The Hanai model for concentrated suspensions

Suppose that the system will finally reach the dielectric constant e when the particle volume fraction reaches cji via the successive addition of infinitesimal small quantities of the dispersed particles, then this process can be summed up as  

Note that the left side of Eq.( 169) is integrated along a certain contour from , to e. It can be evaluated as [T. Hanai, Electrical Properties of Emulsions, in Emulsion Science, P. Sherman, ed.. Academic Press, 1968, p394]  

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

For a special case where the conductivity of the dispersed particle is much higher than that of the dispersing medium, i.e r 7, , the high 

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