Bubble Rise Velocity and Drag Co-efficient at High Reynolds Number in Power-Law Fluids

Transcription

1 Proceeings of the 5th IASME / WSEAS International Conference on Flui Mechanics an Aeroynamics, Athens, Greece, August 25-27, Bule Rise Velocity an Drag Co-efficient at High ynols Numer in Power-Law Fluis * HASSAN, N. M. S., KHAN, M. M. K. AND RASUL, M. G. College of Engineering an Built Environment Faculty of Sciences, Engineering an Health Central Queenslan University Rockhampton, Ql-4702 AUSTRALIA * n.hassan@cqu.eu.au Astract: - Air ules are use in chemical, iochemical, environmental, an foo process for improving the heat an mass transfer. Due to the ominance of non-newtonian liquis use in various process inustries, an unerstaning of ule rise in rheologically complex liquis has grown to e important. An experimental stuy of the ule rise velocity an rag co-efficient at high ynols numer in non-newtonian (Power-Law) fluis are presente in this paper. The main characteristics, namely, the ule velocity an the rag relationship are investigate at high ynols numers (<4000). The experiments were conucte in 125 mm an 400 mm cylinrical column at liqui heights of 1 m, 1.2 m, 1.4 m an 1.6 m y introucing ifferent ule volumes (from 0.1mL to 20.0mL) corresponing to each height. The ule rise velocity an ule size were measure using a comination of non-intrusive (high spee photographic) metho an igital image processing. The parameters that significantly affect the rise of air ule are ientifie. The effect of ifferent liqui heights an ule volumes on the ule rise velocity is analyse an the influence of two ifferent sizes of tues on the ule velocity for various ule volume is iscusse. A correlation of the rag coefficient at high ynols numer is explaine an compare with the results of other analytical an experimental stuies availale in the literature. Key-wors: - Bule rise velocity, ule volume, rags co-efficient, ynols numer, power-law flui, non-intrusive metho 1 Introuction A gas nees to e in contact with a liqui phase in any process where the gas phase is present in the form of ules. The ules fin uses in many applications such as in the sparkling everages, in the cooking processes, in the transfer of heat an mass, in the pipeline transport applications, in polymer processing an activate sluge processes an others [1]. The most significant ynamic ehaviour of air ules are the ule rise velocity, trajectory an the rag coefficient. The rag coefficient correlates the rag force exerte on a moving air ule to its terminal velocity an projecte surface area. The terminal velocity of an air ule is terme as the velocity attaine at steay state conitions where all applie forces are alance. The ule rise velocity an rag coefficient of an air ule are mainly epene on the liqui an the properties of the ule. In small-ynols numers flows, the viscous forces are large relative to internal terms an the viscous shear stresses transmit the motion of the ule far into the flow. So the viscosity forces ominate the terminal motion an terminal rise velocity increases with iameter of the ule at very low ynols numer. An intermeiate region (>1), ules are no more spherical as their size increase an terminal velocity may increase or remain constant or ecrease with equivalent iameter of the ule. In this region, surface tension an inertia forces etermine the terminal rise velocity. On the other han, in very high ynols numer flow the viscous shear stresses only affect the flow close to the wall so a thin ounary layer forms near the ule where the velocity varies from the (relative velocity etween the ule an meium) value to approximately zero. The flow outsie of this layer is cause y the pressure graients. The averse pressure graient forme as the flow moves aroun the ack of the

2 Proceeings of the 5th IASME / WSEAS International Conference on Flui Mechanics an Aeroynamics, Athens, Greece, August 25-27, ule causes the flow to separate from the ule. The location of this flow separation an the nature of the separation (either steay or unsteay) are calculate y the interaction etween the pressure graients outsie the ounary layer an the viscous flow near the ule so it is etermine y the ynols numer of the flow. At high ynols numer, ules are spherical cap or mushroom shape an the motion of the ule is ominate y the inertia forces. In this region, ule rise velocity increases with the equivalent iameter of the ule [15]. A ule rise characteristic in Newtonian liqui has receive consierale attention an is well unerstoo. Due to the ominance of non-newtonian liquis use in various process inustries, an unerstaning of ule rise in rheologically complex liquis has grown to e important. The ynamic of the ule characteristics in a gas-liqui system are still not totally unerstoo. Many researchers have unertaken various stuies to preict the actual phenomena of the ule rise in a column of non-newtonian liqui since last century [1-16]. The relationship etween the terminal velocity an volume for larger gas ules were investigate y Dewsury et al. in non-newtonian powerlaw fluis [3]. Margaritis et al. stuie the rag coefficient variation for ules over a wie range of in ifferent non-newtonian polysaccharie solutions an propose a correlation which matche very well with experimental ata [2]. A new rag correlation for rising spheres in power-law liquis was presente y Dewsury et al. which is vali for 0.1<<25000 an escrie the relationship etween C an in creeping, transitional, turulent an even critical flow regimes[4]. For the case of power-law non-newtonian fluis, it has een shown that the rag curve for air ules followe Haamar- Ryczynski moel rather than Stokes moel for < 5 [5, 6]. But on the other han, Miyahara an Yamanaka reporte for the case of highly viscous non-newtonian liqui that the rag coefficient eviate from the Haamar Ryczynski type equation if the ynols numer increase [6]. Dhole et al. investigate that the rag co-efficient always increase with the increase in power law inex for all values of the ynols numer [7]. There have een limite stuies availale in the literature on ule rise velocity an rag co-efficient of spherical an non-spherical ule at high ynols numers in non-newtonian power-law fluis. More research an in epth analysis on ule rise phenomena in non-newtonian flui is necessary as most of the inustrial fluis are non-newtonian in nature. The aim of this stuy is to investigate the ehaviour of the ule rise velocity an rag co-efficient of spherical an nonspherical ule in non-newtonian power liquis. The correlation of rag co-efficient of the ule at high ynols numer is compare with the results of other analytical an experimental stuies availale in the literature. 2 Experimental an Calculation The experimental set up selecte in this stuy was similar to that use y Dewsury et al. [3]. The experimental apparatus is shown schematically in Fig. 1. Two-test rigs were use for investigating the ule rise characteristics in xanthan gum an polyacrylamie solution. The first rig consiste of a polycaronate tue approximately 1.8 m in height an 125 mm in iameter. The ule insertion mechanism consiste of a lale or spoon that ha a capaility to control the injection of air. The secon rig was esigne with acrylic tue of 400 mm in iameter an 2.0 m in height. Larger sizes of ule 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 move through roughly 1.8 m in height. The variale spee rive of camera lifting apparatus regulates the control of the camera mount evice. This rive allows the camera to e raise at approximately the same velocity as the ule. A high spee igital vieo camera (Panasonic, NV-GS11, 24X optical Zoom, mae in Japan) was mounte on a camera mount evice with a small attachment to the sie of the camera lifting apparatus. Bule rise velocities were compute y a frame y frame analysis of successive images. The ule images were analyse with the software Winows Movie Maker y which the ule rise time was recore an velocity was measure. Bule equivalent iameter was measure from the still frames which were otaine from the vieo image. The still images were then opene using SigmaScan Pro 5.0 commercial software an the ule height ( h ) an the ule with ( w ) were measure in pixels. The pixel measurements woul then e converte to millimetres ase on caliration ata for the camera. The ule equivalent iameter, eq was etermine [10] as ( ) = (1) eq h w where w the long axis length an h is the short axis length of the ule. For this measurement it was

3 Proceeings of the 5th IASME / WSEAS International Conference on Flui Mechanics an Aeroynamics, Athens, Greece, August 25-27, assume that the ule was axi-symmetric with respect to its short axis irection. an w is the iameter of the horizontal projection of ule or long axis length of the ule. Fig. 1 Schematic iagram of experimental apparatus A = Stury Base; B = Rotating Spoon; C = Cylinrical test rig (0.125m or 0.40 m iameter), D = Vieo camera; E = Variale spee motor; F = Pulley; an G = Camera lifting apparatus. Since the flui viscosity varies as a function of the shear rate so the terminal velocity of the ule also changes with the change in shear rate. The average shear rate over the entire ule surface is equal to U / so the apparent viscosity can e written [2, 9] as n 1 µ = K( U ) (2) In the case of spherical ule, the ynols numer for non-newtonian power law flui was efine as n 2 n ρliqu = (3) K For a non-spherical ule with a vertical axis of symmetry, the ynols numer was efine [2, 3, 9, 11] y n 2 n wu ρliq = (4) K The rag co-efficient for spherical ule was calculate y 4g ρ C = (5) 2 3ρliqU In the case of non-spherical ule, the rag co-efficient was compute y 3 4geq ρ C = (6) 2 2 3ρliqwU The rag co-efficient for non-spherical ule was calculate on the asis of the real ule geometry in equation (6), where is the equivalent sphere iameter eq 3 Material Use The xanthan gum an polyacrylamie solutions use in this stuy were a non-newtonian (shear thinning pseuoplastic) flui type. The solutions (polyacrylamie an xanthan gum) with concentration of 0.025% (y weight) were use. The temperature of all solutions in this stuy was maintaine at 25 C. For every solution, the measure ensity of the solution was very close to the ensity of water at 25 C since they were low concentration liquis. Rheological properties of the solutions were measure using an ARES (Avance Rheometric Expansion System) rheometer. The rheological properties for ifferent solutions are summarize in Tale 1. The usual range of shear rates to etermine flui rheology was 1-650s -1. Tale 1 Rheological an physical properties of polymer solutions n Density, Flui Type Concentration K, (%) Pa. s n 3 kg / m Polyacrylamie Xanthan Gum sults an Discussion 4.1 Bule rise velocity The velocity profile of xanthan gum an polyacrylamie solutions for various ule volumes (0.1mL- 5.0mL) at ifferent liqui heights is illustrate in Fig. 2. The Fig. 2 shows that the ule velocity increases with the increase in ule volume for all liquis. The average ule velocity is oserve (0.22 m/sec-0.30 m/sec) at 1.0 m height for a ule volume up to 5mL. The velocity profile of xanthan gum an polyacrylamie solutions for various ule volumes (0.1mL- 20mL) at ifferent liqui heights is illustrate in Fig. 3. The Fig. 3 shows that the ule velocity increases with the increase in ule volume for all liquis. The average ule velocity is oserve (0.22 m/sec-0.41 m/sec) at 1.0 m height for a volume up to 20mL. The ule velocity of xanthan gum is foun slightly lower in comparison with polyacrylamie solution correspons to the larger ule volume of 20mL at 1.0 m height.

4 Proceeings of the 5th IASME / WSEAS International Conference on Flui Mechanics an Aeroynamics, Athens, Greece, August 25-27, Bule velocity (m/sec) Vel. of xanthan sol n at 1.0 m Vel. of poly sol n at 1.0 m Vel. of xanthan sol n at 1.2 m Vel. of poly sol n at 1.2 m Vel. of xanthan sol n at 1.4 m Vel. of poly sol n at 1.4 m Vel. of Xanthan sol n at 1.6 m Vel. of poly sol n at 1.6 m Bule velocity (m/sec) Bule vel. of xanthan sol n at small rig Bule vel. of xanthan sol n at large rig Bule vel. of poly sol n at small rig Bule vel. of poly sol n at large rig Bule volume (ml) Fig. 2 Velocity profile for polyacrylamie an xanthan gum solutions at ifferent heights (small rig) Bule velocity (m/sec) Vel. of xanthan sol n at 1.0 m Vel. of poly sol n at 1.0 m Vel. of xanthan sol n at 1.2 m Vel. of poly sol n at 1.2 m Vel. of xanthan sol n at 1.4 m Vel. of poly sol n at 1.4 m Vel. of Xanthan sol n at 1.6 m Vel. of poly sol n at 1.6 m Bule volume (ml) Fig. 3 Velocity profile for polyacrylamie an xanthan gum solutions at ifferent heights (Large rig). It can e oserve from Fig. 2 an Fig. 3 that the average ule velocity slightly ecreases with the increase in liqui height ut it is not significant, though the pressure changes with the increase in height is very small. Fig. 4 presents the ata otaine from oth test rigs for a liqui column of 1.0 m height. It can e seen from Fig. 4 that the ule velocity ata fall on the same straight line for corresponing ule volume an liqui height. The similar tren can also e foun for all liqui heights. Hence it can e sai that the ule velocity is not epenant on rig size Bule volume (ml) Fig. 4 Velocity profile at ifferent rig size correspons to the same ule volume. 4.2 Drag co-efficient Bule rag coefficients as a function of ynols numer for xanthan gum an polyacrylamie solutions are presente in Fig. 5 an Fig. 6 respectively. No universal rag curve has een evelope yet for the case of rising air ules in non-newtonian power-law fluis. For 0.1, the creeping flow regime, the governing equations can e solve to yiel [12] F = 3πµ U (7) which is a form of Stokes Law. This Stokes moel is given y 24 C = (8) The equation (8) is only vali for soli ule or particle at very low ynols numer ut not suitale for gas ules rising in power-law liquis. The gas ules oey the Haamar-Ryczynski moel at very low ynols numer which is given [6] y 16 C = (9) As expecte, moels (8) an (9) fail in high ynols numer when the current experimental ata was compare. The rag coefficient for soli particles can e etermine [14] y, C = ( ) + (10) ,300 The aove correlation converges to Stokes moel at low numer. A moifie correlation was propose for gas ules in non-newtonian power-law fluis [3], given y

5 Proceeings of the 5th IASME / WSEAS International Conference on Flui Mechanics an Aeroynamics, Athens, Greece, August 25-27, C = ( ) + (11) ,300 The equation (11) converges to the Haamar - Ryczynski equation, at low ynols numer. The following two equations (12) an (13) have een liste for spherical ules [13], 6 21 C 0.28 = + + (12) 0.5 C 0.5 ( ) 2 = (13) The aove equations are vali for (0.1< < 4000) an ( < 6000) respectively. Again, the equation (12) was also use y other researcher for spherical ule [16]. It can e seen from Fig. 5 that the eviation of the experimental C was initially higher in comparison with the equations (10), (11), (12) an (13) ut this eviation appeare to e less with the increase in. For = 3200 an higher, it is seen that the values of the experimental C an the preicte C from these equations are nearly constant. This phenomenon is also oserve in Fig.6 for polyacrylamie solution at = 4000 an higher. C Experimental C By equation 10 By equation 11 By equation 12 By equation Fig. 6 Drag coefficients vs. ynols numer for rising air ule in polyacrylamie solution C Experimental C By equation 10 By equation 11 By equation 12 By equation Conclusions The following conclusions can e reache from this stuy: The average ule rise velocity increases with the increase in ule volume for xanthan gum an polyacrylamie solutions. The average ule velocity slightly ecreases with the increase in liqui height for corresponing ule volume. The ule velocity of xanthan gum is foun slightly lower than the velocity in polyacrylamie solution for corresponing ule sizes. The ule velocity was not epenant on the size of the test rig. The relationship etween C - for non- Newtonian power-law fluis showe acceptale results with the availale analytical an experimental stuies of the literature ut it eserves further stuy for a wie range of ynols numer. Fig. 5 Drag coefficients vs. ynols numer for rising air ule in xanthan gum solution. Nomenclature: [m] ule iameter [m] ule height h [m] projecte iameter onto horizontal plane w

6 Proceeings of the 5th IASME / WSEAS International Conference on Flui Mechanics an Aeroynamics, Athens, Greece, August 25-27, eq [m] equivalent sphere iameter µ [Pa.s] apparent viscosity [-] ynols numer C [-] rag coefficient F [N] rag force g [m/s 2 ] acceleration ue to gravity U [m/s] ule rise velocity n [-] power law inex K [Pa.s n /m 2 ] consistency inex Greek letters ρ [kg/m 3 ] ensity ifference etween liqui an air ule [kg/m 3 ] liqui ensity ρ liq ferences: [1] Shosho, C. an Ryan, Micheal, E., An experimental stuy of the motion of long ules in incline tues, Chemical engineering science, 56, 2001, [2] Margaritis A., te Bokkel, D. W. an Karamanev, D. G., Bule Rise Velocities an Drag Coefficients in non-newtonian Polysaccharie solutions, John Wiley & Sons, Inc., [3] Dewsury, K., Karamanev, D. G. an Margaritis, A., Hyroynamic Characteristics of free Rise of Light soli Particles an Gas Bules in Non-Newtonian Liquis., Chemical engineering Science, vol. 54, 1999, pp [4] Dewsury, K., Karamanev, D. G. an Margaritis, A., Rising soli sphere hyroynamics at high ynols numers in non-newtonian fluis, Chemical Engineering Journal, 87, 2002, [5] Dewsury, 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, [6] Miyahara, T. an Yamanaka, S., Mechanics of Motion an Deformation of a single Bule Rising through Quiescent Highly Viscous Newtonian an non-newtonian Meia, Journal of chemical engineering, Japan, Vol. 26, No. 3,1993. [7] Dhole, S. D., Chhara, R. P. an Eswaran, V., Drag of a Spherical Bule Rising in Power Law Fluis at Intermeiate ynols Numers., In. Eng. Chem. s. 46, 2000, [8] Zheng, Li an Yapa, P. D., Buoyant Velocity of Spherical an Non spherical Bules/Droplets, Journal of Hyraulic Engineering, Vol. 126, No.1, [9] 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, [10] Lima-Ochoterena, R. an Zenit, Visualization of the flow aroun a ule moving in a low viscosity liqui, vista Mexicana De Fisica 49 (4), 2003, [11] Miyhara, T. an Takahashi, T., Drag coefficient of a single ule rising through a quiescent liqui., International Chemical Engineering, Vol. 25, No. 1, [12] Chhara, R. P., Bules, Drops, an Particles in Non-Newtonian Fluis, Taylor & Francis Group CRC Press, [13] Clift, R., Grace, J. R an Weer, M. E., Bules, Drops an Particles; Acaemic Press, 1978, repulishe y Dover, [14] Turton, R., an Levenspiel, O., A short note on the rag correlation for spheres, Power Technology, 4, [15] Kulkarni, A. A. an Joshi, J. B., Bule Formation an Bule Rise Velocity in Gas-Liqui Systems: A view, In. Eng. Chem. s., 44, 2005, [16] Zhang, Y. an ese, J. M., The rag force in twoflui moels of gas soli flows, chemical Engineering Science 58, 2003,