Quemada equation

For concentrated systems, equation 31 loses its appeal because of the difficulties involved in evaluating the higher order coefficients. Instead, semiempirical models are more suitable. Such models are given by the Mooney equation (77), the Krieger-Dougherty equation (67), and the Quemada equation. In particular, the Quemada equation is the most used equation because of its simplicity and utility. The Mooney equation is given by... 

Equation 50 does not agree with the experimentally observed Quemada equation. However, by comparing equations 48 and 49 with 50, we may expect that the longitudinal viscosity of the cylindrical fiber suspension is smaller than the viscosity of spherical particle suspensions at a concentrated state, whereas the transverse viscosity is higher than the viscosity of spherical suspensions. 

Kitano et al. (82) found that the relative viscosity of a suspension of cylindrical rods can be estimated by the Quemada equation with the maximum packing fraction given by... 

By an asymptotic matching of the dilute limit homogeneous viscosity with the Quemada equation, Phan-Thien and Graham (118) obtained the following equation valid for a wide range of solid concentration ... 

Equations 66 and 68 indicate that the droplet behaves like a solid particle only when the viscosity ratio of the dispersed phase to the continuous phase is large. For liquid-in-liquid dispersions, the modified Quemada equation, Krieger-Dougherty equation, and Mooney equation are still applicable provided that the maximum packing limit and the Einstein constant are left as adjustable parameters for a given system. 

Figure 19-13. Relative zero shear stress viscosities of hard sphere silica suspensions as a function of volume fraction. The symbols represent the data of different researchers. The solid line is the Doolittle Eq. (19-13) with 0m = 0.638, and the dashed line is the Quemada equation with 4>m = 0.638 (Marshaall, 1990). Good agreement with the Doolittle equation suggests an exponential increase in viscosity in the glassy regime ( > 0.5).

In diluted solutions Quemada s theory leads to the following equation ... 

To illustrate, solution of problems of this kind in the percolation theory, Quemada [71] cites the analysis of the dependence of electrical conductivity on concentration for a mixture of conducting and nonconducting spherical particles carried out by Fitzpatrick [80] or Clerk [81]. According to Clerk, this dependence is described by an equation similar to Eq. (66) or other similar formulas ... 

Quemada et al. (1985) proposed a viscosity equation for dispersed systems based on zero-shear, tjq, and infinite-shear, i oo> viscosities, and a structural parameter, k, dependent on the shear rate, that may be written as ... 

Quemada et al. (1985) model (Equation 2.11) was used to analyze data on cocoa dispersions (Fang et al., 1996) and the role of cocoa butter replacers (Fang et al., 1997). Selected values of rheological properties of chocolate are given in Table 5-G. In addition, the data of Fang et al. (1996) (Table 5-H) before and after degasification as a function of temperature are note worthy. 

Heuristic derivations of equations 32 and 33 have been presented by Mooney (77), Krieger (67), Ball and Richmond (78), and Stein (79). The Quemada expression is given by... 

The same line of arguments can be made for the Quemada expression, for example, letting e = 1 and making appropriate assumptions when equation 39 is considered. However, one can notice that the Quemada expression is not very accurate for dilute systems where equation 31 may be useful. It is also possible to match equations 31 and 33 asymptotically to give a better equation that is suitable for the entire range of volume fraction,... 

In deriving equation 32 or equation 33, it is assumed that 0max is the solid volume fraction at which the suspended particles cease moving. Thus, the forces, such as shearing, that can disturb the suspension structure and hence improve the mobility of particles will have an effect on the value of max. This is confirmed by the fact that a value of kH = 6.0 is observed at low shear limit, that is, y 0 and at high shear limit, y - oo, kH = 7.1 is found. Typical values of max have been found with the use of Quemada s equation as 0max = 0.63 0.02 in the low shear limit and high shear limit for submicrometersized sterically stabilized silica spheres in cyclohexane (72, 85, 88). 

Jones and co-workers (72, 88) found that the suspension viscosity variation with shear rate can be fitted fairly well by the Cross equation, equation 12, with m = 0.5 — 0.84. Both the low and high shear limit relative viscosities, Mro, Mroo> can be expressed by the Quemada s equation with 0max = 0.63 and 0.71, respectively. 

A similar equation but with the exponent of —2 was advanced by Quemada [31]. For instance, this type of equation was shown to describe well the experimental data obtained on relatively monodisperse dispersions of hydrophobically modified silica particles suspended in cyclohexane [32], i.e., systems that can well be regarded as good model systems for a hard-sphere fluid. Typical values for O , are 0.6-0.7 [32], i.e., values that correspond to the packing fraction of a cubic array of spheres. 

Equation 10.73 can be solved using a regularization procedure under conditions of non-negativity f(E) > 0 at any E and fixed or unfixed regularization parameter determined by statistical analysis of the experimental data and (Quemada and Berli 2002) ... 

In 1977, Quemada [52] derived Equation 11.78 with exponent -2 by applying the minimum energy dissipation principle. The exponent -2 was verihed by several experiments and confirmed by the theoretical work of Brady [53] on the basis of statistical mechanics. 

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