Solids Mixing L. T. FAN, S. J. CHEN, AND C. A. WATSON
olids mixing or solids blending is the operation by which S two or more solid materials in particulate form are scattered randomly in a mixer among each other by the random movement of the particles. I t has long been one of the most widely used industrial, agricultural, and pharmaceutical processes. However, it has been much less developed both theoretically and practically compared to other processes. Therefore, the practice of art often dominates in the operation and design of the mixing equipment and in the assessment of quality of a mixture. The rapid growth of other process industries has made solids mixing a well recognized but not yet well developed operation. A considerable amount of work has been reported in the literature since Weidenbaum (779) made an extensive review in 1958. The development of fast and large memory computers has made digital simulation an important technique in studying solids mixing. The present work surveys the literature (from 1958 to 1969) related to solids mixing, especially the more recent information, and discusses a variety of topics and concepts on solids mixing. The important aspects of solids mixing can be summarized by asking the following questions: What is a homogeneous mixture? How can one tell if a mixture is well mixed? How can a homogeneous mixture be prepared efficiently and economically? A homogeneous solid mixture is one in which the compositions of the constituents are uniform throughout the whole mixture. T o know the homogeneity of a mixture, some criteria of the degree of mixedness have to be defined. Samples, instead of the entire mixture, are analyzed; the samples should be representative of the mixture. The amount of material in a sample and the number of samples to be taken are important and difficult problems in solids mixing. Knowing mechanisms of mixing, characteristics of materials to be mixed, and the characteristics of a mixer is essential for the preparation of a desired homogeneous mixture. In the following sections these topics are discussed in detail.
of mixedness. A useful and general index should be related as closely as possible to specified characteristics of the final mixture, should be independent of the mixing processes, and should be easily determined. Criteria of degree of mixedness. Over 30 different criteria of the degree of mixedness which have been proposed by many investigators in studying different systems are summarized in Table I. Most of these criteria are based on statistical analysis. Others include photometry (58, 779) and kinetic approaches (78, 28). The difference i n the definitions for the criteria reveals the complexity of the mixing process and the uncertainty of various concepts and notions in the field of solids mixing. Because of the random nature of the mixing process, statistical analysis has become the approach most frequently used among investigators. I t concerns primarily the measurement of the standard deviation or the variance of the spot samples taken from a mixture. T h e criteria are then expressed in different forms of the standard deviation of the variance. Most of the statistical criteria deal with binary mixtures, mixtures consisting of two constituents. Table I, a n expanded form of Kanise's (84),includes the criteria which were reviewed extensively by Weidenbaum ( 179) in 1958. Some criteria reported in the literature since then and some criteria not covered by Weidenbaum are reviewed here and are also included in the table. T h e table also lists mixing equipment and methods of sampling and analysis used by various investigators. Some fundamental statistical terms are essential for the following discussions. They are defined below:
A sample arithmetic mean is defined as 5 =
-x 1
"
n
i=l
xi
where n = number of spot samples ith value of x which represents a characteristic of a spot sample such as composition
xi =
Concept of the Degree of Mixedness
A completely ordered arrangement of particles in a mixture was widely regarded as a perfect mixture for a long time. Though spot samples taken from this ordered mixture would give the same exact composition, such a mixture cannot be obtained by means of a conventional mixing operation. Skidmore (762) reported that it may be obtained by other techniques, such as manually placing individual constituents in a definite order, or properly controlling a specially designed mixer to handle individual constituents. For most applications, a statistically random mixture is obtained (767). The most common definition of a perfectly random mixture is one in which the probability of finding a particle of a constituent of the mixture is the same for all points in the mixture. Usually a mixture obtained from a mixing process is not completely random. Unlike a liquid mixture, compositions of spot samples of a solid mixture are different from each other. Therefore, the first important thing to know about a solids mixture is how well it is mixed. To understand the homogeneity of a solids mixture, a mixing index is generally used to measure the degree
A sample variance is defined as = -
n
5
(Xi
- 5)2
(2 1
i=l
An unbiased sample variance is defined as (Tz
= __
5
n - I . 1=1
(Xi
- a)%
(3)
The theoretical mean (or expectation), variance, uV2,are defined as
/I, and the theoretical
= E ( x ) = ZXiF(X)
(4 )
/L
and LT2
= E[(Xi
- p)2]
=
Z(Xi
- /.)ZF(X)
= E(Xi2) - ,uz
(5)
where P ( x ) is the probability distribution function. The summations are carried over all the spot samples in the system.
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53
TABLE I . SUMMARY OF DEFINITIONS O F Degree of mixedness
Material
or varznnce
Investigators
M='
Charlier (see 69)
- uz - orz
U
= degree of mixedness = observed standard de-
.
I
.
-7
=
p
= mean stituent fraction in mixture ,of con-
x*
= mean fraction of con-
CI
M = 1
-X
=
ff X
...
,..
22
100%
Coal-salt 25-100 mesh
Inclined cylinder
30 samples each containing 150 particles were taken with a sample thief
I: nj(xj Hemelrijk (see 69)
.Volniioni
Sampling M
Lexis (1877) as reported (66)
Coulson and Maitra (38)
Experiment Type of mixer
x12
when use
I
random mixture
x
- PI2
j =1 _____P ( l - P)
uo
c
ing = observed viation standard de-
P is known, otherwise
Lacey ( 9 6 )
Bullet tapioca-bullet Wooden sphere tapioca 5000-5000 p
Blumberg and Maritz (7 7 )
Sand-Sand (-34,'
Horizontal cylinder
+48) (-48/+65)
Mixture was partitioned in a Y trav. Particles were x: co;nted in each partition Sample thief was used to take * z at least 160 samples. Colored particles were counted up
Ferrite-pyrite
Tumbler
?riagnetic properties of ferrite were used to separate it from pyrite
Finger prong
Chudzikiewicz (31j
Sand-picric acid tracer 41712'35 -417/295 z. ! 1 . Quartz-ZnO 2 . Quartz-Salt
Weidenbaum and Bonilla (780)
Sand-salt -40/50 mesh
Samples of 1 cc. were taken and analyzed colorimetrically Samples of size ranging from 0.5 g to 10 g were taken a t different time intervals Three sample thieves were inserted horizontally to take 27 spot samples with sample size ranging from 120 to 160 particles
Kramer (see 95)
stituent in samples = percentage taken together unmixed
= number of samples which have approximately same composition of overall mixture = number of samples = number of particles in the ]th sample = fraction of A particles in j t h sample = fraction of A constituent = average number or particles per sample = fraction of key constituent in sample = mean fraction of x in mixture = standard deviation of complete random mixture = standard mixturedeviation before mixof
m
...
viation theoretical viation of standard complete de-
Lacey (95) Smith (763)
PVC NazCO3sal;d-~nnj20n mesh
Yano Kanise and Tahaka (I&)
Masterbatch-polythene 3000-4000 fi
Adams and Baker (1)
Viscose-caustic
Beaudry (8)
Simpson laboratory roll mixer Horizontal cylinder
1. V-mixer 2. Double cone 3. Cubic mixer 4. Horizontal cylinder I . Double cone 2. Ribbon blender 3. Cubic mixer 4. V-mixer Continuous mixer
= degree of mixedness = standard deviation o
c
= standard mixturedeviation
co
= standard deviation of
xi
= particle concentration
p
of ith sample = mean concentration of mixture = number of samples
complete random among samples
mixture before mixing
Samples of 50 g. each were taken from outlet of mixer, and number of black masterbatch particles were counted 30 samples taken
...
Buslik (27)
M C?
n
uo c
1. Witherite-Sand (isoojsoo1500/800 p 2. Witherite-pyrites
Ashton and Valentin (6)
1. Double cone 2 . Airmix 3. Vertical helical im-
Sample thief was used to take 20-100 samples, which were analyzed chemically
(isoojsoo-
180/110 ill 3. Calcite-sand
ur
C 2Tg
(1soojsoo-
Michaels and Puzinaukas (705) Manning (702) Bonilla and Crownover, and Bonilla and Goldsmith (see
.w ,Jvz
M =
=
180/1 10) Dextrose-claywater
5 P(1 - P ) w ___ =
00
W Z'Xi
-i
Samples of 1 cc were taken and analyzed chemically
l
,..
Ai - log 2___ x(l - x j
For a binary mixture: C,2 =
W q,
...
779)
Stange (765)
Finger prong
Batch mixer with horizontal screw-shaped blades
...
P w
...
FV .x\Ii
constituent in the ith sample 3 0. q 1 . .
5
54
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
= degree of mixedness = standard deviation of mixture before mixing = standard deviation among samples = standard deviation of complete random mixture = fraction by weight of mrticle size e = avkrage weighiof particle of size g = average weight of particle of all sizes in mixture = wekilt = standard deviation of the complete random mixture = fraction by weight of one constituent in the mixture = weight of one size particle = sample weight = degree mixedness weight of fraction of key
= mean value of x for .V
spot samples = concentration of the constituents = mean particle weight mix constituents of
DEGREE OF MIXEDNESSa Degree o/ mixedness or variance
Investigators Poole, Taylor, and Wall’s modification of Stange’s definition for a binary mixture
Material Copper-nickel
Experiment Type of mixer Lodige mixer
( 733)
Danckwerts (39)
Sampling
Notations
Random numbers were used to help sample randomly by using a sample thief, Samples of size ranging from 5 to 1000 g were analyzed spectrophotometrically
= standard deviation o
UT
Scale of segregation:
p, q
=
7
= =
ai, a2
=
6
=
S = LmR(r)dr where
ri,
ea2, abZ=
Hyun and Chazal (75)
Horizontal cylinder
Sakaino (750)
TNT-NHaNOs
20 samples were taken randomly with the aid of random numbers after 900 revolutions. A plastic scoop was used to collect samples. Each sample was separated into 3 constituents by screening
Samples were analyzed chemically
the complete random mixture concentration of the constituents sample weight weight fraction of a sample within a size class concentrations measured a t two points in mixture a distance r apart mean concentration of a and b in mixture as a whole variances of point of B concentrations constituents A and from the overall mean concentraspectively tions d and 6, re-
= volume fraction of large particles = ratio of sam le to volume o?large particles y = volume fraction of medium particle sample volume Vsa = sample volume V m + , = volume available to medium and small particles V = volume of medium particles Vez = excluded volume VI = volume of large particles w = ratio of diameter of a medium particle to ticle of a large parthat
x
a
E
= notation for expecta-
tion Leggatt (99)
Niffenegger (774)carried out experiments usi ng mixtures of seeds and compared these four definitions
Westmacott and Linehan (782)
W M E xi
Miles (706) Miles Carter and Shgnberge; (707)
co
= s t before 2 d Z X :mixing iation = standard deviation
M
Gray (58)
Shinnar and Naor
1. Sand-ilmenite
If
(760)
2
2 rrp2rz2, then the mixture is not random x2
Horizontal cylinder
1. Horizontal cylinder
2. AlzOailmenite
2. V-mixer
3. BaSOa-ilmenite
3. Double cone 4. Ribbon-mixer V-mixer
White-black pottery bisque
Micro photographic method was used to measure the intensity of light as mixing proceeded A reflectivity probe was used to measure the light intensity T h e whole mixture was frozen by wax. Samples were taken by slicin the mixture. Shortest tances were measured under a microscope
31s-
If
Gayle, Lacey, and Gary (56)
Dukes (44)
Bosanquet (72)
xrZ
sponding to maximum concentration difference = light intensity, corresponding to maximum concentration difference at the start = probe meter reading = average probe meter reading = Pearson’s chi-squares = average number of particles per unit area = shortest distance (distance of each particle to its nearest neighbor) = observed chi squares for any mixture = ex ected chi-squares
-
xo2 xs2
-
-
x12 x72
x2
= exfor ected random chi-squares mixture
ua2
= variance of samples
na
= nurn%erof agglomer-
P
= equivalent agglom-
12
= intensity of variation
I
7 x2 p
ri
xoZ
x2 5 2 T f i 2 1 2 2 , the mixture is random
Roofin granules, 14/28 mesh
Coal-MgO
Mixing wheel of the Bureau of Mines
...
...
12 samples, each containing
about 40 particles were taken by a sample thief. T h e distribution of the constituents in the first 30 particles was recorded 40 samples were taken. x i s equal to 13,000 and 21,000 for 0.57 g and 1.34 g samples, respectively
...
among samples = degree of mixedness = light intensity, corre-
io Limestone-limestone 12-200 mesh
= degree of mixedness = mean concentration of the constituent in samples = mean concentration of the constituent in
e
i
Oyama (see 729)
= sample weight
For segregated mixture
containing tia ag lomerates
ates
erate size
between agglomerates S2 = measure of segregaHersey (70) tion Potassium diKenwood planetary A sample thief was used to chromate-exsiccated mixer cq = theoretical standard take 10 to 50 samples with sodium sulfate deviation of sample weight ranging from 50 to concentration 500 mg. T h e samples allowed by the mixwere analyzed chemically ture specification, assuming 95 % confidence limits and normal distribua Notations used by various investigators arc often different. For uniformity, one notation is used to denote the same variable in the Tables. tion
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55
A chi-square value is computed from the expression
where xi = observed frequencies fi = expected frequencies k = number of possible outcomes of a n experiment that is performed n times For a solids mixture consisting of particles which are distinguishable only by color--e.g., black and white, the perfectly random state follows the binomial distribution.
F ( ~ AiV) , =
- P)~-%A
(6 )
(~AN)PNA(~
where F ( ~ AN) , is the probability of obtaining n~ black (or white) particles out of the total of N particles i n a sample and P is the concentration of black (or white) particles in the mixture. T h e theoretical mean (or expectation) of this distribution is found as p
= E(nA) = & A F ( n A ,
fv)
(7)
NP
and the variance is uTz =
E [ ( ~A NE')'] = ( n ~ N P ) ' F ( ~ AN , ) =
E(na2) - ( N P ) z = N P ( 1
- P)
(8)
Alternatively, the variance can be obtained by considering the fraction instead of the number of black (or white) particles, that is,
E[
(;
Figure 7.
- P)z]
Schematic relationship betzceen variance
CT:
and sample size
(6, 72, 28, 96, 97, 733)
Combining Equations 7 and 8 gives
(9)
Before mixing starts, the mixture is in a completely segregated state. Among the n spot samples taken, n A particles are black ) white (or black) (or white) and the remaining (n - n ~ are particles. The sample variance can be obtained by using Equation 2, in which
From Equations 9 and 10 it can be seen that for a n intermediate mixture, the variance is a function of the sample size and lies ~ uo2. Schematically the between these two extreme values, u , and relationship between the theoretical variance for the perfectly random mixture and the sample size is shown in Figure 1. Dukes (44) proposed the equivalent agglomerate size as the degree of mixedness. I n this model a mixture is considered as consisting of randomly distributed agglomerates. As mixing proceeds the size of agglomerates diminishes. If P is the fraction of A agglomerates in a perfectly random mixture, then for samples containing n, agglomerates the theoretical variance is as in Equation 9,
If we consider the ,V particles i n the n, agglomerates, the theoretical variance then is
% = P Thus (723), nA
n
- 72.4
=
nP
= n(l
- P)
If dV is the number of particles in each agglomerate (the equivalent agglomerate size), then
N
-
= n,N
(13)
From Equations 11-13, the equivalent agglomerate size can be found as .
n
Taking into account the agglomeration and segregation, Bosanquet ( 1 2 ) proposed the following formula,
where
S = measure of segregation Z = intensity of variation between agglomerates of size
m
If we apply Equation 13, Equation 15 can be rewritten as (28) 56
I N D U S T R I A L A N D E N G I N E E R I N G CHEM'ISTRY
where If there is no segregation and if only unmixed agglomerates exist, then (28)
s2
= 0
Hyun and Chazal (75) pointed out that Buslik’s model is a substantial improvement over Manning’s and gives good results for large particles, but gives poor results for small particles. They studied the case in which a mixture is composed of large, medium, and small particles. If the large particles are randomly distributed, some of the large particles will be so close together that the medium particles cannot fit between them. I n other words, the gaps between these large particles are available only to small particles and are not available to medium-size particles. Thus, the volume in which the medium particles can be randomly distributed is correspondingly smaller. Buslik did not take this into consideration. Hyun and Chazal derived three variance equations for large, medium, and small particles, respectively :
and
I’ = uo? = P ( l
- P)
Equation 16 thus becomes g?
P ( l - P) (N/R) 1
+
=
average weight per particle of size range g average weight of all particles G = weight fraction of particles in the particular size range W = sample weight tijg = @ =
Combining Equations 9 and 17 yields U? _ -- N - (N/R)+ 1
UT2
which is slightly different from the expression obtained by Dukesi.e., Equation 14. When N > fl, Equation 18 reduces to the Dukes equation. Hixson and Tenney (72) studied the mixing of a solid and liquid system of sand and water. The efficiency of mixing of agitators and the homogeneity of the mixture were studied. T h e degree of mixedness was defined as the average value of the percentage mixed of the samples taken from the mixer. T h e percentage mixed was defined as
where
y‘
=
b
=
rC by wt. of sand i n the ith sample
uez2
=
yo by
x a
= volume fraction of large particles i n a mixture = ratio of the sample volume to the volume of large
Y
= volume fraction of medium particles i n a mixture
Si
xz = SO
x
100 - li 100 = ____ 100
- I, x
100
where si = so = I; = I, =
ye by wt. of sand in the total mixture 7@ by wt. of liquid in the ith sample wt. of liquid in the total mixture
v,, vm+s v*
N
st
vez v1
where n is the number of samples. T h e degree of mixedness us. the speed of agitation was studied experimentally. The degree of mixedness was found to increase slowly a t low speeds, to increase sharply at higher speeds, and then to flatten off. Theoretical analysis of a mixture of particles with a size range is less tractable than that of a mixture of uniform particle size. Manning (702) attempted the analysis of a mixture of multisized particles. The variance was expressed up? =
(4.25
8 w.
I’
+ 4 w’)
uz2
particles
Thus the degree of mixedness was defined as
c
[F) +
G ( l - G)w
W
where = average weight per particle i n a particular size range G = weight fraction of particles i n the particular size range W = sample weight
w
E
sample volume volume available to medium and small particles volume of a medium particle excluded volume = volume of large particles = ratio of the diameter of a medium particle to that of a large particle = notation for the expectation = = = =
Hyun and Chazal (75) used a horizontal cylinder for their experiments. T h e mixtures were glass beads and silica gel, each consisting of three different sizes. Standard deviations were computed by using Equations 23-25 and by Buslik’s Equation 22. They found that the standard deviations computed by their equations had better agreement with the experimental data than those computed by Buslik’s equation. Stange (765) derived a n expression for a binary mixture of different particle sizes.
w
where
For a mixture of two particle sizes, the following relation is used: U?2
=
IT12
+
u1+22
(21)
where
=
fraction of one constituent of a mixture
u
= total sample weight = average particle weight of the whole mixture = standard deviation of the particle weights of the mixture
q
=1-p
a
constituents
variance of large particles = variance of small particles uZ2 u 1 + 2 = variance of large and small particles
u12
p W
=
Buslik (27) pointed out that Manning’s relations are not applicable to random samples. H e modified Manning’s model to give
Harnby (64)showed that for a binary mixture in which one of the constituents has a narrow size range, the expressions of Buslik, Equation 22, and Stange, Equation 26, for the variances of a perfectly random mixture of multisized particles, are identical. Poole, Taylor, and Wall (733)modified the above equation to give
VOL. 6 2
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7 JULY 1970
57
then the mixture is not random, and if
where
fa
=
fraction of particles within a size range
n
wa = mean particle weight of a size range
xznla2
They also verified the model experimentally, using mixtures of copper and nickel powders in a centrifugal type batch mixer. Stange (766)further developed a n equation for the variance of a single constituent i n a multiconstituent mixture.
(28) Extending Buslik’s expression, Harnby (64) obtained the same equation as Equation 28. Investigators who have studied the mixing of seed have proposed several criteria for the degree of mixedness. Leggatt ( 9 9 )proposed a chi-square index for determining seed homogeneity with respect to numbers of weed and (or) foreign seeds.
where x = observed concentration of a particular seed 2 = expected concentration
The Leggatt homogeneity test is described i n the 1953, 1956, and 1959 editions of the International Rules for Seed Testing ( 7 8 a ) . Westmacott and Linehan (782) suggested that the Leggatt index should be replaced with an index that measures the extent of heterogeneity.
> 2 =pCr32 i= 1
then the mixture is random where
n
=, values ~ 2 of chi-square distribution with 2 n degrees of freedom, which is exceeded with probability of a = number of particles under investigation
p
=
ri
= shortest distance (distance of each particle to its nearest
~
2
~
average number of points per unit area neighbor)
Hersey (70) compared the criterion for the degree of mixedness suggested by Poole et al. (133) and that by Ashton and Valentin (6). He concluded that both criteria appear to agree with the fact that coarse particles are more likely to result in a randomized mixture than fine particles. Nevertheless, the two criteria appear to show opposing tendencies o n the effect of different concentrations. Hersey (70) studied the effects of concentrations and particle sizes using different proportions of potassium dichromate mixed in sodium sulfate of the mean sizes of 48 p and 84 w , respectively. With these studies as a basis he proposed the measure for the degree of mixedness as (34) where u
=
observed standard deviation
uq = theoretical standard deviation of sample concentration
allowed by the mixture specification, assuming 95y0 confidence limits and a normal distribution.
Mixing parameters. Solids mixing is a complex process in which characteristics of solids and mixing equipment and operating conditions can influence the tendency to mix a n d demix. Therefore, the degree of mixedness and the rate of mixing are functions of many variables related to these characteristics and of operating conditions. T h e variables can be grouped as follows (2, 3 6 ) :
where u = observed standard deviation u, = theoretical standard deviation
Miles (706) recommended the criterion
This criterion replaced Leggatt’s in the 1966 International Rules for Seed Testing. Miles, Carter, and Shenberger (107) proposed the criteria based on the F-distribution
for nonchaffy seeds, and
(33) for chaffy seeds. Niffenegger (774) evaluated these criteria of the degree of mixedness by conducting a series of experiments on seed blending. He concluded that none of them in their present form fulfill the need of the seed industry. Shinnar and Naor (160) emphasized that the design of a test for randomness depends on the specifications of an alternative hypothesis (or a series of alternative hypotheses) about the nonrandom character of the mixture. Perfect randomness of a mixture is taken to be the null hypothesis. For a good test of a null hypothesis, two things are required (760): that it possesses a small probability of rejecting the null hypothesis when it is true that it possesses a large probability of rejecting the null hypothesis when it is false Shinnar and Naor’s (760) model of a random mixture of particles of uniform size was assumed to follow the Poisson distribution in which the fraction of black particles was less than 570. T h e criterion for the degree of mixedness is based upon the shortest distance between particles. If n
58
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
Characteristics of Solids 1. Particle size distribution 2. Particle shape and surface characteristics 3. Bulk density and particle density 4. Moisture content 5. Angle of repose 6. Coefficient of friction of particles 7. Friability 8. State of agglomeration 9. Flowability Characteristics of Mixing EquiFment 1. Mixer dimension and geometry 2. Agitator dimension 3. Construction materials and surface finishes 4. Type, location, and number of loading and emptying devices Operating Conditions I . Weight of each constituent added 2. Ratio of volume of the mixture to that of the mixer 3. Method, sequence, place, and rate of adding constituents 4. Mixer or agitation speed, if any
In solids mixing, each constituent to be mixed usually has a range of particle sizes. Most of the reported results deal with constituents of uniform particle sizes or of two sizes. Some investigators studied mixtures containing a range of particle sizes. Sawahata (752) studied the degree of mixedness of a powder
AUTHORS L. T . Fan and S. J . Chen are with the Department of Chemical Engineering, and C . A . Watson is with the Agricultural Research Service, U. S. Department of Agriculture, Kansas State University, M a n hattan, Kan. 66502
mixture of given size distribution by applying the method of sedimentation for size analysis. Segregation of particles usually occurs when mixing a solids mixture having a size distribution (26, 40-42). Larger particles stay at the top of the mixture, and smaller particles sink to the bottom. If there is a difference in density of particles, segregation will occur. However, for a density ratio of less than 3 : 1 , which is usually encountered in practice, Campbell and Bauer (26) reported that size distribution exerts greater influence on mixing and demixing than the difference in density. Friedrich (53) studied effects of the bulk weight, bulk density, and particle size distribution on dispersion of tracer particles. Shape and surface characteristics affect the flowability of particles. Round and smooth particles flow more readily than rough and irregular ones. Ciborowski (34)studied the influence of moisture content of the particles mixed in a horizontal cylinder. He found that the rate of axial mixing can be enhanced by increasing the moisture content of the particles, provided the mixing mechanism is diffusion. Nevertheless, moist and sticky particles retard the mixing process if they adhere to the walls of the mixer or if they cause agglomeration ( 3 6 ) . Angle of repose, coefficient of friction, and flowability are closely related. Solids particles possessing small angles of repose exhibit good flowability and small coefficient of friction. Good flowability is not necessarily of advantage for mixing. Segregation may occur due to their rapid movement. Friable particles are easily broken into pieces; mixing friable particles can thus increase the range of particle sizes. Dimension and geometry of a mixer and those of an agitator have influence on the particles flow pattern and flow velocities. Interactions between solids particles and the construction materials and surface finishes of a mixer may produce static charges and hence cause agglomeration. Friction between particles and the surface of a mixer has some effect on mixing and demixing. Type, location, and number of loading and emptying devices may enhance or hamper the mixing action. The operating conditions of a mixer greatly affect the degree of mixedness and the rate of mixing. Donald and Roseman (40) and Carley-Macauly, and Carley-Macauly and Donald (27-29) investigated, in detail, the effects of different operating conditions on mixing and demixing. Rose (737) attempted a dimensional analysis by assuming that mixing and demixing occur simultaneously in a mixer. He discussed the relationships between the mixing process and the physical parameters. Parameters considered to be related to mixing were: Diameter of the mixer Speed of rotation of the mixer Mean diameter of the particles Mean coefficient of friction of the particles Gravitational force and those parameters considered to be related to demixing were Diameter of the mixer Speed of rotation of the mixer Mean diameter of the particles Difference of the mean sizes of the two constituents of the mixture Mean density of the mixture Difference of the densities of the two constituents of the mixture This analysis is in good agreement with the experimental data. Experimental data of Coulson and Maitra ( 3 8 ) and those of Weidenbaum and Bonilla (780) are also in good agreement with the analysis. Sampling. Unlike liquid mixing, which can produce a homogeneous mixture, solids mixing always produces an inhomogeneous mixture. T o determine the degree of mixedness, samples are taken and analyzed. Therefore, sampling techniques and the methods of analyzing the samples are indispensable in studying solids mixing. Sampling can be as simple as scooping a quantity of solids from the mixer or it can be complex. It is always desirable that the samples taken should represent the whole mixture. However, there is always a difference between the sample and the mixture. Therefore, details of the sampling procedure are often given in reporting results of a mixing study and should include the method and the locations of sampling, the sample size, and the number of samples (779).
Method of Sampling and Samplers. The choice of a sampling technique depends on the characteristics of the mixer and the mixture. Commonly used methods are suggested in (27,28). (a) Probe sampling: Coulson and Maitra (38),and Blumberg and Maritz (77)devised a sampling thief which consists of a solid rod with radial holes. A sleeve is rotated to uncover these holes when a sample is taken. The thief is designed to cause as little disturbance to the mixture as possible. Ridgway and Wibberley (736) reported that their photoelectrical sampling probe is connected to a computer for fast data gathering. The Prob-A-Vac and the Pneumatic Probe ( 9 2 )use air in conjunction with a probe. A smaller cylinder fits inside a larger cylinder. By use of a negative pressure a t the top of the smaller cylinder, air is forced downward between the two cylinders and back up inside the smaller cylinder. This movement of air pulls the particles into the smaller cylinder and assists in pushing the probe into the bulk mixture. (b) Complete subdivision of the mixture: Lacey ( 9 6 ) used trays divided into many cells to hold the final mixture. Particles are counted one by one in each cell. Though accurate counts can be obtained, it becomes very tedious if the mixture contains many particles. (c) Freezing the mixture and taking a section through it: Shinnar, Kattan, and Steg (759) froze the mixture by pouring a monomer into the mixture, and then polymerizing in an oven. (d) Samples can be taken at an outlet stream from the mixer at a fixed time interval. (e) Sampling by mechanical belt cup-type: These samplers have cups which periodically dip into the material on a moving belt or as the material falls off the end of the belt. The cups are arranged so that they sample at different places across the width of the belt. Performance of the samplers was reported by Kramer (97). ( f ) Sampling by manual cup-type: These samplers are essentially some form of a container that is passed through the stream of free-falling material. The most common one used for grain sampling has a container which is formed in the shape of a pelican’s pouch and is called the Pelican sampler. Kramer (90, 91) compared the performance of the Pelican sampler with a mechanical diverter-type sampler. Other samplers and their performance have been discussed by Johnson (82). Locations where the samples are taken are randomly or orderly distributed throughout the whole mixture. Tables of random numbers are usually used to help locate random positions in a mixture. For orderly sampling, samples are taken a t uniform intervals in the mixture. Theories of both random and orderly sampling have been discussed by Landry (97). Methods which have been used to analyze samples are ( 5 ,27, 29,68, 739): numerical counting, radioactive counting, chemical analysis, reflection of light to a photocell, flame spectrometry, X-ray fluorescence, and magnetic separation. Sample Sire. There are two extremes of the sample size. T h e smallest possible sample size is a single particle which does not give any information of mixing, and the largest possible sample size is the entire mixture which always indicates that mixing is perfect. This of course, is not true. I n the neighborhood of these two extremes, little information about the mixedness of a mixture can be obtained. Too small a sample gives scattered data of variance, and too big a sample gives rise to information which makes the mixture appear to be better than it actually is. No systematic methods have been proposed to determine the optimum sample size. When the final product is a tablet or a package, the convenient sample size is the actual tablet or the package. If a number of samples must be taken, too much of the material should not be removed for each sample. A rough rule (2, 779) is to remove not more than 5 yo of the mixture. Recently some work has been done on the effect of sample size on the mixing index. Figure 1 schematically summarizes the results. Lacey (96) showed that for a completely random mixture the variance of sample composition decays inversely with the sample size, while for a totally segregated mixture it is independent of the sample size. Bourne ( 7 7 ) derived relationships for the variance and the sample size. The relationships were influenced by the type of sampling and the type of correlation inherent in the mixture. Bourne ( 7 6 ) also studied the influence of the shape of samples on variances. The statistical properties of square, two-dimensional samples lie between those of linear and spherical samples. For diffusive mixing, mixing due to the random movement of particles over a freshly developed surface, Lacey ( 9 5 ) stated that VOL. 6 2
NO. 7 J U L Y 1 9 7 0
59
(1 - M ) , where M is the degree of mixedness, should be independent of sample size. However, for the case of convective mixing, mixing due to transfer of groups of adjacent prrticles from one location in the mixture to another, the index depends on the sample size. Poole, Taylor, and Wall (733)found an approximate correlation between the coefficient of variation and the sample size. The experimental data were approximated by a straight line on the logarithmic plot in their range of sample sizes from 2 to 1000 mg. Effect of the sample size on the rate of mixing was also reported. Landry (97) developed detailed statistical methods and applied the methods to the problem of coal sampling in the determination of the ash content. He reported that the change in variance with the sample size gives information about the structure of a mixture. Budryk ( 1 9 ) had a similar discussion on variance and sample size in studying coal and ore sampling. Cornish (37) discussed the problems of coal blending and the methods of sampling in large scale operations. Bourne (74) interpreted the mixing data obtained by Poole, Taylor, and Wall (733, 734) by using Landry's correlation theory. Other results of variance-sample size relationships have been reported by Ashton and Valentin (6). The effect of sample size on the degree of mixedness was investigated by Chazal and Hung (30). They showed that the ratio of actual variance to the random variance is a function of the sample size and the cluster size. (A cluster is a group of particles having the same characteristics.) However, if the samples are approximately spherical or cubic in nature, and if the mixture is uniform, samples containing ten thousand particles will give an index of mixing nearly independent of sample size. Number of Samples. The larger the number of samples, the more knowledge of the mixture we can gain. Statistically speaking, to obtain a higher per cent confidence, a large number of samples are required. I t is, therefore, necessary to optimize the effort and the time in taking and analyzing a large number of samples. The effect of the number of samples on the precision of determination of mixing time was discussed by Stange (764). T h e determination of the number of samples depends primarily upon the mixer and the mixture, the technique of sampling, and the method and cost of analyzing the sample. No systematic methods have yet been developed to determine the optimum number of samples. I t is suggested that 5 to 1 5 samples be taken for both batch and continuous mixing (2).
Fz'gure 2. Flow of particles in simple horizontal cylinder (749)
-
. ...
- - + Centrifuging - - - - + Cataracting ---+ Cascading
+
e O--
Rolling SliF General Flow
Figure 3. Boundary conditions f o r horizontal cylinder (95)
Rate OF Mixing
I n mixing operations it is necessary and important to know the time required to achieve a desired degree of mixedness in a mixer. I t is often difficult to predict systematically the time of mixing because no complete theories of mixing mechanisms have been developed. Rate of mixing is important for understanding the basic mechanisms of mixing and for improving the design of a mixer. I n this section mixing mechanisms and rate equations proposed by various investigators are discussed. Mechanisms of mixing and rate equations. Three mixing mechanisms were assumed by Lacey (95) in developing the theory of the rate of mixing. They are: (a) convective mixing-transfer of groups of adjacent particles from one location to another in the mixture, (b) diffusive mixing-distribution of particles over a freshly developed surface, and (c) shear m i x i n g s e t t i n g up of slipping planes within the mixture. As mixing proceeds, all three mechanisms function to some extent. Different mixers give different predominating mechanisms. A considerable amount of work has been done on the model of diffusive mixing. Identical solid particles differing only in color are loaded in a horizontal cylinder which rotates about its axis. I n the mixer the behavior of particles rolling down the freshly developed surface is similar to ordinary molecular diffusion. Each particle has an equal chance of deflecting to either side on each collision with another particle. Thus within the mixing plane, particles exhibit random movement. Flow of particles in a simple horizontal cylinder is shown in Figure 2. Lacey (95) applied the classical diffusion theory to describe the mixing process. Fick's diffusion equation is written as
bC _
at
a*c - Da52
-
(35)
60
= time
= Da=
distance in the direction of diffusion axial diffusion coefficient
Lacey stated that since radial diffusion is extremely fast, rate of mixing is controlled by axial diffusion. Hence the mixing process can be considered as a one-dimensional diffusion process described by Equation 35. The axial diffusion coefficient was assumed to be constant. Equation 35 was solved with the following initial and boundary conditions
C(0, x )
=
1,
C(t,x ) = 0,
0
concentration of the mixture
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
x
5 z'
(36)
1
where =
average concentration of black (or white) particles
x = fractional distance in the direction of diffusion
t
=
time
Schematically the initial and the boundary conditions are shown in Figure 3. The solution is
where
Dt T = L2 k
=
5
E
