Empirical Viscosity Models

At the highest shear rates shown in Fig. 10.10, the curves tend toward a linear relationship on the log-log plot, implying that a power law can be used to represent the variation of viscosity with shear rate at sufficiently high shear rates. This can be expressed as  

It is important to note that this model contains no characteristic time. It thus implies that the power-law parameters are independent of shear rate. Of course such a model cannot describe the low-shear-rate portion of the curve, where the viscosity approaches a constant value. Several empirical equations have been proposed to allow for the transition to Newtonian behavior over a range of shear rates. It was noted in the discussion of the Weissenberg number earlier in this chapter that the variation of 77 with y implies the existence of at least one material property with units of time. The reciprocal of the shear rate at which the extrapolation of the power-law line reaches the value of tiq is such a characteristic time. Models that can describe the approach to tIq thus involve a characteristic time. Examples include the Cross equation [64] and the Carreau equation [65], shown below as Eqs. 10.55 and 10.56 respectively. 

These models approach power-law behavior at high shear rates, and the dimensionless material constants m and p are simply related to the power law exponent. Hieber and Chiang [66] 

This is often called the Carreau-Yasuda equation. 

We note the appearance in these models of a material constant, A, with units of time. As mentioned above, such a constant is an essential feature of a rational model for the shear rate dependency of viscosity. Elberli and Shaw [68] compared a number of empirical viscosity equations. They observed that time constant values obtained by fitting data to two-parameter viscosity models were less sensitive to experimental error than those based on more complex models. The data at low shear rates and in the neighborhood of the reciprocal of the time constant are most critical in obtaining meaningful values of the parameters, while the high shear rate data are not as important. 

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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 ... 

Rheology. Flow properties of latices are important during processing and in many latex appHcations such as dipped goods, paint, inks (qv), and fabric coatings. For dilute, nonionic latices, the relative latex viscosity is a power—law expansion of the particle volume fraction. The terms in the expansion account for flow around the particles and particle—particle interactions. For ionic latices, electrostatic contributions to the flow around the diffuse double layer and enhanced particle—particle interactions must be considered (92). A relative viscosity relationship for concentrated latices was first presented in 1972 (93). A review of empirical relative viscosity models is available (92). In practice, latex viscosity measurements are carried out with rotational viscometers (see Rpleologicalmeasurement). 

It must, of course, be clearly understood that e and e are not, like v and a, properties of the fluid involved alone but depend primarily on the turbulence structure at the point under consideration and hence on the mean velocity and temperature at this point and the derivatives of these quantities as well as on the type of flow being considered. The use of e and e does not, in itself, constitute the use of an empirical turbulence model. It is only when attempts are made to describe the variation of 6 and eh through the flow field on the basis of certain usually rather limited experimental measurements that the term eddy viscosity turbulence model is applicable. In fact, even when advanced turbulence models are used, it is often convenient to express the end results in terms of the eddy viscosity and eddy diffusivity. 

Various viscosity models have implicitly included the effects of gelation on the chemoviscosity, and these were reviewed in Table 4.2 incorporating gelation-conversion and glass-transition-temperature effects implicitly in the cure effects on chemoviscosity. Explicit models for the expression of gel time versus temperature and time are sparse, with empirical measurements mainly being used. 

A new approach to calculating viscosities of concentrated polymer solutions has been presented. It consists of the derivation of a semi-empirical computational model containing three parameters characteristic of a particular polymer. Once these parameters have been established, the viscosity of any solution of the polymeric material in a solvent or solvent blend may be calculated. The method should be of particular interest to the coatings industry, where they often require a screening estimate of the potential viscosity-reducing power of a new solvent blend. 

Many mathematical expressions of varying complexity and form have been proposed in the literature to model shear-thinning characteristics some of these are straightforward attempts at cmve fitting, giving empirical relationships for the shear stress (or apparent viscosity)-shear rate curves for example, while others have some theoretical basis in statistical mechanics - as an extension of the application of the kinetic theory to the liquid state or the theory of rate processes, etc. Only a selection of the more widely used viscosity models is given here more complete descriptions of such models are available in many books [Bird et al., 1987 Carreau et al., 1997] and in a review paper [Bird, 1976],... 

At a certain time layer T tep the empirical coefficient of viscosity model is calculated by the following formula Cg = Z IV, where T tep = 10 t, r-time... 

Thus, the empirical coefficient of viscosity model for magnetic field at a certain time step Tstep assumes the following form Ds = jY. ... 

The two-viscosity models (eq. 2.5.6) and Papanastasiou s modification (eqs. 2.5.7-2.5.9) are empirical improvements designed primarily to afford a convenient viscoplastic constitutive equation for numerical simulations (Abdali et al., 1992). Figure 2.5.6 compares them with the ideal Bingham model. 

Eor typical polydisperse commercial polymers, data often do not extend into the Newtonian and power-law regions. If the method is applied to a data set that simply cuts off at each end of the accessible shear rate range, pathological results will be obtained. It is thus necessary to extrapolate the experimental results at both ends. The selection of extrapolation procedures is arbitrary one cannot create missing information by means of curve fitting. The objective is rather to make optimum use of the information that is contained in the data. Various empirical viscosity equations have been used to extrapolate at the low shear rate end (see Section 10.7.1.1), while the Vinogradov fluidity (I/77) model has been foimd useful at the high end. 

A) Models based on an empirical transformation from rheological material function to molecular weight distribution (viscosity models and modulus models)... 

Software packages for data interpretation that are provided by rheometer manufacturers sometimes include programs to calculate the parameters of empirical viscosity equations based on experimental data. It is important to keep in mind, however, that these are simple empirical models and that they do not provide an accurate fit of the entire viscosity curves of real materials. Also, there is no unique procedure for inferring parameter values from data. When such equations are fitted to experimental data, information is lost. For example, it is not possible to use such an equation to infer the molecular weight distribution using the methods described in Chapter 8. 

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)... 

A basic theme throughout this book is that the long-chain character of polymers is what makes them different from their low molecular weight counterparts. Although this notion was implied in several aspects of the discussion of the shear dependence of viscosity, it never emerged explicitly as a variable to be investi-tated. It makes sense to us intuitively that longer chains should experience higher resistance to flow. Our next task is to examine this expectation quantitatively, first from an empirical viewpoint and then in terms of a model for molecular motion. 

Additives can alter the rate of wet ball milling by changing the slurry viscosity or by altering the location of particles with respect to the balls. These effects are discussed under Tumbhng Mills. In conclusion, there is still no theoretical way to select the most effective additive. Empirical investigation, guided by the principles discussed earlier, is the only recourse. There are a number of commercially available grinding aids that may be tried. Also, a Idt of 450 surfactants that can be used for systematic trials (Model SU-450, Chem Service... 

Curing of Polyimlde Resin. Thermoset processing involves a large number of simultaneous and interacting phenomena, notably transient and coupled heat and mass transfer. This makes an empirical approach to process optimization difficult. For instance, it is often difficult to ascertain the time at which pressure should be applied to consolidate the laminate. If the pressure is applied too early, the low resin viscosity will lead to excessive bleed and flash. But if the pressure is applied too late, the diluent vapor pressure will be too high or the resin molecular mobility too low to prevent void formation. This example will outline the utility of our finite element code in providing an analytical model for these cure processes. 

Loutfy and coworkers [29, 30] assumed a different mechanism of interaction between the molecular rotor molecule and the surrounding solvent. The basic assumption was a proportionality of the diffusion constant D of the rotor in a solvent and the rotational reorientation rate kOI. Deviations from the Debye-Stokes-Einstein hydrodynamic model were observed, and Loutfy and Arnold [29] found that the reorientation rate followed a behavior analogous to the Gierer-Wirtz model [31]. The Gierer-Wirtz model considers molecular free volume and leads to a power-law relationship between the reorientation rate and viscosity. The molecular free volume can be envisioned as the void space between the packed solvent molecules, and Doolittle found an empirical relationship between free volume and viscosity [32] (6),... 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

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. 

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