Powder Technology,
`
`61 (1990) 255 - 287
`
`255
`
`Recent Developments
`
`in Solids Mixing
`
`L. T. FAN and YI-MING CHEN
`Department of Chemical Engineering, Kansas State University, Manhattan, KS 66506
`
`(U.S.A.)
`
`F. S. LAI*
`U.S. Grain Marketing Research Center, USDA, ARS, 1515 College Avenue, Manhattan, KS 66502
`
`(U.S.A.)
`
`(Received August 18, 1989; in revised form December 28, 1989)
`
`SUMMARY
`
`This review covers the major development
`in solids mixing since 1976. The publications
`on the subject have been divided in to three
`major categories: characterization of states
`of solids mixtures, rates and mechanisms
`of solids mixing processes, and design and
`scale-up of mixers or blenders. Possible future
`work has been proposed.
`
`INTRODUCTION
`
`Solids mixing is a common processing
`operation widely used in industry. It is exten-
`sively employed
`in the manufacture of
`ceramics, plastics, fertilizers, detergents, glass,
`pharmaceuticals, processed food and animal
`feeds, and in the powder metallurgy industry.
`In fact,
`this operation
`is almost always
`practised wherever particulate matter
`is
`processed. We resort to solids mixing to
`obtain a product of an acceptable quality or
`to control rates of heat transfer, mass transfer
`and chemical reaction. It is a common occur-
`rence to read about, or to view on television,
`researchers and technicians blending par-
`ticulate
`ingredients
`for producing super-
`conducting materials.
`The present comprehensive review of solids
`mixing focuses on the published works since
`1976; nevertheless, significant papers not
`identified
`in our previous reviews are also
`included. The works prior to 1976 have
`been extensively reviewed in a number of
`expositional articles and literature surveys,
`
`*Present address: Department of Plastic Engi-
`neering, University of Lowell, Lowell, MA 01854
`(U.S.A.).
`
`[2],
`including those by Lacey [ 11, Scott
`Weidenbaum
`[3], Valentin
`[4], Venkates-
`warlu [ 51, Clump [ 61, Gren [ 71, Fan et al.
`[8], Chen et al. [9], Fan et al. [lo], Fan
`et al. [ll], Fan and Wang [ 121, Cooke et al.
`[13], Hersey [14], Kristensen [15], Rowe
`and Nienow [16] and Williams [17]. Some
`of the works in the last ten years have also
`been included in more recent reviews [18 -
`271.
`In blending or mixing different kinds of
`particulate matter, we need to be concerned
`with three broad aspects. The first is the type
`of mixer selected or designed and the mode of
`its operation. The second is the characteriza-
`tion of state of the resultant mixture, and the
`third is the rate and mechanism of the mixing
`process giving rise to this state. The mixing
`process is influenced profoundly by the flow
`characteristics of the particulate matter to be
`mixed. Recognition of the existence of the
`two types of particulate matter, free flowing
`and cohesive, forms the basis for classifying
`and characterizing mixtures and mixing
`processes. The present review covers the
`classification of mixing equipment,
`the
`characterization of mixtures and the rates
`and mechanisms of mixing processes, and the
`design and scale-up of mixers.
`
`CLASSIFICATION OF MIXING EQUIPMENT
`
`Mixing equipment can be categorized
`relatively simply according to the mixing
`mechanisms prevailing in them. The four
`major
`types are tumbler,
`convective,
`hopper (gravity flow) and fluidized mixers
`[28]. Their main characteristics are given
`in Table 1.
`
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`256
`
`TABLE 1
`Summary of mixer characteristics
`5w of
`mixer
`
`Batch or
`Main
`continuous mixing
`mechanism
`
`[ 1651
`
`Segregation
`(suitability
`for ingredients
`of different
`properties)
`
`Axial
`mixing
`
`Ease of
`emptying
`
`Tendency to
`segregate
`on emptying
`
`Ease of
`cleaning
`
`B
`
`B
`C
`
`B
`
`B
`
`Horizontal
`drum
`Lodige mixer
`Slightly
`inclined drum
`Steeply
`inclined drum
`Stirred vertical
`cylinder
`V mixer
`Y mixer
`Double cone
`Cube
`Ribbon blender
`Ribbon blender
`Air jet mixer
`Nauta mixer
`
`Diffusive
`
`Convective
`Diffusive
`
`Diffusive
`
`Shear
`
`Diffusive
`Diffusive
`Diffusive
`Diffusive
`Convective
`Convective
`Convective
`Convective
`
`Bad
`
`Good
`Fair
`
`Bad
`
`Bad
`
`Bad
`Bad
`Bad
`Poor
`Good
`Good
`Fair
`
`Tumbler mixers
`A tumbler mixer, a totally enclosed vessel
`rotating
`about an axis, causes
`the particles
`within
`the mixer
`to tumble over each other
`on the mixture
`surface.
`In the case of the
`horizontal
`cylinder,
`rotation
`can be effected
`by placing
`the cylinder
`on driving
`rollers.
`In most other cases, the vessel is attached
`to
`a drive shaft and supported
`on one or two
`bearings. Common vessel shapes
`include
`the
`cube, double-cone,
`drum, and V and Y (see
`Fig. 1).
`
`Fig. 1. V-shaped tumbler mixer [28].
`
`Bad
`
`Good
`Bad
`
`Good
`
`Good
`
`Bad
`Bad
`Bad
`Good
`Slow
`Fair
`Good
`Good
`
`Bad
`
`Good
`Good
`
`Bad
`
`Good
`
`Good
`Good
`Good
`Good
`Good
`Good
`Good
`Good
`
`Bad
`
`Good
`Good
`
`Bad
`
`Bad
`
`Bad
`Bad
`Bad
`Bad
`Fair
`Good
`Good
`Good
`
`Good
`
`Bad
`Good
`
`Good
`
`Good
`
`Good
`Good
`Good
`Good
`Fair
`Fair
`Fair
`Bad
`
`is rela-
`radial mixing
`In a tumbler mixer,
`tively
`fast while axial mixing
`is slow and is
`the rate-controlling
`step. Appreciable
`segrega-
`tion can occur
`in this type of mixer.
`If an
`internal
`impeller
`is added,
`its
`impaction
`action
`is likely
`to minimize segregation.
`Convective mixers
`of convective mixer
`In the majority
`designs, an impeller operates within a static
`shell and groups of particles are moved from
`one location
`to another within
`the bulk of the
`mixture
`[ 281. The ribbon blender
`is probably
`the most widely used convective mixer (see
`Fig. 2). A ribbon
`rotates within a static
`trough or open cylinder and the particles are
`relocated
`by
`the moving
`ribbon.
`If the
`powder
`is cohesive
`in nature, mechanical
`
`iv-
`
`Fig. 2. Ribbon mixer [28].
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`Fig. 3. Orbiting-type vertical screw mixer [30].
`
`mixing devices, such as rotating screws (see
`Fig. 3), are normally required. Convective
`mixers are likely to be less segregative than
`mixers having a predominant mechanism
`of diffusion or shear mixing.
`
`Hopper (gravity flow) mixers
`In a hopper mixer, particles flow under the
`influence of gravity and mixing is totally
`energized by gravity flow [28]. A central
`cone is usually installed so that a pronounced
`velocity gradient in the vertical direction is
`produced without causing dead zones (see
`Fig. 4). Depending on the required degree of
`homogeneity, the particles may need to be
`recycled externally, thus causing considerable
`axial mixing. The recycle can be effected by
`either pneumatic or mechanical means. Due
`to percolation, segregation is likely to occur
`in this type of mixer, both on the free surface
`of the hopper and within the bulk of the
`material.
`
`Fluidized mixers
`Mixing in a fluidized bed is energized by
`both convective and gravity effects. In the
`fluidized bed, the powder is subject to a gas
`stream flowing upward against the direction
`of gravity [31]. The weight of the particles is
`counterbalanced by the buoyancy. The
`individual particle mobility, therefore, is
`greatly increased. If the gas flow rate is suf-
`ficiently large, the turbulence within the bed
`
`(a), With multiple 1
`Fig. 4. Gravity flow blender [30]:
`inner hoppers; (b) with single inner cone.
`
`‘Fig. 5. Fluidized-bed blender [30].
`
`will be considerable, and the combination
`of turbulence and particle mobility can
`produce excellent mixing. Figure 5 illustrates
`a fluidized-bed blender.
`
`CHARACTERIZATION OF STATES OF
`MIXTURES
`
`This section reviews publications dealing
`with the characteristics of mixtures. Mixtures
`can be classified into two major groups, one
`only involves free-flowing particles and the
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`258
`
`other contains cohesive or interactive con-
`stituent(s). A free-flowing mixture will
`generally permit individual particulates
`in it
`freedom to move independently, while a
`cohesive mixture generally has some inter-
`particulate bonding mechanism, permitting
`particles to move only with an associated
`cluster of particles. Yet,
`the boundary
`between a free-flowing and a cohesive particle
`is not distinct; instead, it is “fuzzy” [ 28, 321.
`Free-flowing mixtures
`The formation of a mixture involving only
`free-flowing particles
`is a statistical or
`stochastic process in which the rules of
`probability apply. If the free-flowing particles
`are identical in all aspects except color, then
`a completely random mixture can be ob-
`tained. If they are not identical, a partially
`randomized final mixture will be generated
`due to incomplete mixing or segregation
`present in the mixing process. A mixture in
`this group exhibits a skewed distribution of
`the individual particles,
`thus featuring a
`relatively low degree of homogeneity.
`The homogeneity of a solids mixture or
`the distribution of its composition is usually
`quantified by a mixing index. Over thirty
`different mixing indexes have been reviewed
`and summarized [8]. The diversity of the
`definitions
`is indicative of the difficulty
`involved in describing the complex nature of
`the mixing process and that of the resultant
`mixture. Most of the available definitions are
`based on the variance of the concentration of
`a certain component among spot samples
`[15, 33 - 411. Nevertheless, it is difficult to
`discern the significance of these definitions.
`For processes involving contact between
`different solid phases, the mixing rate is
`proportional to the contact points or area
`among particles of the different phases. Thus,
`a definition of a geometric mixing index
`based on the number of contact points
`appears to be of practical significance. Two
`approaches exist for determining the mixing
`index based on the contact number. One
`involves the co-ordination number sampling,
`and the other, the spot sampling [42]. The
`former is effected by selecting a number of
`non-key particles in contact with a randomly
`sampled key particle. The latter is accom-
`plished by obtaining concentrations of a
`certain or key component
`in spot samples.
`
`When a single particle is taken randomly
`from a mixture, the number of all particles
`in contact with this particular particle is
`called the total co-ordination number denoted
`by n*, and the particle is called the key parti-
`cle. Let A,, particles be key particles in a
`binary mixture containing two kinds of parti-
`cles of the same size A0 and AI. The number
`of particles of component AI in contact with
`a key particle is defined as the contact
`number contributed by component AI and is
`denoted by CicO, (see Fig. 6).
`
`JFig. 6. Illustration of the contact number and the
`co-ordination number: Cl(,,) = 3, n* = 4.
`
`An expression has been derived for esti-
`mating the contact number by spot sampling
`of a binary mixture in a completely mixed
`state [43]. Under the assumption that the
`completely mixed state exists in each of the
`spot samples, the population contact number
`can be directly estimated from its concentra-
`tions in spot samples; it is
`
`C,(O) =
`
`total contact no. contributed
`by component AI in k spot samples
`total no. of key particles in 12
`spot samples
`
`i n*xin(l -xi)
`
`i=l
`
`=
`
`+lNl
`
`-xi)
`
`i n*Xi(l -xi)
`i=l
`
`k(1 -fi)
`
`=
`
`(1)
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`where
`xi = concentration of Al particles in the ith
`spot sample,
`f i = sample mean concentration of the parti-
`cles of component Al,
`k = number of spot samples,
`n = number of particles in a spot sample, and
`n * = total co-ordination number, the number
`of particles in contact with the sample parti-
`cle.
`The unbiased e_stimator of the population
`contact number Cl& can be defined from
`the biased estimator C’n,,) obtained in eqn. (1)
`as follows:
`
`8. Cl(O)’ = -E- GO)
`
`n-l
`Based on the contact number evaluated in
`eqn. (l), the mixing index is defined as
`
`(3)
`
`Cl(O)
`M=--Z-
`n*X
`where g denotes the population mean con-
`centration of Al particles.
`Expressions similar to eqns. (2) and (3)
`have been derived for a multicomponent
`solids mixture in the completely mixed state
`[ 441. They are
`
`(jj(o)r = n
`
`n-l
`
`i (n*xdi(xo)i
`
`i=l
`
`kio
`
`j= 1,2,
`
`. . ..p
`
`(4)
`
`and
`
`M
`
`=
`
`I
`
`i
`
`‘it?’
`j=ln*Xj
`
`- xi
`
`l-X0
`
`where
`(Xj)i = concentration of Aj particles in the
`ith spot sample of size n,
`f. = sample mean concentration of the key
`component,
`k = number of spot samples,
`x0 = population mean concentration of the
`key component, and
`= population mean concentration
`of
`Xj
`c_omponent Aj.
`C j(o)’ in eqn. (4) can be referred to as the
`unbiased estimator of mean contact number
`contributed by component Aj in k spot
`samples. The precision of the estimator Cjtoj’
`
`259
`
`in estimating the population mean contact
`number has also been derived through evalua-
`tion of the variance of its distribution.
`Numerical experiments have resulted in a
`smaller relative standard error of the mean
`contact number estimator than that of the
`variance estimator of spot samples. Thus, the
`former is considered to be superior to the
`latter. This new mixing index was employed
`to investigate the transverse mixing in a
`Kenics Motionless Mixer [45]. The mixing
`index has been found to increase exponen-
`tially as the number of helices in the motion-
`less mixer increases; it is also more effective
`in differentiating the quality of a mixture
`than the conventional ones derived from the
`variance of some spot samples. Based on this
`study of the relationship between the co-
`ordination number and compaction
`in the
`mixture through the mixer, it has been
`concluded that the packings of the mixtures
`are between cubic and hexagonal.
`For a binary mixture in the incompletely
`mixed state, the total co-ordination number
`is random and it varies throughout the whole
`mixture; therefore, the population concentra-
`tion is a variable. To deal with the distribu-
`tion of the concentration or inhomogeneity
`among the spot samples, a beta-binomial
`distribution has been introduced as a model
`for an incompletely mixed or semi-random
`binary mixture [ 421. A general expression
`has been developed to estimate the precision
`of the estimation of the population contact
`number from the distribution of the number
`of non-key particles.
`The application of the contact number to
`estimation of the mixing index in an in-
`completely mixed state has been extended
`to a multicomponent mixture [46]. A
`Dirichlet-multinomial model, a multivariate
`generalization of the beta-binomial model,
`has been proposed to describe a multicompo-
`nent mixture in an incompletely mixed state.
`The model gives the distribution of the
`number of particles of component Aj in a
`spot sample of size n. By equating the sample
`mean of each component to its respective
`expectation, an estimator of the model
`parameter has been derived. The parameter
`has been found to define uniquely the mixing
`index based on the contact number.
`In a study of the effect of particle-permea-
`tion on the segregation of a solids mixture in
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`260
`
`a rotating cylinder [ 471, the radial and axial
`segregation indices have been defined, respec-
`tively, as
`
`SR =
`
`C - Cmin
`c max
`
`- Gin
`
`Z(Ci - q2vl: 1’2
`
`xv;: 1
`
`s, =
`i
`
`(6)
`
`(7)
`
`where C is the average volume concentration
`of B particles in the binary mixture of A and
`B particles, sampled in the radial or axial
`direction of the vessel; Cmin and C,,,
`are,
`respectively,
`the minimum and maximum
`concentrations of B particles sampled in the
`radial direction; Ci is the concentration of B
`particles in each sampling section; and Vj is
`the bulk volume of the mixture sampled in
`each section. The radial and axial segregation
`of the mixture has been found to be closely
`related to the permeation effect measured in
`the moving bed. A similar study, conducted
`using a conical vessel, examined the effects of
`the initial load conditions and the speed of
`rotation of the vessel [48]. The same axial
`segregation index S, has been proposed to
`calculate the segregation potential.
`The statistical properties of a multi-
`component solids mixture have been of great
`interest to researchers in the field of solids
`mixing. Multivariate statistics was applied to
`the analysis of mixing processes and mixtures
`of multicomponent particles
`in a drum
`mixer [ 491. The applications include test of
`sampling techniques,
`test of the complete
`random state,
`test of the completely
`segregated state, and definition of a mixing
`index for a multicomponent mixture.
`tech-
`Auto-correlation and cross-correlation
`niques were used to assess the degree of
`mixedness of a multicomponent
`solids mix-
`ture [ 501. The application of the proposed
`technique was demonstrated
`through simple
`mixing experiments (Computer Simulations)
`of a multicomponent
`solids mixture. The
`results derived from the correlogram are
`shown to provide practical indices for the
`degree of mixedness.
`Four approaches have been proposed to
`examine the existing mixing indices of a
`heterogeneous multicomponent mixture [ 511.
`
`These approaches are based on (i) the pseudo-
`binary mixture concept,
`(ii) the pooled
`variance of the whole system, (iii) the deter-
`minant of the sample covariance matrix, and
`(iv) absolute deviations from the population
`means. These approaches have been tested
`in a ternary mixture over a wide range of
`physical properties. The relationships among
`the ten most frequently used indices based on
`the second and third approaches have been
`evaluated. The results indicate that the rela-
`tionship between the mixing indices based on
`different approaches requires further
`in-
`vestigation.
`Nonparametric procedures have been
`proposed for the study of multicomponent
`solids mixing [ 521. Their distinctive feature
`is a lack of dependence on a particular
`distribution
`type. These procedures were
`tested with actual homogeneous and hetero-
`geneous ternary mixtures generated by a
`drum mixer. It has been shown that the
`proposed procedures can be employed to
`test hypotheses concerning
`the sampling
`technique and the significance of treat-
`ment effects
`in multicomponent
`solids
`mixing.
`If the continuous sampling over various
`spots of the mixture is possible, the discrete
`Fourier
`transform
`(DFT),
`a convenient
`orthogonal
`transform,
`can be used to
`interpret the data collected from a sampler
`[ 531. The maximum component of the DFT
`power spectrum can be employed as a mixing
`index which can distinguish random mixtures
`from ordered ones. It has also been confirmed
`theoretically
`that,
`for a mixing process
`obeying the Fickian diffusion equation, the
`logarithmic plot of the variance against the
`maximum DFT power spectrum component
`obeys a linear relationship.
`for pat-
`A feature extraction
`technique
`tern recognition was applied to the iden-
`tification of the homogeneity of a solids
`mixture
`[ 541. The feature extracted
`from
`the mixture pattern has been used to
`characterize
`the homogeneity of the mix-
`ture. The process of forming a striated
`mixture containing particulate solids was
`studied by increasing the number of stria-
`tions [ 551. Four possible classes of mix-
`tures have been identified according to
`the packing arrangements and striated pat-
`terns of the particles.
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`Cohesive mixtures
`A mixture in this group contains one or
`more cohesive constituents; its final state is
`mainly determined by interparticulate forces.
`The term ‘ordered mixture’ has been coined
`to describe mixtures formed by mixing inter-
`active or cohesive particles as compared to
`random mixtures formed by mixing free-
`flowing particles [ 561. When the term is
`applied to mixtures containing both interac-
`tive (tiny) and free-flowing (coarse) particles,
`controversies arise [ 57 - 621. It has been sug-
`gested that two types of terms are needed:
`one to define the level of homogeneity, and
`the other the type of mixture [ 611. To dif-
`ferentiate between mixtures of interactive
`powders and mixtures of free-flowing con-
`stituents, interactive, instead of ordered,
`and non-interactive,
`instead of random,
`should be used. However, it has also been
`contended that the mechanism by which the
`mixture is formed should dictate the name
`given [ 581. Thus, ‘ordered mixtures’ should
`be applied only to the systems in which
`adhesion of the fine component to the surface
`of coarse carrier particles is the dominant
`mechanism of mixture formation since
`adhesion imparts a certain degree of order
`to the system. The term ‘partially ordered
`random mixing’ has been suggested to
`describe the situation where both adhesion
`(ordering) and random (shuffling) mixing
`occur between the components
`[60].
`Another, hybrid version uses a general term
`‘total mix’ to describe all types of powder
`mixture [63]. The relationship between the
`various ‘total mixes’ is depicted in a two-
`dimensional diagram (Fig. 7) showing the
`influence of gravitational versus surface forces
`on the mixture’s homogeneity. Table 2 sum-
`marizes the various versions of terminology.
`The mixing indices reviewed in the
`preceding subsection are suitable only for
`evaluating the homogeneity of mixtures of
`free-flowing particles having similar physical-
`mechanical properties. For a mixture formed
`by adhesion of cohesive fine particles to
`coarser carrier particles, the sample standard
`deviation is the method of choice used as a
`measure of the mixture’s homogeneity
`[ 641. It is defined as
`
`(8)
`
`261
`
`it
`
`Perfect
`
`ordered
`
`Imperfect
`2
`-_________
`
`ordered
`
`Pseudo-random
`
`Portlolly
`ordered
`random
`
`Non-random
`
`Influence
`
`factor
`
`force
`= Gravitational
`Surface
`farce
`
`I
`
`Fig. 7. Relationship between different total mixes
`based on relative influence of gravitational and sur-
`face forces in a given set of particles and on the
`homogeneity of the mix [ 631.
`
`where Xi is the amount of the minor compo-
`nent in each of the n spot samples taken from
`the mixture. One widely used sampling
`method is thief-probe sampling. Nevertheless,
`this method of sampling can yield non-
`representative (biased) samples when the
`material in the mixer is segregated. This is
`especially true whenever the carrier particles
`are polydispersed
`[64]. A number of
`investigators have resorted to the coefficient
`of variation (s/X) to investigate the mixture’s
`homogeneity of cohesive powders [65,66].
`The results of mixing three cohesive drugs
`were compared in different concentrations
`with a fixed concentration of a cohesive
`excipient
`[67]. The relative effects of
`concentration and material type on mixing
`time were evaluated. The data were analyzed
`in terms of the specification index. This
`index is defined as the ratio of the sample
`standard deviation s to the acceptable
`standard deviation cA, which can be calcu-
`lated with 95% confidence within *lo% of
`the mean, x [ 681, that is,
`
`+1.960, = &0.10X
`
`(9)
`The specification indices s/oA and s/oR (based
`on random mixing), have been applied,
`separately, to the evaluation of the degree
`of mixing of a cohesive drug with a cohesive,
`non-cohesive and free-flowing excipient [ 681.
`Both indices have been shown to give similar
`results in evaluating the degree of mixing of
`such mixtures. However,
`for mixing a
`cohesive drug with a cohesive excipient,
`
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`262
`
`TABLE 2
`Comparison of nomenclatures for mixtures [ 621
`
`Reference
`Figure
`(see Fig. 8)
`
`Interaction
`
`Homogeneity
`
`Egermann-Orr
`157,611
`
`Hersey et al.
`1581
`
`Staniforth
`1631
`
`Line AFI
`
`Absent
`
`BEH
`
`Partial
`
`CDJ
`
`Total
`
`Point A
`B
`
`C
`D
`
`E
`
`F
`
`G
`H
`I
`J
`
`K
`
`Absent
`Partial
`
`Total
`Total
`
`Partial
`
`Absent
`
`Total
`Partial
`Absent
`Total
`
`Absent
`
`o>dR
`
`(T<uR
`0 < OR
`0 < (IR
`a=0
`
`u=o
`
`Non-interactive
`mixing
`Interactive
`mixing
`
`Interactive
`mixing
`Random
`Pseudorandom
`
`Pseudorandom
`
`Incomplete
`(segregated)
`
`Ordered
`Ordered
`Ordered
`Ideally
`ordered
`(perfect)
`Ideally
`ordered
`(perfect)
`
`Random mixing
`
`Random mixing
`
`Partially
`ordered
`random mixing
`Ordered mixing
`
`Partially ordered
`random mixing
`
`Ordered mixing
`
`Random
`Partially
`ordered
`Ordered
`Ordered unit
`segregation
`Partially
`randomized
`Partially
`randomized
`
`Perfect
`ordered
`
`Ordered
`mixture
`
`Random
`
`Pseudorandom
`Ordered unit
`segregation
`Partially ordered
`random
`Non-random
`
`Perfect ordered
`
`Ideal random
`
`uPOR
`
`a random mixture the following equation has
`been established for such a system [ 11.
`* _ (x + FY)(Y - FY)
`-
`n
`where oroR2 is the theoretical standard devia-
`tion of mixture concentrations for a partially
`ordered random mixture, x the weight frac-
`tion of carrier particles, y the weight fraction
`of fine particles, and F the weight fraction of
`fine particles adhering to the carrier particles.
`When F approaches 0, which is the condition
`for the random mixture, eqn. (10) becomes
`
`Fig. 8. The homogeneity
`types of mixtures [62].
`
`surface for different
`
`is not a suitable index; instead, S/UA
`s/on
`should be used.
`A partially ordered mixture was formed by
`mixing agglomerates of fine particles with
`ordered units [ 691. Based on the equation for
`
`whereas when F approaches 1, which is the
`condition for the ordered mixture, eqn.
`( 10) becomes
`
`(Jo = 0
`
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`
`

`

`The mixing of ordered units of different
`carrier particles has also been studied [69].
`This has led to the following expression:
`
`(11)
`
`where x and y are weight fractions of two
`different carrier particles, and px and pY their
`respective densities, p the density of the
`mixture of all components, and P, and P,
`the weight proportions of fine particles
`adhering to carrier particles. Equations
`(10)
`and (11) are suitable in evaluating the effects
`of sample size and degree of ordering on the
`homogeneity of a mixture.
`A model has been proposed to estimate
`the degree of mixing of a mixture in which
`coarse spherical particles are coated with a
`single layer of fine spherical particles [ 701.
`When fine particles are in excess, they are
`assumed to agglomerate
`in a similar way
`around a nucleus of a coarse particle. Based
`on this model, the upper and lower bounds
`of the variance of coated mixtures, so2
`and sa2, have been derived, respectively as
`follows:
`
`sQ2=P
`
`,gos
`
`i
`
`m3
`
`fi -P
`
`1
`
`(12)
`
`wi(m3 + nP)P
`
`SR2 =
`
`n3
`(m3 + nP)
`[
`W
`
`-P
`
`I
`
`+ p2[fi
`
`- (m3 + nP)wi]
`W
`
`(13)
`
`where
`so2 = variance of a completely segregated mix-
`ture,
`sR2 = variance of a fully randomized mixture,
`p = mass fraction of the coarse component,
`m = ratio of the radii of the coarse and fine
`particles (m > l),
`n = maximum number of fine particles con-
`tained in the agglomerate,
`i = number of fine particles in the agglom-
`erate,
`fi = relative frequency of occurrence of ag-
`gregates containing i fine particles,
`wf = weight of a fine particle,
`fi = average aggregate mass,
`
`263
`
`P = probability of a site on a coarse particle
`being occupied by a fine particle, and
`W = weight of the sample.
`Equations (12) and (13) take into account
`the relative bonding strengths between coarse
`and fine particles and those between fine and
`fine particles in terms of the probability.
`Although
`these equations describe only
`idealized models of the coating operation,
`they reflect the importance of
`(i) the strength of the interparticulate bond,
`(ii) the absolute size ‘and size range of the
`agglomerating particles, and
`(iii) the need to control the particle size
`distribution if a high-quality mixture is to be
`produced.
`The dependence of the standard deviation
`of sample concentration on sample size has
`been proposed as a criterion to differentiate
`the ordered mixture from random mixture
`[71]. However, large experimental errors
`may mask the mixture’s variances
`[72].
`A re-examination of the issue through the
`variance-sample
`size relationship of an
`ordered powder mixture has confirmed this
`argument, especially for micro-dose-mixing
`[73]. Thus, no simple relationship exists
`between the standard deviation and sample
`size. Consequently,
`the concept of ordered
`mixtures can only be viewed qualitatively
`[71, 741.
`
`in fluidized beds
`Mixtures
`When a bed of binary particles is fluidized,
`one component
`is generally fluidized at the
`lower gas velocity and the other at the higher
`velocity. In general, if the densities of the
`components
`are different,
`the heavier
`component tends to sink while the lighter one
`rises. The former is called ‘jetsam’ and the
`latter ‘flotsam’. The following correlation has
`been obtained for the mixing index for an
`equal-size, density-variant binary mixture in
`a three-dimensional
`fluidized bed [ 751:
`
`M = (1 + e-z)-1
`
`z= u - u~o euluTo
`u-u*
`where
`M = equilibrium mixing index,
`U = superficial gas velocity,
`
`(14)
`
`(15)
`
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`
`

`

`264
`
`Ur = minimum fluidization velocity of the
`flotsam, and
`Uro = take over velocity defined at the value
`U corresponding to M = 0.5.
`The correlation has been found to qualita-
`tively agree with the observations for a
`density-variant system composed of large,
`e.g., 3500-pm, low density solids [76].
`For size-variant, equal-density systems, the
`expression
`
`(16)
`
`has been proposed. In this expression, dp is
`the weight-average particle diameter of the
`entire mixture in the bed, dF is the mean flot-
`sam diameter, and m and 12 are empirical
`constants.
`Most of the previous studies on the subject
`are divided between two- and three-dimen-
`sional beds without interrelating them. To
`remedy this situation a study was carried out
`for the mixing indices of initially segregated
`beds of two radically different geometries
`[ 771. One is a conventional cylindrical three-
`dimensional bed and the other a narrow,
`rectangular-base, two-dimensional bed. Based
`on the equilibrium mixing/segregation data
`obtained with binary mixtures of different-
`sized, equal-density glass beads in these two
`beds, the following correlation has been
`proposed [ 771:
`
`M = (1 + eez*)-l
`z* = f u - UT, 1’2
`u- UFB
`
`I
`
`e”/uTO(fs)l/2
`
`I
`
`for the value of UT,. Its correlations have
`been developed separately for a three-dimen-
`sional and a two-dimensional fluidized bed
`[ 771; they are, respectively,
`
`uTO, 3D = (~JJJB)~~~(~R,)-~*~
`and
`
`(20)
`
`UT,, 2D = (k&h)“*62
`(21)
`where U,, is the minimum bubbling velocity
`of the jetsam, and R, the ratio of the bed-
`height to the bed-diameter (aspect ratio).
`Equation (17) is said to be an improvement
`over eqn. (14); the former resorts to a
`modified gas velocity parameter and an
`explicit term for particle size ratio to predict
`adequately the behavior of different-sized
`equal-density systems. Experimental results
`indicate a much better fit for the data from
`the three-dimensional bed than for the data
`from the two-dimensional bed. This is most
`likely due to the fact that wall effects sig-
`nificantly retarded mixing in the two-
`dimensional bed [77].
`Another widely used mixing index is
`defined as [78]
`
`X*
`M= -
`Xbed
`
`(22)
`
`where x&d iS the OVer~l weight fraction Of
`jetsam in the bed, and X the average weight
`fraction of jetsam in the relatively uniform
`upper section of the bed. The fluidized bed
`is considered to comprise a multiple of thin
`fluidized beds in series, each having a constant
`minimum fluidization velocity, bubble diam-
`eter, wake fraction and model parameters.
`The average weight fraction of jetsam in each
`segment needs be solved iteratively to obtain
`the value for M.
`Solids mixing and segregation in liquid-
`solids fluidized beds containing binary mix-
`tures of spherical particles of different
`densities and sizes were studied for a variety
`of liquid velocities, bulk bed compositions
`and particle properties [ 791. Solids mixing in
`a gas-liquid-solids
`fluidized bed containing
`a binary mixture of particles has been
`analyzed qualitatively based on visual observa-
`tion [80]. The analyses include complete
`segregation, partial intermixing and complete
`intermixing.
`
`(17)
`
`(18)
`
`(19)
`
`(for u> uro)
`
`(for u< uro)
`
`where
`Rd = jetsam to flotsam average particle diam-
`eter ratio, and
`UF~ = minimum bubbling velocity of the
`flotsam.
`The f sign in eqn. (18) refers to U > UT0
`and U< U