SETTLING RATES AND SEDIMENT VOLUMES OF FLOCCULATED KAOLIN SUSPENSIONS

Transcription

1 SETTLING RATES AND SEDIMENT VOLUMES OF FLOCCULATED KAOLIN SUSPENSIONS ALAN S. MICHAELS AND JUSTIN C. BOLGERI Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge 39, Mass. This study investigated the settling rates and sediment volumes of aqueous, flocculated kaolin suspensions as functions of kaolin concentration, container dimensions, and chemical composition of the aqueous phase. Equations which correlated the data were derived, based on a struc- tural model which assumed that in a flocculated suspension, the basic flow units are small clusters of particles (plus enclosed water) called flocs. These flocs retain their identity under the mild forces experienced in gravity settling. At low shear rates, the flocs group into clusters of flocs, called aggregates. The aggregates may form networks which extend to the walls of the container and give the suspension its plastic and structural properties, HE KINETICS of sedimentation of particulate solids in liquid Tmedia and the structural and rheological characteristics of the resulting sediments are of great importance to a broad segment of the chemicals processing and process metallurgical industries. Although sedimentation phenomena have been subjects of research for many decades, and much insight has been gained into the hydrodynamics of sedimentation processes, relatively little is yet known of the role of particleparticle attractive forces in settling and sediment consolidation. The object of this investigation, therefore, was to study the sedimentation behavior of aqueous suspensions of a solid (kaolin) whose surface characteristics are well enough understood to allow analysis and interpretation of sedimentation data in terms of particle-particle interactions. Most previous studies of settling rates have pertained to suspensions of dispersed, nonattracting particles, rather than to flocculated suspensions. In general, these have been attempts to extend Stokes law to cover nonspherical particles (7, 3, 2), to allow for the effect of particle concentration upon settling rate (7, 77, 77, 2), or to calculate the wall effect for settling in containers of finite size (77). The first general study of flocculated suspensions was by Coe and Clevenger (4), who described the various concentration zones which exist within a settling suspension, and who observed that the upward flow of displaced supernatant was a special case of pore flow. Wadsworth and Cutler (22) studied the effects of flocculating agents upon the settling rates of kaolin suspensions. Smellie and La Mer (72, 79) also studying flocculated kaolin suspensions, sbggested that settling rates could be correlated using the Darcy filtration equation to predict the flow rate of displaced supernatant, and proposed an 1 Present Address, AVCO Research and Advanced Development Division, Wilmington, Mass. extrapolation procedure to estimate the ultimate volume of the settled bed. Gaudin, Fuerstenau, and Mitchell (6-8) used x-ray adsorption techniques to measure the local kaolin concentrations in settling beds as a function of time and position. The density profiles so obtained indicated that the displaced supernatant fluid leaves the bed through pores of comparatively large diameter during the early stages of settling and later is expelled via much smaller tubules in the compressive phase of sediment consolidation. These authors also noted that settling rates increased as container height increased. The studies cited above represent important contributions to the understanding of relative movement of solid and liquid during settling and consolidation in flocculated suspensions, but the part played by suspension microstructure in these phenomena remains obscure. Examination of a highly purified, well characterized solid, suspended in a solution phase of controlled composition, appeared to offer promise of analysis in terms of its microstructural features. The clay mineral kaolinite, in aqueous suspension, was selected for this purpose. Kaolinite, a hydrous aluminum silicate of composition A123 2Si2.2H~, occurs in the form of thin, roughly hexagonal platelets, of length-to-thickness ratio of about 1. Under normal conditions, in aqueous suspension, the particles carry a negative charge, distributed over the basal surfaces; this charge arises, it is believed, by occasional substitution of aluminum for silicon in the clay crystal lattice, this charge deficiency being satisfied by retention of foreign cations (commonly hydrogen, aluminum, calcium, magnesium, or sodium) on these surfaces. These ions are mobile in aqueous media, and are responsible for the cation exchange capacity of the clay, which is of the order of 1 to 2 meq. per 1 grams of solid. Under acidic conditions (ph 6), alumina exposed at the edges of the plates apparently binds hydrogen ions, and assumes a positive charge (78); this causes electrostatic attraction between edges and faces, with the formation of highly expanded card-house flocs. Under alkaline conditions, the edges become neutral or negatively charged, and the particles deflocculate, provided the electrolyte concentration in solution is low. At high electrolyte concentrations (at both high and low ph) electrostatic repulsion (or attraction) between particles is reduced because of double-layer compression or ion shielding of the surface charges. Under these conditions, residual valence forces at the particle surface cause the pa1 ticles to adhere to one another along their basal surfaces, forming card-pack flocs. More complete treatments of the surface chemistry of kaolin and of the factors controlling particle interaction forces are given by Brindley (2), Michaels (75), Street and Buchanan (27), and others (73, 78, 22). The ability to control and to predict the magnitude and geometry of particle interactions in kaolin suspensions thus renders it an ideal substance for systematic investigation. 24 I&EC FUNDAMENTALS

2 Figure 1. Floc-aggregate structural model I 1 I 1 The tl c2 Figure 2. Three general types of settling plots Procedure The kaolin used in this study was a well crystallized, acidbleached product of rather narrow particle size distribution [9 weight % between.2- and 2-micron equivalent diameter (g)] provided by the Georgia Kaolin Co., and designated PD-1. The cation exchange capacity of this material was found to be approximately 1. meq. per 1 grams. As received, the solid was found to contain approximately.5 mmole of readily extractable aluminum (as alumina) per 1 grams. Sedimentation data were obtained in a 25' =t.5' C. constant temperature room, using vertical glass cylinders which ranged from 5-ml. graduates up to tubes 12 cm. long and 6.5 cm. in I.D. To investigate the effects of changes in the chemical content of the ambient fluid, five flocculated slurries were studied in detail. SLURRY A. Cntreated kaolin, ph 4. This was PD-1 grade kaolinite, mixed with distilled water in a Waring Blendor. At a kaolin volume fraction. dk >.3, the ph of slurry A was 4.. SLURRY B. Untreated kaolin, ph 6. This was identical to slurry A except that the ph was raised to 6. by the addition of YaOH. SALT-LVASHED SLURRIES C A ~ D. The Georgia Kaolin Co. uses an acid-wash procedure (9) to remove iron oxides and other impurities from the kaolinite. This procedure. however, also dissolves considerable quantities of lattice aluminum, which is subsequently precipitated and deposited upon the platelet faces as positively charged colloidal alumina. By reducing the number of negative charge sites on the platelet faces, this alumina weakens the edge-to-face attractive forces which give rise to the card-house flocs in acid slurries. It is possible to remove most of this surface alumina by repeated extraction of the kaolinite with 111; SaCl at ph 3, according to the procedure described by Martin (73). Slurry C was salt-\vashed twice this way. and then adjusted to a ph of 4.5 and a chloride ion concentration of.1s. Slurry D was identical to slurry C except that the slurry ph was adjusted to 6.5 ivith h-aoh. CARD-PACK SLURRY E. In contrast to the card-house flocced slurries A, B, C. and D, slurry E had a platelet face to platelet face card-pack structure. Slurry E was prepared by raising the ph of slurry A to 9. with KaOH, and then adding enough XaCl to bring the C1 concentration in the supernatant 2 to.6a\r. forces and collisions experienced in gravity settling. Because the flocs are originally formed under conditions of severe agitation in the Waring Blendor, the most probable floc shape is a sphere, the shape most capable of resisting deformation by surface forces. Further, it has been shown (74, 76) that flocs tend to approach a uniform size in any shear field. This size increases as clay concentration increases or as shear rate decreases. At low shear rates the flocs tend to group into clusters of flocs, herein designated as aggregates. These aggregates may join together to form extended networks (as in Figure I), which extend to the walls of the container and give the suspension its plastic and structural properties. Settling Rate in Dilute Suspensions. Settling rates were determined by plotting the height, Z, of the interfacial plane between the slurry and the supernatant as a function of time. There were three general types of plots, as shown in Figure 2. For very dilute suspensions ($K <.7 for untreated kaolin) 2 w- U LT W I- - z U I- - W 3: 6 1 8C Results and Discussion The Model. The model used to interpret the data was based on the premise that in a flocculated suspension, the basic flow units are not the primary particles, but are small clusters of particles (plus enclosed water), herein called flocs. These flocs have a certain amount of mechanical strength and so are able to retain their identity under the very mild surface shear TIME, MINUTES Figure 3. Dilute settling plots Slurry B (1 ml. = 32. cm.) VOL. 1 NO. 1 FEBRUARY

3 % z 3 U 3. ci 1 C Figure 4. Equation 4 Table I..5.1 &<, KAOLIN VOLUME CONCENTRATION Correlation of dilute settling rates by -I - Slurry c - - Slurry D - A - s.urry E \ Effect of Prior Agitation and Chemical Treatment upon Settling Rates Runs 1, 2, 3, and 4, made in quick succession on the same slurry, (all +,y E.5) Chcni icu I L n frcflfed Suit- Washed,.6,V LYa CI, Treatment jh4 ph6 jh 7 ph9 Interface settling Inverted rate, Qo, Inverted cm./hr. Blended Inverted clay remaining Inverted in supernate, Blended after fall of interface Table II. Structural Parameters Obtained from Dilute Settling Rate Data Slurry Treatment VSA, Cm./Hr. d*, hlicrons CAK C Salt-washed, ph A Untreated, ph D Salt-washed, ph 6, B Untreated, ph E.6N NaC1, ph the straight-line plot of Figure 2, a, is obtained. Data taken with slurry B over this concentration range are shown in Figure 3. These dilute settling rates decrease rapidly as clay concentration increases, but are independent of the dimensions of the settling tube as long as the tube I.D. is large compared to the average aggregate diameter. Direct observation, using a 45X microscope mounted against the glass tube wall, showed that the aggregates at these dilute concentrations settled as roughly spherical, individual units, rather than as chains or networks. While the aggregates were not uniform in size, the size range was fairly narrow for untreated kaolin, where the horizontal diameters ranged from about 5 to 4 microns. FACTORS DETERMINING AGGREGATE SIZE. The siie Of the aggregates is not a fundamental property of a flocculated suspension, but is a dynamic property which depends upon the rates at which aggregates grow by collision and are broken down by viscous shear forces. The data of Table I permit some deductions regarding the effect of mixing intenrity upon aggregate size. Inverted means that the settling tube was turned end over end by hand 8 to 1 times to start the settling run. Blended means that the sample was mixed for 1 minute in a Waring Blendor just before being poured into the settling tube. The settling rates for the blended suspensions are higher than for the inverted suspensions for all four systems ; therefore strong mixing must produce large aggregates. Yet the spread in aggregate sizes must also increase as mixing intensity increases. If all aggregates were of the same size and density, then all would settle at the same rate and the interface would be sharp and the supernatant fluid would be clear. Strong agitation in the blender produces large aggregates which settle quickly but also produces many small aggregates which remain behind in the cloudy supernatant. Table I also shows that settling rates are reproducible for each of the cases of inversion of the settling tube, that the aggregates can recover their former size after severe agitation in the blender, and that the low-ph slurry, having the strongest cohesive forces, is affected least by changes in the mixing conditions. Hence, the major factors determining aggregate size are the internal cohesive forces holding the aggregates together and the method of agitation used to start the run. If one uses a standard mixing procedure (in this work, the standard procedure consisted of inverting the tube 8 to 1 times), then for any slurry the aggregates should approach a fairly uniform, - reproducible average equivalent spherical diameter, da. SETTLING RATE EQUATION. The Richardson and Zaki (77) equation for the group settling rate for uniform, spherical particles can be written Q = Vs~e485 (1) If one assumes that da4 is relatively independent of clay concentration over the. dilute range, and that g.4 does not change once settling has begun, the dilute settling rates should be given by Equation 1 in the form From a material balance on the kaolin it follows that Ps - PI7 = +A(PA - Pw) = + dpk - PET) (3) Equation 3 permits the accurate, independent measurement of bk, by measuring ps with a pycnometer, and also permits 26 I&EC FUNDAMENTALS

4 Equation 2 to be rewritten as and Equation 4 can be written QV4.65 = p ssa1!4 G(1 - CAK,$x) (7) where CAK volume of aggregates E 9! = +, volume of kaolin in aggregates If d, is expressed in microns, then VBA, the Stokes settling velocity for a single aggregate, is (5) Therefore, if one plots Q1!4.66 against the corresponding \ alue of $K, a straight line should result. From the ordinate intercept of this line, VsA and d:, can be obtained, and from the abscissa intercept, C, can be determined. Figures 4 and 5 show that the predicted straight line was obtained over the dilute region for each of the five slurry systems studied. The values of CAK, V.yA, and d, calculated from these data lines are collected into Table a. 1. min 1 Density b. 2 min 1 c. 3 min d. 45 min Initial Density 5 E V 4 2 M *( i I/ Io Intensity of,transmitted Radiation/ Incident Radiation Intensity E u -$, 25 d s f. 15 min - g. 43 min h. 115 min r,i I I I I I I I Figure 6. X-ray data taken by Fuerstenau (5) +K =.19,.1 gram CaO per liter VOL. 1 NO. 1 FEBRUARY

5 Initial height Interface Control volume (dashed) Glass tube, Rubber stopper TIME DENSITY Figure 7. Control volume and time-height-density relationships Left. a. Height of interface vf. time b. Height VS. density profile at time t Right. C. Settling tube The value of 2 microns for da for slurry A agrees with the visually observed size spread of 5 to 4 microns, but the value of CAK = 48 is probably considerably higher than the true aggregate volume coefficient. The aggregates are not smooth spheres but are somewhat irregular in shape; therefore the plotting method based on Equation 4 overestimates CAK to some extent. Even allowing for this, the values of CAK in Table I1 indicate that the water-clay ratio within an aggregate is surprisingly high: 3-6 to 1. As the slurry ph is raised, the strength of the edge-to-face bonding between particles is decreased because of reduction in the positive edge charge density. As the data of Table I1 show, this weakens the aggregates so that they break down into smaller fragments during the prerun mixing, and also contain less trapped intra-aggregate water. Conversely, saltwashing the clay, by replacing aluminum on the planar exchange sites with sodium, increases the strength of the edge-toface bond forces and consequently increases the aggregate size and effective volume. In alkaline, saline slurries (slurry E), where relatively dense card-pack flocs are expected to form, rather small and dense aggregates are observed. Thus, the dilute settling rate equation provides a relatively simple method of evaluating the specific effects of chemical treatment upon aggregate structure. Settling Rates at Intermediate Concentrations. At +K higher than the dilute concentrations discussed in the previous section, the aggregates settle as a coherent network, and the settling plots have the shape of curve 6 of Figure 2. The initial settling rates are very low, but Q increases with time, reaching a maximum value of Q1 at t = tl. The rates over all portions of the curve increase as container diameter, Do, or initial slurry height, Z,, increases. Gaudin and Fuerstenau (6-8) have proved, by means of x-ray transviewer data of the type shown in Figure 6, that during the early free-fall settling period there exists a region extending from the interface downward, whose density is constant and equal to the density of the original suspension at time t =. Assuming that this constant density zone also exists for the kaolin slurries studied in this work, Figure 7 shows the time-height-density relationships which will be used to derive a general equation for settling rates at intermediate concentrations. The control volume of Figure 7 may be regarded as a plug of slurry, of length AZ, which is moving downward at a velocity Q. The plug is supported by fluid friction and buoyancy forces. In addition, the aggregate networks have mechanical strength, and extend to the walls and bottom of the container, so that the control volume is also supported by resting on the underlying material and by shear forces at the walls. If uy is defined as the compressive load which the original aggregate network can support without rupturing, then the boundary between the uniform density zone and the underlying compressed zone corresponds to that point at which the weight of the slurry overburden just equals the network compressive yield strength, uy. The total support force exerted by the underlying material on the control volume must be H Under support force = - D$ry 4 Treating the movement of the control volume as pure plug flow, the shear stress at the container wall should just equal the yield stress, T?, of the original slurry. Thus (8) IVall support force =?rd,azr, (9) The weight of the aggregates minus the weight of the displaced liquid is xet weight = - ~?.izg(pa - prr-)4a 4 Combining Equations 3 and 1, (1) Net weight = D!AZg(pf: - pw)+f: (11) 4 The pressure gradient through the control volume resulting from the flow of displaced liquid up through the spaces between the aggregates may be estimated by applying the Kozeny-Carman equation to this case of pore flow. The geometric factors which must be evaluated are the pore shape factor, k,; the pore hydraulic radius, rh; and the tortuosity factor, L,. flow area - e - void fraction rx = --- wetted perimeter S specific surface area L, = length of effective fluid path through pores length of straight-line flow path (12) (13) If Q is the settling velocity of the interface, then the average velocity of the liquid in the pores is Ji,, = 9 L, (14) 28 l&ec FUNDAMENTALS

6 2--?r Substituting these relationships into the Poiseuille equation, the Kozeny-Carman equation predicts that The total upward force exerted on the control volume due to fluid friction is then HD dp Friction force = -D A2-4 dz (16) At equilibrium, =. By combining and rearranging these equations, one obtains the general settling equations for flocculated suspensions where 1 Q = Qr[ (17) Do Zo Physically, Q is the settling rate in an infinitely large container. The significance of D,, the yield diameter, and Z,, the yield height, IS that if Do 5 D, or if Zo 5 Z,, then Q is everywhere zero. This does not mean that the slurry does not settle at all but that there is no accelerating rate period as there is for t < tl in Figure 2, 6. Instead, the interface subsides slowly, at an ever-decreasing rate, and the settling curve resembles the curve of Figure 2, c, for compressive settling. Equation 18 may be used to explain the increasing slope of the settling curves of Fiqure 8. As settling proceeds, the struc- ture of the aggregate network changes with time. Initially the aggregates are interconnected, irregular spheres and the fluid flow paths up between the aggregates are tortuous and contain many expansions and contractions. Drag forces are strong and the initial settling rates are low. As settling proceeds, the aggregates tend to coalesce and line up into vertical rows and the flow paths straighten out. Plastic deformation of the aggregates occurs whenever the viscous shear forces exceed the aggregate yield stress. Contractions and sharp bends are smoothed out. Given enough time, in a vessel of infinite extent the liquid paths tend to approach the configuration of smooth, vertical tubes rising up through the continuous aggregate region and the settling rate increases to its maximum value, Qi, For this limiting liquid-tube and aggregate-shell configuration, the geometric constants in the Kozeny-Carman equation reduce to L, = 1. for smooth, straight cylinders k, = 2. for smooth, straight cylinders The void fraction, e, is the volume of the supernatant liquid outside the aggregates-4.e.) the volume of the flow tubes. Therefore : E = 1 - $A = 1 - CAF4F (21 ) 4E S = n,tdp = - d, The maximum settling rate, in an infinitely large tube, is obtained by substituting these values in Equation 18. This equation may be tested using data taken in finite, laboratory-sized tubes, by using Equation 17 to correct for the effects of height and diameter. DIAMETER EFFECT. The yield diameter, D,, can be calculated directly from Equation 19. For example, for slurry A at $K =.32, 7y as determined from measurements with a TIME, MINUTES Figure! 8. Settling plots Slurry A, ph 4, &rc =.1 92 l/zo, RECIPROCAL OF INITIAL HEIGHT, CM.- Figure 9. Correlation of maximum settling rates by Equation 26 Slurry A VOL. 1 NO. 1 FEBRUARY

7 concentric cylinder viscometer is 3. dynes per sq. cm. fore There- 4 X 3. D = - 98( )(.32).24 cm. (24) This means that there would be no settling in a tube of I.D. equal to or less than.24 cm. But in a tube of I.D. = 6 cm., which was a typical tube size used in this work, the wall effect at $K =.32 would be Therefore, Equation 17 predicts that the settling rates in a 6-cm. I.D. tube would be only 4% less than in a tube of infinite radius. This explains the experimental fact that the diameter effect was negligible for the container diameters (4.8- to 6.5-cm. I.D.) used in this work, While settling rates are not a strong function of container diameter, the data of Figure 8 confirm that Q decreases rapidly as initial slurry height, Z,,, decreases. Because the aggregate network is stronger in compression than in shear, Z, in Equation 17 is much larger than D,. It is not, however, possible to calculate 2, directly from Equation 2, because the height of the control volume, AZ, varies during the settling period from an initial value of 4Z = Z, at t =, to a final value of AZ+ at t+ tl. Fortunately, it is possible to evaluate Z, empirically. At any particular phase of the free settling period-e.g., at t = l/*tlthe value of the ratio AZ/Z, in Equation 2 should depend upon the aggregate concentration and properties, but should be independent of Z,. Thus, for the case of greatest practical importance, in the maximum settling rate equation, one may write Q: = Q: [I - 21 tl. At any fixed clay concentration and - - Z, Z,, at t chemical treatment, Z,, should be a constant. Equations 17, 19, and 26 are important for design purposes. These equations permit settling rates in settling ponds and large tanks to be predicted on the basis of laboratory sized settling experiments. The application of Equation 26 is demonstrated by Figure 9. Maximum settling rates, Q1, were measured for 1 intermediate concentrations of slurry A, from.8 < $K <.382, and at initial heights from 1 to 12 cm. As Equation 26 predicts, when Q: is plotted against I/Z, a straight line results at each $K. Q1 and Z,, may be determined from the intercepts of these lines. ZYl, the yield height, was found to be zero over the dilute settling range, +K <.7, where the aggregate concentration is too low to form a continuous network; but Z,, increases rapidly for $K >.7. The values of Q1 obtained from the intercepts of Figure 9 may be used to check Equation 23. If dp is given in microns, and Q1 in centimeters per hour, Equation 23 may be written The kaolin concentration, OK, is calculated directly via Equation 3 from pycnometer measurement of suspension density. Floc volume concentration. $=, is then obtained from the curve of $p us. $K for slurry A in Figure 13. If the two remaining terms in Equation 27, d,, and Cap, are constant with respect to clay concentration, a plot of Q1 /GK us. $p should give a straight line. Figure 1 shows that the data do give this straight line for floc concentrations between.12 and.275. Therefore, from Figure 1 1 = = 3.64 #p $F FLOC VOLUME CONCENTRATION Figure 1. Plot of Q 1 data from Figure 9 according to Equation 23 Slurry A Figure 11. heights Za$K, CM. Final sediment heights vs. initial Slurry A 3 l&ec FUNDAMENTALS

8 Ox, KAOLIN VOLUME CONCENTRATION Figure 13. c$~ vs. 4K Figure 12. heights z o $KI CM. Fiinal sediment heights vs. initial the lower, uniform density region of the settled bed is also assumed to be.62, the final bed volume should be R Zo$F T DaZf = -,' - f Figure 1 implies that the pore diameter is constant and equal to about 18 niicrons over the region.45 < < 1. The viscous shear forces at the pore walls are proportional to the pore diameter, while the ability of the aggregates to resist plastic deformation is ]proportional to the strength of the flocfloc forces within the aggregates. For strongly flocculated slurries, the attractive forces are strong, the pore diameters are large, and the settling rates, other things being equal, are high. As concentration decreases, neither dd nor CAP should change, because these are fixed by the interparticle bonding intensity, and so the number of pores per unit area must increase. Eventually, at +P <.12 in Figure 1, the pores become so numerous that a continuous aggregate region is no longer possible, and settling rates increase above the line predicted by Equation 23. Density of Settled Beds. If a flocculated suspension is allowed to stand undisturbed, the interface eventually falls to some final height, Z,, at some final time, t,. The x-ray density data of Gaudin and I'uerstenau (5), shown in Figure 6, h, prove that the density of the final settled bed is not uniform with respect to height, but rather consists of a lower, uniform density region plus an upper, nonuniform, decreasing density zone. The density profile of Figure 6, h, can be explained in terms of the structural inodel of Figure 1, which considers that the flocs behave like iiicompressible, rigid spheres at the mild stresses encountered durinq sedimentation under normal gravity. The flocs behave rather like a sack of marbles. Comparatively small forces can compress the sack until the marbles are randomly close packed, but a much larger force would be required for further compression. Unlike the marbles, however, the flocs attract one another. The effect of random adhesions is that the flocs do not tumble into a closepacked array of their own accord, but must be forced in by the weight of the overlying solids. Near the top of the settled bed, the overburden weight is insufficient to force the flocs into this close-packed array, and so the local solids concentration decreases with increasing height over the upper few centimeters of the bed. The solid volume fraction for random, close-packed, uniform spheres is.62 (2). Therefore, if the floc volume fraction in GP is the floc volume fraction originally present in the suspension before any settling occurs and b is the constant additional height added by the presence of the upper, nonuniform, low density zone. Combining Equation 29 with the identity, $P C,,$,, yields CF K 2, = -- Z,$r( Figures 11 and 12 present the results for more than 4 settling tests using slurries A and D. For each concentration studied, the data start out on a curve through the origin? but eventually level off into the straight line predicted by Equation 3 at high values of Zo+K, For Equation 3 to apply, enough clay must be present initially (Zo+K > -.3 cm.) to form both a uniform density zone and an upper nonuniform region in the final settled bed. DETERMINATION OF FLOC CONCESTRATION. Table 111 shows how measurements of the slopes of the straight-line regions of the settling plots of Figures 11 and 12 may be used to convert kaolin volume concentration, $K, into floc The procedure illustrated in Table 111 may be used to construct a plot of +P us. dk for each slurry. A less accurate but less time-consuming procedure is to measure Z, for only two initial heights (Zo1 and Zo*) at each concentration, and then to calculate $= from Equation 3 in the form Figure 13 shows the complete curves of df us. $, with the points either taken from Table I11 or calculated by Equation Table 111. Calculation of Floc Concentration by Equation 3 m 6, CFK? +F> Slurry Figure +K (Slope) Cm..62m CFR$K A 11, A 11, ,42 A 11, A 12, D 12, VOL. 1 NO. 1 FEBRUARY

9 ~ height 31. In the latter cases, the two initial heights used at each concentration were Z,, - 6 cm. and Z2-3 cm. Each curve in Figure 13 is concave to the $K axis, indicating that, C, increases (floc density decreases) as $K decreases for all five slurries, most probably because floc size decreases as clay concentration decreases, as observed by Reich and Vold (76). The relative heights of the curves differ, showing that floc volume and floc water content are functions of liquid phase composition. The order of the curves agrees with the order for the same five slurries in Table 11. DETERMINATION OF $A. According to the floc-aggregate structural model shown in Figure 1, water can be removed from a bed settling under normal gravity by two mechanisms. (In a centrifuge, additional water may be expelled as the flocs deform and compress.) During the early stages of settling, interaggregate water is expelled as the aggregates deform and move closer together. The second, and much slower, mechanism involves the squeezing out of intra-aggregate water as the flocs within the aggregate crowd closer together. The x-ray density data of Gaudin and Fuerstenau illustrate both these mechanisms in Figure 6, e andf, which show an upper decreasing density zone: a central uniform density plateau, and a bottom increasing density region. The original aggregates are crowding closer together in the upper region and are being compressed in the bottom region. The uniform density zone in between corresponds to a region wherein the volume fraction of the original aggregates is just equal to unity. The aggregate density, pa, and the original aggregate volume concentration, da, could be calculated from the fraction of incident radiation which is transmitted through the uniform density zone of Figure 6,f. A less accurate, but more universal, method is based on the observation that the density profile shown in Figure 6, f, occurs at the time of the second break in the settling plot, at t = t2 in Figure 2. If one assumes that the area under the actual profile of Figure 6, f, can be approximated by a squared off rectangular profile of height Zz and width corresponding to pa, then the original aggregate volume is ir/4d2,z~, and the aggregate volume coefficient is The results of a large number of measurements of ZZ as a function of Z, and $F indicate that Equation 32 overestimates, C, for short columns, Z, < 5 to 7 cm. The results for taller columns, however, were found to be independent of either Z, or $p, and to agree with the previous value of CAP = 3.64 for slurry A from Equation 28. Nine tests with slurry A at 2, > 7 cm. gave an average value of CAP = 3.78 with a standard deviation of.15. Table IV shows similar results for slurries B, C! D, and E. Table IV. Estimate of CAF by Equation 32 Slurry +K +F ZO Z2 CAF B C D E From these data, CaF, the ratio of the aggregate to the floc volume concentration, is approximately 3.8 for all five slurry systems studied. Unless the aggregates are both confined and strongly compressed, the flocs always seem to cluster in such a way that the floc fraction within an aggregate is -1/3.8 =.26. This means that for $F <.26, approximately, the aggregates are composed of about 3 volume parts of water to one part of floc, independent of floc concentration or liquid phase composition. For $F >.26, the aggregate volume concentration is unity and there can be no free-fall settling. The slow subsidence of the interface which does occur at high concentrations (Figure 2, c) is due to the slow squeezing out of intra-aggregate water as the bed consolidates. Conclusions For a flocculated suspension, the floc rather than the primary particle is the fundamental structural unit in low shear processes such as gravity sedimentation. The settling rates and sediment volumes observed could be explained by considering the flocs to be rigid spheres which tend to cluster together into weak aggregates. Aggregates grow by collision, break down by shear forces, and are able to form the extended networks which give the suspension its structural properties. Each of the properties studied in this work, including settling rates, sediment volume, and aggregate size, was a function of just two variables: floc volume concentration and the strength of the attractive force between flocs. Nomenclature b = c.. = s = tl = tz = L, = rx = ordinate intercept of straight line of Equation 3, cm. ratio of volume concentration of component i to component j CAF = +A/+F; CAK = +A/+X; CFK = +F/+K da = average (equivalent) aggregate diameter, microns d, = average pore diameter, Equation 27, microns Do = diameter of container tube, cm. D, = yield diameter, defined by Equation 19, cm. g local gravitational acceleration, 98 cm./sec2 I = intensity of x-ray transviewer beam k, = shape factor in Kozeny-Carman equation m slope of straight line predicted by Equation 3 Q = settling rate of slurry-supernatant interface, cm./hr. Q. = initial settling rate, cm.,/hr. QI = maximum settling rate, cm./hr. Q = prime superscript denotes settling rate in infinitely large settling tube specific surface area, sq. cm./cc. time at which free-fall settling period ends time defined by Figure 2 tortuosity factor pore hydraulic radius vs = Stokes settling velocitv. cm./hr. VSA Stokes veloci6 for single aggregate, cm./hr. z = of slurry interface, cm. Z, = initial height of slurry column, cm. Z, = final height of settled bed, cm. Z, = yield height defined by Equation 2, cm. Z,1 = yield height at t = tl, Equation 26 ZQ = height of interface at t = t2, Figure 2 AZ = height of control volume in Figure 8 SUBSCRIPTS. K = kaolin A = aggregates F = flocs f = final condition S = suspension (the slurry taken as a whole) GREEK E fi fiw = void fraction = abbreviation for micron. 1 I.( = lo-* cm. = viscosity of water,.893 cp. at 25 C. 32 l&ec FUNDAMENTALS

10 P = U = density, grams/cc. PIC = 2.58 compressive strength of aggregate network. dynes sq. cm. yield stress, dynes/sq. cm. For a plastic suspension, Ty = T-T~ as shear rate- Q.4 = aggregate volume concentration $ F = floc volume concentration $ K = kaolin volume concentration literature Cited 11) Andreason. A. M. H.. Kolloid Z ). \ I (2j Brindley, G. W., Ceramic Fabrication Processes. Chap. 1, 11. D. Kingery, ed., Wiley, New York, (3) Burgers, J. M., Second Report on Viscosity and Plasticity. pp , VA 51. Academy of Sciences, Amsterdam, (4) Coe. H. S.. Clevenger, G. H., Trans. Am. Znst. Mzning Engrs ). (5) Fuerstinau. M. C., Department of Metallurgy, M.I.T., unpublished data, Mav 196. (6) Gaudin, A. M., Fuerstenau. M. C., Eng. Mining J. 159, 11 /15A\ \-,--/. (7) Gaudin, A. M., Fuerstenau, M. C., Preprint, Intern. Mining Proc. Congr. London, April 196. (8) Gaudin. A. M., Fuerstenau, M. C., Mitchell, S. R., Mining Eng. 77, (1959). (9) Georgia Kaolin Co., Elizabeth. K. J., Georgia Kaolin Handbook, Bull. TSBH-1 (1956). (1) Hawkesley, P. G. W., Some Aspects of Fluid Flow, Chap. 7, Arnold &.Co., London, (11) Kynch, G. J., Trans. Faraday Sic. 48, 166 (1952). 112) La Mer. V. K.. Smellie. R. H.. Jr.. Lee. P. K.. J. Colloid Sci., I 12, 23 (1957). (13) Martin, R. T., Proc. 5th Conf. on Clays and Clay Material, Natl. Acad. Sci.-Natl. Research Council, Publ. 566, 23 (1958). (14) Mason, S. G., Pulp and Paper Mag. Canada 49, 13, 99 (1948). (15) Michaels, A. S.? Ceramic Fabrication Processes, Chap. 2, W. D. Kingery, ed.. Wiley, New York, (16) Reich, I., Vold, R. D., J. Phys. Chem. 63, 1497 (1957). (17) Richardson, 3. F., Zaki, \V. Tu., Trans. Znst. Chem. Engrs. 32, 35 (1954). (18) Schofield, R. K., Samson, H. R., Discussions Faradqy Soc. 18, 135 (1954). (19) Smellie, R. H., LaMer, V. K., J. ColloidSci. 11, 74 (1956). (2) Steinour, H. A., IND. ENG. CHEM. 36, 618, 84, 91 (1944). (21) Street, N., Buchanan, A. S., Australian J. Chem. 9, 4 (1956). (22) Wadsworth, M. E., Cutler, I. B., Mining Eng. 8, 83 (1956). RECEIVED for review November 6, 1961 ACCEPTEDecember 15, 1961 Division of Industrial and Engineering Chemistry, ACS, Symposium on Dynamics of Multiphase Systems, University of Delaware, Newark, Del., December FULLY DEVELOPED TURBULENT PIPE FLOW OF A GAS-SOLID SUSPENSION S. L. S, University of Illinois, Urbana, Ill. Velocity and concentration distributions of solid particles were colculoted for the case of low solid-to-gas mass ratios, small particles, and negligible gravity effect. When these conditions ore sctisfied, the solid particles slip at the woll and lag behind the stream ot the center of the pipe. Dependence of the particle velocity distribution on the concentration distribution is to be exoected. PART from studies of motion of a single solid particle in the A turbulent field of a fluid, studies of dynamics of a gas-solid suspension as a whole have either excluded turbulence (3, 72) or have been completely empirical (7, 77, 79). Studies on the gas dynamics (72) and compressible potential motion (3) have shown that rigorous forniulation of general turbulent compressible motion of a gas-solid suspension over a solid boundary will be very complicated indeed. A solution is not even available for the statistical.lormulation of simple turbulent pipe flow, where the turbulence is nonhomogeneous (7). Even when one assumes an isotropic turbulent field, solution of the problem of motion of a single particle is at best academic (6). The present study proposes an extension of the present semiempirical method of treating turbulent flow through circular pipes-namely, the 1/7th velocity law (77)-to the case of a gas-solid suspension. Data are utilized on distributed mass flow of solids in pipe flow at air velocities from 5 to 1 feet per second, solid particles consisting of 1- to 2-micron diameter glass beads, and solid-to-air mass ratios of.5 to.15 (77). In the range under consideration, compressibility of the fluid phase and the effect of gravity on the density distribution of solid particles are negligible. Concentrations of solid particles are small. such that the velocity distribution of the gas stream is not significantly affected by the solid particles. It is sometimes tempting to suppose that the heterogeneous mixture of solid particles and fluid may behave as a homogeneous fluid, the dense particles playing the role of the heavy component of a homogeneous mixture. This assumption would greatly facilitate the development of a theory but is almost never true in practical situations, for it would require that the suspended particles be extremely small. In fact, for particles as large as those of real interest, the solid-fluid mixture has to be treated as heterogeneous. Then, because the solid particles have greater density than the fluid, the concentration of particles is not uniform over a horizontal pipe, the average velocity of the particles near the wall is not zero even though the velocity of the fluid is zero at the wall, the concentration of particles is velocity-dependent, and the excess of particle velocity over fluid velocity is positive near the wall and negative in the core of fluid, A realistic theory of the motion of solid-fluid suspensions must account for these facts. VOL. 1 NO. 1 FEBRUARY

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