Recent developments in solids mixing

Powder Technology,

255

61 (1990) 255 - 287

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 extensively 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 occurrence to read about, or to view on television, researchers and technicians blending particulate ingredients for producing superconducting 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 Engineering, University of Lowell, Lowell, MA 01854 (U.S.A.). 0032-5910/90/$3.50

including those by Lacey [ 11, Scott [2], Weidenbaum [3], Valentin [4], Venkateswarlu [ 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 characterization 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. @ Elsevier Sequoia/Printed in The Netherlands

256 TABLE 1 Summary of mixer characteristics

5w

mixer

of

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

[ 1651

Batch or continuous

Main mixing mechanism

Segregation (suitability for ingredients of different properties)

Axial mixing

Ease of emptying

Tendency to segregate on emptying

Ease of cleaning

B

Diffusive

Bad

Bad

Bad

Bad

Good

B C

Convective Diffusive

Good Fair

Good Bad

Good Good

Good Good

Bad Good

B

Diffusive

Bad

Good

Bad

Bad

Good

B

Shear

Bad

Good

Good

Bad

Good

Diffusive Diffusive Diffusive Diffusive Convective Convective Convective Convective

Bad Bad Bad Poor Good Good Fair

Bad Bad Bad Good Slow Fair Good Good

Good Good Good Good Good Good Good Good

Bad Bad Bad Bad Fair Good Good Good

Good Good Good Good Fair Fair Fair Bad

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].

In a tumbler mixer, radial mixing is relatively fast while axial mixing is slow and is the rate-controlling step. Appreciable segregation can occur in this type of mixer. If an internal impeller is added, its impaction action is likely to minimize segregation. Convective mixers In the majority of convective mixer 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].

Fig. 4. Gravity flow blender [30]: (a), With multiple 1 inner hoppers; (b) with single inner cone. 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 sufficiently large, the turbulence within the bed

‘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

258

other contains cohesive or interactive constituent(s). A free-flowing mixture will generally permit individual particulates in it freedom to move independently, while a cohesive mixture generally has some interparticulate 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 obtained. 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 accomplished 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 particle. Let A,, particles be key particles in a binary mixture containing two kinds of particles 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 estimating 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 concentrations 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

i =

i=l

-xi)

n*Xi(l -xi) k(1 -fi)

(1)

259

where xi = concentration of Al particles in the ith spot sample, f i = sample mean concentration of the particles 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 particle. The unbiased e_stimatorof the population contact number Cl& can be defined from the biased estimator C’n,,) obtained in eqn. (1) as follows: 8.

GO) Cl(O)’ = -En-l Based on the contact number evaluated in eqn. (l), the mixing index is defined as Cl(O)

M=--Zn*X

(3)

where g denotes the population mean concentration 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*xdi(xo)i

i i=l

kio

n-l

j= 1,2, . . ..p

(4)

and I

M

=

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 Xj = population mean concentration of c_omponent Aj. Cj(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’

in estimating the population mean contact number has also been derived through evaluation 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 exponentially as the number of helices in the motionless 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 coordination 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 concentration is a variable. To deal with the distribution 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 incompletely 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 multicomponent 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-permeation on the segregation of a solids mixture in

260

a rotating cylinder [ 471, the radial and axial segregation indices have been defined, respectively, as

SR =

C-

cmax

Cmin -

Z(Ci - q2vl:

s, = i

(6)

Gin

xv;:

1’2

1

(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 multicomponent 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. Auto-correlation and cross-correlation techniques were used to assess the degree of mixedness of a multicomponent solids mixture [ 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 pseudobinary mixture concept, (ii) the pooled variance of the whole system, (iii) the determinant 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 relationship between the mixing indices based on different approaches requires further investigation. 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 heterogeneous 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 treatment 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. A feature extraction technique for pattern recognition was applied to the identification of the homogeneity of a solids mixture [ 541. The feature extracted from the mixture pattern has been used to characterize the homogeneity of the mixture. The process of forming a striated mixture containing particulate solids was studied by increasing the number of striations [ 551. Four possible classes of mixtures have been identified according to the packing arrangements and striated patterns of the particles.

261

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 interactive or cohesive particles as compared to random mixtures formed by mixing freeflowing particles [ 561. When the term is applied to mixtures containing both interactive (tiny) and free-flowing (coarse) particles, controversies arise [ 57 - 621. It has been suggested that two types of terms are needed: one to define the level of homogeneity, and the other the type of mixture [ 611. To differentiate between mixtures of interactive powders and mixtures of free-flowing constituents, 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 twodimensional diagram (Fig. 7) showing the influence of gravitational versus surface forces on the mixture’s homogeneity. Table 2 summarizes 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 physicalmechanical 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)

it

Perfect

2 Imperfect -_________

ordered ordered Portlolly ordered

Pseudo-random

random

Influence

factor

=

Non-random

I

Gravitational force Surface farce

Fig. 7. Relationship between different total mixes based on relative influence of gravitational and surface forces in a given set of particles and on the homogeneity of the mix [ 631.

where Xi is the amount of the minor component 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 nonrepresentative (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 calculated 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,

262 TABLE 2 Comparison of nomenclatures for mixtures [ 621 Reference Figure (see Fig. 8)

Interaction

Line AFI

Absent

BEH

Partial

CDJ

Total

Point A B

Homogeneity

Total Total

E

Partial

F

Absent

G H I J

Total Partial Absent Total

K

Absent

Hersey et al. 1581

Staniforth 1631

Non-interactive mixing Interactive mixing

Random mixing

Random mixing

Partially ordered random mixing Ordered mixing

Partially ordered random mixing

Random Partially ordered Ordered Ordered unit segregation Partially randomized Partially randomized

Random

Interactive mixing Random Pseudorandom

Absent Partial

C D

Egermann-Orr 157,611

Pseudorandom

o>dR

Incomplete (segregated)

(T
Ordered Ordered Ordered Ideally ordered (perfect) Ideally ordered (perfect)

0 < OR 0 < (IR

a=0

u=o

Ordered mixing

Pseudorandom Ordered unit segregation Partially ordered random Non-random

Perfect ordered

Perfect ordered

Ordered mixture

Ideal random

a random mixture the following equation has been established for such a system [ 11. *_ uPOR

-

(x + FY)(Y - FY) n

where oroR2 is the theoretical standard deviation of mixture concentrations for a partially ordered random mixture, x the weight fraction 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 should be used. A partially ordered mixture was formed by mixing agglomerates of fine particles with ordered units [ 691. Based on the equation for

s/on

whereas when F approaches 1, which is the condition for the ordered mixture, eqn. ( 10) becomes (Jo =

0

263

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: m3 sQ2=P ,gos i

fi -P

wi(m3 + nP)P SR2 =

(12)

1 n3

[ (m3 + nP)

-P

I

W + p2[fi

-

(m3 + nP)wi] W

(13)

where so2 = variance of a completely segregated mixture, 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 contained in the agglomerate, i = number of fine particles in the agglomerate, fi = relative frequency of occurrence of aggregates containing i fine particles, wf = weight of a fine particle, fi = average aggregate mass,

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.

Mixtures in fluidized beds 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 +

z=

e-z)-1

u- u~o

euluTo

u-u*

where M = equilibrium mixing index, U = superficial gas velocity,

(14) (15)

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 qualitatively 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 flotsam diameter, and m and 12are empirical constants. Most of the previous studies on the subject are divided between two- and three-dimensional 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 threedimensional bed and the other a narrow, rectangular-base, two-dimensional bed. Based on the equilibrium mixing/segregation data obtained with binary mixtures of differentsized, equal-density glass beads in these two beds, the following correlation has been proposed [ 771: M = (1 + eez*)-l

z* = f

u-

UT,

(17)

1’2

Iu- I

e”/uTO(fs)l/2

UFB

(18)

(for u> uro)

(19) (for u< uro) where

Rd = jetsam to flotsam average particle diameter ratio, and UF~ = minimum bubbling velocity of the flotsam. The f sign in eqn. (18) refers to U > UT0 and U< UT0, respectively. The validity of the above and any similar correlations appears to depend critically on having a good estimatti

for the value of UT,. Its correlations have been developed separately for a three-dimensional and a two-dimensional fluidized bed [ 771; they are, respectively, uTO,

3D = (~JJJB)~~~(~R,)-~*~

(20)

2D = (k&h)“*62

(21)

and UT,,

where U,, is the minimum bubbling velocity of the jetsam, and R, the ratio of the bedheight 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 significantly retarded mixing in the twodimensional bed [77]. Another widely used mixing index is defined as [78]

M=

X*

-

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 diameter, 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 liquidsolids fluidized beds containing binary mixtures 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 observation [80]. The analyses include complete segregation, partial intermixing and complete intermixing.

265 RATES AND MECHANIS?biS OF MIXING PROCESSES

Mixing processes can be classified ina variety of ways. For example, classification can be based on the modes of solids flow or mixing mechanisms involved. In this section, mixing processes are classified into freeflowing and cohesive systems based on flow characteristics. A comprehensive account is also provided of solids mixing. in fluidized beds, effected mainly by fluidizing media. Free-flowing systems Mixing in a free-flowing system is a process of randomization or ‘shuffling’ of the particles. The randomization may be induced by a variety of mechanisms, such as diffusion, convection or both, according to the type of a mixer employed [28]. The mixing and segregation mechanisms can also be viewed either macroscopically or microscopically. Macroscopically, we visualize the movement of solids as that of coherent clumps or blocks of powders and microscopically as the interparticle percolation in the regions known as failure zones existing between moving blocks [ 811. Diffusion model A random or diffusion-like process has been proposed as a possible mechanism for solids mixing [ 11. Analogous to the molecular diffusion, this mechanism is often described by Fick’s equation

ac -= at

D-

a2c a2

(23)

where C = solids concentration, D = diffusivity, x = distance in the direction of dispersion, and t = time. Experimental data obtained with mixing binary non-spherical particles having rough surfaces indicate that the mixing process can be well characterized by the diffusion model as long as the mean diameter of the two component particles are identical [ 821. The axial mixing of particles of different densities in a horizontal batch mixer was experimentally determined by loading the tracer particles in such a way that they

formed a thin layer perpendicular to the axis [ 831. To describe the motion of particles in the axial direction, the following diffusion equation has been fitted to the data.

azdx,n) = D a2~(x,n) a ax2 an

(24)

where n = number of revolutions of the mixer, x = distance from the left end wall, P(x, n) = C(x, n)/J$(x, n) dx, relative concentration of the particles, C(x, n) = concentration of the particles, and L = mixer length. The mixer can be treated as an infinite cylinder for the short duration of mixing. Under this assumption, the solution of eqn. (24) is obtained as

P(X, nix,,0) =

(x4D::2 $i32ex+

(25) where x, is the initial location of the tracer. This equation can be rewritten in the form of the Gaussian distribution as

(26) A comparison of eqns. (25) and (26) reveals that the variance uX2(n) of the relative concentration distribution is related to the diffusion coefficient, D,, through Einstein’s equation, as (27) The diffusion coefficient Da* with a dimenssion of L2/t is obtained by Da* = D,N

(23)

where N is the rotating speed of the mixer. The experimental results have indicated that the diffusion coefficients of the particles of different densities are larger than those of the particles of the same density. The diffusion model was applied to the radial mixing in a two-dimensional drum mixer [ 841. The radial mixing coefficient has been evaluated based on the assumption that

266

the random motion of the particles is a Markovian process. Accordingly, the mixing coefficient E [equivalent to D in eqn. (23)] is expressed as

the proposed model provides a unified mathematical description of both the homogenization and segregation processes. This model has been verified experimentally

ff371.

where d = thickness of the mixing zones, p = one-step transition probability, and At = time-interval between two consecutive steps. This radial mixing coefficient has been found to be dependent on the path through which particles enter the mixing zone. While the mixing of an ideal system (for example, system with particles of same size and density) is well described by the classical diffusion model, of the form given by eqn. (24), it is not applicable to the mixing of a real system resulting from interaction between the mechanisms of homogenization and segregation [ 851. A mathematical model has been developed for mixing of solids accompanied by segregation [ 861. The model equation has been solved for a two-component system mixed radially in a horizontal rotating drum mixer. The concentration of the key component, C(r, t), is considered to be the sum of two independent component functions called the homogenization and the segregation functions, that is, C(r, 0 = Ch(r, t) + C,(r, t)

(30)

Applying eqn. (23) in the cylindrical coordinate to both Ch(r, t) and C,(r, t) leads to ach(r9

at

t)

=&

a2ch(r9

ar*

t) + 1 ach(r9 t) r

ar

(31)

I

a*C,(r, t) + J_ aCAr, t) r ar ik*

1

(32) where Dh = diffusional homogenization coefficient, D, = diffusional segregation coefficient, and h = constant (dimensionless), and k > 1. These equations are solved for C(r, t) under the assumptions of unity loading and complete unmixing or segregation as the final state of the system. The latter assumption may be somewhat impractical. Nevertheless, in comparison with other empirical models,

A method has been proposed for mixing particles during their pneumatic transport [88]. The mixing process is assumed to be characterized by the particle dispersion coefficient, determined in the tracer experiment from the solution of the diffusion equation. This particle dispersion coefficient has been found to be stabilized at a height of about 20 - 30 cm from the point of indicator injection and remains constant along the pipe radius and with varying gas velocities. The proposed method has been used to neutralize the electric charge on the particles generated during their pneumatic transport. Diffusion-convection model The diffusive-convective mixing mechanism is described by the Fickian diffusion equation with an additional term involving drift velocity:

ac _=+7~

(33)

at

Experimental studies were conducted to examine the mixing of flowing powders over an inclined surface by the diffusiveconvective mixing model [89]. Under steady state conditions, eqn. (33) becomes v!.$D-

d*C

(34)

dx*

A model has been proposed to determine the dispersion coefficient of particles, D, in eqn. (33), based on the experimental results indicating a strong dependence of the dispersion coefficient on shear in the flowing material. An assumption that the dispersion coefficient varies linearly with the velocity gradient in the flowing stream of powder yields D=D*(l+L

g)

(35)

In this expression, D* is the ‘intrinsic’ dispersion coefficient due to random oscillations or fluctuations of the individual particles during flow. The parameter L takes into account the contribution to diffusion by inter-particle

267

collisions enhanced by the presence of velocity gradients. The values of these parameters reportedly are dependent on the nature of the flow system and the characteristics of the flowing powder. The validity of the proposed model has been verified through a series of experiments using different chute lengths and different feeding conditions. The axial mixing of particles was experimentally measured in batch and continuous rotary dryers and coolers [ 901. The diffusion model has been adapted for the batch dryers and coolers and the diffusion-convection model for the continuous dryers and coolers; the influence of the operating variables on diffusion coefficients has been examined. The results show that the effect of the operating variables on both coefficients is essentially the same, and the ratio of the diffusion coefficient for the batch case to that for the continuous case is approximately 0.7 under the same experimental conditions. The deterministic dispersion model, given by eqn. (32), has been employed to describe the axial mixing process in a continuously operated ball mill [ 911. The dimensionless residence time distributions measured by tracer concentrations in the mill exit stream have been found to be in agreement with the predictions of the model which indicates a constant mixing factor along the mill. Band formation can be observed when particles of different sizes are mixed in a horizontal drum mixer. This is attributed to axial variation of the radial velocity profile. For modeling such a complex non-ideal particle system, a stochastic approach appears to be more appropriate than a deterministic one. Accordingly, a stochastic diffusion model has been derived to examine the non-ideal mixing in a horizontal drum mixer [ 921. The model is based on the continuous version of the Markov chain, i.e., the Kolmogorov diffusion equation. It is expressed as af -=--at

a2(m

ax2

a(w)

ax

(36)

In this equation, f = f(x,, t,; x, t) is the probability density function which determines the probability that a key particle originally at x, migrates to x during the time period At = t - t,. D = D(x, t) and V = V(x, t) &e the diffusion coefficient and the drift velocity, respectively. Consider a volume

element in the binary mixture mass. When all the particles are of the same density, the weight of the volume element can be assumed to be approximately constant even though the size of particles are different. Suppose that the weight fraction (or concentration) of the key particles in the volume element is C(x), t,) at given time t, and at axial position x’, and that the fraction is C(x, t) at time t (>t,). Then C(x’, to) and C(x, t) are related by

jC(x!, t,)f(x’,

C(x, t) =

t,;

x,

(37)

t) dx’

0

Multiplying eqn. (37) with V(x, t) and D(x, t) on both sides gives, respectively, D(x, t)C(x, t) = s ‘C(x), t,)D(x, t)f(x’, 0

t,;

x,

t) dx’

(38)

t,;

x,

t) dx’

(39)

V@, W@, 0 =

r

lC(x’, &)V(x,

t)f(x’,

0

Differentiating with respect to x, eqn. (39) once and eqn. (38) twice and combining the resultant expressions with eqns. (36) and (37) yields

Ql

a2WW

ax2 = 1 lc(x’, to;

a[vw, 01 ax to; x,

wf(x’,

t,;x,

t)i

ax

dx

i af(x’,

=

J 0

lC(x’,

t,;x,

01

a2

I

0

-

a2Pf(x’,

x, t)

t)

t . X, t) dx

a;’

1

=

t

Jqx’, t,;

x, t) f(x’,

t,;

x,

t) dx

0

=

ack t) at

(40)

Under the assumption that the mixing process is time stationary, the diffusivity and drift velocity become, respectively [ 921,

D(x,

t) = D(x)

V(x, t) = V(x)

I

(41)

which can be obtained from tracer experiments. The initial condition for eqn. (40) is C(% 0) = C,(x) Assuming that D(x) and V(x) are finite and continuously differentiable with respect to X, the boundaries at x = 0 and x = 1 become the so-called regular boundaries. Consequently, the flux of particles, J, in terms of C, i.e., J=-

;(Dc)+vc

(42)

vanishes at both boundaries. Therefore, the boundary conditions for a closed or batch system can be rewritten as J=-

$(DC)

+ VC

atx=Oandl

= 0

(43)

If the drift velocity is neglected, eqn. (39) becomes

ac _=at

a% ax2

(44)

which reduces to the Fickian diffusion equation, only when D is invariant with respect to x. Thus, the deterministic Fickian diffusion equation can be seen as a special case for the stochastic Kolmogorov diffusion equation. Under the equilibrium or steady state condition, eqn. (44) reduces to (45) Solving this equation subject to eqn. (43) yields

-5

for all x. (46) D(x) where QIis constant. On the other hand, the equilibrium profile obtained by solving the steady-state Fickian diffusion equation subject to no-flux boundary conditions leads to C,(x) =

C,(x) = q(constant)

for all x

(47)

A comparison of eqns. (46) and (47) reveals that the Fickian equation always results in a constant equilibrium profile, while the Kolmogorov equation gives a variable profile

depending on D(x), and the two become identical only when D(X) is constant. For the general case where velocity V(x) is included in the diffusion equation, the theoretical equilibrium profiles are obtained from eqn. (42) at steady state conditions, that is, at J = 0; the resultant expression is

C,(x) =

$j./ exp[gg dx

(48)

where /3is an integration constant to be determined from the initial concentration profiles. As can be seen from eqn. (48), the convective or drift velocity, which varies with respect to the axial position of the mixer, appears to be important in elucidating the non-ideal mixing characteristics. A good agreement between the model and the experimental data has been observed. Convection model A model has been proposed to represent convective mixing by a process of forming a striated mixture [ 551. The process, in which a mass of material is repeatedly subjected to a series of mixing operations, includes division and recombination. The effect of each step of the mixing operation is superimposed on all previous steps. Such repetitive operations illustrate the convective mechanism of mixing. Consider the process of forming a striated RR-mixture (regular packing arrangement mixture with a regular pattern) shown in Fig. 9. Initially, four cells are specified in the two striae of a rectangular block mixture (see Fig. 10). The rectangular block is compressed to one-half of its original height, cut into halves, and then is reassembled into the form of the original rectangular block as indicated. Let JckN) denote the average concentration of the particles of component Al in cell k (k = 1, 2, 3, 4) after N cycles of the mixing operation. With the vector and matrix notations, the average concentration distribution among the cells after N cycles is given by x(N) = p(N)x(O) = [p’ where

l)]N$O)

(49)

269

For the case demonstrated in Fig. 10,

p! 1)

l/2

l/2

0

0

0

0

l/2

l/2

l/2

l/2

0

0

0

0

l/2

l/2

(53)

and g(O)= [O 0 1 l]T

(54)

Hence, x(l)

= pC1’$0’

(d) Fig. 9. Geometric representation the ideally mixed state [55].

of RR-mixture

il 0

in

1

0

= and $2’

2

=

(55)

1

[p(l)]

290)

striae

N=O

4

striae

N=I

8 striae N=2

Fig. 10. Schematic representation of formation of a striated mixture [55].

(56)

This implies that after two cycles, the average concentrations in all the cells become identical, i.e., the distribution of the average concentrations in all the cells becomes uniform. This does not imply that the concentration distribution within each cell is uniform. If a visually uniform mixture is desired, the operation should stop when the striation thickness becomes visually indistinguishable. The mixing process to form the striated layers described above is essentially an idealized representation of convective mixing. It is obvious that a mixture with a desired degree of mixedness can be obtained by this mode of mixing without using other modes of mixing, such as diffusive mixing. This study of the formation of striated mixtures has provided a deeper understanding of certain

270

basic mixing operations. As an example, it has been proposed that a mixer can be synthesized by some combination of the two elementary mixers, the convective model mixer (stratified feeding mixer) and diffusive mixer [93]. The study has also provided a theoretical foundation for the development of several practical mixers and mixing systems, such as various types of motionless mixers and multi-port mixers.

0 ther models A discrete model and a random walk model have been presented to describe the axial mixing of grains in a Sulzer (Koch) motionless mixer [ 941. To validate the model, an experimental system was devised for determining the axial mixing of grains (wheat and sorghum) in this mixer. The progress of mixing was followed with a sampling technique which incorporated a radioactive tracer. The radioactive tracer, the portion of grain irradiated with neutrons, provided a practical means of obtaining estimates of different parameters with greater accuracy than previously possible and without disturbing the mixture. A good agreement between each of the models and data was observed. A modified stochastic coalescence-dispersion model has been developed for the axial mixing of segregating particle systems in a motionless mixer (Kenics mixer) [ 951. In such systems, particles differ in properties and characteristics. In the model, the entire packed mass of mixture is equally divided into a finite number of sections-in-series in the vertical direction. Each section, in turn, is equally divided into a finite number of cells in the horizontal direction. Each cell contains a specified number of particles. The convective mixing in the axial direction is considered to be induced by allowing coalescence between cells in the adjacent sections. The diffusive mixing mechanism is assumed to be induced by collision and redistribution of individual particles between coalescing cells. The proposed stochastic model contains two parameters, distribution ratio of particles and number of coalescences per pass through the mixer. The validity of the model has been tested successfully with the available data. The results show that the model can be applied to improve mixer design.

Nonstationary Markov chains have been applied to the analysis of the mixing and segregation occurring within vibrated polydispersed particulate beds [96]. The advantages of the stochastic approach are that both particle distributions and particle mobilities can be studied simultaneously; moreover, with this approach, the process of mixing and segregation can be examined in detail throughout the system. The model has been tested with a binary mixture of 112.5 g each of 2.3% and 5.56-mm chrome steel balls in a segmented brass cylinder. The transition probabilities calculated have been found to agree with those estimated using tracer particles. The nonstationary Markov chains have also been applied to examine the size segregation of binary mixtures of noncohesive spherical particles subjected to shear in a rotary cell [97]. A discrete steady-state Markov chain model has been derived to describe the axial segregation of solid particles in a motionless mixer [ 941. Systems studied contain particles of different sizes and/or densities. The proposed model can predict concentration profiles, the degree of mixedness, and the equilibrium states. A stochastic model for the process with binary or multicomponent homogeneous solids has been proposed for solids mixing during discharge of granular materials from bins [98,99]. The mixing process can be treated as a sequence of tests joined in a Markov chain. The concentration distribution and the degree of mixing can be calculated with the knowledge of the transition matrix. In the case of multicomponent homogeneous solids, the transition matrix can be obtained from mixing experiments with binary homogeneous particles of the same size, density and shape as those in the multicomponent system. The results calculated from the stochastic model have been compared with those obtained experimentally. In terparticle percolation When particles of different types are mixed together, a random distribution of the components is rarely produced. Under this situation, failure zones of higher porosities than the bulk are generated between blocks of material sliding over each other (see Fig. 11). Interparticle percolation, the drainage of small

271

terizing the axial movement of a percolating particle :

Fig. 11. Schematic representation horizontal failure zone.

of an idealized

particles through the interstices between large ones, takes place in such zones (usually 10 - 15 particle diameters thick). The rate of percolation depends on the effects of gravity and strain on particles of different sizes, shapes or densities. Percolation can be induced under shear or occur spontaneously if the diameter ratio of percolating particles to bulk particles is less than a critical value of 0.155 [loo]. This interparticle percolation controls the microscopic distribution of freeflowing particles and agglomerates of cohesive particles. In an experimental study [ 1011, a bed of particles was sheared so that its shape changed from a rectangle to a parallelogram; thus, the bed was analogous to a failure zone. Percolation rates were determined by measuring the mean residence time taken from a single particle fed to the top of the cell to traverse the bulk and fall out the bottom. The estimated mean residence time t,,, and variance c2 have been evaluated through t, = J@(t) dt

(57)

and u2 = J(t - t,)2p(t) dt

(58)

where p(t) is the probability density of the percolating particles to pass through the bed between time t and t + dt [lOl, 1021. The percolation velocity u is obtained from the mean residence time as L u=

z

(59)

where L is the bed height. The variance of the residence time distribution is combined with the mean residence time to yield the following practical Peclet number, Pe: charac-

where db is the diameter of the bed particles. The percolation velocity and Peclet number have been determined for the size ratios of percolating particle to bed particle in the range of 0.25 - 0.7; the following expression has been fitted to the data.

(61) where t is the strain rate and d, is the diameter of the percolating particle; k, and kb are two arbitrary constants found to be 20 and l/8, respectively. It has been shown that the scale of the experiment could be changed while yielding the same dimensionless percolation rates [loll. These dimensionless percolating rates were found to be controlled primarily by the total shear strain and the particle diameter ratio. The latter may be explained using statistical mechanics by supposing that the large particle has a fluctuating free space and that percolation occurs if the length scale of this space exceeds a certain value into which the small particle can move [103]. The motion of bulk particles has been found to be well described by a diffusion model [ 104,105]. The model yields the self-diffusion coefficients in the SC,y and z directions, respectively as (see Fig. 11)

DX

= 0.055 +db”

(62)

DY = 0.057 +db”

DZ

= 0.051 (64) +db2 This indicates that the motion of the bulk particles is random and isotropic. Only small variations in the diffusion coefficients were obtained by varying the normal load. Shear-induced percolation has been analyzed by modeling the bulk particles as a series of layers in which the percolating particles move from one layer to the next by

272

(a)

(b) Fig. 12. Size segregation [llO]: (a), Initial configuration: (b), configuration obtained after 300 shakes.

a random process [loo]. It has been demonstrated that as a result of the random motion, percolation can be well described by a convection diffusion equation. It also follows that the practical Peclet number is about 2 for a sufficiently small particle, which is in good agreement with experimental results. The model has been found to be similar to the

well-mixed tank-in-series model, and the flow of liquid through a packed bed. A fairly general mathematical model, which is free of adjustable parameters has been developed, for mixing and segregation of particles in failure zones [106]. The model allows for the effects of diffusion and interparticle percolation. The validity of the approach has been confirmed by comparing the model with experimental evidence obtained by following the segregation of differently sized glass beads in an annular shear cell. Although it is generally accepted that a smaller component will drain to the bottom of a failure zone due to interparticle percolation, the converse is not necessarily true; a large particle does not rise to the top through an array of small particles. This has been verified by the experiments conducted in an annular shear cell designed to stimulate a failure zone [107,108]. The experimental results have revealed that the steady-state concentration profiles in the failure zone often show a region of constant concentration. This phenomenon is attributed to particle migration, a tendency for large particles to move to the region of highest strain rate. The mechanism of such a movement has been assessed independently [ 1091. When a granular mixture of particles of different sizes is shaken or jostled, the larger particles do emerge on top of the smaller ones [ 1101. If the size discrepancy is large, the sifting mechanism, in which tiny grains filter down through the interstices between the larger particles, prevails. In the situation where the larger and smaller particles are of comparable sizes, sifting cannot occur. Thus, the larger particles rise to the top solely due to size segregation. The segregation rate and its dependence on particle size and shaken frequency with sifting have been studied [ill]. However, a basic phenomenological understanding of the segregation mechanism remains elusive. To gain a further understanding of size segregation, an adaption of the Monte Carlo method was employed to simulate its dynamics [ 1101. It has been observed that as the particles fall during a shake, smaller particles move easily underneath larger ones whenever gaps open up. The opposite is, however, an unlikely event and therefore, the larger

273

particles move up relatively to the smaller ones. The results of simulation (see Fig. 12) indicate that lateral shaking is relatively ineffective in producing segregation, and that an apparent threshold value exists for the lift distance below which segregation does not occur. An experimental study was conducted to elucidate the mechanism of radial segregation in the bed of a rotary kiln and to determine the size and composition of the segregated core [112]. Two mixtures of sand and one of limestone containing fines were used in the experiment. The fines resulted in having no effect on static or dynamic angle of repose, shear angle slumping frequency, or activelayer thickness. Thus, it has been concluded that the percolation rather than the flow mechanism is responsible for the observed segregation. Fluidized systems Mixing of particles in fluidized beds has received considerable attention in recent years; fluidized beds are mainly used to homogenize free flowing particles [ 301. Mixing is usually easy to achieve for such particles with sizes up to about 1 cm [ 1141. Generally, cohesive powders cannot be fluidized homogeneously because they tend to cause channeling. Effects of bubbles The flow of gas through a gas-solids fluidized bed separates and mixes the particles. The bed is considered to comprise a discontinuous phase of bubbles and a continuous phase (called the dense phase) of gas and solids existing among these bubbles. The movement of bubbles is largely responsible for the mixing of solids [22]. In nearly all cases, mixing exists in both the horizontal and vertical directions. The horizontal or lateral mixing of solids is most readily observed in the splash zone at the bed surface. The same mixing phenomenon within the bed itself, however, is not easily observable. An earlier study has led to the postulation that within the bed all the lateral particle mixing occurs in the bubble wakes, which is yet to be verified experimentally [ 1151. A more recent experimental study has been reported on the mechanism of horizontal or lateral mixing

within the bed itself [ 1163. The latter was conducted in a freely bubbling two-dimensional bed with heated particles as tracers. The bubbling behavior of the bed was recorded by a high-speed video camera and an optical bubble probe. The experimental observations have indicated that particle motion is closely related to the random bubble motion in the bed. Thus, mixing experiments must be repeated numerous times to achieve meaningful results. It has also been indicated that an asymmetrical particle transport exists in the vertical direction, which is characterized by a mixing length of the order of bubble diameter, whereas downward displacement is more uniform, and at a much lower velocity level. Unlike what was postulated in the former study [115], the flow visualization experiments performed in the latter study have suggested that in addition to mixing in the bubble wakes, the horizontal mixing of solids is considerably augmented by the lateral displacement of the bubbles. This lateral bubble motion is caused by interactions between the bubbles prior to their coalescence. Therefore, prediction of solids mixing is intimately related to an accurate representation of bubble behavior. Many of the investigations into solids mixing were made in fluidized beds having diameters less than 10 cm, and the height to diameter (H/D) ratios between 20 and 200. In beds of such geometry, bubbles rapidly coalesce to form gas slugs [ 421. Two distinct types of slugging can occur [ 1171; one is round-nosed and the other squared-nosed slugs. Different slug flow regimes cause very widely differing, mixing rates [118]. Roundnosed slugging gives excellent axial mixing of solids, while square-nosed slugs give much lower mixing rates. Thus, the majority of the data in the literature is not applicable to freely bubbling beds with large diameter and low H/D-values [ 221. Mixing of a binary system in the fluidized bed can be classified into three states [119]: (i) complete mixing, (ii) complete segregation, and (iii) partial mixing (Fig. 13). An empirical equation has been proposed to estimate the minimum fluidization velocity for complete and partial mixing states. For binary systems containing particles different in size and density, however, this minimum fluidization

274

Flotsom

Flotsam

and Jetsom Jetsam

t (a)

(b)

Fig. 13. Typical mixing/segregation

states for a binary fluidized bed system [119]: (a), Completely mixed; (b), completely segregated; (c), partially mixed.

velocity should be replaced by the “true minimum fluidization velocity” at which both coarse and small particles are completely fluidized [120]. A correlation of the true minimum fluidization velocity has been obtained from experimental results. Particles differing only in size were found to mix well even when the size differences were large [ 751. The degree of segregation of mixtures of differing densities was observed to decrease with the increasing air velocity and aspect ratio (H/D). Consequently, a conclusion has been drawn that density is the most important factor in determining the degree of segregation of binary powders in a fluidized bed. A similar conclusion has been obtained from a study of the continuous mixing of particles of different sizes in a continuous fluidized bed [ 1211. Visible segregation of coarse and fine particles is observed only for air velocities less than 1.5 times the minimum fluidizing velocity for the coarse particles. Solids motion was measured in a fluidized bed with a computer-aided radioactive tracer particle tracking facility [122]. In such a facility, the gamma radiation from the tracer was continuously monitored. It has been found that the fluidizing velocity and the distributor plate configuration have a dominating influence on the solids circulation pattern. Results from the bed with a uniform air distributor have shown the existence of two counter-rotating toroidal vortices whose relative sizes and strengths vary with the fluidizing velocity. Diffusion model A tracer experiment was conducted to measure the lateral mixing of solids in a rectan-

gular gas-solids fluidized bed [ 1231. The classical diffusion model has been adopted to analyze the data, which has given rise to a new correlation relating values for D,, to the operating parameters. Three important factors inducing the lateral mixing of solids are taken into account in the proposed correlation. They are: (i) bubble movement through the bed, (ii) bubble burst at the bed surface, and (iii) gross particle circulation in the bed. These factors depend on parameters U,, db, d,, (us - Pf), Pr, cci, K and (u - - Knth thereby yielding the following correlation for DSl: = f(Ub,

41

dbr d,,

Ps-

Pi, Pf, Pi, Ho,

u-

h,f)

(65)

where d, = particle diameter, H mf = static bed height, U = superficial gas velocity, Umf = minimum fluidizing velocity, pf = fluid viscosity, pi = fluid density, and ps = solid density. The parameters &, and db in the above expression can be written in terms of other more basic parameters by ub

= (u-

urn,)

+ Ubr

(66)

with Ubr = 0.711(gdb)1’2

(67)

and db = (U - U,f)-“2(H

+ f&,-3’4g-1’4

(68)

Combining with eqns. (66) through (68), eqn. (65) now only contains six parameters, that is, Qi = f(U-

Umr, d,,

Dimensionless

A

pi,

C(f)

(6%

analysis, in turn, results in

D,i

W-

( U --- U,, )H,i x

-pi,

U,f)d,pf Pf

[5E]b[y]c

1 ’

(70)

Fitting the data with eqn. (70) yields A = 0.46, a = -0.21,

b = 0.24 and c = -0.43

Equation (70) is reported to be valid for particles belonging to group B or D of the

275

particles and bed material were identical except for size; this enabled a simple sieve analysis to be employed for determining tracer concentrations at various locations inside the bed. To develop a correlation for the axial dispersion coefficient of solids D,, the expression from the bubbling bed model [ 1311 was assumed D&la g,,-

(72)

cnf)”

mf

0.5

23

I

5

IO

D,Ikal.)

20

30

50

100

200

[cm*/sec]

Fig. 14. Comparison between the observed and the calculated values of the lateral dispersion coefficient of solids, D,l [123].

Equation (72) has been obtained under the assumption that the axial dispersion of particles is caused by rising bubbles dragging solids in the wake. Using the D,-values obtained from approximately 130 data and regression analyses, the proportional constant and the value of n in eqn. (72) have been determined; this has resulted in [ 1301 D, = 0.051 -&I-

powder classification [ 1241. Figure 14 compares the observed D,i with D,i calculated from eqn. (70). Lateral solids mixing in a rectangular fluidized bed with cylindrical screen packings was investigated through observation of unsteady state behavior of tracer particles [ 1291. Based on investigation of the effect of the superficial gas velocity, packing size and minimum fluidizing gas velocity upon the lateral dispersion coefficient, the following empirical equation has been proposed for D Sl; Dsl = 0.0026D,*2~oU,to*96

u-

X i

ill,,

umf

la4

1

(cm*/s)

(71)

This correlation is said to be valid with the ranges of Dp* = 1.5 - 4.0 (cm) V,r = 3.0 - 8.0 (cm/s) U:= 10 - 80 (cm/s) where Dp* = equivalent diameter of packing, defined as (3dZ/2) 1’3, d = diameter of screen packing or sphere, and I= length of screen packing. The axial mixing of solids in a cylindrical fluidized bed was investigated with a tracer technique [ 1301. In the investigation, tracer

Umf)1*471(cm*/@

(73)

mf

The axial solids mixing and bubble characteristics in columnar gas-fluidized beds with vertical internals were studied experimentally [ 1321. The axial solids diffusivity D, was determined by a tracer technique using the one-dimensional diffusion model. The experimental results have yielded empirical correlations for D, with longitudinal springs or rods as internals; D, decreases in the presence of internals with a reduction in the voidage of the internal. Although axial solids mixing in fluidized beds was conducted mainly in beds operating in the slug flow regime the significant parameters characterizing this phenomenon, such as the slug rise velocity and slug spacing, were often neglected in these studies. To overcome this deficiency a model has been developed to predict the axial solids diffusivity in a slugging fluidized bed containing round nosed slugs [ 1181. The model contains a parameter indicating the fraction of the total interslug space that is perfectly mixed. Different values of this parameter have been given for various operating conditions and for the scaling effect of varying bed diameters. The mixing rates in the radial, axial and tangential directions in a centrifugal fluidized bed were experimentally determined [133]. The radial direction in the fluidized bed refers to the direction of the air flow, axial direction to that of the axis of rotation, and the tangen-

276

tial direction to that perpendicular to the radial direction of the bed. The rate of axial mixing has been expressed in terms of a mixing coefficient with the assumption that axial mixing can be adequately described by the Fickian diffusion model. Based on experimental observations, it has been concluded that the motion of bubbles is the primary mechanism, and the rate of mixing increases with the air flow rate. Tangential mixing, on the other hand, has been found to be strongly influenced by the freeboard vortex. This is due to a temporary entrainment of particles being flung into this region of bursting bubbles. The random walk theory was applied to the investigation of the axial mixing of large spherical particles in a liquid fluidized bed [134]. The axial dispersion coefficient has been recovered from the average passage time of a paramagnetic tracer particle through two detection planes, created by mutual inductors. The dispersion coefficient has been found to be approximated by (74)

D, = O.lU*

where U is the superficial liquid velocity in the range of 0.001 < U< 1 (m/s). A stochastic or statistical mechanical model has been proposed to relate the solids diffusivity to the local or microscopic characteristics of the particle motion [135]. The solids diffusivity has been derived under the steady fluidized state by combining the Ornstein-Uhlenbeck equation of stochastic processes and the partition function of particle velocities; the resultant expression is 2 -+ n i

1 - exp(-Y*) V

i erf(-Y)

(e-l)

+

uzc = minimum value of the instantaneous velocity of an individual particle relative to the upward superficial velocity of the fluidizing medium. The calculated solids diffusivity has been compared with the available data; if the velocity oZ is relatively low, the solids diffusivity based on the model agrees well with the available experimental data. The discrepancy arises when oZ, or the superficial velocity of the fluidizing medium, is high. It has been pointed out that the lack of agreement is due to the circulatory particle flow in a liquid-solids fluidized bed, thus leading to the deviation of the experimental data from the Fickian diffusion equation [ 1351. In contrast to conventional Fickian or deterministic forms of the diffusion equations for batch fluidized beds, the particle mixing or motion in a stochastic model is characterized by both diffusive and convective components [ 1361. It has been suggested that the dispersion coefficient be replaced with an overall particle mixing coefficient in the Fickian diffusion equation. The proposed expression is (%)t=

(&)d

+ (&.)c

(76)

where (Eph = overal mixing coefficient, (Ep)d = mixing coefficient attributable to the diffusive component, and (E,), = mixing coefficient attributable to the convective component. (Ep)d and (Ep)c are defined, respectively, as

(Eph =

I

= (e - 2)oZ

(75)

where a = inverse of relaxation time D, = solids diffusivity

nZ = average velocity of the particle relative to the upward superficial velocity of the fluidizing medium

with a = inverse of the relaxation time, b = variance of the particle velocity, and U40 = initial particle velocity. The experimental results have verified the existence of both the diffusive and convective components of the mixing coefficient. It has been revealed that the mixing coefficient estimated from the deterministic diffusion model simply lumps the two components.

277

Diffusion-convection model Axial segregation and mixing of solids in a batch liquid-solids fluidized bed were investigated with the bed containing a binary mixture of size-variant lead glass spherical particles [ 1371. The mixing coefficient for particles of either size in the mixture was evaluated by fitting the convective-diffusive model to experimental results. Under steadystate conditions, the total flux of solids, which is the sum of those from convection and diffusion, is zero across any horizontal plane. In other words, O=V$-M,d$

OS’s, H,f)

with & = 0.0444goS “sH,,.,f *.125(U - &,, )“*750 (80) The values of Db calculated according to the above equation are compared with the observed values of Db in Fig. 15. A comparison of eqn. (79) with the experimental data in Fig. 16 indicates a good agreement with 22% standard deviation of the mean response.

(77)

where z = vertical (axial) co-ordinate, M, = axial mixing coefficient, and . vz = bulk velocity of solids in the axial direction. This equation has been solved for both large and small particles. By fitting to the experimental data through regression analysis, the values of the mixing coefficients for both particle sizes have been recovered and found to be in the same ranges as those obtained by previous investigators. The lateral mixing coefficients of solids in continuous gas-fluidized beds have been related to those in the equivalent batchfluidized beds [ 1381. The lateral mixing of solids in a fluidized bed with solids flow is caused not only by particle diffusion, as is the case with batch operation, but also by the conve;tive effects, including the effects of gross velocity profile. Combining these factors with the grid zone effect, which is dependent on the bed height, the following relationship has been postulated to determine D,: D, = f(pb,

(79)

(78)

where D, = dispersion coefficient for the continuous operation, Db = dispersion coefficient for the batch operation, . us = average lateral velocity of solids, and H,f = static bed height. Based on the random walk theory and the two phase tKeory of fluidization, the following expression has been obtained:

0.5 0.5

I

235

IO Dbkal.)

20 30

50

100 200

[cm*/sec]

Fig. 15. Comparison between the observed and the calculated values of I),-, [138].

Other models A hyperbolic diffusion equation has been proposed to describe solids mixing in fluidized beds [ 1351. This equation has been derived by linearizing the two-phase flow equations of motion based on nonequilibrium thermodynamic principles. A hydrodynamic model has been developed with nine parameters to characterize the dynamic exchange of tracer solids in a fluidized bed containing horizontal tubes [ 1421. Retaining all the features of the counter-current backmixing model [ 311, the model describes solids mixing as solids convection in the dense phase. The dense phase is considered to contain an upward moving drift (wake) phase and two down flowing phases, one of which is the main dense phase and the other the faster moving

278

8.0 6.0 4.0

0.01

0.02

0.04

0.1

0.2

0.4

1.0

2.0

To characterize axial mixing and segregation of large particles in two-dimensional gas fluidized beds, a non-stationary random walk model has been developed [ 1471. The motion of particles leading to axial mixing is described as a one-dimensional process by equally dividing the entire length of the bed into y fictitious sections in series in the vertical direction. Moreover, it is assumed that (i) the concentration distribution inside a section is spatially uniform, and each section is a discrete state, and (ii) the particle can move only to the nearest sections during a small time interval At. The proposed model can be written as

& “mf Db

Fig. 16. Comparison between the experimental data and eqn. (79) [138].

boundary dense phase flowing near the bed walls parallel to the tubes. The model interpretation of the experimental solids mixing data for gas superficial velocities of less than 1 m/s has shown that in the fluidized bed with horizontal tubes, the circulation rate of solids is much lower and the solids exchange rate of the drift phase is higher than in an empty bed at the same gas velocities. The same experimental data as those employed by Sitnai [ 1421 can be represented reasonably well by a computationally simple numerical model (‘cinematic’ model) [ 1431. Solids mixing of the Geldart Group A powder in a 0.15-m internal diameter fluidized bed was studied with a ferromagnetic tracer [ 1441. The tracer distribution has been found to be well described by the counter-current flow model of solids mixing, and at a relatively high gas velocity, the data have been successfully fitted by a one-dimensional Taylor dispersion model. The axial dispersion coefficient has been reported to increase with the square root of the bed diameter. A backflow cascade model has been applied to the evaluation of the residence time distribution of solid particles in a benchscale fluidized bed reactor [145]. The distribution of solids mixtures in a liquidsolids fluidized bed has been using a new approach, calculating rather than assuming the porosity profile [ 1461.

$ Xi(t) = i Xj(t)Uji i=l

i = 1, 2, . . . . r

(81)

where Xi(t) is the concentration of the key particles in section i. Uij denotes the intensity of transition of a key particle from state i to state i, and is expressed as Uij = 0, if # (i - l), i or (i + 1)

(82)

A two-dimensional random walk model has been proposed to simulate agglomeration process of monodisperse systems in which particles encounter and adhere to one another after impact [ 1481. Particulate mixing systems are stochastic in nature due to the finite, mesoscopic sizes of discrete particles involved. Thus, the master equation has been modeled to the axial mixing of particles in a fluidized bed [ 1491. The physics of mixing at a single particle level can best be seen as a random process since the trajectory of the particle is influenced by many competing factors. A stochastic model retains the unpredictability of individual particle paths while a deterministic model would have to approximate the system as a continuum first and then introduce a phenomenalistic description using a diffusion equation. The proposed stochastic model has been formulated by resorting to probability balance whose solution yields information on the number of particles of each species at any position in the system. The parameter in the model is estimated from the available tracer experimental data. Expressions are derived for the mean tracer concentration and its variance, and the first passage time for a tracer particle to reach the bottom of the bed.

279

A semi-stochastic approach has been proposed to model the mixing behavior between coal and limestone particles in a gassized fluidized bed [ 1501. The dynamic and steady state characteristics of solids mixing have been simulated using the Monte Carlo method, which considers the downward or upward movement of a particle in the bed as being governed by certain probability laws; these laws in turn, are affected by the bubbles. The model has incorporated the effect of time, flow rate and relative size on solids mixing. Cohesive systems With free-flowing mixtures, the scale of segregation is largely a consequence of the movement of individual particles. In the case of cohesive powder mixtures, however, the independent movement of particles is restrained by a variety of interparticulate forces. The nature and strength of these forces will determine the ease or difficulty of forming groupings of similar constituents [28]. The mixing mechanisms discussed earlier for free-flowing particles, therefore, are not relevant for systems of cohesive powders. Interparticulate bonding Three main forces cause the structuring of a mixture, if solid bridge formation between particles is excluded [28]; these forces are induced by (i) moisture, (ii) electrostatic charge, and (iii) van der Waals’ attraction. Moisture bonding is caused by the overlapping of the adsorbed layers of neighboring

Fig. 17. Adsorbed layer bonding between two neighboring spherical particles [ 281.

Stage I Particle

Stage of

Sucrose Agglomerote

salicylic

2

acid

cry.901 of

salicylic

acid

Fig. 18. The process of formation of an ordered mixture [152].

particles (see Fig. 17) when moisture is present as absorbed vapor, or by a liquid bridge formed between particles when moisture exists as free moisture, which is the moisture in excess of that of equilibrium. Electrostatic bonding is attributable to the attractive forces between particles with different signs of surface charge. Van der Waals’ force refers to the attractive forces existing between neutral atoms or molecules separated by a distance larger than their own dimensions. While various bonding forces usually coexist in real systems, the moisture bonding forces can dominant other possible bonding forces and also influence their magnitude. In other words, the humidity of the atmosphere plays a significant role in determining the relative importance of the forces [ 281. Formation of ordered mixtures The formation of ordered/randomized mixtures were studied by several investigators [69,74,151]. By varying the concentration in a binary mixture of salicylic acid and crystalline sucrose, it has been possible to obtain an ordered mixture (0.1 wt.% salicylic acid), a partially ordered randomized mixture (1 wt.% salicylic acid) or a random mixture of salicylic, and agglomerates and ordered salicylic acid/sucrose units (4 wt.% salicylic acid). To study the effect of humidity variations on the formation of ordered powder mixtures, an ordered mixture of a l-1000 mixture of salicylic acid and sucrose is formed in two stages (see Fig. 18) [ 1521. In the first stage, a single fine particle of salicylic acid is produced from its agglomerate. The rate at which the first stage occurs depends on the rate of energy input from the momentum

280

(a)

(b) Fig. 19. Constituent segregation [74]: scale; (b), large scale.

(a), Small

changes of coarse particles (sucrose) and by the mixer. In the second stage, the adsorption of the single fine particle to be the coarse particle depends on the orientation of the fine particle relative to an adsorption site and the magnitude of the adsorption force. Although humidity affects the magnitude of the various interparticulate forces involved in the adsorption process, the results of the mixing experiment have shown that the adsorption forces are sufficient to form a stable ordered mixture at the same rate at three different humidity levels (0%, 55%, 84% RH). Consequently, it can be inferred that the magnitude of the adsorption forces does not affect the mixing rate. Using the same powder system, the rate of formation of the ordered mixture of salicylic acid/sucrose has been shown to be the same in a ball mill and a cube mixer despite the significantly higher energy input of the ball mill [71]. Therefore, it can be concluded that the correct orientation of microfine particles to adsorption sites in stage 2 determines the overall rate of this particular mixing process. A study was carried out on systems containing salicylic acid and three different carriers [64]. The salicylic acid was present in the proportion of 1:lOOO. It has been found that ordered mixtures are formed in all cases and that less segregation of the ordered units has taken place at 0 and 84% RH than at 55% RH, attributed to the interaction between the ordered units resulting from electrostatic forces and moisture bonding.

Fig. 20. Ordered unit segregation [157]: (a), Randomized ordered units; (b), segregated ordered units.

Mechanism of mixing and segregation Two forms of segregation have been identified [ 1531. They may occur within an ordered mixture constituent segregation and ordered unit segregation. Constituent segregation can be viewed as occurring in two stages [ 741: (i) the separation of the adhering particles from their carrier particles (small-scale constituent segregation); and (ii) the attachment of these fine particles to other fine particles (large-scale constituent segregation) or to other carrier particles (Fig. 19). Ordered unit segregation, on the other hand, is caused by the segregation of different sized carrier particles with adhering fine cohesive materials (Fig. 20). Ordered mixtures have been shown to be mechanically stable [65,151,152]; the constituents in an ordered mixture tend not to segregate because of the strong adhesive force acting among them. The phenomenon of ordered unit segregation, therefore, is of primary importance in determining the homogeneity of an ordered mixture. The effect of ordered unit segregation is to increase the mixture heterogeneity [66, 151, 154,155]; thus, the greater the size range of the carrier particles, the more prone

281

the mixture to segregation. Monodisperse carrier particles serve to eliminate segregation, but they are not economically viable in the pharmaceutical manufacturing industry [ 1561. The granulation of an ordered powder mixture has been investigated as a method of preventing segregation in the presence of polydisperse carrier materials [ 1521571. A new method has been proposed to control the segregation of ordered mixture by triboelectrification [ 1581. To mix powders by triboelectrification is to alter the surface electrical properties of constituent particles so as to facilitate inter-particle adhesion towards the formation of a stable ordered mixture. This is different from the conventional approach of employing shear and other forces to split and recombine powders. Mixtures produced by this method have demonstrated higher resistance to segregation than those produced in conventional mixing equipment. Mixtures produced by electrostatic mixing of powders have also been reported [ 1591. A binary mixture of particles having a diameter of 100 pm and above has been discovered to behave mainly like cohesionless mixtures since for this particle size the gravitational force dominates over interparticle electrostatic forces [ 1601. Mixtures of particles having diameters of 40 pm or less tend to be cohesive mixtures because for this size range the interparticle force becomes a decisive factor. The term ‘total mix’ has thus been suggested to describe all types of powder mixes. The behavior of homogeneous total mixes in different vibration conditions was studied [63]. Three different total mixes were formed using direct compression tableting excipients and a model drug, potassium chloride. The mixes were vibrated in conditions similar to those encountered during normal pharmaceutical production. The segregation has been found to be induced through either a percolation or a diffusion mechanism depending upon the vibration frequency and the acceleration force. Different tendencies were observed for different excipients when mixed with the model drug particles. The results have suggested that segregation of a given total mix can be minimized by careful selection of drug/ excipient combinations and by ensuring that processing conditions do not subject

powders to prolonged vibration at frequencies below 100 Hz and to accelerations greater than approximately 2 G. In the investigation of a fluidized-bed mixer producing a microdose powder mixture from granulating phenylbutazone and lactose, both cohesive and formed agglomerates were not broken down during the initial lo-min mixing phase of the granulation; this subsequently led to a poor homogeneity in the final product [161]. It has been revealed that the fluidized-bed mixer may lead to homogeneity problems when the mixing powders have one or more cohesive components. The nature and characteristics of circulation and mixing of cohesive and agglomerative material (flour) in a stirred fluidized bed have been investigated by direct observation of solids movement and by tracer experiments [ 1621. Two dominant features observed are stick-slip flow of solids and the presence of relatively stable horizontal cracks. The former is a consequence of the downward movement of coherent masses of flour in a succession of short jerky movements. The latter is formed in the descending mass of flour and by the passage of rising bubbles. At lower gas velocities, the behavior of the system has been found to be characterized by zone fluidization which leads to an erratic and unreproducible mixing performance. Consistent mixing could be achieved, however, by increasing the gas velocity to that of greater than twice the minimum fluidization velocity or by adding an anti-agglomerant to reduce the minimum fluidization velocity and to promote smooth fluidization.

DESIGN AND SCALE-UP OF MIXERS

More often than not, mixers for particulate solids have been designed heuristically [ 931. The reason for this is the complexity of physical properties of the solids and mixers that can only be adequately described with a large number of parameters. We cannot, however, totally rely on experiences gained from experiments with a pilot-scale mixer for scale-up. As we proceed with scale-up, the hypotheses developed from and verified in the early experiments may become invalid.

282

One of the first procedures proposed for the scale-up of mixers is based on the principle of similarity [ 1631. The three similarities considered in the example of scaling-up a tumbling mixer are geometric, kinematic and dynamic. The geometric similarity involves a constant ratio of the linear dimensions of the prototype and scale-up system. The kinematic similarity requires an equal ratio of velocities between the corresponding points in the two systems in addition to the systems being geometrically similar. The dynamic similarity refers to an equal ratio of forces between the corresponding points in the two systems. Two general methods are available for identifying similarity criteria. When the governing equations are not known, such criteria can be derived by means of dimensional analysis. When the differential equations governing a particular process are known, the equations can be transformed into dimensionless form with the similarity relationships being obtained from the parameters - a procedure that may be termed ‘normalization’. For geometrically and kinetically similar systems with nonsegregating materials and with the Fickian diffusion equation as the governing equation to describe the mixing mechanism in the mixers, the following expressions have been derived: 033) for mixing time, and (84) for power requirement. In these expressions, K = (x2/2)0, mobility coefficient of particles, D = particle diffusivity, L, = effective length of the mixer, P = power requirement for the driving mixer, N = rotational speed of the mixer, y = rotating radius of the mixer, p = true density of the particles, g = gravitational acceleration, and ao, a, = parameters. In application, the mixing time can be determined by eqn. (83), provided that the mobility coefficient can be correlated to operating conditions, such as the fraction of loading or

rotational speed. Such a correlation is available [ 1641. The parameters a, and al in eqn. (84) can be calculated by the linearization and nonlinear least-square methods. In investigating the applicability of scale-up rules to ribbon mixers with segregating materials, it has been found that the geometric and kinematic (but not dynamic) similarities are met for the mixers [ 1641. The measurements of the variance of composition of the mixing products as a function of time have revealed that scale-up rules are not valid due to segregation and agglomeration effects. The similarity rule of fluidization [165] has been extended to the lateral solids mixing in a fluidized bed, based on the results of a transient dispersion experiment of ferritecoated sand tracers and large floating particles in beds of different diameters (50, 150, 300 and 600 mm; tapered and non-tapered) [ 1661. The requirements from the similarity rule to establish a geometric similarity are that U, - U,, and Umt for a fluidized bed m times larger than the pilot bed be

(UIJ-

)plant =

Unlf

(for bubble similarity) (Knf

)plant

=

dmi

(85) (for interstitial

flow similarity)

(86)

It is worth noting that lateral solids mixing is much slower than the axial solids mixing in a fluidized bed. The ratio of centrifugal force to the gravitational force, or Froude number, has been used as the main factor for the scale-up of a tumbling mixer [ 1671. The following scaleup relationship has been proposed

(?i,.b

=

(?)&mt

(87)

where N is the rotating speed (rpm) and R is the rotating radius. Figure 21 shows the relationship between the maximum rotating radius of the mixer, R,,,, and the optimum rotating speed Nopt . Nopt denotes the rotating speed resulting in the maximum degree of homogeneity for the mixture. The average particle diameter & is related to the Froude number with N = Nopt in Fig. 22, presenting the following correlation:

283

100

I

I

10

‘01

/

I

0

Horizontol cylinder

-

n

Double cone

0

V-mwer

H

II

3 50

20

IO

5 Maximum

rotating

radius

100

Rmoxkm)

Fig. 21. Relationship between the optimum rotating speed and the maximum rotating radius of the tumbler mixer [ 167 1.

constant specific power input, as indicated by the scale-up criterion in the case of complete suspension. Homogeneity of the suspension has been found to be better on the large-scale than on the laboratory scale. Other findings are that the thickness of the stirrer blades has an important influence on the mixing performance, and incorrect scaling of the blade thickness will lead to over optimistic figures for the power consumption on a commercial scale. A blade-edge modification has been suggested with which savings in power consumption of 17% can be reached. For additional information on the design and scale-up of mixers, readers are referred to various publications [24,167,169].

CONCLUSIONS

1.0

0.

I

0.05 0.02

0.01

Average

0.05 particle

0.1 diometer

0.2 dp km)

Fig. 22. Relationship between the average particle diameter and the optimum Froude number [ 167 1.

N opt =c

v -

4

R max

038)

where C is a constant characteristic of the mixer with 121 for horizontal drum mixers, 85.2 for cube and double-cone mixers and 78.6 for V-type mixers. Scale-up rules for solids suspension in stirred vessels were investigated by carrying out a series of experiments in a large stirred vessel (4.3 m in diameter) and a small-scale vessel under identical conditions [ 1681. A comparison of the results has revealed a

This review covers the major development in solids mixing since 1976. The publications on this subject have been divided into three major categories of characterization of states of solids mixtures, rates and mechanisms of solids mixing processes, and design and scaleup of mixers or blenders. The homogeneity of a solids mixture or the distribution of its composition is often quantified by a mixing index. Most of the available mixing indexes are based on the variance of the concentration of a certain or key component among spot samples. These mixing indexes can only depict the macroscopic behavior of a solids mixture. It is difficult to visualize the structural details of the mixture (e.g., contact points or area among particles of different solid phases) through the available indexes. Thus, the definition of a geometric mixing index based on the number of contact points between different solid phases has lately received increased attention. Nevertheless, representation of the complex characteristics of a solids mixture through any available mixing index appears to be far from satisfactory. Three major mechanisms, diffusion, convection, and diffusion-convection, have been proposed for solids mixing. Various mathematical models for these mechanisms have been proposed, and numerous mathematical expressions for the rates of solids mixing based on these mechanisms have been developed. While many of the models

284

and expressions are deterministic or macroscopic, some resort to stochastic approaches. This may be attributed to the difficulties in delineating the inherently complex nature of solids mixing processes by means of the deterministic approaches. The trend will probably continue [170] because a particle mixing system is neither macroscopic nor microscopic, but mesoscopic in nature. Although considerable progress has been made in our understanding of solids mixing processes, the design of mixers or blenders for particulate solids has mainly been carried out heuristically, again due to the complexity of solids mixing behavior, describable only by a large number of parameters. Meanwhile, the experience gained with a pilot-scale mixer may not be reliable for scale-up. Therefore, an effective design procedure employing both heuristics and algorithms needs to be developed. Systematic coupling of these entails the development of an expert system. An expert system contains a generalized inference engine and a rule base. It takes input data and assumptions, explores the inferences derivable from the rule base, yields conclusions and advice, and offers to explain its results by retracting its reasoning for the user. The technology for building expert systems is the most advanced subfield of Artificial Intelligence, which has made tremendous progress in recent years. ACKNOWLEDGEMENT

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