experiments on the dynamics of phase growth

Chemical Engineering Science, 197 1, Vol. 26, pp. 675-684. Pergamon Press. Printed in Great Britain.

Experiments on the dynamics of phase growth

M. SADDYi and G. J. JAMESON

Department of Chemical Engineering and Chemical Technology, Imperial College, London S.W.7, England

(First received 3 June 1970; in revisedform 24 July 1970)

Abstract-The growth of vapour bubbles at a prepared nucleation site in uniformly superheated liquids has been observed.

With acetic acid, the bubble growth follows the asymptotic theory:

(radius) a (time)“*.

However for water and solutions of glycerol in water, the exponent was in the range 0.75 to 1 .O. It is suggested that the discrepancy is due to the translatory motion of the bubbles as they grow.

INTRODUCTION

THE GROWTH of gas bubbles in liquids by diffusion-controlled mechanisms is an important feature of certain transfer processes. In nucleate boiling the rate of bubble growth is determined sometimes by diffusion of heat to the evaporating interface, and in the case of multicomponent liquids, mass diffusion may also play a role. Be- cause of the importance of bubble dynamics with phase growth not only in nucleate boiling but also in electrolysis, effervescence and cavi- tation, there have been a number of attempts to solve the equations of motion and energy in order to predict the bubble volume as a function of time. However, this is a formidable problem and many simplifications have to be made in order to enable analytical solutions to be obtained. Compre- hensive reviews of progress to date have been given by Bankoff [ 1,2].

The most successful analyses so far have dealt with the case of simplest geometry, that of spherically symmetric growth of a single bubble in a large body of uniformly superheated or super saturated liquid. The earlier works of Plesset and Zwick [3,4] and Forster and Zuber [5] were primarily concerned with change of phase by simple evaporation. However, in a very thorough work, Scriven[6] extended the theory to cover bubble growth by mass transport and

combined mass and heat transport. This paper also gives a very clear exposition of the simplifications and assumptions involved.

In practice it is virtually impossible to dupli- cate experimentally the conditions required by this theory, because there is always a translatory motion brought about by buoyancy. Several workers have observed bubbles growing in free rise in superheated liquids, with conflicting results. Dergarabedian[7] and Kosky[8] have found that the bubble radius increased like (time)‘/* as predicted by the theories. On the other hand Darby]_9] found a dependence of (time)2’3, so that the bubbles grew faster than predicted. In these works the bubbles were moving relative to the liquid so it is to be expected that the heat transfer to the bubble surface would be enhanced by the forced convection of the translatory motion.

In order to reduce the effects of relative motion, we have carried out experiments in which bubbles are formed at an artificial nucleation site in a uniformly superheated liquid. By taking high speed motion films of the bubbles, the radius can be measured as a function of time from inception and compared with the theory. Since the nucleation site is finite (although small) the initial radius of the bubble is known. This is an important parameter, because the theories are

tPresent address: C.O.P.P.E., Federal University of Rio de Janeiro, Brazil.

675

M. SADDY and G. J. JAMESON

described as “asymptotic”, that is the predictions are only valid if the radius is sufficiently large for certain terms in the equation of motion to be dropped. By choosing nucleation sites of the correct diameter this criterion can be satisfied. Furthermore, while the bubble grows attached to the cavity, its translational velocity is very small, and the conditions envisaged in the theory should come close to realization.

inertial, viscous and surface tension terms to be ignored in the equation of motion. Thus (1) reduces simple to pb = pm, and the bubble growth is determined entirely by the rate at which heat can be conducted through the liquid to provide the latent heat of vaporization at the interface.

Two types of liquid have been used. As single component systems we have used water and acetic acid, and as multi-component mixtures, solutions of glycerol in water at three different concentrations.

Consider a stationary bubble of radius a in a liquid essentially infinite in extent. The liquid is uniformly superheated by an amount AT. Spherical symmetry may be assumed so that the equation of motion may be integrated to give the Rayleigh equation [6]:

where p is a “growth constant”, (Y is the thermal diffusivity and t is time measured from the initia- tion of the motion, assumed to occur when the radius of the bubble is a,, a s a,; to is the time at which a = ao, i.e. to = (ao/2~d’2)z. In the experiments described below, we take a, to be the radius of the nucleation site. Scriven[6] showed that for one-component systems

p -_ (y2 AT

( (PG/PL) WCL + cc,, - CGVCLW) t5j

aii+?g+4yi_Pb-Pm-2h (1) a EPL

676

where pb is the pressure inside the bubble, and pm is the pressure in the liquid a large distance from the bubble; E is the density factor (1 - pc/pr,).

provided p + 1 and the dimensionless superheat cLAT/L is sufficiently large, i.e. generally larger than 1O-2. For binary systems where both heat and mass transport are important, the analogous expression for p is:

The energy equation is, after substituting for the velocity from the equation of continuity:

8T -=N (a, 2dT\__a%iT at -“\a? ’ r Jr/ c r dr

where viscous dissipation and internal tion are taken to be zero or non-existent.

(2)

again valid for p % 1. genera-

The equation of mass flow, again after substituting for velocity from continuity, is:

The Eqs. (1-3) may be solved by approximate analytical methods if it is assumed that the bubble is in an “asymptotic” stage of growth, that is, the radius a is sufficiently large for the

A diagrammatic sketch of the apparatus is shown in Fig. 1. The technique used was similar to that of Dergarabedian[7]. The liquid was placed in a glass vessel between two 250 W infra-red lamps. Power input was controlled by a Variac autotransformer. Temperatures were measured by a mercury-in-glass thermometer, calibrated to @l”C. As has been demonstrated by Dergarabedian, this method of direct radiant heating minimizes temperature gradients and

THEORY

The solution to the equations is of the form

a = Qjal/z(t + to)112 = 2/3&/271’2 (4)

p = (.y

AT L + (pr, - C,)R,T,,,(p,clkl~)“2 cl, c,,C&W2Cm + CPL L Cco)MI

(6)

APPARA’TUS

a Light source

Cu So4 sajutian ~ Nucleation

/ site

/ Boiling

vessel

lamp

High speed

camera

Fig. I. Diagrammatic sketch of apparatus.

therefore free convective currents throughout the liquid. The mercury-in-glass thermometer is convenient, since these are annealed and consequently have a very smooth finish. Thus spontaneous nucleation on the thermometer is not likely. Also, the glass is essentially trans- parent to radiation in the infra-red, while the mercury is highly reflective, and absorbs little of the incident radiation. Dergarabedian carried out the test of putting an aluminum sleeve over the thermometer with the liquid boiling at 102°C. The thermometer reading was not changed by the presence of the shield.

No prepared nucleation sites were used by Dergarabedian, so that he had no practical way of knowing just when or where a bubble would form. In this work, a nucleation cavity was formed on the end of a glass rod, bent in the form of a U so that while the rod was suspended from above, the cavity opening was vertically up- ward. The cavity was made by taking a piece of glass capillary tube, fusing it in the centre to give a cylindrical cavity with one closed end, and then drawing the tube out until the cavity was of the required diameter. The cavity was of the form of a hollow cylinder with a conical end, and the length was reduced to reasonable propor- tions by grinding. Two sizes of cavity were used: (a) 0.002 cm radius, 0.46 mm long with H,O and


acetic acid, and (b) 0.0045 cmradius,0*570cm long for the glycerol solutions. The reason for the difference in size was that when the smaller cavity was tried with water, such large superheats were necessary that spontaneous nucleation took place at many different sites in the boiling vessel. This produced unwanted bubble streams which interfered with the photography of bubble behaviour at the prepared nucleation site. With the more viscous glycerol solutions this was not a problem and so a smaller nucleation cavity could be used. The nucleation site was placed in the centre of the glass container which was 7 cm dia. and 14 cm high. Photographs of bubble behaviour were taken through an optically flat glass window incorporated in the side of the vessel. A Fair- child HS 10 1 16 mm camera of the rotating prism ’ type was used, in the range of 1000 to 2500 fames 1 sec. Accurate timing was achieved with a 100 cps light pulse on the edge of the film. The camera lens system produced X 2 magnification, and the developed film was displayed on a motion picture analyser of X 10 magnification. The vertical and horizontal cross wires could be located to &O.OOl in. For recording purposes, the zero time was taken to occur on the frame on which a = a, was first observed. This often took place between frames so that there is a maximum error of - 0.001 sec.

The following procedure was carried out be- fore each experiment was conducted. The vessel and cavity rod were cleaned with chromic acid, washed and dried. The liquid to be used in the experiment was poured into the vessel to a level about 6 cm above the glass window, and heated at the base by an electric heater. The liquid was boiled for about twenty minutes to (eliminate dissolved gases and to provoke nucleate boiling and nucleation sites on the sides of the vessel and especially in the vicinity of the window. On cooling the liquid to about 10°C below the boiling point, these cavities filled with liquid and became inactive. During this boiling step, the vapour formed was condensed and returned to the vessel by a glass reflux condenser.

The boiling liquid in its container was then placed between the two infrared lamps and

677

M. SADDY and 0. J. JAMESON

heated at the rate of about 1 “C per min, to ensure a uniform temperature distribution. (This rate was found by trial, placing one thermometer near the wall and another at the centreline). When the temperature exceeded the boiling point the artificial nucleation site was immersed and the heating adjusted to give the required superheat.

Profiles of a bubble growing in water boiling at 104.O”C are shown in Fig. 2. It can be seen that the bubbles are very close to being either spherical segments, as in the early stages of growth, or spheres in the later stages. According- ly from the film, the maximum horizontal dimen- sion of the bubble was read as the diameter. After breakoff the bubbles tended to be spher- oidal, but here the dimensions were not recorded.

the bubble radius were taken were determined by two considerations. The lowest limit on the superheat should be such that the approximate analytic solutions given by Scriven[6] are valid, that is, the growth constant p should be large, and also the dimensionless superheat cLATIL. The upper limit was set by the requirement that there should be no coalescence between succes- sive bubbles. Further experimental details are described by Saddy [ 171.

Water

In Fig. 3 are shown the experimental results for water boiling at three temperatures: 102.0, 103.0 and 104*O”C. The data points lie quite closely on a straight line of slope 2/3 indicating that

RESULTS

gf IO a-

20 c

30

44 45 46 48

Fig. 2. Profiles of a bubble growing in water boiling at 104.0” C. The frames are numbered from the frame on which a = a, was first observed. Film speed: 1100 f.p.s. Cavity radius

a, = 0.022 cm.

The one-component liquids used were distilled water (b.p. 100°C) and acetic acid (b.p. 118.O”C). For binary mixtures, solutions of glycerol in water were used at boiling points ranging from 106.1” C to 17O$NZ. The physical data used in the calculations are given in the Appendix. The boiling temperatures at which measurements of

as had been found by Darby[9]. In order to check that the results had not been distorted by the zero time to in T = t + t,,, the data were also plotted in the form (~/2p@)~ vs. the real time t, which was the experimental time measured from the frame on which a = a, was first observed. This graph is shown in Fig. 4. If the

l-

>-

I I IO 0

TXIO? set

Fig. 3. Growth of steam bubbles at superheats of 2.0, 3.0, and 4.O”C, as a function of the adjusted time 7 = I+ lo.

678


+ x I03 5ec

Fig. 4. Linear plot of (~/2po”~)~ against experimental time t for water boiling at three temperatures.

radius did follow a one-half power dependence on time, the points should lie on a straight line. Clearly they do not. At each superheat the points lie on curved lines indicating that the radius depends on the time raised to a power greater than one-half.

Acetic acid

The results for acetic acid are given in Fig. 5 at temperatures of 121-O and 124*O”C corresponding to superheats of 3-O and 5*O”C, and also in

Asymptotic

I theory

Fig. 5. Growth of bubbles in acetic acid at two superheats.

Fig. 7 at 122.O”C (4W’C). In each case the data lie very close to the asymptotic theory. The data are also plotted as (+/2/kP)* vs. time t in Fig. 6. For temperatures of 121-O and 122.O”C the data lie on straight lines verifying the half-power

200

"0 -

x

"- 100

0 -%

% -

0 50 100

t x I03 set

Fig. 6. Linear plot of (a/2~cP)’ against experimental time r for boiling acetic acid.

IOC

“0 - x

- ‘C -ia

O@l k-5

‘!= t c

)-

ovv v

94% Glycerol

Acetic acid

Asymptotic

theory

T x IO”. set

Fig. 7. Growth of bubbles in (a) 94% w/w glycerol solution, b.p. 156.O”C; (b) acetic acid, b.p. 118.O”C.

dependence of cr with 7. Because of the difficulty in obtaining accurate physical data with which to compute (Y and p, no significance should be attached to the differences in slope of the data

679


lines. At 124.O”C the data line is slightly curved. In each case there is a small negative intercept on the time curve, in line with the values of f,, given in Table 4. In fact, we have fitted the data by the method of least squares to an equation of the form

certainty in the determination of the physical data needed to calculate p and (Y.

a =A(f+t,)“2 (7)

where A, to were regarded as variables. The theoretical and experimental values are shown in Table 1. The table also includes values at temperatures of 120-O and 123WC which are not represented in Figs. 5, 6 and 7 for reasons of space. The agreement is very good at the highest superheats but not so at the lowest two values, as is perhaps to be expected.

I I 10

Table 1. Experimental and theoretical growth para- meters for acetic acid (b.p. I 18°C)

Experimental

WI. (7))

Theoretical Fig. 8. Growth of bubbles in glycerol solutions. The boiling points of the solutions are: 104.7”C (45%); 121.O”C (80%);

and 156.O”C (94%).

T A to P A = 2@zP t,, (“C) (cm/sec”2) (set) (Striven (cm/secl@) (set)

[61)

120 0.258 0.0079 3.447 0.178 0.0153 121 0.363 0.0040 5.146 0.265 0.0069 122 0.384 0.0024 6.830 0.352 0.0039

123 0.426 0.0024 8.499 0.438 0.0025 124 0.532 0.0022 10.149 0.522 0.0018

The results show that while the growth of bubbles in boiling acetic acid is accurately described by the theory, the theory fails in the case of water and glycerol-water solutions.

Glycerol solutions

The experimental data are shown in Figs. 7 and 8. In only one of these cases (45% w/w glycerol boiling at 106.1”C (AT = 1*4”C) ) is the time dependence close to +. In the other three cases the exponent is much higher, being almost unity. The nucleation site used with the glycerol solutions was much smaller (a,, = 0.0045 cm) than that used with the water and acetic acid (a0 = O-022 cm). Consequently the displacement times to as given in Table 4 are very small. There is considerable separation between data lines for the various concentrations and superheats, much more than with the water and acetic acid. This is probably a reflection of the greater un-

For water the results indicate a two-thirds power dependence of the radius with time. This agrees with the results of Darby[9] but is in apparent disagreement with the findings of Dergarabedian[7] and Kosky[8]. In the former work the times of observation were limited by the experimental techniques, and were no more than 15 msec, compared with the 100 msec here. Thus it is possible that the exponent was different from the one-half predicted by the theory, but the time span was not sufficient for the difference to show up on the graphs. Inci- dentally, it should be pointed out that Dergara- bedian made no claims as to the value of the exponent. His conclusion was simply that his results were much closer to the theory of Plesset and Zwick[4] than to the “extended Rayleigh” theory assuming isothermal growth. The work of

45% 106~lT 0 60% 127~O'C * 94% 170~0°C +

0

T x lot 5ec

DISCUSSION

680


Kosky was conducted at a much higher superheat than we have used (17°C) at a reduced pressure (0.488 atm).

At this point we should justify, in the light of the results, the application of the asymptotic theory. The major simplification in this theory is the assumption that the inertial, viscous and surface tension terms in Eq. (1) are all negligible, so that (1) reduces to pb = pm, and the volume of the bubble is simply equal to the volume of the evaporated liquid at the pressure pm and the corresponding saturation temperature. Thus we must show that

pLaii 3~9 4ppLci 2u --__ Pm ’ 2pm ’ up, ‘up,

are all 4 1. If we assume that Eq. (4) will be approximately correct, the two inertial terms are of order

pLaci PL(Wa1’2)4 PLh2 _

Pm Pm p-a2 *

The growth factor 2p~rl’~ is about 0.3 for all the liquids used. For water and acetic acid, the mini- mum bubble radius was the radius of the nucleation site, 0.022 cm, so the inertial terms are of order 2 x lop5 (maximum). For glycerol solutions the nucleation cavity radius was 0.0045 cm so the inertial terms are not more than 4 x 10e4. Clearly the inertial terms were always extremely small in these experiments.

If we carry out similar calculations for the surface tension term 2cr/ap,, we find that with water (a = 72 dyn/cm) it is about 10p3, acetic acid about 10V4 and glycerol solution ((T = 52 dyn/cm) about 2 x 10e2 at the beginning of the motion, and of course the term will become smaller with increasing a. Hence the surface tension term is always negligible.

The ratio of the normal viscous stress to the ambient pressure is 4ppLa/p,a, and is of order 2pp,J2/p,a2. For the water (p = 0.25 CP at boiling point) this ratio is very small, about lo-‘, and with acetic acid it is even smaller. For the most viscous glycerol solution, the term is about 10e5, i.e. still negligibly small.

From these calculations it is apparent that the inertial, viscous and surface tension pressure terms in the equation of motion of the surface of the bubble were at all times negligible in these experiments, and that the asymptotic theory should hold.

Another effect which may be important is the presence of the glass tip as shown in Fig. 2. For the beginning of the motion, the temperature field in the liquid between the bubble and the glass may be different from what it would be in an infinite liquid. However the thermal conductivity of glass is not significantly different from that of the liquids used, so that heat would still be conducted to it. When the bubble is large, of course, the effect of the tip on the temperature field would be much reduced, especially after the bubble has started to rise away from it. Also, the experimental results show that the bubbles tended to grow faster than is predicted by the theory. It is difficult to see how the presence of the nucleation site would actually increase the bubble growth rate.

The only effect whose influence is unknown is the translation of the centre of the bubble. During the early stages of bubble growth when the bubble is resting on the nucleation cavity, the centre of the bubble is rising with velocity b. After lift-off has occurred, this velocity in- creases. If the bubble were a perfect sphere of constant diameter moving with steady velocity through the liquid, it could be argued that the translational convective heat transfer would be negligible in comparison with the conductive mechanism assumed here if the product PrRe were very small, for in this case the convective terms would disappear from the energy equation. Now if we put the Reynolds number Re = ha/u and the Prandtl number Pr = v/a, we find

PrRe = 2p2.

In all these experiments /? > 1 so it is impossible for RePr G 1. Obviously the sphere is not in a steady motion nor is its diameter constant. The movement of the phase boundary complicates the problem still further, so that this criterion

681

CES Vol. 26 No. 5 -H


may be far too simple. Until the effect of the translational motion is elucidated, it does appear to be the likely cause for the deviation of experimental results from the theory.

CONCLUSIONS

In one case, that of acetic acid boiling at atmo- spheric pressure, the rate of growth of the bubble followed the asymptotic theory for spherical growth in a large body of liquid.

For boiling water, the radius increased with (time)2’3, in contradiction with the theory, but in agreement with the observations of Darby[9]. For solutions of glycerol in water similar devia- tions from the theory occurred.

The most likely cause of deviation from the theory is the translational motion of the bubble.

Acknowledgments-One of us (M.S.) wishes to express his gratitude to the Conselho National de Pesquisas, Brazil, and to the British Council for financial support during this work.

a

a0

NOTATION

bubble radius bubble radius at inception of motion;

radius of nucleation cavity specific heat of gas, liquid concentration of solute, solvent

9 diffusivity k thermal conductivity L latent heat

L1, L, latent heat of solute, solvent MI, M2 molecular weight of solute, solvent

pb pressure inside bubble pm pressure at a large distance from the

bubble Pr Prandtl number, U/CX R, gas constant Re Reynolds number, ha/v

t real time to (ao/2~a”2)2 T temperature

T sat liquid saturation temperature at pressure pb

AT superheat

Greek symbols

;

E

V

PC, PL

u

7

thermal diffusivity growth constant defined by Eqs. (5),

(6) 1 - PCIPL

kinematic viscosity density of gas, liquid surface tension t-t to

[II 121

;:; VI [61 r71 181 [91

[lOI u11 [121

REFERENCES

BANKOFF S. Cl., Advances in Chemical Engineering, Vol. 5. Academic Press, New York 1965. BANKOFF S. G.,Aduances in Chemical Engineering, Vol. 6. Academic Press, New York 1966. PLESSET M. S. and ZWICK S. A., J. appl. Phys. 1952 23 95. PLESSET M. S. and ZWICK S. A., J. appl. Phys. 1954 25,493. FORSTER H. K. and ZUBER N. J., J. appl. Phys. 1954 25 474 1067. SCRIVEN L. E., Chem. Engng Sci. 1959 IO 3. _ DERGARABEDIAN P.. J. awl. Mech. 1953 20 537. KOSKY P. Cl., Chem. E&g ci. 1964 23 695. DARBY R., Chem. Engng Sci. I964 218 13. BAIN R. W., Steam Tables, National Engineering Laboratory, H.M.S.O. Edinburgh 1964. Handbook ofChemistry and Physics, 46th Edn. Chemical Rubber Co., Cleveland, Ohio 1966. PERRY R. H. (ed.), Handbook ofchemical Engineering, 4th Edn. McGraw-Hill, New York 1963.

[13] International Critical Tables. McGraw-Hill, New York 1926. 1141 TIMMERMANS J., Physical Chemical Constants ofOrganic Compounds. Elsevier, New York 1965. [151 TSEDERBERG N. V., Thermal Conductivity ofGases andLiquids. Arnold, London 1965. 1161 KIRK R. E. and OTHMER D. F., Encyclopedia of Chemical Technology. Interscience, New York 195 1. [171 SADDY M., Ph.D. Thesis, Imperial College, London 1970.

APPENDIX constants fi were taken from Scriven[6], Fig. 4. The growth constants, thermal diffusivities, etc. used in

The physical properties of water at various superheats are the calculations are given in Table 4, together with the time shown in Tables 2 and 3, together with source references. to calculated from In some cases interpolation of literature data were necessary. For the solutions of glycerol and water, the growth to = (aO/2/3a1’P)z.

682


Table 2. Properties of water and water vapour

T VL[ 101 v,[lOl L[lOl kx 103[11] CL1121 CL131 (Y x 103 (“C) (cm”/g) (cm*/g) (cal./g) (Cal/cm2 “C set) (Cal/g “C) (dynlcm) (cm%ec)

100 1.0435 1673.0 539.1 1.598 I.0076 58.85 1.655 102 1.0450 1565.6 537.9 1.599 1.0082 58.47 1.657 103 1.0458 1515.0 537.2 1600 1.0085 58.29 1.659 104 1.0466 1466.3 536.6 1601 1.0088 58.10 1.661

Table 3. Properties of acetic acid

T PLL141 p@ x 1011141 L[131 kx 104[15] CL1131 r[131 ax 103 (“C) (g/cm3) (glcm3) (Cal/g) (Cal/cm2 “C set) (Cal/g “C) (dynlcm) (cm%ec)

118 0.9386 3.15 96.89 3.627 0.5787 18.10 O-6677 120 0.9386 3.30 96.67 3.617 0.5806 17.91 o-6654 121 0.9348 3.41 96.56 3.612 0.5815 17.81 06645 122 0.9335 3.52 96.45 3.607 0.5824 17.71 0.6635 123 0.9321 3.63 96.35 3602 0.5834 17.62 0.6624 124 0.9308 3.74 96.24 3.598 0.5843 17.52 06616

Table 4.

T a x 103 P 2pc+ to (“C) (cm%ec) (set X 103)

Water 102.0 1.657 5.84 0.476 2.15 (b.p. lOO.O”C) 103.0 1.659 8.74 0.713 0.95

104.0 1.661 11.66 0.950 0.54

Acetic acid 120.0 0.665 3.45 0.178 15.3 (118.O”C) 121.0 0.664 5.15 0.265 6.9

122.0 0.664 6.83 0.352 3.9 123.0 0.662 8.50 0.438 2.5 124.0 0.662 10.15 0.522 1.8

Glycerol solutions

x, = 040 106.1 1.431 2.8 0.212 0.45 (104.7”C) 107.8 1.439 6.2 0.470 0.08

xp = 0.80 124.0 1.037 3.0 0.193 0.55 (121~0°C) 127.0 1.039 6.0 0.387 0.13

X* = 0.94 165.0 0.841 2.3 0.133 1.1.5 (156.O”C) 170.0 0.838 3.4 0.197 0.52

Resume-On a observe le developpement de bulles de vapeur a un endroit determine de nucleation dans les liquides uniformement surchauffes.

Avec I’acide acttique, la croissance des bulles suit la theorie asymptotique:

(rayon) = (duree)r’*.

Toutefois, pour l’eau et les solutions de glycerol dans l’eau, I’exposant &tit dans la gamme de 0,75 a 1 ,o.

I1 est suggere que le d&accord est dO au mouvement translatoire des bulles au fur a mesure de leur dkveloppement.

683


Zusammenfassung- Das Anwachsen von Dampfblasen an einem vorbereiteten Keimbildungspunkt in gleichm8Big iiberhitzten Fliissigkeiten wurde beobachtet.

Bei EssigsPure entspricht das Anwachsen der Blasen dem folgenden asymptotischen Gesetz:

(Radius) m (Zeit)‘j2.

Fur Wasser und Losungen von Glyzerol in Wasser variierte der Exponent jedoch von 0,75 bis 1 .O. Es wird die Ansicht vertreten, dag die Diskrepanz durch die translatorische Bewegung der

wachsenden Blasen bedingt ist.

684

experiments on the dynamics of phase growth

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