Vapour bubble collapse

VAPOUR

BUBBLE

COLLAPSE

THE INFLUENCE OF THE INITIAL RADIUS AND OF SUBCOOLING

Laboratoue

H DELMAS and H ANGFLMO Associe 192, lnsbtut du Geme Chumque, Chemm de la Loge, 31078 Toulouse, Cedex, France (Recewed 23 July 1976, accepted 11 October 1976)

Abstract-Expenmental results for nsmg bubbles wrth nutml radn rangmg from 0 06 to 1 cm collapsw m water durmg a short pressure nse are presented Data are analysed m two ways -firstly mveswtmg the mfluence of the lmt~al radius and of subcoolmg on the bubble wall velocity, -secondly experunental data relatmg to bubble radms decrease are compared m dunensloniess form, witi a sphencally symmetnc heat transfer theory A strong tiuence of the rmk+l radms and of subcoolmg contra&ctory to the theory IS &II shown

1 INTRODUCTION

two bubbles are m practically the same untial conditions of temperature and pressure In fact the dtierence m pressure, resultmg from surface tenslon effects, is always less than 2 mm of mercury and the two bubbles can be consldered as subJected to the same pressunsation The mfluence of the nut1a.l radius can therefore be mvestigated m the best con&tions For d*erent ratios RoJRoe of m&al radn of two bubbles m an experiment, the ratios v.J gB of bubble wall velocity, averaged from the m&al radms Ro to 0 7 Ro, were exammed The ratios t*/t~ of the correspondmg collapse times were also exammed The quantity 0 7 was chosen as the hnut of the dnnenslonless radius for two reasons, firstly the ma)onty of bubbles reach it without shattenng and secondly for thus quantity the volume of the bubbles 1s already only about a thud of the irut.& volume The results are shown m Table 1 It can be seen that the large bubbles (A) decrease more qmckly than the small ones (B) but that take longer to reach the same dnnenslonless radius

Various factors enter mto the dynanucs of a vapour bubble, dependmg on the part~ular stage and the particular enwonment m wlch the bubble grows or collapse Among these are the effects of surface tension, vlscoslty, the compresslbfilty of the hqmd, hquld metia and heat transfer from the bubble to the hqmd, due to phase change It 1s now commonly known that only hquld mertla and heat transfer 111 the hqutd need to be taken into conslderatlon, except m the spec& case of very small bubbles or of very viscous hqmds When the hqmd inertia plays a secondary role to heat transfer, the dymumcs of a vapour bubble undergomg a rapid variation of pressure depend, for a gven hqudvapour system, only on the lmtml radms and on the vmatlon through time of the eqmvalent subcoolmg In the expenments presented here, the pressure rrse tnne ranges on the whole from 2 to 5 msec, wluch leads on to study the mfluence of the two remammg parameters the nntml radius and the subcoohng eqmvalent to the pressure rise 2 EXPERIMENTAL The expenmental apparatus has been described elsewhere [l, 41 It allows the cmematograpluc study of the collapse of a steam bubble nsmg m a hquld which 1s quickly subcooled by a rapld pressunsatlon of the system The u~tial pressure ranges from 146 to 620 mm of mercury and the final pressure 1s shghtly greater than the atmospheric pressure The degree of subcoohng eqmvalent to these pressure vanations ranges from 42 to 7°C The films analysed show the evolution of the radius as eqmvalent to that of a sphere of same volume, for bubbles with an uutml equivalent radius of 0 06 cm to lcm 3 RESULTS VARIATION OF THE EQUIVALENT RADIUS THROUGH TIME

ROB/

VBta than one,

Table 1 Expenment

0

2

y

y=l

23

105

105

10

17

16

1 1

14

10

245 27 29 31 34 58

4

723

R oe

The two ratios vA,lvB and tAjfB, greater mcrease with the ratio RJRw 0% 1)

40

3 1 Injluence of the lnlfral radrus In certam experunents two bubbles of dtierent radn form and collapse snnultaneously, when tins happens the

ta

19 20 20

y=07)

H

724

35

15

DELMASand H

ANGELINO

-

-

Fig 1 Muence of the nuti radms on the collapse tune from y = 1 to y = 0 7 reached at a Merence of pressure of 583 mm of mercury, that ISto say a dtfference of temperature of 37 4°C The two velocities mcrease almost lmearly with the subcoohng up to 2S”C, there remam constant to AT, = 4O“C, after wbch they seem to decrease suddenly Indeed for the greatest dtierence m temperature the two veloclhes, average and mstantaneous, are greatly mfenor to the previous ones It must be noted that contrary to the bubble wall velocltles the maxunum pressure contmues to mcrease greatly m proportion to the subcoolmg It must be remembered that only one smgle point supports this fact, nevertheless other experunents, camed out m fields of maxnnum subcoolmg (38_42”C), confirm that the average bubble wall velocities m tis case are shghtly slower than bubble wall velocities recorded for subcoolmg rangmg between 25” and 38°C Thus study of the mfluence of subcoohag on the collapse rate of bubbles, gves evidence of two unforseen results, first of all a lmutahon m the bubble wall velocity after a certam subcoohng-round about 25°C-, then a

3 2 Infiuence of subcoolrng The overall results show up the unportant miluence of

subcoolmg on the collapse rate of the bubbles On the whole the h@er the subcoohng or the dtierence m pressure, the qurcker IS the decrease However it IS not at the greatest Merences m temperature that the lllgher bubble wall velocities are recorded To mvestigate this apparent hmltaltion of the collapse rate for subcoolmg h&er than a @ven hmlt, 10 bubbles of an lrutlal radius from 0 44 to 0 5cm were taken Their average bubble wall velociues between Ro and 0 7 R. as well as their mstantaneous velocitres at y = 0 7 were then compared Fwe 2 shows the average and mstantaneous bubble wall velocities versus subcoohng It can be noted that on tis figure the Instantaneous velocltles are more drspersed than the average velocities Tlus dlsperslon IS the result of the Imprecise determmation of the mstantaneous velocities and especially of the eventual mtluence of oscdlations The maximum average velocity for these ten bubbles IS

140

120 i

0

IO

20

Ai:

30

40

‘=c

F@ 2 Muence of subcoolmg on bubble wall velocity X, mstantaneous velocity at y = 0 7, 0, average velocrty from y=l to y=O7

Vapour bubble collapse

decrease m this velocity In fact the mfluence of the uutml radms and of subcoohng, which are the only

parameters actmg on the collapse of vapour bubble of a aven substance with an mstantaneous pressure nse, have been mvestigated Now, as the length of the pressure nse tune IS m reahty not neghgble, the degree of subcoohng vanes contmuously vvlth the collapse of the bubble Generally speakmg when the final subcoohng IS greater than 25°C the length of tnne that the bubble takes to decrease between R. and 0 7&, becomes less than that of the pressure nse[l] Thus when the bubble reaches the dunenslonless radms 0 7 it has not yet undergone the total subcoolmg taken mto conslderaltlon mFw 2 Fma.lly the decrease m velocities at greatest subcoolmgs could be explamed by the presence of autovaponzation bubbles at the top of the contamer Thus two phase medmm would hmlt the propagation of the pressure wave towards the bubble 4 COMP-N

WITH

Usmg the approxunate

TEEORETICAL

725

heat transfer equation gven by Pbsset and ZHrlck[2], Florschuetz and Chao[3] obtamed the two followmg relations, lmkmg the dnnenslonless radius to the dnnenstonless tune 711 symetrx

TH= (yZ + 2/y - 3)/3 y = 1 -(TH)“*

Y =

In eqn (2) curvature effects are neglected These relations assumed neghgeible mechamcal effects of meti and a step mcrease m the system pressure Because of the real pressure nse we used

ANALYSls

IOc

Fe 3 Compansonwith Florschuetzand Chao’s theory Influence of the uut~alradms Ro

(2)

RI&

solution of the sphencally

Bubble

(1)

(cm)

A 4A

0465

A.4B

0 14

AT, 12 3

V, 10A

0 31

v, 10B

0 755

292

q, 17A

0 93

1.17B

0 575

0.2OA

0775

l , 20B

0 25

394

336

726

H

and H

DELMAS

The experunental data have been plotted mto the system of dunenslonless coordmates y, rH, and have been compared to the theoretical curves described by the eqns (1) and (2) Although the actual length of pressure vanatlons 1s taken mto cons.lderaUon m determuung the dunenslonless time, the results are very dlspersed[l, 41 and an mfiuence of the uutml radius and of subcoohng can S~IU be noted, which 1s contradictory to the theory of Florschuetz and Chao (Figs 3 and 4) 4 1 Influence of the mrtral radms To estabhsh this mfiuence the results of four expenments where two bubbles collapsed sunultaneously have been plotted m Fig 3 It can be seen that under the same experimental conditions the dlmenslonless radius of large bubbles decreases more qmckly 111dunenslonless tune than that of the small ones, and that the dtierence between the expenmental curves 1s even greater when the uuk+l radu are dflerent There may be three reasons for these dtierences The Peclet Number (Pe = (2U&o)/a), which characterlzes the unportance of convective transfer due to bubble translation relative to conduction, increases wtth the radius T~s then would explam a more rapid collapse of iarge bubbles Nevertheless comparmg such results to the solution of the complete heat transfer equation, takmg mto account the translation, It was shown[l] that

ANGELINO

tlus supplementary convection importance of the divergences, The B number[3]

does

4

8

10

12

T xloz %

Fig 4 Comparison mth Florschuetz and Chao’s theory Influence of subcoolmg

A,2 +, 4A q, 29B 0, 12 a 19 0, 15 w, 16B

Ro (cm) 048 0445 0500 0445 044 0445 044

the

4 2 The mfluence of subcoolrng and of JACOB Number In Fig 4 the hlstones of 7 bubbles of sunilar radius are compared It appears that the bubbles 2, 4 A, 29 B corresponding to the lowest subcoolmgs, thus to the smallest JACOB Numbers, decrease m relative regular manner in spite of brief and small fiuctuatlons When the dunenslonless radius 1s mferlor to 0 85 the slope of these curves are close together and almost constant, but about two tunes greater than the slopes of

6

Bubble

explam

which characterizes the mfluence of metia relative to heat transfer 1s mversely proporttonal to uutml radms It can therefore be sad that, m the chosen dunenslonless representation, the retardmg effect of the hqmd mertla IS greater on the small bubbles than on the large ones -Fmally, and above all, the large bubbles are more distorted and present a larger speck exchange area at the Interface It must also be noted that the determination of the dunenslonless tune and of the radius 1s less precise for small bubbles than for large ones

08

2

not

AT,

Ja

101 123 174 234 353 37 4 409

29 36 51 67 106 113 123

Vapour

bubble collapse

727

theoretical curves for the same value of y It must be emphasized that the last expenmental pomts of each bubble pnor to shattemg pomt are not evaluated with great preclslon For the four other bubbles taken mto conslderatlon (12, 15, 16 B and 19) with greater correspondmg subcoolmg, and therefore also h&er JACOB Numbers, the experunental pomts are closer to the theoretical curves (1) and (2), but the general form of these curves IS less regular, the vacations m the slopes are greater, especmlIy for bubble 16 B subJected to the greatest subcoolmg It would seem that the oscdlatmg pattern of these curves IS due to the mertla phenomenon and vapour compresslbkty It can be deduced from tins comparison that the decrease m dunensionless radius y vs the dunenslonless tune TH IS even more rapld when the subcoohng and the JACOB Number are low For Ja < 50, the real collapse rate 1s much greater than that antlclpated by the equations of Florschuetz and Chao, wbch seems to underestunate heat transfer For .7u> 70, the expenmental data are more m agreement but hqurd mertra begms to mterfere considerably

Acknowledgement-The from the Comrmssanat

5 CONCLUSION analysis of experimental data 1s approached m two ways Fast of all, directly mvestlgatmg the mfluence of the mltial radius and of subcoohng on the bubble wall velocity Secondly the data relatmg to experunental bubble radms decrease are presented m a dlmenslonless form and compared with a theoretical analysis based on the solution of the hqmd heat transfer problem m the spherically symetrlc case Tlus compmson shows evidence for the inadequacy of tlus theory to correctly predict the collapse of rlsmg vapour bubbles which are not spheric The dlstortlon of the bubbles due to translation mcreases the exchange area and seems to be the mam reason for the excessive collapse rates recorded for large bubbles or for low subcooled cases A local study El, 51 has specified the influence of the dlstortlon of bubbles on the collapse speed of the mterface and on the stab&y of the bubbles

p’*

The

authors wish to acknowledge a 1’Energe Atomque

support

NOTATION

B

dimensionless number defined as

(Pcj-gy(&y” speck

G

heat, L2T-*fI-’

Jacob Number defined as w

Ja

heat of vaponsatlon, L*T-’

L Pe

2U& Peclet number defined as -

R t u, V

bubble radius, L tune, T bubble translational velocity, L T-’ average bubble wall velocity, LT-’

Greek

(Z y Ap AT p

symbols

thermal dfiuslvlty, L2T-’ dlmenslonless bubble radius defined as R/R0 pressure vartation, ML-‘i? equivalent subcoohng, 0 liquid density, ML-3 saturated vapour density, MLw3

.7H dlmenslonless

time defined as %(wy$

Subscripts

A B

c j 0

large bubble small bubble corrected final mltlal

Delmas H , T&se Docteur ln&ueur. Toulouse (1976) Plesset M. S and Zwck S A, J &pl Phys ‘1952.23 95 Florschuetz L W and Chao B T , Trans A SM E (0 (1965) a? 209 Delmas H and Angeimo H to be H Augehno H ,