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Non-colloidal sedimentation compared with Kynch theory
POWD TIECHNOWGY ELSEVIER
Powder Technology 92 (1997) 81-87
Letter
Non-colloidal sedimentation compared with Kynch theory D. Chang
a..,
T. Lee ", Y. Jang b, M. Kim b, S. Lee b
*Department of Chemical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Kusung-dong. Taejon 305-701, South Korea b Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Kusung-dong, Taejon 305-701, South Korea Received 9 October 1996
Abstract
Instantaneous concentration profiles of the batch sedimentation of non-colloidal hard spheres were measured for various initial suspension concentrations from 0.04 to 0.55 using the magnetic resonance imaging (MPd) technique. Measured profiles commonly had two distinct interfaces, the upper one between the clarified fluid and the settling suspension, and the lower one between the settling suspension and the sediment. The upper interface was found to keep spreading due to polydispersity,especially for suspensions with low initial concentration. It was observed that the lower interface pattern is highly dependent on the initial concentration. If the pattern is considered to be spread due to the broadening effects of hydrodynamicdiffusion and polydispersity,its overall behavior seems to be consistent with Kynch's prediction. For all initial concentrations the sediment was incompressible with packingconcentration 0.60_+0.01. Three transition concentrations separating four regions of different profile patterns were determined by fitting the experimental results to the Richardson-Zaki formula. The validity of the first (0.16) and third (0.46) transition concentrations was cross-checked by various criteria based on the analysis of the experimental results. The second transition concentration (0.33) determined from the flux curve analysis could not be confirmed experimentally. Keywords: Kynch theory; Sedimentation; Hydrodynamic diffusion; Polydispersity;Magnetic resonance imaging; Tomography
1. Introduction
During a batch sedimentation process, the column consists of three layers: upper clarified fluid, settling suspension, and sediment. Kynch [ 1] suggested a mathematical description of t~e batch sedimentation of an initially homogeneous susi~nsioa ¢~asid~,~ng only the hindered settling effect. This mathematic;a! description, referred to as Kynch theory, is well documented by Rhee et al. [2] and Russel et al. [3] at the extreme of the negligible diffusion effect. The theory states that in sedimentation of an in~lially homogeneous suspension a concentration shock or discontinuity should develop at the upper interface between the clarified layer and the suspension, and a continuous profile as well as a discontinuity at the lower interface between the suspension and the sediment. According to the theory, the structure of the lower interface is governed by the initial concentration and flux curve. Recently, it has been found that the upper interface attains a smooth pattern due to the polydispersity stemming from the distribution in physical properties of particles and hydrodynamic diffusion from fluctuation of the settling velocity [49]. The effect of hydrodynamic diffusion for a monodispe~se * Corresponding author. Tel.: + 82 42-869 3966; fax: + 82 42-869 3910; e-mail: [email protected]. 0032-5910/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved Pil S0032-5910 (97) 03211-7
suspension can be represented as a diffusion process in the mass conservation equation of particles [ 3 ]. The equation was solved for the colloidal suspension numerically [ 10] and asymptotically [ 11 ]. Auzerais et al. [ 12], observing the concentration profile of an entire column of colloidal suspension, found that the pattern of the profile resembles that of the Kynch solution if the particles do not tend to flocculate. In order to take into account ~ e polydispersity effect, the dependence of the hindered settling velocity and hydrodynamic diffusion on the particle size is essential. For a dilute polydisperse suspension, Batchelor [ 13] derived expressions for the sedimentation coefficients. Masliyah [ 14], Zimmels [ 15], and AI-Naafa and Selim [ 16] suggested hindered settling expressions for polydisperse concentrated suspensions. Based on these expressions, Chang et al. [ 17] proposed a polydisperse model for the upper interface, and showed that the polydispersity effect estimated from experimental observation is quantitatively consistent with the results of numerical simulation of the model. The microstructure and concentration of the sediment are dependent on particle size, particle shape, degree of flocculation, accumulation time, distance from the top of the sediment, weight of solids, pH of the fluids, and so forth [ 182! ]. For a compressible sediment, additional consideration should be paid to explaining the transient compression of the
82
D. Chang et al. / Powder Technology 92 (1997) 81-87
sediment due to stress caused by accumulating particles. TypicaUy, momentum balance equations with Darcy's law are developed to describe the compression [22-26]. Relatively large particles of narrow size range, however, tend to form incompressible sediments with packing concentration around 0.55--0.64 [ 27,281. The dynamics of the rising interface above the sediment plays an important role in many practical applications. As Kynch indicated, if the flux curve is convex at the concentrated region, and the packing concentration is relatively high, the concentration profile above the sediment should be continuous or stretched. Dixon et al. [29], however, showed that the inertia cannot be ignored in the simulation, and that incompressible sediments do not follow the Kynch type sedimentation. Bacri et al. [ 30] measured the falling velocity of the upper interface and the rising velocity of the lower interface by an acoustic method. They concluded that there is no mass accumulation between the two interfaces, and that the result is consistent with the fact that the flux curve is nearly straight at the concentrated region rather than convex. Hoyos etal. [31 ] showed in an experimental study on bidisperse suspensions that mutual hindrance results in a velocity reduction of large particles and in an enhancement of small ones, as compared with a monodisperse suspension. The concern of the present research is to investigate the batch sedimentation process of non-colloidal particles, especially to verify whether Kynch's prediction is plausible or not. In order to explore the validity of Kynch theory under the influence of the broadening effects of hydrodynamic diffusion and polydispersity, it is essential to measure precisely the instantaneous concentration profile of the entire sedimentation column without any significant disturbances by measurement tools. Since suspensions are usually opaque, even at relatively low concentrations, the concentration range measurable by optical methods is severely limited [ 32]. Recently, it has become possible to measure the concentration profiles of these opaque suspensions with the aid of non-invasive imaging techniques. The X-ray CAT scan [ 33 ], X-ray computerized tomography [ 12 ], and magnetic resonance imaging (MRI) [8] have been utilized to investigate the transient behavior of the batch sedimentation column. In this work, the concentration profiles are measured using MRI for various initial concentrations.
where c = c(z, t) is the volume concentration of particles at position z along the direction of settling and time t, and q is the volume flux of particles. Kynch considered the purely convective sedimentation
q:f
V(c)c
for C Cm
1.o
(2)
where v(c) is the hindered velocity of a settling particle at local concentration c, and Cmis the packing concentration of the sediment. Kynch theory states that the concentration profile in the rising lower interfacial region shows different patterns depending on Co and the hindered settling function f(c), which is the ratio of the settling velocity v(c) to the Stokes velocity to. Four different patterns exist when the hindered settling velocity obeys the Richardson-Zaki fommla [34]
f(c)=(l-c)"
(3)
with n : 5, and the packing concentration of the sediment, Cm, is 0.6. These values are very close to those found experimentally as discussed later. The flux curve, plotted in Fig. 1 as a function of concentration, has an inflection point at c2. A solid straight line originating from (Cm, O) is tangential to the curve at c3 and crosses the flux curve at c:. These three transition concentrations c~, c2, and c3 define the four consecutive intervals of concentration, named regions I-IV in the figure. For a given flux curve, the shape of the lower interface depends uniquely on Co for an initially uniform suspension, as shown in Fig. 2. The initial concentrations Co of Fig. 2(a)-(d) correspond to regions I-IV, respectively. According to Kynch's model, the interface velocity for each region can be obtained from the shock structure analysis [35]. For all regions, the settling velocity of the upper interface, us(co), is equal to the settling velocity in the free settling suspension layer:
us(co) - -Uof(Co)
(4)
•Here, the upward direction is taken positive. At the lower rising interface of regions I and IV a single concentration i R e g i o n li
II
/If
IV
g
i Q
2. Sedimentation model For further consistent notation, Kynch theory and its modifications are briefly stated. For the suspension of monodisperse particles, the concentration profile of the batch settling column can be described by the following one-dimensional mass conservation equation:
ac.
01
(1)
02
o*
cs
cm
c
Fig. I. Flux curve and definition of regions for the Richardson-Zaki formula with n = 5 and Cm~--- 0.6.
D. Chang et al. / Powder Technology 92 (1997) 81-87
where D(c) is the concentration-dependent diffusivity. The solution to Eqs. (1) and (8) with no flux conditions at both the top and bottom of the column has been numerically [ 10] and asymptotically [ 11 ] solved for colloidal sedimentation. The main effect of the diffusion term is to make the interfaces smooth, although the detailed shape of the profile may vary with the concentration dependence of the diffusivity. Roughly speaking, the simulated profiles keep the characteristics of the Kynch solution if Pe = 104 or larger (note that Pe is scaled with the suspension height and not with the particle size). If Pe = 100 or smaller, the simulated profiles no longer carry the traces of discontinuities [ 10,12].
i
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-(o)
-(b)
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!
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i
-(d)
o.6
83
J
0.4.
3. Experimental
0.2 1
I
0.0 0.0
0.5
1.0
0.0
0.5
1.0
~/L Fig. 2. Calculated concentration profiles using the Richardson-Zaki formula with n = 5 . 0 and Cm=0.6: (a) Co--0.08 and t=O.5LIvo; (b) co-0.20 and tffi 1.5LI vo; (c) co=0.36 and t = l .SLI vo; (d) Co=0.51 and t= 2LI vo.
discontinuity, denoted by Ja in Fig. 2, develops and moves with Ua(Co)"
UdCo)=V(Co)Co
(5)
Cm-- Co
If Co belongs to region III, a continuous concentration profile Jb for Co< c < c3 develops and a concentration shock Ja, jumping from c3 to Cm. The rising velocity of the isoconcentration slice in Jb is given by ul,( c ) =
d(v(c)c) dc
(6)
and the discontinuity J,, moves with ua(c3). The lower interface of region II type is composed of the discontinuity jumping from Co to c*, followed by the continuous increase from c* to c3, followed by the discontinuity jumping from ca to Cm, denoted respecti'¢ely by Jc, Jb, and J~ in Fig. 2(b). The discontinuity Jc moves upward with uc(co), given by
v(c*)c*-v(Co)Co uc(co) -
(7)
c* - Co
where c* is the tangential point concentration of a line which crosses the flux curve at Co in region II. An example of such a line is shown as the dashed line in Fig. 1 for Co- 0.22. The rising velocities of J~ and Jb in region II are also given by u~(ca) and Ub(C), respectively, for c* < c < ca. The complete description of the theory is found elsewhere [ 2,3]. The effect of diffusion can be included in the conservation equation by expressing the effect as a diffusion process and replacing the flux term in Eq. (2) by
0c
q - v ( c ) c - D(c) OZ
(8)
In the experiment, suspensions are prepared by mixic~g a lubricant oil and glass beads. The glass beads are class V precision grade microbeads manufactured by Cataphote. The size distribution of the particles is characterized by optical microscope pictures (magnification × 100). The average radius and its standard deviation are 69.4 and 7.4 ~m, respectively. Hollow particles are not observed in the photographs, and the density of the particles is 2.46 g/cm 3. The hydrocarbon composite oil used for the liquid phase material is Visco 2000 manufactured by Kyungin Energy. The specific gravity and viscosity of this oil are 0.89 and 240 cP, respectively, at 24°C. An acrylic column of inner diameter 5.2 cm is used as the container of the glass beads and the hydrocarbon oil mixture. The bulk concentration of the suspensions is increased from 0.04 to 0.55 by intervals of 0.02-0.04, and about 20 suspensions with different initial concentration are prepared separately. The suspension height L is so long that the completion of sedimentation takes one hour or so. Magnetic resonance imaging (MRI), more often called nuclear magnetic resonance computerized tomography (NMR-CT), is used to measure the concentration profiles over the entire container. This is a kind of non-destructive computerized tomography, recently introduced to detect the physical properties of fluids, together with X-ray [ 33] and acoustic methods [30]. Quantum magnetic properties of atoms, such as the nucleus density, spin-spin relaxation time, spin-lattice relaxation time, chemical shift, degree of magnetization, and the rate of change of the nucleus density, can be used to detect the temperature, velocity, and density distributions. Since a pulse of electromagnetic waves of radio frequency range is input to the suspension, the sedimentation container is free from external disturbance. Since the attainability of a measurement depends on the magnetic prope~ies of the object atoms, it does not matter whether the object is optically transparent or not. The proton signal of the oil is detected in this experiment, so the signal-to-noise ratio increases with bulk concentration. The spatial resolution is set at a value between 193 and 386 I~m depending on the sedimentation velocity. Detailed theoretical principles are explained elsewhere [ 8].
84
D, Chang et al./ Powder Technology 92 (1997) 81-87 0.04 .
After the container is repeatedly inverted up and down to attain a homogeneous suspension, it is placed vertically in the equipment. Then, the MRI equipment starts to detect the concentration profiles over the entire container with a fixed time interval of 20 to 30 s. The acquisition time for a profile, usually less than 10 s, is taken to be small enough for the particles in a suspension not to exceed the spatial resolution of the MRI equipment. Since the completion of sedimentation takes one hour or so, 100-200 profiles are collected from each batch of sediment.
,
,
,
,
,
0.3
0.4
0.5
0 0.03
0.02
0.01
0.00 0.0
4. Results and discussion
0.1
0.2
0.6
O
A sample contour map depicted on the t-z plaice (Fig. 3) shows how the concentration profile evolves in time. This map, constructed fiom 168 concentration profiles measured every 25 s for Co= 0.34, plots the concentration values ranging from 0 to 0.60 with a concentration step of 0.04. Though the lower interface keeps spreading with time, the bottom of the lower interface, considered as the discontinuity J,, rises with constant velocity after an early stage until it closely approaches the upper interface which also falls with constant velocity. From this contour map, the falling velocity of the upper interface, u,, and the rising velocity of the discontinuity, u,, can be precisely determined. The settling velocity of the upper interface, us, is estimated from the plot of the position of the interface versus time. It is found that the experimental data align so well that they consist of a nearly straight line, whose slope corresponds to u,. This is also the case for the lower interface velocity u,. The falling velocity of the upper interface, u, (Co), is equal to u (Co), and the exponent of the Richardson-Zaki formula as well as the Stokes velocity are estimated from the linear regression of a plot of In[ u,(Co) ] versus in( 1 - Co), The experimental results show good agreement with the formula, with a correlation coefficient of 0.98 and n = 5.06. The value of n - 5 is taken in this study for further discussion. 20
Fig. 4. R u x curve obtained from the falling velocities of the upper interfaces. The solid line is the theoretical curve based on the Richardson-Zaki formula with n = 5.
Fig. 4 presents the experimental results. It is noted that the error bars are smaller than the physical size of the objects in the figure. The solid curve represents the flux curve based on f ( c ) = ( 1 - c) s. Small scattering of the data points near the curve seems to be caused by day-to-day temperature variations. It is found that the sediment, precisely the part lower than ~, is incompressible, and that its packing concentration Cm remains constant at 0.60+0.01 regardless of Co. With Cm= 0.60 and the Richardson-Zaki formula with n = 5, it can be calculated that cn = 0.16, c2 = 0.33, and c3 = 0.46. The validity of ct and c3 can be verified by examining Ua, the rising velocity of,/,. The rising velocity u, was determined from the contour maps for various co, and the result is depicted in Fig. 5. The rising velocities calculated from Eqs. (5), (6), and (7) with the Richardson-Zaki formula and n = 5 are also represented graphically in the same figure. The solid line represents the rising velocity u, which has a constant value ua(cz) in regions II and III. The dashed line is the rising velocity uc of the discontinuity Jc which is found only in the region II type profile• The dotted line is the upper bound of 0.12
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0.04 0.02 0.00 0.0
I
I
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0•1
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D. Chang et al. / Powder Technology 92 (1997) 81-87
the rising velocity Ub in the region III type profile. The rising velocity of the continuous profile Jb is bounded by the dashed and solid lines in region II, and by the dotted and solid lines in region III. The measured rising velocity of the sediment surface, ua, agrees well with the theoretical curve. This supports the existence of transition concentrations c] = 0.16 and c3 = 0.46 estimated from the flux curve analysis. Although the fundamental structure of the experimental concentration profile of non-colloidal particles can be analyzed basically by Kynch theory, the concentration profiles of the lower interface are different from those in Fig. 2 due to hydrodynamic diffusion and polydispersity. The concentration profiles of an entire suspension column at various instances are shown in Fig. 6 ( a ) - ( f ) for Co= 0.10, 0.15, 0.22, 0.34, 0.45, and 0.55, respectively. The flux cutve analysis indicates that the profiles (a) and (b) are region I type, (c) is region II type, (d) and (e) are region III type, and (f) is region IV type. The profiles do not show sharp concentration changes in either the upper or lower interfaces because of the broadening effects of diffusion and polydispersity. The thickness of the upper interface decreases with increasing Co as shown in the figure. The higher the bulk concentration, the larger the velocity difference across the upper interface, i.e. the thinner the upper interface. Consequently, I
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Fig. 6. Measured concentration profiles at various Co: (a) Co=0.10 and measured at t= 330, 930, 1515, 2122, and 2497 s; (b) Co=0.15 and measured at t= 319, 919, 1519,2119, and 3199 s, (c) co= 0.22 and measuredat t= 353,953, 1553,2334, and 4309 s; (d) Co= 0.34 and measuredat t= 320, 920, 1520, 2420, and 4249 s; (e) co=0.45 and measured at t=311,881, 1456, 2026, and 3706 s; (f) Co=0.55 and measured at t=339, 639, 939, 1539, and 2739 s.
85
Co can be considered as the rough measure of the self-sharpening effect at the upper interface.The velocity fluctuation relative to the mean settling velocity remains constant or decreases with increasing concentration [6,7]. Since the thickness of the upper interface in a monodispcrse suspension is determined from a balance of the hindered settling and the velocity fluctuation, it decreases with the initial concentration. The lower interface of the concentration profile for Co=0.10 exhibits a sharp change compared with the other profilesin Fig. 6. This profileis region I type, but the discontinuityJa becomes smooth due to the broadening effects.The magnitude of era-Co is a rough measure of the self-sharpening effect for the lower interface in region I and IV type profilesas Co is for the upper interface.This agrees with the observation that the lower interfaceof profile (b) is broader than that of (a). The profilefor Co = 0.55 shown in Fig. 6(f) is region IV type. The magnified window in Fig. 6(f) shows that the concentration profileof the lower interfaceis almost symmetric sigmoidal in shape, which is quite similar to that in Fig. 6(a). This implies that the velocity fluctuationis not negligible near the packing concentration. Compared with the profilesin Fig. 6(a) and (f), the profilesin (c) and (d) have longer tailsstretchedto the suspension region. In the absence of the velocity fluctuationeffect, the difference between region Ill and IV type profilesis that the region Ill type profilehas a continuous Jb profile at the lower interface.In the actual profilefor Co = 0.34, the continuous part is more stretchedby the velocity fluctuationeffect. Since a small stretched tailJb is also observed in the profile for Co = 0.45 (Fig. 6(e)), it is concluded that the transition point between regions Illand IV exists at Co and is a littlebit larger than 0.45. This is consistent with the previous result Ca = 0.46 obtained from the flux curve analysis in Fig. 4 and the risingvelocity curve analysis in Fig. 5. The transition concentration c3 can be identified more clearly from the evolution of region III type concentration profiles. In the range of Ca < c < c,,, the effects of hindered settling and :'elocityfluctuationcounterbalance each other, so that a steady shape is developed. But, in the range of Co < c < c3, b~)thof them contribute to the interfacebroadening.These two differenteffectsof the hindered settlingvelocity on the transientbehavior of the concentration profilecan be depicted visually by the concentration profile in the Lagrangian coordinate system. In Fig. 7, the concentration profiles c(z+ Vt, t) for Co=0.34 are depicted as a function of z+ Vt for t= I020, 1520, 2020, and 2549 s. The dashed line represents the transition concentration Ca =0.46. The velocity V of the moving coordinate system is chosen as Ua=0.00115 cm/s, obtained from Fig. 5. Fig. 7 shows that the profile above the dashed line is steady, whereas the profile below the line keeps spreading. The significance of this figure is that the transition concentration c3 can be identified from the transient behavior of the concentration profile for a single Co in region III.
D. Chang et aL / Powder Technology 92 (1997) 81-87
86 '
'
'
l
06
0.5 c
0.4
0.3
L
z
I
z
8
9
10
11
12
z+V~o orn Fig. 7. Lagrangian description of the transient behavior of the concentration profiles for Co--0.34 measured at 1020, 1520, 2020, and 2549 s, denoted respectively by hollow circles, filled circles, hollow triangles, and filled triangles. The dashed line represents c3 calculated from the Richardson7..aki formula with n = 5.
Fig. 6(c) shows the concentration profile for Co--0.22 which is presumed to be region II type. Based upon the flux curve analysis, the range of region H is 0.16 < Co<0.33, and region II type profiles are expected to show some indications of the two discontinuities la and J¢. But the actual profile (c) does not show any trace of the discontinuity Jc, and it is similar to the region III type profile in (d). Actually, all the concentration profiles for Co in regions II and III look similar to each other, and it is almost impossible to find the transition concentration c2 from the difference in their shapes. Three reasons for this are plausible: one due to a non-convex flux curve at the intermediate concentration range, another to polydispersity, and the third to diffusion. If the flux curve is non-convex and does not have the inflection point c2 in the intermediate concentration range (i.e. near c - 0 . 3 ) , region II does not exist any more. This interpretation is consistent with the results ofBacriet al. [ 30]. For the non-convex flux curve, the line originating from (Cm, 0), being tangential to the curve, nearly overlaps the curve itself. In this case, the existence of region III is dependent on Cm, that is, it would exist for large Cm, but does not for small Cm. As is apparent in Fig. 6, the packing concentration Cm is large enough for region Ill to be present over a wide range of concentration. It should be noted that the measurement of the velocity of the rising sediment, as done in the study by Bacri et al. [ 30], may not be able to verify whether region III exists or not. When the lower interface becomes spread, its rising velocity would not be determined precisely unless the interface were divided into two parts, as in the present study. Even though the flux curve is convex, the polydispersity effect causes the discontinuity Jc to disappear. It was found in previous experimental investigations on gradient diffusion that for the upper interface the polydispersity effect is comparable to the diffusion effect [5,9,17]. Additionally, the dependence of hindered settling on concentration is similar
to that of diffusivity [4,7]. The self-sharpening measure for Jc is the concentration difference across Jc, or c*-Co. In consequence, provided the polydispersity effect is considerable in the upper interface with concentration jump Co, it is plausible that the effect is also significant if c* -Co is comparable to Co. That is, it is possible that in a polydisperse suspension Jc would be erased by polydispersity, and that in a monodisperse suspension J~ would be observable with the initial concentration of region II. The third reason for region II type or the discontinuity J~ to be unobservable can be attributed to diffusion overriding the self-sharpening effect of hindered settling. The concentration profiles can be simulated numerically if the hydrodynamic diffusivity is expressed as a function of concentration. • In previous experimental investigations for gradient and selfdiffusion [4-9], it was found that diffusivity scaled with particle size, and the settling velocity remains in the same order over a wide range of concentration. This indicates that the dependence of diffusivity on concentration is very similar to that of settling velocity, as expressed by the equation
D(c) --Dof(c)
(9)
This expression is similar to the diffusivity used to describe the suspension of colloidal particles [ lO,11 ] except for the fact that the compressibility factor is set to unity since noncolloidal particles are used in this work. Though the diffusivity expression does not completely support the experimental data at relatively low and high concentrations, it can qualitatively describe the decreasing tendency of velocity fluctuation in region II. For the diffusivity by Eq. (9), cm=O.60, and the hindered settling functionf(c) given by the Richardson-Zaki formula with n = 5, a continuous solution to Eqs. ( l ) and (8) exists if the discontinuity in 0c/0zjust above the sediment is admitted. The Peclet number for the present system, which is defined by V(co)L/Do (note that L is the suspension height), is about 100 [ 9]. The numerically calculated concentration profiles for Pe= 100 (solid line) and 1000 (dashed line) are depicted for Co- 0.22 with the corresponding experimental data in Fig. 8. The figure clearly shows that the calculated profile for Pe = 100 does not show any indication of the discontinuity J~, whereas the profile for Pe = 1000 still keeps a trace of the two discontinuities. The simulated profile for Pe = 100 agrees qualitatively with the experimental profile, but the discrepancy between them indicates that the real diffusivity is different from the simple assumption in Eq. (9). There is another qualitative disagreement between the experimental data and the present simulated profile: the concentration gradient of the simulated profile is not continuous at c - Cmbut the experimental data looks smooth everywhere° The profile simulated for the sedimentation of the colloidal suspension is smooth since the compressibility factor diverges at C--Cm [ 11,12]. The most probable reasons for this disagreement are the thermodynamic effect and the wall effect. In the colloidal suspension, the thermodynamic effect is very important even at the intermediate concentration, say
D. Chang et al. / Powder Technology 92 (1997) 81-87 It
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;3, c m Fig. 8. Experimental (dots) and simulated concentration profiles for Co= 0.22. The solid and dashed lines represent the simulated curves for Pe = 100 and Pe = 1000, respectively.
c = 0.3 or more when Pe = 100. This effect, however, i.~ negligible for the non-colloidal system in the present study even when c is close to Cm. Fig. 6 supports this fact since the sediment of c = Cmforms immediately after the sedimentation begins. The discontinuity of aclaz at c = Cmin the sirnulated profile is caused by the discontinuity in the flux curve at c-- Cm. In order to reflect the wall effect of the sediment, the flux curve in Fig. 4 should be modified so that it decreases faster near Cmand vanishes at c - Cm.
5. Conclusions The batch sedimentation process for monodisperse noncolloidal particles is governed by the hindered settling velocity, polydispersity, and hydrodynamic diffusion. As the interfaces do not form discontinuities or shocks due to the hydrodynamic diffusion and polydispersity, they keep the characteristics of Kynch sedimentation. The concentration profiles exhibit different patterns of transient behavior depending on the initial concentration of the suspension. The transition concentrations were determined from the flux curve and cross-checked by three other analyses: the rising velocity of the lower interface, direct comparison of concentration profiles with different initial concentrations, and the evolution of the concentration profile in the lower interfacial region. The distinction of region II from region III is not clear from the pattern of the concentration profiles. Plausible reasons for this are a non-convex flux curve in the intermediate region, the polydispersity effect and/or the diffusion effect.
[ 1] G.J. Kynch, Trans. Faraday Soc.. 48 (1952) 166. [2] H.-K. Rhee, R. Aris and N.R. Amundson, First Order Partial Differential Equations: Volume !. Theory and Applications of Single Equations, Prentice-Hall, Englewood Cliffs, NI, 1987. [ 3 ] W.B. Russei, D.A. Saville and W.R. Schowalter, ColloidalDispersion, Cambridge University Press, London, 1989. [4] J.M. Ham and G.M. Homsy, Int. J. Multiphase Flow, 14 (1988) 533. [5] R.H. Davis and M.A. Hassen, J. Fluid Mech., 196 (1988) 107. [6] J.-Z. Xue, E. Herbolzheimer, M.A. Rutgers, W.B. Russel and P.M. Chaikin, Phys. Rev. Lett., 69 (1992) 1715. [7] H. Nicolai, B. Herzhaft, E. Hinch, L. Oger and E. Guazzelli, Phys. Fluids, 7 (1995) 12. [8] S. Lee, Y. tang, C. Choi and T. Lee, Phys. FluidsA, 12 (1992) 2601. [9] D. Chang, T. Lee, S. Lee and Y. Jang, in J.S. Lee, S.H. Chung and K.H. Kim (eds.), Proc. Sixth Int. Syrup. Transport Phenomena, Seoul, South Korea, 1993, Vol. 1, p. 585. [ 10] F.M. Auzemis, R. Jackson and W.B. Russel, J. Fluid Mech., 195 (1988) 437. [ 11 ] K.E. Davis and W.B. Russel, Phys. Fluids A, 1 (1989) 82. [ 12] F.M. Auzerais, R. Jackson, W.B. Russel and W.F. Murphy, J. Fluid Mech., 221 (1990) 613. [13] G.K. Batchelor, J. Fluid Mech.. 119 (1982) 379. [ 14] J.H. Masliyah, Chem. Eng. Sci., 34 (1979) ! 166. [ 15] Y. Zimmels, Powder Technol., 29 (1983) 669. [ 16] M.A. AI-Naafa and S. Selim, AIChEJ., 38 (1992) 1618. [ 17] D. Chang, T. Lee, S. Lee and Y. Jang, Int. J. Multiphase Flow, (1996) submitted. [ 18] A.S. Michaeis and J.C. Bolger, Ind. Eng. Chem. Fundam., I (1962) 24. [ 19] E.M. Tory and P.T. Shannon, Ind. Eng. Chem. Fundam., 4 (1965) 194. [20] I.E. Melton and B. Rand, J. Colloidlnterface Sci., 60 (1977) 331. [21 ] F.M. Tiller and Z. Khatib, J. Colloid Interface Sci., 100 (1984) 55. [22] B. Fitch, AIChEJ., 29 (1983) 940. [23] M.C. Bustos and F. Concha, AIChEJ., 34 (1988) 859. [24] R. Font, AIChEJ., 34 (1988) 229. [ 25 ] KJ. Scott, Ind. Eng. Chem. Fundam., 5 (1966) ! 09. [26] K. Stamataski and C. Tien, Powder Technoi., 72 (1992) 227. [27] R.F. Benenati and C.B. Brosilow, AIChEJ., 8 (1966) 359. [28] P.T. Shannon, E. Stroupe and E.M. Tory, Ind. Eng. Chem. Fundam., 2 (1963) 203. [29] D.C. Dixon, P. Sourer and J.E. Buchanan, Chem. Eng. Sci., 31 (1976) 737. [30] J.C. Bacri, C. Frenois, M. Hoyos, R. Perzynski, N. Pakotomalala and D. Salin, Europhys. Lett., 2 (1986) 123. [31 ] M. Hoyos, J.C. Bacri, J. Martin and D. Salin, Phys. Fluids (1995) submitted. [32] J. Abbott, L.A. Mondy, A.L. Graham and H. Brenner, in M.C. Roco (ed.), Particulate Two-Phase Flow, Butterworth-Heinemann,Boston, MA, 1993, p. 3. [331 P. Somasundran, Y.B. Huang and C.C. Gryte, Powder Technol., 53 (1987) 73. [341 J.F. Richaxdson and W.N. Zaki, Trans. Inst. Chem. Eng., 32 (1954) 35. [351 G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
Powder Technology 92 (1997) 81-87
Letter
Non-colloidal sedimentation compared with Kynch theory D. Chang
a..,
T. Lee ", Y. Jang b, M. Kim b, S. Lee b
*Department of Chemical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Kusung-dong. Taejon 305-701, South Korea b Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Kusung-dong, Taejon 305-701, South Korea Received 9 October 1996
Abstract
Instantaneous concentration profiles of the batch sedimentation of non-colloidal hard spheres were measured for various initial suspension concentrations from 0.04 to 0.55 using the magnetic resonance imaging (MPd) technique. Measured profiles commonly had two distinct interfaces, the upper one between the clarified fluid and the settling suspension, and the lower one between the settling suspension and the sediment. The upper interface was found to keep spreading due to polydispersity,especially for suspensions with low initial concentration. It was observed that the lower interface pattern is highly dependent on the initial concentration. If the pattern is considered to be spread due to the broadening effects of hydrodynamicdiffusion and polydispersity,its overall behavior seems to be consistent with Kynch's prediction. For all initial concentrations the sediment was incompressible with packingconcentration 0.60_+0.01. Three transition concentrations separating four regions of different profile patterns were determined by fitting the experimental results to the Richardson-Zaki formula. The validity of the first (0.16) and third (0.46) transition concentrations was cross-checked by various criteria based on the analysis of the experimental results. The second transition concentration (0.33) determined from the flux curve analysis could not be confirmed experimentally. Keywords: Kynch theory; Sedimentation; Hydrodynamic diffusion; Polydispersity;Magnetic resonance imaging; Tomography
1. Introduction
During a batch sedimentation process, the column consists of three layers: upper clarified fluid, settling suspension, and sediment. Kynch [ 1] suggested a mathematical description of t~e batch sedimentation of an initially homogeneous susi~nsioa ¢~asid~,~ng only the hindered settling effect. This mathematic;a! description, referred to as Kynch theory, is well documented by Rhee et al. [2] and Russel et al. [3] at the extreme of the negligible diffusion effect. The theory states that in sedimentation of an in~lially homogeneous suspension a concentration shock or discontinuity should develop at the upper interface between the clarified layer and the suspension, and a continuous profile as well as a discontinuity at the lower interface between the suspension and the sediment. According to the theory, the structure of the lower interface is governed by the initial concentration and flux curve. Recently, it has been found that the upper interface attains a smooth pattern due to the polydispersity stemming from the distribution in physical properties of particles and hydrodynamic diffusion from fluctuation of the settling velocity [49]. The effect of hydrodynamic diffusion for a monodispe~se * Corresponding author. Tel.: + 82 42-869 3966; fax: + 82 42-869 3910; e-mail: [email protected]. 0032-5910/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved Pil S0032-5910 (97) 03211-7
suspension can be represented as a diffusion process in the mass conservation equation of particles [ 3 ]. The equation was solved for the colloidal suspension numerically [ 10] and asymptotically [ 11 ]. Auzerais et al. [ 12], observing the concentration profile of an entire column of colloidal suspension, found that the pattern of the profile resembles that of the Kynch solution if the particles do not tend to flocculate. In order to take into account ~ e polydispersity effect, the dependence of the hindered settling velocity and hydrodynamic diffusion on the particle size is essential. For a dilute polydisperse suspension, Batchelor [ 13] derived expressions for the sedimentation coefficients. Masliyah [ 14], Zimmels [ 15], and AI-Naafa and Selim [ 16] suggested hindered settling expressions for polydisperse concentrated suspensions. Based on these expressions, Chang et al. [ 17] proposed a polydisperse model for the upper interface, and showed that the polydispersity effect estimated from experimental observation is quantitatively consistent with the results of numerical simulation of the model. The microstructure and concentration of the sediment are dependent on particle size, particle shape, degree of flocculation, accumulation time, distance from the top of the sediment, weight of solids, pH of the fluids, and so forth [ 182! ]. For a compressible sediment, additional consideration should be paid to explaining the transient compression of the
82
D. Chang et al. / Powder Technology 92 (1997) 81-87
sediment due to stress caused by accumulating particles. TypicaUy, momentum balance equations with Darcy's law are developed to describe the compression [22-26]. Relatively large particles of narrow size range, however, tend to form incompressible sediments with packing concentration around 0.55--0.64 [ 27,281. The dynamics of the rising interface above the sediment plays an important role in many practical applications. As Kynch indicated, if the flux curve is convex at the concentrated region, and the packing concentration is relatively high, the concentration profile above the sediment should be continuous or stretched. Dixon et al. [29], however, showed that the inertia cannot be ignored in the simulation, and that incompressible sediments do not follow the Kynch type sedimentation. Bacri et al. [ 30] measured the falling velocity of the upper interface and the rising velocity of the lower interface by an acoustic method. They concluded that there is no mass accumulation between the two interfaces, and that the result is consistent with the fact that the flux curve is nearly straight at the concentrated region rather than convex. Hoyos etal. [31 ] showed in an experimental study on bidisperse suspensions that mutual hindrance results in a velocity reduction of large particles and in an enhancement of small ones, as compared with a monodisperse suspension. The concern of the present research is to investigate the batch sedimentation process of non-colloidal particles, especially to verify whether Kynch's prediction is plausible or not. In order to explore the validity of Kynch theory under the influence of the broadening effects of hydrodynamic diffusion and polydispersity, it is essential to measure precisely the instantaneous concentration profile of the entire sedimentation column without any significant disturbances by measurement tools. Since suspensions are usually opaque, even at relatively low concentrations, the concentration range measurable by optical methods is severely limited [ 32]. Recently, it has become possible to measure the concentration profiles of these opaque suspensions with the aid of non-invasive imaging techniques. The X-ray CAT scan [ 33 ], X-ray computerized tomography [ 12 ], and magnetic resonance imaging (MRI) [8] have been utilized to investigate the transient behavior of the batch sedimentation column. In this work, the concentration profiles are measured using MRI for various initial concentrations.
where c = c(z, t) is the volume concentration of particles at position z along the direction of settling and time t, and q is the volume flux of particles. Kynch considered the purely convective sedimentation
q:f
V(c)c
for C
1.o
(2)
where v(c) is the hindered velocity of a settling particle at local concentration c, and Cmis the packing concentration of the sediment. Kynch theory states that the concentration profile in the rising lower interfacial region shows different patterns depending on Co and the hindered settling function f(c), which is the ratio of the settling velocity v(c) to the Stokes velocity to. Four different patterns exist when the hindered settling velocity obeys the Richardson-Zaki fommla [34]
f(c)=(l-c)"
(3)
with n : 5, and the packing concentration of the sediment, Cm, is 0.6. These values are very close to those found experimentally as discussed later. The flux curve, plotted in Fig. 1 as a function of concentration, has an inflection point at c2. A solid straight line originating from (Cm, O) is tangential to the curve at c3 and crosses the flux curve at c:. These three transition concentrations c~, c2, and c3 define the four consecutive intervals of concentration, named regions I-IV in the figure. For a given flux curve, the shape of the lower interface depends uniquely on Co for an initially uniform suspension, as shown in Fig. 2. The initial concentrations Co of Fig. 2(a)-(d) correspond to regions I-IV, respectively. According to Kynch's model, the interface velocity for each region can be obtained from the shock structure analysis [35]. For all regions, the settling velocity of the upper interface, us(co), is equal to the settling velocity in the free settling suspension layer:
us(co) - -Uof(Co)
(4)
•Here, the upward direction is taken positive. At the lower rising interface of regions I and IV a single concentration i R e g i o n li
II
/If
IV
g
i Q
2. Sedimentation model For further consistent notation, Kynch theory and its modifications are briefly stated. For the suspension of monodisperse particles, the concentration profile of the batch settling column can be described by the following one-dimensional mass conservation equation:
ac.
01
(1)
02
o*
cs
cm
c
Fig. I. Flux curve and definition of regions for the Richardson-Zaki formula with n = 5 and Cm~--- 0.6.
D. Chang et al. / Powder Technology 92 (1997) 81-87
where D(c) is the concentration-dependent diffusivity. The solution to Eqs. (1) and (8) with no flux conditions at both the top and bottom of the column has been numerically [ 10] and asymptotically [ 11 ] solved for colloidal sedimentation. The main effect of the diffusion term is to make the interfaces smooth, although the detailed shape of the profile may vary with the concentration dependence of the diffusivity. Roughly speaking, the simulated profiles keep the characteristics of the Kynch solution if Pe = 104 or larger (note that Pe is scaled with the suspension height and not with the particle size). If Pe = 100 or smaller, the simulated profiles no longer carry the traces of discontinuities [ 10,12].
i
0.6
-(o)
-(b)
0.4
J°
dc
!
0.2 0.0 !
i
-(d)
o.6
83
J
0.4.
3. Experimental
0.2 1
I
0.0 0.0
0.5
1.0
0.0
0.5
1.0
~/L Fig. 2. Calculated concentration profiles using the Richardson-Zaki formula with n = 5 . 0 and Cm=0.6: (a) Co--0.08 and t=O.5LIvo; (b) co-0.20 and tffi 1.5LI vo; (c) co=0.36 and t = l .SLI vo; (d) Co=0.51 and t= 2LI vo.
discontinuity, denoted by Ja in Fig. 2, develops and moves with Ua(Co)"
UdCo)=V(Co)Co
(5)
Cm-- Co
If Co belongs to region III, a continuous concentration profile Jb for Co< c < c3 develops and a concentration shock Ja, jumping from c3 to Cm. The rising velocity of the isoconcentration slice in Jb is given by ul,( c ) =
d(v(c)c) dc
(6)
and the discontinuity J,, moves with ua(c3). The lower interface of region II type is composed of the discontinuity jumping from Co to c*, followed by the continuous increase from c* to c3, followed by the discontinuity jumping from ca to Cm, denoted respecti'¢ely by Jc, Jb, and J~ in Fig. 2(b). The discontinuity Jc moves upward with uc(co), given by
v(c*)c*-v(Co)Co uc(co) -
(7)
c* - Co
where c* is the tangential point concentration of a line which crosses the flux curve at Co in region II. An example of such a line is shown as the dashed line in Fig. 1 for Co- 0.22. The rising velocities of J~ and Jb in region II are also given by u~(ca) and Ub(C), respectively, for c* < c < ca. The complete description of the theory is found elsewhere [ 2,3]. The effect of diffusion can be included in the conservation equation by expressing the effect as a diffusion process and replacing the flux term in Eq. (2) by
0c
q - v ( c ) c - D(c) OZ
(8)
In the experiment, suspensions are prepared by mixic~g a lubricant oil and glass beads. The glass beads are class V precision grade microbeads manufactured by Cataphote. The size distribution of the particles is characterized by optical microscope pictures (magnification × 100). The average radius and its standard deviation are 69.4 and 7.4 ~m, respectively. Hollow particles are not observed in the photographs, and the density of the particles is 2.46 g/cm 3. The hydrocarbon composite oil used for the liquid phase material is Visco 2000 manufactured by Kyungin Energy. The specific gravity and viscosity of this oil are 0.89 and 240 cP, respectively, at 24°C. An acrylic column of inner diameter 5.2 cm is used as the container of the glass beads and the hydrocarbon oil mixture. The bulk concentration of the suspensions is increased from 0.04 to 0.55 by intervals of 0.02-0.04, and about 20 suspensions with different initial concentration are prepared separately. The suspension height L is so long that the completion of sedimentation takes one hour or so. Magnetic resonance imaging (MRI), more often called nuclear magnetic resonance computerized tomography (NMR-CT), is used to measure the concentration profiles over the entire container. This is a kind of non-destructive computerized tomography, recently introduced to detect the physical properties of fluids, together with X-ray [ 33] and acoustic methods [30]. Quantum magnetic properties of atoms, such as the nucleus density, spin-spin relaxation time, spin-lattice relaxation time, chemical shift, degree of magnetization, and the rate of change of the nucleus density, can be used to detect the temperature, velocity, and density distributions. Since a pulse of electromagnetic waves of radio frequency range is input to the suspension, the sedimentation container is free from external disturbance. Since the attainability of a measurement depends on the magnetic prope~ies of the object atoms, it does not matter whether the object is optically transparent or not. The proton signal of the oil is detected in this experiment, so the signal-to-noise ratio increases with bulk concentration. The spatial resolution is set at a value between 193 and 386 I~m depending on the sedimentation velocity. Detailed theoretical principles are explained elsewhere [ 8].
84
D, Chang et al./ Powder Technology 92 (1997) 81-87 0.04 .
After the container is repeatedly inverted up and down to attain a homogeneous suspension, it is placed vertically in the equipment. Then, the MRI equipment starts to detect the concentration profiles over the entire container with a fixed time interval of 20 to 30 s. The acquisition time for a profile, usually less than 10 s, is taken to be small enough for the particles in a suspension not to exceed the spatial resolution of the MRI equipment. Since the completion of sedimentation takes one hour or so, 100-200 profiles are collected from each batch of sediment.
,
,
,
,
,
0.3
0.4
0.5
0 0.03
0.02
0.01
0.00 0.0
4. Results and discussion
0.1
0.2
0.6
O
A sample contour map depicted on the t-z plaice (Fig. 3) shows how the concentration profile evolves in time. This map, constructed fiom 168 concentration profiles measured every 25 s for Co= 0.34, plots the concentration values ranging from 0 to 0.60 with a concentration step of 0.04. Though the lower interface keeps spreading with time, the bottom of the lower interface, considered as the discontinuity J,, rises with constant velocity after an early stage until it closely approaches the upper interface which also falls with constant velocity. From this contour map, the falling velocity of the upper interface, u,, and the rising velocity of the discontinuity, u,, can be precisely determined. The settling velocity of the upper interface, us, is estimated from the plot of the position of the interface versus time. It is found that the experimental data align so well that they consist of a nearly straight line, whose slope corresponds to u,. This is also the case for the lower interface velocity u,. The falling velocity of the upper interface, u, (Co), is equal to u (Co), and the exponent of the Richardson-Zaki formula as well as the Stokes velocity are estimated from the linear regression of a plot of In[ u,(Co) ] versus in( 1 - Co), The experimental results show good agreement with the formula, with a correlation coefficient of 0.98 and n = 5.06. The value of n - 5 is taken in this study for further discussion. 20
Fig. 4. R u x curve obtained from the falling velocities of the upper interfaces. The solid line is the theoretical curve based on the Richardson-Zaki formula with n = 5.
Fig. 4 presents the experimental results. It is noted that the error bars are smaller than the physical size of the objects in the figure. The solid curve represents the flux curve based on f ( c ) = ( 1 - c) s. Small scattering of the data points near the curve seems to be caused by day-to-day temperature variations. It is found that the sediment, precisely the part lower than ~, is incompressible, and that its packing concentration Cm remains constant at 0.60+0.01 regardless of Co. With Cm= 0.60 and the Richardson-Zaki formula with n = 5, it can be calculated that cn = 0.16, c2 = 0.33, and c3 = 0.46. The validity of ct and c3 can be verified by examining Ua, the rising velocity of,/,. The rising velocity u, was determined from the contour maps for various co, and the result is depicted in Fig. 5. The rising velocities calculated from Eqs. (5), (6), and (7) with the Richardson-Zaki formula and n = 5 are also represented graphically in the same figure. The solid line represents the rising velocity u, which has a constant value ua(cz) in regions II and III. The dashed line is the rising velocity uc of the discontinuity Jc which is found only in the region II type profile• The dotted line is the upper bound of 0.12
I~v, •
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0 0
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I~ BeO Fig. 3. Isoconcentration contour map for Co= 0.34 with a concentration step of 0.04.
0.04 0.02 0.00 0.0
I
I
I
I
I
0•1
0.2
0•3
0.4
0.5
0.6
o0 Fig. 5. Sediment rising velocities estimated from experimental results (circles) and calculated from the Richardson-Zaki formula with n = 5 (lines). The solid, dashed, and dotted lines represent, respectively, ua in all regions, uc in region II, and the upper bound of uu in region III.
D. Chang et al. / Powder Technology 92 (1997) 81-87
the rising velocity Ub in the region III type profile. The rising velocity of the continuous profile Jb is bounded by the dashed and solid lines in region II, and by the dotted and solid lines in region III. The measured rising velocity of the sediment surface, ua, agrees well with the theoretical curve. This supports the existence of transition concentrations c] = 0.16 and c3 = 0.46 estimated from the flux curve analysis. Although the fundamental structure of the experimental concentration profile of non-colloidal particles can be analyzed basically by Kynch theory, the concentration profiles of the lower interface are different from those in Fig. 2 due to hydrodynamic diffusion and polydispersity. The concentration profiles of an entire suspension column at various instances are shown in Fig. 6 ( a ) - ( f ) for Co= 0.10, 0.15, 0.22, 0.34, 0.45, and 0.55, respectively. The flux cutve analysis indicates that the profiles (a) and (b) are region I type, (c) is region II type, (d) and (e) are region III type, and (f) is region IV type. The profiles do not show sharp concentration changes in either the upper or lower interfaces because of the broadening effects of diffusion and polydispersity. The thickness of the upper interface decreases with increasing Co as shown in the figure. The higher the bulk concentration, the larger the velocity difference across the upper interface, i.e. the thinner the upper interface. Consequently, I
I
I
!
0.6 0.4. j'i:.
0.2 6~,~---'~
0.0
(a) ,
(b)-
I
I
I
I
I
I
!
I
0.6 0.4
I'
i
!
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~.~-
•
!
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!
'
' i
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t
.
-
.
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. . . . •.:..~ ~.~
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: ,:.:.
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!
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.
;L.'.
.." ~r • .
• o ~.~-.,.,
(e) l
.'. t.
""
:
"
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z, cm
Fig. 6. Measured concentration profiles at various Co: (a) Co=0.10 and measured at t= 330, 930, 1515, 2122, and 2497 s; (b) Co=0.15 and measured at t= 319, 919, 1519,2119, and 3199 s, (c) co= 0.22 and measuredat t= 353,953, 1553,2334, and 4309 s; (d) Co= 0.34 and measuredat t= 320, 920, 1520, 2420, and 4249 s; (e) co=0.45 and measured at t=311,881, 1456, 2026, and 3706 s; (f) Co=0.55 and measured at t=339, 639, 939, 1539, and 2739 s.
85
Co can be considered as the rough measure of the self-sharpening effect at the upper interface.The velocity fluctuation relative to the mean settling velocity remains constant or decreases with increasing concentration [6,7]. Since the thickness of the upper interface in a monodispcrse suspension is determined from a balance of the hindered settling and the velocity fluctuation, it decreases with the initial concentration. The lower interface of the concentration profile for Co=0.10 exhibits a sharp change compared with the other profilesin Fig. 6. This profileis region I type, but the discontinuityJa becomes smooth due to the broadening effects.The magnitude of era-Co is a rough measure of the self-sharpening effect for the lower interface in region I and IV type profilesas Co is for the upper interface.This agrees with the observation that the lower interfaceof profile (b) is broader than that of (a). The profilefor Co = 0.55 shown in Fig. 6(f) is region IV type. The magnified window in Fig. 6(f) shows that the concentration profileof the lower interfaceis almost symmetric sigmoidal in shape, which is quite similar to that in Fig. 6(a). This implies that the velocity fluctuationis not negligible near the packing concentration. Compared with the profilesin Fig. 6(a) and (f), the profilesin (c) and (d) have longer tailsstretchedto the suspension region. In the absence of the velocity fluctuationeffect, the difference between region Ill and IV type profilesis that the region Ill type profilehas a continuous Jb profile at the lower interface.In the actual profilefor Co = 0.34, the continuous part is more stretchedby the velocity fluctuationeffect. Since a small stretched tailJb is also observed in the profile for Co = 0.45 (Fig. 6(e)), it is concluded that the transition point between regions Illand IV exists at Co and is a littlebit larger than 0.45. This is consistent with the previous result Ca = 0.46 obtained from the flux curve analysis in Fig. 4 and the risingvelocity curve analysis in Fig. 5. The transition concentration c3 can be identified more clearly from the evolution of region III type concentration profiles. In the range of Ca < c < c,,, the effects of hindered settling and :'elocityfluctuationcounterbalance each other, so that a steady shape is developed. But, in the range of Co < c < c3, b~)thof them contribute to the interfacebroadening.These two differenteffectsof the hindered settlingvelocity on the transientbehavior of the concentration profilecan be depicted visually by the concentration profile in the Lagrangian coordinate system. In Fig. 7, the concentration profiles c(z+ Vt, t) for Co=0.34 are depicted as a function of z+ Vt for t= I020, 1520, 2020, and 2549 s. The dashed line represents the transition concentration Ca =0.46. The velocity V of the moving coordinate system is chosen as Ua=0.00115 cm/s, obtained from Fig. 5. Fig. 7 shows that the profile above the dashed line is steady, whereas the profile below the line keeps spreading. The significance of this figure is that the transition concentration c3 can be identified from the transient behavior of the concentration profile for a single Co in region III.
D. Chang et aL / Powder Technology 92 (1997) 81-87
86 '
'
'
l
06
0.5 c
0.4
0.3
L
z
I
z
8
9
10
11
12
z+V~o orn Fig. 7. Lagrangian description of the transient behavior of the concentration profiles for Co--0.34 measured at 1020, 1520, 2020, and 2549 s, denoted respectively by hollow circles, filled circles, hollow triangles, and filled triangles. The dashed line represents c3 calculated from the Richardson7..aki formula with n = 5.
Fig. 6(c) shows the concentration profile for Co--0.22 which is presumed to be region II type. Based upon the flux curve analysis, the range of region H is 0.16 < Co<0.33, and region II type profiles are expected to show some indications of the two discontinuities la and J¢. But the actual profile (c) does not show any trace of the discontinuity Jc, and it is similar to the region III type profile in (d). Actually, all the concentration profiles for Co in regions II and III look similar to each other, and it is almost impossible to find the transition concentration c2 from the difference in their shapes. Three reasons for this are plausible: one due to a non-convex flux curve at the intermediate concentration range, another to polydispersity, and the third to diffusion. If the flux curve is non-convex and does not have the inflection point c2 in the intermediate concentration range (i.e. near c - 0 . 3 ) , region II does not exist any more. This interpretation is consistent with the results ofBacriet al. [ 30]. For the non-convex flux curve, the line originating from (Cm, 0), being tangential to the curve, nearly overlaps the curve itself. In this case, the existence of region III is dependent on Cm, that is, it would exist for large Cm, but does not for small Cm. As is apparent in Fig. 6, the packing concentration Cm is large enough for region Ill to be present over a wide range of concentration. It should be noted that the measurement of the velocity of the rising sediment, as done in the study by Bacri et al. [ 30], may not be able to verify whether region III exists or not. When the lower interface becomes spread, its rising velocity would not be determined precisely unless the interface were divided into two parts, as in the present study. Even though the flux curve is convex, the polydispersity effect causes the discontinuity Jc to disappear. It was found in previous experimental investigations on gradient diffusion that for the upper interface the polydispersity effect is comparable to the diffusion effect [5,9,17]. Additionally, the dependence of hindered settling on concentration is similar
to that of diffusivity [4,7]. The self-sharpening measure for Jc is the concentration difference across Jc, or c*-Co. In consequence, provided the polydispersity effect is considerable in the upper interface with concentration jump Co, it is plausible that the effect is also significant if c* -Co is comparable to Co. That is, it is possible that in a polydisperse suspension Jc would be erased by polydispersity, and that in a monodisperse suspension J~ would be observable with the initial concentration of region II. The third reason for region II type or the discontinuity J~ to be unobservable can be attributed to diffusion overriding the self-sharpening effect of hindered settling. The concentration profiles can be simulated numerically if the hydrodynamic diffusivity is expressed as a function of concentration. • In previous experimental investigations for gradient and selfdiffusion [4-9], it was found that diffusivity scaled with particle size, and the settling velocity remains in the same order over a wide range of concentration. This indicates that the dependence of diffusivity on concentration is very similar to that of settling velocity, as expressed by the equation
D(c) --Dof(c)
(9)
This expression is similar to the diffusivity used to describe the suspension of colloidal particles [ lO,11 ] except for the fact that the compressibility factor is set to unity since noncolloidal particles are used in this work. Though the diffusivity expression does not completely support the experimental data at relatively low and high concentrations, it can qualitatively describe the decreasing tendency of velocity fluctuation in region II. For the diffusivity by Eq. (9), cm=O.60, and the hindered settling functionf(c) given by the Richardson-Zaki formula with n = 5, a continuous solution to Eqs. ( l ) and (8) exists if the discontinuity in 0c/0zjust above the sediment is admitted. The Peclet number for the present system, which is defined by V(co)L/Do (note that L is the suspension height), is about 100 [ 9]. The numerically calculated concentration profiles for Pe= 100 (solid line) and 1000 (dashed line) are depicted for Co- 0.22 with the corresponding experimental data in Fig. 8. The figure clearly shows that the calculated profile for Pe = 100 does not show any indication of the discontinuity J~, whereas the profile for Pe = 1000 still keeps a trace of the two discontinuities. The simulated profile for Pe = 100 agrees qualitatively with the experimental profile, but the discrepancy between them indicates that the real diffusivity is different from the simple assumption in Eq. (9). There is another qualitative disagreement between the experimental data and the present simulated profile: the concentration gradient of the simulated profile is not continuous at c - Cmbut the experimental data looks smooth everywhere° The profile simulated for the sedimentation of the colloidal suspension is smooth since the compressibility factor diverges at C--Cm [ 11,12]. The most probable reasons for this disagreement are the thermodynamic effect and the wall effect. In the colloidal suspension, the thermodynamic effect is very important even at the intermediate concentration, say
D. Chang et al. / Powder Technology 92 (1997) 81-87 It
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References
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-
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;3, c m Fig. 8. Experimental (dots) and simulated concentration profiles for Co= 0.22. The solid and dashed lines represent the simulated curves for Pe = 100 and Pe = 1000, respectively.
c = 0.3 or more when Pe = 100. This effect, however, i.~ negligible for the non-colloidal system in the present study even when c is close to Cm. Fig. 6 supports this fact since the sediment of c = Cmforms immediately after the sedimentation begins. The discontinuity of aclaz at c = Cmin the sirnulated profile is caused by the discontinuity in the flux curve at c-- Cm. In order to reflect the wall effect of the sediment, the flux curve in Fig. 4 should be modified so that it decreases faster near Cmand vanishes at c - Cm.
5. Conclusions The batch sedimentation process for monodisperse noncolloidal particles is governed by the hindered settling velocity, polydispersity, and hydrodynamic diffusion. As the interfaces do not form discontinuities or shocks due to the hydrodynamic diffusion and polydispersity, they keep the characteristics of Kynch sedimentation. The concentration profiles exhibit different patterns of transient behavior depending on the initial concentration of the suspension. The transition concentrations were determined from the flux curve and cross-checked by three other analyses: the rising velocity of the lower interface, direct comparison of concentration profiles with different initial concentrations, and the evolution of the concentration profile in the lower interfacial region. The distinction of region II from region III is not clear from the pattern of the concentration profiles. Plausible reasons for this are a non-convex flux curve in the intermediate region, the polydispersity effect and/or the diffusion effect.
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