Power Law viscosity model

The power law viscosity model was developed by Ostwald [28] and de Waele [29]. The model has been used in the previous sections of this chapter and it has the form shown in Eq. 3.66. The model works well for resins and processes where the shear rate range of interest is in the shear-thinning domain and the log(ri) is linear with the log (7 ). Standard linear regression analysis is often used to relate the log... 

Although the analysis here was performed using a power law viscosity model, other models could be used. For other viscosity models, the power law value n would be calculated using two reference shear rates, one higher and one lower than the shear rate calculated using Eq. 7.41. These high and low shear rates and viscosity data would be used to determine a local n value as follows ... 

Derive the equations for pressure driven flow through a slit using a shear thinning power law viscosity model. 

Solve the above problem using a power law viscosity model. 

The die land thickness differences can be compensated by using different land lengths such that the speed of the emerging melt is constant, resulting in a uniform product. If we assume a power-law viscosity model, a uniform pressure in the manifold and an isothermal die and melt, the average speed of the melt emerging from the die is... 

Predicting pressure profiles in a disc-shaped mold using a shear thinning power law model [1]. We can solve the problem presented in example 5.3 for a shear thinning polymer with power law viscosity model. We will choose the same viscosity used in the previous example as the consistency index, m = 6,400 Pa-sn, in the power law model, with a power law index n = 0.39. With a constant volumetric flow rate, Q, we get the same flow front location in time as in the previous problem, and we can use eqns. (6.239) to (6.241) to predict the required gate pressure and pressure profile throughout the disc. 

The material properties of the PE-LD to be used in the calculations are for a power law viscosity model and thermal properties independent of temperature and pressure, except for the melt density ... 

Figure 11.21 Comparison of lubrication approximation solution and RFM solution of the pressure profiles between the rolls for a bank-to-nip ratio of 10, and several power law indices using power law viscosity model.

Figure 11.23 RFM rate of deformation profile (1/s units) between the rolls using a power law viscosity model with a power law index, n, of 0.5.

To include the velocity change in polymer flooding, we have to consider the velocity-dependent viscosity in the Darcy equation 5.45. For the power-law viscosity model, the polymer viscosity is defined by Eq. 5.3, and the shear rate is defined by Eq. 5.23. Then Eq. 5.45 becomes... 

The dynamics of the molten material in the extrusion head is a quite complex process involving the melting point of the materials coupled with flow processes and stresses which are in part related to the viscosity and surface energy as well as the design of the extruder. Most 3D printers use shear thinning materials that follow power-law viscosity models (7.4) ... 

For this work, the Roelands viscosity equation Is chosen to model the temperature Influence on viscosity. When substituted Into the reduced pressure transformation, however, the form of this equation does not Integrate to a slinple algebraic relationship. To remedy this situation the Power Law viscosity model can be fitted to approximate the Roelands equation very well. For the constants provided In the notation, the two viscosity relations differ by less than 4% over the Important pressure range of 0.05 GPa to 0.7 GPa. The resulting expression for q Is... 

However, polymer fluids are not Newtonian and the Newtonian models will not accurately estimate the true metering rate from a screw. Real polymer melts shear thin, which is often estimated with a power law viscosity model in the model development in... 

The data generated for each resin were fit to a power law viscosity model and the resulting curves are shown in the Figure 5. Even though the magnitudes of the resins viscosities are significantly different at comparable shear rates, the slopes of the curves are very similar for these polyethylene resins. The slopes of these curves are related to the power law index, n. The power law indices for these resins at 190 C are 0.36, 0.35, and 0.31 for Resins A, B, and C, respectively. 

The effect of shear thinning on die swell was also studied for the trilobal profile. The power-law viscosity model was chosen. As shown in Figure 6, the die swell effect becomes less as n decrease (or shear-thinning increases). When n = 0, there is no die swell and shape change. This limiting case essentially results in a plug flow. 

Vilchis et al. [81] presented a new idea to achieve better control of the particle size distribution by the synthesis in situ of a water-soluble copolymer of acrylic acid-styrene as suspension stabilizer without additional inorganic phosphate. Publications describe increasing the particle formation by using a physical (population balance, Maxwell fluid, power law viscosity, compartment mixing) modeling approach [22,60,98,105]. 

In this case, the system does not show a yield value rather, it shows a limiting viscosity ri 6) at low shear rates (that is referred to as residual or zero shear viscosity). The flow curve can be fitted to a power law fluid model (Ostwald de Waele)... 

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. 

Non-Newtonian behavior of complex fluids is usually governed by various constitutive laws which relate the viscosity of liquids to the rate of shear. The power-law constitutive model is used in most instances due to its ability to predict rheological behaviors of a wide range of non-Newtonian liquids. The power-law model is characterized with a flow behavior index, n, and a flow consistency index, m. Specifically, n=l corresponds to Newtonian fluids whose viscosity is constant, n< corresponds to shear-thinning fluids whose viscosity decreases with increasing the rate of shear, and n>l... 

Derezinski solved the problem numerically with the barrel temperature as the boundary condition and a power law viscosity [337], Later the analysis was extended to include a Carreau-Yasuda viscosity model [338],... 

Moat lubricants which contain polymers will show a constant viscosity Newtonian behavior up to a critical shear rate. Above that shear rate, the viscosity of these polymer solutions decreases with increasing shear rate and as the first approximation this Shear-thinning region can be represented by a power-law rheological model. At very high shear rates, polymer solutions will also characteristically approach an upper Newtonian limit. 

The most effective model for non-Newtonian flow is the Ostwald-de Waele two-parameter, power-law fluid model [13]. The popularity of this model is easily traced to its Iractability in mathematical manipulations. In the power-law model, the apparent viscosity y = j. l CM Idy = r/y of a polymer fluid subject to a simple steady shear flow is given by... 

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... 

Dilatant fluids (also known as shear thickening fluids) show an increase in viscosity with an increase in shear rate. Such an increase in viscosity may, or may not, be accompanied by a measurable change in the volume of the fluid (Metzener and Whitlock, 1958). Power law-type rheologicaJ equations with n > 1 are usually used to model this type of fluids. 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

Step 3 - using the calculated velocity field, find the shear rate and update viscosity using the power law model. 

Here /, r and, v are unequal integers in the set 1, 2, 3. As already mentioned, in the thin-layer approach the fluid is assumed to be non-elastic and hence the stress tensor here is given in ternis of the rate of deforaiation tensor as r(p) = riD(ij), where, in the present analysis, viscosity p is defined using the power law equation. The model equations are non-dimensionalized using... 

VISCA Calculates shear dependent viscosity using the power law model. 

Rheology. Flow properties of latices are important during processing and in many latex appHcations such as dipped goods, paint, inks (qv), and fabric coatings. For dilute, nonionic latices, the relative latex viscosity is a power—law expansion of the particle volume fraction. The terms in the expansion account for flow around the particles and particle—particle interactions. For ionic latices, electrostatic contributions to the flow around the diffuse double layer and enhanced particle—particle interactions must be considered (92). A relative viscosity relationship for concentrated latices was first presented in 1972 (93). A review of empirical relative viscosity models is available (92). In practice, latex viscosity measurements are carried out with rotational viscometers (see Rpleologicalmeasurement). 

The other models can be appHed to non-Newtonian materials where time-dependent effects are absent. This situation encompasses many technically important materials from polymer solutions to latices, pigment slurries, and polymer melts. At high shear rates most of these materials tend to a Newtonian viscosity limit. At low shear rates they tend either to a yield point or to a low shear Newtonian limiting viscosity. At intermediate shear rates, the power law or the Casson model is a useful approximation. 

The power law model can be extended by including the yield value r — Tq = / 7 , which is called the Herschel-BulMey model, or by adding the Newtonian limiting viscosity,. The latter is done in the Sisko model, 77 +. These two models, along with the Newtonian, Bingham, and Casson... 

Same definitions as 5-26-M. ILff = effective viscosity from power law model, Pa-s. <3 = surface tension liquid, N/m. 

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