Shear thinning parameter

Table 7.1 The viscosities of the polyether polyol-CNT mixtures at a low shearing rate of 4.45s (t[l) and a high shear rate of 159.8 s (t h), and shear thinning parameter (n) at room temperature.

Figures 13.37 and 13.38 demonstrate that the shear thinning parameters t]o, , S, Kc, and X have straightforward dependences on c and M. The dependences are in part system dependent. One infers that measurements of r] K) at fixed M and a limited number of values of c, followed by determination of the c-dependence of the fitting parameters, should allow quantitative calculation of t](K) at intermediate concentrations not studied experimentally, and similarly if the roles of M and c are interchanged. Extrapolation as always is fraught with additional uncertainty. Interpolation for t](k) via interpolation on its parameters is the modem equivalent of the master curve approach of the last century, except that parametric interpolation can be effective in systems in which a master curve would be inappropriate.

Equation 5.2 is found to hold well for non-Newtonian shear-thinning suspensions as well, provided that the liquid flow is turbulent. However, for laminar flow of the liquid, equation 5.2 considerably overpredicts the liquid hold-up e/,. The extent of overprediction increases as the degree of shear-thinning increases and as the liquid Reynolds number becomes progressively less. A modified parameter X has therefore been defined 16 171 for a power-law fluid (Chapter 3) in such a way that it reduces to X both at the superficial velocity uL equal to the transitional velocity (m )f from streamline to turbulent flow and when the liquid exhibits Newtonian properties. The parameter X is defined by the relation... 

Equation 5.2, with the modified parameter X used in place of X, may be used for laminar flow of shear-thinning fluids whose behaviour can be described by the power-taw model. 

Many materials are conveyed within a process facility by means of pumping and flow in a circular pipe. From a conceptual standpoint, such a flow offers an excellent opportunity for rheological measurement. In pipe flow, the velocity profile for a fluid that shows shear thinning behavior deviates dramatically from that found for a Newtonian fluid, which is characterized by a single shear viscosity. This is easily illustrated for a power-law fluid, which is a simple model for shear thinning [1]. The relationship between the shear stress, a, and the shear rate, y, of such a fluid is characterized by two parameters, a power-law exponent, n, and a constant, m, through... 

As the behaviour becomes more complicated, more parameters are required to fit the experimental curves. To illustrate this, consider two common equations used to describe the shear-thinning behaviour observed in viscometers. Figure 1.5 shows these two responses. 

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... 

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. 

Since homogenous melts are covered in a later account of pressure build-up and power input in the extruder (Chapter 7), this chapter confines itself to the flow behavior of homogenous unfilled polymer melts and on the introduction of the most important rheological parameters such as viscosity, shear thinning, elasticity, and extensional viscosity. The influence of these rheological properties on simple pressure- and drag flows is demonstrated, while the influence of rheological parameters on pressure build-up and power input in the extruder is described in more detail in Chapter 7. 

It contains the free parameters zero shear viscosity r]0l a critical shear rate yc, at which shear thinning starts, and the gradient m at high shear rates. The reciprocal value of the critical shear rate Yc gives a time constant . At high shear rates, Eq. 3.7 can be approximated by Eq. 3.5 with a constant gradient of m= 1-n. 

The rotational speed, which only appeared as a parameter in the linear Eqs. 7.1 and 7.4, forms now an independent dimensionless parameter in the form of the Deborah number n . While the dimensionless pressure generation and dimensionless energy only depend on the kinematic parameter of flow for Newtonian liquids, the dimensionless revolution speed appears as an additional influencing variable for shear thinning. This is plausible if we consider that the rotational speed is a measure of the shear stresses on the material, and thus influences the effective viscosity of the material. It is also to be expected that the interaction will assume a non-linear form since the flow curve is already non-linear. 

A method for predicting pressure drop and volume fraction for non-Newtonian fluids in annular flow has been proposed by Eisen-berg and Weinberger (AlChE J., 25, 240-245 [1979]). Das, Biswas, and Matra (Can. J. Chem. Eng., 70,431 37 [1993]) studied holdup in both horizontal and vertical gas/liquid flow with non-Newtonian liquids. Farooqi and Richardson (Trans Inst. Chem. Engrs., 60, 292-305, 323-333 [1982]) developed correlations for holdup and pressure drop for gas/non-Newtonian liquid horizontal flow. They used a modified Lockhart-Martinelli parameter for non-Newtonian liquid holdup. They found that two-phase pressure drop may actually be less than the single-phase liquid pressure drop with shear thinning liquids in laminar flow. 

One may try to avoid the problem by the use of the upper-convected derivative, which ensures the coincidence of the principal axes of stress and strain. But doing that, it appears that any kinetics based on the stress amplitude is improper, since materials which exhibits thickening behaviour in elongation are, to the contrary, shear-thinning. Consequently no unique dependence can be expected for these two kinematics. The determination of a single set of parameters in various flows in then bound to be a compromise. 

The preferred units of viscosity are Pa s or mPa s. Some of the older units and their relation to the SI unit are given in Table 1-1. It is clear that the shear rate employed in the calculation must be specified when the magnitude of apparent viscosity is discussed. Apparent viscosity has many useful applications in characterizing a fluid food in particular, in the characterization of shear-thinning fluids, the apparent viscosity at low shear rates, called the zero-shear rate viscosity ( jo), is a useful parameter. 

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