Mixing and Segregation in Powders : Evaluation , Mechanisms and Processes †

Mixing in powders generally results from relative motion of groups of particles ‒ convective mixing ‒ or of individuals ‒ dif fusive mixing. Segregation or demixing occurs when the motion of individual particles is biased according to their particular characteristics ‒ size, shape, composition etc. In the absence of such bias, individual motion invariably leads to homogenization of the mixture. Relationships between mixing/segregation processes and the external and interparticle forces responsible for causing or opposing relative motion are reviewed. Specific examples of mixing and segregation in flow over surfaces, in rotating cylinders and other applications are described.


Introduction
Mixing of particulate solids is an important process in its own right, as in blending of the components of a mixed powder; it can also play a critical role in the performance of other particle processing operations such as grinding, granulation, classification and chemical treatment.In single-phase systems such as gases or miscible liquids, mixing proceeds spontaneously and irreversibly so that actions such as stirring are needed only to speed up the process.In contrast, solids mixing processes are neither spontaneous nor irreversible some kind of mechanical agitation is required for mixing to occur and a tendency for different kinds of particles to segregate is commonly observed.Segregation is typically described as a problem.While this is clearly appropriate for simple blending operations, there are cases where it is actually desirable by providing some degree of classification.Segregation due to agitation or flow of a powder is rarely an effective means of classification on its own but it can be a useful adjunct to other classification processes.For example, continuous screening op-erations commonly involve movement of a layer of vibrated particles over the screen surface.Segregation of coarse (oversize) particles towards the top of the layer can improve screening efficiency by reducing blockage of the screen apertures.Agglomeration processes take advantage of segregation of large agglomerates to promote preferential discharge of the product from balling drums or pans.
It is important to distinguish between the results of active segregation processes and those due simply to inadequate mixing.If some of the components of a mixture are added in agglomerated form, complete mixing requires break-up and redistribution of this material.Low-shear devices such as tumblers may be incapable of performing this action.Indeed, they may actually enhance segregation by encouraging agglomerate formation and growth.Another distinction can be made in the context of mixing as an element of some other process such as a continuous grinding operation.In liquid-phase reactors, "perfect" mixing refers to uniform distribution of the components throughout the vessel intimate mixing at the molecular level usually follows automatically.The latter is often not true for particle systems.Spatial uniformity may be achieved while the components remain in the form of clumps.However, uniformity of this kind may be quite sufficient to regard the reactor as fully mixed for estimating process parameters such as residence time distributions.This paper is intended as an overview of the subject rather than a comprehensive review of published work in the area.

Assessment of Mixture Homogeneity
A generally accepted procedure for evaluating the relative homogeneity of a particulate mixture is to take a number of samples, at random, and determine the variance of their composition: the more homogeneous the mixture, the lower the expected variance.Because of the discrete nature of the particles, the expected variance of a random mixture is ultimately limited by the chance of different kinds of particles, especially those of minor components, being included in a sample.Based on simple statistics, it has been shown 1) that the expected variance, σ 2 of the relative composition of samples containing N particles each taken from a completely random binary mixture is given by: where p is the overall number fraction of one of the components.The simple treatment has been extended to the more practical case of the expected variance in the mass fraction of a given component in samples of equal mass taken from a multi-component mixture [2][3][4][5][6] . Te equivalent to Equation 1 then becomes where (σ 2 m ) i is the expected variance of the mass concentration of component i in samples taken from a random mixture of all of the components; f i is the overall mass fraction of component i in the mixture; wi is the mass of a single particle of that component and w is the overall mean particle mass, defined by: An observed sample variance that is greater than the expected value as determined using Equations 1 or 2 implies that the mixture is heterogeneous (nonrandom).It is possible, however, for such a result to be obtained, by chance, for a random mixture.
Comparison of the ratio of the observed to expected variance using, for example, the simple F-test 7) gives an indication of whether or not the difference is significant statistically.Danckwer ts 8) used the terms micromixing and macromixing to distinguish between shor t-range homogeneity, i.e., at the individual particle level, and long-range variations within a batch of material.The expected, randomly mixed variance described above refers to micromixing.Since the observed variance also includes the effects of errors in sample assay, it is necessary increase the expected variance by using small samples to ensure that incomplete micromixing is not masked by such errors,.At the same time, however, the increase in the expected variance can mask the effects of small, but significant long-range variations.For example, in a study of axial mixing in a rotating cylinder 9) it was found, using samples of about 500 particles, that the observed sample variance became equal to the expected value after about 70,000 revolutions of the drum.At that stage, however, there remained a small but detectable and significant variation in composition along the length of the cylinder.
To detect both long-range and short-range variations, it is necessar y to sample at more than one level small samples to identify local clustering of components and larger samples to test for the long-range variations.Typically, the latter can be minimized by further mixing while the former may require alternative procedures such as increased shear.Procedures for characterizing particulate mixtures have been described in greater detail in a recent publication by the author 10) .

Mixing Mechanisms
Lacey 11) proposed the following basic mechanisms for solids mixing: convective mixing which involves the collective transfer of groups of particles from one location to another; diffusive mixing defined as the distribution of particles over a freshly developed surface and shear mixing due to slipping planes set up within the mixture.Further consideration, however, suggests that, for free-flowing powders, the fundamental mechanisms are simply convection essentially as defined by Lacey and dif fusion defined more broadly as the random motion of individual particles.Shear mixing can be regarded as a combination of convection and diffusion resulting from the redistribution of material across slipping planes.Shear is an important feature of the mixing process by providing the driving force for convection and diffusion.A third mechanism transfer between agglomerates is important for cohesive particles and is discussed below.Typically, diffusion is a rather slow process while convection proceeds rapidly.

Segregation Mechanisms
Micromixing by diffusion in a free-flowing powder requires that the motion of individual particles in response to agitation etc. be truly random and independent of the type of particle concerned.In other words, all particles in the system should have the same mobility.When there are differences in mobility among the components of a mixture, segregation becomes a possibility.Particle motion results from the effects of mechanical actions such as tumbling or stirring, and external forces, primarily gravity.Particle mobility is determined by the response to these forces and the resistance to motion due to other particles, their environment and the mixing device itself.In many ways, segregation in powders is analogous to differential settling of particles in a fluid.Convective mixing does not normally depend on individual particle mobility; the mixing effect tends to be more or less random and is unlikely to contribute directly to segregation.At the same time, however, the actions that lead to convective mixing generally promote individual motion which can cause segregation.Because of restricted mobility, segregation is usually less prevalent in cohesive powders than in free-flowing material.Whereas convection and diffusion are complementar y in mixing processes, they generally work in opposition in segregation.
Segregation is caused by differences in the basic particle characteristics which can be defined in terms of size, shape, composition and structure 10) .As far as segregation is concerned, structure the distribution of phases within a single particle is unlikely to cause differences in mobility and can probably be discounted.Particle size and shape affect mobility largely through their effect on the resistance to motion whereas density (determined by particle composition) is more likely to influence the actual motion resulting from applied forces.

Size Segregation
Variations in individual particle mobility with size typically lead to segregation in a bed of particles subjected to vibration or flow.However, the form of the resulting segregation can vary widely depending on the actual conditions.This has lead to a number of proposed mechanisms which are actually different manifestations of the same basic phenomenon: the ability of small particles to penetrate the bed more readily than coarser material.
Imposed vibration or particle-particle collisions in a flow stream cause some dilation of the bed and increased particle mobility.Gravitational forces lead to a net downward motion of the smaller particles which can readily pass through the voids in the expanded bed.This simple mechanism is especially important when the fine material is a minor component of the mixture and is commonly referred to as percolation or sieving 12,13,14) .Segregation can also occur, by the same basic mechanism, regardless of the overall composition of the mixture.If the coarse particles are a minority each surrounded by the finer components their ability to move upwards is constrained only by the weight of the overlying material while downward motion requires either compaction of the material below or the unlikely possibility of encountering a sufficiently large void.At the same time, the void created by upward motion of the particle is easily filled by percolation of adjacent fines, which prevents a return to its original position 15) .The net result is a ratcheting upwards of the coarser particles.This effect has been described as displacement segregation 12,13) or the "Brazil nut effect" 16) .
Size segregation is also widely obser ved when particles are poured onto a heap or in the filling of a container.Again the basic mechanism involved is the difference in the ability of small and large particles to penetrate a bed of material, although the result ap-pears to be quite different.In contrast to a vibrated or agitated bed, a heap is essentially static with flow occurring only in the surface layers.Particles flow over the surface until they become trapped in a surface void.Since the finer material is more likely to encounter a sufficiently large void, the result is that coarse particles travel further down the slope and accumulate around the lower edges of the heap leaving excess fines in the core.This effect has often been called rolling segregation 17) .
The term trajectory segregation 18) has been used to describe separation of coarse and fine particles projected from the end of a chute or fast-moving belt.
Retardation of the lateral motion by aerodynamic drag is greater for small than for large particles.Projection of the particles individually would certainly lead to segregation.In practice, however, material is normally projected as a fairly dense, continuous stream carr ying the adjacent air along with it.As a result, aerodynamic drag acts on the stream as a whole rather than on individual particles, thereby minimizing the segregation effect.Obser vation of segregation in such systems can probably be attributed to existing segregation, prior to projection.This effect was demonstrated by Augenstein 19,20) and is illustrated in Fig. 1.Mixtures of coarse and fine particles projected from chute surfaces were collected in a sampling grid to determine the distributions of horizontal displacement.For smooth chute surfaces, the range of displacements was found to be narrow, indicating a narrow range of velocities, and the composition was essentially uniform (Fig. 1a).When the surface was rough, on the other hand, a much broader range of displacements was observed and a progressive increase in the concentration of coarse particles with displacement was found (Fig. 1b).
The extent of size segregation is generally limited by packing considerations.In the case of a binar y mixture, segregation can be expected to lead to the development of two regions: one consisting essentially of the major component alone and the other containing both components under roughly optimum packing conditions.

Shape Segregation
Particle shape also affects mobility and differences are, therefore, a potential cause of segregation.It is generally agreed, however, that segregation by shape becomes significant only when the differences are substantial smooth spheres from irregular particles, flakes or needles from roughly isometric particles, for example 21,22) .In part, this may be an impression gained due to the difficulties involved in quantifying small dif ferences in shape.Tang and Puri 13) have noted that combinations of size and shape differences may lead to enhanced segregation.

Density Segregation
Differences in particle density are known to cause segregation, which might be anticipated from potential energy considerations.However, it is generally agreed that the effects are usually substantially less than those due to size dif ferences 23) .This may be partly due to the smaller range of densities found in particle systems a 5:1 range in density is considered to be large while 1000:1 is a quite common size range.The analogy to sedimentation is again appropriate here.The (gravitational) force acting on a particle is affected by both size and density while the fluid drag (resistance to motion) is independent of density.It is likely that the same is largely true for segregation in powders.As noted above, a primary cause of segregation is the relative ability to penetrate a bed of particles; increased density provides little advantage in this respect, especially in fairly compact systems such as vibrated or agitated beds or flow streams.A "push-away" mechanism 24) has been proposed in which a heavy particle is able to force its way between lighter components.Such an effect is unlikely to be significant in relatively compact beds since the weight of a single particle is generally much smaller than the force needed to shear the bed.
For highly dilated beds subject to vigorous agitation or vibration, and especially for fully fluidized beds, a mechanism of this kind may indeed be viable, leading to an accumulation of the denser component(s) in the lower regions of the bed 25) .These conditions could also have a significant effect on size segregation.By reducing the resistance to downward motion, a more open bed could actually reverse the segregation effect by allowing settling of the large particles.This could, in fact, account for some seemingly contradictory observations noted in the literature.

Segregation Due to Particle Interactions
In some circumstances interaction forces, either attractive or repulsive, can promote segregation.Selective agglomeration of certain components is an obvious example and could occur if particles of one component specifically attract particles of the same kind or repel other kinds.Typically, agglomeration is most prevalent among the fine, cohesive components of a mixture.Selective agglomeration is quite common in liquid suspensions but the close confinement of particles in dry or semi-dry powders is not generally conducive to selectivity.Most cases of segregation by clumping of fine components probably reflect incomplete micromixing due to low particle mobility, rather than a spontaneous segregation process.Segregation due to interaction forces is possible when the forces are strong enough, or of sufficiently long range to affect relatively large, mobile particles.Magnetic particles are an obvious example.Other kinds of forces can produce similar effects as was obser ved by this author.For a mixing study using particles differing only in color, a narrow sieve fraction (40×50 US mesh) of crushed dolomite was dyed using a commercial food color.Attempts to prepare uniform mixtures of these particles with the same size fraction of the same dolomite, but not dyed, for use as calibration standards proved to be remarkably unsuccessful.Even when a reasonably homogeneous mixture was obtained by careful stirring of small quantities, any slight disturbance, by tapping or shaking the container, led to an immediate visible segrega-tion.Apparently interaction forces, introduced by the surface coating, were sufficient to cause segregation of these relatively coarse (-0.35 mm) particles.In this particular study, the problem was circumvented by switching to a different color.

Role of Cohesion
By restricting the mobility of individual particles, cohesion generally serves to reduce the tendency for segregation to occur.However, cohesion in dry powders is normally associated with fine (e.g., < 20μm) particles.Other than perhaps acquiring a coating of fines, coarser particles in a mixture can retain their mobility and be subject to segregation.Furthermore, the additional restriction of fine par ticle mobility may actually compound the problem.Agitation of a cohesive powder commonly leads to the formation of agglomerates which readily become segregated from the bulk (fine) powder.This effect is widely exploited in granulation processes.
Cohesion due to the presence of liquid phases generally extends over a broader range of particle sizes than that due to fine components and, in some situations, may provide a means of minimizing segregation problems.

Mixing and Segregation Processes
Mixing of free-flowing powders can generally be regarded as proceeding through some combination of the diffusion and convection mechanisms, convection being primarily responsible for macromixing while intimate micromixing relies on diffusion.The relative contributions of these two depend mostly on the geometry and operating conditions of the mixing device but may also be influenced by powder characteristics.

Mixing in Rotating Cylinders
The simple horizontal drum mixer has been studied extensively.For non-segregating powders (equal size, shape and density), it has been established that axial mixing proceeds through essentially pure diffusion 9,11) while transverse mixing is dominated by convection with superimposed diffusion [26][27][28][29] . Abed of powder in a horizontal, rotating cylinder exists as two, fairly distinct regions: a "static" region in which particles follow the rotation with no relative motion and a "shear zone" where particles flow down the free surface of the bed.Conditions in the shear zone depend on the Froude number: Fr = Rω 2 /g where R is the cylinder radius, ω is the angular velocity and g is the acceleration due to gravity.At low rotational speeds (Froude numbers less than about 0.01), velocities in the shear zone are much higher than those in the static region so that the shear zone is thin and the free surface remains essentially planar.As the speed is increased, velocities in the static region increase relative to those in the shear zone, which are mostly determined by gravity; the zone increases in thickness and assumes a characteristic S-shape.With further increase in speed, cateracting occurs where particles are projected from the bed and follow a freeflight trajector y before re-entering the bed.At the critical speed, the centrifugal force acting on the layer of particles adjacent to the shell just balances that due to gravity (Rω 2 = g) and centrifuging occurs.The critical speed ωc is defined by: And corresponds to Fr = 1.Under steady flow conditions, continuity requires that a particle entering the shear zone at some axial location will, on the average, re-enter on the same transverse plane.Inter-particle collisions during flow down the surface cause the random lateral displacements that lead to axial diffusion 30,31) .Convection in the transverse plane results primarily from differences in the time taken for particles at different radial locations to complete one pass through the static region.Random displacement of particles during flow through the shear zone provides the diffusive component.

Axial Mixing
Because of the absence of any contribution from convection, axial mixing in a simple drum proceeds very slowly.Experimental values of the diffusion coefficient for particles of about 0.2 mm in size rotated at relatively low speeds in cylinders with diameters in the 5-10 cm range are typically found to be about 1 mm 2 /revolution 9,32,33) .
It has been shown 9) that, for a simple cylinder loaded end-to-end with equal quantities of two, essentially identical components, the expected sample variance due to incomplete macromixing can be estimated from: where D is the diffusion coefficient, N is the number of revolutions of the cylinder and L is its length.Using the above value of D, Equation 4 predicts that more than 20,000 revolutions would be required to reduce the variance from an initial value of 0.25 to 0.0025 for a cylinder 10 cm long.This required time would increase with the square of mixer length.
Experimental studies by Rao et al. 33) indicated that the diffusion coefficient generally decreases with increased filling of the cylinder and increased particle size and increases with increased rotational speed.
Higher diffusion coefficients at low filling levels were attributed to an increase in the number of passes through the shear zone per revolution, partially offset by the corresponding reduction in the path length in the zone.The particle size effect was thought to result from higher velocities in the shear zone due to higher dynamic angle of repose obser ved for finer particles.The increase with speed, even on a per revolution basis was also attributed to increases in the dynamic angle of repose.The approximate analysis presented by Rao et al. suggests that the diffusion coefficient (per revolution) should increase with the square of the cylinder diameter d for the same rotational speed.This would also be consistent with models used to describe axial transport through horizontal cylinders [34][35][36] . Fr conditions of dynamic similarity (constant Froude number) the diffusion coefficient (per time) should vary with d 3/2 .Unfortunately, systematic investigations of the relationship between axial diffusion and cylinder diameter appear not to have been carried out.
The addition of mixing aids in the form of relatively large balls (usually plastic or rubber) has been shown to increase axial mixing rates significantly [35][36][37] . Swahata 37) and Chaudhuri and Fuerstenau 38) found that the axial diffusion coefficient was increased by a factor of about 3 for mixing aid additions comprising about 10% of the mixer volume but became constant after further addition.Shoji et al. 39) attributed the leveling off to segregation of the balls towards the central region of the cylinder and correspondingly high diffusion rates in that region. As he ball filling was increased further, their concentration in the central region became more or less constant while their distribution along the cylinder became more uniform.
When the concentration of balls had become essentially uniform along the length of the cylinder, further ball additions again led to enhanced diffusion of the particles up to the condition where the volume of particles was just sufficient to fill the voids in the bed of balls.Abouzeid and Fuerstenau 40) demonstrated that the presence of mixing aids led to substantial reduction of size segregation of particles in a rotating drum.

Transverse Mixing
The convection process that dominates mixing in the transverse plane of a rotating cylinder results from the different rates at which particles at different radial locations circulate the particle bed 26,27) .As shown in Fig. 2, a particle at a radial distance r from the center of the cylinder travels through an angle 2θ in each pass through the static region.For a cylinder rotating at an angular velocity ω, the time to pass though the static region is equal to 2θ/ω.If the filling of the cylinder with powder is less than 50%, the angle and the circulation time decrease with decreasing radial distance from the center.Thus, particles close to the center circulate several times in the time taken for a particle adjacent to the shell to circulate once.
The result of the difference in circulation time is that particles in the bed undergo a progressive angular displacement leading to the transformation of an initial side-by-side arrangement of two components into a series of spiral bands as illustrated in Fig. 3.
The pattern shown in the figure was determined using the assumption that particles entering the shear zone on the upper part of the free surface reappear instantaneously at the same radial position on the lower part, i.e., the time spent in the shear zone is negligible 27) .Modifying the calculations by allowing a finite time for passage through the shear zone leads to a retardation of the development of the pattern but to only minor changes in its form.
As the filling of the cylinder approaches 50%, the difference in circulation times is reduced until, ideally, at exactly 50%, the free surface passes through the center of rotation, the time for passage through the static region becomes the same for all radial locations and no convective mixing occurs.The time taken for particles to pass through the shear zone may actually assist the mixing process for this case.Due to the greater distances involved, particles entering the zone close to the cylinder wall require a longer time than those near the center.This effect again leads to an angular displacement of particles at different radial positions, resulting in a spiral pattern similar to those shown in Fig. 3 but taking substantially longer to develop.The patterns shown in the Fig. 3 are the result of convection alone.In practice, random interchange between particles in different layers as they pass through the shear zone (diffusion) causes a blurring of the boundaries and a rapid approach to complete mixing.Hogg and Fuerstenau 27) presented a simplified model for the overall process, treating the diffusion component as equivalent to simple diffusion between the alternate bands produced by convection.More recently, the analysis has been extended to include a description of the actual mixing process in the shear zone 28,29) .Prigozhin and Kalman 29) presented a model which included an implicit assumption of perfect mixing in the shear zone which is essentially equivalent to the treatment by Inoue et al. 26) who assumed that particles entering the shear zone are redistributed at random as they reenter the rotating bed.Kharkhar et al, 28) developed a simplified model for flow in the shear zone and described mixing in the zone as a process of diffusion between layers.Their results suggest that particles are not completely mixed in the shear zone and that description as a diffusion process is appropriate.Each of the models agrees qualitatively with experimental obser vation but, because of the many approximations involved

Segregation in Rotating Cylinders
Both axial and transverse segregation are commonly obser ved in rotating cylinder systems 41) .
Transverse segregation usually proceeds rapidly and appears to be a necessary precursor to axial segregation 42) .

Transverse Segregation
Segregation of particles in the transverse plane of a horizontal rotating cylinder occurs during flow through the shear zone.Segregation by size, density and shape has been observed.Size segregation results from differences in resistance to flow.Large particles flow readily over the surface but encounter considerable resistance to penetration into the bed of flowing material.For smaller par ticles, on the other hand, trapping effects restrict their ability to flow over the surface but they are able to penetrate into and through voids in the bed.The combination of these two mechanisms leads to accumulation of the coarser components around the periphery of the rotating material and concentration of fines in a central core.Segregation by shape may also occur due to differences in resistance to flow.Again, it might be expected that particles that exhibit the greater resistance would accumulate in the central core, but experimental data in this area are ver y limited indeed.In the case of density segregation, the greater force of gravity acting on the denser particles causes a bias in the relative motion leading to accumulation of these particles in the central core 43) .The density effect may be enhanced at high rotational speeds where the shear zone becomes partially fluidized.
Transverse segregation occurs rapidly, approaching a steady state in a few revolutions of the cylinder [44][45][46] . I practice, however, variations in the distribution of particles across the transverse plane may continue for substantially longer times due to the effects of simultaneous, but much slower, axial segregation.The limiting segregation pattern seems to be relatively independent of rotational speed and volumetric filling of the cylinder, except when a "dead zone" exists for fillings greater than 50%.Some attenuation of the segregation may be expected at higher rotational speeds due to enhanced diffusive mixing.The size distribution of the powder is obviously an important factor.Most research in the area has been carried out on simple binary mixtures and has shown that even size differences as small as 20% can cause segregation.For such small differences, the concept of percolation seems inappropriate.Nevertheless, the differences in resistance to motion noted above still apply and lead to segregation.As might be expected, larger size differences promote faster segregation 45) .

Axial Segregation
Separation of dissimilar components in the form of alternate bands along the axis of the cylinder is commonly observed 41,47) .It appears that the process is a consequence of axial transport of particles caused by small variations in the inclination of the rotating bed due to friction at the end-walls and differences in internal friction of the mixture 41,[48][49][50] . Aial transport due to the end-effect is illustrated in Fig. 4 which shows schematically the expected trajectories of particles from different locations in the bed as they pass through the shear zone.The curved paths are the result of motion in the direction of steepest slope and are estimated using modification of the models developed for continuous flow through rotating cylinders 34,35,51) .As expected, particles originating from close to the outer shell of the cylinder are subject to greater axial displacement than those closer to the center of the bed.Since each of the trajectories shown returns to its original axial location, these estimates, based on independence of the individual paths, would predict no net axial displacement.However, the figure also shows that, especially for

Transverse Position
the outer layers, the paths of particles starting from different distances from the end-wall converge during passage through the shear zone.Obviously, therefore, the individual particle paths cannot really be independent, but there must nevertheless be a tendency for accumulation of those particles in that axial location.In order to maintain uniform loading of particles along the cylinder, migration of material away from that region must occur from all layers in the shear zone.Since there is less replenishment into the inner layers, the result is a reduction in the relative amount of that material.If transverse segregation has led to concentration of the finer (or heavier) particles in the central core and of the coarser (or lighter) material in the outer layers, the net effect of the axial transport described above is to form a band enriched in the outwardly segregating material close to but slightly removed from the end-wall, with corresponding regions of depletion on both sides.Axial variations in the composition of the mixture due to the end-effects or even random fluctuations may lead to corresponding changes in inclination of the rotating charge and consequent axial migration by similar mechanisms to those described above and the formation of alternating bands along the length of the cylinder.
It is clear that the mechanisms of axial segregation are complex and the convective processes described here can only represent part of the picture.Individual particle mobility can be expected to vary with size and shape and perhaps with density.It follows that dif fusivity should be similarly variable and could either promote or oppose segregation.In the case of segregation due to variations in inclination of the bed, Savage 52) has shown that both migratory convection and random diffusion fluxes could be roughly proportional to the concentration gradient.Mathematically, the process could then be represented by a simple diffusion equation with an effective diffusion coefficient given by the difference between contributions from convection and true diffusion.Under some circumstances, this could result in a negative effective diffusion coefficient meaning that the process would lead to enhanced rather than reduced concentration gradients, i.e., to segregation.Simulation of the process using a cellular automaton showed that combinations of random (diffusion) and biased (convection) interchange of particles in an initially uniform mixture can indeed result in band formation and exhibit the kinds of instabilities observed in real systems.
It has been suggested that band formation away from the ends of the cylinder may result from statistical fluctuations in bed depth or inclination 48,49,53,54) .However, it seems unlikely that such perturbations would be of sufficient magnitude and persistence to cause significant axial segregation.Propagation of disturbances from the end-walls appears to offer a more a more realistic explanation.

Mixing in Simple Flow
Both mixing and segregation in powders are generally promoted by flow of the material.Flow in powders can be considered to fall into one or other of two basic types: rapid flow as in the shear zone of a rotating cylinder or in the formation of a heap and slow flow as in internal motion during slow discharge of a hopper.Slow flow appears to be characterized by shearing at specific slip planes between other wise static layers of powder.In rapid flow, on the other hand, the process involves the existence of a continuous velocity gradient, akin to that in flow of a viscous fluid.
Flow over an inclined sur face such as a chute is a simple example of rapid flow.If the surface is smooth, motion occurs by sliding and there is little or no shearing within the powder.Since there is little relative motion of individual particles under such conditions, mixing effects are minimal.However, if the surface is roughened, to an extent equal to or greater than the surface of the powder bed itself, shear occurs within the powder and liquid-like flow occurs, provided of course the inclination of the surface exceeds the angle of friction between powder and surface.Particle-particle collisions due to shear provide a driving force for diffusional mixing.If diffusion is promoted by shear, it is reasonable to expect that the rates should increase with increasing shear rate 55,56) .A simple linear relationship has been proposed 56) of the form: where D is the diffusion coefficient, ∂v / ∂y is the velocity gradient perpendicular to the flow direction, Do and α are constants.Experiments were carried out to evaluate the mixing of sand with garnet powder as a tracer fed as initially separate layers, one on top of the other onto the surface of an inclined chute whose sur face was roughened using an attached layer of the same sand.Concentration profiles were determined for several distances of flow down the chute by a splitting procedure followed by magnetic separation of the components 56) .Velocity profiles were estimated using the trajector y procedure described by Augenstein and Hogg 57) .An example of the distribution of tracer through the depth of the bed is shown in Fig. 5 for increasing distance along the inclined surface.The solid lines are model predictions based on Equation 6with fixed values of the parameters D0 and α of 1.15×10 -2 cm 2 /sec and 7.03× 10 -3 sec.respectively.The agreement between model and experiment is very good.
The results shown in Fig. 5 are for the case where the tracer was added above the layer of sand.Because of the dependence of the diffusion coefficient on the velocity gradient, more extensive mixing should be expected if the tracer was added below the sand, due to the higher gradient closer to the chute surface.A comparison of the results for tracer addition above and below the sand layer is shown in Fig. 6.It is clear that placing the tracer in the high-shear region adjacent to the chute surface does indeed lead to enhanced mixing.Furthermore, the fact that the results could be fitted using the same values of the parameters D0 and α provides strong support for the applicability of Equation 6 to this system.It appears that agitation of the powder due to flow is sufficient to promote diffusive mixing and is further enhanced by velocity gradients within the flow stream.

Segregation in Simple Flow
The conditions that promote mixing during simple flow of non-cohesive powders also favor segregation of particles with different characteristics.Some results of experiments carried out on initially homogeneous mixtures consisting of 10% coarse (20×30 US mesh) and 90% fine (50×70 US mesh) sand are shown in Fig. 7.It appears that segregation occurs rapidly during the initial stages of the flow, leading to significant depletion of the coarse particles close to the chute surface and corresponding enhancement near the upper, free surface of the stream.Further downstream, the process continues more slowly.Hwang 20,58) proposed that the data were consistent with a simple extension to the diffusive mixing model described above with segregation assumed to occur through a constant upward convective velocity of the coarse particles.The solid lines shown in Fig. 7 were obtained using this model.The slowing down of the segregation process following the initially high rate is attributed to the effects of diffusive mixing which assume greater importance as the concentration gradient develops.Tracer fed as a thin layer at the bottom of the steam.Points are experimental; lines are calculated based on Equation 6.Data of Hwang 58) .
Fig. 7 Segregation of coarse (20×30 US mesh) and fine (50×70 US mesh) sand particles during flow of an initially homogeneous mixture over a rough surface.Data of Hwang 58) .Savage and Lun 59) conducted similar experiments using mixtures of polystyrene beads but with the finer particles as the minor component.These authors proposed two segregation mechanisms: a "random fluctuating sieve" mechanism and a "squeeze expulsion" mechanism.The former corresponds to size-dependent percolation while the latter refers to the ejection of particles, by mechanical forces, from one layer to another and was assumed to be sizeindependent with no preferential direction.This combination represents a more realistic treatment of the segregation mechanisms than the simple convection process described above, in particular by providing the means to maintain mass conservation across the bed.Analysis of the two processes provides some predictive capability for the effects of variables such as particle size ratio on segregation rates.Diffusive mixing was not included in the model, leading to the rather unlikely conclusion that the process ultimately results in complete separation of the coarse and fine components.Dolgunin et al. 60,61) conducted experiments similar to those described by Hwang et al., 20,58) .They concluded that the bulk density was highest in the central region of the flowing stream as were the concentrations of the larger or denser particles in the mixture.It should be noted that these experiments were restricted to relatively thin layers typically around 4 particle diameters which might account for the form of the bulk density profiles.The authors correlated the observed segregation patterns to the bulk density profile which may also explain the differences in their obser vations from those typically seen in other studies: migration of coarse particles towards the upper region of the flowing bed.

Fraction of Bed Thickness
Theoretical analyses have been used [62][63][64][65][66] to predict the effects of particle and system variables size, density, elasticity, chute inclination, etc., on segregation in flow over inclined surfaces. Dfferent procedures appear to give equivalent results and to be in qualitative agreement with experimental data.

Segregation in Heap Formation
Flow over the surface of a heap has much in common with flow down a chute.However, the process is more complex due to the finite extent of the surface in the flow direction and the continuous development of the surface itself.Because the surface is inclined at essentially the angle of repose of the powder, flow can readily become unstable and subject to temporary build-up of particles followed by avalanching.
It is generally agreed 22) that large par ticles encounter less resistance to flow over the surface than does finer material which is more easily trapped in surface voids.As a result the coarser components in a mixture can move further down the surface and tend to accumulate around the bottom of the heap.At the same time, build-up of the finer fractions in the upper regions of the surface can lead to periodic avalanching producing patches or streaks of fines within the accumulated coarse material.In the case of particles of similar size but different density, the heaver particles tend to be concentrated in the center of the heap with the lighter components towards the periphery 22,24) .The segregation patterns formed in binary mixtures have been shown to depend primarily on the relative sizes and densities of the components, their concentration and the velocity of impingement on the surface of the heap 12,22) .Studies of the formation of two-dimensional heaps from binar y mixtures with relatively small dif ferences in size have shown the development of a regular pattern of striations over the entire heap, except at the central core formed during the initial stages of the process [65][66][67] . Hwever, there is reason to suspect that the highly regular pattern may be an artifact of the materials used and the nature of the test procedure.
In practice, pouring generally leads to the formation of three-dimensional, roughly conical heaps.This adds further complication to the formation process and any resulting segregation 12) .Avalanching, due to local variations in slope, is unlikely to occur simultaneously around the heap leading to asymmetric radial distributions of the different particles.Flow over the surface is divergent, causing downstream thinning of the stream and possibly increased opportunity for trapping of particles.

Segregation in Bins and Hoppers
Segregation can be a serious problem in the storage of granular materials in bins or silos.While some segregation may occur across failure planes during flow through a bin 68) , such effects are probably small relative to surface segregation during filling and/or discharge.The problem of segregation is most pronounced in funnel-flow bins in which flow takes place through a channel formed within the mass of powder.Mass-flow bins, where material flows essentially en masse by sliding at the walls, provide little opportunity for segregation 69) .Segregation can occur for both types if the material is fed into the bin so as to form a heap leading, for example, to accumulation of coarse particles close to the walls, with a central core of finer particles.Segregation can also result from air entrainment of very fine particles during feeding 12) .This effect can cause deposition of fines close to the walls leaving excess coarse material in the central region.In a mass-flow bin, the segregation pattern remains more or less intact during passage through the bin but becomes recombined in the converging hopper section at the discharge.The situation is more complicated for funnel-flow bins.Since flow is confined to an internal channel, the initial discharge comes from the finesrich central core.As flow continues, a funnel-shape depression develops in the upper region of the bin contents.Coarse particles from the region close to the walls then begin to flow preferentially towards the center of the bin, reversing the segregation process.As a result, the discharge can switch from excess fines to excess coarse material.In many applications, bins are used to provide surge capacity with alternate or simultaneous feeding or discharge.The segregation patterns will then change according to whether feeding or discharge dominates and cycling of the discharge composition can occur.
Johanson 70) and Carson et al., 71) have described procedures for minimizing segregation in storage bins.A principal recommendation is to ensure that mass-flow conditions prevail.This should certainly be a priority in the design of new installations.However, despite more than 40 years of awareness of the advantages of such designs, the widespread use of funnel-flow bins remains.Conversion to or replacement by massflow systems is often considered to be economically impractical, especially for large-scale systems.Partial conversion can be achieved through the use of properly designed and placed inserts 70) .It should be noted that complete drainage should be avoided, even for mass-flow bins; a minimum filling level, somewhat above the converging hopper, should be maintained to ensure uniform flow of the material.Segregation during bin filling, due to heap formation, can be reduced by distributing the feed over the bin cross section.Air entrainment effects can be largely eliminated by minimizing free-fall height during filling.A useful summary of procedures for suppressing segregation in storage systems has been presented by Tang and Puri 13) .

Concluding Remarks
Mixing of solid par ticles generally proceeds through combinations of two basic mechanisms: Convection and diffusion.Other proposed mechanisms such as "shear mixing" represent special cases or specific combinations of the two.Convection provides for rapid macromixing but is considerably less effective in promoting micromixing.Diffusion, on the other hand, is the primary mechanism for producing intimate micromixing.However, because the process occurs at the individual particle scale, mixing by diffusion alone proceeds very slowly.Convective mixing is controlled largely by the geometry and mechanical action of a mixing system while diffusion is more dependent on particle mobility and is severely hindered by cohesion of particles.Segregation, i.e., de-mixing, is possible whenever different kinds of particles var y in mobility or response to external forces.Segregation by size can usually be attributed to differences in individual particle mobility fines have higher mobility within a bed of particles while coarser material is more mobile on free surfaces.Shape effects can be expected to follow a similar pattern, with isometric particles having greater mobility than plates or needles, for example.Density differences can promote segregation due to the different gravitational force experienced by heavy or light particles.The actual form of segregation top to bottom or side to side, for example depends on the specific conditions of mechanical agitation of the system.Variations in both size and density can lead either to enhancement or reduction in segregation tendency depending on specific conditions.Because segregation is normally a result of differences in individual particle mobility, its occurrence is most common in free-flowing powders.However, segregation of coarse particles is possible even in the presence of cohesive fine material.Aggregation of fines due to cohesion can also lead to segregation.
Procedures for minimizing segregation tend to be highly system-specific; there is no universal solution.Avoidance of mechanical agitation is generally appropriate the use of completely, rather than partially, filled containers, for example.Introducing cohesion by liquid addition, where practical, can be effective but will, of course, exacerbate problems in mixing and storage.

Fig. 2
Fig. 2 Geometric arrangement used in the analysis of convective mixing in the transverse plane of a rotating cylinder.

Fig. 4
Fig. 4 Schematic illustration of particle trajectories in the surface layers adjacent to the end-walls of a rotating cylinder.Axial Position 0.00 0.10 0.20

FractionFig. 6 Fig. 5
Fig. 6 Comparison of mixing behavior for tracer fed above and below the main flow stream.Calculated lines are based on the same values of the diffusion parameters.Data of Hwang 58) .