Carreau model solution

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

Table 4-2 Carreau Model Parameters for Locust Bean Gum Solutions (ionic strength 0.1 M, pH 7.0) (Lopes da Silva et al., 1992)...

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 ... 

C. CONCENTRATED SOLUTION/MELT THEORIES 1. The Bird-Carreau Model... 

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. 

In Fig. 23, the Bird-Carreau model is compared to the experimental data of a 1.0% guar solution in the frequency/shear rate range of 0.1 to 100 sec . 

Chauveteau also studied flow of biopolymers in porous rock. By using well filtered biopolymer solutions, he determined apparent viscosities as a function of Darcy velocity in Fountainbleau sandstone over permeabilities ranging from 3.3 md to 256 md. Polymer retention was low and it was possible to restore the permeability of the rock to its prepolymer value after each polymer flow experiment. Apparent viscosities were fitted with the Carreau Model A. Analysis of the experimental data yields pairs of apparent viscosity and Darcy velocity. Conversion of Darcy velocity to apparent shear rate in the porous rock was done using Equation 14. 

Figure 3 illustrates the relationship between steady shear viscosity and shear rate for PBTA homopolymer solutions in NMP/4% LiCl with various concentrations. This figure clearly revels the shear-thinning effect for isotropic (C C r) solutions and anisotropic (C > Ccj-) solutions with the most shear rate region. Meanwhile, a Newtonian plateau appears in a low shear rate region for anisotropic solutions, especially for C = 6 wt% and C = 6.5 wt%. Furthermore, the experimental data could be fitted with theoretical non-Newtonian fluid model. Among which, power-law model was applied for isotropic solutions and Carreau model (22) for anisotropic solutions, as shown below ... 

Table II. Zero-Shear-Viscosity, and Power-Law and Carreau Model Parameters for Solutions of PBTA in NMP-4%LiCl at 30"C...

Table IV. Vq, and power-law and Carreau Model Parameters for Solutions of CO3-40R in NMP-3%LiCl at 30 C...

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. 

Example 5,1—Application of Carreau Model. Viscometric data are obtained to evaluate pol3rmer solutions over a range of shear rates. A data set for a 1,500-ppm solution of Pusher 500 in 53 meq/L... 

Nad brine at 72.3°F is presented in Table 5.2. Fit these data to the Carreau model (Eq. 5.8) and determine and n. Assume that the viscosity of the poiymer solution at high shear rates, > is the viscosity of the soivent. In this case /to = 1.0 cp. 

There is substantial loss in solution viscosity caused by mechanical degradation. Table 5.18 summarizes Carreau model parameters and screen factors for these solutions. The reduction in solution viscosity is a clear indication that the average molecular weight of the polymer was reduced by mechanical degradation. This reduction was confirmed by size-exclusion chromatography when a decrease in the molecular weight of about 68% was determined for the effluent from the flow experiment at a frontal-advance rate of 278 ft/D. 

Reduced reciprocal of force versus time to squeeze to half-thickness k is the time constant from the Carreau model, eq. 2.4.16, or the Maxwell time constant when n = 1. Hatched region is the range of experimental data on several polymer solutions from Leider (1974). The experiments deviate from the power law model (dashed line) at short times. 

Example 5.4 Flow of a Non-Newtonian Fluid. Write a general MATLAB function for solution of a boundary value problem by the shooting method using the Newton s technique. Apply this function to find the velocity profile of a non-Newtonian fluid that is flowing through a circular tube as shown in Fig. E5.4a. Also calculate the volumetric flow rate of the fluid. The viscosity of this fluid can be described by the Carreau model [5] ... 

Given here are shear viscosity data at 20°C for a 500-ppm solution of a high-molecular-weight polyacrylamide in distilled water. Obtain the best-fit parameters for the power-law and Carreau models. ... 

The advantage of these models is that they predict a Newtonian plateau at low shear rates and thus at low shear stresses. We will see back these models in Chap. 16 where an extra term 7700 is added to the equations to account for the viscosity of polymer solutions at high shear rates. At high shear rates the limiting slopes at high shear rates in log r) vs. log y curves are for the Cross, the Carreau and the Yasuda et al. models —m, (n-1) and (n-1), respectively. 

FIG. 23. Comparison of predictions of the Bird-Carreau constitutive model and experimental data for 1% guar solution (Kokini et al., 1984). 

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