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Bulk-solids mixing: overview
Handbook of Conveying and Handling of Particulate Solids A. Levy and H. Kalman (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
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Bulk-solids mixing: overview L.T. Fan Department of Chemical Engineering, Kansas State University Manhattan, Kansas 66506, USA A survey is given of various industrial applications of solids mixing and the role it plays in our daily lives. This is followed by a review of the history and current status of the technology of solids mixing. Finally, a proposal is presented for further development of this technology and concomitant fundamental research. 1. INTRODUCTION This overview focuses on both the fundamental and practical aspects of solids mixing. Particulate solids are regularly found in numerous chemical, agrochemical, pharmaceutical, food and other products. The majority of such products comprise a multitude of components whose homogeneous distribution over any part or the entire batch of each of these products is of the utmost importance. This homogeneous distribution, often accomplished by blending or mixing, obviously plays a significant role in bulk-solids handling and processing that are ubiquitous in diverse industries and in our daily lives. Various fundamental aspects of solids or powder mixing and the equipment and devices to effect it, therefore, need be explored to gain insight into the homogenization and its reverse, segregation, induced by mixing The importance of solids or powder mixing can be readily seen in laboratory manuals and handbooks in different disciplines of pure and applied sciences and in engineering, of which the most recent edition of Perry's Chemical Engineers' Handbook [1] is an example. Solids mixing or powder blending is frequently a stand-alone operation in various processes and technologies, e.g., in manufacturing feeds, insecticides, fertilizer, glass batches, packaged foods, plastics, and cosmetics. It is also frequently carried out simultaneously with other processes or operations, some of which include cooling or heating solid materials, e.g., cooling sugar or preheating plastics, prior to calendering; drying or roasting cereal grains and ores; carrying out heterogeneous chemical reactions; agglomerating powder; and surface coating or enlarging particles common in the preparation of pigments, dyes, pharmaceuticals, detergents, fertilizer, coated seeds and candy. Reducing the size of particles is usually accompanied by solids mixing. 2. CHARACTERIZATION OF MIXTURES The ability to characterize the object or system of interest is essential for any undertaking. Solids mixing is no exception; the object of interest here is a mixture of particulate materials.
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2.1. Uniformity and homogeneity Among the characteristics of a mixture, uniformity or homogeneity in the composition is of primary importance because specifying the composition is one of the most crucial requirements for product quality. Nevertheless, the required uniformity in the composition of the mixture is application-dependent or object-oriented. For example, the composition of a human medicine must be strictly prespecified to ensure its effectiveness and safety. Thus, the range of deviations tolerable is narrowly defined. On the other hand, the deviations are of less concern for solid foodstuffs, e.g., dried vegetables and prepackaged soup ingredients. The deviations can be even greater for solids mixtures serving as building materials such as premixed concrete and plaster. Fan et al. [2] have defined a homogeneous mixture as a particle system in which the concentrations of all constituents are uniform throughout. Under ideal situations, the spatial distribution of the constitutive particles in a mixture of two components can be characterized such that all the particles of a component are regularly or evenly distributed among the particles of the other component in any part or direction of the mixture. This arrangement can be visualized readily if the sizes and numbers of the component particles are equal. A mixture with regularly arranged components, i.e., an ordered mixture, however, can be formed even if the sizes and numbers of the component particles are different. An ideally ordered mixture can be defined as a mixture where individual particles of a given component, whose number is equal to or less than the other component, are evenly dispersed in the matrix of the latter, and the distance between the particles of the former is identical in all directions. An ideally ordered multicomponent mixture can be defined analogously by identifying the key component and by lumping the remaining components as one component. As such, the particles of either component must be arranged according to a regular spatial pattern. In spite of this, not every mixture with a regular spatial pattern is necessarily an ideally ordered mixture because a striated arrangement of the components represents a regular pattern but it does not always correspond to the highest degree of mixedness attainable. Naturally, the particles of the component of interest can be regularly arranged variously to suit different classes of materials. For instance, ideally ordered states should be achieved or be closely approximated for controlled-drug delivery from the matrix of inert or support materials. To characterize the forms and extent of the approach to the ideally ordered states, several terms have come into being, examples of which are regimented, structured, layered, and incompletely or imperfectly ordered mixtures (see, e.g., Hersey and Cook [3], Orr and Shotton [4], Kaye [5], Staniforth [6], Fan et al. [7]). For an ideally ordered mixture, the degree of homogeneity measured on the basis of any mixing index must be the highest. For example, the standard deviation or variance of the sample concentrations must be zero or nearly zero in such a mixture: the edge effect and finiteness of the size and number of particles of any individual component in a sample prevent the variance to vanish. In practice, it is extremely difficult to obtain perfectly or ideally ordered mixtures of freely moving particles by ordinary mixing processes: any disturbance causing relative displacements of the particles would diminish the degree of homogeneity. At the same time, it is possible to generate almost ideally ordered, stable mixtures for specific purposes by applying unique methods such as agglomeration, coating, and micro-encapsulation, none of which, however, is a mixing process in the conventional sense. Contrary to an ideally ordered mixture, the distribution of particles of any component in a completely random mixture is totally randomly dispersed among the particles of the other components. This implies that if individual particles of any component were marked by an increasing or decreasing series of numbers prior to mixing, the
649 order of such numbers in any direction throughout the mixture would be totally random when the completely random state is attained. The probability of finding a particle of any given component in the completely random mixture is identical in every location and is equal to the global, volumetric ratio of this component. In his comprehensive works, Kaye [5, 8] has pointed out that in contrast to an ordered mixture, the composition within a random mixture can vary to a certain extent from site to site and from experiment to experiment. The extent of this inherent variation depends on the overall composition of the mixture and also on the "sample size" with which the local concentrations of the constituents are determined. Unlike the ideally ordered mixture, some particles of a given component in the random mixture are clustered; in other words, they are not separated from each other by the particles of the other components regardless of the mixture' s composition. In some of the numerical examples given for the characteristics ofthe randomly formed clusters in powder mixtures, Kaye [5,8] has emphasized that in evaluating the randomness of a solids mixture, the concentration of any component can be measured in terms of either the weight or volume fraction. The particles compete for space and are distributed along spatial coordinates in the bulk of the mixture; therefore, it is more meaningful to measure the volume fraction than the weight fraction. Two obstacles hinder the total randomization of the particles. One is attributable to the geometry of particles. If the component particles are significantly different in size, the smaller particles can be concentrated in some voidages among the larger ones, thereby causing the particles of different components to segregate and preventing them to be completely randomly distributed. The other obstacle is attributable to possible surface interactions, e.g., adhesion and electric static attraction. If the surface interactions are profound between the particles of different components, these particles tend to form partially ordered arrangements, thus reducing the possibility of forming a completely random mixture. The appreciable surface effects among the particles of the same component can cause segregation similar to those induced by the particlesize differences. Noticeable density or weight differences between the component particles can also lead to segregation. In general, the extent of segregation is greatly affected by the concentrations of components. 2.2. Degree of mixedness Any mixture at an intermediate state between the totally segregated and completely random states must be deemed an incomplete mixture irrespective of its initial state: it can be generated by mixing initially separated components or by spontaneous segregation of a completely random mixture. The actual state between these two extremes is characterized by the degree ofmixedness measured by various mixing indices. The term, "partially segregated mixture," applies to a mixture which is yet to be fully homogenized and to a mixture homogenized once but experienced subsequent segregation. In both cases, the terms, "scale of segregation" and "intensity of segregation," serve to measure the actual state between the completely mixed and totally segregated states. In a totally segregated mixture, the components are unequivocally separated from each other in different and distinct regions in a batch of particles. This is usually the situation prior to mixing. To form a mixture, e.g., a stratified one, a given component is fed or charged into the mixer in the form of layers separated by other components. The relative positions, sizes and numbers of these layers as well as the distances between them appreciably affect the attainable rate of mixing. The configuration of the layers can be characterized by the scale of segregation.
650 The degree of homogeneity of a mixture expressing the extent of approach to perfectness has often been estimated by the uniformity of sample concentrations. To illustrate this, the standard deviation of the concentrations of a key component in a mixture is given by
O" =
(1)
i=l
N
where o- is the standard deviation of the sample concentrations; xi, the concentration in the i-th sample; ~-, the average ofxi,s; and N, the number of samples yielded by dividing the entire batch of the mixture. In a perfectly homogeneous fluid mixture, cris zero; if the components are totally segregated, the value of o-is maximal. In a conventional mixing process, the maximum achievable degree of mixedness corresponds to a completely random arrangement of different particles. Based on the assumption that the particle size of an individual component is identical, Lacey [9,10] has demonstrated that the minimal possible standard deviation of sample concentrations for a binary mixture can be expressed as
O'r = ~/PQY/pP)
=I s
/'/p
(2)
where P is the overall proportion of the particles of a given component in the whole mixture, which is equal to the average concentration, ~-, and np is the number of particles in each sample. In principle, the values of the standard deviation even lower than o;,.can be achieved by regularly arranging the particles, i.e., by forming an ideally ordered mixture. In practice, this can only be realized by means of unique processes such as surface adhesion or agglomeration, as discussed earlier. When the particles can move relative to each other without much hindrance, this ideally ordered arrangement is unstable. Naturally, the value of o-for an incomplete mixture is greater than ~.. The greater the value of the standard deviation, the closer the mixture to the totally segregated state. The standard deviation for the totally segregated mixture, denoted by Cro,is maximal; it depends on the average concentration of the key component of interest as shown below. o 0 = x/g0 - 2)
(3)
During mixing, the standard deviation of an incomplete mixture must be in the range between the two extremes given by Eqns. 2 and 3. It is not always the case for an ordered mixture because the lower bound, as characterized by the standard deviation of the completely randomized mixture, can be exceeded by such a mixture. Measuring the quality of the incomplete mixture during mixing, therefore, is crucial in controlling and optimizing the process. It is frequently impossible, however, to evaluate the quality by measuring the concentrations in all parts of the mixture, i.e., to determine the standard deviation from the entire sample. This means that uncertainty arises in estimating the value of the standard deviation due to the finiteness of the number of spot samples, n, which gives the sample standard deviation, s, defined in the following.
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I Z~_,n-1(x, _~)2 S=
n-1
(4)
In contrast to a fluid mixture, the quality of a solids mixture is difficult to determine because of the discontinuous nature of the particulate systems as well as the finiteness of particle and sample sizes: The sample concentrations and their standard deviations can be affected by the error caused by the propensity of any component' s particle or particle cluster to straddle the boundaries of the sample containing it. It follows that the measured standard deviation depends on the average concentration of the given key component as well as on the relative sample size. For a completely randomized mixture whose component particles are identical in size and density, this uncertainty can be determined mathematically. For other cases, this is much more complicated. To account for the effects of the sample number and size, the general rule is to take a sufficient number of samples from well-distributed points in representative regions of the mixture. Each sample must also contain a sufficient number of particles; such a number can be determined from practical points of view.
2.3. Mixing indices The standard deviation of sample concentrations relative to the mean composition, given by Eqn. 4, is the most common indicator for characterizing the mixture quality. Based on the standard deviation, s, or on the variance, s 2, various mixing indices have been defined, the majority of which varies between zero and unity as the mixture quality changes from the totally segregated state to the completely randomized or ideally ordered state. Fan and Wang [ 11 ] and Poux et al. [12] have reviewed and compiled over 30 mixing indices, and Boss [13] has collected nearly 40 mixing indices based on the concepts of sample variance or standard deviation and has determined the interrelations among these indices. The influence of the sample size is taken into account in defining some of these mixing indices. The enormous diversity of mixing indices due to the variety and structural complexity of solids can be a source of confusion. Consequently, a single, unambiguous mixing index suitable for all applications is difficult to achieve. An overview is provided here on the relations among the important properties of mixtures and various mixing indices to aid selection of an index appropriate for a specific situation. Fan and Wang [11] as well as Boss [13] have compared a number of mixing indices and derived conversion formulas among them. Table 1 lists some of the frequently adopted mixing indices in terms of the statistical analysis of sample concentration. Note that certain indices are affected by the sample size. As such, a value of zero or unity for one of the two extreme states of the mixture quality can be obtained only if the sample size is sufficiently large. The sensitivities of mixing indices to the concentration variance or standard deviation differ at various stages of mixing; hence, it is necessary to compare their behavior. It is imperative that the quality of multi-component mixtures be evaluated and controlled because the components may behave differently at specific periods of mixing. This implies that at a given stage, one of the components may be well homogenized, but simultaneously, other components may continue to be partially segregated. Thus, the mixture as a whole does not meet the necessary homogeneity specification. The majority of mixing indices defined for multicomponent mixtures is based on the concentration variance as summarized by Too et al. [14].
652 3. M E C H A N I S M S AND MODELS OF SOLIDS MIXING
The particulate materials in solids cause mixing to be complex: The particles in such materials are small but finite in size; moreover, they are isolated from each other, i.e., they are discrete. These characteristics render particulate materials mesoscopic. In general, the behavior of mesoscopic materials is describable neither by the firmly established laws of continuum mechanics valid for macroscopic materials, e.g., steel beams or large ice cubes, nor by the wellknown principles of statistical mechanics applicable to microscopic materials, e.g., air. Any attempt to rigorously portray the motion of a countless number of interacting particles in a mixture by particle dynamics would be futile, as evidenced by the difficulty of portraying the motion of interacting particles as few as three. Although efforts have been made to circumvent the complexity involved in establishing the continuum mechanical, statistical mechanical or particle dynamical description of solids mixing, these efforts have focussed on the behavior of particles in a mixture in specific types of mixers through the experimental determination of the controlling mechanism or mechanisms of mixing or its opposite, segregation. As will be delineated later, segregation is often unavoidable in mixing particulate materials. 3.1. Mechanisms
Similar to fluid mixing, three major mechanisms have been identified for solids mixing convective, shear and diffusive mixing. Convective mixing involves bulk movement or gross displacement of particles within the mixture. Shear mixing is induced by sliding displacement of successive layers of particles. Diffusive mixing is caused by the random motion of particles. Generally, the attainable rate of homogenization (mixing) is greatest with convective mixing and least with diffusive mixing, whereas the degree of local homogenization attainable is greatest with diffusive mixing and least with convective mixing. Another mechanism, chaos, has been identified lately as contributing substantially to the mixing of particles and powder. Chaotic mixing of these and other materials exhibits highly complex patterns. Nevertheless, such mixing does not lead to random dispersion of particles: Chaos is a deterministic phenomenon. Hence, chaotic mixing results in irregularly interwoven and interacting convective and shear mixing. It is common knowledge that the mixing of solids mixtures is frequently accompanied by demixing (segregation) of the particulate components being mixed. Segregation does not occur when the particulate components have identical physical properties and geometrical characteristics but differ only in chemical composition. Segregation is caused by the percolation of a particulate component among the interstices of the remaining component or components. The propensity for different particles to segregate is strictly due to the fact that these particles are mesoscopic: the smaller particles tend to drop or percolate through voids ifthe size of the voids is sufficiently large or enlarged temporarily by mechanical vibration and/or aeration. The trend to segregate is magnified by the increase in the density of smaller particles over the larger particles; it is also affected by the differences in shape and surface characteristics. 3.2. Models
Sundry mathematical models can be easily conceived for each mechanism of mixing or segregation in the light of the number of methods available. Obviously, these models are too numerous to list individually; therefore, they are only classified broadly below.
653 Table 1 Some important mixing indices based on the variance or standard deviation of sample concentration.*
No.
Equation 0-
1
Value in the totally segregated state or=or 0
Value in the ideally ordered mixture cr=O
Value in the completely random mixture ~=~,.
2
M 1 -
1
0
0"02
i ;7
(3"
2
M2-
1
0
~o
47;,
0-2
3
M3=l-~
0-02
0
1
1 1-~7
O"
4
M4 = 1 - ~ 0-0
0
1
! l--~n
0
n n-1
1
0-02 - O . 2
5
Ms -
6
M6-
o'02 -0-r2 cro - o cr0
-o r
log cr0 7
-
0
"s/-s ~n 1
log cr
M7 = log o"0 - log crr
0
oO
0-2 M8 - -2
1-~
0
x
1
y
1
!-2 .u ;7
0-
9
Mg-_
* For the original sources, refer to Fan and Wang [ 11 ] and Boss [ 13].
Deterministic models. Analogous to fluid mixing, the deterministic mathematical models of solids mixing are derived from the continuity (mass conservation) of a key component being mixed in a certain volume element in the mixer. If this volume element is finite in size, a lumped model is obtained, and if it is infinitesimally small, a continuous model. Under steady-state conditions prevailing in a continuous mixer, the lumped model manifests itself as a set of difference equations, and the continuous model, a set of ordinary differential equations; naturally, a combined or hybrid model is expressed in the form of a set of differential-difference equations. The corresponding mathematical expressions under unsteady-state conditions prevailing in a batch mixer are sets of ordinary differential equations, partial differential equations, and ordinary differential-partial differential equations.
654 As mentioned earlier, chaos is a deterministic phenomenon in spite of the fact that it is complex and apparently irregular. In fact, a mathematical expression of chaos must comprise at least three nonlinear ordinary differential equations. As a result, it is logical for any mathematical model of chaotic mixing to comprise three or more ordinary differential equations. A deterministic mathematical model of segregation, essentially induced by particle percolation, can probably be derived by regarding solids mixing as a reversible process whose forward and reverse steps are homogenization and segregation, respectively. This is similar to a reversible chemical reaction. Stochastic models. It should be obvious by this time that solids mixing is an extremely complex phenomenon influenced by parameters ranging from physical, geometrical and surface characteristics of the particles involved to the design features and operating conditions of the mixer adopted. By definition, no mechanistic model of solids mixing takes into account "all" parameters. The failure to consider any of the influencing parameters gives rise to the internal noises in the behavior of the mixer of interest; the performance or experimental data of the mixer invariably exhibit appreciable scattering from the model. A simplistic way of taking into account the fluctuations of a mixer's performance or behavior is to add a term representing the noise or fluctuations to any of the deterministic models derived for the mixer. This yields a stochastic model of the mixer as the Langevin equation ubiquitous in the literature of turbulence and communication theories. Unfortunately, the noise term in any Langevin equation represents fluctuations induced by external forces that are not inherent in the system. Another approach to the stochastic modeling of solids mixing is to exploit the theorems and algorithms of stochastic processes. These theorems and algorithms make it possible to explicitly incorporate from the outset only the inherent fluctuations into modeling in terms of the probabilities of the system to be at various states. Markov processes, a highly applicable class of stochastic processes, have been used successfully by this author as well as by others in modeling solids mixing in a variety of mixers. Those wishing to explore the mechanisms and mathematical modeling of solids mixing can consult the review articles by Fan et al. [2, 7] as well as the monograph by Kaye [5]. Those with a particular interest in the chaos aspects of solids mixing will benefit from reading the exposition on chaos in fluid mixing by Ottino et al. [ 15] and Ottino [ 16, 17]. 4. DESIGN AND SCALE-UP OF MIXERS
The design and scale-up of mixers appear to be the least developed among various aspects of solids mixing because of the mesoscopic nature of materials being homogenized, as mentioned previously. In practice, the design of such mixers and blenders has generally been carried out heuristically; moreover, the mixer or blender has been scaled up mainly through successive experimentation with ever larger mixers or blenders, including the bench-scale, semi-pilot-scale and pilot-scale units. The paucity of publications attests to the difficulty in the design and scaleup procedures for particle mixers or powder blenders (see, e.g., [7]). The principle of similarity is one of the well-known procedures for the scale-up of various process systems including mixers. A mixer as simple as the tumbling mixer has three similarities: geometric, kinematic and dynamic similarities. The prototype and scaled-up mixers are geometrically similar if the ratios of their corresponding linear dimensions are constant; they are kinematically similar if the ratios of the corresponding points' velocities are constant; and they are dynamically similar if the ratios of the corresponding points' forces are constant. Two general
655 methods are available for deriving the similarity criteria in terms of the dimensionless numbers composed of parameters significantly infuencing the performance of the mixers. These methods are the dimensionless analysis and the normalization of the governing equations, both of which have been widely applied in other fields. In recent years, substantial progress has been made in establishing algorithmic approaches to the systematic design and scale-up of solids mixers on the basis of the similarity criteria identified for them. In spite of this, heuristics will continue to play a vital role because new types of mixers are constantly being conceived, and novel or exotic particulate materials requiring blending are introduced with ever-increasing frequency. Nevertheless, it is highly risky to extend the procedures established for the existing mixtures and available mixers to any mixture of new particulate materials and/or any novel mixers. 5. MIXERS AND MIXING SYSTEMS
Powder or particulate materials are homogenized by mixers in which one or more of the mixing mechanisms occur, as outlined in the preceding section. Mixers for particulate materials and powder can be divided into 3 classes: batch, continuous, and semi-batch (or semicontinuous), similar to almost any other processing equipment. Mixers can also be classified logically as active and passive; the former is equipped with moving parts while the latter is not [5]. Listed below are various commercial mixers with the predominating mechanisms given in square brackets. 5.1. Batch Mixers
Typical batch mixers include: planetary mixer [shear]; blade (Helen) mixer [convection; shear]; blade (double-arm) mixer [convection; shear]; paddle (pan-type) mixer [shear]; paddle (Marion) mixer [shear; diffusion]; tumbler (drum) mixer [shear; diffusion]; tumbler (double-cone) mixer [shear]; tumbler (twin-shell) mixer [shear]; Muller mixer [shear]; mill mixer [shear]; ribbon mixer [convection; diffusion]; vertical spiral mixer [convection; diffusion]; Nautal mixer [convection; diffusion]; Banbury mixer [shear]; airmix mixer [convection; diffusion]; Lodige mixer [shear; diffusion]; gravity mixer [convection]; and vibratory mixer [convection; shear]. 5.2. Continuous Mixers
Typical continuous mixers include: Farrel mixer [convection; shear]; zig-zag mixer [convection; shear]; fluidized-bed mixer [convection; shear; diffusion]; spouted-bed mixer [convection; shear; diffusion]; and packed-bed mixer [convection; shear; diffusion]. 5.3. Semi-Batch (Semi-Continuous) Mixers
Many of the batch mixers can be modified to operate in the semi-batch (semi-continuous) mode by feeding the material to be blended continuously or by withdrawing the blended material continuously. By the same token, many of the continuous mixers can be made to operate in the semi-continuous or semi-batch mode. The mixers described in this section will probably be modified for specific applications. The modifications are accomplished mainly by changing the geometry of housings or shape and size of agitating devices as well as type and size of packings. For example, static or motionless mixers can serve as packings for packed-bed continuous mixers or internals of tumbler mixers (see, e.g., Chen et al. [18, 19], Gyenis et al. [20]). The mixers can also be modified by combining the
656 features of two or more mixers. For instance, various vibrating devices can be attached to tumbler, fluidized-bed or packed-bed mixers. As expected, any modification tends to alter the extent of contribution by the three mixing mechanisms. It is worth emphasizing that powder and particulate materials can be mixed without using selfstanding mixers. Convective mixing of batches of materials not in containers can be carried out by switching the positions or locations of different portions of the materials within the batches by means of conveyors, haulers or other transport tools. To attain the desired homogeneity, this convective mixing is usually followed by shear and diffusive mixing induced by agitating the batches mechanically or manually. The scenario described is common in the mining and construction industries and in our daily activities ranging from the preparation of food in our home kitchens to that of samples in our laboratories. As such, it is not the mixing by mixers in the conventional sense; instead, the mixing is accomplished by mixing operations. Engineering handbooks (see, e.g., [ 1]), extensive review articles (see, e.g., [2, 7], Weidenbaum [21 ]), and the monograph by Kaye [5] on solids mixing provide comprehensive information on solids mixers and blenders. Detailed information on individual mixers or classes of mixers can be found in articles in technical and trade journals; academic theses and dissertations; and commercial publications by the makers of mixers. Many of the technical articles, theses, and dissertations are cited in the extensive reviews mentioned earlier; they can also be identified by searching through publications such as Chemical Abstracts and Engineering Indexes. 6. CONCLUDING REMARKS This overview has hardly touched several important topics of current interest pertaining to the mixing of particulate materials and blending of powder. Such topics include the formation of ordered or partially ordered mixtures and prevention of segregation due to percolation in preparing mixtures capable of maintaining specified compositions by various means (see, e.g., [5], [7]). These mixtures are often involved when medicines are produced in the form of pills, tablets, and controlled release devices as well as when advanced materials, e.g., composites and semiconductors, are manufactured. Numerous research problems of practical and industrial importance and of theoretical and academic interest await solution in the preparation, characterization and application of ordered mixtures and composition-stable mixtures. Scant attention is paid here to these subjects because they are regarded as being outside the scope of bulk-solids mixing in the conventional sense, which usually is concemed with mixing and blending of free-flowing particles and powder in relatively large quantities. It is worth reiterating that much remains to be done to expand our understanding of the mechanisms of solids mixing and to estimate their rates; to characterize the quality of resultant mixtures; to design and scale-up mixers; and to determine suitable operating conditions of the mixers. To accomplish these, we should resort increasingly to modem paradigms such as those based on fractals, chaos, computer-aided particle simulation, and expert systems (see, e.g., Kaye [5], Fan et al. [7, 22], Ysuji et al. [23]). For brevity, no attempt has been made to cite every original or primary source of information or data. Nevertheless, every secondary source cited - monographs, handbooks, expositional review articles - includes a compendium of the original references.
657 REFERENCES
1. R.H.. Perry, D.W. Green and J.O. Malony (editors), Perry's Chemical Engineers Handbook, 7 th ed., McGraw Hill, New York, pp. 19.7-19.23, 1997. 2. L.T. Fan, S.J. Chen and C.A. Watson, Solids Mixing, in: Annual Review of Ind. Eng. Chem. 62 pp. 53-69, 1970. 3. J.A. Hersey and P.C. Cook, A Homogeneity of Pharmaceutical Dispersed Systems, J. Pharmacy and Pharmacology 26, pp.126-133, 1974. 4. N.A Orr and E. Shotton, The Mixing of Cohesive Powders, Chem. Eng. 269, pp. 12-19, 1973. 5. B.H.Kaye, Powder Mixing, Chapman & Hall, London, 1997. 6. J.N. Staniforth, Ordered Mixing of Spontaneous Granulation, Powder Technol. 45, pp.73-77, 1985. 7. L.T. Fan, Y.M. Chen and F.S. Lai, Recent Developments in Solids Mixing, Powder Technol. 61, pp. 255-287, 1990. 8. B.H. Kaye, A Random Walk through Fractal Dimensions, VCH Publishers, Weinheim, Germany, 1989. 9. P.M.C. Lacey, The Mixing of Solid Particles, Paper Read before the Graduates and Students of the Institution of Chemical Engineers, 21, pp. 53-59, February 26, 1943. 10. P.M.C. Lacey, Developments in the Theory of Particle Mixing, J. Appl. Chem. 4, pp. 257-268, 1954. 11. L.T. Fan and R.H. Wang, On Mixing Indices, Powder Technol. 11, pp. 27-32, 1975. 12. M.P. Poux, P. Fayolle, J. Bertrand, D. Bridoux, and J. Bousquet, Powder Mixing: Some Practical Rules Applied to Agitated Systems, Powder Technol. 68, pp. 213-234, 1991. 13. J. Boss, Mieszanie Materialow Ziarnistych (Polish), P.W.N. Warszawa- Wroclaw, pp.62-65, 1987. 14. J.R. Too, L.T. Fan, F.S. Lai, Mixtures and Mixing of Multicomponent Solid Particles-A Review, J. Powder & Bulk Solids Technol. 2, pp. 2-8, 1978. 15. J.M. Ottino, C.W. Leong, H. Rising and P.D. Swanson, Morphological Structures Produced by Mixing in Chaotic Flows, Nature 333(6172), pp. 419-25, 1988. 16. J.M. Ottino, The Mixing of Fluids, Scientific American 258(1), pp. 56-67, 1989. 17. J.M. Ottino, The Kinematics of Mixing." Stretching Chaos and Transport, Cambridge University Press, Cambridge, 1990. 18. S.J. Chen, L.T. Fan and C.A. Watson, The Mixing of Solid Particles in a Motionless MixerA Stochastic Approach, AIChE J. 18, pp. 984-989, 1972. 19. S.J. Chen, L.T. Fan and C.A. Watson, Mixing of Solid Particles in Motionless Mixer--Axial Dispersed Plug Flow Model, I & EC Process Design and Development 12, pp. 42-47, 1973. 20. J.Gyenis, J. N6meth and J. Arva, Gravity Mixing of Solid Particle Systems in Steady-State Static Mixer Tubes, Swiss Chem. 13(5), pp. 51-55,1991. 21. S.S. Weidenbaum, Solids Mixing, in Advances in Chemical Engineering II, T.B. Drew and J.W. Hoopes, Jr. (editors), Academic, New York, 1958. 22. L.T. Fan, D. Neogi and M. Yashima, Elementary Introduction to Spatial and Temporal Fractals, Lecture Note in Chemistry, Springer, Berlin, 1991. 23.Y.T. Tsuji, T. Kawaguchi and T. Tanaka, Discrete Particle Simulation of Two-dimensional Fluidized Bed, Powder Technol. 77, pp. 79-87, 1993.
658 ACKNOWLEDGMENT The contribution of Dr. J. Gyenis is duly acknowledged: Some portion of this lecture is an excerpt from the manuscript of a monograph on solids mixing, of which he is a co-author. This is contribution No. 00-288-B, Department of Chemical Engineering, Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506.
647
Bulk-solids mixing: overview L.T. Fan Department of Chemical Engineering, Kansas State University Manhattan, Kansas 66506, USA A survey is given of various industrial applications of solids mixing and the role it plays in our daily lives. This is followed by a review of the history and current status of the technology of solids mixing. Finally, a proposal is presented for further development of this technology and concomitant fundamental research. 1. INTRODUCTION This overview focuses on both the fundamental and practical aspects of solids mixing. Particulate solids are regularly found in numerous chemical, agrochemical, pharmaceutical, food and other products. The majority of such products comprise a multitude of components whose homogeneous distribution over any part or the entire batch of each of these products is of the utmost importance. This homogeneous distribution, often accomplished by blending or mixing, obviously plays a significant role in bulk-solids handling and processing that are ubiquitous in diverse industries and in our daily lives. Various fundamental aspects of solids or powder mixing and the equipment and devices to effect it, therefore, need be explored to gain insight into the homogenization and its reverse, segregation, induced by mixing The importance of solids or powder mixing can be readily seen in laboratory manuals and handbooks in different disciplines of pure and applied sciences and in engineering, of which the most recent edition of Perry's Chemical Engineers' Handbook [1] is an example. Solids mixing or powder blending is frequently a stand-alone operation in various processes and technologies, e.g., in manufacturing feeds, insecticides, fertilizer, glass batches, packaged foods, plastics, and cosmetics. It is also frequently carried out simultaneously with other processes or operations, some of which include cooling or heating solid materials, e.g., cooling sugar or preheating plastics, prior to calendering; drying or roasting cereal grains and ores; carrying out heterogeneous chemical reactions; agglomerating powder; and surface coating or enlarging particles common in the preparation of pigments, dyes, pharmaceuticals, detergents, fertilizer, coated seeds and candy. Reducing the size of particles is usually accompanied by solids mixing. 2. CHARACTERIZATION OF MIXTURES The ability to characterize the object or system of interest is essential for any undertaking. Solids mixing is no exception; the object of interest here is a mixture of particulate materials.
648
2.1. Uniformity and homogeneity Among the characteristics of a mixture, uniformity or homogeneity in the composition is of primary importance because specifying the composition is one of the most crucial requirements for product quality. Nevertheless, the required uniformity in the composition of the mixture is application-dependent or object-oriented. For example, the composition of a human medicine must be strictly prespecified to ensure its effectiveness and safety. Thus, the range of deviations tolerable is narrowly defined. On the other hand, the deviations are of less concern for solid foodstuffs, e.g., dried vegetables and prepackaged soup ingredients. The deviations can be even greater for solids mixtures serving as building materials such as premixed concrete and plaster. Fan et al. [2] have defined a homogeneous mixture as a particle system in which the concentrations of all constituents are uniform throughout. Under ideal situations, the spatial distribution of the constitutive particles in a mixture of two components can be characterized such that all the particles of a component are regularly or evenly distributed among the particles of the other component in any part or direction of the mixture. This arrangement can be visualized readily if the sizes and numbers of the component particles are equal. A mixture with regularly arranged components, i.e., an ordered mixture, however, can be formed even if the sizes and numbers of the component particles are different. An ideally ordered mixture can be defined as a mixture where individual particles of a given component, whose number is equal to or less than the other component, are evenly dispersed in the matrix of the latter, and the distance between the particles of the former is identical in all directions. An ideally ordered multicomponent mixture can be defined analogously by identifying the key component and by lumping the remaining components as one component. As such, the particles of either component must be arranged according to a regular spatial pattern. In spite of this, not every mixture with a regular spatial pattern is necessarily an ideally ordered mixture because a striated arrangement of the components represents a regular pattern but it does not always correspond to the highest degree of mixedness attainable. Naturally, the particles of the component of interest can be regularly arranged variously to suit different classes of materials. For instance, ideally ordered states should be achieved or be closely approximated for controlled-drug delivery from the matrix of inert or support materials. To characterize the forms and extent of the approach to the ideally ordered states, several terms have come into being, examples of which are regimented, structured, layered, and incompletely or imperfectly ordered mixtures (see, e.g., Hersey and Cook [3], Orr and Shotton [4], Kaye [5], Staniforth [6], Fan et al. [7]). For an ideally ordered mixture, the degree of homogeneity measured on the basis of any mixing index must be the highest. For example, the standard deviation or variance of the sample concentrations must be zero or nearly zero in such a mixture: the edge effect and finiteness of the size and number of particles of any individual component in a sample prevent the variance to vanish. In practice, it is extremely difficult to obtain perfectly or ideally ordered mixtures of freely moving particles by ordinary mixing processes: any disturbance causing relative displacements of the particles would diminish the degree of homogeneity. At the same time, it is possible to generate almost ideally ordered, stable mixtures for specific purposes by applying unique methods such as agglomeration, coating, and micro-encapsulation, none of which, however, is a mixing process in the conventional sense. Contrary to an ideally ordered mixture, the distribution of particles of any component in a completely random mixture is totally randomly dispersed among the particles of the other components. This implies that if individual particles of any component were marked by an increasing or decreasing series of numbers prior to mixing, the
649 order of such numbers in any direction throughout the mixture would be totally random when the completely random state is attained. The probability of finding a particle of any given component in the completely random mixture is identical in every location and is equal to the global, volumetric ratio of this component. In his comprehensive works, Kaye [5, 8] has pointed out that in contrast to an ordered mixture, the composition within a random mixture can vary to a certain extent from site to site and from experiment to experiment. The extent of this inherent variation depends on the overall composition of the mixture and also on the "sample size" with which the local concentrations of the constituents are determined. Unlike the ideally ordered mixture, some particles of a given component in the random mixture are clustered; in other words, they are not separated from each other by the particles of the other components regardless of the mixture' s composition. In some of the numerical examples given for the characteristics ofthe randomly formed clusters in powder mixtures, Kaye [5,8] has emphasized that in evaluating the randomness of a solids mixture, the concentration of any component can be measured in terms of either the weight or volume fraction. The particles compete for space and are distributed along spatial coordinates in the bulk of the mixture; therefore, it is more meaningful to measure the volume fraction than the weight fraction. Two obstacles hinder the total randomization of the particles. One is attributable to the geometry of particles. If the component particles are significantly different in size, the smaller particles can be concentrated in some voidages among the larger ones, thereby causing the particles of different components to segregate and preventing them to be completely randomly distributed. The other obstacle is attributable to possible surface interactions, e.g., adhesion and electric static attraction. If the surface interactions are profound between the particles of different components, these particles tend to form partially ordered arrangements, thus reducing the possibility of forming a completely random mixture. The appreciable surface effects among the particles of the same component can cause segregation similar to those induced by the particlesize differences. Noticeable density or weight differences between the component particles can also lead to segregation. In general, the extent of segregation is greatly affected by the concentrations of components. 2.2. Degree of mixedness Any mixture at an intermediate state between the totally segregated and completely random states must be deemed an incomplete mixture irrespective of its initial state: it can be generated by mixing initially separated components or by spontaneous segregation of a completely random mixture. The actual state between these two extremes is characterized by the degree ofmixedness measured by various mixing indices. The term, "partially segregated mixture," applies to a mixture which is yet to be fully homogenized and to a mixture homogenized once but experienced subsequent segregation. In both cases, the terms, "scale of segregation" and "intensity of segregation," serve to measure the actual state between the completely mixed and totally segregated states. In a totally segregated mixture, the components are unequivocally separated from each other in different and distinct regions in a batch of particles. This is usually the situation prior to mixing. To form a mixture, e.g., a stratified one, a given component is fed or charged into the mixer in the form of layers separated by other components. The relative positions, sizes and numbers of these layers as well as the distances between them appreciably affect the attainable rate of mixing. The configuration of the layers can be characterized by the scale of segregation.
650 The degree of homogeneity of a mixture expressing the extent of approach to perfectness has often been estimated by the uniformity of sample concentrations. To illustrate this, the standard deviation of the concentrations of a key component in a mixture is given by
O" =
(1)
i=l
N
where o- is the standard deviation of the sample concentrations; xi, the concentration in the i-th sample; ~-, the average ofxi,s; and N, the number of samples yielded by dividing the entire batch of the mixture. In a perfectly homogeneous fluid mixture, cris zero; if the components are totally segregated, the value of o-is maximal. In a conventional mixing process, the maximum achievable degree of mixedness corresponds to a completely random arrangement of different particles. Based on the assumption that the particle size of an individual component is identical, Lacey [9,10] has demonstrated that the minimal possible standard deviation of sample concentrations for a binary mixture can be expressed as
O'r = ~/PQY/pP)
=I s
/'/p
(2)
where P is the overall proportion of the particles of a given component in the whole mixture, which is equal to the average concentration, ~-, and np is the number of particles in each sample. In principle, the values of the standard deviation even lower than o;,.can be achieved by regularly arranging the particles, i.e., by forming an ideally ordered mixture. In practice, this can only be realized by means of unique processes such as surface adhesion or agglomeration, as discussed earlier. When the particles can move relative to each other without much hindrance, this ideally ordered arrangement is unstable. Naturally, the value of o-for an incomplete mixture is greater than ~.. The greater the value of the standard deviation, the closer the mixture to the totally segregated state. The standard deviation for the totally segregated mixture, denoted by Cro,is maximal; it depends on the average concentration of the key component of interest as shown below. o 0 = x/g0 - 2)
(3)
During mixing, the standard deviation of an incomplete mixture must be in the range between the two extremes given by Eqns. 2 and 3. It is not always the case for an ordered mixture because the lower bound, as characterized by the standard deviation of the completely randomized mixture, can be exceeded by such a mixture. Measuring the quality of the incomplete mixture during mixing, therefore, is crucial in controlling and optimizing the process. It is frequently impossible, however, to evaluate the quality by measuring the concentrations in all parts of the mixture, i.e., to determine the standard deviation from the entire sample. This means that uncertainty arises in estimating the value of the standard deviation due to the finiteness of the number of spot samples, n, which gives the sample standard deviation, s, defined in the following.
651
I Z~_,n-1(x, _~)2 S=
n-1
(4)
In contrast to a fluid mixture, the quality of a solids mixture is difficult to determine because of the discontinuous nature of the particulate systems as well as the finiteness of particle and sample sizes: The sample concentrations and their standard deviations can be affected by the error caused by the propensity of any component' s particle or particle cluster to straddle the boundaries of the sample containing it. It follows that the measured standard deviation depends on the average concentration of the given key component as well as on the relative sample size. For a completely randomized mixture whose component particles are identical in size and density, this uncertainty can be determined mathematically. For other cases, this is much more complicated. To account for the effects of the sample number and size, the general rule is to take a sufficient number of samples from well-distributed points in representative regions of the mixture. Each sample must also contain a sufficient number of particles; such a number can be determined from practical points of view.
2.3. Mixing indices The standard deviation of sample concentrations relative to the mean composition, given by Eqn. 4, is the most common indicator for characterizing the mixture quality. Based on the standard deviation, s, or on the variance, s 2, various mixing indices have been defined, the majority of which varies between zero and unity as the mixture quality changes from the totally segregated state to the completely randomized or ideally ordered state. Fan and Wang [ 11 ] and Poux et al. [12] have reviewed and compiled over 30 mixing indices, and Boss [13] has collected nearly 40 mixing indices based on the concepts of sample variance or standard deviation and has determined the interrelations among these indices. The influence of the sample size is taken into account in defining some of these mixing indices. The enormous diversity of mixing indices due to the variety and structural complexity of solids can be a source of confusion. Consequently, a single, unambiguous mixing index suitable for all applications is difficult to achieve. An overview is provided here on the relations among the important properties of mixtures and various mixing indices to aid selection of an index appropriate for a specific situation. Fan and Wang [11] as well as Boss [13] have compared a number of mixing indices and derived conversion formulas among them. Table 1 lists some of the frequently adopted mixing indices in terms of the statistical analysis of sample concentration. Note that certain indices are affected by the sample size. As such, a value of zero or unity for one of the two extreme states of the mixture quality can be obtained only if the sample size is sufficiently large. The sensitivities of mixing indices to the concentration variance or standard deviation differ at various stages of mixing; hence, it is necessary to compare their behavior. It is imperative that the quality of multi-component mixtures be evaluated and controlled because the components may behave differently at specific periods of mixing. This implies that at a given stage, one of the components may be well homogenized, but simultaneously, other components may continue to be partially segregated. Thus, the mixture as a whole does not meet the necessary homogeneity specification. The majority of mixing indices defined for multicomponent mixtures is based on the concentration variance as summarized by Too et al. [14].
652 3. M E C H A N I S M S AND MODELS OF SOLIDS MIXING
The particulate materials in solids cause mixing to be complex: The particles in such materials are small but finite in size; moreover, they are isolated from each other, i.e., they are discrete. These characteristics render particulate materials mesoscopic. In general, the behavior of mesoscopic materials is describable neither by the firmly established laws of continuum mechanics valid for macroscopic materials, e.g., steel beams or large ice cubes, nor by the wellknown principles of statistical mechanics applicable to microscopic materials, e.g., air. Any attempt to rigorously portray the motion of a countless number of interacting particles in a mixture by particle dynamics would be futile, as evidenced by the difficulty of portraying the motion of interacting particles as few as three. Although efforts have been made to circumvent the complexity involved in establishing the continuum mechanical, statistical mechanical or particle dynamical description of solids mixing, these efforts have focussed on the behavior of particles in a mixture in specific types of mixers through the experimental determination of the controlling mechanism or mechanisms of mixing or its opposite, segregation. As will be delineated later, segregation is often unavoidable in mixing particulate materials. 3.1. Mechanisms
Similar to fluid mixing, three major mechanisms have been identified for solids mixing convective, shear and diffusive mixing. Convective mixing involves bulk movement or gross displacement of particles within the mixture. Shear mixing is induced by sliding displacement of successive layers of particles. Diffusive mixing is caused by the random motion of particles. Generally, the attainable rate of homogenization (mixing) is greatest with convective mixing and least with diffusive mixing, whereas the degree of local homogenization attainable is greatest with diffusive mixing and least with convective mixing. Another mechanism, chaos, has been identified lately as contributing substantially to the mixing of particles and powder. Chaotic mixing of these and other materials exhibits highly complex patterns. Nevertheless, such mixing does not lead to random dispersion of particles: Chaos is a deterministic phenomenon. Hence, chaotic mixing results in irregularly interwoven and interacting convective and shear mixing. It is common knowledge that the mixing of solids mixtures is frequently accompanied by demixing (segregation) of the particulate components being mixed. Segregation does not occur when the particulate components have identical physical properties and geometrical characteristics but differ only in chemical composition. Segregation is caused by the percolation of a particulate component among the interstices of the remaining component or components. The propensity for different particles to segregate is strictly due to the fact that these particles are mesoscopic: the smaller particles tend to drop or percolate through voids ifthe size of the voids is sufficiently large or enlarged temporarily by mechanical vibration and/or aeration. The trend to segregate is magnified by the increase in the density of smaller particles over the larger particles; it is also affected by the differences in shape and surface characteristics. 3.2. Models
Sundry mathematical models can be easily conceived for each mechanism of mixing or segregation in the light of the number of methods available. Obviously, these models are too numerous to list individually; therefore, they are only classified broadly below.
653 Table 1 Some important mixing indices based on the variance or standard deviation of sample concentration.*
No.
Equation 0-
1
Value in the totally segregated state or=or 0
Value in the ideally ordered mixture cr=O
Value in the completely random mixture ~=~,.
2
M 1 -
1
0
0"02
i ;7
(3"
2
M2-
1
0
~o
47;,
0-2
3
M3=l-~
0-02
0
1
1 1-~7
O"
4
M4 = 1 - ~ 0-0
0
1
! l--~n
0
n n-1
1
0-02 - O . 2
5
Ms -
6
M6-
o'02 -0-r2 cro - o cr0
-o r
log cr0 7
-
0
"s/-s ~n 1
log cr
M7 = log o"0 - log crr
0
oO
0-2 M8 - -2
1-~
0
x
1
y
1
!-2 .u ;7
0-
9
Mg-_
* For the original sources, refer to Fan and Wang [ 11 ] and Boss [ 13].
Deterministic models. Analogous to fluid mixing, the deterministic mathematical models of solids mixing are derived from the continuity (mass conservation) of a key component being mixed in a certain volume element in the mixer. If this volume element is finite in size, a lumped model is obtained, and if it is infinitesimally small, a continuous model. Under steady-state conditions prevailing in a continuous mixer, the lumped model manifests itself as a set of difference equations, and the continuous model, a set of ordinary differential equations; naturally, a combined or hybrid model is expressed in the form of a set of differential-difference equations. The corresponding mathematical expressions under unsteady-state conditions prevailing in a batch mixer are sets of ordinary differential equations, partial differential equations, and ordinary differential-partial differential equations.
654 As mentioned earlier, chaos is a deterministic phenomenon in spite of the fact that it is complex and apparently irregular. In fact, a mathematical expression of chaos must comprise at least three nonlinear ordinary differential equations. As a result, it is logical for any mathematical model of chaotic mixing to comprise three or more ordinary differential equations. A deterministic mathematical model of segregation, essentially induced by particle percolation, can probably be derived by regarding solids mixing as a reversible process whose forward and reverse steps are homogenization and segregation, respectively. This is similar to a reversible chemical reaction. Stochastic models. It should be obvious by this time that solids mixing is an extremely complex phenomenon influenced by parameters ranging from physical, geometrical and surface characteristics of the particles involved to the design features and operating conditions of the mixer adopted. By definition, no mechanistic model of solids mixing takes into account "all" parameters. The failure to consider any of the influencing parameters gives rise to the internal noises in the behavior of the mixer of interest; the performance or experimental data of the mixer invariably exhibit appreciable scattering from the model. A simplistic way of taking into account the fluctuations of a mixer's performance or behavior is to add a term representing the noise or fluctuations to any of the deterministic models derived for the mixer. This yields a stochastic model of the mixer as the Langevin equation ubiquitous in the literature of turbulence and communication theories. Unfortunately, the noise term in any Langevin equation represents fluctuations induced by external forces that are not inherent in the system. Another approach to the stochastic modeling of solids mixing is to exploit the theorems and algorithms of stochastic processes. These theorems and algorithms make it possible to explicitly incorporate from the outset only the inherent fluctuations into modeling in terms of the probabilities of the system to be at various states. Markov processes, a highly applicable class of stochastic processes, have been used successfully by this author as well as by others in modeling solids mixing in a variety of mixers. Those wishing to explore the mechanisms and mathematical modeling of solids mixing can consult the review articles by Fan et al. [2, 7] as well as the monograph by Kaye [5]. Those with a particular interest in the chaos aspects of solids mixing will benefit from reading the exposition on chaos in fluid mixing by Ottino et al. [ 15] and Ottino [ 16, 17]. 4. DESIGN AND SCALE-UP OF MIXERS
The design and scale-up of mixers appear to be the least developed among various aspects of solids mixing because of the mesoscopic nature of materials being homogenized, as mentioned previously. In practice, the design of such mixers and blenders has generally been carried out heuristically; moreover, the mixer or blender has been scaled up mainly through successive experimentation with ever larger mixers or blenders, including the bench-scale, semi-pilot-scale and pilot-scale units. The paucity of publications attests to the difficulty in the design and scaleup procedures for particle mixers or powder blenders (see, e.g., [7]). The principle of similarity is one of the well-known procedures for the scale-up of various process systems including mixers. A mixer as simple as the tumbling mixer has three similarities: geometric, kinematic and dynamic similarities. The prototype and scaled-up mixers are geometrically similar if the ratios of their corresponding linear dimensions are constant; they are kinematically similar if the ratios of the corresponding points' velocities are constant; and they are dynamically similar if the ratios of the corresponding points' forces are constant. Two general
655 methods are available for deriving the similarity criteria in terms of the dimensionless numbers composed of parameters significantly infuencing the performance of the mixers. These methods are the dimensionless analysis and the normalization of the governing equations, both of which have been widely applied in other fields. In recent years, substantial progress has been made in establishing algorithmic approaches to the systematic design and scale-up of solids mixers on the basis of the similarity criteria identified for them. In spite of this, heuristics will continue to play a vital role because new types of mixers are constantly being conceived, and novel or exotic particulate materials requiring blending are introduced with ever-increasing frequency. Nevertheless, it is highly risky to extend the procedures established for the existing mixtures and available mixers to any mixture of new particulate materials and/or any novel mixers. 5. MIXERS AND MIXING SYSTEMS
Powder or particulate materials are homogenized by mixers in which one or more of the mixing mechanisms occur, as outlined in the preceding section. Mixers for particulate materials and powder can be divided into 3 classes: batch, continuous, and semi-batch (or semicontinuous), similar to almost any other processing equipment. Mixers can also be classified logically as active and passive; the former is equipped with moving parts while the latter is not [5]. Listed below are various commercial mixers with the predominating mechanisms given in square brackets. 5.1. Batch Mixers
Typical batch mixers include: planetary mixer [shear]; blade (Helen) mixer [convection; shear]; blade (double-arm) mixer [convection; shear]; paddle (pan-type) mixer [shear]; paddle (Marion) mixer [shear; diffusion]; tumbler (drum) mixer [shear; diffusion]; tumbler (double-cone) mixer [shear]; tumbler (twin-shell) mixer [shear]; Muller mixer [shear]; mill mixer [shear]; ribbon mixer [convection; diffusion]; vertical spiral mixer [convection; diffusion]; Nautal mixer [convection; diffusion]; Banbury mixer [shear]; airmix mixer [convection; diffusion]; Lodige mixer [shear; diffusion]; gravity mixer [convection]; and vibratory mixer [convection; shear]. 5.2. Continuous Mixers
Typical continuous mixers include: Farrel mixer [convection; shear]; zig-zag mixer [convection; shear]; fluidized-bed mixer [convection; shear; diffusion]; spouted-bed mixer [convection; shear; diffusion]; and packed-bed mixer [convection; shear; diffusion]. 5.3. Semi-Batch (Semi-Continuous) Mixers
Many of the batch mixers can be modified to operate in the semi-batch (semi-continuous) mode by feeding the material to be blended continuously or by withdrawing the blended material continuously. By the same token, many of the continuous mixers can be made to operate in the semi-continuous or semi-batch mode. The mixers described in this section will probably be modified for specific applications. The modifications are accomplished mainly by changing the geometry of housings or shape and size of agitating devices as well as type and size of packings. For example, static or motionless mixers can serve as packings for packed-bed continuous mixers or internals of tumbler mixers (see, e.g., Chen et al. [18, 19], Gyenis et al. [20]). The mixers can also be modified by combining the
656 features of two or more mixers. For instance, various vibrating devices can be attached to tumbler, fluidized-bed or packed-bed mixers. As expected, any modification tends to alter the extent of contribution by the three mixing mechanisms. It is worth emphasizing that powder and particulate materials can be mixed without using selfstanding mixers. Convective mixing of batches of materials not in containers can be carried out by switching the positions or locations of different portions of the materials within the batches by means of conveyors, haulers or other transport tools. To attain the desired homogeneity, this convective mixing is usually followed by shear and diffusive mixing induced by agitating the batches mechanically or manually. The scenario described is common in the mining and construction industries and in our daily activities ranging from the preparation of food in our home kitchens to that of samples in our laboratories. As such, it is not the mixing by mixers in the conventional sense; instead, the mixing is accomplished by mixing operations. Engineering handbooks (see, e.g., [ 1]), extensive review articles (see, e.g., [2, 7], Weidenbaum [21 ]), and the monograph by Kaye [5] on solids mixing provide comprehensive information on solids mixers and blenders. Detailed information on individual mixers or classes of mixers can be found in articles in technical and trade journals; academic theses and dissertations; and commercial publications by the makers of mixers. Many of the technical articles, theses, and dissertations are cited in the extensive reviews mentioned earlier; they can also be identified by searching through publications such as Chemical Abstracts and Engineering Indexes. 6. CONCLUDING REMARKS This overview has hardly touched several important topics of current interest pertaining to the mixing of particulate materials and blending of powder. Such topics include the formation of ordered or partially ordered mixtures and prevention of segregation due to percolation in preparing mixtures capable of maintaining specified compositions by various means (see, e.g., [5], [7]). These mixtures are often involved when medicines are produced in the form of pills, tablets, and controlled release devices as well as when advanced materials, e.g., composites and semiconductors, are manufactured. Numerous research problems of practical and industrial importance and of theoretical and academic interest await solution in the preparation, characterization and application of ordered mixtures and composition-stable mixtures. Scant attention is paid here to these subjects because they are regarded as being outside the scope of bulk-solids mixing in the conventional sense, which usually is concemed with mixing and blending of free-flowing particles and powder in relatively large quantities. It is worth reiterating that much remains to be done to expand our understanding of the mechanisms of solids mixing and to estimate their rates; to characterize the quality of resultant mixtures; to design and scale-up mixers; and to determine suitable operating conditions of the mixers. To accomplish these, we should resort increasingly to modem paradigms such as those based on fractals, chaos, computer-aided particle simulation, and expert systems (see, e.g., Kaye [5], Fan et al. [7, 22], Ysuji et al. [23]). For brevity, no attempt has been made to cite every original or primary source of information or data. Nevertheless, every secondary source cited - monographs, handbooks, expositional review articles - includes a compendium of the original references.
657 REFERENCES
1. R.H.. Perry, D.W. Green and J.O. Malony (editors), Perry's Chemical Engineers Handbook, 7 th ed., McGraw Hill, New York, pp. 19.7-19.23, 1997. 2. L.T. Fan, S.J. Chen and C.A. Watson, Solids Mixing, in: Annual Review of Ind. Eng. Chem. 62 pp. 53-69, 1970. 3. J.A. Hersey and P.C. Cook, A Homogeneity of Pharmaceutical Dispersed Systems, J. Pharmacy and Pharmacology 26, pp.126-133, 1974. 4. N.A Orr and E. Shotton, The Mixing of Cohesive Powders, Chem. Eng. 269, pp. 12-19, 1973. 5. B.H.Kaye, Powder Mixing, Chapman & Hall, London, 1997. 6. J.N. Staniforth, Ordered Mixing of Spontaneous Granulation, Powder Technol. 45, pp.73-77, 1985. 7. L.T. Fan, Y.M. Chen and F.S. Lai, Recent Developments in Solids Mixing, Powder Technol. 61, pp. 255-287, 1990. 8. B.H. Kaye, A Random Walk through Fractal Dimensions, VCH Publishers, Weinheim, Germany, 1989. 9. P.M.C. Lacey, The Mixing of Solid Particles, Paper Read before the Graduates and Students of the Institution of Chemical Engineers, 21, pp. 53-59, February 26, 1943. 10. P.M.C. Lacey, Developments in the Theory of Particle Mixing, J. Appl. Chem. 4, pp. 257-268, 1954. 11. L.T. Fan and R.H. Wang, On Mixing Indices, Powder Technol. 11, pp. 27-32, 1975. 12. M.P. Poux, P. Fayolle, J. Bertrand, D. Bridoux, and J. Bousquet, Powder Mixing: Some Practical Rules Applied to Agitated Systems, Powder Technol. 68, pp. 213-234, 1991. 13. J. Boss, Mieszanie Materialow Ziarnistych (Polish), P.W.N. Warszawa- Wroclaw, pp.62-65, 1987. 14. J.R. Too, L.T. Fan, F.S. Lai, Mixtures and Mixing of Multicomponent Solid Particles-A Review, J. Powder & Bulk Solids Technol. 2, pp. 2-8, 1978. 15. J.M. Ottino, C.W. Leong, H. Rising and P.D. Swanson, Morphological Structures Produced by Mixing in Chaotic Flows, Nature 333(6172), pp. 419-25, 1988. 16. J.M. Ottino, The Mixing of Fluids, Scientific American 258(1), pp. 56-67, 1989. 17. J.M. Ottino, The Kinematics of Mixing." Stretching Chaos and Transport, Cambridge University Press, Cambridge, 1990. 18. S.J. Chen, L.T. Fan and C.A. Watson, The Mixing of Solid Particles in a Motionless MixerA Stochastic Approach, AIChE J. 18, pp. 984-989, 1972. 19. S.J. Chen, L.T. Fan and C.A. Watson, Mixing of Solid Particles in Motionless Mixer--Axial Dispersed Plug Flow Model, I & EC Process Design and Development 12, pp. 42-47, 1973. 20. J.Gyenis, J. N6meth and J. Arva, Gravity Mixing of Solid Particle Systems in Steady-State Static Mixer Tubes, Swiss Chem. 13(5), pp. 51-55,1991. 21. S.S. Weidenbaum, Solids Mixing, in Advances in Chemical Engineering II, T.B. Drew and J.W. Hoopes, Jr. (editors), Academic, New York, 1958. 22. L.T. Fan, D. Neogi and M. Yashima, Elementary Introduction to Spatial and Temporal Fractals, Lecture Note in Chemistry, Springer, Berlin, 1991. 23.Y.T. Tsuji, T. Kawaguchi and T. Tanaka, Discrete Particle Simulation of Two-dimensional Fluidized Bed, Powder Technol. 77, pp. 79-87, 1993.
658 ACKNOWLEDGMENT The contribution of Dr. J. Gyenis is duly acknowledged: Some portion of this lecture is an excerpt from the manuscript of a monograph on solids mixing, of which he is a co-author. This is contribution No. 00-288-B, Department of Chemical Engineering, Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506.