Casson model

The other models can be appHed to non-Newtonian materials where time-dependent effects are absent. This situation encompasses many technically important materials from polymer solutions to latices, pigment slurries, and polymer melts. At high shear rates most of these materials tend to a Newtonian viscosity limit. At low shear rates they tend either to a yield point or to a low shear Newtonian limiting viscosity. At intermediate shear rates, the power law or the Casson model is a useful approximation. 

The extrapolated yield stress gives 0.06 Pa and a plastic viscosity of 3.88 mPas. We can use this to estimate the force between the particles, which gives 425kBT/a, in fair agreement with the value determined using pair potential curves. Here the Casson model has been used to partially linearise a pseudoplastic system rather than a system with a true yield stress. 

Both the Casson model and that of Michaels and Bolger form a class... 

Figure 6.13 A flow curve partially linearised using the Casson model for a weakly attractive system with the pair potential shown in Figure 5.9. The curvature at low stresses is indicative of a viscoelastic liquid. The Casson model successfully linearised the data where the particles can be visualised as aligning with the applied shear field. This would suggest almost complete breakdown of the aggregates...

The power law model can be extended by including the yield value r — r0 = kyn, which is called the Herschel-BulHey model, or by adding the Newtonian limiting viscosity,. The latter is done in the Sisko model, rf + ky71-1. 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. 

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) ... 

Three empirical models were utilized to fit the rheologic characteristics of the wet grain slurries power law, Herschel-Bulkley, and Casson. The power law and Casson models are two-parameter models and are ideal for... 

The Casson model incorporates yield stress and is written as... 

Table 6 summarizes the Herschel-Bulkley model parameters for each slurry concentration. Figure 4 represents the Casson model regression for 21, 23, and 25 /o grain slurries. Table 7 summarizes the Casson model parameters for each slurry concentration. 

Experimental rheologic data were fit to the power law, Herschel-Bulkley, and Casson models. The power law model does not predict yield stress. Yield stress for 21% grain slurries predicted by the Herschel-Bulkley model was a negative value, as shown in Table 6. Yield stress values predicted by the Herschel Bulkley model for 23 and 25% solids were 8.31 and 56.3 dyn/cm2, respectively. Predicted yield stress values from the Casson model were 9.47 dyn/cm2 for 21% solids, 28.5 dyn/cm2 for 23% solids, and 44.0 dyn/cm2 for 25% solids. 

There are numerous other GNF models, such as the Casson model (used in food rheology), the Ellis, the Powell-Eyring model, and the Reiner-Pillippoff model. These are reviewed in the literature. In Appendix A we list the parameters of the Power Law, the Carreau, and the Cross constitutive equations for common polymers evaluated using oscillatory and capillary flow viscometry. 

As of the time of this writing, the corresponding equations for the Casson model have been developed but have not been tested against experimental data. Therefore, we cannot include any results. 

For non-Newtonian materials that have a yield stress, the Casson or Hershel-Bulkley models can be used. The Casson model is represented by the equation,... 

Kmy Mizrahi and Berk (1972) model is a modification of the Casson model... 

The Casson model (Equation 2.6) is a structure-based model (Casson, 1959) that, although was developed for characterizing printing inks originally, has been used to characterize a number of food dispersions ... 

For a food whose flow behavior follows the Casson model, a straight line results when the square root of shear rate, (y), is plotted against the square root of shear stress, (cr) , with slope Kc and intercept Kqc (Figure 2-2). The Casson yield stress is calculated as the square of the intercept, ctoc = (Kocf and the Casson plastic viscosity as the square of the slope, r]ca = The data in Figure 2-2 are of Steiner (1958) on a chocolate sample. The International Office of Cocoa and Chocolate has adopted the Casson model as the official method for interpretation of flow data on chocolates. However, it was suggested that the vane yield stress would be a more reliable measure of the yield stress of chocolate and cocoa products (Servais et al., 2004). 

Figure 2-2 Plot of versus for a Food that Follows the Casson Model. The Square of the intercept is the yield stress and that of slope is the casson plastic viscosity.

The time constant tc is related to the rate of aggregation of particles due to Brownian motion. For highly concentrated dispersed systems, oc will be much lower than t]o, so that (i oo/i o)yield stress, and Equation (2.11) reduces to the Casson model (Equation 2.6) (Tiu et al., 1992) with the Casson yield... 

For foods, such as chocolates, that can be described by the Casson model (Equation 2.5), Steiner (1958) chose not to develop an explicit expression for the non-Newtonian shear rate, but related j>n to the Casson model parameters Koc,Kc, and shear stress, (7. Steiner s approach is valid for values of a = (n/ro) between 0.5 and 0.9, and j>n values greater than 0.1 s when a — 0.9 and 0.01 s when a = 0.5. 

While application of structure-based models to rheological data, such as the Casson model, provides useful information, structure-based analysis can provide valuable insight in to the role of the structure of a dispersed system. Bodenstab et al. (2003)... 

Table 5-G Casson Model Parameters of Commercial Chocolate Samples (Chevalley, 1991)... 

Table 5-H Casson Model Parameters Yield Stress (croc) nd Plastic Viscosity (rjoo) of Cocoa Mass as a Function of Temperature (Fang et ai., 1996)...

Note that the Casson model (the third model in Table 6.3) fairly well describes various varnishes, paints, blood, food compositions like cocoa mass, and some other fluid disperse systems [443]. 

In Equation (2), n is the flow behavior index (-),K is the consistency index (Pa secn), and the other terms have been defined before. For shear-thinning fluids, the magnitude of nshear-thickening fluids n>l, and for Newtonian fluids n=l. For PFDs that exhibit yield stresses, models that contain either (Jo or a term related to it have been defined. These models include, the Bingham Plastic model (Equation 3), the Herschel-Bulkley model (Equation 4), the Casson model (Equation 5), and the Mizrahi-Berk model (Equation 6). 

COJ of 65 °Brix is a mildly shear-thinning fluid 160) with magnitudes of flow behavior index of the power law model (n) (Equation 2) of about 0.75 that is mildly temperature dependent. In contrast, the consistency index (K) is very sensitive to temperature for example, Vital and Rao (hi) found for a COJ sample magnitudes of 1.51 Pa sec11 at 20 °C and 27.63 Pa secn at -19 °C. Mizrahi and Firstenberg (hi) found that the modified Casson model (Equation 5) described the shear rate-shear stress data better than the Herschel-Bulkley model (Equation 4). 

Figure 10 compares the pressure dependence of the rheological parameters of the Casson model (high shear viscosity k 2), the Bingham... 

Herschef-Bulkley model Bingham model Casson model... 

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