Other Empirical Equations

Still other equations for concentrated suspensions depart from the above forms. There are a number of variations on the Mooney equation [393], 

A comparison of some of these models with a range of experimental results is given by Liu and Masliyah [376]. These equations (6.43 and 6.44) and other, similar relations, have also been used to describe the viscosity of both suspensions and emulsions [376,382], 

---

In practical terms the above analysis is tcx) simplistic, particularly in regard to the assumption that the stresses in the fibre and matrix are equal. Generally the fibres are dispersed at random on any cross-section of the composite (see Fig. 3.8) and so the applied force will be shared by the fibres and matrix but not necessarily equally. Other inaccuracies also arise due to the mis-match of the Poisson s ratios for the fibres and matrix. Several other empirical equations have been suggested to take these factors into account. One of these is the Halpin-Tsai equation which has the following form... 

In the absence of any equations based on fundamental material behaviour, equations such as these should be used for extrapolation, although it is generally dangerous to extrapolate in this way by more that than one decade in time. Other empirical equations may be used, provided that Occam s principle is observed, i.e., the theory should be no more complicated than is necessary to fit the observations. Higher order polynomials in time should be avoided, since they have a tendency to diverge rapidly towards infinity, or minus infinity, when they are extrapolated beyond the range of measurements. 

A linear decrease of KIc with an increase in crosslink density was reported for model PU based on triisocyanate and diols of various molar masses (Bos and Nusselder, 1994), and for epoxy networks (Lemay et al., 1984). It was suggested that the dilational stress field at the crack tip may induce an increase in free volume and a devitrification of the material. A linear relationship between GIc and M XJ2 was verified for these systems, although other empiric equations were found in other cases (Urbaczewski-Espuche et al., 1991). 

Other empirical equations have been used by Ferry (1942) ... 

Other empirical equations with different activation energies , such as... 

Note that the term Arrhenius activation energy is to be used only for the empirical quantity defined in the table. Other empirical equations with different activation energies , such as k(T) = A Tnexp( — E JRT), are also being used. 

When heterovalent exchange (e.g., exchange involving both mono- and bivalent ions) takes place, which is frequently seen in soil-solution systems, the quantitative treatment runs into further difficulties. In this case, the determination of mole fractions also becomes problematic. The Vanselow equation assumes that the monovalent and bivalent ions are equivalent when calculating mole fractions. Other empirical equations, however, simply introduce some factors for the ions with different valencies. For example, according to Krishnamoorthy and Overstreet (1949), this factor is 1 for monovalent ions, 1.5 for divalent ions, and 2 for trivalent ions, which is not in agreement in stoichiometry. Another model... 

For the reactions in solid state, several other empirical equations as well as numerical methods were proposed [21] to better simulate the dependence on temperature of the reaction rate (see e.g. [16]). 

There are other empirical equations relating gas exchange to wind speed, such as those given by O Connor (1983) and Yu and Hamrick (1984). 

I would like to make a few quick comments about some other empirical equations of state which are used from time to time. One is the Dieterici equation of state ... 

Of lesser use in physical adsorption is the Jovanovich equation. It is applicable to mobile and localised adsorption (Hazlitt et al., 1979). Although it is not as popular as the other empirical equations proposed so far, it is nevertheless a useful empirical equation ... 

Other empirical equations in the sintering literature include one due to Tikkanen and Makipirtti (34) ... 

The moisture content dependence of the diffusivity can be introduced in the Arrhenius equation by considering either the activation energy or the Arrhenius factor as an empirical function of moisture. Both modifications can be considered simultaneously. Other empirical equations not based on the Arrhenius equation can be used. 

The defining Equations 3.22 and 3.23 produced by Huggins (1942) and Kraemer (1938) apply only to polymers at low concentrations or, more strictly, polymers in the dilute regime, in which interactions between polymer molecules are negligible. Many other empirical equations have been proposed which describe the variation of the polymer viscosity with the concentration at higher concentrations where polymer-polymer molecule interactions are more significant. For example Lyons and Tobolsky (1970) proposed the following empirical form ... 

This equation—sometimes referred to as the Martin equation—has been applied by Chauveteau (1982) in correlating the viscosity-concentration behaviour of xanthan in the semi-dilute regime. A number of other empirical equations which correlate polymer viscosity over a wide range of concentrations are reviewed by Ott (1955). 

---