with the boundary conditions
f(N,t) = f ( - N , t ) = 0 f ( k , 0 ) = 0,k # 0
Miyauchi and Vermeulen (5) have used a similar approach to a related problem. The solution is straightforwardly obtained by the method of images, and is m
The distribution of solute when unit solute is injected at every transfer is given by (9)
b ( x , t ) =+rd,j
The asymptotic distribution can be obtained from Equation
9 and is, when x 2 0,
an expression which could have been obtained from elementary considerations. The expressions for b ( x , t ) andf(x,t) appear complex, but their computation is simplified by noting that only the first two or three terms are appreciably different from zero, unless t is very large. I t is evident that any characteristics of the process can be calculated from Expressions 8, 9, IO, and 11. From Equations 8 and 9 it can be seen that if b ( x , t , A , p ) , ( A = pq, p = mq - np), is the quantity of solute in cell x after t transfers, when the parameters of the separation are m, n, p, and q, then
This identity greatly reduces the computation when several separations with different parameters are to be carried out. Expressions 8, 9, 10, and 11 need only trivial modifications if the cell train is of length M and solute is injected at cell I. Similarly, if differing quantities of solute are added at each transfer, Equation 8 can be modified accordingly. literature Cited
when x
5
(1) Alderweireldt, F., Bull. Soc. Chim. Belg. 67, 225-55 (1950). ( 2 ) Alderweireldt, F., Verzele, M., Ibid., 7 0 , 703-44 (1961). (3) Compere, E. L., Ryland, A. L., Ind. Eng. Chem. 46, 24-34
0,
When mq
(1954). (4) Craig, L. C., “Techniques of Organic Chemistry,” Vol. 111, Interscience, New York, 1950. (5) Miyauchi, T., Vermeulen, T., IND.ENG.CHEM.FUNDAMENTALS 2, 304-9 (1963).
- np
=
RECEIVED for review November 15, 1965 ACCEPTEDMay 26, 1966
0, b(x,t) is not divergent, and
VISCOSITY OF SUSPENSIONS OF
GAS-FLUIDIZED SPHERES T H O M A S H A G Y A R D A N D A L B E R T M. S A C E R D O T E l Department of Chemical Engineering, University of Canterbury, Christchurch, ,Vew Zealand
The apparent viscosity of air-fluidized beds of spherical particles has been measured with a torsion pendulum viscometer especially designed to minimize disturbance of the bed. Beds were made of closely sized uniform spheres or of binary mixtures of these. Sphere diameters ranged from 125 to 305 microns. Superficial air velocities ranged up to 10 cm. per second. A general viscosity correlation for uniform beds of this particle density has been obtained. The measurements tend to indicate Newtonian behavior, in which case the observations would b e of a true kinematic viscosity. Evidence that fluidized beds can assume certain crystal-like structures has been found.
attempts have been made to measure the viscosity A number of workers (6, 70, 72-15, 20) used rotating viscometers iihose immersed bodies rotated too quickly, in that the immersed body peripheries moved at linear velocities greater than or only slightly less than the linear superficial velocity of the fluidizing fluid. Thus, bed conditions near the immersed body differed markedly from those in the bulk of the bed. A quantitative description of this differEVERAL
Sof fluidized beds.
1
Present address, Positrode Associates, 209 Bullard St., Wal-
pole, Mass. 500
l&EC FUNDAMENTALS
ence cannot be given because the viscometer normally moves in a plane perpendicular to the flow of fluidizing fluid, and because the relative amount of fluidizing fluid which flows as bubbles and as continuous medium is usually not specified. The rotational viscometers of others (5,8 ) were not described in enough detail to permit a similar check of validity of results. Falling sphere viscometers have been used by some (4, 27), but no results concerning gas-fluidized beds, expressed in common viscosity units, have been reported. Schugerl, Merz, and Fetting (79) have presented a comprehensive and seemingly valid study of the apparent viscosity of
gas-fluidized beds. Their experiments were performed with a rotating cylinder viscometer, and viscosity was determined from the torque required to maintain the rate of rotation of the immersed body. They found that it was necessary to roughen the surfaces of the immersed body to minimize slip between that body and the fluidized bed. They noted the importance of slow immersed body rotation, but also operated with faster rotation to study bed behavior under these conditions. When rotation rates were slow enough for measurements to be meaningful, apparent viscosity seemed to be virtually independent of rotation rate. Viscosity determinations were not precise when their immersed body was rotated very slowly and the fluidized particles were small. Hagyard et al. (2) constructed and calibrated a torsion pendulum viscometer designed for use with fluidized beds. T h e solid cylindrical immersed body was not totally immersed, so defluidization above the immersed body could not occur. The viscometer wa? designed so that the fastest motion of the immersed body was slow compared with the superficial velocity of the fluidizing fluid. Nimmo (76) used high-speed photography to measure the slip a t a solid boundary moving inside a fluidized bed. He concluded, for a slowly moving boundary, that, if slip occurred, it was small. Caesar (3) used Hagyard’s (2) viscometer to measure the apparent kinematic viscosity of fluidized beds. Caesar showed that the viscosity determined when a smooth immersed body surface was used was less than that determined with a rough immersed body surface. Thus, some slip occurred a t a smooth boundary moving slowly inside the bed. Caesar used in these experiments an immersed body coated with a single layer of the particles of which the bed was made. Woodhams (23) corroborated Caesar’s (3) results with a coated immersed body. The work described here extends these ohservations to a series of beds of uniform spheres having mean diameters of 304.8, 223.8, 160.8, and 125.4 microns and to a serie? of binary mixtures of these sizes.
Both were axially tiue within 0.001 cm. and were mounted as required on a shaft true to 0.002 cm. which was, in turn, clamped to a 0.041-cm. diameter hard-drawn phosphor bronze torsion wire. Neither vessel size and neither immersed body size affected the results. T o eliminate slip, the cylindrical part of the immersed body was first coated with a layer of rubber latex 0.005 cm. thick to which a single layer of the spheres was caused to adhere by dipping the body into the barely fluidized bed. The coating spheres did not, in general, touch each other, but were separated by much less than one particle diameter. Each fluidized bed consisted of graphite-coated shellac spheres of any of four sizes, or of any of several binary mixtures of the largest size available and one other size. Each “size” was really a narrow- size range; standard deviation of the single observation of particle diameter was, in each case, about 3.5% of the mean pa1 tick diameter. Figure 1 is a photomicrograph of spheres of the largest diameter used, and Figure 2 is a photomicrograph of spheres of ail diameters used, 304.8, 223.8, 160.8, and 125.4 microns. Figure 1 was especially chosen to show a nonspherical partide. Particles of a t least this degree of nonsphericity occurred to the extent of about 0.25% in beds of the largest particles, and more often in beds of smaller particles, to a maximum extent of 1.5% in beds of the smallest particles. The density of graphite-coated shellac spheres a t 25’ C. was 1.15 grams per ml. Shellac spheres were used only because they were easily made and easily coated with graphite.
Experimental
The viscometer described by Hagyard et 01. (2) was wed to make the measurements reported. I t consisted, essentially, of a cylindrical body partially immersed in the fluidized bcd, and suspended by a . vertical torsion wire. The immersed body was given a momentary impulse to start it oscillating torsionally about its vertical axis; and the rate of decay of amplitude af oscillation was measured, and was converted directly into kinematic viscosity units through the use of the correlation reported by Hagyard et al. The amplitude of oscillation could be determined at any instant, without dis.. turbing the oscillations, by means of an optical beam. The maximum tangential velocity was less than 0.3 cm. per second. Beds of the spheres described below were fluidized with dry air a t 25.0’ C. The fluidized bed was contained in a vertical cylindrical vessel whose internal diameter was 6.6 or 7.6 cm. The defluidized bed depth was about 5.7 cm. The bottom of the vessel was fitted with a sintered Diakon (Imperial Chemical Industries, Ltd.) plate which distributed fluidizing air uniformly across the bed cross section. The apparatus was contained within a glass-sided cabinet and maintained a t 25.0’ C. and about 5% relative humidity. Saunders (78) reported that the properties of the torsion wire were significantly dependent on temperature and so affected the accuracy of the readings. The experiments were performed a t 60 feet above sea level; daily variations of barometric pressure had no measurable effects. Fluidizing air, provided by a small blower, was drawn from within the cabinet. The immersed bodies used were accurately machined solid brass right circular cylinders, 0.500 inch in diameter by 4 inches long, and 0.750 inch in diameter by 3 inches long.
Figure 1.
Spheres of 304.8-micron diameter
Figure 2. Spheres of 304.8-, 223.8.. 160.8-, and 125.4micron diameter VOL. 5
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The graphite coating on the spheres was immeasurably thin. Its purpose was to aid the dissipation of static electricity. The spheres were coated by tumbling them with finely ground graphite until, ,with all excess graphite removed, they passed two electrical tests. The graphited spheres were vigorously shaken inside a glass bottle. A thief sample was xvithdraivn and placed in a metal box electrically connected to a charged gold-leaf electroscope. If the spheres were adequately graphited, no change of electroscope charge was noted. The second, more sensitive, test consirted of inserting brass probes into an unfluidized bed of spheres to measure its electrical resistance. The bed was placed in a 3-inch diameter vessel, and the probes were inserted 1 inch into the bed at opposite ends of a diameter of the vessel. Bed resistance to 1.5 volts d. c . was required to be below the arbitrary value of 100,000 ohms. Beds of the smaller particles barely passed this test. The resistance of a bed of the largest particles used was 12,000 ohms. (When fluidized, such a bed had substantially infinite resistance.) As a precautionary measure, the bed !vas always earthed through brass earthing probes inserted into the bed at the vessel wall and also through the uncoated bottom of the immersed body, via the torsion wire. These precaution? substantially eliminated the possibility that static electricity \vas a factor in the results reported herein. The viscometer operating procedure consisted, essentially, of measuring the rate of decay of amplitude of oscillation of the immersed body a t two different immersions. Before measurements were begun, the containing vessel was raised and clamped into position so that the immersed body extended I , ’ ? inch into the defluidized bed. The vessel was raised an inch by inserting a 1.000-inch gage block betLveen the bottom of the vessel and its support and the change in submergence was calculated for the diameters of bob and vessel used. Fluidizing air flow rate was set high and then slowly reduced to the desired value. The viscometer suspension was given a momentary impulse to start torsional oscillations. The time required for the half-amplitude of oscillation to decay from 0.14 to 0.07 radian was measured repeatedly, 10 to 40 times according to circumstances. The fluidizing vessel \vas then lo\vered 1.000 inch by removing the gage block, and another series of amplitude decay half-times was measured. Bed density, shoivn by Hagyard et ul. (2) to be the only bed property which enters into these viscosity calculations, was calculated from the pressure difference across a differential p o b e located \vhen needed near the center of the bed. I t \Taried \vith air velocity and lay between 0.53 and 0.65 gram per ml. Experimental Results
Beds of Single Size of Sphere. Figure 3 shows, for beds of uniform spheres whose diameter was about 300 microns, bed viscosit>- as a function of fluidizing air velocity. Here “viscosity” means apparent kinematic viscosity of the fluidized bed; “air velocity” means linear superficial air velocity-i.e., the upward air velocity which would have prevailed in the annulus between the immersed body and the fluidizing vessel wall if the bed particles had not been present. The curves of Figure 3 have the general characteristics observed \\ith all beds, even those containing particles of more than one size. At fluidizing air velocities not far above the minimum fluidizing velocity calculated from the correlation of Leva (77), viscosity was high. With increasing air velocity, it dropped markedly until it reached a constant terminal value near 130 centistokes. The average viscosity of the vigorously fluidized beds is 130 cs. .411 the relevant data fall within the 95% confidence limits d r a u n around 130 cs. Some other viscosity, not far from 130 cs., is equally probable. The 957, confidence limits on the value of viscosity were Xvider a t high and low air flow rates, and narrower at inter502
l&EC FUNDAMENTALS
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Bed viscosity as a function of fluidizing air
300-micron spheres
-Caesar (3), 3 1 0-micron polystyrene spheres
- - _ _Woodhams (23),294-micron polystyrene spheres This work, 305-micron shellac spheres - - Minimum fluidization velocity, this work, Correlation of Leva /////
( 7 1) 95% confidence limits, this work
mediate flon rates. .4t low flow rates, the sintered Diakon air distribution plate did not distribute air uniformly across the bed cross section; air flow tended to channel, and the places where flow occurred shifted with time. Thus, the immersed body was, a t times, in a defluidized area. Slugging occurred at high flow rates; large air bubbles violently agitated the bed, causing the immersed body to sway with irregular amplitude and direction; it is evident that this type of viscometer is unsuited to slugging beds. Both effects tended to increase the width of the 95% confidence interval on the measured viscosity. The points plotted in Figure 3 hint at a shallow minimum a t about 7.5 cm. per second, as do the points for other size fractions shown in Figure 4. However, the variance of the observations was too large for such a minimum to be statistically significant and smooth curves without minima were equally probable. Shuster and Haas (20) predicted and observed a similar minimum. Schugerl (79) also observed a similar minimum. The data of Caesar (3) and Woodhams (23) are compared with those reported in this paper in Figure 3. Caesar‘s experiments were performed with a bed of graphite-coated polystyrene spheres, specific gravity 1.05. Average sphere diameter ;)as 310 microns, and the size range was very narrow. Woodhams used a similar bed whose average sphere diameter was 294 microns. The data, shown in Figure 3. for the experiments reported herein concern a narrow size range of graphitecoated shellac spheres, specific gravity 1.15, average diameter 304.8 microns. At high air velocities, the viscosity values reported by Caesar (3) and Woodhams (23) lie well within the 9574 confidence limits in Figure 3. As air velocity was reduced, their data also seem to show a viscosity upturn. The upturn for beds of their polystyrene spheres occurred a t air velocities substantially lower than those for our shellac spheres, probably because of the density difference of nearly 10%.
0
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Bed viscosity as a function of fluidizing air velocity
Beds of uniform spheres 305-micron spheres A 224-micron spheres 0 1 6 1 -micron spheres A 125-micron spheres Maximum tangential velocity of immersed cylinder
--
Figure 4 shows bed viscosity as a function of fluidizing air flou rate for uniform beds of each of four particle sizes. At high air velocities. the viscosity was approximately equal to 130 cs. but it increased markedly with further decrease in air velocity, bel015 a certain value The marked viscosity increase occurred, for different particle sizes, a t significantly different air rates The air velocities a t which the viscosity upturns occurred M ere proportional to particle diameter. These results are plotted in Figure 5 as a viscosity against a “reduced velocity” obtained by dividing the actual superficial velocity by the minimum fluidization velocity, as calculated from the Leva (77) correlation. I n these curves the sudden increase in viscosity does not occur a t the same reduced velocity. If the air velocities a t which the upturns occur on Figure 4 are plotted against particle diameter on a log-log plot, it is found that upturn air velocity is directly proportional to the first power of particle diameter. Hence a plot of bed viscosity us. u/D, might be expected to yield a general correlation. Figure 6 is such a plot and includes data on particles of all four sizes. I t seems a reasonable correlation of the data. The characteristics of the curve in Figure 6 are similar to those of viscosity us. ail velocity plots for single sizes. The data points become very scattered in the lou viscosity portion of the general correlation curve, no doubt because of slugging disturbances in the bed. The variables plotted as ordinate and abscissa on Figure 6 are not dimensionless, as might be expected in a general correlation. Further wcrk is needed to establish a dimensionlesb correlation. In such \$ark particle density nould have to be varied. whereas in Figure 6 it was constant. If the curve in Figure 6 be taken as a general correlation for the viscosity of air-fluidized uniform beds Ivithin the range of variables studied, some conclusions may be drawn. At high air velocities, apparent kinematic viscosity is independent of particle diameter and air velocity; it is approximately equal to I30 cs. At low air velocity, it increases markedly with slight decrease in air velocity. T h e air velocity a t which the
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Bed viscosity as a function of reduced air velocity H 305-micron spheres 224-micron spheres 0 161-micron spheres A 125-micron spheres
A
viscosity increase becomes rapid is directly proportional to the first power of particle diameter, and might be considered a new definition of minimum fluidization velocity, different from that given by the Leva correlation. Beds of Binary Mixtures. Each of the beds made with two particle sizes (mixed beds) was a mixture of spheres of 304.8-micron diameter and of one other size. Most measurements were made with a fluidizing air velocity of 8.5 cm. per second, sufficiently great to assume that the viscosity of beds of one particle size (uniform beds) and of mixed beds is independent of air velocity. At this air velocity the only over-all trend noted with mixed beds was that, for most mixtures, the viscosity lay very close to that of uniform beds. No mixtures studied had viscosities appreciably below those of VOL. 5
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A few mixtures had viscosities significantly hose of uniform beds. These viscosity maxima reted in terms of bed StNcture as follows. photography by Hagyard and by Aird ( I ) , ntaining wall surrounding a fluidized bed made N size fraction of graphited polystyrene spheres, has shown that about 80% of the spheres, even in a rapidly moving bed, are arranged in hexagonal blocks. Figure 7 is one such photograph. I n no case are the axes of the structures exactly vertical, although in some cases they are nearly so.
in the hope that structure investigations might lead to an understanding of the mechanism of flow in the interior of the bed and indeed it has indicated that flow involves the breaking up and reforming of apparently hrxasonal block9 of oriented spheres within the bed. Graton and Fraser (7)discuss1:d the six ways in which a regular packing of equal-sized spheres can be arranged. If 1 . ”.,. 3 . a regular packing of spheres occurrea in a nuiaizea oea, IT would be expected to show as niuch symmetry as possible, especially if the orientation of “CF ystal” grains were such that symmetry is exhibited by 1
._
experiments, the two possible cubic close-packed lattices were considered identical. Because their void fractions and, in a sense, void sizes and void shapes are identical, the viscosity measurements described below could not distinguish between the geometrical differences of these two packings. The general plan fo:r proving the existence of a crystal-like lattice was to attempt to fill the lattice voids with the small spheres which would exactly fit into them. Presumably, if the voids were filled, a marked change in bed viscosity would occur. I t \vas believed that, if there were a tendency to form regular "crystal" grains in mixed beds, any small spheres which could fit into the voids of the lattice \iould go into them. The 125.4-micron spheres very nearly fit into the largest voids betxieen the 304.8-micron spheres when the latter are arranged in a cubic closest packe'd lattice, and were therefore used in the binary mixtures. The least stable regular packing of spheres would be a form of cubic packing for which the unit cell is a cube with a sphere a t each corner. Even in a packed bed such a condition, where each sphere has only six tangent neighbors, would not be very stable. If the 304.8-micron spheres \vere packed in a cubic lattice, the 223.8-micron spheres would very nearly fit into the voids betlieen the larger spheres. These spheres Itwe also tested in binary mixtures. Orthorhombic structure Xvould also be unlikely in the interior of a fluidized bed, as this structure has very little symmetry. Its unit cells are formed by placing the centers of spheres at the vertices #of a right prism whose bases are equilateral triangles. Thus, the packing might be prepared by taking hexagonal layers of spheres and placing these one upon the other so that successive layers of spheres are directly over each other. Each sphere has eight tangent neighbors. If the 304.8-micron spheres were packed in this fashion, the 160.8-micron spheres \.iould very nearly fit into the largest voids in the packing. Consequently this size was also tested. Table I summarizes the packing geometries described above. Figure 8 sho\is the packing arrangements. Figure 9 sho\vs the viscosity of mixtures of 304.8- and 125.4micron diameter spheres as a function of percentage of fines: the per cent by number of particles. All data for Figures 9, 10, and 11 refer to an air velocity of 8.5 cm. per second, high enough so that bed viscosity was independent of air velocity, The viscosity was close 1:o the uniform bed viscosity, except for a sharp viscosity maximum near 36% fines. There is photographic evidence that i n a uniform bed 80% of the particles are arranged in apparently hexagonal "crystals" and, exactly to fill a hexagonal lattic'e involving 80% of the coarse particles present, 40% fines are required. The maximum a t 36% Table I. Filling of Intlerstitial Voids in Regular Packing of Spheres of Diameter D, Ratio of A70. of Large to Small 5;bheres in Unit Cell ( ~ ~ ~Diameter , ~ of ~Small ~ t Diameter of Voidr amon,p . Spheres, Microns Smaller ,Sphere Large Spheres TheoreJYhich Exactlv Filled with tical. Fits into Relecant D, = .Vame of Largest Voids Small 304.8 Packing in Packing Spheres) microns Actual ( 4- 1 ) Cubic closest 1:l 126.2 125.4 0,
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fines could be taken as strong evidence of the existence of a hexagonal structure within the bed. The evidence is weakened, however, when the orthorhombic lattice is considered. Figure 10 is a plot of the viscosity of mixtures of 304.8and 160.8-micron spheres, as a function of per cent of fines. The ratio of particle diameters is appropriate for the formation of orthorhombic lattices. Viscosity is constant a t the value observed iiith uniform beds, except for a rapidly rising, slowly declining maximum at 677, fines. The maximum occurs at the per cent of fines appropriate for filling the voids in the orthorhombic lattice. If the orthorhombic lattice did occur, every coarse particle could not, because of bed turbulence, be in the regular array. Therefore, the maximum would have been expected a t a less-rich-than-ideal mixture, as observed \vith the 125.4-micron particles in Figure 9. The maximum probably would occur at a lower per cent of fines if some unknown factor did not act to pull it toTvard the right-hand side of i of the Figure 10. This pull is evidenced by the s l o ~ decline maximum after its peak. Because a viscosity maximum occurs with the approximate amount of fines needed to fill the voids of both the hexagonal and orthorhombic lattices, it could be concluded that either packing is equally likely. More probably, the addition of fines of the correct size could have forced the bed to assume the appropriate geometry. LVhen the relative sphere sizes were appropriate for the formation of the cubic lattice with 223.8-micron spheres, results as in Figure 11 were obtained. Viscosity was greater, but not significantly greater, than 130 cs. For these observations the 9570 confidence interval a n viscosity was, in general, unusually wide, indicating poor quality of fluidization. Poor quality of fluidization !vas> in fact, observed. Figure 12 shoxvs a plot of bed viscosity as a function of air velocity for different mixtures of 304.8- and 125.4-micron spheres. For 98.656 small particles the point of viscosity upturn is presumably appropriate to that of 125 microns and VOL. 5
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Figure 9.
Bed viscosity as a function of per cent of small spheres Mixtures o f 125- and 305-micron spheres 95% confidence limits Viscosity of uniform beds Theory for hexagonal lattice
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Mixtures of 161 and 305-micron spheres 95% confidence limits Viscosity of uniform beds Theory for orthorhombic lattice
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Bed viscosity as a function of per cent of small spheres Mixtures of 224- and 305-micron spheres 95% confidence limits Viscosity of uniform beds Theory for cubic lattice
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Bed viscosity as a function of fluidizing air velocity Mixtures of 1 2 5 - and 305-micron spheres 1 0 0 % 305-micron spheres A 36.6% 125-micron spheres 0 4 5 . 9 % 125-micron spheres 1 98.6% 125-micron spheres
beyond the range of the observations. For 0 and 48.9y0 fines the curves are similar to one another, and it is presumed that this is generally true for not excessive additions of fines to the 304-micron spheres. However, for 36.6% fines, corresponding to the maximum in Figure 7, the abnormally high viscosity seems to be retained a t all air velocities and may reflect the difficulty :in forming and reforming hexagonal “crystals” when the major voids are filled by 125-micron particles. ‘rwo types of bed ,segregation were observed with these binary mixed beds. Segregation, if present near the fluidizing vessel wall (which was transparent), was easily noted because insignificant differenceis in the graphite coating on particles caused color differences. When the bed was barely fluidized, fines collected at the top of the bed. This type of segregation, described by Katz ( 9 ) : increased when the fine particle size decreased, probably because smaller particles could more easily find their way u p narrow flow channels. T o minimize the effect of this segregation at low air velocities, the bed was momentarily fiuidized vigorously a t brief intervals. When the bed was well fluidized, a few fines collected momentarily near the wall? a t the tDp of the bed, and were eventually caught in the downivard circulation near the ivall. This type of segregation seemed to increase as the diameter of fines increased. Both types would result in a discrepancy between the actual bed composition and that calculated, but the effect was, no doubt, minor. The magnitude of the standard deviation on the measured value of apparent kinematic viscosity might be used, in a given system, as an index of the quality of fluidization. Sacerdote (77) showed that the va.riations in the rate of decay of oscillations of the immersed body \\.ere the only significant source of error in the viscosity measurements described herein. At high air velocities, these variations were caused primarily by the random s\vay of the immersed body which was induced by the violently slugging bed, At loiv air velocities, channeling occurred, so that the immersed body was, a t times, in a defluidized area. At intermediate air velocities, when fluidization appeared to be moat nearly uniform-Le., of good quality -the standard deviation of viscosity measurements \vas least.
Discussion
For beds of closely sized uniform shellac spheres fluidized with air a t 25” C.. the appaient viscosity falls rapidly from very high values near the conventional minimum fluidization velocity to a constant value of about 130 cs. a t several times the minimum fluidization velocity. This constant value is independent of particle size from 305 down to 125 microns. For different particle diameters. kinematic viscosity does not correlate M ith a reduced velocity, obtained by dividing the superficial air velocity by the conventional minimum fluidization velocity for that size of particle. A common curve for all the single sizes of particle studied can be obtained by plotting kinematic viscosity against superficial velocity divided by particle diameter. as shown in Figure 6. There are no published data for beds composed of such closely sized particles except those in which the measurements are open to criticism in that the immersed body, by too rapid motion, probably interfered Xvith the state of fluidization. For binary mixtures of 305-micron spheres with one other size the behavior is more complex. While most mixtures exhibit behavior identical with that of single sized particles, viscosity is abnormally high at all air velocities for the mixture containing 36.6y0 of 125-micron particles. This might be because the smaller particles fill the voids in the hexagonal “crystalline” regions of the bed already observed to exist in uniform beds. Perhaps, the inclusion of the correct size and number of fine spheres causes the formation of difficultly broken agglomerates. Similarly, particles of 160 microns, capable of filling the major voids in an orthorhombic lattice, show a maximum viscosity at 67%, and may indicate a tendency towards this structure in the presence of fine particles of a suitable size. Using 224-micron fines, no similar tendency towards a cubic structure was detected. For mixtures other than these special ratios the viscosity at the higher air rates always became constant a t appioximately 130 cs. Comparison has been made with the \+ark of Schugerl et al. (79), who used a rotating and roughened immersed body. For 40- to 60-micron glass spheres, a t loiv rates of revolution, VOL. 5
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comparable with the lov~rates of shear used in the present work, the viscosity observed by them has been calculated to be 150 cs. This calculation is, however, only approximate, as bed density had to be assumed and the air rate was not directly ascertainable. Results obtained with the rotational viscometer used by them are roughly comparable with those here described obtained with the oscillating viscometer. There is some evidence of Newtonian behavior of the bed under the low shear stress involved in these measurements. The torsional viscometer correlation ( 2 ) is a unique function of several system variables, such as immersed body diameter and period of oscillation. When one of the system variables was altered, such as bob diameter or period, a different portion of the viscometer correlation curve became relevant. Calculations based on this different portion invariably led to substan tially the same viscosity value, whereas the mean rate of shear had been perceptibly altered, thus giving indirect evidence that the observed viscosity seems to be independent of the rate of shear. Further evidence is that, as nearly as could be ascertained in these experiments, the logarithmic rate of decay of amplitude was independent of the amplitude, as it is for a Neittonian fluid. The data of Schugerl et al. (79) for their rotational viscometei and for slow rates of immersed body rotation, also indicated an independence between viscosity and rate of shear. The theoretical predictions of Trawinski (27) are not confirmed by the results here described, in that bed viscosity did not decrease lvith particle size. Trawinski (22) also predicted a minimum viscosity a t such proportion of fines as isould coat a large particle with a “lubricating” layer of small ones. KO such minimum was observed in the present nork.
literature Cited (1) Aird, R. M., B. E. (Hons) (Chem) thesis, Canterbury Universitv CollePe. 1957. (2) Ashk,in, B.”S:, Hagyard, T., Saunders, I. C. B., Young, T. E., J. Sci.Znstr. 37, 480 (1 960). (3) Caesar, M. B., B.E. (Chem) Project Report, University of Canterbury, 1960. (4) Daniels, T. C., in “Rheology of Disperse Systems,” C. C. Mill, ed., Pergamon Press, London, 1959. (5) Diekman, R., Forsythe, LV. L., Ind. Eng. Chem. 45, 1174 (1953). (6) Furukawa, J., Ohmae, T., Ibid., 50, 821 (1958). ( 7 ) Graton, L. C., Fraser, H. J., J . Geol. 43, 785 (1935). (8) Jottrand, R., J . A$@. Chem. 2, SI7 (1952). ( 9 ) Katz, H. M., .4rgonne National Laboratories, Publ. ANL5725 (1957). (IO) Kramers, H., Chem. Eng. Sci. 1, 35 (1951). (11) Leva, M., Chem. Eng. 64 (111, 266 (1957). (12) Long, F. J., M. Sc. (Eng) thesis, Princeton University, 1952. (13) Martin, J., Chirn. Znd. 69, 57 (1953). (14) Martin, J., unpublished paper, “Mesure de la viscosite d‘un Lit, Fluidis6 en Liason avec le PhtnomPne de Slugging,” Services Laboratories de la SociCtC d’Etudes Chimiques pour 1’Industrie e 1’Agriculture (France), Sept. 14, 1951. (15) Matheson, G. L., Herbst, W. A., Holt, P. H., Ind. Eng. Chem. 41, 1099 (1949). (16) Nimmo, N., B. E. (Chem) Project Report, Canterbury University College, 1958. (17) Sacerdote, A. M., M.E. (Chem) thesis, University of Canterbury, 1962. (18) Saunders, I. C. B., B.E. (Hons) (Chem) thesis, Canterbury Lniversity College, 1956. (19) Schugerl, K., Merz, M., Fetting, F., Chem. Eng. Sci. 15, 1 (1961). (20) Shuster, LV. LV., Haas, F. C., J . Chem. Eng. Data 5 , 525 (1960). (21) Trawinski, H., Chem. Ing. Tech. 25, 201 (1953). (22) Ibid., p. 229. ( 2 3 ) \Voodhams, D. J., B.E. (Chem) Project Report, University of Canterbury, 1961. RECEIVED for review September 8, 1965 ACCEPTED June 6, 1966
EFFECT OF INTERPARTICLE ADHESIVE FORCES
ON FLUIDIZATION OF FINE PARTICLES M A N F R E D BAERNS’ Chemical Engineering Division, Argonne National Laboratory, Argonne, Ill.
The feasibility of gaseous fluidization of particles in the size range of less than 50 microns was investigated, The ratio of the incipient fluidization velocity, calculated by a conventional relationship without accounting for interparticle forces, to the incipient fluidization velocity, determined by pressure drop and heat transfer measurements, was used as an index, FI, describing the fluidizability of a particulate material. The results indicate that FI is closely related to the interparticie adhesive force. The limitations of the feasibility of fluidization depend on the ratio of the weight of a particle to the sum of its weight and adhesive force; no fluidization could be obtained when this ratio was less than 1O-3.
FLUIDIZATIOS of fine particles is becoming of interest in various chemical processes such as the fluorination of uranium Us08, U02F2, UF,-the preparation of compounds-e.g., uranium monocarbide, and the coating of reactor fuel particles. Fine particles are in general difficult to fluidize. This difficulty is related to interparticle forces (76, 77) which are greater than those transmitted to the particles by the fluidizing gas. The interparticle forces cause agglomeration of these Present address, Institut fur Technische Chemie der Technischen Hochschule Hannover, 3 Hannover, Germany. 508
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particles and bridging between such agglomerates. The agglomeration and bridging favor the formation of channels through which the gas passes without fluidizing the fine particles. There is evidence that fluidization can generally be achieved only when the particle size of the solids is above a certain limit (70, 72, 76, 22, 2 3 ) ; the lower size limit of the particles \shich can be fluidized varies for different materials. There is general agreement in the literature upon the molecular nature of the interparticle forces ( 7 , 3, 4, 6 , 7 , 9, 7 7 , 73, 75, 78, 79, 27, 28). These forces that affect the fluidization of fines comprise mainly the interparticle adhesive force
