Page 185 - Percolation Models for Transport in Porous Media With

11.1 TEMPERATURE EFFECTS 183

a is determined from the normalization condition for f(r).
For a* » a. we have

a= (i- 1)a:- 1 , Tc = a.~;- 1 /(i- 1 )
</>(rl) = (i + 1)/(i- 1) = </>o, ,P(rl) = (i + 3)/(i- 1) = 1/Jo
1
1
dn(rt)jdr1 = no(a./rd[(a./rd- - (a./rc)i- ] (11.4)
1
Io(r1) = Eo(1 </>o7rr~, k(r1) = (1rj8)rt'lf1o
q(r, rt) = E5(1'(rtfr) </>~
4
We shall now use the solution to the problem (8.2) presented in §8.1. This
problem deals with the temperature distribution in a long cylindrical capillary
with radius r in a boundless medium (with temperature conductivity the same as
that of the fluid in the capillary) as electric current J(t) = J0i(t) passes through it.
After determining q 0 according to (11.2) and substituting the obtained expression
in the relationship (8.3) for q(t'), after transformations we obtain the following
1
1
T(t,z) = 1/2+ j qo(xt){1-exp[(-y'(1-x)- ]}
0
x{1 + q>[zr- 1 (y'(1- x))- 1 1 2 ]}dx (11.5)
Here y' is determined according to (8.4), x is a dummy integration variable,
and q>( ·) is the error integral defined as

a;
q>(x) = Jrr J exp( -e)d~ (11.6)
0
After differentiating (11.5) with respect to z and t, we obtain, respectively, the
temperature gradient along the capillary
1
T~(t,z) = ~ 1
j qo(xt){1- exp[(-y'(1- x))- ]}
0
x exp{ -z 2 [r 2 y'(1- x)t 1 }(1- x)- 1 1 2 dx (11.7)

and the rate of the temperature change in the capillary
1
T(t, oo) = q 0(t)- ;, j q 0 (x, t)exp{[-y'(1- x)- ]}(1- x)- dx (11.8)
1
2
0
When q 0 = Qp = const, where Qp = I6/(ciPi(1 7r r ) is the current density
1 2 4
amplitude, it follows from that (11.5)- (11.8) that
1
T(t, oo) = Qpt{1- exp[-(y')- 1 ]- y'- Ei[-(y')- 1 ]} (11.9)