Production of air bubbles of a specified size

Chemtcol

Engrneenng

Scwtce

1977 Vol

32 pp

1109-1112

Pergamon

Press

Pnnted

m Great Bntam

Shorter Communications . . . . . ..*..........................................*............................................ Production of air bubbles of a specified size (Recerved 27 October 1976, accepted 10 February 1977) Studies m adsorptive bubble separatlon[l], and m the water-toau transfer of blologlcal materlals[2-4], requue techmques for the productlon of au bubbles In many of these studies it has been necessary to produce au bubbles m water one at a time and of umform size Although numerous papers have been published on the productron of smgle bubbles&121, most of them treat bubbles > 1 mm radius We are not aware of any experunents where the sizes of bubbles m the sub-mllhmeter range were successfully predlcted from the geometry of the capdlary tips that produced them Those who have worked with such small bubbles have used a trial-and-error method, which necessitated makmg many tips untlt one was obtained to produce a bubble of the size desired It would be good to have an objective method of making a capillary trp to produce a bubble of a speclfred size We report here on such a method

By neglectmg pa (smce It 1s only about low3 JL), by letting g be 980 cm set?, and y = 72 4 dynes cm-’ (surface tensron of water at 22C), we can solve the equation to get

Rb = 223 R:13 where both &

TEEDRY If a capillary top IS placed m water with the tip vertically upward, a bubble forming slowly on the end of the tip Increases m size untd the buoyant force equals the surface tenslon force that holds the bubble on the tip At that moment the bubble rises from the tip If the end of the tip is clean the water wets the glass (Fig l), and the bubble attaches to the mslde of the capdlary By equatmg the two forces we get 4 T#(P,

- P.,)B = 27%~

(1)

AIR

Fig 1 An air bubble on the end of a glass capdiary tip m water The dashed hne shows the posltlon of mmlmum radrus as the bubble IS formmg

and R, are gven

(2)

m micrometers

Accordmg to (2) the size of a bubble produced by a capillary tip m water can be predtcted if we know the Inner radius R, of the tip TIE was checked by determmmg the size of the bubble produced by caplllary tips of known size Thuteen tips with mner radu rangmg from about 0 8 to 14OO~m were used m the experiments Those with radu < 2001.lm were drawn from capillary tubing of about 6 7mm o d and 1 2 mm 1 d For the larger tips stock capdlary tubmg of the desued radms was used as LS Generally speakmg, the pulling of the tips 1s best done as a two-step process The first step consists of heatmg the tubmg m a relatively hot flame, and then qmckly pulhng it out to a fine taper The flame 1s then turned down, the tubmg reheated at a pomt 2 to 3 cm down the taper, and drawn out agam The drstance from the pomt the caplllary begins to taper to the top should be at least about 3 cm If less, the capillary may not provide the necessary pressure drop for the bubble to form slowly at the end of the tip With but one exception all of the trps used m thus study were so constructed that the bubbles reqmred at least a second to form and detach With less times one cannot assume that the quasi-static balance assumed m derlvmg (2) will hold It 1s essential that the breaking of a fine capillary tip be done such that the end of the tip be reasonably flat and about 90 degrees to the axis of the trp, as shown m Fig 1 Scissors must not be used to cut the tips, for more often than not they produce a Jagged break that bears no resemblance to Fig I In such case the bubble size is variable and bears no obvious relation to the inner radius of the capillary A clean cut can be obtamed most of the time If the tip, at the pomt where the cut 1s desued, 1s hghtiy scribed with a diamond or tungsten carbide pencd If the capdlary tip 1s held m a mlcromampulator this LSeasdy done under the mlcroscope It can then be cut by gently holding It between the fingers and applymg a combmed bendmg and pulling force m exactly the same manner as one does to cut standard size glass tubing at the pomt where a scratch-mark has been made with a file Smce the four largest tips were not pulled out to a fine pomt and thus did not provide the internal resistance to air flow that 1s requued to form bubbles slowly, the au could not be obtamed from an au pump or pressurized contamer A modlficatlon of the au supply system was necessary Fortunately tlus turned out to be relatively easy To understand thus simple modlficatlon It ~111 be necessary to consider the pressure requued to form a bubble at the end of a capillary tip The well-known Laplace equation states that the pressure dlfferentlal P across an m-water mterface for a spherical au bubble or bubble cap of radius R IS

P = 2ylR 1109

(3)

Shorter Commumcattons

1110

ir

Refer now to Rg 1 The an pressure wtthm the captllary to Just mamtam a honzontal an-water mterface (R = m) at the end of the ttp must equal the atmospheric pressure plus the hydrostattc pressure If the au pressure 1s mcreased slowly a bubble starts to form on the ttp The bubble cap radms decreases unhl a mnnmum IS reached when the bubble ts a hemtsphere with radtus equal to that of the capdlary (the dotted lure m Ftg 1) The maximum au pressure 1s attatned at this point Smce the pressure 1s mcreastng and the radius R 1s decreasing durmg thts tune the process is one of stable equtltbnum However, once R exceeds the captllary radtus the bubble radius increases rapidly The au pressure LSnow too high to satrsfy the Laplace equation and the au bubble grows m an explostve-hke manner and detaches from the tip A steady stream of bubbles will contmue unttl the an pressure 1s lowered To prevent this tt 1s necessary to arrange the geometry of the system such that the au pressure wtll drop automattcally to a suffictently low value when the bubble radnts exceeds the captllary radius Thts 1s done by drasttcaliy reducing the volume of au m the supply system that leads to the capillary Thus, when the bubble forms tt increases the total volume sufficiently to cause the requued pressure drop This can be done quite easily by attachmg a syringe to a capillary up wtth about 10 cm of tygon or flextble plastic tubing With the syringe the au pressure ts mcreased until bubbles are about to be produced A screw clamp IS then attached to the tubing By slowly closing the clamp an au bubble can be detached from the tip If the bubble sttll forms too rapidly the clamp IS closed to pmch off the tubing, and a second clamp IS used to produce a bubble from the now-reduced au volume If necessary the total au volume can be reduced further by moving the clamps closer to the captllary The au volume ur the tubing can eastly be reduced to less than 1 cm3 by partmlly filling it wrth water By trial and error one can qurckly determtne the proper geometry necessary to prevent large bubbles from growing qutckly on the ttp A detailed analysts of the sumlar problem of large bubbles produced from holes m flat plates was recently carried out by Marmur and Rubm [8,9] The radius of the bubble produced from a captlhuy ttp was determined from the buoyant force of the bubble This force was measured (see Ftg 2) by collecting a number of bubbles w~thtn an inverted, long-stemmed glass cup that was pa&ally submerged in the water The upper end of the glass cup was attached to the arm of a standard torsion wue balance To provide controlled vertical adjustment to about 100&m the balance was attached to the rack and pnuon of a cathetometer The difference between readings before and after a known number of bubbles were collected tn the cup gave the weight of water dtsplaced by the bubbles Since the denstty of the water LS one, the bubble radius was quickly calculated Figure 2 is drawn approximately to scale The capillary tips were Inserted through hollow polyethylene stoppers placed m the bottom of a short length of glass tube The tips were about 3 cm beneath the surface of the water, the bottom of the glass cup about 15 cm All tips were cleaned by inserting them in chromtc acid for about 30mm prtor to the expenment To prevent the acrd from going mto the tips the au pressure was kept hrgh enough so that bubbles were produced This was followed by rmsmg m hot tap water and dlsttlled water The glass tube,

-

OAGNCE

AiR Ftg

stopper, and cup were washed m a hot detergent soluhon, folIowed by a rmsmg m hot tap water and dtstdled water Dtstrlled water was used to fill the tube m all the expertmeats Since the vertical force caused by the memscus on the glass stem as tt enters the water IS comparable to the buoyant force produced by the bubbles, tt was necessary that the water surface be very clean (to insure a known and reproductble surface tensron), and that the stem was well-wetted by the water The former was assured by overflowing the surface Just prior to an expenment, the latter by using the cathetometer to lower the cup, then ratsmg tt 2 to 3 mm to the posttton used m the expenment Dependmg upon bubble size, from two to several hundred bubbles were collected m the cup Typtcally thts produced between 10 and 15 degrees of twist on the torsion wue The scale attached to the wire could be read to about 0 2 degree By caltbratton it was found that a wire twrst of one degree was caused by a change of 3 53 x IO-’g of water dtsplaced beneath the cup At the start of each expertment four independent readings were obtamed for the equtltbrmm positton of the glass cup After the bubbles were released the torsion wire was readjusted, which returned the cup to the same posrtton Four more readings were obtained The repeat readings seldom vaned by more than 0 2 degree Each experrment was done at least twrce The temperature of the water ranged from about 20 to 22C RESULTS AND DJSCUSSION The results are shown m Ag 3 where bubble radius ts plotted as a function of capillary radms The lute IS the theorehcal relattonshlp grven by eqn (2) The cucles gve the results of the experiments with the 13 capillary tips It ts apparent that the agreement between expertment and theory IS quite good The small drfference between observed and predtcted bubble radms found m some of the expertments cannot be explamed by random scatter, for the reproductbrhty of each data point was quite

EXPERIMENT

I

IO CAPILLARY

Ftg

3 Companson

2 Apparatus used to measure the buoyant force, and thus the size, of air bubbles m water

THEORY 0

01

TO TORSION

of theory and experiment

100 RADlUS

IOCO

p”

for the srze of a bubble produced

by a glass capillary tip m water

1111

Shorter Communications remarkable For examule, su( successive determmatlons of the bubbb size produced by .the 622 pm radms top gave values of 1790, 1790, 1785, 1795, 1800 and 1795 tirn radius This 1s about SLX percent less than the predlcted value-of 1900 pm Smce the time reqmred for the bubble to form and detach from the trp was vaned between about 0 5 and 5 set It LSapparent that the bubble growth time was not a factor m determmmg bubble size The experImental values of bubble radn for all four tips > 100 pm are shghtly less than the theoretical values We beheve this 1s caused by a dlstortlon of the bubble while It 1s growmg on the tip Just p~lor to detachment It has a shape somewhat hke an mverted teardrop It does not detach precisely at the trp but at a narrow neck of au Just above the tip Consequently, a small amount of au which theory predicts should be carried away m the bubble remains behmd to start the formatlon of the next bubble on the tip This bubble distortIon IS not observed on the smaller bubbles where the surface-to-volume ratios are hrgher As mentioned earlier If the bubbles form too rapldly acceleratlve forces are involved, and the simple theory cannot apply The bubbles tend to be larger than theory predicts For example, the 160 pm tip produced bubbles of a radius only four per cent less than predicted by theory when the bubble growth time was between 2 and 8 set, but about 10% greater when the bubble growth tune was decreased to t0 2 set The vertical onentatlon of the tops was done by eye The small random deviations from the vertical that must occur as ddTerent tops are tested did not appear to have any observable effect on bubble size Three of the tips were tested twice for bubble size, on dtierent days with different batches of water No observable difference was found Ddferences between theory and experiment were found d the plane of the end of the tip was more than a few degrees from being normal to the long axls of the tip For example, two ups of about 15 pm which deviated from this Ideal by about 10 degrees produced bubbles whose size was about eight per cent less than that predlcted by theory Two other tips, around 4pm radms, deviated by 25-30 degrees and produced bubbles of a size 18% less than predlcted These data are not shown m Fig 3 This marked influence of tip geometry on bubble size probably played a role m causmg the large dtierences between theory and experiment that were found by prior mvestlgators[l3-151 Some of their data were as much as 230% from theory They also encountered dticultles m using photographlc techmques to get accurate measurements of bubble size Because of these problems we have not included then data m Fig 3 Tbe excellent agreement between theory and expenment found m this work allows one to use Fig 3 to select the capdlary radius necessary to produce a bubble of a predetermined size If, for example, one wishes to construct a tip to produce a bubble of 800 pm radius then the capdlary tip radius must be 46 Frn A long tapermg tip can be pulled out, as described earlier, and observed m a honzontal posltlon under the mlcroscope The Inside diameter can be seen, but due to the lens effect of the glass It wdl appear larger than it really IS One must be careful and not depend too much on a measure of the outside diameter to predict the mslde dmmeter, as this ratio generally increases with decreasmg size and varies from tip to tip It IS best to break the tip at a point where It 1s known to be smaller than that required, and observe the discarded section end-on to determme the actual radms, and so determine the ratio of outer-to-mner diameter If this 1s done at a second point, and perhaps a thud pomt, the rate of change of this ratio 1s determined, and one can move up to a tlucker portion of the tip and locate the point where the Internal radms 1s 46 pm At that pomt the break should be made If a micromampuIator 1s used to hold the tip, the entue operation of scribing the tip wrth the diamond pencil and cuttmg It can be performed under the mlcroscope A variety of simple techmques can be devised to hold the outer part of the tip securely to the microscope stage, whde the mlcromampulator can be used to provide the bendmg of the tip necessary to produce the cut at the point where It has been scmbed The data for Fig 3 were obtamed with capillary tips m dlstdled water, but we find no measurable difference when seawater 1s used Tlus LSto be expected Using the values of surface tension and density of seawater, one can show from eqn (1) that the CESVol

32 No

9-l

bubble radms from a gven tip m seawater should be only 0 997 hmes that m dlstdled water Nor do we find any ddference m bubble size as a function of the time the tip has been m the water, at least for times of up to 10 hr when the tests were ended Apparently the approximately 1 ppm of dissolved orgamc materra1 m seawater IS not sufficient to modify wtthm 10 hr the wetting properties of the end of the capillary trp However, this IS not the case when the bps are tested m nutrient broth solutions of concentrations of 1,000 to 3,OOOppm, as are sometimes used m water-to-an transfer studies of bactena[4] In these solutions the bubble size IS equivalent to that for the dlstdled water or seawater case only for a few mmutes after the tip IS immersed in the solution The bubble size then Increases steadily with time, reachmg a steady state size some SIX to seven per cent greater than the untlal size after about two to three hours This increase, though relatively small, IS easdy measurable No satisfactory explanation has been found, though one way to account for it is to assume that a small portion of the end of the capillary tip has become non-wettmg Although the largest tip used m this work was only 1400 pm radms, there appears to be no reason why even larger tips could not be used for bubble production, though one would expect the bubble size to be less than that predicted by theory At the other end of the Size range, however, there are problems We found it dlffrcult to draw trps with Internal radn
Theoretical predlctlons of bubble size ( i about 1 mm radms) as a function of capillary size have been shown by expenment to be valid for glass capdlary tips m both dlstdled water and seawater In solutions contarnmg appreciable amounts of dlssolved orgamc material the bubble size IS several per cent larger than predicted by theory If one pays close attention to the geometry of the capillary tip, It 1s possible to prepare tips to produce bubbles of almost any size desired, one at a time, and upon demand Acknowledgement-This research was sponsored Science Foundation Grant GA-23413 Atmospherrc Sczences Research Center State Unrversrty of New York at Albany Albany, NY 12222, US A

by Natlonal

DUNCANC BLANCHARD LAWRENCE D SYZDEK

Shorter Commumcatmns

1112 NOTATION

g R P Rb R,

acceleration of gravity, 980 cm see-’ bubble cap radms, cm air pressure, dynes cm-’ bubble radius, cm m eqn (1) and pm m eqn (2) inner radms of caplllary, cm m eqn (1) and pm

Greek symbols y surface tenslon of water, dynes p. density of au, g cm-” pW denlsty of water, g cm-’

in eqn (2)

cm-’

REFERENCES

[l] Lemhch Academic [2] Blanchard [3] Bezek H 566 [4] Blanchard 8 529

R , Adsorptroe

Bubble Separatzon Techncques Press, New York 1972 D C , Prog Oceanog 1%3 1 71 F and Carluccl A F Llmn Ocennog 1972 17

D , J de Rech Atmos

D C and Syzdek L

PropertIes of liqmd-vapour

1974

compositlon surfaces at azeotropic pomts

(Recemed

19 January

The propertles of multlcomponent equdlbrmm phase surfaces can be deduced from classical thermodynamics by means of the theory of eqmhbrmm displacements If a C-component llquldvapour mixture IS uutlally at equdlbnum and IS displaced to a new equdlbrmm state, the basic equations descnbmg the change are 8pgL(T, P, XI,

xc-,) = &y(T, r=l,

P, YI,

151 Khurana A K and Kumar R , Chem Engng Scr 1%9 24 1711 C61 McCann D J and Prmce R G H , Chem Engng Scr 1%9 24 801 171 McCann D J and Prmce R G H , Chem Engng SCL 197126 1.505 I81 Marmur A and Rubm E , Chem Engng Scr 1973 23 1455 191 Marmur A and Rubrn E , Chem Engng Scr 1976 31 453 [lOI Ramaknshnan S , Kumar R and Kuloor N R ,Chem Engng SCl I%9 24 731 [Ill Satyanarayan A, Kumar R and Kuloor N R , Chem Engttg Scr 1969 24 749 [I21 Wraith A E , Chem Engng Scr 1971 26 1659 [131 Maer C G , Umted States Bureau of Mines, Bull No 260 1927, 63 Kniill H , Kollord Zertschnft I940 90 189 Guyer A and Peterhans E, Chzmrcu Acto 1943 26 1099 Blanchard D C and Syzdek L D , J Geophy Res 1972 77 5087 [I71 Brown K T ‘and Flammg D G , Science 1974 185 693 iI81 Mdler C W and Kessler W V , Tellus 1974 26 609

1977)

however, the procedure yields equations which contam these derlvahves lmphcltly and as a consequence their propertles have remamed largely unknown The notable exceptlon being bmary mixtures, for which It LSpossible to show that >O

, Yc-I)

,c

and

VXlEKA11 (1)

After expansion and algebraic mampulatlon It IS possible to derive expresstons for the followmg derlvatrves along the equihbrmm surfaces,

See Malesmskl[l] (p 59) In the special case when the quantltles and are evaluated at an azeotroptc pomt, It IS possible to assess their propertles and the remamder of thts paper wdl be devoted to this task Define the matnx, YpAZ. such that

where Xl.

subscrlpt x’ mdlcates that the X,-V, x,+*3 , xc_, are held constant

mole fractions For example,

x~,

J

~c--L,x(c--I,

,c-1

1=1,

(2)

where Ah, = h.” - h.‘The propertles of the bodmg surface are, therefore, dictated by this equation and in particular, the Gibbs-Konovalov laws may be deduced by mspectlon A most comprehensive account of the theory of equlllbrmm displacements IS given by Malesmskl[l] (pp 54-71) In prmclple, we may deduce simdar expresslons for the quantitles and

where the superscrlpt AZ means “evaluated at an azeotroplc pomt” as the matnx of isothermal partld Slmdarly, define Y=derlvatlves

The followmg

theorem

can now be proved

Theorem 1 The matnces Y,” and Y,^’ are equal and possess (C - 1) dlstmct, posltlve, real, non-zero elgenvalues, from which It fol-