again, so long as the surface tension is normal when the bubble is two or three times 

 its initial size, the bubble growth will subsequently be that given by the present analysis. 



The bubble growth as found for our initial conditions may be divided into three 

 phases: A delay period in which the growth of the bubble from equilibrium is very slow 

 being initiated by the slow rise in the liquid temperature; there is essentially no tempera- 

 ture drop in the bubble vapor during this "delay" period. When the bubble has reached 

 approximately twice its initial radius, it has acquired a small velocity and the surface 

 tension effect is reduced. For superheats of the order of 10°C the time t d required for 

 this delay period is about 0.01 millisec; for a superheat of 2°C, t d is 0.07 millisec. The 

 bubble growth appears to begin abruptly near t — t d . In the second or early phase of 

 growth, the bubble velocity increases rapidly and continues to increase until the cooling 

 effect becomes significant. The growth velocity goes through a maximum and thereafter 

 continues to decrease as the third, or asymptotic, phase of growth is reached. At the 

 beginning of the early or second phase of the growth, the heat source term which initiates 

 the growth is negligible and makes no contribution to the subsequent bubble behavior. 

 From a physical standpoint, this means that the observable bubble behavior is inde- 

 pendent of the particular conditions initiating its growth. 



It is clear that liquid inertia effects are important in determining the bubble 

 growth in the second phase, during which the bubble experiences its maximum radial 

 acceleration. After this phase, however, the rate of bubble growth is controlled by a 

 balance between the rate of evaporation and the rate of cooling it produces. It is evi- 

 dent that the bubble must continue to grow since a stationary bubble would be at the 

 temperature of the superheated liquid and therefore would have an excess internal pres- 

 sure. Hence, the temperature at the bubble wall must continue to drop because of 

 evaporation. But the temperature cannot drop below the boiling point and still maintain 

 the pressure difference necessary for growth. It follows that the temperature of the 

 bubble wall must approach the limit T b as f— > co , and this fact is sufficient to characterize 

 the asymptotic phase of bubble growth. 



We may approximate the leading term in the asymptotic phase by simple physical 

 arguments as follows. At a time when R^>R , the difference between the temperature 

 in the liquid at the bubble wall and that in the liquid at a distance is only slightly less 

 than T -T b . This temperature drop takes place principally in a liquid layer around the 

 bubble of approximate thickness given by the diffusion length \/Dt. The heat flow 

 into the bubble per unit time is therefore given roughly by 



k(T - T b ) 



Q « — — 4tt# 2 . (26) 



VDt 



The heat requirement per unit time for evaporation, on the other hand, is 



d /4tt \ 



Q = L — [ — R* P ' ) « 4:ttR 2 RL p '. (27) 



dt\ 3 / 



The approximation indicated in Eq. (27) is based on the fact that the heat flow require- 

 ment because of the volume change is considerably greater than that arising from the 

 relatively small change in vapor density. When (27) is equated to (26), there results 



k (T - T b ) 



R^ — — . (28) 



V VDt 



Analytically one finds that the leading term in the asymptotic region is 



~3 k (T - T b ) 



w Lp Vol 



30 



—^r~ • (29) 



