PowderTechnology, 48 (1986) 205  216
205
Concentration Profiles of Solids Suspended in a Stirred Tank H. YAMAZAKI*,
K. TOJO**
and K. MIYANAMI
Departmentof ChemicalEngineering, Universityof OsakaPrefecture, Sakai, Osaka(Japan) (Received February 5,1986;
in revised form June 1, 1986)
SUMMARY
The solids concentration profiles in a slurry mixing tank with mechanical agitation have been measured by using a photoelectric method. The concentration profile has been explained by the sedimentationdispersion model in which the slurry flow in a stirred tank with a flatdisk turbine or a marine propeller is assumed to be dominated mainly by the upflow. The model parameters, modified Peclet numbers, Pe, in terms of the falling velocity of the solid particles and Pei in terms of the liquid flow velocity, have been markedly influenced by the amount of the solids suspended. The Peclet number Pe, is proportional to the solids concentration in the stirred tank under the present experimental conditions, while the parameter ratio Pet/Pe,, which coincides with the ratio of the liquid flow velocity to the falling velocity of the solid particles, approaches a constant value as the concentration of solid particles is increased to 2Ovol.% or more. INTRODUCTION
Slurry operations have been used widely among the chemical and allied industries because of their advantages; excellent homogeneity of temperature and concentration throughout the tank, high reaction rate due to large interfacial area and vigorous turbulence between the phases, and so on. Typical slurry operations are dissolution of a solid, crystallization, suspension of a catalyst and suspension polymerization. In these slurry operations, the solid particles in the stirred tank are usually required to be *Research Fellow of Engineering Research Laboratory, Fujisawa Pharmaceutical Co. Ltd., 2l6, Kashima, Yodogawaku, Osaka 532 (Japan). **Associate Professor of Controlled Drug Delivery Research Center, Rutgers University, Busch Campus, P.O. Box 789, Piscataway, NJ 08854 (U.S.A.) 00325916/86/$3.50
suspended completely throughout the tank in order to attain the maximum interfacial area between the solids and liquid phases and to avoid the accumulation of solids on the bottom of tank. The operating conditions to achieve the complete suspension have been extensively investigated by many researchers [ 1  31. In some continuous operations such as one with solid particles of a wide size distribution, the homogeneous suspension in which the solid particles are uniformly dispersed throughout the tank is preferable. However, the homogeneous suspension cannot be achieved easily in the industrialscale operation due to high energy consumption. It is obvious that for estimating the mass transfer or chemical reaction rates in slurry mixing tanks, flow characteristics of all the phases involved should be taken into account. In particular, the concentration profile of solid particles is an important parameter which affects the rate of solidliquid chemical reaction and the interfacial areas. In slurry mixing tanks, the concentration profiles of the suspended solids depend largely upon the operating conditions of a mixing tank such as tank or agitator geometry, agitator speed and flow characteristics. In this work, the concentration profile of solids suspended in the slurry tank has been measured in high solids concentration regions, using a photoelectric method [4, 51. The experimental data have been analyzed by a sedimentationdispersion model [ 5  81. MODEL OF THIS STUDY It is of great importance to elucidate the flow behavior of the suspended solid particles following the vigorous flow of the liquid phase generated by the impeller. In this study, we consider two phases of liquid and solids, and a flow model with respect to these two phases has been devel
0 Elsevier Sequoia/Printed in The Netherlands
206
oped. The energy for solids suspension generated by the impeller rotation is almost dissipated around the impeller in a slurry. Little energy is transferred to points far from the impeller positions. Liquid flows are vigorous near the impeller but slow at points far from the impeller position. Figures l(a) and l(b) show the wellknown bulk flow patterns in vessels with baffles agitated by axial and radial flow impellers, respectively. At positions h,, h,i and hcz, the vigorous flow regions are separated from the moderate flow regions. In the range of Z/H > h, in Fig. l(a), the flow is moderate. In the range of Z/H > hcl or Z/H < hc2 in Fig. l(b), the flow is moderate. In developing a flow model, liquid is assumed to flow through four regions in the slurry mixing tank with an axial flow impeller as shown in Fig. 2; these are two upflow regions (A) and B)) and two downflow regions (C) and D)). In the same way, it is assumed that liquid flows through eight regions in the vessels agitated by a radial flow impeller; these are four upflow regions (A), B), G) and H) in Fig. 3) and four down flow regions (C), D), E) and F) in Fig. 3). In the vessel agitated by a radial flow impeller, the flow in the regions below the impeller is in the reverse direction to that in the regions above the impeller. The regions below the impeller can be treated by the same procedures as the regions above the impeller. Therefore, liquid is assumed to flow through eight regions in the slurry tank with a radial flow impeller as shown in Fig. 3. The material balance with respect to the solids concentration in a differential volume element of each flow region of the reactor is shown in Table 1. In Table 1, u is the liquid superficial velocity, 8, is the solid dispersion
(a)Axial
flow
agitator
(b)Radlal
flow agitator
Fig. 1. Bulk flow patterns in vessels agitated by impellers. Upflow regions = shaded parts, downflow regions = unshaded parts.
5

.
E
* ___%2
_ P3 _
_“I c3_ !If_ _“I E P3 __r_yL____yL_;~___
______ E
u3
Uf
_ _?
_
_
_I?
u2

_

_
_
Z=fi
Uf
fb3
P3
_
“e2_
t
EP2 
t “e4________.&__fe_


Z=hc
____ E P4
%4 
EP1
__z=~
=1 D/2
Fig. 2. Schematic diagram of the sedimentationdispersion model for an axial flow impeller.
_ 1
z, =Hh,
%2
t”3 __ tUf
A?_ 

_fP’ T”;cL!?_ _ c;
_ _t”r _ 
__,“P_ _ Uf t
FbiDl
$1
U4 Uf L__
_t_ E _!4Fu4___rI’
c4
3 ‘%l ” t’
tUf
G,cJ
;!I Uf
m!2d1er
Uf

+!!__t”e
__L”eTkz_LU5_t!k
_E~__~~~__~~__E’_~i~~ t
I

%4’
D/2
T%
I
Fig. 3. Schematic diagram of the sedimentationdispersion model for a radial flow impeller.
coefficient and uf is the solid falling velocity, which is assumed to be constant. The appropriate boundary conditions are expressed in Tables 2 and 3. C,, and C,s are the solids concentrations at the top of the flow regions. C,, and Cb6 are the solids concentrations at the bottom of the flow regions. Cbl, C,,s, C,, and C,, are the solids concentrations at the boundary regions of flow patterns in the tank. h,, h,, and hc2 are the critical heights separating the flow regions, which are determined by fitting this model to the experimental results. The dimensionless forms of eqns. (1) to (12) are shown in Table 4, and the dimensionless forms of eqns. (13) to (36) are shown in Tables 5 and 6.
207
TABLE 1 Material balance with respect to the solids concentration Region
in a differential volume element
Axial flow impeller
Radial flow impeller
ac, ac, a zQ +zQ + at az az az acZ acZ a ac, =u2 +uf + at az az az
ac1 =
Upflow

(5)
ac2 ac, a ac2 “u2+uf.+ at az, az, az, ac, ac, a acl at =u7a22 +Ufz
(2)

ac,
ac8 at Downflow
ac3
ac3
ac,
=““aZ,
a
ac,
(7)
a (3)
+Ufz
ac3
(3)
ac,
a (2)
(10)
(11)
(12)
TABLE 2 Boundary conditions for the upflow regions Axial flow impeller E
dC1

”
dZ
=ul(Cb,Cl)
dcl %IdZ =ufcb, dc2
Ep2
dZ
=u2(cbzcz)ufc$!
dc2 Ep2 =ufcb, dZ
Radial flow impeller ufCl
at Z = 0
atZ=h,0
(13)
E,,~=rr,(Cb~C1)ufC1
atZI=O
(17)
(14)
Epl ;
atZr=h,r0
(18)
atZI=h,,+O
(19)
at Zl=Hhi
(20)
at Z2 = 0
(21)
atZ=h,+O
(15)
atZ=H
(16)
1
=u&l
1 dc2 EpzdZ =u~(cb2c~)ufcz 1 dc2 Epz dZ = ufcb2 1
ED,=
dG
= u7(Cb7

CT) 
zqc7
2 dc7 atZ2=hti0 Ep7z =ufcb7 2 dcs =u8(Cbsc8)ufC8atZ2=hc2+0 EM
(22)
(23)
dZ2
dcs
Ep8z
2
=uf cb8
at Z = hi
(24)
208 TABLE 3 Boundary conditions for the downflow
regions
Axial flow impeller
dG Ep3
E

dZ
Radial flow impeller atZ=h,+O
= _UfCb3
(25)
Ep3z
=ufCb3
at Z1 = h,r + 0
(29)
at.Zr=Hhi
(30)
atZr=O
(31)
at Z1 = h,,  0
(32)
at Z2 = hc2+ 0
(33)
at Z2 = hi
(34)
atZ2=0
(35)
1
dC3
=U~(C~~C~)UfC~ p3 dZ
atZ=H
(26)
Ep3 2
= u3(Cb3
C,) 
ufC3
1
atZ=O
(27)
~~2
=ufCb4 1
dG
*TG
= uq(Cb4 
C,)

Uf c4
at2 =h,0
(28)
= u4(Cb4
Eti;
C,)
ufC4
1
E+
= ufC,,, dZ2
dcs Ep5

= %(CbS

‘%

WC5
dZ2
dC6
Eti
=u&&j
u2
dC6
=
u6(cb6

C6) 
uf c6
at Z2 = hc2 0 (36)
TABLE 4 Dimensionless forms of eqns. (1)  (12) Region
Axial flow impeller
Upflow
~+$)$+_?(__&?S)
(37)
~=(I$)~+&~)
(41)
%=(I$)f+&$f)
(38)
~=(I;)~+;(&~)
(42)
2=(l$$+&z)
(43)
z=(l$)~+&($.J
(44)
Z=(l+$)Z+&$)
(45)
Downflow
Radial flow impeller
(39)
:=(I+$$+;(&:)
(40) ~=(I+&,)$+&~)
$=(l+;)z+;(&$)
(46)
(48)
209
TABLE 5 Dimensionless forms of eqns. (13)  (24) Radial flow impeller
Axial flow impeller
dX1 = (Pef, Pe,l)X, dt
a1

dt
dG

d5 fl2

d8
Pef,
= Pesl = (Pef2 Pe,z)Xz
Pef2
= Pes2
at !j = 0
(49)
dX1 = (Peil Peil)XI db
at E = 6  0
(50)
dX1 = Pe:l dtl
at t; = 6 + 0
(51)
dh
= (Pei2 Pei2)X2
atg1
(52)
dh
u2
= Pei2
at El = 0
(53)
at&=610
(54)
at El = 61 + 0
(55)
at [I= 1
(56)
dX3 = (Pe;,  Pe:,)X3  Pe&
at [2 = 0
(57)
dX3 = Pe.& db
at.$2=620
(58)
= (Peis  Pe&)X4Peis
at t2 = a2 + 0
(59)
= Pe&
at & = 1
(60)
at [I = Sl+ 0
(65)
at El = 1
(66)
at El = 0
(67)
at cl = sl  0
(68)
at [2 = 82 + 0
(69)
at f2 = 1
(70)
at .$2= 0
(71)
at t2 = 62  0
(72)
dG
Peil
Pei2
dh
a4

at.2 a4

3‘ at,
TABLE 6 Dimensionless forms of eqns. (25)  (36) Radial flow impeller
Axial flow impeller
W = Pe,3
dY1
at[=6+0
(61)
= Pei, dF1
dY, = (Pef3 + Pe,,)Yl + Pef3
at t = 1
(62)
dh
dYz = Pe,4
at[=O
(63)
f Pe.& dEl
dYz = _(Pef4 + Pe,4)Y2 + Pef4
att=60
(64)
dtl
dt
dt
dt
d&i
dY1
= (Pei3
+ Pei3)Yl + Pei3
dY?
dY2
= (Ped4 + Pei4)Y2 + Pe/4
dY3 = Pe& db dY3 = (Pets dt2
+ Peis)Y3 + Peis
dY4 = Pe& dt2 dY4 = (Pei, dtz
+ PeiB)Y4 + Pe&
210 TABLE 7 Analytical solution of eqns. (37)  (48) Region
Axial flow impeller
Upflow
x1 =
x1 = Q& {&I’explJ+&(Q1’
~{Q~exp[Pe.,(QIl)(p6)1} 1
x2 =
Radial flow impeller
at 0 < i < 6
~{Q*exp[Pes2(Q*l)(~ 2
at&
1
(73)
1)l)
1X& &)I1
at 0 < ,$I < 61
X2 = ~{&2’exp[p,:2(Q2’1)(~, 2
(74) x3=
x,=
& 7
(77)
1)l)
at h1 < fl < 1
(73)
(97)exp[Pel,(Q,'l1)(~2%)I} at 0 < t2 < S2
(79)
~{Q81exp[Pe:g(Qgl1)(~21)1} 8
at S2 < t2 < 1
Downflow
Y1= Q13+
1{Q3+exdPe,dQ3+1)([email protected])
Yz=__ Q14+ 1”
at6
4+
Yz = &IQ4 (76)
+ exdPei3(&3’
+ l)(b

&)I}
at S1 < [I < 1
(75)
exdPedQ4+ Ml) atO<.$
Y, = &{Q3
(80)
+exp[Pe:4(Qd
(31)
+ 1)&l}
at 0 < E1 < a1
+exdPq&(Qs'+Uh
(82) &)I}
at 62 < .$ < 1
(83)
+expiP&(Q~'+1)&1) at 0 < & < 62
Under the steadystate conditions, eqns. (37) to (48), which are subjected to the boundary conditions, eqns. (49) to (72), can be solved analytically as shown in Table 7. As can be expected from the eqns. (73) to (84), the solids concentration profiles in the upflow regions (A), B), G), H)) and in the downflow regions (C), D), E), F)) are convex and concave, respectively. The symbols used in Table 4, Table 5, Table 6 and Table 7 are summarized in Table 8. EXPERIMENTAL
A schematic diagram of the experimental setup is shown in Fig. 4. A flatbottomed cylindrical tank 0.3 m (I.D.) X 0.3 m (liquid height) was employed.
(84)
Two types of agitator, a sixblade flatdisk turbine and a fourblade marine propeller, were used. Known masses of solids and water were fed into the tank, and then the contents mixed under complete suspension conditions were verified by careful observation. After reaching a steady state, the solids concentrations at several positions in the tank were measured both radially and vertically, using a lightreflection method. The principle of measurement was described in our previous papers [4, 51. The details of the probe for measuring the solids concentrations are shown in Fig. 4. Figure 5 shows the sampling positions in the tank. The experimental conditions are listed in Table 9.
211 TABLE 8 Symbols used in Tables 4  7 Axial flow impeller
Radial flow impeller
x1 =
x1
= C1/Cblr
x2
= c2/cb2,
x3
= c,/cb,,
x4
= cS/cbS
yl
= C3/Cb3,
y2 = c,/cb,,
y3
= c5/cb5,
y4 = C&b6
yl
cl/cbl,
= c3/cb3,
x2
y2
= c,/cb,
= c,/cb,
Pe,j = ufHlE,j
(j = 1, 2, 3, 4)
Pei, = uf(H hi)lE,, Pei,, = uf hilEp,
(m = 1, 2, 3, 4) (n = 5, 6, 7, 8)
Pefj = UjH/Epj
(j = 1, 2, 3,4)
Pei,,, = u,,,(H  hi)lEpm Pei, = unhi lE,,
(m = 1, 2, 3, 4) (n = 5, 6, 7,8)
Qj = PefjlPe,j
(j = 1,2,
3,4)
9,’
(m=l,2,3,4,5,6,7,8)
Tj = tl(Hluj)
(j = 1,2,
3,4)
7,’ = tl[(H7,’ = tl(hi I+)
hi)lu,l
.$ = ZlH
FI = zll(H
hi), t2 = Z2lhi
6 = h,lH
61~ hcll(H
= Pe&,lP&
(m = 1, 2,3,4) (n = 5,6,7,8)
hi), 62 = hc2lhi
\ _l.O
Sampling pints 0 ;r/D=o.x A
;r/D=O.ZC
.
;r/D=O.OZ

5=0.85
o

5=0.75
.A
o

CO.65
.A
0

5=0.55
.A
o

GO.45

5=0.35

5=0.25
A
D
Fig. 4. Schematic diagram of experimental apparatus. 1, Motor; 2, tachometer; 3, torquemeter; 4, amplifier; 5, recorder; 6, tank; 7, photometer, 8, optic fiber probe.
RESULTS AND DISCUSSION
Solids concentration profiles The radial concentration profiles of solids in the stirred tank are plotted in Fig. 6. It can be seen from Fig. 6 that radial concentrations are nearly constant. Therefore, the radial concentration profiles do not have to be taken into account in what follows. The axial concentration profiles of solids suspended in the stirred tank with a flatdisk
60.95
.A
.A0
El

o
=O
__5=0.15
.A0
__ 5=0.05 
5=0
I
Fig. 5. Positions of sampling points in the slurry mixing tank.
turbine and a marine propeller are plotted in Figs. 7  10. The lines in these figures were computed by adjusting the model parameters, Pe, and Pef, so as to agree with the experimental profiles. It can be clearly seen from these figures that the solids concentration profile approaches uniform suspension with increasing speed of agitation. It is also interesting to observe that the concentration near the surface decreases markedly as the amount of solids suspended increases and the solids falling velocity becomes higher. This finding can be mainly attributed to the turbulence damping in the liquid phase caused by an
212 TABLE 9
1
Experimental conditions Inside diameter of tank, (m) Impeller diameter : gblade disk turbine, (m) 4blade marine propeller, (m) Width of baffles, (m) Impeller speed, (1 /s) Impeller height, () Liquid depth, (m) Solids concentration, (vol.%)
D = 0.3
Solid particles
Free settling velocity us (m/s)
Mean particle size d, (Pm)
Glass beads (sphere) Glass beads (sphere) Glass beads (sphere) Toyoura silica sands
r 1z13.3l/s
d = 0.07 d = 0.10 B, = 0.03 n=520 h,lH = 0.3 H = 0.3 c=o30
Density
pP (kg/m3 1
87
2.37 x lo3
0.56 x 1O2
135
2.47 x lo3
1.22 x 102
264
2.48 x lo3
3.26 x 1O2
230
2.62 x lo3
2.91 x102
h~~fH'0.3 C =Svol% a"
7 LIF
cI.5
3
impelk?I
0.5
Fig. 7. Axial concentration profiles of solid particles (glass beads, d, = 87 pm) in the stirred tank with a flatdisk turbine. 0, r/D = 0.35; A, r/D = 0.20; 0, calculated using eqns. (74) and r/D = 0.05; , (76).
0.5
1.0 "Ca"
r/D Il
l
. . . ..o/
I
I II
..ooe~4. sampling point 2=0.35H
0
I
1
0.1
I
I
I
0.2
I
0.3
I
1.0 c/ca"
.
I
I
0.5
Fig. 8. Axial concentration profiles of solid particles (glass beads, d, = 87 pm) in the stirred tank with a marine propeller. 0, r/D = 0.35; A, r/D = 0.20;0, calculated using eqns. (74) and r/D = 0.05; , (76).
I I I
sampling point 2=0.65H
1.0
c/cav
I 1 0.4
I
I 0.5
r/D I1
Fig. 6. Radial concentration profiles of solid particles (glass beads, d, = 135 pm, C, = 15vol.%) in the stirred tank with baffles.
increased interference among solids particles, since the energy for solids suspension is generated by the impeller and little energy is transferred to points far from the impeller position. Einenkel [9] has shown a distinct peak in a solids concentration distribution just above the impeller. But no peak is indicated in this paper. Baldi et al. [lo, 111 also reported the same results as ours. The disagreement of these results may be caused
213
(a)
1.0
0.5
c/c
0.5
a"
1.0
0.5
1.0 c/c
"ca"
a"
Fig. 9. Effect of impeller speed on axial concentration profiles of solid particles (glass beads, d, = 135 Pm, C, = 15vol.%) in the stirred tank. (a), Flatdisk turbine; (b), marine propeller; 0, r/D = 0.35; l, r/D = 0.05;, calculated using eqn. (74).
__________
1.0
0.5
c/c
1.0
0.5 c/c
a"
________
0.5
1.0 "ca"
a"
The solids concentration profiles are influenced largely by the solidliquid relative velocity. When solid particles with a large falling velocity were used, we observed a slight peak in the solids concentration profiles above the impeller. Figure 11 shows axial concentration profiles of the solid particles (dp = 264 pm, U, = 3.26 X 10e2 m/s) suspended in the stirred tank. As can be seen from this figure, a peak in solids concentrations above the impeller was observed. Therefore, our model cannot be applied for the large particle, the falling velocity of which is greater than U, = 3 X 10e2 m/s under the present experimental conditions. From observations with both types of impeller, the upflow dominates the upper part of the tank and the downflow dominates the lower part of the tank. For an axial flow impeller, the main flow regions seem to be at B) and D) in Fig. 2. On the other hand, the main flow regions for a radial flow impeller are considered to be at B) and F) in Fig. 3. If the value of 6, in eqn. (78) is assumed to be 0 (h,i = 0), and the value of 6, in eqn. (84) is assumed to be 1 (hC2= hi), then eqns. (78) and (84) can be replaced by eqns. (74) and (76), respectively, by setting 6 = hi/H. Therefore, the flow model for the stirred tank with a radial flow impeller can be handled by the same procedure as the one with an axial flow
Q2=1.2
lo\
Q2=1.6
n=20 l/S 1.0 C’%”
1.0
0.5 “ca”
1.0
0.5 “ca”
hi/H=0.3 C,,=l
OMl%
0
Fig. 10. Effect of solids concentration on axial concentration profiles of solid particles (glass beads, d, = 135 pm) in the stirred tank. (a), Flatdisk turbine; (b), marine propeller; 0, r/D = 0.35; l, r/D = calculated using eqn. (74). 0.05; ,
by the different method of measuring the solids concentration in the stirred tank. In this study, the solids concentrations in the stirred tank were measured directly, using a photoelectric method. Einenkel [ 91, however, measured the solids concentrations in the stirred tank using a simple sampling of the slurry.
01
I
’
II
II
I
1.
0.5 c/ca"
(a) flat disk turbine
c/c_&
10 *
(bl nmine pmpzller
Fig. 11. Axial concentration profiles of solid particles (glass beads, d, = 264 pm) in the stirred tank. 0, calculated Experimental data (r/D = 0.35); , using eqn. (74).
I
214
impeller. The axial concentration profiles can be expressed simply by eqns. (74) and (76) for both types of impeller under the present experimental conditions. The validity of this simplified flow model was also supported by the direct measurement of solids concentrations in the tank. In Figs. 7 and 8, concentration profiles calculated by eqns. (74) and (76) are plotted for comparison. As can be seen, the calculated profiles agree fairly well with the experimental data; the concentration profiles in the upflow and downflow regions can be described by the present sedimentationdispersion model. It can be also seen that the flow pattern of the upper region in the tank is clearly different from that of the lower region. The upflow region was separated from the downflow region at the position oft = F as shown in Figs. 7 and 8. The dimensionless critical height 6 in the tank was then given by the following expression: +
(35)
where h, is the critical height from the tank bottom and g is the dimensionless height from the tank bottom. In this work, the value of 6 is estimated to be 0.3 < S < 0.5 by the direct measurement of solids concentrations in the tank. For the radial flow impeller, the value of 6 was assumed to be 6 = 0.3, which was equal to the height of the impeller (h,). For the axial flow impeller, the value of 6 was determined by curve fitting (6 = 0.4). It is interesting to point out that, when t = 0.3 0.6, C/C,, is nearly constant (pseudohomogeneity). Under the present experimental conditions, the solids concentrations near the bottom of the tank do not increase so significantly. Therefore, the axial concentration profiles of solids suspended in the stirred tank with the radial and axial flow impellers can be simply expressed by eqn. (74) with 6 = 0 under the present experimental conditions. In Figs. 9 and 10, the axial concentration profiles calculated from eqn. (74), where the value of 6 is assumed to be 0, are also plotted for comparison. The solids concentration profiles in a stirred tank are found to be represented satisfactorily by eqn. (74). Axial concentration profiles in a slurry mixing tank depend largely on both the operating variables and the tank design, and the effect of these variables on the
concentration profiles can be interpreted approximately by the present model. Correlation of model parameters The effects of solids concentration on the model parameters, Pei and Pe,, are shown in Figs. 12 and 13, respectively. As can be seen, the Peclet number Pe, increases linearly with an increase in the solids concentration. The ratio of the parameters, Pef/Pe,, which coincides with the ratio of the liquid flow velocity to the falling velocity of solid particles, u/z+, is found to decrease initially as the solids concentration increases and then to approach a constant value. This is mainly due to the decreased upflow velocity in the liquid phase caused by the strong interference with the solid particles. If uf is assumed to be equal to the free settling velocity, the values of u are from 0.01 to 0.2 m/s. These values are in agreement with the data reported in the literature [12, 131. The effect of operating variables and solids properties on the model parameters Pe, or Pef/Pe, can be correlated as a function of nd/u,, which is the ratio of
Glass
beads
0 :dp=B7 arm
10
20
c
10
30
20
c 3(
c lVOl%l
[vol%l
Fig. 12. Effect of solids concentrations on the model parameters, Pe, and Pef /Pe,, in the stirred tank with a flatdisk turbine (n = 16.6 l/s).
20 Glass _
teads
15
2”
x d
1c
c rvol%l
Fig. 13. Effect of solids concentrations on the model parameters, Pe, and Pef /Pe,, in the stirred tank with a marine propeller (n = 13.3 l/s).
215
the tip speed to the free settling velocity. However, the influence of the particle diameter on the model parameters cannot be ignored under the present experimental conditions. In this study, the effect of the particle diameter has been corrected by multiplying (c&/d) by nd/u,. Pe, and Pef/Pe, have been analyzed with (nd/u,)(d,/d), as shown in Figs. 14 and 15. The Peclet number Pe, approaches a constant value at (rid/u,,, (d,/d) = 0.2 or greater under the present experimental conditions, while the parameter ratio Pe,/Pe, is approximately proportional to (ndlu,)(d,ld).
The data reported in the literature [14] are also plotted in Fig. 15 for comparison, though the agreement is qualitative. The trend of the correlation curves of the model parameter, Pe, implies that the agitation with the radial flow impeller is more effective for solids suspension in comparison with the agitation with the axial flow impeller on an impeller speed basis, although the efficiency of agitation should be compared on an energy basis. We shall discuss this point in more detail in our future work.
CONCLUSION
solid particles O,.:Glass hads,$=87um ~,~:Glass
keads,dp=135um
4
15
7
2

’
0.16
0.18
0.20
0.22
0.24
0.26
0.28
1
O
hd/usl(dp/d) i1
Fig. 14. Correlation of model parameters, Pe, and Pef/Pe,, in the stirred tank with a flatdisk turbine (C, = lOvol.%).
solid particles O,.:Glass
teads,$=87um
&&G~SS
beads,dp=1351rm
d
q ,~:'Ityourasands,dp=230m
A 0
[email protected] s
J
0
:E&net and Niemak’s
data
:thiswork
The local concentration profiles of solid particles in the slurry tank with mechanical agitation have been measured, using a photoelectric method. The flow pattern in the slurry mixing tank is too complicated to be described by a rigorous mathematical model for fluid flow. In this study, we have employed the sedimentationdispersion model [5  81 to describe the solids concentration profiles in the slurry mixing tank for solids particles with a free settling velocity of u, < 3.0 X 10e2 m/s. The concentration profile has then been explained in terms of the proposed sedimentationdispersion model. The model parameters have been found to be remarkably influenced by the amount of solids suspended. It has been found that the ratio of the liquid flow velocity to the falling velocity of the solid particles, u/z+, which coincides with the model parameter ratio Pef/Pe,, is approximately proportional to the impeller speed under the complete suspension conditions. LIST OF SYMBOLS
C, C, c a” D d
hd/us)(dp/d) C1
Fig. 15. Correlation of model parameters, Pe, and Pef/Pe,, in the stirred tank with a marine propeller (C, = lOvol.%).
d* ED H
solids concentration in tank, m3/m3slurry average solids concentration, m3/m3slurry inside diameter of tank, m impeller diameter, m particle diameter, pm axial dispersion coefficient for solid particles, m2/s liquid depth in tank, m
216 hc hi ;Js Pef Pe,
Q
r t
U
uf us
X Y 2
critical height of flow regions, m impeller height from tank bottom, m impeller speed, l/s minimum speed of impeller for complete suspension, l/s Peclet number based on fluid velocity (= WE,), Peclet number based on falling velocity of solid particles (= UfH/E,), Peflpe,
(=
Of
1, 
radial distance from center of tank, m time, s superficial velocity of liquid phase, m/s falling velocity of solid particles, m/s free settling velocity of solid particles, m/s dimensionless solids concentration dimensionless solids concentration height, m
Greek symbols dimensionless critical height dimensionless height ! density of solid particles, kg/m3 PP 7 dimensionless time
REFERENCES
1 T. H. Zwietering, Chem. Eng. Sci., 8 (1958) 244. 2 A. W. Nienow, Chem. Eng. Sci., 23 (1968) 1453. 3 C. M. Chapman, A. W. Nienow, M. Cooke and J. C. Middleton, Chem. Eng. Res. Des., 61 (1983) 71. 4 K. Tojo and K. Miyanami, Znd. Eng. Chem. Fundam., 21 (1982) 214. 5 H. Yamazaki, K. Tojo and K. Miyanami, Bull. Univ. Osaka Prefecture, Series A, 31 (1982) 159. 6 W. B. Argo and D. R. Cova, Znd. Eng. Chem. Proceap Des. Dev., 4 (1965) 352. 7 D. R. Cova, Znd. Eng. Chem. Process Des. Dev., 5 (1966) 21. 8 K. Tojo, K. Miyanami and T. Yano, Powder TechnoZ., 12 (1975) 239. 9 W. D. Einenkel, Ger. Chem. En& 3 (1980) 118. 10 G. Baldi and R. Conti, Proc. Znt. Symp. on Mixing, Mons, Belgium, Feb. 21  24, 1978. 11 G. Baldi, R. Conti and A. Gianetto, AZChE J., 27 (1981) 1017. 12 Y. Nagase, T. Iwamoto, S. Fujita and T. Yoshida, Chem. Eng. (Tokyo), 38 (1974) 519. 13 T. Sato and I. Taniyama, Chem. Eng. (Tokyo), 29 (1965) 153. 14 M. Bohnet and G. Niesmak, G. Chem. Zng. Tech., 51 (1979) 314.