Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
Special Symbols phase average Over volume v of quantity ,J, asso($) ciated with fluid; see eq 8 intrinsic phase average over volume V, of quantity ($)f $ associated with fluid; ($)' = @-l($) fluctuation about intrinsic phase average $ Literature Cited Batchelor, G. K. J . Nuid Mech. 1970, 41, 545-570. Brinkman, H. C. Appl. Sci. Res. 1947, A I , 27-34. Gray, W. G. Chem. Eng. Sci. 1975, 30, 229-233. Gray, W. G.; Lee, P. C. Y. Dit. J . Mulriphase Now 1977, 3 , 333-340. Happei, J.; Brenner, H. "Low Reynolds Number Hydrodynamics", Prentice-Hall: Englewood Cliffs, N.J., 1965.
45
Poreh, M.; Elata, C. Isr. J . Techno/. 1966, 4(3), 214-217. Saffman, P. G. Stud. APPi. Math. 1971, 50(2), 93-101. Serrin, J. In "Handbuch der Physik", Bd. VIIIII, Springer: Berlin, 1959; pp 125-263. Slattery, J. C. AIChE J , 1969, 15(6),866-872. Slattery, J. C. "Momentum, Energy and Mass Transfer in Continua", Mc&aw-Hill: New York, 1972. Whitaker. S. Ind. Eng. Chem. 1969, 61(12), 14-28.
Received for review March 29, 1978 Accepted September 28, 1978 This research W a s supported by grant number NSF-ENG75-16072 from the National Science Foundation.
Accelerated Settling by Addition of Buoyant Particles Ralph H. Welland' and Rodney R. McPherson Departrnent of Chemical Engineering, University of Queensiand, St. Lucia, Queensiand, Australia 406 7
Buoyant particles, when added to a settling suspension, cause all heavy and light material to quickly segregate laferal'y into separate fast-moving streams. These streams convect particles vertical& at up to six times normal settling rates. Enhancement is greatest with the most concentrated suspensions.
Introduction The settling of suspensions of small particles is a notoriously slow process requiring large diameter equipment. A means of speeding it up would be beneficial in reducing the size of new settlers and thickeners and increasing the handling capacity of existing ones. Most recent technology has been concerned with improving settling rates by flocculation of the suspended particles. However, in attempting to study the interactions between particles falling in a concentrated suspension, Whitmore (1955) found that just by adding neutrally buoyant particles to the slurry, he could greatly increase its settling rate for a t least part of the settling period; the lighter particles werle not entrapped in the thickened sludge. Despite its potential benefits, this surprising result does not appear to have been explored further. If one were to contemplate adding buoyant particles to a sedimenting suspension, the common-sense expectation would be greatly increased hindrance and a marked reduction in settling rates. However, the benefits found when particles of neutral density were used lead one naturally to enquire about the possibility of achieving even greater enhancement by the use of positively buoyant particles (which might be separated from the slurry a t the same time as the heavy ones) and to enquire about the mechanism responsible. This is the motivation for the current work. We have investigated the effect of varying the concentration of both settling and buoyant particles under two different density driving forces. Concentrated suspensions usually settle out with a rather sharp interface separating the clear fluid which lies above the settling mass; there may also be a zone below the suspension in which settled solids consolidate into an arrangment of closer packing. The addition of buoyant particles to such a suspension results in quite a different picture. We now have the possibility of two interfaces, one falling and the other 0019-7874/79/1018-0045$01.00/0
rising, as shown schematically in Figure 1. Between the two interfaces is a mixture of both heavy and light particulates undergoing separation and settling. In the space above the upper interface, which would normally contain clear fluid, one now finds a mass of buoyant rising particles, while below the lower interface, the light particles have departed. Here one finds only heavy particles falling through the supporting fluid, just as in normal settling. As sedimentation continues, the two interfaces move progressively closer together until eventually they pass through each other. In so doing, they cease to demarcate a suspension of two solids from one of only heavy or light particles. Instead we find a settling suspension of heavy material a t the lower end of the vessel, separated from a rising suspension of light material a t the top by a region of clear fluid. The heavy and light particles now continue to move as they would in normal settling, the situation at the top end of the vessel being an inverted image of the lower end. Consolidation may occur a t both the top and bottom. Experimental Section A glass cylinder 25 mm in inside diameter and 315 mm long was used for containing the suspensions. It was graduated in 1-mm divisions and was made vertical with the aid of a cathetometer and plumb bob (Whitmore reports that displacement from the vertical of 1 in 200 causes a just-discernible difference in settling rate). The stirrer was made from 1-mm brass rod, twisted into a loop at the bottom and passing through a rubber stopper at the top. With the tube completely filled with fluid, air entrainment was prevented during stirring. The tube was illuminated by a fluorescent lamp, but a t concentrations of the rising particles greater than 1070,it was found necessary to use the more intense lighting of a projector type of microscope lamp. In general it was not possible to see the top (or bottom) interface which formed between the region of rising solids 0 1979 American
Chemical Society
46
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979 Time (s)
0
t1
Fluid
500
1000
1500
I
I
I
I
.
Figure 1. Schematic of counter-current settling.
-1
17mm
I
Figure 2. Schematic of diver.
only and the region containing both rising and falling particles. The position of this interface was made “visible” with the aid of a diver whose effective density could be adjusted to be intermediate between that of the settling suspension (below the interface) and the combination of fluid and rising particles (above the interface). Of course, a similar interface existed near the bottom of the tube but here it marked the plane below which there were no rising particles. The velocity of rise of the light particles was not measured. The diver was an annulus of high density polyethylene and had the dimensions and shape given in Figure 2. It was so shaped to prevent the deposition of particles onto the diver as this would have changed its effective density. Copper wire was used to adjust the density of the diver, the wire being placed so as to ensure that the annulus fell with its axis along the vertical. A similar device was used by Whitmore (1955), who found that the settling rates determined from independent measurements of the position of the diver and the head of falling particles were virtually identical, but that by using the diver, scatter in position-time measurements was reduced. The solids used were polystyrene (PS) spheres and poly(viny1chloride) (PVC) particles obtained from unsized material by dry mechanical sieving on new B.S.S. sieves. The fraction in the sieve size range from +90 to -106 pm was used and microscopy analysis gave a projected-area mean diameter of 100 pm for both materials. The polystyrene particles were quite spherical, whereas the PVC material was irregular although of well-rounded form and without sharp edges. The density of these materials was bracketed using the sink-float method in sucrose solutions for PS and in calcium nitrate solutions for PVC and was found to be 1.04 and 1.38 gm/cm3, respectively, with an error of *0.005 g/cm3. Aqueous solutions of sucrose, with the addition of 0.2% Triton X-100 to inhibit flocculation, were used as the suspending medium. Since sugar solutions are subject to biological degradation, they were freshly prepared daily. Solutions of density 1.10 and 1.15 g/cm3 (*0.1%) were used and their respective viscosities, measured with a Contraves viscometer, were 1.93 and 3.42 CPat 26 “C. All experiments were conducted a t a constant temperature of 26 “C. The general procedure was to start with a suspension of PVC a t the desired concentration and measure its settling velocity. Then polystyrene powder was added in
20 L Figure 3. Typical settling curves, 15% PVC, fluid density 1.15 g/cm3.
increments of 5% concentration (by volume) and the measurement of distance fallen against time was repeated. Between additions of P S an appropriate amount of fluid was removed from the vessel so that the PVC concentration would not be diluted by the addition of the next increment of polystyrene. In this manner the PS concentration varied between 0% and 20% and the PVC concentration between 5% and 20%, all in 5% increments for two fluid densities. Between each set of measurements (corresponding to a particular combination of particle concentrations) the diver had to be removed from the fluid to have its density adjusted. To ensure that particles of one density were not preferentially removed over others, the suspension was first stirred. Total particle loss was below detectable limits over a series of experiments a t a given PVC concentration. Results and Discussion The primary data were records of distance settled against time and a typical plot is shown in Figure 3 for a suspension of 15% PVC with various amounts of PS, the rising material, in aqueous sucrose solution of density 1.15 g/cm3. At any nominated time, the heavy particles settled further, the greater the amount of light material added, but the rate of sedimentation was not the same at all times. The graph may be broken into three distinct time regions. Early in the settling process the addition of any amount of PS apparently leaves the sedimentation velocity of PVC unchanged. This quickly gives way to an accelerated rate period during which the settling velocity is greatly dependent on the concentration of PS. The final phase of settling takes place a t the same rate as in the absence of buoyant particles. In the present example, the top of the diver came to rest a t approximately the 16 cm mark; beyond this point consolidation of sediment is presumed to occur. The crucial visual observation (and one that we are currently documenting in a quantitative way) which explains the results reported here is as follows: in the initial few moments of settling, the particles underwent a lateral migration in addition to their normal vertical movement. A situation quickly established itself in which the light polystyrene particles were found to be moving upward in thin streams while the heavy poly(viny1 chloride) particles moved downward in adjacent streams. The whiteness of the PS and the dark grey color of the PVC made it easy to see that a complete lateral separation had been made in that the dark particles were excluded from the upward moving streams and light ones from the descending streams. The streams were between 1 and 2 mm wide. They did not always follow vertical, straight-line paths but occasionally meandered. We therefore conclude that the addition of sufficient numbers of buoyant particles destabilizes a uniformly settling suspension and sets up a new stable flow pattern in the form of vertical streaming which convects particles a t greatly increased rates over those found in normal settling.
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979 6 r
47
6r
0 5 Yo PVC 0 10 8 15 0
0 50PS 0
10 15
0
20
0
20
2
1 0
5
10 Oh
15
20
0 PIC
Figure 6. Dependence of settling velocity on PVC concentration, fluid density 1.10 g/cm3.
PS
Figure 4. Dependence of settling velocity on PS concentration, fluid density 1.10 g/cm3.
d
/@’
I
: : : :FI :
2
/
21&:o / 1 0
1 0
5 4,
5
10
15
20
0 PS
Figure 5. Dependence of settling velocity on PS concentration, fluid density 1.15 g/cm3.
Returning now to Figure 3, it is tempting to ascribe the slowness of settling dluring the initial period to the time needed to set up streaming motion, Le., an induction period. Indeed, Whitmore (1955) has found that for neutrally buoyant particles, as much as 60% of the total settling time was used in establishing streams, although his experiments were not reproducible in that respect. However, the distance fallen by the diver during this initial period is very nearly the same as its own height (7 mm), so the induction period may represent an “end” effect. Whatever the explanation, it is apparent that streaming flow is established very quickly and persists until the ascending interface (below which light particles are absent) and the descending interface (above which heavy particles are absent) pass through each other and we are left with single components “settling” a t either end of the tube. This marks the end oE the accelerated rate period and the return of settling to normality as evidenced by parallel settling curves. Streaming motion did not persist unless both heavy and light particles were present together. The
0
,
,
10
15
20
PIC
Figure 7. Dependence of settling velocity on PVC concentration, fluid density 1.15 g/cm3.
transition from streaming to uniform settling appears to be quite gradual in Figure 3, but this is probably a result of the large diver being unable to reflect accurately the sudden passing of two interfaces. The extent to which buoyant particles increase the settling rate of a suspension can be seen in Figures 4 to 7 , where we have scaled the accelerated settling velocity with the settling rate measured in the absence of buoyant particles. Figures 4 and 5 compare the effect of adding increasing amounts of PS to settling suspensions of PVC in fluids of density 1.10 and 1.15 g/cm3, respectively. The more concentrated the settling component, the greater is the improvement to be had by adding buoyant particles, and the same is nearly always true for increasing concentrations of PS. It is to be noted, however, that when the PVC concentration is high, there is an optimum amount of PS above which no further improvement in settling rate is to be had. This is explained most easily by thinking in terms of the streams of light and heavy material rising and falling through the bulk. As the amount of PS is increased, the effective density of the rising stream is lowered so that the
48
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
difference in densities between rising and falling streams is increased. It is this difference which drives the flows and as a result they tend to accelerate. Countering the increase in density driving force with added buoyant material, however, is the increase in the apparent viscosity of the suspension, that is, a greater resistance to flow. At low concentrations, increased buoyancy wins but at high concentrations the greater flow resistance dominates over buoyancy; thus, the existence of an optimum amount of PS is to be expected. Alternatively, adding increasing amounts of PS will eventually result in so little fluid remaining that motion is not possible at all; hence, one must expect some optimum PS concentration. When the density of the supporting fluid is itself low, adding PS does not greatly lower the effective density of the rising stream and the optimum amount of P S is soon reached (Figure 4). For a greater fluid density, however (Figure 5), the effect is stronger so that even a t 20% PVC and 20% P S (40% particulates), maximum enhancement has still not been reached. A higher density difference between fluid and light material gives a stronger effect. We surmise that the greatest improvement in settling rate can be had when the density differences between fluid and buoyant particles and heavy material and fluid are about equal, although this has not been demonstrated experimentally in the present work. Our experiments are certainly far from such a condition, so it is highly probable that the maximum enhancement is far greater than we report here. Figures 6 and 7 show the effect of increasing PVC concentration with the concentration of light material as parameter. Again the existence of a maximum in some of these curves is to be anticipated from the arguments already outlined. Whitmore (1955) has reported similar effects by adding neutrally buoyant particles. For reasons to be made clear, we have not quantitatively compared our data with his. If his added particles were truly neutrally buoyant, it is difficult to explain the formation of streaming flows (which he too observed), since the addition of neutral particles would seem to serve merely to increase the effective viscosity and so hinder settling. It provides no obvious mechanism for generating and maintaining vertical streams. Indeed, for many of Whitmore’s experiments, particularly at high concentrations, settling was greatly retarded for much of the settling period. It was only toward the end of settling that the particles had fallen as far as they would have without the addition of neutrally buoyant particles a t all. He also found it impossible to reproduce his settling curves, the time during which increased hindering was observed being quite variable. Whitmore made the further important observation that for concentrations exceeding about 15%, streaming was present to a small extent even when no foreign particles were added, nor could relevelling of the tube eliminate them. This seems to suggest that uniform settling is inherently unstable and that the use of buoyant particles accentuates the instability, driving the system toward another more stable condition. It is well known that groups of particles fall more rapidly than separate, isolated ones (Timbrell, 1954). Crowley (1971) has shown theoretically that a one-dimensional horizonal row of evenly spaced particles falling in Stokes flow is an unstable configuration. A slight displacement of one particle ahead of the others in the row causes neighboring particles to move closer to it through lateral migration. He has confirmed experimentally that such a line of particles eventually separates into clumps whose spacing depends on the ratio of initial particle spacing to
particle size when this ratio is large. More recently, Crowley (1976,1977) has reported that a horizontal sheet of uniformly spaced particles is most unstable to disturbances of infinite wavelength but that finite wavelengths are most dangerous for vertical sheets of particles. His theory depends on the assumption that particles interact linearly, Le., Stokes flow fields are additive, and interestingly, he found that all particles had to be considered if meaningful predictions were to be obtained. His analysis has not yet been extended to three-dimensional arrays of particles, even with uniform spacing. Noting the strong variations he has found between rows and horizontal and vertical sheets of particles, it does not appear possible to even surmise about what might result from three-dimensionality. The effects of a random initial distribution (within the approximation of linear interaction) and the presence of both positively and negatively buoyant particles in roughly equal numbers and each in a random initial configuration further complicate the picture. Finally, we are dealing with concentrated suspensions, whereas Crowley’s theories hinge crucially on the assumption of linearity which can only be valid for dilute suspensions. Nevertheless, his theories do suggest that sedimentation of suspensions of a single solid component may be unstable; an extension to even the simplest three-dimensional case would be worthwhile. It is perhaps noteworthy that once particles leave the streaming region they return immediately to a uniformly distributed configuration with no trace of their former convected state. Thus, uniform settling appears to be quite stable if only heavy particles are present. But buoyant particles have a powerful influence in destabilizing uniformity and creating a beneficial vertical streaming; their removal results in immediate destruction of these streams. Thus, even after setting up very large disturbances in the form of directed streams, uniform settling is easily regained. In a recent note, Thacker and Lavelle (1978) have reported that theoretically a uniformly settling dense suspension is unconditionally unstable to infinitesimal disturbances but that disturbances in the form of horizontal waves play no role in determining stability. It appears that we have a case here in which all linear theories predict instability where none exists in practice. The factors which control the width of the streams in counter-current settling are unknown, but it is noted that these plumes are about 10 or 15 particle diameters across, so they are not what one would call wide compared t o particle size or spacing. We are presently investigating the effect of particle size on stream geometry and are extending the experiments to tubes of larger diameter and smaller size particles, more closely resembling the usual situations in sedimentation. Particle size dispersion effects may be important and need to be examined. The influence of size distribution does not appear to be well documented even for settling of single components; the work of Steinour (1944) and Hawksley (1951), for example, deals only with monodisperse particles in dense suspension. Perhaps one can view size distribution effects in terms of the type of distribution, its mean, and its standard deviation, as Sohn and Moreland (1968) have done for the voidage and flow resistance of filter cakes. Clearly, much remains to be done; we are reporting our first efforts. However, by adding substantial amounts of buoyant particles to a settling suspension we have found that the apparent settling velocity can be increased by a t least six times and the batch settling time reduced by a factor of about 2.5. It is most improbable that these figures
Ind. Eng. Chem. Fundam., Vol.
represent the ultimate enhancement. The mechanism responsible is the rapid setting up of a fast vertical streaming motion whose intensity is governed by density differences between h o y a n t and settling particles and suspending fluid. Nomenclature
uwl = particle velocity relative to wall when only heavy
particles are present
18, No. 1, 1979
49
Literature Cited Crowley, J. M., J . Fluid Mech., 45, 151 (1971). Crowley, J. M., Phys. Fluids, 19, 1296 (1976). Crowley, J. M., Phys;, Fluids, 20, 339 (1977). Hawksley, P. G. W., Conference on Some Aspects of Fluid Flow, 1950", p 114 Edward Arnold and Co., London, 1951. Sohn, H. Y.. Morland, C., Can. J. Chem. Eng., 46, 162 (1968). Steinour, H. H., Ind. Eng. Chem., 36, 618 (1944). Thacker, W. C., Lavelle. J. W., Phys. Fluids. 21, 291 (1978). Timbrell, V., Brit. J. Appl. Phys. Sup. No. 3 , 5, SI2 (1954). Whitmore, R . L., Brit. J. Appl. Phys., 6, 239 (1955).
uw2= particle velocity relative to wall when heavy and light
Received for review April 13, 1978 Accepted October 21, 1978
particles are present
A Third Parameter for Use in Generalized Thermodynamic Correlations Michael G. Kesler and Byung I k Lee Mob/ Research and Development Corporation, Princeton, New Jersey 08540
Stanley I. Sandler Department of Chemical Engineering, University of Delaware, Newark, Delaware 1971 1
Thermodynamic correlations based on corresponding states generally use T,, P,, and w. For heavy hydrocarbons, which decompose at temperatures far below the critical, T,, P,, and w cannot be measured and have been correlated as a function of only T, and SG. This work introduces a third parameter, w,, similar to w but calculated from vapor pressures near T,, without knowledge of T, and P,. Correlations of T,, P,, w , heats of vaporization at the normal boiling point, and vapor pressures in terms of T,, SG, and w, of a number of hydrocarbons show significant improvement over existing correlations. The work is based on general concepts of perturbation theory and, using n-alkane as reference, correlates departures of the other hydrocarbons from n-alkane behavior.
Thermodynamic properties correlations, developed in the past 20 years and based on the principle of corresponding states, generally use three correlating parameters. A widely accepted set is the critical temperature T,, the critical pressure P,, and Pitzer's acentric factor w. The introduction of this :latter parameter has significantly improved the prediction of volumetric and thermodynamic properties of pure coimponents and mixtures as well as vapor-liquid equilibria (Pitzer et al., 1955; Pitzer and Curl, 1957; Curl and Pitzer, 1958; Pitzer and Hultgren, 1958; Barner et al., 1966; Wilson, 1964; Chao et al., 1971; Chao and Greenkorn, 1971; Starling and Han, 1972; Soave, 1972; Carruth and Kobayasihi, 1972; Lu et al., 1973; Lee and Kesler, 1975). Such correlation schemes rely on the availability of T , and P, values for com:ponents and mixtures. In fact, the main limitation of these correlations results from the difficulties of measuring T , and P,; many industrially important components, such as heavy hydrocarbons, decompose a t temperatures far below the critical. For such components and petroleum cuts, T,, P,, and w have been correlated as functions of normal boiling point (Tb),specific gravity (SG),and/or the Watson Characterization Factor (WCF) defined as
W C F = (Tb)'/3/SG
(1)
0019-7874/79/1018-0049$01.00/0
where Tb is in degrees Rankine. Thus, any thermodynamic properties correlations for such components, based on T,, P,, and w , are reduced to correlations in the two parameters Tb and SG. The main objective of this work is to introduce a third, easily measurable parameter which, together with Tb and SG can be used to develop accurate predictions for T,, P,, and w. Another, broader objective is to use the new parameter together with Tb and SG to develop thermodynamic properties correlations for heavy components and petroleum cuts. This second objective has only in part been accomplished. D e f i n i t i o n o f a Third P a r a m e t e r The third parameter proposed here has, in analogy with Pitzer's acentric factor, been defined as follows w , = a[ln P.,,(T = 0.85Tb) + b ] (2) where P,(T = 0.85Tb) is the vapor pressure of the liquid in atmospheres a t a temperature equal to 0.85 of Tb in the absolute scale. The constants a and b were chosen to make the values of w, most similar to w for the n-alkanes from C4 to C16 and Czo,for which reliable vapor pressure data are available. Thus we find that w, = -1.46456[1n P,,(T = 0.85Tb) '- 1.74731 = -3.37227tlog pv,(T = 0. jTb) + 0.758841 (3) 0 1979 American Chemical Society
