Shorter Communications

Chemical Engineering Science, 1972, Vol. 27, pp. 837-840. Pergan~on Press. Printed in Great Britain

Mass transfer to bubbles

(Receioed 28 June 197 1)

MANY processing operations demand the passage of a gas into a liquid. Following entrance into the liquid, bubbles may grow or shrink, and undergo changes in mean concentration due to mass transfer into or out of the bubbles. Danckwerts [l] gave detailed physical considerations and examined the case for transfer of a single component, with and without a non-transferring second component in the bubble. He found that the insoluble gas did not affect the overall absorption rate of the transferring component substantially. Subse- quently, Loudon, Calderbank and Coward [2] removed many of the restrictions of the first treatment and showed that the conclusion remained valid.

More recently, Hirose and Moo-Young[3] considered the case when both components may be transferred between the gas and liquid. It is the purpose of the present work to con- sider, once again, the effects stemming from the transfer of both components.

Following the assumptions implicit in the treatment of Hirose and Moo-Young [3], consider mass transfer between a spherical bubble of diameter dB in which the gas concentra- tions of components A and B are CGA and Ccs (moles/vol), in contact with a liquid in which the concentrations of A and B, C,, and CLB (moles/vol) are held constant. Assume that the mass transfer coefficients for components A and B (kU and kLB) are constant, the gas in the bubble is well mixed, with a negligible gas side resistance to mass transfer, the bubble is a perfect gas or a gas with a constant compressi- bility factor, and that the bubble is subject to a constant pressure. A mass balance on components A and B yields

where A”, AB are vapour-liquid equilibrium ratios based on [moles/vol],,,/[moles/vol],,,.ld and t denotes time.

Also, CGA + CGB = CGT (3)

where CGT is the total molar concentration of the gas, which remains constant. Rearranging the simultaneous Eqs. (1) and (2) yields, in dimensionless form,

~=2(R-l)C~~+I(C;~-_R(L-C;J) (4)

dCZ* _ 6 ---K~-R)C,*:+(R(~-C~)-~-C~)C,??+C~~).

dt* d,* 0)

c* +c* =l GA GB . (6)

Note that

In these equations d,* = dB/dBo, where d,, is the initial bubble diameter, t* = hAkLAtldBO, R = ABkLB/AAkLA. C& =

CGAICGT. C,*, = CLAIAACGTI C,*, = CLBIABCGT. For computa- tional convenience, define component A so that 0 G R s 1. This covers all possible cases. When R = 0, B will not transfer. and when R = 1, A will behave the same as B under identical conditions, e.g. isotopic mass transfer.

The boundary conditions are:

t* = 0, d; = 1 (7a)

t*=o c* cc* ’ GA Cd0 (7b)

i.e. the initial diameter and gas concentration are specified. Equations (4) and (5) with boundary conditions (7) were

so&d numdncally ‘dsing a four&-order Runge-Kutta method[4] on an IBM 360. Tables of C& and df vs. t* were produced. Since Eqs. (4) and (5) are not severely non- linear, it was possible to choose a step size for t* which gave answers that were sensibly independent of step size without appreciable roundoff errors. The results were checked using

mass balance techniques and were found to be accurate to beyond the fourth decimal place.

The asymptotic values, where these arise, of d,* and C& were checked analytically, as was the case R = 0, C,*, = 0. which corresponds to B not transferring, the case examined by Danckwerts[l]. The present results are in agreement with his analytical result, but it should be noted that the earlier numerical presentation of the results is in error. A revised version of the results is given in Fig. 1. Here is plot- ted, as in the original, the fraction of A absorbed from an impure bubble divided by the fraction absorbed from a pure bubble (y/y’) vs. dimensionless time, t/t,, . tp is the time it takes the pure bubble to disappear. Now

t t* _=_ tp P

t* (8)

where t; is the value of t* at which an initially pure bubble vanishes, which may be shown numerically to be 0.5. Danck- werts’ volume parameter, relating the volume u at time t to the initial volume Q,, (U/L+,) is equal to (d,*)3. Hence

Y -=

1 -(d,*,$

Y’ 11 -_(l -wP*))31C~A,

when t*/r,* =Z 1. When r*/tz > 1, the term [l-(1-@*/ t,*))“] is replaced by unity. It must be noted that the con- clusions in the analysis are in no way affected by the error. In fact, his remark that y/y’ is not very dependent upon impurity is strengthened.

It should be noted that an analytical, if somewhat untidy, relationship between d,* and C& may be obtained by division of Eqs. (4) and (5) and integration. Singularities may arise when dC&ldt* = 0 or dd,*/dt* = 0.

837

Fig. 1. y/y’ vs. f*/tt.

It is considered that the solution due to Hirose and Moo-Young[3] is not valid. In effect they added Eqs. (1) and (2). and then generated an extra equation that demanded that the driving force ratio (C& - c,*,)/(C& -C,*,) be constant for a given system. Use of this relationship with Eq. (6) demands that C& and C& have specified values that do not change with t*. Substitution of these particular values into Eqs. (1) and (2) yields, save in a trivial case, two different expressions for dd,/dt.

The total vapour pressure that would be exerted by the bulk liquid is, in dimensionless form, C,*, + C,*,. The behaviour of the system may be illustrated by considering three values of this group.

(i) CTA + C& = 1.0. The total vapour pressure equals the ambient pressure (Figs. 2, 3). The bubbles generally attain a constant diameter and have a composition given by that in equilibrium with the bulk liquid, C& = C&, C* = C&. Note, however, that if the bubble were initially A r&e, and no B could pass into it, the bubble would vanish unless CL*A = 1. The ultimate values of d,* are determined by R, which includes within it the interphase equilibrium coefficients. The lack of symmetry of the C& versus t* plots about C& = 0.5 stems from the change of bubble volume.

(ii) C,*, + C& = 0.9. Generally, in this instance, the bubble will ultimately vanish since the total vapour pressure exerted by the bulk liquid is less than the ambient pressure. An ex- ception arises when B is insoluble (R = O), d,* becoming constant. The behaviour of the system for three values of CL; is shown at CGA0 = 0.1, R = 0.5, i.e. A may be regarded as transferring twice as readily as B. (Figs. 4, 5). As expec- ted, d,* falls as t* increases. Note, however, that initial growth may occur due to more rapid initial transfer of A into the bubble than B out of the bubble. It may also be seen that C& may change from being less than C,*, to greater than CT”, the mass transfer direction for A being reversed, to be in the same direction as B. This is a result of B not being at equilibrium when A is. At high t*, whatever the initial conditions, C& may settle to a particular value before d,* reaches zero, i.e. dC&/dr* = 0.

(iii) C& + C,*, = 1.1. The total vapour pressure exceeds the amblent pressure. Consider C&= 0.9, R = 0.5 (Figs. 6,7). The effects are much the same as for the previous case, save now the bubble generally suffers indefinite growth. The initial conditions may temporarily cause a bubble to shrink rather than grow since A initially leaves faster than B can replace it. Change from absorption to desorption behaviour for A (see CL*A = 0.5). is also found.

o.55 0 I.0 I.4

1”

Fig. 2. d$ vs. t *. Dimensionless bubble diameter vs. dimen- sionless time when bulk liquid vapour pressure equals

ambient pressure. Cz+ + C& = 1.

I.0

c:, + c:, = I.0

0 I -0

t*

Fig. 3. C& vs. t*. Dimensionless bubble concentration vs. dimensionless time when C,*, + C& = 1.

838

Shorter Communications

I

*, I.,

O-'

IO-

xl

goO t*

Fig. 4. dB vs. t* for C& + C$ = O-9.

t* Fig. 5. C,& vs. I * for C& + C& = 0.9.

,-

t*

Fig.6.dfvs.t*forC&+C&.= 1.1.

I.0 t*

Fig.7.C,$vs.t* forC,*A+C&= 1.1.

Shorter Communications

Both A and B cannot be at equilibrium with the liquid if C& +C& is not equal to 1.0. However, they will often reach a balance where C& and C&, do not change. At this point they both experience mass transfer in the same direc- tion, either into the bubble or out, at such a ratio of rates so that their relative amounts will not change. This will usually happen, if the bubble has not disappeared, by a time of 2t*.

Solving Eq. (5) when dC&/dr* = 0 gives the limiting value of C&. The solution has the quadratic form

c* =-EE(E~-~C~~(~-R))“* CA 2(1-R) (10)

where E=R(l-C,*,)-C&-l.

When C,*, + C& = 1, Eq. (10) yields C& = C,*,. It is found by numerical test in the region 0 s C,*, + C& < 2, the zone of most practical interest, that E* > 4C,*,( 1 -R). It is interesting to note that the final value of C,& is independent of its initial value.

In conclusion, it may be noted that Eqs. (4J, (5), (7), can be integrated numerically quite readily and simply. The analy- tical results of Hirose and Moo-Younar31 are in error. In more complex practical cases the e&s analyzed by Loudon et al. [2] and the physical discussion of Danckwerts [l] should always be borne in mind.

Department of Chemical Engineering J. BRIDGWATER? The University of British Columbia G. S. McNAB Vancouver 8 British Columbia Canada

tAddress after 1 October 1971: Department of Engineer- ing Science, University of Oxford, Parks Road, Oxford OX 1 3 PJ, England.

CG CO CL cc* ci? 4s d,* E

kL M R t

t* t,

t* P

v YIY’

NOTATION concentration in gas phase, ML3 total concentration in gas phase, ML3 concentration in liquid phase, MLS C&Z,,. Dimensionless gas phase concentration C,/A CGp Dimensionless liquid phase concentration bubble diameter, L d,/d,,, dimensionless bubble diameter R(l-C&)-C&-l liquid side mass transfer coefficient, LT-’ Danckwerts parameter = C &, when C *LA = 0, R = 0 hBkLBIAAkbA. A is defined so that 0 < R s 1 time, T )kkLAt/dSO. Dimensionless time time at which bubble of pure A disappears when there

is no A in the liquid and no transfer of B, T t* at which bubble of pure A disappears when there is

no A in the liquid and no transfer of B = 0.5 bubble volume, L3 Danckwerts parameter = fraction of A absorbed from

impure bubble/fraction of A lost from a pure bubble of A at the same time

Greek symbols

A = CL/C,, at equilibrium

Subscripts

A componentA (defined so that 0 s R < 1) B component B (defined so that 0 c R < 1) 0 initial condition

REFERENCES [l] DANCKWERTS P. V., Chem. Engng Sci. 20 78.5 (1965). [23 LOUDON J. R., CALDERBANK P. H. and COWARD I., Chem. Engng Sci. 21614 (1966). [3] HIROSE T. and MOO-YOUNG M., Chem. Engng Sci. 25 729 (1970). [4] MCCRACKEN D. D. and DORN W. S., Numerical Methods and Fortran Programming. John Wiley, New York 1964.

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