Page 191 - Percolation Models for Transport in Porous Media With

11.2 MOVEMENT AND GROWTH OF BUBBLES 189

On the other hand, conservation of the total energy of the phases yields

Lpdmp + (q~ + qf )dt = 0 {11.28)
_ 0 I
qf- -qf- qf
q/ = -kt(8Tp/8rp)jrp=a+O · 47ra 2
where qf is the heat flow through the boundary of the liquid phase; qJ is an
additional heat flow due to the release of the joule heat in the fluid as the electric
current passes through it.
Since the heat conductivity of vapor kp is much less than the heat conductivity
of liquid k" it follows that lqtl ~ lqpl·
Transform {11.28) to get

{11.29)
After substituting ~12 from {11.29) into {11.27) and changing Pp/Pp for {11.23)
with regard to {11.24) and qp for {11.26) with regard to {11.25), we obtain

[ 3 a
3q1
Pp = -xo - + -:----::-7- ] {11.30)
-
Pp a 47ra 3 Lppp
xo = /'p[1 + (l'p -1){1-1jr.) t 1
2
The movement of the bubble boundary is described by the Rayleigh-Lamb
equation [89, 90]

2
a ~t (a- 62/ P!) + ~(a- 62/ P! )
= (pp- Pt- 2x/a)/ Pt- 4J.t(a- ~12/ Pt )/(ap,) {11.31)
Due to the fact that the liquid is virtually incompressible, and its radial velocity
is relatively small, we can set p 1 = Poo.
The heat flow qf must be determined from the solution of the exterior heat
conductance problem for rp > 0 using the initial and boundary conditions

{11.32)

Here the effective temperature growth rate T in a capillary, defined by the
expression {11.10), can be taken as €(t)

2
b(t) = qp[1- exp( -r /(4Ktt))] (11.33)