Non-Newtonian Power-law Models

In power-law fluids, the relationship between the shear stress and shear rate is nonlinear and a finite amount of stress will initiate flow. The mathematical model (sometimes called the Ostwald-de-Waele equation) 

In power-law fluids, an apparent viscosity, / app/ is defined in a similar manner to a Newtonian fluid. 

Most slurries are shear-thinning. It is h5q othesized that this shear-thinning behaviour is due to the formation of particulate aggregates which provide a lower resistance to flow than fully dispersed particles. 

2 Pressure Drop Prediction for Slurries Exhibiting Power-law Rheology 

The fluid momentum balance applied to the case of laminar, fully developed flow of a power-law fluid in a horizontal pipe of diameter D yields the following expression for the relationship between the pressure drop, AP/L, and the average flow velocity, vav- 

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For non-Newtonian Power Law model fluids, these ratios are... 

Favelukis et al. (37,38) dealt with the problem of droplet deformation in exten-sional flow with both Newtonian and non-Newtonian Power Law model fluids, as wellas bubble breakup. For the Newtonian case, they find that as an inviscid droplet (or bubble) deforms, the dimensionless surface area is proportional to the capillary number... 

Wolf and White (21) verified the theoretical RTD function experimentally with radioactive tracer methods. Figure 9.12 gives some of their results, indicating excellent agreement with theory. RTD functions in extruders using non-Newtonian Power Law model fluids have also been derived (22,23). 

Chapter 5 deals with the aspects of the flow behaviour of polymer melts which are relevant to the processing methods. The models are developed for both Newtonian and Non-Newtonian (Power Law) fluids so that the results can be directly compared. 

Chapter 4 describes in general terms the processing methods which can be used for plastics and wherever possible the quantitative aspects are stressed. In most cases a simple Newtonian model of each of the processes is developed so that the approach taken to the analysis of plastics processing is not concealed by mathematical complexity. Chapter 5 deals with the aspects of the flow behaviour of polymer melts which are relevant to the processing methods. The models are developed for both Newtonian and Non-Newtonian (Power Law) fluids so that the results can be directly compared. 

This energy-based method of calculation of viscosity rj is due to Einstein [87], who considered hydrod)mamic dissipation in a very dilute suspension of non-interacting spheres. Tanaka and White [86] base their calculations on the Frankel and Acrivos [88] cell model of a concentrated suspension, but use a non-Newtonian (power law) matrix. The interaction energy is considered to consist of both van der Waals-London attractive forces and Coulombic interaction, i.e. 

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. 

A Newtonian liquid of viscosity 0.1 N s/m2 is flowing through a pipe of 25 mm diameter and 20 m in lenglh, and the pressure drop is 105 N/m2. As a result of a process change a small quantity of polymer is added to the liquid and this causes the liquid to exhibit non-Newtonian characteristics its rheology is described adequately by the power-law model and the flow index is 0.33. The apparent viscosity of the modified fluid is equal to ihc viscosity of the original liquid at a shear rate of 1000 s L... 

For non-Newtonian fluids, any model parameter with the dimensions or physical significance of viscosity (e.g., the power law consistency, m, or the Carreau parameters r,]co and j/0) will depend on temperature in a manner similar to the viscosity of a Newtonian fluid [e.g., Eq. (3-34)]. 

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. 

A non-Newtonian fluid, described by the power law model, is flowing through a thin slit between two parallel planes of width W, separated by a distance H. The slit is inclined upward at an angle 0 to the horizontal. 

Basic Protocol 2 is for time-dependent non-Newtonian fluids. This type of test is typically only compatible with rheometers that have steady-state conditions built into the control software. This test is known as an equilibrium flow test and may be performed as a function of shear rate or shear stress. If controlled shear stress is used, the zero-shear viscosity may be seen as a clear plateau in the data. If controlled shear rate is used, this zone may not be clearly delineated. Logarithmic plots of viscosity versus shear rate are typically presented, and the Cross or Carreau-Yasuda models are used to fit the data. If a partial flow curve is generated, then subset models such as the Williamson, Sisko, or Power Law models are used (unithi.i). 

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... 

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. 

To calculate the shear rate constant, k, a relationship must be established between shear rate and viscosity of a non-Newtonian calibration fluid. A cone-and-plate viscometer is used to determine a correlation between shear rate and viscosity that can be fit to a power law model. The power law correlation is then applied to viscosity data calculated from the impeller viscometer and Eq. 4. The shear rate constant can be calculated as follows ... 

This disagreement between theory and practice must therefore partly be due to the non-Newtonian shear-thinning viscosity. This conclusion is supported by the work of Kiparissides and Vlacopoulos (35), who showed that for a Power Law model fluid, lower n values reduce the disagreement between theory and experiments, as illustrated in Fig. 6.27. They used the FEM for computing the pressure profile, which eliminates the geometrical approximations needed in the Gaskell model. 

Non-Newtonian Flow between Jointly Moving Parallel Plates (JMP) Configuration Derive the velocity profile for isothermal Power Law model fluid in JMP configuration. 

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. 

The former vanishes when the velocity of the moving plate is zero, and the latter vanishes in the absence of a pressure gradient, (a) Explain on physical and mathematical grounds why the solution of the same flow problem with a non-Newtonian fluid, for example, a Power Law model fluid, no longer leads to the same type of expressions, (b) It is possible to define a superposition correction factor as follows... 

The most common non-Newtonian fluid characterization in polymeric and biological applications is the power-law model... 

The radial film thickness profiles at different durations of spinning and for the four different rheological models (Newtonian, Power-law, Carreau and Viscoplastic) were obtained by numerical methods (12-13). Figure 3 shows representative film thickness profiles for Newtonian (n = 1) and non-Newtonian (n < 1) liquids. 

In contrast to a cone and plate geometry to be discussed next, the shear rate of non-Newtonian foods cannot be determined from a simple expression involving the angular velocity and often one must use a suitable relationship between rotational speed and shear stress to correct for non-Newtonian behavior. More complex equations are needed to describe the flow of non-Newtonian fluids in concentric cylinder geometry. For example, for fluids that can be described by the power law model, an expression presented by Krieger and Elrod (Van Wazer et al., 1963) has been used extensively in the literature ... 

For non-Newtonian fluids that can be described by the power law model, Lin (1979) showed that the RTD is given by the expression ... 

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 ... 

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