Meredith and Tobias

In addition, a model derived by Meredith and Tobias [51] applies to a cubic array of spheres inside a matrix. Consequently, it cannot be used for volumetric concentration above 52% since the spheres will touch at that point. However, their model predicts the thermal conductivity very well up to 40% by volume of particle concentration. When mixing several materials the following variation of Knappe s model applies... 

The electrical properties of heterogeneous media have been modeled for over 100 years. Meredith and Tobias [1962], Mitoff [1968], and McLachlan et al. [1990] have given clear accounts of their scope and vahdity. However, since these articles cover the case where the conductivity or permittivity are real, which normally means dc conductivity or permittivity of loss-free dielectrics, they are not directly apphca-ble to IS. AC properties have been discussed by Wimmer, Graham, and Tallan [1974] with special reference to ceramics. The dielectric literature has been reviewed by van Beek [1965], while Dukhin and Shilov [1974] have described models that include the effects of the interfacial double layer. Sihvola [1999] has produced a comprehensive survey of the properties of mixed phase systems with coverage of the historical and theoretical background. 

Meredith and Tobias noted that Bruggeman s equation overcorrects in the concentrated ranges and devised another approach called the Distribution Model by considering only two size fractions. As in the Bruggeman equation, the smaller size fraction is added first and then is considered as part of a continuous medium having its own bulk conductivity when the larger size fraction of bubbles is added ... 

The parameter a is the ratio of the conductivity of the dispersed phase to that of the continuous phase. Meredith and Tobias extended this result to higher-order terms. Zuzovsky and Brenner used a multipole expansion technique to calculate the effective conductivity of simple cubic, body-centered cubic, and face-centered cubic arrays of spheres. Their technique allowed for fourfold symmetry in the arrays, while those of previous authors did not. McPhe-dran and McKenzie and McKenzie, McPhedran, and Derrick extended Rayleigh s method for calculating the conductivities of lattices of spheres. Their method includes the effects of multipoles of arbitrarily high order specifically, their equation gives the numerical value of the / -order term referred to by Zuzovsky and Brenner. Sangani and Acrivos also used a fourfold potential to calculate effective conductivities of simple cubic, body-centered cubic, and face-centered cubic lattices to 0(/ ). They corrected a numerical slip in the work of Zuzovsky and Brenner. Their equation is... 

The application of this formula, and subsequent studies on the extension of Maxwell s work to more condensed media (i.e. media with large /) in which the shape of the solid is not necessarily spherical has been discussed in detail by Meredith and Tobias [2]. If one tries to represent the Maxwell formula by an array of prisms as shown in Figure 1, in which the electrical current passes vertically through a cube of width and height 1 cm, application of Ohm s law leads indeed to an expression for the conductivity, k, of the suspension in terms of the conductivities k and k of the... 

R. E. Meredith and C. W. Tobias [1962] Conduction in Heterogeneous Systems, in Advances in Electrochemistry and Electrochemical Engineering, ed. 

Meredith RE, Tobias CW. Conduction in heterogeneous systems. In Tobias CW, editor. Advances in electrochemical science and engineering, Vol. 2. New York Interscience, 1962 15 7. 

Figure 8. Comparison of equations predicting the reduced conductivity dispersions of spheres of unequal sizes with data. Bruggemann Eq. (10), Meredith/Tobias Distribution Model Eq. (11). Also included are Maxwell Eq. (6) and Prager Eq. (8) for comparison.

Predictions of the Heterogeneous Conductivity of Simple Cubic Arrays, by Rayleigh s Corrected Equation (12), Meredith/Tobias/ and the Equation of Sangani and Acrivos (13)c... 

Void fraction Rayleigh, corrected Meredith/Tobias Sangani and Acrivos... 

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