On the diffusion from a slender bubblein an extensional flow

This article was downloaded by: [University of Waterloo]On: 04 November 2014, At: 12:05Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK

Chemical EngineeringCommunicationsPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/gcec20

On the diffusion from a slenderbubblein an extensional flowMoshe Favelukis a & Sim Ai Chiam aa Department of Chemical and EnvironmentalEngineering , National University of Singapore ,SingaporePublished online: 09 Sep 2010.

To cite this article: Moshe Favelukis & Sim Ai Chiam (2003) On the diffusion from aslender bubblein an extensional flow, Chemical Engineering Communications, 190:4,475-488, DOI: 10.1080/00986440302085

To link to this article: http://dx.doi.org/10.1080/00986440302085

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of theContent.

http://www.tandfonline.com/loi/gcec20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/00986440302085

http://dx.doi.org/10.1080/00986440302085

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,sub-licensing, systematic supply, or distribution in any form to anyone isexpressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

http://www.tandfonline.com/page/terms-and-conditions

ONTHEDIFFUSIONFROMASLENDERBUBBLEINANEXTENSIONAL FLOW

MOSHEFAVELUKISSIMAICHIAM

Department of Chemical and Environmental Engineering,National University of Singapore, Singapore

Mass transfer between a slender bubble and a Newtonian liquid in a simple

extensional and creeping flow has been theoretically studied. The analytical

steady-state solution when the Peclet number is zero has been obtained using

the works of Szego and Payne, who studied the electrostatic capacity of a

spindle surface in a bispherical coordinate system. The result shows, as ex-

pected, that the modified Sherwood number increases as the capillary number

increases since the surface area of the bubble also increases.

Keywords: Bubble deformation; Extensional flow; Mass transfer; Slender

bubble

INTRODUCTION

The problem of mass transfer between a bubble and a liquid has beenwidely studied in the literature because of its importance to the chemicalprocess industry. High viscosity liquids, such as polymer melts, foods,and biological materials, are usually processed in rotary machines, underconditions in which shear or extensional flows exist. A bubble present inthese types of flows will deform and, at some conditions, elongatedbubbles with pointed ends can be obtained.

Address correspondence to Moshe Favelukis, Department of Chemical and Environ-

mental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore

119260. E-mail: [email protected]

Chem. Eng. Comm.,190: 475�488, 2003

Copyright# 2003 Taylor & Francis

0098-6445/03 $12.00+ .00

DOI: 10.1080/00986440390192591

475

Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

Mass transfer between a bubble and a liquid is proportional to thesurface area of the bubble. When a bubble is deformed, its surface areaincreases with respect to that of a spherical bubble of the same volume.Therefore, we can expect slender bubbles to be very efficient in masstransfer operations. Another important parameter is the velocity of theflow; however, in this work, we shall examine the specific case in whichconvection is negligible compared to diffusion.

In a previous paper, the steady-state mass transfer between a slenderbubble and a liquid in an extensional and creeping flow has been theo-retically studied (Favelukis and Semiat, 1996). Using the methoddescribed in Levich (1962), an analytical solution has been obtained whenthe resistance to mass transfer is only in a thin concentration boundarylayer in the liquid phase. The present work treats the other asymptoticregime, where the concentration boundary layer is thick.

BUBBLEDEFORMATION

Consider a slender bubble, with a local radius R(z) and a half-length L,placed in a simple (axisymmetric) extensional flow (Figure 1) defined incylindrical coordinates by:

vr ¼ � 1

2Gr ð1Þ

vz ¼ Gz ð2Þ

where G is the strength of the flow; it will be positive in this work.The solution to the deformation problem in a Newtonian liquid and

in creeping flow has been obtained by Taylor (1964), Buckmaster (1972),

Figure 1. A slender bubble in a simple extensional flow. R(z) is the local radius and L is the

half-length of the bubble.

476 M. FAVELUKIS AND S. A. CHIAM

Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

and Acrivos and Lo (1978). The following dimensionless bubble para-meters can be obtained:

R� ¼ RðzÞa

¼ 1

4Ca1� z�

L�

� �2" #

ð3Þ

L� ¼ L

a¼ 20Ca2 ð4Þ

A� ¼ A

4pa2¼ 10

3Ca ð5Þ

where z*¼ z/a, a is the equivalent radius (radius of a sphere of equalvolume), A is the surface area, and Ca is the capillary number defined as:

Ca ¼ mGa

sð6Þ

Here m is the viscosity of the liquid and s is the surface tension. Thecapillary number is the ratio of the viscous force, which tends to deformthe bubble, to the surface tension force, which tends to keep the bubblespherical. Note that the slender bubble has a parabolic shape withpointed ends, and that the bubble is slender (R/L� 1) when Ca3� 1.Two excellent reviews on bubble or drop deformation in viscous flowswere published by Rallison (1984) and Stone (1994).

MASS TRANSFER

A differential mass balance for our binary axisymmetric system, assumingsteady state, constant density, and diffusion coefficient (D), reduces to:

vr@c

@rþ vz

@c

@z¼ D

@2c

@r2þ 1

r

@c

@rþ @2c

@z2

� �ð7Þ

where c is the molar concentration of the solute that diffuses in the liquid.Uniform and constant concentrations at the surface of the bubble (cs)and far away from the bubble (c1 ) will be assumed. Equation (7) isgoverned by a single dimensionless number, the Peclet number, defined inthis problem as:

Pe ¼ Ga2

Dð8Þ

ON THE DIFFUSION FROM A SLENDER BUBBLE IN AN EXTENSIONAL FLOW 477

Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

The Peclet number describes the ratio of convection to diffusion. At highPeclet numbers (Pe�Ca2), the thin concentration boundary layerapproximation can be applied. In this case Favelukis and Semiat (1996)obtained the steady-state solution and Favelukis (1998) obtained theunsteady solution. It is the purpose of this work to obtain the solution forthe other regime of low Peclet numbers. We shall solve here the asymp-totic case of Pe¼ 0, which refers to mass transfer by diffusion alonethrough a thick concentration boundary layer. In this case Equation (7)reduces to the well-known Laplace equation:

@2c

@r2þ 1

r

@c

@rþ @2c

@z2¼ 0 ð9Þ

which has to be solved with the following boundary conditions:

c ¼ cs at r ¼ RðzÞ ð10Þ

c ¼ c1 at r ¼ 1 ð11Þ

The other two boundary conditions can be obtained by the requirementof an axisymmetric problem. Clearly, the cylindrical coordinate system isnot a natural system for the surface of the bubble since according toEquation (10), R(z) 6¼ constant.

BISPHERICALCOORDINATES

Figure 2 describes a bispherical coordinate system (Z, y, c) as given byMoon and Spencer (1971). The surface of the two spheres is given byZ¼ � constant. The surfaces’ y¼ constant represent an apple-shapedsurface if y< p/2, a sphere at y¼ p/2, and a spindle if y> p/2. Finally, theangle c is not shown in the figure as it is of little interest in our axi-symmetric problem. The range of the coordinates is 71 < Z< 1 ,0 � y < p, and 0 � c < 2p.

The following relations exist between the Cartesian and the bisphe-rical system:

x ¼ b sin y cosccosh Z� cos y

ð12Þ

y ¼ b sin y sinccosh Z� cos y

ð13Þ


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

z ¼ b sinh Zcosh Z� cos y

ð14Þ

where b is the distance from the z¼ 0 (Z¼ 0) plane to the edge of thespindle. Note that infinity is defined in the bispherical system as Z¼ y¼ 0.

The surface y¼ constant is described by:

x2 þ y2 þ z2 � 2b x2 þ y2� �1=2

cot y ¼ b2 ð15Þ

Imagine our slender bubble to be a spindle (y> p/2). Substituting L¼ band R2¼ x2þ y2 in Equation (15) results in:

R2 þ z2 � 2RL cot y ¼ L2 ð16Þ

For a slender bubble, R/L�1, Equation (16) in its dimensionless formcan be simplified (except near z¼ 0) to:

R� ¼ � 1

2L� tan y 1� z�

L�

� �2" #

ð17Þ

Comparing the last equation with Equation (3) we conclude that, at firstapproximation, the parabolic shape of a slender bubble in an extensionalflow can be represented by a slender spindle. Together with the expres-sion for the half-length of the bubble given by Equation (4) we obtainthat the surface of the bubble is given by:

tan y ¼ � 1

40Ca3ð18Þ

Figure 2. The bispherical coordinate system.


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

However, for a slender bubble, Ca3� 1, y! p, and the surface of thebubble, at first approximation, can be simplified to:

y ¼ p� 1

40Ca3ð19Þ

Another useful relation, when y! p, can be derived from Equation (14):z/L¼ z*/L*� tanh(Z/2), and the radius of the bubble given by Equation(3) can be approximated in the bispherical system to:

R� ¼ 1

4Casech2

Z2

� �ð20Þ

If we define a dimensionless concentration as:

c� ¼ c� c1cs � c1

ð21Þ

then the Laplace equation in our axisymmetric problem takes the form of(Moon and Spencer, (1971)):

sin y@

@Z1

cosh Z� cos y@c�

@Z

� �þ @

@ysin y

cosh Z� cos y@c�

@y

� �¼ 0 ð22Þ

where the boundary conditions, given by Equations (10)�(11), reduce to:

c� ¼ 1 at y ¼ ys ð23Þ

c� ¼ 0 at Z ¼ y ¼ 0 ð24Þ

Here y¼ ys¼ constant denotes for the surface of the bubble and, again,the other boundary conditions can be obtained by the requirement ofsymmetry.

RESULTSANDDISCUSSION

Fortunately, the problem defined by Equations (22)�(24) has been solvedby Neumann (1881), Szego (1945), and Payne (1952) in an integral form,and using infinite series by Macdonald (1900), who studied the electro-static capacity of a spindle. We will take the advantage of the analogybetween electrostatics and diffusion to obtain our solution. The notationgiven in Payne (1952) will be used to present our results.


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

The dimensionless concentration profile is given by:

c� Z; tð Þ ¼ffiffiffi2

pcosh Z� tð Þ1=2

Z 1

0

Kað�tsÞKaðtÞ cos aZð ÞKaðtsÞ cosh apð Þ da ð25Þ

where t¼ cos y, ts¼ cos ys, and Ka(t) is the conal function defined by:

KaðtÞ ¼2

pcosh apð Þ

Z 1

0

2 cosh uþ 2tð Þ�1=2 cos auð Þdu ð26Þ

The conal function is actually the Legendre function Pn(t) where n isreplaced by the following complex expression: 7 1/2þ ia; see alsoHobson (1931). Note that, for the case of a slender bubble, the termKa(7ts) can be substituted by 1 in Equation (25) without practicallyaffecting the numerical results.

Figure 3 is a contour plot of the dimensionless concentration, in thebispherical coordinate system, for different values of the capillary num-ber. From the figure we can see that as the capillary number increases, thelength of the bubble increases, and the concentration boundary layerthickness decreases, leading to a higher mass flux. A similar plot, but in amore convenient form of cylindrical coordinates, is shown in Figure 4. Itis interesting to mention here that there is no external solution to theLaplace equation for an infinite cylinder. The results presented heresuggest that in the case of an infinite cylinder the mass flux is infinite.

The local molar flux, N(Z,Ca), from a slender bubble to the liquid isdefined and given, from the definition of a gradient in bispherical coor-dinates, at first approximation (ys! p) as:

NðZ;CaÞ ¼ kðZ;CaÞ cs � c1ð Þ ¼ Dcosh Zþ 1ð Þ

L

@c

@y

� �y¼ys

ð27Þ

where k(Z,Ca) is defined as the local mass transfer coefficient. Oncek(Z,Ca) is obtained, the local Sherwood number can be calculated:

ShðZ;CaÞ ¼ kðZ;CaÞaD

¼ cosh Zþ 1ð ÞL�

@c�

@y

� �y¼ys

ð28Þ

The local Sherwood number describes the ratio of the total mass transfer(convection þ diffusion) to the diffusional mass transfer. Figure 5 showsa plot of the local Sherwood number, according to Equation (28), forfour different values of Ca. We observe that as the capillary numberincreases the local Sherwood number also increases. This conclusion was


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

expected from the results presented in Figures 3 and 4. Note also that thelocal molar flux does not change much along the length of the bubbleexcept near the end when it tends to infinity.

The average molar flux is defined and given by:

�NðCaÞ ¼ �kðCaÞ cs � c1ð Þ ¼ 2

A

Z L

0

Nðz;CaÞ2pRðzÞdz ð29Þ

where �kðCaÞ is the average mass transfer coefficient.The total quantity of solute transferred to or from the slender bubble

is proportional to the average flux times the area of the bubble. The finalsolution, in a dimensionless form, is:

ShA� ¼�ka

DA� ¼ 2L�

Z 1

0

Ka �tsð ÞKa tsð Þ coshðapÞ da ð30Þ

Figure 3. The dimensionless concentration profile, in bispherical coordinates, according to

Equation (25). (a) Ca¼ 1, (b) Ca¼ 2, (c) Ca¼ 5, and (d) Ca¼ 10.


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

where the product ShA� will be referred to here as the modified Sherwoodnumber. Here again, for slender bubbles, the term Ka(7ts) can replacedby 1. For a spherical bubble, drop, or solid particle at zero Peclet num-ber, and regardless of the type of flow, A*¼ 1 and Sh ¼ Sh ¼ ShA� ¼ 1.Clearly, we should expect higher values than that of the sphere since thesurface area of a deformed slender bubble is much greater. Equation (30)can also be compared with other similar results reviewed by Clift et al.(1978) for the electrostatic capacity of slender bodies but with roundedends. Substituting L/R(0)� 1 instead of the aspect ratio we obtain for aslender prolate spheroid (Szego, 1945):

ShA� ¼ L�

ln 2 L�

R�ð0Þ

� � ¼ 20Ca2

ln 160Ca3� � ð31Þ

Figure 4. The dimensionless concentration profile, in cylindrical coordinates, according to

Equation (25). (a) Ca¼ 1, (b) Ca¼ 2, (c) Ca¼ 5, and (d) Ca¼ 10.


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

and a very similar result for a needle-like body (Miles, 1967):

ShA� ¼ L�

ln 4 L�

R�ð0Þ

� �� 1

¼ 20Ca2

ln 320Ca3� �

� 1ð32Þ

Another interesting formula can be obtained from an earlier work ofSzego, who proved the following theorem (Polya and Szego, 1951):

ShA� � M

4pa2e

ln 1þe1�e

ð33Þ

where M is Minkowski’s M or the surface integral of the mean curvatureof the solid with principal radii R1 and R2:

M ¼ 1

2

ZZ1

R1þ 1

R2

� �dA ð34Þ

and e is defined as:

e ¼ 1� 4pAM2

� �1=2

ð35Þ

Figure 5. The local Sherwood number as a function of z/L, according to Equation (28), for

different values of the capillary number.


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

Substituting our bubble parameters in Equations (33)�(35), we obtain atfirst approximation:

ShA� � 20Ca2

ln 120Ca3� � ð36Þ

The values of the modified Sherwood number, calculated from the lastequation, are a little higher than the exact values calculated fromEquation (30). This was explained by Polya and Szego (1951), who statedthat the capacity of a convex solid is never larger than the capacity of aprolate spheroid, the major semiaxis of which is the mean radius of thesolid (M/4p) and the minor semiaxis the surface radius (A/4p)1/2.

Among the three approximated formulas for the modified Sherwoodnumber, the closest one to the exact result is Equation (31). It is com-pared with the exact result, given by Equation (30), in Figure 6. It can beseen from the graph that the values of the modified Sherwood numbergiven by the two equations are very close to each other.

Finally, the result for the modified Sherwood number given byEquation (30) should be compared with the other asymptotic case of highPeclet numbers (Pe�Ca2) obtained by Favelukis and Semiat (1996):

Figure 6. The modified Sherwood number as a function of the capillary number. The solid

line is the exact expression given by Equation (30), the dashed line is an approximated ex-

pression given by Equation (31).


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

ShA� ¼ 5

ffiffiffiffiffiffi2

3p

rCaPe1=2 ¼ 2:30CaPe1=2 ð37Þ

Clearly, higher values of the modified Sherwood number can be obtainedat high Peclet numbers, since then convection also contributes to the masstransfer rate.

CONCLUSIONS

Mass transfer between a slender bubble and a Newtonian liquid in asimple extensional and creeping flow has been theoretically studied. Inorder to simplify the problem, the shape of the bubble, originallydescribed in a cylindrical coordinate system, has been reformulated inbispherical coordinates. The analytical steady-state solution, at zeroPeclet number, has been obtained using the method presented by Szegoand Payne, for the electrostatic capacity of a spindle surface. The resultshows, as expected, that the modified Sherwood number increases as thecapillary number increases since then the surface area of the bubble alsoincreases. Furthermore, the result obtained for the case of a slenderprolate spheroid can be used as an excellent approximation in our casewhere the ends of the bubble are pointed.

ACKNOWLEDGMENT

This research was supported by a grant from the National University ofSingapore.

NOMENCLATURE

a equivalent radius

A surface area

b half-length of the spindle

c molar concentration of solute

Ca capillary number

D diffusion coefficient

G strength of the flow

k mass transfer coefficient

Ka conal function

L half-length of the bubble

M surface integral of mean curvature

N molar flux

Pn Legendre function

Pe Peclet number

r coordinate in the cylindrical system

R local radius

Sh Sherwood number

t function of y


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

u dummy variable

v velocity

x coordinate in the Cartesian system

y coordinate in the Cartesian system

z coordinate in the Cartesian or cylindrical system

Greek letters

a dummy variable

e dimensionless parameter

Z coordinate in the bispherical system

y coordinate in the bispherical system

m viscosity

s surface tension

c coordinate in the bispherical system

Subscripts

r radial direction

s at the surface of the bubble

z axial direction

1 far away from the bubble

Superscripts

* dimensionless

REFERENCES

Acrivos, A. and Lo, T. S. (1978). J. Fluid. Mech. 86, 641.

Buckmaster, J. D. (1972). J. Fluid. Mech. 55, 385.

Clift, R., Grace, J. R., and Weber, M. E. (1978). Bubbles Drops and Particles,

Academic Press, New York.

Favelukis, M. (1998). Can. J. Chem. Eng. 76, 959.

Favelukis, M. and Semiat, R. (1996). Chem. Eng. Sci. 51, 1169.

Hobson, E. W. (1931). The Theory of Spherical and Ellipsoidal Harmonics, Uni-

versity Press, Cambridge.

Levich, V. G. (1962). Physicochemical Hydrodynamics, Prentice-Hall, Englewood

Cliffs, N.J.

Macdonald, H. M. (1900). In: Memoirs Presented to the Cambridge Philosophical

Society on the Occasion of the Jubilee of Sir George Gabriel Stokes, 292�297,

University Press, Cambridge.

Miles, J. W. (1967). J. Appl. Physics 38, 192.

Moon, P. and Spencer, D. E. (1971). Field Theory Handbook, 2nd ed., Springer-

Verlag, New York.

Neumann, C. (1881). Math. Ann. 18, 195.

Payne, L. E. (1952). Q. Appl. Math. 10, 197.


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

Polya, G. and Szego, G. (1951). Isoperimetric Inequalities in Mathematical

Physics, In: Annals of Mathematics Studies, E. Artin and M. Morse, eds., v.

27, Princeton University Press, Princeton, N.J.

Rallison, J. M. (1984). Ann. Rev. Fluid. Mech. 16, 45.

Stone, H. A. (1994). Ann. Rev. Fluid. Mech. 26, 65.

Szego, G. (1945). Bull. Am. Math. Soc. 51, 325.

Taylor, G. I. (1964). In: Proceedings of the 11th International Congress on Applied

Mechanics, Munich, 790�796. Springer, Berlin.


Dow

nloa

ded

by [

Uni

vers

ity o

f W

ater

loo]

at 1

2:05

04

Nov

embe

r 20

14

Documents

On the diffusion from a slender bubblein an extensional flow