Viscosity models Ellis model

Ellis rjQ = zero shear rate viscosity (i) Ellis model is expressed in terms... 

If r]oo <3C ( /. j/o), the Carreau model reduces to a three-parameter model ( 0,k, and p) that is equivalent to a power law model with a low shear limiting viscosity, also known as the Ellis model ... 

As discussed in Chapter 3, the Carreau viscosity model is one of the most general and useful and reduces to many of the common two-parameter models (power law, Ellis, Sisko, Bingham, etc.) as special cases. This model can be written as... 

The Ellis model (41), is a three-parameter model, in which the non-Newtonian viscosity is a function of the absolute value of the shear stress tensor, x,... 

The three parameters are a, which is the slope of the curve log(/y/— 1) vs. log(r/Tj /2 ) ii/2, which is the shear stress value, where r = t 0/2 and t]0, which is the zero shear viscosity. Thus the Ellis model matches the low shear Newtonian plateau and the shearthinning region. 

The Ellis model is in general given as a shear stress dependent viscosity r/([Pg.547]

Electrophorus, 329 Electrostatic interaction, 275 Elliptic anisotropy, 289 Ellis model, 547 Elongation/Elongational, 459 at break, 454, 475 flow, 532,585 rate, 459 viscosity, 585 at yield, 457 Emeraldine, 344 End-to-end distances, 246, 248 End groups, 7, 8 End of pyrolysis, 765... 

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 power law does not describe the regions of the viscosity curve near y = 0 and y —> oo. To this end, the Ellis model at low shear rates and the Sisko model at high shear rates can be used (2). The models are given by... 

A plot of viscosity vs. shear rate (Figure 5.5) shows that the fluid approaches Newtonian behavior at low shear rates and is shear thinning at higher shear rates, with power law-type behavior. As shown in Figure 5.5, the Ellis model gives a good representation of these data. 

Ellis model See viscosity, non-Newtonian flow Ellis model. 

The two viscosity equations presented so far are examples of the form of equation (1.11). The three-constant Ellis model is an illustration of the inverse form, namely, equation (1.10). In simple shear, the apparent viscosity of an Ellis model fluid is given by ... 

The test for dynamic frequency-dependency of complex viscosity q was used as the measuring method for each substance. Relaxation time was calculated from G to G above the intersection point, and the zero shear viscosity was mathematically derived by means of the so-called Ellis-fit (Fig. 32). The viscosity model, according to Ellis allows the calculation of zero shear viscosity using the following formula (compare Fig. 32) ... 

Zero shear viscosity represents a function of The following Figure 32 relates the Ellis model coefficient to the viscoelastic data. 

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. 

The Ellis fluid parameters allow the construction of a time constant given by tio/t /2). One of the advantages of the Ellis model is that the low to high shear rate range can be spanned continuously, without the need to patch together two separate equations (cf. TPL model). On the other hand, the shear rate dependence of viscosity now becomes implicit (via the magnitude of the shear stress, r). 

For flow in a narrow gap viscometer, the energy equation and the momentum balance are coupled together by the temperature-dependent viscosity. These equations have been solved for the equilibrium temperature profile and the effect on shear stress by Gavis and Laurence (1968) for a power-law fluid and by Turian (1969) with the Ellis model. For the power law model, the effect on torque in a narrow gap instrument can be expressed in terms of a power series in the Brinkman number (see Example 2.6.1, eq. 2.6.15). The first term of the series is helpful to the experimentalist to indicate where shear heating can affect data. 

FIGURE 2.6 Viscosity versus shear rate (In-ln plot) for LLDPE at 170 °C. The data were obtained by various types of rheometers as indicated in the figure. (—) Graphical fit of power law (—) fit of power law using regression analysis. Insert shows plot of (h/ho) — 1 versus shear stress used to obtain Ellis model parameters. 

B.3 Forced Convection Heat Transfer in Tubes-Short Contact Times. A polymeric fluid whose viscosity function is described by the Ellis model is flowing through the tube as shown in Figure 5.26. Determine the temperature profile and the wall heat flux for the... 

Zero-shear viscosity Shear-thinning exponent Upper and lower Newtonian Random coil strnctnre Relaxation time End-to-end distance Deborah nnmber Weissenberg nnmber Constitntive eqnation Power-law llnid Ellis model Cross model... 

Table 2 also shows rheological parameters obtained by fitting the experimental data of dynamic viscosity vs. frequency to an Ellis model [9]. The viscosity extrapolated to zero frequency values can be considered proportional to the weight average molecular weight of the polymers. Indeed, the values are inversely proportional to the MFI values reported in Table 1. cot is the frequency onset at which pseudoplastic effects start... 

The Cross and the temperature-dependent Cross-WLF model (42) is an often used GNF-type model accounting for, like the Ellis and Carreau fluids for the viscosity at both low and high shear rates,... 

The utility of mathematical models incorporating power-law Eq 10.2 is limited to either a small range of shear rate, or to its high values. To improve the description of the viscosity dependence on shear rate, several alternative relations were proposed, e.g., by Ellis, Bueche, Eyring, Carreau [Bird et al., I960]. The latter relation ... 

The shear-dependent viscosity of a commercial grade of polypropylene at 403 K can satisfactorily be described using the three constant Ellis fluid model (equation 1.15), with the values of the constants fiQ = 1.25 x lO Ea s, Ti/2 = 6900 Pa and a = 2.80. Estimate the pressure drop required to maintain a volumetric flow rate of 4cm /s through a 50 mm diameter and 20 m long pipe. Assume the flow to be laminar. 

In this model proposed by Ellis and discussed by Reiner [1], the apparent viscosity versus shear rate relationship is given in the following form ... 

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