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Settling velocities of particulate systems. Part 8. Batch sedimentation of polydispersed suspensions of spheres
International Journal of Mineral Processing, 35 ( 1992 ) 159-175
159
Elsevier Science Publishers B.V., Amsterdam
Settling velocities of particulate systems. Part 8. Batch sedimentation of polydispersed suspensions of spheres F. Concha, Ch.H. Lee and L.G. Austin Department of Mineral Engineering, The Pennsylvania State University, 115 Mineral Science Bldg., University Park, PA 16802, USA (Received 17 May 1991; accepted after revision 30 September 1991 )
ABSTRACT Concha, F., Lee, Ch.H. and Austin, L.G., 1992. Settling velocities of particulate systems, 8. Batch sedimentation of polydispersed suspensions of spheres, Int. J. Miner. Process., 35:159-175. As an extension of previous work by one author on the settling of ideal suspensions, we analyze the settling behavior of a suspension with two particle sizes. The solid flux density as a function of both particle concentrations is constructed. Using a finite difference method three initial and boundary value problems were solved showing the settling curves and the concentration profiles for each particle size. We conclude the work by extending the concept of a Kynch Sedimentation Process to a mixture of any number of ideal suspensions, each having a different particle size. An example for six particle sizes is analyzed in detail. Again the result is given as sedimentation curves and concentration profiles for each particle size.
INTRODUCTION
Although the batch sedimentation of ideal suspensions has been extensively studied (Kynch, 1952; Aris and Amundson, 1973; Bustos and Concha, 1988; Concha and Bustos, 1991 ), the equivalent work for suspensions having a particle size distribution is just starting. Several authors have considered the constitutive equation for the settling velocity of suspensions having two particle sizes (Richardson and Meikle, 1961; Smith, 1965, 1966; Lockett and A1-Habbooby, 1973, 1974; Mirza and Richardson, 1979; Malisyah, 1979; Selim et al., 1983; Zimmels, 1983, 1985; Epstein and LeClair, 1985 ). The early works assumed that the settling velocity of any particle was a function of the total local concentration only, ignoring the motion of the nearby particles. In the more recent works the role of Correspondence to: F. Concha, Department of Metallurgical Engineering, University of Concepci6n, Casilla 53-C, Correo 3, Fax: 56-41-230759, Concepci6n, Chile.
0301-7516/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
~0()
[ (I)N(H\|-J
\1
the return fluid and of the small particles on the settling velocity has been considered. An analysis of this phenomena was also made by Shih et al. (1987). Schneider (1985) studied the sedimentation of mixtures of two particle sizes emphasizing the composition of the resulting sediment. The works of Greenspan and Ungarish (1982) and that of Shish et al. ( 1987 ) use the theory of mixtures to develop mass and momentum balances appropriate for suspensions with a particle size distribution. The aim of both works is to predict the falling and rising interfaces within the suspension, although the report of Shih et al. also gives concentration profiles for one time during settling. It is the purpose of this present paper to analyze the batch sedimentation of mixtures of ideal suspensions. To do so we extend the concept of a Kynch Sedimentation Process to suspensions with two particle sizes, and then, to multi-particle sizes. We develop constitutive equations for the flux-density functions and solve the initial and boundary value problem numerically. KYNCH SEDIMENTATION PROCESS FOR A SUSPENSION HAVING TWO PARTICLE SIZES
Consider a mixture of two ideal suspensions (Concha and Bustos, 1991 ) each having a different particle size. The suspension will be characterized by the volume fraction of solid and the flux-density functions of each particle size: ~01,~02,f~ ,rE. We will say that these four field variables constitute a Kynch Sedimentation Process (KSP) if, in regions £2= { (z, t ) 10 _
0~2
=U
(i)
OA
~T+~-z =0
(2)
and at points of discontinuity they obey the jump conditions:
¢i:~f+--f/--~
i=1,2
(3)
where f~ = ¢~vl, f2 = ~2v2, vl and v2 are the settling velocities of sizes dl and d2 respectively and t7s is the displacement velocity of the discontinuity j having ¢~ on one side and ~7 on the other side. To complete the kinematical process, constitutive equations must be postulated for the flux-density functions f~(~,¢2) and f2(¢~,~2) and suitable boundary conditions should be specified.
BATCHSEDIMENTATIONOF POLYDISPERSEDSUSPENSIONSOF SPHERES
161
CONSTITUTIVE EQUATIONSFOR THE FLUX-DENSITYFUNCTIONS
We know that for batch sedimentation the volume average velocity q of the suspension is zero. Then: (4)
q= ( 1 - ¢ 1 - ~2)Uf"[-~1UI-[- (f12U2= 0 that is:
vr+ , (v,-vf) +
(v2- vf) =0
In terms of the relative solid-fluid velocities:
(5)
Vf=--(~IUI"[-~2U2)
The flux-density functionsf~ and f2 in terms of the relative solid-fluid velocities are fl = ~l Ul + tp~vf and f2 = ~2u2 + ~2vf. Therefore, substituting the value of vr from equation ( 5 ) yields: fl =~01( 1 - ~ l )Ul-~01~02u2
(6)
A = ~ 0 2 ( 1 -- ~02) U2-- ~01~02U1
(7)
The constitutive equations for f~ and f2 depend on the selection of an appropriate equation for the relative solid-fluid velocities. For this purpose we can use the Richardson and Zaki equation as proposed by Malisyah ( 1979 ): ui=u~i(1--q~) n for0
i=1,2
(8)
where
I (p,-p)gd~ uoo,- 18
~/f
i= 1,2
(9)
and p = p , ~ + p f ( 1- ~ )
(10)
and where Ps and pf are the densities of the solid particles and the fluid, respectively, and tp= ~Pl+tp2. Any other equation such as that of Concha and Almendra (1979) can be used instead ofeqs. (8) to (10). Figures 1 and 2 show the flux density function according to eqs. (6) to (10), where dl =0.05 cm, d2=0.0125 cm, ps=2,79 g/cm 3, pf= 1.208 g/cm 3, #r= 0.242 poises and n = 2.7 (Schneider et al., 1985 ). The magnitude of these functions depend on the combination of values of tp~ and ~2. The flux-density fl of the bigger particles is always negative but for the smaller particles f2 is positive for large values oftpt and negative for small values of¢~ and ~2.
162
! ('(tN(
H'~E1%l
FI
-0.000 ~
/
"
~ ~ ~,
~,
x ~" -.
"~ ~ "I
,. /'~ i ~- /
-
--0.010 -0.020 -0.030 --0,040 -0.050 -0.060
-0 •0700 •~
~ 0.4 0.4
~
~
O ¢1
.8 0.5
~
~
0.2
0
~ 0" I
.
I
3
"
~2
~-,,,F,-~O.0 0.0
Fig. 1. Flux-density function for the larger particles, with diameter d~ in a two-particle mixture, versus concentration of each particle size, where d, = 0.0496 cm. d2 = 0.0125 cm, p~ = 2.790 g/ cm 3, pf= 1.208 g / c m 3,/~F= 0.242 poises and n = 2.7. V2
0.0040 0.0030
0.0020 0.0010
0.0000 --0.0010 --0.0020
--0.0030
-0.0040
O,
"8
v
.
v
O.0
Fig. 2. Flux-density function for the smaller particles, with diameter d2 in a two-particle mixture, versus concentration of each particle size, where dt = 0.0496 cm, d2 = 0.0125 cm, Ps = 2.790 g / c m 3, pr= 1.208 g / c m 3,/4-= 0.242 poises and n = 2.7.
163
BATCH SEDIMENTATION OF POLYDISPERSED SUSPENSIONS OF SPHERES
SOLUTION TO THE INITIAL AND BOUNDARY VALUE PROBLEM
To solve eqs. ( l ) to (3), initial and boundary conditions must be stipulated. We will consider the case of a uniformly mixed dispersion at time zero and a packed bed of concentration tPmforming at the base, that is:
~oi(z,O)=~Oo/ for 0 < z < L ~(0,l)=~0 m
i=1,2
(11)
for 0
(12)
Unfortunately, at this stage we cannot give an analytical solution to this initial and boundary value problem and the equation will be solved numerically. In order to show some of the possible solutions we will select some different initial concentrations for the settling problem. To do so some results in the settling of monosized spherical particles will be recalled. Analyzing the sedimentation of ideal suspensions, Bustos and Concha ( 1988 ) and Concha and Bustos ( 1991 ) established that, for suspensions with a constant initial concentration, the different modes of sedimentation (MS) possible for a Kynch Sedimentation Process depend on the constitutive equation for the flux density function and on the initial concentration ~0o. For example, for a flux density function with two inflexion points only five different modes of sedimentation occur depending on the initial concentration ~0o.These are settling plots consisting of two regions of constant concentration sepa-
1.0 0,9 0.8
.~ 0 . 7
~'~
A
_o
~ 0.6
REGION
$1
A B C 0 E
0 0 0.05 0.5] 0
~'2 0 0.04 0.05 0.17 0.68
2 0.5 ~ 0.4 ~ 0.3 o5=0
0.2 0
0,1
4
= 0.061 oils
0.0
.0
=
0
"
0
0
~
E
0 0.5
06=0 1,0
DIMENSIONLESS
1.5
2,
TIME t*
Fig. 3. Discontinuities in the settling of a mixture of two ideal suspensions with initial concentration ~0o~=0.05 and fo2=0.05, where dj=0.0496 cm, d2=0.0125 cm, ps=2.790 g/cm 3, pf= 1.208 g/cm 3, #f= 0.242 poises and n = 2.7.
104
i ~t~N( i t \ l
[ \1
rated by: MS-I, a shock wave: MS-I1, a contact discontinuity [ollowed by a rarefaction wave; MS-III, a rarefaction wave: MS-IV, two contact discontinuities and MS-V, a rarefaction wave followed by a contact discontinuit? ~. By using a finite difference computational scheme, three cases have been solved for different initial concentrations of a mixture of two sizes with a m a x i m u m concentration q~m= 0.68. These initial concentrations were chosen based on the knowledge of the sedimentation of monosized particles with the hope of obtaining different modes of sedimentation. The three cases are: case l , (fl01=~002=0.05: case 2, ~0o~=0.2 and ~o2=0.05: case 3, ~0o~=0.35 and ~Oo2=0.05 and the other conditions are as used in Figs. 1 and 2. The solution to case 1 is given in Fig. 3. It is convenient to represent height in the dimensionless form z/L and time in the dimensionless form t / ( L / tl~.2), where u.~2 is the velocity of the smallest particle relative to the fluid if it settles alone in an infinite fluid. As was expected each line in this figure is a shock wave. Line 1, with velocity o-~= - 0 . 0 4 2 2 cm/s, is the boundary between the water and a suspension of constant concentration of ~0~= 0 and ~02= 0.04 and represents the top of the suspension of the small particles. Line 2, with velocity o-2= - 0 . 5 5 7 c m / s , is the boundary between a suspension with the initial concentration and the region with concentration (p~= 0 and ~0:= 0.04 and represents the top of the suspension of larger particles. Line 3, with velocity o-3=0.061 cm/s, is the boundary between the suspension with initial concentration and the top of the settled bed with composition ~0~=0.507 and ~0:= 0.173. Line 4, with velocity o4=0.026 cm/s, is the boundary between the suspension with concentration ~ot= 0 and ~02=0.04 and the sediment with concentration q~L= 0 and ~0~=0.68, representing the top of the sediment of small particles. Line 5, with velocity o-5= 0 c m / s , is the boundary between the water and the sediment of small particles and line 6, with velocity o-6=0 c m / s, is the boundary between the sediment of small particles and the sediment with composition ~0~=0.51 and (02=0.17. The fact that in case 1 all particles move downward can be checked by inspecting the flux density functions in Figs. 1 and 2 at the initial concentrations of q~o~=~0o2=0.05. The solution to case 2 is given in Figs. 4 to 6. Here the values of the initial concentrations were chosen so that the results could be compared with experimental data (Schneider et al., 1985 ). The heavy lines in Fig. 4, as in Fig. 3, represent the predicted discontinuities with velocities o-~= - 0 . 0 3 9 3 cm/s, ch = - 0 . 2 3 9 cm/s, o-3= 0.093 c m / s , o-4= 0.005 c m / s and o-5= o-6= 0 cm/s, and the points are Schneider's experimental values. The correspondence is better than that shown by Schneider et al. in their Fig. 3, because they assumed that all the lines corresponded to shock waves and it can be seen that this is not the case. Figure 4 also shows the sedimentation behavior of the larger particles (the steps in the lines are the consequence of the finite steps used in the numerical solution). Figure 5 gives the corresponding concentration profiles. The fig-
BATCHSEDIMENTATIONOF POLYDISPERSEDSUSPENSIONSOF SPHERES 1
165
.00
0.75
4 I
~
A
"
8
0"50 I
eI
e2
0 0 0.20 0.65 0
0 0.07 0.05 0.03 0.68
•
C
(no~ 0.25
REGION A B C D E
l-
\
,
I A 0"264-?' 05.~19=.10o6-6'~00,"6400"400 ~1 = 0.63 %0.64 ~'0.65
0.00
. . . . . . . . . 0.0
~.
- ULII6ER PARTICLES
D
, 0.5
"~0.'65 . . . . . .
1 .'0 . . . . . . . . .
D t MENS I ONLESS
1 .'5 . . . . . . . .
T I ME
2.0
t*
Fig. 4. Settling curve and discontinuities for the larger particles in a mixture of two ideal suspensions with initial concentration ~o, = 0.20 and tPo2= 0.05, where d, = 0.0496 cm, d2 = 0.0125 cm, ps= 2.790 g / c m 3, pf= 1.208 g / c m 3, p f = 0.242, poises and n = 2.7.
1.0
0.8 N 0
t* : 0.1
~ 0.6
w o m 0.4 z
0.2
O,O O.
0.1
0.2 VOLUME
0.3 FRACTION
0.5
0.4 OF
SOLIDS
OF
SIZE
0.6
0.7
l~e 1
Fig. 5. Concentration profiles for the large particles in a mixture of two ideal suspensions with initial concentration (0o~= 0.20 and ~o2 = 0.05, where d, = 0.0496 cm, d2 = 0.0125 cm, Ps = 2.790 g / c m 3, pf= 1.208 g / c m 3,/tf= 0.242 poises and n = 2.7.
166
t
('()NLH,XLI
\L
ures show clearly that the regions of constant concentration with ~0~= 0.2 and ~p2=0.05 and the region of final concentration with ~0~=0.65 and ~o2=0.03 (qm =0.68 ), at short time, are separated by two contact discontinuities. First there is a rarefaction wave from ~p=0.2 to qh =0.29, followed by a discontinuity from qh =0.29 to ~0,=0.54 and then a new rarefaction wave from ~0~=0.54 to ~0,=0.65. This settling behavior is different from any of the modes of sedimentation possible for a suspension with a single particle size. The sediment composition at the bottom of the settling column is ~0,=0.65, ~02=0.03 and at the top ~Pt=0.63 and q~2=0.05, although there are minor deviations from ~0, =0.65, ~02=0.03 due to the upward velocity of size 2 particles. The finite difference computation is not sufficient accurate to follow the variations precisely in this region. At longer times the settled bed has a discontinuity at z*=0.33 where ¢~ changes from 0.65 to 0. Figure 6 gives the settling behavior of the smaller particles of the mixture. The first thing worthwhile to notice is that in the triangular region C of constant concentration equal to the initial concentration the small particles move upward, so that four regions of constant concentration are established in the settling plot. Region C of initial concentration ~p2=0.05, a region D of depleted concentration ~02= 0.03 at the bottom, an upper region B of concentration ~02= 0.07 where no big particles exist and a region E of m a x i m u m concentration ~p2=0.68 at the top of the thick layer of sediment of depleted I
.OO
. . . . . . . . . . . . . . . .
•~ N
A [~ 0.75
•
II
"~
[] 0.0]
(] []
C
0.05
0.20
~
o03
065
g
\'2
:
o.o3o
\'2
:
0.4 K,2 : $MA~LL R P RT] LE
0.0
0.5
1 .O DIMENSIONLESS
1 ,5 TIME
2.0
t*
Fig. 6. Settling curve for the smaller particles in a mixture of two ideal suspensions with initial concentration q~o~=0.20 and (0o2=0.05, where dz =0.0496 cm, d2 =0.0125 cm,ps= 2.790 g/cm 3, pr= 1.208 g/cm 3,/#=0.242 poises and n = 2.7.
BATCH SEDIMENTATION
167
OF POLYDISPERSED SUSPENSIONS OF SPHERES
1.00
N O
0.75
REGION
4,1
4'2
A B C D E
0 0 0.35 0.65 0
0 0.10 - 0.30 0.05 0.03 0.68
Ld O
0.50 -4, z = 0 . 4 Z "---0.46
~'1 =
-0.50 0.54 0.65
E3
O" ~-5 " ~
LARGER PARTICLES
D
~'d' " .
'~ . 0 0 . c,.~,
•
J
. . . . . . . . .
i
1 .0
1 .5
DIMENSIONLESS
TIME
. . . . . . . . .
2.0
t*
Fig. 7. Settling curve for the larger particles in a mixture of two ideal suspensions with initial concentration ~o~ =0.35 and q~o2=0.05, where d~0.0496 cm, d2=0.0125 cm, ps=2.790 g/cm 3, ,of= 1.208 g/cm 3,/~f= 0.242 poises and n=2.7. 1 .00
REGION
t., o
A B C
A 0
• 75
4'2 0 0.10-0.30 0.05
D E
0.03 0.68
4'1
0 0 0.35 0.65 0
w C
O
0.50
l 4
1
.65
,,,,,,~-/p~
o.o~
,?~,/7'/¢--o.o42 ,~;'//JF~ o.o4o 0.038 0.032
O. 25
D
+2
0 .00 0.0
O.
¢2 = SMALLER PARTICLES
= 0.30
1 .O DIMENSIONLESS
1 .5 TIME
2.0
t*
Fig. 8. Settling curve for the smaller particles in a mixture of two ideal suspensions with initial concentration qTo]= 0.35 and tpo2= 0.05, where d] = 0.0496 cm, d2 = 0.0125 cm, Ps = 2.790 g / c m 3, pf= 1.208 g / c m 3,/~f= 0.242 poises and n = 2 . 7 .
concentration. Regions C and I) and regions B and C are separated by rarefaction waves and regions B and E and E and D are separated by a shock. The solution to case 3 is given in Figs. 7 and 8. Figure 7 shows the different discontinuities that form in this case, corresponding to the top of the suspension of each particle size and to the top of the sediment for each particle size and for the mixture, and gives the settling behavior of the larger particles. Here two regions of constant concentration C and D form with concentration ~0~= 0.35 and ~0~= 0.65 respectively. They are separated by a rarefaction wave. Figure 8 shows the settling behavior of the smaller particles. Here again the small particles move upwards for short times so that three regions of constant concentration appear. Region C with the initial concentration ~02= 0.05, region D of sediment with concentration ~02=0.03 and a region E of m a x i m u m concentration ~02= 0.68. Regions C and D and C and B are separated by rarefaction waves and regions D and E by a shock. Region B is not constant in this case and the concentration ~02 varies from ~02=0.10 to ~2=0.30. Regions B and E are separated by a shock wave going from concentrations ~02= 0.3 to ~o2=0.68. KYNCH SEDIMENTATION PROCESS FOR A SUSPENSION WITH N PARTICLE SIZES
The extension of the description of sedimentation of two particle sizes to any number of sizes is done in the following way. Consider a suspension made up by a mixture of Nideal suspensions with different particle sizes [d]V= [dl, d2..... dx], each with concentration and flux density functions given by [~0]x= [~01, ~02..... ~0N] and [ f ] l = ~ , f 2 ...... fx] respectively. These field variables constitute a KSP if, in regions where the variables are continuous, they obey the system of field equations: 0[~]0t ~ - ~ - ~ = [ 0 ]
(13)
and at points of discontinuity they obey the j u m p condition:
a-[f+-~f-~1 '-Lo -o;
with i - 1,2,
N
(14)
....
Also in this case we have to postulate constitutive equations for the flux-density functions [f] and find suitable boundary conditions. CONSTITUTIVE EQUATIONS FOR THE FLUX DENSITY FUNCTIONS
To complete the kinematical process represented by the field variables and eqs. ( 13 ) to ( 15 ), constitutive equations must be postulated for all flux-den-
BATCH SEDIMENTATIONOF POLYDISPERSEDSUSPENSIONSOF SPHERES
169
sity functionsf. For batch sedimentation the volume average velocity is zero, therefore: N (1-
N
~i ~i)Vf'~- ~i ~tvi=O
so that: N
vr=- E
(vi-vr)
(15)
i
The relative velocity between the solid of size i and the fluid is us = v,- vr, and in terms of ui the previous equation becomes: N
vr= - ~ q)iui
( 16 )
Substituting back into the relative particle fluid velocity gives: N
Vi~. bli- ~i~iUi
i = 1, 2 ..... N
and therefore the flux density function for a suspension of particles of size i in a mixture of N sizes is: N
f = ~il~ti- ~i
E ~ i~li
i = 1, 2 ..... N
( 17 )
i=1
where ui is given by eqs. ( 8 ) to (10):
u~=uo~(1-~o)" 1 (p~-p)gd 2 u ~ i - 18 /.lf
for
O<~o<~Om, i = l , 2 , . . . , N i= 1, 2, ...,N
(8) (9)
where p is the density of the mixture and Ps and Pr are the densities of the solid particles and the fluid respectively, and where N
~p= Eq~,
(18)
i=l S O L U T I O N TO T H E I N I T I A L A N D B O U N D A R Y V A L U E P R O B L E M
The initial and boundary conditions to solve the system of eqn. ( 13 ) with the flux-density function given by eq. ( 17 ) are:
[¢(z,0) ] = [¢ol
(19)
F ('()N(:H~x E[ 41
170
[l ][¢]
=~,.,
(20)
where [ 1 ] is the vector with all terms equal to 1. Using a finite difference computational scheme (Lee, 1989), the result shown in Figs. 9 to 13 were obtained for a suspension having six sizes with the following properties (Greenspan and Ungasrish, 1982 ): [d]V= [75, 53, 37.5, 26.5, 19.0. 13.5]
( all expressed in micrometers )
[~00]T= [0.0104, 0.0505, 0.0923, 0.0934, 0.0419, 0.0114]
p.~=2.34g/cm 3, p f = l g/cm 3, ~tf=0.00890poises n=2.7 Figure 9 shows the discontinuities corresponding to the top of the suspension for each particle size and the top of the sediment for each particle size and for the mixture. Figure 10 shows the settling curve for the second largest particles. A similar result exist for the largest particles. Both present a simple settling behavior with a shock wave as the top of the sediment, having the same speed in both cases. Figure 11 shows the settling behavior for the third largest particle sizes. The transition from the initial concentration to higher concentration occurs through a rarefaction wave originating from the bottom of the container, propagating upwards through curved lines and finishing in a contact discon1 .00
L
0.75
~J 0 . 5 0
°Ii
0
o /
0.25
Og
o8
0.00 0
300
600
TIME.
900
1200
S
Fig. 9. Discontinuities in the settling of a suspension with particles of six different sizes, where dt =0.0075 cm, d2 =0.0053 cm, d3 = 0.0037.5 cm, d4=0.00265 cm, ds= 0.0019 cm, d6=0,00135 cm, Ps = 2.43 g/cm 3, Pr= 1.00 g/cm 3, #r= 0.00890 poises, and initial concentrations (aot = 0.0104, ¢o2 =0.0505, ¢o3=0.0923, ¢o4=0.0934, ~o5 =0.0419, ~o6=0.0114: n = 2 . 7 .
171
BATCH S E D I M E N T A T I O N O F P O L Y D I S P E R S E D S U S P E N S I O N S O F SPHERES I .00
i
~
~ LL.
o.75 "-~
~.~
~2
'l'l
A c
o o.osos
o o.oto,~ o.oel
o
C
o. 2 5
REGION
A
Z= " ~...,~:
-~
'1'2 = SECOND I.AI~EST PARTICLES
:...:czcccc:::::,:::c;:::::::::::_:,___
= = = = = = = = = = = = = = = = = = = = = = = = = =
....................
z., = 0.198
~ .....................................
.... :;,!:";
/02=0.198
0
............................
J
O
.........................................
0.00
o.,~
,---.---,---,,_,---,--,---,--,---,,,-.---,
300
600
900
TIME.
1200
S
Fig. 10. Settling curve for particle size 2, in a mixture of six different particle sizes. For conditions see Fig. 9.
\
1 .00
0.75
AREA
t3
A C
0 0.0923
t 3
THIR9 LARGEST PARTICLES
=
0.50 l*3=O.nO
3=0.105 _ .............
.3._-" . . . . . . . . . .
/// /j~__~3-o.17s j
0
. 0 0
.
0
.
.
.
.
f/~3=0.175 ,
300
.
.
.
.
.
,
.
.
.
.
.
600
,
900
TIME.
.
.
.
.
.
1200
S
Fig. 11. Settling curve for panicle size 3, in a mixture of six different panicle sizes. For conditions see Fig. 9.
[72
I ('ON('I-t,Xt:I Xl
I
I/I)
t
280 !;
t
600 s
0.8
O ~
0.6
bJ O o10.4" z a 0.2
0.O
.0
0.1
0.2
0.3
VOLUME
FRACTION
0,4
0.5
OF S O L I D S
OF S I Z E
0.6
0.7
3~ ¢3
Fig. 12. Concentration profile for particles size 3 in a mixture of six particle sizes. For conditions see Fig. 9. 100
].
t~
.
.
1 !
.
.
.
.
.
.
.
.
.
.
.
.
.
_ ( -,
I
o ©
"5
]
\\ "X \,
L4
L..
O
¢6=0.060
t~ '~
¢6=0. 0114
',,
O
"m~,~
~6=0. 04 0 ~-
¢6=0.080 \-
~-
¢6~o.12o ,(,6:(o.14o-o.2oo)
. . . .
¢6=0.100
i
025
[
0.00 0
300
600
900
TIME,
120C
S
Fig. 13. Settling curve for particle size 6, in a mixture of six different particle sizes. For conditions see Fig. 9.
tinuity that curves, becoming horizontal at a certain level of the vessel. In this way the sediment has an inverse concentration gradient in these sizes, as shown in Fig. 12. In spite of the fact that this behavior is easy to understand intui-
BATCH SEDIMENTATION OF POLYDISPERSED SUSPENSIONS OF SPHERES
173
tively we cannot at this point give an analytical solution to the problem and thus we are not able to predict the Modes of Sedimentation. Similar results exist for all smaller sizes, with a very small size giving regions as shown in Fig. 13. CONCLUSIONS
The following conclusions may be drawn from this work: ( 1 ) The batch settling of a mixture of N ideal suspensions can be described as a Kynch Sedimentation Process. (2) The flux density function for each particle size in the suspension depends on the concentration and on the relative solid fluid velocity of all particle sizes. (3) For the case of the settling of a suspension formed by the mixture of spherical particles of two sizes, the predicted results are in agreement with experimental data presented by Schneider et al. ( 1985 ). (4) However, the two particle size system shows behavior which does not fit the Modes of Sedimentation obtained from analysis of settling of a single particle size. ( 5 ) At present it is not possible to describe analytically the different Modes of Sedimentation admissible for a suspension of N particle sizes. This would require an analytical solution by the method of characteristics. (6) The numerical solution presented in the form of settling curves and concentration profiles for the suspension with 6 particle sizes will help the understanding of the settling behavior of mixtures, which is a prior step to obtaining an analytical solution. NOTATION
di f~ g n q t t* vi vf u, u~o~ z z* L
Diameter of a sphere of size i Flux-density function for particles of size i Gravity constant Experimental parameter in eq. (8) Volume average velocity Time in s t~ ( L / uo~2 ) , dimensionless time Settling velocity of particles of size i Fluid velocity Particle (of size i)-fluid relative velocity Particle-fluid relative velocity for one particle in an infinite medium Height in the settling column z / L , dimensionless height Initial height of the suspension
174 ~7
0,
Ps ,Of
P ay
F. U()N(H,k ET At.
Total number of panicle sizes in the suspension Concentration of panicles of size i, expressed as volume fraction of solids Initial concentration for particles of size i Total volume fraction of solids Maximum total volume fraction of solids Density of the solid particles Density of the fluid Density of the suspension (solid-fluid mixture ) Displacement velocity of the discontinuityj
ACKNOWLEDGEMENTS
We thank the National Science Foundation, the Utah Generic Mineral Technology Center in Comminution, the University of Concepci6n and Fundaci6n Andes for partial financial support of F. Concha as visiting professor at the Department of Mineral Engineering of the Pennsylvania State University. We also acknowledge support from the Organization of American States through the project of Polymetallic ores. This work was performed under NSF grant No. INT 8610400 and Utah Comminution Center grant No. G- 1125149.
REFERENCES Aris, R. and Amundson, N.R., 1973. Mathematical Methods in Chemical Engineering, Vol. 2. Prentice Hall, Englewood Cliffs, NJ. Bustos, M.C. and Concha, F., 1988. On the construction of global weak solutions in the Kynch theory of sedimentation. Math. Meth. Appl. Sci., 10: 245-264. Concha, F. and Almendra, A.R., 1979. Settling velocities of particulate systems, 2. Settling velocities of suspensions of spherical particles. Int. J. Miner. Process., 6:31-41. Concha, F. and Bustos, M.C., 1991. Settling velocities of particulate systems, 6. Kynch sedimentation processes, Int. J. Miner. Process., 32:193-212. Epstein, N. and LeClair, B.P., 1985. Liquid fluidization of binary particle mixtures, II. Bed Inversion. Chem. Eng. Sci., 40(8): 1517-1526. Greenspan, H.P. and Ungarish, M., 1982. On hindered settling of particles of different sizes. Int. J. Multiphase Flow, 8(6): 587-404. Kynch, G.J., 1952. A theory of sedimentation. Trans. Faraday Soc., 48: 166-176. Lee, Ch.H., 1989. Modeling of Batch Hindered Settling. Ph.D. Thesis, College of Earth and Mineral Sciences, The Pennsylvania State University. Lockett, M.J. and A1-Habbooby, H.M., 1973. Differential settling by size of two particle species in a liquid. Trans. IChE, 51: 281-292. Lockett, M.J. and A1-Habbooby, H.M., 1974. Relative particle velocities in two-species settling. Powder Technol., 10:67-71. Masliyah, J.H., 1979. Hindered settling in a multi-species particle system. Chem. Eng. Sci., 34: 1166-1168.
Mirza, S. and Richardson, J.F., 1979. Sedimentation of suspensions of particles of two or more sizes. Chem. Eng. Sci., 34: 447-454.
BATCH SEDIMENTATION OF POLYDISPERSED SUSPENSIONS OF SPHERES
175
Richardson, J.F. and Meickle, R.A., 1961. Sedimentation and fluidization. Part III. The sedimentation of uniform fine particles and of two component mixtures of solids. Trans. IChE, 39: 348-356. Schneider, W., Anestis, G. and Schaflinger, U., 1985. Sediment composition due to settling of particles of different sizes. Int. J. Multiphase Flow, 11 (3): 419-423. Selim, M.S., Kothari, A.C. and Turian, R.M., 1983. Sedimentation of multisized particles in concentrated suspensions. AIChE J., 29 (6): 1029-1038. Shish, Y.T., Gidaspow, D. and Wasan, D.T., 1987. Hydrodynamics of sedimentation of multisized particles. Powder Technol., 50:201-215. Smith, T.N., 1965. The differential sedimentation of particles of two different species. Trans. IChE, 43: T69-T73. Smith, T.N., 1966. The sedimentation of particles having a dispersion of sizes. Trans. IChE, 44: T153-T157. Zimmels, Y., 1983. Theory of hindered sedimentation of polydispersed mixtures. AIChE J., 29(4): 669-676. Zimmels, Y., 1985. Theory of density separation of particulate systems. Powder Technol., 43: 127-139.
159
Elsevier Science Publishers B.V., Amsterdam
Settling velocities of particulate systems. Part 8. Batch sedimentation of polydispersed suspensions of spheres F. Concha, Ch.H. Lee and L.G. Austin Department of Mineral Engineering, The Pennsylvania State University, 115 Mineral Science Bldg., University Park, PA 16802, USA (Received 17 May 1991; accepted after revision 30 September 1991 )
ABSTRACT Concha, F., Lee, Ch.H. and Austin, L.G., 1992. Settling velocities of particulate systems, 8. Batch sedimentation of polydispersed suspensions of spheres, Int. J. Miner. Process., 35:159-175. As an extension of previous work by one author on the settling of ideal suspensions, we analyze the settling behavior of a suspension with two particle sizes. The solid flux density as a function of both particle concentrations is constructed. Using a finite difference method three initial and boundary value problems were solved showing the settling curves and the concentration profiles for each particle size. We conclude the work by extending the concept of a Kynch Sedimentation Process to a mixture of any number of ideal suspensions, each having a different particle size. An example for six particle sizes is analyzed in detail. Again the result is given as sedimentation curves and concentration profiles for each particle size.
INTRODUCTION
Although the batch sedimentation of ideal suspensions has been extensively studied (Kynch, 1952; Aris and Amundson, 1973; Bustos and Concha, 1988; Concha and Bustos, 1991 ), the equivalent work for suspensions having a particle size distribution is just starting. Several authors have considered the constitutive equation for the settling velocity of suspensions having two particle sizes (Richardson and Meikle, 1961; Smith, 1965, 1966; Lockett and A1-Habbooby, 1973, 1974; Mirza and Richardson, 1979; Malisyah, 1979; Selim et al., 1983; Zimmels, 1983, 1985; Epstein and LeClair, 1985 ). The early works assumed that the settling velocity of any particle was a function of the total local concentration only, ignoring the motion of the nearby particles. In the more recent works the role of Correspondence to: F. Concha, Department of Metallurgical Engineering, University of Concepci6n, Casilla 53-C, Correo 3, Fax: 56-41-230759, Concepci6n, Chile.
0301-7516/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
~0()
[ (I)N(H\|-J
\1
the return fluid and of the small particles on the settling velocity has been considered. An analysis of this phenomena was also made by Shih et al. (1987). Schneider (1985) studied the sedimentation of mixtures of two particle sizes emphasizing the composition of the resulting sediment. The works of Greenspan and Ungarish (1982) and that of Shish et al. ( 1987 ) use the theory of mixtures to develop mass and momentum balances appropriate for suspensions with a particle size distribution. The aim of both works is to predict the falling and rising interfaces within the suspension, although the report of Shih et al. also gives concentration profiles for one time during settling. It is the purpose of this present paper to analyze the batch sedimentation of mixtures of ideal suspensions. To do so we extend the concept of a Kynch Sedimentation Process to suspensions with two particle sizes, and then, to multi-particle sizes. We develop constitutive equations for the flux-density functions and solve the initial and boundary value problem numerically. KYNCH SEDIMENTATION PROCESS FOR A SUSPENSION HAVING TWO PARTICLE SIZES
Consider a mixture of two ideal suspensions (Concha and Bustos, 1991 ) each having a different particle size. The suspension will be characterized by the volume fraction of solid and the flux-density functions of each particle size: ~01,~02,f~ ,rE. We will say that these four field variables constitute a Kynch Sedimentation Process (KSP) if, in regions £2= { (z, t ) 10 _
0~2
=U
(i)
OA
~T+~-z =0
(2)
and at points of discontinuity they obey the jump conditions:
¢i:~f+--f/--~
i=1,2
(3)
where f~ = ¢~vl, f2 = ~2v2, vl and v2 are the settling velocities of sizes dl and d2 respectively and t7s is the displacement velocity of the discontinuity j having ¢~ on one side and ~7 on the other side. To complete the kinematical process, constitutive equations must be postulated for the flux-density functions f~(~,¢2) and f2(¢~,~2) and suitable boundary conditions should be specified.
BATCHSEDIMENTATIONOF POLYDISPERSEDSUSPENSIONSOF SPHERES
161
CONSTITUTIVE EQUATIONSFOR THE FLUX-DENSITYFUNCTIONS
We know that for batch sedimentation the volume average velocity q of the suspension is zero. Then: (4)
q= ( 1 - ¢ 1 - ~2)Uf"[-~1UI-[- (f12U2= 0 that is:
vr+ , (v,-vf) +
(v2- vf) =0
In terms of the relative solid-fluid velocities:
(5)
Vf=--(~IUI"[-~2U2)
The flux-density functionsf~ and f2 in terms of the relative solid-fluid velocities are fl = ~l Ul + tp~vf and f2 = ~2u2 + ~2vf. Therefore, substituting the value of vr from equation ( 5 ) yields: fl =~01( 1 - ~ l )Ul-~01~02u2
(6)
A = ~ 0 2 ( 1 -- ~02) U2-- ~01~02U1
(7)
The constitutive equations for f~ and f2 depend on the selection of an appropriate equation for the relative solid-fluid velocities. For this purpose we can use the Richardson and Zaki equation as proposed by Malisyah ( 1979 ): ui=u~i(1--q~) n for0
i=1,2
(8)
where
I (p,-p)gd~ uoo,- 18
~/f
i= 1,2
(9)
and p = p , ~ + p f ( 1- ~ )
(10)
and where Ps and pf are the densities of the solid particles and the fluid, respectively, and tp= ~Pl+tp2. Any other equation such as that of Concha and Almendra (1979) can be used instead ofeqs. (8) to (10). Figures 1 and 2 show the flux density function according to eqs. (6) to (10), where dl =0.05 cm, d2=0.0125 cm, ps=2,79 g/cm 3, pf= 1.208 g/cm 3, #r= 0.242 poises and n = 2.7 (Schneider et al., 1985 ). The magnitude of these functions depend on the combination of values of tp~ and ~2. The flux-density fl of the bigger particles is always negative but for the smaller particles f2 is positive for large values oftpt and negative for small values of¢~ and ~2.
162
! ('(tN(
H'~E1%l
FI
-0.000 ~
/
"
~ ~ ~,
~,
x ~" -.
"~ ~ "I
,. /'~ i ~- /
-
--0.010 -0.020 -0.030 --0,040 -0.050 -0.060
-0 •0700 •~
~ 0.4 0.4
~
~
O ¢1
.8 0.5
~
~
0.2
0
~ 0" I
.
I
3
"
~2
~-,,,F,-~O.0 0.0
Fig. 1. Flux-density function for the larger particles, with diameter d~ in a two-particle mixture, versus concentration of each particle size, where d, = 0.0496 cm. d2 = 0.0125 cm, p~ = 2.790 g/ cm 3, pf= 1.208 g / c m 3,/~F= 0.242 poises and n = 2.7. V2
0.0040 0.0030
0.0020 0.0010
0.0000 --0.0010 --0.0020
--0.0030
-0.0040
O,
"8
v
.
v
O.0
Fig. 2. Flux-density function for the smaller particles, with diameter d2 in a two-particle mixture, versus concentration of each particle size, where dt = 0.0496 cm, d2 = 0.0125 cm, Ps = 2.790 g / c m 3, pr= 1.208 g / c m 3,/4-= 0.242 poises and n = 2.7.
163
BATCH SEDIMENTATION OF POLYDISPERSED SUSPENSIONS OF SPHERES
SOLUTION TO THE INITIAL AND BOUNDARY VALUE PROBLEM
To solve eqs. ( l ) to (3), initial and boundary conditions must be stipulated. We will consider the case of a uniformly mixed dispersion at time zero and a packed bed of concentration tPmforming at the base, that is:
~oi(z,O)=~Oo/ for 0 < z < L ~(0,l)=~0 m
i=1,2
(11)
for 0
(12)
Unfortunately, at this stage we cannot give an analytical solution to this initial and boundary value problem and the equation will be solved numerically. In order to show some of the possible solutions we will select some different initial concentrations for the settling problem. To do so some results in the settling of monosized spherical particles will be recalled. Analyzing the sedimentation of ideal suspensions, Bustos and Concha ( 1988 ) and Concha and Bustos ( 1991 ) established that, for suspensions with a constant initial concentration, the different modes of sedimentation (MS) possible for a Kynch Sedimentation Process depend on the constitutive equation for the flux density function and on the initial concentration ~0o. For example, for a flux density function with two inflexion points only five different modes of sedimentation occur depending on the initial concentration ~0o.These are settling plots consisting of two regions of constant concentration sepa-
1.0 0,9 0.8
.~ 0 . 7
~'~
A
_o
~ 0.6
REGION
$1
A B C 0 E
0 0 0.05 0.5] 0
~'2 0 0.04 0.05 0.17 0.68
2 0.5 ~ 0.4 ~ 0.3 o5=0
0.2 0
0,1
4
= 0.061 oils
0.0
.0
=
0
"
0
0
~
E
0 0.5
06=0 1,0
DIMENSIONLESS
1.5
2,
TIME t*
Fig. 3. Discontinuities in the settling of a mixture of two ideal suspensions with initial concentration ~0o~=0.05 and fo2=0.05, where dj=0.0496 cm, d2=0.0125 cm, ps=2.790 g/cm 3, pf= 1.208 g/cm 3, #f= 0.242 poises and n = 2.7.
104
i ~t~N( i t \ l
[ \1
rated by: MS-I, a shock wave: MS-I1, a contact discontinuity [ollowed by a rarefaction wave; MS-III, a rarefaction wave: MS-IV, two contact discontinuities and MS-V, a rarefaction wave followed by a contact discontinuit? ~. By using a finite difference computational scheme, three cases have been solved for different initial concentrations of a mixture of two sizes with a m a x i m u m concentration q~m= 0.68. These initial concentrations were chosen based on the knowledge of the sedimentation of monosized particles with the hope of obtaining different modes of sedimentation. The three cases are: case l , (fl01=~002=0.05: case 2, ~0o~=0.2 and ~o2=0.05: case 3, ~0o~=0.35 and ~Oo2=0.05 and the other conditions are as used in Figs. 1 and 2. The solution to case 1 is given in Fig. 3. It is convenient to represent height in the dimensionless form z/L and time in the dimensionless form t / ( L / tl~.2), where u.~2 is the velocity of the smallest particle relative to the fluid if it settles alone in an infinite fluid. As was expected each line in this figure is a shock wave. Line 1, with velocity o-~= - 0 . 0 4 2 2 cm/s, is the boundary between the water and a suspension of constant concentration of ~0~= 0 and ~02= 0.04 and represents the top of the suspension of the small particles. Line 2, with velocity o-2= - 0 . 5 5 7 c m / s , is the boundary between a suspension with the initial concentration and the region with concentration (p~= 0 and ~0:= 0.04 and represents the top of the suspension of larger particles. Line 3, with velocity o-3=0.061 cm/s, is the boundary between the suspension with initial concentration and the top of the settled bed with composition ~0~=0.507 and ~0:= 0.173. Line 4, with velocity o4=0.026 cm/s, is the boundary between the suspension with concentration ~ot= 0 and ~02=0.04 and the sediment with concentration q~L= 0 and ~0~=0.68, representing the top of the sediment of small particles. Line 5, with velocity o-5= 0 c m / s , is the boundary between the water and the sediment of small particles and line 6, with velocity o-6=0 c m / s, is the boundary between the sediment of small particles and the sediment with composition ~0~=0.51 and (02=0.17. The fact that in case 1 all particles move downward can be checked by inspecting the flux density functions in Figs. 1 and 2 at the initial concentrations of q~o~=~0o2=0.05. The solution to case 2 is given in Figs. 4 to 6. Here the values of the initial concentrations were chosen so that the results could be compared with experimental data (Schneider et al., 1985 ). The heavy lines in Fig. 4, as in Fig. 3, represent the predicted discontinuities with velocities o-~= - 0 . 0 3 9 3 cm/s, ch = - 0 . 2 3 9 cm/s, o-3= 0.093 c m / s , o-4= 0.005 c m / s and o-5= o-6= 0 cm/s, and the points are Schneider's experimental values. The correspondence is better than that shown by Schneider et al. in their Fig. 3, because they assumed that all the lines corresponded to shock waves and it can be seen that this is not the case. Figure 4 also shows the sedimentation behavior of the larger particles (the steps in the lines are the consequence of the finite steps used in the numerical solution). Figure 5 gives the corresponding concentration profiles. The fig-
BATCHSEDIMENTATIONOF POLYDISPERSEDSUSPENSIONSOF SPHERES 1
165
.00
0.75
4 I
~
A
"
8
0"50 I
eI
e2
0 0 0.20 0.65 0
0 0.07 0.05 0.03 0.68
•
C
(no~ 0.25
REGION A B C D E
l-
\
,
I A 0"264-?' 05.~19=.10o6-6'~00,"6400"400 ~1 = 0.63 %0.64 ~'0.65
0.00
. . . . . . . . . 0.0
~.
- ULII6ER PARTICLES
D
, 0.5
"~0.'65 . . . . . .
1 .'0 . . . . . . . . .
D t MENS I ONLESS
1 .'5 . . . . . . . .
T I ME
2.0
t*
Fig. 4. Settling curve and discontinuities for the larger particles in a mixture of two ideal suspensions with initial concentration ~o, = 0.20 and tPo2= 0.05, where d, = 0.0496 cm, d2 = 0.0125 cm, ps= 2.790 g / c m 3, pf= 1.208 g / c m 3, p f = 0.242, poises and n = 2.7.
1.0
0.8 N 0
t* : 0.1
~ 0.6
w o m 0.4 z
0.2
O,O O.
0.1
0.2 VOLUME
0.3 FRACTION
0.5
0.4 OF
SOLIDS
OF
SIZE
0.6
0.7
l~e 1
Fig. 5. Concentration profiles for the large particles in a mixture of two ideal suspensions with initial concentration (0o~= 0.20 and ~o2 = 0.05, where d, = 0.0496 cm, d2 = 0.0125 cm, Ps = 2.790 g / c m 3, pf= 1.208 g / c m 3,/tf= 0.242 poises and n = 2.7.
166
t
('()NLH,XLI
\L
ures show clearly that the regions of constant concentration with ~0~= 0.2 and ~p2=0.05 and the region of final concentration with ~0~=0.65 and ~o2=0.03 (qm =0.68 ), at short time, are separated by two contact discontinuities. First there is a rarefaction wave from ~p=0.2 to qh =0.29, followed by a discontinuity from qh =0.29 to ~0,=0.54 and then a new rarefaction wave from ~0~=0.54 to ~0,=0.65. This settling behavior is different from any of the modes of sedimentation possible for a suspension with a single particle size. The sediment composition at the bottom of the settling column is ~0,=0.65, ~02=0.03 and at the top ~Pt=0.63 and q~2=0.05, although there are minor deviations from ~0, =0.65, ~02=0.03 due to the upward velocity of size 2 particles. The finite difference computation is not sufficient accurate to follow the variations precisely in this region. At longer times the settled bed has a discontinuity at z*=0.33 where ¢~ changes from 0.65 to 0. Figure 6 gives the settling behavior of the smaller particles of the mixture. The first thing worthwhile to notice is that in the triangular region C of constant concentration equal to the initial concentration the small particles move upward, so that four regions of constant concentration are established in the settling plot. Region C of initial concentration ~p2=0.05, a region D of depleted concentration ~02= 0.03 at the bottom, an upper region B of concentration ~02= 0.07 where no big particles exist and a region E of m a x i m u m concentration ~p2=0.68 at the top of the thick layer of sediment of depleted I
.OO
. . . . . . . . . . . . . . . .
•~ N
A [~ 0.75
•
II
"~
[] 0.0]
(] []
C
0.05
0.20
~
o03
065
g
\'2
:
o.o3o
\'2
:
0.4 K,2 : $MA~LL R P RT] LE
0.0
0.5
1 .O DIMENSIONLESS
1 ,5 TIME
2.0
t*
Fig. 6. Settling curve for the smaller particles in a mixture of two ideal suspensions with initial concentration q~o~=0.20 and (0o2=0.05, where dz =0.0496 cm, d2 =0.0125 cm,ps= 2.790 g/cm 3, pr= 1.208 g/cm 3,/#=0.242 poises and n = 2.7.
BATCH SEDIMENTATION
167
OF POLYDISPERSED SUSPENSIONS OF SPHERES
1.00
N O
0.75
REGION
4,1
4'2
A B C D E
0 0 0.35 0.65 0
0 0.10 - 0.30 0.05 0.03 0.68
Ld O
0.50 -4, z = 0 . 4 Z "---0.46
~'1 =
-0.50 0.54 0.65
E3
O" ~-5 " ~
LARGER PARTICLES
D
~'d' " .
'~ . 0 0 . c,.~,
•
J
. . . . . . . . .
i
1 .0
1 .5
DIMENSIONLESS
TIME
. . . . . . . . .
2.0
t*
Fig. 7. Settling curve for the larger particles in a mixture of two ideal suspensions with initial concentration ~o~ =0.35 and q~o2=0.05, where d~0.0496 cm, d2=0.0125 cm, ps=2.790 g/cm 3, ,of= 1.208 g/cm 3,/~f= 0.242 poises and n=2.7. 1 .00
REGION
t., o
A B C
A 0
• 75
4'2 0 0.10-0.30 0.05
D E
0.03 0.68
4'1
0 0 0.35 0.65 0
w C
O
0.50
l 4
1
.65
,,,,,,~-/p~
o.o~
,?~,/7'/¢--o.o42 ,~;'//JF~ o.o4o 0.038 0.032
O. 25
D
+2
0 .00 0.0
O.
¢2 = SMALLER PARTICLES
= 0.30
1 .O DIMENSIONLESS
1 .5 TIME
2.0
t*
Fig. 8. Settling curve for the smaller particles in a mixture of two ideal suspensions with initial concentration qTo]= 0.35 and tpo2= 0.05, where d] = 0.0496 cm, d2 = 0.0125 cm, Ps = 2.790 g / c m 3, pf= 1.208 g / c m 3,/~f= 0.242 poises and n = 2 . 7 .
concentration. Regions C and I) and regions B and C are separated by rarefaction waves and regions B and E and E and D are separated by a shock. The solution to case 3 is given in Figs. 7 and 8. Figure 7 shows the different discontinuities that form in this case, corresponding to the top of the suspension of each particle size and to the top of the sediment for each particle size and for the mixture, and gives the settling behavior of the larger particles. Here two regions of constant concentration C and D form with concentration ~0~= 0.35 and ~0~= 0.65 respectively. They are separated by a rarefaction wave. Figure 8 shows the settling behavior of the smaller particles. Here again the small particles move upwards for short times so that three regions of constant concentration appear. Region C with the initial concentration ~02= 0.05, region D of sediment with concentration ~02=0.03 and a region E of m a x i m u m concentration ~02= 0.68. Regions C and D and C and B are separated by rarefaction waves and regions D and E by a shock. Region B is not constant in this case and the concentration ~02 varies from ~02=0.10 to ~2=0.30. Regions B and E are separated by a shock wave going from concentrations ~02= 0.3 to ~o2=0.68. KYNCH SEDIMENTATION PROCESS FOR A SUSPENSION WITH N PARTICLE SIZES
The extension of the description of sedimentation of two particle sizes to any number of sizes is done in the following way. Consider a suspension made up by a mixture of Nideal suspensions with different particle sizes [d]V= [dl, d2..... dx], each with concentration and flux density functions given by [~0]x= [~01, ~02..... ~0N] and [ f ] l = ~ , f 2 ...... fx] respectively. These field variables constitute a KSP if, in regions where the variables are continuous, they obey the system of field equations: 0[~]0t ~ - ~ - ~ = [ 0 ]
(13)
and at points of discontinuity they obey the j u m p condition:
a-[f+-~f-~1 '-Lo -o;
with i - 1,2,
N
(14)
....
Also in this case we have to postulate constitutive equations for the flux-density functions [f] and find suitable boundary conditions. CONSTITUTIVE EQUATIONS FOR THE FLUX DENSITY FUNCTIONS
To complete the kinematical process represented by the field variables and eqs. ( 13 ) to ( 15 ), constitutive equations must be postulated for all flux-den-
BATCH SEDIMENTATIONOF POLYDISPERSEDSUSPENSIONSOF SPHERES
169
sity functionsf. For batch sedimentation the volume average velocity is zero, therefore: N (1-
N
~i ~i)Vf'~- ~i ~tvi=O
so that: N
vr=- E
(vi-vr)
(15)
i
The relative velocity between the solid of size i and the fluid is us = v,- vr, and in terms of ui the previous equation becomes: N
vr= - ~ q)iui
( 16 )
Substituting back into the relative particle fluid velocity gives: N
Vi~. bli- ~i~iUi
i = 1, 2 ..... N
and therefore the flux density function for a suspension of particles of size i in a mixture of N sizes is: N
f = ~il~ti- ~i
E ~ i~li
i = 1, 2 ..... N
( 17 )
i=1
where ui is given by eqs. ( 8 ) to (10):
u~=uo~(1-~o)" 1 (p~-p)gd 2 u ~ i - 18 /.lf
for
O<~o<~Om, i = l , 2 , . . . , N i= 1, 2, ...,N
(8) (9)
where p is the density of the mixture and Ps and Pr are the densities of the solid particles and the fluid respectively, and where N
~p= Eq~,
(18)
i=l S O L U T I O N TO T H E I N I T I A L A N D B O U N D A R Y V A L U E P R O B L E M
The initial and boundary conditions to solve the system of eqn. ( 13 ) with the flux-density function given by eq. ( 17 ) are:
[¢(z,0) ] = [¢ol
(19)
F ('()N(:H~x E[ 41
170
[l ][¢]
=~,.,
(20)
where [ 1 ] is the vector with all terms equal to 1. Using a finite difference computational scheme (Lee, 1989), the result shown in Figs. 9 to 13 were obtained for a suspension having six sizes with the following properties (Greenspan and Ungasrish, 1982 ): [d]V= [75, 53, 37.5, 26.5, 19.0. 13.5]
( all expressed in micrometers )
[~00]T= [0.0104, 0.0505, 0.0923, 0.0934, 0.0419, 0.0114]
p.~=2.34g/cm 3, p f = l g/cm 3, ~tf=0.00890poises n=2.7 Figure 9 shows the discontinuities corresponding to the top of the suspension for each particle size and the top of the sediment for each particle size and for the mixture. Figure 10 shows the settling curve for the second largest particles. A similar result exist for the largest particles. Both present a simple settling behavior with a shock wave as the top of the sediment, having the same speed in both cases. Figure 11 shows the settling behavior for the third largest particle sizes. The transition from the initial concentration to higher concentration occurs through a rarefaction wave originating from the bottom of the container, propagating upwards through curved lines and finishing in a contact discon1 .00
L
0.75
~J 0 . 5 0
°Ii
0
o /
0.25
Og
o8
0.00 0
300
600
TIME.
900
1200
S
Fig. 9. Discontinuities in the settling of a suspension with particles of six different sizes, where dt =0.0075 cm, d2 =0.0053 cm, d3 = 0.0037.5 cm, d4=0.00265 cm, ds= 0.0019 cm, d6=0,00135 cm, Ps = 2.43 g/cm 3, Pr= 1.00 g/cm 3, #r= 0.00890 poises, and initial concentrations (aot = 0.0104, ¢o2 =0.0505, ¢o3=0.0923, ¢o4=0.0934, ~o5 =0.0419, ~o6=0.0114: n = 2 . 7 .
171
BATCH S E D I M E N T A T I O N O F P O L Y D I S P E R S E D S U S P E N S I O N S O F SPHERES I .00
i
~
~ LL.
o.75 "-~
~.~
~2
'l'l
A c
o o.osos
o o.oto,~ o.oel
o
C
o. 2 5
REGION
A
Z= " ~...,~:
-~
'1'2 = SECOND I.AI~EST PARTICLES
:...:czcccc:::::,:::c;:::::::::::_:,___
= = = = = = = = = = = = = = = = = = = = = = = = = =
....................
z., = 0.198
~ .....................................
.... :;,!:";
/02=0.198
0
............................
J
O
.........................................
0.00
o.,~
,---.---,---,,_,---,--,---,--,---,,,-.---,
300
600
900
TIME.
1200
S
Fig. 10. Settling curve for particle size 2, in a mixture of six different particle sizes. For conditions see Fig. 9.
\
1 .00
0.75
AREA
t3
A C
0 0.0923
t 3
THIR9 LARGEST PARTICLES
=
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3=0.105 _ .............
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.
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300
.
.
.
.
.
,
.
.
.
.
.
600
,
900
TIME.
.
.
.
.
.
1200
S
Fig. 11. Settling curve for panicle size 3, in a mixture of six different panicle sizes. For conditions see Fig. 9.
[72
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t
280 !;
t
600 s
0.8
O ~
0.6
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0.O
.0
0.1
0.2
0.3
VOLUME
FRACTION
0,4
0.5
OF S O L I D S
OF S I Z E
0.6
0.7
3~ ¢3
Fig. 12. Concentration profile for particles size 3 in a mixture of six particle sizes. For conditions see Fig. 9. 100
].
t~
.
.
1 !
.
.
.
.
.
.
.
.
.
.
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.
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o ©
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¢6=0.060
t~ '~
¢6=0. 0114
',,
O
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~6=0. 04 0 ~-
¢6=0.080 \-
~-
¢6~o.12o ,(,6:(o.14o-o.2oo)
. . . .
¢6=0.100
i
025
[
0.00 0
300
600
900
TIME,
120C
S
Fig. 13. Settling curve for particle size 6, in a mixture of six different particle sizes. For conditions see Fig. 9.
tinuity that curves, becoming horizontal at a certain level of the vessel. In this way the sediment has an inverse concentration gradient in these sizes, as shown in Fig. 12. In spite of the fact that this behavior is easy to understand intui-
BATCH SEDIMENTATION OF POLYDISPERSED SUSPENSIONS OF SPHERES
173
tively we cannot at this point give an analytical solution to the problem and thus we are not able to predict the Modes of Sedimentation. Similar results exist for all smaller sizes, with a very small size giving regions as shown in Fig. 13. CONCLUSIONS
The following conclusions may be drawn from this work: ( 1 ) The batch settling of a mixture of N ideal suspensions can be described as a Kynch Sedimentation Process. (2) The flux density function for each particle size in the suspension depends on the concentration and on the relative solid fluid velocity of all particle sizes. (3) For the case of the settling of a suspension formed by the mixture of spherical particles of two sizes, the predicted results are in agreement with experimental data presented by Schneider et al. ( 1985 ). (4) However, the two particle size system shows behavior which does not fit the Modes of Sedimentation obtained from analysis of settling of a single particle size. ( 5 ) At present it is not possible to describe analytically the different Modes of Sedimentation admissible for a suspension of N particle sizes. This would require an analytical solution by the method of characteristics. (6) The numerical solution presented in the form of settling curves and concentration profiles for the suspension with 6 particle sizes will help the understanding of the settling behavior of mixtures, which is a prior step to obtaining an analytical solution. NOTATION
di f~ g n q t t* vi vf u, u~o~ z z* L
Diameter of a sphere of size i Flux-density function for particles of size i Gravity constant Experimental parameter in eq. (8) Volume average velocity Time in s t~ ( L / uo~2 ) , dimensionless time Settling velocity of particles of size i Fluid velocity Particle (of size i)-fluid relative velocity Particle-fluid relative velocity for one particle in an infinite medium Height in the settling column z / L , dimensionless height Initial height of the suspension
174 ~7
0,
Ps ,Of
P ay
F. U()N(H,k ET At.
Total number of panicle sizes in the suspension Concentration of panicles of size i, expressed as volume fraction of solids Initial concentration for particles of size i Total volume fraction of solids Maximum total volume fraction of solids Density of the solid particles Density of the fluid Density of the suspension (solid-fluid mixture ) Displacement velocity of the discontinuityj
ACKNOWLEDGEMENTS
We thank the National Science Foundation, the Utah Generic Mineral Technology Center in Comminution, the University of Concepci6n and Fundaci6n Andes for partial financial support of F. Concha as visiting professor at the Department of Mineral Engineering of the Pennsylvania State University. We also acknowledge support from the Organization of American States through the project of Polymetallic ores. This work was performed under NSF grant No. INT 8610400 and Utah Comminution Center grant No. G- 1125149.
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Mirza, S. and Richardson, J.F., 1979. Sedimentation of suspensions of particles of two or more sizes. Chem. Eng. Sci., 34: 447-454.
BATCH SEDIMENTATION OF POLYDISPERSED SUSPENSIONS OF SPHERES
175
Richardson, J.F. and Meickle, R.A., 1961. Sedimentation and fluidization. Part III. The sedimentation of uniform fine particles and of two component mixtures of solids. Trans. IChE, 39: 348-356. Schneider, W., Anestis, G. and Schaflinger, U., 1985. Sediment composition due to settling of particles of different sizes. Int. J. Multiphase Flow, 11 (3): 419-423. Selim, M.S., Kothari, A.C. and Turian, R.M., 1983. Sedimentation of multisized particles in concentrated suspensions. AIChE J., 29 (6): 1029-1038. Shish, Y.T., Gidaspow, D. and Wasan, D.T., 1987. Hydrodynamics of sedimentation of multisized particles. Powder Technol., 50:201-215. Smith, T.N., 1965. The differential sedimentation of particles of two different species. Trans. IChE, 43: T69-T73. Smith, T.N., 1966. The sedimentation of particles having a dispersion of sizes. Trans. IChE, 44: T153-T157. Zimmels, Y., 1983. Theory of hindered sedimentation of polydispersed mixtures. AIChE J., 29(4): 669-676. Zimmels, Y., 1985. Theory of density separation of particulate systems. Powder Technol., 43: 127-139.