A Study of Bubble Trajectory and Drag Co-efficient in Water and Non- Newtonian Fluids

Transcription

1 A Stuy of Bubble Trajectory an Drag Co-efficient in Water an Non- Newtonian Fluis N. M. S. HASSAN *, M. M. K. KHAN, AND M. G. RASUL College of Engineering an Built Environment Faculty of Sciences, Engineering an Health Central Queenslan University Rockhampton, Ql-4702 AUSTRALIA * n.hassan@cqu.eu.au ; nmshassan@gmail.com Abstract: - A new experimental set-up was use to analyze the characteristics of the bubbles rising in water an three ifferent concentrations of xanthan gum solutions for higher ynols number. The bubble size, bubble rise velocity, an bubble trajectory were measure using a combination of non-intrusive-high spee photographic metho an igital image processing. The results of bubble trajectory for various bubbles in water an ifferent xanthan gum solutions are presente an iscusse. In trajectory analysis, it is seen that the smaller bubbles show helical or zigzag motion an larger bubbles follow spiral motion for water. In xanthan gum solutions, small bubbles experience less horizontal motion than that in water. Larger bubbles prouce more spiral motion with the increase in xanthan gum concentration. Drag coefficients for air bubbles at higher ynols number are reporte. It is seen that the experimental rag coefficient increases with the increase in xanthan gum concentration corresponing to the same bubble volume. Key-wors: - Bubble trajectory, bubble volume, rag co-efficient, ynols number, polymer solution, nonintrusive metho 1 Introuction The bubble rise characterization is very important for the esign of heat an mass transfer operations. Air bubbles are use in chemical, biochemical, environmental, an foo process for improving the heat an mass transfer. The overall mass transfer is affecte by the bubble size, pressure insie the gas phase, interaction between bubbles, rise velocity an trajectory [1]. The most significant ynamic behaviour of air bubbles are the bubble rise velocity, trajectory an the rag coefficient. After formation, a bubble quickly accelerates to its terminal velocity. The terminal velocity of an air bubble is terme as the velocity attaine at steay state conitions where all applie forces are balance. The terminal rise velocity of a single bubble rising in a liqui epens on the volume of the bubble an on the physical properties of the bubble an the liqui. The bubble rising in non- Newtonian polymer solutions also epens on the rheological properties of the liqui. The bubbles will experience a lift force if the liqui is sheare, an they move perpenicular to the velocity shear an the bubble trajectory is affecte by the turbulence motions of the bubble in the liqui. This effect is being strongest for bubbles whose rise is less than the turbulence velocity scale. When a bubble rises through a liqui, the way it travels is ifferent from the path of soli particles either rising ue to buoyancy or sinking. Soli particles o not circulate internally an ten to remain rigi; however a soli particle coul have the same shape as a bubble. But a gas bubble has constant motion internally. A bubble tries to follow the path of least resistance uring its motion. As a bubble rises upwars through liqui, the most resistance will be irectly on top. However, if the bubble moves slightly to one sie, less total resistance is experience. It is reaily observe that bubbles commence a helical, or spring shape path as they rise in a column of liqui. ISSN: Issue 3, Volume 3, July 2008

2 Generally, the ynamics of bubble rise are nonlinear an the egree of the nonlinearity increases with bubble size [2]. Single air bubbles rising through a liqui have been stuie extensively. It has been foun that when the bubbles are very small, surface tension, which is preominant over the internal force an the buoyant force, makes the bubble spherical an they ten to retain the spherical shape as long as their rising velocity, thus ynols number () remains small. In most practical situation, all three factors, inertia effect, viscosity, an surface tension shoul be regare in that the bubbles are not spherical in shape an they move in an oscillatory manner. For low- flows, the viscous forces are large relative to internal terms. So the viscosity forces ominate the terminal motion an terminal rise velocity increases with increase of iameter of the bubble at very low. At intermeiate region (>1), bubbles are no more spherical as their size increases an terminal velocity may increase or remain constant or ecrease with equivalent iameter of the bubble. In this region, surface tension an inertia forces etermine the terminal rise velocity. At high, bubbles are spherical cap or mushroom shape an the motion of the bubble is ominate by the inertia forces. In this region, bubble rise velocity increases with the equivalent iameter of the bubble [2]. The ynamics of the bubble characteristics in a gas-liqui system are still not totally unerstoo. From most of the stuy, it was seen that a small bubble rises through water in a straight line at its terminal velocity until it finishes its journey. The paths of larger bubbles were not stable an starte to zigzag an much larger bubbles followe spiral motion [3-8]. Wu an Gharib [4] stuie the bubble trajectories for spherical an ellipsoial bubbles in clean water an they showe that the ellipsoial bubbles exhibit a spiralling path instability, while the spherical bubbles follow a zigzagging path instability when the bubble iameter exceee 0.15 cm. Saffman [8] observe only the zigzag motions as the bubble rises in water when the raius of the bubble was less than 1 mm but bubbles of larger raius shows either zigzag or spiral motion epening upon ifferent factors. Feng an Leal [9] verifie various possible trajectories in ifferent shape regimes. A single bubble can pursue a zigzag path at 600, accompanie with vortex sheing behin the bubble. Uner the same experimental conitions, Yoshia an Manasseh [10] suggeste that the bubbles can also pursue spiral trajectory without vortex sheing. Shew an Pinton [11] presente that the commencement of path instability for smaller bubble size change remarkably in the case of polymer solution an it also appeare that the split to path instability for increasing bubble size was less rapi for the polymer solution in comparison with water. Dewsbury et al. [12] investigate the relationship between the terminal velocity an volume for larger gas bubbles in non-newtonian power-law fluis. Tsuge an Hibino [13] reporte that the trajectories of rising spherical an ellipsoial gas bubbles at higher were ientical. Dewsbury et al. [14] etermine experimentally that the rag coefficient for a rising soli sphere in non-newtonian pseuo plastic liquis were significantly affecte by its trajectory. A new rag correlation for rising spheres in non-newtonian power-law liquis was presente by Dewsbury et al. [15]. It escribe the relationship between an in creeping, transitional, turbulent an even critical flow regimes an it is vali for 0.1<< Margaritis et al. [16] stuie the rag co-efficient variation for bubbles over a wie range of in ifferent non-newtonian polysaccharie solutions an propose a correlation which matche very well with experimental ata. For the case of power-law non- Newtonian fluis, it has been shown that the rag curve for air bubbles followe Haamar-Rybczynski moel rather than Stokes moel for < 5 [14, 17]. On the other han, Miyahara an Yamanaka [17] reporte for the case of highly viscous non-newtonian liqui that the rag coefficient eviate from the Haamar Rybczynski type equation if the increase. Dhole et al. [18] investigate that the rag co-efficient always increase with the increase in power law inex for all values of the. The stanar rag curves are well establishe for Newtonian liqui. However, no such vali correlation of the rag coefficient of the gas bubble for non- Newtonian liquis at high exists. Clift et al. [19] liste a large number of correlations of rag coefficient for Newtonian liquis at a wie range of. On the other han, Dewsbury et al. [12, 14], Margaritis et al. [16], Miyhara an Yamanaka [17], Chhabra [20], Karamanev [21] presente the correlations for rag coefficient of gas bubble in non- Newtonian power-law liquis concerning soli particles an spherical bubbles. However, no universal rag curve for the case of rising air bubbles in non- Newtonian Power-Law fluis have been evelope yet in the available literature. ISSN: Issue 3, Volume 3, July 2008

3 The aim of this stuy is to measure the bubble trajectory, an the rag on the bubble as it rises through water an ifferent concentration xanthan gum (shear thinning) solutions an investigate the influence of various parameters namely, the bubble sizes, flui properties on the bubble rise characteristics (trajectory, rag etc). A new set of experimental ata of rag coefficient for spherical an non spherical air bubble with a wie range of ynols number are obtaine for both water an ifferent concentrations of xanthan gum solutions.these new experimental ata are compare with the results of other analytical an experimental stuies available in the literature. 2 Experimental Set-up an Proceure 2.1 Experimental test rig The experimental apparatus is shown schematically in Figure 1. Two-test rigs were use for investigating the bubble rise characteristics in water an three ifferent concentration of xanthan gum solution. The first rig consiste of a polycarbonate tube approximately 1.8 m in height an 125 mm in iameter. A height of 1.8 m is require to allow the bubble when it is release at the bottom of the rig to reach its terminal velocity. It contains two holes near the base. One is to facilitate the removal of the liqui containe in it an the other is to make possible the insertion of gas bubbles into the test rig. The insertion mechanism consists of a lale or spoon with a small pipe running own the centre that has a capability to control the injection of air. The air is injecte through this pipe into the upsie-own lale using a syringe. The cup will then nee to be twiste to allow the bubble to rise. The secon rig was esigne with acrylic tube of 400 mm in iameter an 2.0 m in height. Larger sizes of bubble were teste in this rig to eliminate the wall effect. The camera lifting apparatus stans approximately 2.0 m high which allows the movement of the camera mount evice to rise through 1.8 m in height. The variable spee rive of camera lifting apparatus regulates the control of the camera mount evice. This rive allows the camera to be raise at approximately the same velocity as the bubble. A high spee igital vieo camera was mounte on a evice with a small attachment to the sie of the camera lifting apparatus. A = Stury Base; B = Rotating Spoon; C = Cylinrical test rig (0.125m or 0.40 m iameter), D = Vieo camera; E = Variable spee motor; F = Pulley; an G = Camera lifting apparatus. Fig. 1 Schematic iagram of experimental apparatus 2.2 Bubble rise velocity measurement A known volume of air bubble was injecte from injection apparatus close to the bottom of the test rig. The injection apparatus was esigne in such a way that allows controllable quantities of air into the test rig. A high spee igital vieo camera was use to recor the bubble motion as they rose through liquis. These bubble vieos clips were analyse by Winows Movie Maker where various bubble rise times were note. Bubble rise velocities over these times were calculate since the istance travelle was known. 2.3 Bubble iameter measurement Bubble equivalent iameter was measure from the still images which were obtaine from the vieo clips. The still images were then opene using commercial software SigmaScan Pro 5.0 an the bubble height ( h ) an the bubble with ( w ) were measure in pixels. The pixel measurements were converte to millimetres base on calibration ata for the camera. The bubble equivalent iameter, eq was calculate [22] as ( ) = (1) eq h w 2.4 Bubble trajectory measurement Trajectory was etermine from the still images collecte from the igital vieo camera. Bubble trajectory was compute from the still frames obtaine from the vieo image. The still frames were then opene in commercial software, SigmaScan Pro ISSN: Issue 3, Volume 3, July 2008

4 which was capable of showing pixel location on an image. The pixel coorinates (X an Y) of the bubbles centre were note an recore into spreasheet. X coorinate correspons to the istance from the left ege an Y coorinate correspons to the istance from the top ege respectively. The pixel line running through the centre of the bubble release point was known. The eviation of the bubble centre from the release point was compute by subtracting the X of the bubble centre from the X of the bubble release point. 2.5 Calculation of an rag coefficient The terminal velocity of the bubble changes with the change in shear rate as the flui viscosity changes as a function of the shear rate. The average shear rate over the entire bubble surface is equivalent to U b / b so the apparent viscosity can be written [16, 23] as n 1 μ = K( Ub b) (2) In the case of spherical bubble, the for non-newtonian power law flui was rearrange from equation (2) as n 2 n ρliqu b b = (3) K For a non-spherical bubble with a vertical axis of symmetry, the was efine [12, 16, 23, 24] by n 2 n ρliqu w b = (4) K The rag co-efficient for spherical bubble was calculate by 4gbΔρ C = (5) 2 3ρ U liq b In the case of non-spherical bubble, the rag coefficient was compute by 3 4geqΔρ C = (6) 2 2 3ρ U liq w b 2.6 Test fluis Three ifferent concentrations of xanthan gum solutions use in this stuy were a non-newtonian (shear thinning) flui. Water an xanthan gum with ifferent concentrations of 0.025%, 0.05% an 0.1% (by weight) were use. These non-newtonian solutions were prepare by mixing xanthan gum by weight of each concentration with water in the test rig an stirring it for long hours (5-7 hrs). The temperature of water an all solutions in this stuy o was maintaine at 25 C. For every solution, the measure ensity was very close to the ensity of water at 25 0 C since they were mae with low concentrations of xanthan gum in the solution Flui Characterization The rheological properties of the solutions were measure using an ARES (Avance Rheometric Expansion System) rheometer. The range of shear rates to etermine flui rheology was 1 s s -1. The rheological properties for ifferent concentration of xanthan gum solutions teste are illustrate in Fig. 2 an summarize in Table 1. Figure 2 shows that the three ifferent xanthan gum solutions exhibit non-newtonian shear-thinning pseuoplastic behaviour which is aequately escribe by Power-Law moel given below. η n 1 = K & γ (7) The K an n values for the xanthan gum solutions were etermine from this response curve an are shown in the Table 1. Table 1 Rheological an physical properties of xanthan gum solutions. Flui Type Concentration (%) K, Pa. s n n Density, 3 kg / m Xanthan gum Xanthan gum Xanthan gum K represents the consistency of the flui behaviour i.e. the higher the value of K, the more viscous the flui an n enotes power law inex which is a measure of the extent of non-newtonian behaviour. For shearthinning pseuoplastic liquis, the power law inex, n, lies between zero an unity with values further remove form unity emonstrating a more pronounce non-newtonian behaviour. As seen, the viscosity of xanthan gum increases with the increase in liqui concentration. On the other han, power law inex ecreases with the increase in liqui concentration. The xanthan gum solution with 0.1% concentration has the highest viscosity an low power law inex in comparison with other concentrations use in this stuy. ISSN: Issue 3, Volume 3, July 2008

5 Viscosity, Pa.s Shear rate vs viscosity at 0.025% Xanthan Shear rate vs viscosity at 0.05% Xanthan Shear rate vs viscosity at 0.1% Xanthan Shear rate, 1/s Fig. 2 Viscosity vs. shear rate of xanthan gum solutions emonstrating the pseuoplastic behaviour. 3 sults an Discussion 3.1 Comparison between bubble rise velocity an bubble equivalent iameter The current results of bubble rise velocity as a function of the bubble equivalent iameter was compare with the result of Haberman an Morton [25] an Zheng an Yappa [26], see Figure 3. Figure 3 inicates that an increase in rise velocity with the increase of bubble iameter. As seen, the current experimental ata agree well with these publishe ata. Bubble rise velocity, cm/sec Haberman an Morton (1954) Current experiment Zheng an Yappa (2000) 5 isplays the schematic pictures of the bubble trajectories of ifferent sizes of air bubbles in water. Figure 4 shows the eviation of bubble from its release point as it rises through water. The general tren was for the bubble to remain close to the release centre, when the bubble was release an as it rose through water, it sprea out as the height increases. For water, the smaller bubble (0.1mL) eviate more horizontally with respect to the bubble release centre an the bubble starte to rise with helical or zigzag motion. It was seen that the path instability occurs from the equivalent iameter of 3.38 mm which equates to a Weber number This phenomenon agrees well with the finings of Duinevel [27] an Leal [28]. With increasing bubble size, the bubble surface oscillations change from a simple oscillation to higher orer moes, the trajectory changes from a simple helix to more complex trajectories. For larger bubbles, it was seen from Figure 4 an Figure 5 that initially, the bubble followe straight path, attaine its terminal velocity an shape, then it switche to spiral path. The larger bubble of 5.0mL, at high of 4331 an Weber number of 11.77, initially eviate horizontally with zigzag motion an it then, travelle in straight path an finally, followe a spiral path. Distance, mm Bubble release point 0.1mL bubble 0.2mL bubble 2.0mL bubble 5.0mL bubble 10.0mL bubble Bubble aimeter, mm Deviation, mm Fig. 3 Bubble rise velocity vs. bubble equivalent iameter. 3.2 Bubble trajectory The trajectory results of water are shown in Figure 4 for ifferent bubble sizes when measure over a height of 1.0 m from the point of air injection an Fig. Fig. 4 Rise trajectories of ifferent sizes of bubbles in water. The 10mL bubble, in the beginning, followe the straight path, an it then followe spiral path an finally, straight path until it finishe its journey. ISSN: Issue 3, Volume 3, July 2008

6 0.1mL bubble 2mL bubble 5mL bubble 10mL bubble alese point 800 Distance, mm Deviation, mm 0.1mL 5mL 10mL Fig.5 Schematic rise trajectories of ifferent sizes of air bubbles in water. It was seen that the spiral motion never change into the zigzag motion in this stuy that was also foun by saffman [8] in the literature. Clift et al. [19] an Duinevel [27] investigate the smaller bubbles less than 2 mm in iameter rise in straight or linear path but the linear trajectory was not observe in this stuy as the bubble equivalent iameter of this stuy was more than 3 mm. For smaller bubble at low ynols number, the rising bubble showe a zigzag trajectory [5, 6, 29, 30]. The larger bubbles at high ynols number, isplaye a spiral trajectory because the effect of wake sheing influence the bubble to inuce a spiraling rising motion. The trajectory results of three ifferent concentrations of xanthan gum solution are shown respectively in Figure 6, Figure 8 an Figure 9 for ifferent bubble sizes. The schematic iagram of rise trajectories of 0.025% xanthan gum solution for ifferent bubble volumes are shown in Figure 7. It is seen from Figure 6 an Figure 7, the horizontal eviation of the smaller bubble (0.1mL) is less than that of water. This is ue to increase viscosity of the solution an less friction acting upon their surface compare to the larger bubbles an so the smaller bubbles experience less resistance to vertical movement. The larger bubbles (5.0mL an 10mL) initially choose straight path an it then eviate horizontally an finally, they switche to a spiral path. But 10mL bubble exhibite more spiral motion than that by 5mL bubble. The larger bubbles however experience more resistance on top an eform as their size increases which results in spiral motion. Fig. 6 Rise trajectories of ifferent sizes of bubbles in 0.025% xanthan gum solution. 0.1mL 5mL 10mL Fig.7 Schematic rise trajectories of ifferent sizes of air bubbles in 0.025% xanthan gum. Distance, mm mL bubble 2mL bubble 5mL bubble 10mL bubble lease point Deviation, mm Fig. 8 Rise trajectories of ifferent sizes of bubbles in 0.05%% xanthan gum solution. ISSN: Issue 3, Volume 3, July 2008

7 It is observe from Figure 8 an Figure 9 that the horizontal movement for smaller bubbles was less with the increase in xanthan gum concentration. Therefore, smaller bubble of high concentration (0.1%) xanthan gum solution exhibits a less horizontal eviation. On the other han, larger bubbles of high concentration (0.1%) xanthan gum prouce more spiral motion in comparison with other concentrations of xanthan gum use in this stuy By equation 8 By equation 9 By equation 10 Experimental Distance, mm mL bubble 2mL bubble 5mL bubble 10mL bubble lease point Deviation, mm Fig. 9 Rise trajectories of ifferent sizes of bubbles in 0.1%% xanthan gum solution. 3.3 Drag co-efficient Bubble rag coefficients as a function of for water are presente in Fig. 10. For 0.1, the creeping flow regime, the well known Stokes moel is given by 24 C = (8) For low (<0.1), the bubble velocity is epenant on the viscosity of the flui an the gas bubble follows Haamar-Ryczynski moel at very low rather than Stokes moel ue to the internal circulation of the gas bubble which is given [17] by 16 C = (9) For any, the following equation (10) was suggeste [31] for spherical bubble = 1 + ( ) 2 1 (10) As seen, the equations (8), (9) an (10) give a reasonable fit at high when the current experimental ata were compare e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 Fig. 10 Drag coefficients vs. ynols number for rising air bubble in water. Many authors have preicte the rag relationship which focuse on either spherical bubble or soli particle. There have been limite stuies available in the literature on rag co-efficient of spherical an nonspherical bubble at high in non-newtonian powerlaw fluis. The most wiely accepte correlation of rag coefficient for soli particles was evelope by Turton an Levenspiel [32] an is given by C = ( ) + (11) ,300 The above correlation converges to Stokes moel at low. A moifie correlation propose for gas bubbles in non-newtonian power-law fluis [12], is given by C = ( ) + (12) ,300 The equation (12) converges to the Haamar - Rybczynski equation, at low. The following equation (13) was also suggeste for bubbles [33], = (13) The above equation (13) is vali for (10-2 < <3*10 5 ). It is seen from Figure 10 that the experimental of 0.025% xanthan gum solution shows a goo fit at higher in comparison with the equations (11), (12) an (13).The experimental in Fig.11 shows the exact match with the equation (13) when it was compare with the above mentione equations. The same phenomenon is also observe in Fig.12 for 0.1% xanthan gum solution. ISSN: Issue 3, Volume 3, July 2008

8 seen, these new ata are agree well with the publishe equations mentione above By equation ( 11) By equation ( 12) By equation ( 13) Experimental Fig.10 Drag coefficients vs. ynols number for rising air bubble in 0.025% xanthan gum solution By equation (11) By equation (12) By equation (13) Experimetal Conclusion The bubble rise characteristics, namely, bubble velocity, trajectory an rag coefficient prouce acceptable an consistent results. The bubble equivalent iameter in water observe a reasonable fit with the publishe literature. The trajectory analysis showe that small bubbles followe a helical or zigzag motion while larger bubbles followe a spiral motion for water. For water, the horizontal motion observe was less when the bubble size increase. In the case of xanthan gum solutions, the small bubbles experience less horizontal motion than that of water. On the other han, larger bubbles prouce more spiral motion with the increase in xanthan gum concentration. The experimental ata of rag coefficient increases with the increase in xanthan gum concentration for corresponing bubble volume. The relationship between - for ifferent concentration xanthan gum solutions showe acceptable results with the available analytical an experimental stuies in the literature with a wie range of ynols numbers. Fig.11 Drag coefficients vs. ynols number for rising air bubble in 0.05% xanthan gum solution By equation (11) By equation (12) By equation (13) Experimental Fig.12 Drag coefficients vs. ynols number for rising air bubble in 0.1% xanthan gum solution The reporte experimental ata of rag coefficient increases with the increase in xanthan gum concentration for corresponing bubble volume. As Nomenclature: b [m] bubble characteristic iameter h [m] bubble height or short axis length w [m] projecte iameter onto horizontal plane or long axis length eq [m] equivalent sphere iameter μ [Pa.s] apparent viscosity [-] ynols number [-] rag coefficient F [N] rag force g [m/s 2 ] acceleration ue to gravity U b [m/s] bubble rise velocity n [-] power law inex K [Pa.s n ] flui consistency inex g [m/s 2 ] gravitational acceleration Greek letters ISSN: Issue 3, Volume 3, July 2008

9 Δ ρ [kg/m 3 ] ensity ifference between liqui an air bubble ρ [kg/m 3 ] liqui ensity liq γ& [s -1 ] shear rate η [Pa.s] non-newtonian viscosity, Acknowlegments: The authors gratefully acknowleges the financial support from Sugar search Institute, Queenslan, Australia an the technical assistance provie by Mr. Ray Kerney in the experimental works. N.M.S. Hassan is thankful to Central Queenslan University for the awar of a research scholarship to pursue this stuy. ferences: [1] Shosho, C. an Ryan, Micheal, E., An experimental stuy of the motion of long bubbles in incline tubes, Chemical engineering science, 56, 2001, [2] Kulkarni, A. A. an Joshi, J. B., Bubble Formation an Bubble Rise Velocity in Gas- Liqui Systems: A view, In. Eng. Chem. s., 44, 2005, [3] Shew, W. L., Poncet, S. an Pinton, J. F., Viscoelastic effects on the ynamics of a rising bubble, Journal of statistical Mechanics, P01009, [4] Wu, Mingming. an Gharib, Morteza., Experimental Stuies on the Shape an Path of Small Air Bubbles Rising in clean Water, Physics of Fluis, vol.14, no.7, [5] Hassan, N. M. S., Khan, M. M. K., Rasul, M. G. an Rackemann, D.W., An Experimental Stuy of Bubble Rise Characteristics in non Newtonian (Power-Law) Fluis, Proceeings of the 16 th Australasian Flui Mechanics Conference, Gol Coast, Australia, 2007, [6] e Vries, A. W. G., Biesheuvel, A. an van Wijngaaren, L., Notes on the path an wake of a gas bubble rising in pure water, Int. J. Multiph. Flow 28, 1823, [7] Mougin, G. an Magnauet, J., Phys. v. Lett. 88, , [8] Saffman, P.G., On the rise of small air bubbles in water. Journal of Flui Mechanics, Digital Archive, 1956, 1: , Cambrige University Press. [9] Feng, Z. C. an Leal, L.G., Nonlinear bubble ynamics, Annu. v. Flui mech., 29, 1997, 201. [10] Yoshia, S. an Manasseh, R., Trajectories of rising bubbles, the 16 th Japanese Multiphase Flow Symposium, Touha, Hokkaio, July, [11] Shew, W. L. an Pinton, J. F., Dynamical Moel of Bubble path Instability, PRL 97, , 2006 [12] Dewsbury, K., Karamanev, D. G. an Margaritis, A., Hyroynamic Characteristics of free Rise of Light soli Particles an Gas Bubbles in Non- Newtonian Liquis, Chemical engineering Science, vol. 54, 1999, [13] Tsuge, H., an Hibino, S., The motion of Single Gas Bubbles Rising in Various Liquis, Kagaku Kogaku, 35, 65, [14] Dewsbury, K., Karamanev, D.G. an Margaritis, A., Dynamic Behavior of Freely Rising Buoyant Soli Spheres in Non-Newtonian Liquis, AIChE Journal, Vol. 46, No. 1, [15] Dewsbury, K., Karamanev, D. G. an Margaritis, A., Rising soli sphere hyroynamics at high ynols numbers in non-newtonian fluis, Chemical Engineering Journal, 87, 2002, [16] Margaritis A., te Bokkel, D. W. an Karamanev, D. G., Bubble Rise Velocities an Drag Coefficients in non-newtonian Polysaccharie solutions, John Wiley & Sons, Inc., [17] Miyahara, T. an Yamanaka, S., Mechanics of Motion an Deformation of a single Bubble Rising through Quiescent Highly Viscous Newtonian an non-newtonian Meia, Journal of chemical engineering, Japan, Vol. 26, No. 3, [18] Dhole, S. D., Chhabra, R. P. an Eswaran, V., Drag of a Spherical Bubble Rising in Power Law Fluis at Intermeiate ynols Numbers, In. Eng. Chem. s. 46, 2007, [19] Clift, R., Grace, J. R an Weber, M. E., Bubbles, Drops an Particles; Acaemic Press, 1978, republishe by Dover, [20] Chhabra, R. P., Bubbles, Drops, an Particles in Non-Newtonian Fluis, Taylor & Francis Group CRC Press, [21] Karamanev, D. G., Rise of gas Bubbles in Quiescent Liquis, AIChE Journal, Vol. 40, 1994, No.8. [22] Lima-Ochoterena, R. an Zenit, Visualization of the flow aroun a bubble moving in a low ISSN: Issue 3, Volume 3, July 2008

10 viscosity liqui, vista Mexicana De Fisica 49 (4), 2003, [23] Lali, A. M., Khare, A. S., Joshi, J. B. an Nigam, K. D. P., Behaviour of Soli Particles in viscous Non-Newtonian Solutions: Settling Velocity, Wall Effects an Be Expansion in Soli-Liqui Fluiize Bes, Power Technology, 57, 1989, [24]Miyhara, T. an Takahashi, T., Drag coefficient of a single bubble rising through a quiescent liqui., International Chemical Engineering, Vol. 25, No. 1, [25] Haberman, W. L. an Morton, R. K., An Experimental Stuy of Bubbles moving in Liquis, ASCE, 387, 1954, pp [26] Zheng, Li an Yapa, P.D., Buoyant Velocity of Spherical an Non spherical Bubbles/Droplets, Journal of Hyraulic Engineering, Vol. 126, No. 11, [27] Duinevel, P. C., The Rise Velocity an Shape of Bubbles in pure water at high ynols Number, J. Flui Mech. vol. 292, 1995, [28] Leal, L. G., Velocity transport an wake structure for bluff boies at finite ynols number, Physc. Fluis A 1, 1989, [29] N. M. S. Hassan, M. M. K. Khan an M. G. Rasul, Characteristics of air Bubble Rising in Low Concentration Polymer Solutions, WSEAS TRANSACTIONS on FLUID Mechanics, ISSN: , Issue 3, Vol. 2, [30] Krishna, R., van Baten, J.M., Rise Characteristics of Gas Bubbles in a 2D ctangular Column: VOF Simulations vs. Experiments, Int. Comm. Heat Mass Transfer, vol. 26, no. 7, 1999, [31] Mei, R. an Klausner, J. F., Unsteay force on a spherical bubble at finite with small functions in the free stream velocity, Phys. Fluis. A, 4, 1992, 63. [32] Turton, R., an Levenspiel, O., A short note on the rag correlation for spheres, Power Technology, 4, [33] Khan. A. R. an Richarson, J. F., The resistance to motion of a soli sphere in a flui, Chem. Eng. Comm., vol. 62, 1987, ISSN: Issue 3, Volume 3, July 2008