Remarks on the behaviour of single oscillating droplets

The Chemical Engineering Journal, 13 (1977) 185-189 @Elsevier Sequoia S.A., Iausanne. Printed in the Netherlands

Remarks on the Behaviour of Single Oscillating Droplets VACLAV VACEK Institute of Inorganic Chemistry, CzechoslovakAcademy of Sciences, 25068 i?e?, near Prague (Czechoslovakia)

PROKOP NEKOVAR Chemical Engineering Department, Technical University, 16628 Prague (Czechoslovakia) (Received 2 April 1975; in final form 15 November 1976)

Abstract An investigation of drop shapes, drop velocity and

interfacial area was carried out for single oscillating drops falling through a stationary continuous liquid phase. The behaviour of single oscillating drops is highly reproducible in pure liquid-liquid systems. During the oscillation period the drop shapes are much more complex than is assumed in the spheroid approximation. The variation in the interfacial area wasfound to be significantly greater than expected from the spheroid approximation. This should be taken into account in evaluations of experimental as well as theoretical mass transfer data.

but an empirical correlation for the velocity of oscillating or circulating drots in pure systems has been given by Thorsen et al. No methods enabling the direct prediction of the am litude of the oscillations have been published so far sp. The effect of a third and transferred components on the frequency and amplitude of the oscillations as well as the time dependence of the drop interfacial area A during the oscillation period have not yet been systematically investigated. It is generally assumed’ that the following relation holds: A(t)=A,{l

+esin”(nt/T)}

t
n= 1,2

(1)

where t is the time, T the period of oscillation and A0 the area of a sphere with volume equal to that of the examined drop. The value of the parameter INTRODUCTION e = (&ax - A,)/A, is based on the maximum area A of the spheroid. The volume V and the parameter In certain cases of drop motion through a motionETzhich is defined as the ratio of the maximum horiless continuous phase the drop shape changes periodiczontal to the maximum vertical diameter) are conally. Drop shapes are complicated and reveal a vigorsidered to be the same for the spheroid and for the ous flow pattern inside as well as outside the drop. given drop. From the general theoretical treatment of It is known that oscillating drops give high mass transmass transfer between oscillating drops and their surfer rates and therefore the drop behaviour has been roundings given by Brunson and Wellek’, we conclude the object of extended research work. In spite of this that a time-dependent area should be incorporated into relatively little is known about oscillating drop hydrothese models to improve agreement between empirical dynamics. No theoretical formalism is available to and theeretical data. determine whether a given drop will oscillate or not In this paper some experimental results on oscilor to predict the frequency and the amplitude of these lating drops are presented. Two-component twooscillations (see the reviews of Winnikow and Chao’ phase systems without mass transfer as well as a and of Edge and Grant’). Frequency is generally system with a third component were studied. In correlated by the empirical relationships given by addition to conventionally measured quantities such as the average terminal velocity ut of the drop motion Edge and Grant’ and Schroeder and Kintner3 which and the oscillation period T, the time variations of the are based on the theoretical analysis by Lamb4 of instantaneous drop velocity u, the parameter E and infinitesimally small oscillations of a motionless the interfacial area A during one oscillation period liquid sphere. Miller and Striven’ and Subramanyam6 were determined. Mass transfer coefficients averaged presented a more general theoretical treatment for over one period were also measured. The measured small amplitude oscillations of a fluid sphere at rest data are compared with the results obtained on approxiin another fluid. No theoretical predictions are mating the drop shape by a spheroid. available for the velocity of the oscillating drops 185

V. VACEK, P. NEKOVzik

186 THEORETICAL

with boundary

c(r < rr ) t > 0) = Cl

The

dependences of the steady state velocity of an oblate and/or a prolate spheroid on the parameter E were given by Luiz 10 for high Reynolds numbers and low viscosities. Assuming that the velocity of the oscillating drop at every moment is equal to the velocity of a solid spheroid with the same volume and the same value of E (referred to as the equivalent spheroid) moving in an ideal fluid, then lo u(t) = Q(B/F{E(t))

and initial conditions

c(r>r2,t>O)=c2 c(r1
(9

has the following solution l1 -13 :

2

c=-

-

rz(c2 - :,)cos..)

(2)

where

(r2c2

x exp

F(E) = (1 - E2)h - E arcsin (1 - E2)k 1_ arcsin(1

-E2)t

(E2 - 1)) -E

vdr-rd xr

(6) The average interfacial

arccosh E

flux fi is defined by

to+T

E ’ ’

t,+T

s

N(t)dt

to+T

s

=;

to

E(t)dt

j

DA’(t) %/,=,

dt (7)

to

When c2 = 0 we have

to and

&

t,+T

1 llt = T

-

E
- E(1 - E’)tE

F(E) = {arccosh E - E(E2 - 1); )E

sin [ N;-r,,)

=r n=r CI

I

5F

I=r2

u(t)dt s to

(

c1 r,t2zI x r2

-

exp(-nyfDT))

(64

From the measured values of the function A(t) we can calculate the flux fi as well as the mass transfer coefficient K, which is defined by,

The spheroid area is given by (1 +E’(l

=

K =@,A

-E’)-+-x

(8)

where x arctanh (1 - E2)k} 1 +Ea(Ea

E>l

E<1 _ l)- karcsin

to + T

1 II @!$.)!

(3)

We assume that the oscillating drop and its surroundings are ideally mixed and that the concentrations inside and outside the drop are cr and ca , respectively (cr is the equivalent concentration, i.e. the actual concentration multiplied by a distribution coefficient). At the drop interface a film of thickness x = r2 - rl exists in which only molecular diffusion can occur. For a spherical drop of radius ra, the diffusion equation

a2c ac ar=Dar’+;;iT

2

ac (4)

A(t)dt J‘ to

The interfacial film thickness can be estimated from Levich’s erjuation l4 as was suggested for a similar case by Marsh : x = 25qlvtp EXPERIMENTAL

The shapes of oscillating drops and their velocities were determined from motion pictures of the drops moving through a glass column. The experimental apparatus has been described previously15J6. All liquids used (originally of analytical quality) were

(9)

187

THE BEHAVIOUR OF SINGLE OSCILLATING DROPLETS

redistilled twice and the apparatus was maintained scrupulously clean. The results were obtained mainly for the system benzene (continuous phase)water (dispersed phase)-acetic acid (component being transferred from water to benzene). Drop volumes were 0.080-0.120 cm3 depending on the acid concentration. At a predetermined position in the column the drop was captured and isolated in a Teflon bowl of volume 0.43 cm3. Electrodes were located at the bottom of the bowl for a conductometric determination of the average acetic acid concentration. The root mean square deviation of ten measurements was about f 5%. This method, even though it needs a number of calibrations, allows a relatively rapid and precise determination of the concentration of the transferring component inside the drop, for an arbitrary position in the column.

0 0: 63 G4 20

a5

" cmls

06

0

10

7

1.2 A

RESULTS AND DISCUSSION

A, 10

The measured frequencies of drop oscillation in the two-phase equilibrium systems were best fitted by the empirical correlation of Edge and Grant* (Fig. 1); measured values of one period of oscillation were compared with calculated values. It should be noted that this correlation was derived for water as a continuous phase2. It was foun,d later14 that it also fits the dependence of the oscillation frequency on drop volume. The drop shapes which occur during one period of oscillation are shown in Fig. 2. This figure relates to

Fig. 1. Comparison of the oscillation periods measured in this work (TB) with the data (Tc) calculated according to Lamb4 (o), Miller and &riven’ (o), Schroeder and Kintner3 (0) and Edge and Grant* (0). The systems used were (1) waterbenzene; (2) water-n-hexane; (3) water-n-heptane; (4) waterdiethylether.

0

05

t/T

'

Fig. 2. Variation of the instantaneous drop velocity v, the interfacial surface area A and the parameter E during one measured values; - - - period of oscillation: values calculated with the spheroid approximation. The numbers in the upper part of the figure refer to the instantaneous drop shapes which are shown on the right-hand side.

a water drop (volume 0.120 cm3) falling through benzene. The same type of changes in drop shape was observed also in other liquid-liquid systems of low viscosity, e.g. water-n-hexane, water-n-heptane; water-cyclohexane, water-toluene and carbon tetrachloride-water (the phase mentioned first is the dispersed phase). The smaller the drop, the smaller the oscillation amplitude. At a certain critical size the oscillation becomes indiscernible. The amplitude of oscillation is very sensitive to the purity of the system used, but for our systems with twice redistilled liquids (originally of analytical purity) the amplitudes were found to be perfectly reproducible. As can be seen from Fig. 2, the drop shapes are very complex and we have not attempted to describe them analytically. The measured time dependence of the parameter E characterizing the drop shape is also shown in Fig. 2. Here again, the smaller the drop, the smaller the difference between the maximum and minimum E values. In the water-benzene system, no amplitude damping was observed over the first six periods. However, according to the theory of Miller and Striven’ , the amplitude damping is

188

V. VACEK, P. NEKOVAA

3l Qe

VCC [cm/s]

c .

20

0

IO

/.i 15

vtC[cm/s]

25

Fig. 3. Comparison of measured (UtE, this work) and calculated (utC, according to ref. 7) terminal velocities of oscillating drops. The systems used were: o water-benzene; owater-n-heptane; o water-diethylether; e water-cyclohexane; l water-n-hexane; 4 carbon tetrachloride-water.

small and within the limits of experimental error for these systems (the linear dimensions of the drops were measured with an error of 4%). Measured values of the drop terminal velocity ut were well described by the empirical relation of Thorsen et al.’ (Fig. 3). This correlation is based only on experiments with water as the continuous phase. A comparison of the measured instantaneous drop velocity (full line) with that (broken line) calculated according to relation (2) is given in Fig. 2. The similarity of the two lines indicates that the velocity of the drop is largely determined by the instantaneous drop shape. These results are described in greater detail elsewhere 16. TABLE 1 The effect of the concentration

The measured time dependence of the drop interfacial area, determined by numerical integration of the drop meridional profile, is shown in Fig. 2 (full line). The corresponding broken line represents the interfacial area of the equivalent spheroid, which was calculated from eqn. (3). It is evident that the approximation of the drop shape by a spheroid significantly underestimates the real values. The average real value is 1.14 cm2 whereas the average value based on the spheroid approximation is only 1.04 for this case. Moreover, the shape of the function A(t) differs considerably from that given by eqn. (1). Nevertheless, the mass transfer coefficients published so far are based on the spheroid approximation of drop shapes. The addition of a third component changes the physical properties of the system. Mass transfer across the interface can also contribute to changes in drop behaviour. The measured effects of the acetic acid concentration on the characteristics of the drop behaviour are summarized in Table 1. The maximum values of E decrease almost linearly with increasing concentration of the third component, whereas the minimum values of E as well as the oscillation frequency remain almost constant. The terminal velocity ut decreases with increasing acetic acid concentration. This is a consequence of the opposing effects of decreasing drop size7v15 and increasing mass transfer rate across the interface”. The drop shapes and their changes remain almost constant up to an initial concentration of 3 mol 1-l . For higher concentration levels the shapes become more spherical. Mass transfer coefficients averaged over one oscillation period were measured for three concentration

of acetic acid on drop behaviour

Initial concentration c of acetic acid in drop (mol 1-l)

Volume V of drop

TermiMl drop velocity q

(cm3 )

Z266 0:397

0.105 0.096 0.092

(cm s-l)

Perbd T of one oscillation (s)

E max

18.5 18.6 18.3

0.104 0.102 0.098

1.30 1.27 1.24

Drop shape characteristics Emin

E 0.82 0.80 0.79 0.78 0.75 0.65 0.61 0.55 0.50

0.606

0.086

17.9

0.094

1.21

0.45 0.44 0.44 0.43

1.04 1.98 2.98 3.82 4.59

0.070 0.055 0.041 0.035 0.030

17.8 17.4 17.2 16.8 16.6

0.093 0.090 0.092 0.091 0.092

1.17 0.90 0.82 0.69 0.56

0.43 0.44 0.43 0.44 0.45

All drops were formed at the tip of a nozzle of inner diameter (wetted) 0.1375 cm. The flow rate of the dispersed phase was a constant 0.0121 cm3 s-l for all cases. Emax and Emin represent the maximum and minimum values of the characteristic E during an oscillation period; the time averaged value is 8.

THJ? BEHAVIOUR OF SINGLE OSCILLATING DROPLETS TABLE 2 Mass transfer coefficients in the benzene-water-acetic system Mass transfer coefficient K (cm s-l)

acid

D K ;

Origin

r

rl based on measured values of A(t) based on spheroid approximation

0.0131 0.0128

This work; measured value This work; calculated value

0.0144 0.0124

This work; measured value This work; calculated value

0.012 0.012 0.011 0.016 0.020

Angelo et al.“; theory ’ Brunson and Wellek’ ; theory Brunson and Wellek’; an alternative theory Brunson and Wellek’; modified Higbie theory Brunson and Wellek’ ; modified HandlosBaron theory Wellek and Skelland9*19; empirical correlation Ellis9*‘*; semiempirical correlation

0.030 0.059

189

r2

t T U vt V X

diffusion coefficient, cm2 s-1 overall mass transfer coefficient, cm s-l constant mass flux, mol s-l radial coordinate, cm r2 -x,cm radius of the sphere, cm time, s period of drop oscillation, s instantaneous drop velocity, cm s-l average terminal drop velocity, cm s-l volume of a drop, cm3 film thickness, cm

Greek symbols defined in the text E viscosity of the dispersed phase, g s cm-’ 17 density of the dispersed phase, g cmm3 P REFERENCES

levels: 0.27,0.40 and 0.61 mol 1-l . All results are approximately equal with a value of K = 0.0 13 1 cm s-l for the actual average interfacial area of the drop. The spheroid approximation leads to a mass transfer coefficient K = 0.0144 cm s-’ . The coefficients calculated from eqns. (6a), (7) and (8) and from the experimental curve ofA imply that K = 0.0128 cm s- ‘. When the values of A(t) from the spheroid approximation are used, a value of K = 0.0124 cm s-l is obtained. From a comparison of the calculated and measured coefficients it is evident that the experimental values based on the measured function A(t) are in better agreement with the simple model involving the same function than with the spheroid approximation. This approximation incorporates errors into the experimental as well as the calculated coefficients so that the agreement between the measured and calculated values becomes worse. The mass transfer coefficients calculated for our case accordin to relations recommended in the literature’*‘*- 9’ are given in Table 2. The simple model presented here seems to give better results than the other models considered.

NOMENCLATURE

A Ao c

interfacial area of a drop, cm* surface of the sphere, cm2 concentration, mol 1-l

S. Winnikow and B. T. Chao,Phys. Fluids, 9 (1966) 50.

: R. M. Edge and D. C. Grant, Chem. Eng. Sci., 26 (1971)

1001. 3 R. R. Schroeder and R. C. Kintner, A.Z.Ch.E.J., I1 (1965) 5. 4 H. Lamb, Hydrodynamics, Dover, New York, 1932. 5 A. C. Miller and L. E. Striven, J. Fluid Mech., 32 (1968) 417. 6 S. V. Subramanyam, J. Fluid Mech., 37 (1969) 715. I G. Thorsen, R. M. Stordalen and S. G. Terjesen, Chem. Eng. Sci., 23 (1968) 413. 8 R. J. Brunson,A.Z.Ch.E. J., 19 (1973) 858. 9 R. J. Brunson and R. M. Wellek, Can. J. Chem. Eng., 48 (1970) 267. 10 A. M. Luiz, Chem. Eng. Sci., 22 (1967) 1083;24 (1969) 119. 11 J. Crank, Muthametics of Diffusion, Clarendon Press, Oxford, 1964. 12 B. D. Marsh, Ph.D. Thesis, University of Washington, Seattle, 1966. 13 B. D. Marsh and W. J. Heideger, Znd. Eng. Chem. Fundam. 4 (1965) 129. 14 V. G. Levich, Physicochemical Hydrodynamics, PrenticeHall, Englewood Cliffs, N. J., 1962. 15 V. Vacek and P. Nekovif, Paper presented at The IVth CHISA Conaress. Praae. I9 72. 16 P. Nekovii and v. Vacek, Paper presented at The XVIII Conf. CHISA, Brno, 1973 (in Czech), in Collect. Czech. Chem. Commun., 41 (1976) 3528. 17 H. Sawistowski, in C. Hanson (ed.), Recent Advances in Liquid-Liquid Extraction, Pergamon, Oxford, 1971. 18 W. B. Ellis, Ph.D. Thesis, University of Maryland, 1966. 19 A. H. P. Skelland and R. M. Wellek, A.I.Ch.E. J., 12 (1966) 751. 20 J. B. Angelo, E. N. Lightfoot and D. W. Howard, A.1.Ch.E J., 10 (1964) 491.