A simple parameterization for the rising velocity of bubbles in a liquid pool

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A simple parameterization for the rising velocity of bubbles in a liquid pool. This article suggests a simple parameterization for the gas bubble rising velocity as a function of the volume-equivalent bubble diameter; this parameterization does not require prior knowledge of bubble shape. This is more convenient than previously suggested parameterizations because it is given as a single explicit formula.

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A simple parameterization for the rising velocity of bubbles in a liquid pool. This article suggests a simple parameterization for the gas bubble rising velocity as a function of the volume-equivalent bubble diameter; this parameterization does not require prior knowledge of bubble shape. This is more convenient than previously suggested parameterizations because it is given as a single explicit formula..

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  1. N u c l e a r E n g i n e e r i n g a n d T e c h n o l o g y 4 9 ( 2 0 1 7 ) 6 9 2 e6 9 9 Available online at ScienceDirect Nuclear Engineering and Technology journal homepage: www.elsevier.com/locate/net Original Article A Simple Parameterization for the Rising Velocity of Bubbles in a Liquid Pool Sung Hoon Park a,*, Changhwan Park b, JinYong Lee b, and Byungchul Lee b a Department of Environmental Engineering, Sunchon National University, 255 Jungang-ro, Suncheon, Jeonnam 57922, South Korea b FNC Technology, Co., Ltd., 32F Heungdeok IT Valley, 13 Heungdeok 1-ro, Yongin, Gyeonggi 16954, South Korea article info abstract Article history: The determination of the shape and rising velocity of gas bubbles in a liquid pool is of great Received 28 July 2016 importance in analyzing the radioactive aerosol emissions from nuclear power plant ac- Received in revised form cidents in terms of the fission product release rate and the pool scrubbing efficiency of 11 November 2016 radioactive aerosols. This article suggests a simple parameterization for the gas bubble Accepted 13 December 2016 rising velocity as a function of the volume-equivalent bubble diameter; this parameteri- Available online 3 January 2017 zation does not require prior knowledge of bubble shape. This is more convenient than previously suggested parameterizations because it is given as a single explicit formula. It is Keywords: also shown that a bubble shape diagram, which is very similar to the Grace's diagram, can Bubble Rising Velocity be easily generated using the parameterization suggested in this article. Furthermore, the Bubble Shape boundaries among the three bubble shape regimes in the Eo eRe plane and the condition for EoeRe Plane the bypass of the spheroidal regime can be delineated directly from the parameterization Pool Scrubbing formula. Therefore, the parameterization suggested in this article appears to be useful not Radioactive Aerosol Emissions only in easily determining the bubble rising velocity (e.g., in postulated severe accident analysis codes) but also in understanding the trend of bubble shape change due to bubble growth. © 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/ 4.0/). 1. Introduction efficiency of pool scrubbing is dependent on the size, shape, and rising velocity of the bubbles that contain particles. Pool scrubbing has been used in a variety of applications to The shape and rising velocity of bubbles have another remove particulate air pollutants, and in particular for significance in the analysis of the emissions of radioactive removing radioactive aerosols [1e4]. In the pool scrubbing aerosols in nuclear power plant accidents. A considerable process, aerosol particles are collected on the bubble surface fraction of fission product species contained in radioactive mainly because of gravitational sedimentation, inertial aerosol particles results from product species vaporization impaction, and Brownian diffusion. The particle removal from the molten core pool into bubbles formed during the molten coreeconcrete interaction process [5]. Mass transfer * Corresponding author. E-mail address: shpark@sunchon.ac.kr (S.H. Park). http://dx.doi.org/10.1016/j.net.2016.12.006 1738-5733/© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
  2. N u c l e a r E n g i n e e r i n g a n d T e c h n o l o g y 4 9 ( 2 0 1 7 ) 6 9 2 e6 9 9 693 between the molten core and the bubbles is highly dependent spherical-cap regimes were visualized in the diagram. More- on the bubble size, shape, and rising velocity. Determination over, the condition for the direct conversion of spherical of the bubble shape and rising velocity for a given bubble bubbles to spherical-cap bubbles, without passing through the volume, therefore, is an important procedure in the analysis spheroidal regime, was given simply as “M >  10.” Never- of radioactive aerosol emissions in terms not only of the pool theless, the determination of Re using this diagram may be scrubbing efficiency but also of the fission product release troublesome and liable to error because an interpolation for M rate. All the postulated severe accident analysis codes is required. currently used worldwide contain their own schemes to In this article, a simple parameterization that expresses Re determine the bubble shape and rising velocity. The devel- as an explicit function of Eo and M is suggested. This is more opment of an efficient scheme to determine the bubble shape convenient than the previously suggested parameterizations and rising velocity is one of the key issues in enhancing the because it is given as a single explicit formula. In addition, this performance of the accident analysis code in terms of the parameterization can be used to produce a bubble-shape di- fission product release. agram similar to Grace's diagram. Several different theories for determining bubble rising velocity are available in the literature [6e13]. Most of those theories deal with a particular bubble shape type, e.g., sphere, 2. Theories of bubble rising in a liquid pool spheroid, and spherical (or spheroidal) cap; the bubble shape must be determined in advance to decide which theory to use. Gas bubbles rising in a liquid pool can be categorized based on The problem, however, is that the bubble shape cannot be their shape into one of the following three groups: sphere, determined without information on the bubble rising velocity. spheroid, and spherical cap [10,17]. Bubbles are spherical This implies that iteration is needed to simultaneously when they are so small that the inertial force is much smaller determine both the bubble shape and the rising velocity. than the surface tension or the viscous force. As the bubble Wallis [14] suggested 10 different bubble rising velocity sizedand hence, the rising velocitydincrease, the bubbles equations that depend on the size, shape, and rigidity of the change into oblate spheroid shapes because of the resistance bubbles. The study of Jamialahmadi et al [15] was apparently imposed by the liquid medium. When the bubbles are suffi- the first effort to suggest a universal formula to determine the ciently large, they tend to have flat and often indented bases, bubble rising velocity, but they neglected the effect of inertial breaking the upedown symmetry of the bubble shape. This force for spherical bubbles. Bozzano and Dente [12] suggested shape is called the spherical cap. a method to determine the bubble shape and rising velocity The shape and rising velocity of a bubble given a volume- simultaneously without iteration. They determined the bub- equivalent diameter have long been an important subject of ble drag coefficient by assuming that a rising bubble would fluid mechanics. It is well known that the bubble rising ve- have such a shape that the total energy (potential locity depends on the bubble shape, which in turn is deter- energy þ surface energy þ kinetic energy) is minimized. The mined according to three dimensionless numbers [9,17]: the results of numerical minimization were approximated into a gr d2 Eotvos number Eo ¼ sLL e , the Reynolds number Re ¼ rL vmbL de , and parameterization, which was a function of two dimensionless gm4 the Morton number M ¼ r sL3 , where g is the gravitational ac- parameters: the Eotvos number Eo and the Morton number M. L L celeration, rL is the density of the liquid medium, de is the Using the drag coefficient, the bubble rising velocity was given volume-equivalent diameter of the bubble, sL is the surface as a solution of a second-order equation. tension of the liquid medium, vb is the terminal rising velocity By analyzing experimental data obtained from 21 different of the bubble, and mL is the viscosity of the liquid medium. Eo is liquids with a very wide range of physical properties, Grace [9] the ratio between body forces and surface tension forces and showed that the size, shape, and rising velocity of a single Re is the ratio between inertial forces and viscous forces. M, bubble in infinite liquid can be deduced from a diagram in roughly speaking, increases with increasing viscous forces which the relation among three dimensionless numbers, Eo , and decreasing surface tension forces. M, and the Reynolds number Re , is given graphically. In this When the inertial force is negligible compared to the diagram (hereafter referred to as Grace's diagram), Re (repre- viscous force ðRe <  1Þ, the terminal rising velocity of a senting bubble velocity) was plotted as a function of Eo (rep- spherical bubble is given by [6,18]: resenting bubble size) for different values of M (representing   liquid properties). Dividing the particle shape into three re- gðrL  rG Þd2e 1 þ k vb;vis ¼ ; (1) gimes (sphere, spheroid, and spherical cap), Grace [9] con- 6mL 2 þ 3k verted the relationships between particle size and rising where rG is the density of the gas, mG is the viscosity of the gas, velocity in the spherical regime (i.e., for small particles) and in and k ¼ mG =mL . Because for most liquids and gases mG ≪mL (kz0) the spherical-cap regime (large particles) into relationships and rG ≪rL , Eq. (1) becomes: between Eo and Re . Then, for the spheroidal regime (i.e., be- tween those 2 size limits), cross-plotting was used to fill the grL d2e vb;vis ¼ : (2) gap. Grace et al [16] extended the work of Grace [9] to single 12mL liquid drops moving in another liquid medium. Eq. (2) is not valid when Re is significantly larger than 1 Grace's diagram was very useful and provided great insight because inertial force is not negligible. Wallis [14] suggested into the bubble behavior. The boundaries between the the following formula for spherical bubbles with non- spherical and spheroidal regimes, between the spheroidal and negligible inertial force in an Re range of  1 < Re <  100: spherical-cap regimes, and between the spherical and
  3. 694 N u c l e a r E n g i n e e r i n g a n d T e c h n o l o g y 4 9 ( 2 0 1 7 ) 6 9 2 e6 9 9  2=3 rL vb;in ¼ 0:14425g5=6 e : d3=2 (3) mL For spheroidal bubbles, vb is determined by [7,17]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:14sL vb;spheroid ¼ þ 0:505gde : (4) rL de For spherical-cap bubbles, the following formula was sug- gested for vb [8,17]: qffiffiffiffiffiffiffiffi vb;cap ¼ 0:721 gde : (5) Fig. 1 compares Eqs. (2e5) as a function of de . Pure water and air were chosen as the liquid and gas for preparing this figure and the following similar figures in which vb is plotted against de . The rising velocity of a spherical bubble increases with increasing bubble size because increased body force (buoyancy) dominates over increased friction in this shape regime. As the bubble shape changes to spheroid, however, the rising velocity begins to decrease with increasing bubble size because increased friction becomes greater than increased buoyancy owing to the effect of flattening. After the aspect ratio, the ratio of the longer axis length to the shorter axis length, becomes sufficiently large, no further flattening Fig. 2 e General formula for the rising velocity of bubbles occurs and the rising velocity begins to increase again with with internal circulation compared to experimentally increasing bubble size. When the bubbles become too large, measured data. the bubbles finally change into the spherical cap shape. pffiffiffiffiffiffiffiffi vb ¼ 0:711 gde , which is almost the same as Eq. (5). Actually, pffiffiffiffiffiffiffiffi 3. Parameterization for bubble rising velocity vb ¼ 0:711 gde was suggested for spherical-cap bubbles by for entire bubble shape range Clift et al [17] when Eo  40 and Re  150. Therefore, it is sug- gested to use Eq. (4) not for spheroidal bubbles only but for all In this section, a new parameterization that involves all the nonspherical bubbles. formulas for the three bubble shape regimes (Eqs. 2e5) is Second, to let Eq. (2), (3), or (4) be selected automatically for suggested. appropriate bubble size and shape, the following equation is The first step is to unify the two regimes for nonspherical suggested for bubbles with arbitrary size and shape. bubbles. Fig. 1 shows that Eqs. (4) and (5) exhibit very similar   trends for large bubble size. When de is sufficiently large vb ¼ min vb;vis ; vb;in ; vb;spheroid : (6) (0:505gde [2:14s rL de , L i.e., Eo [4:24) Eq. (4) converges to The reason for using the minimum value in Eq. (6) can be easily seen in Fig. 1. One important advantage of using Eq. (6) is that it is not necessary to identify the bubble shape in advance. Rather, the bubble shape is identified automatically when the bubble rising velocity is determined. One shortcoming of Eq. (6), however, is that there is an abrupt change in the derivative of the bubble rising velocity when the bubble sizeeshape regime changes (e.g., from sphere to spheroid). Actually, the transition from sphere to spheroid happens gradually, and the boundary between the two shape regimes is defined somewhat arbitrarily, e.g., by an Sphere in creeping flow aspect ratio of about 1.1 [9,10,17], indicating that the abrupt Sphere with inertial force bending at the shape regime boundaries is not natural. Spheroid Therefore, we suggest the following equation to bridge Eqs. (2), Spherical cap (3), and (4) smoothly: 1 1 vb ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 v2b;vis þ v21 þ v2 1 144m2L 4=3 mL b;in b;spheroid g2 r2 d4 þ 2 5=3 4=3 3 þ 2:14sL 1 L e 0:14425 g rL d e rL de þ0:505gde Fig. 1 e Comparison of the bubble rising velocities for three (7) bubble shape regimes: sphere, spheroid, and spherical cap.
  4. N u c l e a r E n g i n e e r i n g a n d T e c h n o l o g y 4 9 ( 2 0 1 7 ) 6 9 2 e6 9 9 695 Eq. (7) can be regarded as a combination of the two for- no internal circulation occurs, whereas it is 1 when the bub- mulas for spherical and nonspherical bubbles: bles are not contaminated or are sufficiently large and hence internal circulation fully develops. The value of fsc in real sit- 1 vb ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (8) uations varies between 1 and 1.5 depending on the specific v 2 1 þ v2 1 contaminants present and their concentrations. b;sp b;nonsp In the same way, the effect of surface contaminants needs where vb;sp and vb;nonsp are the rising velocities of spherical to be taken into account also for spherical bubbles with non- and nonspherical bubbles, respectively, given by: negligible inertial force, resulting in the following equation: 1 1 vb;sp ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi (9) 1 1 1 þ 1 144m2L 4=3 mL vb;sp ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (12) 2 vb;vis 2 vb;in þ 1 þ v21 144m2L 4=3 mL fsc g2 r2 d4 þ 4 4=3 g2 r2 d L e 2 5=3 0:14425 g rL d e 3 v2 b;vis b;in 2 5=3 4=3 3 L e 0:14425 g rL de and vb;nonsp ¼ vb;spheroid . Applying this method to Eq. (7), we have: Fig. 2 compares the bubble rising velocity formulas for spherical and nonspherical bubbles with the general formula 1 1 vb ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi: (Eq. 7) as well as with experimentally measured data 1 þ 1 144m 2 mL 4=3 v2b;sp v2b;nonsp 2 [8,10,19e28] reproduced from Clift et al [17]. Despite the gen- f sc g2 r2 dL4 þ 4=3 3 þ 2:14s 1 L e 2 5=3 0:14425 g rL de L rL de þ0:505gde eral agreement, it is observed that the scatteredness of the (13) data is large and that Eq. (7) tends to overestimate the bubble rising velocity, especially for small bubbles. This can be The value of fsc must have the following three properties. attributed, at least partly, to the effect of surface contami- First, it must be 1.5 for very small bubbles, i.e., for very small nants contained in the liquid, which is the subject of the next Eo . For instance, Bond and Newton [30] argued that internal section. gas circulation does not occur when Eo  4. However, the measurements shown in Fig. 2 indicate that internal gas cir- culation does not vanish completely for Eo as small as 0.01 (deq z0:3 mm). Therefore, we assume here that fsc ¼ 1:5 for 4. Effects of surface contaminants Eo  0:001. Second, it must be 1 for very large bubbles. It is assumed here that fsc ¼ 1 for Eo  10; i.e., bubbles always have Eqs. (2e4) and their combination, Eq. (7), are based on the internal circulation when body forces dominate over surface assumption that internal gas circulation is fully developed tension. Third, it must decrease monotonously with when a bubble rises by momentum transfer through the liq- increasing particle size (i.e., with increasing Eo ) from 1.5 to 1 in uidegas interface. It has often been observed in experiments, the range of 0:001  Eo  10. Although several different highly however, that small spherical bubbles move with a lower ve- sophisticated methods to determine the value of fsc as a locity given by the following Stokes equation, which indicates function of Eo were suggested previously [31e33], differences that they behave like rigid bodies with no internal circulation: among the methods are relatively large, and their agreements with experimental data are only qualitative. Therefore, a grL d2e vb;vis ¼ : (10) much simpler parameterization for fsc is suggested here: 18mL 0:5 The bubble rising velocity predicted by Eq. (10) is 33% lower fsc ¼ 1 þ  : (14) log Eo þ1 than that predicted by Eq. (2) because the suppression of gas 1 þ exp 0:38 circulation inside the bubble increases the friction imposed by liquid on the gas bubbles. Frumkin and Levich [13] and Levich and Technica [29] attributed this to the presence of surface- active substances in the liquid medium. According to their 1.5 explanation, the surface-active substances accumulating at the liquidegas interface (bubble surface) reduce the surface 1.4 tension. As the bubbles rise, the surface-active substances are dragged to the bubble bottom, building a surface tension gradient. This gradient creates tangential stress, which sup- 1.3 presses the fluid motion at the interface. The strength of the fsc effect of the surface tension gradient increases with 1.2 decreasing particle size. By taking this effect of surface contaminants into account, the rising velocity of spherical bubbles can be expressed by: 1.1 1 grL d2e vb;vis ¼ $ ; (11) fsc 12mL 1.0 0.001 0.01 0.1 1 10 where fsc is a factor accounting for the suppressed internal gas circulation due to surface contaminants. The value of fsc is 1.5 Eo when the bubbles are very small and contaminated and hence Fig. 3 e Parameterization for f sc as a function of Eo .
  5. 696 N u c l e a r E n g i n e e r i n g a n d T e c h n o l o g y 4 9 ( 2 0 1 7 ) 6 9 2 e6 9 9 The value of fsc calculated using Eq. (14) is plotted as a function of Eo in Fig. 3. It can be clearly seen that Eq. (14) satisfies all the above-mentioned three properties. Eq. (13) combined with Eq. (14) is the general formula for bubble rising velocity suggested to be approximately valid for any bubble size and shape. Fig. 4 compares it with Eq. (12) (for spherical bubbles), Eq. (4) (for nonspherical bubbles), and Eq. (7) (general formula with fsc ¼ 1) as well as with measured data (the same as those shown in Fig. 2) as a function of de . For Eq. (12), fsc ¼ 1 or 1:5 was assumed depending on whether internal circulation was taken into account. Consideration of the effect of surface contaminants led to better agreement between the parameterization and the measured data obtained with contaminated water (represented by the symbols located lower in the scatter plot), whereas Eq. (7) gives better agree- ment with the measured data obtained with pure water (represented by the symbols located higher in the scatter plot). Therefore, the scatteredness of the measured data indicates Fig. 5 e Comparison of the parameterization suggested in that real systems can fall into various degrees of the present study with previous parameterizations found contamination. in the literature. Eq. (13) combined either with fsc ¼ 1 or with Eq. (14) can be used for “uncontaminated” and “highly contaminated” liq- uids, respectively. The bubble rising velocity in “slightly parameterizations show reasonable agreement with contaminated” liquid may lie between those two limits measured data. However, the parameterization in the present depending on the degree of contamination. Unfortunately, study has a couple of distinct advantages over the others. there is no theory available to represent the factor fsc as a First, it is more convenient than the parameterizations sug- function of the specific contaminants and their concentra- gested by Wallis [14] and by Bozzano and Dente [12] because it tions. In practical applications of pool scrubbing, water is is given as a single explicit formula. Second, it can be used to inevitably contaminated. In addition, polar liquids are known very easily produce a bubble-shape diagram similar to Grace's to be more sensitive to the effect of contamination than diagram that was suggested based on experimental observa- nonpolar liquids [17]. Therefore, it is expected that Eq. (13) tions (without theoretical justification). This will be the sub- combined with Eq. (14) can be used in most pool scrubbing ject of the next section. applications. The parameterization suggested in this study is compared with the previous parameterizations found in the literature 5. Discussion [12,14,15] in Fig. 5. Except that of Jamialahmadi et al [15], which significantly overestimates the rising velocity of In the work of Grace [9], the boundaries among the spherical, spherical bubbles because it neglects the inertial force, all the spheroidal, and spherical-cap regimes were delineated in the Eo eRe plane, in which the bubble size changes for different M values were depicted by parallel lines. Grace's diagram can be Sphere without internal circulation produced easily using the results of this study. To do so, it is Sphere with internal circulation Nonsphere required to express the equations shown in the previous General formula with internal circulation sections in terms of dimensionless numbers. General formula with fsc By multiplying both sides of Eqs. (12) and (4) by rL de =mL and rearranging the equations, the following two equations are obtained for spherical and nonspherical bubbles, respectively: 1 Re ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for spherical bubbles; (15) þ M 2 5=2 144M 5=6 fsc E3 o 0:14425 Eo  0:25 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eo Re ¼ 2:14 þ 0:505Eo for nonspherical bubbles: (16) M Applying the method of bridging two bubble shape regimes to Eqs. (15) and (16), we have: 1 Re ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi: (17) Fig. 4 e Comparison of the general formula [Eq. (13) 2 f sc 144M E3o þ M5=6 5=2 þ 1=2 M1=2 combined with Eq. (14)] with the formulas for different 0:144252 Eo Eo ð2:14þ0:505Eo Þ shape regimes as well as with measured data.
  6. N u c l e a r E n g i n e e r i n g a n d T e c h n o l o g y 4 9 ( 2 0 1 7 ) 6 9 2 e6 9 9 697 It should be noted that Eq. (17) can be directly obtained bubbles increases with decreasing Eo because strong surface from Eq. (13) by multiplying both sides by rL de =mL . tension forces tend to minimize the bubble surface area, and In Fig. 6, Eq. (17) is plotted for different values of M ranging with decreasing Re because deformation (from spherical from 1014 to 108. The lines plotted in this figure using Eq. (17) shape) is caused by inertial forces and bubbles tend to be are almost the same as those created by Grace [9]. Besides this, spherical under strong viscous forces (relative to inertial the boundaries between the bubble shape regimes are forces) [34]. included in this figure. Detailed discussion of these bound- aries is given below. 5.2. Boundary between spheroid and spherical cap 5.1. Boundary between sphere and spheroid In Section 3, Eq. (5) for spherical-cap bubbles was regarded as a limiting case of Eq. (4) (for Eo [4:24). The same is obtained With the analogy that was used to unify the formulas for from Eq. (16) when the second term in the square root is much bubble rising velocity for spherical and nonspherical bubbles, larger than the first term. Eo ¼ 40 suggested by Grace [9] based the boundary between sphere and spheroid can be defined as on observations agrees very well with the result of this study, the point where Eq. (16) (rising velocity of spheroid) begins to where Eo [4:24. The boundary between spheroid and spher- be smaller than Eq. (15) (rising velocity of sphere). When Re is ical cap in Fig. 6 was plotted using Eo ¼ 40. sufficiently large (at the boundary between sphere and spheroid regimes), Eq. (15) converges to: 5.3. Boundary between sphere and spherical cap 5=4 0:14425Eo Considering that the spherical-cap regime can be regarded as Re ¼ : (18) fsc M5=12 a limiting case of the nonspherical regime, with Eo [4:24, as Therefore, by equating the right-hand sides of Eqs. (18) and mentioned above, the boundary between sphere and spherical (16), rearranging the resulting equation in terms of M, and cap can be expressed by a limiting case of Eq. (19) with a very combining it with either Eq. (18) or Eq. (16), the following large value of Eo , which would result in: relation between Re and Eo is obtained. Re ¼ 7:77: (20)  5=4 4:24 This result is somewhat different from that suggested by Re ¼ 7:77fsc 3=2 1þ : (19) Eo Grace [9] (Re ¼ 1:2), but the two cases are similar in that the boundary between sphere and spherical cap is given as a The curve appearing as the boundary between sphere and constant Re value (i.e., it does not depend on Eo ). spheroid in Fig. 6 was plotted using Eq. (19). The sphericity of Another important aspect of the boundary between sphere and spherical cap is that it exists only with a value of M larger than a certain value (e.g., ~10, suggested by Grace). When M is smaller than this value, the bubble passes through the spheroid region. This phenomenon can be explained using Fig. 7, in which there are two different cases of the intersec- tion of the rising velocity lines for spherical bubbles and nonspherical bubbles. In the first case (with low M), the rising velocity line for spherical bubbles intersects with that for Fig. 7 e Intersection of the rising velocity line for spherical Fig. 6 e Diagram showing different bubble shape regimes bubbles with low M and high M with that of nonspherical in the Eo eRe plane. bubbles.
  7. 698 N u c l e a r E n g i n e e r i n g a n d T e c h n o l o g y 4 9 ( 2 0 1 7 ) 6 9 2 e6 9 9 nonspherical bubbles, where Eo is smaller than 40. In this case, Acknowledgments the bubble passes through spheroid when it grows. In the second case (with high M), by contrast, the spherical bubble This work was supported by the Nuclear Research & Devel- line intersects with the nonspherical bubble line where Eo is opment of the Korea Institute of Energy Technology and larger than 40. In this case, the bubble shape converts directly Planning (KETEP) grant funded by the Korea government from sphere to spherical cap. Therefore, whether or not a Ministry of Trade, Industry and Energy (No. 2011T100200045). spheroidal shape appears when a bubble grows can be determined based on the value of Eo at the intersection of the two lines represented by Eqs. (15) and (16). Again, we use Eq. references (18) instead of Eq. (15) because the conversion from sphere to nonsphere occurs at Re [1 (see Fig. 5). In addition, fsc can be assumed to be 1 because here we are interested in the phe- [1] A.T. Wassel, A.F. Mills, D.C. Bugby, R.N. Oehlberg, Analysis of nomenon for Eo of around 40. By equating the right-hand sides radionuclide retention in water pools, Nucl. Eng. Des. 90 of Eqs. (18) and (16), we have: (1985) 87e104. [2] S.M. Ghiaasiaan, G.F. Yao, A theoretical model for deposition E2o  24:27M1=3 Eo  102:8M1=3 ¼ 0: (21) of aerosols in rising spherical bubbles due to diffusion, convection, and inertia, Aerosol Sci. Technol. 26 (1997) The solution of Eq. (21) is dependent on the value of M and 141e153. increases with increasing M. The value of M with which the [3] C. Gabillet, C. Colin, J. Fabre, Experimental study of bubble solution of Eq. (21) is Eo ¼ 40 can be found easily to be M ¼ 3:3, injection in a turbulent boundary layer, Int. J. Multiphase which is a factor of 3 smaller than the value (~10) estimated Flow 28 (2002) 553e578. [4] T.S. Laker, S.M. Ghiaasiaan, Monte-Carlo simulation of graphically from Grace's diagram [9]. It should be noted that aerosol transport in rising spherical bubbles with internal the factor of 3 difference is not significant considering that circulation, J. Aerosol Sci. 35 (2004) 473e488. Eo ¼ 40 is a rough estimation for the distinction between [5] H. Allelein, A. Auvinen, J. Ball, S. Guentay, L.E. Herranz, spheroid and spherical cap. A. Hidaka, A.V. Jones, M. Kissane, D. Powers, G. Weber, State- of-the-art report on nuclear aerosols, 2009, p. 5. OECD/NEA/ CSNI; 2009. Report nr NEA/CSNI/R. 6. Conclusions [6] J.S. Hadamard, Mouvement permanent lent d’une sphere liquide et visqueuse dans un liquide visqueux, Comp. Rend. A simple parameterization for the gas bubble rising velocity in Acad. Sci. 152 (1911) 1735e1738 [in French]. [7] H.D. Mendelson, The prediction of bubble terminal velocities a liquid pool was suggested. The parameterization formula from wave theory, AIChE J. 13 (1967) 250e253. was given as an explicit function of the volume-equivalent [8] R.M. Davies, G. Taylor, The mechanics of large bubbles rising diameter of a rising bubble. It is not required to identify the through extended liquids and through liquids in tubes, Proc. bubble shape in advance in using this parameterization to R. Soc. Lond. Ser. A, Math. Phys. Sci. 200 (1950) 375e390. determine the bubble rising velocity. [9] J.R. Grace, Shapes and velocities of bubbles rising in infinite A bubble-shape diagram, which is very similar to Grace's liquids, Trans. Inst. Chem. Eng. 51 (1973) 116e120. [10] T. Tadaki, S. Maeda, On the shape and velocity of single air diagram, was generated using the parameterization suggested bubbles rising in various liquids, Kagaku Kogaku 25 (1961) in this study. The boundaries among the three bubble shape 254e264 [in Japanese]. regimes in the Eo eRe plane were delineated directly from [11] M. Ishii, N. Zuber, Drag coefficient and relative velocity in  5=4 bubbly, droplet or particulate flows, AIChE J. 25 (1979) the parameterization formula: Re ¼ 7:77 1 þ 4:24Eo for the 843e855. boundary between sphere and spheroid; Eo [4:24 (practically [12] G. Bozzano, M. Dente, Shape and terminal velocity of single bubble motion: a novel approach, Comput. Chem. Eng. 25 Eo ¼ 40) for the boundary between spheroid and spherical cap; (2001) 571e576. and Re ¼ 7:77 for the boundary between sphere and spherical [13] A. Frumkin, V.G. Levich, On surfactants and interfacial cap. These formulas for the shape regime boundaries showed motion, Zh. Fiz. Khim. 21 (1947) 1183e1204. good agreement with those suggested by Grace [9] based on [14] G.B. Wallis, The terminal speed of single drops or bubbles in experimental observations. Moreover, the condition for an infinite medium, Int. J. Multiphase Flow 1 (1974) 491e511. bypassing the spheroidal regime (i.e., direct conversion from [15] M. Jamialahmadi, C. Branch, H. Mu¨ller-Steinhagen, Terminal sphere to spherical cap) was derived from the parameteriza- bubble rise velocity in liquids, Chem. Eng. Res. Des. 72 (1994) 119e122. tion and found to be M  3:3, which is in order-of-magnitude [16] J.R. Grace, T. Wairegi, T.H. Nguyen, Shapes and velocities of agreement with that estimated roughly from Grace's dia- single drops and bubbles moving freely through immiscible gram (M  10). Therefore, the parameterization appears to liquids, Trans. Inst. Chem. Eng. 54 (1976) 167e173. be useful not only in easily determining the bubble rising ve- [17] R. Clift, J.R. Grace, M.E. Weber, Bubbles, Drops, and Particles, locity (e.g., in postulated severe accident analysis codes) but Academic Press, New York (NY), 1978. also in understanding the trend of bubble shape change ac- [18] W. Rybczynski, On the translatory motion of a fluid sphere in a viscous medium, Bull. Int. Acad. Pol. Sci. Lett. Cl. Sci. Math. cording to changes in Eo and Re values due to bubble growth. Nat., Ser. A (1911) 40e46. [19] R.L. Datta, D.H. Napier, D.M. Newitt, The properties and behaviour of gas bubbles formed at circular orifices, Trans. Conflicts of interest Inst. Chem. Eng. 28 (1950) 14e26. [20] W.L. Haberman, R.K. Morton, An experimental investigation All authors have no conflicts of interest to declare. of the drag and shape of air bubbles rising in various liquids,
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