Page 149 - Percolation Models for Transport in Porous Media With

8.1 THRESHOLD VALUES. IMPULSE CURRENT 145

Introduce the cylindrical coordinates r, 4>, z with the origin at the center of the
capillaries and with the z-axis coinciding with the axis of the capillaries. In this
case the current density and the energy release density rate (expressed in units
of degree per second) vanish outside the capillaries and are determined by the
expressions
(8.1)
inside the capillaries.
Here Ci, Pi, u~ are, respectively, the specific heat, the density and the electric
conductivity of the fluid or cement in a capillary (depending on what it is filled
with); i = 1, 2.
The heat equation, as well as the boundary and initial conditions in the cylin-
drical coordinates with regard to the independence of temperature and other quan-
tities on the angle 4> and the fact that there is no fluid flow, for such a complex
capillary has the following form,
2
ar
a T
1 a ( aT))
(
Itt az2 + r ar r ar + q(t) = at' (8.2)
I arl
arl
arl
T t=O- ar r=O- az z=-oo - az z=oo - -o .
--
--
-
Here the term that has to do with the source, q(t), is determined from the rela-
tionship (8.1).
Green's function for the problem (8.2), with regard to (8.1), is [83]
2rr
1
[
12
1 1 1 [ (z- Z ) + r + r ] Jo 4~tt(t- t') ]
1 2
2
G(r , z, t , r, z, t) = exp 4 ,.,t(t _ t1 ) [ 4 7rKt(t _ t1)]3/2
Here r 1 , z 1 , and t 1 are the current values of radius, axial coordinate and time;
Jo ( ·) is Bessel's function of a purely imaginary argument of the zero order.
Consequently temperature is determined from the expression
dz
T(z,t) = j dt q(t ) [ J ] dr 27rr G(r ,z ,t ,r,z,t) {8.3)
1
1
1
1
1
1
1
1
0 -oo 0
+ jdz J 2n G(r ,z ,t ,r,z,t)l
dr
1
1
1
1
1
1
0 0
Distribution of temperature for an infinitely long capillary can be found from
(8.3) after setting r 1 = r2 = r and integrating with respect to z and r'. The
1
greatest temperature on the capillary axis (r = 0) at the moment t is
T(t) = w(Io, r, t) A(y') A0 1 (8.4)