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The effect of mixing aids on the kinetics of mixing in a rotating drum
The
Effect of Mixing
Aids on the Kinetics
P. K. CHAUDHURI
AND
of Mixing
SUMMARY The batch mixing of particulate solids in a rotating drum mixer has been investigated with various numbers of Lucite balls added as mixing aids. Two experimental systems hate been studied. One invo!ved the mixing of-dolomite particles of two sizes, 28 x 32 and 42 x 48 mesh, while the other consisted of mixing calcite and silica of the same size range, namely 65 x 100 mesh. The progress
of mi.xing was followed
by a simple sampling and assaying technique-which pemlits accurate determination of the composition of the powder in sections along the axis of the mixer. The role of mixing aids on the mixing process was quantified by evaluating their effect on the dz@tsion coefficient and the degree of mixedness An empirical relationship between the difision coeficient and the number of balls added to the mixer was obtained.
INiRODUCTION
addition of mixing aids to a drum mixer is known to increase the rate of mixing of particulate solids. However, to date only one qualitative discussion on the effect of mixing aids has been published’. The system used in that study involved sand grains dyed with two different colors, using rubber balls as mixing aids The purpose of our paper is to present the results of a systematic investigation of the effect of mixing aids on the kinetics of mixing. To understand how mixing aids might affect the process let us first consider briefly how particuIate solids are transported and mixed within a blending device_ Two basic mechanisms have been identified2T3_One is operative when individual particles of one component are redistributed throughout the The
* Thispaperis based on the thesissubmitted by P. K. Chaudhuri for the MS. degree in the College of Engineering, University of California, Berkeley_ Powder Technology - Elsevia
Sequoia SA,
Lausanne
Rotating
Dram*
D. W. FUERSTENAU
unirersitrof Cal~omia.Berx-eler, calif_ 947m (Received
in a
[U.S.A.)
July 4, 1970)
batch as a result of random motion of the individual particIes caused by the tumbling motion within the mixer_ This process, which is very much akin to dillusion of metallic atoms through vacancies, has been called “micromixing” or “diffusive mixing”3_ The second process consists of the collective movement of groups of particles. This has been called “macromixing” or “convective mixing”3. Thus, the first process involves the movement of individual particles, and the second the movement of groups of particles. For an axially loaded drum mixer, diffusion is the mechanism by which mixing takes place3. Since no component of the motion is parallel to the axis of the mixer4_ random movement or diffusion takes place along the axis, and analysis of the
mixing process can be carried out in terms of a “diffusion” coeffxcient. If the materials are loaded in layers on top of each other, mixing occurs by convection primarily, with uhimate dispersion being controlled by diffusion_ To date no theoretical analysis has been given of the very complicated layer-loaded drum mixer in which mixing occurs by the two mechanisms. Thus, an axially loaded system must be investigated in order to make a quantitative study of the effect of mixing aids on the kinetics of the process. To quantify the effect of mixing aids on the kinetics of mixing, analysis of mixing results must be carried out in terms of the variance of mixture samples For the entire mixer bed, the sample variance is given by
i$ G$ =
[ci-c]2
S-l
(1)
where Ci is the concentration at the i th sample position, c is the mean concentration, and S is the total number of samples taken. The criterion used forexpressingthestateofmixednessofthemixercontents is the parameter MN defined by
KlXSl-iCS
OF MIXING
where 6 is the unmixed variance3_ As shown previously3, solution of Fick’s second law of diffusion equation for the mixing of equal proportions of two identical powders yields the following expression for larger numbers of mixer revolutions, N I &=-$exp(-Fj
(3)
where D is the diffusion coefficient and L the mixer length. For small values of N,
From eqns. (2), (3) and (4), expressions for the kinetics of mixing can be derived for relating the degree of mixedness nrfs to the number of mixer revolutions For small N.
and, for large N. since the unmixed variance is l/4: log [l -M,]
= log $
-
;;O;;
(6)
The specific objective of this paper is to present the results of a study of the effect of adding balls as mixing aids on the batch mixing of particulate solids. Their effect on the mixing process has been quantiliecl by evaluating how the presence of the balls affects the diffusion coefficient and the degree of mixeciiless.
EXPERlhENTAl_
?.ElHOD
Ah?,
hlATERLUS
The mixer and sampling methods used in this investigation have been described in a recent publication5_ Briefly, the mixer was constructed from a Lucite tube 10 in. in length and 5 in. in diameter. The mixer was cut in half along its length so that sampling the contents could be carried out by replacing one half of the mixer with a chamber tha had been divided into 20 compartments spaced 3 in. apart After each mixing run, the top half of the mixer was replaced by the sampling unit. This assembly was rotated one halfturn, thereby dividing the mixer contents into 20 individual samples5.
I47
IN A DRUM
The choice of balls to be used as mixing aids was critical in that comminution was to be avoided_ Preliminary experimentation showed that $-in. Lucite balls (specitic _mvity about unity) were suitable. Even after 7000 revolutions of the mixer, no appreciable grinding was found to have taken pIace_Also, these balls remained submerged in the bed of particles without floating_ In loading the mixer, a partition was placed in the center of the mixer normal to the mixer axis and the two components placed on either side of it. The bed was leveled, the partition removed, and the balls placed uniformly on the surface of the charge After the cover was put in place, the mixer was rotated the desired number of revolutions at 22: i of the critical speedTo simplify analysis of the contents of the mixer. two different particulate systems were used. The first system consisted of dclomite of two size fractions_ Equal quantities by weight or 2S x 32 and 42x48 mesh particles were mixed, and assaying was performed by sieving on a 35 mesh screen. The diffusion coefficients of the two sizes were very nearly the same, differing from each other by abcut 5-7 %_ In the second system, equal amounts of 65 x 100 mesh calcite and 65 x 100 mesh quartz were mixed. These materials have very similar densities, and consequently segregation would not be a problem. Simple dissolution of calcite in hydrochloric acid was used for analysis of the mixture compositions_ For the system consisting of the 28 x 32 mesh and the 42 x48 mesh dolomite, the drum was a!ways iilled to approximately 23 o/Oof its volume, i Q the total volume occupied by particles plus balls was approximately constant With no balls present, the total weight of dolomite was 900 g and the amount of powder was reduced appropriately to maintain the same bulk volume as the number of balls was increased_ When 160 balls were added, the amount of dolomite was 800 S For the 65 x 100 mesh quartz-calcite system, the mixer filling wz maintained approximately constant at 22% of the total bulk volume. In the absence of balls, the feed charge was 1060 8 and this was reduced accordingly as the number of balls was increased. With 160 balls added, the feed charge was 800 g
first set of experiments involved determining the mixing characteristics of the two different parti-
The
P&r&r
TeM_4
(197Oj71) 146-150
P. Ii. CHAUDIiURI,
148
D. W.
FLERSENAU
01
0
0
2
4 NUMBER
6
8
10
I2
OF REVOLUTIONS.
14
IO J~vHBER
IS
I
,
20
30
I
I
40
o.= REvoLuTIohS.
50
fi
N=IO-’
Fig I. Standard deviation of samples taken from the mixer as a function of the number of revolutiotls after long mixing times (without balk).
Fig 3_ The variance of samples taken from mixer cs the square root of the number of revolutions for the mixing of 65 x 100 mesh quartz and calcite in the prcsene of balk
culate systems in the absence of balls. For the evaluation of the diffusion coefficient in ‘the above system, the mixer was rotated for a larger number of revolutions than in the experiments with balls. Figure 1 presents the logarithm of the standard deviation as a function of the number of revolutions of the mixer, plotted in accordance with eqn. (3) the slope these straight lines, average diffusion of in dolomite system to 0.60 x inx/rev and 10e3 S/rev for particles in the quartz-calcite system. Figures 2 and 3 present the variation of variance of the mixture with the number of revolutions for the dolomittiolotnite and quartz-calcite system,
respectively, in the presence of balls. In order to minimizegrindingeffects,mixingwaslimited to7000 revolutions of the mixer. Thus eqn. (4) is applicable for the interpretation of the data. Observations of these two figures shows that straight lines are also obtained for C; versus jiV plots when balls are present inside the drum, indicating that particle transport can still be interpreted by a random movement process However, the increase in the magnitude of the slope of the line indicates that the diffusion coefficient is increased as more balls are
DOLOMITE
0
10 ,,NuHBER
20
40
30
OF REVOLUTIONS.
50
fi
Fi_e 2 The variance of -pies takenfrommixercs. thesquare root of the number of revolutions-for the mixing of dolomite in tbepresenceofballsarmixingaids.
01
0
1
1
40
80 NUMBER
OF
1
I
I20
160
BALLS
Fig 4. The effect of the number of balk added a~ mixing aids on tbc diGdon cueffxients for the mixing of skits-quartz and dolomit+dolomitc systems.
Ponder
Technol,
4
(1970/7I) 146450
OF MIXING IN
KIkl-iCS
added to the system From the slopes of these lines, the diffusion coefficients are calculated for the different additions of balls to the system, and in Fig 4 these diffusion coefficients have been plotted ZXXYZLS the number of balls. It is interesting to note that the manner of variation of the diffusion coeflicient with the number of balls is similar in the two systems investigated. At first the increase is very rapid up to an addition of about 100 balls, after which the value of D becomes nearly constant. The maximum diffusion coefficient obtainable with balls is also reasonably constant at a value 3-3.5 times that without added balls. For the conditions used in these experiments, the maximum diffusion coelkient is obtained when 3 “/D of the internal volume of the mixer is filled with balls An empirical equation relating the diffusion coefficient to the number of balls in the mixer is D,-D -kb D,-D,=e
A
149
DRWi
617 x 10eJ in’/rev for the quartz-calcite system and 225 x 10m3 in’jrev for the dolomite system. The value of k for any mixer-particulate system will depend on the characteristics of the materials being mixed and on such operating variables as the speed and filling of the mixer (for the quartz-calcite system k=O.O089 and for the dolomite system k= 0_0059). These results clearly show that the diffusion coefficients of the smSller (65 x 100 mesh) and lighter (specific gravity=27) quartz-calcite particles are considerably greater than that of the larger and heavier dolomite particles (specific gravity=29)_ Addirig balls to the system similarly has a greater
where D, is the diffusion coefficient with infinite number of balls, D is the diffusion coefficient with b balls, D, is the diffusion coellicient without added balls, k is a constant, and b equals the number of balls. The loaarithm of (D, - D) has been plotted against b in F& 5 for both the systems and straight lines are obtained_ The values of D, are iound to be 10
0
30
20
,/-zil
Q
REVOL”TIDNS.
40 JTi
Fig 6. The effect of balls added to the mixer on the rate of miring in the tno-size dolomite system as expressed in terms of the degree of mixdness
1
I-
L
/I 0 Fig 5. (D, -0)
1
1
40
80
-NUMBER
I
120
OF BALLS
I
160
I
a!3 a function of the number of balls in the dolomite-dolomite sys:cms
1
CAI_CtTE-OUCRTZ
-,-
0
0
/-
*
,hJMER
--1
,
1
20
30
40
OF
REVOLUTIONS.
SO
dki
Fig 7. The effectof balls on the rate of mixing in thecakitc-quartz system as cxprcscd in tams of the degree of mixedness_
Pan-der Techndl,:
(197OPt) IS-150
150
P. K_ CHAUDHURl.
effect on the quartz-calci:e system than in the dolomite system The random movement of balls, thus, must also have a greater effect on the random movement of the finer particulates within the mixer. Finally, let us briefly look at the effects of the addition of balls on the kinetics of the mixing process For small numbers of revolutions of the mixer, eqn. (5) shows that the degree of mixedness M,shculd be directly proportional to the square root of the number of revolutions of the mixer Accordingly, in Figs. 6 and 7, MN is plotted t‘ersus JN for the dolomite and the quartz-calcite system, respectively_ These figures show that the addition of balls increases the slope of the M, rxnus JN plots. In the ease of both mixing systems, the maximum effect of added balls on mixing kinetics, as expressed in terms of M,, is reached when at least 96 balls have been added to the drum. The further addition of bails has no further effect on the degree of mixedness. The linearity of the relation between M, and ,/N in these last two figures also indicates that random motion or diffusion continues to play a dominant role in the mixing process in the case of mixing in the presence of balls In an end-loaded barrel mixer, increasing the number of balls changes
D. W_
EWERSENAU
the diffusion coefficient, but the mixing mechanism remains substantially the same. ACKNOWJiEDGEMENTS
The authors wish to thank the National Science Foundation, Engineering Division, for support of this research We also thank Dr. Richard Hogg for discussions during the course of the work REFERENCES 1 Y. SAWAHATA, On the mixing of solids in mixing aids, ffigakuKagaku (abridged edition). 4 (1966) 270 2 P. M. C. LAM, Developments in the theory of particle mixing. J. Appl. Chem., 4 (1954) 257. 3 R H~GG. D. S. CAHN, T_ W_ H-Y AXE D. W. F-AU, Diiffcsional z&sing in an ideal system. Chm. fig_ ScL, 21 (1966) 1025. 4 D_S.C,uis,D.W_F -AU, T_ W_ Hwy AS- R HOGG. Diffusional mechanism of solid-solid mixing A’arure. 109 (1966) 494. 5 R. HCIGG. G. MESPEL AFZP D. W_ F uEaslEsAU, The mixing of trace quantities m particulate solids, Poder TechmI_ 2 (1968/69) 223. 6 D_S_CAHNA~DD_W_F UERSTESAU. Simulation of diffusional mixing of particulate solids by Monte Carlo techniques, Pow&r Technd. I (1967) 174.
Pomier Tc?dznoL.C i1970/71)
146-150
Effect of Mixing
Aids on the Kinetics
P. K. CHAUDHURI
AND
of Mixing
SUMMARY The batch mixing of particulate solids in a rotating drum mixer has been investigated with various numbers of Lucite balls added as mixing aids. Two experimental systems hate been studied. One invo!ved the mixing of-dolomite particles of two sizes, 28 x 32 and 42 x 48 mesh, while the other consisted of mixing calcite and silica of the same size range, namely 65 x 100 mesh. The progress
of mi.xing was followed
by a simple sampling and assaying technique-which pemlits accurate determination of the composition of the powder in sections along the axis of the mixer. The role of mixing aids on the mixing process was quantified by evaluating their effect on the dz@tsion coefficient and the degree of mixedness An empirical relationship between the difision coeficient and the number of balls added to the mixer was obtained.
INiRODUCTION
addition of mixing aids to a drum mixer is known to increase the rate of mixing of particulate solids. However, to date only one qualitative discussion on the effect of mixing aids has been published’. The system used in that study involved sand grains dyed with two different colors, using rubber balls as mixing aids The purpose of our paper is to present the results of a systematic investigation of the effect of mixing aids on the kinetics of mixing. To understand how mixing aids might affect the process let us first consider briefly how particuIate solids are transported and mixed within a blending device_ Two basic mechanisms have been identified2T3_One is operative when individual particles of one component are redistributed throughout the The
* Thispaperis based on the thesissubmitted by P. K. Chaudhuri for the MS. degree in the College of Engineering, University of California, Berkeley_ Powder Technology - Elsevia
Sequoia SA,
Lausanne
Rotating
Dram*
D. W. FUERSTENAU
unirersitrof Cal~omia.Berx-eler, calif_ 947m (Received
in a
[U.S.A.)
July 4, 1970)
batch as a result of random motion of the individual particIes caused by the tumbling motion within the mixer_ This process, which is very much akin to dillusion of metallic atoms through vacancies, has been called “micromixing” or “diffusive mixing”3_ The second process consists of the collective movement of groups of particles. This has been called “macromixing” or “convective mixing”3. Thus, the first process involves the movement of individual particles, and the second the movement of groups of particles. For an axially loaded drum mixer, diffusion is the mechanism by which mixing takes place3. Since no component of the motion is parallel to the axis of the mixer4_ random movement or diffusion takes place along the axis, and analysis of the
mixing process can be carried out in terms of a “diffusion” coeffxcient. If the materials are loaded in layers on top of each other, mixing occurs by convection primarily, with uhimate dispersion being controlled by diffusion_ To date no theoretical analysis has been given of the very complicated layer-loaded drum mixer in which mixing occurs by the two mechanisms. Thus, an axially loaded system must be investigated in order to make a quantitative study of the effect of mixing aids on the kinetics of the process. To quantify the effect of mixing aids on the kinetics of mixing, analysis of mixing results must be carried out in terms of the variance of mixture samples For the entire mixer bed, the sample variance is given by
i$ G$ =
[ci-c]2
S-l
(1)
where Ci is the concentration at the i th sample position, c is the mean concentration, and S is the total number of samples taken. The criterion used forexpressingthestateofmixednessofthemixercontents is the parameter MN defined by
KlXSl-iCS
OF MIXING
where 6 is the unmixed variance3_ As shown previously3, solution of Fick’s second law of diffusion equation for the mixing of equal proportions of two identical powders yields the following expression for larger numbers of mixer revolutions, N I &=-$exp(-Fj
(3)
where D is the diffusion coefficient and L the mixer length. For small values of N,
From eqns. (2), (3) and (4), expressions for the kinetics of mixing can be derived for relating the degree of mixedness nrfs to the number of mixer revolutions For small N.
and, for large N. since the unmixed variance is l/4: log [l -M,]
= log $
-
;;O;;
(6)
The specific objective of this paper is to present the results of a study of the effect of adding balls as mixing aids on the batch mixing of particulate solids. Their effect on the mixing process has been quantiliecl by evaluating how the presence of the balls affects the diffusion coefficient and the degree of mixeciiless.
EXPERlhENTAl_
?.ElHOD
Ah?,
hlATERLUS
The mixer and sampling methods used in this investigation have been described in a recent publication5_ Briefly, the mixer was constructed from a Lucite tube 10 in. in length and 5 in. in diameter. The mixer was cut in half along its length so that sampling the contents could be carried out by replacing one half of the mixer with a chamber tha had been divided into 20 compartments spaced 3 in. apart After each mixing run, the top half of the mixer was replaced by the sampling unit. This assembly was rotated one halfturn, thereby dividing the mixer contents into 20 individual samples5.
I47
IN A DRUM
The choice of balls to be used as mixing aids was critical in that comminution was to be avoided_ Preliminary experimentation showed that $-in. Lucite balls (specitic _mvity about unity) were suitable. Even after 7000 revolutions of the mixer, no appreciable grinding was found to have taken pIace_Also, these balls remained submerged in the bed of particles without floating_ In loading the mixer, a partition was placed in the center of the mixer normal to the mixer axis and the two components placed on either side of it. The bed was leveled, the partition removed, and the balls placed uniformly on the surface of the charge After the cover was put in place, the mixer was rotated the desired number of revolutions at 22: i of the critical speedTo simplify analysis of the contents of the mixer. two different particulate systems were used. The first system consisted of dclomite of two size fractions_ Equal quantities by weight or 2S x 32 and 42x48 mesh particles were mixed, and assaying was performed by sieving on a 35 mesh screen. The diffusion coefficients of the two sizes were very nearly the same, differing from each other by abcut 5-7 %_ In the second system, equal amounts of 65 x 100 mesh calcite and 65 x 100 mesh quartz were mixed. These materials have very similar densities, and consequently segregation would not be a problem. Simple dissolution of calcite in hydrochloric acid was used for analysis of the mixture compositions_ For the system consisting of the 28 x 32 mesh and the 42 x48 mesh dolomite, the drum was a!ways iilled to approximately 23 o/Oof its volume, i Q the total volume occupied by particles plus balls was approximately constant With no balls present, the total weight of dolomite was 900 g and the amount of powder was reduced appropriately to maintain the same bulk volume as the number of balls was increased_ When 160 balls were added, the amount of dolomite was 800 S For the 65 x 100 mesh quartz-calcite system, the mixer filling wz maintained approximately constant at 22% of the total bulk volume. In the absence of balls, the feed charge was 1060 8 and this was reduced accordingly as the number of balls was increased. With 160 balls added, the feed charge was 800 g
first set of experiments involved determining the mixing characteristics of the two different parti-
The
P&r&r
TeM_4
(197Oj71) 146-150
P. Ii. CHAUDIiURI,
148
D. W.
FLERSENAU
01
0
0
2
4 NUMBER
6
8
10
I2
OF REVOLUTIONS.
14
IO J~vHBER
IS
I
,
20
30
I
I
40
o.= REvoLuTIohS.
50
fi
N=IO-’
Fig I. Standard deviation of samples taken from the mixer as a function of the number of revolutiotls after long mixing times (without balk).
Fig 3_ The variance of samples taken from mixer cs the square root of the number of revolutions for the mixing of 65 x 100 mesh quartz and calcite in the prcsene of balk
culate systems in the absence of balls. For the evaluation of the diffusion coefficient in ‘the above system, the mixer was rotated for a larger number of revolutions than in the experiments with balls. Figure 1 presents the logarithm of the standard deviation as a function of the number of revolutions of the mixer, plotted in accordance with eqn. (3) the slope these straight lines, average diffusion of in dolomite system to 0.60 x inx/rev and 10e3 S/rev for particles in the quartz-calcite system. Figures 2 and 3 present the variation of variance of the mixture with the number of revolutions for the dolomittiolotnite and quartz-calcite system,
respectively, in the presence of balls. In order to minimizegrindingeffects,mixingwaslimited to7000 revolutions of the mixer. Thus eqn. (4) is applicable for the interpretation of the data. Observations of these two figures shows that straight lines are also obtained for C; versus jiV plots when balls are present inside the drum, indicating that particle transport can still be interpreted by a random movement process However, the increase in the magnitude of the slope of the line indicates that the diffusion coefficient is increased as more balls are
DOLOMITE
0
10 ,,NuHBER
20
40
30
OF REVOLUTIONS.
50
fi
Fi_e 2 The variance of -pies takenfrommixercs. thesquare root of the number of revolutions-for the mixing of dolomite in tbepresenceofballsarmixingaids.
01
0
1
1
40
80 NUMBER
OF
1
I
I20
160
BALLS
Fig 4. The effect of the number of balk added a~ mixing aids on tbc diGdon cueffxients for the mixing of skits-quartz and dolomit+dolomitc systems.
Ponder
Technol,
4
(1970/7I) 146450
OF MIXING IN
KIkl-iCS
added to the system From the slopes of these lines, the diffusion coefficients are calculated for the different additions of balls to the system, and in Fig 4 these diffusion coefficients have been plotted ZXXYZLS the number of balls. It is interesting to note that the manner of variation of the diffusion coeflicient with the number of balls is similar in the two systems investigated. At first the increase is very rapid up to an addition of about 100 balls, after which the value of D becomes nearly constant. The maximum diffusion coefficient obtainable with balls is also reasonably constant at a value 3-3.5 times that without added balls. For the conditions used in these experiments, the maximum diffusion coelkient is obtained when 3 “/D of the internal volume of the mixer is filled with balls An empirical equation relating the diffusion coefficient to the number of balls in the mixer is D,-D -kb D,-D,=e
A
149
DRWi
617 x 10eJ in’/rev for the quartz-calcite system and 225 x 10m3 in’jrev for the dolomite system. The value of k for any mixer-particulate system will depend on the characteristics of the materials being mixed and on such operating variables as the speed and filling of the mixer (for the quartz-calcite system k=O.O089 and for the dolomite system k= 0_0059). These results clearly show that the diffusion coefficients of the smSller (65 x 100 mesh) and lighter (specific gravity=27) quartz-calcite particles are considerably greater than that of the larger and heavier dolomite particles (specific gravity=29)_ Addirig balls to the system similarly has a greater
where D, is the diffusion coefficient with infinite number of balls, D is the diffusion coefficient with b balls, D, is the diffusion coellicient without added balls, k is a constant, and b equals the number of balls. The loaarithm of (D, - D) has been plotted against b in F& 5 for both the systems and straight lines are obtained_ The values of D, are iound to be 10
0
30
20
,/-zil
Q
REVOL”TIDNS.
40 JTi
Fig 6. The effect of balls added to the mixer on the rate of miring in the tno-size dolomite system as expressed in terms of the degree of mixdness
1
I-
L
/I 0 Fig 5. (D, -0)
1
1
40
80
-NUMBER
I
120
OF BALLS
I
160
I
a!3 a function of the number of balls in the dolomite-dolomite sys:cms
1
CAI_CtTE-OUCRTZ
-,-
0
0
/-
*
,hJMER
--1
,
1
20
30
40
OF
REVOLUTIONS.
SO
dki
Fig 7. The effectof balls on the rate of mixing in thecakitc-quartz system as cxprcscd in tams of the degree of mixedness_
Pan-der Techndl,:
(197OPt) IS-150
150
P. K_ CHAUDHURl.
effect on the quartz-calci:e system than in the dolomite system The random movement of balls, thus, must also have a greater effect on the random movement of the finer particulates within the mixer. Finally, let us briefly look at the effects of the addition of balls on the kinetics of the mixing process For small numbers of revolutions of the mixer, eqn. (5) shows that the degree of mixedness M,shculd be directly proportional to the square root of the number of revolutions of the mixer Accordingly, in Figs. 6 and 7, MN is plotted t‘ersus JN for the dolomite and the quartz-calcite system, respectively_ These figures show that the addition of balls increases the slope of the M, rxnus JN plots. In the ease of both mixing systems, the maximum effect of added balls on mixing kinetics, as expressed in terms of M,, is reached when at least 96 balls have been added to the drum. The further addition of bails has no further effect on the degree of mixedness. The linearity of the relation between M, and ,/N in these last two figures also indicates that random motion or diffusion continues to play a dominant role in the mixing process in the case of mixing in the presence of balls In an end-loaded barrel mixer, increasing the number of balls changes
D. W_
EWERSENAU
the diffusion coefficient, but the mixing mechanism remains substantially the same. ACKNOWJiEDGEMENTS
The authors wish to thank the National Science Foundation, Engineering Division, for support of this research We also thank Dr. Richard Hogg for discussions during the course of the work REFERENCES 1 Y. SAWAHATA, On the mixing of solids in mixing aids, ffigakuKagaku (abridged edition). 4 (1966) 270 2 P. M. C. LAM, Developments in the theory of particle mixing. J. Appl. Chem., 4 (1954) 257. 3 R H~GG. D. S. CAHN, T_ W_ H-Y AXE D. W. F-AU, Diiffcsional z&sing in an ideal system. Chm. fig_ ScL, 21 (1966) 1025. 4 D_S.C,uis,D.W_F -AU, T_ W_ Hwy AS- R HOGG. Diffusional mechanism of solid-solid mixing A’arure. 109 (1966) 494. 5 R. HCIGG. G. MESPEL AFZP D. W_ F uEaslEsAU, The mixing of trace quantities m particulate solids, Poder TechmI_ 2 (1968/69) 223. 6 D_S_CAHNA~DD_W_F UERSTESAU. Simulation of diffusional mixing of particulate solids by Monte Carlo techniques, Pow&r Technd. I (1967) 174.
Pomier Tc?dznoL.C i1970/71)
146-150