Williamson viscosity model

The Ellis (46), Carreau (47), and Cross-Williamson (48) models are three-parameter models. The Ellis model gives the viscosity as a function of T o, the shear rate cr, the parameter ai/2, which is the shear rate for which the viscosity is rio/2, and a dimensionless parameter a. The equation is... 

The Williamson equation is useful for modeling shear-thinning fluids over a wide range of shear rates (15). It makes provision for limiting low and high shear Newtonian viscosity behavior (eq. 3), where T is the absolute value of the shear stress and is the shear stress at which the viscosity is the mean of the viscosity limits TIq and, ie, at r = -H... 

If some or all of this curve is present, the models used to fit the data are more complex and are of two types. The first of these is the Carreau-Yasuda model, in which the viscosity at a given point (T ) as well as the zero-shear and infinite-shear viscosities are represented. A Power Law index (mi) is also present, but is not the same value as n in the linear Power Law model. A second type of model is the Cross model, which has essentially the same parameters, but can be broken down into submodels to fit partial data. If the zero-shear region and the power law region are present, then the Williamson model can be used. If the infinite shear plateau and the power law region are present, then the Sisko model can be used. Sometimes the central power law region is all that is available, and so the Power Law model is applied (Figure H. 1.1.5). 

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). 

For the investigated emulsions, i oo tio Under this condition, the Peeck-Mak-Lean-Williamson equation transforms into the Ferry equation and, therefore, is not suitable for description of the viscosity of extracting emulsions. In the Meter equation, the term (P/Pav) ( oo/T o) approaches 1 at Tjoo i1o- Thus, Equation 8 is transformed into the Ellis equation. Values of P1/2 and the exponential coefficient A for the Ellis model are presented in Figure 8. It should be noted that the value for A is constant and equal to 6 in the equation which describes the rheological curves of the extracting emulsions for the indicated range of dispersed phase content. 

Quantitative comparisons were made by fitting the Williamson model to the flow curve data. The zero-shear viscosity and rate index are two parameters obtained from the Williamson model fits. The zero-shear viscosity is the viscosity valne extrapolated to infinitely low shear rate, while the rate index is the slope of the viscosity versus shear rate relationship in the shear-thinning region at high shear rates. These two parameters are very sensitive to differences in the molecular weight distributions of the resins. Table 15-Tl lists the zero-shear viscosities and rate indexes of the two resin samples. 

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