Transverse migration of single bubble这个有升力公式来源

Chemical Engineering Science 57 (2002) 1849–1858 www.elsevier.com/locate/ces Transverse migration of single bubbles in simple shear %ows Akio Tomiyamaa ; ∗, Hidesada Tamaia, Iztok Zunb, Shigeo Hosokawaa aGraduate School of Science and Technology, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan bFaculty of Mechanical Engineering, University of Ljubljana, Askerceva 6, 1000, Ljubljana, Slovenia Received 24 October 2001; received in revised form 18 February 2002; accepted 20 February 2002 Abstract Trajectories of single air bubbles in simple shear %ows of glycerol–water solution were measured to evaluate transverse lift force acting on single bubbles. Experiments were conducted under the conditions of −5:56 log10 M6 − 2:8, 1:396Eo6 5:74 and 06 |dVL=dy|6 8:3 s−1, where M is the Morton number, Eo the E7otv7os number and dVL=dy the velocity gradient of the shear %ow. A net transverse lift coe8cient CT was evaluated by making use of all the measured trajectories and an equation of bubble motion. It was con:rmed that CT for small bubbles is a function of the bubble Reynolds number Re, whereas CT for larger bubbles is well correlated with a modi:ed E7otv7os number Eod which employs the maximum horizontal dimension of a deformed bubble as a characteristic length. An empirical correlation of CT was therefore summarized as a function of Re and Eod. The critical bubble diameter causing the radial void pro:le transition from wall peaking to core peaking in an air–water bubbly %ow evaluated by the proposed CT correlation coincided with available experimental data. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Multiphase %ow; Bubble; Lateral migration; Lift force; Simple shear %ow 1. Introduction Accurate prediction of developing bubble %ows in verti- cal pipes cannot be carried out without su8cient knowledge of a transverse lift force acting on a bubble, the force which governs the direction of transverse migration of a bubble in a shear :eld. It has been clari:ed through a number of exper- iments that the lateral migration strongly depends on bub- ble size, i.e., small bubbles tend to migrate toward the pipe wall which causes a wall-peak bubble distribution, whereas large bubbles tend to migrate toward the pipe center which results in a core-peak bubble distribution. For an air–water system at atmospheric pressure and room temperature, bub- bles ranging from about 1 to 5 mm in sphere-volume equiv- alent diameter d might correspond to small bubbles, and bubbles larger than about 5 mm to large bubbles (Liu, 1993; Grossetete, 1995; Sakaguchi, Ijiri, Tabasaki, & Shakutsui, 1996). The migration of small bubbles toward the pipe wall can be explained with the so-called shear-induced lift force ∗ Corresponding author. Fax: +81-78-803-1131. E-mail address: tomiyama@mech.kobe-u.ac.jp (A. Tomiyama). model (Zun, 1980; Auton, 1987; Drew and Lahey, 1987) given by FLF =−CLF�L �d 3 6 (VG − VL)× rotVL; (1) where the subscripts G and L denote the gas and liquid phases, respectively, FLF is the shear-induced lift force, CLF the lift coe8cient, � the density and V the velocity. Zun (1980) and Lance and Lopez de Berodano (1994) reported that CLF for small bubbles in an air–water system takes a value ranging from 0.25 to 0.3. As for the migration to- ward the pipe center, Serizawa and Kataoka (1994) surveyed available experimental data and presumed that the direction of lateral migration would be governed by complex inter- action between a bubble wake and a shear :eld about the bubble. The validity of their presumption was partly con- :rmed by Tomiyama, Zun, Sou, and Skaguchi (1993) and Tomiyama, Sou, Zun, Kanami, and Sakaguchi (1995). They carried out interface tracking simulation of single bubbles in a Poiseuille %ow and pointed out that the migration toward the pipe center relates to the presence of a slanted wake be- hind a deformed bubble, which is apparently caused by the interaction between the wake and shear :eld. In addition, 0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0009 -2509(02)00085 -4 1850 A. Tomiyama et al. / Chemical Engineering Science 57 (2002) 1849–1858 their simulation indicated that another kind of transverse lift force, FTL, is induced by the complex interaction. In spite of these previous studies, our knowledge on the lateral migration is still insu8cient due to the lack of rele- vant experimental data. Fundamental experiments on lateral migration such as the measurement of bubble trajectories in a simple shear %ow conducted by Kariyasaki (1987) are de:nitely indispensable for improving transverse lift force models. Trajectories of single air bubbles in simple shear %ows of viscous liquids were therefore measured in this study to provide an experimental database and to present an empirical correlation of a net transverse lift force. The applicability of the proposed correlation to bubbles in low liquid viscosity systems was also discussed. 2. Experiments The Morton number M , which is a property group of the two phases, the E7otv7os number Eo, which is the ratio of buoyancy to surface tension forces, and the intensity of a velocity gradient ! of a simple shear %ow were selected as the parameters of experiments. They are de:ned by M = g(�L − �G)�4L �2L�3 ; (2) Eo= g(�L − �G)d2 � (3) and != |rotVL|= ∣∣∣∣dVLdy ∣∣∣∣ ; (4) respectively. Here g denotes the acceleration of gravity, � the viscosity, � the surface tension, VL the liquid velocity in the vertical (z) direction and y the horizontal coordinate. Note that in the case of a two-dimensional simple shear %ow heading in the z direction, the x, y and z components of the liquid velocity vector VL are given by (0; 0; VL(y)), and thereby the liquid velocity gradient ! = |dVL=dy| is equivalent to |rotVL|. These quantities were varied within the ranges of −5:56 log10M6 − 2:8, 1:396Eo6 5:74 and 06!6 8:3 s−1. Fig. 1 shows a schematic of the experimental setup. Glycerol–water solution at atmospheric pressure and room temperature was :lled in the acrylic tank, the height, width and depth of which were 0.9, 0.45 and 0:152 m, respec- tively. The seamless belt, 0:15 m in width, was rotated by the servomotor at a constant speed, the value of which was regulated by a controller within the range of 0–0:27 m=s. The position of the rotating belt was stabilized by the two guide plates and two pulleys so as to prevent its bending and %uttering. A simple shear %ow with a constant velocity gradient ! was thus realized in the 0:03 m gap between the belt and sidewall of the tank. Distilled water was used for making the glycerol–water solution to avoid the eJect of Fig. 1. Schematic of experimental apparatus. 10 20 30 20 40 60 80 100 0 y [mm] V L [m m/ s] z =100 mm z = 0 mm Fig. 2. An example of measured liquid velocity pro:les. surfactants on lateral migration. A single air bubble was re- leased from a nozzle made of a brass tube. The nozzle tip was positioned at the elevation where the liquid %ow estab- lished a simple shear %ow. Fig. 2 is an example of liquid ve- locity pro:les measured with a hot :lm probe at the nozzle location (z = 0 m) and at 0:1 m downstream of the nozzle location (z=0:1 m). Five diJerent tubes with 0.14, 0.5, 2.0, 3.0 and 4:0 mm in inside diameter were used for the nozzle to release various bubbles. The liquid density was evaluated as the ratio of a measured mass of the solution to its volume. The liquid viscosity and surface tension were measured with a rotational viscometer and a capillary tube tensiometer, re- spectively. Bubble shapes and trajectories were recorded us- ing a high-speed video camera (shutter speed = 1=1000 s, frame rate=400 frame=s). Enlarged video images were used A. Tomiyama et al. / Chemical Engineering Science 57 (2002) 1849–1858 1851 Fig. 3. Two examples of consecutive images of single bubbles in a simple shear %ow: log10M =−5:3; ! = 3:8 s−1. to evaluate d and its aspect ratio. The uncertainties esti- mated at 95% con:dence for measured �L; �L; �; d and ! were 0.6%, 2.0%, 3.3%, 0.7% and 2.3%, respectively. The measurement error of bubble position was±0:3 mm. All the measured %uid properties are summarized in Table 1 in the appendix. Fig. 3 shows two examples of consecutive images of sin- gle bubbles in a shear %ow. The small bubble in Fig. 3(a) migrated toward the stationary sidewall, whereas the large bubble in Fig. 3(b) migrated toward the moving belt. Sim- ilar images were taken for various combinations of d, ! and M . As a result, a database consisting of 116 trajectories was obtained. As an example, a dataset for log10M =−5:3 is shown in Fig. 4. The data for the other Morton number systems are summarized in the appendix. The y∗ and z∗ in the :gure are the dimensionless horizontal and vertical coordinates, normalized by the gap width D = 30 mm, i.e., 0 2 4 6 8 y*=0: Belt location y*=1: Wall location =0 [s-1] =3.8 =5.7 ω ω ω ω =6.2 Eqs. (7)-(9) d=2.84 mm d=3.52 mm d=4.16 mm d=4.85 mm d=5.54 mm 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 2 4 6 0 z* 0 0.2 0.4 0.6 0.8 1 y* 0 y* Fig. 4. Measured bubble trajectories for log10M =−5:3. y∗ = y=D and z∗ = z=D. This :gure clearly shows that (1) the direction of lateral migration under a constant Morton number is not aJected by !, but by the bubble diameter d, in other words, by the E7otv7os number Eo, and (2) the lateral migration length increases with !. As shown in Figs. 11 and 12 in the appendix, bubbles in the other Morton number systems exhibited the same tendencies. 3. Evaluation of lift coe�cient under a simple shear �ow Bubble trajectories were calculated using a one-way bub- ble tracking method to evaluate the transverse lift force act- ing on a bubble. Since the measured lateral migration length was proportional to ! both for the bubbles migrating toward the moving belt and for the bubbles migrating toward the stationary wall, it was assumed in the calculations that FTL caused by the slanted wake has the same functional form as that of the shear-induced lift force FLF , that is, FTL =−CTL�L �d 3 6 (VG − VL)× rotVL; (5) where CTL is the transverse lift coe8cient due to the slanted wake. Hence the net transverse lift force FT was assumed to be given by FT = FLF+FTL=−(CLF+CTL)�L �d 3 6 (VG−VL)× rotVL =−CT�L �d 3 6 (VG − VL)× rotVL; (6) where CT is the net transverse lift coe8cient, i.e. the sum of shear- and wake-induced lift coe8cient. The equation of bubble motion was therefore given by (�G + 0:5�L) dVG dt =− 3CD�L 4d |VR|VR − CT�LVR × rotVL + (�L − �G)g; (7) 1852 A. Tomiyama et al. / Chemical Engineering Science 57 (2002) 1849–1858 0 2 4 6 8 10 -0.4 -0.2 0 0.2 0.4 ω [s-1] C T d=2.80 d=3.56 d=4.23 d=4.85 d=5.68[mm] (a) log10M=-5.5 0 2 4 6 8 10 -0.4 -0.2 0 0.2 0.4 ω [s-1] C T d=3.19 d=3.40 d=4.19 d=4.93 d=5.64[mm] (b) log10M=-3.6 Fig. 5. EJects of liquid velocity gradient ! on CT : (a) log10M =−5:5, (b) log10M =−3:6. where VR is the relative velocity (=VG −VL); g the accel- eration due to gravity (=(0; 0;−g)) and CD the drag coef- :cient, which was evaluated by (Tomiyama, Kataoka, Zun, & Sakaguchi, 1998b) CD = �max [ min { 16 Re (1 + 0:15Re0:687); 48 Re } ; 8 3 Eo Eo+ 4 ] ; (8) where � is a tuning factor to make calculated bubble ve- locities just equal to measured values, and Re the bubble Reynolds number de:ned by Re = �L|VR|d �L : (9) The value of CT was adjusted so as to yield the best :ts to the measured bubble trajectories. As shown in the solid curves in Fig. 4, we could con:rm that all the measured trajecto- ries were well reproduced with Eq. (7), which implies the validity of the employed assumption for the functional form of FT . It should be also noted that the acceleration of bubble velocity, dVL=dt, was negligibly small for all the measured bubbles, and thereby, even if we neglected the left-hand side of Eq. (7), there was no diJerence in the evaluated CT . In other words, the value of the virtual mass coe8cient, 0.5, in Eq. (7) had no substantial eJects on the evaluation of CT . Fig. 5 shows thus evaluated net transverse lift coe8cients CT for two Morton number systems, (a) log10M = −5:5 0 10 20 30 40 50 60 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 log10M=-2.8 log10M=-3.6 log10M=-4.2 log10M=-5.0 log10M=-5.3 log10M=-5.5 Eq. (11) Re C T Fig. 6. CT for small bubbles. 0 5 10 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 log10M=-2.8 log10M=-3.6 log10M=-4.2 log10M=-5.0 log10M=-5.3 log10M=-5.5 Eq. (11) Eod C T Fig. 7. CT for large bubbles. and (b) log10M=−3:6. As shown in these :gures, CT takes positive values for small bubbles, whereas it takes nega- tive values for large bubbles. In addition, CT for a constant bubble diameter is not aJected by the liquid velocity gradi- ent ! and is more or less constant. As shown in Figs. 6 and 7, we also found that CT for small bubbles is well correlated with Re, whereas CT for intermediate and large bubbles is a function of a modi:ed E7otv7os number Eod, which is de:ned by using the maximum horizontal dimension of a bubble as a characteristic length as follows: Eod = g(�L − �G)d2H � : (10) The resulting empirical correlation of CT is given by CT ={ min[0:288tanh(0:121Re); f(Eod)] for Eod¡4; f(Eod) for 46Eod610:7; (11) A. Tomiyama et al. / Chemical Engineering Science 57 (2002) 1849–1858 1853 where f(Eod) = 0:00105Eo3d − 0:0159Eo2d −0:0204Eod + 0:474: (12) Eq. (11) yields 0¡CT 6 0:288 for small bubbles migrating toward the wall, which coincides with available experimen- tal data of CT for small bubbles in an air–water system, e.g., CT =0:3 by Zun (1980) and CT =0:25 by Lance and Lopez de Berodano (1994). On the other hand, Eq. (11) gives neg- ative values for large bubbles, and thereby large bubbles in a pipe %ow would migrate toward the pipe center due to the net transverse lift force. 4. Application of CT correlation to an air–water system The proposed CT correlation is based on the experimen- tal data obtained in a high-viscosity system, so that in prin- ciple it is not applicable to a low-viscosity system such as air–water and vapor–water systems. However, as noted above, the correlation coincidentally yields the same value of CT with experimental data for a small bubble in an air–water system. In view of this coincidence and the lack of CT correlations for a low-viscosity system, it would be worth applying the proposed correlation to a bubble in an air–water system to examine whether or not it can explain the tendency of bubble lateral migration in a low-viscosity system. 4.1. Bubble lateral migration in a bubbly up8ow In an air–water system under atmospheric pressure and room temperature, the proposed CT correlation yields the bubble diameter dependency shown in Fig. 8. To evaluate dH in the de:nition of Eod, we made use of the following empirical correlation of the aspect ratio E for spheroidal Wall Regime: 0.4 < d < 5mm migration toward near wall region Neutral Regime: 0< d <0.4mm 5< d <6mm affected by turbulence, bubble residence time, etc. Core Regime: 6 mm < d migration toward pipe center µL-controlled σ and g -controlled Possible Three Regimes 0 1 2 3 4 5 6 7 8 9 10-0.3 -0.2 -0.1 0 0.1 0.2 0.3 d [mm] C T Fig. 8. CT in an air–water system and postulated regimes of lateral migration. bubbles in a fully contaminated system (Wellek, Agrawal, & Skelland, 1966): E = dV dH = 1 1 + 0:163Eo0:757 ; (13) where dV is the maximum vertical dimension of a bubble. For d¡ 4:4 mm, CT is controlled by the bubble Reynolds number, i.e. by a viscous force. On the other hand, it is gov- erned by the E7otv7os number for d¿ 4:4 mm. CT changes its sign at d = 5:8 mm from positive to negative. We can, therefore, postulate three regimes for the bubble lateral migration in a bubbly up%ow in a vertical pipe. The :rst ( 16d¡ 5:8 − 2 where k (mm) is a small value) is a wall regime in which CT takes a large positive value, and as a result, bubbles would migrate toward the pipe wall. The second (d¿ 5:8 + 3) is a core regime in which CT takes a large negative value, and thereby bubbles would migrate toward the pipe center. The third (5:8 − 26d¡ 5:8 + 3 and 0¡d¡ 1) is a neutral or intermediate regime in which the bubble lateral migration might be strongly af- fected by many other factors such as the magnitude of bulk liquid turbulence and a bubble residence time in a %ow domain because the net transverse lift force in this regime keeps a low value due to the low magnitude of CT . Although the above-mentioned three regimes are based on the CT correlation for single bubbles in viscous shear %ows, they again coincide with available experimental data on radial void pro:les in air–water turbulent bubbly up- %ows in vertical pipes. As an example, radial void pro:les measured by Sakaguchi et al. (1996) are replotted in Fig. 9, in which R (=15:4 mm) is the pipe radius, r the radial coordinate, 〈J 〉 the area-averaged volumetric %ux, and 〈$G〉 the area-averaged void fraction. They measured the position and size of each bubble using an image processing method and classi:ed bubbles into several groups in terms of their sizes. Then they obtained radial void pro:les $G(r; d) for each bubble group (A–F or a–f in the :gure) and $G(r) for all the bubbles (G or g). As shown in the :gure, bubbles less than 5 mm (groups A–C or a–c) constitute the wall peak- ing, bubbles of 5–6 mm (group D or d ) correspond to the intermediate pro:le, and bubbles larger than 6 mm (groups E and F or e and f) form the core peaking. Liu (1993) also measured bubble sizes and void pro:les in air–water tur- bulent bubbly %ows using a vertical pipe of R = 28:6 mm, and concluded that the critical bubble diameter causing the void pro:le transition from wall peaking to core peaking is about 5–6 mm. Though not to mention all, many ex- perimental data have indicated that the pro:le transition in air–water bubbly %ows occurs when bubbles are larger than about 5 mm (Grossetete, 1995; Zun, 1988). These facts imply that the net transverse lift coe8cient in an air–water system is not so much diJerent from the proposed CT correlation. 1854 A. Tomiyama et al. / Chemical Engineering Science 57 (2002) 1849–1858 0 1 0 1 0 1 0 1 2 r/R G α (r) /<α G > d=2-3mm 0 2 4 0 1 2 0.2 0.4 0.6 0.8 1 1 2 0 A(8) B(122) C(810) D(445) E(76) F(5) G(1466) =0.504m/s =0.017m/s <αG>=2.18% d=3-4mm d=4-5mm d=5-6mm d=6-7mm d=7-8mm d=2-8mm 0 1 0 1 0 1 0 1 2 r/R 0 2 4 6 0 1 2 3 0.2 0.4 0.6 0.8 1 1 2 0 a(4) b(231) c(895) d(342) e(44) f(2) g(1518) =1.01m/s =0.017m/s <αG>=1.34% Fig. 9. Measured radial void pro:les as a function of d; data are quoted from Sakaguchi et al. (1996). 4.2. Most probable radial position of a bubble in a turbulent bubbly up8ow Once we establish a reliable equation for the balance of lateral forces acting on a bubble, it would be possible to predict the most probable radial position rmp of a bubble in an air–water bubbly %ow in a vertical pipe. The lateral motion of a bubble in a turbulent bubbly %ow would be, of course, aJected by many factors such as the transverse lift force, bulk liquid turbulence, wall eJect, non-rectilinear bubble path intrinsic to deformed bubbles, bubble collision and so on. However, for the purpose of evaluating the most probable radial position, we might be able to neglect the fol- lowing eJects: bubble collision, %uctuating bubble motion and bulk liquid turbulence, since these phenomena are more or less stochastic. In any case, a sort of restraining condi- tions to bubble lateral migration is indispensable to account for wall eJects. At this stage of examining whether or not the CT correlation can explain the tendency of rmp in a tur- bulent bubbly %ow, we might be able to employ a wall force model proposed by Tomiyama et al. (1995) as one repre- sentation of wall eJects. Other possible way to