Herschel/Bulkley

Thus, equation 3.127, which includes three parameters, is effectively a combination of equations 3.121 and 3.125. It is sometimes called the generalised Bingham equation or Herschel -Bulkley equation, and the fluids are sometimes referred to as having/n/re body. Figures 3.30 and 3.31 show shear stress and apparent viscosity, respectively, for Bingham plastic and false body fluids, using linear coordinates. 

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 ... 

Atapattu, D. D., Chhabra, R. P. and Uhlherr, P. H. T. J. Non-Newt. Fluid Mech. 59 (1995) 245. Creeping sphere motion in Herschel - Bulkley fluids Flow field and drag. 

ReHB Reynolds number for spherical particle in a Herschel-Bulkley — —... 

Herschel-Bulkley model Herschel effect Hertz equation... 

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

Two protocols are presented for non-Newtonian fluids. Basic Protocol 1 is for time-independent non-Newtonian fluids and is a ramped type of test that is suitable for time-independent materials. The test is a nonequilibrium linear procedure, referred to as a ramped or stepped flow test. A nonquantitative value for apparent yield stress is generated with this type of protocol, and any model fitting should be done with linear models (e.g., Newtonian, Herschel-Bulkley unithit). 

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) ... 

Four models (Power Law, Herschel-Bulkley, Casson, and Bingham) were used to fit the experimental data and to determine the yield stress of... 

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. 

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... 

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. 

Herschel-Bulkley Parameters for Distiller s Grain Slurries... 

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. 

Overall, each model accurately represented the experimental data for shear rates >5 s4. At shear rates between 0 and 5 s, the empirical models disagree. Since no experimental data exist at low shear rates, it is impossible to determine which empirical model best represents the actual properties of the grain slurries. Further, all of the empirical models have comparable regression coefficients, as shown in Tables 5-7. Regression coefficient (R2) ranges were 0.992-1.000 (power law), 0.998-1.000 (Herschel-Bulkley), and 0.997-0.998 (Casson). 

Newtonian viscosity density shear stress Casson parameters Herschel-Bulkley parameters... 

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... 

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