Ind. Eng. Chem. Res. 2008, 47, 9767–9772
9767
An Empirical Correlation of Drag Coefficient for a Single Bubble Rising in Non-Newtonian Liquids Li Zhang,† Chao Yang,†,‡ and Zai-Sha Mao*,† Key Laboratory of Green Process and Engineering, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China, and Jiangsu Institute of Marine Resource Exploitation, Lianyungang, Jiangsu 222005, China
The motion of a single bubble rising unsteadily and steadily in a quiescent non-Newtonian liquid was investigated experimentally. By using a charged coupled device camera to follow the rising bubble, the sequences of the recorded frames were digitized and analyzed using image analysis software, and the measurements of the acceleration and steady motion of bubbles were obtained. Using the experimental data, we proposed an empirical correlation to predict the total drag coefficient calculated from the accelerating motion to the steady motion with the added mass force and history force included. This correlation is an extension of our previous work with non-Newtonian fluids. This new correlation represents very well the experimental data of the total drag force in a wide range covering both unsteady accelerating motion and steady motion in non-Newtonian fluids. 1. Introduction The occurrence of bubbles in liquids in chemical, biochemical, and processing applications is ubiquitous. Typical examples include bubble column reactor, extraction column, fermentation, mineral processing, foaming, polymerization, sewage sludge treatment, and so forth. In all these above-mentioned applications, bubble swarms rather than single bubbles are often encountered, but the understanding of a single bubble hydrodynamics can provide important insights and often serve as a useful starting point to undertake the modeling of applications involving bubble swarms. For a given type of equipment, the knowledge of free rise bubble velocity permits the estimation of the mean residence time of gas phase, or conversely, if the desired mean residence time is known from other considerations (such as the rate of reaction, etc.), it helps to size the process equipment. This gave the motivation for intensive research in this particular field, which has been the subject of several reviews and books.1-3 As compared to Newtonian fluids, because of the nonlinear viscosity terms, it is not possible to obtain a full picture about the motion of gas bubbles rising freely in a clean non-Newtonian fluid even when the nonlinear inertial terms are neglected. In the past, several efforts have been made to develop the standard drag curve for single bubbles rising in power-law fluids. Because of the role of surface active agents, wall effect, and deviation from spherical shape, experimental work in this field is also much more challenging than that for rigid particles. It is thus not surprising that the experimental results reported by different investigators seldom agree with each other. Despite these inherent difficulties, some efforts have been made to develop universally applicable drag correlations.4-8 The previous investigations were mainly on the bubble terminal velocity in steady motion, drag coefficient, and/or bubble shape by empirical, semiempirical, and theoretical approaches. However, relatively fewer attempts were made on the unsteady forces on bubbles or particles in Newtonian * Corresponding author. Tel.: +86-10-62554558. Fax: +86-1062561822. E-mail:
[email protected]. † Chinese Academy of Sciences. ‡ Jiangsu Institute of Marine Resource Exploitation.
fluids9-15 and particles in viscoelastic fluids16-21 and shearthinning fluids.22,23 On the basis of the preceding discussion, it is safe to conclude that there is a lack of knowledge about the accelerating motion of a bubble in shear-thinning fluids. The added mass force is important for a bubble in unsteady motion when it accelerates or decelerates (e.g., near the nozzle or in a turbulent flow with swirl). Without the added mass force included in mathematical formulations, the bubble acceleration reaches an unrealistic value because of the relatively strong buoyant forces and the small mass of the bubbles.24,25 The history force (also called the Basset force) represents the fading memory effect of the relative acceleration between the bubble and the surrounding liquid. The history force formulation involves a history integral which makes its estimation computationally intensive, and it is usually neglected for the liquid of low viscosity26 because it is difficult to measure directly the added mass force and the history force exerted on a bubble and there exists no general drag coefficient correlation suitable for bubbles in both accelerating and steady motions in nonNewtonian liquids with the added mass force and the history force considered. In this article, consideration is given to the transient motion of a bubble accelerating under the influence of gravity in nonNewtonian liquids. A force balance equation was used for analyzing the force exerted on a bubble rising rectilinearly. The contribution of drag force, added mass force, and history force retarding the bubble motion was accounted for as a total drag force. A correlation for the total drag coefficient of single bubbles undergoing acceleration or moving steadily through a stagnant non-Newtonian liquid was proposed. This correlation could represent the present experimental data with good accuracy. 2. Experimental Section 2.1. Apparatus. Experiments were conducted at room temperature under atmospheric pressure in a way similar to that in previous work.15 A Lucite column was constructed with a square cross section (side length 0.21 m) and 0.6-m high (Figure 1), which was thought wide enough to minimize the wall effect.2 The general experimental procedure is briefly described as
10.1021/ie8010319 CCC: $40.75 2008 American Chemical Society Published on Web 11/08/2008
9768 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008
Figure 1. Schematic diagram of the experimental setup.
follows. With a precision syringe pump, air at room temperature was injected at a low rate through a nozzle into the column and single bubbles were formed and released successively at the stainless steel nozzle with a flat opening. The nozzle was located 3 cm above the center of the bottom. The size of single bubbles was mainly decided by the nozzle inner diameter. The time interval between successive bubbles was sufficiently long (>100 s) to minimize the mutual influence between them. Since the single bubbles in the present experiment were roughly spherical and axisymmetric, rising rectilinearly, the image of rising bubbles was recorded using a single high-speed charged coupled device camera with the shutter speed variable between 1/60 and 1/100 000 s. The instantaneous positions of a bubble were read out from each frame of images with the resolution of 752 × 582 pixels against a precision scale along the column axis. Aqueous solutions of four different polysaccharides were used in this study: aqueous carboxymethyl cellulose (CMC) sodium salt, sodium polyacrylate (PAAS), xanthan gum, and hydroxylethyl cellulose (HEC). All were of analytical purity, available on the market. CMC: type 7MF, supplied by Sinopharm Chemical Reagent Co., nominal viscosity (25 °C, 20 g/L aqueous) 300-800 mP · s. Xanthan gum: type KX-F supplied by Kaijin Bio-Tech Co., nominal viscosity (25 °C, 1% aqueous) 1200-1600 mP · s. HEC: type QP15000H supplied by Beijing Sinmaya Chemicals Co., nominal viscosity (25 °C, 1% aqueous) 1100-1500 mP · s. PAAS: type G, Mw 4-5 MD, Changzhou Board Chemical Co., nominal viscosity (30 °C, 0.2% aqueous) 540-760 mP · s. The solutions were prepared by adding weighed amounts of polysaccharide to distilled water with very low conductivity of 0.2 µS/cm. The homogenization was assured by mechanical mixing. Solutions were maintained at 21 ( 1 °C during the experiments. Small (100 mL) samples were used for the study of rheological and physical properties. The liquid viscosities were measured using a Brookfield viscometer (DV-III) at experimental temperature. The surface tension of each mixture was measured with a tensiometer (the du Nouy Ring method). The liquid densities were measured using a balance hydrometer, and none were found to be significantly different from water because the concentration of polysaccharide is very low. The surface tension of the solution did not change significantly as a function of polysaccharide concentration within the concentration ranges used in this study and remained close to that of water (72 g/cm3). For the physical properties of the bubbles, Fg ) 1.2 kg/m3 and µg ) 1.8 × 10- 5 Pa · s. 2.2. Bubble Velocity and Diameter. The video images were analyzed frame-by-frame on the monitor screen. The increment between successive positions xi of a bubble was measured
against the calibrating scale. The local velocity ub of the bubble at a given vertical distance x ) (xi + xi-1)/2 from the nozzle tip was calculated by ub )
xi - xi-1 ∆t
(1)
where xi is the coordinate of the successive positions of a bubble because the bubbles rose rectilinearly, and ∆t is the time interval between the two successive pictures. The terminal velocity of a bubble, UT, was calculated as the mean value of instantaneous velocities when bubble velocity had approached its asymptotic value. The data of ub(t) was smoothed with a second-order polynomial to get rid of random measurement error, so that a more accurate estimate of the acceleration a ) dub/dt could be retrieved. The equivalent diameter de was calculated from the volumetric flowrate of air V and the bubble frequency Nb by counting the bubbles generated in a time interval: 3
de )
6V πNb
(2)
2.3. Total Drag Coefficient. The rectilinear motion of a bubble is described by Newton’s second law: mb
dub ) FB - (FD + FA + FH) dt
(3)
where mb ) FgVb represents the mass of a bubble with density Fg and volume Vb. The forces on the right are buoyancy FB, drag force FD, added mass force FA, and history force FH, respectively. The drag force, added mass force, and history force are summed as a total drag force, and the total drag coefficient CDA is defined as: dub FD + FA + FH 4 (Fl - Fg)g - Fg dt CDA ) ) de (4) 1 1 3 Flub2 Flub2 πde2 2 4 When the bubble rises steadily (dub/dt ) 0), the conventional drag coefficient CD results:
(
)
4 (Fl - Fg)g CD ) de · 3 FU 2 l
(5)
T
Via dimensional analysis, it can be shown that the drag coefficient is a function of the Reynolds number and the nonNewtonian parameters. The latter depends on the choice of a specific rheological model.
Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9769 Table 1. Rheological and Physical Properties of Different Solutions solution CMC
xanthan gum HEC PAAS a
CW (wt %)
µ0 (mPa · s)
λ (s)
s (-)
ε (%)a
K (mPa · sn)
n (-)
ε (%)a
0.25 0.5 0.75 1.0 0.036 0.06 0.3 0.5 0.01 0.03
13.17 34.76 57.93 122.02 25.65 115.49 47.0 315.9 17.94 162.73
93.8 32.08 0.376 0.207 0.505 0.984 0.062 0.261 1.36 0.522
0.98 0.96 0.95 0.90 0.63 0.49 0.79 0.68 0.53 0.37
0.5 0.51 1.18 0.94 1.13 1.21 1.83 0.84 2.73 2.53
11.8 29.5 61.9 138.3 30.1 112.6 63.5 376.0 51.8 218.2
0.99 0.95 0.93 0.90 0.65 0.50 0.86 0.77 0.52 0.41
1.06 0.9 4.2 3.12 5.01 5.56 3.47 2.89 10.9 12.48
ε: Relative fitting errors.
Figure 2. Viscosity as a function of the shear rate of non-Newtonian solutions (symbols represent measurements, whereas the lines for the prediction are based on the Carreau equation).
3. Results and Discussion 3.1. Rheological Characteristics of Test Fluids. In most cases, it was found that the three-parameter Carreau viscosity equation27 provided an adequate representation of the shear viscosity of the fluids of interest. The three-parameter Carreau viscosity equation is written as: µ(γ˙ ) ) µ0[1 + (λγ˙ )2](s-1)⁄2
Figure 3. Drag coefficients of bubbles versus Reynolds number for steady motion.
formulation of the Reynolds number should satisfy the following requirements:28 (1) the shear-thinning effect is reflected in the Reynolds number, and (2) the Reynolds number of a shearthinning fluid is identical to one for a Newtonian fluid if the shear-thinning effect vanishes. They defined ReM by introducing 2UT/de as a representative shear rate over the system:
(6)
where µ0 is the zero-shear viscosity, λ is a characteristic time of the fluid, and s is the slope of the shear stress-shear rate curve in the shear-thinning region. This model reduces to the Newtonian relation and describes inelastic, as well as viscoelastic, behavior depending on the magnitude of the parameter λ. Note that a Newtonian fluid is a special case of the present Carreau fluid with s ) 1 or λ ) 0. It is also worthwhile to note here that the Carreau viscosity equations contain the powerlaw model as a special case. For instance, for (λ˙ γ˙ )2 . 1, eq 6 reduces to µ ) (µ0λs-1)γ˙ s-1, which is identical to the powerlaw fluid model with K ) µ0λs-1. The resulting values of the parameters (µ0, λ, s) along with the other physical properties are listed in Table 1, where a wide range of rheological conditions are seen to be encompassed. For comparison, the corresponding values of the power-law parameters are also presented. Apparently, the Carreau model gives smaller relative fitting errors (ε). Figure 2 shows the representative apparent viscosity-shear rate data for some test fluids. The predictions of the model are plotted as solid lines in Figure 2 for four of the test fluids. An excellent fit is obtained. 3.2. Steady Drag Coefficient. Since the fluid viscosity varies as a function of the shear rate, the first problem in developing the drag curve is the definition of Reynolds number since it contains both fluid viscosity and bubble velocity. The ideal
ReM )
FldeUT µ0 ⁄ {1 + [λ(2UT ⁄ de)] }
2 (1-s)⁄2
)
FldeUT Cµ0
(7)
with C )
1 {1 + [λ(2UT ⁄ de)]2}(1-s)⁄2
(8)
Cµ0 means the effective viscosity over the system. If C ) 1.0, the bubble motion can be regarded as a Newtonian flow case. The drag coefficients versus Reynolds numbers ReM of bubble rise steadily in non-Newtonian solutions are presented in Figure 3. Figure 3 clearly shows the PAAS solutions’ lack of agreement with other test solutions. The PAAS solutions showed shearthinning and viscoelastic behavior while the xanthan gum, HEC, and CMC solutions displayed shear-thinning behavior only. Figure 3 suggests an additional dependence of CD on the nonNewtonian characteristics of the fluids. A modified Reynolds number based on the zero shear viscosity (Re0), De, and s is introduced:29 ReC ) Re0(1 + Des-1)
(9)
The Deborah number, De, and the Reynolds number, Re0, based on the zero shear viscosity, are given by De ) λUT/de and Re0 ) FlUTde/µ0.
9770 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008
Figure 4. Velocity number versus flow number curve using RePL (s: Rodrigue, 2002; 9: Experimental data).
Figure 5. Velocity number versus flow number curve using ReMC (s: Rodrigue, 2002; 9: Experimental data).
Therefore, the Reynolds numbers are calculated using eq 7 for xanthan gum, HEC, and CMC solutions, while the PAAS solution uses eq 9: ReMC )
{
ReM for shear-thinning liquids (10) ReC for shear-thinning viscoelastic liquids
Rodrigue7 collected the literature data and extended his previous model for predicting the rise velocities of gas bubbles in purely Newtonian liquids to bubbles in power-law liquids in terms of 1 F 12 V ) (1 + 0.018)3⁄4 where the new coordinates V and F are defined as V ) (ReWe)1⁄3 and
(
(11)
(12)
)
Bo3Re4 1⁄3 F ) (13) We2 where de is the equivalent bubble diameter. The Reynolds number is calculated on the basis of the power-law model, namely, FlUT2-nden (14) K Figures 4 and 5 compare eq 11 with the experimental data using the Reynolds numbers of RePL and ReMC, respectively. It can be seen that the data in Figure 5 fit eq 11 better than those in Figure 4. Figure 6 shows the results calculated using the combination of modified Reynolds number ReMC for the nonNewtonian liquids. On the basis of the 28 experimental data for bubble steady motion, a simple correlation is proposed in the form of: RePL )
CD )
16 (1 + 0.12ReMC0.6) ReMC
(15)
The value of the drag coefficient in eq 15 converges to CD ) 16/Re at very low values of Reynolds number (i.e., reaches the values predicted by the Hadamard-Ribczynky model30 of a bubble with a shear-free interface). It can be seen that eq 15 shows good agreement with the experiments with the average deviation of 15.9%. 3.3. Total Drag Coefficient Correlation. In the presence of acceleration, the total drag coefficient depends additionally
Figure 6. Relationship between CD and ReMC in steady rising.
on acceleration as quantified by the acceleration number Ac proposed by our previous work.15 In this section, only shearthinning non-Newtonian liquids are treated and the PAAS solution is excluded for its viscoelastic behavior. By fitting the measured values of CDA, Ac() dea/ub2), and Ar() de3Fl2g/ [µ(γ˙ )]2) with the least-squares method, the following correlation for the total drag coefficient results on the basis of 246 data (146 of steady motion and 100 in transient motion): CDA )
16 (1 + 0.12ReM0.6) (1 + 0.196Ac0.767Ar0.381) (16) ReM
The average relative deviation between the predicted and measured total drag coefficient is about 25.4%, which is acceptable for the large range of rheological parameters involved in the experiments. The validity of eq 16 to single bubbles in shear-thinning non-Newtonian liquids is also shown by plotting CDA/(1 + 0.196Ac0.767Ar0.381) of various systems against ReM in Figure 7. These data for non-Newtonian liquids cover a range of flow index s from 0.37 to 0.97 corresponding to the Reynolds numbers from 0.15 to 261. In these wide ranges, the present correlation approximates and applies to both accelerating and steady motion of rising bubbles. 4. Conclusions Single bubbles rising in quiescent non-Newtonian liquids were investigated experimentally. When the contributions of added-
Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9771
Figure 7. Correlation of transient CDA against ReM.
mass force, drag force, and history force acting on a single bubble were combined, the total drag coefficient was determined using Newton’s second law from the experimental measurements of unsteady motion of rising single bubbles released from a nozzle. On the basis of our previous work, a single correlation was proposed to predict the total drag coefficient of bubbles in accelerating and steady motions: CDA )
16 (1 + 0.12ReM0.6) (1 + 0.196Ac0.767Ar0.381) ReM
in which the modified Reynolds number is defined as: ReM )
[ ( )]
FldeU 2U 1+ λ µ0 de
2 (1-s)⁄2
The correlation is in good agreement with experimental measurements in clean aqueous systems covering a wide range of liquid properties and bubble sizes. Acknowledgment The financial support from the National Natural Science Foundation of China (Nos. 20236050, 20576133, 20490206), 973 Program (Nos. 2004CB217604, 2007CB613507), and 863 Project (2007AA060904) is gratefully acknowledged. Nomenclature Roman Letters a ) bubble acceleration, m/s2 Ac ) Acceleration number, Ac ) dea/ub2 Ar ) Archimedes number, Ar ) de3Fl2g/[µ(γ˙ )]2 Bo ) Bond number, Bo ) de2Flg/σ CD ) drag coefficient, dimensionless CDA ) total drag coefficient, dimensionless CW ) concentration of solute (mass fraction) de ) volume equivalent bubble diameter, m De ) Deborah number, De ) λUT/de FA ) added mass force, N FB ) buoyancy force, N FD ) drag force, N FH ) history force, N g ) gravitational acceleration, m/s2 K ) power-law consistency coefficient, Pa · sn
mb ) mass of bubble, kg n ) power-law index, dimensionless Nb ) bubble frequency, s-1 ReMC ) modified Reynolds number (eq 10), dimensionless Re0 ) Reynolds number based on zero shear viscosity, Re0 ) FlUTde/µ0 ReC ) modified Reynolds number (eq 9), dimensionless ReM ) Reynolds number based on the Carreau model (eq 7), dimensionless RePL ) Reynolds number based on the power-law model, RePL ) FlUT2-nden/K s ) Carreau model parameter, dimensionless t ) time, s ub ) bubble local velocity, m/s Vb ) volume of bubble, m3 U ) bubble rise velocity, m/s UT ) terminal rise velocity of bubble, m/s V ) volumetric flowrate, cm3/min We ) Weber number, We ) FlUT2de/σ ∆t ) change in time, s Greek Symbols µ ) apparent viscosity, Pa · s µ0 ) zero-shear viscosity, Pa · s λ ) Carreau model parameter, s γ ) shear rate, s-1 F ) density, kg/m3 σ ) surface tension, mN/m Subscripts g ) gas phase l ) liquid phase b ) bubble
Literature Cited (1) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particle; Academic Press: New York, 1978. (2) Kulkarni, A. A.; Joshi, J. B. Bubble formation and bubble rise velocity in gas-liquid system: A review. Ind. Eng. Chem. Res. 2005, 44, 873–5931. (3) Chhabra, R. P. Bubbles, Drops, and Particles in Non-Newtonian Fluids, 2nd ed.; CRC Press: Boca Raton, FL, 2006. (4) Margaritis, A.; Te Bokkel, D. W.; Karamanev, D. G. Bubble rise velocities and drag coefficients in non-Newtonian polysaccharide solutions. Biotechnol. Bioeng. 1999, 64, 257–266. (5) Dewsbury, K.; Karamanev, D. G.; Margaritis, A. Hydrodynamic characteristics of free rise of light solid particles and gas bubbles in nonNewtonian liquids. Chem. Eng. Sci. 1999, 54, 4825–4830. (6) Dziubinski, M.; Orczykowska, M.; Budzynski, P. Average value of shear stress and shear rate at the surface of gas bubble rising in non-Newtonian liquids. Inz. Chem. Procesowa 2002, 23, 341–348. (7) Rodrigue, D. A simple correlation for gas bubbles in power law non-Newtonian liquids. Can. J. Chem. Eng. 2002, 80, 289–292. (8) Dhole, S. D.; Chhabra, R. P.; Eswaran, V. Drag of a spherical bubble rising in power law fluids at intermediate Reynolds numbers. Ind. Eng. Chem. Res. 2007, 46, 939–946. (9) Odar, F.; Hamilton, W. S. Forces on a sphere accelerating in a viscous liquid. J. Fluid Mech. 1964, 18, 302–314. (10) Odar, F. Verification of the proposed equation for calculation of the forces on a sphere accelerating in a viscous fluid. J. Fluid Mech. 1966, 25, 591–592. (11) Lovalenti, P. M.; Brandy, J. F. The force on a bubble, drop, or particle in arbitrary time-dependent motion at small Reynolds number. Phys. Fluids A 1993, 5, 2104–2116. (12) Mei, R.; Klausner, J. F. Unsteady forces on spherical bubbles at finite Reynolds number with small fluctuations in the free-stream velocity. Phys. Fluids A 1992, 4, 63–70.
9772 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 (13) Mei, R.; Klausner, J. F.; Lawrence, C. J. A note on the history force on a spherical at finite Reynolds number. Phys. Fluids 1994, 6, 418– 420. (14) Magnaudet, J.; Eames, I. The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. ReV. Fluid Mech. 2000, 32, 659– 708. (15) Zhang, L.; Yang, C.; Mao, Z.-S. Unsteady motion of a single bubble in highly viscous liquid and empirical correlation of drag coefficient. Chem. Eng. Sci. 2008, 63, 2099–2106. (16) Thomas, R. H.; Walters, K. The unsteady motion of a sphere in an elastico-viscous liquid. Rheol. Acta 1966, 5, 23–27. (17) Frater, K. R. Drag on a sphere oscillating in an elastico-viscous fluid. Z. Angew. Math. Phys. 1967, 18, 798–803. (18) King, M. J.; Walters, N. D. The unsteady motion of a sphere in an elastico-viscous liquid. J. Phys. D: Appl. Phys. 1972, 5, 141–150. (19) Lai, R. Y. S. Drag on a sphere accelerating rectilinearly in a Maxwell fluid. Int. J. Eng. Sci. 1974, 12, 645–655. (20) Lai, R. Y. S. Accelerating motion of a sphere in a Maxwell fluid. Appl. Sci. Res. 1975, 30, 208–220. (21) Gupta, R. K. Unsteady motion of a spheroid in an elastic-viscous liquid. Z. Angew. Math. Phys. 1976, 27, 273–279. (22) Bagchi, A.; Chhabra, R. P. Accelerating motion of spherical particles in power law type non-Newtonian liquids. Powder Technol. 1991, 68, 85– 90.
(23) Chhabra, R. P.; Soares, A. A.; Ferreira, J. M. A numerical study of the accelerating motion of a dense rigid sphere in non-Newtonian powerlaw fluids. Can. J. Chem. Eng. 1998, 76, 1051–1055. (24) Park, W. C.; Klausner, J. F.; Mei, R. Unsteady forces on spherical bubbles. Exp. Fluids 1995, 19, 167–172. (25) van den Hengel, E. I. V.; Deen, N. G.; Kuipers, J. A. M. Application of coalescence and breakup models in a discrete bubble model for bubble columns. Ind. Eng. Chem. Res. 2005, 44, 5233–5245. (26) Buwa, V. V.; Deo, D. S.; Ranade, V. V. Eulerian-Lagrangian simulations of unsteady gas-liquid flows in bubble columns. Int. J. Multiphase Flow 2006, 32, 864–885. (27) Carreau, P. J. Rheological equations from molecular network theories. Trans. Soc. Rheol. 1972, 16, 99–127. (28) Ohta, M.; Iwasaki, E.; Obata, E.; Yoshida, Y. A numerical study of the motion of a spherical drop rising in shear-thinning fluid system. J. Non-Newtonian Fluid Mech. 2003, 116, 95–111. (29) Rodrigue, D.; Chhabra, R. P.; de Kee, D. Drag on non-spherical particles in non-Newtonian fluids. Can. J. Chem. Eng. 1994, 72, 588–593. (30) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962.
ReceiVed for reView July 2, 2008 ReVised manuscript receiVed October 3, 2008 Accepted October 8, 2008 IE8010319
