The motion of a bubble in a vertically oscillating liquid - Science Direct

Chemical Engineering Science, 1966. Vol. 21, pp. 29-34. Pergamon Press Ltd., Oxford. Printed in Great Britain.

The motion of a bubble in a vertically oscillating liquid: theory for an inviscid liquid, and experimental results G.

J. JAMESON? and J. F. DAVIDSON

Department of Chemical Engineering, Pembroke Street, Cambridge (Receiued10 February1965) Abstract-A theory is given to describe the motion of a small spherical bubble in an inviscid liquid which is in forced oscillation with an up and down movement. The theory gives the frequency at which the downward force due to oscillation balances the buoyancy force; in this condition there is no net movement of the bubble, and the theory predicts that it should oscillate about a fixed point with an amplitude three times. the amplitude imposed on the liquid. These results are only roughly in agreement with experimental data; it is clear that with all conceivable experiments in which a bubble is held stationary by vibration, the effect of viscous forces is

important.

WHENa mass of liquid is forced to oscillate with an up and down motion, a small bubble contained within the liquid may, under certain conditions, move down against the buoyancy force. This phenomenon was studied by BUCHANAN et al. [l] who used the theory of BJERKNES[2]; the latter had considered the hypothetical case of a balloon in a vibrating liquid, and derived a theory to show that there is a steady force in such a case. BUCHANAN et al. observed experimentally the frequency of vibration required to stabilize a bubble in a liquid so that the downward force due to oscillation just balanced the buoyancy force, and the bubble appeared to remain stationary in the liquid. The observed frequencies were in reasonable agreement with theory. HARBAUMand HOUGHTON [3,4] measured the effect of vibration on the rate of absorption of gas bubbled through a liquid, and observed peaks in the value of k,a, where kL is the transfer coefficient, and a the interfacial area per unit volume. BAIRD[5] suggested that these peaks are due to resonance between the vertically applied vibration and the natural frequency of the column of bubbles and liquid. However it is noticeable that the first and most pronounced peak in the curve relating k,a and frequency is at a frequency of 40-60 c/s, when the rising velocity of a single bubble

is low or zero. BRETSZNAJDER and PASIUK [6,7] also observed bubbles moving down in a vertically vibrating column of liquid and the rate of CO2 absorption showed pronounced maxima at frequencies similar to those noted by the other workers [3-51. The present paper gives a theory for the motion of a single bubble, based on the assumption of potential flow within the liquid around the bubble. The most important deductions from the theory are (i) the frequency necessary to stabilize the bubble, and (ii) the amplitude of the bubble’s motion as compared with the amplitude imposed on the bulk liquid. Result (i) is a criterion similar to that derived by BUCHANAN et al [I]; their theory did not give the amplitude of the bubble’s motion. The theory herein bears some resemblance to that of HOUGHTON[8] who considered the motion of a particle or bubble in a vibrating liquid ; however, he included the effect of drag forces, assuming that these are of the same kind as in steady motion. The motion of a particle in a vibrating viscous liquid was also considered by ANDRADE[9]. The experiments described in the present paper were similar to those of BUCHANANet al. [l], but were carried a step further; in addition to measuring the frequency for no net motion of the bubble we have also observed the amplitude and phase of the

t Present address: Department of Chemical Engineering and Chemical Technology, Imperial College, London. 29

G. J. JAMESON and J. F. DAVIDKIN bubble’s motion for comparison imposed on the liquid.

The pressure p at any point in the liquid is given by BERNOULLI’Sequation, 2 P ad -=- 5 + E’(t) + 5 +

with the motion

THEORY

P

Figure 1 shows the axes and co-ordinates to describe the motion around a bubble in a liquid which is forced to oscillate vertically in simple harmonic motion. The bubble is assumed to be at a mean depth h below the surface, which is maintained at pressure PA. The displacement of the bulk of the liquid is A sin nt about any fixed point, where A is the amplitude, n is the circular frequency and t is time. We wish to find the displacement z of the bubble relative to fixed axes x, y. It will be assumed that surface tension forces maintain the bubble in a spherical shape. Under these conditions the velocity potential for the liquid is Cj=-

RR2 UR3 - cos 0 + xnA cos nt 2r2 r

( at ) X,Y

P

+ g(h + A sin nt - x)

F(1) = n2A2(2 cos2 nt - 1) - n2hA sin rtt

(4)

using (I), it being assumed that r 9 R at the top. To calculate &$/at, we need the derivatives of r and 8 at a fixed point in space; these are

(arjat>,,y = (1)

where R is the radius of the bubble, and r and 8 are polar co-ordinates as shown in Fig. 1. The dot denotes differentiation with respect to time. The first term in equation (1) accounts for the pulsation of the bubble, and the second for the relative motion, with velocity U, between the bubble and the liquid, so that U = i - nA cos nt (2)

(3)

where q is the liquid velocity at any point, h is the mean head of liquid above the bubble, as shown in Fig. 1, p is the liquid density and g is the acceleration due’ to gravity. To satisfy the requirement that p = PA at the top surface of the liquid we must have

-2

cos

r(0f3jat),,Y = i sin

e

e

1

(5)

Using equations (1) and (5) we evaluate &#@t, q2 from (1) and substituting into equation (3) with r= R gives the pressure pR at the surface of the bubble

+ xn2A sin nt + g

(9 cos 20 - 1) -

where z is the co-ordinate shown in Fig. 1. n2A2 - 2 cos2 nt + F(t) + PA P+

+ g(h + A sin nt - x).

I-

&-or

Line

__

z

fixed

Here i has been eliminated, using equation (2). Equation (6) is similar to an equation given by BJERKNE~[2]. We can then evaluate the upward force Q on the bubble

in the

Q= -

x

--_

t

s

Ip,2nR2 sin d cos 0 dO

(7)

We note that the only terms from equation (6) which contribute to equation (7) are those containing cos 0; the latter equation thus simplifies to

Y\ Line fixed space

(6)

in

Q

- = I/g - Vn2A sin nt - $ P

FIG. 1. Bubble in an oscillating liquid. 30

(8)

The motion of a bubble in a verticallyoscillatingliquid where V = 4&/3 is the bubble volume. It is easy to grasp the physical significance of the terms in equation (8); the first represents the ordinary buoyancy force; the second term represents the mass-acceleration of the liquid displaced by the bubble; and the third is the familiar term representing the momentum of the liquid surrounding the bubble. We may also use equation (6) to calculate the way in which the bubble volume varies with time; for this purpose we require the mean pressure ?l p,2nR2 sin 8 dl?/&R’ PM = s0

should be noted that the error introduced by neglecting surface tension is small, since the difference between the pressures inside and outside the bubble is small compared with the atmospheric pressure. Using Boyle’s law, as above, with equation (9) gives sin nt * Vo(l + B sin nt) (10) provided B< 1, where B = pn’Ah/P,. Substitu& for Y from equation (10) into equation (8), with Q = 0 for zero net force on the bubble, gives after integration

This integration eliminates the terms in cos 8, and eliminating also second-order terms we get the following approximate result, using equation (4). +

P&f - = 5 + gh - n*Ah sin nt P P

nAB Bg sin 2nt - - cos nt + FIA cos nt. 4 n

(11)

The constant of integration has been omitted because it represents a steady velocity of the bubble which would be damped out by viscous forces in an experiment. If the bubble is to oscillate about a fixed mean position, the term containing t in equation (11) must be zero, so that

The second-order terms omitted from equation (6) in deriving equation (9) are as follows: (i) The terms due to radial motion of the liquid, containing li2 and Rfj, can be omitted because the resonant frequency of the bubble, for radial osciln4A2ph n2AB lation of the type considered by MINNAERT[lo], is 1 M =-=--_= (12) 29 2gPo about 5000 rad/sec which is much larger than the applied frequency n, about 180 rad/sec. This and this is similar to the condition given by frequency is much higher than the value calculated by BUCHANANet al. [l]. Equation (11) then gives the BRETSZNAJDER and PASIUK[7] for small oscillations relative velocity U between the bubble and the liquid of, the bubble about the spherical form. In our as a combination of harmonic terms. However, experiments it was clear from the photographs that equation (11) can be simplified since B < 1, and the bubble was always spherical and therefore the therefore all the terms except the first and last can higher frequency from the theory of MINNAERT[IO] be neglected, giving for radial oscillations is relevant in predicting the U = 2nA cos nt (13) importance of the terms A2 and m; it is thus clear which shows that the total amplitude of the bubble’s that these terms are negligible. (ii) Since A + h, the retained term n2hA sin nt motion is three times the amplitude A of the motion from equation (4) is larger than the omitted terms imposed on the liquid; this agrees with the result containing n2AZ and xn2A in equations (4) and (6). given by RSCHEVKIN[12] for the motion of a Similarly the U2 terms can be neglected because Uis spherical gas bubble due to a plane sound field in of order nA, and the gravity term in equation (6) a liquid. Equation (13) also shows that the motion of the bubble should be in phase with that of the can likewise be simplified. For the frequencies with which we are concerned, liquid. PLESSETand HSIEH [II] have shown that volume EXPERIMENTAL changes take place isothermally, so that pMV = The liquid was contained in a vertical Perspex PoVo where PO = PA + pgh is the mean pressure within the bubble, and V, is its mean volume. It cylinder to which vibration was imparted by means 31

G. J. JAMESON and J. F. DAVIDSON

To constant pressure

Vorioble

’

single air bubble was introduced into the liquid via a fine capillary tube. Once a bubble had been introduced, the motor speed was adjusted to bring the bubble within the Perspex box shown in Fig. 2. This box was mounted round the tube to minimise optical distortion when the bubble was photographed. Cine photographs were taken in the range 1000-2000 frames/set and a frame by frame analysis gave the displacement-time curve for the liquid and the bubble. Three different liquids were used, namely water and two glycerol solutions. The densities of the glycerol solutions were measured, and the viscosity corresponding to each density was obtained from the tables of MINER and DALTON [13].

t f+

speed eccentric

FIG. 2. Diagrammatic arrangement (not to scales.

moss

RESULTS AND DISCUSSION

of the vibrator

From the high-speed film, the displacement of the bubble and of the bulk liquid were plotted as functions of time, as shown in Fig. 3. This diagram

of the double beam resonator shown in Fig. 2. The steel base with the Perspex cylinder and the liquid together had the same mass as the motor and eccentric mass attached to the lower beam. The knife edges AA’ formed the hinges to allow vibration of the two beams, and were maintained in contact by external clamps applied to the knife edges BB’ (these clamps are not shown in Fig. 2). Since the clamps were at nodal points they were also convenient for supporting the apparatus. The knife edges and anvils were fixed to the beams in such a way that the effective length AA’ could easily be altered to vary the natural frequency of the system. The maximum amplitude which could be produced was about 0.8 cm. The top of the Perspex cylinder was attached to a loosely fitting guide, to keep the full length of the cylinder vibrating in a vertical direction. The vibration source was a l/20 h.p. variable speed motor (maximum speed 3000 rev/min), with an eccentric mass attached to its flywheel. The motor speed was varied by means of a Variac controller, and measured to 0.5 per cent by a stroboscope. The liquid level in the Perspex cylinder was maintained just below the top and the pressure in the air space above was regulated via a flexible connexion to a constant pressure air supply. A

Ql I rll

FIG. 3. Displacement of liquid and bubble.

32

The motion of a bubble in a vertically oscillating liquid

necessary to modify the theory to allow for viscous effects, since the experiments could not exclude them; a modified analysis is given by JAMESON [14] and is published in the following paper.

Table 1. Experimental conditions for Fig. 3

& pgmlml h cm n rad/sec. A cm M

also

Water

Dilute gZycero1

1.0

18.1 1.176 42.6 202 Q.214 1.65

74.3 1.215 38.7 200 0.212 1.41

bubble

displacement,

4::: 207 0.169 2.82

shows

the

theoretical

Concentrated glycerol

NOTATION A B F g h M n

from equation (13), and the observed values of the ratio A4 at which the bubble had no net upward or downward motion. These values of M are somewhat different from the value-of unity predicted by equation (12), and Fig. 3 also shows that the observed amplitude of the bubble’s motion is much less than the theoretical. These discrepancies are thought to be due to viscous effects; it would obviously be desirable to carry out experiments in which viscous effects were negligible but this is not practicable because with real liquids it is necessary to study large bubbles in order to minimise viscous effects; and experiments show that large bubbles are unstable in a vibrating liquid. It was, therefore,

PA

Amplitude of liquid movement p&ih/Po Function of time defined bv eauation (4) Acceleration of gravity - _ . . Mean depth of bubble n2AB/2g Circular frequency of oscillation Pressure applied to liquid surface

PO PA + pgh P Pressure at any point in the liquid PM Mean pressure on the bubble pn Pressure at the bubble surface Q Net force on the bubble R Bubble radius r Polar co-ordinate Time : Relative velocity between bubble and liquid V Bubble volume = 4rR3/3 vo Mean bubble volume x Vertical co-ordinate Y Horizontal co-ordinate Distance moved by bubble itJ Polar co-ordinate EL Liquid viscosity Liquid density Velocity potential in the liquid

REFERENCES

:;; 131 [r:; t:; [81 [91 I101 [ill t121 1131 [141

BUCHANANR. H., JAME~ONG. and OEDJOED., Zndustr. Engng Chem. (Fundamentals) 1962 182. 1909. HARBAUMK. L. and HOUGHTONG., Chem. Engng Ski. 1960 13 90. HARBAUMK. L. and HOUGHTONG., J. Appl. Chem. 1962 12 234. BAIRD M. H. I., Chem. Engng Sci. 1963 18 685. BRET~ZNAJ~ER S. and PASIUKW., Bull. Acad. Polon. Sci. Ser. Sci. Chim. 1963 11 107. BRET~ZNAJDER S. and PASIUK W., Znt. Chem. Engng 1964 4 61. HOUGHTONG., Proc. Roy. Sot. 1963 A272 33. ANDRADE E. N. da C.. Proc. ROY. Sot. 1932 A134 445. MINNAERTM., Phil. iag. 1933 iSer. 7) 16 235. PLESSETM. S. and HSIEH D. Y., Phys. FZuids 1960 3 882. RSCHEVKIN S. N., A course of lectures on the theory of soundp. 385. Pergamon Press, Oxford, 1963. MINER C. S. and DALTONN. N., Glycerol. Reinhold, New York, 1953. JAMEXING. J., Ph.D. dissertation, Cambridge 1963. BJERKNFSV., Die Kraftfelder p. 16. Vieweg, Braunschweig,

RBsum&L’auteur expose une theorie decrivant le mouvement d’une petite bulle sphtrique dam un liquide non visqueux anime d’un mouvement oscillant ascendant et descendant. La thtorie donne la frequence pour laquelle la force, engendree par ce mouvement alternatif et dirigee vers le bas, Cquilibre la poussee; dans ces conditions il n’y a aucun mouvement net de la bulle, et la theorie prevoit qu’elle oscillerait autour dun point fixe avec une amplitude trois fois plus grande que celle imposee au liquide. Ces resultats sont approximativement en accord avec les donnees experimentales; il est evident que dans toutes les experiences dans lesquelles une bulle est maintenue fixe g&e aux vibration impos&s au liquide, l’effet des forces de viscosite est important. B

33

G. J. JAMESON and J. F. DAVIDSON Zusammenfasstmg-Die Bewegung einer kleinen kugeligen Gasblase in einer niedrig viskosen Fliissigkeit wird fiir jenen Fall theoretisch beschrieben, bei dem die Blase einer aufgezwungenen Schwingung in senkrechter Richtung unterliegt. Aus dieser Theorie wird eine bestimmt Schwingfrequenz abgeleitet, bei der die abwarts gerichtete Schwingung durch den Auftrieb gerade kompensiert wird; dabei tritt dann keine Ortsveranderung der Blase auf. Nach der vorliegenden Theorie schwingt die Blase dabei urn einen Ruhepunkt, und zwar mit einer dreifach grBl3eren Amplitude als jener, die auf die Fliissigkeit einwirkt. Dieses theoretische Ergebnis stimmt nur grbl3enordnungsmlRig mit den experimentellen Befunden iiberein. Im Verlauf aller Versuche. bei denen die Gasblase durch Vibration im station&en Zustand gehalten wird, spielt natiirlich der Ei&uD der Reibungskriifte eine bedeutende Rolle.

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