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Assessment of mixing mechanism on the basis of concentration pattern
Chemical Engineering and Processing 38 (1999) 665 – 674 www.elsevier.com/locate/cep
Assessment of mixing mechanism on the basis of concentration pattern J. Gyenis * Pannon Uni6ersity of Agricultural Sciences, Research Institute of Chemical and Process Engineering, Egyetem u. 2, P.O. Box 125, Veszpre´m, H-8201, Hungary Received 30 March 1999; received in revised form 23 April 1999; accepted 23 April 1999
Abstract The method of intermingling of solid particles in a mixing operation has great significance in finding suitable equipment and conditions to render the process more effective. After a survey of various mixing mechanisms, this paper deals with the principal possibilities for their identification, and gives several methods to evaluate artificial concentration patterns resulting from simulating diffusive-type and convection-shear mixing. In addition to the frequency distribution of the sample concentrations, the correlation technique, fractal analysis, phase-space technique, etc., are discussed. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Solids mixing; Concentration pattern; Mixing mechanisms; Auto-correlation; Fractal analysis; Phase-space technique
1. Introduction Solids mixing is a complex process to obtain a uniform mixture of ingredients distributed among each other as uniformly as possible. To fulfill this requirement, it is necessary to understand the elementary processes acting during this operation. The main questions are how, at what rate, and to what extent is the mixing going on. To answer these questions, the mechanism and kinetics of the process have to be resolved. Mechanism in this respect can be considered as a qualitative feature, which characterizes the way of intermingling of components. Visual observation is the simplest tool to achieve information on mechanism. But, in the case of different mixing mechanisms acting simultaneously, it is important to assess their relative contribution to the global result of operation, or to determine the dominant ones. To achieve such quantitative characteristics, detailed analysis of data obtained either by measurements or simulation should be carried out. This also helps to resolve the kinetics of the process, i.e. to determine the rate of mixing and equilibrium homogeneity. Dedicated to Professor Em. Dr-Ing. Dr h.c. mult. E.-U. Schlu¨nder on the occasion of his 70th birthday. * Tel.: +36-88-425-206; fax: + 36-88-424-424. E-mail address: [email protected] (J. Gyenis)
The mechanism significantly influences the rate of the process and also the achievable homogeneity. Therefore information on mechanism gives a possibility to improve and control the operation. Mechanism also affects the type and parameters of the descriptive kinetic equations. Hence mechanism, kinetics and modeling are closely related to each other. In this paper some promising methods are outlined, which can be used as diagnostic tools to resolve mechanism.
2. Mixing mechanisms It has been more than 40 years since Lacey [1] assumed the action of three different mechanisms. Namely: (i) convective mixing—which means the transfer of larger particle groups from one location to another, (ii) diffusive mixing—defined as the distribution of particles over a freshly developed surface, and (iii) shear mixing—setting up of slipping planes within the mixture. Later on, it was shown that diffusive-type mixing takes place not only on free surfaces, but within particle beds, too. It has also been revealed that the situation in real systems is more complex, because in the majority of mixers, several mechanisms take place simultaneously.
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Regarding these concepts, it should be emphasized that convective and shear mechanisms principally cannot be totally separated from each other [2]. Convection, especially in dense particle beds, always causes certain shear between the adjacent regions moving at different velocities, and vice versa: shear cannot exist without relative displacements, i.e. convection. Naturally, the relation between them depends on the material properties and on the actual conditions of the particle bed. In a loose particle bed for example, convection takes place with a smooth velocity profile and low shear rate, i.e. there is no sharp slipping plane and significant shear-force present. In turbulent fluidized beds, shear mainly occurs at the particle level. In a dense particle bed, convection and shear occur together, as pointed out by the theory of bulk solids mechanics. In the majority of most bulk solids mixers, dense particle regions and velocity differences occur together, at least in some locations. Therefore, it is better to speak about the complex convective-shear mixing mechanism instead of pure convection or shear. Diffusive-type particle mixing is analogous to molecular diffusion, taking place in fluids, with respect to the random walk of particles. This analogy is not faultless, because, in addition to great differences between their size and physical properties, the crucial difference is that molecular diffusion takes place spontaneously, while particles should be energized to move. In some particle systems, random movements of particle groups may also occur, together with their dispersion and combination, analogously to turbulent dispersion or eddy-diffusion in fluids. Diffusive and convective-shear mixing mechanisms may also act together. They can take place simultaneously in given regions of a mixer, or throughout its entire volume. In circulating fluidized beds for example, convection and diffusive particle movements are present in the whole mixture. In rotating drum mixers, definite diffusive-type mixing takes place on the free surface of the particle bed, but diffusive-type motion may also happen within the bulk. Inside the bed, mixing proceeds mainly by convection along transversal streamlines [3]. In non-ideal particle systems, the relative rates of mixing and segregation, thus the equilibrium homogeneity are also influenced by these mechanisms [4 – 9]. Shear and diffusion may cause segregation besides mixing, while convection generally acts against segregation [10,11]. Fundamental powder mixing mechanisms can also be categorized on a more sophisticated basis, making for example, a distinction between macromixing and micromixing [2]. Macromixing is related to broad residence time distribution caused by velocity differences between macroscopic regions of particle flow, while micromixing can be interpreted as a process causing microscopic distribution of individual particles among
each other. Interparticle percolation is a similar mechanism, which controls the microscopic distribution of cohesionless particles by their random walks throughout the gaps between the larger particles. Because nonhomogeneous or non-isotropic fields of forces, e.g. gravity can influence these movements, it may lead to segregation besides mixing, depending on process conditions. Micromixing and interparticle percolation are closely related to diffusive mixing, while macromixing is related to convection and shear. Mixing processes can also be distinguished according to their systematic or stochastic feature [11]. In a macroscopic sense, stochastic mixing is a process where the different regions of a particle bed are rearranged at random, i.e. not in a predetermined manner. It corresponds therefore to a chaotic convection-shear mechanism. In the microscopic sense, stochastic mixing is identical to the diffusive-type mixing caused by chaotic movements of individual particles. Systematic mixing is generally carried out at the macroscopic level only, for example in certain types of blending silos and motionless mixers, where the streams of material are repeatedly divided and rearranged in space and time, according to predetermined regular patterns. Shear and convection, therefore, are often involved in these processes. Production of microscopically ordered mixtures was also ranked by Sommer to systematic mixing [11], but regarding their manufacturing methods, they belong rather to stochastic than systematic processes. Kaye emphasized the importance of chaotic particle motion and turbulent convection in bulk solids, as basic mixing mechanisms to achieve random, i.e. the best achievable distribution of a component [12]. To perform a really efficient mixing, simultaneous actions of different mechanisms are desirable. Convection and shear alone cannot produce a high degree of homogeneity at the particle level within a reasonable time. Although in the early stages of a process, and in some regions of the particle bed, diffusive mixing produces a rapid increase of mixture quality, later on this improvement slows down considerably. From some respects, it is also true for the convective-shear mechanism. However, if diffusive-type mixing acts together with convection and shear, it has a strong synergetic effect, making possible a high mixing rate, and to achieve a high mixture quality within a short time, due to the reduction of the necessary diffusion paths of component particles. Particulate solids are highly intricate systems with very complex dynamical flow and mixing behavior during the interactions between their constituents and the various parts of the mixture. McCarthy et al. [13] have emphasized that while fundamental mixing mechanisms had been studied energetically by investigators for years, clearing up these complex processes remained a great challenge. Therefore, to find new tools for iden-
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tification of mixing mechanism has great importance, because it opens new ways for process improvement.
3. Identification of mixing mechanisms There are two principal possibilities to identify the way of mixing: (i) direct methods, by observing the movement of the particle bed or individual constituents during operation, and/or (ii) indirect methods, by analyzing the spatial distribution of components after given mixing times. In principle, spatial distribution is the consequence of the relative movements of constituents, therefore the results obtained by these methods can be mutually reconstructed from each other and must be equivalent. Selection between them depends on the available measurement techniques and on the nature of the studied system. From another respect, investigation very often has to be carried out at different levels to elucidate microscopic and macroscopic mechanisms. These studies can be carried out either by real experiments or simulations, or by both.
3.1. Mo6ement of the constituents Applying the direct method of investigation, relative displacements of different particle regions and particles have to be traced. Transient or steady state velocity field, flow streamlines, particle velocities and trajectories offer the most important information. Velocity profiles in any sections of the particle bed are obtained by measuring the instantaneous bulk velocities and their time averages at different locations. To investigate the large-scale behavior, generally it is not necessary to know the velocities of individual particles and their statistical distributions, but this may be a tool to determine global values. Non-invasive measurements are applied for this purpose, such as laser Doppler anemometry (LDA) in loose transparent particle beds, or video-recording of colored components in two-dimensional model experiments. Other techniques, such as applying light reflection probes or radioactive tracers are also available. From these measurements either the velocity profiles, streamlines, or deformation rate can be reconstructed. Independent random motions of particles indicate diffusive mixing, while relative displacements of larger bulk solids regions indicate convection and shear. Motion according to regular patterns indicates systematic mixing, while high temporal variation of local velocities indicates chaotic behavior. Long distance trajectories of individual particles, being either tendentious or stochastic, indicate convection or shear. Random variation of length and direction of these movements refers to large-scale stochastic phenomena. Microscopic variation indicates diffusive mixing. Such particle motions were noted by Broadbent et
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al. [14], using a radioactive tracing technique. Very often, particle trajectories involve both large-scale and microscopic variation. Kaye [12] has defined this phenomenon as a pseudo Levy-walk, which is characteristic for chaotic mixing mechanisms.
3.2. Concentration field The relative position of the constituents observed after a given mixing time can indicate the way and history of movements which have led to this situation. Therefore, observation of colored particles, for example just after the start of the mixing is generally enough to get qualitative assessment on the mixing mechanism. From macroscopic and microscopic arrangements, conclusions can be drawn both for large-scale and particlescale mixing phenomena. Large-scale deformations and relative displacements of various regions of the particle bed, their fragmentation and shuffling correspond to the results of convection and shear. Starting from regularly layered components, the consequences of these actions are easily observable in two-dimensional physical models. Mapping up the spatial concentration distribution in a three-dimensional particle bed by sample analysis is also possible and gives valuable information [15,16]. This method has a great disadvantage, because sampling and analysis destroy the structure of the particle bed, therefore it does not remain available for further experiments. The appearance of individual particles within the contiguous regions of other components, observable for example on microscopic photos indicates particle dispersion, i.e. a diffusive-type mechanism. Simultaneous deformation and/or fragmentation of contiguous regions, together with particle scattering, refer to composed mechanisms. Although these observations can give a clear picture on the nature of mixing, unfortunately they give valuable information only at the initial stage of operation. Another disadvantage is that this knowledge is rather qualitative, i.e. it is not expressed quantitatively by a characteristic parameter or correlation. However, there is a possibility to obtain quantitative information from the relative arrangements of constituents, because the structure of a measured concentration profile often preserves the fingerprint of the ruling mechanism for a long time. For example, sharp macroscopic jumps along a concentration profile refer to shear and convection, while gradual and tendentious change may be the sign of diffusive mixing. Although it also may be the consequence of a continuous velocity profile caused by convection, measurements along different cross-sections may help to distinguish these mechanisms. On the other hand, regular concentration patterns indicate systematic mixing, while random variations suggest the action of
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stochastic phenomena. Fortunately, these fingerprints can be recognized even at rather advanced stages of mixing and can be quantified. Therefore, evaluation of concentration patterns has great importance in assessing the mechanisms of mixing. The traces of different mechanisms are not always obvious at first sight. Therefore, to discover hidden information from various concentration patterns, special techniques should be applied. To find suitable evaluation methods and to test them, special artificial concentration patterns were created in this study, characteristic of different mixing mechanisms. For this reason, numerical simulations were carried out.
4. Applied method of simulation Besides experimental investigation, the mixing of particles can also be traced by simulation. Sophisticated methods based either on Eulerian or Lagrangian modeling philosophies have been developed. The first method generally focuses on macroscopic phenomena occurring within a fixed coordinate system, while the second type of modeling investigate the behavior of individual particles at a microscopic scale, following their positions along their changing coordinates. One of the most dynamically developing methods belonging to this latter approach is the discrete particle simulation (DPS), also known as distinct element modeling (DEM). By statistical evaluation of the results, particle level simulation can also give information on large-scale phenomena. Both methods can be used to investigate either the velocity field or the resulted concentration field. Apart from these sophisticated methods, very simple one-dimensional numerical experiments were carried out in this study, to create various artificial concentration patterns by simple interchange of particles. During this, two types of interchange were performed to imitate diffusive-type and convective-shear mixing mechanisms, respectively. The aim of the obtained artificial concentration patterns was to test the applicability of various evaluation methods in revealing the consequences of these mechanisms. During these numerical experiments 1000 particles were taken into consideration, and lined up in a row. Initially, i.e. in totally segregated state, the first 500 particles were considered as the first, and the remaining 500 particles as the second component. In a series of ‘experiments’, these components were subjected to ‘diffusive mixing’ by random interchanges of individual particles being relatively close to each other. Span and direction of an interchange were randomly selected for each particle within the limits from 1 to 10 particle distances. During a basic ‘mixing step’, the number of random interchanges was varied from 5000 to 50 000
for each of the 1000 particles, in order to simulate different mixing rates. In one numerical experiment, from 10 to 50 basic mixing steps were carried out and evaluated. Another series of numerical studies was carried out to investigate concentration patterns obtained by a convective-shear mixing mechanism. For this purpose, various sections of a particle row were shifted by given distances in any direction within the particle row. The lengths of these sections were randomly selected between 20 and 250 particles. Initial and target positions were also random variables, allowing any starting location, distance and direction of displacements, naturally taking the limited length of particle row into consideration. Particles pushed out from target positions were transferred to the free places deliberated at the initial location of the shifted sections. Overlap between starting and target sections was allowed. By these random interchanges, the initially contiguous regions of components were gradually fragmented into smaller and smaller sections, similarly as it happens to real convective-shear mixing. The number of these convectiveshear actions in a basic ‘mixing step’ was varied between 5 and 20, to simulate different mixing rates. In a simulation, the maximum number of basic mixing steps was varied between 10 and 50. During simulations, the actual state of the ‘particle mixture’ was evaluated after each basic ‘mixing step’ by dividing the total particle row into ‘samples’. Sample ‘concentrations’ were calculated from the ratio of particles belonging to different components in a sample. The resultant concentration patterns along the series of samples after given times were evaluated by various methods. The primary aim was to discover the fingerprints of different mixing mechanisms on the resultant concentration patterns, but in addition the influence of sample size on the obtained quantitative characteristics was also studied. Besides the usual calculation of concentration variances, other assessment methods, namely the frequency distribution of sample concentrations, auto-correlation functions, fractal analysis, phase-space technique, etc. were also used to evaluate the concentration pattern.
5. Results and discussion
5.1. Frequency distribution and 6ariance of sample concentrations By simulating different mixing mechanisms outlined above, quite different concentration patterns were obtained. Diffusive-type mixing gave smooth concentration profiles with relatively narrow frequency distributions during the whole time of mixing, illustrated by Fig. 1a. On the other hand, concentration
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patterns produced by convective-shear type mixing showed a high variation of sample concentrations along the whole particle row, with a much broader frequency distribution (Fig. 1b). The variance of sample concentrations was influenced by sample size in both cases, but in a different manner. It is clearly seen from Fig. 2a,b, where the values of 1−M are plotted versus reciprocal sample size after different mixing times for diffusive and convective-shear mixing mechanisms respectively, where M is the Kramers’ mixing index, given by Eq. (1), based on the variance of sample concentrations M=
s 20 − s 2 s 20 − s 2r
(1)
These plots confirm the opinion of Lacey [1], demonstrating that in the case of a diffusive mechanism, this mixing index does not depend on reciprocal sample size, while there is strong dependence in convectiveshear mechanism in a considerable range of sample sizes.
5.2. Correlation technique The difference between the results of these mixing mechanisms is even more pronounced by comparing auto-correlograms, determined from the resultant concentration patterns by Eq. (2) as
Fig. 1. Typical frequency distributions of sample concentrations after different mixing times; (a) diffusive-type mixing; (b) convective-shear mixing.
Fig. 2. Dependence of Kramers’ mixing indices on reciprocal sample size; (a) diffusive-type mixing; (b) convective-shear mixing.
R(Di)=
(xi − x¯ )(xi + Di − x¯ ) (xi − x¯ )2
(2)
where xi and xi + Di are the concentrations along a series of samples denoted by running serial numbers i and i +Di, respectively. Correlation step Di can also be expressed in reduced length l, relative to the total length of mixer. Plotting the correlation coefficient R(l) versus l, important information can be obtained on the macro- and microstructure of concentration patterns. Fig. 3a,b shows correlograms characteristic of the studied mixing mechanisms after different mixing times. In these examples, individual particles were used as samples (n= 1) to reveal the microstructure of the simulated concentration patterns. The most important differences between the correlograms obtained for different mixing mechanisms were as follows: In diffusive mixing, shown by Fig. 3a, the correlation coefficients extrapolated to l= 0 are gradually diminishing from unity to zero with increasing mixing times. At small l values, the slope of these curves is almost zero and apart from minor statistical variations, is continuously changing till the intersection with the horizontal axis. In convective-shear mechanism (Fig. 3b), the slopes are much steeper at l= 0, evidencing significant reduction in the scale of segregation [17], just at the beginning of the process. The extrapolated intersections with the vertical axis remain close to unity, while the slopes become steeper and steeper with increasing mixing time. Comparing random variation
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along these correlograms, it is relatively low for diffusive mixing and much higher for convective-shear mechanism. During this study, it was also noted that sample size had a great influence. It has been proved that the correlation technique had the best sensitivity in the case of small sample sizes, both to evaluate the degree of homogeneity and to reveal the microstructure of concentration patterns, especially at the early stages of the process.
5.3. Characterization by fractal analysis Fractal analysis has been advantageously used for a long time to characterize various natural objects and technical phenomena [18 – 20]. This technique is based on the fact that the boundary length of an object having a rugged contour can be quite different, if it is measured with different scales. When the measured length L(l) is plotted on a log – log diagram against the size of the measuring scale l, called a Richardson plot, generally a straight line is obtained [19] whose slope is characteristic of the fractal properies. This dependence is expressed as L(l) :l − dF + 1
(3)
where dF is the fractal dimension by definition. To analyze the micro-scale concentration patterns obtained by one-dimensional simulation, a modified fractal method can be proposed. Considering a situa-
Fig. 3. Correlograms of simulated concentration patterns; (a) diffusive mixing; (b) convective-shear mixing.
tion when the size of samples is sufficiently small, for example close to the size of one particle, a rugged concentration pattern is obtained. A reduced distance kl between any two points of this curve can be calculated from the concentration difference Dx between the samples being at distance l from each other as kl =
)
l 2 + Dx 2 l
)
(4)
If there is no concentration difference between the samples, i.e. Dx =0, then kl is equal to unity. In the case of any concentration difference, kl is greater than unity. The average of these kl values over the whole range of concentration pattern analyzed is %kl K(l)=
N
N
(5)
In the completely mixed state when the particles of a constituent are well distributed among the others, and in the case of small sample size, e.g. if n=1, the number of concentration jumps between the neighboring samples is maximal, therefore this mixture gives the greatest K(l) value. In the totally segregated state, there is no concentration jump along a large range of l values, i.e. the average fractal length is much lower, close to unity. Therefore the increment of K(l) relative to the totally segregated state is DK(l)$ K(l)− 1
(6)
which well characterizes the actual state of mixture. Similarly to the conventional fractal length L(l), the value of DK(l) also decreases as l increases, and vice versa. Plotting DK(l) as a function of l on a log–log scale, multifractal properties, i.e. not totally linear correlations were experienced for the studied concentration patterns. But the first sections of these curves were straight lines, similar to the usual Richardson plots. It was also proved that slopes and intersections with the vertical axis were diminishing with increasing mixing times. Sample size had a great influence: higher slopes and intersections were obtained with smaller samples. Maximal sensitivity was experienced with sample size equal to one particle. But, it should be emphasized that in the case of large sample sizes this technique does not remain unambiguous, therefore this method is recommended only to characterize the microstructure of a mixture. From the results obtained by this fractal analysis, it can be concluded that this technique is also suitable to characterize the mixing mechanism. Fig. 4a,b shows the modified Richardson plots of concentration patterns resulting from diffusive-type and convective-shear mechanisms, respectively. In diffusive mixing, the slopes and intersections of lines obtained after different mixing times are rapidly increasing at the beginning of
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Dx xi − xi − Di = f1(xi − Di ) = Di Di
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(8)
or in a simpler way as xi = f2(xi − Di )
Fig. 4. Modified Richardson plot of concentration patterns obtained by simulation; (a) diffusive-type mixing; (b) convective-shear mixing.
the process (Fig. 4a). Then, this change slows down considerably as the concentration pattern approaches the completely randomized state, because of the decreasing rate of development of the final microstructure of the mixture. In convective-shear mixing mechanism, however, fractal properties change much more slowly even at the beginning and remain nearly uniform during the whole process. It means that it is almost impossible to achieve a random mixture by this mechanism at the particle level, even after a very long mixing time. From these differences it follows that studying fractal properties offers a suitable tool to assess the main mechanism of a mixing process.
(9)
where Di is the temporal or spatial difference between the serial numbers of samples, xi − Di and xi are the concentrations of the (i −Di )th and ith samples running along the concentration pattern. Usually, Di is fixed and is equal to one. If the microstructure of a mixture has to be revealed, the sample size should be suitably small but to get an informative picture, it has to be composed of at least several particles. To study the feature of medium- or large-scale structures, sample size can be considerably larger. Useful information is obtained on the structure of concentration patterns created by different mixing mechanism by plotting attractors or phase-portraits after different time intervals. Fig. 5 gives demonstrative examples for different stages of diffusive mixing, showing the transformation from segregated state to an almost completely randomized mixture. It can be seen that at an early stage of mixing the phase-portrait of a mixture corresponds to a winding curve (Fig. 5a) extending diagonally between two extreme concentration values, revealing tendentious change along a characteristic direction. Later on, these signatures become less and less elongated (Fig. 5b,c), also indicating that differences between sample concentrations are gradually
5.4. Phase-space technique Recently, the phase-space technique has frequently been used for chaos analysis [21 – 23]. In this data representation method, the derivative dj/dt of a temporal signal is plotted against the signal value j(t), which clearly characterizes the dynamic nature of a chaotic phenomena, resulting in an attractor function f(j(t)) as dj = f(j(t)) dt
(7)
Using finite differences in the spatial coordinate system, the attractor function of a concentration pattern can also be defined as
Fig. 5. Phase-portraits obtained by diffusive-type mixing (sample size corresponds to 20 particles); (a, b, c, d) signatures in order of increasing mixing times (t1 Bt2 Bt3 Bt4).
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sive mixing, because the resultant fragments will be even smaller.
5.5. Spectral density technique
Fig. 6. Phase-portraits obtained by convective-shear mixing (sample size corresponds to 20 particles); (a, b, c, d) signatures in order of increasing mixing times (t1 B t2 B t3 B t4).
diminishing. The almost globular curve in Fig. 5d corresponds to an almost completely randomized mixture. Fluctuation and loops indicate the stochastic nature of the process. It also has to be recognized that the area covered by a signature is characteristic of the actual state of mixture. A smaller area corresponds to less intensity of segregation. This area also depends on the size of samples, because it decreases as the sample size increases. Therefore, for a given aim of investigation, the sample size should be fixed in advance, depending on whether the micro- or the macrostructure is the most important. Concentration patterns obtained by the convectiveshear mechanism show quite different phase-portraits. At an early stage of mixing there is no tendentious elongation (Fig. 6a). Another important feature is that, instead of the generally continuous curve characterizing the diffusive-type mixing also depending on sample size, a number of joined straight lines are obtained with randomly varying directions and lengths. These lines correspond to the change of concentrations in samples taken successively from the fragmented and randomly distributed component regions. As fragmentation proceeds during convective-shear mixing, the number of these lines increases, together with the reduction of their sizes. After a suitably long time, these phase-portraits become quite similar to those obtained by diffu-
Other pattern recognition methods can also be used to discover the mixing mechanism. Among them, the spectral density technique is a promising method. Power spectra of spatial or temporal concentration signals can be determined by using Fast Fourier Transformation (FFT). In real mixtures such signals can be measured for example, by light reflection or image analysis. The concentration pattern used for the spectral density technique should be determined from samples as small as possible, to evaluate the micro-scale structure of a mixture. As with fractal analysis, the best sensitivity in this regard can be obtained by measurements on the particle level. In the case of n= 1, the concentration signal consists of square waves having two values: zero or unity. In a random arrangement of the constituent particles, the maximal spectral density is obtained at the high frequency range, while in a more segregated state this characteristic peak is shifted to higher wavelengths, i.e. to low frequency. For uncompleted mixtures, spectral densities represent intermediate states transforming from the segregated to randomized state. The mechanism of mixing influences the way of transformation. In the first stages of mixing, the diffusive mechanism creates the concentration pattern, which has high frequency concentration variation within a short region near the initial border of components, while there is no concentration variation in its remaining parts. Therefore, power spectra produced by this mechanism show a relatively small characteristic peak at the high frequency range. This peak increases as the mixed diffuse zone spreads out between the pure components, indicating the gradual development of microstructure along the concentration pattern. In convective-shear mechanism, however, concentration jumps appear randomly between contiguous regions of pure components, at an ever-growing number of locations along the length of the concentration pattern. The sizes of these regions gradually decrease, while the numbers of concentration jumps are increasing. Therefore, at the beginning of the mixing process the spectral density will be higher at low frequency regions, and later on it will be gradually shifted towards higher frequencies as the process advances. Although not too much detailed work has been done till now in studying this latter method, it offers an additional tool for mixture characterization. To resolve its potential in evaluation of mixtures or mixing mechanisms, more detailed investigation is necessary in the future.
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6. Summary and conclusions The way of mixing highly influences the rate of the process and also the achievable homogeneity. The simultaneous action of diffusive and convective-shear mixing has advantages in both respects, due to their useful synergetic effects. Therefore, to find suitable tools to discover the main mechanisms has great importance both for equipment design and process improvement. Investigation of macro- and microstructures of the resultant concentration patterns may be a possible way for this. In this study, various artificial concentration patterns were created by one-dimensional numerical simulation of diffusive and convective-shear mixing. Concentration patterns obtained by these mechanisms are highly different, both in their macroscopic and microscopic characteristics, indicating the different method and route of formation. To investigate the applicability of different pattern evaluation methods, in addition to the spatial and frequency distribution of sample concentrations, autocorrelation functions, fractal analysis, phase-space technique and spectral density were tested. It was proved that all of these methods are suitable to trace the improvement of mixture homogeneity, and to distinguish the mechanisms of the process. In addition, all have certain specialties, which can advantageously be used from different respects. Frequency distribution of sample concentration and its variance as a function of sample size may help to indicate the right sample size, beside giving information on homogeneity and the mixing mechanism. The correlation technique can be applied to assess both the micro- and macrostructure of concentration patterns, giving information on the characteristics of microscopic and large-scale processes. During this study, it has been proved that the correlation technique had the best sensitivity in assessing mixing mechanism in the early stages of the process, by applying small sample sizes. A special kind of fractal analysis was also tested. It has been proved that this technique can be used advantageously to assess mixing mechanism. However it is unambiguous only at small sample size, and showed the best sensitivity if the sample size is equal to one particle. Using larger samples may also give information on the rate of process and mixture quality, but in this case the fractal dimension of concentration patterns can also be diminishing during the process, and the special fingerprint of diffusion cannot be recognized. The phase-space technique is a very illustrative method to indicate both the micro- and macrostructure of concentration patterns by a single phase-portrait. Therefore it can be used effectively to recognize mixed mechanisms, i.e. in the case of simultaneous action of diffusion and convection-shear. This method is not very
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sensitive to sample size, therefore this situation gives more freedom in selecting the right scale of scrutiny corresponding to the actual purpose of investigation. The spectral density technique is a promising method to assess mixing mechanism, but more detailed investigation on its features is necessary in the future. In real processes, different mixing mechanisms may be acting simultaneously, meaning that they cannot be observed in sterile forms. However, the structure of concentration patterns preserves the traces of both mechanisms. Therefore, these pattern recognition methods can help to answer which is the dominant mechanism, and if they are acting together or not. In principle, these assessment methods can be used not only for solids mixtures, but also to resolve mixing mechanism in fluids and pseudo-plastic materials, if a suitable technique is found to measure their spatial or temporal concentration pattern. 7. Nomenclature dF f(j(t)) f1,2(xi−Di) Di kl K(l) DK(l) L(l) M n N R(Di ), R(l) t x Dx Greek symbols l s2 Indices 0 i r
fractal dimension, according to Eq. (3) attractor function defined by Eq. (7) attractor functions, defined by Eq. (8) and Eq. (9) correlation step, or the increment in serial numbers of samples distance between two points of concentration curve depending on scale length l average of kl values along the whole concentration pattern increment of K(l) relative to the segregated state total length of rugged curve measured by a scale of length l Kramers’ mixing index, according to Eq. (1) number of particles in a sample number of samples auto-correlation function, Eq. (2) time sample concentration along the concentration pattern concentration difference between two samples
reduced length, correlation step, or scale used to determine fractal length variance of sample concentrations initial or totally segregated state serial number of a sample completely randomized state
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Acknowledgements The author wishes to acknowledge the support of the Hungarian National Foundation for Fundamental Research (OTKA No. T 29313) and the help of the US – Hungarian Joint Fund (JF No. 593/96) also serving as a background for these results.
References [1] P.M.C. Lacey, Developments in the theory of particle mixing, J. Appl. Chem. 4 (1954) 257–268. [2] J. Bridgwater, Fundamental powder mixing mechanism, Powder Technol. 15 (1976) 215–236. [3] Y. Inoue, K. Yamaguchi, K. Sato, Motion of particle and mixing process in a horizontal drum mixer, Kagaku Kogaku 34 (12) (1970) 1323 – 1329. [4] M.B. Donald, B. Roseman, Mechanisms in a horizontal drum mixer, Br. Chem. Eng. 7 (1962) 749–753. [5] L.T. Fan, S.H. Shin, Stochastic diffusion model of non-ideal mixing in a horizontal drum mixer, Chem. Eng. Sci. 34 (1979) 811 – 820. [6] K.M. Hill, J. Kakalios, Reversible axial segregation of rotating granular media, Phys. Rev. E 52 (4) (1995) 4393–4400. [7] K.M. Hill, A. Caprihan, J. Kakalios, Axial segregation of granular media rotated in a drum mixer: pattern evolution, Phys. Rev. E 56 (4) (1997) 4386–4393. [8] K.M. Hill, A. Caprihan, J. Kakalios, Bulk segregation in rotated granular material measured by magnetic resonance imaging, Phys. Rev. Let. 78 (1) (1997) 50–53. [9] L. Prigozhin, H. Kalman, Radial mixing and segregation of a binary mixture in a rotating drum: model and experiment, Phys. Rev. E 57 (2) (1998) 2073–2080. [10] J.C. Williams, The mixing of dry powders, Powder Technol. 2
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(1968) 13 – 20. [11] K. Sommer, Stochastic and systematic mixing mechanisms of powders (A review), Proc. 1st Eur. Congress on Chem. Eng., 4 – 7 May, 1997, Florence, Italy, pp. 953 – 954. [12] B.H. Kaye, Powder Mixing, Chapman & Hall, London, 1997. [13] J.J. McCarthy, T. Shinbrot, G. Metcalfe, J.E. Wolf, J.M. Ottino, Mixing of granular materials in slowly rotated containers, AIChE J. 42 (12) (1996) 3351 – 3363. [14] C.J. Broadbent, J. Bridgwater, D.J. Parker, S.T. Keningley, P. Knight, A phenomenological study of a batch mixer using a positron camera, Powder Technol. 76 (1993) 317 – 329. [15] J. Gyenis, J. Arva, Mixing mechanism of solids in alternately revolving mixers. Part I. Change of local concentrations and concentration profiles, Powder Handling Proc. 1 (3) (1989) 247– 253. [16] J. Gyenis, J. Arva, Mixing mechanism of solids in alternately revolving mixers. Part II. The role of convection and diffusion mechanisms, Powder Handling Proc. 1 (4) (1989) 365 –371. [17] P.V. Danckwerts, The definition and measurement of some characteristics of mixtures, Appl. Sci. Res. (Hague) A3 (1952) 279 – 296. [18] B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, New York, 1982. [19] L.F. Richardson, The problem of contiguity: an appendix of statistics of deadly quarrels, Gen. Syst. Yearbook 6 (1961) 139 – 187. [20] L.T. Fan, D. Neogi, M. Yashima, Elementary introduction to spatial and temporal fractals, Lecture Notes in Chemistry, Springer, Berlin, 1991. [21] J.X. Buillard, A.L. Miller, Experimental investigations of chaotic hydrodynamic attractors in circulating fluidized beds, Powder Technol. 79 (1994) 211 – 215. [22] D.V. Pence, D.E. Beasley, Multi-fractal signal simulation of local, instantaneous pressure in a bubbling gas fluidized bed, in: Gas-Particle Flows, FED-Vol. 228, ASME 1995, 217 –225. [23] J. Gyenis, Z. Ulbert, J. Szepvolgyi, T. Tanaka, Y. Tsuji, Is discrete particle simulation a perfect experiment in particulate systems? In: Proc. 8th International Summer School of Chem. Eng., 3 – 9 June 1998, Sozopol, Bulgaria, pp. 139 – 158.
Assessment of mixing mechanism on the basis of concentration pattern J. Gyenis * Pannon Uni6ersity of Agricultural Sciences, Research Institute of Chemical and Process Engineering, Egyetem u. 2, P.O. Box 125, Veszpre´m, H-8201, Hungary Received 30 March 1999; received in revised form 23 April 1999; accepted 23 April 1999
Abstract The method of intermingling of solid particles in a mixing operation has great significance in finding suitable equipment and conditions to render the process more effective. After a survey of various mixing mechanisms, this paper deals with the principal possibilities for their identification, and gives several methods to evaluate artificial concentration patterns resulting from simulating diffusive-type and convection-shear mixing. In addition to the frequency distribution of the sample concentrations, the correlation technique, fractal analysis, phase-space technique, etc., are discussed. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Solids mixing; Concentration pattern; Mixing mechanisms; Auto-correlation; Fractal analysis; Phase-space technique
1. Introduction Solids mixing is a complex process to obtain a uniform mixture of ingredients distributed among each other as uniformly as possible. To fulfill this requirement, it is necessary to understand the elementary processes acting during this operation. The main questions are how, at what rate, and to what extent is the mixing going on. To answer these questions, the mechanism and kinetics of the process have to be resolved. Mechanism in this respect can be considered as a qualitative feature, which characterizes the way of intermingling of components. Visual observation is the simplest tool to achieve information on mechanism. But, in the case of different mixing mechanisms acting simultaneously, it is important to assess their relative contribution to the global result of operation, or to determine the dominant ones. To achieve such quantitative characteristics, detailed analysis of data obtained either by measurements or simulation should be carried out. This also helps to resolve the kinetics of the process, i.e. to determine the rate of mixing and equilibrium homogeneity. Dedicated to Professor Em. Dr-Ing. Dr h.c. mult. E.-U. Schlu¨nder on the occasion of his 70th birthday. * Tel.: +36-88-425-206; fax: + 36-88-424-424. E-mail address: [email protected] (J. Gyenis)
The mechanism significantly influences the rate of the process and also the achievable homogeneity. Therefore information on mechanism gives a possibility to improve and control the operation. Mechanism also affects the type and parameters of the descriptive kinetic equations. Hence mechanism, kinetics and modeling are closely related to each other. In this paper some promising methods are outlined, which can be used as diagnostic tools to resolve mechanism.
2. Mixing mechanisms It has been more than 40 years since Lacey [1] assumed the action of three different mechanisms. Namely: (i) convective mixing—which means the transfer of larger particle groups from one location to another, (ii) diffusive mixing—defined as the distribution of particles over a freshly developed surface, and (iii) shear mixing—setting up of slipping planes within the mixture. Later on, it was shown that diffusive-type mixing takes place not only on free surfaces, but within particle beds, too. It has also been revealed that the situation in real systems is more complex, because in the majority of mixers, several mechanisms take place simultaneously.
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Regarding these concepts, it should be emphasized that convective and shear mechanisms principally cannot be totally separated from each other [2]. Convection, especially in dense particle beds, always causes certain shear between the adjacent regions moving at different velocities, and vice versa: shear cannot exist without relative displacements, i.e. convection. Naturally, the relation between them depends on the material properties and on the actual conditions of the particle bed. In a loose particle bed for example, convection takes place with a smooth velocity profile and low shear rate, i.e. there is no sharp slipping plane and significant shear-force present. In turbulent fluidized beds, shear mainly occurs at the particle level. In a dense particle bed, convection and shear occur together, as pointed out by the theory of bulk solids mechanics. In the majority of most bulk solids mixers, dense particle regions and velocity differences occur together, at least in some locations. Therefore, it is better to speak about the complex convective-shear mixing mechanism instead of pure convection or shear. Diffusive-type particle mixing is analogous to molecular diffusion, taking place in fluids, with respect to the random walk of particles. This analogy is not faultless, because, in addition to great differences between their size and physical properties, the crucial difference is that molecular diffusion takes place spontaneously, while particles should be energized to move. In some particle systems, random movements of particle groups may also occur, together with their dispersion and combination, analogously to turbulent dispersion or eddy-diffusion in fluids. Diffusive and convective-shear mixing mechanisms may also act together. They can take place simultaneously in given regions of a mixer, or throughout its entire volume. In circulating fluidized beds for example, convection and diffusive particle movements are present in the whole mixture. In rotating drum mixers, definite diffusive-type mixing takes place on the free surface of the particle bed, but diffusive-type motion may also happen within the bulk. Inside the bed, mixing proceeds mainly by convection along transversal streamlines [3]. In non-ideal particle systems, the relative rates of mixing and segregation, thus the equilibrium homogeneity are also influenced by these mechanisms [4 – 9]. Shear and diffusion may cause segregation besides mixing, while convection generally acts against segregation [10,11]. Fundamental powder mixing mechanisms can also be categorized on a more sophisticated basis, making for example, a distinction between macromixing and micromixing [2]. Macromixing is related to broad residence time distribution caused by velocity differences between macroscopic regions of particle flow, while micromixing can be interpreted as a process causing microscopic distribution of individual particles among
each other. Interparticle percolation is a similar mechanism, which controls the microscopic distribution of cohesionless particles by their random walks throughout the gaps between the larger particles. Because nonhomogeneous or non-isotropic fields of forces, e.g. gravity can influence these movements, it may lead to segregation besides mixing, depending on process conditions. Micromixing and interparticle percolation are closely related to diffusive mixing, while macromixing is related to convection and shear. Mixing processes can also be distinguished according to their systematic or stochastic feature [11]. In a macroscopic sense, stochastic mixing is a process where the different regions of a particle bed are rearranged at random, i.e. not in a predetermined manner. It corresponds therefore to a chaotic convection-shear mechanism. In the microscopic sense, stochastic mixing is identical to the diffusive-type mixing caused by chaotic movements of individual particles. Systematic mixing is generally carried out at the macroscopic level only, for example in certain types of blending silos and motionless mixers, where the streams of material are repeatedly divided and rearranged in space and time, according to predetermined regular patterns. Shear and convection, therefore, are often involved in these processes. Production of microscopically ordered mixtures was also ranked by Sommer to systematic mixing [11], but regarding their manufacturing methods, they belong rather to stochastic than systematic processes. Kaye emphasized the importance of chaotic particle motion and turbulent convection in bulk solids, as basic mixing mechanisms to achieve random, i.e. the best achievable distribution of a component [12]. To perform a really efficient mixing, simultaneous actions of different mechanisms are desirable. Convection and shear alone cannot produce a high degree of homogeneity at the particle level within a reasonable time. Although in the early stages of a process, and in some regions of the particle bed, diffusive mixing produces a rapid increase of mixture quality, later on this improvement slows down considerably. From some respects, it is also true for the convective-shear mechanism. However, if diffusive-type mixing acts together with convection and shear, it has a strong synergetic effect, making possible a high mixing rate, and to achieve a high mixture quality within a short time, due to the reduction of the necessary diffusion paths of component particles. Particulate solids are highly intricate systems with very complex dynamical flow and mixing behavior during the interactions between their constituents and the various parts of the mixture. McCarthy et al. [13] have emphasized that while fundamental mixing mechanisms had been studied energetically by investigators for years, clearing up these complex processes remained a great challenge. Therefore, to find new tools for iden-
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tification of mixing mechanism has great importance, because it opens new ways for process improvement.
3. Identification of mixing mechanisms There are two principal possibilities to identify the way of mixing: (i) direct methods, by observing the movement of the particle bed or individual constituents during operation, and/or (ii) indirect methods, by analyzing the spatial distribution of components after given mixing times. In principle, spatial distribution is the consequence of the relative movements of constituents, therefore the results obtained by these methods can be mutually reconstructed from each other and must be equivalent. Selection between them depends on the available measurement techniques and on the nature of the studied system. From another respect, investigation very often has to be carried out at different levels to elucidate microscopic and macroscopic mechanisms. These studies can be carried out either by real experiments or simulations, or by both.
3.1. Mo6ement of the constituents Applying the direct method of investigation, relative displacements of different particle regions and particles have to be traced. Transient or steady state velocity field, flow streamlines, particle velocities and trajectories offer the most important information. Velocity profiles in any sections of the particle bed are obtained by measuring the instantaneous bulk velocities and their time averages at different locations. To investigate the large-scale behavior, generally it is not necessary to know the velocities of individual particles and their statistical distributions, but this may be a tool to determine global values. Non-invasive measurements are applied for this purpose, such as laser Doppler anemometry (LDA) in loose transparent particle beds, or video-recording of colored components in two-dimensional model experiments. Other techniques, such as applying light reflection probes or radioactive tracers are also available. From these measurements either the velocity profiles, streamlines, or deformation rate can be reconstructed. Independent random motions of particles indicate diffusive mixing, while relative displacements of larger bulk solids regions indicate convection and shear. Motion according to regular patterns indicates systematic mixing, while high temporal variation of local velocities indicates chaotic behavior. Long distance trajectories of individual particles, being either tendentious or stochastic, indicate convection or shear. Random variation of length and direction of these movements refers to large-scale stochastic phenomena. Microscopic variation indicates diffusive mixing. Such particle motions were noted by Broadbent et
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al. [14], using a radioactive tracing technique. Very often, particle trajectories involve both large-scale and microscopic variation. Kaye [12] has defined this phenomenon as a pseudo Levy-walk, which is characteristic for chaotic mixing mechanisms.
3.2. Concentration field The relative position of the constituents observed after a given mixing time can indicate the way and history of movements which have led to this situation. Therefore, observation of colored particles, for example just after the start of the mixing is generally enough to get qualitative assessment on the mixing mechanism. From macroscopic and microscopic arrangements, conclusions can be drawn both for large-scale and particlescale mixing phenomena. Large-scale deformations and relative displacements of various regions of the particle bed, their fragmentation and shuffling correspond to the results of convection and shear. Starting from regularly layered components, the consequences of these actions are easily observable in two-dimensional physical models. Mapping up the spatial concentration distribution in a three-dimensional particle bed by sample analysis is also possible and gives valuable information [15,16]. This method has a great disadvantage, because sampling and analysis destroy the structure of the particle bed, therefore it does not remain available for further experiments. The appearance of individual particles within the contiguous regions of other components, observable for example on microscopic photos indicates particle dispersion, i.e. a diffusive-type mechanism. Simultaneous deformation and/or fragmentation of contiguous regions, together with particle scattering, refer to composed mechanisms. Although these observations can give a clear picture on the nature of mixing, unfortunately they give valuable information only at the initial stage of operation. Another disadvantage is that this knowledge is rather qualitative, i.e. it is not expressed quantitatively by a characteristic parameter or correlation. However, there is a possibility to obtain quantitative information from the relative arrangements of constituents, because the structure of a measured concentration profile often preserves the fingerprint of the ruling mechanism for a long time. For example, sharp macroscopic jumps along a concentration profile refer to shear and convection, while gradual and tendentious change may be the sign of diffusive mixing. Although it also may be the consequence of a continuous velocity profile caused by convection, measurements along different cross-sections may help to distinguish these mechanisms. On the other hand, regular concentration patterns indicate systematic mixing, while random variations suggest the action of
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stochastic phenomena. Fortunately, these fingerprints can be recognized even at rather advanced stages of mixing and can be quantified. Therefore, evaluation of concentration patterns has great importance in assessing the mechanisms of mixing. The traces of different mechanisms are not always obvious at first sight. Therefore, to discover hidden information from various concentration patterns, special techniques should be applied. To find suitable evaluation methods and to test them, special artificial concentration patterns were created in this study, characteristic of different mixing mechanisms. For this reason, numerical simulations were carried out.
4. Applied method of simulation Besides experimental investigation, the mixing of particles can also be traced by simulation. Sophisticated methods based either on Eulerian or Lagrangian modeling philosophies have been developed. The first method generally focuses on macroscopic phenomena occurring within a fixed coordinate system, while the second type of modeling investigate the behavior of individual particles at a microscopic scale, following their positions along their changing coordinates. One of the most dynamically developing methods belonging to this latter approach is the discrete particle simulation (DPS), also known as distinct element modeling (DEM). By statistical evaluation of the results, particle level simulation can also give information on large-scale phenomena. Both methods can be used to investigate either the velocity field or the resulted concentration field. Apart from these sophisticated methods, very simple one-dimensional numerical experiments were carried out in this study, to create various artificial concentration patterns by simple interchange of particles. During this, two types of interchange were performed to imitate diffusive-type and convective-shear mixing mechanisms, respectively. The aim of the obtained artificial concentration patterns was to test the applicability of various evaluation methods in revealing the consequences of these mechanisms. During these numerical experiments 1000 particles were taken into consideration, and lined up in a row. Initially, i.e. in totally segregated state, the first 500 particles were considered as the first, and the remaining 500 particles as the second component. In a series of ‘experiments’, these components were subjected to ‘diffusive mixing’ by random interchanges of individual particles being relatively close to each other. Span and direction of an interchange were randomly selected for each particle within the limits from 1 to 10 particle distances. During a basic ‘mixing step’, the number of random interchanges was varied from 5000 to 50 000
for each of the 1000 particles, in order to simulate different mixing rates. In one numerical experiment, from 10 to 50 basic mixing steps were carried out and evaluated. Another series of numerical studies was carried out to investigate concentration patterns obtained by a convective-shear mixing mechanism. For this purpose, various sections of a particle row were shifted by given distances in any direction within the particle row. The lengths of these sections were randomly selected between 20 and 250 particles. Initial and target positions were also random variables, allowing any starting location, distance and direction of displacements, naturally taking the limited length of particle row into consideration. Particles pushed out from target positions were transferred to the free places deliberated at the initial location of the shifted sections. Overlap between starting and target sections was allowed. By these random interchanges, the initially contiguous regions of components were gradually fragmented into smaller and smaller sections, similarly as it happens to real convective-shear mixing. The number of these convectiveshear actions in a basic ‘mixing step’ was varied between 5 and 20, to simulate different mixing rates. In a simulation, the maximum number of basic mixing steps was varied between 10 and 50. During simulations, the actual state of the ‘particle mixture’ was evaluated after each basic ‘mixing step’ by dividing the total particle row into ‘samples’. Sample ‘concentrations’ were calculated from the ratio of particles belonging to different components in a sample. The resultant concentration patterns along the series of samples after given times were evaluated by various methods. The primary aim was to discover the fingerprints of different mixing mechanisms on the resultant concentration patterns, but in addition the influence of sample size on the obtained quantitative characteristics was also studied. Besides the usual calculation of concentration variances, other assessment methods, namely the frequency distribution of sample concentrations, auto-correlation functions, fractal analysis, phase-space technique, etc. were also used to evaluate the concentration pattern.
5. Results and discussion
5.1. Frequency distribution and 6ariance of sample concentrations By simulating different mixing mechanisms outlined above, quite different concentration patterns were obtained. Diffusive-type mixing gave smooth concentration profiles with relatively narrow frequency distributions during the whole time of mixing, illustrated by Fig. 1a. On the other hand, concentration
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patterns produced by convective-shear type mixing showed a high variation of sample concentrations along the whole particle row, with a much broader frequency distribution (Fig. 1b). The variance of sample concentrations was influenced by sample size in both cases, but in a different manner. It is clearly seen from Fig. 2a,b, where the values of 1−M are plotted versus reciprocal sample size after different mixing times for diffusive and convective-shear mixing mechanisms respectively, where M is the Kramers’ mixing index, given by Eq. (1), based on the variance of sample concentrations M=
s 20 − s 2 s 20 − s 2r
(1)
These plots confirm the opinion of Lacey [1], demonstrating that in the case of a diffusive mechanism, this mixing index does not depend on reciprocal sample size, while there is strong dependence in convectiveshear mechanism in a considerable range of sample sizes.
5.2. Correlation technique The difference between the results of these mixing mechanisms is even more pronounced by comparing auto-correlograms, determined from the resultant concentration patterns by Eq. (2) as
Fig. 1. Typical frequency distributions of sample concentrations after different mixing times; (a) diffusive-type mixing; (b) convective-shear mixing.
Fig. 2. Dependence of Kramers’ mixing indices on reciprocal sample size; (a) diffusive-type mixing; (b) convective-shear mixing.
R(Di)=
(xi − x¯ )(xi + Di − x¯ ) (xi − x¯ )2
(2)
where xi and xi + Di are the concentrations along a series of samples denoted by running serial numbers i and i +Di, respectively. Correlation step Di can also be expressed in reduced length l, relative to the total length of mixer. Plotting the correlation coefficient R(l) versus l, important information can be obtained on the macro- and microstructure of concentration patterns. Fig. 3a,b shows correlograms characteristic of the studied mixing mechanisms after different mixing times. In these examples, individual particles were used as samples (n= 1) to reveal the microstructure of the simulated concentration patterns. The most important differences between the correlograms obtained for different mixing mechanisms were as follows: In diffusive mixing, shown by Fig. 3a, the correlation coefficients extrapolated to l= 0 are gradually diminishing from unity to zero with increasing mixing times. At small l values, the slope of these curves is almost zero and apart from minor statistical variations, is continuously changing till the intersection with the horizontal axis. In convective-shear mechanism (Fig. 3b), the slopes are much steeper at l= 0, evidencing significant reduction in the scale of segregation [17], just at the beginning of the process. The extrapolated intersections with the vertical axis remain close to unity, while the slopes become steeper and steeper with increasing mixing time. Comparing random variation
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along these correlograms, it is relatively low for diffusive mixing and much higher for convective-shear mechanism. During this study, it was also noted that sample size had a great influence. It has been proved that the correlation technique had the best sensitivity in the case of small sample sizes, both to evaluate the degree of homogeneity and to reveal the microstructure of concentration patterns, especially at the early stages of the process.
5.3. Characterization by fractal analysis Fractal analysis has been advantageously used for a long time to characterize various natural objects and technical phenomena [18 – 20]. This technique is based on the fact that the boundary length of an object having a rugged contour can be quite different, if it is measured with different scales. When the measured length L(l) is plotted on a log – log diagram against the size of the measuring scale l, called a Richardson plot, generally a straight line is obtained [19] whose slope is characteristic of the fractal properies. This dependence is expressed as L(l) :l − dF + 1
(3)
where dF is the fractal dimension by definition. To analyze the micro-scale concentration patterns obtained by one-dimensional simulation, a modified fractal method can be proposed. Considering a situa-
Fig. 3. Correlograms of simulated concentration patterns; (a) diffusive mixing; (b) convective-shear mixing.
tion when the size of samples is sufficiently small, for example close to the size of one particle, a rugged concentration pattern is obtained. A reduced distance kl between any two points of this curve can be calculated from the concentration difference Dx between the samples being at distance l from each other as kl =
)
l 2 + Dx 2 l
)
(4)
If there is no concentration difference between the samples, i.e. Dx =0, then kl is equal to unity. In the case of any concentration difference, kl is greater than unity. The average of these kl values over the whole range of concentration pattern analyzed is %kl K(l)=
N
N
(5)
In the completely mixed state when the particles of a constituent are well distributed among the others, and in the case of small sample size, e.g. if n=1, the number of concentration jumps between the neighboring samples is maximal, therefore this mixture gives the greatest K(l) value. In the totally segregated state, there is no concentration jump along a large range of l values, i.e. the average fractal length is much lower, close to unity. Therefore the increment of K(l) relative to the totally segregated state is DK(l)$ K(l)− 1
(6)
which well characterizes the actual state of mixture. Similarly to the conventional fractal length L(l), the value of DK(l) also decreases as l increases, and vice versa. Plotting DK(l) as a function of l on a log–log scale, multifractal properties, i.e. not totally linear correlations were experienced for the studied concentration patterns. But the first sections of these curves were straight lines, similar to the usual Richardson plots. It was also proved that slopes and intersections with the vertical axis were diminishing with increasing mixing times. Sample size had a great influence: higher slopes and intersections were obtained with smaller samples. Maximal sensitivity was experienced with sample size equal to one particle. But, it should be emphasized that in the case of large sample sizes this technique does not remain unambiguous, therefore this method is recommended only to characterize the microstructure of a mixture. From the results obtained by this fractal analysis, it can be concluded that this technique is also suitable to characterize the mixing mechanism. Fig. 4a,b shows the modified Richardson plots of concentration patterns resulting from diffusive-type and convective-shear mechanisms, respectively. In diffusive mixing, the slopes and intersections of lines obtained after different mixing times are rapidly increasing at the beginning of
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Dx xi − xi − Di = f1(xi − Di ) = Di Di
671
(8)
or in a simpler way as xi = f2(xi − Di )
Fig. 4. Modified Richardson plot of concentration patterns obtained by simulation; (a) diffusive-type mixing; (b) convective-shear mixing.
the process (Fig. 4a). Then, this change slows down considerably as the concentration pattern approaches the completely randomized state, because of the decreasing rate of development of the final microstructure of the mixture. In convective-shear mixing mechanism, however, fractal properties change much more slowly even at the beginning and remain nearly uniform during the whole process. It means that it is almost impossible to achieve a random mixture by this mechanism at the particle level, even after a very long mixing time. From these differences it follows that studying fractal properties offers a suitable tool to assess the main mechanism of a mixing process.
(9)
where Di is the temporal or spatial difference between the serial numbers of samples, xi − Di and xi are the concentrations of the (i −Di )th and ith samples running along the concentration pattern. Usually, Di is fixed and is equal to one. If the microstructure of a mixture has to be revealed, the sample size should be suitably small but to get an informative picture, it has to be composed of at least several particles. To study the feature of medium- or large-scale structures, sample size can be considerably larger. Useful information is obtained on the structure of concentration patterns created by different mixing mechanism by plotting attractors or phase-portraits after different time intervals. Fig. 5 gives demonstrative examples for different stages of diffusive mixing, showing the transformation from segregated state to an almost completely randomized mixture. It can be seen that at an early stage of mixing the phase-portrait of a mixture corresponds to a winding curve (Fig. 5a) extending diagonally between two extreme concentration values, revealing tendentious change along a characteristic direction. Later on, these signatures become less and less elongated (Fig. 5b,c), also indicating that differences between sample concentrations are gradually
5.4. Phase-space technique Recently, the phase-space technique has frequently been used for chaos analysis [21 – 23]. In this data representation method, the derivative dj/dt of a temporal signal is plotted against the signal value j(t), which clearly characterizes the dynamic nature of a chaotic phenomena, resulting in an attractor function f(j(t)) as dj = f(j(t)) dt
(7)
Using finite differences in the spatial coordinate system, the attractor function of a concentration pattern can also be defined as
Fig. 5. Phase-portraits obtained by diffusive-type mixing (sample size corresponds to 20 particles); (a, b, c, d) signatures in order of increasing mixing times (t1 Bt2 Bt3 Bt4).
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sive mixing, because the resultant fragments will be even smaller.
5.5. Spectral density technique
Fig. 6. Phase-portraits obtained by convective-shear mixing (sample size corresponds to 20 particles); (a, b, c, d) signatures in order of increasing mixing times (t1 B t2 B t3 B t4).
diminishing. The almost globular curve in Fig. 5d corresponds to an almost completely randomized mixture. Fluctuation and loops indicate the stochastic nature of the process. It also has to be recognized that the area covered by a signature is characteristic of the actual state of mixture. A smaller area corresponds to less intensity of segregation. This area also depends on the size of samples, because it decreases as the sample size increases. Therefore, for a given aim of investigation, the sample size should be fixed in advance, depending on whether the micro- or the macrostructure is the most important. Concentration patterns obtained by the convectiveshear mechanism show quite different phase-portraits. At an early stage of mixing there is no tendentious elongation (Fig. 6a). Another important feature is that, instead of the generally continuous curve characterizing the diffusive-type mixing also depending on sample size, a number of joined straight lines are obtained with randomly varying directions and lengths. These lines correspond to the change of concentrations in samples taken successively from the fragmented and randomly distributed component regions. As fragmentation proceeds during convective-shear mixing, the number of these lines increases, together with the reduction of their sizes. After a suitably long time, these phase-portraits become quite similar to those obtained by diffu-
Other pattern recognition methods can also be used to discover the mixing mechanism. Among them, the spectral density technique is a promising method. Power spectra of spatial or temporal concentration signals can be determined by using Fast Fourier Transformation (FFT). In real mixtures such signals can be measured for example, by light reflection or image analysis. The concentration pattern used for the spectral density technique should be determined from samples as small as possible, to evaluate the micro-scale structure of a mixture. As with fractal analysis, the best sensitivity in this regard can be obtained by measurements on the particle level. In the case of n= 1, the concentration signal consists of square waves having two values: zero or unity. In a random arrangement of the constituent particles, the maximal spectral density is obtained at the high frequency range, while in a more segregated state this characteristic peak is shifted to higher wavelengths, i.e. to low frequency. For uncompleted mixtures, spectral densities represent intermediate states transforming from the segregated to randomized state. The mechanism of mixing influences the way of transformation. In the first stages of mixing, the diffusive mechanism creates the concentration pattern, which has high frequency concentration variation within a short region near the initial border of components, while there is no concentration variation in its remaining parts. Therefore, power spectra produced by this mechanism show a relatively small characteristic peak at the high frequency range. This peak increases as the mixed diffuse zone spreads out between the pure components, indicating the gradual development of microstructure along the concentration pattern. In convective-shear mechanism, however, concentration jumps appear randomly between contiguous regions of pure components, at an ever-growing number of locations along the length of the concentration pattern. The sizes of these regions gradually decrease, while the numbers of concentration jumps are increasing. Therefore, at the beginning of the mixing process the spectral density will be higher at low frequency regions, and later on it will be gradually shifted towards higher frequencies as the process advances. Although not too much detailed work has been done till now in studying this latter method, it offers an additional tool for mixture characterization. To resolve its potential in evaluation of mixtures or mixing mechanisms, more detailed investigation is necessary in the future.
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6. Summary and conclusions The way of mixing highly influences the rate of the process and also the achievable homogeneity. The simultaneous action of diffusive and convective-shear mixing has advantages in both respects, due to their useful synergetic effects. Therefore, to find suitable tools to discover the main mechanisms has great importance both for equipment design and process improvement. Investigation of macro- and microstructures of the resultant concentration patterns may be a possible way for this. In this study, various artificial concentration patterns were created by one-dimensional numerical simulation of diffusive and convective-shear mixing. Concentration patterns obtained by these mechanisms are highly different, both in their macroscopic and microscopic characteristics, indicating the different method and route of formation. To investigate the applicability of different pattern evaluation methods, in addition to the spatial and frequency distribution of sample concentrations, autocorrelation functions, fractal analysis, phase-space technique and spectral density were tested. It was proved that all of these methods are suitable to trace the improvement of mixture homogeneity, and to distinguish the mechanisms of the process. In addition, all have certain specialties, which can advantageously be used from different respects. Frequency distribution of sample concentration and its variance as a function of sample size may help to indicate the right sample size, beside giving information on homogeneity and the mixing mechanism. The correlation technique can be applied to assess both the micro- and macrostructure of concentration patterns, giving information on the characteristics of microscopic and large-scale processes. During this study, it has been proved that the correlation technique had the best sensitivity in assessing mixing mechanism in the early stages of the process, by applying small sample sizes. A special kind of fractal analysis was also tested. It has been proved that this technique can be used advantageously to assess mixing mechanism. However it is unambiguous only at small sample size, and showed the best sensitivity if the sample size is equal to one particle. Using larger samples may also give information on the rate of process and mixture quality, but in this case the fractal dimension of concentration patterns can also be diminishing during the process, and the special fingerprint of diffusion cannot be recognized. The phase-space technique is a very illustrative method to indicate both the micro- and macrostructure of concentration patterns by a single phase-portrait. Therefore it can be used effectively to recognize mixed mechanisms, i.e. in the case of simultaneous action of diffusion and convection-shear. This method is not very
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sensitive to sample size, therefore this situation gives more freedom in selecting the right scale of scrutiny corresponding to the actual purpose of investigation. The spectral density technique is a promising method to assess mixing mechanism, but more detailed investigation on its features is necessary in the future. In real processes, different mixing mechanisms may be acting simultaneously, meaning that they cannot be observed in sterile forms. However, the structure of concentration patterns preserves the traces of both mechanisms. Therefore, these pattern recognition methods can help to answer which is the dominant mechanism, and if they are acting together or not. In principle, these assessment methods can be used not only for solids mixtures, but also to resolve mixing mechanism in fluids and pseudo-plastic materials, if a suitable technique is found to measure their spatial or temporal concentration pattern. 7. Nomenclature dF f(j(t)) f1,2(xi−Di) Di kl K(l) DK(l) L(l) M n N R(Di ), R(l) t x Dx Greek symbols l s2 Indices 0 i r
fractal dimension, according to Eq. (3) attractor function defined by Eq. (7) attractor functions, defined by Eq. (8) and Eq. (9) correlation step, or the increment in serial numbers of samples distance between two points of concentration curve depending on scale length l average of kl values along the whole concentration pattern increment of K(l) relative to the segregated state total length of rugged curve measured by a scale of length l Kramers’ mixing index, according to Eq. (1) number of particles in a sample number of samples auto-correlation function, Eq. (2) time sample concentration along the concentration pattern concentration difference between two samples
reduced length, correlation step, or scale used to determine fractal length variance of sample concentrations initial or totally segregated state serial number of a sample completely randomized state
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Acknowledgements The author wishes to acknowledge the support of the Hungarian National Foundation for Fundamental Research (OTKA No. T 29313) and the help of the US – Hungarian Joint Fund (JF No. 593/96) also serving as a background for these results.
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