Powder Technology 108 Ž2000. 74–87 www.elsevier.comrlocaterpowtec
On upper rarefaction waves in batch settling R. Burger ¨ b
a,)
, E.M. Tory
b,1
a Institute of Mathematics A, UniÕersity of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany Department of Mathematics and Computer Science, Mount Allison UniÕersity, SackÕille, Canada NB, E4L 1E8
Received 2 August 1999; received in revised form 29 October 1999; accepted 2 November 1999
Abstract We present a complete solution of the batch sedimentation process of an initially homogeneous ideal suspension where the Kynch batch flux density function is allowed to have two inflection points. These inflection points can be located in such a way that during the sedimentation process, the bulk suspension is separated from the supernate by a rarefaction wave or concentration gradient. This observation gives rise to two new modes of sedimentation as qualitative solutions of the batch sedimentation problem that had not been considered in previous studies. A reanalysis of published experimental data indicates that several observed upper concentration gradients can actually be interpreted as a rarefaction wave, and therefore be included in the framework of Kynch’s theory. A numerical example shows an upper rarefaction wave in the settling of a flocculated suspension, to which Kynch’s theory applies if the solid particles are in hindered settling. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Batch sedimentation; Ideal suspension; Hyperbolic conservation law; Rarefaction wave; Flux plot
1. Introduction Kynch’s sedimentation theory w1,2x is a well-known mathematical model for the one-dimensional batch and continuous sedimentation of ideal suspensions of monosized spheres under the influence of gravity. The principal assumption of this kinematical theory is that the local settling velocity or solid phase velocity is a function of the local solids concentration f only. The resulting governing equation is a scalar first-order partial differential equation or conservation law. Bustos and Concha w3x and Concha and Bustos w4x studied the problem of sedimentation of an initially homogeneous ideal suspension in the framework of Kynch’s theory. For flux density functions with two inflection points, they obtained five qualitatively different solutions or modes of sedimentation, depending on the initial concentration and geometrical properties of the Kynch batch flux density function. All of these modes had been seen in experimental studies of suspensions of closely sized spheres w5x. We show that two additional modes of sedimentation
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are theoretically possible and find experimental evidence that these modes do occur. In the solutions constructed by Bustos and Concha w3x and Concha and Bustos w4x, the supernate is always separated from the settling suspension by one shock-type concentration discontinuity. However, the two inflection points in the flux density function can also be located in such a way that the clear fluid section is separated from the settling suspension by a rarefaction wave. Such waves will be denoted in short as ‘upper rarefaction waves’, by analogy to the term ‘upper discontinuity’ used by Font et al. w6x. It is the purpose of this contribution first to complete the solution of the problem studied by Bustos and Concha. Then, we show that the two new modes of sedimentation, or upper rarefaction waves in general, provide explanations of actually observed sedimentation behaviour that is not usually assumed to fall within the scope of Kynch’s theory. This paper is organized as follows. In Section 2, we recall some basic mathematical properties of Kynch’s equation and briefly review the method of characteristics employed to determine exact entropy solutions of the batch sedimentation problem. In Section 3, this method is employed to obtain the complete solution of this problem for a flux density function with two inflection points.
0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 2 5 7 - 0
R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨
The concept of upper rarefaction waves not only completes Bustos and Concha’s study w3x, but also widens the scope of application of Kynch’s theory, since there is strong evidence from stochastic sedimentation models and settling experiments for upper rarefaction waves. This is shown in Sections 4 and 5. In Section 6, we examine some experimental data that raise the possibility of a family of modes of sedimentation different from those considered by Bustos and Concha and extended by us. This, in turn, leads to a discussion of the limitations of Kynch’s theory. Our results may also be useful for compressible slurries since upper rarefaction waves may occur within sedimentation models that rely on Kynch’s theory for zones in which the solids are in hindered settling and do not yet touch each other. This is demonstrated in Section 7 by a hypothetical numerical example. Conclusions that can be drawn from our study are summarized in Section 8.
2. Kynch’s sedimentation model and mathematical preliminaries In its original formulation, Kynch’s kinematical sedimentation theory w2x states that the solid–fluid relative velocity, or drift velocity, is a function of the volumetric solids concentration only. The volume average velocity of the mixture is given by q s f Õs q Ž 1 y f . Õf ,
Ž 1.
where f is the solids volumetric concentration and Õs and Õf are the respective solid and fluid phase velocities. In one space dimension, this quantity is always determined by boundary conditions and vanishes for batch settling, and Kynch’s assumption can be restated by assuming that the solid phase velocity Õs is a function of f only. Defining the Kynch batch flux density function f bk Ž f . s f ÕsŽ f ., we can rewrite the solids local mass balance for batch settling as the scalar conservation law Ef q Et
E f bk Ž f . Ez
s 0, 0 F z F L, 0 F t F T .
75
determined by Shannon et al. w5,7x for a suspension of small glass beads Žsee Fig. 1Ža.., and the flux density function proposed by Barton et al. w8x,
ž
f bk Ž f . s Õ 0 f 1 y
f 0.9
C
/
q Õ 1 f 2 Ž 0.9 y f .
Ž 4.
with parameters Õ 0 s y1.18 = 10y4 mrs, Õ1 s y1.0 = 10y5 mrs and C s 5 Žsee Fig. 1Žb... For batch sedimentation, Eq. Ž2. is solved together with the initial condition
° ¢
0 f Ž z ,0 . s f 0 Ž z . fmax
~
for for for
z ) L, 0 F z F L, z - 0.
Ž 5.
Note that, due to the assumptions on f bk , the initial condition Ž5. could be replaced by the initial condition f Ž z,0. s f 0 Ž z . for 0 F z F L, combined with the boundary conditions f Ž0,t . s fmax and f Ž L,t . s 0 for 0 - t F T. To construct the solution of the initial value problem Ž2., Ž5., the method of characteristics is employed. This method is based on the propagation of f 0 Ž z 0 ., the initial X Ž f 0 Ž z 0 .. value prescribed at z s z 0 , at constant speed f bk in a z vs. t diagram. These lines might intersect, which makes solutions of Eq. Ž2. discontinuous in general. This is due to the nonlinearity of the flux density function f bk . In fact, even for smooth initial data, a scalar conservation law with a nonlinear flux density function may produce discontinuous solutions, as the well-known example of Burgers’ equation illustrates Žsee Ref. w9x.. On the other hand, one particular theoretically and practically interest-
Ž 2.
Here, t is time and z is the space variable. The flux density function is usually assumed to satisfy f bk Ž f . - 0 X Ž0. - 0 and for 0 - f - fmax , f bk Ž0. s f bk Ž fmax . s 0, f bk X f bk Ž fmax . ) 0, where fmax is the maximum solids concentration. Initially, we shall restrict the discussion to flux Y Ž0. ) 0. density functions satisfying f bk Two examples having all these properties are the flux density function f bk Ž f . s Ž y0.33843f q 1.37672 f 2 y 1.62275f 3 y0.11264f 4 q 0.902253f 5 . = 10y2 mrs
Ž 3.
Fig. 1. Flux density function with two inflection points a and b, Ža. after Shannon et al. w5,7x, Žb. after Barton et al. w8x.
R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨
76
ing initial value problem for a scalar conservation law is the Riemann problem, where an initial function
fq 0 f0 Ž z . s y f0
½
for for
f Ž z ,t .
z ) 0, z-0
Ž 6.
y
s Ž f ,f . s
f bk Ž fq . y f bk Ž fy .
fqy fy
.
f bk Ž f . y f bk Ž fy .
G s Ž fq, fy . G
q 0
y1 X s Ž f bk . Ž zrt .
¢f
y 0
for
X z ) f bk Ž fq0 . t ,
for
X X f bk Ž fy0 . t F z F f bk Ž fq0 . t ,
for
X z - f bk Ž fy0 . t
Ž 10 . X y1 X . where Ž f bk is the inverse of f bk restricted to the y q interval w f 0 , f 0 x. This solution is called a rarefaction waÕe and is the entropy weak solution of the Riemann problem. For details on entropy weak solutions of scalar conservation laws, we refer to the books by Le Veque w9x, w12x, Serre w13x, Godlewski and Raviart w10,11x, Kroner ¨ Smoller w14x or Toro w15x.
Ž 7.
However, discontinuous solutions satisfying Eq. Ž2. at points of continuity and the Rankine–Hugoniot condition Ž7. at discontinuities are, in general, not unique. For this reason, an additional selection criterion or entropy principle is necessary to select the physically relevant discontinuous solution, the entropy weak solution. One of these entropy criteria, which determine the unique entropy weak solution, is Oleinik’s jump entropy condition requiring that
f y fy
°f ~
consisting just of two constants is prescribed. Obviously, the initial value problem Ž2., Ž5. consists of two adjacent Riemann problems producing two ‘fans’ of characteristics and discontinuities, which in this case, start to interact after a finite time t 1. At discontinuities, Eq. Ž2. is not satisfied and is replaced by the Rankine–Hugoniot condition w4x, which states that the local propagation velocity s Ž fq, fy . of a discontinuity between the solution values fq above and fy below the discontinuity is given by q
q be constructed between fy 0 and f 0 . In that case, the Riemann problem has a continuous solution
f b k Ž f . y f bk Ž fq .
f y fq
for all f between fy and fq
Ž 8.
is valid. This condition has an instructive geometrical interpretation: it is satisfied if and only if, in an f bk vs. f plot, the chord joining the points Ž fq, f bk Ž fq.. and Ž fy, f bk Ž fy.. remains above the graph of f bk for fq- fy and below the graph for fq) fy. Discontinuities satisfying both Eqs. Ž7. and Ž8. are called shocks. If, in addition, X X f bk Ž fy . s s Ž fq, fy . or f bk Ž fq . s s Ž fq , fy .
Ž 9.
is satisfied, the shock is called a contact discontinuity. In that case, the chord is tangent to the graph of f bk in at least one of its endpoints. A piecewise continuous function satisfying the conservation law Ž2. at points of continuity, the initial condition Ž5., and the Rankine–Hugoniot condition Ž7. and Oleinik’s jump entropy condition Ž8. at discontinuities is unique. Consider Eq. Ž2. together with the Riemann data Ž6.. If Y q Ž . we assume Žfor simplicity. that fy 0 - f 0 and that f bk f y q ) 0 for f 0 F f F f 0 , it is easy to see that no shock can
3. Solutions of the batch sedimentation problem We now consider entropy weak solutions of the problem of Eq. Ž2. with the initial condition Ž5., where we set f 0 ' const. corresponding to an initially homogeneous suspension and consider a flux density function f bk with at most two inflection points. Using the method of characteristics and applying the theory developed by Ballou w16x, Cheng w17,18x, and Liu w19x, Bustos and Concha w3x and Concha and Bustos w4x construct entropy weak solutions of this problem in the class of piecewise continuous functions, in which zones of constant concentrations are separated by shocks, rarefaction waves or combinations of these. Therefore, it is necessary to solve the two Riemann problems given at t s 0 at z s L and at z s 0 and to treat the interaction of the two solutions at later times. For flux density functions with at most two inflection points, they obtain five qualitatively different entropy weak solutions or modes of sedimentation. Unfortunately, their classification is not complete, since they consider only functions with two inflection points that are similar to our Fig. 1. However, two inflection points can also be located in a different way, producing two additional modes of sedimentation. In these two modes of sedimentation, the supernate–suspension interface is not a sharp shock, but a rarefaction wave. The mathematically rigorous, detailed construction of the complete set of seven modes of sedimentation ŽMS. MS-1 to MS-7 is presented in Chap. 7 of Ref. w20x. Here, the construction of the seven different entropy solutions is outlined in Figs. 2 and 3. The simplest case is that of an MS-1, in which the supernate–suspension and the suspension–sediment interfaces are both shocks Žsee Fig. 2Ža... These shocks meet at the critical time t c to form a stationary clear water–sediment interface. In an MS-2, the rising shock is replaced,
R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨
77
Fig. 2. Modes of sedimentation MS-1 to MS-3. From the left to the right, the flux plot, the settling plot showing characteristics and shock lines, and a representative concentration profile taken at time t s tU are shown for each mode. Chords in the flux plots and shocks in the settling plots having the same slopes are marked by the same symbols. Slopes of tangents to the flux plots occurring as slopes of characteristics in the settling plots are also indicated.
from top to bottom, by a contact discontinuity followed by a rarefaction wave Žsee Fig. 2Žb... In the flux plot, the contact discontinuity is represented by a chord joining the points Ž f 0 , f bk Ž f 0 .. and Ž f U0 , f bk Ž f U0 .. that is tangent to the graph of f bk in the second point. If we take the same flux function and still increase f 0 , this chord can no longer be drawn and the contact discontinuity becomes a line of continuity. This situation corresponds to an MS-3 Žsee Fig. 2Žc... Note that the modes of sedimentation MS-2 and MS-3 can occur only with a flux density function f bk with exactly one inflection point Žwe recall that we always Y Ž0. ) 0.. assume that f bk With a flux density function f bk having exactly two inflection points, four additional modes of sedimentation are possible, that are collected in Fig. 3. Clearly, an MS-1 is also possible with two inflection points. Bustos and Concha w3x and Concha and Bustos w4x obtained two of the four additional modes of sedimentation, namely the modes MS-4 and MS-5 shown in Fig. 3Ža. and Žb.. These modes are similar to an MS-2 and MS-3, respectively, but the second inflection point produces an additional contact discontinuity, denoted by C2 in Fig. 3Ža. and by C1 in
Fig. 3Žb., which separates the lower rarefaction wave Ž R1 . from the rising sediment. Of course, the shape of a given flux density function determines which modes of sedimentation are actually possible. In Chap. 7 of Ref. w20x, a corresponding geometrical criterion to decide this is presented. It is quite obvious that flux density functions that are similar to that shown in Fig. 1 make an MS-4 or MS-5 possible. However, it is also possible to place the inflection points a and b in such a X X Ž a. - 0, f bk Ž b . - 0, and that there exists a way that f bk ˜ point a - ft - b such that the tangent to the graph of f bk at Ž f˜ t , f bk Ž f˜ t .. also goes through the point Ž0, f bk Ž0. s 0.. This situation is shown in the flux plots of Fig. 3Žc. and Žd.. If the initial concentration f 0 is then chosen between f˜ t and the second inflection point b, a new mode of sedimentation, called MS-6, is produced Žsee Fig. 3Žc..: the supernate–suspension interface is no longer sharp, that is, a shock; rather, an upper rarefaction wave R1 emerges from z s L at t s 0, which is separated from the supernate by a contact discontinuity and from the bulk suspension by a line of continuity. That line of continuity meets the rising sediment–suspension interface, the shock S 1 , at t s t 1.
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R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨
Fig. 3. Modes of sedimentation MS-4 to MS-7.
After that, a curved, convex shock S 2 forms, separating the rarefaction wave from the sediment. At the critical time t s t c , the shock S 2 meets the upper end of the rarefaction wave, that is, the contact discontinuity C1 , and the stationary shock S 3 forms, which denotes the sediment– supernate interface located at the sediment height z c . A similar construction applies if f 0 ) b is chosen such X Ž f U0 . s that there exists a point f˜ t - f U0 - b with f bk U s Ž f 0 , f 0 . Žsee Fig. 3Žd... The resulting entropy weak solution is an MS-7, differing from an MS-6 in that the bulk suspension is separated from the upper rarefaction wave by a contact discontinuity instead of a line of continuity.
All modes of sedimentation terminate in a stationary sediment of the maximum concentration fmax and of height z c s f 0 Lrfmax . This stationary state is attained at the critical time t c .
4. Anomalous rates of fall of the slurry–supernate interface Kynch’s theory w2x is based on the approximation that all particles at a given local solids concentration settle with the same velocity. In a uniformly mixed slurry, therefore, the particles at the top of the dispersion should settle with
R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨ Table 1 Velocity ratios of the slurry–supernate interface Žafter Di Felice w25x and Oliver w23x.
f0 0.0033 0.0050 0.0066 0.0100 0.0200
Di Felice
Oliver
X u r u0
u r u0
X
u r u0
0.979 0.968 0.957 0.935 0.870
0.864 0.843 0.808 0.789 0.732
0.848 0.827 0.793 0.774 0.718
the same velocity as those in the interior. If so, the rate of fall of the slurry–supernate interface would be the unique velocity of the particles at that concentration. Long before Kynch, it was tacitly assumed that this was true. Interface velocities are easy to measure and are constant for a considerable period of time. Indeed, Tory and Pickard w21x found that the velocity of the interface was remarkably constant. Linear regression of height vs. time showed a correlation coefficient of better than y0.9995 in every case and better than y0.9999 in 13 of 22 runs. This enables interface velocities to be determined with considerable precision. For example, two trials carried out at the same temperature and solids fraction gave 95% confidence intervals of Žy8.828 = 10y4 , y9.055 = 10y4 . and Žy8.842 = 10y4 , y8.935 = 10y4 . mrs. Consequently, it is not surprising that these velocities have long been taken to represent the unique velocity of the dispersion at a given concentration. In those cases where the interface was diffuse and some particles were clearly being left behind, it was assumed Žwith some justification. that these particles were smaller andror lighter. Despite the reproducibility of interface velocities for a particular solids concentration in a particular settling tank, experimental measurements differ widely on the relationship between these velocities and the initial solids fraction f 0 . According to classical analyses, the settling rate of a monodisperse suspension decreases monotonically with the solids concentration w22x. Measurements of interface velocities of very dilute dispersions of closely sized spheres divide roughly into two classes w23–25x. For one group, the decrease in magnitude with increasing concentration is very rapid w23,24x, being roughly proportional to f 01r3 w23,25x; for the other, the change in velocity is proportional to f 0 w25x. We examine three studies w23–25x in which the suspensions were essentially monodisperse. Detailed information
79
about these experimental systems is given in the Appendix A. The important observation for the present analysis is that all three authors note that a slurry–supernate interface was clearly Õisible. This indicates that there was a discontinuity at the top of the slurry and reflects the sharpness of the size distribution Žsee Appendix A.. Nevertheless, these suspensions exhibited very different behaviours. We can compare the concentration dependence of interface velocities of different monodisperse suspensions by examining the corresponding values of uru 0 . Batchelor w22x derived the equation
m Ž f 0 . ru 0 s 1 y k f 0
Ž 11 .
for the mean velocity of spheres in a very dilute dispersion. Di Felice found that all his data Ž uŽ f 0 . vs. f 0 . fitted a linear equation and that the u intercept agreed closely with the experimental single particle terminal settling velocity, uX0 Žsee Appendix A.. Thus, his data can be represented as u Ž f 0 . ruX0 s 1 y k f 0 .
Ž 12 .
His Fig. 6 and his Eq. Ž12. both indicate that k f 6.5, which agrees closely with Batchelor’s value of 6.55 for identical spheres, but is higher than Batchelor’s value of 5.5 for very closely sized spheres. ŽTory and Kamel w26x argue that the limit for the latter must equal the value of the former, whatever that value may be.. Nevertheless, Di Felice’s data are typical of those found by many others w25x. Values calculated from Eq. Ž12. and shown in Table 1 contrast sharply with the values obtained by Oliver for the same concentrations. ŽWhen values were obtained with two different viscosities, we have averaged the values.. Di Felice suggests that the problem lies with the determination of the Stokes velocity. However, Oliver’s calculated and experimental values of uX0 agree closely Žsee Appendix A.. Also, it is clear that the dependence of u on f 0 is distinctly nonlinear. Given the narrowness of Oliver’s distribution, we would expect little or no segregation by particle size. Oliver verified this by showing that spheres at the top of his suspensions had values of uX0 that did not differ appreciably from those determined for randomly selected particles w23x. Table 2 shows unpublished data obtained by Verhoeven w24x for a similar range of concentrations. ŽBoth Oliver and Verhoeven studied a much broader range, but only dilute dispersions are relevant here.. A comparison of the two tables shows that Verhoeven’s values lie roughly midway
Table 2 Velocities of the slurry–supernate interface Žafter Verhoeven w24x. X
f0
uŽ f 0 . w10y4 mrsx
u Žsmoothed. w10y4 mrsx
f bk Ž f 0 . w10y5 mrsx
uru 0
uru 0
0.0046 0.0096 0.0147 0.0197
y7.133 y7.000 y6.583 y6.483
y7.133 y6.833 y6.583 y6.500
y0.3283 y0.6567 y0.9683 y1.2800
0.904 0.866 0.834 0.824
0.867 0.831 0.800 0.790
R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨
80
between those of Di Felice and Oliver. As noted earlier in this paper, Kynch’s theory holds that the value of f just below the interface determines its rate of fall. This suggests that the solids concentration just below the interface may be appreciably different when the nominal concentrations, f 0 , are the same. This possibility is discussed further in Sections 5 and 6.
5. Evidence for upper rarefaction from experiments and stochastic sedimentation models All observations of individual particles Žsummarized in Ref. w27x. show that the velocities of spheres in the interior of a dispersion vary greatly. In some cases, such as in the experiments of Di Felice, the mean velocity calculated from Eq. Ž11. is essentially the same as the interface velocity. In many cases, however, the mean velocity exceeds Žin absolute value. that of the interface and, in some cases, the Stokes velocity w27x. Tory and Pickard w21x measured the settling rate of black Lucite spheres in a dispersion of translucent spheres of the same size. They used exactly the same system as that of VerhoeÕen w24x, but their readings were taken at a higher temperature and only for f 0 s 0.02. They also measured interface velocities at that temperature and initial solids concentration. Using height vs. time readings for 32 of the 109 independent data sequences, they found that the mean velocity of the individual spheres was 1.095 times the velocity of the interface. We want to compare their mean velocity with the interface velocities measured by Verhoeven for the initial concentrations shown in Table 2. Because interface velocities for a giÕen system are very reproducible, we can relate Tory and Pickard’s mean velocities to Verhoeven’s interface velocities by using their ratio of 1.095. From Verhoeven’s Fig. 3 or our Table 2, the interface velocity of a dispersion with f 0 s 0.02 at 248C is y6.483 = 10y4 mrs. Thus, the mean velocity of spheres under these conditions would be
m Ž 0.02 . s 1.095 = Ž y6.483 . = 10y4 mrs s y7.099 = 10y4 mrs
tion w21x. However, their criteria for selecting suitable data were flawed. Further analysis Žw27x, Fig. 2. showed that some of the marker spheres were in the region of reduced concentration and should not have been grouped with the others Žsee the explanation below.. Also, the inappropriate exclusion of data weakened the strength of their statistical analysis. Thus, it seemed advisable to reanalyse their data. First, we construct a height vs. time diagram that shows the position of all interfaces. Fig. 1 of Tory and Pickard shows that it would take only 14 min for the lower boundary of the dilute region to fall the same distance that the interface would fall in 19. Thus, the velocity of the former is Ž19r14. Žy6.483. = 10y4 mrs s y8.798 = 10y4 mrs. The construction of their column made it impossible to see its bottom w27x and they did not attempt to follow the rise of the packed bed during the sedimentation of such a dilute dispersion. However, the final solids fraction for runs with much higher values of f 0 was fmax s 0.576. The initial height of the dispersion was L s 1.23 m. Thus, the final height is calculated to be z c s Lf 0rfmax s 1.23 = 0.02r0.576 m s 0.0427 m Ž 14 . and the total settling time is tc s
zc y L y
1.23 y 0.0427 s
s Ž 0, f .
6.483 = 10y4
s s 1831.4 s f 30.5 min,
Ž 15 . where s is the Žnegative. velocity of the slurry–supernate interface and fy is the Žunknown. solids fraction just below it. From their recorded data for individual spheres settling near the bottom of the cylinder, we estimate that the height of the packed bed when it reached the bottom of the dilute region was approximately z 1 s 0.035 m. Since 0.035 ) 0.0427 = 14r19s 0.0315, this is consistent with the fact that the solids flux into the packed bed is reduced as spheres settle out from the dilute region. Then, the time for the packed bed to reach the bottom of that region is t1 s
z1 y L q
s Ž f ,f 0 .
1.23 y 0.035 s
8.798 = 10y4
s s 1358.3 s f 22.6 min
Ž 16 . Ž 13 .
and f bk Ž0.02. s y1.420 = 10y5 mrs at 248C. Tory and Pickard observed that a concentration gradient formed below the interface and that this gradient expanded with time, but they did not measure its position. Using all the data from their experiments, they found that there was a deficit of marker Žblack. spheres in the region near the top and their Fig. 1 shows that this region expanded linearly with time. They also found that the variance of the particle velocity fell quickly in the first few minutes before stabilizing at a roughly constant value. Thus, they attempted to base their mean velocity on spheres that were still at the original concentration, were far from the packed bed, and had time to reach a steady-state velocity distribu-
since, until t 1 , f 0 is the solids fraction just above the packed bed and fq is the value at the bottom of the dilute region. These values of z and t are shown in Fig. 4, which is similar to the settling plot in Fig. 3Žd. and to Fig. 7.19Žb. in Bustos et al. w20x. Note that we have constructed this diagram entirely from observations of the interfaces. We now use it to select the regions from which to select representative particles. From the laboratory notebooks of Tory and Pickard Žsee Ref. w21x., we chose 68 data sequences Žheight vs. time readings. for individual spheres which were always within the leftmost triangular region Ži.e., at the original concentration.. Only data from the interior of the sequence were eligible for selection. That is, spheres that had just come
R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨
81
Thus, it is virtually impossible that the difference arose by chance. This establishes unequivocally that there are two distinct regions. Tory et al. w27x speculated that the velocities of spheres within a concentration gradient would be much less variable than those of spheres in a region of uniform density. The values of s 12 and s 22 are consistent with this conjecture. On the other hand, the lower value may be due solely to the lower concentration. A one-tailed test for the equality of the two means Žwhen the variances of the two populations are different. has the alternative hypothesis w28x Ha : < m 1 < ) < m 2 < with n s 87. Then texp s 1.675 ) t 0.05 Ž 87 . f 1.665.
Fig. 4. Concentration gradient and rise of the packed bed obtained from reanalyzing measurements by Tory and Pickard w21x.
from the dilute region, or were about to return to it, were judged to be transitional and not representative of spheres in the interior of the original dispersion. Similarly, spheres that were close to the packed bed were excluded. These measures also guarded against the possibility of small errors in estimating the positions of the boundaries between the regions. From the eligible sequences, we randomly selected velocities for consecutive intervals Že.g., 5th and 6th min.. We excluded the first 4 min to let the system evolve to a quasi-steady state w21x. Using the procedure outlined by Tory and Pickard w21x, we obtained the following maximum likelihood estimates of the three parameters for the denser region:
m 1 s 1.2227 = Ž y6.483 . = 10y4 mrs
Ž 20 .
Thus, the difference is significant. Although the mean velocity calculated by Tory and Pickard w21x was higher Žin absolute value. than the interface velocity u, there was almost a 10% probability that this difference could occur by chance. A one-tailed test Žusing the 68 pairs of data in our reanalysis. for equality has the alternative hypothesis Ha : m 1ru ) 1 with n s 67. Then texp s 2.457 ) t 0.01 Ž 60 . s 2.390.
Ž 21 .
Thus, the difference is highly significant. Nevertheless, the 95% confidence interval for m Ž0.02. includes the original Tory–Pickard value of y7.1 = 10y4 mrs Ždespite the contamination of their data by spheres in the lower part of the dilute region.. Using the value f bk Ž0.02. s y1.583 = 10y5 mrs and the slopes of the two upper lines of Fig. 4, we construct a flux plot shown in Fig. 5 which is similar to the flux plot of Fig. 3Žd. or to Fig. 7.19Ža. in Ref. w20x. The lines f bk Ž f . s y6.483 = 10y4f mrs and f bk Ž f . q 1.583 = 10y5 mrs s y8.8 = 10y4 Ž f y 0.02. mrs intersect at the point Ž0.00763, y4.95 = 10y6 mrs., which
s y7.927 = 10y4 mrs,
s 12 s 2.349 = 10y7 m2rs 2 , and b 1 s 0.008252rs.
Ž 17 .
Similarly, an analysis of 22 pairs above the discontinuity yields the following estimates for the upper zone:
m 2 s 1.0435 = Ž y6.483 . = 10y4 mrs s y6.765 = 10y4 mrs,
s 22 s 2.986 = 10y8 m2rs 2 , and b 2 s 0.01935rs.
Ž 18 .
The difference between the variances is the most striking. A one-tailed test for the equality of the two variances has the alternative hypothesis Ha : s 12 ) s 22 with n 1 s 67 and n 2 s 21. Then F s 7.87 4 F0.005 Ž 60,21 . s 2.84.
Ž 19 .
Fig. 5. Flux plot constructed from the slopes of the two upper lines of Fig. 4 Ždashed lines.. The black dots Ž Ø . correspond to the values given in Table 2.
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R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨
must lie above the flux plot because the two discontinuities are separated by a rarefaction wave Žsee Section 3, especially Fig. 3Žd... This value is shown as a small open circle Ž(.. The values of f bk Ž f . from Table 2 are shown as black dots Ž Ø .. However, these values of f are nominal in that they do not take into account the probable formation of a dilute upper region such as that which occurred in the experiment reported here. The explanation of Fig. 5 requires that we generalize Kynch’s theory. We will use the Markov model developed by Pickard and Tory w29x. In some circumstances, this model yields essentially the same results as Kynch’s w2x, as shown in Ref. w30x, but it provides explanations for additional phenomena. According to this model, the velocity distribution of the spheres in the interior of the dispersion evolves to a steady state. There is a great deal of experimental evidence that the mean of the steady-state distribution is greater than that of the initial. A ‘‘demixing’’ of the dispersion Žin which clusters form and settle rapidly. results in this increase w27,31x. In cylinders with Õery large diameters compared to those of the spheres, clusters form at very low concentrations. The mean velocity exceeds the Stokes velocity and continues to increase up to about f s 0.01. It then decreases, slowly at first and then rapidly at higher concentrations. As the velocities in a very dilute dispersion approach steady state, the spheres in the interior of the dispersion pull away from those at the top, producing a concentration gradient. Thus, the rate of fall of the interface is less than the mean velocity in the interior and the concentration just below the interface is less Žperhaps much less. than the concentration in the interior w27x. At the diameter ratio in the experiments of Tory and Pickard w21x, cluster settling was less extensive Žbut still very noticeable.. We assume that the value of m Ž0.005. for spheres is equal to the rate of fall of the interface for a dispersion with that initial concentration ŽFig. 5., that for m Ž0.01. lies slightly below Žalgebraically., and those for m Ž0.015. and m Ž0.02. lie well below their corresponding interface velocities. Following Pickard and Tory w30x, we generalize Kynch’s theory by setting Õs s m , the steady-state value of the mean velocity. Consequently, according to our analysis of this situation, the dispersion evolves to an MS-7. According to Fig. 5, the concentration at the top of the interface is about 0.0064 for an initial concentration of 0.02. Thus, what Tory and Pickard w21x recorded as the interface velocity is probably the velocity of spheres at f s 0.0064. The concentration at the bottom of the gradient can be calculated by material balance as f s 0.00857, but the rather large uncertainties for m 1 and m 2 imply that this value is very uncertain. If the marker spheres had been uniformly distributed over the entire height of the column, we could have estimated the mean concentration in the dilute phase by counting the number of marker spheres there. However, Tory Žcited in Tory et al. w27x. noted that fulfilling the
goals of their study created a bias for spheres in the upper region. Our subsequent analysis shows that this bias was specifically for marker spheres very near the top of the dispersion. Though the initial mixing process was judged to produce a uniform concentration w21x, it is likely that there was a gradient in a very small region at the top of the dispersion. Then these spheres were already in or near this concentration gradient and most of them remained in, or soon entered, the dilute zone since the velocity of its lower boundary exceeded Žin absolute value. both m 1 and m 2 . ŽA fortuitous result was that the surplus of marker spheres in the dilute region improved the precision of the values of the parameters of that zone.. A count of marker spheres yields an upper bound of somewhere between 0.013 and 0.016 for f . In the first case, borderline spheres are excluded and, in the second case, included. Fig. 5 indicates a concentration at the bottom of the upper zone of roughly 0.0095, which lies well within the region of the most probable values. It appears that Žas we move up the column. f changes rapidly from 0.02 to 0.0095, then gradually to 0.0064, and finally suddenly to 0. Of course, the parameter values Ž m 2 , s 22 and b 2 . for the dilute region are averages over the gradient. The situation is complicated by the fact that conditions in the gradient are constantly changing. As the gradient expands, spheres enter and leave it, and a steady state may not be reached there. However, the velocity of the interface remains constant w21x. In any event, the flux plot does indicate an MS-7, which is consistent with the observation of a distinct upper region and a readily discernible interface. Although Tory and Pickard w21x did not measure the velocities of individual spheres in dispersions that were initially more dilute, these velocities are known to be variable w23,31x. Thus, the values of m Ž f . will probably lie below Žalgebraically. the interface velocities as we have suggested in Fig. 5. Since the experiments of Tory and Pickard w21x were designed for another purpose, the concentration of their upper zone can be estimated only very roughly. It would be desirable to use the information from their experiment to design another to determine the concentration of the upper zone more precisely. According to Fig. 5, f 0 s 0.01 would produce an MS-6. These two modes of sedimentation ŽMS-6 and MS-7. can be expected when < m Ž f .< initially decreases and then increases as f increases from zero. This, in turn, occurs only when drD is small enough to permit cluster settling. As f continues to increase, < m Ž f .< decreases once again. This can result in a flux plot like that in Fig. 3Žc.. The suspensions studied by Oliver w23x, Verhoeven w24x, and Tory and Pickard w21x are unusual in being essentially monodisperse. Nevertheless, the diversity of individual velocities would be expected to blur the discontinuities in concentration w32,33x and change their propagation velocities w33,34x. These effects can be studied by stochastic simulation w35,36x or, equivalently, by incorporating a term for ‘self-induced hydrodynamic diffusion’ w33,37x into Eq.
R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨
Ž2. to make it parabolic. Given that the extended Kynch theory Ž Õs s m . closely approximates the observed behaviour, the abrupt changes in concentration would need to be handled carefully in a numerical scheme Žsee Section 7.. The suspensions used in most studies are at least slightly polydisperse w33,37x. At the very low concentrations at which rarefaction waves occur, interface spreading from polydispersity is also important w32,33x. Davis and Hassen w33x measured local values of f in the concentration gradient Žsee our Fig. 4. of a slightly polydisperse suspension with f 0 s 0.02 Žand other values., but their analysis is based on the usual assumption that m Ž0.02. ) u 0 Žalgebraically.. By eliminating this restriction and taking account of rarefaction waves, it might be possible to gain more insight into the behaviour of slightly polydisperse suspensions.
6. Limitations of Kynch’s theory Koglin w38x observed an expanding concentration gradient near the interface and measured concentration values within that gradient. His Fig. 1 suggests that the difference between the velocities of the upper and lower boundaries of the concentration gradient, plotted against the nominal initial concentration f 0 of the suspension, assumes a distinct maximum at a positive concentration value and vanishes when f 0 tends to zero or exceeds 0.05. The values presented in this figure suggest the flux plot shown in Fig. 6. The maximum width Žin terms of the concentration gradient. is obtained at f 0 s 0.007, which identifies this point with the inflection point a shown in our Fig. 6. Kynch’s theory predicts that an upper rarefaction wave is produced if the initial concentration does not exceed f LUU f 0.048. This value is obtained as the positive intersection of the tangent to the graph of f bk through the origin with that of the graph itself. For even larger initial concentration
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values, the slurry–supernate interface propagates as a sharp concentration discontinuity. For larger values Žthe measurements permit the construction of the flux function up to 0.5., the flux function is similar to Richardson and Zaki’s w39x. We point out that Koglin’s data and the corresponding flux plot provide evidence for upper rarefaction waves. The flux plot constructed in Fig. 5 belongs to Y Ž0. ) 0, for which seven modes of the family with f bk sedimentation are known. In contrast, Fig. 6 implies a different family of modes of sedimentation. Though the application of Kynch’s theory to this flux plot accounts qualitatively for the observed behaviour, the settling velocity of spheres in the upper region appears to be less than that predicted from the flux plot. ŽWe have not been able to construct a flux plot that is consistent with Koglin’s Fig. 2.. The isolated spheres that are left behind during cluster settling w27x approximate a lattice structure. It is known that such a spatial distribution leads to a decrease in m that is proportional to f 1r3 w22x. Thus, it is not surprising that Oliver’s data can be fitted with the equation
s Ž 0, fy . ru 0 s Ž 1 y k f 0 . Ž 1 y K f 01r3 . .
Ž 22 .
Using data from a wide variety of sources, he determined k s 2.15 and K s 0.75, but this equation oÕerestimates the magnitude of his own settling rates w23x. When drD is extremely small, as in Koglin’s experiments, cluster settling can occur at very small solids concentrations w27x, leaving behind spheres that are very widely spaced. Koglin’s Fig. 1 shows that spheres in the interior of a suspension where f s f 0 s 0.007 settle much faster than a single particle. When f 0 s 0.02, the mean, m Ž0.02., of the steady-state velocity distribution also exceeds the Stokes velocity and produces a rarefaction wave. With this initial concentration Žaccording to Koglin’s Fig. 1., the mean velocity of spheres in a region where f s 0.007 is less than m Ž0.02. even though m Ž0.007. ) m Ž0.02.. This anomaly can be understood by recognizing that the spatial distribution of particles in the interior differs from that in the rarefaction wave. Hence, the steady-state velocity distribution cannot be applied there. However, this problem can be treated stochastically w32x. Experimentally, the polydispersity of suspensions becomes more troublesome as f 0 in the rarefaction wave.
™
7. Application to the phenomenological sedimentationconsolidation model
Fig. 6. Approximate flux plot obtained from measurements by Koglin Žw38x, Fig. 1..
Most industrial slurries form a compressible sediment layer showing curved isoconcentration lines w40–42x and therefore cannot be regarded as an ideal suspension over the whole range of concentration values. For these mix-
R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨
84
To demonstrate an upper rarefaction wave in the theory of sedimentation with compression, we use the flux density function
°
y6.0=10y5f Ž 1y4f . 8
f
4
f bk Fig. 7. Flux density function with three inflection points given by Eq. Ž25..
tures, Kynch’s model is assumed to be valid until the flocs start to touch each other, that is, where the concentration reaches the critical value or the gel point fc . In the compressible sediment layer, the presence of effective solid stress requires the treatment by a generalized Darcy’s law. Since upper rarefaction waves occur within very dilute regions of hindered settling, they are also relevant to sedimentation with compression, which we show by a hypothetical numerical example. A unified treatment of both the sedimentation and the compression zones is possible by means of the degenerate hyperbolic–parabolic equation Žsee Refs. w43–46x. Ef q
E fbk Ž f .
Et
E sy
Ez
Ez
ž
f bk Ž f . seX Ž f . Ef D pg f
Ez
/
,
Ž 23 .
where D p ) 0 denotes the solid–fluid density difference, g the acceleration of gravity, and seŽ f . is the effective solid stress function, which is usually assumed to satisfy
seX Ž f . s
d se df
½
s0 )0
for for
f F fc , f ) fc .
Ž 24 .
Here fc denotes the critical concentration at which the solid flocs touch each other.
12
y9.6 f ž 1y Ž f . s~ 0.9 /
¢
y9.6=10y2f 4 1y
ž
mrs
f 0.9
for
0 F f - 0.25,
for
0.25F f F 0.9,
12
/
mrs
Ž 25 . which is plotted in Fig. 7. This flux density function was designed in such a way that the upper rarefaction wave becomes well visible. Of course, with more realistic flux density functions Žsuch as that drawn in Fig. 5., the width of the concentration gradient is smaller, but the effect remains the same. This flux density function is combined with an effective solid stress function of the common power law type
°0 ~
se Ž f . s
f
a1
¢
ž / fc
for
f F fc ,
for
f ) fc ,
a2
y1
Ž 26 .
where the parameters a 1 s 5 Pa, a 2 s 8 and fc s 0.22 were chosen. Fig. 8 shows the simulated sedimentationconsolidation behaviour of an initially homogeneous flocculated suspension of concentration f 0 s 0.105 using these constitutive functions. The numerical method used is a finite-difference operator splitting scheme Žsee Ref. w47x. where a particular conservative scheme ensures that the right entropy weak solution Žin this case, an upper rarefaction wave. is approximated wherever Eq. Ž23. is of hyperbolic type, that is, above the isoconcentration line corresponding to fc .
Fig. 8. Settling plot for sedimentation with compression including an upper rarefaction wave. The isoconcentration lines correspond to the annotated values.
R. Burger, E.M. Tory r Powder Technology 108 (2000) 74–87 ¨
8. Conclusions We have shown in this paper that the classification of Bustos and Concha w3x and Concha and Bustos w4x of sedimentation processes with flux density functions having two inflection points can be completed by the concept of upper rarefaction waves, which were at first a purely theoretical construct. However, the analysis of experimental data shows that this concept can explain the spreading of the slurry–supernate interface of dilute suspensions within Kynch’s theory. Since upper rarefaction waves occur in regions of small concentrations only, this finding is also relevant to sedimentation theories which reduce to the kinematical model for subcritical Žsmall. concentrations.
9. List of symbols Latin symbols a, b inflection points of f bk Žy. C parameter in the flux density function defined by Eq. Ž4. Žy. C1 , C2 symbols indicating contact discontinuities d sphere diameter Žm. D cylinder diameter Žm. f bk Kynch batch flux density function Žmrs. F test statistic variable Žy. F0.005 quantile of the F-distribution Žy. g acceleration of gravity Žmrs 2 . Ha alternative hypothesis Žy. L initial height Žm. NRe Reynolds number Žy. q volume-average velocity Žmrs. R1 symbol indicating a rarefaction wave S 1 , S 2 , S 3 symbols indicating shocks t time Žs. tU time at which the profiles in Figs. 2 and 3 are taken Žs. t 0.05 , t 0.01 quantiles of the t-distribution Žy. t1 time at which the first wave interaction occurs Žs. tc critical time Žs. texp test statistic variable Žy. T maximum time Žs. u interface velocity Žmrs. u0 Stokes velocity Žmrs. X u0 velocity of a single sphere settling along the axis of a cylinder Žmrs. Õ 0 , Õ1 parameters in the flux density function de fined by Eq. Ž4. Žmrs. Õs , Õf solidrfluid phase velocity Žmrs. z space variable Žheight. Žm. z1 sediment height at time t 1 Žm. zc final sediment height Žm.
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Greek symbols a 1, a 2 parameters in the effective solid stress function ŽPa, y. b1, b 2 values of the parameter b in the Markov model Ž1rs. r fluid density Žkgrm3 . rs solid density Žkgrm3 . Dp solid–fluid density difference Žkgrm3 . f volumetric solids concentration Žy. fq, fy approximate limits of f aboverbelow a discontinuity Žy. f0 initial concentration distribution Žy. f U0 concentration value constructed from f 0 , see Figs. 2 and 3 Žy. y fq constants of a Riemann problem Žy. 0 , f0 fc critical concentration Žy. fmax maximum volumetric solids concentration Žy. ft , f˜ t tangential points, see Fig. 3 Žy. ld coefficient of variation of sphere diameters Žy. m mean velocity of individual spheres Žmrs. md mean diameter of spheres Žm. m1, m 2 values of m in the Markov model Žmrs. n , n 1, n 2 sample sizes Žy. nf kinematic viscosity of fluid Žm2rs. s jump propagation velocity Žmrs. s 12 , s 22 values of the variance s 2 in the Markov model Žm2rs 2 . sd standard deviation of sphere diameters se effective solid stress function ŽPa.
Acknowledgements We acknowledge support by the Collaborative Research Programme ŽSonderforschungsbereich. 404 at the University of Stuttgart.
Appendix A From the many studies of interface velocities, we have chosen those of Oliver w23x, Verhoeven w24x, and Di Felice w25x. Unlike many other studies in which the spheres were merely ‘‘closely sized’’, they used particles that were essentially monodisperse. Verhoeven settled Lucite Žmethyl methacrylate polymer. spheres in Polyglycol P4000 Ža highly viscous polymer.. He stated that the mean sphere diameter was 0.1175 in. However, his 55 recorded values indicate a mean of 0.1172 and a variance of 1.988 = 10y3 in. His distribution is slightly skewed toward smaller values with a minimum of 0.111 in. and a maximum of 0.120. However, the ‘‘left tail’’ consists of only five spheres. Oliver states that 85% of his Kallodoc Žmethyl methacryl-
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ate polymer. spheres were within 3% of the mean diameter. Thus Pr Ž < d y md
