Cross viscosity model

T critical stress where r transitions from ijg to the power law regime for the Cross viscosity model... 

Wan and Isayev (1996) examined a hybrid approach of control-volume finite-element and finite-difference modelling of injection moulding of rubber compounds. The effect of vulcanization on viscosity and yield stress during cavity filling is reported. On comparing two versions of the modified Cross viscosity models - with and without the effect of cure - the use of a viscosity model that accounts for the cure was found to improve the accuracy of the cavity-pressure-prediction models. When the modified Cross model was further extended to include the yield stress and was implemented in the simulation program a significant improvement in the prediction of cavity pressure was obtained in the case of low injection speed. 

The Cross and the temperature-dependent Cross-WLF model (42) is an often used GNF-type model accounting for, like the Ellis and Carreau fluids for the viscosity at both low and high shear rates,... 

Our first task is to evaluate the validity of the conventional concept about the mobility control requirement using a simulation approach. This model uses the UTCHEM-9.0 simulator (2000). The dimensions of the two-dimensional XZ cross-section model are 300 ft x 1 ft x 10 ft. One injection well and one production well are at the two extreme ends in the X direction, and they are fully penetrated. The injection velocity is 1 ft/day the initial water saturation and oil saturation are 0.5. The displacing fluid is a polymer solution. The purpose of using the polymer solutuion in the model is to change the viscosity of the displacing fluid. Therefore, polymer adsorption, shear dilution effect, and so on are not included in the model. To simplify the problem, it is assumed that the oil and water densities are the same that the capillary pressure is not included that the relative permeabilities of water and oil are straight lines with the connate water saturation and residual oil saturation equal to 0 and that the water and oil viscosity is 1 mPa s. Under these assumptions and conditions, we can know the fluid mobilities at any saturation. The model uses an isotropic permeability of 10 mD. 

NatureWorks s injection moldable grade of Ingeo in the capillary rheometer test has shown a good fit into the Cross-WLF viscosity model (see Table 6.2). There are seven coefficients in the model and it is readily embedded into Moldflow software for injection molding simulation (Moldflow Plastic Labs 2007). Moldflow is computer software that is widely used across the plastic injection molding industry to predict and optimize the... 

Viscosity models are sometimes used to estimate the zero-shear viscosity when no experimental data are available at shear rates sufficiently low that the viscosity is constant. However, this is an unreliable procedure, as there is no fundamental basis for any of these equations, and the resulting value of t/q should be deemed at best a rough estimate. For example, Kataoka and Ueda [70] found that the Cross equation yielded extrapolated values of t]q that were about 50% less than measured values. 

Fig. 1 Shear viscosity data (symbols) and Cross-WLF model fit (curves) to the viscosity data for the ABS resin at 190,230,and270°C.

A pressure rheometer was built to measure rheological properties of polymers at elevated pressures and temperatures. However the viscosity at high deformation rates decreases more than the predicted theoretical value represented by the Cross-WLF model. This paper studies the role of viscous heating in the resulting viscosity deviations by means of simulation nsing the Radial Functions Method (RFM). 

Modeling of the melt viscosity of polyethylene and random copolymers of ethylene and a-olefins has been extensively dealt with in the past. Empirical viscosity models of the form of the Generalized Cross/Carreau models can and have been fitted to viscosity data for INSITE Technology Polymers. The shear viscosity data is usually measured at 190 °C in the molten regime from 0.1 - 100 rad/s. Of the several models available, the Cross model provides a good fit to the data with a minimum number of fitting parameters. The Cross model is of the form ... 

Fig. 4. Graphic representations (viscosity vs shear rate) of Cross model with different values for d.

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

Under conditions of steady fully developed flow, molten polymers are shear thinning over many orders of magnitude of the shear rate. Like many other materials, they exhibit Newtonian behaviour at very low shear rates however, they also have Newtonian behaviour at very high shear rates as shown in Figure 1.20. The term pseudoplastic is used to describe this type of behaviour. Unfortunately, the same term is frequently used for shear thinning behaviour, that is the falling viscosity part of the full curve for a pseudoplastic material. The whole flow curve can be represented by the Cross model [Cross (1965)] ... 

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