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The influence of approach velocity on bubble coalescence
Chemical
Engineering
Science,
THE
1974, Vol. 29, pp. 2363-2373.
Pergamon
Press.
in Great
Britain
INFLUENCE OF APPROACH VELOCITY BUBBLE COALESCENCE R. D. KIRKPATRICK
Department
Printed
of Chemical
and M. J. LOCKETT
Engineering, University of Manchester Institute P.O. Box 88, Manchester I, England (Received
29 October
ON
1973; accepted
21 June
of Science
and Technology,
1974)
Abstract-Experiments have been carried out in which a cloud of air bubbles has been prevented from rising by downflowing water in a tube. High speed photography revealed an almost complete absence of bubble coalescence. This has been attributed to the large approach velocities of bubbles in the cloud. Further experiments in which a single bubble has been allowed to coalesce with a plane air-water interface have demonstrated the effect more clearly. Two basic types of bubble coalescence have been recognised depending on the approach velocity of the bubbles. At a low approach velocity, bubble coalescence is rapid, but coalescence times are considerably increased at large approach velocities. For pure liquids, a theory is put forward which shows that at low approach velocities film rupture can occur before the approaching bubbles are brought to rest. At large approach velocities the bubbles are brought to rest before rupture occurs. In the latter case bubble bounce can occur and the total
coalescence
time is thereby considerably
increased.
in a stationary bubble cloud, it is suggested that large approach velocities in a bubble column may be an important factor in limiting bubble coalescence. Based
on observed
approach
velocities
INTRODUCTION
The loss of interfacial area caused by bubble coalescence is important in mass transfer equipment, for example on distillation plates and in bubble columns, while the coalescence dependent transition between the bubble and slug flow regimes is important in two phase flow in pipes. In spite of considerable research, however, the mechanism of bubble coalescence is still unclear. Most previous studies can be classified into the following broad categories: (1) Coalescence in a bubble cloud where the gas distributor has a dominant influence on coalescence [l-3]. (2) Coalescence in a bubble cloud where coalescence is free from any gas distributor influence [4-61. (3) Coalescence studies in agitated tanks [7,8]. (4) Wake coalescence studies [9, 101. (5) Coalescence experiments using two adjacently formed bubbles [l l-131. (6) Theoretical descriptions of coalescence [14,15] and also in many of the papers listed above. In general the bulk of previous work has been performed with air-water and air-electrolyte SYSterns, although some work has been carried out with liquids containing surface active additives and with organic solutions [ 1,171.
The results of the papers in category 1 show that the gas distributor has a considerable influence on coalescence when the distributor hole size and pitch allow interaction between the bubbles, either at formation or just above the distributor. For this reason it was felt desirable to eliminate the effect of the gas distributor in the present work. Studies in category 2 show a wide variation in the results reported. There is little agreement on coalescence frequency. Lee and Ssali [6] for example quote values which are a factor of ten lower than the corresponding values of Calderbank et al. [4]. Towel1 et al. [S], on the other hand, observed no change in bubble size and suggested that the coalescence rate is exactly equal to the break-up rate. Marrucci and Nicodemo[l] also observed little coalescence in a bubble cloud apart from that occurring at the gas distributor. The papers in category 2 also show that, at a high gas holdup, coalescence rates are difficult to measure. A gas holdup of 0.12 is the highest reported although theoretical considerations show that the maximum gas throughput occurs at a much higher holdup. Using the Wallis relationship [ 161for a cloud of identical bubbles in the absence of liquid circulation, G = Ubmc(1- E)A the maximum gas throughput
2363
is found to occur at a
2364
R. D.
KIRKPATRICK
holdup of O-5. Industrially it is an advantage to operate at or close to the point at which maximum gas throughput is achieved, so that the region of industrial importance for bubble coalescence is at a gas holdup of greater than O-12. For example a typical gas holdup on distillation plates is 0.75. Wallis’s equation neglects liquid recirculation. In practice for a continuously bubbling system where the bubbles rise as a free cloud, liquid recirculation becomes appreciable as the holdup increases above 0.2. At a holdup approaching 0.5 liquid recirculation is excessive and attempts to observe bubble coalescence at holdups of this order are difficult. Another aim of this paper was therefore to try and study bubble coalescence at higher gas holdups than had previously been obtained while keeping liquid recirculation at a minimum. A further limitation experienced by previous workers is the limited time over which an individual bubble in a bubble cloud may be observed. This results from the small residence time of the gas in most gas-liquid contactors. For this reason it was felt desirable to increase the gas residence time in the present study. The initial aims of the present work can be summarised as follows: a study of bubble coalescence under the following conditions(i) No gas distributor influence. (ii) Operation over a wide range of gas holdups, approaching a holdup of 0.5. (iii) Observation keeping liquid recirculation at a minimum. (iv) Observation of the bubbles in a particular bubble cloud over a longer time than had previously been achieved. All these aims were met using apparatus similar to that developed by Davidson and Kirk[l8] in which a bubble or swarm of bubbles may be stabilised in a tube by downflowing liquid. EXPERIMENTS WITH STATIONARY BUBBLE CLOUDS
The apparatus is shown diagramatically in Fig. 1. The test section was a vertical inverted glass rotameter tube of length 450 mm and mean diameter 80 mm (having its largest cross-section at the bottom). Water flowed down through the tube under gravity from a header tank above. By adjusting the downflowing water velocity an air bubble or cloud of bubbles could be held stationary in the test section. By ‘stationary’ we mean that in any direction the average velocity of the bubbles was zero. The water was collected in a tank after passing through the test section and then pumped
and
M.
J.
LOCKET
Velocity profile modifier
Test section I\
Fig. 1. Stationary bubble cloud apparatus.
back to the header tank. A velocity profile modifier was placed above the test section in the pipe connecting the header tank to the test section. Its purpose was to flatten the normal velocity profile so that bubbles did not preferentially rest at the tube wall. The velocity profile modifier used in these experiments was constructed from 600 polypropylene straws and was similar to that described in[18]. Readers are referred to[18] for further details of this experimental technique. Air was injected into the tube from an air bottle via a l/16 in. nozzle to form the bubble cloud. The procedure for creating a high holdup cloud was to start with a high downflowing water velocity sufficient to hold the initial bubbles in the test section. As the holdup slowly increased with injection of further bubbles, the downflowing water velocity was slowly reduced because the velocity required to hold a cloud of bubbles stationary decreases as the holdup increases. When the desired holdup and cloud volume had been achieved the air supply was shut off so that observation of the cloud could be made without gas distributor influence. The mean gas holdup was measured by suddenly shutting off the water flow and allowing the air bubbles to rise to the top of the apparatus where they were collected in a measuring cylinder. RESULTS OF EXPERIMENTS USING A STATIONARY BUBBLE CLOUD
The bulk of the experiments
were carried out
The influence of approach velocity on bubble coalescence
with tap water at 20-22°C. A few observations were also made in which the apparatus was filled with distilled water. The results were indistinguishable from the experiments using tap water. The bubbles studied in the experiments had an equivalent spherical dia of 5 mm, and holdups could be achieved between 0.5 and 50 per cent. A typical bubble cloud having a holdup of 50 per cent is shown in Fig. 2. At high holdups (typically 20-50 per cent) observations over a long period of time (exceeding 1 hr in cases) showed no change in the size of the bubbles in the cloud. A series of 16 mm movie films taken of the bubble cloud at a film speed of 100 frameslsec showed that no coaIescence was occurring. At low holdups (typically 0.5-20 per cent) observations again showed no increase in the bubble size of the cloud. However, the movie films taken at lower holdups showed that in two of the several thousand contacts observed coalescence, or more probably cohesion, occurred. The large bubbles so formed did not persist in the bubble cloud and appeared to be broken down in subsequent contacts. These observations at low holdup are similar to those of Towel1 et al. [5] who state that coalescence and break-up rates are equal for air-water systems. However, the coalescence frequency observed in this present work is several
2365
orders of magnitude lower than that reported in[5]. Careful observation of the films taken at 50 per cent holdup showed that the bubbles which formed the cloud were not stationary but oscillated quite rapidly. These oscillations were not oscillations of the bubbles as a whole but rather shape oscillations of the individual bubbles, i.e. the bubble centre moved slowly in a random manner while the actual bubble shape oscillated rapidly. The shape oscillations were such that the majority of the bubbles changed their linear dimension in any given direction by approximately a of the bubble diameter every 0.01 sec. Occasionally a bubble remained approximately constant in shape for 0.03-0.04 sec. The bubble-bubble contact time was in the range of 0.01-0.12 set with an average of 0.06 sec. Observations of the films taken at lower holdups showed similar results. In general as the holdup decreased, the shape oscillations of the individual bubbles decreased and the random linear velocity of the individual bubbles increased. The following table summarises the measurements taken from the films. Some 50 individual measurements were made and the figures in brackets represent the standard deviations. Table 1. Gas holdup (W
Relative contact velocity (cmlsec)
Bubble-bubble contact time (msec)
50 30 15
20.7(6.2) l&5(5.4) 27.0(4*1)
62(27) 53(20) 39(14)
DISCUSSION OF THE STATIONARY BUBBLE CLOUD EXPERIMENTS
Fig. 2. Bubble cloud (50 per cent hold-up).
The virtually complete absence of bubble coalescence is surprising in view of the widely reported observation that bubble coalescence is very rapid in pure water. Marrucci et al. [12] for example quote a coalescence time for two bubbles in pure tap water of 10-3-10-4 sec. We initially considered it possible that impurities in the water were hindering coalescence[l9], and as has been frequently reported for electrolyte solutions [ I, 2,12,13]. To test this idea a separate series of experiments were performed similar to those reported by Marrucci et al. [12] and Lessard and Zieminski[ 131. Bubbles were formed at the ends of two adjacent 1 mm i.d. tubes (in stationary water) and allowed to touch as they formed. The results obtained were similar to those of[12] and [13]. Coalescence was found to be very rapid and in a sample of the water used in the
2366
R. D.
KIRKPATRICK
stationary bubble cloud experiments coalescence times were less than 0.01 sec. Ageing of the bubbles by forming them apart, holding them apart for 20 sets and then gently bringing them together was found to have no effect. They still coalesced immediately on touching. It appears that there is a fundamental difference between coalescence of two bubbles formed at adjacent tubes and coalescence in a bubble cloud. Even in pure water coalescence in a bubble cloud is very infrequent. The major difference between experiments using two bubbles and those performed on a bubble cloud is that in the latter the bubbles usually have a significant velocity of approach at contact. In the section below we report experiments in which the dependence of coalescence times on approach velocity is investigated.
EXPERIMENTS WITH BUBBLES HAVING A FINITE APPROACH VELOCITY
The apparatus consisted of a glass 3 in. dia. bell immersed in a vessel of water so that air was trapped to form a flat interface at the mouth of the bell. The sides of the vessel were plane glass to enable undistorted observation to be made. Water was obtained from the normal laboratory distilled water supply and experiments were carried out at room temperature (20-22°C). A 4 mm i.d. glass tube was arranged to enable an air bubble having an equivalent spherical dia. of 5 mm to be formed at its tip. A diagram of the apparatus is shown in Fig. 3. The separation between the bubble injection tube and the air-water interface, x, could be varied so
Fig. 3. Bubble-interface apparatus.
and M.
J.
L~CKETT
that the following types of bubble interface contact could be achieved: (a) Contact before the bubble was fully formed. (b) Contact just as the bubble became fully formed. (c) Contact just as the neck, connecting the air injection tube and the rising bubble, ruptured. (d) Contact at larger separations where the bubble at contact was rising freely at velocities ranging from 0.5 to 1.0 Ub, (where Ub, is the terminal rise velocity of the bubble). Bubble coalescence was recorded on 16mm Kodak 4-X negative film using a Hycam movie camera operated between 20 and 500 frameslsec. The films were analysed using a motion analyser and bubble size, separation x, contact area and coalescence times were measured. The coalescence time is the time from contact between the bubble and interface to the rupture of the contact film. A large number of observations were made[20] and the results are summarised in Table 2. Each result represents an average of approximately ten readings. The figures in brackets are the standard deviations of the coalescence times. The letters in brackets indicate the film speed at which the individual results were obtained: (a) 500 frameslsec, (b) 20 frameslsec, (c) 100 frameslsec, (d) 50 frameslsec. The estimated error in any reading is *I frame, i.e. at 500 frames/set the estimated error is *0+02 sec. Table 2. Equivalent spherical bubble diameter when fully formed = 5 mm. Separation x (cm)
Coalescence time T (set)
0.45 0.55 0.61 0.67 0.87 1.35 1.50 2.70 3.50
0~005-0~01(a) 0~005-0~01 (a) O-O.15 (b) 0.14(0.04) (b) O.lS(O.02) (c) O.H(O.02) (d) 0.16(0.03) (c) 0.18(0.03) (c) 0.18(0.02) (c)
Figure 4 shows the variation of coalescence time with separation. There are three distinct regions on the graph. Region I-where the bubble is still attached to the formation tube as it contacts the interface. The velocity of approach of the interfaces in this region was measured from the films and was found to be less than 1.0 cmlsec. The contact film radius just before film rupture was quite small (approximately
The influence of approach velocity on bubble coalescence 0.20
I
0.16-
Y "
I
I
T/+4+
I
I
I
II
I PO
I 2.5
I 3.0
I_ 3.5
2367
/'-
0.12-
I- o.oe-
0.04-
0
v 0.5
I I.0
I I.5 x.
cm
Fig. 4. Coalescence time vs separation.
0.05 cm). Figure 5 shows a typical coalescence event for Region I. Region II-is a transition region for separations between 0.55 and 0.67 cm. It was occasionally found that rapid coalescence (as found in Region I) occurred in this region when the bubble had clearly detached from the formation tube. This leads us to believe that the distinction between Regions I and III does not depend on whether or not the bubble was supported by the formation tube at contact. Region III-the neck joining the bubble to the formation tube is able to break at separations greater than about 0.6cm. Above this separation the bubbles are rising freely as they contact the interface. The rise velocity at contact can be calculated from the gradient of Fig. 6 in which the distance travelled by the bubble is plotted against time-both measured from the instant of neck rupture. It is evident from the graph that at neck rupture the bubbles have a high velocity, 10 cmlsec, and quickly accelerate to their terminal velocity. Detailed observation of the sequence of events which occurred in Region III revealed that no deceleration of the bubble was apparent prior to the bubble-interface contact. After contact the bubble could be seen to sink into the interface without coalescence and then to oscillate at the surface before finally coalescing. The time for the first oscillation fell within the range 0.04-0.06 set for separations x, between 0*87-3.50cm. The bubble almost completely disappeared during the first contact and then usually oscillated twice before coalescing. Figure 7 shows the initial stages of a Region III coalescence event. At coalescence the radius of the contact film was approximately equal to the bubble radius.
Fig. 5. Region I coalescence (100frameslsec).
To summarize the above results, it appears that coalescence is very rapid when the interface contact velocity is less than l.Ocm/sec, but when the contact velocity is greater than 10 cmlsec coalescence times are considerably increased. It should be noted that the above results are specific to the bubble size used in these experiments. We do not believe that impurities in the water had any significant effect as results obtained with tap water and doubly distilled water were virtually identical.
2368
R. D. KIRKPATRICK and
M. J. LOCKET
3.2
2.4$
2.0-
d k
1.6-
f i5
1.2-
0
0.04
0.06
0.12
Time,
set
0.16
0.20
Fig. 6. Distance travelled vs time for a rising bubble, both measured fromtheinstantthebubbleneckruptures.
A few experiments were carried out with an ionic solution of NaCl in water[20]. At low concentrations the same general trends were observed. As the concentration was increased to 0.6 M the distinction between Region I and III disappeared and coalescence times were large (0.7 set) and independent of separation and hence contact velocity. This confirms the widely quoted observation that coalescence is inhibited in concentrated electrolyte solutions. The effect of approach velocity is unimportant in such systems and the remainder of this paper is inapplicable to concentrated electrolyte solutions. In the following section we present a mathematical model which predicts an increase in coalescence time with approach velocity. In addition we use the model to indicate the maximum approach velocity for which Region I behaviour may occur. Fig. 7. Initial stages of Region III coalescence (100 frameslsec.). THEORETICAL DEVELOPMENT OF BUBBLE COALESCENCE WITH AN INTERFACE-HAVING A FINITE APPROACH VELOCITY
When a bubble approaches an interface with a finite approach velocity, the area of the contact film increases with time because the two surfaces press against each other. At the same time film drainage occurs, because of the overpressure in the film, which tends to reduce the film thickness. The rate of drainage of the film depends on its area and decreases as its area increases. Thus these two
effects compete against each other. The mechanism of bubble-interface coalescence may be subdivided into three types depending upon the approach velocity of the bubble. Region I. Low approach velocities. The rate of increase of the contact film area with time is sufficiently slow to allow the contact film to drain to the rupture thickness before the bubble is brought to rest. Region II. This is the region for moderate
2369
The influence of approach velocity on bubble coalescence approach velocities and is a transition region between Regions I and III. Region III. Large approach velocities. The rate of increase of the contact film area with time is sufficiently rapid to retard film thinning, to the extent that the contact film has insufficient time to drain to the rupture thickness before the bubble is brought to rest. The strain ‘energy stored in the deformed bubbles then tends to reverse the bubble’s motion and the contact film starts to thicken[23]. Coalescence at large approach velocities is no longer a simple process but depends on bubble oscillation after contact and the complex shape of the drainage film formed between the two interfaces. PRESSURE
PROFILE
IN THE FILM
Consider the bubble-interface geometry at some time t after contact as shown in Fig. 8. For small bubbles, the static liquid pressure PL can be taken as everywhere constant and equal to the pressure above the interface P,,. The internal pressure within the bubble is assumed constant at PL + 2 y lb. The shape of the deformed bubble is difficult to predict for a moving bubble, and so this will be estimated from the quasi-static situation where the pressures at point G in the film are assumed to balance. At G,
the film surfaces and hence there are no frictional losses. Benoulli’s equation between G and H is then po/pr. + uo2/2 = PHlPL + UH72.
At the stagnation point G, UG= 0 and from Eqs. (1) and (2) it follows that PO = PL + y/b. At the edge of the film H, the pressure is PL and the velocity is u@. From Eq. (3), ylb = p~ue2/2.
4
+2r_z_r
L b
t,liquidinfilm=8Pb2(1-Cos0)h
At t = t + at, liquid in film
p’
Since the film thickness is negligible compared with p, it follows that at the nose of the bubble p +q +2b.
(4)
To calculate the pressure at intermediate points within the film, it is necessary to make simplifying assumptions about the local radius of curvature and the film thickness if complex calculations are to be avoided. We therefore take a model geometry of a constant radius of curvature 2b at all points in the film and a constant film thickness h. Equation (4) still applies to this proposed geometry but a pressure balance over the whole bubble surface will no longer be satisfied because of the two simplifying assumptions made above. The proposed model is shown in Fig. 9. The surface area of the film is $0”2~ 2b Sin +.2bd+ = &rb ‘( 1 - Cos 6). Consider a mass balance on the film over a time increment 6t. Att=
p am +zr,p
=g?rb’[l-Cos(B+$t)](h+$St).
(2)
Restricting subsequent arguments to pure liquids, it follows that in the flowing film there is no shear at
Fig.8. A modelfor bubble-interfacecoalescence.
CES Vol. 29, No. 12-H
(3)
Fig.9. Proposedmodel
for bubble-interface
coalescence.
R. D. KIRKPATRICK and M. J. L~CKETI
2370
the bubble F
The liquid flowing out of the film in St is 237 2b Sin 8h(ue - (de/dt)) Neglecting & -0,
St.
second order terms and simplifying,
as hence
dh ueh Sin 8 dt = -2b(l -Cos 19)’
Sin’ e + (I - COS e)’
Applying Bernoulli’s equation to the film between a general angle 4 and 8,
x[2ln2-2ln(l+Cose)-2(1-c0s8)+~Sin2e] P+ IPL + ufl2 = ue2l2+ PL IPL.
(6) The bubble deceleration
Equation (5) may also be written for any general angle 4, so that combining with Eqs. (6) and (4) we have PO - PL = y/b. (I-
[;;;
~~l~“c”o”s~;]2).
(7)
Equation (7) is an equation for the pressure profile in the film. DRAINAGE
Eliminating
d2s 96~ dt2= -llb2p,
Sir? e 2+
is therefore Sin2 e (1 - COS e)r
x[2ln2-2ln(l+CostI)-2(1-Cose)+fSin*e]
1 (10)
where e = Sin-’ ((b’-$“i).
EQUATION
1.
(11)
ue from Eqs. (4) and (5) we have CALCULATION
; = 2b’pL
(8)
The change in film thickness, from an initial thickness hi to a final thickness hf, which occurs in a film of constant area during a drainage time At is obtained by integrating Eq. (8) to give
“’
At
. BUBBLE
(9)
PROCEDURE
The calculation procedure adopted was to assume that at t = 0, s = b, (d’sldt’) = 0, U = Uo. Taking a series of equal time increments of 0.0001 set, the new value of s at the end of each increment could be calculated from s,+~*= st - LJ,At. Using Eqs. (10) and (11) the deceleration of the bubble could be calculated and hence the velocity at the end of the increment. Equation (9) was used to determine the change in the film thickness over the time increment. The calculations were repeated until the bubble velocity fell to zero.
DECELERATION
As the bubble contacts the interface, it slows down because of the excess pressure on its upper surface. Deformation occurs but this is hard to allow for. We can make an estimate of the deceleration of the bubble by assuming it to behave as a sphere with an effective mass near to an interface of % 3 rb’pL (21). The restraining force is taken as the excess pressure force in the contact film. The buoyancy force is taken to be exactly equal to the drag force. As the bubble slows down this assumption probably leads to an overestimation of the deceleration and to a conservative estimate of the inhibition of coalescence due to approach velocity. Integrating the vertical component of Eq. (7) over the contact film gives the restraining force on
THEORY FOR BUBBLE-BUBBLE COALESCENCE WITH FINITE APPROACH VELOCITY
The bubbles are assumed to be of equal size and their centres to have equal velocities U/2 towards each other along their line of centres. For small bubbles Pr can be assumed constant and the pressure in the film at the bubble nose is PL + 2y/b. Since the pressure at the edge of the film is P,_, application of Bernoulli’s equation along the film yields p~u~=l2= 2ylb
(12)
As for the bubble-interface model, it is now necessary to make assumptions about the local
2371
The influence of approach velocity on bubble coalescence
radius of curvature along the film and the film thickness. We assume a flat film of uniform thickness h as shown in Fig. 10. As before these simplifying assumptions make it impossible to satisfy a complete pressure balance over the distorted bubble surface.
equation In (2)
= 4(--&)“2$.
The bubble restraining force F is obtained integrating Eq. (15) over the contact film F=
I
R
by
2?rr(p~-pr)=~=%(b2-s’).
0
Taking $3 rb3pL as the effective bubble, the deceleration is d’s
12
Z’_fiYbrpL.
mass
(b2-s’)
of the
(17)
The calculation procedure for bubble-bubble contact is the same as that for bubble-interface contact. RESULTS
Bubble-interface
Fig. 10.Proposed model for bubble-bubblecoalescence.
A material balance on the film within a radius r gives dh -=dt
-2urh r’
(13)
Bernoulli’s equation between a radius r in the film and the edge is P, IpL + u,2/2 = lb72 + PL IPL.
(14)
From Eqs. (12), (13) and (14),
(15) A derivation leading to an equation similar to Eq. (15) has also been given by Farooq[22]. Writing Eq. (13) for the complete film and combining with Eq. (12) gives
This equation may be integrated to give a drainage
OF THE CALCULATIONS AND DISCUSSION
coalescence
The calculated thinning of the contact film with time as a 5 mm diameter bubble makes contact with an interface is shown in Fig. 11. Calculations for various initial approach velocities are shown. The rupture thickness has been taken as 2.5 x 10”cm which is a typical value assumed by previous workers. The termination point for each line is when the bubble velocity is zero. The time for the bubble to decelerate from 22 cm/set to zero velocity is seen from the theoretical curve (Fig. 11) to be 0.037 sec. This value is in good agreement with half the experimentally observed time for the first oscillation (0*020-0~030 set). The calculations indicate that at approach velocities greater than 5 cmlsec the bubble will come to rest before film rupture occurs. It seems probable that on coming to rest the stored strain energy in the film will cause the bubble to reverse its motion and to move down away from the interface. Buoyancy forces acting on the bubble will subsequently cause it to slow down and eventually to reverse its motion so that it again approaches the interface. This sequence of events is repeated with the approach velocity being reduced after each bounce until the approach velocity is small. The calculations indicate that bubbles having an approach velocity of less than about 4cm/sec coalesce with the interface before they are brought to rest. The relatively simple theory proposed above is not capable of making predictions about the mechanism of bubble bounce and it only indicates whether coalescence would be expected on the first approach.
R. D. KIRKPATRICK and
5mm bubble-interface
\
3 cm /set
I
I
0.012
0.024
Time,
contact
1 ox
set
Fig. 1I. Filmthicknessvs time. The experiments we have reported for bubbleinterface coalescence indicated a transition velocity lying between 1 and 10 cmlsec for 5 mm dia bubbles and this is in agreement with these predictions. Figure 12 shows the predicted changes in film thickness with time for different size bubbles. The
M. J. LOCKETT
bubbles are all initially assumed to be moving at their terminal rise velocity as was found for 5 mm dia bubbles. Bubble bounce is predicted for 5 mm dia and 1 mm dia bubbles whereas 0.5 and 0.25 mm bubbles are not expected to bounce. Bubbles of 0.75 mm dia are just on the border line. It is of interest to compare these predictions with the recently reported data of Farooq [22] in which small bubbles were allowed to coalesce with an interface under very carefully controlled conditions. Farooq found that 1.0 and 0.75 mm dia bubbles bounced and 0.5 and 0.25 mm dia bubbles did not. The predictions of the model are thus in agreement with the available experimental data which is remarkable in view of the quite severe assumptions which are inherent in the model. Farooq [22] has suggested that bubble bounce might be attributable to film immobilisation due to surface tension gradients, which can arise if the film temperature falls under adiabatic stretching. Such a mechanism might operate in addition to the one we propose and their relative importance deserves further study. Bubble-bubble
coalescence
Predictions for bubble-bubble contact of 5 mm dia bubbles having various initial relative approach velocities are shown in Fig. 13. The calculations show that if the approach velocity of the bubbles is less than about 12cm/sec, the bubbles would be expected to coalesce on first contact. At higher approach velocities the bubbles are expected to
5mm Bubble-bubble
contact
Rupture
0.25 ICP 0
-
mm I 0.6
I I.6
Time,
msec
Fig. 12. Film thickness vs time.
I 2.4
lo-7l 0
0.004
0.006
Time,
0 012
set
Fig. 13. Film thickness vs time.
0.016
The influence of approach velocity on bubble coalescence
bounce. As noted earlier, no predictions can be made with this model about the total coalescence time if, after bounce occurs, the bubbles come together again. A very approximate estimate for bubble-bubble coalescence under these conditions can be made from the bubble interface experiments reported earlier and this is O-15 sec. Table 1 shows contact velocities which we have observed in a bubble cloud and they are large, which indicates that bubble bounce or Region III type coalescence would be expected. The rest times which we observed (about 0.05 set) are smaller than the coalescence time (0.15 set) which is required for Region III coalescence. Thus bubble coalescence is not expected to be important in a bubble swarm, even for pure liquids. An exception to this may be where the bubbles are held together for comparatively long periods in clusters. We intend to deal with this in a subsequent paper. It must be emphasized that in order to make quantitative predictions about the effect of approach velocity on bubble coalescence, we hme had to make a number of quite sweeping assumptions in the model. Considerable refinement is possible but more detailed models are unlikely to change the broad conclusion of this paper that bubble approach velocity is important and can prevent bubble coalescence. NOTATION
A
cross sectional area of vessel, cm* b bubble radius, cm F force, dynes G volumetric gas flow rate, cm’lsec h film thickness, cm P pressure, dynes/cm2 P radius of curvature, cm radius of curvature, cm z film radius, cm r radial distance, cm s distance of bubble centre from interface or from edge of film defined in Figs. 9 and 10, cm T coalescence time, set t time, set u relative approach velocity, cmlsec Ub, bubble terminal rise velocity, cmlsec U liquid velocity in film, cmlsec separation-Fig. 3, cm X surface tension, dynes/cm Y E gas holdup angle defining edge of film-Fig. 9, rads e liquid density, g/cm3 PL 4 angle defined in Fig. 9, rads
2373
Subscripts
atm G H ; L R r 0
f
value above the plane interface value at stagnation point value of edge of film initial value final value value in the liquid value at r = R value at r value at t = 0 value at edge of film value at angle 4 REFERENCES
[1] Marrucci G. and Nicodemo L., Chem. Engng Sci. 1967 22 1257. [2] Zieminski S. A. and Whittemore R. C., Chem. Engng Sci. 1971 26 509. [3] Bowonder B. and Kumar R., Chem. Engng Sci. 1970 25 25. [4] Calderbank P. H., Moo-Young M. B. and Bibby R.. Proc. Third Symp. on Chem&zl Reactor E&g.; p. 91, Pergamon Press, London 1964. PI Towel1 G. D., Strand G. P. and Ackerman G. H., Proc. A.LCh.E.-I. Chem. Engng Joint Meeting, ~97, London 1965. I61 Lee J. C. and Ssali G. W. K., Joint Meeting on Bubbles and Foams, Sl-6.1, Verfahrenstechnische Gesellschaft im VDI and the Inst. of Chem. Engr, Nurnberg 1971. [71 Reith T. and Beek W. .I.. Trans. Inst. Chem. Enar. 1970 48 T63. PI Lee J. C. and Meyrick D. L., Trans. Inst. Chem. Engr 1970 48 T37. [91 Crabtree .I. R. and Bridgwater J., Chem. Engng Sci. 1971 26 839. WJI De Nevers N. and Wu J., A.LCh.E. JI. 1971 17 182. 1111 Foulk C. W. and Miller J. N., Ind. Engng Chem. 1931 23 1283. WI Marrucci G., Nicodemo L. and Acierno D., Inf. Symp. on Research in Co-current Gas-Liquid Flow, p. 95, Plenum Press, New York 1969. u31 Lessard R. R. and Zieminski S. A., Ind. Engng. Chem. Fund. 1971 10 260. I _ Sci. 1968 r141 Lee J. C. and Hodeson T. D.. Chem. Enana 23 1375. WI Marrucci G., Chem. Engng Sci. 1969 24 975. tW Wallis G. B., Intern. Heat Transfer, p. 319, Am. Sot. Mech. Engr. 1961. I171 Delmar R. R. and Zieminski S. A., A.1.Ch.E. JI. 1971 17 57. 1181 Davidson 3. F. and Kirk F. A., Chem. Engng Sci. 1969 24 1529. 1191 Anderson J. L. and Quinn J. A., Chem. Engng Sci. 1970 2s 373. ml Kirkpatrick R. D., M. SC. Report U.M.I.S.T. 1973. PII Milne-Thomson L. M., Theoretical Hydrodynamics, p. 528 Macmillan 1968. WI Farooq S. Y. Ph.D. Thesis, University College of Swansea, (University of Wales), 1972. r231 Jeffreys G. V. and Hawksley J. L., A.I.Ch.E. JI 1965 11 413.
Engineering
Science,
THE
1974, Vol. 29, pp. 2363-2373.
Pergamon
Press.
in Great
Britain
INFLUENCE OF APPROACH VELOCITY BUBBLE COALESCENCE R. D. KIRKPATRICK
Department
Printed
of Chemical
and M. J. LOCKETT
Engineering, University of Manchester Institute P.O. Box 88, Manchester I, England (Received
29 October
ON
1973; accepted
21 June
of Science
and Technology,
1974)
Abstract-Experiments have been carried out in which a cloud of air bubbles has been prevented from rising by downflowing water in a tube. High speed photography revealed an almost complete absence of bubble coalescence. This has been attributed to the large approach velocities of bubbles in the cloud. Further experiments in which a single bubble has been allowed to coalesce with a plane air-water interface have demonstrated the effect more clearly. Two basic types of bubble coalescence have been recognised depending on the approach velocity of the bubbles. At a low approach velocity, bubble coalescence is rapid, but coalescence times are considerably increased at large approach velocities. For pure liquids, a theory is put forward which shows that at low approach velocities film rupture can occur before the approaching bubbles are brought to rest. At large approach velocities the bubbles are brought to rest before rupture occurs. In the latter case bubble bounce can occur and the total
coalescence
time is thereby considerably
increased.
in a stationary bubble cloud, it is suggested that large approach velocities in a bubble column may be an important factor in limiting bubble coalescence. Based
on observed
approach
velocities
INTRODUCTION
The loss of interfacial area caused by bubble coalescence is important in mass transfer equipment, for example on distillation plates and in bubble columns, while the coalescence dependent transition between the bubble and slug flow regimes is important in two phase flow in pipes. In spite of considerable research, however, the mechanism of bubble coalescence is still unclear. Most previous studies can be classified into the following broad categories: (1) Coalescence in a bubble cloud where the gas distributor has a dominant influence on coalescence [l-3]. (2) Coalescence in a bubble cloud where coalescence is free from any gas distributor influence [4-61. (3) Coalescence studies in agitated tanks [7,8]. (4) Wake coalescence studies [9, 101. (5) Coalescence experiments using two adjacently formed bubbles [l l-131. (6) Theoretical descriptions of coalescence [14,15] and also in many of the papers listed above. In general the bulk of previous work has been performed with air-water and air-electrolyte SYSterns, although some work has been carried out with liquids containing surface active additives and with organic solutions [ 1,171.
The results of the papers in category 1 show that the gas distributor has a considerable influence on coalescence when the distributor hole size and pitch allow interaction between the bubbles, either at formation or just above the distributor. For this reason it was felt desirable to eliminate the effect of the gas distributor in the present work. Studies in category 2 show a wide variation in the results reported. There is little agreement on coalescence frequency. Lee and Ssali [6] for example quote values which are a factor of ten lower than the corresponding values of Calderbank et al. [4]. Towel1 et al. [S], on the other hand, observed no change in bubble size and suggested that the coalescence rate is exactly equal to the break-up rate. Marrucci and Nicodemo[l] also observed little coalescence in a bubble cloud apart from that occurring at the gas distributor. The papers in category 2 also show that, at a high gas holdup, coalescence rates are difficult to measure. A gas holdup of 0.12 is the highest reported although theoretical considerations show that the maximum gas throughput occurs at a much higher holdup. Using the Wallis relationship [ 161for a cloud of identical bubbles in the absence of liquid circulation, G = Ubmc(1- E)A the maximum gas throughput
2363
is found to occur at a
2364
R. D.
KIRKPATRICK
holdup of O-5. Industrially it is an advantage to operate at or close to the point at which maximum gas throughput is achieved, so that the region of industrial importance for bubble coalescence is at a gas holdup of greater than O-12. For example a typical gas holdup on distillation plates is 0.75. Wallis’s equation neglects liquid recirculation. In practice for a continuously bubbling system where the bubbles rise as a free cloud, liquid recirculation becomes appreciable as the holdup increases above 0.2. At a holdup approaching 0.5 liquid recirculation is excessive and attempts to observe bubble coalescence at holdups of this order are difficult. Another aim of this paper was therefore to try and study bubble coalescence at higher gas holdups than had previously been obtained while keeping liquid recirculation at a minimum. A further limitation experienced by previous workers is the limited time over which an individual bubble in a bubble cloud may be observed. This results from the small residence time of the gas in most gas-liquid contactors. For this reason it was felt desirable to increase the gas residence time in the present study. The initial aims of the present work can be summarised as follows: a study of bubble coalescence under the following conditions(i) No gas distributor influence. (ii) Operation over a wide range of gas holdups, approaching a holdup of 0.5. (iii) Observation keeping liquid recirculation at a minimum. (iv) Observation of the bubbles in a particular bubble cloud over a longer time than had previously been achieved. All these aims were met using apparatus similar to that developed by Davidson and Kirk[l8] in which a bubble or swarm of bubbles may be stabilised in a tube by downflowing liquid. EXPERIMENTS WITH STATIONARY BUBBLE CLOUDS
The apparatus is shown diagramatically in Fig. 1. The test section was a vertical inverted glass rotameter tube of length 450 mm and mean diameter 80 mm (having its largest cross-section at the bottom). Water flowed down through the tube under gravity from a header tank above. By adjusting the downflowing water velocity an air bubble or cloud of bubbles could be held stationary in the test section. By ‘stationary’ we mean that in any direction the average velocity of the bubbles was zero. The water was collected in a tank after passing through the test section and then pumped
and
M.
J.
LOCKET
Velocity profile modifier
Test section I\
Fig. 1. Stationary bubble cloud apparatus.
back to the header tank. A velocity profile modifier was placed above the test section in the pipe connecting the header tank to the test section. Its purpose was to flatten the normal velocity profile so that bubbles did not preferentially rest at the tube wall. The velocity profile modifier used in these experiments was constructed from 600 polypropylene straws and was similar to that described in[18]. Readers are referred to[18] for further details of this experimental technique. Air was injected into the tube from an air bottle via a l/16 in. nozzle to form the bubble cloud. The procedure for creating a high holdup cloud was to start with a high downflowing water velocity sufficient to hold the initial bubbles in the test section. As the holdup slowly increased with injection of further bubbles, the downflowing water velocity was slowly reduced because the velocity required to hold a cloud of bubbles stationary decreases as the holdup increases. When the desired holdup and cloud volume had been achieved the air supply was shut off so that observation of the cloud could be made without gas distributor influence. The mean gas holdup was measured by suddenly shutting off the water flow and allowing the air bubbles to rise to the top of the apparatus where they were collected in a measuring cylinder. RESULTS OF EXPERIMENTS USING A STATIONARY BUBBLE CLOUD
The bulk of the experiments
were carried out
The influence of approach velocity on bubble coalescence
with tap water at 20-22°C. A few observations were also made in which the apparatus was filled with distilled water. The results were indistinguishable from the experiments using tap water. The bubbles studied in the experiments had an equivalent spherical dia of 5 mm, and holdups could be achieved between 0.5 and 50 per cent. A typical bubble cloud having a holdup of 50 per cent is shown in Fig. 2. At high holdups (typically 20-50 per cent) observations over a long period of time (exceeding 1 hr in cases) showed no change in the size of the bubbles in the cloud. A series of 16 mm movie films taken of the bubble cloud at a film speed of 100 frameslsec showed that no coaIescence was occurring. At low holdups (typically 0.5-20 per cent) observations again showed no increase in the bubble size of the cloud. However, the movie films taken at lower holdups showed that in two of the several thousand contacts observed coalescence, or more probably cohesion, occurred. The large bubbles so formed did not persist in the bubble cloud and appeared to be broken down in subsequent contacts. These observations at low holdup are similar to those of Towel1 et al. [5] who state that coalescence and break-up rates are equal for air-water systems. However, the coalescence frequency observed in this present work is several
2365
orders of magnitude lower than that reported in[5]. Careful observation of the films taken at 50 per cent holdup showed that the bubbles which formed the cloud were not stationary but oscillated quite rapidly. These oscillations were not oscillations of the bubbles as a whole but rather shape oscillations of the individual bubbles, i.e. the bubble centre moved slowly in a random manner while the actual bubble shape oscillated rapidly. The shape oscillations were such that the majority of the bubbles changed their linear dimension in any given direction by approximately a of the bubble diameter every 0.01 sec. Occasionally a bubble remained approximately constant in shape for 0.03-0.04 sec. The bubble-bubble contact time was in the range of 0.01-0.12 set with an average of 0.06 sec. Observations of the films taken at lower holdups showed similar results. In general as the holdup decreased, the shape oscillations of the individual bubbles decreased and the random linear velocity of the individual bubbles increased. The following table summarises the measurements taken from the films. Some 50 individual measurements were made and the figures in brackets represent the standard deviations. Table 1. Gas holdup (W
Relative contact velocity (cmlsec)
Bubble-bubble contact time (msec)
50 30 15
20.7(6.2) l&5(5.4) 27.0(4*1)
62(27) 53(20) 39(14)
DISCUSSION OF THE STATIONARY BUBBLE CLOUD EXPERIMENTS
Fig. 2. Bubble cloud (50 per cent hold-up).
The virtually complete absence of bubble coalescence is surprising in view of the widely reported observation that bubble coalescence is very rapid in pure water. Marrucci et al. [12] for example quote a coalescence time for two bubbles in pure tap water of 10-3-10-4 sec. We initially considered it possible that impurities in the water were hindering coalescence[l9], and as has been frequently reported for electrolyte solutions [ I, 2,12,13]. To test this idea a separate series of experiments were performed similar to those reported by Marrucci et al. [12] and Lessard and Zieminski[ 131. Bubbles were formed at the ends of two adjacent 1 mm i.d. tubes (in stationary water) and allowed to touch as they formed. The results obtained were similar to those of[12] and [13]. Coalescence was found to be very rapid and in a sample of the water used in the
2366
R. D.
KIRKPATRICK
stationary bubble cloud experiments coalescence times were less than 0.01 sec. Ageing of the bubbles by forming them apart, holding them apart for 20 sets and then gently bringing them together was found to have no effect. They still coalesced immediately on touching. It appears that there is a fundamental difference between coalescence of two bubbles formed at adjacent tubes and coalescence in a bubble cloud. Even in pure water coalescence in a bubble cloud is very infrequent. The major difference between experiments using two bubbles and those performed on a bubble cloud is that in the latter the bubbles usually have a significant velocity of approach at contact. In the section below we report experiments in which the dependence of coalescence times on approach velocity is investigated.
EXPERIMENTS WITH BUBBLES HAVING A FINITE APPROACH VELOCITY
The apparatus consisted of a glass 3 in. dia. bell immersed in a vessel of water so that air was trapped to form a flat interface at the mouth of the bell. The sides of the vessel were plane glass to enable undistorted observation to be made. Water was obtained from the normal laboratory distilled water supply and experiments were carried out at room temperature (20-22°C). A 4 mm i.d. glass tube was arranged to enable an air bubble having an equivalent spherical dia. of 5 mm to be formed at its tip. A diagram of the apparatus is shown in Fig. 3. The separation between the bubble injection tube and the air-water interface, x, could be varied so
Fig. 3. Bubble-interface apparatus.
and M.
J.
L~CKETT
that the following types of bubble interface contact could be achieved: (a) Contact before the bubble was fully formed. (b) Contact just as the bubble became fully formed. (c) Contact just as the neck, connecting the air injection tube and the rising bubble, ruptured. (d) Contact at larger separations where the bubble at contact was rising freely at velocities ranging from 0.5 to 1.0 Ub, (where Ub, is the terminal rise velocity of the bubble). Bubble coalescence was recorded on 16mm Kodak 4-X negative film using a Hycam movie camera operated between 20 and 500 frameslsec. The films were analysed using a motion analyser and bubble size, separation x, contact area and coalescence times were measured. The coalescence time is the time from contact between the bubble and interface to the rupture of the contact film. A large number of observations were made[20] and the results are summarised in Table 2. Each result represents an average of approximately ten readings. The figures in brackets are the standard deviations of the coalescence times. The letters in brackets indicate the film speed at which the individual results were obtained: (a) 500 frameslsec, (b) 20 frameslsec, (c) 100 frameslsec, (d) 50 frameslsec. The estimated error in any reading is *I frame, i.e. at 500 frames/set the estimated error is *0+02 sec. Table 2. Equivalent spherical bubble diameter when fully formed = 5 mm. Separation x (cm)
Coalescence time T (set)
0.45 0.55 0.61 0.67 0.87 1.35 1.50 2.70 3.50
0~005-0~01(a) 0~005-0~01 (a) O-O.15 (b) 0.14(0.04) (b) O.lS(O.02) (c) O.H(O.02) (d) 0.16(0.03) (c) 0.18(0.03) (c) 0.18(0.02) (c)
Figure 4 shows the variation of coalescence time with separation. There are three distinct regions on the graph. Region I-where the bubble is still attached to the formation tube as it contacts the interface. The velocity of approach of the interfaces in this region was measured from the films and was found to be less than 1.0 cmlsec. The contact film radius just before film rupture was quite small (approximately
The influence of approach velocity on bubble coalescence 0.20
I
0.16-
Y "
I
I
T/+4+
I
I
I
II
I PO
I 2.5
I 3.0
I_ 3.5
2367
/'-
0.12-
I- o.oe-
0.04-
0
v 0.5
I I.0
I I.5 x.
cm
Fig. 4. Coalescence time vs separation.
0.05 cm). Figure 5 shows a typical coalescence event for Region I. Region II-is a transition region for separations between 0.55 and 0.67 cm. It was occasionally found that rapid coalescence (as found in Region I) occurred in this region when the bubble had clearly detached from the formation tube. This leads us to believe that the distinction between Regions I and III does not depend on whether or not the bubble was supported by the formation tube at contact. Region III-the neck joining the bubble to the formation tube is able to break at separations greater than about 0.6cm. Above this separation the bubbles are rising freely as they contact the interface. The rise velocity at contact can be calculated from the gradient of Fig. 6 in which the distance travelled by the bubble is plotted against time-both measured from the instant of neck rupture. It is evident from the graph that at neck rupture the bubbles have a high velocity, 10 cmlsec, and quickly accelerate to their terminal velocity. Detailed observation of the sequence of events which occurred in Region III revealed that no deceleration of the bubble was apparent prior to the bubble-interface contact. After contact the bubble could be seen to sink into the interface without coalescence and then to oscillate at the surface before finally coalescing. The time for the first oscillation fell within the range 0.04-0.06 set for separations x, between 0*87-3.50cm. The bubble almost completely disappeared during the first contact and then usually oscillated twice before coalescing. Figure 7 shows the initial stages of a Region III coalescence event. At coalescence the radius of the contact film was approximately equal to the bubble radius.
Fig. 5. Region I coalescence (100frameslsec).
To summarize the above results, it appears that coalescence is very rapid when the interface contact velocity is less than l.Ocm/sec, but when the contact velocity is greater than 10 cmlsec coalescence times are considerably increased. It should be noted that the above results are specific to the bubble size used in these experiments. We do not believe that impurities in the water had any significant effect as results obtained with tap water and doubly distilled water were virtually identical.
2368
R. D. KIRKPATRICK and
M. J. LOCKET
3.2
2.4$
2.0-
d k
1.6-
f i5
1.2-
0
0.04
0.06
0.12
Time,
set
0.16
0.20
Fig. 6. Distance travelled vs time for a rising bubble, both measured fromtheinstantthebubbleneckruptures.
A few experiments were carried out with an ionic solution of NaCl in water[20]. At low concentrations the same general trends were observed. As the concentration was increased to 0.6 M the distinction between Region I and III disappeared and coalescence times were large (0.7 set) and independent of separation and hence contact velocity. This confirms the widely quoted observation that coalescence is inhibited in concentrated electrolyte solutions. The effect of approach velocity is unimportant in such systems and the remainder of this paper is inapplicable to concentrated electrolyte solutions. In the following section we present a mathematical model which predicts an increase in coalescence time with approach velocity. In addition we use the model to indicate the maximum approach velocity for which Region I behaviour may occur. Fig. 7. Initial stages of Region III coalescence (100 frameslsec.). THEORETICAL DEVELOPMENT OF BUBBLE COALESCENCE WITH AN INTERFACE-HAVING A FINITE APPROACH VELOCITY
When a bubble approaches an interface with a finite approach velocity, the area of the contact film increases with time because the two surfaces press against each other. At the same time film drainage occurs, because of the overpressure in the film, which tends to reduce the film thickness. The rate of drainage of the film depends on its area and decreases as its area increases. Thus these two
effects compete against each other. The mechanism of bubble-interface coalescence may be subdivided into three types depending upon the approach velocity of the bubble. Region I. Low approach velocities. The rate of increase of the contact film area with time is sufficiently slow to allow the contact film to drain to the rupture thickness before the bubble is brought to rest. Region II. This is the region for moderate
2369
The influence of approach velocity on bubble coalescence approach velocities and is a transition region between Regions I and III. Region III. Large approach velocities. The rate of increase of the contact film area with time is sufficiently rapid to retard film thinning, to the extent that the contact film has insufficient time to drain to the rupture thickness before the bubble is brought to rest. The strain ‘energy stored in the deformed bubbles then tends to reverse the bubble’s motion and the contact film starts to thicken[23]. Coalescence at large approach velocities is no longer a simple process but depends on bubble oscillation after contact and the complex shape of the drainage film formed between the two interfaces. PRESSURE
PROFILE
IN THE FILM
Consider the bubble-interface geometry at some time t after contact as shown in Fig. 8. For small bubbles, the static liquid pressure PL can be taken as everywhere constant and equal to the pressure above the interface P,,. The internal pressure within the bubble is assumed constant at PL + 2 y lb. The shape of the deformed bubble is difficult to predict for a moving bubble, and so this will be estimated from the quasi-static situation where the pressures at point G in the film are assumed to balance. At G,
the film surfaces and hence there are no frictional losses. Benoulli’s equation between G and H is then po/pr. + uo2/2 = PHlPL + UH72.
At the stagnation point G, UG= 0 and from Eqs. (1) and (2) it follows that PO = PL + y/b. At the edge of the film H, the pressure is PL and the velocity is u@. From Eq. (3), ylb = p~ue2/2.
4
+2r_z_r
L b
t,liquidinfilm=8Pb2(1-Cos0)h
At t = t + at, liquid in film
p’
Since the film thickness is negligible compared with p, it follows that at the nose of the bubble p +q +2b.
(4)
To calculate the pressure at intermediate points within the film, it is necessary to make simplifying assumptions about the local radius of curvature and the film thickness if complex calculations are to be avoided. We therefore take a model geometry of a constant radius of curvature 2b at all points in the film and a constant film thickness h. Equation (4) still applies to this proposed geometry but a pressure balance over the whole bubble surface will no longer be satisfied because of the two simplifying assumptions made above. The proposed model is shown in Fig. 9. The surface area of the film is $0”2~ 2b Sin +.2bd+ = &rb ‘( 1 - Cos 6). Consider a mass balance on the film over a time increment 6t. Att=
p am +zr,p
=g?rb’[l-Cos(B+$t)](h+$St).
(2)
Restricting subsequent arguments to pure liquids, it follows that in the flowing film there is no shear at
Fig.8. A modelfor bubble-interfacecoalescence.
CES Vol. 29, No. 12-H
(3)
Fig.9. Proposedmodel
for bubble-interface
coalescence.
R. D. KIRKPATRICK and M. J. L~CKETI
2370
the bubble F
The liquid flowing out of the film in St is 237 2b Sin 8h(ue - (de/dt)) Neglecting & -0,
St.
second order terms and simplifying,
as hence
dh ueh Sin 8 dt = -2b(l -Cos 19)’
Sin’ e + (I - COS e)’
Applying Bernoulli’s equation to the film between a general angle 4 and 8,
x[2ln2-2ln(l+Cose)-2(1-c0s8)+~Sin2e] P+ IPL + ufl2 = ue2l2+ PL IPL.
(6) The bubble deceleration
Equation (5) may also be written for any general angle 4, so that combining with Eqs. (6) and (4) we have PO - PL = y/b. (I-
[;;;
~~l~“c”o”s~;]2).
(7)
Equation (7) is an equation for the pressure profile in the film. DRAINAGE
Eliminating
d2s 96~ dt2= -llb2p,
Sir? e 2+
is therefore Sin2 e (1 - COS e)r
x[2ln2-2ln(l+CostI)-2(1-Cose)+fSin*e]
1 (10)
where e = Sin-’ ((b’-$“i).
EQUATION
1.
(11)
ue from Eqs. (4) and (5) we have CALCULATION
; = 2b’pL
(8)
The change in film thickness, from an initial thickness hi to a final thickness hf, which occurs in a film of constant area during a drainage time At is obtained by integrating Eq. (8) to give
“’
At
. BUBBLE
(9)
PROCEDURE
The calculation procedure adopted was to assume that at t = 0, s = b, (d’sldt’) = 0, U = Uo. Taking a series of equal time increments of 0.0001 set, the new value of s at the end of each increment could be calculated from s,+~*= st - LJ,At. Using Eqs. (10) and (11) the deceleration of the bubble could be calculated and hence the velocity at the end of the increment. Equation (9) was used to determine the change in the film thickness over the time increment. The calculations were repeated until the bubble velocity fell to zero.
DECELERATION
As the bubble contacts the interface, it slows down because of the excess pressure on its upper surface. Deformation occurs but this is hard to allow for. We can make an estimate of the deceleration of the bubble by assuming it to behave as a sphere with an effective mass near to an interface of % 3 rb’pL (21). The restraining force is taken as the excess pressure force in the contact film. The buoyancy force is taken to be exactly equal to the drag force. As the bubble slows down this assumption probably leads to an overestimation of the deceleration and to a conservative estimate of the inhibition of coalescence due to approach velocity. Integrating the vertical component of Eq. (7) over the contact film gives the restraining force on
THEORY FOR BUBBLE-BUBBLE COALESCENCE WITH FINITE APPROACH VELOCITY
The bubbles are assumed to be of equal size and their centres to have equal velocities U/2 towards each other along their line of centres. For small bubbles Pr can be assumed constant and the pressure in the film at the bubble nose is PL + 2y/b. Since the pressure at the edge of the film is P,_, application of Bernoulli’s equation along the film yields p~u~=l2= 2ylb
(12)
As for the bubble-interface model, it is now necessary to make assumptions about the local
2371
The influence of approach velocity on bubble coalescence
radius of curvature along the film and the film thickness. We assume a flat film of uniform thickness h as shown in Fig. 10. As before these simplifying assumptions make it impossible to satisfy a complete pressure balance over the distorted bubble surface.
equation In (2)
= 4(--&)“2$.
The bubble restraining force F is obtained integrating Eq. (15) over the contact film F=
I
R
by
2?rr(p~-pr)=~=%(b2-s’).
0
Taking $3 rb3pL as the effective bubble, the deceleration is d’s
12
Z’_fiYbrpL.
mass
(b2-s’)
of the
(17)
The calculation procedure for bubble-bubble contact is the same as that for bubble-interface contact. RESULTS
Bubble-interface
Fig. 10.Proposed model for bubble-bubblecoalescence.
A material balance on the film within a radius r gives dh -=dt
-2urh r’
(13)
Bernoulli’s equation between a radius r in the film and the edge is P, IpL + u,2/2 = lb72 + PL IPL.
(14)
From Eqs. (12), (13) and (14),
(15) A derivation leading to an equation similar to Eq. (15) has also been given by Farooq[22]. Writing Eq. (13) for the complete film and combining with Eq. (12) gives
This equation may be integrated to give a drainage
OF THE CALCULATIONS AND DISCUSSION
coalescence
The calculated thinning of the contact film with time as a 5 mm diameter bubble makes contact with an interface is shown in Fig. 11. Calculations for various initial approach velocities are shown. The rupture thickness has been taken as 2.5 x 10”cm which is a typical value assumed by previous workers. The termination point for each line is when the bubble velocity is zero. The time for the bubble to decelerate from 22 cm/set to zero velocity is seen from the theoretical curve (Fig. 11) to be 0.037 sec. This value is in good agreement with half the experimentally observed time for the first oscillation (0*020-0~030 set). The calculations indicate that at approach velocities greater than 5 cmlsec the bubble will come to rest before film rupture occurs. It seems probable that on coming to rest the stored strain energy in the film will cause the bubble to reverse its motion and to move down away from the interface. Buoyancy forces acting on the bubble will subsequently cause it to slow down and eventually to reverse its motion so that it again approaches the interface. This sequence of events is repeated with the approach velocity being reduced after each bounce until the approach velocity is small. The calculations indicate that bubbles having an approach velocity of less than about 4cm/sec coalesce with the interface before they are brought to rest. The relatively simple theory proposed above is not capable of making predictions about the mechanism of bubble bounce and it only indicates whether coalescence would be expected on the first approach.
R. D. KIRKPATRICK and
5mm bubble-interface
\
3 cm /set
I
I
0.012
0.024
Time,
contact
1 ox
set
Fig. 1I. Filmthicknessvs time. The experiments we have reported for bubbleinterface coalescence indicated a transition velocity lying between 1 and 10 cmlsec for 5 mm dia bubbles and this is in agreement with these predictions. Figure 12 shows the predicted changes in film thickness with time for different size bubbles. The
M. J. LOCKETT
bubbles are all initially assumed to be moving at their terminal rise velocity as was found for 5 mm dia bubbles. Bubble bounce is predicted for 5 mm dia and 1 mm dia bubbles whereas 0.5 and 0.25 mm bubbles are not expected to bounce. Bubbles of 0.75 mm dia are just on the border line. It is of interest to compare these predictions with the recently reported data of Farooq [22] in which small bubbles were allowed to coalesce with an interface under very carefully controlled conditions. Farooq found that 1.0 and 0.75 mm dia bubbles bounced and 0.5 and 0.25 mm dia bubbles did not. The predictions of the model are thus in agreement with the available experimental data which is remarkable in view of the quite severe assumptions which are inherent in the model. Farooq [22] has suggested that bubble bounce might be attributable to film immobilisation due to surface tension gradients, which can arise if the film temperature falls under adiabatic stretching. Such a mechanism might operate in addition to the one we propose and their relative importance deserves further study. Bubble-bubble
coalescence
Predictions for bubble-bubble contact of 5 mm dia bubbles having various initial relative approach velocities are shown in Fig. 13. The calculations show that if the approach velocity of the bubbles is less than about 12cm/sec, the bubbles would be expected to coalesce on first contact. At higher approach velocities the bubbles are expected to
5mm Bubble-bubble
contact
Rupture
0.25 ICP 0
-
mm I 0.6
I I.6
Time,
msec
Fig. 12. Film thickness vs time.
I 2.4
lo-7l 0
0.004
0.006
Time,
0 012
set
Fig. 13. Film thickness vs time.
0.016
The influence of approach velocity on bubble coalescence
bounce. As noted earlier, no predictions can be made with this model about the total coalescence time if, after bounce occurs, the bubbles come together again. A very approximate estimate for bubble-bubble coalescence under these conditions can be made from the bubble interface experiments reported earlier and this is O-15 sec. Table 1 shows contact velocities which we have observed in a bubble cloud and they are large, which indicates that bubble bounce or Region III type coalescence would be expected. The rest times which we observed (about 0.05 set) are smaller than the coalescence time (0.15 set) which is required for Region III coalescence. Thus bubble coalescence is not expected to be important in a bubble swarm, even for pure liquids. An exception to this may be where the bubbles are held together for comparatively long periods in clusters. We intend to deal with this in a subsequent paper. It must be emphasized that in order to make quantitative predictions about the effect of approach velocity on bubble coalescence, we hme had to make a number of quite sweeping assumptions in the model. Considerable refinement is possible but more detailed models are unlikely to change the broad conclusion of this paper that bubble approach velocity is important and can prevent bubble coalescence. NOTATION
A
cross sectional area of vessel, cm* b bubble radius, cm F force, dynes G volumetric gas flow rate, cm’lsec h film thickness, cm P pressure, dynes/cm2 P radius of curvature, cm radius of curvature, cm z film radius, cm r radial distance, cm s distance of bubble centre from interface or from edge of film defined in Figs. 9 and 10, cm T coalescence time, set t time, set u relative approach velocity, cmlsec Ub, bubble terminal rise velocity, cmlsec U liquid velocity in film, cmlsec separation-Fig. 3, cm X surface tension, dynes/cm Y E gas holdup angle defining edge of film-Fig. 9, rads e liquid density, g/cm3 PL 4 angle defined in Fig. 9, rads
2373
Subscripts
atm G H ; L R r 0
f
value above the plane interface value at stagnation point value of edge of film initial value final value value in the liquid value at r = R value at r value at t = 0 value at edge of film value at angle 4 REFERENCES
[1] Marrucci G. and Nicodemo L., Chem. Engng Sci. 1967 22 1257. [2] Zieminski S. A. and Whittemore R. C., Chem. Engng Sci. 1971 26 509. [3] Bowonder B. and Kumar R., Chem. Engng Sci. 1970 25 25. [4] Calderbank P. H., Moo-Young M. B. and Bibby R.. Proc. Third Symp. on Chem&zl Reactor E&g.; p. 91, Pergamon Press, London 1964. PI Towel1 G. D., Strand G. P. and Ackerman G. H., Proc. A.LCh.E.-I. Chem. Engng Joint Meeting, ~97, London 1965. I61 Lee J. C. and Ssali G. W. K., Joint Meeting on Bubbles and Foams, Sl-6.1, Verfahrenstechnische Gesellschaft im VDI and the Inst. of Chem. Engr, Nurnberg 1971. [71 Reith T. and Beek W. .I.. Trans. Inst. Chem. Enar. 1970 48 T63. PI Lee J. C. and Meyrick D. L., Trans. Inst. Chem. Engr 1970 48 T37. [91 Crabtree .I. R. and Bridgwater J., Chem. Engng Sci. 1971 26 839. WJI De Nevers N. and Wu J., A.LCh.E. JI. 1971 17 182. 1111 Foulk C. W. and Miller J. N., Ind. Engng Chem. 1931 23 1283. WI Marrucci G., Nicodemo L. and Acierno D., Inf. Symp. on Research in Co-current Gas-Liquid Flow, p. 95, Plenum Press, New York 1969. u31 Lessard R. R. and Zieminski S. A., Ind. Engng. Chem. Fund. 1971 10 260. I _ Sci. 1968 r141 Lee J. C. and Hodeson T. D.. Chem. Enana 23 1375. WI Marrucci G., Chem. Engng Sci. 1969 24 975. tW Wallis G. B., Intern. Heat Transfer, p. 319, Am. Sot. Mech. Engr. 1961. I171 Delmar R. R. and Zieminski S. A., A.1.Ch.E. JI. 1971 17 57. 1181 Davidson 3. F. and Kirk F. A., Chem. Engng Sci. 1969 24 1529. 1191 Anderson J. L. and Quinn J. A., Chem. Engng Sci. 1970 2s 373. ml Kirkpatrick R. D., M. SC. Report U.M.I.S.T. 1973. PII Milne-Thomson L. M., Theoretical Hydrodynamics, p. 528 Macmillan 1968. WI Farooq S. Y. Ph.D. Thesis, University College of Swansea, (University of Wales), 1972. r231 Jeffreys G. V. and Hawksley J. L., A.I.Ch.E. JI 1965 11 413.