Power law model parameters

Using the same equipment as in problem 3.10, the following height-time data have been obtained for a coal slurry of density 1135 kg/m. Evaluate the power-law model parameters for this slurry (Assume streamline flow under all conditions)... 

There are a number of models in common use, a few of which will be listed here. The most common two-parameter model is the power law model (parameters n and k), whose form is given in Eq. (15). This model corresponds to a straight line on a log-log plot, and fits viscosity data well at high shear rates. It is the simplest model to account for shear thinning. In logarithmic form, the equation for the model becomes... 

A useful two-parameter model is the power-law model, or Ostwald-de Waele law to identify its first proponents. The relation between shear stress and shear rare is given by ... 

In order to overcome the shortcomings of the power-law model, several alternative forms of equation between shear rate and shear stress have been proposed. These are all more complex involving three or more parameters. Reference should be made to specialist works on non-Newtonian flow 14-171 for details of these Constitutive Equations. 

If it is known that a particular form of relation, such as the power-law model, is applicable, it is not necessary to maintain a constant shear rate. Thus, for instance, a capillary tube viscometer can be used for determination of the values of the two parameters in the model. In this case it is usually possible to allow for the effects of wall-slip by making measurements with tubes covering a range of bores and extrapolating the results to a tube of infinite diameter. Details of the method are given by Farooqi and Richardson. 21 ... 

The Arrhenius plots for both sets of kinetic parameters together with experimental points are shown in Fig. 5.4 -32. Experimental points scatter uniformly on both sides of the straight lines indicating that the power-law model with the evaluated rate constant can be satisfactorily used to describe the kinetic experiments under consideration. 

If r]oo <3C ( /. j/o), the Carreau model reduces to a three-parameter model ( 0,k, and p) that is equivalent to a power law model with a low shear limiting viscosity, also known as the Ellis model ... 

You would like to determine the pressure drop in a slurry pipeline. To do this, you need to know the rheological properties of the slurry. To evaluate these properties, you test the slurry by pumping it through a 1 in. ID tube that is 10 ft long. You find that it takes a 5 psi pressure drop to produce a flow rate of 100 cm3/s in the tube and that a pressure drop of 10 psi results in a flow rate of 300cm3/s. What can you deduce about the rheological characteristics of the slurry from these data If it is assumed that the slurry can be adequately described by the power law model, what would be the values of the appropriate fluid properties (i.e., the flow index and consistency parameter) for the slurry ... 

Corresponding expressions for the friction loss in laminar and turbulent flow for non-Newtonian fluids in pipes, for the two simplest (two-parameter) models—the power law and Bingham plastic—can be evaluated in a similar manner. The power law model is very popular for representing the viscosity of a wide variety of non-Newtonian fluids because of its simplicity and versatility. However, extreme care should be exercised in its application, because any application involving extrapolation beyond the range of shear stress (or shear rate) represented by the data used to determine the model parameters can lead to misleading or erroneous results. 

The disadvantage of the power law model is that it cannot predict the viscosity in the zero-shear viscosity plateau. When the zero-shear viscosity plateau is included, a nonlinear model must be specified with additional fitting parameters. A convenient model that includes the zero-shear viscosity and utilizes an additional parameter is the Cross model [30] ... 

The focus of this evaluation is on the results that were reported using four different resins [52] PC resin, LLDPE resin, EAA copolymer, and an LDPE resin. The shear viscosities for the resins at selected processing temperatures are shown in Pig. 7.17 and were modeled using the power law model provided by Eq. 7.42. The parameters for the model are given in Table 7.3. As shown in Pig. 7.17 and the n values in Table 7.3, the PC resin shear-thinned the least while the EDPE resin shear-thinned the most. The LLDPE and EAA resins have n values between those for the PC and LDPE resins. The melt density for the LDPE and LLDPE resins at 240 °C is 735 kg/mT The melt density of the EAA resin at 220 °C was 785 kg/m and the melt density of the PC resin at 280 °C was 1073 kg/mT... 

In general, the use of Langmuir-Hinshelwood-Hougen-Watson (LHHW)-type of rate equation for representing the hydrogenation kinetics of industrial feedstocks is complicated, and there are too many coefficients that are difficult to determine. Therefore, simple power law models have been used by most researchers to fit kinetic data and to obtain kinetic parameters. 

If some or all of this curve is present, the models used to fit the data are more complex and are of two types. The first of these is the Carreau-Yasuda model, in which the viscosity at a given point (T ) as well as the zero-shear and infinite-shear viscosities are represented. A Power Law index (mi) is also present, but is not the same value as n in the linear Power Law model. A second type of model is the Cross model, which has essentially the same parameters, but can be broken down into submodels to fit partial data. If the zero-shear region and the power law region are present, then the Williamson model can be used. If the infinite shear plateau and the power law region are present, then the Sisko model can be used. Sometimes the central power law region is all that is available, and so the Power Law model is applied (Figure H. 1.1.5). 

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. 

The Power Law model (excluding temperature dependence) is a two-parameter empirical model proposed by Ostwald and de Waele (39). It is based on the experimental observation that by plotting In tj(y)vs. In(7), a straight line is obtained in the high shear rate region for... 

Distributed Parameter Models Both non-Newtonian and shear-thinning properties of polymeric melts in particular, as well as the nonisothermal nature of the flow, significantly affect the melt extmsion process. Moreover, the non-Newtonian and nonisothermal effects interact and reinforce each other. We analyzed the non-Newtonian effect in the simple case of unidirectional parallel plate flow in Example 3.6 where Fig.E 3.6c plots flow rate versus the pressure gradient, illustrating the effect of the shear-dependent viscosity on flow rate using a Power Law model fluid. These curves are equivalent to screw characteristic curves with the cross-channel flow neglected. The Newtonian straight lines are replaced with S-shaped curves. 

Fig. 9.6 Computed curves of dimensionless flow rate versus dimensionless pressure gradient for isothermal flow of a power law model fluid in shallow screw channels with the power law exponent n as a parameter, for helix angles 6f as follows O, 30° A, 20° , 10° solid curves are for a helix angle 30°. Note that for n < 1, the reduced flow rate is less than 1, with the deviation diminishing with decreasing of the helix angle. [Reprinted with permission from R. M. Griffith, Fully Developed Flow in Screw Extruders, Ind. Eng. Chem. Fundam., 1, 180-187 (1962).]...

The calculated SBPs in Figs. 9.34—9.36 are based on a model that is no different in concept from the one discussed previously, except that some of the simplifying assumptions were eliminated. In particular, a Power Law model fluid, with a temperature dependent parameter, replaces the Newtonian constant viscosity fluid assumption. 

We acknowledge Dr. Dongyun Ren, who, with the help of Mr. B. J. Jeong and Drs. J. Guo and Linjie Zhu, evaluated, via mathematical regression, the Power Law model, Carreau model, and Cross model parameters. 

In general, a reaction kinetics following a LHHW model is suitable, but the identification of parameters remains demanding. For some catalysts power-law models may be appropriate, for others not. For example, reaction orders identical with stoichiometric coefficients were suitable for Pd/Al203 doped with different metals. On the contrary, for Pd/MgO reaction orders with respect to phenol ranging from -0.5 to 0.5 were observed [17]. However, the bibliographic search was not able to find a quantitative kinetic model for Pd-type catalysts suitable for reactor design. 

The operating methods were tested with two relevant model reaction. Teh kinetic data obtained were fitted to simple power-law models as well as more complicated ones and parameters estimated by the least-square method, activation energies and volumes could be determined and an adequately accuracy in the reproduction of experimental results was always achieved. 

A new mathematical model was developed to predict TPA behaviors of hydrocarbons in an adsorber system of honeycomb shape. It was incorporated with additional adsorption model of extended Langmuir-Freundlich equation (ELF). LDFA approximation and external mass transfer coefficient proposed by Ullah, et. al. were used. In addition, rate expression of power law model was employed. The parameters used in the power model were obtained directly from the conversion data of hydrocarbons in adsorber systems. To get numerical solutions for the proposed model, orthogonal collocation method and DVODE package were employed. 

Because it contains only two parameters (K and n) that can describe shear rate-shear stress data, the power law model has been used extensively to characterize fluid foods. 

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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