Hanai equation

The basis for interpreting the transport data is mixture theory, which relates the transport properties of the bulk suspension to those of the continuous and dispersed phases. Of the many mixture relations that have been proposed, we employ those of Maxwell and Hanai (Equations 1 and 2, respectively) ... 

Figure 4. Water self-diffusion coefficient D of the micro-emulsions and PEO solutions, normalized to that of the pure liquid water. The need for the additional factor (1-p) is described in the text. Also shown are predictions of the Maxwell and Hanai equations.

The so-called Hanai equation (8,9) gives the complex (frequency dependent) permittivity of an emulsion as... 

Calame [16] has proposed a numerical simulation method to solve an equation of the similar form as Hanai equation with power of w-th fraction, as shown in equation 9.29, and has described a method of empirically determining the value of n using least-squares fitting. Equation of the same form as equation 9.29 has also been discussed by Tuncer et al. [17] in context of effect of inclusion shape factor on the permittivity of the mixture. 

There are a number of formulas for the determination of the effechve dielectric constant Maxwell Garneh (MG) formula, Bruggeman (BG) equation, Hanai-Bruggeman (HBG) formulas, empirical Lichtenecker mixture equahons, and so on. 

T. Hanai, Quantitative structure-retention relationships of phenolic compounds without Hammett s equations,/ Chromatogr., A, 2003, 985, 343-349. 

Hanai s equation gives rise to dispersion curves that are broader than the Maxwell-Wagner equation (Takashima, 1989). Similar work had also earlier been done by Bmggeman (1935). 

Eq.(171) is called Hanai s equation, which can be considered as the complex form ofBruggeman s equation [76], shown in Eq. (171a). 

Several authors have derived formula for dielectric permittivity of a two component mixture from differential analysis. Most of them considered the excess polarization due to a small sphere introduced to the effective medium where the volume of the sphere can become infinitesimal enabling establishment of a differential equation for the effective permittivity of the mixture [30], The particular solutions obtained for a 3D case by three of the most notable authors contain a fractional power of a third. These are Looyenga formula [31], Bruggeman [29] and Hanai formula [32] and formula by Sen et al. [33] given in equations 9.25, 9.26 and 9.27, respectively. 

The Hanai-Bmggeman equation (11.7) can be applied on carbonate rocks with a matrix porosity and fracture porosity. If fractures have a preferred orientation, conductivity is a tensor. The tensor simplifies if the axes of the... 

Coutanceau-Monteil and Jacquin (1993) investigated the permittivity of sedimentary rock samples in the frequency range from 20 to 1000 MHz. A comparison of model equations and experimental results shows that the GRIM model fits the data for water-saturated quarry limestones, whereas the Hanai-Bruggeman equation fits the data for water-saturated Fontainebleau sandstones fairly well. 

Thus, the GRIM equation is identical with the generalized Lichtenecker and Rother (1931) equation with a = 0.5 and with the Hanai-Bruggeman equation for L = 033. 

Aligned ellipsoids can be used for modelling of fractured rocks with a preferred fracture orientation. For aligned ellipsoids, Sen et al. (1981) show that the differential effective medium estimate for an ellipsoidal inclusion in a host material is (Hanai-Bmggeman equation) ... 

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