Page 188 - Percolation Models for Transport in Porous Media With

186 CHAPTER 11 GAS COLMATATION IN ELECTRIC ACTION

The bubbles can merge at this stage of the process due to the attraction forces
that act when the distance between the bubbles is of the order of two to three
times their radius. Attraction between the bubbles is caused by the overlap of the
low-pressure domains in the surrounding fluid. These domains are formed once
the balance between the surface tension and the vapor and gas pressure inside a
bubble is established. When a0 < w- 7 m attraction forces can be significant,
since the pressure outside the bubbles becomes negative.
It is evident from (11.2) and (11.6) that the first to be filled with bubbles are
the thinnest r 1-capillaries in r 1-chains. For probability density functions of the
form (11.3), we have the following, with regard to (11.4)

48/.L"-t 1 (a*) 2 (11.17)
Tg = (-x')E5u'¢>~ ao ~

Note that during the period of"' r 9 , the density of bubbles in the thin capil-
laries increases from the initial value n. to no ~ n.(a* fr) 2 , since a thin capillary
is most likely to borrow almost all capillaries from the adjacent pore. However,
adjacent capillaries separated by a pore are not likely to compete in the future,
since an own "warm" capillary is more attractive for any bubble than perhaps a
"hotter" one, but separated from it by a big "cold" pore.
For a bubble in the fluid being heated by the electric current, two essentially
different phases of growth can be specified.
At the first stage, when the temperature of the fluid T < n(p 00 ), where Poo
is the fluid pressure at a large distance from the bubbles, Tb is the boiling point
which corresponds to the pressure p 00 , the bubble is in the quasi-steady state. At
the second stage, when T = n(Poo), the bubble is not in a steady state, and its
limitless growth begins. The rate of this growth is determined only by the rate of
the energy supply to the bubble.
Estimates show that for the actually achievable values q 0 of the energy release
rate in micro capillaries in different media, the pattern of a homogeneous steady
state vapor bubble can be used. In this pattern, the velocities of the radial motion
of the phases and the surfaces of the bubbles are considered much less than the
velocities of the molecular thermal motion or the velocity of sound.
At the vapor-liquid surface, there is a thin boundary layer, where the phase
transitions to and from vapor take place. For a balanced system, it is assumed that
the temperature Tp inside the bubble near the boundary layer, the temperature
T,p in the boundary layer, and the temperature Ta in the fluid near the boundary
layer are all equal
Tp = T.y, = Ta
The vapor temperature conductivity "-v is much greater than that of the liquid
""t. and therefore the time for establishing the temperature balance outside the