Page 192 - Percolation Models for Transport in Porous Media With

190 CHAPTER 11 GAS COLMATATION IN ELECTRIC ACTION

The equations (11.31), (11.32) are good for describing the growth of a single
bubble in a boundless medium. However there may be many bubbles in a capillary
and the average distance between them may be relatively small compared to their
size (la ~ (10 -10 3 )a). Therefore it is necessary to take into account the interaction
of thermal fields of the bubbles.
Suppose that at the time dt, an amount of energy equal to 7rr 2 le(t) was released
in a capillary. (In the general case, heat losses caused by the outflow through the
capillary surface must be taken into account in e(t).) Furthermore, suppose that
this amount of energy was distributed uniformly among 7rr 2 ln0 bubbles in the
capillary. Now, taking into account QJ ~ qp, we obtain
(11.34)

If we assume that not all, but only a part of the energy e(t) was used for
supplying heat to the bubbles, then the rest of the energy will be used for the fluid
overheating. The latter will make heat losses through the capillary surface larger,
and therefore will decrease e(t). In its turn, this will create an additional flow q/

in (11.28), which is going to partially comensate the mentioned loss, and therefore
is likely to make the quantity QJ closer to e(t)fn0 in its value. It is possible to
estimate the additional flow qf, which may cause deviation from equality (11.34),
as follows. If there is no heat discharge, the temperature at a distance ""' la from
a bubble must grow by approximately AT = e(t)rk/(Cfpf) in a characteristic
period of temperature equalizing

Tk = l~/Kt

Therefore q/ ~ kJATl; 1 47ra 2 ~ 10a 2 lae(t).
In this case the ratio
(11.35)
since (afla) « 1.
Initial conditions

a(O) = ao, Pp(O) =PI+ 2xfao
and the equations (11.27), (11.26), (11.31), (11.33), (11.34) define the relations
a(t) and Pp(t) completely.
Introduce the notations

Ao = no(4/3)7rag, A= Ao(no/n.), X= Qp/(PJLp),
1
H = XoRpTpPJP~ , Z = 2xf(aopoo)
and the dimensionless variables

O. = Xt, w. = afao, G. = Pp/Poo