A Study of the Motion and Eruption of a Bubble at the Surface of a Two

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A Study of the Motion and Eruption of a Bubble at the Surface of a Two-Dimensional Fluidized Bed Using Particle Image Velocimetry (PIV) Christoph R. Mu1 ller,* John F. Davidson, John S. Dennis, and Allan N. Hayhurst Department of Chemical Engineering, UniVersity of Cambridge, Cambridge CB2 3RA, United Kingdom

Particle image velocimety (PIV) was used to study the motion of a single bubble as it approached and then broke through the top surface of a two-dimensional (2D) gas-fluidized bed. A digital camera recorded the motion of individual particles; this led, via computer analysis, to a complete set of particle velocity vectors around the bubble, particularly its roof, as the bubble broke through the bed surface. These results revealed the following: (1) The vorticity around the bubble is close to zero, except for the wake region at the bottom of the bubble, justifying the use of potential flow theory to predict the bubble motion around the bubble, apart from the wake. (2) Potential flow analysis, for a cylinder moving through infinite fluid, was used to predict the particle velocities for the bubble roof as the bubble broke through the surface of the bed. There is good agreement between theory and experiment for the roof velocities. (3) The experiments suggest that bubble eruption occurs when the thickness of the roof is about three particle diameters. (4) Even after eruption, the potential flow theory gives quite a good prediction of the velocities of particles from the ruptured roof of the bubble. 1. Introduction Fluidized beds have various applications in industry, such as fluid catalytic cracking (FCC), gasification, and combustion of coal and Fischer-Tropsch synthesis.1 Despite the fact that fluidized beds have been used in industry since the 1920s and good progress has been made in numerical simulations using two-fluid2 or discrete element models,3 some aspects of fluidized bed hydrodynamics are still far from fully understood. One aspect worthy of further study is the behavior at the top of the bed, that is, bubble eruption and collapse and the associated elutriation and entrainment of particles. An understanding of the underlying physics of these processes is not only important from an academic point of view but is also of industrial relevance. The rate of particle entrainment affects the operation of a fluidized bed reactor; the height of entrainment is an important design parameter. This paper is concerned with these phenomena. It has been suggested that particles are ejected into the freeboard either from the roof4,5 or the wake of a bubble.6,7 Pemberton and Davidson8 emphasized the importance of the effects of the walls on the ejection mechanism, finding that in a 2D bed, ejection from the roof of the bubble dominated, whereas in a 3D bed ejection from the wake of a bubble was favored. Besides the influence of the walls, the pattern in which bubbles burst determines whether particles are ejected from the wake or the roof as well as the maximum height reached by the ejected particles.9 Hatano and Ishida9 divided the patterns of bubbles bursting at the top of a fluidized bed into four categories: (1) isolated bubbles, (2) successively rising bubbles, (3) coalescing bubbles, and (4) successively coalescing bubbles. In the case of the eruption of an isolated bubble, particles were ejected from the bubble’s roof, whereas particles were ejected from the wake of the leading bubble in the case of coalescing bubbles.6,9 Various models for the trajectories of the ejected particles have been proposed, for example, refs 4 and 8. However, to calculate the trajectories of the ejected particles, their initial * To whom correspondence should be addressed. Tel.: +44 1223 762962. Fax: +44 1223 334796. E-mail address: [email protected].

velocity has to be known. Different assumptions have been reported with regard to the variation with respect to the eruption angle θ. There is some ambiguity in the definition of the eruption angle in refs 10 and 11. Here, a rigorous definition of θ is given by means of Figure 1. The origin is at the center of the chord defined by the bed surface above the bubble. It is believed that this definition is in agreement with the schematic definition given in refs 10 and 11. Fung and Hamdullahpur10 assumed a linear decrease of the particle velocity, V, with θ; Demmich12 assumed an exponential distribution in terms of θ. However, none of these studies gave experimental evidence to support their assertions. According to Fung and Hamdullahpur,10 the initial velocity of the ejected particles is given by

(

V θ ) 1Vy,max θmax

)

(1)

where Vy,max is the maximal vertical particle velocity, V is the particle velocity, and θmax is the maximal eruption angle. Studies of particle elutriation and entrainment have mainly used either optical probes13 or laser Doppler anemometry,10,14 both giving nonintrusive measurements. To investigate the gas velocity in the freeboard of a gas-fluidized bed, Pemberton and Davidson15 used a hot wire anemometer. However, both hot wire and laser Doppler anemometry have the shortcoming of only providing point measurements. In contrast, particle image velocimetry (PIV), as used in this study, provides simultaneous velocity measurement in an entire plane. “Traditional” PIV consists of laser sheeting, one or two CCD cameras, for 2D or 3D resolution, respectively, and image processing software to track the seed particles. In opaque systems, such as a gassolid fluidized bed, a laser sheet cannot be applied and observations are limited to the layers closest to the boundaries of the bed. In a 2D fluidized bed, however, it is assumed that particles along the depth of the bed move identically, that is, the movement of particles near the wall of the bed is representative of the movement of particles within the bed. The question whether or not the movement observed near the wall of a 2D fluidized bed is representative of the flow throughout the entire thickness of the bed has been widely discussed, for example,

10.1021/ie0611397 CCC: $37.00 © 2007 American Chemical Society Published on Web 02/03/2007

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Figure 2. Sketch of the experimental setup. Table 1. Details of the Particles Used Figure 1. Definition of the eruption angle θ. The origin is at the center of the chord defined by the bed surface above the bubble. Particles 1. Diameter of bubble Db ) 70 mm.

ref 16. However, Raso et al.17 presented images of the displacement of black tracer particles in a bed of white sand after the passage of a bubble. The tracer particles appeared black at the wall and gray if the tracer particle was further away from the wall. They confirmed that the movements of the tracer particles inside the bed (gray) and at the wall (black) were identical. Link et al.18 used discrete elemental model (DEM) simulations to quantify the differences in the particle flux in a 2D spouted bed. They observed barely any difference between the wall and inside the bed in the dense region, that is, outside of the spout. There was only a small difference at the center of the spout. In the study reported here, we are only interested in the eruption of particles at the top of the bed and the flow of particles around a bubble, but not inside a bubble. Thus, the difference between the velocity measured at the wall and the center of the bed will be negligible. As the ratio of bubble diameter to thickness of the bed was much larger than unity, at the top of a bubble any effects of curvature across the thickness would be negligible, anyway. Experiments to confirm Link et al.’s18 observations are currently being conducted in a 2D bed using magnetic resonance velocity imaging (cf. refs 19, 20). PIV studies of gas-solid-liquid fluidized beds have been reported in refs 21 and 22. Previous PIV studies of gas-solid fluidized beds have mainly concentrated on the flow patterns of gas above an erupting bubble and the determination of the flow field of the gas in the freeboard.23,24 In a recent study, Santana et al.11 used PIV to investigate the spatial distribution of the particle ejection velocity at the top of an erupting bubble in a 2D gas-solid fluidized bed. They calculated the particle ejection velocities at the point at which the bubble breaks through the surface and compared it with the correlation of Fung and Hamdullahpur.10 Santana et al.11 observed good agreement of only the vertical component of the particle velocity, Vy, with the correlation of Fung and Hamdullahpur,10 whereas their measurements of the horizontal component of particle velocity, Vx, revealed a significant difference from this correlation. Santana et al.’s11 results indicate that the magnitude of the particle velocity probably does not decrease linearly with θ. However, Santana et al.11 did not fully examine the entire process of bubble eruption and collapse, starting from the point at which the bubble is well below the surface. In other areas, Sousa et al.25 used PIV to study the wake of Taylor bubbles in a liquid. For low-viscosity liquids, they observed two closed vortices in the wake of a rising bubble. However, on increasing the viscosity of the liquid beyond a certain value, it was found that closed vortices were not formed. The transition took place at viscosities ∼ 0.36 Pas. The objectives of the present paper are (1) to study the entire process of bubble eruption and collapse using PIV, that is,

parameter

particles 1

particles 2

material particle size (µm) (sieve analysis) particle density (kg/m3) measured Umf (mm/s) Umf (mm/s) predicted by Wen and Yu30 Geldart’s31 classification

sand 500-710 2600 225 262 B

aluminum-oxide 425-600 3840 305 278 borderline B/D

extending the study of Santana et al;11 (2) to present a new model, describing the initial velocity of particles as a bubble erupts at the top of a fluidized bed; and (3) to study the wake of a bubble before, during, and after eruption of the bubble. In addition, plots of the vorticity around a bubble in a gas-fluidized bed have been measured for the first time. 2. Experimental Apparatus Observations were made on a 2D fluidized bed of height, width, and horizontal thickness 500 mm, 194 mm, and 10 mm, respectively. The beds were constructed of 3-mm-thick glass sheets. Perforated plates (aluminum of 2.5-mm thickness) served as distributors. The distributors were designed so that the pressure drop over the distributor was equal to, or larger than, the weight of the bed per unit cross-sectional area. The bed was fluidized by compressed air, the flow rate of which was regulated by a calibrated rotameter. To avoid electrostatic effects, the air stream was humidified using a large water bubbler. The bed was illuminated with two 500 W lamps, each placed as shown in Figure 2 at an angle of 45° to provide homogeneous illumination. Images of the bubbling and slugging fluidized beds were recorded by a Kodak Motion Corder Analyzer SR series CCD camera (512 × 480 pixels), using frame rates of 125 and 250 Hz. The backside of the 2D bed was covered with a black paper. Velocity vectors, that is, Vx and Vy, were calculated using MATPIV1.6.1,26 where x and y indicate, respectively, horizontal and vertical direction of the 2D fluidized bed. The size of the interrogation window was iteratively reduced from 64 × 64 to 16 × 16 pixels, using four iterations. After the calculation of the velocity vectors, (1) a signal-to-noise (s/n) filter (s/n ) 1.3, as suggested in ref 27) and (2) a global and a local mean filter were applied. Less than 5% of the calculated vectors were outliners, that is, calculated velocity vectors rejected by the applied filters. Of these, 8090% of the rejected vectors did not pass the s/n filter. To reduce the number of outliners and to improve the signal-to-noise ratio, mixed black and white colored sand as well as mixed white and brown aluminum-oxide were used rather than particles of just one color as used by others.11,18 Special care was taken to get tracer particles of identical density and particle diameter to avoid segregation effects. The properties of the particles used are summarized in Table 1. Using these black-white or brownwhite mixtures of particles, patterns were produced which could be tracked. Different mixture ratios of colored particles were

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Figure 3. Bubble approaching the top of a 2D gas-fluidized bed, U/Umf ) 1.26, Particles 1, Db ) 56 mm. (a) The bubble and the particulate phase as seen by the camera, with arrows indicating the particle velocity. (b) The same arrows with colors to indicate the magnitude of the velocity.

tested. Very good results, that is, high s/n-ratios, were achieved when two to three black particles were present per interrogation window. This is in agreement with ref 28. Furthermore, it is an advantage that the particles were larger than traditional PIV particles. This enabled very sharp images and consequently high s/n-ratios to be obtained. A typical ratio of pixel size to particle diameter was ∼1.9 for sand and ∼1.6 for aluminum-oxide. The maximum bubble diameters of bubbles presented in this paper are 71 mm for sand and 47 mm for aluminum-oxide particles, giving ratios of bubble diameter to thickness of the bed of 7.1 and 4.7, respectively. Link et al.18 used only large white particles, dp ) 2.45 mm, while covering the backside of their 2D spout-fluid bed with a black sheet. Although this setup also made it possible to generate patterns, the thickness of the 2D bed used18 was only 6 × dp, which might well have caused particle bridging effects. Grace29 suggested that the ratio of bed thickness to particle diameter should exceed 10. Santana et al.11 used fine white sand and also covered the backside of their 2D fluidized bed with black paper. However, they reported velocity vectors only for the layers of particles close to the top of the bed and at positions where the black paper was (partly) visible, that is, regions covered by bubbles. The use of mixtures of black and white particles in this study made it possible to apply a rigorous s/n-filter to the vectors calculated by the software, therefore giving reliable results. Furthermore, as reported by Raso et al.,17 black tracer particles, which are not directly at the wall but deeper inside the bed, appear gray and these also contribute to generated velocity vectors. 3. Theory Pemberton and Davidson8 used potential flow theory to predict particle flow from the dome formed by an erupting bubble. This theory will be extended to predict the initial velocity of particles ejected by an erupting bubble. Assuming that the particulate phase is incompressible and inviscid, and using the continuity and momentum equations, the velocity potential function, Φ, is governed by Laplace’s equation:32

∇2Φ ) 0

(2)

For a 2D system, the velocities in the x- and y-direction can be derived from Φ using the following identities:

Vx )

∂Φ ∂x

(3)

Vy )

∂Φ ∂y

(4)

As a first approximation, the bubble in a 2D fluidized is assumed to be of cylindrical shape. The velocity potential function of a cylinder, of radius R, rising with velocity Ub in a stagnant infinite volume is given as32

( )

Φx,y ) -Uby

R2 x + y2 2

(5)

The rise velocity and the radius of the bubble can be deduced from images, recorded by the high-speed camera, by identifying the bubble boundary in the images. A circle was fitted to the identified bubble boundary using a best-fit algorithm. The coordinates, x and y, at which Vx and Vy are calculated, are given by the surface of the dome formed by the erupting bubble. The center of the bubble, that is, the center of the circle, is the origin. 4. Results Figure 3 shows a typical image of the eruption of a single bubble at the top of a 2D fluidized bed and shows some interesting features. First, particles do not rain down from the top of a bubble’s surface. Instead, there is a location inside the bubble (Db/5 - Db/4 from the top), at which the particle velocity is zero. Above that position, particles rise; below it, particles fall through the bubble. Second, a region of high velocity can be seen at the bottom of the bubble. This region forms the wake, usually expected to rise with the same velocity as the bubble; a more detailed discussion of the wake region will follow. Bubble eruption is dynamic, so a single snapshot provides insufficient information for a thorough understanding of the process. To obtain an insight into the dynamics, the velocities Vx and Vy of the particles in the roof were measured from successive images like Figure 3 as bubbles approached and burst through the surface. Values of Vy and Vx were then plotted against the angle θ, defined in Figure 1, giving the results in Figures 4 and 5, covering a total time of 104 ms. The origin of time t was chosen arbitrarily, when the nose of the bubble was some distance from the top of the bed (∼2030 mm). A total time of 104 ms is covered in Figures 4 and 5. It is apparent in Figure 5 that the horizontal component of particle velocity, Vx, does not change significantly during bubble eruption and collapse. The Vx-θ relationship is roughly sinusoidal, with maximum and minimum velocities at θ ∼ (60°. As expected Vx is zero at θ ∼ 0°. Vx is also close to zero at


Figure 4. Vertical velocity Vy of particles in the roof of a bubble breaking through the surface of a fluidized bed of particles 1, with U/Umf ) 1.2 and Db ) 64 mm. The angle θ is defined in Figure 1.

θ ) (90°. The behavior of the vertical velocity, Vy, is quite different, as seen in Figure 4. For early times, corresponding to t < 32 ms in this case, a plot of Vy against θ has a cosine form, as seen in Figure 4a and b. Vy becomes negative for eruption angles close to (90°. For 0 < t < 32 ms, Vy is almost constant but starts to decrease when t > 32 ms for all values of θ. Accompanying the decrease of Vy is a change in the shape of the Vy-θ relation. With increase of time beyond 32 ms, new minima in the value of Vy develop at angles of ∼ -75° and ∼65°, whereas Vy becomes almost zero at θ ( 90°. From t J 64 ms (Figure 4c-e), the particles at θ ) 0° start to have a negative velocity, which indicates the start of bubble collapse. With further progress of time, the particles accelerate downward leading to larger negative velocities. To validate and compare models, it is crucial to define carefully the time at which the bubble eruption takes place and particles are first ejected. Pemberton and Davidson8 defined this point as that at which the particle layer around the bubble is only about 1 particle diameter thick. A very similar definition was given by Do et al.4 To gain more information about the process of bubble eruption, the position of the top of the dome was tracked from images recorded by the CCD camera. On the

basis of these, the vertical rise velocity of the dome was calculated by

Udome(t) )

Xtop(t + 1) - Xtop(t - 1) 2∆t

(6)

where Xtop(t) is the vertical position of the top of the dome, measured from an arbitrary fixed point. Both the change of the position of the top of the dome and its rise velocity are plotted in Figure 6. It can be seen that the rise velocity of the dome increases to a maximum value, remains constant for about 30 ms, and then decreases. To validate the criterion of Pemberton and Davidson8 and Do et al.,4 successive images of bubble eruption were analyzed. It was observed that the dome thickness was not uniform but increased toward θ ) (90°, being usually thinnest around the top of the dome, that is, at θ ) 0°. However, it was often observed that the thinnest part of the dome was not exactly at θ ) 0° but to its right or left. One of the objectives of the present work was to determine if the initial velocity of the particles at the roof of a bubble can be better predicted by a theory based on potential flow theory than by existing models. Figure 7 shows (1) the tracked surface of the dome (+++), (2) the boundary

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Figure 5. Horizontal velocity Vx of particles in the roof of a bubble breaking through the surface of a fluidized bed of particles 1, with U/Umf ) 1.2 and Db ) 64 mm. The angle θ is defined in Figure 1.

Figure 6. Position and rise velocity of the top of the dome of an erupting bubble, U/Umf ) 1.2 and Db ) 64 mm, Particles 1.

of the bubble ° ° °, (3) the circle which is the best fit to the boundary points °, and (4) the center of the circle, marked ×. The rise velocity of the bubble, Ub, necessary for comparison with the potential flow model, was calculated by tracking the position of the nose of the bubble. Figure 8 compares PIV measurements of Vx and Vy with a published correlation and with potential flow theory at three different times during a bubble eruption. The corresponding images of the erupting

Figure 7. Tracked position of the surface of the dome (+) and of the boundary of the bubble (O). The circle is a best fit to the boundary points and its center is indicated by ×. Particles 2 and Db ) 47 mm.

bubbles are shown in Figure 9. To compare the correlation of Fung and Hamdullahpur10 with the experimental data, the maximum vertical velocity, Vy,max, was deduced from the PIV measurements. Especially in the upper part of a bubble the position of the interface between the bubble and the dense phase is sometimes not clear. This problem is caused by particles falling through the bubble, that is, the bubble is not a perfect


Figure 8. Measured rise velocities of particles at the roof of the dome compared with the correlation of Fung and Hamdullahpur10 and with the potential flow theory, from eq 6, at different times; U/Umf ) 1.9, Particles 2 and Db ) 45 mm. Measurements and predictions of Vx and Vy are given at the following times: (a, b) 16 ms before the bubble erupts t ) 40 ms, (c, d) at bubble eruption t ) 56 ms, and (e, f) 16 ms after bubble eruption t ) 72 ms. See also Figure 9.

void. To give an estimate of the error introduced, the interface of a bubble and the dense phase was tracked successively 10 times. The results were averaged and a 95% confidence interval based on student’s t-distribution was calculated. The error bars in Figure 8 on the predictions of the potential-flow theory represent the calculated 95% confidence intervals. It can be seen that the error introduced by the uncertainty of the interface between the bubble and the dense phase is not significant. Figure 8 shows that the model based on potential flow theory can predict the experimental measurements very well at times t ) 40 and 56 ms, whereas the model of Fung and Hamdullahpur10 cannot capture the change in shape of the Vy profile from t ) 40 ms to t ) 56 ms. In addition, the model of Fung and Hamdullahpur10 significantly underpredicts the horizontal velocity, Vx. However, at time t ) 72 ms (Figure 8f), the potential flow theory slightly overpredicts the experimental measurements of Vy. A detailed discussion of the validity of the potential flow theory model is given below. To put the model of Fung and Hamdullahpur10 and the potential flow model into the context of the entire process of bubble eruption and collapse, plots of Vx and Vy at t ) 4-100 ms are given in Figure 10b and d. Figure 10 is similar to the measure-

ments shown in Figures 4 and 5, that is, Vx (Figure 10 a and c) remains almost unchanged with time, whereas the plots of Vy reveal changes. The maximum of Vy increases until t ∼ 36-52 ms. For t > 52 ms (Figure 10c), the maximum in Vy decreases, in fact, reaching negative values at θ ) 0° for t > 100 ms. As mentioned above, Levy et al.6 and Pemberton and Davidson8 observed that, for the case of single bubble eruption, the ejected particles originate from the roof of the dome and not from the wake. The velocity profiles measured by PIV, as presented in this paper, make it possible to study the fate of the wake in more detail, especially its flow pattern and vorticity during bubble eruption and collapse. Starting from the calculated particle velocities, it is possible to calculate the vorticity Ω ) ∇ × V. For the 2D case, which is investigated in this study, the magnitude of the vorticity of the particulate phase, ω, is calculated from

ω)

(

)

∂Vy ∂Vx ∂x ∂y

(7)

Information on the vorticity in a fluidized bed is of great importance, since various correlations are based on potential

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eruption, that is, Figures 12a and b. At the end, Figure 12d, the disturbance left by the erupted bubble can still be observed, although much damped. It is interesting to see that the flow profile in the residual wake, at the top of the bed after the bubble has erupted, is very similar to the flow profile in the wake at the bottom of a bubble inside a fluidized bed. 5. Discussion

Figure 9. Sequence of images showing a bubble erupting at the top of the bed. Times: (a) t ) 40 ms, (b) t ) 56 ms, (c) t ) 72 ms; U/Umf ) 1.9, Particles 2 and Db ) 45 mm. See also Figure 8.

flow theory, for example, the rise velocity of bubbles33 and slugs,34 which required ω ) 0. Figure 11 shows the flow field around a bubble approaching the top of a bed. In Figure 11a, the velocity, that is, the magnitude and direction, is given by the vectors, whereas the color indicates the magnitude of the velocity, that is, V )

xVx2+Vy2. In Figure 11b, the same velocity vectors are shown,

but the color gives the magnitude of the vorticity, calculated from eq 7 using the measured velocity field as a function of x and y. In Figure 11, the area of high particle velocity and high vorticity is marked by an arrow. In Figure 11, it is very interesting that, except for the wake region, the fluidized bed has a vorticity close to zero. The blue and red regions (Figure 11b) indicate high vorticities with rotation in opposite directions, rather like Hill’s spherical vortex.35 Figure 12 gives images, flow fields, the magnitude of the particle velocity, and the vorticity of a rising bubble during eruption and collapse. It can be observed that during the rise of the bubble, Figure 12a and b, the area of high velocity and vorticity at the bottom of a bubble prevails, despite a change in bubble shape. During bubble eruption, the bubble seems at first to elongate a bit but then to become broader at the end of the eruption process. While the bubble collapses, Figure 12c and d, a region of relatively high velocity and vorticity at the top surface of the bed can be observed. However, the absolute values of both the velocity and vorticity are small compared with those observed at bubble

For a proper understanding of the events during bubble eruption and collapse, it is necessary to consider the velocity profiles, Vy and Vx, in Figures 4 and 5, and the rise velocity and position of the top of the dome, Figure 6. Figures 4 and 5 show, as noted earlier, that Vx does not change significantly during bubble eruption and collapse, whereas the magnitude and functional form of Vy with θ alters considerably with time. Between t ) 0 and t ) 32 ms, before the bubble has erupted, both the shape and magnitude of Vy remain almost constant. During this period, the rise velocity of the dome has reached its maximum value and remains almost constant, as seen in Figure 6. However, at the time of maximum rise velocity, the dome has not yet reached its maximum height as shown in Figure 6. Also, at eruption angles close to (90°, Vy is negative as shown in Figure 4. As time increases beyond t ∼ 40 ms, both the functional form and magnitude of Vy change. The magnitude of Vy, most notably between -45° e θ e 40°, decreases and two minima, at θ ≈ -70° and θ ≈ 65°, develop. In contrast, Vx does not not change much in this period. This might be expected, since most of the change in Vy occurs at θ ≈ 0°, where Vx ≈ 0 and is therefore not affected by any change in dome shape and so forth. When t > 72 ms, Vy is negative for all θ, and the dome has passed its maximum height as seen in Figure 6, indicating the start of bubble collapse. To compare the model based on eq 6 for predicting the initial velocities of ejected particles with models from the literature,10,12 the time at which bubble eruption occurs needs to be determined; this is the time when the dome finally bursts and particles are ejected into the freeboard. Conflicting views on this matter are expressed in the literature. Pyle36 assumed that, at eruption, the final thickness of the dome, δ, is equal to the particle diameter, dp. Pemberton and Davidson8 applied the same criterion, but noted, reporting the results of Saxena and Mathur,5 that in the case of 2D beds, δ might be about several particle diameters. Figure 9 shows that the bubble surface starts to break locally, when t ) 56 ms; from Figure 10c, this corresponds to a decrease in Vy, that is, Vy is less at t ) 68 ms than at t ) 52 ms. At these positions of “breakthrough”, the dome thickness is only about three particle diameters, whereas other parts of the dome are significantly thicker, as seen in Figure 9b. The surface of the dome, previously smooth in shape, becomes rougher after surface breakage: at t ) 72 ms (Figure 9c), large parts of the top of the dome are rough and the position of the nose of the erupting bubble is ill-defined. In this study, the instant of bubble eruption was defined as the time when Vy reaches its maximum. From the images in Figure 9, this coincides with local breakthrough of the dome, when the dome thickness δ ≈ 3 × dp. The potential flow model for the initial particle velocity was compared with experimental measurements and with the model of Fung and Hamdullahpur10 in Figure 8. To compare Fung and Hamdullapur’s10 model with experimental measurements, Vy,max was deduced from the PIV measurements. Figure 8a-d shows clearly that the potential flow model predicts both Vx and Vy very well. It also captures the change in shape of Vy between t ) 40 ms and t ) 52 ms (Figure 8a and b and Figure 8c and d). The correlation of Fung and Humdullahpur10 roughly


Figure 10. PIV measurements of the rise velocity of particles at the roof of the dome of an erupting bubble. The time is the middle value between the two images used for the PIV analysis; U/Umf ) 1.9, Particles 2 and Db ) 39 mm

Figure 11. Position and rise velocity of the top of the dome of an erupting bubble, U/Umf ) 1.96 and Db ) 66 mm. In a, the colors indicate the magnitude of the velocity. In b, the colors indicate the magnitude of the vorticity with red ) clockwise rotation and blue ) anticlockwise rotation.

captures the shape of the Vy-θ relationship but strongly underpredicts Vx as seen in Figure 8b, d, and f, as has been previously reported by Santana et al.11 However, at t ) 72 ms, Figure 8e and f, the potential flow theory overestimates Vy at -50° e θ < + 50°. Figure 9c shows that at t ) 72 ms, there is no clearly defined dome at -50° e θ < + 50°. For times t > 72 ms, the particle motion is not expected to be governed by potential flow theory, since the particles at this time do not behave as a single phase but must move as single particles in a stream of gas. The validity of using potential flow theory for times up to 52 ms, Figure 8, is manifested by the fact that vorticity occurs only in the wake of the bubble, remote from events at the dome, as seen in Figure 11b. In general, the potential flow model can be used only until the point of roof collapse. Turning now to the wake region, it is interesting to note that the vortices in Figure 11b are not closed ones. This is consistent with the observation of Sousa et al.,25 who reported closed

vortices in the wake of a rising Taylor bubble in liquid, provided the viscosity of the liquid was less than ∼0.36 Pa s. A fluidized bed containing particles of similar size to those used in this study has a viscosity of ∼1.2 Pa s,37,38 which may explain vortices that are not closed. It is often assumed that the wake of a bubble is the region roughly delineated by drawing a circular extension to its “flat” bottom, so that a single circle encloses both the bubble and its wake. From Figure 11b, this assumption appears to be a good approximation. Figure 12 shows the fate of the wake during bubble eruption and collapse. The region of high velocity and vorticity in the wake at the bottom of the erupting bubble prevails during the entire process of eruption and collapse. The flow profile at the surface of the fluidized bed at the end of the collapse, Figure 12d, is still similar to that at the bottom of a rising bubble with the vortex pair, shown by red and blue colors, prevailing after the bubble roof has disappeared. However, a damping effect of the fluidized bed during bubble collapse, owing to its viscosity,

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Figure 12. The wake region of a bubble during eruption and collapse at different times (a) t ) 0 ms, (b) t ) 40 ms, (c) t ) 100 ms, (d) t ) 148 ms. (i) Left column, vector plots. (ii) Center column, vector plot, color indicating magnitude of velocity V ) indicating vorticity. U/Umf ) 1.96, Particles 1 and Db ) 66 mm.

is clearly visible. The absolute values of the velocity and the vorticity are continuously decreasing from Figure 12b to d, whereas no change can be observed during bubble eruption, that is, Figure 12a and b. The flow pattern during bubble collapse has some similarity to the eruption of a gas bubble in a liquid. Comparing measurements made in beds of particles 1 and 2 reveals that the vertical and horizontal velocities of the particles at the roof of each erupting bubble behave in the same manner. As the diameter of the bubble in Figure 10 (particles 2) is smaller than that of the bubble in Figures 4 and 5 (particles 1), a smaller maximum value of the vertical velocity is expected, as larger bubbles rise faster than smaller ones. Potential flow theory gives good predictions of the velocity of the roof of an erupting bubble for beds of both particles 1 and 2. Also, the appearance of a region of high vorticity only in the wake of a bubble, the rest of the bed being close to zero vorticity, can be observed for

x(Vx2+Vy2). (iii) Right column, vector plot, color

both particles. A similarity between the beds of particles 1 and 2 is expected, as both particles fall into the same group of Geldart’s31 classification, namely, B and borderline B/D. Usually the differences between particles of Geldart’s31 groups B and D are less pronounced compared to differences between B and the other two groups, that is, A and C. 6. Conclusion Particle image velocimetry (PIV) was used to follow the particle velocities when a bubble broke through the surface of a two-dimensional gas-fluidized bed. The motion of colored tracer particles added to the bed gave, from PIV, a complete set of particle velocity vectors during the process of bubble eruption, leading to the following conclusions: • The velocity field around a bubble, apart from its wake, is irrotational, that is, the vorticity is zero in the particle field


around the bubble, including the roof of the bubble as it erupts at the bed surface. • In agreement with Pemberton and Davidson,8 ejected particles appear to originate predominantly from the roof of a bubble rather than from the wake. • The irrotational nature of the particle model suggested the use of potential flow theory. This led to a theoretical model, giving prediction of the particle movements in the bubble roof as the bubble broke though the top surface of the fluidized bed. These predictions agree well with the experiments and are appreciably better than those of published models. • The potential flow theory predicts how the upward velocity, Vy, and the horizontal velocity, Vx, of the roof particles vary with eruption angle θ. The theory predicts, and the experiments confirm, that the Vy-θ relation changes during eruption of the bubble, whereas the Vx-θ relation does not change during eruption. • The experiments suggest that the bubble breakthrough occurs when the roof thickness is about 3 times the particle diameter. • The particle velocity measurements in the bubble wake show high vorticity. The wake appears to be a double vortex, consisting of two vortices, rising together, the right-hand vortex clockwise, the left-hand vortex counterclockwise. This flow structure survives until the wake reaches the top of the bed, and then bubble roof particles scatter over the surface of the bed by eruption of the bubble. Acknowledgment The authors wish to thank EPSRC (Rutherford Appleton Laboratory) for the loan of equipment and J. Kristian Sveen (DAMTP) for providing the code of MATPIV1.6.1. One of the authors (C.R.M.) acknowledges funding from the Deutscher Akademischer Austauschdienst (DAAD) and the Cambridge European Trust. Nomenclature dp ) particle diameter (m) Db ) bubble diameter (m) R ) radius of bubble (m) t ) time (s) Ub ) rise velocity of bubble (m/s) Udome ) rise velocity of dome (m/s) Umf ) minimum fluidization velocity (m/s) V ) particle velocity (m/s) Vmax ) maximum value of V (m/s) Vx ) particle velocity in horizontal direction (m/s) Vy ) particle velocity in vertical direction (m/s) Vy,max ) maximum velocity of Vy,max (m/s) Xtop ) position of the top of the dome (m) x ) horizontal coordinate (m) y ) vertical coordinate (m) Greek Letters δ ) dome thickness (m) θ ) eruption angle (°) θmax ) maximum value of θ (°) ω ) vorticity (1/s) Φ ) velocity potential function Literature Cited (1) Kunii, D.; Levenspiel, O. Fluidization Engineering; ButterworthHeinemann: Newton, MA, 1991.

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ReceiVed for reView August 30, 2006 ReVised manuscript receiVed November 6, 2006 Accepted November 20, 2006 IE0611397

A Study of the Motion and Eruption of a Bubble at the Surface of a Two

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