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Solids suspension and distribution in liquids under turbulent agitation
Chemi.d
Engineering Science, Vol. 44. No.
Printed in Great
3, pp. 529-542,
OOO!-2509/89 %3.00+0.00 ,Q 1989 Pergamon Press pk
1989.
Britain.
SOLIDS
SUSPENSION AND DISTRIBUTION UNDER TURBULENT AGITATION
IN LIQUIDS
P. AYAZI Department
of Chemical
(First
SHAMLOU+ and E. KOUTSAKOS and Biochemical Engineering, University College London, London WClE 7JE, U.K.
received 13 October
1987; accepted
in revised form
10 August
Torrington
Place,
1988)
Abstract-Experimental data are provided for axial concentration gradients formed during particle suspension in liquids under turbulent agitation. A simple model based on a turbulent diffusion support mechanism is developed in order to interpret the observed longitudinal concentration profiles. The model is based on KolmogorolT’s theory of isotropic turbulence and the settling characteristics of the particles. The reasonable agreement between the theory and the measurements indicates a general applicability for the
proposed
model.
INTRODUCTION
Mechanically agitated vessels have been adopted widely in the chemical, biochemical and allied industries for solid-liquid contacting operations; a particularly important group of these operations includes heterogeneous reactions between negatively buoyant particles and liquids. In these systems the so-called “just-suspension” speed of the impeller ensures that all particles are kept fully in suspension (Ayazi Shamlou and Zolfagharian, 1987; Zwietering, 1958; Nienow, 1968; Baldi et al., 1978). In many industrial units however, e.g. heterogeneous reactors, it is also important to have information on the distribution of solid particles within the agitated liquid. But unlike the condition for just-suspension which has attracted considerable attention in the past (Nienow, 1985; 1987; Ayazi Shamlou and Molerus and Latzel, Zolfagharian, 1987; Baldi et al., 1981) there have been few reported studies in which the distribution of particles in the agitated liquid has been investigated systematically. This is the aim of the present investigation. Local solid concentrations have been measured at different impeller speeds using a non-intrusive optical
technique.
The
effects
of tank
diameter
and
impeller configuration on the distribution of monosized particles with different physical properties have been studied. A one-dimensional steady-state model based on a turbulent diffusion mechanism is proposed to describe the axial distribution of the suspended particles in the bulk of the agitated liquid. The model assumes that the axial movement of the suspended particles is caused by the longitudinal dispersion of the liquid and the settling of the solid particles. The longitudinal dispersion coefficient of the particles is assumed to be proportional to the liquid dispersion coefficient which is, in turn, related to the turbulent characteristics of the agitated fluid.
‘To
whom correspondence
should be addressed.
Fig. 1. Schematic diagram of experimental facility: (I) motor, (2) torque transducer, (3) speed controller, (4) speed/torque/ power indicator, (5) outer tank, (6) turbine impeller, (7) inner conical vessel, (8) conical deflector, (9) baffle (10) laser transmitter, (11) safety cutoff filter, (12) photocell receiver, (13) noise filter, (14) A/D converter, (15) digital voltage display, (16) microcomputer, (17) chart recorder.
EXPERIMENTAL
A schematic drawing of the experimental equipment is shown in Fig. 1. Experiments were carried out in a 0.225 m diameter, spherical-bottom, cylindrical glass vessel with a height to diameter ratio of approximately 3. Agitation was provided by a series of 45” pitch529
530
P. AYAZI Table
SHAMLOU
1. Diameters of impellers used in this work
Impeller No. 1 2 3
Impeller diameter D (cm)
and E. KOUTSAKOS Sampling position Sampliw position
B -YT 0.044T A
I
irl
10 8.5 6.5 --c
+-O.lT 3T
0.29D
Fig. 3. Sampling positions and geometrical details of experimental vessel. Fig. 2. Details of impellers used in this work.
bladed open turbines (Table 1 and Fig. 2) driven by a l.l-kW motor with infinitely variable speed control within the range 0-2ooO rpm. The vessel was also equipped with four flat baffles, each of a width equal to 10% of the tank diameter, equally spaced and positioned firmly against the inner wall of the tank. Moreover, to prevent the formation of dead zones in the spherical bottom of the vessel, a conical deflector was fixed directly beneath the impeller (Jones and Mullin, 1973). The power input to the impeller was measured using a shaft-mounted torque transducer (Ayazi Shamlou and Edwards, 1985). Results on power measurements will be reported separately. The experimental set up for the measurement of the local concentration of solids (Fig. 1) consisted of a 5mW He-Ne laser transmitter with an expanded beam diameter of 9.0 mm. The intensity of the beam, after passing through the suspension, was measured by a photovoltaic receiver. Since the response of the photocell was linear, the voltage output from it was used directly as a measure of the light intensity. For mono-sized spherical particles, constant optical path length and low concentration of solids, the relationship between light transmittance and concentration was found by calibration using well-defined and nearhomogeneous suspensions (Ayazi Shamlou and Koutsakos, 1986). The effect of particle diameter and optical path length on light transmittance was found by a simple modification of the Beer-Lambert law (Bos and Heerens, 1982). The laser transmitter and receiver were fixed relative to each other but could be adjusted in the vertical and horizontal planes. This enabled mean concentration of solids to be measured at various horizontal
planes from the bottom and center of the vessel. The output voltage from the receiver was connected directly to a microcomputer which greatly enhanced the speed of collection, analysis and presentation of data. The glass tank was placed inside a rectangular transparent container filled with water to reduce optical distortions. Water and glass ballotini, narrowly classified to provide a range of particle sizes, were the main materials used in this study. Further details on materials, equipment and experimental procedure may be found elsewhere (Ayazi Shamlou and Koutsakos, 1986; Koutsakos, 198X). RESULTS
Figure 3 shows the sampling positions in the liquid. During the initial runs, mean concentrations of solids were measured at two radial locations at each of which measurements were made at various heights from the bottom of the tank (Fig. 3). It is emphasised that each individual reading is the mean value of the solids concentration along a horizontal plane and not a point concentration. In the present study this mean concentration is referred to as the local concentration in that horizontal plane. Initial results (Kontsakos, 1988) indicated that in the majority of cases the biggest variation in concentration of solid particles in the “core of the liquid” was in the axial direction. The “core of the liquid” refers to the bulk of the agitated fluid away from the walls, the region near the liquid surface and the discharge zone of the impeller. Examples of axial distributions of solids in the steady state during some typical runs are shown in Figs Q-10. The data are plotted as the ratio of the mean concentration of solids for a given horizontal plane to the total concentration in the tank, E/c,,
Solids suspensionand distribution in liquids under turbulent agitation Key . -0
0 0
531
hkm) 7.0 22.0
* 0
30.0 40.0
.
53.0
I
I
100
900
Impeller speed (rpm)
11 30
Fig. 4. Solids concentration vs impeller speed for differentsample heights (& = 390 pm, p. = 2900 kg/m”, ~=lOOOkg/m~, C=13cm,D=lOcm,H=65cm,C,=l wt%).
1.5
1
u”
;3
0.5
0
100
200
300
400
600
600
700
600
900
'II
lmpellerspeed(rpm)
Fig. 5. Solids concentration vs impeller speed for differentsample heights (a,, = 390 pm, ps = 2900 kg/m”, p=1000kg/m3,C=13cm,D=iOrm,W=65cm,C,=2~t%).
against impeller speed, N, for different axial positions, h, measured from the base of the tank. The curves indicate a number of distinct zones. At low impeller speeds, below about 200 r-pm in the case of the data shown in Figs 4-10, most solid particles remain on the bottom of the vessel with little, if any, motion. At higher speeds an intermediate zone is formed characterized by a comparatively steep curve indicating a sharp increase in solids concentration with speed as more and more particles are picked up from the base of the tank and carried into the bulk of the agitated liquid. The rate of rise and the absolute value of the local concentration vary with speed and sampling
position. It is important to note that in each case shown in Figs 4-10 a peak in concentration occurs at a critical impeller speed which has been identified as the just-suspension speed (Ayazi Shamlou and Koutsakos, 1986; Musil et al., 1984; Bourne and Sharma, 1974; Machon et al., 1982; Ohiaeri, 1981). Three correlating equations, typical of those published in the literature, are used to predict values of N, based on the tank/impeller configurations used in this work. The results are tabulated in Figs 4-10 and serve to highlight the degree of deviation that exists between values of Nj. predicted by these equations. However, these deviations are not surprising given the ambi-
532
P. AYAZI SWAMLOU and E. KOUTSAKOS Key
h(cm)
I
7.0
* 0
22.0 30.0 40.0
.
53.0
1-O
Impellerspeed
(rpm)
Fig. 6. Solids concentration vs impeller speed for different sample heights (a_ = 390 pm, p. = 2900 kg/ma, p = 1000 kg/m3, C = 13 cm, D = 10 cm, H = 65 cm, C, = 3 wt’?/Q).
4-
2-
O0 Fig. 7. Solids concentration vs impeller speed for different sample heights (a, = 390 pm, p, = 2900 kg/m”, p = 1000 kg/m3, C = 13 cm, D = 10 cm, H = 65 cm, C, = 6 wt%).
guity
in the definition
of and
the experimental
un-
in the measurement of N, reported in most of the published investigations. With regard to the shape of the curves, the rate of change of the local concentration of solids with speed is very gradual for sampling positions near the surface of the liquid. Moreover as the total hold-up of solids increases, the point of maximum concentration becomes more difficult to identify. For values of total concentration of solids below about 4% the experimental data indicate that any sampling location below the position of the impeller may be used effectively to identify the critical speed at which the peak in concentration occurs.
certainties
As the speed of the impeller increases beyond the point of maximum concentration a third zone is formed in which the local concentration of solids remains constant or decreases slightly with increase in speed. The exact behaviour of the suspended particles depends upon many factors including particle size and density and the size and position of the impeller (Figs 8-10). Similar observations have been reported by previous workers (Musil, 1976; Boume and Sharma, 1974; Machon et nl., 1982). Figures 11-15 show the variation in the concentration of solids as a function of height for different impeller speeds, total solids hold-up and particle size. The curves indicate that the concentration of solids
Solids suspension and distribution in liquids under turbulent agitation
lmpellerspeed
533
(rpm)
Fig. 8. Solids concentration vs impeller speed for different sample heights (ri, = 390 pm, ps = 2900 kg/ma. p = 1000 kg/mJ. C = 11.3 cm, D = 10 cm, H = 37 cm, C, = 1 wt%).
'.
0
0
20.2
.
25.8 30.3
100
200
300
Impellerspeed
400
500
6
(rpm)
Fig. 9. Solids concentration vs impeller speed for different sample heights (zP = 390 pm, p. = 2900 kg/m’, p=1000kg/m3,C=11.3cm,D=8.5cm,H=37cm,C,=1 wt%).
increases nearly exponentially with depth from a region just below the surface except in the zone close to the impeller blades where, at high impeller speeds, the concentration of solids appears to oscillate about a mean value. The amplitude of these oscillations is dependent upon the speed of the impeller and, to some extent, on the radial position. These oscillations in concentration in the discharge zone of the impeller must be! related to the periodic passage of the impeller and thd centrifugal forces generated as the impeller rotates. At high impeller speeds these forces are sufficiently strong to overcome the forces tending to increase the uniformity in concentration.
DISCUSSION
The axial distribution of solid particles at a steady state in the bulk of the agitated liquid may be interpreted in terms of a one-dimensional balance between the net flux of particles caused by longitudinal turbulent diffusion down the concentration gradient (i.e. upwards in the vessel) to the downward rate of settling of particles under the action of the gravitational force (Kay, 1957; Ayazi Shamlou and Koutsakos, 1986; Barresi and Baldi, 1986). Thus, from a mass b$lance on the particles over a thin horizontal section of the liquid in the vessel, and assuming that there is no accumulation or depletion of particles, the following equation
534
P. AVAZI SHAMLOU
and E. Kours~os
8.10 15.2 20.2 l
25.8
impeller Fig.
speed (rpml
10. Solids concentrationvs impellerspeed for differentsample heights (&, = 390 W, p = 1000 kg/m3, C = 11.3 cm, D = 6.5 cm, H = 37 cm, C, = 1 WV!!).
Key
NIrPm)
.
300 400
0 P 0 .6cG *
450 500 700 800
*
I
0.3
I
t
I
0.6
0.9
I
I 1.2
pa = 2900 kg/m3.
I
1.5
WC0
Fig. 11. Dimensionless solid concentration ratio vs dimensionless height ratio for different impeller speeds (d, = 390 pm, pS = 2900 kg/m3, p = 1000 kg/m’, D = 10 cm, C = 13 cm, H = 65 cm, C, = 1 wt%).
may be written:
where 5, and U, are, respectively, the dispersion coefficient and settling velocity of the particles in the agitated suspension. Assuming that, for a fixed system geometry and a given operating condition, the mean values of these parameters remain constant in the bulk of the agitated liquid, eq. (1) may be rearranged to give d (In c)
-=
dh
-us/<,.
(2)
Thus, a plot of In E against height, h, is expected to be a straight line with a slope of - U,/<, _ Figures 16-18 are typical plots showing the variation in the mean local concentration of solids as a function of axial position. h. The lines indicate that, within the range of operating variables studied, eq. (2) describes the experimental data adequately. Similar results were obtained for other values of total solids concentration and impeller diameter (Koutsakos, 1988). In order to study the effect of some of the more important variables, such as the impeller diameter and the speed and physical properties of the particles,
Solids suspensionand distributionin liquids under turbulentagitation N(rpml
Key
.
300
0
400 500 560 600 700 800
P
0 0 P *
0
0
I
0.3
I
I
I
I
0.6
535
I
0.9
1.2
E/c,
Fig. 12. Dimensionless solid concentration vs dimensionlessheight ratio for different impeller speeds (C, = 2 wt%, all other experimentaldetails as in Fig. 11).
Key
N
.
(rpm)
*
300 400 500
0 . *
620 650 700
0
A + + + l *
0 I
I 0.3
I 0.6
I
I 0.9
2
E/C,
Fig. 13. Dimensionless solid concentration vs dimensionless height ratio for different impeller speeds (C, = 3 wt%, all other experimentaldetails as in Fig. 11).
upon the distribution of particles in the agitated liquid it is first necessary to understand the effect that these variables have on U, and 5,. Assuming that relatively small particles in suspension behave in the same way as the agitated liquid then the particle diffusion coefficient, tp, may be expected to coincide closely with the liquid diffusion coefficient,
agitated vessel, both of these turbulent parameters may be expressed in terms of the size of the impeller, D, and the rate of energy dissipation, +. . Thus, in the fully turbulent region: Ed a N3 DS/T2H.
(3)
It is assumed that away from the discharge zone of impeller, the mean rate of energy dissipation in the core of the vessel, E*, is directly proportional to +_ Moreover, for small eddies in the Kolmogoroff range,
536
P.AYMI
SHAMLOU
andE.
KOUTSAKOS
Key Ntrpm) I P
300 350 400
0 a n
420 450 500
0
B f + + + + +
0 al 0
I 0.2
I 0.4
I 0.6
I 0.8
I 1
2
e/c,
Fig. 14. Dimensionless
solid concentration vs dimensionless height ratio for different impeller speeds (a, = 530 ,um, p, = 2900 kg/m3, p = 1000 kg/ma, D = 10 cm, C = i 1.3 cm, H = 37 cm, C, = 1 wt%).
Key
Fig. 15. Dimensionless (C, = 3 wt%,
N (rPm)
I
300
0
* 0 . n
400 450 500 600 700
*
800
solid concentration vs dimensionless height ratio for different impeller speeds samples taken at position B: other experimental details as in Fig. 11).
L,, the fluctuating velocity, U ‘, may be written as U’ K (&TL,)“3.
1972): (4)
In the present analysis, it is assumed that eq. (4) may be applied to the larger energy containing eddies of scale L, (Davies, 1972) that are responsible for keeping the particles in suspension. The turbulent liquid diffusion coefficient, c,, has also been related to the fluctuating velocity and to the critical eddy size by the following equation (Davies,
5, a U’L,.
(5)
Assuming that the diffusion coefficient of solid particles is of the same order as that of the fluid, eqs (4) and (5) may be used to give 5, oz <, oc U’L,
cc (E7Le)1’3L,.
(W
With L, proportional to the impeller diameter, D, but practically independent of the speed of the impeller
Solids suspension and distribution in liquids under turbulent agitation
537
Key N(rpm) 300 400 500 600 700 800
.
0 * 0 . *
Concentration,
._
C (g/l)
Fig. 16. Semilogarithmic plot of solid concentration vs sample height for different impeller speeds (zP= 390 pm, p, = 2900 kg/m3, p = 1000 kg/m”, D = 10 cm, C = 11.3 cm, H = 37 cm, C, = 1 wt%).
N (rpm)
Kev .
300 350
0
01 0.1
I
I
IllIll
I
Concent Aion,
10
E(glll
Fig. 17. Semilogarithmic plot of solid concentration profile YS sample height at differentimpeller speeds (zp = 530 pm, p. = 2900 kg/m’, p = 1000 kg/m’, H = 37 cm, D = 10 cm, C = 11.3 cm, C, = 1 wt%).
and of the viscosity in the fully turbulent region, eq. (5a) reduces to 5, a ND3/(T2H)‘j3.
(6)
The relative settling velocity of particles in suspension, U,, is a complex function of particle properties, solids concentration and the intensity of fluid turbulence (Clift and Gauvin, 1971). As far as the effect of solids concentration is concerned, for voidage values approaching unity, which is the case in the present study, the particles may be assumed to follow the pattern adopted in free settling. McCabe and Smith (1976) have proposed a convenient criterion for the estimation of the terminal velocity in the still fluid, U,, ces
44:3-P
using a single factor, K, defined as K = zP(gpAp/p2)“3. For
3.3<
(7)
K < 43.6
0.153~0.7’~~.‘4~~0.7~ U*= P”.29PoL43
(8)
and for 43.6 < K < 2360
u, = (gzPAp/p)“2.
(9)
For K -K 3.3, the Stokes’ equation may be used to estimate U,. Turning now to the effect of turbulence, some studies (Clift and Gauvin, 1971; Schwartzberg and Treybal, 1968) have shown that under turbulent
538
P. AYAZI
Fig.
SHAMLOU
and E. KOUTSAKOS
18. Semilogarithmicplot of local solids concentrationYSsample heightat differentparticlediametersat N = 500 rpm, C, = 1 wt% (for remaining experimentaldetails see Fig. 17).
conditions the fall velocity of a single particle in an agitated liquid, U, , can deviate greatly from the terminal velocity in the still fluid. Experimental data obtained over a range of particle sizes and densities indicate that, to a first approximation, the ratio of U,,/U, falls in the range 0.3-0.5. Schwartzberg and Treybal(l968) recommended a mean value of 0.4 for this ratio. In the light of this, either U,, or U, may be used to characterize the behaviour of the solid particles within the agitated liquid. From what has been said above, it seems reasonable to assume that the ratio U,/c, in eq. (2), i.e. the reciprocal of the slope of the lines in Figs 1618, may be written as U,/<,
a U,/[ND3/(T2H)“3].
(10)
Equation (10) provides the relationship between the particle dispersion coefficient and some of the important parameters affecting it. E#ect of impeller speed upon solids concentration profiles For a fixed tank/impeller geometry, eq. (10) reduces to (11) Equation (11) may be expressed in dimensionless form by using a Peclet number defined as l-J,&,/<, and employing the impeller tip speed ND. Thus U&l<,
cc U,IND.
(lla)
Equation (11) is verified experimentally in Figs 16 and 17 which show semi-log plots of local concentration of solids against height for various impeller speeds. In these plots, absolute homogeneity is represented by a vertical line. It is interesting to note that the slope of the lines can be used as a direct measure of the quality of the suspension.
The slopes of the lines in Fig. 16 are used to calculate the dimensionless Peclet number, U,zJc,. In Fig. 19, experimental data from Fig. 16 have been plotted on log-log graph paper as the reciprocal of the dimensionless Peclet number against speed, N. A slope very close to unity is observed in accordance with the proposed model [eq. (1 l)]. Efict of particle diameter upon solids concentration profiles For a given tank/impeller geometry and a fixed impeller speed, i.e. constant t,, an increase in particle diameter leads to an increase in U, and hence to a decrease in the homogeneity of the suspension (Fig. 18). Local concentrations of solids as a function of height for various diameters of glass particles are shown in Fig. 18 and confirm the well-established experimental fact that, for a given tank and impeller configuration and for known operating conditions, uniformity of solids concentration increases with decreasing particle size. Figure 20 shows data from Fig. 18 plotted as the dimensionless Peclet number against terminal velocity of a single particle. The slope of the line is proportional to the 1.6th power of U,, indicating that the relationship between U, and U, is more complicated than that predicted by the present model [eq. (1 l)]. This is perhaps not surprising because both the bulk motion and the turbulent nature of the agitated liquid can be expected to have profound effects upon U,. It is however difficult to quantify these effects at the present stage of development. With regard to Figs 1618 it may be noted that absolute homogeneity is not reached even at impeller speeds several times greater than the complete suspension speed. Indeed, above a critical impeller rotation, centrifugal forces created by the action of the impeller
Solids suspension and distribution in liquids under turbulent agitation Key
Impellerspeed,
539
a,(&m) 175 390 530 780
N (rps)
Fig. 19. Effect of impeller speed on the ratio {,,/7_J,zP (D = l_Ocm, C, = 1 wt%, C = 11.3 cm, H = 37 cm; for & = 39,175,530 pm, p. = 2900 kg/m3; for d, = 780 Frn p. = 3800 kg/m’).
10-Z
10-4
1
0.01
Key I .
ps (kg/m31 2900 3800
I
I Terminal
I
L
velocity,
I1111 U,
0.1
(m/s)
Fig. 20. Effectof terminal velocity on the ratio lJ,&J5, (N = 500 rpm, D = 10 cm, C, = 1 wt%, C = 11.3 cm).
dominate in a substantial proportion of the vessel causing large fluctuations in local solids concentration, and consequently deviations from homogeneity are observed. Therefore, the best that can be achieved in practice is an optimum homogeneity often referred to as the “pseudo-homogeneous” condition.
Efict of total solids hold-up upon concentration gradients Figure 21 shows the experimental results plotted as
local concentration of solids against height for various values of total solids hold-up in the vessel. The data presented in Fig. 21 are for particles with a diameter of
390 pm suspended in water by a turbine impeller of 0.1 m diameter and rotating at 300rpm. It may be observed that, although the absolute value of the concentration of solids at any location depends upon the total solids hold-up, the distribution of particles for c, < 4% is practically independent of it as indicated by the constant slope of the lines. This suggests that, within the range of variables studied, the hold-up concentration has little effect on the ratio of the particle diffusion coefficient to the particle settling velocity. It is however expected that, as the total solids concentration in the tank increases, particle--particle interference will increase, influencing both <,, and U, and, as a result, the ratio <,/US.
540
P.
1
AYAZI
I
I
SHAMLOU
,
and E. KOUTSAKOS
Illll
I
I
I,,,,
I
100
10
Concentration,
C (g/l)
Fig. 21. Effect of mean hold-up concentration upon solids concentration gradient (zp = 390 pm, ps = 2900 kg/m3, p = 1000 kg/m ‘,D=lOcm,C=15cm,H=65 cm, N = 300 rpm, sampling position R).
Key . .
Concentihm, Fig.
Effect
D lcm) 10.0 8.50
i: (g/l)
22. Semilogarithmicplot of solid concentration vs sample height for different impeller diameters at N =500 t-pm, CO= 1 wt% (all other experimental details as in Fig. 17).
of impeller
diameter
upon
solids
concentration
pro@ les
Equation (6) indicates that for a fixed impeller speed and a given fluid/particle and tank configuration, increasing impeller diameter, D, will increase the diffusivity of the particles and, as a result, better uniformity should be attained within the bulk of the agitated liquid. This is demonstrated in Fig. 22 in which data have been plotted as the dimensionless concentration against height for three different impeller diameters at a fixed impeller speed. Figure 23 shows the data in Fig. 22 plotted as the dimensionless Peclet number against impeller diameter. The lines have a slope of 2.8 which compares favourably with the expected value of 3 [eq. (6)].
From the results presented so far, it may be concluded that the dimensionless Peclet number is a correlating parameter that can bring together experimental data from various runs. Moreover, the Peclet number may be related to impeller speed and the terminal settling velocity of the particles. All the results presented in this article are replotted in Fig. 24, data for the various runs have been brought together in a single plot confirming the proposed approach. An important assumption in the development of the turbulent dispersion model proposed in this work is the existence of homogeneous and isotropic turbulence in the agitated fluid. The question of whether turbulence in a mixing vessel is isotropic or not has been a matter of contention between investigators in
Solids suspension and distribution in liquids under turbulent agitation
541
./ . //
l /
:/
v /
I
.
/
I
r
IllI
I1
I
0.1 bmpeller
Fig.
23. Effect
of impeller
Fig. Fig. Fig. Fig. Fig. Fig.
diameter
on
the
diameter,
ratio
c =
(,/USaP
D (m)
(zP = 530pm,
D =
IOcm, C, = 1 wt%,
11.3 cm).
19,$=175pm 19, d, = 390 km 20 23, N = 500 rpm 23, N = 400 rpm 23, N = 300 rpm
P
Fig. 19, d,, = 530 pn dp = 780 pm, ljs = 3800
kg/m3
1 I
I .
Fig. 24. Dimensionless
ratio
U,~pj~, as a function of system parameters.
this field, e.g. early studies (Davies, 1972) indicate that turbulence is spatially highly variable throughout the vessel while some recent measurements (Mersmann and Laufhuette, 1985; Laufhuette and Mersmann, 1985) of U’ using the LDA technique have shown that, in the core of the vessel, i.e. away from the discharge zone of the impeller and the immediate vicinity of the walls of the vessel and the surface of the liquid, turbulence is nearly isotropic. In the discharge zone of the impeller the fluctuating velocity is confounded by the periodic passage of the impeller. These workers have also attempted to separate the total fluctuating
velocity
in the discharge
components
which
zone
of the impeller
into two
a;e assumed to be additive, i.e. u total= u ’+ Upcr
(12)
where 7-Jperis the contribution due to the periodic passage of the impeller and U ’ is the turbulent part. Measurements of the local concentration of solids presented in this work (Figs 16-18) appear to support the general observations of Mersmann and Laufhuette (1985) and Laufhuette and Mersmann (1985); nonisotropic turbulence and hence a variable 5, in the core of the vessel would be expected to give non-linear
542
P. AYAZI
SHAMLOU
plots. Moreover, the observed oscillations in the local concentration of solids in the discharge zone of the impeller (Figs 1l-l 5) are clearly due to the action of the impeller and, thus, indirectly support the view that the nature of fluid turbulence in the discharge zone is confounded by the periodic passage of the impeller. It is therefore important to appreciate that the concept of a constant mean eddy/particle diffusivity as proposed in this work is only valid in the bulk region of the vessel and its extension to the discharge zone of the impeller is an approximation.
CONCiUSIONS
Measurements have been provided of solids concentration profiles in the bulk of a mechanically agitated fluid in the fully turbulent flow regime. Over the range of variables studied the data indicate a near-exponential decay in the concentration of solids with axial height from the surface of the liquid in the vessel. These solids concentration profiles have been interpreted in terms of a steady-state turbulent diffusion support mechanism. The distribution of particles results from the balance between the random motion of the liquid in the vessel and the action of gravity on the particles. In this way, the distribution of particles in the agitated liquid is characterized by a single parameter, nameIy the ratio of the turbulent diffusion coefficient of the particles to their terminal settling velocity. All other properties of the system, such as particle size and density, impeller diameter and speed and fluid properties exert their effects only through the value of this ratio.
NOTATION
mean axial concentration of solids, g/l total concentration of solids in the tank, g/l diameter of the impeller, m mean particle diameter, m acceleration due to gravity, m’/s height of liquid in the tank, m height of sampling point, m constant in eq. (7) size of the critical energy containing eddies, m size of eddies in the Kolmogoroff range, m impeller speed, rpm fluctuating velocity, m/s fall velocity of a solid particles in suspension, m/s fall velocity of a single particle in turbulent flow, 111/S
fall velocity Greek &b
&T
p r/
of a single particle in still fluid, m/s
letters
bulk or mean energy dissipation
per unit mass,
W/kg total energy input per unit mass, W/kg liquid viscosity, kg/m s liquid diffusion coefficient, m/s particle diffusion coefficient, m/s liquid density, kg/m3
and E. KOUTSAKOS Ps
solid density, kg/m3
AP
P.--P REFERENCES
Ayazi Shamlou, P. and Edwards M F., 1985, Power consumption of helical ribbon mixers in viscous Newtonian and non-Newtonian fluids. Chem. Engng Sci. 40,
1773-1781. Ayazi Shamlou, P. and Koutsakos, E., 1986, Solids suspension in mechanically agitated systems, in Proceedings ofthe International Conference on Mechanical Agitation, Toulouse, France, pp 3-9-3- 16. Ayazi Shamlou, P. and Zolfagharian, A., 1987, Incipient solid motion in liquids in mechanically agitated vessels, in IChemE Symposium, Fluid Mixing III, Bradford University, UK, pp. 195-208.
Baldi, G., Conti, R. and Alaria, E., 1978, Complete suspension of particles in mechanically agitated vessels. Chem.
Engng Sci. 33,21-25. Baldi, G., Conti, R. and Gianetto A., 1981, Concentration profiles for solids suspended in a continuous agitated J. 33, 21-25. reactor. A.I.Ch.E. Bos, A. S. and Heerens, J. J., 1982, Light backed-scattering as a technique to measure solids particle size and concentration in suspensions. Chem. Engng Commun. 16,301-309. Bourne, J. R. and Sharma, R. N., 1974, Suspension characteristics of solid particles in propeller-agitated tanks, in First European Conference on Mixing and Centrifugal Separation, BHRA, Cambridge, UK, pp. B3-25. Barresi, A. and Baldi, G., 1986, Solids suspension in an agitated vessel, in Proceedings of the International Conference on Mechanical Agitation, Toulouse, France, pp 334-3-41. Clift, R. and Gauvin, W. H., 1971, Motion of entrained particles in gas streams. Proc. Powtechnol. 71, 47-57. Davies, J. T., 1972, Turbulence Phenomena. Academic Press, New York. Jones, A. G. and Mullin. J. W., 1973, The design of a drafttube agitated vessel. Chem. Ind. 387-389. Koutsakos, E., 1988, PhD thesis, University College London (in preparation). and Heat Transfer, 1st Kay, J. M., 1957, Fluid Mechanics Edition. Cambridge University Press, Cambridge. Laufhuette, H. D. and Mersmann, A. B., 1985, Dissipation of Conference on powers in stirred vessels, in 5th European Mixing, Wurzburg, Germany, pp. 331-341. Machon, V., Fott, I. and Skrivahek, J., 1982, Local solids distribution in the space of a stirred vessel, in 4th European Conference on Mixing, Netherlands, pp. 289-302. McCabe, W. L. and Smith, J. C., 1976, Unit Operations of Chemical Engineering. 3rd Edition. McGraw-Hill, New York. Mersmann, A. and Laufhuette, H. D., 1985, Scale-up of agitated vessels for different mixing processes, in 5th European Conference on Mixing, Wurzburg, Germany, pp. 273-285. Molerus, 0. and Latzel, W., 1987, Suspension of solid particles in aaitated vessels-I. Archimedes numbers 540. Chem. EnQng-Sci. 42, 1423-1430. Musil. L.. 1976, The hvdrodvnamics of mixed crvstallizers. _ Colln Czech. them. dommk 41, 839-852. Musil, L., Vlk, J. and Jirondkora, H., 1984, Suspending solid particles In an agitated tank with axial-type impellers. Chem. Engng Sci. 39, 621-628. Nienow, A. W., 1968, Suspension of solid particles in turbine agitated baffled vessels. Chem. Engng SC;. 23, 1453-1459. Nienow, A. W., 1985, in Mixing in the Process Industries (Edited by N. Hamby, M. F. Edwards and A. W. Nienow), Chap. 16. Butterworths, London. Ohiaeri, I. N., 1981, PhD thesis, University of Bradford. Schwartzberg, H. G. and Treybal, R. E., 1968, Fluid and narticle motion in turbulent stirred tanks. Ind. Enana ” Y ‘Chem..Fundam. 7, 1-12. Zwietering, T. N., 1958, Suspending of solid particles in liquid by agitators. Chem. Engng Sci. 8, 244253.
Engineering Science, Vol. 44. No.
Printed in Great
3, pp. 529-542,
OOO!-2509/89 %3.00+0.00 ,Q 1989 Pergamon Press pk
1989.
Britain.
SOLIDS
SUSPENSION AND DISTRIBUTION UNDER TURBULENT AGITATION
IN LIQUIDS
P. AYAZI Department
of Chemical
(First
SHAMLOU+ and E. KOUTSAKOS and Biochemical Engineering, University College London, London WClE 7JE, U.K.
received 13 October
1987; accepted
in revised form
10 August
Torrington
Place,
1988)
Abstract-Experimental data are provided for axial concentration gradients formed during particle suspension in liquids under turbulent agitation. A simple model based on a turbulent diffusion support mechanism is developed in order to interpret the observed longitudinal concentration profiles. The model is based on KolmogorolT’s theory of isotropic turbulence and the settling characteristics of the particles. The reasonable agreement between the theory and the measurements indicates a general applicability for the
proposed
model.
INTRODUCTION
Mechanically agitated vessels have been adopted widely in the chemical, biochemical and allied industries for solid-liquid contacting operations; a particularly important group of these operations includes heterogeneous reactions between negatively buoyant particles and liquids. In these systems the so-called “just-suspension” speed of the impeller ensures that all particles are kept fully in suspension (Ayazi Shamlou and Zolfagharian, 1987; Zwietering, 1958; Nienow, 1968; Baldi et al., 1978). In many industrial units however, e.g. heterogeneous reactors, it is also important to have information on the distribution of solid particles within the agitated liquid. But unlike the condition for just-suspension which has attracted considerable attention in the past (Nienow, 1985; 1987; Ayazi Shamlou and Molerus and Latzel, Zolfagharian, 1987; Baldi et al., 1981) there have been few reported studies in which the distribution of particles in the agitated liquid has been investigated systematically. This is the aim of the present investigation. Local solid concentrations have been measured at different impeller speeds using a non-intrusive optical
technique.
The
effects
of tank
diameter
and
impeller configuration on the distribution of monosized particles with different physical properties have been studied. A one-dimensional steady-state model based on a turbulent diffusion mechanism is proposed to describe the axial distribution of the suspended particles in the bulk of the agitated liquid. The model assumes that the axial movement of the suspended particles is caused by the longitudinal dispersion of the liquid and the settling of the solid particles. The longitudinal dispersion coefficient of the particles is assumed to be proportional to the liquid dispersion coefficient which is, in turn, related to the turbulent characteristics of the agitated fluid.
‘To
whom correspondence
should be addressed.
Fig. 1. Schematic diagram of experimental facility: (I) motor, (2) torque transducer, (3) speed controller, (4) speed/torque/ power indicator, (5) outer tank, (6) turbine impeller, (7) inner conical vessel, (8) conical deflector, (9) baffle (10) laser transmitter, (11) safety cutoff filter, (12) photocell receiver, (13) noise filter, (14) A/D converter, (15) digital voltage display, (16) microcomputer, (17) chart recorder.
EXPERIMENTAL
A schematic drawing of the experimental equipment is shown in Fig. 1. Experiments were carried out in a 0.225 m diameter, spherical-bottom, cylindrical glass vessel with a height to diameter ratio of approximately 3. Agitation was provided by a series of 45” pitch529
530
P. AYAZI Table
SHAMLOU
1. Diameters of impellers used in this work
Impeller No. 1 2 3
Impeller diameter D (cm)
and E. KOUTSAKOS Sampling position Sampliw position
B -YT 0.044T A
I
irl
10 8.5 6.5 --c
+-O.lT 3T
0.29D
Fig. 3. Sampling positions and geometrical details of experimental vessel. Fig. 2. Details of impellers used in this work.
bladed open turbines (Table 1 and Fig. 2) driven by a l.l-kW motor with infinitely variable speed control within the range 0-2ooO rpm. The vessel was also equipped with four flat baffles, each of a width equal to 10% of the tank diameter, equally spaced and positioned firmly against the inner wall of the tank. Moreover, to prevent the formation of dead zones in the spherical bottom of the vessel, a conical deflector was fixed directly beneath the impeller (Jones and Mullin, 1973). The power input to the impeller was measured using a shaft-mounted torque transducer (Ayazi Shamlou and Edwards, 1985). Results on power measurements will be reported separately. The experimental set up for the measurement of the local concentration of solids (Fig. 1) consisted of a 5mW He-Ne laser transmitter with an expanded beam diameter of 9.0 mm. The intensity of the beam, after passing through the suspension, was measured by a photovoltaic receiver. Since the response of the photocell was linear, the voltage output from it was used directly as a measure of the light intensity. For mono-sized spherical particles, constant optical path length and low concentration of solids, the relationship between light transmittance and concentration was found by calibration using well-defined and nearhomogeneous suspensions (Ayazi Shamlou and Koutsakos, 1986). The effect of particle diameter and optical path length on light transmittance was found by a simple modification of the Beer-Lambert law (Bos and Heerens, 1982). The laser transmitter and receiver were fixed relative to each other but could be adjusted in the vertical and horizontal planes. This enabled mean concentration of solids to be measured at various horizontal
planes from the bottom and center of the vessel. The output voltage from the receiver was connected directly to a microcomputer which greatly enhanced the speed of collection, analysis and presentation of data. The glass tank was placed inside a rectangular transparent container filled with water to reduce optical distortions. Water and glass ballotini, narrowly classified to provide a range of particle sizes, were the main materials used in this study. Further details on materials, equipment and experimental procedure may be found elsewhere (Ayazi Shamlou and Koutsakos, 1986; Koutsakos, 198X). RESULTS
Figure 3 shows the sampling positions in the liquid. During the initial runs, mean concentrations of solids were measured at two radial locations at each of which measurements were made at various heights from the bottom of the tank (Fig. 3). It is emphasised that each individual reading is the mean value of the solids concentration along a horizontal plane and not a point concentration. In the present study this mean concentration is referred to as the local concentration in that horizontal plane. Initial results (Kontsakos, 1988) indicated that in the majority of cases the biggest variation in concentration of solid particles in the “core of the liquid” was in the axial direction. The “core of the liquid” refers to the bulk of the agitated fluid away from the walls, the region near the liquid surface and the discharge zone of the impeller. Examples of axial distributions of solids in the steady state during some typical runs are shown in Figs Q-10. The data are plotted as the ratio of the mean concentration of solids for a given horizontal plane to the total concentration in the tank, E/c,,
Solids suspensionand distribution in liquids under turbulent agitation Key . -0
0 0
531
hkm) 7.0 22.0
* 0
30.0 40.0
.
53.0
I
I
100
900
Impeller speed (rpm)
11 30
Fig. 4. Solids concentration vs impeller speed for differentsample heights (& = 390 pm, p. = 2900 kg/m”, ~=lOOOkg/m~, C=13cm,D=lOcm,H=65cm,C,=l wt%).
1.5
1
u”
;3
0.5
0
100
200
300
400
600
600
700
600
900
'II
lmpellerspeed(rpm)
Fig. 5. Solids concentration vs impeller speed for differentsample heights (a,, = 390 pm, ps = 2900 kg/m”, p=1000kg/m3,C=13cm,D=iOrm,W=65cm,C,=2~t%).
against impeller speed, N, for different axial positions, h, measured from the base of the tank. The curves indicate a number of distinct zones. At low impeller speeds, below about 200 r-pm in the case of the data shown in Figs 4-10, most solid particles remain on the bottom of the vessel with little, if any, motion. At higher speeds an intermediate zone is formed characterized by a comparatively steep curve indicating a sharp increase in solids concentration with speed as more and more particles are picked up from the base of the tank and carried into the bulk of the agitated liquid. The rate of rise and the absolute value of the local concentration vary with speed and sampling
position. It is important to note that in each case shown in Figs 4-10 a peak in concentration occurs at a critical impeller speed which has been identified as the just-suspension speed (Ayazi Shamlou and Koutsakos, 1986; Musil et al., 1984; Bourne and Sharma, 1974; Machon et al., 1982; Ohiaeri, 1981). Three correlating equations, typical of those published in the literature, are used to predict values of N, based on the tank/impeller configurations used in this work. The results are tabulated in Figs 4-10 and serve to highlight the degree of deviation that exists between values of Nj. predicted by these equations. However, these deviations are not surprising given the ambi-
532
P. AYAZI SWAMLOU and E. KOUTSAKOS Key
h(cm)
I
7.0
* 0
22.0 30.0 40.0
.
53.0
1-O
Impellerspeed
(rpm)
Fig. 6. Solids concentration vs impeller speed for different sample heights (a_ = 390 pm, p. = 2900 kg/ma, p = 1000 kg/m3, C = 13 cm, D = 10 cm, H = 65 cm, C, = 3 wt’?/Q).
4-
2-
O0 Fig. 7. Solids concentration vs impeller speed for different sample heights (a, = 390 pm, p, = 2900 kg/m”, p = 1000 kg/m3, C = 13 cm, D = 10 cm, H = 65 cm, C, = 6 wt%).
guity
in the definition
of and
the experimental
un-
in the measurement of N, reported in most of the published investigations. With regard to the shape of the curves, the rate of change of the local concentration of solids with speed is very gradual for sampling positions near the surface of the liquid. Moreover as the total hold-up of solids increases, the point of maximum concentration becomes more difficult to identify. For values of total concentration of solids below about 4% the experimental data indicate that any sampling location below the position of the impeller may be used effectively to identify the critical speed at which the peak in concentration occurs.
certainties
As the speed of the impeller increases beyond the point of maximum concentration a third zone is formed in which the local concentration of solids remains constant or decreases slightly with increase in speed. The exact behaviour of the suspended particles depends upon many factors including particle size and density and the size and position of the impeller (Figs 8-10). Similar observations have been reported by previous workers (Musil, 1976; Boume and Sharma, 1974; Machon et nl., 1982). Figures 11-15 show the variation in the concentration of solids as a function of height for different impeller speeds, total solids hold-up and particle size. The curves indicate that the concentration of solids
Solids suspension and distribution in liquids under turbulent agitation
lmpellerspeed
533
(rpm)
Fig. 8. Solids concentration vs impeller speed for different sample heights (ri, = 390 pm, ps = 2900 kg/ma. p = 1000 kg/mJ. C = 11.3 cm, D = 10 cm, H = 37 cm, C, = 1 wt%).
'.
0
0
20.2
.
25.8 30.3
100
200
300
Impellerspeed
400
500
6
(rpm)
Fig. 9. Solids concentration vs impeller speed for different sample heights (zP = 390 pm, p. = 2900 kg/m’, p=1000kg/m3,C=11.3cm,D=8.5cm,H=37cm,C,=1 wt%).
increases nearly exponentially with depth from a region just below the surface except in the zone close to the impeller blades where, at high impeller speeds, the concentration of solids appears to oscillate about a mean value. The amplitude of these oscillations is dependent upon the speed of the impeller and, to some extent, on the radial position. These oscillations in concentration in the discharge zone of the impeller must be! related to the periodic passage of the impeller and thd centrifugal forces generated as the impeller rotates. At high impeller speeds these forces are sufficiently strong to overcome the forces tending to increase the uniformity in concentration.
DISCUSSION
The axial distribution of solid particles at a steady state in the bulk of the agitated liquid may be interpreted in terms of a one-dimensional balance between the net flux of particles caused by longitudinal turbulent diffusion down the concentration gradient (i.e. upwards in the vessel) to the downward rate of settling of particles under the action of the gravitational force (Kay, 1957; Ayazi Shamlou and Koutsakos, 1986; Barresi and Baldi, 1986). Thus, from a mass b$lance on the particles over a thin horizontal section of the liquid in the vessel, and assuming that there is no accumulation or depletion of particles, the following equation
534
P. AVAZI SHAMLOU
and E. Kours~os
8.10 15.2 20.2 l
25.8
impeller Fig.
speed (rpml
10. Solids concentrationvs impellerspeed for differentsample heights (&, = 390 W, p = 1000 kg/m3, C = 11.3 cm, D = 6.5 cm, H = 37 cm, C, = 1 WV!!).
Key
NIrPm)
.
300 400
0 P 0 .6cG *
450 500 700 800
*
I
0.3
I
t
I
0.6
0.9
I
I 1.2
pa = 2900 kg/m3.
I
1.5
WC0
Fig. 11. Dimensionless solid concentration ratio vs dimensionless height ratio for different impeller speeds (d, = 390 pm, pS = 2900 kg/m3, p = 1000 kg/m’, D = 10 cm, C = 13 cm, H = 65 cm, C, = 1 wt%).
may be written:
where 5, and U, are, respectively, the dispersion coefficient and settling velocity of the particles in the agitated suspension. Assuming that, for a fixed system geometry and a given operating condition, the mean values of these parameters remain constant in the bulk of the agitated liquid, eq. (1) may be rearranged to give d (In c)
-=
dh
-us/<,.
(2)
Thus, a plot of In E against height, h, is expected to be a straight line with a slope of - U,/<, _ Figures 16-18 are typical plots showing the variation in the mean local concentration of solids as a function of axial position. h. The lines indicate that, within the range of operating variables studied, eq. (2) describes the experimental data adequately. Similar results were obtained for other values of total solids concentration and impeller diameter (Koutsakos, 1988). In order to study the effect of some of the more important variables, such as the impeller diameter and the speed and physical properties of the particles,
Solids suspensionand distributionin liquids under turbulentagitation N(rpml
Key
.
300
0
400 500 560 600 700 800
P
0 0 P *
0
0
I
0.3
I
I
I
I
0.6
535
I
0.9
1.2
E/c,
Fig. 12. Dimensionless solid concentration vs dimensionlessheight ratio for different impeller speeds (C, = 2 wt%, all other experimentaldetails as in Fig. 11).
Key
N
.
(rpm)
*
300 400 500
0 . *
620 650 700
0
A + + + l *
0 I
I 0.3
I 0.6
I
I 0.9
2
E/C,
Fig. 13. Dimensionless solid concentration vs dimensionless height ratio for different impeller speeds (C, = 3 wt%, all other experimentaldetails as in Fig. 11).
upon the distribution of particles in the agitated liquid it is first necessary to understand the effect that these variables have on U, and 5,. Assuming that relatively small particles in suspension behave in the same way as the agitated liquid then the particle diffusion coefficient, tp, may be expected to coincide closely with the liquid diffusion coefficient,
agitated vessel, both of these turbulent parameters may be expressed in terms of the size of the impeller, D, and the rate of energy dissipation, +. . Thus, in the fully turbulent region: Ed a N3 DS/T2H.
(3)
It is assumed that away from the discharge zone of impeller, the mean rate of energy dissipation in the core of the vessel, E*, is directly proportional to +_ Moreover, for small eddies in the Kolmogoroff range,
536
P.AYMI
SHAMLOU
andE.
KOUTSAKOS
Key Ntrpm) I P
300 350 400
0 a n
420 450 500
0
B f + + + + +
0 al 0
I 0.2
I 0.4
I 0.6
I 0.8
I 1
2
e/c,
Fig. 14. Dimensionless
solid concentration vs dimensionless height ratio for different impeller speeds (a, = 530 ,um, p, = 2900 kg/m3, p = 1000 kg/ma, D = 10 cm, C = i 1.3 cm, H = 37 cm, C, = 1 wt%).
Key
Fig. 15. Dimensionless (C, = 3 wt%,
N (rPm)
I
300
0
* 0 . n
400 450 500 600 700
*
800
solid concentration vs dimensionless height ratio for different impeller speeds samples taken at position B: other experimental details as in Fig. 11).
L,, the fluctuating velocity, U ‘, may be written as U’ K (&TL,)“3.
1972): (4)
In the present analysis, it is assumed that eq. (4) may be applied to the larger energy containing eddies of scale L, (Davies, 1972) that are responsible for keeping the particles in suspension. The turbulent liquid diffusion coefficient, c,, has also been related to the fluctuating velocity and to the critical eddy size by the following equation (Davies,
5, a U’L,.
(5)
Assuming that the diffusion coefficient of solid particles is of the same order as that of the fluid, eqs (4) and (5) may be used to give 5, oz <, oc U’L,
cc (E7Le)1’3L,.
(W
With L, proportional to the impeller diameter, D, but practically independent of the speed of the impeller
Solids suspension and distribution in liquids under turbulent agitation
537
Key N(rpm) 300 400 500 600 700 800
.
0 * 0 . *
Concentration,
._
C (g/l)
Fig. 16. Semilogarithmic plot of solid concentration vs sample height for different impeller speeds (zP= 390 pm, p, = 2900 kg/m3, p = 1000 kg/m”, D = 10 cm, C = 11.3 cm, H = 37 cm, C, = 1 wt%).
N (rpm)
Kev .
300 350
0
01 0.1
I
I
IllIll
I
Concent Aion,
10
E(glll
Fig. 17. Semilogarithmic plot of solid concentration profile YS sample height at differentimpeller speeds (zp = 530 pm, p. = 2900 kg/m’, p = 1000 kg/m’, H = 37 cm, D = 10 cm, C = 11.3 cm, C, = 1 wt%).
and of the viscosity in the fully turbulent region, eq. (5a) reduces to 5, a ND3/(T2H)‘j3.
(6)
The relative settling velocity of particles in suspension, U,, is a complex function of particle properties, solids concentration and the intensity of fluid turbulence (Clift and Gauvin, 1971). As far as the effect of solids concentration is concerned, for voidage values approaching unity, which is the case in the present study, the particles may be assumed to follow the pattern adopted in free settling. McCabe and Smith (1976) have proposed a convenient criterion for the estimation of the terminal velocity in the still fluid, U,, ces
44:3-P
using a single factor, K, defined as K = zP(gpAp/p2)“3. For
3.3<
(7)
K < 43.6
0.153~0.7’~~.‘4~~0.7~ U*= P”.29PoL43
(8)
and for 43.6 < K < 2360
u, = (gzPAp/p)“2.
(9)
For K -K 3.3, the Stokes’ equation may be used to estimate U,. Turning now to the effect of turbulence, some studies (Clift and Gauvin, 1971; Schwartzberg and Treybal, 1968) have shown that under turbulent
538
P. AYAZI
Fig.
SHAMLOU
and E. KOUTSAKOS
18. Semilogarithmicplot of local solids concentrationYSsample heightat differentparticlediametersat N = 500 rpm, C, = 1 wt% (for remaining experimentaldetails see Fig. 17).
conditions the fall velocity of a single particle in an agitated liquid, U, , can deviate greatly from the terminal velocity in the still fluid. Experimental data obtained over a range of particle sizes and densities indicate that, to a first approximation, the ratio of U,,/U, falls in the range 0.3-0.5. Schwartzberg and Treybal(l968) recommended a mean value of 0.4 for this ratio. In the light of this, either U,, or U, may be used to characterize the behaviour of the solid particles within the agitated liquid. From what has been said above, it seems reasonable to assume that the ratio U,/c, in eq. (2), i.e. the reciprocal of the slope of the lines in Figs 1618, may be written as U,/<,
a U,/[ND3/(T2H)“3].
(10)
Equation (10) provides the relationship between the particle dispersion coefficient and some of the important parameters affecting it. E#ect of impeller speed upon solids concentration profiles For a fixed tank/impeller geometry, eq. (10) reduces to (11) Equation (11) may be expressed in dimensionless form by using a Peclet number defined as l-J,&,/<, and employing the impeller tip speed ND. Thus U&l<,
cc U,IND.
(lla)
Equation (11) is verified experimentally in Figs 16 and 17 which show semi-log plots of local concentration of solids against height for various impeller speeds. In these plots, absolute homogeneity is represented by a vertical line. It is interesting to note that the slope of the lines can be used as a direct measure of the quality of the suspension.
The slopes of the lines in Fig. 16 are used to calculate the dimensionless Peclet number, U,zJc,. In Fig. 19, experimental data from Fig. 16 have been plotted on log-log graph paper as the reciprocal of the dimensionless Peclet number against speed, N. A slope very close to unity is observed in accordance with the proposed model [eq. (1 l)]. Efict of particle diameter upon solids concentration profiles For a given tank/impeller geometry and a fixed impeller speed, i.e. constant t,, an increase in particle diameter leads to an increase in U, and hence to a decrease in the homogeneity of the suspension (Fig. 18). Local concentrations of solids as a function of height for various diameters of glass particles are shown in Fig. 18 and confirm the well-established experimental fact that, for a given tank and impeller configuration and for known operating conditions, uniformity of solids concentration increases with decreasing particle size. Figure 20 shows data from Fig. 18 plotted as the dimensionless Peclet number against terminal velocity of a single particle. The slope of the line is proportional to the 1.6th power of U,, indicating that the relationship between U, and U, is more complicated than that predicted by the present model [eq. (1 l)]. This is perhaps not surprising because both the bulk motion and the turbulent nature of the agitated liquid can be expected to have profound effects upon U,. It is however difficult to quantify these effects at the present stage of development. With regard to Figs 1618 it may be noted that absolute homogeneity is not reached even at impeller speeds several times greater than the complete suspension speed. Indeed, above a critical impeller rotation, centrifugal forces created by the action of the impeller
Solids suspension and distribution in liquids under turbulent agitation Key
Impellerspeed,
539
a,(&m) 175 390 530 780
N (rps)
Fig. 19. Effect of impeller speed on the ratio {,,/7_J,zP (D = l_Ocm, C, = 1 wt%, C = 11.3 cm, H = 37 cm; for & = 39,175,530 pm, p. = 2900 kg/m3; for d, = 780 Frn p. = 3800 kg/m’).
10-Z
10-4
1
0.01
Key I .
ps (kg/m31 2900 3800
I
I Terminal
I
L
velocity,
I1111 U,
0.1
(m/s)
Fig. 20. Effectof terminal velocity on the ratio lJ,&J5, (N = 500 rpm, D = 10 cm, C, = 1 wt%, C = 11.3 cm).
dominate in a substantial proportion of the vessel causing large fluctuations in local solids concentration, and consequently deviations from homogeneity are observed. Therefore, the best that can be achieved in practice is an optimum homogeneity often referred to as the “pseudo-homogeneous” condition.
Efict of total solids hold-up upon concentration gradients Figure 21 shows the experimental results plotted as
local concentration of solids against height for various values of total solids hold-up in the vessel. The data presented in Fig. 21 are for particles with a diameter of
390 pm suspended in water by a turbine impeller of 0.1 m diameter and rotating at 300rpm. It may be observed that, although the absolute value of the concentration of solids at any location depends upon the total solids hold-up, the distribution of particles for c, < 4% is practically independent of it as indicated by the constant slope of the lines. This suggests that, within the range of variables studied, the hold-up concentration has little effect on the ratio of the particle diffusion coefficient to the particle settling velocity. It is however expected that, as the total solids concentration in the tank increases, particle--particle interference will increase, influencing both <,, and U, and, as a result, the ratio <,/US.
540
P.
1
AYAZI
I
I
SHAMLOU
,
and E. KOUTSAKOS
Illll
I
I
I,,,,
I
100
10
Concentration,
C (g/l)
Fig. 21. Effect of mean hold-up concentration upon solids concentration gradient (zp = 390 pm, ps = 2900 kg/m3, p = 1000 kg/m ‘,D=lOcm,C=15cm,H=65 cm, N = 300 rpm, sampling position R).
Key . .
Concentihm, Fig.
Effect
D lcm) 10.0 8.50
i: (g/l)
22. Semilogarithmicplot of solid concentration vs sample height for different impeller diameters at N =500 t-pm, CO= 1 wt% (all other experimental details as in Fig. 17).
of impeller
diameter
upon
solids
concentration
pro@ les
Equation (6) indicates that for a fixed impeller speed and a given fluid/particle and tank configuration, increasing impeller diameter, D, will increase the diffusivity of the particles and, as a result, better uniformity should be attained within the bulk of the agitated liquid. This is demonstrated in Fig. 22 in which data have been plotted as the dimensionless concentration against height for three different impeller diameters at a fixed impeller speed. Figure 23 shows the data in Fig. 22 plotted as the dimensionless Peclet number against impeller diameter. The lines have a slope of 2.8 which compares favourably with the expected value of 3 [eq. (6)].
From the results presented so far, it may be concluded that the dimensionless Peclet number is a correlating parameter that can bring together experimental data from various runs. Moreover, the Peclet number may be related to impeller speed and the terminal settling velocity of the particles. All the results presented in this article are replotted in Fig. 24, data for the various runs have been brought together in a single plot confirming the proposed approach. An important assumption in the development of the turbulent dispersion model proposed in this work is the existence of homogeneous and isotropic turbulence in the agitated fluid. The question of whether turbulence in a mixing vessel is isotropic or not has been a matter of contention between investigators in
Solids suspension and distribution in liquids under turbulent agitation
541
./ . //
l /
:/
v /
I
.
/
I
r
IllI
I1
I
0.1 bmpeller
Fig.
23. Effect
of impeller
Fig. Fig. Fig. Fig. Fig. Fig.
diameter
on
the
diameter,
ratio
c =
(,/USaP
D (m)
(zP = 530pm,
D =
IOcm, C, = 1 wt%,
11.3 cm).
19,$=175pm 19, d, = 390 km 20 23, N = 500 rpm 23, N = 400 rpm 23, N = 300 rpm
P
Fig. 19, d,, = 530 pn dp = 780 pm, ljs = 3800
kg/m3
1 I
I .
Fig. 24. Dimensionless
ratio
U,~pj~, as a function of system parameters.
this field, e.g. early studies (Davies, 1972) indicate that turbulence is spatially highly variable throughout the vessel while some recent measurements (Mersmann and Laufhuette, 1985; Laufhuette and Mersmann, 1985) of U’ using the LDA technique have shown that, in the core of the vessel, i.e. away from the discharge zone of the impeller and the immediate vicinity of the walls of the vessel and the surface of the liquid, turbulence is nearly isotropic. In the discharge zone of the impeller the fluctuating velocity is confounded by the periodic passage of the impeller. These workers have also attempted to separate the total fluctuating
velocity
in the discharge
components
which
zone
of the impeller
into two
a;e assumed to be additive, i.e. u total= u ’+ Upcr
(12)
where 7-Jperis the contribution due to the periodic passage of the impeller and U ’ is the turbulent part. Measurements of the local concentration of solids presented in this work (Figs 16-18) appear to support the general observations of Mersmann and Laufhuette (1985) and Laufhuette and Mersmann (1985); nonisotropic turbulence and hence a variable 5, in the core of the vessel would be expected to give non-linear
542
P. AYAZI
SHAMLOU
plots. Moreover, the observed oscillations in the local concentration of solids in the discharge zone of the impeller (Figs 1l-l 5) are clearly due to the action of the impeller and, thus, indirectly support the view that the nature of fluid turbulence in the discharge zone is confounded by the periodic passage of the impeller. It is therefore important to appreciate that the concept of a constant mean eddy/particle diffusivity as proposed in this work is only valid in the bulk region of the vessel and its extension to the discharge zone of the impeller is an approximation.
CONCiUSIONS
Measurements have been provided of solids concentration profiles in the bulk of a mechanically agitated fluid in the fully turbulent flow regime. Over the range of variables studied the data indicate a near-exponential decay in the concentration of solids with axial height from the surface of the liquid in the vessel. These solids concentration profiles have been interpreted in terms of a steady-state turbulent diffusion support mechanism. The distribution of particles results from the balance between the random motion of the liquid in the vessel and the action of gravity on the particles. In this way, the distribution of particles in the agitated liquid is characterized by a single parameter, nameIy the ratio of the turbulent diffusion coefficient of the particles to their terminal settling velocity. All other properties of the system, such as particle size and density, impeller diameter and speed and fluid properties exert their effects only through the value of this ratio.
NOTATION
mean axial concentration of solids, g/l total concentration of solids in the tank, g/l diameter of the impeller, m mean particle diameter, m acceleration due to gravity, m’/s height of liquid in the tank, m height of sampling point, m constant in eq. (7) size of the critical energy containing eddies, m size of eddies in the Kolmogoroff range, m impeller speed, rpm fluctuating velocity, m/s fall velocity of a solid particles in suspension, m/s fall velocity of a single particle in turbulent flow, 111/S
fall velocity Greek &b
&T
p r/
of a single particle in still fluid, m/s
letters
bulk or mean energy dissipation
per unit mass,
W/kg total energy input per unit mass, W/kg liquid viscosity, kg/m s liquid diffusion coefficient, m/s particle diffusion coefficient, m/s liquid density, kg/m3
and E. KOUTSAKOS Ps
solid density, kg/m3
AP
P.--P REFERENCES
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