The Casson Model

This is a semi-empirical linear parameter model that has been applied to fit the flow curves of many paints and printing ink formulations [10], 

a plot of versus y should give a straight line from which and can 

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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 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)... 

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). 

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

Casson models were used to compare their yield stress results to those calculated with the direct methods, the stress growth and impeller methods. Table 2 shows the parameters obtained when the experimental shear stress-shear rate data for the fermentation suspensions were fitted with all models at initial process. The correlation coefficients (/P) between the shear rate and shear stress are from 0.994 to 0.995 for the Herschel-Bulkley model, 0.988 to 0.994 for the Bingham, 0.982 to 0.990 for the Casson model, and 0.948 to 0.972 for the power law model for enzymatic hydrolysis at 10% solids concentration (Table 1). The rheological parameters for Solka Floe suspensions were employed to determine if there was any relationship between the shear rate constant, k, and the power law index flow, n. The relationship between the shear rate constant and the index flow for fermentation broth at concentrations ranging from 10 to 20% is shown on Table 2. The yield stress obtained by the FL 100/6W impeller technique decreased significantly as the fimetion of time and concentration during enzyme reaction and fermentation. 

The rheology of a polymer solution can be approximated reasonably well by either a power-law or a Casson model over the shear rate range of 20-100 s . If the power law consistency coefficient, m, is lOPa s" and the flow behaviour index, n, is 0.2, what will be the approximate values of the yield stress and the plastic viscosity in the Casson model ... 

Calculate the pressure drop using the power-law model when this polymer solution is in laminar flow in a pipe 200 m long and 40 mm inside diameter for a centreline velocity of 1 m/s. What will be the calculated centreline velocity at this pressure drop if the Casson model is used ... 

An additional well-known mathematical expres.sion for viscoplastic fluids is the Casson model ... 

Workers who handle molten chocolate favour the Casson model blood is also believed to conform to this model. 

Casson (1959) proposed an alternate model to describe the flow of viscoplastic fluids. The three-dimensional form is left as an exercise to the reader (hint see eq. 2.5.9), but the one-dimensional form of the Casson model is given by. 

Comparison of Bingham and Casson fits to the iron oxide suspension data over the entire range of experimental data obtained parameters for the Bingham model are t) = 0.25 Pa-s and T,. 1.66 Pa, while for the Casson model they are O.IS Pa-s and 1.66 Pa, respectively. 

The iron oxide suspension data, including even lower shear rates, are shown in Figure 2.5.4. Curves of the Bingham and Casson models are also shown Bingham model parameters fix>m Figure 2.5.3b are used instead of a best fit. Note that values of the parameters for the Casson model also depend on the range of shear rates considered. 

As illustrated in Figure 5.7, the Casson model corresponds indeed to a t (Y) function without Newtonian plateau and an upward curvature in the low shear rate region significantly depending on the magnitude of the yield stress but the shear thinning behavior does not correspond to observations on filled polymer systems. Similar comments are made for the 11(0) fimction. 

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