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Relative particle velocities in two-species settling
Powder Technoiogy, IO (1974) 67-71 @ Ekvier Sequoia S-A., Lausanne - Printed in The Netherlands
Relative Particle Ve?ocities in Two-Species Settling M.J. MCKE’IT
and H-M. ALHABBOOBY
Department of Chemical I (Gt. Britain)
Engineering,
(Received November 2.1973;
University
of Manchester
Institute
of Science
and Technology.
Manchester
in revised form January 21, 1974)
SUMMARY
Settling velocities have been measured for two particle fractions settling together through a liquid. The two fractions differed only in particle density but not in particle size. It is shown that the settling velocity of each fraction may be determined from the Richardson-Zaki equation, if the total particle concentration is used to correct the terminal fall velocity of the particles in each fraction. The concept of a critical concentration in binary sedimentation has been examined. At this concentration the two fractions appear to settle at the same velocity. Previous authors have attributed the effect to mechanical interlocking of the particles, but the present work shows that an alternative hydrodynamic argument can account for the phenomenon.
INTRODUCI’ION
Most workers dealing with vertically flowing liquid - particle systems have chosen to work with uniform particles as far as practically possible. Reliable cdcuIation methods have consequently onIy been available for systems of this typo Cl]. However, the present authors have shown recentIy how differences in particle size can be taken into account [ZJ . For a mixture of two particle species of different size, but of equal density, flowing down a tube against an upflow of liquid, it was shown that the fall velocity of each species could be obtained from the Richardson and Zaki equation if the tota particle tioncentration was used to correct the terminal fail velocity of e&h species. The foI-
lowing equations were found to appIy (expressions defined in the List of SymboIs): For the larger particles (fraction 1) u+ul=ul_
(l--ol,-a,)+
U)
For the smaller particles {fraction 2)
In eqns. (1) and (2), 7 and u, can be obtained for each species from batch sedimentation or batch fluidisation experiments on that species alone. These equations were found to hold for or less than about 0.4. At higher concentrations, the small particles generally were found to fall faster and the large particles to fall slower than given by eqns. (1) and (2). The purpose of the present paper is’ to extend the work in two ways: (i) To investigate the effect of a difference in particle density on the relative settling veIocity of two particle fractions. (ii) To examine the concept of the critical concentration in batch sedimentation of a binary particle mixture, at which both boundaries appear to settle at the same velocity.
SmLING OF PARTICLE DIFFERENT DENSITIES
FRACTIONS
HAVING
In this work two particle fractions of spherical ballotini were prepared by sieving, one from lead glass and the other from soda glass. The surface mean diameter of each fraction was the same as far as possible, but the particIe density differed for each fraction. The properties of the &actions are given in Table 1.
68 TABLE
1
Surface mean diameter (pm)
p, (glcm3)
ucn
7)
(.;rm) Lead glass ballotini
600 - 710
682
2.99
10.60
3.23
Soda glass ballotini
600 - ‘710
689
2.48
8.24
3.28
Fraction
Size range
u, and v were obtained for each fraction separately, by measuring the expansion of a water-fluidised bed (1 in. diam.) and plotting log (interstitial water velocity) uers’sus log (voidage fraction). The technique is a wellknown one and has been explained in more detail in 123. For determination of the settling velocities of each fraction through an upflow of water, a very similar technique to that described in [Z] was used. For the experimental details the reader is referred to that paper. The experimental arrangement is shown schematically in Fig. 1, where the important variables are shown- For various mixtures of the two fractions, S, , S,, Q, er, and Q, were measured. Using the equations ‘h
Q u =A(1
--a)*
‘h
=yA’
Sl U, = ---Q
the slip velocities u + or., and u + ui of the two fractions were determined. Figure 2 shows the slip velocity of each fraction plotted against the total particle holdup. For comparison, the curves for each fraction alone, obtained from separate batch fludisation experiments on each fraction, are shown. The agreement between the results and the respective batch lines is sufficiently good to indicate that eqns. (1) and (2) are also valid for this form of
(cm/s)
Fig. 2. Holdup U.S.sIip veIocity results for mixtures of the light and heavy fractions_ X Heavy fraction, e light fraction , _ slip velocity = u, (1 - J_Y)~ - ’ _ 11 and v, obtained Table 1.
for each fraction alone and shown in
settling, where fraction (1) and fraction (2) denote the light and heavy fractions respectively. Because the mean diameters of the two fractions are virtually identical, settling because of size differences can be excluded. It appears from these results that the settling velocity of each species can be obtained from eqns. (1) and (2) when settling arises solely as a result of density differences between the species. This finding considerably extends the usefulness of eqns. (1) and (2). It seems safe to assume that they apply when the terminal
fall velocities of the particles comprising the two species are different, whether this arises because of size (as shown in f23) or density differences or both. We now turn our attention to an application of eqns. (1) and (2) in sedimentation.
BATCH SEDIMENTATION CLE MIXTURE Fig. 1. Schematic representation of continuous twospecies settling through an upflowing liquid.
OF A BINARY PARTI-
Consider two particle fractions, the particles in each fraction being as far as possible
69 0.020 \
E $
0.0150.010
*
*
\
\\
>
in simple
batch sedimentation
6 2
\
0.005-
“\
2
Fraction
Size range (lun)
u, (cm/s)
rl
1 2 3
152 - 180 180 - 210 250 - 300
0.0155 0.0345 0.075
4.64 5.41 5.14
glcm3,
pr = 0.883
-
3
identical. Suppose that the terminal fall velocity of the particles comprising one fraction differs from that of the particles comprising the other fraction. When a mixture of the two particle fractions, homogeneously dispersed in a viscous liquid in a tube, is allowed to settle, two boundaries are usually observed if the fractions are of different colours. In Fig. 3 the upper zone (zone 1) consists of particles having the lower terminal fall velocity, whereas zone 2 consists of both particle fractions. The concentration of each fraction in zone 2 is taken to be the same as the initial concentration of each fraction in the homogeneous mixture before settling commences. In support of this assumption, the settling velocities we shall deal with are the values just after settling begins. Experiments have been described in 12 3 for spherical glass particles settling in liquid paraffin_ Each fraction had a different mean particle size but the same particle density. Details of the fractions used are summarised in Table 2. In Fig. 4 experimental values of the fall velocity of each boundary are shown as a function of the initial total particle concentration for a mixture of fractions (2) and (3). Also shown are the calculated boundary velocities based on eqns. (1) and (2) using the theory which has been given in [2 J _ The velocity of the lower boundary, which
ps = 2.93
I ‘\
0-J 0
TABLE
I
. -
3 0
Fig. 3. Settling zones of a binary mixture.
’
glcm3,
,ur = 2.21
poise.
0.1 .a-Total
02 paride
0.3
04
0.5
concentmtion
Fig. 4. Boundary settling velocities for mixtures of fractions (2) and (3). x Lower boundary, o upper predicted velocities. Concentration boundary, _ of fraction (2) constant at 0.05.
is the velocity of the large particles in zone 2, is greater than the upper boundary velocity, which is the velocity of the small particles in zone 1_ The length of zone 1 thereforz increases as settling proceeds. Table 3 shows calculated values for the particle concentrations in each zone. It is evident that the particle concentration in zone 1 is always less than the total particle concentration in zone 2. As shown by eqns. (1) and (2), the particle slip velocity can be represented by slip velocity = terminal fall velocity X (1 - concentration)“-’
(3)
We can consider the small particles in zone 1 and the large particles in zone 2, since it is their velocities which determine the boundary velocities. Although the small particles have a lower terminal fall velocity than the large particles, the particle concentration which causes hindered settling is lower in zone 1 than in zone 2. As indicated by eqn. (3), increases in terminal fall velocity and in particle concenTABLE
3
Concentration of the small particles (fraction 2) in zone 1
Total particle concentration zone 2
0.118 0.158 0.204 0.253 0.305 0.357 0.413
0.20 0.25 0.30 0.35 0.40 0.45 0.50
in
70
tration compensate. Thus it is possible, if a suitable concentration difference exists between zones 1 and 2, for the small particles in zone 1 and the large particles in zone 2 to fall at the same velocity. The argument presented above is slightly oversimplified because it neglects the upward interstitial velocity of the displaced fluid. It does, nevertheless, bring out the principle of two compensating effects in each zone. A fuller mathematical treatment has been given in
r21-
The experimental results shown in Fig. 4 support the above arguments. As the total particle concentration is increased, the boundary velocities approach each other. For this particular experiment, at a total concentration of O-4, the difference between the velocities of the two boundaries is difficult to detect and there appears to be a common boundary. The theory does in fact still predict a very small relative velocity between the boundaries at this concentration (0.00088 cm/s). Jn general the lower bo-undary is always predicted to fall faster than the upper boundary although the difference in velocity can be extremely small.
TXE CRITICAL CONCENTRATION The argument outlined above may be contrasted with a theory put forward by Davies [3,43 _ In this, the explanation for the apparently common velocity of the two boundaries at a critical concentration does not rely on any hydrodynamic argument, but is attributed to interlocking of the particles. Davies proposed that at the critical concentration, the smaller particles are physically unable to pass through the matrix of the larger particles so that two zones are unable to form. The critical concentration calculated by Davies’ method for the example shown in Fig. 4 is 0.403, which does in fact agree with experimental observation_ It is suggested, however, that the agreement is fortuitous for the reasons discussed below. The critical concentration calculated from Davies’ theory is independent of the relative proportion of each fraction in the mixture, but depends only upon their size ratio. Boundary settling velocities calculated by the present authors’ model do, however, depend
o-
0.2
0.1
0
a-Tot01
parttie
03
0.4
0.5
contentmtim
Fig_ 5. Boundary settling velocities for mixtures of fractions (1) and (3). X Lower boundary, 0 upper predicted velocities. Concentration boundary, _ of fraction (1) constant at 0.15.
on the proportion of each fraction and this therefore affords a sensitive test of the two theories. By carefully choosing the fractions and their proportion, it is possible to demonstrate the importance of the proportion of each fraction _ Experiments have been carried out using a mixture of fractions (1) and (3). In Fig. 5 the concentration of fraction (1) is held constant at 0.15 and in Fig. 6 at 0.02. In both cases the boundary settling velocities are accurately predicted by our model. A common boundary velocity is observed only in Fig. 6 at a total concentration of 0.4, and in Fig. 5 it is apparent that the boundary velocities are significantly different even at a high particle concentration. Davies’ theory, on the other hand, predicts a critical concentration of 0.438 for
01
0
0.1
02
~~-Tolal particle
03
0.0
1
0.5
concentration
Fig. 6. Boundary settling velocities for mixtures of fractions (1) and (3). x Lower boundary, 0 upper predicted velocities. Concentration boundary, _ of fraction (1) constantat 0.02.
71
both cases, indicating that in both cases, at or above this concentration, a common settling velocity is expected. These results suggest that the explanation lies in the hydrodynamics of the system rather than in considerations of geometrical packing. A critical concentration does not exist, and the observation of a common boundary velocity simply arises from the experimental difficulty of detecting the small difference between the boundary velocities. It is quite probable, however, that a mechanism such as that prcposed by Davies plays some part in accounting for the deviations from eqns. (1) and (2) which were observed in [Zj - For a total concentration greater than about 0.4 it was found that the smaller particles fell faster and the larger particles fell slower than given by these equations_ It must be emphasised, however, that even up to concentrations of 0.5, the deviations were minor and a considerable difference in particle velocities was always observed_
LIST OF SYMBOLS
tube cross-sectional area, cm2 exponent in Richardson and Zaki eqnation volumetric liquid upflow, cm”/s volumetric particle downflow, ems/s upward interstitial liquid velocity, cm/s particle fall velocity, cm/s terminal particle fall velocity, cm/s total volume of particles per unit volume of tube (holdup) volume of fraction (1) per unit volume of tube volume of fraction (2) per unit volume of tube volume of heavy fraction per unit volume of tube volume of light fraction per unit volume of tube fluid viscosity, poise fluid density, g/cm3 particle density, g/cm3
CONCLUSION
Su bscrfp
In summary, the experimental observation of a common boundary velocity is believed to depend on the hydrodynamics of the system rather than on physical interlocking of the particles for the following reasons: (i) The hydrodynamic model is able to accurately predict the boundary velocities over a wide concentration range. Settling at high concentrations, where the boundary velocities are very nearly the same, has been shown to be only a special case of a more general situation. (ii) The experimental observation of a common boundary velocity has been shown to depend on the relative proportion of each fraction in the mixture, which is not predicted by the interlocking model.
11
1 1 2
fs
heavy particles light particles fraction (1) fraction (2)
REFERENCES G.B. Wallis, One-Dimensional Two-Phase Flow, McGraw-Hill, New York, 1969. M.J_ Lockett and H.M. Al-Habbooby, Trans. Inst. Chem. Engrs., 51 (1973) 281. R. Davies, Powder Technol., 2 (1968/69) 43 - 51. R. Davies and B.H. Kaye, Powder Technol., 5 (1971172) 61 - 68.
Relative Particle Ve?ocities in Two-Species Settling M.J. MCKE’IT
and H-M. ALHABBOOBY
Department of Chemical I (Gt. Britain)
Engineering,
(Received November 2.1973;
University
of Manchester
Institute
of Science
and Technology.
Manchester
in revised form January 21, 1974)
SUMMARY
Settling velocities have been measured for two particle fractions settling together through a liquid. The two fractions differed only in particle density but not in particle size. It is shown that the settling velocity of each fraction may be determined from the Richardson-Zaki equation, if the total particle concentration is used to correct the terminal fall velocity of the particles in each fraction. The concept of a critical concentration in binary sedimentation has been examined. At this concentration the two fractions appear to settle at the same velocity. Previous authors have attributed the effect to mechanical interlocking of the particles, but the present work shows that an alternative hydrodynamic argument can account for the phenomenon.
INTRODUCI’ION
Most workers dealing with vertically flowing liquid - particle systems have chosen to work with uniform particles as far as practically possible. Reliable cdcuIation methods have consequently onIy been available for systems of this typo Cl]. However, the present authors have shown recentIy how differences in particle size can be taken into account [ZJ . For a mixture of two particle species of different size, but of equal density, flowing down a tube against an upflow of liquid, it was shown that the fall velocity of each species could be obtained from the Richardson and Zaki equation if the tota particle tioncentration was used to correct the terminal fail velocity of e&h species. The foI-
lowing equations were found to appIy (expressions defined in the List of SymboIs): For the larger particles (fraction 1) u+ul=ul_
(l--ol,-a,)+
U)
For the smaller particles {fraction 2)
In eqns. (1) and (2), 7 and u, can be obtained for each species from batch sedimentation or batch fluidisation experiments on that species alone. These equations were found to hold for or less than about 0.4. At higher concentrations, the small particles generally were found to fall faster and the large particles to fall slower than given by eqns. (1) and (2). The purpose of the present paper is’ to extend the work in two ways: (i) To investigate the effect of a difference in particle density on the relative settling veIocity of two particle fractions. (ii) To examine the concept of the critical concentration in batch sedimentation of a binary particle mixture, at which both boundaries appear to settle at the same velocity.
SmLING OF PARTICLE DIFFERENT DENSITIES
FRACTIONS
HAVING
In this work two particle fractions of spherical ballotini were prepared by sieving, one from lead glass and the other from soda glass. The surface mean diameter of each fraction was the same as far as possible, but the particIe density differed for each fraction. The properties of the &actions are given in Table 1.
68 TABLE
1
Surface mean diameter (pm)
p, (glcm3)
ucn
7)
(.;rm) Lead glass ballotini
600 - 710
682
2.99
10.60
3.23
Soda glass ballotini
600 - ‘710
689
2.48
8.24
3.28
Fraction
Size range
u, and v were obtained for each fraction separately, by measuring the expansion of a water-fluidised bed (1 in. diam.) and plotting log (interstitial water velocity) uers’sus log (voidage fraction). The technique is a wellknown one and has been explained in more detail in 123. For determination of the settling velocities of each fraction through an upflow of water, a very similar technique to that described in [Z] was used. For the experimental details the reader is referred to that paper. The experimental arrangement is shown schematically in Fig. 1, where the important variables are shown- For various mixtures of the two fractions, S, , S,, Q, er, and Q, were measured. Using the equations ‘h
Q u =A(1
--a)*
‘h
=yA’
Sl U, = ---Q
the slip velocities u + or., and u + ui of the two fractions were determined. Figure 2 shows the slip velocity of each fraction plotted against the total particle holdup. For comparison, the curves for each fraction alone, obtained from separate batch fludisation experiments on each fraction, are shown. The agreement between the results and the respective batch lines is sufficiently good to indicate that eqns. (1) and (2) are also valid for this form of
(cm/s)
Fig. 2. Holdup U.S.sIip veIocity results for mixtures of the light and heavy fractions_ X Heavy fraction, e light fraction , _ slip velocity = u, (1 - J_Y)~ - ’ _ 11 and v, obtained Table 1.
for each fraction alone and shown in
settling, where fraction (1) and fraction (2) denote the light and heavy fractions respectively. Because the mean diameters of the two fractions are virtually identical, settling because of size differences can be excluded. It appears from these results that the settling velocity of each species can be obtained from eqns. (1) and (2) when settling arises solely as a result of density differences between the species. This finding considerably extends the usefulness of eqns. (1) and (2). It seems safe to assume that they apply when the terminal
fall velocities of the particles comprising the two species are different, whether this arises because of size (as shown in f23) or density differences or both. We now turn our attention to an application of eqns. (1) and (2) in sedimentation.
BATCH SEDIMENTATION CLE MIXTURE Fig. 1. Schematic representation of continuous twospecies settling through an upflowing liquid.
OF A BINARY PARTI-
Consider two particle fractions, the particles in each fraction being as far as possible
69 0.020 \
E $
0.0150.010
*
*
\
\\
>
in simple
batch sedimentation
6 2
\
0.005-
“\
2
Fraction
Size range (lun)
u, (cm/s)
rl
1 2 3
152 - 180 180 - 210 250 - 300
0.0155 0.0345 0.075
4.64 5.41 5.14
glcm3,
pr = 0.883
-
3
identical. Suppose that the terminal fall velocity of the particles comprising one fraction differs from that of the particles comprising the other fraction. When a mixture of the two particle fractions, homogeneously dispersed in a viscous liquid in a tube, is allowed to settle, two boundaries are usually observed if the fractions are of different colours. In Fig. 3 the upper zone (zone 1) consists of particles having the lower terminal fall velocity, whereas zone 2 consists of both particle fractions. The concentration of each fraction in zone 2 is taken to be the same as the initial concentration of each fraction in the homogeneous mixture before settling commences. In support of this assumption, the settling velocities we shall deal with are the values just after settling begins. Experiments have been described in 12 3 for spherical glass particles settling in liquid paraffin_ Each fraction had a different mean particle size but the same particle density. Details of the fractions used are summarised in Table 2. In Fig. 4 experimental values of the fall velocity of each boundary are shown as a function of the initial total particle concentration for a mixture of fractions (2) and (3). Also shown are the calculated boundary velocities based on eqns. (1) and (2) using the theory which has been given in [2 J _ The velocity of the lower boundary, which
ps = 2.93
I ‘\
0-J 0
TABLE
I
. -
3 0
Fig. 3. Settling zones of a binary mixture.
’
glcm3,
,ur = 2.21
poise.
0.1 .a-Total
02 paride
0.3
04
0.5
concentmtion
Fig. 4. Boundary settling velocities for mixtures of fractions (2) and (3). x Lower boundary, o upper predicted velocities. Concentration boundary, _ of fraction (2) constant at 0.05.
is the velocity of the large particles in zone 2, is greater than the upper boundary velocity, which is the velocity of the small particles in zone 1_ The length of zone 1 thereforz increases as settling proceeds. Table 3 shows calculated values for the particle concentrations in each zone. It is evident that the particle concentration in zone 1 is always less than the total particle concentration in zone 2. As shown by eqns. (1) and (2), the particle slip velocity can be represented by slip velocity = terminal fall velocity X (1 - concentration)“-’
(3)
We can consider the small particles in zone 1 and the large particles in zone 2, since it is their velocities which determine the boundary velocities. Although the small particles have a lower terminal fall velocity than the large particles, the particle concentration which causes hindered settling is lower in zone 1 than in zone 2. As indicated by eqn. (3), increases in terminal fall velocity and in particle concenTABLE
3
Concentration of the small particles (fraction 2) in zone 1
Total particle concentration zone 2
0.118 0.158 0.204 0.253 0.305 0.357 0.413
0.20 0.25 0.30 0.35 0.40 0.45 0.50
in
70
tration compensate. Thus it is possible, if a suitable concentration difference exists between zones 1 and 2, for the small particles in zone 1 and the large particles in zone 2 to fall at the same velocity. The argument presented above is slightly oversimplified because it neglects the upward interstitial velocity of the displaced fluid. It does, nevertheless, bring out the principle of two compensating effects in each zone. A fuller mathematical treatment has been given in
r21-
The experimental results shown in Fig. 4 support the above arguments. As the total particle concentration is increased, the boundary velocities approach each other. For this particular experiment, at a total concentration of O-4, the difference between the velocities of the two boundaries is difficult to detect and there appears to be a common boundary. The theory does in fact still predict a very small relative velocity between the boundaries at this concentration (0.00088 cm/s). Jn general the lower bo-undary is always predicted to fall faster than the upper boundary although the difference in velocity can be extremely small.
TXE CRITICAL CONCENTRATION The argument outlined above may be contrasted with a theory put forward by Davies [3,43 _ In this, the explanation for the apparently common velocity of the two boundaries at a critical concentration does not rely on any hydrodynamic argument, but is attributed to interlocking of the particles. Davies proposed that at the critical concentration, the smaller particles are physically unable to pass through the matrix of the larger particles so that two zones are unable to form. The critical concentration calculated by Davies’ method for the example shown in Fig. 4 is 0.403, which does in fact agree with experimental observation_ It is suggested, however, that the agreement is fortuitous for the reasons discussed below. The critical concentration calculated from Davies’ theory is independent of the relative proportion of each fraction in the mixture, but depends only upon their size ratio. Boundary settling velocities calculated by the present authors’ model do, however, depend
o-
0.2
0.1
0
a-Tot01
parttie
03
0.4
0.5
contentmtim
Fig_ 5. Boundary settling velocities for mixtures of fractions (1) and (3). X Lower boundary, 0 upper predicted velocities. Concentration boundary, _ of fraction (1) constant at 0.15.
on the proportion of each fraction and this therefore affords a sensitive test of the two theories. By carefully choosing the fractions and their proportion, it is possible to demonstrate the importance of the proportion of each fraction _ Experiments have been carried out using a mixture of fractions (1) and (3). In Fig. 5 the concentration of fraction (1) is held constant at 0.15 and in Fig. 6 at 0.02. In both cases the boundary settling velocities are accurately predicted by our model. A common boundary velocity is observed only in Fig. 6 at a total concentration of 0.4, and in Fig. 5 it is apparent that the boundary velocities are significantly different even at a high particle concentration. Davies’ theory, on the other hand, predicts a critical concentration of 0.438 for
01
0
0.1
02
~~-Tolal particle
03
0.0
1
0.5
concentration
Fig. 6. Boundary settling velocities for mixtures of fractions (1) and (3). x Lower boundary, 0 upper predicted velocities. Concentration boundary, _ of fraction (1) constantat 0.02.
71
both cases, indicating that in both cases, at or above this concentration, a common settling velocity is expected. These results suggest that the explanation lies in the hydrodynamics of the system rather than in considerations of geometrical packing. A critical concentration does not exist, and the observation of a common boundary velocity simply arises from the experimental difficulty of detecting the small difference between the boundary velocities. It is quite probable, however, that a mechanism such as that prcposed by Davies plays some part in accounting for the deviations from eqns. (1) and (2) which were observed in [Zj - For a total concentration greater than about 0.4 it was found that the smaller particles fell faster and the larger particles fell slower than given by these equations_ It must be emphasised, however, that even up to concentrations of 0.5, the deviations were minor and a considerable difference in particle velocities was always observed_
LIST OF SYMBOLS
tube cross-sectional area, cm2 exponent in Richardson and Zaki eqnation volumetric liquid upflow, cm”/s volumetric particle downflow, ems/s upward interstitial liquid velocity, cm/s particle fall velocity, cm/s terminal particle fall velocity, cm/s total volume of particles per unit volume of tube (holdup) volume of fraction (1) per unit volume of tube volume of fraction (2) per unit volume of tube volume of heavy fraction per unit volume of tube volume of light fraction per unit volume of tube fluid viscosity, poise fluid density, g/cm3 particle density, g/cm3
CONCLUSION
Su bscrfp
In summary, the experimental observation of a common boundary velocity is believed to depend on the hydrodynamics of the system rather than on physical interlocking of the particles for the following reasons: (i) The hydrodynamic model is able to accurately predict the boundary velocities over a wide concentration range. Settling at high concentrations, where the boundary velocities are very nearly the same, has been shown to be only a special case of a more general situation. (ii) The experimental observation of a common boundary velocity has been shown to depend on the relative proportion of each fraction in the mixture, which is not predicted by the interlocking model.
11
1 1 2
fs
heavy particles light particles fraction (1) fraction (2)
REFERENCES G.B. Wallis, One-Dimensional Two-Phase Flow, McGraw-Hill, New York, 1969. M.J_ Lockett and H.M. Al-Habbooby, Trans. Inst. Chem. Engrs., 51 (1973) 281. R. Davies, Powder Technol., 2 (1968/69) 43 - 51. R. Davies and B.H. Kaye, Powder Technol., 5 (1971172) 61 - 68.