Ind. Eng. C h e m . Res. 1990,29, 1346-1348
1346
Separation of Fine Particles by Laminar Flow in a Tilted Slender Tube Slowly Rotating around the Longitudinal Axis Masatsugu Kimura,* Kotaro Shirane, and Yoshiko Yamaguchi Biophysics Laboratory, Osaka City University Medical School, Osaka 545, Japan
Ichiro Aoki Omichi Clinic Laboratory, Sakae-machi, Yao, Osaka 581, Japan
Takayuki Tokimoto Department of Physiology, Osaka City University Medical School, Osaka 545, J a p a n
Fractionation of fine (- 10-pm) particles by their size, density, or both was studied. A laminar flow of suspended particles through a tilted slender tube that slowly rotates around the tube axis separated fine particles into two fractions according to the sedimentation velocity of the particles. The threshold of separation was mainly governed by the rotation period, which depends on the particle size. Nearly spherical particles of silica gel were separated experimentally. A two-step operation yielded a fraction of particles of uniform size (half-width, -0.7 pm) with a mean diameter of - 5 pm. The particle-size threshold was close t o the theoretical value. Fine particles (with diameters of about 10 pm or less) of uniform size are often needed as fillers for high-pressure liquid chromatography, as standard samples for instruments that classify particles, and so on. For such purposes, commercially available fine particles generally require fractionation. There are various methods to separate fine particles, such as with a hydraulic cyclone, spinning coils (Papanu et al., 1986),and coil planet centrifuge (Ito et al., 1966), but they do not give fractions with narrow size distributions. A slowly rotating tube can be used for the classification of fine particles (Aoki and Shirane, 1973; Aoki et al., 1986, 1988). Here, in a modification of that method, we describe the continuous separation of fine particles into two fractions according to their size, density, or both. Principles The apparatus is shown in Figure 1. A slowly rotating tube (inner radius A and constant angular velocity wo) with laminar liquid flow (liquid density p’, viscosity 17, and mean flow speed Ul) is set a t an angle 0 (0’ < 0 < 90’). The movement of a particle (density p and sedimentation velocity Vo)in the tube can be analyzed as two parts, one being movement with respect to the cross section of the tube and the other being movement along the tube axis. In the cross-sectional movement, the particle in the tube describes a circle (or an arc) around the center ( Vi/wo,O) (Figure 2 ) , where V i = Vocos 8. This path is explained by constant particle sedimentation through the rotating liquid treated as a rigid body on the assumption of a small wo, which assures that the centrifugal force is small (Aoki et al., 1988; Kimura et al., 1988). Because a particle with Vd/woL A does not rotate in the steady state (Figure 2a), only the case of Vd/wo< A (Figure 2b) is discussed below. The radius of the circle depends on the initial position of the particle. Let that position be (fA,O) (-1 < f < 1) by setting an initial time when the y component = 0. Depending on whether IfA - Vd/wol 1 A - Vo’/woor not, the particle will or will not touch the tube wall, in paths shown by a solid line and broken line, respectively, in Figure 2b. Then, radii in the steady state are given as follows. When -1 < f I2V0’/Aw0- 1. R = A - V~/WO (1) When 2V0‘/Aw0- 1 < f < 1,
R = IfA - Vo’/wol
(2)
With laminar flow from the lower end of the tube to the upper end, the particle drifts along the tube axis. The particle descends with Vosin 0 with respect to the liquid, which obeys Poiseuille’s flow: flow speed u(r) = 2U1(1? / A 2 ) (r = ( x 2 + y2)l/, and Ul is the mean flow speed). The speed of the upward drift of the particle averaged for one revolution is estimated by l / T L * u ( r ( t ) ) dt - V osin 0 ( T , period) where
+
r2 = ( V , , ’ / W ~ ) ~ R2
+ 2R(Vor/oO)cos (coot)
Then the speeds, U and Upll of the upper drift, corresponding to eqs 1 anc! 2 , are given by Up = 4UI(Vo COS O/Awo)(l - Vo COS 8 / A ~ o )V o sin 0 (-1 < f I2V0 cos O/Awo - 1) (3) and
U,l = 2Ul(1 - 2( Vo COS O/AUO)~+ 2f( Vo COS B/Awo) f 2 1 - V osin 0 (2V0cos O/Awo - 1 < f < 1) (4) where V i is replaced by V ocos 8. Of the particles with the same Vo,those with the smaller R drift faster. Especially, the maximum of U i , Ui’, is found at f = V ocos O/Awo (i.e., R = 0) and is given by
Upr’= 2UJ1 - ( V ocos O / A U ~ )-~Vo ] sin 0 Among these three speeds, there is a relationship: u;, 1 Up’1 up
(5) (6)
If Up > 0, then Vo < (Awo/cos O)(l - Aoo tan 0/4Ul) = V,, where U , is implicitly assumed to be large. If Ui’ < 0, then VO> (Awo/cos O)[{l + (Awotan 0/4U,)2}1/2 Awo tan O/4Ul] = V,, Then, a particle with Vo< V,, ascends and a particle with Vo> V,, descends. We have discussed the case of Vg)/oo
0888-5885/90/2629-1346$02.50/0 0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1347
I
z I
I
I
I
I
i
F.
valve
s t i r r i n g wing
0
4.57
5.76
6.50
(100)
Figure 1. Diagram of the apparatus.
7.26
(200)
7.02
8.31
pm
(300) ( f l )
D i a m e t e r (Volume) Figure 3. Size distributions of the original sample and the three fractions obtained. (a) Sample. (b) Three fractions. 1 fL = cm3. Points: h = 4.1 pm, i = 5.3 pm,j = 5.8 pm, and k = 6.8 pm.
Experimental Method
Figure 2. Path of movement of particles with respect to the cross section of a rotating tube filled with liquid. (a) V,l 2 Awo. (b) V,l Vmz. When V,, < V, < Vmz,the particle descends or ascends depending on its f values. However, this transition region is negligible when U, >> Awo tan 6
(7)
This condition helps to make the separation more rapid and is easily attained in practice. Thus, the threshold of sedimentation velocity is approximately given by v t h = Vml. It corresponds roughly to the rotation period (T)as
T = 2xA/ v t h COS 6
(8)
More precise relations can be derived but are of little practical use (see Results section). When a particle is spherical, Vth is related to the threshold radius (ath)as vth
= 2ath2(p - P' )g/91
(9)
By substituting eq 9 into eq 8, T is related to T = 9nqA/{ath2(p- p')g
COS
6)
ath
as (10)
If a small amount of fine particles and a laminar flow of liquid (U,>> Awo tan 6 ) are present in a tilted slender tube that rotates at period T, particles with radius < a t h are collected from the upper end of the tube and the remaining particles from the lower end. T is sensitively affected by ath, so a fraction with a narrow distribution is easily extracted by a two-step operation with two different
TS.
In the apparatus shown in Figure 1, the rotating glass tube with an inner diameter of 3.4 mm and that is 550 mm long is set at the angle 6 = 40". About 50 mm of a glass pipet (tip, 1.4-mm0.d.) is inserted into the upper end of the tube. In the stirring region, six sets of stirring wings, each of which is 2 cm long and is fixed to the tube, are aligned 2 cm apart. In a practical test, spherical fine particles of silica gel with diameters of about 10 pm (LC-IOK, Wako Pure Chemical Industries, Ltd., Osaka, Japan) and with a density estimated experimentally to be 2.15 g/cm3 were used. The liquid was ethanol (p' = 0.793 g/cm3 and q = 1.19 CPat 20 "C), which was injected into the lower end of the tube. Particles soaked in ethanol were introduced from the upper end at the rate of about 1mL/h. Particles flowing out of the tip of the pipet seemed to drop down to the tube wall in a thin line without being influenced by the liquid flow, so they were introduced continuously. The upper fraction (F,) of the original sample was initially collected at T = 126 s with Ul = 2.0 cm/min at 23 "C. This fraction was then separated a t T = 224 s into an upper fraction (F,,) and a lower fraction (Fud) with Ul = 2.5 cm/min at 21.5 "C. Each fraction was analyzed by a microcell counter (Sysmex, F-800, Toa Medical Electronics Inc., Kobe, Japan), which operates on the same principles as a Coulter counter. The mode diameters of fractions F, and Fudwere also found by measurement of the diameters of more than 100 particles in photographs. These diameters were compared with peaks of the size distributions found by the counter.
Results The size distributions of the original sample and the three fractions (Fu,F,,, and Fud)are shown in Figure 3 (a and b, respectively). The curves have been treated as if they have the same peak value. The abscissa is linear to the particle volume (given in parentheses). In Figure 3b, the half-value of the peaks is shown by the straight line. Mode diameters were 5.5 pm for F,, 5.0 pm (5.2 pm) for F,,, and 5.9 pm (5.9 pm) for Fud,where the values in parentheses are from the photographs. The closeness of
1348 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
the values from the photographs and from the microcell counter suggests that the counter can be used for this purpose. The distribution of Fud(Figure 3b) is independent of that of the original sample, because this fraction contains a large part of F,, which is a small part of the original sample. The fraction had a narrow distribution; that is, the half-width of the particle diameters was 0.7 pm. The distributions shown in Figure 3b tail off in the direction of the larger particles, perhaps because some large particles ascended just after they dropped out of the pipet, although they seemed to be unaffected by the liquid flow. The experimental particle-size thresholds were compared with the theoretical values. For the separation a t T = 224 s, the substitution of A = 0.17 cm, e = 40°, p = 2.15 g/cm3, p' = 0.793 g/cm3, and 7 = 1.16 CP(at 21.5 "C) into eq 10 gives a theoretical threshold diameter (=2ath)of 9.8 pm. This is larger than the experimental threshold of 5.5 pm given from Figure 3b. For the separation a t T = 126 s, the threshold diameter is threoreticdy 13.0 pm when 7 = 1.13 CP(at 23 " C ) and was experimentally found to be about 6.5 pm. The theoretical threshold diameters were larger than the experimental values. The processing rate was about 0.2 g/h. V , of a particle with a diameter of 5 pm is about 0.2 cm/min. When gravitational sedimentation is small, the processing rate was not rapid. Discussion Our method is based on the relation between the particle-size threshold and the rotation period of the tube. The theoretical threshold should coincide with the experimental one if a few particles are supplied into the tube. When the supply of particles is abundant, however, the assumption of the rotation of liquid like a rigid body no longer applies, because about one-fourth of the wall is covered with particles that do not rotate but slide downward. This makes liquid rotation slower, so the experimental threshold should be smaller than the theoretical one. However, eq 8 or eq 10 can be used in practice to decide the experimental conditions. In a preliminary experiment, we separated fine particles without use of stirring wings. The upper fraction contained only small particles, but some small particles slid down to join the large ones. Stirring wings helped to separate the particles clearly into two fractions. These wings shift particles toward the middle of the tube, where liquid rotation is not so disturbed by particles sliding downward. If fine particles are completely suspended before being supplied into the tube, particles together with the liquid can be injected into the lower end of the tube, which should improve fractionation a t the upper end. The minimum length of the tube can be decided as follows. Some of the descending particles occasionally ascend, immediately after they are stirred by the stirring wings. They later descend after their movements become steady. Pathways of the particles become steady within one revolution or sooner. The length between the stirring region and the top of the pipet should be 2TU, or more so that such particles stay in the tube for T a t least, where 2Ul is the maximum flow speed of the liquid. In our second
separation, with T = 224 s and U, = 2.5 cm/min, 2TU1 = 18.7 cm. The length of our apparatus was set a t 21 cm. The processing rate increased when larger particles (10-100 pm) were separated by this method, because of larger gravitational sedimentation. We separated nearly spherical but somewhat deformed particles of silica gel measuring -30 pm and nonspherical particles of 400-mesh Amberlite, among others, a t the rate of 1-10 g/h. Centrifugal force might be used for still higher processing rates. Nomenclature a = radius of the particle a t h = threshold radius A = inner radius of the tube f = parameter of the initial position of the particle R = radius of the circular pass of the particle r = distance from the longitudinal axis of the tube t = time T = rotation period of the tube U , = mean flow speed of the liquid Up = mean speed of the drift of a particle when it touches the tube wall U i = mean speed of the drift of a particle when it does not touch the tube wall U " = maximum U,/ for f u&) = flow speed of the liquid at r Vo = sedimentation velocity of the particle V ',, = V, cos = cross-sectionalcomponent of V,,in the tube Vth = threshold sedimentation velocity Greek Letters 7 =
liquid viscosity
0 = angle of tilt of the tube p = particle density p' = liquid density oo = angular velocity
of the tube
Subscripts
1 = liquid p = particle
th = threshold u = upward
d = downward Literature Cited Aoki, I.; Shirane, K. A New Separation Method of Particles. Jpn. J . Appl. Phys. 1973, 12, 487-488. Aoki, I.; Shirane, K.; Tokimoto, T.; Nakagawa, K. Separation of Fine Particles Using Rotating Tube with Alternate Flow. Rev. Sci. Instrum. 1986,57, 2859-2861. Aoki, I.; Shirane, K.; Tokimoto, T.; Kimura, M. Rapid Separation of Fine Particles with Narrow Size Distribution. Reu. Sci. Znstrum. 1988,59, 484-485. Ito, Y.; Weinstein, M. A.; Aoki, I.; Harada, R.; Kimura, E.; Nunogaki, K. The Coil Planet Centrifuge. Nature (London) 1966, 212, 985-987.
Kimura, M.; Shirane, K.; Aoki, I.; Tokimoto, T. Method for Measurement of Viscosity and Density of Liquid with a Slowly Rotating Column. Reo. Sci. Instrum. 1988, 59, 967-970. Papanu, J. S.; Adler, R. J.; Gorensek, M. B.; Menon, M. M. Separation of Fine Particle Dispersions Using Periodic Flows in a Spinning Coiled Tube. AIChE J . 1986, 32, 798-808.
Received for review April 13, 1989 Revised manuscript received January 30, 1990 Accepted February 26, 1990
