Shape and terminal velocity of single bubble motion: a novel approach

European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.

649

Shape and Terminal Velocity of Single Bubble Motion" a Novel A p p r o a c h G. Bozzano a and M. Dente a aCIIC Department, Politecnico di Milano, Piazza L. da Vinci, 32, 20131 Milano, Italy

1. ABSTRACT The terminal relative rising velocity of a single gas bubble, moving into a liquid phase, is determined by its size, by the interfacial tension, by the density and viscosity of the surrounding liquid. Both shape and velocity are strongly interacting. Several methods have been presented along the years, for solving the problem of bubble deformation and relative rising velocity, at least in connection with some specific regimes of motion and/or shape of bubble. In this work a new approach is proposed. 2. INTRODUCTION The understanding of bubble motion mechanism is essential for many gas-liquid operations, not only related to chemical processes applications. Even if in practical applications the overall motion regards bubbles swarms, the behavior of the single bubble (i.e. spoiled of the interactions with the other surrounding bubbles) can support a better knowledge of the overall. Some theories have been presented in the literature covering specific aspects of the problem (ref. 1 to 5). This work has been concentrated on the single bubble behavior: by itself it is a quite complex problem, particularly because the purpose has been to cover a wide range o f properties. The first aspect of the proposed approach is to consider that the approximate shape assumed by the rising bubble is that one that minimizes the total energy associated with the bubble The second one is constituted by the (approximated) generalization of the drag coefficient

3. APPROXIMATED SHAPE AND RELATED TOTAL ENERGY The selected shape is constituted by the superposition of two oblate semi-spheroids (figure 1) having in common the larger semi-axis. This shape asymptotically can degenerate towards something resembling a spherical-cap. The total energy associated to the bubble is the following: (1)

Eto t - Epo t + Esu p + Eki n

where

Epot

-

potential

energy

with

(PL--Pc~)'g" V, "(3" bl + 5" b2)/8

reference

to

the

upper

pole

=

650 Esup = surface energy = cy SB =

Esup -

(7

(

1 b~ 2 rt a 2 + - r c - - l n 2 e1

I I b2 ;l+e ll l+e~ 1-el

1 "eln +-re " 2 e2 \l-e~

= kinetic energy of the virtual mass of adherent liquid displaced by the bubble motion. It is, from a theoretical point of view, an extremely complex function to be evaluated (because of the bubble shape). However, and as a first approximation, it can be obtained by extending the expression with a correction for spherical bubbles. At sufficiently high Reynolds numbers Ekin

3

it can be estimated (in a first approximation) as: EKi n - 2/3. rc0La U

2

Of course the bubble volume is given by: V B - 4/3. rcR 3 - 2/3. rca 2 (b 1 + b2)

b2 = 0

Fig. 1 - Basic shape of the bubble and asymptotic degenerations

The total energy has to be minimized as a function of the two geometrical parameters bl/a and bJa.

4. D R A G C O E F F I C I E N T AND E X T E N D E D E X P R E S S I O N When the bubble reaches the steady state motion the balance of the forces gives: . . . . . . (PL. Pa)gVl~

pLU2

. r~a2 . . .2

f

(2)

By neglecting the gas density in comparison with that of the liquid, the previous equation gives Ur

4 g-D o

-S cD

(3)

651 With a good approximation, and as a consequence of interpolating the minimization procedure, the generalized friction factor that is proposed is the following (taking into account the effect of the bubble deformation): 48 ( 1 + 12. Mo 1/3] Eo f - Re 1+36 ~ J +085 1.8. (1 + 30. Mo 1/6) + Eo

deformation factor =

I~0_l 2 = 3.4(l+30.Mol/6)+3.1-Eo .4(1 + 3 30. Mo 1/6) + Eo

(4)

(5)

So that the drag coefficient is given by the product of equations 4 and 5, and therefore it is an explicit expression ofEo, Mo and Re numbers:

C D - f.

(<12a

(6)

By substituting equation 6 into equation 3 a simple second order equation is generated. 5. WALL EFFECT Most of the experiments on bubble rising velocity have been performed inside limited diameter tubes. For relatively large bubbles (D0/Dt >0.2+0.3) the presence of thew tube wall has the effect of reducing the absolute rising velocity (compared to that attainable in an infinite environment). A first approximation of this {u effect can be estimated as follows. The flowrate entrained by the bubble wake is about CD/2(rc/4. D2) 9U. For continuity, an equal flowrate has to descend crossing the restricted section between the tube wall and the bubble equator. This (negative) contribution gives place to a maximum absolute descending velocity (VD, see figure 2) equivalent to: Fig.2 CD DO VD= 2 D 2 - D 2 U

(7)

Therefore the relative velocity (at the equator of the bubble) is:

uo- u + VD ~ U- U~,'""I

CD/2 l 1+ (Dt/Do)2 _ DEF

U0 is assumed equal to that one of the same bubble rising in infinite environment.

(8)

ouozuoqo.~l!N u! solqqn~[ -qV t7 s ! d (tua) .~olotue!( I lUOleA!nt)~l 001,

01,0

I, U U

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0~

<

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9

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o

poluoso~ao~ pue op!~ s! 'lopotu oql Xq p0.IOAO3 S0ttlI~'O.I o!tueuXp-p!nlJ ,,Io 'oouonbosuoo e se 'pue 'uo!suotu!p olqqn q jo o~ue.~ oql leql osaosqo ol olq!ssod s! 1! s v (/, u!seo iopolAI .~OlXe• p!AeG oql) u!seq oz!s o:&IeI e s e ~ snle.mdde ieluotugodxo oqk Iopom poluoso.ld oql jo sllnso.~ oql pue '.tole~ m solqqn q .he ol OA!lelO-~ '(9) uolaolAI pue uetu.~oqeH jo elep Ieluotu!-~odxo IlOA~

~UOU113 uostJt~dllIo3 ottl slAodoJ s oJn~!~ opl~tlI uooq OAl~tI 1131.]1suostJedttlo3 fdolol~ots!ll3s 113101

oql jo oldtues e oJe Xoqz suo!l!puo3 jo oo~ueJ Op!A~ e ~IIIJOAO3 'elep lelUOtU!Jodxo oJnleJOl!l ql!st poJedtuoo 'lopotu posodoJd oql jo luotuooJ~e pou!mqo oql A~oqs soJn~ U ~U!A~OIIOJ 0 q i ~dL l l l ~ ~ l l ~

9

653

9 Experimental ~, r]3

9 Experimental

Theoretical j

100

10o

Theoretical I

.... ! :

r

>.-, +..a .,..~ O

9

.,--,

o >

10

10

> E

.,-.~

E

r

1

~D

~"

I _

0.01

0.10

1.00

0.10

1.00

Equivalent Diameter (cm)

Equivalent Diameter (cm)

Fig. 5' Air Bubbles in Pyridine

Fig. 6: Air Bubbles in Cottonseed Oil

Figures 4 to 6, are relative to the experiment of Peebles and Garber (8) obtained in a 2.62 cm diameter tube. Also in this case the agreement is satisfactory.

9

Experimental

Theoretical}

9 Experimental

%-,

100

100

E o

_

-:

o o

;>

o

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Equivalent Diameter (cm)

Equivalent Diameter (cm)

Fig. 7 Air Bubbles in Glycerol 90.6%

Fig. 8' Air Bubbles in Glycerol 99%

Finally Fig. 7 and 8 show the comparison of simulation results with the data of Calderbank (9). The system is constituted respectively by 90.6 Glycerol in Water and 99% Glycerol in Water. The tube diameter is 10.6 cm. The physical data of liquids that have been used for the experiments are reported in Table 1. A wide range of viscosity has been covered. Table 1 Pyridine Density (g/cm 3) 0.987 Surface Tension (g/s z) 36.6 Viscosity (g/cm/s) 0.0085

Nitrobenz. 0.987 42.5 0.0167

Cot. seed Oil 0.910 35.5 0.59

Glycerol 90.6 1.235 64.0 1.8

Glycerol 99 1.260 63.0 7.75

654 7. C O N C L U S I O N S The presented new approach has allowed to obtain a unified model for the description, in extended fluid-dynamic conditions, of a single bubble motion. Both terminal velocity and shape of the bubble can be determined. The comparison with different literature experimental data, covering a wide range of physical properties and bubble sizes, are satisfactory. NOMENCLATURE Reynolds number: Re -

9LDc)Uo/btL

EOtvosnumber:Eo-(gL-gg) gD2/cs Morton number: Mo - g bt~/9LCS3 el, e2 " eccentricity of the two semi-spheroids defined as e - -~/1- b : / a : f = friction factor U = absolute bubble terminal velocity U0 = bubble terminal velocity in infinite environment VD = absolute descending velocity CD = drag coefficient a = bubble major semi-axis bl, b2 = bubble minor semi-axes L - liquid, g = gas Do = diameter of the equivalent spherical bubble D = equator diameter of the spheroid Dt = tube diameter Ro = radius of the equivalent spherical bubble (s = surface tension ACKNOWLEDGEMENT

The authors wish to thank Rita Bizzozzero and Christina Ktihlwetter for their contributions to the computational efforts of this work. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9.

Batchelor, G. K., Cambridge University Press, (1970) Bhaga D. and Weber, M. E., J. Fluid Mech., 105 (1981) 61-85 Grace, J. R., Trans. Instn. Chem. Engrs., 51 (1973) 116-120 Mendelson, H. D.,A.I.Ch.E. Journal, 13 (1960) 250-253 Sadhal, S. S., Ayyaswamy, P. S. and Chung, J. N., Springer New York (1997) Haberman, W. L. and Morton, R. K., Soc. Civil Eng. Trans., 121 (1956) 227-251 Bryn, T., David Taylor Model Basin Transl., Rep. No 132, 1949 Peebles, F. N., Garber, H. J., Chem. Eng. Progr., 49 (1953) 88-97 Calderbank, P. H., Johnson, D. S. L. and Loudon, J., Chem. Eng. Sci., 25 (1970) 235-256