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Mass transfer enhancement induced by a steady interfacial cellular convection
The Chemical Engineering Journal, 17 (1979) 237 - 243 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
237
Short Communication Mass transfer enhancement induced by a steady interfacial cellular convection
G. ANTONINI
and A. ZOULALIAN
Dbpartement de GEnie Chimique, Technologie de Compikgne, B.P. Compikgne (France)
Universite’ de 233, 60206
G. GUIFFANT L.B.H.P., Universite’ de Paris VII - 2, Place Jussieu, 75005 Paris (France) (Received
15 July 1978)
A hydrodynamical model of interfacial mass transfer is presented. The solution is obtained numerically by solving the full form of the diffusion-convection equation and shows the dependence of the Sherwood number for both free and solid-fluid interfaces on the Peclet number. Results are discussed with reference to classical renewal models.
1. Introduction The role that interfacial convection may play in the mass transfer at free or solid boundaries between two phases (gas-liquid or solid-fluid interfaces) has received considerable attention during recent years. This convection may arise from the interface itself (“surface-engine”) or may result from the bulk turbulence of the liquid phase impinging on the interface. The first case is referred to as free-type interfacial turbulence and the second as forced-type interfacial turbulence. Both types are known to contribute to a large extent to the enhancement of mass transfer in systems in which the phenomenon commonly called interfacial turbulence occurs. However, the term turbulence is certainly more appropriate for interfacial phenomena involving disordered convection or highly irregular transient flows than for the ordered-flow interfacial convection also referred to as interfacial instability occurring in those systems.
Ordered-flow interfacial convection is generally restricted to a thin layer of fluid adjacent to the interface and can be viewed as two- or three-dimensional steady roll-cells such as also occur in evaporating layers [l] . It has also been stressed that, in some conditions, trains of pulses due to the bulk turbulence impinging on the interface can lead to the formation of such steady rotating structures in its vicinity [ 21 . Hydrodynamical models of mass transfer at interfaces, especially roll-cell models compatible with the observed well-orderedflow interfacial convection, are certainly needed, despite the complexity of integrating the convective-diffusion equation in recirculative flows. Furthermore, in this type of approach the renewal is not assumed, as in classical [3, 41 or more recent penetration models [ 5 - lo], but is a consequence of the selected velocity distribution. Mass transfer in a steady roll-cell model of the interface has been considered by Fortescue and Pearson [ 111 in their study of the influence of turbulence on gas absorption into a turbulent liquid. They assumed that, although turbulent flows contain a wide spectrum of eddies of constantly varying nature, their mean mass-transfer properties could be well modelled by steady square rollcells touching the surface (large-eddy model). Arguments have been adduced by Ruckenstein [ 12 - 141 in support of a roll-cell model to account for enhanced mass transfer in systems in which agitation of the interface may arise because of either bulk turbulence or free interfacial convection. This approach, based on idealized eddy structures of interfacial convection, has been further developed by Lamont and Scott [15] with a small-eddy model and more recently by Theofanous et al. [16] who include transient effects in mass transfer. Mass transfer with buoyancy-driven convective cells has been studied recently by Javdani [ 171 who suggests that it accounts for the high mass-transfer rate observed when a cellular motion occurs.
238
It is the purpose of this communication to calculate the mass-transfer resistance using eddy-cell models of the interfacial convection near both free and solid limits. Mass-transfer results, as governed by a convective interfacial motion, will be obtained numerically by solving the full form of the convectivediffusion equation for the Peclet number varying from 0 to 100. Close attention will be payed to the low Peclet number limit which has been somewhat neglected in the literature. It is also shown that this type of hydrodynamical model can be used to derive the typical power-law dependence of interfacial mass transfer. 2. Formulation of the problem We consider a liquid layer having a plane free fluid or solid-fluid interface undergoing steady cellular convection. The steady roll-cell pattern for a free fluid interface will be represented by a bidimensional velocity field analogous to the one first proposed by Fortescue and Pearson [ 111 and subsequently utilized by Ruckenstein [12] and by Theofanous et al. [16]. The flow pattern within the fluid layer is thus represented by a sequence of square roll-cells with alternating vorticity, each presenting an up-welling motion on one side and a corresponding down-current on the other. Thus, by using characteristic length and velocity scales L and uo, the corresponding stream function in dimensionless form can readily be written as 1
$ = - - sin 71x sin ny n forO
and
with a velocity
field
u, = - ? -sin ay
71x cos 7ry
(1)
u,(x, 1) = 0. A possible corresponding function can be written as
(2)
ati u, = - G = cos nx sin ny In an analogous way, the stream function representing the periodic steady roll-cell pattern for a fluid-solid interface has been chosen in order to satisfy the solid-fluid hydrodynamical boundary conditions, that is: u,(x, 0) = u,(x, 0) = 0 and, at the limit of the bulk, au,(x, l)/ay = 0 together with
(3)
The coefficients a,, have been calculated numerically to satisfy the previous boundary conditions together with the condition of zero shear rate at the limits x = 0 and x = 1 and uX($, a) = 1, this last condition giving an estimate of the position of the vortex within the cell. The a, are: a2 = 47r, a3 = -15.09, a4 = -10.64, a5 = 20.19, and as = -7.02. Again, the corresponding velocity field components can be readily obtained using u, = a$/ay, U, = - a+/ax with (3). . A typical eddy-cell structure for (1) or (3) is shown in Fig. 1 with coordinate system and length scales. c-0
30 Y
CZI
Y
Fig. 1. Typical eddy-cell structure near the interface with coordinate system.
The full form of the dimensionless convection-diffusion equation to be solved is
ac 2
O
6 x anyn tl=Z
1 4 =--sinnx x
stream
+(v’v)c-
-
$
v2c
where v is obtained either using (1) or (3) and Pe is the Peclet number Pe = Luo/D, D being the molecular diffusion coefficient. The concentration boundary conditions to be used are: c=l
for
y=O
c=O
for
y=l
In both cases the eddy is exposed at time t = 0 (with c = 0 everywhere) and participates in mass transfer until a steady state is reached. In the case of a free fluid interface (that is, absorption of a gas at the interface) a zero gas-phase resistance to mass transfer is
239
assumed, which is equivalent to considering that a thermodynamical equilibrium is reached at the surface y = 0. This is also generally assumed in the case of a solid-fluid interface. At y = 1, the boundary condition implies a complete and rapid mixing with the bulk. We note that this set of boundary conditions is similar to those used in the filmpenetration theory [ 5, 61. Owing to the symmetry of the flow pattern, a zero mass flux will be assumed across the vertical cell boundaries for both solid-fluid or free fluid conditions: Ar ar -_ G (O,Y) = E (13) = 0 (6) 3. Numerical method and results Equation (4) was solved by a finite-difference method. Spatial derivatives were all approximated by three-point central differences and time derivatives by two-point forward differences. For the non-linear term, we used a modified form of forward or backward difference, first proposed by Torrance [ 181, which preserves the stability of the numerical scheme. A stability criterion for this scheme has also been derived which is dependent on the Peclet number. In order to test the accuracy of the numerical method, various runs with different mesh sizes were performed for each Pe and the results corrected correspondingly. Convergence of the method was obtained with mesh sizes varying from 21 X 21 for the lower Peclet numbers to 21 X 101 for higher ones. This mesh-size refinement in the ydirection for the higher values of Pe was necessary to preserve the convergence of the numerical solution when increasing compression of the concentration contours occurred because of convection within the cell. The numerical investigations were carried out for Peclet numbers varying from 0 to 100, for the two flow field conditions (free fluid and solid-fluid interface). The mass transfer across the cell is described using a Sherwood number averaged over the length of the cell for each value of the Peclet number: Sh=/
E(.r,O)dr
(7)
this expression being evaluated when no significant modification of the concentration distribution within the cell is observed (AC/C < 10e6). This definition of the Sherwood number is appropriate to situations in which the flow pattern is maintained for a sufficiently long time. However, for situations in which an exposure time (lifetime) is to be considered, a time-averaged Sherwood number would be more appropriate to describe the net transfer during time of exposure: f%(t)
= ;- /
Sh(t’) dt’
0
where t is the exposure time. It should be noticed that SK(t) decreases much slower than Sh( t) and that, when Sh( t) has reached the stationary value Sh, defined in (7), a strong discrepancy between & and Sh is still observed, especially for the lower values of the Peclet numbers (-40% discrepancy when Pe = 0.8, Fig. 2; and only -16% when Pe = 25, Fig. 3). Nevertheless, for sufficiently long times Sh and sh converge to the same value. The Sherwood number Sh thus defined will be considered to be representative of the mass transfer across the stationary convective cell. One can note two main modes of behaviour for the steady mass flux uersus Peclet number. For the lower Peclet numbers the Sherwood number has been found to increase as Pe3/’ for both solid-fluid and free fluid conditions (Fig. 4), whereas for higher values of Pe the Sherwood number has been found to vary as Pe1j2 (i.e. k - D1j2) for the free fluid interface and Peli3 for the fluid-solid interface (i.e. k D2’3). Between these two regions, the transition is essentially linear (Fig. 5). The low Peclet number mass transfer has been the subject of only a few investigations using the hydrodynamical approach [ 12,131 (except in the related problems of transfer in two-phase particulate systems) owing to the presence of recirculative flows occurring in or around the spherical drops or solid spheres of these systems [ 19 - 231. Our result at the low Peclet number limit is that there is no essential discrepancy between Sherwood numbers for solid-fluid or free fluid interfaces, which is similar to the result of Bowman et al. [20]. Moreover, the explicit
240
Pez.8
I .2
.I
.3
.4
Fig. 2. Transient Sherwood numbers Sh(t) and m(t)
I
, 1
2
3
4
5
b
7
I
Fig. 3. Transient Sherwood numbers Sh(t) and a(t)
dependence of Sh on Pe3/’ in this limit was obtained by Leal [23] in his study of the effective conductivity of a dilute suspension of drops undergoing a simple shear with an imposed transverse temperature gradient. In his study, the role of drops was to disturb the main velocity field by creating recirculative flows somewhat similar to those considered in the present study. In the low Peclet number limit, the results of the hydrodynamical model of liquid-liquid interfacial mass transfer developed by
.5
.6
I
for Pe = 0.8 (free fluid interface).
9
IO
II
12
t
for Pe = 26 (free fluid interface).
Ruckenstein [ 12,131 were obtained by using finite expansions. His results, although very similar to those presented here, exhibit a certain discrepancy when numerical values are calculated. It is felt that the method used by Ruckenstein is the cause of this discrepancy. 4. Discussion In the case of a free fluid interface and for large values of the Peclet number, the main result is that the Pe ‘I2 dependence of the Sherwood number, already derived by means
241
.2
.4
.6
.8
l.
Pe
Fig. 4. Plot of the Sherwood number versus Peclet number 0 < Pe < 1, for both free and solid-fluid interface.
Fig. 5. Plot of the Sherwood number versus Peclet number 1 < Pe < 100; curve A corresponds to free fluid results and curve B to solid-fluid results.
of renewal models, is obtained without the renewal assumption but is a consequence of the velocity field near the interface within the cell. A power law Sh = 0.6Pe112 is suitable when Pe reaches values of about 25. However, it should be kept in mind that the velocity uo, used to obtain a dimensionless form of the diffusion equation to be solved when using hydrodynamical models of interfacial mass transfer, does not generally correspond to the mean effective velocity for the renewal within the cell. In order to illustrate this point, numerical results for the free fluid interface obtained for
Pe > 25 have been plotted in Fig. 6 together with a possible expression of the enhancement factor R [lo] derived by Dobbins [ 51 and Toor and Marchello [6] : R = da coth da
(9)
where a = L2SdlD, Sd being an average renewal rate of the disturbed system. One may consider a to be proportional to the Peclet number, i.e. a = ape. An estimate of (Ycan be obtained by fitting the numerical results (Fig. 5) for Pe > 25 with (9) in Fig. 6. This leads to 01 = 0.36, corresponding to a mean effective velocity uo/2.7 for the renewal within the cell.
242
Sh 6_
1. .... .... .... ..: 1'
2
3
4'
h
6
7
8
9
Fig. 6. Plot of the Sherwood number versus JPe
lo
11 '
",@e
together with expression (9).
Jn the case of a solid-fluid interface and for large values of the Peclet number, a power law Sh = 0.93 Pe113 is appropriate and is in good agreement with classical results. Similarly to the free fluid interface results, an expression of the enhancement factor equivalent to (9) can be used, which leads to a mean effective velocity ue/1.24 for the renewal within the cell. Although the ultimate mechanism of interfacial transfer must be by molecular diffusion, the influence of convective motions on the diffusion process is quite pronounced, especially for high values of the Peclet number. The effect of interfacial convection is to control the resistance to transfer by sharpening the concentration gradients near the interface, thus leading to the enhancement in the mass-transfer rate previously described as a function of the Peclet number. It should be stressed that it has been possible to account for the large differences observed in the mass-transfer rate between the free and solid interfaces only by modifying the hydrodynamical boundary conditions at the interface. That is, the fundamental difference is that for a solid interface all velocity components of the fluid must vanish at the interface allowing a non-zero shear rate at this interface, whereas in the case of a free fluid interface a zero shear rate can be assumed. Stationary kinematical models of recirculating flows near the interface can be used that account for the typical power-law dependence of inter-facial mass transfer as
long as the physical boundary conditions are fulfilled. However, in a forthcoming paper the generation of flow fields compatible with the momentum transport equation will be considered, in order to obtain a better differentiation in interfacial mass-transfer results between the hydrodynamical parameters and those characteristic of molecular diffusion only. Nomenclature a a,
; k L Pe R Sd Sh t V vo X Y
L2Sd/D, defined by eqn. (9) numerical coefficients defined by eqn. (3) volumetric concentration in the fluid molecular diffusion coefficient mass-transfer coefficient characteristic length scale of a cell Lu,/D, Peclet number enhancement factor defined by eqn- (9) average renewal rate kL/D, Sherwood number; s?; defined in (8) dimensionless time dimensionless velocity field within the cell with components u,., u, characteristic velocity scale for cellular convection dimensionless coordinate along the interface dimensionless coordinate normal to the interface
243
Greek symbols numerical coefficient a=aPe stream function ti
ar
defined
by
References J. C. Berg, M. Boudart and A. Acrivos, J. Fluid Me&., 24 (1966) 721. E. Ruckenstein and D. G. Suciu, Chem. Eng. Sci., 24 (1969) 1395. R. Higbie, Trans. A.I.Ch.E., 31 (1935) 365. P. V. Danckwerts, Znd. Eng. Chem., 43 (1951) 1460. W. E. Dobbins, in McCabe and Eckenfelder (eds.), Biological Treatment of Sewage and industrial Wastes, Part 2-1, Reinhold, New York, 1956. H. L. Toor and J. M. Marchello, A.I. Ch.E. J., 4 (1958) 97. N. G. Maroudas and H. Sawistowski, Chem. Eng. Sci., I9 (1964) 919. V. Linek, Chem. Eng. Sci., 27 (1972) 627. V. Linek, V. Machon and Z. Gall, Sci. Pap. Inst. Chem. Tech., K7 (1972) 217.
10 N. Imaishi, M. Hozawa and K. Fujinawa, J. Chem. Eng. Jpn., 9 (1976) 499. 11 G. E. Fortescue and J. R. A. Pearson, Chem. Eng. Sci., 22 (1967) 1163. 12 E. Ruckenstein and C. Berbente, Chem. Eng. Sci., 25 (1970) 475. 13 E. Ruckenstein, Can. J. Chem. Eng., 49 (1971) 62. 14 E. Ruckenstein, Chem. Eng. J., 2 (1971) 1. 15 J. C. Lamont and D. S. Scott, A.I.Ch.E. J., 16 (1970) 513. 16 T. G. Theofanous, R. N. Houze and L. K. Brumfield, Int. J. Heat Mass Transfer, 19 (1976) 613. 17 K. Javdani, Chem. Eng. J., 11 (1976) 1. 18 K. E. Torrance, J. Res. Nat. Bur. Stand., 72 B (1968) 281. 19 S. K. Friedlander, A.1.Ch.E. J., 3 (1957) 43. 20 C. W. Bowman, D. M. Ward, A. I. Johnson and 0. Trass, Can. J. Chem. Eng., 39 (1961) 9. 21 P. L. T. Brian and H. B. Hales,A.I.Ch.E. J., 15 (1969) 419. 22 I. Yaron and B. Gal-or, A.1.Ch.E. J., 19 (1973) 662. 23 L. G. Leal, Chem. Eng. Commun., 1 (1973) 21.
237
Short Communication Mass transfer enhancement induced by a steady interfacial cellular convection
G. ANTONINI
and A. ZOULALIAN
Dbpartement de GEnie Chimique, Technologie de Compikgne, B.P. Compikgne (France)
Universite’ de 233, 60206
G. GUIFFANT L.B.H.P., Universite’ de Paris VII - 2, Place Jussieu, 75005 Paris (France) (Received
15 July 1978)
A hydrodynamical model of interfacial mass transfer is presented. The solution is obtained numerically by solving the full form of the diffusion-convection equation and shows the dependence of the Sherwood number for both free and solid-fluid interfaces on the Peclet number. Results are discussed with reference to classical renewal models.
1. Introduction The role that interfacial convection may play in the mass transfer at free or solid boundaries between two phases (gas-liquid or solid-fluid interfaces) has received considerable attention during recent years. This convection may arise from the interface itself (“surface-engine”) or may result from the bulk turbulence of the liquid phase impinging on the interface. The first case is referred to as free-type interfacial turbulence and the second as forced-type interfacial turbulence. Both types are known to contribute to a large extent to the enhancement of mass transfer in systems in which the phenomenon commonly called interfacial turbulence occurs. However, the term turbulence is certainly more appropriate for interfacial phenomena involving disordered convection or highly irregular transient flows than for the ordered-flow interfacial convection also referred to as interfacial instability occurring in those systems.
Ordered-flow interfacial convection is generally restricted to a thin layer of fluid adjacent to the interface and can be viewed as two- or three-dimensional steady roll-cells such as also occur in evaporating layers [l] . It has also been stressed that, in some conditions, trains of pulses due to the bulk turbulence impinging on the interface can lead to the formation of such steady rotating structures in its vicinity [ 21 . Hydrodynamical models of mass transfer at interfaces, especially roll-cell models compatible with the observed well-orderedflow interfacial convection, are certainly needed, despite the complexity of integrating the convective-diffusion equation in recirculative flows. Furthermore, in this type of approach the renewal is not assumed, as in classical [3, 41 or more recent penetration models [ 5 - lo], but is a consequence of the selected velocity distribution. Mass transfer in a steady roll-cell model of the interface has been considered by Fortescue and Pearson [ 111 in their study of the influence of turbulence on gas absorption into a turbulent liquid. They assumed that, although turbulent flows contain a wide spectrum of eddies of constantly varying nature, their mean mass-transfer properties could be well modelled by steady square rollcells touching the surface (large-eddy model). Arguments have been adduced by Ruckenstein [ 12 - 141 in support of a roll-cell model to account for enhanced mass transfer in systems in which agitation of the interface may arise because of either bulk turbulence or free interfacial convection. This approach, based on idealized eddy structures of interfacial convection, has been further developed by Lamont and Scott [15] with a small-eddy model and more recently by Theofanous et al. [16] who include transient effects in mass transfer. Mass transfer with buoyancy-driven convective cells has been studied recently by Javdani [ 171 who suggests that it accounts for the high mass-transfer rate observed when a cellular motion occurs.
238
It is the purpose of this communication to calculate the mass-transfer resistance using eddy-cell models of the interfacial convection near both free and solid limits. Mass-transfer results, as governed by a convective interfacial motion, will be obtained numerically by solving the full form of the convectivediffusion equation for the Peclet number varying from 0 to 100. Close attention will be payed to the low Peclet number limit which has been somewhat neglected in the literature. It is also shown that this type of hydrodynamical model can be used to derive the typical power-law dependence of interfacial mass transfer. 2. Formulation of the problem We consider a liquid layer having a plane free fluid or solid-fluid interface undergoing steady cellular convection. The steady roll-cell pattern for a free fluid interface will be represented by a bidimensional velocity field analogous to the one first proposed by Fortescue and Pearson [ 111 and subsequently utilized by Ruckenstein [12] and by Theofanous et al. [16]. The flow pattern within the fluid layer is thus represented by a sequence of square roll-cells with alternating vorticity, each presenting an up-welling motion on one side and a corresponding down-current on the other. Thus, by using characteristic length and velocity scales L and uo, the corresponding stream function in dimensionless form can readily be written as 1
$ = - - sin 71x sin ny n forO
and
with a velocity
field
u, = - ? -sin ay
71x cos 7ry
(1)
u,(x, 1) = 0. A possible corresponding function can be written as
(2)
ati u, = - G = cos nx sin ny In an analogous way, the stream function representing the periodic steady roll-cell pattern for a fluid-solid interface has been chosen in order to satisfy the solid-fluid hydrodynamical boundary conditions, that is: u,(x, 0) = u,(x, 0) = 0 and, at the limit of the bulk, au,(x, l)/ay = 0 together with
(3)
The coefficients a,, have been calculated numerically to satisfy the previous boundary conditions together with the condition of zero shear rate at the limits x = 0 and x = 1 and uX($, a) = 1, this last condition giving an estimate of the position of the vortex within the cell. The a, are: a2 = 47r, a3 = -15.09, a4 = -10.64, a5 = 20.19, and as = -7.02. Again, the corresponding velocity field components can be readily obtained using u, = a$/ay, U, = - a+/ax with (3). . A typical eddy-cell structure for (1) or (3) is shown in Fig. 1 with coordinate system and length scales. c-0
30 Y
CZI
Y
Fig. 1. Typical eddy-cell structure near the interface with coordinate system.
The full form of the dimensionless convection-diffusion equation to be solved is
ac 2
O
6 x anyn tl=Z
1 4 =--sinnx x
stream
+(v’v)c-
-
$
v2c
where v is obtained either using (1) or (3) and Pe is the Peclet number Pe = Luo/D, D being the molecular diffusion coefficient. The concentration boundary conditions to be used are: c=l
for
y=O
c=O
for
y=l
In both cases the eddy is exposed at time t = 0 (with c = 0 everywhere) and participates in mass transfer until a steady state is reached. In the case of a free fluid interface (that is, absorption of a gas at the interface) a zero gas-phase resistance to mass transfer is
239
assumed, which is equivalent to considering that a thermodynamical equilibrium is reached at the surface y = 0. This is also generally assumed in the case of a solid-fluid interface. At y = 1, the boundary condition implies a complete and rapid mixing with the bulk. We note that this set of boundary conditions is similar to those used in the filmpenetration theory [ 5, 61. Owing to the symmetry of the flow pattern, a zero mass flux will be assumed across the vertical cell boundaries for both solid-fluid or free fluid conditions: Ar ar -_ G (O,Y) = E (13) = 0 (6) 3. Numerical method and results Equation (4) was solved by a finite-difference method. Spatial derivatives were all approximated by three-point central differences and time derivatives by two-point forward differences. For the non-linear term, we used a modified form of forward or backward difference, first proposed by Torrance [ 181, which preserves the stability of the numerical scheme. A stability criterion for this scheme has also been derived which is dependent on the Peclet number. In order to test the accuracy of the numerical method, various runs with different mesh sizes were performed for each Pe and the results corrected correspondingly. Convergence of the method was obtained with mesh sizes varying from 21 X 21 for the lower Peclet numbers to 21 X 101 for higher ones. This mesh-size refinement in the ydirection for the higher values of Pe was necessary to preserve the convergence of the numerical solution when increasing compression of the concentration contours occurred because of convection within the cell. The numerical investigations were carried out for Peclet numbers varying from 0 to 100, for the two flow field conditions (free fluid and solid-fluid interface). The mass transfer across the cell is described using a Sherwood number averaged over the length of the cell for each value of the Peclet number: Sh=/
E(.r,O)dr
(7)
this expression being evaluated when no significant modification of the concentration distribution within the cell is observed (AC/C < 10e6). This definition of the Sherwood number is appropriate to situations in which the flow pattern is maintained for a sufficiently long time. However, for situations in which an exposure time (lifetime) is to be considered, a time-averaged Sherwood number would be more appropriate to describe the net transfer during time of exposure: f%(t)
= ;- /
Sh(t’) dt’
0
where t is the exposure time. It should be noticed that SK(t) decreases much slower than Sh( t) and that, when Sh( t) has reached the stationary value Sh, defined in (7), a strong discrepancy between & and Sh is still observed, especially for the lower values of the Peclet numbers (-40% discrepancy when Pe = 0.8, Fig. 2; and only -16% when Pe = 25, Fig. 3). Nevertheless, for sufficiently long times Sh and sh converge to the same value. The Sherwood number Sh thus defined will be considered to be representative of the mass transfer across the stationary convective cell. One can note two main modes of behaviour for the steady mass flux uersus Peclet number. For the lower Peclet numbers the Sherwood number has been found to increase as Pe3/’ for both solid-fluid and free fluid conditions (Fig. 4), whereas for higher values of Pe the Sherwood number has been found to vary as Pe1j2 (i.e. k - D1j2) for the free fluid interface and Peli3 for the fluid-solid interface (i.e. k D2’3). Between these two regions, the transition is essentially linear (Fig. 5). The low Peclet number mass transfer has been the subject of only a few investigations using the hydrodynamical approach [ 12,131 (except in the related problems of transfer in two-phase particulate systems) owing to the presence of recirculative flows occurring in or around the spherical drops or solid spheres of these systems [ 19 - 231. Our result at the low Peclet number limit is that there is no essential discrepancy between Sherwood numbers for solid-fluid or free fluid interfaces, which is similar to the result of Bowman et al. [20]. Moreover, the explicit
240
Pez.8
I .2
.I
.3
.4
Fig. 2. Transient Sherwood numbers Sh(t) and m(t)
I
, 1
2
3
4
5
b
7
I
Fig. 3. Transient Sherwood numbers Sh(t) and a(t)
dependence of Sh on Pe3/’ in this limit was obtained by Leal [23] in his study of the effective conductivity of a dilute suspension of drops undergoing a simple shear with an imposed transverse temperature gradient. In his study, the role of drops was to disturb the main velocity field by creating recirculative flows somewhat similar to those considered in the present study. In the low Peclet number limit, the results of the hydrodynamical model of liquid-liquid interfacial mass transfer developed by
.5
.6
I
for Pe = 0.8 (free fluid interface).
9
IO
II
12
t
for Pe = 26 (free fluid interface).
Ruckenstein [ 12,131 were obtained by using finite expansions. His results, although very similar to those presented here, exhibit a certain discrepancy when numerical values are calculated. It is felt that the method used by Ruckenstein is the cause of this discrepancy. 4. Discussion In the case of a free fluid interface and for large values of the Peclet number, the main result is that the Pe ‘I2 dependence of the Sherwood number, already derived by means
241
.2
.4
.6
.8
l.
Pe
Fig. 4. Plot of the Sherwood number versus Peclet number 0 < Pe < 1, for both free and solid-fluid interface.
Fig. 5. Plot of the Sherwood number versus Peclet number 1 < Pe < 100; curve A corresponds to free fluid results and curve B to solid-fluid results.
of renewal models, is obtained without the renewal assumption but is a consequence of the velocity field near the interface within the cell. A power law Sh = 0.6Pe112 is suitable when Pe reaches values of about 25. However, it should be kept in mind that the velocity uo, used to obtain a dimensionless form of the diffusion equation to be solved when using hydrodynamical models of interfacial mass transfer, does not generally correspond to the mean effective velocity for the renewal within the cell. In order to illustrate this point, numerical results for the free fluid interface obtained for
Pe > 25 have been plotted in Fig. 6 together with a possible expression of the enhancement factor R [lo] derived by Dobbins [ 51 and Toor and Marchello [6] : R = da coth da
(9)
where a = L2SdlD, Sd being an average renewal rate of the disturbed system. One may consider a to be proportional to the Peclet number, i.e. a = ape. An estimate of (Ycan be obtained by fitting the numerical results (Fig. 5) for Pe > 25 with (9) in Fig. 6. This leads to 01 = 0.36, corresponding to a mean effective velocity uo/2.7 for the renewal within the cell.
242
Sh 6_
1. .... .... .... ..: 1'
2
3
4'
h
6
7
8
9
Fig. 6. Plot of the Sherwood number versus JPe
lo
11 '
",@e
together with expression (9).
Jn the case of a solid-fluid interface and for large values of the Peclet number, a power law Sh = 0.93 Pe113 is appropriate and is in good agreement with classical results. Similarly to the free fluid interface results, an expression of the enhancement factor equivalent to (9) can be used, which leads to a mean effective velocity ue/1.24 for the renewal within the cell. Although the ultimate mechanism of interfacial transfer must be by molecular diffusion, the influence of convective motions on the diffusion process is quite pronounced, especially for high values of the Peclet number. The effect of interfacial convection is to control the resistance to transfer by sharpening the concentration gradients near the interface, thus leading to the enhancement in the mass-transfer rate previously described as a function of the Peclet number. It should be stressed that it has been possible to account for the large differences observed in the mass-transfer rate between the free and solid interfaces only by modifying the hydrodynamical boundary conditions at the interface. That is, the fundamental difference is that for a solid interface all velocity components of the fluid must vanish at the interface allowing a non-zero shear rate at this interface, whereas in the case of a free fluid interface a zero shear rate can be assumed. Stationary kinematical models of recirculating flows near the interface can be used that account for the typical power-law dependence of inter-facial mass transfer as
long as the physical boundary conditions are fulfilled. However, in a forthcoming paper the generation of flow fields compatible with the momentum transport equation will be considered, in order to obtain a better differentiation in interfacial mass-transfer results between the hydrodynamical parameters and those characteristic of molecular diffusion only. Nomenclature a a,
; k L Pe R Sd Sh t V vo X Y
L2Sd/D, defined by eqn. (9) numerical coefficients defined by eqn. (3) volumetric concentration in the fluid molecular diffusion coefficient mass-transfer coefficient characteristic length scale of a cell Lu,/D, Peclet number enhancement factor defined by eqn- (9) average renewal rate kL/D, Sherwood number; s?; defined in (8) dimensionless time dimensionless velocity field within the cell with components u,., u, characteristic velocity scale for cellular convection dimensionless coordinate along the interface dimensionless coordinate normal to the interface
243
Greek symbols numerical coefficient a=aPe stream function ti
ar
defined
by
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