A MECHANISM OF SEDIMENTATION B R Y A N T FlTCH Dorr-Olioer, Inc., Stamford, Conn.
A mechanism i s deduced to explain a wide range of sedimentation behavior unaccounted for in existing theory. Voids or lean-phase bubbles may spontaneously segregate and channel through the pulp where the curvature of a plot of solids flux vs. concentration i s positive (concave upward). Evidence of such phase behavior i s presented based on fluidization, batch settling, and continuous thickening.
c
and Clevenger (7) postulated two regimes of sedimentation thickening. [n one (free settling) the weight of particles is borne solely by hydraulic forces. I n the other (compression) particles receive some mechanical support from layers below, and thus exert a squeeze or compressive force on them. There is no reason to doubt that two such regimes can exist, although many writers (3, 9, 70, 7476, 78, 79) ascribe a n insignificant role to compression in the performance of Dorr-type thickeners. But a t least some observed behavior (2) is incompatible with the idea of free settling, and is not explainable on the assumption that partial mechanical support is slowing do\vn subsidence (compression). I t remains to be explained in some other way. This paper develops the hypothesis that there is a third mechanism of subsidence, “phase settling.” I t is conceived as a species of aggregative fluidization, distinguished by breakup or separation of pulp into two phases. A low-concentration phase bubbles or channels through the pulp, producing shortcircuiting and hence a n (augmentedporosity for flow of escaping fluid (2). This leads to free settling behavior in which the over-all subsidence rate, 8, is not necessarily a function only of over-all concentration, 6. although locally in either phase R is still R ( C ) .
H e also teaches that when lower concentration loci overrun ones of higher concentration. a discontinuity is formed. Figure 1A is a typical G us. C plot such as used by Kynch. Figure 1B is a plot of dG/dC us. C derived from 1 A . Since dG/dC = P. it measures the propagation velocity of zones of concentration C. (The positive direction is downward, in the direction of settling.) Figure 1C shows two reverse concentration gradients, a-b and e-f, distributed over height H,-H2 between concentrations C I - C ~and Cs-Cq, respectively. I n the range CI-C~, d2G/dC2is negative. A reverse gradient such as a-b will after a time have moved to some position c-d. The gradient becomes less steep and the concentration front disperses. O n the other hand, where d2G/dC2is positive, lower concentrations propagate more rapidly than higher ones. The original gradient becomes ever steeper with time, finally propagating to a reverse discontinuity as represented by g-h. Once a reverse discontinuity has formed, any one of three things may happen. Kynch deduced that, if it is propagating in the direction of settlement with a velocity 6,
Mechanism
I t may d o this. If it does, it remains only a tiny local discontinuity propagating upward through the pulp. Nothing of interest happens. But there is nothing in the Kynch derivation which requires that a newly formed discontinuity must propagate as indicated by C3. Just as consistent with free settling postulates is the following: Above the discontinuity, particles are present in higher concentration and settle less rapidly. Below it, the concentration is lower and the settling rate greater. Unless particles are forced across the discontinuity, those of lower concentration below simply settle away from those above. Pulp falls apart a t the discontinuity, leaving a void or cavity. The single discontinuity breaks up into two, a reverse or roof one above the parvoid, a normal or floor one below, each propagating according to the demands of its own material balance. A third possibility is that particles flow across the discontinuity with a flux between zero and that permitted by CS. I n this case the concentration below the discontinuity will adjust, yielding a bubble of lowered concentration. Because the flow of solids into the bubble is restricted from above, it will be filled with a “dilute phase” as defined by Mertes and Rhodes (7). Thus if the discontinuity propagates as shown by Equation 2, nothing notable happens. I t leaves a local wake of C3, which
OE
I t has been reported in fluid bed studies that cavities do arise spontaneously ( 6 ) . Formation of cavities in thickening pulp has not been explicitly noted, but is readily inferred from observed behavior. On occasion water channels upward in intermittent, rapidly moving streams, which show as “volcanos” (7) on the upper interface of the settling pulp. It is difficult to conceive of this rapid flow suddenly percolating out of the walls of a channel; it must have its source in collapsing reservoirs or pockets or already-separated liquid. Cavities or parvoids (4) do form, and our purpose is to explain them. Settling of particles is usually treated as a steady linear motion, and the average motion must be essentially so. There is, however, random movement superimposed upon average settling velocity, which would be expected to cause local fluctuations in concentration. These result in local concentration gradients which in places will be negative in sign (more concentrated region above). Kynch (51, using continuity arguments, deduced that if a concentration gradient exists, loci of constant concentration within it will propagate in the direction of settlement with a velocity: = dG/dC
(1)
8 = - Gq
Cq
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G3 - AG
- C3 - -AC
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is little different from the average concentration. If it propagates upward more slowly, then by the demands of material balance it leaves behind it a zone of concentration lower than C3. If it propagates a t the settling velocity of C4,a void forms beneath it. This is the central concept of the proposed hypothesis. There is nothing in Kynch's arguments which establishes what the propagation rate will be. This depends upon a balance between forces tending to drive particles over the discontinuity and those tending to stabilize it. Stabilizing Forces. Two sorts of forces act to stabilize roof discontinuities, hydraulic or Bernoulli effects, and interparticle or yield value forces. As fluid passes from the more open voids of a cavity into the restricted passages between particles in the more concentrated pulp above, it must speed up. This creates a pressure drop. Particles at the discontinuity then have to overcome this added pressure gradient to escape. In other words, the front particles show a somewhat lowered settling rate. Such hydraulic forces a t a boundary were studied by Rowe and Henwood (73) and found to be substantial. Yield value is a consequence of interparticle cohesion. If particles a t a discontinuity resist displacement from their structure, the roof of a cavity will be stabilized. Driving Forces. Forces inhibiting transfer across a discontinuity are opposed by others tending to drive particles over the boundary. Fluid turbulence exerts randomly unbalanced forces on particles. The resultant is available for driving particles over the boundary force barrier, much as activation energies operate in chemical kinetics. Thus one criterion for phase behavior is the amount of turbulent momentum transferred to solids particles. A reverse discontinuity is hydrostatically unstable. The region of lowered concentration has a lowered density, and tends to rise as a bubble. This creates shear forces around the bubble which might be expected to erode it away. But if the pulp as a whole exhibits yield value, low concentration or void bubbles will not rise until they attain some size. Growth of Lean Phase. At this point the spontaneous generation and stability of cavities or lean-phase bubbles have been rationalized. Presumably such cavities or bubbles, as formed, would be very small since the expected extent of the original concentration deviation could not be large; but once formed, they would tend to grow. If, for example, the original concentration deviation was positive, so that a nodule of slightly over average concentration is formed above a stabilized discontinuity, it would settle more slowly than the surrounding pulp. I t would then not only persist, but would increase in height through accumulation of particles settling into it from above. Furthermore, if the nodule grew vertically, it would flow outwardly a t its bottom, extending the supporting discontinuity laterally. Also particles above a roof discontinuity are subject to augmented drag forces, and may settle more slowly than particles above. This tends to back up layers of ever-increasing concentration, augmenting the effect. Thus the parvoid below the discontinuity would grow laterally through spreading of its roof discontinuity, and vertically because its roof subsides less rapidly than its floor. I t would grow until density instability caused it to break through its roof and channel to a higher level, until upward flotation produced flows around it which eroded away its growth, or until it was overrun by higher concentrations propagating up from below. 130
l&EC FUNDAMENTALS
Phase Saturation. I t is postulated that a dense phase will not carry an unlimited quantity of bubbles, but in effect may become saturated with them. Perhaps as the lean-phase bubbles increase in number, they cause increasing agitation of the dense phase, which impairs stability of roof discontinuities. At some flux a n equilibrium is reached. At saturated levels over-all subsidence becomes again a function of over-all concentration. Saturated phase behavior superficially will closely resemble zone settling, the significant difference being that sufficient pulp must exist below the saturated level to generate the required amount of lean phase. Effect of Gradients. Over-all concentration gradients will affect both generation and growth of lean-phase bubbles. If the gradient is large-if it increases rapidly with increasing depth-in order to produce a local region of reverse gradient there would have to be abrupt deviations in local concentration. Also the pulp a t the floor discontinuity would be increasing in concentration because of propagation up to it of higher concentration zones. Unless the cavity reaches flotation or break-away size first, the floor of the parvoid will eventually subside less rapidly than the roof, and it Mill dwindle. In general, however, a normal concentration gradient should not completely prevent a local reverse discontinuity, if it formed, from producing a subjacent bubble, since it takes a finite time for concentration a t the floor discontinuity to reach that a t the roof. Thus, presence of normal concentration gradients should depress phase behavior, but not necessarily prevent it.
Evidences of Phase Behavior
A possible mechanism has now been deduced to explain phase behavior. We recognize experimentally that certain types of such behavior exist (aggregative fluidization, channeling), but there is as yet no direct proof that it is caused by the postulated mechanism. The problem of obtaining such proof appears formidable. Therefore we support the theory by circumstantial evidence. I t accounts for a wide range of sedimentation phenomena observed over the years which are otherwise inexplicable. A few instances will be presented. Fluidized Beds. Aggregative fluidization occurs, the only regime normally encountered when fluidizing with a gas. The question is whether or not the observed behavior is consistent with the postulated mechanism. In most fluidizing experiments spontaneous separation of a lean phase from pulp is not necessary, since cavities or bubbles may be generated a t the constriction plate. However, if d2C/dC2is negative in the dense phase, it was argued that roof discontinuities \vi11 tend to blend out (Figure l ) , and thus that the lean phase will disappear. Unless there are extraordinarily strong boundary-stabilizing forces, phase behavior should not persist with a dense phase for which d2C/'dG2is negative. Unfortunately in gas-fluidized beds the local fluid flux and solids concentration are not usually determined. However, all agree that the dense phase is barely fluidized; hence its local concentration is near the upper end of the flux curve. When the fluidizing medium is water, particulate behavior (zone settling) is obtained. Wilhelm and Kwauk (17) and Richardson and Zaki ( 7 7 ) have found empirically for particulate fluidization of closely sized particles Ro(l - C ) n
(3)
G = RoC(l - C)"
(4)
R
=
and since G = RC,
+
This function has a maximum a t l / ( n l ) , a n inflection point l ) , and a locus of maximum curvature a t 3 / ( n 1). at 2/(n The value of exponent n varies from 2.39 for spheres a t high Reynolds numbers, to above 4.65 a t low ones. Nonspherical particles show increased values of n. At the minimum observed value of n there will be positive curvature above a concentration of 0 . 5 9 . The range of C in such cases extends to from 0 . 6 1 5 to 0 . 6 3 5 . Even in this limiting case, there is a small domain a t the high end of the range in which d2G/dC2 is positive. The domain becomes larger a t higher values of n. Unless there is a change in settling mechanism, Richardson and Zaki’s dimensionless correlations should remain valid for particulate behavior whatever the fluidizing medium. Therefore Equation 4 is believed a t least approximately valid locally for the separate phases in gas fluidization. A region of positive curvature should exist a t the upper end of all such flux curves, which is the necessary condition under the proposed hypothesis. Observed aggregative fluidization is consistent with the postulated mechanism. There is also definite indication of phase behavior in water fluidization data. Figure 2 sho1vs a flux plot for sea sand (77, Table 2). The curve has a maximum a t a concentration of about 0 . 2 5 . If extrapolated by Equation 4 it would have an inflection a t about 0.50. The data depart from the equation someuhere just below C = 0.50 and show a pronounced twohumped curve of the sort reported by Shannon, Stroupe, and Tory (74). Figures 3 and 4 show similar plots for large and small spheres. In each case the flux plot breaks upward above the values given by Eqiiation 4 or a t near the predicted point of inflection.
+
I
I
I
I
I
I I
II
I
I
! 1
I
“I
1
B
+
I
I I
I
I
C 8 e
I ’
I ‘I I
I Figure 1 .
0
I
I
I I
C
I
1
‘
I
I
1
‘
I
I
I
Propagation of concentration gradients
0.4 0
0.20
Flux plot for sea sand fluidized with water Wilhelrn and Kwauk, Table 2
0,40 C
0.60
Figure 3. Flux plot for large Socony beads fluidized with water Wilhelm and Kwauk, Table 5
0.60
C Figure 2.
0.20
0
0.20
0 Figure 4.
0.40
0.60
Flux plot for small glass beads
Wilhelm and Kwauk, Table 2 6
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The water flux, R (identical to the subsidence rate as defined), is largert han predicted. This indicates augmented porosity and supports the idea of phase behavior. I t could not be caused by compression, since particle support would decrease R below prediction. I t seems probable that any lean phase occurring in water fluidization consists not of voids but rather of bubbles with a concentration not too far from that of the dense phase. Void bubbles probably would have been observed. Bubbles of reduced concentration, particularly if they build only to a small size, would be evident only as an apparent turbulence or swirling of the pulp, which usually is seen. There is practical evidence of phase behavior in fluid bed classification. In commercial sizers particle size separation is poor unless the teeter bed is expanded to a t least about 60% voids. Figure 2 indicates why. Below 60% voids phase behavior would cause some mixing and declassification. Furthermore, greater bed expansion is required for good operation a t finer meshes of separation. Fine solids operate a t lower Reynolds number, higher n, and a n inflection point a t a lower value of C. Higher bed expansions are then necessary to avoid phase behavior. Thus phase behavior appears to exist generally in water fluidization in regions where positive flux plot curvature is anticipated. I t is probably characterized by small, lean-phase bubbles of relatively high concentration, present in considerable numbers. Such behavior is different from the all-out aggregative nature of air fluidization. Limited vs. All-out Phase Behavior. In air fluidization it appears that roof-stabilization forces dominate. The limit on lean-phase flux is high. Thus a flux plot might appear as in Figure 5 , in which the solid curve represents over-all apparent flux us. over-all concentration, and the dashed curve the true or local R us. C. Line o-a corresponds to some fixed fluid flux, R,. I t crosses the solids flux curve a t points a, b, and c. Operation a t point b is unstable because any upward perturbation in concentration permits a higher flux than exists, so the pulp collapses immediately to point a. Operation a t point c is unstable because if phases a and c are both present, there will always be local areas a t their interface in which the dense phase of a underlies zone c. I n these local areas the air flux is very low, and zone c will collapse into the dense phase of zone a. Stable operation is confined to point a. As over-all air flux is increased, it may reach a maximum such as point d, after which operation would jump to condition e.
In flux plots showing limited phase behavior, such as Figure 3, there is no fluid flux line giving multiple intersections with the solids flux plot. As fluid flux is increased, the operating point moves continuously along the solids flux curve, and transition from phase to zone behavior is smooth. Criteria for Aggregative Fluidization. The proposed mechanism is supported by what is known of criteria for aggregative fluidization. Settling of a single particle through an unbounded quiescent fluid is a function of the following variables:
Purely dimensional arguments lead to the dimensionless correlation
If, from some nondimensional knowledge of the physical situation, it is specified that ps enters only in the functional group - p f ) (as is usually done), the system loses a degree of freedom, and
g(p,3
This leads to the familiar Reynolds number-drag coefficient correlation for a settling particle. If the system consists of a uniform (at least stochastically) array of similar particles, one further degree of freedom is added. I t can be characterized by the already dimensionless variable, C. Then the state of the system (functional relationship between variables) is fixed by the dimensionless correlation
(7)
Again, if it is stipulated that ps enters only as the functional group g ( p s - p f ) (which is equivalent to specifying uniform rectilinear settling velocity u ) , Equation 7 reduces to
I
d
0 Figure 5. behavior 132
C Postulated flux plot showing aggregative
I&EC FUNDAMENTALS
Richardson and Zaki's correlation rests in the last relationship. If values of any two groups in it are specified, the state of the system should be fixed. The area representing possible sets (Re, C) should cover all states of the system. Some subspace within this area should represent conditions under which aggregative fluidization takes place. Experimentally it does not. Therefore Relationship 8 is inadequate. Either we have missed a variable, or the group p s / p f enters in some other way than stipulated. The latter alternative seems more reasonable, and is supported by the publications of Zenz (20)
I
Thus we fall back on general dimensional relationship (Equation 7 ) . State of the system is fixed by specifying any three of its dimensionless numbers. Then the space representing all sets of the chosen three comprises all states of the system. Some subspace represents conditions under which aggregative fluidization takes place. The boundaries of this subspace can be represented by projecting constant value contours for any one of the groups down its axis onto the plane of the other two. The result will be a “phase diagram” or map of the aggregative fluidization regime (20). Since the group ps,’p,. enters, motion of particles cannot be considered rectilinear. By interaction with fluid, particles must be given oscillation or other accelerated motion which involves particle inertia and such flutter must be a significant factor in determining whether or not aggregative fluidization takes place. The ideas of random variations in local concentrations and flutter to drive particles over a discontinuity are key factors in our hypothesis. Continuous Thickening. Fluidization is characterized by a Lvater flux, R, which is constant a t all levels. In steady-state thickening it is the solids thioughput flux, G,, which is fixed. From material balance, through all levels below the feed in a cylindrical thickener (76)
and since G = RC
(9)
Equation 9 defines a n operating line on a Kynch flux plot. Any concentration present a t steady state must plot on this line, and also must be on the flux curve for the pulp in question. Therefore only concentration zones can exist for which the flux curve and operating line coincide. Thickeners operate with a bed of solids ranging from a high concentration a t the bottom to a much lower one a t the top. Over this range 1:he flux curve and the operating line must coincide. But they still operate with a bed of pulp when underflow rate U ,and throughput Gt are changed, giving a different operating line. I n Figure 6, a-6 and c-d represent two such operating lines. The flux plot must coincide with each of these. I t could not do so if G = f(C). T h e difference might be ascribed to cornpression, but in some actual measurements compression was not found. The bed of solids must be explained in some other way. Figure 6 shows a postulated settling flux plot for steady-state thickening (idealized by omitting the possibility of a true compression regime). The lower curve represents local or true settling flux, the upper one the limiting or saturation flux permitted by phase behavior. Once phase behavior becomes possible, a given layer of pulp will handle any over-all flux between the two curves, by appropriate adjustment of the amount or depth of sludge available to release lean phase beneath the layer. According to the postulated mechanism, within the shaded area between curves sludge depth adjusts so
(3
C Figure 6.
d
b
Postulated flux plot for thickening
that the flux curve and operating line coincide. Thus phase settling resolves the dilemma. The above requires that local flux curves have positive curvature over the range of concentrations concerned. This appears reasonable in the light of existing theory. hfost correlations for settling rate of impermeable particles, including that of Carman-Kozeny which has some basis in fluid dynamics theory, show singly concave flux plots (74). Only the completely empirical correlation of Shannon and Tory predicts a doubly concave plot, and it presents no data in a second concave region, arriving a t its form through a n assumption entailing that pulp reach zero porosity a t a solids volume concentration, C, of 0.64. Actual experimental data (Figure 2) depart abruptly from all predictions, and the most rational explanation is a change of mechanism. Floccules should settle much as nonporous particles until they reach compression (8, 72). Batch Settling. In batch settling, particularly of metallurgical pulps, there is frequently a range of initial concentrations over which a zone of some critical lower concentration forms a t the top. One then observes about the same surface subsidence rate over the initial concentration range. The lower concentration propagates downward from the interface more rapidly than its settling rate. From Equation 1 :
> R , dR/dC must be positive. In other words, Lvherever such a layer appears, subsidence rate must increase when concentration is increased-a difficult thing to explain unless short-circuiting or phase behavior is postulated. If p
Summary
A mechanism has been presented to explain the observed fact that lean-phase bubbles or channels under some conditions segregate from a column of sedimenting suspension. VOL. 5
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133
A necessary (but not sufficient) condition is positive curvature in a Kynch plot. Such phase behavior exists in more forms and is more prevalent than formerly conceived. I t accounts for a wide range of formerly inexplicable sedimentation behavior. Nomenclature
A
=
C
= local solids concentration, volume of solids per volume
6
= = =
D, G
G, = g n
R
= = =
I?
=
U
p
= = =
6
=
pf p. fi
= = =
u
Re -
area of thickener or settling column of pulp over-all concentration of solids in phase settling diameter of settling particles solids volume flux past locus of zero total flux (solids fluid), volume of solids per unit area per unit time value of G a t axis intercept of thickener operating line, equal to total solids flux through thickener acceleration of gravity exponent local solids subsidence rate past locus of zero total flux, also equals volume flux of water upward with respect to solids over-all subsidence rate in phase settling thickener underflow rate, volume/time settling rate propagation rate of locus of constant concentration (Kynch zone) measured in direction of sedimentation propagation rate of Kynch discontinuity in direction of sedimentation density of fluid density of solid viscosity of fluid
+
RECEIVED for rekfiew March 18, 1965 ACCEPTED October 15, 1965 Correction
T H E R M O D Y N A M I C S OF SOLUTIONS. EQUATIONOFSTATEANDVAPOR PRESSURE
Fr = - U2 D,e = 46g(pspf)
3!J2Pf
(drag coefficient)
literature Cited
(1) Coe, H. S., Clevenger, G. H., Trans. Am. Inst. Mining Engrs. 5 5 . 356 (1916). (2) eitch, B., Ibid., 223, 129 (1962).
134
/,ne(17JJ).
(19) Yoshioka, N., Hotta, Y., Tanaka, S.,Naito, S., Tsugami, S., Ibid.. 21. 66 (1957). (20) Zenz; F. 24., Othrner, D., “Fluidization and Fluid Particle Systems,” Reinhold, iYew York, 1960.
DPupf
U
/D
(3) Gaudin, A. M., Fuerstenau, M. C., Inst. Mining and AMetallurgy (British), Paper 6, Group 11, 1960. (4) Hassett, N. J., Ind. Chemist 34, 116, 169, 489 (1958); 37, 25 I1961). (5)’ Kydch, G. J., Trans. Faraday SOC. 48, 161 (1952). (6) Massimilla, L., IVestwater, V. I$’.,A.I.Ch.E. J . 6 , 134 (1960). (7) Mertes, T. S., Rhodes, H. B., Chem. Eng. Progr. 51, 429 (1955). (8) Michaels, A. S., Bolger, J. C., IND.ENG.CHEM. FUXDAMENTALS 1. 24 (1962). (9) ’Mon‘tcrieff,H. G., Bull. Inst. M i n i n g Met. 73, 729 (1964). (10) Porter, J. L., Scandrett, H. F., International Aluminum Symposium, Am. Inst. Mining Engrs., Sew York, February 1962. (11) Richardson, J. F., Zaki, I$’. N., Trans. Inst. Chem. Engrs. 32, 35 (1954). (12) Roberts, E. J., Mining Eng. 1, 61 (1949). (13) Rowe, P. N.,Henwood, G. A , , Trans. Inst. Chem. Engrs. 39, 431 (1961). (14) Shannon, P. T., Stroupe, E., Tory, E. M., IND.Esc. CHEM. FUNDAMENT.ALS 2,2C3 (1963). (15) Tory, E. M., Ph.D. thesis, Purdue University, 1961. (16) Xt‘allis, G. B., Proceedings of Symposium in Interaction between Fluids and Particles, Inst. Chem. Engr., London, Vol. A, p. 9, 1963. (17) \Vilhelm, R. H., Kwauk, M., Chem. Eng. Progr. 44, 201 (1948). (18) Y?:hioka, J., Hotta, Y.,Tanaka, S., Kagaku Kopa”u 19, 616
l&EC FUNDAMENTALS
In this article by Otto Redlich, F. J. .4ckerman, R . D. Gunn, Max Jacobson, and Silvanus Lau [IND.ESG. CHEM. F u m A m w A L s 4, 369 (1965) 1, there is a n error on page 373, first column, Equation 12. T h e equation should read
ZO3- Zo2 (1
+ [0.427481 PTTT-*.Z- 0.0866404 P,T,-1
X
+ 0.0866404 PiT7-l)]Z,- 0.0370371 P1*Tr-3.5 = 0
