Velocidad Terminal de subida de burbuja en un liquido newtoniano

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Revista Mexicanade Ingenierıa Quımica

1

Academia Mexicana de Investigacion y Docencia en Ingenierıa Quımica, A.C.

Volumen 11, Numero 2, Agosto 2012

ISSN 1665-2738

1Vol. 11, No. 2 (2012) 269-278

RISING VELOCITY FOR SINGLE BUBBLES IN PURE LIQUIDS

VELOCIDAD DE ASCENSO DE BURBUJAS AISLADAS EN LIQUIDOS PUROSS. Baz-Rodrıguez1∗, A. Aguilar-Corona2 and A. Soria1

1Departamento de Ingenierıa de Procesos e Hidraulica, Universidad Autonoma Metropolitana - Iztapalapa, SanRafael Atlixco 186, Col. Vicentina, Mexico, D. F. CP. 09340, Mexico.

2Facultad de Ingenierıa Mecanica, Universidad Michoacana de San Nicolas de Hidalgo, Francisco J. Mujica s/n,Ciudad Universitaria, CP. 58040, Morelia, Michoacan, Mexico.

Received October 9, 2011; Accepted March 10, 2012.

AbstractAn equation to predict the terminal rise velocity of single bubbles in stagnant Newtonian liquids is proposed. Theformulation combines a force balance obtained from the boundary layer theory for non-distorted bubbles and ananalytic equation coming from a mechanic energy balance. Without including geometric parameters, which aredifficult to assess, it is assumed that the weighting of dominant forces is enough to adequately predict the terminalvelocity in both the intermediate and inertial motion regimes. The proposed equation shows good agreement withexperimental data from bubbles rising in pure liquids. Moreover, for bubbles rising in clean water, the effect ofhelical trajectories was estimated from experimental data trends and included in the formulation as a correctionfactor for the terminal velocity.

Keywords: single bubble motion, terminal velocity equation, oscillatory bubble path, pure liquids.

ResumenSe propuso una ecuacion para predecir la velocidad terminal del ascenso de burbujas aisladas en lıquidosNewtonianos. La formulacion combina un balance de fuerzas obtenido de la teorıa de capa lımite para burbujasesfericas con una ecuacion analıtica proveniente de un balance de energıa mecanica. Sin la inclusion de parametrosgeometricos que son difıciles de determinar, se asume que la ponderacion de las fuerzas dominantes es suficientepara predecir adecuadamente la velocidad terminal en los regımenes de movimiento intermedio e inercial. Laspredicciones de la ecuacion propuesta presentan buen ajuste con datos experimentales para burbujas ascendiendo enlıquidos puros. Ademas, para burbujas ascendiendo en agua pura, se estimo el efecto de las trayectorias helicoidalesa partir de la tendencia de datos experimentales y se incluyo en la formulacion como un factor de correccion de lavelocidad terminal.

Palabras clave: movimiento de burbujas aisladas, ecuacion de velocidad terminal, trayectoria oscilatoria deburbujas, lıquidos puros.

1 Introduction

Motion analysis of a single bubble rising in quiescentliquid is useful to understand and describe gas-liquidbubbly flows. Let us consider a train of bubbles risingunconfined through still liquid. Its rise velocity canbe approximated by that of a single bubble plus the

velocity defect caused by the wakes (Marks, 1973). Ifone is dealing with bubble columns, the slip velocitybetween phases can be expressed in terms of a singlebubble terminal velocity and the gas volume fraction(Shah et al., 1982; Ruzicka et al., 2001). These factsemphasize the importance of having equations thatpredict the terminal rise velocity of single bubbles, sin-

∗Corresponding author. E-mail: [email protected].: +52 55 58044648 Ext. 237

Publicado por la Academia Mexicana de Investigacion y Docencia en Ingenierıa Quımica A.C. 269

Baz-Rodrıguez et al./ Revista Mexicana de Ingenierıa Quımica Vol. 11, No. 2 (2012) 269-278

ce it is a fundamental parameter in gas-liquid flows.The shape and terminal velocity of isolated

bubbles vary as the equivalent diameter increases,according to three regimes (Tomiyama et al., 2002):1) dominant viscosity, 2) intermediate region, whereviscous, surface tension and inertial effects shouldbe taken into account, and 3) dominant inertia. Outof the three regimes, the intermediate one presentsmajor difficulties since the force interactions andtheir effect on the terminal velocity is not yet wellunderstood. Moreover, the terminal velocity isparticularly sensitive to the presence of contaminantsand even to the initial deformation induced byinjection (Tomiyama et al., 2002; Celata et al., 2007).

The intermediate regime spans an equivalentdiameter range from 0.07 to 0.65 cm for air bubblesascending through pure water (Mendelson, 1967;Loth, 2010). This corresponds to particle Reynoldsnumbers (Re) between 80 and 1500 for normalpressure and temperature conditions, approximately.This working interval is usual in many industrialsystems. When Re > 200, bubble shapes are oblateellipsoidal with fore-aft symmetry. This symmetrygradually brakes as Re increases, until the bubblemotion attains the inertial spherical cap regime.Besides, for Re ∼ 650-685, the onset of path instabilityoccurs, and a transition from rectilinear to zig-zagor helical trajectories is observed (Duineveld, 1995;Sanada et al., 2008; Veldhuis et al., 2008).

There exist three basic approaches to establish theterminal velocity (Kulkarni and Joshi, 2005):

• Force balance. The terminal velocity equationsresult from a force balance between drag andbuoyancy. The most representative solutions areobtained from drag in creeping and potentialflows. This approximation can be adequate topredict the rise velocity of small bubbles whenviscosity effects are still dominant.

• Dimensional analysis. Dimensionless groupsare determined from the leading variables thatgovern the bubble motion. A functional relationis proposed for these groups, and the adjustableparameters are fitted from experimental ornumerical data (Rodrigue, 2001).

• Wave analogy. Interfacial disturbances areassumed for bubbles whose dynamics is similarto propagating waves in an ideal fluid, andthen the terminal velocity is estimated as afunction of the equivalent diameter and the fluidproperties (Mendelson, 1967). Even if some

doubt has been cast on the original approachbecause of its lack of physical basis (Lehrer,1976; Fan and Tsuchiya, 1990; Bozzano andDente, 2001), the wave analogy approach canspan some later investigations, which obtainedequations with the same basic form (Lehrer,1976; Zudin, 1995). Predictions using thisapproach fit the main trend after the bubblesize corresponding to the terminal velocity localmaximum is attained (0.14-0.18 cm for air inwater, see Fig. 1).

Other formulations for bubble terminal velocities havebeen proposed from some detailed bubble shape andexternal flow considerations (Moore, 1965; Bozzanoand Dente, 2001; Tomiyama et al., 2002). However,these approaches often require additional data suchas aspect ratio (between minor and major axes)and (hemispherical) bubble distortion factor. Theseparameters are difficult to obtain. Some investigationsreport the aspect ratio from experimental results onpure and contaminated liquids as a function of theEotvos number (Eo) (Wellek et al., 1966; Okawa etal., 2003; Sanada et al., 2008). Sometimes the Webernumber (We) is used instead (Moore, 1965; Wellek etal., 1966), which seems more appropiate to establishfunctional relations with this geometrical parameter(Celata et al., 2007). Nevertheless, expressingthe aspect ratio in terms of We involves its owndependency on the terminal velocity. Besides, forpure liquids it is difficult to estimate aspect ratiosor distortion factors even experimentally (Tomiyamaet al., 2002), in contrast with the case of liquidscontaminated with surfactants, which act by dampingbubble shape oscillations (Celata et al., 2007).

The inclusion of drag coefficients with non-linearRe dependency, or aspect ratio that depends on Wein any terminal velocity formulation implies solvingnon-linear equations, often with fractional exponents.This is inconvenient when the terminal velocity is notthe main objective of an engineering calculation. Forthis reason, a simple equation to predict the isolatedbubble terminal velocity is proposed in the presentwork. If geometrical considerations are avoided, webelieve that the proper weighting of the dominantforces is enough to assess this important parameter forthe intermediate and inertial regimes.

The terminal velocities reported in literaturecorrespond to the axial component of the bubblecenter of mass velocity vector. However, for thecase of air rising through water, it is important totake into account that rectilinear trajectories become

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unstable above some critical Re. For this reason, andconsidering a helical trajectory, a correction factor wasobtained as a function of the mean slope of the helix.This procedure improves the fit to experimental datain the equivalent diameter range that correspond to theoscillatory motion of the bubble.

2 Terminal velocity formulationConsidering the intermediate region, inertial effectsare combined with viscosity and surface tensioneffects. This suggests that an equation for the bubbleterminal velocity, which is valid when viscous effectsare still important, can be combined with another onethat is valid when surface tension effects are significant(Fan and Tsuchiya, 1990; Jamialahmadi et al., 1994).In the former case it is reasonable to propose anequation developed from the force balance approach.In the later it would be sensible to use a wave analogyapproach. Proceeding this way, an equation can beobtained, which is valid for a wide range of equivalentbubble diameters.

The proposed formulation for the terminal velocityis given here by the following combination of thecorresponding effects.

VT =1√

1V2

T1+ 1

V2T2

(1)

where VT1 is the rise velocity when viscous effects arestill important and VT2 is the corresponding velocitywhen surface tension effects are significant. The formof Eq. (1) was suggested by Jamialahmadi et al.(1994) and remains in use (Kulkarni and Joshi, 2005;Cai et al., 2010; Duangsuwan et al., 2011); also, asimilar form but with fitting parameters was previouslyproposed by Fan and Tsuchiya (1990). When viscouseffects are important and the bubble diameter is small,Eq. (1) ensures that the contribution of VT2 isneglected. This is important because the terminalvelocity equations from wave analogy yield valuestoo high below the intermediate velocity maximum,and they are preferably valid until significant surfacetension effects appear. The opposite occurs with VT1 atlarger bubble diameter. In the intermediate region, Eq.(1) is just the terminal rising velocity coupled betweenthe two velocities VT1 and VT2.

Jamialahmadi et al. (1994) proposed the equationdeveloped by Hadamard (1911) and that of Mendelson(1967) for VT1 and VT2, respectively. Nevertheless,Hadamard equation is only valid for Reynolds

numbers smaller than unity (Tomiyama et al., 2002)and Mendelson equation seems not to have anadequate basis to justify its deduction (Lehrer, 1976;Bozzano and Dente, 2001).

When viscous effects are losing their influence, theterminal velocity for a bubble in potential flow can beused (Levich, 1962)

VT pot =1

36∆ρgd2

e

µL(2)

where ∆ρ is the density difference between the liquidand gaseous phases, de is the diameter of a sphericalbubble of equivalent volume, µL is the dynamicviscosity of the liquid and g is the accelerationof gravity. Contrary to Hadamard solution, whichoverestimates the terminal velocity for moderate Re,the potential solution underestimates them. Usingthe governing equations of motion for the boundarylayer on a spherical bubble, Moore (1963) obtaineda correction for the potential solution of drag forcewhich fits particularly well for moderate regimemotion (50 < Re < 200), before the surface tensionand inertial effects start to cause deviations from thespherical bubble shape. Consequently, the boundarylayer solution must be more appropriate when thebubble rise at moderate Re.

Buoyancy and drag as expressed by Moore (1963)gives the following equilibrium:

6πµLdeVT1

1 − 2.21

Re1/21

=16πd3

e ∆ρg (3)

whereRe1 =

deVT1ρL

µL(4)

Rearranging Eq. (3), the solution for the terminalvelocity for potential flow appears

1

V1/2T pot(gde)1/2

V3/2T1 −

V1/2T pot

(gde)1/2 V1/2T1 − 0.36833 = 0 (5)

Expanding a Taylor series around the potential flowsolution, and truncating to the second order term, anequation for the terminal velocity is obtained. Theabove allows obtaining an approximate solution for thebubble velocity avoiding numerical solving of Eq. (5).The resulting equation is as follows

VT1 = VT pot

[1 + 0.73667

(gde)1/2

VT pot

]1/2

(6)

Eq. (6) is valid for spherical isolated bubbles risingunder the effect of a hydrodynamic boundary layer andbuoyancy force.

www.rmiq.org 271


When surface tension effects are important, Lehrer(1976) states a mechanical energy balance generatedby bubble displacement in the liquid:

16πd3

e12ρLν

2 = σπd2e +

16πd3

e ∆ρgde (7)

where σ is the surface tension and ν is the liquidvelocity when it is displaced by the bubble passing by.

The right hand side of Eq. (7) is the potentialenergy increase of the displaced liquid due to surfacetension and buoyancy during the bubble motion over adistance de. This potential energy is then transformedin kinetic energy [left hand side of Eq. (7)] as thebubble moves, and finally it is dissipated by the wake.The liquid motion is considered uniformly accelerated.Assuming that the necessary time for the potentialenergy to be transformed into kinetic energy is thatcorresponding to a displacement de of the bubblemoving at a steady velocity, the following equation isobtained (Lehrer, 1976).

VT2 =

(3σρLde

+gde∆ρ

2ρL

)1/2

(8)

Eq. (8) can predict the terminal velocity whenthe dominant effects come from surface tension andinertia. According to the combination proposed by Eq.(1) it is possible to calculate the terminal velocity fora large range of bubble sizes in pure liquids for theintermediate and inertial regimes.

3 Results and discussionPredictions from Eqs. (6) and (8), combined accordingto Eq. (1), were compared with experimental datafrom Haberman and Morton (1953), Peebles andGarber (1953), Okazaki (1964), Aybers and Tapucu(1969), Duineveld (1995), Blandın-Arrieta (1997),Leifer et al. (2000), Okawa et al. (2003), Talaia (2007)and Sanada et al. (2008) for gas bubbles ascendingthrough various pure liquids, including distilled andultra-purified water.

Fig. 1 shows the comparison of the experimentaldata for distilled water case in a wide equivalentdiameter range with the terminal velocity predictedfrom equations. The proposed formulation in thiswork properly predicts the typical behavior of theterminal velocity along the whole range. Predictionsof three other formulations are also presented. Twoof them propose a weighting similar to that of Eq.(1) (Fan and Tsuchiya, 1990; Jamialahmadi et al.,1994), and the other one was obtained by dimensionalanalysis (Rodrigue, 2001).

0 . 1 10

1 0

2 0

3 0

4 0

5 0

6 0

��!-(�)��) ��*-/*)��!-(�)��) ��*-/*)��!!�'!.��) ��-�!-��&�3�&%��2�!-.��) ��+0�0��'�) 4)��--%!/�� !%"!-�!/��'��&�1��!/��'��'�%�� ,��1%/$��,.��) ��)��) ��.0�$%2��(%�'�$(� %�!/��'��* -%#0!��

V T (cm/s

)

d e ( c m )Fig. 1. Comparison between predictions of equationsand experimental data for bubble terminal velocitythrough distilled water (7.4 < Re < 32500; 0.002 <We < 245; 0.01 < Eo < 475) at 20-28 ◦C.Markers correspond to experimental data, and lines topredictions from the equations.

The prediction of terminal velocity byJamialahmadi et al. (1994) equation fails before thelocal maximum, and does not fit the experimental dataat this maximum. This happens because the solutionproposed by Hadamard -and used by Jamialahmadi etal. (1994) for dominant viscous effects-, overestimatesthe terminal velocity value for Re larger than one.A similar situation occurs with Fan and Tsuchiya(1990) equation, which is based on a parameterfit. Additionally, Rodrigue (2001) equation resultsparticularly inadequate to predict the existence of localextrema on the typical terminal velocity curve.

Distilled water can still contain contaminants thataffect the rising motion of a single bubble. Thepresence of such pollutants, especially surfactants,changes the interface characteristics, rigidifying itand reducing its shape oscillations during rise.Duineveld (1995) and Sanada et al. (2008)determined experimentally the rise velocity of bubblesmoving through ultra-purified water with rectilineartrajectories (20 < Re < 685). Fig. 2 showsthe comparison of those experimental data withpredictions of the proposed formulation and someother from the literature, as the equation developed byTomiyama et al. (2002). The latter equation requiresthe knowledge of the aspect ratio. For small equivalentdiameters it is still acceptable to relate this parameter

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to the Eo (Celata et al., 2007). Here, Sanada et al.

0 . 0 4 0 . 0 8 0 . 1 2 0 . 1 6 0 . 2 00

1 0

2 0

3 0

4 0

D u i n e v e l d ( 1 9 9 5 ) S a n a d a e t a l . ( 2 0 0 8 ) E q . ( 1 ) w i t h E q s . ( 6 ) a n d ( 8 ) ( % E r r = 3 . 2 ) F a n a n d T s u c h i y a ( 1 9 9 0 ) ( % E r r = 2 1 . 7 ) J a m i a l a h m a d i e t a l . ( 1 9 9 4 ) ( % E r r = 3 5 . 1 ) T o m i y a m a e t a l . ( 2 0 0 2 ) ( % E r r = 9 . 2 ) R o d r i g u e ( 2 0 0 1 ) ( % E r r = 2 4 . 4 )

V T (cm/s

)

d e ( c m )Fig. 2. Terminal velocity from experiments onultra-purified water (25 < Re < 685; 0.02 <We < 3.36; 0.02 < Eo < 0.5) at 20-23 ◦C.Markers correspond to experimental data, and lines topredictions from the equations.

(2007) equation (Eo < 0.5) was used

E =1

1 + 6.5Eo1.925 (9)

In Fig. 2 one can observe that the relative averageerror (percentage, % Err in the figures) is significantlysmaller for the formulation proposed in this work(3.2 % absolute). Tomiyama et al. (2002) equation,which is the next better fit, obtain a relative averageerror of 9.2 %. This seems to confirm that viscouseffects are concentrated in a thin boundary layer for thebubble size ranging from 0.04 to 0.18 cm of equivalentdiameter, according to the approximation from Eq.(6).

As the bubble diameter increases, up to Eo about0.5, the onset of path instability occurs (Duineveld,1995; Sanada et al., 2008; Velduis et al., 2008),and trajectories become zig-zagging or helical. Thecorresponding bubble size range spans between thelocal extrema of the typical terminal velocity behavior(see Fig. 1). For pure water these extrema are betweende ∼ 0.15 − 0.65 cm, approximately (Mendelson,1967; Loth, 2010), and according to the proposedformulation between de ∼ 0.17 − 0.67 cm. Theselocal extrema are important because they indicatesignificant transitions in the bubble rise mechanics.The local maximum coincides with the path instabilityonset, while the local minimum indicates the pointwhere inertial effects become dominant over surfacetension effects in Eq. (8).

0 . 1 0 . 2 0 . 3 0 . 49

1 8

2 7

3 6

A y b e r s a n d T a p u c u ( 1 9 6 9 ) E q . ( 1 ) w i t h E q s . ( 6 ) a n d ( 8 )

|V| (c

m/s)

d e ( c m )Fig. 3. Comparison between predictions of Eq. (1)and the single bubble velocity vector magnitude fromAybers and Tapucu (1969).

For larger bubble size, the rise regime is dominated byinertia, and bubbles adopt the characteristic sphericalcap shape, with the trajectory becoming approximatelyrectilinear again.

For the terminal velocity between the localextrema, three bubble diameters are possible. Thelowest diameter corresponds to a motion withrectilinear path, the middle diameter is embedded inthe intermediate regime with oscillatory trajectory,and the largest diameter corresponds to a preferablyinertial regime and spherical cap shape for the bubble.In this sense, the description of the bubble motionwithin the oscillating trajectory range is a verycomplex task, and it remains a research topic inrecents numerical and experimental works (de Vrieset al., 2002; Mougin and Magnaudet, 2006; Shew andPinton, 2006a; Veldhuis et al., 2008).

Even if the predictions proposed in this work fit theexperimental data with pure water for many practicalapplications (Re between 25 and 685) best than otherpredictions, they overestimate slightly the terminalvelocity for larger bubbles. This occurs until theinertial regime is reached, as shown in Fig. 1. Thisoverestimation might be caused by the presence ofcontaminants in the water or by oscillatory trajectories.

Aybers and Tapucu (1969) measured both thevelocity magnitude and its axial component (terminalvelocity) for bubbles in the equivalent diameter rangebetween 0.1-1 cm following helical paths. Fig. 3shows a comparison between the predictions of Eq.(1) [with substitutions of Eqs. (6) and (8)] to theobtained data for the velocity magnitude. The percentrelative error has an average of 5.5 %, as compared

www.rmiq.org 273


with experimental data by Aybers and Tapucu (1969).This allows us to state that if the approximation fitsinto Eq. (1) (with VT ≈ |V |), then its axial componentcan be determined from the bubble trajectory. For thecase of small bubbles (de < 0.17) and large ones (de >0.67) the velocity module corresponds approximatelyto its axial component, or terminal, velocity. Yetit is necessary to know the trajectory to determinethe velocity magnitude and its axial component foroscillating motion.

Helical trajectories are much more common thatzig-zagging ones (Shew and Pinton, 2006a), andcorrespond to the observed motion well above thenozzle (Aybers and Tapucu, 1969). Zig-zaggingtrajectories are associated to an unstable wake, sothese trajectories usually become helical at large(Ellingsen and Risso, 2001; Shew and Pinton, 2006a).As a result, it is advisable to assume that helicaltrajectories characterize the whole range of oscillatorymotion, but with a stationary terminal velocity duringthe bubble ascent (Shew and Pinton, 2006a; Shew etal., 2006).

From the vectorial description of a helical path,Aybers and Tapucu (1969) obtained the followingrelationship between the axial velocity component andits vectorial magnitude

VT = |V| sinϕ (10)

where VT is the axial component, |V| is the velocitymagnitude and ϕ is the helix angle with respect to thehorizontal plane, which is given by

ϕ = tan−1(

hπDH

)(11)

h corresponds to the vertical bubble displacement thatcorresponds to a full oscillation cycle and DH is thehelix diameter. In terms of amplitude and frequencyof oscillation, one has

ϕ = tan−1(

VT f2πA

)(12)

where f is the frequency and A is the amplitude.Even if there are some investigations that explore

analytically the effect of the non-steadiness of theflow in unconfined single rising bubbles, there arestill no adequate mathematical descriptions of thebubble oscillatory motion and its perturbation effecton the surrounding liquid. This is because the helixangle determination in Eq. (10) had to be done fromempirical data.

0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 72 0

2 4

2 8

3 2��!,'�(��( ��),.)(��!!�&!-��( ��,�!,��%�1�%$�� 0�!,-��( ��*/�/�� &�( 2(��,,$!.��!$"!,�!.��&��+��,,��(��( ��-/�#$0��,,��'$�&�#'� $�!.��&��,,��

V T (cm/s

)

d e ( c m )Fig. 4. Comparison between predictions of equationsand experimental data for bubble terminal velocity indistilled water (560 < Re < 1500; 2.3 < We <5.9; 0.5 < Eo < 5.9) a 20-28 ◦C. Markers correspondto experimental data, and lines to predictions from theequations.

There are several investigations that studied theoscillatory motion of single unconfined bubbles(Aybers and Tapucu, 1969; Blandın-Arrieta, 1997;Ford and Loth, 1998; Brucker, 1999; Ellingsen andRisso, 2001; Wu and Gharib, 2002; Shew and Pinton,2006a,b; Shew et al., 2006; Veldhuis et al., 2008). Ithas been observed experimentally that the oscillationsamplitude A ∼ 0.0035 m and frequency f ∼ 5 Hz(Loth, 2010). When these values are substituted intoEq. (12) a correction for the terminal velocity isobtained in Eq. (10). This correction depends onthe amplitude, frequency and experimental values forthe terminal velocity. The following equation wasconsequently fitted

ϕ = 71.388 − 12.054de (13)

Taking into account all the aforementioned facts, Fig.4 plots the predictions of Eq. (10) considering thehelical paths followed by the bubbles. It should beremarked that when de < 0.17 and de > 0.67, thenϕ = 90◦ and the trajectory is rectilinear (sinϕ = 1).The corrected fit is similar to that of Fan and Tsuchiya(1990) and Jamialahmadi et al. (1994) equations, allwith a relative average error of about 5 % (absolute).

Experimental data for other liquids than water arescarce. Aqueous solutions and water are most relevantfor practical purposes. Even, Eq. (1), along with Eqs.(6) and (8) can also be used to calculate the bubbleterminal velocity for pure liquids.

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Table 1. Critical equivalent diameter (dec) for the onset of path instability of rising bubbles in pure liquids at20-22 ◦C.

Liquid dec (cm) dec (cm) Reference[Eq. (1) with (6) and (8)] (Experimental data)

Pure water 0.17 0.18 Duineveld (1995)Methanol 0.12 0.13 Hartunian and Sears (1957)Ethanol 0.16 0.17 Hartunian and Sears (1957)Benzene 0.12 0.12 Hartunian and Sears (1957)Silicone oil (DMS-T00) 0.11 0.13 Zenit and Magnaudet (2008)Silicone oil (DMS-T02) 0.19 0.22 Zenit and Magnaudet (2008)

0 . 1 1- 2 0

0

2 0

4 0

6 0

8 0

1 0 0 A n i l i n e E t h y l E t h e r A c e t o n e M e t h a n o l B e n z e n e T o l u e n e E t h y l A c e t a t e I s o p r o p y l A l c o h o l A c e t i c A c i d M i n e r a l O i l V a r s o l T u r p e n t i n e M e t h a n o l

Relat

ive er

ror (%

)

d e ( c m )Fig. 5. Relative error (%) for the predictions of theproposed formulation compared to experimental datafor various pure liquids.

In Fig. 5 the relative errors for the formulationpredictions of this work with respect to experimentaldata by Haberman and Morton (1953) and Peeblesand Garber (1953) are shown. Even if the errorvalues are not as small as for pure water, 72 % ofthe points have a relative error less than 25 %. Theseresults are similar to those obtained by using Fanand Tsuchiya (1990) and Jamialahmadi et al. (1994)equations. With respect to the path instability criticaldiameter prediction, the formulation of this work isgood for bubbles rising in pure liquids of low viscosity,according to the comparison against experimentalresults, shown in Table 1. For bubbles rising in viscousliquids, it is possible that the oscillatory regime couldnot be clearly observed, since it can be inhibited byviscous effects. Anyway, more thorough studies forpure liquids, other than water, are still necessary toexplain satisfactorily the motion of bubbles and theircorresponding regimes.

ConclusionsAn equation was obtained for the terminal velocityof bubbles rising in pure liquids, particularly validfor the intermediate and inertial regimes of bubblemotion. The formulation uses a weighting of the forcebalance obtained from boundary layer theory for non-distorted bubbles and an analytical equation comingfrom a mechanical energy balance. Predictionsobtained this way fit better the available experimentaldata for bubbles rising through pure water (de >0.04 cm), improving the results obtained by othersimilar formulations that do not require bubble shapegeometrical parameters. Oscillatory motion is theonly situation that required the introduction of acorrection, assuming helical rising paths. In general,the formulation proposed in this work showed tobe adequate to predict terminal velocity of risingbubbles in pure liquids. Finally, the local extrema,which bound the onset of the path instability (localmaximum) and the point where inertial effects beginto dominate (local minimum), are well predicted bythe proposed formulation.

AcknowledgmentsThe authors respectfully acknowledge the ConsejoNacional de Ciencia y Tecnologıa (CONACyT -Mexico) for their support through a Graduatescholarship (SBR), as well as through Grant CB-2005-C01-50379-Y.

www.rmiq.org 275


NomenclatureA amplitude of mean oscillation cycle for

helical bubble motion (m).de diameter of a spherical bubble of equivalent

volume (m).dec critical equivalent diameter for onset of

path instability in bubble motion (m).DH helix mean diameter for spiraling path in

bubble motion (m).E aspect ratio between minor and mayor axes

of oblate ellipsoidal bubbles.f frequency of mean oscillation cycle for

helical bubble motion (Hz).g acceleration of gravity (m/s2).h vertical bubble displacement that

corresponds to a mean oscillation cycle forhelical bubble motion (m).

|V| magnitude of the bubble motion velocityvector (m/s).

VT terminal velocity of single bubbles (m/s).VT1 bubble rise velocity when viscous effects

are important (m/s).VT2 bubble rise velocity when surface tension

effects are important (m/s).Greek letters

∆ρ density difference between liquid andgaseous phases (kg/m3)

µL dynamic viscosity of liquid phase (Pa s)ρL density of liquid phase (kg/m3).σ surface tension (N/m)ν liquid phase velocity due to bubble

displacement (m/s).ϕ helix mean angle for spiraling path in

bubble motion (◦).Dimensionless numbers

Eo Eotvos number (∆ρd2bg/σ).

Re particle Reynolds number (ρLdbVT /µL).We Weber number (ρLdbVT2/σ).

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