Mass transfer in a particle bed with oscillating flow

Clwmico, EnytneerinyScience, Vol. 44, No. 10, pp. 2107 211 I, 1989. Printed in Great Bnlain.

@IO9 2509/89 $3.OO+D.o0 0 1989 Pergamon Press plc

MASS TRANSFER IN A PARTICLE OSCILLATING FLOW

BED WITH

J. S. CONDORET I.N.S.A., Avenue de Rangueil, 31077 Toulouse, France and J. P. RIBA and H. ANGELINO+ E.N.S.I.G.C., Chemin de la Loge, 31078 Toulouse, Frlnce (Receiued 5 March 1987; accepted for publication

27 February

1989)

Abstract-Mass transfer in a particle bed with oscillating flow has been experimentally studied by an electrochemical method. A correlation for avcragc mass transfer over a period has been established. Instantaneous mass transfer exhibits a phase lag with a velocity that has been explained by the time response of the concentration boundary layer.

INTRODUCTION

Scientific works dealing with liquid-solid mass transfer in the presence of an oscillating flow are mainly concerned with oscillating flow around an isolated sphere. Altaweel and Landau (1976) have given an exhaustive bibliography concerning the subject. It is useful to recall the different kinds of physical phenomena that have been pointed out in the previous works. The frequency ‘7” is high and the amplitude “a” is small compared (particle or tube sponds

to a characteristic dimension L diameter). This field of study corre-

to

Mass transfer is then controlled by “acoustic streaming” which is the pattern of steady flow that prevails close to the sphere. Theoretical studies of flow pattern have been made by Lane (1955), Landau et al. (1973) and Davidson and Riley (1971). Jameson (1964) pointed out differences between liquids and gases for mass transfer in such a flow arrangement. The frequency is low (a few Hertz) and the amplitude has the same order of magnitude as the characteristic dimension. Mass transfer is only dependent on the oscillating velocity component. Mora et al. (1969) have established a theoretical expression for mass transfer based upon the hypothesis of the quasisteady-state model. Recent works of Sudhakar et al (1979) and Ohmi and Usuit (1982) must be added to the review given by Altaweel and Landau. The limit of these two fields has been pointed out by Gibert and

These results do not allow predicting mass transfer when the particle is located in a particle bed because the flow arrangement in the interparticular void is rather different. Besides, diffusion boundary layers around particles could be organized in a different way. The only available works in a particle bed, done by Krasuk and Smith (1964) considered the same parameters used in the case of an empty tube (Krasuk and Smith, 1963). These last works have been realised with a pulsed flow (oscillating + steady flow). This Iack of information leads us to an experimental work to obtain the needed parameters. The final aim of our work is to obtain the necessary data for the modelling of liquid-solid pulsed reactors. In the case of liquid-solid reactions when mass transfer is an important or a controlling step, increasing agitation and, in this way, mass transfer could be obtained by superposition of a sinusoidal or periodic movement upon a steady flow. The range of operation of such reactors belongs generally to the second case described

here, that is :>0.75.

ence of a steady

flow becomes rapidly negligible and pulsed flow can be approximated by oscillating flow. All experimental works for either the isolated sphere or the particle bed have concerned the average value over a period of the mass transfer coefficient. We intend here to develop an experimental method that leads to the instantaneous value of the mass transfer coefficient. Mechanisms of liquid-solid mass transfer could be better pointed out and the validity of the quasi-steady-state approximation be studied. EXPERIMENTAL

Angelino

(1972) as :=0.75.

An explanation

was given

by Altaweel and Landau (1976) and is related presence or not of a downstream wake.

‘Author lo whom correspondence

to the

should be addressed.

In this range the influ-

STUDY

The experimental work has been devoted to the study of mass transfer around an active sphere located in a bed of inert spheres. This has been done by an electrochemical method. The technique uses the diffusion of an ion in a dilute solution and its reaction at the surface of an electrode 2107

2108

J. S.

CONDORET

(i.e. the active sphere). The sphere is polarised within the bed of inert spheres to a potential so that the surface eIectrochemicaI reaction is only controlled by mass transfer. The domain of potential which can be used is determined by the intensity YS potential curve and corresponds to the flat part of the curve. This allows direct measurement of the mass transfer coefficient. The apparatus is presented on Fig. 1. It consists of an X-cm diameter altuglass column. The lower part is the “calming section” which is filted with glass spheres (4 mm in diameter). The main purpose of this section is to eliminate any channelling and to obtain a flat profile for the velocity distribution. The bed under study, situated above, has been filled with glass spheres (4 or 6 mm in diameter). The two beds are separated by a grid. A fixed grid at the top of the bed prevents any fluidization phenomena. Two sets of four probes (golden brass spheres) are located at 12 and 21 cm above thebottom grid (Fig. 2). One probe is in the axis, while the others are radially distributed at r= 2 cm (Fig. 2). We have compared the signals delivered by the four probes at the same level: the difference was less than 5%. We have interconnected the probes in order to obtain an average signal. We also established that the maximum difference between probes located at two different levels was less than 5.4%. The pulsation has been generated using a piston and a direct current motor: the amplitude and frequency can be adjusted up to 2.5 cm and 3.7 Hz. A circuit using a proximity detector and a mark on the motor axe allows one to follow the piston position and in consequence the instantaneous average velocity of the fluid. The signal given by this detector has been treated to give an accurate value of the frequency. The particle arrangement in the bed has been kept constant and reproducible by using the following method: the bed is held under oscillation without the top grid for several hours under such conditions that partial fluidization occurs. The bed is then blocked and it can be seen at the wall that the bed presents a perfectly organized arrangement. This explains the

I

I

et al. SOmm

I--

300mm

90mm

I 120mm

4

1

200

Botto‘m

mm

1 Longltudlnal

view

Radial

= colmmg section = proximity detector = blucklng grrd = bed under study = drrect-current motor = Tel ton p,ston

small influence of the axial and radial positions probe upon mass transfer. INSTANTANEOUS

MASS TRANSFER

of the

RESULTS

General description Figure 3 shows a typical signal delivered by the probes. The mass transfer signal presents a succession of regular arches. The signal from the detector, whose breaking of is related to the bottom dead center of the piston, exhibits a phase lag qn with the mass transfer signal. The intensity of the mass transfer signal is in the range of a few milliamperes. The shape of the signals suggests an equation of the type k=k,+AJsin(2nft+yl)/.

(1)

The absolute value in this formula corresponds to the fact that mass transfer phenomena only take into account the absolute value of the velocity. The presence of a residual value k, and a phase lag cp does not agree with the quasi-steady-state hypothesis that

1 0 t

Fig. 1. Experimentalapparatus.

v,ew

Fig. 2.

Proxlmlty

C D G L M S

grid

(51

Fig. 3. Experimental signal.

detector \

sgnol

Mass transfer in a particle bed with oscillating flow predicts a zero transfer at the velocity inversion and a phase lag between velocity and transfer. The values of k,, A and q have been identified from experimental results with a Gauss-Newton method of minimization of quadratic criteria.

I*1 I01 [ml (01 (11 (Al

zero

of the phase lag The phase lag can be explained by use of the model established by Reiss and Hanratty (1963). This model is based upon the existence of a viscous sublayer close to the wall and a concentration boundary layer whose thickness is small compared to the viscous layer. This last property is valid for high Schmidt numbers (liquid case). Using different simplifying hypothesis these authors have established a relationship between the axial velocity component fluctuation (ur) and the mass transfer coefficient fluctuation (k/j:

25

SC = SC = SCSC= SC= SC-

2109

8000. a=001 8000; Q -0.02 11300.0=0.01 1800,o=0.02m, l800,a-0.01 1800,o-002~11,

m. d, =0006m m; d; = 0.006 m m, d, =O.O06m dp =0.006m m, dp =O.O04m dp -0004m

r

Study

O(11111 00

0.5

1.0

1.5 f

Fig. 5. Experimental

2.0

25

3.0

3.5

(Hz1

phase lag of the instantaneous

transfer

mass

coefficient.

tained. Figure 5 presents experimental values of the phase lag expressed as a percentage of the period

r=_

ar(4/3)92’3

~

2

l/3

Le’

_ (

-1 DifPZ

That time constant implies a capacitance effect of the concentration boundary layer. When a sinusoidal velocity fluctuation is considered the model predicts for the mass transfer coefficient response a phase lag cp and an amplitude S,. (see detailed calculations in Appendix):

(2)

The experimental phase lag increases with frequency and particle diameter. These variations are in agreement with eq. (2) when electrode length is considered as equivalent to particle diameter. The augmentation of 50 with a decrease in diffusivity (i.e. an augmentation of the Schmidt number) is also predicted by eq. (2). The decrease in cpwith an increasing amplitude could be explained by the hypothesis of an increase in S, with amplitude. of k, and A Experimental results show an increase in A and k, (parameters of eq. (1)) with frequency f (Fig. 6). Augmentation of A is slower and values seem to tend towards a limit that we did not reach in our experimental domain. The residual value of k, is probably due to turbulence that does not disappear at velocity inversion. This phenomenon is emphasized by an increasing Study

We present in Fig. 4 the results of the average wall gradient (S,) obtained from experimental values of cp and eq. (2). This last equation allows one to predict different tendencies of CJJwith the parameters. Specific experiments on the phase lag were conducted after the addition of polyethylene glycol to decrease the ion diffusivity and increase the viscosity. Schmidt number values up to 8000 were ob-

“r

.

4 3000,

:

t

0

.

2500dp’4nm 2000

-

.

0

o

sc-ll3oc

0

0

.

.

7

” Ii-

. 1500-

.

0 0

.

l

,*i

IOOO-

500

,.,

.

c

08 0.0

.

a-

Icnl

(01

A

x IO4 m/s

o-*cm 0

l

0 0 5

1.0

15

2.0 f

2.5

3.0

(Hz)

Fig. 4. Wall gradient from experimental phase lag.

3.5

I I

I 2 f

Fig. 6. Amplitude

and residual

I 3

1 4

[Hz)

value of instantaneous

transfer.

mass

2110

J. S. CONDORET

freqeuncy and constitutes the main contradiction with the quasi-steady-state hypothesis. This residual value is not predicted by Reiss and Hanratty’s model. An inweasing

amplitude

S, [eq.

(3)] is predicted

but S,

CONCLUSION

k,+A=S,. Experimental values of A (= S, - k,) tend towards a limiting value that means that k, and S, present the same type of evolution. VALUE

The increase in mass transfer observed in the particle bed is due to the presence of other spheres that modify favourably the organisation of diffusion and hydrodynamic boundary layers.

to A from eq. (1) but to the sum

does not correspond

AVERAGE

a al.

OF MASS TRANSFER

OVER A PERIOD

The influence of the amplitude and frequency pulsation have been studied in the ranges 0.5iac2.5

of the

cm

0.2
to a”fp

leads to the value of a=0.501 and fi=O.SOO. The equality of these two exponents suggests the use of the vibrational Reynolds number (Re,) already used by previous workers (Gibert and Angelino, 1972). The influence of the Schmidt number (SC) has been pointed out by varying the temperature of the solution between 17 and 27°C. Parametric identification of the vibrational Sherwood number (Sh,) gives Sh, = 1.87Rez.444

Better knowledge of mass transfer in a particle bed with oscillating flow has been given by experimental work that leads to a correlation for the average mass transfer coefficient. This coefficient appears to be related to the vibrational Reynolds number. The average mass transfer was found to be higher than in the case of the isolated sphere. Instantaneous mass transfer was also studied and presents characteristics such as residual value and phase lag that disagree with the quasi-steady-state hypothesis. The use of a liquidPsolid mass transfer model from Reiss and Hanratty allows one to relate the phase lag to the capacitance effect of the diffusion boundary layer. This work will be one of the necessary steps for modelling and analysing pulsed reactors because pulsation modifies other parameters such as residence time and mechanical energy consumption that must be known to achieve that purpose. NOTATION

a A Diff

s~O.33~.

For the rest of the study the exponent of the Schmidt number was taken as l/3 according to boundary layer theory. The experiments conducted with 4- and 6-mm particles lead to the following correlation:

.P ko k

Sh,= 1.75Re;.45g SC”~ 20 < Re, -=c700, 1300 -c SC < 2700 and the results are presented correlation is also plotted:

in Fig. 7 where Gibert’s

kf K L

Sh, = 0.661 Refs3*

for

;>

1.5.

I Le Sx St S*

t T u/ u u, Re, Fig. 7. Experimental

average mass transfer results.

amplitude of the pulsation, m amplitude of the sinusoidal mass transfer coefficient, m s ’ diffusion coefficient m2 s ’ particle diameter, A frequency of pulsation, s-l persistent value of instantaneous mass transfer coefficient, m s-l instantaneous mass transfer coeficient, ms-’ fluctuation of mabs transfer coefficient, ms-’ average value over a period of mass transfer coefficient, m s ’ characteristic length, m direction of instantaneous flow, m electrode length, m average value of wall gradient in main flow direction, s I wall gradient in the instantaneous flow direction s - ’ amplitude of Hanratty’s response model time, s pulsation period, s fluctuation of velocity parallel to main flow, m s-l average value of velocity parallel to main flow, rns-’ instantaneous velocity of the flow, m s- ’

Greek letters exponents = 0.893

4 P I-(4/3)

Mass transfer

il IL

wavelength

v

kinematic viscosity, m2 s- 1 specific weight, kg m- 3

dynamic

of pulsation, viscosity,

P z

time

rp w

phase lag, rad angular velocity

constant,

in a particle

m

kg m -

flow

between

quantities

fluctuating

2111 of k and u:

’ s- 1

s (2nJ),

rad

T=

s- 1

Dimensionless

Rev Sk SC

bed with oscillating

numbers vibrational Reynolds number (4afd,p/p) vibrational Sherwood number (I?d,/&,) Schmidt number (v/D,,,)

We have computed the response of this model to a sinusoidal input by using Fourier’s transform: S(t) = k,/K

E(t) =uJu

so that

XE(t)l

REFERENCES

Altaweel, A. M. and Landau, J., 1976, Can. J. them. Engng 54, 533. Davidson, B. J. and Riley, N., 1971, J. Sound, Vibr. 15, 217. Gibert, H. and Angelino, H., 1973, Can. J. &em. Engng 51, 319. Jameson, G. J., 1964 Chem. Engng Sci. 19, 793. Krasuk, J. H. and Smith, J. M. 1963, Chem. Engng Sci. 18,591. Krasuk, J. H. and Smith, J. M. 1964, A.1.Ch.E. J. 10, 759. Landau, J., Dim, A. and Houlihan, R., 1973, Metall. Trans. 4, 2827. Lane, C. A., 1955, J. Acoust. Sot. Am. 27, 1802. Mora, Y., Imabayashi, M., Higikata, K. and Yoshida, Y., 1969, Int. J. Heat Mass Transfer 12, 571. Ohmi, M. and Usuit, T., 1982, Trans. ISIJ 22, 600. R&s, L. P. and Hanratty, T. J., 1963, A.I.Ch.E. J. 9, 154. Sudhakar, B., Venkateswarlu, P., Bhaskara Sarma, 0. and Jagannadha Raju, G. J. V. 1979, Indian J. Technol. 17,253.

and

3S(t) = E(t) -p. By using Fourier’s

and the transfer

at

transform

function

1s

H*(p) = S’(p),E*(p)=+ Since there is no influence of velocity direction transfer, fluid pulsation is considered by a model U =2nfaj

sin

on mass like

Znftl.

The response of the model for such an input function is rather difficult to compute and we have taken the following approximation, U = 4af+ 2af sin Znff, so that u, = 2af sin 2lcft

APPENDIX

Reiss and Hanratty (1963) have established a model from hydrodynamic and mass transfer equations when the concentration boundary layer thickness is small compared to the hydrodynamic boundary layer. Therefore, a linear variation of the average velocity in the axis of the Row can be used. With some simplifying hypothesis the model gives 8C

PC at+s,,;=D,,,-. w This equation has been solved using the quasi-steady-state assumption. The authors obtained the following relationship

and 1

E(t)= i sin 27cft. The use of the transfer with

function

gives S(t) = S, sin (2rcfi + p)