Viscosity models Carreau model

Carreau =[l+(Ay)2]< - riQ = zero shear rate viscosity (i) Carreau model provides a very good... 

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)). 

The Carreau model (Carreau, 1972) is very useful for describing the viscosity of structural fluids ... 

Figure 3-8 Viscosity data and Carreau model fit of polyacrylamide solutions. (From Darby and Pivsa-Art, 1991.)...

If r]oo <3C ( /. j/o), the Carreau model reduces to a three-parameter model ( 0,k, and p) that is equivalent to a power law model with a low shear limiting viscosity, also known as the Ellis model ... 

A stress-dependent viscosity model, which has the same general characteristics as the Carreau model, is the Meter model (Meter, 1964) ... 

ARe>s is the Reynolds number based on the solvent properties, /zs is the solvent viscosity, D is the pipe diameter, F is the velocity in the pipe, and A is the fluid time constant (from the Carreau model fit of the viscosity curve). 

As discussed in Chapter 3, the Carreau viscosity model is one of the most general and useful and reduces to many of the common two-parameter models (power law, Ellis, Sisko, Bingham, etc.) as special cases. This model can be written as... 

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]. 

A characteristic relaxation time for the Carreau viscosity model or relaxation... 

Fitting a Bird-Carreau model to viscosity data. Table 7.5 shows the measurements of Ballenger et. al [5] of the viscosity as a function of shear rate for polystyrene at 453 K. 

Fit a Bird-Carreau model to the viscosity curve given in Fig. 7.24. To solve this problem, download a non-linear fitting program from the world wide web. 

The viscosity was modeled using a Carreau model with an Arrhenius temperature... 

Carreau model parameter Carreau model parameter Zero shear rate viscosity Arrhenius law parameter Reference temperature Density Specific heat Thermal conductivity... 

We can generalize to include fluids for non-constant viscosity to obtain further dimensionless characteristic values. Two examples are given in Fig. 6.12, and a numerical example for an extruder with a product whose viscosity can be described by the Carreau model is given in Chapter 6. 

The viscosity of polymers is often described using the Carreau model (Eq. 15.1) ... 

Yoo et al. (1994) also found that the Cross and Carreau models were satisfactory for the most part in describing the data that covered the zero-shear and the power law regions of mesquite seed gum. Also in agreement with Lopes da Silva et al. (1992), the Carreau model described well the viscosity data shown in Figure 4-3 over a wide range of shear rates except in the power law region at high shear rates in concentrations > 1.4 g 100 mL . ... 

The Carreau model not only described well the flow data of LB gum solutions, but the magnitudes the time constant (Ac) were in good agreement with those of Rouse time (tr) constant derived from solution viscosity data while the Cross (oc) time constants were lower in magnitudes both the Carreau and the Cross time constants followed well power relationships with respect to the concentration (c) of the solutions (Lopes da Silva et al., 1992) ... 

A typical viscosity characteristic of many non-Newtonian fluids (e.g., polymeric fluids, flocculated suspensions, colloids, foams, gels, etc.) is illustrated by the curves labeled structural viscosity in Figures 5.2 and 5.3. These flnids exhibit Newtonian behavior at very low and very high shear rates, with shear thinning or pseudoplastic behavior at intermediate shear rates. This can often be attributed to a reversible structure or network that forms in the rest or eqnilibrinm state. When the material is sheared, the structure breaks down, resnlting in a shear-dependent (shear thinning) behavior. This type of behavior is exhibited by flnids as diverse as polymer solutions, blood, latex emulsions, paint, mud (sediment), etc. An example of a useful model that represents this type of behavior is the Carreau model ... 

The Bird-Carreau model is an integral model which involves taking an integral over the entire deformation history of the material (Bistany and Kokini, 1983). This model can describe non-Newtonian viscosity, shear rate-dependent normal stresses, frequency-dependent complex viscosity, stress relaxation after large deformation shear flow, recoil, and hysteresis loops (Bird and Carreau, 1968). The model parameters are determined by a nonlinear least squares method in fitting four material functions (aj, 2, Ai, and A2). 

The Bird-Carreau model employs the use of four empirical constants (ai, a2, Ai, and A2) and a zero shear limiting viscosity (770) of the solutions. The constants a, az, Ai, and A2, can be obtained by two different methods one method is using a computer program which can combine least square method and the method of steepest descent analysis for determining parameters for the nonlinear mathematical models (Carreau etal, 1968). Another way is to estimate by a graphic method as illustrated in Fig. 20 two constants, Q i and A], are obtained from a logarithmic plot of 77 vs y, and the other two constants, az and A2, are obtained from a logarithmic plot of 77 vs w. 

Using these empirical equations in conjunction with the predictions of the Bird-Carreau model, it is possible to predict t/ and An example of such a plot is shown in Fig. 24 for a 1.0% CMCguar blend (3 1). Experimental data are superimposed on these plots to judge the aptness of the model. The steady shear viscosity 17 and the dynamic viscosity 17 are well predicted in the shear rate range of 0.1 to 100 sec . The experimental data, as well as the theoretical prediction, portray commonly observed behaviors by polymeric dispersions. In this instance, 17 and rf for this blend ratio tend to some value, a property suggested by the Bird-Carreau model at low shear rate (Kokini et al, 1984). 

FIG. 27. Experimental values and those predicted using the Bird-Carreau model of apparent viscosity (tj) as a function of shear rate for hard flour dough sample (Dus and Kokini, 1990). 

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 ... 

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],... 

Based on the molecular network considerations, Carreau [1972] put forward the following viscosity model which incorporates both limiting viscosities fio and Mco ... 

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