreau model, respectively. Copolymer fibers possess tensile strength and modulus located between those of PBTA fibers and PI fibers. Moreover, the tensile strength and modulus of Col fibers increase with an increase in PBTA content. Besides, increasing the draw ratios would give rise to an increase in the mechanical properties of copolymer fibers... 

Viscometric data for dilute and semidilute poly(acrylamide) solutions can often be fit to a Carreau model (63,64). It is wise to remember the cautions that were cited previously about mechanical degradation of the high molecular weight components of a polyacrylamide sample when analyzing rheological data. 

The rheology of blends of linear and branched PLA architectures has also been comprehensively investigated [42, 44]. For linear architectures, the Cox-Merz rule relating complex viscosity to shear viscosity is valid for a large range of shear rates and frequencies. The branched architecture deviates from the Cox-Merz equality and blends show intermediate behavior. Both the zero shear viscosity and the elasticity (as measured by the recoverable shear compliance) increase with increasing branched content. For the linear chain, the compliance is independent of temperature, but this behavior is apparently lost for the branched and blended materials. These authors use the Carreau-Ya-suda model. Equation 10.29, to describe the viscosity shear rate dependence of both linear and branched PLAs and their blends ... 

Table 6.5 Rheological Parameters of the Modified Carreau Model [Eq. (6.1)] for Different Generic-Type Thermoplastics...

Tables 6.S-6.8 list the model constants and the range of applicability based on the modified Carreau model, the modified Ellis model, the modified Ostwald-de Waele power-law model, and the General Rheological [11] model, respectively, for the master rheograms of most of the polymers discussed in Chapter 4.

During the experiments, the solid concentration was increased to 20% by volume. Except for suspensions with plastic particles, the suspensions showed a Newtonian behavior up to volume contents of 15 %. Suspensions with glass beads and s = 0.2 as well as all examined suspensions with plastic particles showed a shear thinning behavior. Considering the non-Newtonian behavior of these suspensions in the calculation of the time steady flow based on Eqs. (5.9-5.21), the viscosity of the suspension had to be described by a model depending on the deformation speed y. A Carreau-Yasuda model according to Eq. (5.52) fitted well to measurements carried out with a Couette system. The parameters Hq, a, n, and X were determined by the rheological measurements. 

In all these cases of simulation inside extruders, the major focus has been the development of efficient geometry modules to describe in a quick and user-friendly way the complicated geometrical characteristics of screws and extruder charmels. The question of adequately describing the rheology of the polymer melt is usually resolved with good viscosity data as a function of shear rate and temperature (Eq. (4.19)). The Carreau model (Eq. (4.6)) for the former and the exponential model (Eq. (4.9)) or Arrhenius model (Eq. (4.10)) for thelatter are sufficientin these computations, where viscoelasticity does not seem to be of importance or has not been attempted in any meaningful way. The predominance of shear flow inside the extruder seems to be the justification for that. 

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