I n d . Eng. Chem. Res. 1992,31,1366-1372
1366
Greek Letters y = dimensionless slope for a variable distribution coefficient X = eigenvalue
Kim, J. I.; Stroeve, P. Mass transfer in separation devices with reactive hollow fibers. Chem. Eng. Sci. 1988,41, 247-257. Kim, J. I.; Stroeve, P. Uphill transport in mass separation devices with reactive membranes: counter-transport. Chem. Eng. Sci. 1989,44, 1101-1111.
Subscripts
Lonsdale, H. K. The growth of membrane technology. J. Membr. Sci. 1982, IO, 81-181. Noble, R. D. Two-dimensional permeate transport with facilitated transport membranes. Sep. Sci. Technol. 1984,19,469-478. Noble, R. D.; Way, J. D. Application of liquid membrane technology. Literature Cited ACS Symp. Ser. 1987, No. 347, 110-122. Abrams, D. S.; Prausnitz, J. M. Distribution of phenolic solutes Rudisill, E. N.; Levan, M. D. Analytical approach to mass transfer between water and non-polar organic solvents. J. Chem. Therin laminar flow in reactive hollow fibers and membrane devices modyn. 1975, 7,61-72. with nonlinear kinetics. Chem. Eng. Sci. 1990,45, 2991-2994. Belfort, G. Membrane separation technology: an overview. AdSideman, S.;Luss, D.; Peck, R. E. Heat transfer in laminar flow in vanced Biochemical Engineering; Bungay, H. R., Belfort, G., a.; circular and flat conduits with (constant) surface resistance. Wiley Interscience: New York, 1987. Appl. Sci. Res. 1964/65,14, 157-171. Brown, G. M. Heat or mass transfer in a fluid in laminar flow in a Urtiaga, A. M. Applicacidn de las membranaa liquidas soportadas a circular or flat conduit. AIChE J. 1960,6,179-183. la recuperacidn de fenol en m6duloa de fibras huecas. Ph.D. Cuasler, E. L. Diffwion; Cambridge University Press: London, 1986, Thesis, Universidad del Pais Vasco, Bilboa, Spain, 1991. Chapter 15. Urtiaga, A. M.; Ortiz, M. I.; Irabien, A. J. Phenol recovery with Danesi, P. R. A simplified model for the coupled transport of metal supported liquid membranes. I. Inst. Chem. Eng. Symp. Ser. ions through hollow fiber supported liquid membranes. J. 1990, No. 119,35-46. Membr. Sci. 1984,20, 231-248. Won, K. W.; Prausnitz, J. M. Distribution of phenolic solutes beDavis, H. R.; Parkinson, G. V. Mass transfer from small capillaries tween water and polar organic solvents. J. Chem. Thermodyn. with wall resistance in the laminary flow regime. Appl. Sci. Res. 1975, 7,661-670.
j = index in the z direction n = order of the eigenvalues and constants
1970, 2, 20-30.
Kiezyck, P. R.; MacKay, D. The screening and selection of solvents for the extraction of phenol from water. Can. J. Chem. Erg. 1973, 51, 741-745.
Received for review July 22, 1991 Revised manuscript received November 26, 1991 Accepted February 11, 1992
GENERAL RESEARCH Settling Velocity of a Sphere Falling between Two Concentric Cylinders Filled with a Viscous Fluid Guang-HongZheng, Robert L. Powell, and Pieter Stroeve* Department of Chemical Engineering, University of California, Davis, California 95616
The method of reflections is used to analyze experimental data for spheres falling between two concentric cylinders filled with a viscous fluid. The method of reflections is shown to predict the settling velocity of a sphere to within 5 % or less if the sphere is a distance more than three sphere radii away from either the inner or outer walls of concentric cylinders. Near the inner cylinder the method of reflections overpredicts the settling velocity of the sphere, while near the outer cylinder the method underpredicts the velocity. Introduction The problem of a sphere moving in a viscous fluid is of interest in connection with various problems such as the flow of slurries in pipes, the settling of a suspension in a container, and the mixing of an emulsion. M a n y studies have been made to determine the forces acting on a sphere in both bounded and unbounded fluids. In bounded fluids, the geometry of the container walls can play an important role in determining the interaction of a sphere with the wall. There have been considerable theoretical and experimental investigations of a sphere falling in the axial direction inside a single cylinder (Ladenburg, 1907; FGxen, 1922, 1923; Suge, 1931; Francis, 1933; Paine and Sherr, 1975; Happel and Byme, 1954; Haberman, 1956; Haberman and Sayre, 1958; Bohlin, 1960; Fidleris and Whitmore,
1961). Brenner and Happel (1958) used the method of reflections to calculate the frictional force in the axial direction (the k direction) experienced by a sphere falling parallel to the axis of a cylinder but situated in an arbitrary position with the cylinder. T h e y f o u n d
F=
- k ( 6 w ~UJ
(1)
1-f(:);
where Uo is the sphere settling velocity in an off-center position in the cylinder and f(b/Ro)is the eccentricity function which has been evaluated by Famularo (1962), Greenstein (1967), and Hirschfeld et al. (1984). The other symbols used in eq 1 are d e f i i e d in the Nomenclature
0888-5885/92/2631-1366$03.00/00 1992 American Chemical Society
Ind. Ehg. Chem. Res., Vol. 31, No.5, 1992 1367 Lid
Guidg t u b
Part
Lid
Guidg lube
Port
constant tern-
Figure 1. Single cylinder apparatus
Figure 2. Concentric cylinder apparatus.
section. This result can he recast hy comparing the velocity ofa sphere falling along the ads with that of a sphere falling at a distance a from the axis:
where U,is the sphere settling velocity at the center of the cylinder. Other theoretical treatments of this problem have been given by Tineren (1982, 1983) using perturbation techniques and by Graham et al. (1989) using numerical techniques. Hasimoto [1976] determined the vectorial components of the force on a sphere moving with an arhidirection within a cylinder. Hirschfeld et aL (1984), whose used the method of reflections, determined the wall effectson an arhitrarily-shaped,-positioned, and -oriented particle in a cylinder. Falade and Brenner (1985, 1988) gave the asymptotic solutions for particles moving near cylindrical walls. Other problems for spheres moving relative to walls have been considered. Solutions are available for a sphere moving near a plane wall (e.g., Brenner, 1961; Goldman et al., 1967; Malysa and Van De Ven, 1986; h l i et al., 1989) and between two parallel planes (Ho and Leal, 1974). Alam et al. (1980) and Shinohara and Hasimoto (1980) determined the wall effects on a particle moving outside a cylinder. Fukumoto (1985) reported the fust-order wall effeets for a sphere moving between two concentric circular cylinders using the method developed hy Hasimoto (1976) and gave the following expression for frictional force acting on the sphere in the axial direction:
F = -k(67rwUo)(1+ (a/R,)K,)
(3) where K, is the wall correction factor. Fukumoto (1985) compared the K, in eq 3 graphically with the cases of the sphere motion between parallel plane walls (Ho and Leal, 1974)and the sphere motion inside (Hasimoto, 1976) and outside (Alam et al., 1980) a cylinder. Experimental data for spheres falling between two concentric cylinders are scarce. In this work we report experimental results for a sphere settling in the axial direction between two concentric cylinders. The data are compared to the theory of Fukumoto (1985) and an expression that we developed using the method of reflectiona We also report criteria for when the method of reflections is expected to give accurate predictions of the settling velocity.
Figure 3. Definition sketch Table I. Spheren Used in the Exwriments sphere no.
a, mm
1 2 3 4 5
1.605
2.010 2.400 2.790 3.195
Ps. g l d 7.570 7.530 7.605 7.640 7.648
Experimental Section In order to test the experimental techniques, we first conducted experiments with a sphere in a cylinder and the results were compared with the prediction hy Brenner and Happel (1958) as given by eq 2. The single-cylinder a p paratus, as shown in Figure 1,consisted of a cylinder with a lid at the top and a Plexiglas water jacket around it. In the lid there were several ports for inserting a guide tube that specified the radial position of the falling sphere. The guide tube protruded a few millimeters in the liquid contained within the cylinder. The lengths of the cylinders were about 600 mm. Various size guide tubes were used with the inner diameter of the guide tubes being just slightly larger than the diameters of the falling spheres. Thii arrangement ensured that a sphere of any size could always he dropped at the desired radial position. The apparatus using concentric cylinders is shown in Figure 2. The smaller cylinder was put concentrically in the bigger cylinder and was fixed in position with a Plexiglas base. The cylinders were leveled until they were oriented vertically. Experimental runs were made by using various combinations of the inner and the outer cylinders. A defiition sketch of the experimental apparatus is given in Figure 3. Note that R. is the inner radius of the outer cylinder and Ri is the outer radius of the inner cylinder. Commercially available, steel spheres were used as falling spheres. The properties of the spheres are given in Table I. After each experimental run, the spheres a t the bottom of the cylinder were removed with a magnet. The Newtonian fluids used for the experiments were pure 0 gycol) and a solution UCON oil 75-H-90,00(polyalkylene
1368 Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 Table 111. Cylinder Combinations Used in the Experiments Ri, liquid Ro, no. sphere no. range of Re code" mm mm 1 3,4,5 1.54 X 104-1.26 X SC.1 141.85 0.00 2 1, 3, 5 1.54 X 10-*-1.15 X IO-' SC.2 71.05 0.00 I I, 2 , 3 , 4 , 5 1.26 x 10-4-1.11 x 10-3 SC.3 40.00 0.00 1 3, 4, 5 1.54 X 10-'-1.24 X CC.1 141.85 43.05 2 1, 3, 5 1.51 X 10-2-1.08 X lo-' CC.2 71.05 18.75 1 1, 2, 3, 4, 5 1.35 X 104-1.04 X CC.3 40.00 7.00 1 1 , 2 , 3 , 4 , 5 1.28 x 10-4-1.02 x 10-3 CC.4 40.00 12.35 1 1, 2, 3, 4, 5 1.38 X 104-8.89 X CC.5 40.00 22.00
Table 11. Properties of the Liquids Used in the Exaeriments liquid no. T,OC P , P PL, g/mL 1. UCON oil (75-H-90.000) 30 274 1.086 2. water solution of UCON oil 40 25.86 1.084 ~
a
1.12 I 0
,
-a/R,- 0.01692
0
,
-*
1
alR,= 0.01967
SC = single cylinder; CC = coqcentric cylinder.
'.04
-4
I
86
- Fukumoto's results 0
?
method of reflections
gN 0.0
0.1
0.2 0.3
0.4
0.5
0.6
0.7
0.8
0.9
blR
0.65 0 . 8 0 0.66 0 . 7 0 0 . 7 6 0 . 8 0 0 . 8 5 0 . 9 0 0
b/R, Figure 6. Comparison between Fukumoto's results (1985) and the method of reflections for R,/Ri = 2.0.
o . o r l . , . , . I 0.0 0.1 0.2 0.3
.
,
, . , . , . , . I
.
0.4
0.5
0.6
0.7
0.8
0.0
blR,
O
, -a/R,=
1
0.04013
0
?
s
0.0
0.1
0.2
0.3
0.4
0.5
0.8
0.7
0.8
0.9
blR, Figure 4. (a) Comparison of experimental data (symbols) for experiments SC.l and theoretical results (curves) given by eq 2. (b) Comparison of experimental data (symbols) for experiments SC.2 and theoretical results (curves) given by eq 2. (c) Comparison of experimental data (symbols) for experiments SC.3 and theoretical results (curves) given by eq 2.
of UCON oil with water. The viscosities and densities of the fluids are listed in Table 11. Conditions for the eight experimental runs are tabulated in Table 111. The Reynolds numbers (Re) given in Table I11 are the sphere
Reynolds numbers. The sphere Reynolds numbers vary between 10-4 and lW, which is in the creeping flow regime. The temperature of the system was maintained at 30 or 40 "C during the experiment by a thermostated water jacket. A thermometer was used to measure the temperatures at different points in the fluid to determine if the temperatures were within 0.1 "C. During the experiments the beaker, in which the steel spheres were kept, was immersed in the water of the constant-temperature tank to allow the spheres to equilibrate to the same temperature as the fluid before each experiment. A stopwatch which could be read to 0.1 s was used to measure the falling time of the spheres between two marked lines (200 mm apart) on the cylinders. Parallax error was eliminated by using cathometers focused on each marked line. From the measured time the settling velocity in the axial direction was obtained. Reproducibility between replicate runswas better than k0.5%. To eliminate entrance and end effects, the measurements were taken in the middle section, which was about 200 mm from the entrance and end sections. Separate experiments were conducted to verify that end effecta were negligible in this region. Results and Discussion The experimental data for spheres falling in a single cylinder are shown in Figure 4. The data are compared to the theoretical results from eq 2. The ratio of the centerline velocity to the off-center velocity is plotted versus the ratio of the distance from the centerline divided by the cylinder radius, b/R,, which is called the eccentricity ratio. In Figure 4a there are wall effects when the eccentricity ratio is greater than 0.60. The cylinder diameter used in Figure 4a is larger than that in Figure 4b,c. The data and theory agree very well in Figure 4a for eccentricity ratios up to 0.9. Agreement in Figure 4b is also reasonable. However, disagreement between data and theory is ap-
a
1.20 I
1
1:
d 1.6
e 1,s
,: O
L
: 0
a0
. A
, -dR0-0.05025
I:I
.-----~Ro-0.06000 , - * - * - ‘ d R o =0.06975
1.
I-
A
I
1 ;
?
U.
a
DU
RJRi = 3.239
0.4
0.3
0.5
0.6
0.7
0.8
1 . O l . I . 0.60 0.65
0.9
1
0.70
blR,
.
1
0.75
c
-
, 0.80
,
.
0.85
. I 0.90
blR,
b
e 1.4
I
0.03370
-a’Ro=
----- a/Ro, 0.04497 RJR, = 3.789
1.2
3
.J
*.
L
l
0 ..7
0.6
l
0.8
4
0.9
blR,
1.0
I
0.70
0.76
0.80
0.85
0.90
blR,
C
. A
A
0.3
, --..--.dRo= 0.06000 , “““.a/R, 0.06975 , - - - ‘ ~ R o =0.07988
0.4
0.5
0.6
0.7
0.8
0.9
blR, Figure 6. (a) Comparison of experimental data (symbols) for experiments CC.l and theoretical results (curves) given by eq 5. (b) Comparison of experimental data (symbols) for experiments CC.2 and theoretical results (curves) given by eq 5. (c) Comparison of experimental data (symbols) for experiments CC.3 and theoretical results (curves) given by eq 5. (d) Comparison of experimental data (symbols) for experiments CC.4 and theoretical results (curves) given by eq 5. (e) Comparison of experimental data (symbols) for experiments CC.5 and theoretical results (curves) given by eq 5.
parent in Figure 4c when the eccentricity ratio is greater than 0.65. It is known that the method of reflections is limited to small spheres (Happel and Brenner, 19831, and in Figure 4c the diameters of most of the spheres are large relative to the cylinder diameter. However, for the smallest sphere, when a/Ro = 0.040 13, agreement between theory and data is good up to an eccentricity ratio of 0.83. In analyzing the data for spheres falling in the axial direction between the gap in concentric cylinders, the theory of Fukumoto (1985) given by eq 3 can be used. Our preference was to use the method of reflections to obtain
the frictional force in the axial direction exerted on the sphere. The method of reflections is well-known (Brenner and Happel, 1983),and we have summarized our derivation in the Appendix. The frictional force on a sphere is
where the terms f l and f i are functions of b, R,, and Ri. Comparison of eq 4 with eq 3 shows that the term f i + (b/Ro)f2is analogous to the term K,by Fukumoto (1985).
1370 Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992
-8 8
cation of the reliability of the method of reflections to predict the settling velocity of spheres in single and in concentric cylinders.
21 19 17 15
t 13 2 11 5
9
0 ”
7
2 % *
*
Single cylinder
0
Concentric cylinder
Conclusions
5 3 1 -1
0
5
10 1 5 2 0 2 5 30 35 4 0 4 5 5 0 5 5
LIa
Figure 7. Comparison of absolute difference (in percentages) between the method of reflections and experimental data.
Numerical comparisons indicate that in fact K , and f l + (b/R,)f2are equal. An example for the ratio R o / R i = 2.0 is given in Figure 5, where it is seen that Fukumoto’s (1985) theory gives equivalent results when compared to the method of reflections. The expression for the ratio of the velocity of a sphere falling along the centerline of a single cylinder to the velocity of a sphere falling in the gap between concentric cylinders can be obtained from the method of reflections as
while Fukumoto’s (1985) results give
c= -u UO
1 - 2.10444-
U
RO
a
(6)
1- -K,
RO
Both theories are equivalent and we have made all calculations with the use of eq 5. Data for spheres falling in the axial direction between concentric cylinders are compared to the theory in Figure 6. A sphere falling at the centerline of a single cylinder has a higher velocity than a sphere falling in the gap between two concentric cylinders, and consequently, for concentric cylinders, the minimum value of the velocity ratio is greater than 1. The experimental data show good agreement with the theory as long as the sphere radius is small relative to the radii of the inner and outer cylinder. In Figure 6a, the theory and data agree reasonably well for moderate values of b / R o up to 0.9. In this case, the radius of the outer cylinder is large relative to the sphere radius. However, the radius of the inner cylinder is sufficiently small that the theory overpredicts the experimental results. As shown in Figure 6, the method of reflections overpredicts the velocity ratio near the inner cylinder, but underpredicts the velocity ratio near the outer cylinder if the cylinders become sufficiently small. In the middle section of the gap the method of reflections gives good agreement with experimental data. The comparison of the experimental results with the method of reflections is summarized in Figure 7, where the absolute difference between the data and theory, in percent, is plotted. When the sphere is five sphere radii from the nearest wall, the method of reflections gives an excellent prediction of the velocity. If L is the distance between the sphere center and the nearest cylinder wall, Figure 7 shows that for Lla 1 3 the absolute difference is less than 5%. Thus Figure 7 can be used as an indi-
The method of reflections shows good agreement with experimentaldata when the sphere falls at a distance three times its radius from the nearest cylinder wall. When the sphere is closer than this distance to a cylinder wall, the prediction from the method of reflections deviates from the experimental data. For a single cylinder, the method of reflections tends to underpredict the data, while for concentric cylinders the method of reflections tends to overpredict the data near the inner cylinder wall and underpredict the data near the outer cylinder wall.
Acknowledgment Part of this study was supported by a U.C. Faculty Research Grant to P.S.
Nomenclature u = sphere radius
b = distance of sphere center to cylinder axis f = eccentricity function f 1 = function defined in the Appendix f 2 = function defined in the Appendix I k = modified Bessel function of the first kind of order k & = modified Bessel function of the second kind of order k K, = wall correction factor L = distance from the sphere center to the nearest wall Re = 2aU,r/p, Reynolds number Ri = outer radius of inner cylinder Ro = inner radius of outer cylinder Uc= velocity of a sphere falling at the center of a single cylinder Uo= velocity of a sphere falling at the off-center of cylinder(s) U , = sphere velocity in an unbounded medium pL = fluid density ps = sphere density p = fluid viscosity
Appendix The functions of f l and f 2 are defined as follows:
where
Ind. Eng. Chem.Res., Vol. 31, No. 5, 1992 1371
X5k
= Ik(mi), Y 5 k = Kk(mi),
S5k
= Ik(mo),
z5k x6k
=
= Kk(mo) (A271
m$k(mi)+ Ik(mi), Y 6 k = m&k(mi) + Kk(mi), s 6 k = mdi(mo) + Ik(mo), z6k
=
m&k(mo) + Kk(mo) (A281
Literature Cited Alam,S.; Ishii, K.; Hasimoto, H. Slow Motion of a Small Sphere Outside of a Circular Cylinder. J. Phys. SOC.Jpn. 1980, 49, 405-408.
Ascoli, E. P.; Dandy, D. S.; Leal, L. G. Low Reynolds Number Hydrodynamic Interaction of a Solid Particle with a Planar Wall. Znt. J. Numer. Methods Fluids 1989,9,651-688. Bohlin, T. On the Drag on a Rigid Sphere Moving in a Viscous Liquid Inside a Cylindrical Tube. Trans. R. Znst. Technol. 1960, No. 155, 1-63. Brenner, H. The Slow Motion of a Sphere Through a Vicous Liquid Toward a Plane Surface. Chem. Eng. Sci. 1961, 16, 242-251. Brenner, H.; Happel, J. Slow Viscous Flow Past a Sphere in a Cylindrical Tube. J . Fluid Mech. 1958,4,195-213. Falade, A.; Brenner, H. Stokes Wall Effects for Particles Moving Near Cylindrical Boundaries. J. Fluid Mech. 1985,145,145-162. Falade, A.; Brenner, H. First-Order Wall Curvature Effecta Upon the Stokes Resistance of a Spherical Particle Moving Close Proximity to a Solid Wall. J. Fluid Mech. 1988,193, 533-568. Famularo, J. Theoretical Study of Sedimentation of Dilute Suspensions in Creeping Motion. D. Eng. Sci. Dissertation, New York University, 1962. Fax& H. The Resistance to the Motion of a Solid Sphere in a Viscous Liquid Enclosed Between Parallel Walls. Ann. Phys. 1922,68,89-119.
FaxBn, H. Die Bewegung Einer Starren Kugel langs der Ache Eines mit Zaher Flussigkeit Gefullten Rohres. Ark. Mat. Astron. Fys. 1923,17 (27), 1-28.
Fidleris, V.; Whitmore, R. L. Experimental Determination of the Wall Effect for Spheres Falling Axially in Cylindrical Vessels. Br. J. Phys. 1961,12,490-494. Francis, A. W. Wall Effect in Falling Method for Viscosity. Physics 1933,4,403-406.
Fukumoto, Y. Slow Motion of a Small Sphere in a Viscous Fluid Between Two Concentric Circular Cylinders. J. Phys. SOC.Jpn. 1985,54, 1322-1328.
Goldman. A. J.: Cox. R. G.: Brenner. H. Slow Motion of a .%here Parallel to a Plane Wall.‘ I. Motion Through a Quiescent Fluid. Chem. Eng. Sci. 1967,22,637-651. Graham, A. L.; Mondy, L. A.; Miller, J. D.; Wagner, N. J.; Cook, W. A. Numerical Simulations of Eccentricity and End Effects in Falling Ball Rheometry. J . Rheol. 1989,33,1107-1128. Greenstein, T. Theoretical Study of the Motion of One or More Spheres and a Fluid in an Infinitely Long Circular Cylinder. Ph.D. Dissertation, New York University, 1967. Haberman, W. L. Wall Effect for Rigid and Fluid Spheres in Slow Motion. Ph.D. Dissertation, University of Maryland, 1956. Haberman, W. L.; Sayre, R. M. ‘Wall Effect for Rigid and Fluid Spheres in Slow Motion”; David Taylor Model Basin Report No. 1143, Washington, DC; U.S. Navy Department: Washington, DC, 1958.
Happel, J.; Byrne, B. J. Motion of a Sphere and Fluid in a Cylindrical Tube. Znd. Eng. Chem. 1954,46, 1181-1186; Corrections 1957,49,1029.
Hasimoto, H. Slow Motion of a Small Sphere in a Cylindrical DoJpn. 1976,41,2143-2144. main. J. Phys. SOC. Hirschfeld, B. R.; Brenner, H.; Falade, A. First- and Second-Order Wall Effect Upon the Slow Visous Asymmetric Motion of an Arbitrarily-Shaped, -Positioned and -Oriented Particle with a Circular Cylinder. PhysicoChem. Hydrodyn. 1984,5,99-133. Ho, B. P.; Leal, L. G. Inertial Migration of Rigid Spheres in TwoDimensional Unidirectional Flows. J . Fluid Mech. 1974, 65, 365-400.
Ladenburg, R. Uber den Einfluss von Waanden auf die Bewegung Emer Kugel in Einer Reibenden Flussigkeit. Ann. Phys. 1907,23, 447-458.
Malysa, K.; Van De Ven, T. G. M. Rotational and Translational Motion of a Sphere Parallel to a Wall. Znt. J. Multiphase Flow 1986, 12, 459-468.
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1372
Paine, P. L.; Sherr, P. Drag Coefficients for the Movement of Rigid Spheres Through Liquid-Filled Cylindrical Pores. Biophys. J. 1975,15, 1087-1091. Shinohara, M.; Haaimoto, H. Force on a Small Sphere Sedimenting in a Viscous Fluid Outside of a Circular Cylinder. J. Phys. SOC. Jpn. 1980,49, 1162-1166. Suge, Y. An Experimental Study on the Effect of a Wall upon a Sphere Falling in a Viscous Fluid. Bull. Inst. Phys. Chem. Res.,
Tokyo 1931,10, 146-156. Tineren, H. Torque on Electric Spheres Flowing in Tubes. J. Appl. Mech. 1982,49, 279-283. Theren, H. Drag on Eccentrically Positioned Spheres Translating and Rotating in Tubes. J . Fluid Mech. 1983,129, 77-90. Received for review July 22, 1991 Accepted January 28, 1992
Solvent Extraction of Holmium and Yttrium with Bis(2-ethylhexy1)phosphoric Acid Kazuharu Yoshizuka,* Yoshitsugu Sakamoto, Yoshinari Baba, and Katsutoshi Inoue Department of Applied Chemistry, Saga University, Saga 840, Japan
Fumiyuki Nakashio Department of Organic Synthesis, Kyushu University, Fukuoka 812, Japan
A kinetic study on the solvent extraction of holmium(II1) and yttrium(II1) with bis(2-ethylhexy1)phosphoricacid (DBEHPA) from nitrate media was conducted a t 303 K using a hollow fiber membrane extractor. Also studied were the distribution equilibria of these metals and interfacial adsorption equilibria of DBEHPA and ita metal complexes between the organic and aqueous phases. It was found that the metals (M3+)were extracted with D2EHPA (HR) as MRg3HR into the organic phase, and the extraction equilibrium constants were evaluated. Furthermore, it was established that dimeric DBEHPA can be adsorbed a t the interface between the organic and aqueous phases, while the interfacial activities of DBEHPA-metal complexes were negligibly small. The apparent orders 2, 1, and 2 of the permeabilities for the extraction of both metals were found with respect to the pH of the aqueous solution and the concentrations of the metal ion and dimeric DBEHPA, while the orders 1, 1,and -1 of the permeabilities for the stripping of both metals were found with respect to the hydrogen ion activity and the concentrations of the metal complex and dimeric DBEHPA, respectively. The diffusional effects were reasonably explained by the diffusion model accompanied by an interfacial reaction, taking into account the velocity distributions of the aqueous and organic phases through the inner and outer sides of a hollow fiber.
Introduction Dialkylphosphoricacids have found extensive use in the hydrometallurgical recovery and refining processes of rare metals such as cobalt, nickel, zinc, actinides, and lanthanides. The separation and refining processes of rare earth elementa have been mainly carried out by solvent extraction with dialkylphosphoric acids, especially bis(2ethylhexy1)phosphoric acid (henceforce, abbreviated to DBEHPA), and many fundamental studies on the extraction of rare earth elements have been conducted (Bauer and Lindstrom, 1971; Bauer et al., 1972; Bhattacharyya and Ganguly, 1986;Coleman and Roddy, 1971;Danesi and Cianetti, 1982;Huato and Toguri, 1989, Imai and Furusaki, 1987;Inoue and Nakashio, 1982;Kolaric et al., 1974; Pierce and Peck, 1963; Sato, 1975,1989). However, most of the previous investigations on the separation of rare earth elements were concerned with the extraction equilibrium, while the studies on the extraction kinetics are few (Danesi and Cianetti, 1982; Hirato and Toguri, 1989; Imai and Furusaki, 1987; Kolaric et al., 1974). Recently, the application of hollow fiber modules to membrane extraction processes has been attempted for metal recovery (Alexander and Callahan, 1987; Kim, 1984) and protein extraction (Dahuron and Cussler, 1988). In order to improve the membrane extraction processes, the kinetics and mechanism of extraction and stripping through the hollow fiber membrane must be elucidated because these processes are governed by kinetics rather than equilibria. If the difference between the extraction and stripping rates of the metals of interest is greater than 0888-5885/92/2631-1372$03.00/0
that of the distribution ratios at equilibrium, the separation using the membrane extraction processes could represent an attractive alternative to the conventional solvent extraction processes based on distribution equilibrium. However, only a few investigations have been reported that deal with the membrane extraction processes since the mechanism is strongly dependent on the combination of metals and the extractanta (Danesi and Cianetti, 1982). In the present study, the extraction and stripping of holmium(III) and yttrium(III) ions, classified to the heavy rare earth element group, with D2EHPA were carried out by a hollow fiber membiane extractor (Yoshizuka et al. 1986b,c), together with the measurements of the distribution equilibria of these metals with DBEHPA and the interfacial adsorption equilibria of D2EHPA and its metal complexes, in order to obtain more detailed information on the kinetics and equilibria of rare earth elements with D2EHPA. 1. Experimental Section Reagents. DP-8R, a commercial metal extraction reagent, was used as D2EHPA (denoted by HR for monomeric species and as Hz& for dimeric species hereafter) as delivered from Daihachi Chemical Industry Co. Ltd. (Lot No. K20801). The purity of the reagent was found to be above 98% by means of neutralization titration with alcoholic potash using phenolphthalein as an indicator. Reagent grade holmium nitrate and yttrium nitrate were also used as received without further purification since the purities of both metal nitrates are above 99.99%. Reagent 0 1992 American Chemical Society
