Modeling of Axial Dispersion in a Rotating-Bucket Contactor

A rotating-bucket contactor is an interesting equipment used in liquid−liquid extraction. Of course, as in other contactors, axial dispersion widely...
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Ind. Eng. Chem. Res. 2004, 43, 4761-4767

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Modeling of Axial Dispersion in a Rotating-Bucket Contactor Mario Dente and Giulia Bozzano* Department of Chemistry, Materials and Chemical Engineering, “Giulio Natta” Politecnico di Milano, Piazza Leonardo Da Vinci, 32-20133 Milano, Italy

A rotating-bucket contactor is an interesting equipment used in liquid-liquid extraction. Of course, as in other contactors, axial dispersion widely affects the extraction efficiency. A theoretical approach to the problem is presented here. The paper reports also experimental data obtained in a contactor of conventional geometry. The data are analyzed and compared with model previsions. The agreement with the experiments is satisfactory. The obtained correlations adequately describe also some literature experimental data related to smaller equipment. Analysis and identification of the main fluid-dynamic aspects affecting axial dispersion can suggest modifications of the internal geometry of the buckets and compartments. In this way, the detrimental effects of the dispersion can be reduced, and the contactor efficiency can be improved. Introduction The importance of axial dispersion phenomena in liquid-liquid extraction is well recognized. The term “axial” essentially refers to the prevailing direction of contact between the two phases. The unavoidably adverse effects of such dispersion phenomena are encountered in all types of extractors. They often cause a drastic reduction in the overall efficiency of the apparatus, so much that the equipment dimensions can become considerably larger than those that would have been required in their virtual absence. The mixing and mutual dispersion of the phases obtained in the extraction equipment by means of specific devices increase the mass transfer per unit volume of equipment but also result in important axial dispersion, partially penalizing the desired improvement. As a consequence, a proper estimation of axial dispersion coefficients, with the purpose of modeling the extractors and, eventually and hopefully, optimizing their behavior, is essential. The scientific literature has produced, in the past decades, an abundance of experimental data and of correlations for the main diffused types of extractors, characterized by empirical or theoretical bases. On the contrary, in the case of the rotating-bucket contactor, the attention in the literature has been scant.1,2 In a previous paper,3 a preliminary modeling of such equipment has been reported. The present paper is devoted to a deeper analysis of axial dispersion and its consequences on the overall efficiency of the extractor. Also new experiments have been performed in a wider range of operating conditions and compared with the model results. The interest in this kind of equipment is due to the possibility of application in different fields where liquidliquid extraction is fundamental. Some experimental data can also be found in the literature, but they were obtained with a very small equipment (150 mm diameter) so as to cast doubts on * To whom correspondence should be addressed. Tel.: +39 02 23993295. Fax: +39 02 70638173. E-mail: Giulia.Bozzano@ polimi.it.

the reliability of their extrapolation to commercial-size equipment. Some considerations on this will appear in the next paragraph. The new and more extended experimental data presented here have been obtained with a larger diameter (450 mm), conventional geometry, contactor. They are related to the axial dispersion in the aqueous phase and have been interpreted and theoretically modeled: the obtained correlations are shown to adequately describe also the literature experimental data related to the smaller equipment.1 Description of the Equipment and Contact Modalities of the Liquid Phases Since their origin, rotating-bucket contactors were conceived for handling difficult systems. For instance, they can be used in the washing section of diluted extract coming from the recovery of bitumen from oil sands4 or in systems containing small amounts of finely dispersed solids. Their mechanical design is simple and relatively low cost. Other advantages, like the horizontal configuration, make them attractive for different fields of application. The sketch and construction details of the mentioned equipment are reported in Figures 1 and 2. The experimental work reported in this paper has been performed in a contactor that can have from 6-10 compartments. The geometrical dimensions are as follows: shell internal diameter, 450 mm; shell internal length, 900 mm; nominal width of the peripheral gap, 7 mm; bucket internal diameter, 75 mm; length of each compartment, 75 mm; number of buckets for each compartment, 8; shaft diameter, 45 mm; diameter of the inlet and outlet openings, 5 mm. The configuration is usually quite horizontal. The rotor consists of a shaft supporting a series of disks and having a diameter close to that of the shell. The resulting peripheral gap allows the fluid to pass from one to the next compartment. Buckets are placed among the disks toward their periphery. The contacting zone is constituted of the rotating assembly of disks and buckets: it is preceded and followed by two empty regions. The latter behave, for each phase, at the inlet as a mixing zone and after the contacting region as a

10.1021/ie0307277 CCC: $27.50 © 2004 American Chemical Society Published on Web 06/08/2004

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Figure 1. Sketch of a rotating-bucket contactor.

Some other important aspects of the equipment behavior should be emphasized. The drops are formed by the liquid discharged from the buckets, whether by direct drop formation or by the breakup of rivulets or jets. They are relatively large and close to the spontaneous dropping size, i.e., depending only on the interfacial tension and the difference between the specific gravities of the two phases. Notwithstanding the large size of the drops (in the range of 5-15 mm), an efficient mass transfer can be obtained thanks to drop internal circulation and/or oscillation. Moreover, they coalesce rapidly at the interface between the two continuous phases, provided the viscosity of the surrounding continuous phase is not particularly high. The detailed fluid dynamics inside the contactor is too much complex to be accurately described: acceptable simplifications can be adopted. Macroscopically, the most evident phenomena disturbing the plug flow are caused by (a) the buckets movement and their periodic crossing of the coalescence surface (about equatorial, in the optimal conditions), (b) the formation, movement, and coalescence of the dispersed drops, and (c) the jetting of the two continuous phases through their pertinent position of the peripheral gap and the induced recirculation field. Then the buckets and the drops develop a function similar to that of a slow-moving mixer with a horizontal axis. The above-mentioned aspects suggest to simplify the fluid dynamics inside the extractor by means of the superposition of an apparent axial dispersion contribution onto a unidirectional, plug-flow motion of the two phases. Of course, the former contribution has to include all of the essential components of those fluid-dynamic disturbances that promote deviation from ideal plug flow. Experimental Section

Figure 2. Constructional details of a 6-compartment contactor.

calm zone. Macroscopically, the two liquid phases flow horizontally in a countercurrent manner, with the interface surface being located approximately at the equator. The fluids pass from one compartment to the next through the peripheral gap between the disk and the shell. Thanks to the slow movement of the buckets, each phase is discharged through the other in the form of drops. In the upper part of the compartments, the heavier phase falls through the lighter one as a rain of drops, while in the bottom part, the light phase, dispersed in drops, crosses the heavy one. The effect is that, in the compartment and in the two different parts of it, both phases are at the same time dispersed and continuous (while in the more diffused extractors, only one phase is dispersed in the other).

As traditional and convenient in these cases, the axial dispersion coefficient was deduced by measuring the concentration profiles of a tracer (a 3% nickel sulfate solution) along the time and length of the contactor. The two phases were constituted of a kerosene fraction as the organic phase [specific gravity at 15 °C ) 0.8 g/cm3, viscosity at 15 °C ) 1.93 × 10-3 kg/(m‚s)] and tap water as the aqueous phase (kerosene was externally recycled to the contactor). The apparatus was used with both 10 and 6 compartments, maintaining the same length of the static shell. Consequently, the two versions had different lengths of the initial and final nonagitated zones. A sample of 100 cm3 was injected pulsewise (similar to a δ(t) Dirac function) into the aqueous phase upstream at the contactor inlet. The cylindrical shell was provided with sample points at 13 positions along its length. Depending on the effective number of contacting compartments, only a few of the positions were active. From each of these, samples of 10 cm3 were drawn at predetermined times. The withdrawn material was small enough not to alter significantly either the flow rate through the apparatus or the amount of nickel transported thereby. The accuracy of the method of analysis was estimated as (2 mg of Ni/L. The equipment and procedure are the same as those used in the previous work.3 However, a quite more extended range of operating conditions has been adopted

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(rotor speed from 3.1 to 28 rpm, water flow rate from 60 to 480 L/h, and kerosene flow rate kept constant at 120 L/h). During the course of the experiments, almost inevitably, the formation of some stabilized drops of the organic phase surrounded by an aqueous film was observed. In the smaller equipment (i.e., 150 mm diameter), this phenomenon can build up a top-side foam lattice layer. The phenomenon has also been observed previously.1 Nevertheless, in comparison with the 150-mm-diameter contactor used there, the presently adopted size of the apparatus (450 mm) makes this phenomenon negligible during the operation because of the significant reduction in the ratio of surface to volume. In addition, the increase of the peripheral gap size among disks and the shell allows these drops to move through the compartments, avoiding the accumulation of any significant top-side foam layer. Another “parasitic” phenomenon resulting in alteration of the apparatus behavior (however, common to all kinds of extractors) but mainly affecting the rotatingbucket contactor is the possible accumulation of drops at the interface. Their coalescence, in order to reconstitute the original continuous phase, requires some time. They reside in a layer close to the interface, where they become more densely packed, assuming the arrangement of a three-dimensional network. The thickness of this liquid-liquid foam depends on several parameters, among which the most important two are the viscosity of the continuous phase and the difference between the densities of the two phases. The interfacial foam shows two layers, an upper one consisting of drops of the denser phase that have fallen through the upper part of the equipment and a lower one consisting of drops of the less dense phase that have risen through the lower part. Because the two layers must coalesce in opposite directions, they cause a mutual disturbance to each other and subtract volume to the free-falling drops. However, if the velocity of rotation of the contactor is moderate, the viscosities of the two phases are modest, and if the difference between their densities is sufficiently great, the average thickness of the two layers can resemble the diameter of a single drop (e.g., about 10 mm). This size does not significantly affect the operation of the adopted device in comparison to that of smaller dimensions.1 Mechanisms of Axial Dispersion As said before, in the contactor several more or less disordered fluid-dynamic mechanisms are present. They superimpose on the mean translation motion of the two phases, thereby causing axial dispersion. In the previous work, three kinds of dispersion mechanisms were reported. A deeper analysis of the new data made available by the experiments was brought to modify them and to introduce a new one, mainly affecting the larger dimension contactors. Their approximate quantification is presented in the following paragraph. A common feature of the expressions that will be proposed for the four mechanisms is that the numerical coefficients can be theoretically predicted only as an order of magnitude. Their final values are consequent to the application of adaptive multiplying factors, not so far from unity, for improving the agreement with the experimental data. Dispersion Due to the Jet Originating through the Peripheral Gap. A jet is originated through the peripheral gap. In the usual operating conditions, it is

laminar. As a good approximation, it behaves as the half of a free jet issuing from a slot. The volume rate of entrainment per circumferential unit length of the jet therefore is (for example, following Schlichting5) equal to 0.5[3.3(νv02 × 2l0x)1/3] (m2/s). The maximum virtual entrainment, in one compartment, occurs at a distance x ) L (where L is the length of one compartment of the contactor). Half of this maximum is roughly the amount of both the entrained and recycled flow rates. The volume flow rate, on the total circumferential length and in each phase, is therefore

π 0.5 Qentrained ) Dc [3.3(νv02 × 2l0L)1/3] = 2 2 1.63Dc(νv02l0L)1/3 (m3/s) (1) The entrained fluid is picked up from the central zone of the phase concerned and recirculated within each compartment. The mean axial velocity of this circulation can be evaluated by supposing it occurs, for both outward and return flows, over one-quarter of the crosssectional area of the whole contactor. Then

vrec )

Qentrained 8.31 ) (νv02l0L)1/3 (m/s) 1π 2 Dc D 44 c

(2)

Introducing the total flow rate of the phase concerned v0 ) 2(w/πl0Dc) (m/s), from which

vrec ≈ 6.16(w2νL/l0Dc5)1/3 (m/s)

(3)

By analogy with similar phenomena, the axial dispersion coefficient can be estimated as dax/rec ≈ (1/2)vrecLrec (m2/s), where Lrec in this case equals L. Thus, the corresponding contribution to the axial diffusivity is

dax/rec ≈ 3.08(w2νL4/l0Dc5)1/3 (m2/s)

(4)

It should be noted that eq 4 is valid when the interface surface is at the equatorial plane of the contactor. Otherwise, it must be multiplied by a correction factor of (π/2θ)2/3[1 - sin(2θ)/2θ], where θ is the semiangle subtended by the interface surface at the axis of the cylindrical shell. Dispersion Due to the Buckets Crossing the Interface. Without going into great detail, it is enough to recall here that the overall kinetic energy of the fluid surrounding a body moving at constant velocity, vb, can be expressed as

Ek )

∫V

vb2 1 1 Fsurr dV ) VadhFsurrvb2 ) madhvb2 (J) surr 2 2 2 (5)

where Vsurr is the volume of fluid space surrounding the body in motion (for a single isolated body, Vsurr w ∞); madh ) VadhFsurr is known as the adherent mass.6 The physical significance of madh is as if the body were surrounded by a moving mass of fluid solidly connected to the body itself and having mass madh. When the body suffers an acceleration, the adherent mass exhibits inertia; the same happens when the body leaves one fluid phase to enter another immiscible one. When a bucket rotates across the interface surface, the adherent fluid mass is released. The equatorial interface is thus periodically deformed by these released

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fluid volumes as each rejoins its original phase. In doing so, these fluid elements distribute themselves in a manner that is not easily accurately described but that can be considered as about isotropic in proximity of the equatorial interface. The same thing happens in reverse as the new adherent mass is formed when a bucket enters the new phase: the new mass is withdrawn substantially in an isotropic way from the surrounding fluid and proceeds to move solidly with the bucket. Therefore, both the volume of the fluid previously released and the volume withdrawn from the new phase create displacement of fluid: these displacement motions will also pass through the peripheral gap between disks and the shell, both toward the preceding compartment and toward the following one. This phenomenon contributes to the axial dispersion. The adherent volume that follows the bucket in each compartment can be estimated (for cylindrical bucket) as Vadh ) 0.6Db2L. The coefficient 0.6 can be significantly reduced by adopting different bucket geometries (one way to reduce axial dispersion in the contactor is, in fact, to change the bucket design). The number of adherent volumes released and withdrawn in each phase compartment in unit time is equal to 2nbN. The “volumetric flow rate” of the phase displaced is, then, 1.2nbNDb2L, while the average velocity of release (and withdrawal) is 1.2nbNDb2L/LDc. The fraction of fluid elements that pass through the peripheral gap is approximately 4l0/D0, so that their velocity is

vFS ≈

1.2nbNDb2L l0 4l0/Dc ) 4.8nbNDb2 2 (m/s) (6) LDc D c

The corresponding contribution to the axial diffusivity is

(

)

2 2.4nbNDb2l0L 1 4.8nbNDb l0 ) (m2/s) (7) dax/FS = L 2 2 2 Dc Dc

As in the previous case, the validity of eq 7 is restricted to the presence of the interface at the contactor equatorial plane: in the other positions, it must be multiplied by a factor of approximately sin θ[1 - sin(2θ)/2θ]. Dispersion Due to Formation, Movement, and Coalescence of Drops. From a certain point of view, this mechanism presents an analogy with the previous one. When the drop forms (from the bucket), it displaces an equal volume of the surrounding continuous phase: the same volume is restored (in the opposite direction) when the drop coalesces at the equatorial interface. Moreover, during its movement, the drop is surrounded by an “adherent” volume, which is withdrawn during the acceleration of the drop, just after its detachment. The adherent fluid is released at the moment of coalescence. The virtual related volume is estimated to be about half of the volume of the drop. The volumetric flow rate associated with these phenomena is therefore 1.5 times the one poured out by the buckets. The usable internal volume of one bucket is about 0.66LDb2. Following the same method as that in the previous case, the coefficient of axial dispersion due to the present mechanism becomes

(

)

2 1.98nbNDb2l0L 1 3.96nbNDb l0 dax/drop = L ) (m2/s) 2 Dc2 Dc2 (8)

This mechanism is active only if a sufficient average distance (at least three or four drop diameters) between the formation and coagulation of the drops is present. This means that this mechanism is practically absent in the case of the 150-mm-diameter contactor. Of course, also in this case a correction factor in the case of the nonequatorial interface has to be applied (the same as that in the previous mechanism). Dispersion Due to Turbulence Produced by the Buckets. Independently of the specific disturbances caused by their adherent mass, the bucket movements cause energy dissipation and turbulence within each phase. To evaluate the resulting diffusion contribution, a drastically simplifying assumption is made consisting of assuming turbulence as isotropic: this is justified by the aim of the exercise, which is to estimate the order of magnitude of the phenomenon. The power dissipated in each compartment can be so estimated from

f W ) nbFvb3LbDb (W) 2

(9)

The value of “f” behind a cylinder is approximately unit. The tangential velocity of the bucket vb, ignoring the correction related to the peripheral gap, can be expressed approximately by πN(Dc - Db). Therefore, W, expressed per unit mass of fluid, is

(

 ) 2π2nbN3DcDb 1 -

)

Db Dc

3

(W/kg)

(10)

The resulting turbulent diffusivity is as an order of magnitude

1 dturb ) 1/3λmax4/3 (m2/s) 7

(11)

The macroscale of turbulence behind λmax is estimated to be about equal to Db/4. It follows that

dturb ) 0.06nb1/3NDc1/3Db5/3(1 - Db/Dc) (m2/s) (12) The fraction of the cross-sectional area of the peripheral gap through which the turbulent dispersion can effectively occur in the axial direction is 4l0/Dc. Then the effective axial turbulent diffusion is

dturb ) 0.24l0nb1/3NDb5/3(1 - Db/Dc)/Dc2/3 (m2/s) (13) When the interface is displaced from the equatorial plane, eq 13 must by multiplied by a factor of about 2θ/ π. Total Axial Dispersion. The overall axial dispersion coefficient results from the superposition of the four described contributions, which are considered to be additive. Then

dax ≈ dax/rec + dax/FS + dax/drop + dax/turb (m2/s)

(14)

Modeling of Non-Steady-State Dispersion in the Contactor The three different regions of the contactor have been taken into account (indicated in Figure 2). A few of these

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considerations have already been proposed in a previous paper.3 The main aspects are presented here. Region I. The fluid enters in a volume of the contractor without buckets and preceding the first disk. The ratio of the diameter of the inlet nozzle to that of the contactor is such that a narrow round jet of high mixing intensity is generated. As a consequence, in this volume complete mixing can be approximately assumed. However, it must be observed that the instantaneous flow rate leaving this precontacting stage is the sum of both convective and diffusive contributions. The equation wherefrom concentration distribution can be deduced is therefore

VI

{

( )}

∂CI ∂CII ) Sc vax[CF(t) - CI(t)] + dax ∂t ∂x

0

(15)

under the following condition: for t e 0, CI ) 0. Region II. The real and efficient contact between the two phases takes place in this region, which is compartmented and in which the buckets mounted on the rotating-disk assembly are operative. Of course, the previously described mechanisms of axial dispersion mainly affect this region. The latter can be considered as interested by a continuous contacting: the axial dispersion is superimposed to the translation motion of the phases. In the absence of mass transfer between the phases, as in the experiments with the tracer, the equation describing the transport in this region can be simplified to

( ) 2

∂CII ∂ CII ∂CII + vax ) dax ∂t ∂z ∂z2

(16)

under the following conditions:

(Pξ2 )

C h II(ξ,s) ) C h F(s) exp

A cosh[A(1 - ξ)] + (P/2) sinh[A(1 - ξ)] (1 + sτm)A cosh(A) + [(P/2)(1 + sτm) + s] sinh(A) (18) where A ) (P2/4 + sP)1/2 and 0 e ξ e 1. By means of the inversion of the transform represented in eq 18, it is possible to deduce the complete solution. Of course, it has the form of an infinite series involving natural sine and cosine. For practical purposes, it is more convenient to introduce a simplifying assumption, based on the fact that most of the experimental data are related to a range of sampling times (t) such that Lc2/daxt . 1. This means that in the Laplace transform equation (18) it can be assumed that P|s| . 1 and |A| . 1, and then the equation can be simplified to its asymptotic behavior:

(Pξ2 )A +exp(-Aξ) P(1/2 + sτ )

h F exp C h II ) PC

(19)

m

By inversion of this asymptotic transform

mvaxP

{[

]

2 -1 2Lc(1 - Pτm) Pτm ξ 1 τ P 1/2 exp + -1 erfc τm Pτm τm 4τ 2 P 1/2 ξ+ - 1 τ - exp(Pξ) erfc (ξ + τ) Pτm 4τ

CII(ξ,t) ≈

[ ( [ ( ) ]}

) ] {( )

]}

[( )

3

kmol/m (20)

for t e 0, CII ) 0 for z ) 0, CII ) Cm(t) for z ) Lc, (∂CII/∂z)Lc ) 0 Region III. In this section, like in region I, disks and buckets are not present. Here the separation of the drops still dispersed in the continuous phase takes place. It is equivalent to a calm zone, containing baffles for the control of the internal level of the denser phase. The prevailing motion is convective; therefore, it does not affect the two preceding sections (under non-steadystate dispersion conditions). The third region simply behaves as a time delay in the dynamic response of the contactor. Here the continuity equation is

∂CIII ∂CIII + vax )0 ∂t ∂z

The Laplace transform of the concentration profile in the contacting region (II), which is the most interesting for evaluating the coefficients of axial dispersion, is

(17)

with the following conditions:

for t e 0, CIII ) 0 for z ) Lc, CIII ) (CII)Lc The system of differential equations and corresponding initial and boundary conditions can be solved rigorously, for instance, by means of the method of Laplace transforms.

For all of the adopted experimental conditions, the rigorous solution and the approximated equation (20) give practically the same numerical results. It is worth noting that the solutions represented by eqs 19 and 20 are the same as would be obtained for a contactor of infinite length: this corresponds to the intuitive concept that, for relatively short sampling times, the concentration profiles of the tracer in the contacting zone are quite insensitive to the outlet boundary conditions. Results and Comparisons The obtained experimental data were compared with the results of the reported equations. All of the data available for all of the sampling stations along the contactor were used; the estimation of the tracer concentration was made by use of eq 20. Typical obtained results are represented in Figures 3 and 4, which show a very good agreement between experimental and predicted values. In both figures, the concentration of the tracer inside the aqueous phase was measured along the time. Operating conditions for Figure 3 are as follows: water, 240 L/h; kerosene, 120 L/h; 7.2 rpm. Operating conditions for Figure 4 are as follows: water, 120 L/h; kerosene, 120 L/h; 14 rpm. Figure 5 presents the experimental values of axial dispersion coefficients versus the calculated ones in a 450-mm-diameter (6 and 10 compartments) contactor. The agreement is quite satisfactory. The experiments were performed by varying both the flow rate and contactor rotational speed. The analyzed situation is the

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Figure 3. Comparison of experimental and calculated tracer concentrations in a 6-compartment contactor.

Figure 4. Comparison of experimental and calculated tracer concentrations in a 10-compartment contactor.

Figure 5. 450-mm-diameter contactor: axial dispersion coefficient (exptl vs calcd).

following: 7.2 and 14 rpm with 60, 120, 240, and 480 L/h of water; 3.1 and 28 rpm with 120 and 240 L/h of water. The flow rate of kerosene was in all of the cases 120 L/h. To demonstrate the validity of the developed theory to other equipment sizes with the same geometry, also the experimental data reported in the previous work1 and related to a 150-mm-diameter contactor (25 compartments) have been compared with the model previsions. Figure 6 presents such a comparison. It has to be pointed out that the axial dispersion coefficients presented in that work were deduced through

Figure 6. 150-mm-diameter contactor: axial dispersion coefficient (exptl vs calcd).

elaboration of the final tracer concentration data as observed at the outlet of the contactor, considering the whole length, without any special distinction for the effect of the three different zones. In other words, a uniform dispersion coefficient was assumed for the entire contactor. However, if the same data were to be examined on the basis of three regions of the contactor, the coefficients of axial dispersion equivalent to those reported in that work would be somewhat larger than the original ones (in practice, and as an average, a relative 8% more). The satisfactory agreement between the theoretical predictions and the experimental data is evident. The presented equations allows the estimation of dax in a large range of operating conditions. The generalized form of the theoretical expressions for the prediction of dax make them convenient for the extension to nonconventional internal geometrical arrangements of the rotating-bucket contactor. The mechanistic bases of the model justify its application to both of the phases inside this liquid-liquid contactor, even though the validation of the model has been obtained through experiments performed only on the aqueous phase. Specifically regarding this last point, it seems important to observe that in the paper of Wang et al.2 data on axial diffusion in the organic phase (kerosene) are reported, again still obtained in a 150mm-diameter contactor. Those data showed some apparent anomalies for the axial diffusion coefficient in the kerosene phase if compared with the analogous data obtained from the aqueous phase in the same range of conditions. The average value of those data did not demonstrate any real change in the order of magnitude of dax, as far as could be predicted by the fluid-dynamic model. Minor anomalies in dax, not influencing its order of magnitude, can be explained on the basis of wall effects, related to the different wetting properties of the two phases (which influences, for instance, the transition from the dropping to the rivulet discharge regime of the buckets or the passage of the phases through the peripheral gap and so on). Of course, the quantitative amount of these effects strictly depends on the surfaceto-volume ratio (this means on the size of the contactor). The smaller diameter contactor is the most sensitive to surface effects: of course, quite larger diameter contactors are notably less sensitive, so that the abovementioned anomalies should not be expected. As a matter of fact, single-cell contactors of 450 and 1000 mm

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diameter, used for visual observations, did not show the anomalous behavior observed in the smaller contactor. In particular, the dropping or jetting characteristics, and the size of the formed drops, were observed to be about the same for both of the phases. An interesting feature of the formulas for the axial dispersion coefficients is that related to the effect of the size of the apparatus. For geometrically similar contactors, both dax/FS and dax/turb vary as RNDc2, while dax/rec varies as β(vax2νDc2)1/3 (with R and β being nondimensional parameters depending on the internal geometry of the contactor section). As a consequence, when the diameter of the contactor rises, the dominating dispersion mechanism becomes more and more related to the bucket rotation rate. Therefore, in the usual size contactors, practically the only important contributions are dax/FS and dax/turb. It is also on these bases that modifications to the internal geometry of the buckets and of the compartments have been suggested,4 in order to reduce the importance of the axial dispersion on commercial size equipment. Conclusions The most important phenomena governing the axial dispersion in a rotating-bucket liquid-liquid contactor have been considered. The presented theoretical fluiddynamic model has indicated mainly four mechanisms responsible for axial dispersion. The interpretation of the experimental data, obtained in a quite wide range of operating conditions, and an interpretation of other data already presented in the literature have shown good agreement with the model. The identification of the main fluid-dynamic aspects affecting axial dispersion can allow one to propose new buckets and compartment geometries, reducing the relative importance of the dispersion itself and then allowing one to increase the equipment efficiency. Nomenclature C ) concentration D ) diameter d ) diffusion/dispersion coefficient erfc ) complementary error function F ) friction factor L ) (i) length in general, (ii) length of a single compartment l0 ) width if the slot is equivalent to the peripheral gap

m ) mass of tracer per unit of volumetric feed flow rate N ) rotation rate of the contactor rotor n ) number (droplets, buckets) P ) Pe´clet number (vaxDc/dax) s ) Laplace transform variable S ) cross-sectional area t ) time V ) volume v ) velocity v0 ) velocity of the slot jet at the origin w ) volume flow rate of one liquid phase x ) distance over which the jet causes entrainment z ) distance along the contactor ν ) kinematic viscosity F ) specific gravity σ ) interfacial tension τm ) Vm/Vc τ ) vt/Lc ) dimensionless time  ) z/Lc )dimenionless length Subscripts ax ) axial b ) bucket c ) (i) contactor (diameter), (ii) contacting zone (length) F ) feed m ) mixing region I 0 ) initial value

Literature Cited (1) Sheikh, A. R.; Ingham, J.; Hanson, C. Axial mixing in a Graesser Raining Bucket liquid-liquid Contactor. Trans. Inst. Chem. Eng. 1972, 50, 199. (2) Wang, P. S. M.; Ingham, J.; Hanson, C. Further Studies on the Performance of a Graesser Raining Bucket liquid/liquid Contactor. Trans. Inst. Chem. Eng. 1977, 55, 196. (3) Dente, M.; Meotti, C.; Corti, A.; Falcon, J. Rotating Bucket Contactor for Liquid-liquid extraction. Ing. Chim. Ital. 1988, 24 (3-4), Mar-Apr. (4) Robinson, L. F.; Porcari, G. Contactor. U.S. Patent 4,416,858, 1979. (5) Schlichting, H. Boundary Layer Theory, 7th ed.; McGrawHill: New York, 1979. (6) Batchelor, G. K. An Introduction to Fluid Dynamics; Cambridge University Press: Cambridge, U.K., 1970.

Received for review September 17, 2003 Revised manuscript received March 17, 2004 Accepted March 24, 2004 IE0307277