Herschel-Bulkley model, viscosity

A more convenient extrapolation technique is to approximate the experimental data with a viscosity model. The Power Law, shown in Eq. 6, is the most commonly used two-parameter model. The Bingham model, shown in Eq. 7, postulates a linear relationship between x and y but can lead to overprediction of the yield stress. Extrapolation of the nonlinear Casson model (1954), shown in Eq. 8, is straightforward from a linear plot of x°5 vs y05. Application of the Herschel-Bulkley model (1926), shown in Eq. 9, is more tedious and less certain although systematic procedures for determining the yield value and the other model parameters are available (11) ... 

The parameter c was found to be a linear function of Reynolds number with regression coefficients between 0.98 and 1.00. The shear rate constant, k, was within 10% of the values found by Donnelly (15) and Rieth (16) for a double-helical ribbon impeller. Furthermore, the Power Law could be used to describe corn stover suspension viscosity with correlation coefficients above 0.99 for all four concentrations tested. Finally, the yield stress predicted by direct data extrapolation and by the Herschel-Bulkley model was similar for each concentration of com stover. 

Figure 10. Pressure dependence of parameters from various models of the rheology of invert emulsion oil-based drilling fluids at various temperatures. Casson high shear viscosity Bingham plastic viscosity consistency, power law exponent, and yield stress from Herschel-Bulkley model. (Reproduced with permission from reference 69. Copyright 1986 Society of Petroleum Engineers.)...

Figure 13. Pressure dependence of the rheological parameters from the Herschel-Bulkley model (n, K, r0, and a high shear rate viscosity tjhb) for a weighted water-based drilling fluid at 40 °C. (Reproduced with permission from reference 72. Copyright 1988 Society of Petroleum Engineers.)...

The power law model can be extended by including the yield value r—tq = Ay , which is called the Herschel-Bulkley model, or by adding the Newtonian limiting viscosity j]oo- The latter is done in the Sisko model, r]oo + Ay . These two models, along with the Newtonian, Bingham, and Casson models, are often included in data-fitting software supplied for the newer computer-driven viscometers. 

Chilton and Stainsby (1998) indicated that the accuracy of the Herschel-Bulkley model deteriorated at high shear rates. However, this may or may not be significant, depending on the application. At high strain rates the model predicts that the viscosity tends to zero, which is obviously incorrect. 

Using a critical shear rate rather than shear stress as a yield criteria makes application to numerical calculations much easier (Beverly and Tanner, 1989). Equation 2.5.6 with stress yield criteria (eq. 2.5.3) is known as Herschel-Bulkley model (Herschel and Bulkley, 1926 Bird et al., 1982). From Figure 2.5.5 we see that the two-viscosity models will better describe the iron oxide suspension data illustrated here. 

This model accounts for a yield stress combined with power law behavior in stress as a function of shear rate. Besides, this model predicts a viscosity that diverges continuously at low shear rates and is infinite below the yield stress. When n = 1, the Herschel-Bulkley model reduces to the Bingham fluid model where the flow above the yield stress would be purely Newtonian and the constant k would represent the viscosity [28]. 

In other terms, above a critical shear stress, it flows as a Newtonian fluid of (constant) viscosity t). It follows that a fluid obeying the Herschel-Bulkley model is sometimes called a generalized Bingham fluid, since with n=1 and K=r in Equation 5.3, one obviously obtains Equation 5.4. The three fit parameters of the Herschel-Bulkley equation can be reduced to two, when considering that n=0.5. This was in fact the approach used by Casson in proposing the following model ... 

Alderman et al. (72) modeled the temperature and pressure variation of the high shear rate Herschel-Bulkley viscosity tjhb using... 

Herschel-Bulkley viscosity model to represent the behaviour of these suspensions. 

With increasing interparticle collisions the probability of formation of floes from dispersed (nonflocculated) particles increases. Thus the horizontal axis can also be interpreted to mean a change from weakly flocculated particles on the left to increasingly flocculated particles toward the right. An outcome is that, irrespective of the degree of interparticle interaction, at low values of cp the viscosity rises slowly, but tends to increase rapidly when particle packing becomes dense.For randomly packed spheres this change occurs at about 95 = 0.60. A simple viscoplastic model is the Herschel-Bulkley equation... 

Flow behavior was modeled through Ostwald and Herschel-Bulkley equations. Experimental data fit adequately to the first one, showing no tq as well as apparent viscosity (r]a) (shear rate 20 s ) values ranging from 0.018 to 0.027 Pa.s, trend that showed a weak thickening effect of these fractions. Aqueous systems showed pseudoplastic behavior with values of exponential index of 0.85 and 0.73 for R 2-2 and R 2-3, respectively (Table 6). 

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