Bird-Carreau model

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. 

For this particular data we can assume that Voo = 0, which reduces the Bird-Carreau model to,... 

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. 

C. CONCENTRATED SOLUTION/MELT THEORIES 1. The Bird-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 (Bird and Carreau, 1968 Carreau et al., 1968) prediction for tj is... 

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 . 

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 high shear viscosities are not significantly differing for different drop sizes. At elevated shear rates, higher secondary drop deformation for larger drop diameter equilibrates the degree of viscous friction between the emulsion drops of different size [54]. In all cases, the Bird-Carreau model ((23.1) dashed line) was an excellent fit to the experimental results (symbols). 

Here r]o is the zero shear viscosity and tm is the value of Ty when r] = j rjo- Actually most polymeric fluids exhibit a constant viscosity at low shear rates and then shear thin at higher shear rates (see Fig. 2.5). A model that is used often in numerical calculations, because it fits the full flow curve, is the Bird-Carreau model. 

The Bird-Carreau-Yasuda Model. A model that fits the whole range of strain rates was developed by Bird and Carreau [7] and Yasuda [72] and contains five parameters ... 

Bogue (Bird-Carreau) (3.3-19) Good fit depends on model for M Good fit depends on model for M Yes Yes... 

Constitutive equations were applied to simulate viscoelasticity of concentrated food polymer dispersions. Some fundamental and empirical models have been discussed in Section V. Among them, the Bird-Carreau constitutive model [Eqs. (89-94)] have been used for food polymer dispersions (Kokini et ai, 1984 Kokini and Plutchok, 1987b Plutchok and Kokini, 1986). 

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

Dus, S. J., and Kokini, J. L. (1990). Prediction of the non-linear viscoelastic properties of a hard wheat flour dough using the Bird-Carreau constitutive model. J. Rheol. 34(7), 1069-1084. 

Figure 3.8 shows the improved behaviour of the Carreau model compared with the power law model. Several workers have reported that the Carreau model gives a much improved fit to their viscosity/shear rate data (Abdel-Khalik et al, 1974 Bird et al, 1974 Chauveteau and Zaitoun, 1981). 

This is equivalent to the Cross model with a fifth fitting parameter, Carreau model (Bird et al., 1987, p. 171). 

Other models have been used (Bird et al., 1987, p. 228), but most studies have concentrated on the power law or the Cross (Carreau) models. Once one has chosen a numerical method, any of the general viscous models that depend on I ho can 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)... 

A more general model is the Carreau equation (Carreau, 1972 Bird et al., 1987),... 

It should be noted that as t becomes large the lowest order term in the coefficient of the K-term is just 60, that is one half the zero-shear-rate value of the primary normal stress function. A similar result was obtained by Bird and Marsh (7) and by Carreau (14) from the slowly varying flow expansions of two continuum models. Hence the time-dependent behavior of the shear stress is related to the steady-state primary normal stress difference in the limit of vanishingly small shear rate. 

The utility of mathematical models incorporating power-law Eq 10.2 is limited to either a small range of shear rate, or to its high values. To improve the description of the viscosity dependence on shear rate, several alternative relations were proposed, e.g., by Ellis, Bueche, Eyring, Carreau [Bird et al., I960]. The latter relation ... 

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],... 

This relation can be derived from several phenomenological models and also from molecular theory. According to the constitutive equation of Bird and Carreau, it holds for finite 7 even though equations 67 and 74 are limited to small 7. With large deformations or large strain rates, other nonlinear phenomena will... 

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