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The mixing of liquids by a plume of low-Reynolds number bubbles
Chemical Engineering Science 55 (2000) 2585}2594
The mixing of liquids by a plume of low-Reynolds number bubbles M. H. Chen, S. S. S. Cardoso* Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, CB2 3RA, UK Received 23 March 1999; accepted 18 October 1999
Abstract When bubbles are continuously released from a localised source at the bottom of a liquid layer, a plume is produced. As the bubble plume rises due to its buoyancy, it entrains surrounding liquid, which is carried upward with the stream of bubbles. In the present work, we investigate the motion of a plume of low-Reynolds number bubbles in a strati"ed liquid consisting of two homogeneous layers of di!erent densities. The liquid environment is of "nite lateral extent. We develop a theoretical model for the #ow of the bubble plume and the surrounding liquid. The full equations are solved numerically. The mixing at the interface is quanti"ed and the time-evolution of the density pro"les in both layers is calculated. The model also predicts the rate of rise of the density interface. We develop an analytical solution for the problem in the limit of strong strati"cations. Our theoretical predictions are compared with new experimental results using plumes of small bubbles generated by electrolysis of an aqueous solution of sodium chloride and with previous experimental results (McDougall (1978), Journal of Fluid Mechanics, 85, 655}672; Baines & Leitch (1992), Journal of Hydraulic Engineering, 118(4), 559}577). 2000 Elsevier Science Ltd. All rights reserved. Keywords: Bubble-plume; Strati"cation; Mixing; Two-phase #ow
1. Introduction Buoyant plumes produced by a source of bubbles in a liquid medium are used for a variety of purposes. Bubble breakwaters operate because of the surface jet produced by the bubble plume (Taylor, 1955). Oil slicks on water surfaces can be contained by bubble plumes (Jones, 1972). Bubble plumes have also been used to prevent parts of the surface of a river or lake from freezing (Baines, 1961). During an underwater oil-well blow-out, a plume of bubbles, oil droplets and sea water develops; the extent of the damage to marine life depends on whether all the oil rises to the surface or spreads out horizontally at some intermediate depth (McDougall, 1978). In industry, bubble plumes are used with two objectives: enhancing mass transfer and mixing. As a mixing technique, the use of bubbles is attractive because it is very simple and cheap to operate. The most common applications of bubble-driven mixing are encountered in wastewater treatment, in the delicate mixing of pharma-
* Corresponding author. Tel.: #1223-331863; fax: #1223-334796. E-mail address: silvana}[email protected] (S. S. S. Cardoso).
ceutical and food products, in the mixing of very hot or toxic liquids (Leitch & Baines, 1989) and in the destrati"cation of lakes and reservoirs (Schladow, 1992; Wuest, Brooks & Imboden, 1992). Over the last 20 years, there has been considerable research on the structure of bubble plumes and their interaction with the surrounding liquid environment. Laboratory and "eld observations have led to the development of two types of models of bubble plumes (McDougall, 1978; Leitch & Baines, 1989). The simpler approach considers the plume to behave as a one-phase buoyant plume. It assumes that the bubbles producing the buoyancy for motion are evenly dispersed within the liquid in the plume and behave passively (Fig. 1a). The more elaborate modelling regards the bubble plume as being composed of two parts: an inner circular plume containing all the bubbles and an outer annular plume containing only liquid (Fig. 1b). The buoyancy is concentrated within the central region and momentum is imparted to the surrounding liquid via turbulent shear stresses. However, given a speci"c problem, it is not clear from the previous literature (Wilkinson, 1979; McDougall, 1978; Lemkert & Imberger, 1993) whether one can use the simple model or needs to recur to the more complex double-core model.
0009-2509/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 5 3 1 - X
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Nomenclature a A B
e!ective radius of plume cross-sectional area of the tank buoyancy #ow rate in the plume at source, at z"0 bubble diameter EoK tvoK s number of bubble e$ciency of the conversion of the kinetic energy of the plume at the interface into potential energy of the lower layer gravitational acceleration height of the lower layer initial height of the lower layer initial height of the upper layer momentum #ow rate in the plume at the source, at z"0 volumetric #ow rate in the plume at the source, at z"0 equivalent radius of the tank critical Rayleigh number, taken as approximately 10 Reynolds number of the bubble Richardson number initial Richardson number of the two-layer strati"cation arbitrary time uniform downward velocity of the environmental liquid in the lower layer
d @ Eo @ f
g hl hl h M Q R Ra A Re @ Ri Ri t ;
Finite size bubbles exhibit a slip velocity relative to the liquid rising in the plume and this leads to the formation of a bubble-rich core surrounded by bubble-depleted rising liquid. The e!ect of this slip velocity will be greater for weak plumes i.e. for small volumetric #ow rates of gas. A quantitative criterion determining the structure of a bubble plume may be obtained by comparing a scale for the velocity in the plume = with the slip velocity of individual bubbles u . Dimensional arguments suggest @
= & u @
ol !o Q g ol hl , u @
(1)
where ol is the density of the liquid, o is the density of the gas bubbles, Q is the volumetric #ow rate of gas at the source, g is the acceleration of gravity and hl is the typical vertical length scale of the plume. In small-scale experiments performed in the laboratory, the #ow rates are smaller than those found in industrial and natural bubble plumes, whilst the bubble sizes remain approximately the same, and hence =/u is typically smaller. This @ explains why most laboratory observations hitherto have revealed a double-structure plume. As a consequence,
u @ = w z
slip velocity of a bubble scale for the velocity in the plume vertical velocity at the plume axis height above point source
Greek letters a D Dl Dl D D k o o @J ol ol o o
entrainment coe$cient, taken to be constant and equal to 0.09 buoyancy of the plume evaluated at the plume axis buoyancy in the lower layer initial buoyancy in the lower layer buoyancy in the upper layer initial buoyancy in the upper layer viscosity of the liquid liquid density as a function of depth z density of the bubble layer just above interface density of the lower layer initial density of the lower layer, at t"0 density of the upper layer initial density of the upper layer, at t"0
Subscripts 0 b l up
initial state bubble lower layer upper layer
most studies have concentrated on these double-structure plumes. However, for at least some of the largerscale plumes found in industrial processes, the bubbles are distributed widely across the plume. It is therefore of interest to study plumes of very small bubbles in the laboratory. Large bubbles possess wakes where liquid from deeper levels in the environment may be carried and then detrained at shallower levels. This is a relatively e!ective transport mechanism. However, the work of Milgram (1983) suggests that mixing induced by small bubbles should be more e$cient than that by large bubbles. Indeed, a plume of small bubbles entrains a larger quantity of surrounding liquid and this e!ect dominates over the wake transport mechanism exhibited by large bubbles. Nevertheless, very little is known about the dynamics of plumes of low-Reynolds number bubbles, whose ascent speed is very small compared to the convective velocities. It is therefore relevant to investigate the mechanism by which such small bubbles drive mixing in a liquid and to assess its industrial potential. Some insight into the behaviour of plumes of small bubbles may be gained from previous studies of one-phase buoyant plumes (Morton, Taylor & Turner, 1956; Cardoso &
M. H. Chen, S. S. S. Cardoso / Chemical Engineering Science 55 (2000) 2585}2594
Woods, 1993), but there are some fundamental di!erences, including bubble separation in strati"ed liquids and the formation of sequences of bubble plumes in the liquid layer. In the present work, we present new experimental results using bubbles of approximately 150 lm diameter generated by electrolysis; these bubbles are at least one order of magnitude smaller than bubbles used by previous researchers (see Table 1). A bubble plume rising in a density-strati"ed environment will transport dense liquid upward from the deeper regions for some distance; a fraction of this liquid then leaves the plume and spreads out horizontally at its own
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density level. In an environment of "nite lateral extent, the plume induces downward advection of the surrounding liquid, and hence the density distribution in the environment evolves with time. The continuing plume, rising through the changing strati"cation, will in turn transport liquid to varying depths. This interaction between the plume and the environment has been modelled numerically by Patterson and Imberger (1989), Leitch and Baines (1989) and Schladow (1992). The e!ectiveness of bubble-driven mixing in such complex systems has been measured by the e$ciency of conversion of the input bubble energy into potential energy of the strati"cation. The bubble energy at the source is measured by the work required to produce the air #ow via a compressor. Although peak e$ciencies of up to 15% have been achieved for certain design and operating con"gurations (Asaeda & Imberger, 1993; Schladow, 1992), the e$ciency generally decreases with time as the strati"cation weakens, and the average e$ciency of the complete mixing process can be as low as 3}5%. However, comparison with e$ciencies of mixing by one-phase plumes suggests that there is scope for improvement of bubble plume e$ciencies up to approximately 15% (Zilitinkevitch, 1991; Cardoso & Woods, 1993). In this paper, we discuss how the e$ciency of mixing of a plume is a!ected by the bubble size. In the following sections, we develop a novel approach to the modelling of the dynamics and mixing induced by a bubble plume, appropriate in the limit of small bubbles. We focus on the mixing driven by a continuous release of bubbles in a strati"cation consisting of two layers of liquids of di!erent densities. The liquid environment is of "nite lateral extent. We calculate the evolution of the density pro"le in both layers. Our theoretical predictions are validated by comparison with both new and previous experimental observations.
2. Experimental procedure and qualitative observations 2.1. Experimental procedure Fig. 1. Structure of a bubble plume rising in a homogeneous #uid: (a) simple plume when =/u 1 and (b) double-core plume when =/u 1. @ @
The experiments were carried out in two Perspex tanks, tank A of square cross section 0.2 m;0.2 m and
Table 1 Comparison of the size and structure of the bubbles used in the present work and in previous studies Experiment
d (m) @
u (m/s) @
Re @
McDougall (1978) Leitch and Baines (1989) and Baines and Leitch (1992) Present work
1.25;10\ 3.20;10\
0.258 0.203
252 630
4.20;10\ 1.50;10\
0.224 0.0115
1100 1.61
Eo @
Bubble shape
Volume of wake (m) =/u @
0.222 1.40
Spherical Ellipsoidal
1.90;10\ +9.80;10\
0.6 0.4
4.40 3.08;10\
Ellipsoidal Spherical
'9.80;10\ 0.0
0.6 4.1
The bubble shape and the volume of the wake were obtained from Clift, Grace & Weber (1978).
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M. H. Chen, S. S. S. Cardoso / Chemical Engineering Science 55 (2000) 2585}2594
height 0.4 m and a larger tank, tank B, of 0.4 m;0.4 m and height 0.6 m. A two-layer strati"cation was set up with a lower layer of NaCl aqueous solution and a top layer of fresh water; the two layers were separated by a sharp interface with approximately 0.5 cm thickness. The top layer was coloured with food dye to enhance the visibility of the interface. A photograph of the experimental system is shown in Fig. 2.
Bubbles were generated at the bottom of the tank via electrolysis of the NaCl solution. The electrodes consisted of a mesh of "ne platinum wires (Fig. 3), with area 1.5 cm;1.5 cm, connected to a d.c. power supply. Bubbles of hydrogen were released at the cathode and of chlorine at the anode (Creighton & Koehler, 1944). The electrodes were vibrated magnetically to avoid coalescence of bubbles and hence ensure the detachment of
Fig. 2. Photograph of experimental set-up illustrating the structure of a plume of small bubbles in a two-layer strati"cation. White arrows indicate the #ow of liquid.
Fig. 3. Electrodes of platinum used to generate bubbles of hydrogen at the cathode and chlorine at the anode.
M. H. Chen, S. S. S. Cardoso / Chemical Engineering Science 55 (2000) 2585}2594
uniformly sized bubbles. The size of the bubbles was determined in both layers using a laser analyser. The bubble size distribution was found to be independent of height in the tank and approximately Gaussian. An average size of 150$50 lm was measured (Chen, 1998). Such a constant bubble size with height, indicates a low coalescence tendency, which is characteristic of bubbles in concentrated salt solutions (Soo, 1967). Although small bubbles do tend to coalesce in fresh water, our laboratory observations and bubble size measurements suggest that coalescence was also negligible in the upper layer. We believe that the thin "lm of NaCl solution surrounding the bubbles moving into the upper layer hindered coalescence. Calibration of the #ow rate of gas generated versus the intensity of the current was performed directly by sealing the top of the tank and measuring the gas #ow rate with a bubble soap meter. Although chlorine is relatively soluble in water, our #ow rate measurements indicate a constant #ow rate for a given current intensity. This may be a consequence of either the mass of dissolved chlorine being negligible compared to the total gas generated or of saturation being achieved very fast. Nevertheless, since we have measured directly the #ow rate of gas generated, the dissolution of chlorine does not a!ect our experimental results. The position of the interface and the motion in the lower layer were monitored visually. The density distribution in the tank was determined by withdrawing small samples of the liquid at di!erent levels with a syringe. The density was measured by refractometry. The experimental conditions for each run are summarised in Table 2. 2.2. Qualitative observations The typical #ow pattern induced by a bubble plume rising in a strati"ed liquid consisting of two homogeneous layers of di!erent densities is shown in Fig. 4. The lower and upper-layer #uids have both been dyed for visualisation. As the bubble plume rises across the density interface, its buoyancy decreases. As a result, a fraction of the dense, lower layer liquid carried in the plume is detrained above the interface, where the momentum of
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the liquid reaches zero; the detrained dense liquid mixes with upper layer #uid and then spreads radially, forming a gravity current of intermediate density. The bubbles continue to rise through the upper layer, forming a new bubble plume. The surrounding less dense liquid in the upper layer is entrained and mixes with the residual dense liquid in the plume. The mixture of the two liquids is carried to the free surface where the bubbles burst and a liquid current spreads radially. As the tank is of "nite lateral extent, the continual entrainment of surrounding ambient #uid into the plume leads to a downward #ow in the environment. Hence, a front of liquid, which has been in the plume in each layer, descends with time; this can be seen in the sequence of photographs in Fig. 4 as the liquid circulated through the plume becomes discoloured. The liquid above the front in the lower layer is less dense than the original liquid below due to the entrainment of less dense upper layer liquid at the interface. Therefore, the plume will arrive at the interface even lighter. As time evolves, a stable density distribution develops, gradually "lling up the original lower layer. In the upper layer, the liquid detrained at the free surface is denser than the original upper layer liquid. It may therefore drive buoyant convection and mixing in the upper layer. Our measurements of the evolution of the density pro"le in the tank for a typical experiment are shown in Fig. 5. These results suggest that very little lower layer liquid is carried into the upper layer by the rising bubbles. This observation may be explained partially by the fact that small bubbles of diameter 150 lm do not have wakes, as shown in Table 1, and hence are unable to trap and carry lower layer liquid upward. In addition, the buoyancy force associated with the bubbles moving across the interface is relatively large compared to the viscous force that can drag lower layer liquid into the upper layer. Therefore, the contribution of viscous entrainment of lower layer liquid to the #ow rate and density of the plume in the upper layer is small. We conclude then that the dominant mixing mechanism at the interface is associated with the overshoot and collapse of the dense liquid transported in the plume. In Section 3, we present a theoretical model that builds upon these experimental observations.
Table 2 The experimental conditions. Labels A and B refer to tank used (See Section 2.1) Experiment
Q (m/s)
D (m/s)
hl (m)
h (m) S
B (m/s)
Ri
1 2 3 4 5 6 7
7.1;10\ 5.7;10\ 5.7;10\ 7.1;10\ 4.2;10\ 7.1;10\ 7.1;10\
!0.151 !0.230 !0.267 !0.366 !0.256 !0.423 !0.151
0.131 0.130 0.132 0.138 0.143 0.137 0.269
0.152 0.157 0.156 0.152 0.150 0.149 0.289
6.95;10\ 5.55;10\ 5.55;10\ 6.95;10\ 4.14;10\ 6.95;10\ 6.95;10\
0.9 1.6 2.0 2.5 2.6 2.8 3.1
A A A A A A B
Fig. 4. Sequence of photographs showing the evolution of the mixing induced by a bubble plume in a two-layer density strati"cation.
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M. H. Chen, S. S. S. Cardoso / Chemical Engineering Science 55 (2000) 2585}2594
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Here, z denotes the vertical co-ordinate, increasing upward, with its origin at the source; a is the e!ective radius of the plume; w is the vertical velocity at the plume axis; a is the entrainment coe$cient; and D and Dl are, respectively, the buoyancy of the plume evaluated on the plume axis and the buoyancy of the liquid surrounding the plume in the lower layer, de"ned by
Fig. 5. Evolution of the density pro"le with time for Ri "2.5. Data points have been interpolated to aid visualisation.
3. Theory 3.1. Full model In the light of our experimental observations, we shall assume that the upper layer remains uniform, with density equal to that at the beginning of the experiment. The mixing in the lower layer is modelled in detail by looking at both the motion in the plume and in the surrounding environment. Consider a plume rising in the lower layer of a step strati"cation. The cross-sectional area of the region is A"pR, where R is an equivalent radius. We assume that the depth of the lower layer is initially hl . Owing to the continuous variation of the ambient properties described above, the behaviour of the plume is time-dependent. However, if the aspect ratio R/hl is large, for all times, then the time rates of change of the plume properties can be shown to be su$ciently small to be neglected in the relevant conservation equations (Manins, 1979). The plume is then of small radius compared to R, and the ambient behaves quasi-steadily as far as the plume is concerned. Using the Boussinesq approximation and assuming a constant coe$cient of entrainment, conservation of volume, momentum and buoyancy for the plume are expressed by the following equations, respectively: d (aw)"2aaw, dz
(2)
d 1 aw "aD, dz 2
(3)
d 1 *Dl awD "aw . dz 2 *z
(4)
g(ol !o) D" , ol
(5)
g(ol !ol ) Dl " , ol
(6)
where, ol is the density of the lower layer and g is the acceleration due to gravity. We have taken the reference density to be equal to the initial density of the lower layer, ol . Gaussian pro"les of equal width have been assumed for the vertical velocity and buoyancy distributions in the plume. Eqs. (2)}(4) were originally derived by Morton et al. (1956) and have been discussed in detail elsewhere (e.g. Turner, 1979). The entrainment into the plume causes the environmental liquid outside the plume to move downward. We shall assume that the density di!erence between the liquid mixture spreading at the interface and the lower layer liquid just beneath it is su$ciently large so that this downward motion is approximately described by a uniform velocity ;. Continuity may then be written as (7)
!nR;"naw,
where it has been assumed that R a. The buoyancy "eld in the lower layer is governed by *Dl *Dl "!; . *t *z
(8)
Molecular di!usion and mixing have been neglected here. The initial and boundary conditions for the governing equations are as follows. The initial condition is that the strati"cation consists of two uniform layers. Hence, Dl "0,
g(o !ol ) D " ol
at t"0,
(9)
where o is the initial density of the upper layer. The boundary conditions specify the #ow rates of volume, momentum and buoyancy at the source naw"Q , (10a) naw"M at z"0, (10b) nawD"B . (10c) We require one more continuity condition, quantifying the rate of entrainment of the upper layer liquid at the interface. We shall assume that a constant fraction f of
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the kinetic energy of the plume at the interface is converted into potential energy of the lower layer. This energetic conversion is expressed by (Cardoso & Woods, 1993) dhl f B (Dl !D ) " . dt A
(11)
This equation is equivalent to an entrainment law of the type (1/w) dhl /dtJRi\, where Ri is the Richardson number of the strati"cation, de"ned by (Dl !D )hl . Ri" w
(12)
Cardoso and Woods (1993) have shown that such a mixing law is valid for 0.2(Ri(3. The e$ciency of this energetic conversion, f, has been found to lie in the range 0.2}0.3 in penetrative convection studies (Zilitinkevich, 1991). Eqs. (2)}(4) and (7)}(9), (10a)}(10c) and (11) were solved numerically using f"0.3; the theoretical results are compared with the experimental measurements in the next section. 3.2. A simplixed model for strong stratixcations For strong strati"cations, the rate of rise of the interface is very much smaller than the rate at which the environmental #uid is circulated through the plume. We may therefore assume that after the descending front reaches the bottom of the tank, the environment surrounding the plume is approximately homogeneous. In this limit, the full model equations may be simpli"ed to an equation for the conservation of buoyancy in the lower layer, d [hl (Dl !D )]"0 dt
4. Results and discussion Fig. 5 illustrates a typical evolution of the density pro"le with time. Several stages may be identi"ed: in the "rst stage (t"0}10 mins), the lower layer becomes strati"ed, evolving into a region where the density increases with height. This is followed by the main stage (t"10}110 mins) where the density di!erence between the upper and the lower layer is reduced and a homogeneous density is eventually recorded at t"120 min. The homogenisation of the solution occurs soon after the interface disappears, normally in the space of 1}2 min as recorded in nearly all the experiments. It can be seen from the graph that the density in the upper layer has undergone only minor changes until the last stage. It can therefore be concluded that the homogenisation of the solution is mainly due to the increase of the depth of the lower layer and that only a very small amount of liquid from the lower layer is carried across the interface by the bubbles. No strati"cation prevails in the lower layer soon after the "rst stage, showing that the plume drives good circulation in the "nite size tank. Our theoretical predictions for the evolution of the density pro"le have been plotted as solid lines in Fig. 6. The agreement between the model predictions and the experimental results is good for the lower layer. For large times, a small inversion in the density is observed at the top of the upper layer. The model does not capture this small e!ect as expected. Fig. 7 shows the evolution of the density pro"le in an experiment by Baines and Leitch (1992). The pro"les predicted by our new theoretical model, valid for small Reynolds number bubbles, are also shown. The di!erence between the model and experimental results suggests that a plume of small bubbles is more e$cient at destratifying the lower regions of the tank, whilst a plume of large bubbles is more e$cient at transporting lower layer
(13)
with boundary condition hl
fB d (Dl !D )"! . dt A
(14)
The analytical solution of Eqs. (13) and (14) gives the rate of rise of the interface
hl "hl exp
fB t Ahl (Dl !D )
(15)
and the evolution of the density in the lower layer
!f B t (Dl !D )"(Dl !D ) exp Ahl (Dl !D
)
.
(16)
Eqs. (15) and (16) are valid for large Richardson number strati"cations, that is for Ri 1.
Fig. 6. Evolution of the density pro"le: theoretical predictions (solid lines) and experimental results for an initial Richardson number, Ri "2.5.
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Fig. 7. Evolution of the density pro"le: comparison between our theoretical predictions for a plume of small bubbles (solid lines) and the experimental data of Baines and Leitch (1992) for a plume of large bubbles; Ri "0.8.
liquid into the upper layer and thereby increasing the density at the higher levels in the tank. These observations may be explained by comparing the rate of descent of the density front in the lower layer for mixing driven by a plume of small bubbles with that observed for large bubbles. Such a comparison is made in Fig. 8. Our model predictions are in excellent agreement with our experimental results for plumes of small bubbles. However, for plumes of larger bubbles (Baines & Leitch, 1992), our model overpredicts the rate of descent of the front. This di!erence is related to the "nite slip velocity of the larger bubbles. We may conclude that a plume of small bubbles entrains more surrounding liquid, leading to a large re-circulation #owrate of liquid through the plume. A plume of large bubbles exhibits a smaller entrainment capacity, but is able to transport lower layer liquid in the individual bubble-wakes into the upper layer. The results of the numerical integration for the position of the interface as a function of time are shown graphically in Fig. 9. Experimental data have also been plotted. It is seen that the agreement is good for a range of strati"cations and plume buoyancy #uxes. In Fig. 10, we compare the predictions of the full model with those of the simpli"ed model valid for strong strati"cations. It is shown that the simple analytical model given by Eqs. (15) and (16) estimates the position of the interface very well for strong strati"cations, i.e. Ri *1. As mentioned in Section 3, we have assumed in our theoretical predictions that the parameter f is constant and equal to 0.3. The good agreement between our experimental results and the theory suggests then that 30% of the kinetic energy of the plume at the interface is converted into potential energy of the lower layer, as less dense #uid is transported down into the layer of dense #uid against the restoring buoyancy force. Such
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Fig. 8. Non-dimensional position of the descending front as a function of time. The solid lines are theoretical predictions for an entrainment constant a"0.09.
Fig. 9. Non-dimensional position of the interface as a function of time: theoretical predictions (solid lines) and experimental results.
Fig. 10. Non-dimensional position of the interface versus time: comparison between the predictions of the full and approximate models.
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conversion is well within the range reported in penetrative convection studies (Zilitinkevich, 1991). Energy losses are due to viscous dissipation and internal wave radiation. From an engineering point of view, it is important to know how much of the power supplied in the gas stream is converted into potential energy of the density distribution. As the kinetic energy of the plume at the interface is equal to approximately half of the energy of the bubbles at the source (Cardoso & Woods, 1993), we may conclude that the e$ciency of conversion of the bubble energy at the source into potential energy is f /2+15%.
on a large scale application, but alternative methods involving saturation of the liquid with a gas at high pressure followed by decompression can be explored. Also, in non-aqueous systems, small bubbles may absorb a signi"cant amount of vapour from the process liquid, and thus further recovery may be required. However, we note that as long as the criterion =/u 1 is satis"ed, the @ plume will exhibit a structure similar to a plume of lowReynolds number bubbles. We conclude that, from the view point of energy e$ciency and absence of mechanical moving parts, plumes of small bubbles have important engineering potential as a mixing technique.
5. Conclusions
References
We have presented a new experimental and theoretical study on the mixing induced by a plume of small bubbles in a two-layer density strati"cation. Our new experimental results show that very little dense liquid from the lower layer is transported across the interface by the rising bubbles. The density in the upper layer therefore remains constant and equal to that at the beginning of the mixing process. As a result, the mixing in the twolayer system consists of turbulent entrainment of the liquid in the upper layer by the plume liquid detrained at the interface, and the re-circulation of environmental liquid in the lower layer through the plume. A detailed numerical model has been developed. The model predicts the position of the interface as a function of time and the evolution of the density pro"le in the two-layer system. The agreement between our model predictions and experimental results is excellent. We have also shown, that in the limit of strong strati"cations, the density pro"le in the lower layer remains approximately uniform. We have developed a simple analytical solution for the evolution of the interface position and the density di!erence across the interface. We have shown that this simpli"ed approach is valid for Ri *1. Our study has also shown that a plume of small Reynolds number bubbles is e$cient at destratifying the lower regions of the tank, whilst a plume of large bubbles is more e$cient at transporting lower layer liquid into the upper layer and thereby increasing the density at the higher levels in the tank. The overall e$ciency of conversion of the energy released at the source of bubbles into potential energy is approximately 15% for the plume of small bubbles used in our work. This is signi"cantly larger than the average e$ciency of mixing of a plume of large bubbles, which has been found by previous authors to be of order of 3}9% (Schladow, 1992). There are some di$culties associated with the use of very small bubbles in industry. In particular, electrolysis cannot be competitively used to produce small bubbles
Asaeda, T., & Imberger, J. (1993). Structure of bubble plumes in linearly strati"ed environments. Journal of Fluid Mechanics, 249, 35}57. Baines, W. D. (1961). The principles of operation of bubbling systems, Proceedings of the Symposium on Air Bubbling, Ottawa. Baines, W. D., & Leitch, A. M. (1992). Destruction of strati"cation by bubble plume. Journal of Hydraulic Engineering, 118(4), 559}577. Cardoso, S. S. S., & Woods, A. W. (1993). Mixing by a turbulent plume in a con"ned strati"ed region. Journal of Fluid Mechanics, 250, 277}305. Chen, M. H. (1998). Bubble-driven mixing (Internal Report), Dept. of Chemical Engineering, University of Cambridge, UK. Clift, R., Grace, J. R., & Weber, M. E. (1978). In Bubbles, drops and particles (pp. 143}144). New York: Academic Press, Inc. (London) Ltd. Creighton, H. J., & Koehler, W. A. (1944). In Electrochemistry, vol. II: Applications (pp. 275}279). New York: Wiley. Jones, W. T. (1972). Air barriers as oil-spill containment devices. J. Soc. Pet. Eng., 12, 126}142. Leitch, A. M., & Baines, W. D. (1989). Liquid volume #ux in a weak bubble plume. Journal of Fluid Mechanics, 205, 77}98. Lemkert, C. J., & Imberger, J. (1993). Energetic bubble plumes in arbitrary strati"cation. Journal of Hydraulic Engineering, 119(6), 680}703. Manins, P. C. (1979). Turbulent buoyant convection from a source in a con"ned region. Journal of Fluid Mechanics, 91, 765}781. McDougall, T. J. (1978). Bubble plumes in strati"ed environments. Journal of Fluid Mechanics, 85, 655}672. Milgram, J. H. (1983). Mean #ow in round bubble plumes. Journal of Fluid Mechanics, 133, 345}376. Morton, B. R., Taylor, G. I., & Turner, J. S. (1956). Turbulent gravitational convection from maintained and instantaneous sources. Proceedings of the Royal Society, A234, 1}23. Patterson, J. C., & Imberger, J. (1989). Simulation of bubble plume destrati"cation systems in reservoirs. Aquatic Sciences, 51(1), 3}18. Schladow, S. G. (1992). Bubble plume dynamics in a strati"ed medium and the implications for water quality amelioration in lakes. Water Resources Research, 28(2), 313}321. Soo, S. L. (1967). In Fluid dynamics of multiphase systems (p. 96). New York: Blaisdell Publishing Company. Wilkinson, D. L. (1979). Two-dimensional bubble plumes. Journal of Hydraulic Division ASCE, 105(2), 139}154. Wuest, A., Brooks, N., & Imboden, D. M. (1992). Bubble plume modelling for lake restoration. Water resources research, 28(12), 3235}3250. Taylor, G. I. (1955). The action of a surface current, used as a breakwater. Proceedings of the Royal Society of London, A 231, 466}478. Turner, J. S. (1979). In Buoyancy ewects in yuids (pp. 165}206). Cambridge: Cambridge University Press. Zilitinkevich, S. S. (1991). In Turbulent penetrative convection (pp. 1}77). Avebury Technical.
The mixing of liquids by a plume of low-Reynolds number bubbles M. H. Chen, S. S. S. Cardoso* Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, CB2 3RA, UK Received 23 March 1999; accepted 18 October 1999
Abstract When bubbles are continuously released from a localised source at the bottom of a liquid layer, a plume is produced. As the bubble plume rises due to its buoyancy, it entrains surrounding liquid, which is carried upward with the stream of bubbles. In the present work, we investigate the motion of a plume of low-Reynolds number bubbles in a strati"ed liquid consisting of two homogeneous layers of di!erent densities. The liquid environment is of "nite lateral extent. We develop a theoretical model for the #ow of the bubble plume and the surrounding liquid. The full equations are solved numerically. The mixing at the interface is quanti"ed and the time-evolution of the density pro"les in both layers is calculated. The model also predicts the rate of rise of the density interface. We develop an analytical solution for the problem in the limit of strong strati"cations. Our theoretical predictions are compared with new experimental results using plumes of small bubbles generated by electrolysis of an aqueous solution of sodium chloride and with previous experimental results (McDougall (1978), Journal of Fluid Mechanics, 85, 655}672; Baines & Leitch (1992), Journal of Hydraulic Engineering, 118(4), 559}577). 2000 Elsevier Science Ltd. All rights reserved. Keywords: Bubble-plume; Strati"cation; Mixing; Two-phase #ow
1. Introduction Buoyant plumes produced by a source of bubbles in a liquid medium are used for a variety of purposes. Bubble breakwaters operate because of the surface jet produced by the bubble plume (Taylor, 1955). Oil slicks on water surfaces can be contained by bubble plumes (Jones, 1972). Bubble plumes have also been used to prevent parts of the surface of a river or lake from freezing (Baines, 1961). During an underwater oil-well blow-out, a plume of bubbles, oil droplets and sea water develops; the extent of the damage to marine life depends on whether all the oil rises to the surface or spreads out horizontally at some intermediate depth (McDougall, 1978). In industry, bubble plumes are used with two objectives: enhancing mass transfer and mixing. As a mixing technique, the use of bubbles is attractive because it is very simple and cheap to operate. The most common applications of bubble-driven mixing are encountered in wastewater treatment, in the delicate mixing of pharma-
* Corresponding author. Tel.: #1223-331863; fax: #1223-334796. E-mail address: silvana}[email protected] (S. S. S. Cardoso).
ceutical and food products, in the mixing of very hot or toxic liquids (Leitch & Baines, 1989) and in the destrati"cation of lakes and reservoirs (Schladow, 1992; Wuest, Brooks & Imboden, 1992). Over the last 20 years, there has been considerable research on the structure of bubble plumes and their interaction with the surrounding liquid environment. Laboratory and "eld observations have led to the development of two types of models of bubble plumes (McDougall, 1978; Leitch & Baines, 1989). The simpler approach considers the plume to behave as a one-phase buoyant plume. It assumes that the bubbles producing the buoyancy for motion are evenly dispersed within the liquid in the plume and behave passively (Fig. 1a). The more elaborate modelling regards the bubble plume as being composed of two parts: an inner circular plume containing all the bubbles and an outer annular plume containing only liquid (Fig. 1b). The buoyancy is concentrated within the central region and momentum is imparted to the surrounding liquid via turbulent shear stresses. However, given a speci"c problem, it is not clear from the previous literature (Wilkinson, 1979; McDougall, 1978; Lemkert & Imberger, 1993) whether one can use the simple model or needs to recur to the more complex double-core model.
0009-2509/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 5 3 1 - X
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Nomenclature a A B
e!ective radius of plume cross-sectional area of the tank buoyancy #ow rate in the plume at source, at z"0 bubble diameter EoK tvoK s number of bubble e$ciency of the conversion of the kinetic energy of the plume at the interface into potential energy of the lower layer gravitational acceleration height of the lower layer initial height of the lower layer initial height of the upper layer momentum #ow rate in the plume at the source, at z"0 volumetric #ow rate in the plume at the source, at z"0 equivalent radius of the tank critical Rayleigh number, taken as approximately 10 Reynolds number of the bubble Richardson number initial Richardson number of the two-layer strati"cation arbitrary time uniform downward velocity of the environmental liquid in the lower layer
d @ Eo @ f
g hl hl h M Q R Ra A Re @ Ri Ri t ;
Finite size bubbles exhibit a slip velocity relative to the liquid rising in the plume and this leads to the formation of a bubble-rich core surrounded by bubble-depleted rising liquid. The e!ect of this slip velocity will be greater for weak plumes i.e. for small volumetric #ow rates of gas. A quantitative criterion determining the structure of a bubble plume may be obtained by comparing a scale for the velocity in the plume = with the slip velocity of individual bubbles u . Dimensional arguments suggest @
= & u @
ol !o Q g ol hl , u @
(1)
where ol is the density of the liquid, o is the density of the gas bubbles, Q is the volumetric #ow rate of gas at the source, g is the acceleration of gravity and hl is the typical vertical length scale of the plume. In small-scale experiments performed in the laboratory, the #ow rates are smaller than those found in industrial and natural bubble plumes, whilst the bubble sizes remain approximately the same, and hence =/u is typically smaller. This @ explains why most laboratory observations hitherto have revealed a double-structure plume. As a consequence,
u @ = w z
slip velocity of a bubble scale for the velocity in the plume vertical velocity at the plume axis height above point source
Greek letters a D Dl Dl D D k o o @J ol ol o o
entrainment coe$cient, taken to be constant and equal to 0.09 buoyancy of the plume evaluated at the plume axis buoyancy in the lower layer initial buoyancy in the lower layer buoyancy in the upper layer initial buoyancy in the upper layer viscosity of the liquid liquid density as a function of depth z density of the bubble layer just above interface density of the lower layer initial density of the lower layer, at t"0 density of the upper layer initial density of the upper layer, at t"0
Subscripts 0 b l up
initial state bubble lower layer upper layer
most studies have concentrated on these double-structure plumes. However, for at least some of the largerscale plumes found in industrial processes, the bubbles are distributed widely across the plume. It is therefore of interest to study plumes of very small bubbles in the laboratory. Large bubbles possess wakes where liquid from deeper levels in the environment may be carried and then detrained at shallower levels. This is a relatively e!ective transport mechanism. However, the work of Milgram (1983) suggests that mixing induced by small bubbles should be more e$cient than that by large bubbles. Indeed, a plume of small bubbles entrains a larger quantity of surrounding liquid and this e!ect dominates over the wake transport mechanism exhibited by large bubbles. Nevertheless, very little is known about the dynamics of plumes of low-Reynolds number bubbles, whose ascent speed is very small compared to the convective velocities. It is therefore relevant to investigate the mechanism by which such small bubbles drive mixing in a liquid and to assess its industrial potential. Some insight into the behaviour of plumes of small bubbles may be gained from previous studies of one-phase buoyant plumes (Morton, Taylor & Turner, 1956; Cardoso &
M. H. Chen, S. S. S. Cardoso / Chemical Engineering Science 55 (2000) 2585}2594
Woods, 1993), but there are some fundamental di!erences, including bubble separation in strati"ed liquids and the formation of sequences of bubble plumes in the liquid layer. In the present work, we present new experimental results using bubbles of approximately 150 lm diameter generated by electrolysis; these bubbles are at least one order of magnitude smaller than bubbles used by previous researchers (see Table 1). A bubble plume rising in a density-strati"ed environment will transport dense liquid upward from the deeper regions for some distance; a fraction of this liquid then leaves the plume and spreads out horizontally at its own
2587
density level. In an environment of "nite lateral extent, the plume induces downward advection of the surrounding liquid, and hence the density distribution in the environment evolves with time. The continuing plume, rising through the changing strati"cation, will in turn transport liquid to varying depths. This interaction between the plume and the environment has been modelled numerically by Patterson and Imberger (1989), Leitch and Baines (1989) and Schladow (1992). The e!ectiveness of bubble-driven mixing in such complex systems has been measured by the e$ciency of conversion of the input bubble energy into potential energy of the strati"cation. The bubble energy at the source is measured by the work required to produce the air #ow via a compressor. Although peak e$ciencies of up to 15% have been achieved for certain design and operating con"gurations (Asaeda & Imberger, 1993; Schladow, 1992), the e$ciency generally decreases with time as the strati"cation weakens, and the average e$ciency of the complete mixing process can be as low as 3}5%. However, comparison with e$ciencies of mixing by one-phase plumes suggests that there is scope for improvement of bubble plume e$ciencies up to approximately 15% (Zilitinkevitch, 1991; Cardoso & Woods, 1993). In this paper, we discuss how the e$ciency of mixing of a plume is a!ected by the bubble size. In the following sections, we develop a novel approach to the modelling of the dynamics and mixing induced by a bubble plume, appropriate in the limit of small bubbles. We focus on the mixing driven by a continuous release of bubbles in a strati"cation consisting of two layers of liquids of di!erent densities. The liquid environment is of "nite lateral extent. We calculate the evolution of the density pro"le in both layers. Our theoretical predictions are validated by comparison with both new and previous experimental observations.
2. Experimental procedure and qualitative observations 2.1. Experimental procedure Fig. 1. Structure of a bubble plume rising in a homogeneous #uid: (a) simple plume when =/u 1 and (b) double-core plume when =/u 1. @ @
The experiments were carried out in two Perspex tanks, tank A of square cross section 0.2 m;0.2 m and
Table 1 Comparison of the size and structure of the bubbles used in the present work and in previous studies Experiment
d (m) @
u (m/s) @
Re @
McDougall (1978) Leitch and Baines (1989) and Baines and Leitch (1992) Present work
1.25;10\ 3.20;10\
0.258 0.203
252 630
4.20;10\ 1.50;10\
0.224 0.0115
1100 1.61
Eo @
Bubble shape
Volume of wake (m) =/u @
0.222 1.40
Spherical Ellipsoidal
1.90;10\ +9.80;10\
0.6 0.4
4.40 3.08;10\
Ellipsoidal Spherical
'9.80;10\ 0.0
0.6 4.1
The bubble shape and the volume of the wake were obtained from Clift, Grace & Weber (1978).
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M. H. Chen, S. S. S. Cardoso / Chemical Engineering Science 55 (2000) 2585}2594
height 0.4 m and a larger tank, tank B, of 0.4 m;0.4 m and height 0.6 m. A two-layer strati"cation was set up with a lower layer of NaCl aqueous solution and a top layer of fresh water; the two layers were separated by a sharp interface with approximately 0.5 cm thickness. The top layer was coloured with food dye to enhance the visibility of the interface. A photograph of the experimental system is shown in Fig. 2.
Bubbles were generated at the bottom of the tank via electrolysis of the NaCl solution. The electrodes consisted of a mesh of "ne platinum wires (Fig. 3), with area 1.5 cm;1.5 cm, connected to a d.c. power supply. Bubbles of hydrogen were released at the cathode and of chlorine at the anode (Creighton & Koehler, 1944). The electrodes were vibrated magnetically to avoid coalescence of bubbles and hence ensure the detachment of
Fig. 2. Photograph of experimental set-up illustrating the structure of a plume of small bubbles in a two-layer strati"cation. White arrows indicate the #ow of liquid.
Fig. 3. Electrodes of platinum used to generate bubbles of hydrogen at the cathode and chlorine at the anode.
M. H. Chen, S. S. S. Cardoso / Chemical Engineering Science 55 (2000) 2585}2594
uniformly sized bubbles. The size of the bubbles was determined in both layers using a laser analyser. The bubble size distribution was found to be independent of height in the tank and approximately Gaussian. An average size of 150$50 lm was measured (Chen, 1998). Such a constant bubble size with height, indicates a low coalescence tendency, which is characteristic of bubbles in concentrated salt solutions (Soo, 1967). Although small bubbles do tend to coalesce in fresh water, our laboratory observations and bubble size measurements suggest that coalescence was also negligible in the upper layer. We believe that the thin "lm of NaCl solution surrounding the bubbles moving into the upper layer hindered coalescence. Calibration of the #ow rate of gas generated versus the intensity of the current was performed directly by sealing the top of the tank and measuring the gas #ow rate with a bubble soap meter. Although chlorine is relatively soluble in water, our #ow rate measurements indicate a constant #ow rate for a given current intensity. This may be a consequence of either the mass of dissolved chlorine being negligible compared to the total gas generated or of saturation being achieved very fast. Nevertheless, since we have measured directly the #ow rate of gas generated, the dissolution of chlorine does not a!ect our experimental results. The position of the interface and the motion in the lower layer were monitored visually. The density distribution in the tank was determined by withdrawing small samples of the liquid at di!erent levels with a syringe. The density was measured by refractometry. The experimental conditions for each run are summarised in Table 2. 2.2. Qualitative observations The typical #ow pattern induced by a bubble plume rising in a strati"ed liquid consisting of two homogeneous layers of di!erent densities is shown in Fig. 4. The lower and upper-layer #uids have both been dyed for visualisation. As the bubble plume rises across the density interface, its buoyancy decreases. As a result, a fraction of the dense, lower layer liquid carried in the plume is detrained above the interface, where the momentum of
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the liquid reaches zero; the detrained dense liquid mixes with upper layer #uid and then spreads radially, forming a gravity current of intermediate density. The bubbles continue to rise through the upper layer, forming a new bubble plume. The surrounding less dense liquid in the upper layer is entrained and mixes with the residual dense liquid in the plume. The mixture of the two liquids is carried to the free surface where the bubbles burst and a liquid current spreads radially. As the tank is of "nite lateral extent, the continual entrainment of surrounding ambient #uid into the plume leads to a downward #ow in the environment. Hence, a front of liquid, which has been in the plume in each layer, descends with time; this can be seen in the sequence of photographs in Fig. 4 as the liquid circulated through the plume becomes discoloured. The liquid above the front in the lower layer is less dense than the original liquid below due to the entrainment of less dense upper layer liquid at the interface. Therefore, the plume will arrive at the interface even lighter. As time evolves, a stable density distribution develops, gradually "lling up the original lower layer. In the upper layer, the liquid detrained at the free surface is denser than the original upper layer liquid. It may therefore drive buoyant convection and mixing in the upper layer. Our measurements of the evolution of the density pro"le in the tank for a typical experiment are shown in Fig. 5. These results suggest that very little lower layer liquid is carried into the upper layer by the rising bubbles. This observation may be explained partially by the fact that small bubbles of diameter 150 lm do not have wakes, as shown in Table 1, and hence are unable to trap and carry lower layer liquid upward. In addition, the buoyancy force associated with the bubbles moving across the interface is relatively large compared to the viscous force that can drag lower layer liquid into the upper layer. Therefore, the contribution of viscous entrainment of lower layer liquid to the #ow rate and density of the plume in the upper layer is small. We conclude then that the dominant mixing mechanism at the interface is associated with the overshoot and collapse of the dense liquid transported in the plume. In Section 3, we present a theoretical model that builds upon these experimental observations.
Table 2 The experimental conditions. Labels A and B refer to tank used (See Section 2.1) Experiment
Q (m/s)
D (m/s)
hl (m)
h (m) S
B (m/s)
Ri
1 2 3 4 5 6 7
7.1;10\ 5.7;10\ 5.7;10\ 7.1;10\ 4.2;10\ 7.1;10\ 7.1;10\
!0.151 !0.230 !0.267 !0.366 !0.256 !0.423 !0.151
0.131 0.130 0.132 0.138 0.143 0.137 0.269
0.152 0.157 0.156 0.152 0.150 0.149 0.289
6.95;10\ 5.55;10\ 5.55;10\ 6.95;10\ 4.14;10\ 6.95;10\ 6.95;10\
0.9 1.6 2.0 2.5 2.6 2.8 3.1
A A A A A A B
Fig. 4. Sequence of photographs showing the evolution of the mixing induced by a bubble plume in a two-layer density strati"cation.
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M. H. Chen, S. S. S. Cardoso / Chemical Engineering Science 55 (2000) 2585}2594
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Here, z denotes the vertical co-ordinate, increasing upward, with its origin at the source; a is the e!ective radius of the plume; w is the vertical velocity at the plume axis; a is the entrainment coe$cient; and D and Dl are, respectively, the buoyancy of the plume evaluated on the plume axis and the buoyancy of the liquid surrounding the plume in the lower layer, de"ned by
Fig. 5. Evolution of the density pro"le with time for Ri "2.5. Data points have been interpolated to aid visualisation.
3. Theory 3.1. Full model In the light of our experimental observations, we shall assume that the upper layer remains uniform, with density equal to that at the beginning of the experiment. The mixing in the lower layer is modelled in detail by looking at both the motion in the plume and in the surrounding environment. Consider a plume rising in the lower layer of a step strati"cation. The cross-sectional area of the region is A"pR, where R is an equivalent radius. We assume that the depth of the lower layer is initially hl . Owing to the continuous variation of the ambient properties described above, the behaviour of the plume is time-dependent. However, if the aspect ratio R/hl is large, for all times, then the time rates of change of the plume properties can be shown to be su$ciently small to be neglected in the relevant conservation equations (Manins, 1979). The plume is then of small radius compared to R, and the ambient behaves quasi-steadily as far as the plume is concerned. Using the Boussinesq approximation and assuming a constant coe$cient of entrainment, conservation of volume, momentum and buoyancy for the plume are expressed by the following equations, respectively: d (aw)"2aaw, dz
(2)
d 1 aw "aD, dz 2
(3)
d 1 *Dl awD "aw . dz 2 *z
(4)
g(ol !o) D" , ol
(5)
g(ol !ol ) Dl " , ol
(6)
where, ol is the density of the lower layer and g is the acceleration due to gravity. We have taken the reference density to be equal to the initial density of the lower layer, ol . Gaussian pro"les of equal width have been assumed for the vertical velocity and buoyancy distributions in the plume. Eqs. (2)}(4) were originally derived by Morton et al. (1956) and have been discussed in detail elsewhere (e.g. Turner, 1979). The entrainment into the plume causes the environmental liquid outside the plume to move downward. We shall assume that the density di!erence between the liquid mixture spreading at the interface and the lower layer liquid just beneath it is su$ciently large so that this downward motion is approximately described by a uniform velocity ;. Continuity may then be written as (7)
!nR;"naw,
where it has been assumed that R a. The buoyancy "eld in the lower layer is governed by *Dl *Dl "!; . *t *z
(8)
Molecular di!usion and mixing have been neglected here. The initial and boundary conditions for the governing equations are as follows. The initial condition is that the strati"cation consists of two uniform layers. Hence, Dl "0,
g(o !ol ) D " ol
at t"0,
(9)
where o is the initial density of the upper layer. The boundary conditions specify the #ow rates of volume, momentum and buoyancy at the source naw"Q , (10a) naw"M at z"0, (10b) nawD"B . (10c) We require one more continuity condition, quantifying the rate of entrainment of the upper layer liquid at the interface. We shall assume that a constant fraction f of
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M. H. Chen, S. S. S. Cardoso / Chemical Engineering Science 55 (2000) 2585}2594
the kinetic energy of the plume at the interface is converted into potential energy of the lower layer. This energetic conversion is expressed by (Cardoso & Woods, 1993) dhl f B (Dl !D ) " . dt A
(11)
This equation is equivalent to an entrainment law of the type (1/w) dhl /dtJRi\, where Ri is the Richardson number of the strati"cation, de"ned by (Dl !D )hl . Ri" w
(12)
Cardoso and Woods (1993) have shown that such a mixing law is valid for 0.2(Ri(3. The e$ciency of this energetic conversion, f, has been found to lie in the range 0.2}0.3 in penetrative convection studies (Zilitinkevich, 1991). Eqs. (2)}(4) and (7)}(9), (10a)}(10c) and (11) were solved numerically using f"0.3; the theoretical results are compared with the experimental measurements in the next section. 3.2. A simplixed model for strong stratixcations For strong strati"cations, the rate of rise of the interface is very much smaller than the rate at which the environmental #uid is circulated through the plume. We may therefore assume that after the descending front reaches the bottom of the tank, the environment surrounding the plume is approximately homogeneous. In this limit, the full model equations may be simpli"ed to an equation for the conservation of buoyancy in the lower layer, d [hl (Dl !D )]"0 dt
4. Results and discussion Fig. 5 illustrates a typical evolution of the density pro"le with time. Several stages may be identi"ed: in the "rst stage (t"0}10 mins), the lower layer becomes strati"ed, evolving into a region where the density increases with height. This is followed by the main stage (t"10}110 mins) where the density di!erence between the upper and the lower layer is reduced and a homogeneous density is eventually recorded at t"120 min. The homogenisation of the solution occurs soon after the interface disappears, normally in the space of 1}2 min as recorded in nearly all the experiments. It can be seen from the graph that the density in the upper layer has undergone only minor changes until the last stage. It can therefore be concluded that the homogenisation of the solution is mainly due to the increase of the depth of the lower layer and that only a very small amount of liquid from the lower layer is carried across the interface by the bubbles. No strati"cation prevails in the lower layer soon after the "rst stage, showing that the plume drives good circulation in the "nite size tank. Our theoretical predictions for the evolution of the density pro"le have been plotted as solid lines in Fig. 6. The agreement between the model predictions and the experimental results is good for the lower layer. For large times, a small inversion in the density is observed at the top of the upper layer. The model does not capture this small e!ect as expected. Fig. 7 shows the evolution of the density pro"le in an experiment by Baines and Leitch (1992). The pro"les predicted by our new theoretical model, valid for small Reynolds number bubbles, are also shown. The di!erence between the model and experimental results suggests that a plume of small bubbles is more e$cient at destratifying the lower regions of the tank, whilst a plume of large bubbles is more e$cient at transporting lower layer
(13)
with boundary condition hl
fB d (Dl !D )"! . dt A
(14)
The analytical solution of Eqs. (13) and (14) gives the rate of rise of the interface
hl "hl exp
fB t Ahl (Dl !D )
(15)
and the evolution of the density in the lower layer
!f B t (Dl !D )"(Dl !D ) exp Ahl (Dl !D
)
.
(16)
Eqs. (15) and (16) are valid for large Richardson number strati"cations, that is for Ri 1.
Fig. 6. Evolution of the density pro"le: theoretical predictions (solid lines) and experimental results for an initial Richardson number, Ri "2.5.
M. H. Chen, S. S. S. Cardoso / Chemical Engineering Science 55 (2000) 2585}2594
Fig. 7. Evolution of the density pro"le: comparison between our theoretical predictions for a plume of small bubbles (solid lines) and the experimental data of Baines and Leitch (1992) for a plume of large bubbles; Ri "0.8.
liquid into the upper layer and thereby increasing the density at the higher levels in the tank. These observations may be explained by comparing the rate of descent of the density front in the lower layer for mixing driven by a plume of small bubbles with that observed for large bubbles. Such a comparison is made in Fig. 8. Our model predictions are in excellent agreement with our experimental results for plumes of small bubbles. However, for plumes of larger bubbles (Baines & Leitch, 1992), our model overpredicts the rate of descent of the front. This di!erence is related to the "nite slip velocity of the larger bubbles. We may conclude that a plume of small bubbles entrains more surrounding liquid, leading to a large re-circulation #owrate of liquid through the plume. A plume of large bubbles exhibits a smaller entrainment capacity, but is able to transport lower layer liquid in the individual bubble-wakes into the upper layer. The results of the numerical integration for the position of the interface as a function of time are shown graphically in Fig. 9. Experimental data have also been plotted. It is seen that the agreement is good for a range of strati"cations and plume buoyancy #uxes. In Fig. 10, we compare the predictions of the full model with those of the simpli"ed model valid for strong strati"cations. It is shown that the simple analytical model given by Eqs. (15) and (16) estimates the position of the interface very well for strong strati"cations, i.e. Ri *1. As mentioned in Section 3, we have assumed in our theoretical predictions that the parameter f is constant and equal to 0.3. The good agreement between our experimental results and the theory suggests then that 30% of the kinetic energy of the plume at the interface is converted into potential energy of the lower layer, as less dense #uid is transported down into the layer of dense #uid against the restoring buoyancy force. Such
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Fig. 8. Non-dimensional position of the descending front as a function of time. The solid lines are theoretical predictions for an entrainment constant a"0.09.
Fig. 9. Non-dimensional position of the interface as a function of time: theoretical predictions (solid lines) and experimental results.
Fig. 10. Non-dimensional position of the interface versus time: comparison between the predictions of the full and approximate models.
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conversion is well within the range reported in penetrative convection studies (Zilitinkevich, 1991). Energy losses are due to viscous dissipation and internal wave radiation. From an engineering point of view, it is important to know how much of the power supplied in the gas stream is converted into potential energy of the density distribution. As the kinetic energy of the plume at the interface is equal to approximately half of the energy of the bubbles at the source (Cardoso & Woods, 1993), we may conclude that the e$ciency of conversion of the bubble energy at the source into potential energy is f /2+15%.
on a large scale application, but alternative methods involving saturation of the liquid with a gas at high pressure followed by decompression can be explored. Also, in non-aqueous systems, small bubbles may absorb a signi"cant amount of vapour from the process liquid, and thus further recovery may be required. However, we note that as long as the criterion =/u 1 is satis"ed, the @ plume will exhibit a structure similar to a plume of lowReynolds number bubbles. We conclude that, from the view point of energy e$ciency and absence of mechanical moving parts, plumes of small bubbles have important engineering potential as a mixing technique.
5. Conclusions
References
We have presented a new experimental and theoretical study on the mixing induced by a plume of small bubbles in a two-layer density strati"cation. Our new experimental results show that very little dense liquid from the lower layer is transported across the interface by the rising bubbles. The density in the upper layer therefore remains constant and equal to that at the beginning of the mixing process. As a result, the mixing in the twolayer system consists of turbulent entrainment of the liquid in the upper layer by the plume liquid detrained at the interface, and the re-circulation of environmental liquid in the lower layer through the plume. A detailed numerical model has been developed. The model predicts the position of the interface as a function of time and the evolution of the density pro"le in the two-layer system. The agreement between our model predictions and experimental results is excellent. We have also shown, that in the limit of strong strati"cations, the density pro"le in the lower layer remains approximately uniform. We have developed a simple analytical solution for the evolution of the interface position and the density di!erence across the interface. We have shown that this simpli"ed approach is valid for Ri *1. Our study has also shown that a plume of small Reynolds number bubbles is e$cient at destratifying the lower regions of the tank, whilst a plume of large bubbles is more e$cient at transporting lower layer liquid into the upper layer and thereby increasing the density at the higher levels in the tank. The overall e$ciency of conversion of the energy released at the source of bubbles into potential energy is approximately 15% for the plume of small bubbles used in our work. This is signi"cantly larger than the average e$ciency of mixing of a plume of large bubbles, which has been found by previous authors to be of order of 3}9% (Schladow, 1992). There are some di$culties associated with the use of very small bubbles in industry. In particular, electrolysis cannot be competitively used to produce small bubbles
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