Page 189 - Percolation Models for Transport in Porous Media With

11.2 MOVEMENT AND GROWTH OF BUBBLES 187

sphere is much greater than inside it. Thus we can consider the temperature of
vapor inside the bubble to be constant and equal to the temperature T. of the
saturated vapor which corresponds to the vapor pressure p 11 inside the bubble

{11.18}
Thus we can consider the properties of the vapor to be observed only by changing
its pressure.
According to the first law of thermodynamics, we have

6Q 11 = du 11 + p 11d(1/ p 71 ) (11.19}
where oQ,, du 11 are, respectively, the amount of heat absorbed by and the change of
the internal energy of, a unit mass of vapor in the bubble. The correlation between
the heat flow q~ through the surface of the boundary layer and the quantity oQ,
is determined by the following equations

4
3
31l'a p 716Q 71 = q 11 dt ( 11.20 )
I
q~ = -k11(8T,f8r,)lr,.=a-o · 41l'a 2
The equation of state for vapor is

p11 = p 11R,T 11 , R11 = Rf tt11 (11.21}
In the above, R = 8.31 ·10 3 kilojoule per mole-kelvin is the universal molar gas
constant; /Lp is the mass of one mole of vapor; k 11 is the heat conductivity of the
vapor; r, is the radial coordinate with the origin in the center of the bubble; and
a(t) is the current value of the latter's radius. Total internal energy of the vapor
is
Up = C~T, + Upa
where u,o is the initial energy of the vapor and C~ is the specific heat of the
vapor at a constant volume. It follows from (11.18) that dT 8 = dT 11 , and finally
we obtain the following

(11.22)
where c: is the specific heat of the vapor at a constant pressure and -y 71 is the
isentropic exponent for the vapor.
After differentiating the equation of state (11.21) with respect to T 11 , we obtain
(11.23)

Using the Clausius-Clapeyron equation with p 11 = Pa