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Cavity Formation, Growth, and Dispersion behind Rotating Impeller Blades Geoffrey M. Evans,* Shaun A. Manning, and Graeme J. Jameson School of Engineering, The University of Newcastle, Callaghan, New South Wales 2308, Australia
Ventilated gas cavities were observed when air was sparged directly to the rear of one blade of a multi-bladed disc-turbine impeller. For a constant gas rate, it was found that for low impeller speeds no cavities were formed. As the impeller speed was increased, large, tapered cavities were formed and increased in length with increasing impeller speed. At a critical speed, however, the cavities reached a maximum length, beyond which the cavity length decreased with increasing impeller speed. For each gas ventilation rate, it was found that the internal pressure of the cavity decreased only slightly for speeds up to the corresponding critical value. Beyond this point the internal pressure decreased significantly with further increases in impeller speeds. A gas flow model was developed based on the assumption that the gas exited the cavity through an opening (orifice) located randomly on the gas-liquid interface. Introduction When gas is introduced into a mechanically agitated vessel, it is drawn into the low pressure regions at the trailing faces of the impeller blades. At certain combinations of gas ventilation rate and impeller speed, the concentration of accumulated gas is sufficiently high to form gas cavities. As gas continuously enters and exits a cavity, by a not well-understood mechanism, the cavity is said to be ventilated. A number of studies have shown that multi-bladed impellers tend to support ventilated cavities of different shapes on different blades.1-4 The different shapes are believed to arise from an uneven distribution of gas between blades due to local variations in hydrodynamic conditions and pressure distributions in the flow field. In an attempt to identify the best operating conditions for efficient gas dispersion and aeration, the present study examines features and formation conditions of individual ventilated cavities. The experimental scheme involved sparging gas directly to one blade of a multibladed impeller. Simultaneous gas injection and pressure readings allowed cavity formation and cavity characteristics to be analyzed over a wide range of operating conditions. In this study, gas ventilation rates are reported in dimensional form, principally because the dimensionless gas flow number is known to be scale dependent.5 This study is restricted to large cavities as defined elsewhere.6 Experimental Section The main experimental apparatus7 is shown in Figure 1. It consisted of a square, flat-bottomed Perspex vessel with sides 0.58 m in length, equipped with four fulllength wall baffles of width 0.058 m. A standard discturbine8 of diameter 0.185 m was located 0.185 m from the base of the vessel. The blades were rectangular of height 37 mm, width 46 mm, and thickness 3 mm. The impeller was driven by a 1.5 kW three-phase induction * To whom correspondence should be addressed. Tel: +61 2 49 215897. Fax: +61 2 49 216920. E-mail: geoffrey.evans@ newcastle.edu.au.
Figure 1. Experimental apparatus: 1, support bearing; 2, load cell; 3, motor; 4, sealed bearing; 5, hollow shaft; 6, impeller; 7, bearing; 8, plastic tubing; 9, pressure transmitter; 10, chart recorder; 11, rotameter; 12, compressor; 13, inclined mirror; 14, video camera.
motor mounted above the tank on a low friction bearing. A 0.5 N load cell connected to the motor by a lever arm provided accurate torque measurements. A small number of tests were performed using a geometrically scaled larger experimental rig. It was also a baffled square, flat-bottomed Perspex vessel with sides of length 1.2 m. The blades were rectangular of height 0.8 m, width 0.1 m, and thickness 6 mm. The liquid depth for both tanks was equal to the length of the sides. The working fluids for all experiments were filtered tap water and air. Both 2- and 6-bladed disc-turbines were used. Air was supplied to the rear of one of the blades through a copper tube, 3 mm in diameter, attached to the shaft at a point just above the disk. Air entered the top of the hollow shaft through a low-friction bearing, leaving through the copper tube whose open-end was positioned at the center rear of one of the blades. The air ventilation rate was measured using a rotameter, enabling the sparging rate to an individual cavity to be metered. By sparging air to only one of the blades, a relatively small number of bubbles were produced, which enhanced visual observations of the cavities. The sizes and shapes of gas cavities were determined visually by placing a mirror at 45° beneath the clear
10.1021/ie0491600 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/25/2005
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Figure 2. Dimensionless cavity length versus impeller tip speed for 2-bladed disc-turbine at different gas ventilation rates.
Figure 3. Dimensionless cavity length versus impeller tip speed for 2- and 6-bladed disc-turbines.
bottom of the vessel. A high-speed video (NAC model HSV-400 color high speed video, with a frame speed capability of up to 400 fps) was then directly focused on the image on the mirror. This allowed direct observation of the cavity structure, including the shape and mean length, which is reported as the dimensionless cavity length (L) defined as the ratio of (cavity length)/ (blade height). A line connected to a differential pressure cell joined the gas line supplying air to the impeller shaft. Simultaneous air injection and pressure measurements were made. Cavity Length. The length of large cavities is known to be a complex function of impeller tip speed and gas ventilation rate.6,9-15 In the present study, the relationships between cavity length, ventilation rate, and impeller tip speed were examined for a single blade on a 2- and 6-bladed disc-turbine impeller. Preliminary experiments revealed some hysteresis in the cavity length data when cavities were formed by increasing and then decreasing impeller speeds at fixed gas ventilation rates. This potential source of uncertainty in measuring the cavity length was eliminated by first establishing the ventilation rate and increasing the impeller speed from zero to the required value. During the measurements the cavities were observed to fluctuate in length, which was consistent with published observations.16 For this reason, the characteristic cavity lengths were obtained by continuously monitoring the length of each cavity for a period of approximately 30 s and recording an average value. In Figure 2, the dimensionless cavity length (L) has been plotted as a function of impeller tip speed (vtip) for three different gas ventilation rates (QG). It can be seen that for all ventilation rates studied cavities did not form when the impeller tip speed was below about 1 m/s, corresponding to a Froude number of about 0.055. This observation was in good agreement with the results from previous studies.17 Beyond tip speeds of about 1 m/s, the dimensionless cavity length was found to increase with increasing tip speed, until a critical speed (vc) was reached, beyond which the cavity length decreased with further increases in tip speed. It can be seen that both the critical speed and the corresponding dimensionless cavity length increased with increasing gas ventilation rate. However, for the highest ventilation rate the curve reached a plateau at a dimensionless
cavity length of about 8, which approximately corresponded to the spacing between the two blades. The effect of decreasing the blade spacing on cavity behavior is shown in Figure 3, where the dimensionless cavity length has been plotted as a function of impeller tip speed for both 2- and 6-bladed impellers for the same gas ventilation rate. It can be seen that for the 6-bladed impeller a maximum dimensionless cavity length of approximately 2 was reached, which corresponded to the spacing between adjacent blades. It was also found that, at lower tip speeds when the cavity was not being strongly influenced by the presence of the trailing blade, the increase in cavity length was similar for both impellers. Moreover, the critical speed was similar in both cases. The results presented in Figures 2 and 3 are consistent with earlier studies, where the gas was introduced via a ring or porous sparger placed below the rotating impeller. For example, it has been reported19 that large cavities reached a maximum length, which increased for higher gas ventilation rates. It was also found that once the maximum cavity length was reached the ventilation rate could be increased considerably before the impeller became flooded. Similarly, it has been found18 that when using a 6-bladed impeller and a constant gas ventilation rate, the cavity length increased with increasing impeller speed until a maximum length was attained. For further increases in impeller speed, the cavity length decreased and smaller bubbles were generated resulting in a uniform dispersion throughout the vessel. Cavity Shape. Visual observations using the highspeed video camera revealed that the length and shape of cavities were essentially independent of the liquid depth, which was consistent with the observations of others19 on the flooding-loading transition for discturbines. In the region where the cavity length increased with increasing impeller tip speed, a tapered cavity developed as indicated in Figure 4. Beyond the critical tip speed a blunt cavity was developed that was similar in appearance to so-called ragged cavities18 when the impeller became flooded. The difference, however, was that the blunt cavities were observed to form from large cavities. Also the blunt cavities generated fine bubbles and a uniform gasliquid dispersion, unlike that generated by ragged
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assumed that at the cavity interface the liquid was travelling at approximately the impeller tip speed, while the gas velocity was simply the superficial velocity of the gas, defined as the volumetric gas ventilation rate (QG) divided by the cross-sectional area of the impeller blade (AB). Since both the liquid and the gas are travelling in the same direction, the critical impeller tip speed at which Kelvin-Helmholtz instabilities occur is given by Figure 4. Change in cavity shape for increasing impeller tip speed.
cavities present with flooded impellers. The reason for the formation of blunt cavities beyond the critical tip speed was not clear. It is possible that the shape is the result of erosion resulting from the liquid flow generated across the front surface of the trailing blade. However, this is unlikely to be the sole cause since at low gas ventilation rates the maximum cavity length was found to be considerably less than the spacing between the blades. The high-speed video images revealed that at tip speeds below the critical speed the wake region behind the tapered cavities produced little local mixing and bubbles of several millimeters in diameter, which were able to escape the low pressure wake region, rising with strong vertical motion. However, beyond the critical impeller speed the wake region behind the ventilated cavity was vigorously mixed, and fine bubbles, of order 1 mm in diameter, were generated and discharged radially from the impeller. In terms of operational performance, it would be desirable to operate at speeds beyond the critical impeller tip speed. Critical Impeller Tip Speed. To be able to predict the critical impeller tip speed, it is necessary to understand the underlying phenomena that control the behavior of ventilated cavities in both the tapered and the blunt regimes as well as the mechanism responsible for the transition from one cavity type to another. First, for the tapered cavity regime the pressure in the wake, as defined by a separating streamline, is less than that of the surrounding liquid. As the impeller speed is increased, the pressure inside the wake decreases, causing the ventilated cavity to expand in a manner which follows the separating streamline. Consequently, the length of the cavity increases with increasing impeller speed. However, a point is reached whereby the reduction in pressure inside the wake begins to be influenced by higher pressure in the fluid in front of the following impeller blade. The relative increase in pressure in the fluid prevents the cavity from increasing in length. It also causes the cavity shape to change from being tapered to becoming blunt. In some instances, it was observed that the transition took place at lengths well before the rear of the cavity approached the following blade. Under these conditions, it would be unlikely that the cavity growth was being strongly influenced by the pressure field of the following blade. The transition in cavity types may have been due to the propagation of instabilities at the interface resulting from the relative motion between the two phases. The possibility of a Kelvin-Helmholtz instability20 was considered. The prediction of Kelvin-Helmholtz instabilities requires a knowledge of the relative velocity between the two fluids. It was not possible to measure the relative velocity between the liquid and gas phases adjacent to the walls of ventilated gas cavities attached to impeller blades. To test the theory, however, it was
vc ) 6.4 +
( ) (QG)c AB
(1)
where the value of 6.4 m/s is the original relative velocity value reported by Lord Kelvin. Equation 1 predicts critical impeller tip speeds in the range 6.4-7 m/s, which is much higher than the experimentally observed values of around 2-4 m/s. Moreover, the theory does not predict the apparent sensitivity of the transition liquid velocity to the gas ventilation rate. The result is hardly surprising given the complexity of the flow and higher levels of turbulence in the mechanically agitated system when compared with the original model assumptions for the Kelvin-Helmholtz theory. The discrepancy is typical of applications of Kelvin-Helmholtz instability to real problems and has been attributed to surface tension and viscous effects, airflow separations,20,21 and the difficulty in accurately determining the relative velocity between the two phases.22 Even after taking these factors into consideration, it was still considered unlikely that the transition in cavity behavior was due to the generation of Kelvin-Helmholtz instabilities on the gas-liquid interface. An alternative approach to determining relationships between the critical impeller tip speed and the gas ventilation rate is demonstrated in Figure 5a, where the gas ventilation rate has been plotted as a function of Dvc2 for both the small and large vessels. It can be seen that the data for both vessels approximately follow a linear relationship and intercept the horizontal axis at a value of 0.52, which correspond to critical impeller tip speeds of 1.68 and 1.14 m/s for the small and large vessels, respectively. These values represent the minimum tip speed required for the transition from tapered to blunt cavities at zero gas ventilation rate into the vessel. At nonzero ventilation rates, higher critical impeller tip speeds are needed for the transition to take place. The data in Figure 5a can be re-expressed, after manipulation, in terms of the dimensionless gas ventilation rate QG/D2vc as a function of the difference between the critical impeller rotational speed, Nc () vc2D/πD2vc), and the critical impeller speed at zero gas ventilation rate, NQG)0 () vc2D/πD2vc)QG)0, as shown in Figure 5b. It can be seen that the data for both the small and large vessels collapse onto a single line passing through the origin. This result would suggest that, for impeller speeds greater than Nc, the critical impeller speed corresponding to the transition from tapered to blunt cavities is a linearly related to the gas ventilation rate. Previous studies have reported the effect of Reynolds number on the change in wake behavior. For solid bluff objects20 in a single liquid phase, the critical Reynolds number (Rec) is typically in the range Rec > 1000. For mechanically agitated sparged systems, numerous studies6,23-28 have found that the power draw, which is
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Figure 5. Gas ventilation rate versus impeller speed.
known to be influenced by cavity shape, is a function of liquid physical properties, such as density and viscosity. Experiments were carried out in the small vessel to determine the influence of absolute viscosity on the critical impeller tip speed. It was found that the critical impeller tip speed increased with increasing viscosity, consistent with previous work..28 The reason for this dependence was thought to be due to the increase in the velocity of the fluid ahead of the rotating blades of the impeller due to the increased viscous fluid drag. Consequently, for a given rotational speed, the velocity of the moving blade relative to the fluid would be less. Therefore, a higher impeller tip speed would be required in order to maintain the same critical impeller tip speed at higher liquid viscosities. The influence of surface tension, in the range of 51-72 mN/m, on the critical impeller tip speed was also examined and found to have no effect. Gas Ventilation Rates. The analysis thus far has focused on determining the critical impeller tip speed, which marks the transition from tapered to blunt cavities for a given system and gas ventilation rate. The reason it is important to be able to predict this transition is because one would ideally want to operate the vessel in the blunt cavity regime in order to obtain fine bubbles and gas dispersion. Moreover, it would be very useful to be able to predict the gas ventilation rate as a function of impeller speed, particularly for the blunt cavity regime. It is assumed that the gas exits the rear of the ventilated cavity through an opening (orifice), which satisfies the orifice flow equation, that is:
QG ) CoAo
x
2(Pcavity - PL) FG
(2)
where Co is the orifice discharge coefficient, Ao is the cross-sectional area of the orifice, Pcavity is the pressure inside the cavity, PL is the pressure in the liquid at the orifice exit, and FG is the density of the gas. Assume that the (gauge) pressure in the liquid is given by
PL ) FLgH - 1/2FLvt2
(3)
where H is the submersion depth of the impeller and vt is the impeller tip speed. The corresponding (gauge) pressure inside the cavity is plotted as a function of impeller tip speed in Figure 6. It can be seen that two distinct regions of behavior were observed. First, at
Figure 6. Cavity internal (gauge) pressure versus impeller tip speed for 2-bladed disc-turbine and different gas ventilation rates.
relatively low tip speeds, the cavity pressure differed only marginally from the static pressure in the liquid. However, once the critical tip speed for each gas ventilation rate was reached, the internal cavity pressure decreased rapidly in response to further increases in the tip speed. The data in Figure 6 can be used in conjunction with eq 3 to determine whether the gas ventilation rate can be reasonably modeled by eq 2. To test this model, it was further assumed that the orifice was circular and that the discharge coefficient was unity. On this basis, an orifice (equivalent circular) diameter was calculated for each of the measured cavity internal pressure values, using the expression:
Do )
x x 4QG πCo
FG
2(Pcavity - PL)
(4)
The calculated orifice diameter has been plotted as a function of the difference between the impeller tip speed and the critical impeller tip speed for each gas ventilation rate (vt - vc) in Figure 7. It can be seen that for impeller tip speeds below the critical value the calculated diameter decreased with increasing tip speed and was different for each gas ventilation rate. However, once the critical impeller tip speed was reached, the data collapsed onto a single line corresponding to an orifice diameter of 2.1 mm, which is approximately the same size as the bubbles produced at the rear of the cavity. Although there is some doubt regarding the absolute values of the orifice diameter shown in Figure 7, given
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Figure 7. Calculated (eq 4) orifice diameter versus impeller tip speed for 2-bladed disc-turbine and different gas ventilation rates.
dent of the number of blades and the depth of submersion of the impeller. For each gas ventilation rate, it was found that the internal pressure of the cavity decreased only slightly for speeds up to the corresponding critical value. Even though the critical impeller speed was different for each gas ventilation rate, the pressure inside the cavity was the same at this point, and beyond which the internal pressure decreased significantly with further increases in impeller speeds. Based on the assumption that the gas exited the cavity through an opening (orifice) located randomly on the gas-liquid interface, it was found that an orifice equation gave good agreement with the data. For the gas ventilation rates tested, the equivalent diameter of the orifice was found to be 2.2 mm, based on the assumption of an orifice discharge coefficient of unity. This diameter was consistent with the general observation that the blunt cavities produced relatively small bubbles of a few millimeters in diameter and relatively good gas dispersions. Nomenclature A ) area, m2 C ) orifice discharge coefficient, D ) impeller diameter, m g ) acceleration due to gravity, m/s2 H ) submersion depth of impeller, m L ) cavity length/blade height, F ) density, kg/m3 N ) impeller speed, rps P ) pressure (gauge), Pa Q ) volumetric ventilation rate, m3/s v ) velocity, m/s
Figure 8. Pressure difference versus impeller tip speed for 2-bladed disc-turbine and different gas ventilation rates.
the assumptions made in calculating the pressure in the liquid and the discharge coefficient, the trends in the data could possibly be used to explain the difference in gas dispersion characteristics between tapered and blunt cavities. The larger bubbles and poor dispersion characteristics in the tapered cavity regime could be due to the larger discharge orifice diameter at the rear of the cavity when compared with those in the blunt cavity regime. At this stage it is not clear why the data should collapse onto the same orifice diameter for the blunt cavity regime. In Figure 7 it was shown that the velocity difference (vt - vc) could be used to collapse data for different gas ventilation rates onto the same curve. The same approach has been used in Figure 8, where the pressure driving force (Pcavity - PL) based on using vt - vc for the velocity term in eq 3 has been plotted as a function of the velocity difference. It can be seen that the pressure data for all three ventilation rates collapses onto a single curve. Conclusions The experimental study found that for a given gas ventilation rate and low impeller tip speeds tapered cavities were formed and increased in length with increasing impeller speed. At a critical speed, however, the cavities reached a maximum length, beyond which the cavity length decreased with increasing impeller speed. It was found that the critical impeller speed marking this transition increased with increasing gas ventilation rate and liquid viscosity but was indepen-
Subscripts and Superscripts b ) blade t ) blade tip cavity ) cavity c ) critical G ) gas h ) hydrostatic L ) liquid o ) orifice
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(20) Birkhoff, G.; Zarantonello, E. H. Jets, Wakes and Cavities; Academic Press Inc.: New York, 1957. (21) Kellog, O. D. Foundations of Potential Theory; Springer: Berlin, 1930. (22) Proudman, J. Dynamical Oceanography; John Wiley & Sons Inc.: New York, 1953. (23) Paca, J.; Ettler, P.; Gregr, V. Hydrodynamic behavior and oxygen transfer fermenter rate in a pilot plant. J. Appl. Chem. Biotechnol. 1976, 26, 309. (24) Ranade, V. R.; Ulbrecht, J. Gas dispersion in agitated viscous inelastic and viscoelastic liquids. Second European Conference on Mixing; Cambridge, U.K., 1977; F6-83. (25) Kipke, K. Gas dispersion in non-Newtonian liquids. International Symposium on Mixing; Mons, France, 1978; C5. (26) Hocker, H.; Langer, G.; Werner, U. Power consumption of stirrers in non-Newtonian liquids. Chem.-Ing.-Tech. 1980, 52, 916. (27) Desplanches, H.; Essayem, F.; Chevalier, J. L.; Bruxelmane, M.; Delvosalle, C. Hydrodynamics and heat transfer in agitated, aerated, Newtonian and highly viscous liquids, in the transitional flow regime. Sixth European Conference on Mixing; Pavia, Italy, 1988; p 479. (28) Ozcan, N. G.; Decloux, M.; Bruxelmane, M. Effect of viscosity on cavity formation and power characteristics of an agitated aerated Newtonian fluid. Trans. Inst. Chem. Eng. 1990, 68, 63.
Received for review September 3, 2004 Revised manuscript received February 13, 2005 Accepted February 18, 2005 IE0491600
