Power-law parameters

Table 5.2. Power-law parameters for flocculated kaolin suspensions...

Figures 35.39 and 35.40 show a tremendous dependency on both power-law parameters. The pressure buildup and nip force very much depend on the viscous behavior of the rubber compound. Although not calculated one can simply understand that a varying feedstock temperature will cause variations in the nip force because the viscosity of rubber compounds very much depends on temperature.

Other schemes have been proposed in which data are fit to a lower, even order polynomial [19] or to specific rheological models and the parameters in those models calculated [29]. This second approach can be justified in those cases when the range of behavior expected for the shear viscosity is limited. For example, if it is clear that power-law fluid behavior is expected over the shear rate range of interest, then it would be possible to calculate the power-law parameters directly from the velocity profile and pressure drop measurement using the theoretical velocity profile... 

Hydraulic fracturing fluids are solutions of high-molecular-weight polymers whose rheological behavior is non-Newtonian. To describe the flow behavior of these fluids, it is customary to characterize the fluid by the Power Law parameters of Consistency Index (K) and Behavior Index (n). These parameters are obtained experimentally by subjecting the fluid to a series of different shear rates (y) and measuring the resultant shear stresses (t). The slope and Intercept of a log shear rate vs log shear stress plot yield the Behavior Index (n) and Consistency Index (Kv), respectively. Consistency Indices are corrected for the coaxial cylinder viscometers by ... 

The slope of this function is the Rabinowitsch [12] correction for the shear rate in terms of the power law parameter n ... 

Power-law expressions are still nsed to describe snch polymer-fiber melts. Typical power-law parameters for selected fiber-polymer systems are shown in Table 4.7. Semiempirical expressions based on Eq. (4.23) have also been developed, as well as models based on energy dissipation. A complete review of these correlations is beyond the scope of this text, and the interested reader is referred to reference 9 for a more complete review of viscosity in fiber-reinforced polymer melts. 

Table 4.7 Power-Law Parameters for a Fiber-Filled Polyamide...

The DIERS Power Law Scaling Method121 is more robust than those given above and is applicable to fluids which are not Newtonian. The fluid is assumed to obey a power law and the power law parameters are assumed constant along the length of the relief line. Several experiments, at different L/D are required to fit the power law correlation, which may then be used to obtain G. Full details are given in reference... 

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. 

Index Entries Com stover rheological measurement shear stress shear rate non-Newtonian fluids Power Law parameters. 

The Power Law parameters for the corn stover suspensions could not be located in previous studies. Dronawat (13). studied a similar system using filamentous particles (Solka-Floc with a fiber length 215 pm in... 

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. 

Example 5.1 Thermal Degradation Characterization Thermal degradation is characterized by two temperature-dependent parameters, the induction time 6 (7) for the onset of degradation, as shown in Fig. E5.1(a) for unplasticized Polyvinyl chloride (PVC), and the rate of degradation. The latter is shown in Fig. E5.1(b) in terms of the rate of change of the consistency index of the Power Law parameter, as a function of time and temperature, and can be expressed by the following equations ... 

Their experimental results are shown in Fig. E6.14b, which plots dimensionless halftime versus dimensionless reciprocal force. Clearly, the Scott equations describe the experimental results given earlier as ti /nX = 1. They recommend that the choice of the parameter X be made on the basis of the Power Law parameters m and n and a similar Power Law relationship of the primary normal stress difference function 4 1 (y) = n 1 as... 

Fig. 13.17 Comparison of advancing front shapes for Newtonian and Power Law fluids (planar geometry). Power Law parameters m — 10,000 Pa s , n — 0.5, yw = 500s-1. [Reprinted by permission from H. Mavridis, A. N. Hrymak, and J. Vlachopoulos, A Finite Element Simulation of Fountain Flow, Polym. Eng. Sci., 26, 449 (1986).]...

However, they may not indicate the true bulk viscosity of a suspension that forms a thin layer of the continuous phase (e.g., serum of tomato juice) around the immersed probe or when the probe is covered by a higher viscosity gel due to fouling. Vibrational viscometers are suitable for measuring viscosities of Newtonian fluids, but not the shear-dependent rheological behavior of a non-Newtonian fluid (e.g., to calculate values of the power law parameters). 

Because the power law model (Equation 2.3) is used in determination of pumping and mixing power requirements, literature values of the power law parameters of... 

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