Power law model equations

A wide range of temperatures are encountered during processing and storage of fluid foods, so that the effect of temperature on rheological properties needs to be documented. The effect of temperature on either apparent viscosity at a specified shear rate (Equation 2.42) or the consistency index, K, of the power law model (Equation 2.43) of a fluid can be described often by the Arrhenius relationship. The effect of temperature on apparent viscosity can be described by the Arrhenius relationship ... 

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... 

As stated previously, for non-Newtonian foods, the simple power law model (Equation 2.3) can be used to describe shear rate (y) versus shear stress (cr) data at a fixed temperature ... 

As the power law model [Equation (20.3)] fits the experimental results for many non-Newtonian systems over two or three decades of shear rate, this model is more versatile than the Bingham model, although care should be taken when applying this model outside the range of data used to define it. In addition, the power law fluid model fails at high shear rates, whereby the viscosity must ultimately reach a constant value - that is, the value of n should approach unity. 

The power law model (Equation 2) has found extensive use in the food literature ... 

Qiu and Rao (Qiu, C. G. and Rao, M. A. J. Texture Stud., submitted) determined slip coefficients and slip velocities for apple sauce in a concentric cylinder viscometer as well as the effect of insoluble solids content on them. Three concentric cylinder units specified in the theory of Mooney (42.) were employed. Rotational speeds were determined with the different concentric cylinder systems at the same magnitude of torque. Figure 2 shows, for one sample of apple sauce, the shear rates uncorrected and corrected for slip plotted against the shear stress. The magnitudes of the flow behavior index of the power law model (Equation 2) did not change significantly due to correction for wall slip however, the magnitudes of the consistency index increased due to wall slip corrections. 

The simple concept of an average mixer shear rate has been widely used in laboratory and industrial work and in most applications it has been assumed that the shear rate constant, k, is only a function of impeller type. Research is continuing on the possible influence of flow behaviour index and elastic properties, and also on procedures necessary to describe power consumption for dilatant fluids. It should be noted that in all aspects of power prediction and data analysis, power law models (equation 8.12) should only be used with caution. Apparent variability of k, may be due to inappropriate use of power law equations when calculations are made it should be ascertained that the average shear rates of interest (y = k N) lie within the range of the power law viscometric data. 

The limited information reported so far suggests that the appment viscosity-shear rate data often result in linear plots on double logarithmic coordinates over a limited shear rate range and the flow behaviom may be represented by the power-law model, equation (1.13), with the flow behaviom index, n, greater than one, i.e. 

Non-Newtonian characteristics are introduced by expressing the wall shear in the capillary tube as an equivalent shear derived from a rheological model such as the power-law model (Equation 1) or the Carreau Model A (Equation 2). Derivations of polymer flow models based upon power-law and Carreau Model A are found in references 6 and 7. Equation 7... 

Figure 22 shows the effect of polymer concentration on the flow curves of Statoil polymer in deionized water. At polymer concentrations 2,000 ppm, the apparent viscosity was constant at low shear rates (Newtonian behavior) and decreased at higher shear rates. The Carreau model. Equation 8, predicts the experimental data for this polymer concentration range fairly well. At polymer concentrations > 2,000 ppm, the flow curves showed a shear thinning behavior only. The power-law model. Equation 9, predicts the data fairly well at shear rates > 1 s. ... 

We use the power law model. Equation (7-2), along with information from the pnih-lem statement that the reaction is first order in methanol. (B). i.e.. = I to obtain... 

A common choice of functional relationship between shear viscosity and shear rate, that u.sually gives a good prediction for the shear thinning region in pseudoplastic fluids, is the power law model proposed by de Waele (1923) and Ostwald (1925). This model is written as the following equation... 

Empirical Models. In the case of an empirical equation, the model is a power law rate equation that expresses the rate as a product of a rate constant and the reactant concentrations raised to a power (17), such as... 

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. 

In a series of experiments on the flow of flocculated kaolin suspensions in laboratory and industrial scale pipelines(26-27-2Sl, measurements of pressure drop were made as a function of flowrate. Results were obtained using a laboratory capillary-tube viscometer, and pipelines of 42 mm and 205 mm diameter arranged in a recirculating loop. The rheology of all of the suspensions was described by the power-law model with a power law index less than unity, that is they were all shear-thinning. The behaviour in the laminar region can be described by the equation ... 

The power of this technique is two-fold. Firstly, the viscosity can be measured over a wide range of shear rates. At the tube center, symmetry considerations require that the velocity gradient be zero and hence the shear rate. The shear rate increases as r increases until a maximum is reached at the tube wall. On a theoretical basis alone, the viscosity variation with shear rate can be determined from very low shear rates, theoretically zero, to a maximum shear rate at the wall, yw. The corresponding variation in the viscosity was described above for the power-law model, where it was shown that over the tube radius, the viscosity can vary by several orders of magnitude. The wall shear rate can be found using the Weissen-berg-Rabinowitsch equation ... 

Akande, et al.40 in the reforming of crude ethanol containing H20/EtOH ratio of about 18 employed a power law model and expressed the rate equation as shown in eqn (16) ... 

Even though the two-site model has shortcomings, it is excellent for fitting intensity quenching curves. Thus, it has excellent predictive and calibration properties, has a chemically sound basis, and (at least for inorganic complex sensors) is preferable to the less accurate power law calibration equation. 

The simplest generalized Newtonian constitutive equation is the power law model that assumes the viscosity has the following dependence on shear rate,... 

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. 

The mechanism of the formation of phosgene according to the reaction, CO (A) + C12(B) =4 C0C12(C), is to be checked with given data at 30.6 C (Potter Baron, CEP 47 473, 1951). Six Langmuir-Hinshelwood equations and a power law model are examined. The rate equations are analyzed in linearized forms. Those that have negative constants are not physically realistic. 

The derivation of the fiber spinning equations for a non-Newtonian shear thinning viscosity using a power law model are also derived. For a total stress, axx, in a power law fluid, we write the constitutive relation... 

Example 3.5 The CEF Equation in Steady, Fully Developed Flow in Tubes The viscosity functions in both the Power Law model GNF fluid and the CEF fluid are expected to be... 

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