Viscosity 3.4 power law

Temperature coefficient of viscosity (power-law consistency index) at constant shear Specific heat... 

Melt Viscosity n The resistance to shear in a molten resin, quantified as the quotient of shear stress divided by shear rate at any point in the flowing material. Elongational viscosity, which comes into plan in the drawing of extrudates, is analogously defined. In polymers, the viscosity depends not only on temperature and, less strongly, on pressure, but also on the level of shear stress (or shear rate). See Viscosity, Power Law, and Pseudoplastic Fluid. 

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. 

When m = 1.0, as in Fig. 2.5, the exponent becomes zero and the viscosity is independent of 7 when m = 0.7, a factor of 10 change in 7 results in a decrease of viscosity by a factor of 2. This is approximately the case for the data in Fig. 2.5 for 7 values between 10" and 10" sec". Equation (2.14) and its variations are called power laws. Relationships of this sort are valuable empirical tools for extrapolating either F/A or t over modest ranges of 7. In such an application, the exponent m - 1 and the proportionality constant are... 

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). 

For non-Newtonian fluids the correlations in Figure 35 can be used with generally acceptable accuracy when the process fluid viscosity is replaced by the apparent viscosity. For non-Newtonian fluids having power law behavior, the apparent viscosity can be obtained from shear rate estimated by... 

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... 

Polymer melts are frequendy non-Newtonian. In this case the earlier expression given for the shear rate at the capillary wall does not hold. A correction factor (3n + 1)/4n, called the Rabinowitsch correction, must be appHed in such a way that equation 21 appHes, where 7 is the tme shear rate at the wall and nis 2l power law factor (eq. 22) determined from the slope of a log—log plot of the tme shear stress at the wad, T, vs 7. For a Newtonian hquid, n = 1. A tme apparent viscosity, Tj, can be calculated from equation 23. 

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

When log (viscosity) is plotted against log (shear rate) or log (shear stress) for the range of shear rates encounterd in many polymer processing operations, the result is a straight line. This suggests a simple power law relation of the type... 

Assuming that the melt viscosity is a power law function of the rate of shear, calculate the percentage difference in the shear stresses given by the two methods of measurement at the rate of shear obtained in the cone and plate experiment. 

Universal SEC calibration reflects differences in the excluded volume of polymer molecules with identical molecular weight caused by varying coil conformation, coil geometry, and interactive propenies. Intrinsic viscosity, in the notation of Staudinger/ Mark/Houwink power law ([77]=fC.M ), summarizes these phenom-... 

For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

For a shear-thickening fluid the same arguments can be applied, with the apparent viscosity rising from zero at zero shear rate to infinity at infinite shear rate, on application of the power law model. However, shear-thickening is generally observed over very much narrower ranges of shear rate and it is difficult to generalise on the type of curve which will be obtained in practice. 

Some fluids exhibit a yield stress. When subjected to stresses below the yield stress they do not flow and effectively can be regarded as fluids of infinite viscosities, or alternatively as solids. When the yield stress is exceeded they flow as fluids. Such behaviour cannot be described by a power-law model. 

As in the case of Newtonian fluids, one of the most important practical problems involving non-Newtonian fluids is the calculation of the pressure drop for flow in pipelines. The flow is much more likely to be streamline, or laminar, because non-Newtonian fluids usually have very much higher apparent viscosities than most simple Newtonian fluids. Furthermore, the difference in behaviour is much greater for laminar flow where viscosity plays such an important role than for turbulent flow. Attention will initially be focused on laminar-flow, with particular reference to the flow of power-law and Bingham-plastic fluids. 

When the fluid behaviour can be described by a power-law, the apparent viscosity for a shear-thinning fluid will be a minimum at the wall where the shear stress is a maximum, and will rise to a theoretical value of infinity at the pipe axis where the shear stress is zero. On the other hand, for a shear-thickening fluid the apparent viscosity will fall to zero at the pipe axis. It is apparent, therefore, that there will be some error in applying the power-law near the pipe axis since all real fluids have a limiting viscosity po at zero shear stress. The procedure is exactly analogous to that used for the Newtonian fluid, except that the power-law relation is used to relate shear stress to shear rate, as opposed to the simple Newtonian equation. 

The rheological properties of a particular suspension may be approximated reasonably well by either a power-law or a Bingham-plastic model over the shear rate range of 10 to 50 s. If the consistency coefficient k is 10 N s, /m-2 and the flow behaviour index n is 0.2 in the power law model, what will be the approximate values of the yield stress and of the plastic viscosity in the Bingham-plastic model ... 

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... 

Two liquids of equal densities, the one Newtonian and the other a non-Newtonian power law fluid, flow at equal volumetric rates down two wide vertical surfaces of the same widths. The non-Newtonian fluid has a power law index of 0.5 and has the same apparent viscosity as the Newtonian fluid when its shear rate is 0,01 s-1. Show that, for equal surface velocities of the two fluids, the film thickness for the non-Newtonian fluid is 1.125 times that of the Newtonian fluid. 

Uncoupled Rate Constants. An initial evaluation of polymerization kinetics is presented in Figure (2), constrained by viscosity invariant rate constants K. The slopes of these straight lines give initial estimates of Rgg/Kp according to Equation (14). Figure 3 presents graphically a power law relationship between K g/Kp and viscosity at 21°C and at 16.6 C. More scatter In Yu s data may be attributed to the use of an older GPC instrument of relatively low resolution. The ratio Kgq/Kp is temperature-sensitive a change of the order or five times is observed if the temperature is reduced by 4.4°C and viscosity is kept constant. 

It is also observed in SFA experiments that the effective viscosity declines in a power law, as the shear rate increases. The observations of the dynamic shear response of conhned liquid imply that the relaxation process in thin hlms is much slower and the time for the conhned molecules to relax can increase by several orders. 

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.

---