Herschel, Bulkley model

Herpes simplex vaccine, 25 498-499 Herpes simplex viruses, 3 136 Herpesviruses, 3 136 Herpes zoster vaccine, 25 496-497 Herschel-Bulkley model, 21 705 Herschel effect, 19 204 Herz compounds, 23 643... 

Some materials can be modelled well by modifying the power law to include a yield stress this is known as the Herschel-Bulkley model ... 

For a fluid whose rheological properties may be represented by the Herschel-Bulkley model discussed in Volume 1, Chapter 3, the shear stress r is a function of the shear rate y or ... 

Herschel-Bulkley model Herschel effect Hertz equation... 

Herschel-Bulkley model stress = yield stress +... 

Comparison with Eq. (6.24) shows that the Herschel-Bulkley model is the power-law model with the addition of a yield stress. Another such derivative model is the Robertson-Stiff [379] model ... 

Corn stover, a well-known example of lignocellulosic biomass, is a potential renewable feed for bioethanol production. Dilute sulfuric acid pretreatment removes hemicellulose and makes the cellulose more susceptible to bacterial digestion. The rheologic properties of corn stover pretreated in such a manner were studied. The Power Law parameters were sensitive to corn stover suspension concentration becoming more non-Newtonian with slope n, ranging from 0.92 to 0.05 between 5 and 30% solids. The Casson and the Power Law models described the experimental data with correlation coefficients ranging from 0.90 to 0.99 and 0.85 to 0.99, respectively. The yield stress predicted by direct data extrapolation and by the Herschel-Bulkley model was similar for each concentration of corn stover tested. 

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 yield stress values given in Table 3 demonstrate that the yield stresses determined with the Herschel-Bulkley model were lower than the yield stresses determined with all the other methods at equal concentrations. The yield stress predicted by direct data extrapolation and by the Herschel-Bulkley model was similar for each concentration of corn stover. 

Figure 2 shows the power law fits for the wet grain slurries. Table 5 summarizes the power law parameters for each slurry concentration. Figure 3 presents the experimental data fit to the Herschel-Bulkley model. 

Fig. 3. Herschel-Bulkley model fit for distiller s grain slurries.

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. 

The parameter Ca is called the Casson number and is analogous to the Hedstrom number He for the Bingham plastic and Herschel-Bulkley models. 

FIGURE 11 Generalized correlation of drag coefficient for Herschel-Bulkley model fluids Qff is defined by Eq. (165) and reduces to appropriate parameters for Bingham plastic, power law, and Newtonian fluid limits. 

This parameter is defined to accommodate Herschel-Bulkley model fluids. In the limit To = 0, it reduces to an equivalent power law particle Reynolds number. In the limit n = 1, it reduces to a compound parameter involving the Bingham plastic particle Reynolds number and particle Hedstrom number. In both limits it reduces to the Newtonian particle Reynolds number. This correlation permits... 

Table 3.1 Special cases of the Herschel-Bulkley model, Eq. 3.6...

A model to study thixotropic behavior of foods exhibiting yield stress was devised by Tiu and Boger (1974) who studied the time-dependent rheological behavior of mayonnaise by means of a modified Herschel-Bulkley model ... 

Note that in the equations for the Herschel-Bulkley model m = (1 /dh) Bingham plastic model [Pg.429]

In the MEB equation, kinetic energy losses can be calculated easily provided that the kinetic energy correction factor a can be determined. In turbulent flow, often, the value of a = 2 is used in the MEB equation. When the flow is laminar and the fluid is Newtonian, the value of a = 1 is used. Osorio and Steffe (1984) showed that for fluids that follow the Herschel-Bulkley model, the value of a in laminar flow depends on both the flow behavior index ( ) and the dimensionless yield stress ( o) defined above. They developed an analytical expression and also presented their results in graphical form for a as a function of the flow behavior index ( ) and the dimensionless yield stress ( o)- When possible, the values presented by Osorio and Steffe (1984) should be used. For FCOJ samples that do not exhibit yield stress and are mildly shear-thinning, it seems reasonable to use a value of a = 1. 

A comprehensive example for sizing a pump and piping for a non-Newtonian fluid whose rheological behavior can be described by the Herschel-Bulkley model (Equation 2.5) was developed by Steffe and Morgan (1986) for the system shown in Figure 8-2 and it is summarized in the following. The Herschel-Bulkley parameters were yield stress = 157 Pa, flow behavior index = 0.45, consistency coefficient = 5.20 Pas". 

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). 

Yield stresses can also be obtained by extrapolation of shear rate-shear stress data to zero shear rate according to one of several flow models. The application of several models was studied by Rao et al. (AS.) and Rao and Cooley (Al) The logarithm of the yield stresses predicted by each model and the total solids (TS) of the concentrates were related by quadratic equations. The equations for the yield stresses predicted by the Herschel-Bulkley model (Equation 4) which described very well the flow data of Nova and New Yorker tomato cultivars were ... 

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). 

Mitchell (7), Blauer, and co-workers (7, 40) modeled foams as Bingham plastic. Foams were modeled as power law fluids by Patton et al. (41). A more rigorous model is the Herschel—Bulkley model, which combines both the Bingham plastic and Power law models (2,11, 42, 43). 

By using the shear—stress relationship that the Herschel—Bulkley model follows, the pressure loss for foamed fluids in pipe can be deter-... 

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.)...

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