Page 34 - Percolation Models for Transport in Porous Media With

2.1 CONDUCTIVITY OF GRAINED MEDIA 25

Here w is the angular frequency (O ~ w < 10 radian/s), Ti is the character-
5
istic period of ion diffusion in the channels. The function ~(w) characterizes the
relation between resistance and frequency: ~(0) = 1, ~(oo) = 0. Resistance is
determined only by the quantity Po which characterizes the non-polarized two-
component medium

(2.3)

where li is the length of the i-th channel. The conductivity of the i-th channel is

1rr~ , ( >..' )
O'i = _, C1 1 + 2- .
li u'ri
IT does not depend on frequency and is a function of the channel parameters
and of the equivalent ion transfer numbers ni± in the solution, which depend on
the ratio >..' f(u'ri)
n = (nl+n2- - nl-n2+) 2 .
CT1n1+n1-/a1 + u2n2+n2-/a2
S. M. Scheinmann used the formula (2.2) to calculate the effective conductivity
of the medium and its frequency dependence. The conductivity was then averaged
with regard to the channel size distribution function. In the discussed approach,
the structure of the conducting infinite cluster formed in the medium is determined
using this distribution function. Then the effective electric conductivity of the
medium is calculated with regard to structure of the conducting chains. As it
was showed in §1.2, conductivity of the IC is determined from parallel conducting
chains whose distribution with respect to r 1 is described by the function F(r1 ).
Given the mean values < u( r1) > of electric conductivities for the chains, a formula
equivalent to (2.1) can be obtained for calculation of the electric conductivity of
the medium

2
2
I}= 2-yvl- (1- P:)- " 1 [7 f(r) dr]" < u(rl) > /(rl) dr1 • (2.4)
0 r1
Taking account of the relationship (2.2), one can find< u(r1 ) >for a chain of
successive resistances

< u(r1) > -l= z- 1 i [po(r) + IT~(w)]f(r) dr ( 7 f(r) dr) - 1 (2.5)
~ ~1

Formulas (2.4) and (2.5) determine the effective electric conductivity of a
grained medium. Focus on the calculation of < u(r1) >. Formula (2.2) was
obtained for the case of two cylindrical channels connected to each other. How-
ever the geometry of channels in a grained medium can be different. Yet due to