Ellis equation

Show that the Ellis equation corresponds, practically, to a newtonian fluid at low shear rates and a power-law fluid at high shear rates. Show what constant or combination of constants in the Ellis equation corresponds to /a in Fig. 15.6. 

Better fits to the portions of the flow curve that are near y Q and y oo are the Ellis (Equations 6.26a and 6.26b) and Sisko (Equation 6.27) models, respectively ... 

Fit the power law, the Ellis equation, the Eyring equation, and the Reiner-Phil-lipoff equation to the data of Figs. 3-19 and 3-21. 

Bird and Sadowski [16] developed equations to treat the flow of polymer solutions through porous media using the Ellis equation. Later work found pressure drop and flow data for molten polymers through porous media [10,15]. This later work used the power law as its rheological base. Check to determine whether the Ellis equation approach can describe these data. 

The creep results of most solid materials, in the customarily observed linear followed by power-law regions, can be fairly well described by the empirical Ellis equation (omitting the usual high-shear-stress asymptotic viscosity), viz. 

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 equation (and consequently the Meter equation) adequately describe the experimental dependence of the emulsion viscosity on the shear stress over the entire range of dispersed phase contents because the values of the Fisher criterion do not exceedthe tabular value (Table I). Thus, the Ellis equation allows the variation in emulsion viscosity with in the indicated range for different values of shear stress to not only be described, but also predicted. 

Figure 8. The (1) Average Shear Stress Pjyj and (2) Coefficient A for the Ellis Equation Versus the Content of the Dispersed Phase in the Emulsion.

Table I. Values of the Fisher Criterion for the Emulsion Viscosity Using the Ellis Equation...

Burghes, D. N. and Borris, M. S. (1981) Modelling with Differential Equations. Ellis Horwood. 

Huntley, I. D. and Johnson, R. M. (1983) Linear and Nonlinear Differential Equations, Ellis Horwood. 

Harry B. Gray and Walther Ellis,13 writing in Chapter 6 of reference 13, describe three types of oxidation-reduction centers found in biological systems. The first of these, protein side chains, may undergo oxidation-reduction reactions such as the transformation of two cysteine residues to form the cystine dimer as shown in equation 1.28 ... 

Ellis(36) presented the following dimensionally consistent equation for the HETP (Zt) of packed columns using 25 and 50 mm Raschig rings ... 

Brown, P.L. Sylva, R.N. Ellis, J. An equation for predicting the formation constants of hydroxo-metal complexes. J. Chem. Soc., Dalton Trans. 1985, (4), 723-730. 

There are numerous other GNF models, such as the Casson model (used in food rheology), the Ellis, the Powell-Eyring model, and the Reiner-Pillippoff model. These are reviewed in the literature. In Appendix A we list the parameters of the Power Law, the Carreau, and the Cross constitutive equations for common polymers evaluated using oscillatory and capillary flow viscometry. 

At t = 1, the values of x should yield the difficult solution F(x) = 0. This technique resembles the relaxation method, but only requires modifying the independent MESH functions to get the derivatives with respect to one term t. This is a purely mathematical approach and Ellis et al. (78) state that it can give negative flow rates at intermediate values of t, something that if-value and enthalpy routines may not tolerate. An alternative is a homotopy function that is rooted in the MESH equations themselves. 

Several [CoL(NH3)5]3+ and m-[CoL(X)(en)2]2+ complexes have been prepared (Table 26) and some crystal structures are available (Table 25). Pentaammine complexes are easily made from [Co(DMSO)(NH3)5](C104)3, and there is no need to use a better leaving group.340,341 The physical properties ( H NMR, visible-UV, 13C NMR, pATa) for both systems (L = imH) have been comprehensively documented.341,342 The acidity of the N3 proton in (65) is enhanced by coordination and the imidazolate anion is readily accessible (pXa 10.0 vs. 14.1 for imH). Several deprotonated bimetallic bridged species have been prepared and structure (66) has been put forward as a model for heterobiometallic protein centres.343 The tetrazole complexes (68) exist predominantly in the anionic form (pA 1-2)351 and preparation normally gives the N(2)-bonded form. However Ellis and Purcell352 have prepared the N(l) isomer by reacting a coordinated nitrile with N3 ion at pH 5 (equation 52). Properties of both isomers are available.351... 

Ellis and coworkers have extensively studied the chemistry of hexacarbonyltitanate(2-), [Ti(CO)6] . Reductive car-bonylation see Reductive Carbonylation) of Ti(CO)4 (trmpe) with (cryptand 2,2,2)potassium naphthalenide at -70 °C followed by warming to room temperature gives the thermally robust [K(cryptand 2.2.2)]2[Ti(CO)6] in quantitative yield. Treatment of Ti(CO)6 with azobenzene gives [Ti(PhN=NPh)(CO)4] (equation 3) in 40-65% yield. Hydrolysis of [Ti(PhN=NPh)(CO)4] " gives 1,2-diphenyUiydrazine and the hydroxo-carbonyl titanium complex, [Ti2(/x-OH)2(CO)8] (equation 3), which was strac-turally characterized as the [K(18-crown-6)]+ salt. ... 

The standard procedure in LCAO calculations has been to use Gaussian Type Orbitals, GTO, or Slater Type Orbitals, STO, which are normally used in HF based codes but also in DFT codes. With such an approach the H and S could be given in closed forms. This was also partly the procedure used in the first molecular calculations using the DVM method by Baerends, Ellis and Ros [77]. The DVM method [22-24] is however quite general and very suitable for use of numerical basis functions, as first introduced by Averill and Ellis in 1973 [78]. They proposed that a first choice of basis functions for molecular calculations would be to solve the Schrodinger equation to self-consistency for the isolated atoms in a molecule, which will give... 

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... 

Thermodynamic consistency tests are well known, and have been frequently used for vapour-liquid equilibrium data in binary mixtures (for reviews one can see Gmehling and Onken, 1977 Acree, 1984 Prausnitz et al., 1986). These tests are based on theGibbs-Duhem equation and allow one to grade the experimental data for vapor-liquid equilibrium in binary mixtures. A more difficult problem is the consistency of data regarding vapor-liquid equilibrium in ternary or multicomponent mixtures. However, several thermodynamic consistency tests, also based on the Gibbs-Duhem equation, were suggested for vapor-liquid equilibrium in ternary or multicomponent mixtures (Li and Lu, 1959 McDermott and Ellis, 1965). 

Although the preparation of aryl cyanides from the corresponding halogen compounds is described by the very simple equation (8), the process does not correspond to a simple aromatic substitution reaction. In a series of comprehensive articles various aspects of this transformation are discussed. An excellent review, written by Ellis and Romney-Alexander, should be mentioned in particular, as it provides the literature leading into the modem status of the field. 

Despite this drawback, a number of studies based almost entirely upon the use of rats have shown that dietary fibers from various sources may impair iron absorption (Table 1). The fiber sources examined most often were fiber-rich mill fractions of wheat, generally bran. The use of isolated fiber components is confined largely to cellulose. Although wheat bran is rich in phytate, the work of Morris and Ellis (34) supported by subsequent publications from the same source, indicates that interference by phytate with iron absorption is not appreciable, and can be disregarded. Liebman and Driskell (35) and Hunter (36) have also found no interference by phytate with iron metabolism, confirming the earlier report of Cowan et al (22). However, Simpson et al (61) very recently recently reported that a phosphate-rich extract of dephy-tinized bran inhibited iron absorption. Thus, it is not permissible to equate bran with fiber completely. 

Wesselingh, J. A. and Krishna, R., Mass Transfer, Ellis Horwood, Chichester, England, 1990. Whitaker, S., Role of Species Momentum Equation in the Analysis of the Stefan Diffusion Tube, ... 

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