USING THE METHODS OF GEOSTATIC FUNCTION AND MONTE CARLO IN ESTIMATING THE RANDOMNESS OF DISTRIBUTION OF A TWO-COMPONENT GRANULAR MIXTURE DURING THE FLOW MIXING

Jolanta Królczyk, Marek Tukiendorf
Department of Agriculture and Forest Technology, Opole University of Technology, Poland

ABSTRACT

Mixing granular materials has a signficant importance in a lot of industrial branches such as food and agricultural ones. For that reason, it is of extreme importance to get to know the phenomena determining this prcoess. Mixing non-uniform, two-component grnaular structures is a complex phenomenon with a number of factors affecting its course. Obtaining a uniform mixture is dependent on the properties of the elements, type of the mixer as well as the conditions in which the process takes place.

The granular structure used in the studies was that of lead – mustard with the relation of diameters d₁/d₂ = 1.55 and the relation of density ρ₁/ρ₂ = 8.25. The proportion of volume was 1:9. The components were mixed using the funnel-flow method in a laboratory model of flow mixer. The focus of interest was distribution of tracer (lead) on the surface of particular cross-sections of the mixer obtained by means of successive flows.

The first step in estimating the randomness of the quality of the tracer was to use a computer image analysis. An analysing program PATAN was used to this aim, on the basis of which the coefficients of the location of the tracer’s grains were determined on the surface of the container’s segments. The coefficients were next used to analyze the spatial distribution given in program S-PLUS. The estimation of the randomness of distribution of a two-component granular mixture was performed on the basis of geostatic function G(y) and Monte Carlo method.

The paper attempts to adapt these methods in the estimation of the share of randomness of the tracer’s distribution. All photographed cross-sections were subjected to analysis. The method based on geostatic function G(y) used for the estimation of relations between the points is based on the distance to the closest neighbour. Monte Carlo method was used paralelly. After a simulation of the coefficients of the tracer’s location and with a definite number of simulations, two lines of reference were obtained: maximum and minimum ones, which were then compared with the line formed on the basis of empirical data. Basing on the suggested statistical method, the purpose of the studies was to determine whether the mixing process is dependent on the properties of the mixed materials or it is a purely random process where the movement of grains is only accidental.

It was found out that for the studied structure this process is mainly dependent on the properties of the components. This is testified to by the graphds obtained by means of function G(y) and Monte Carlo method. However, one should not neglect random events in the movement of grains, which is shown in the limits.

Key words: granular materials, granular mixture, geostatic function, Monte Carlo method, computer image analysis.

INTRODUCTION

Mixing granular materials is a process of dispersing a few elements in a mixture through a chaotic, accidental movement of grains, where both the elements’ characteristics, type of the mixer and the conditions in which the process takes place have a significant impact on its course. This is shown in better mixing of the components or their secondary segregation [1]. That is why the achieved effect of mixing and its speed are the functions of a number of parameters, referring both to the mixed materials and the conditions in which the process takes place. Parameters characterizing the properties of the mixed materials include, for example, distribution of the dimensions of particles, shape of the grains’ area, proper and flow density, content of moisture and the way of its binding [1, 11]. Important features of the mixer are its dimensions, shape of the mixer and blender and the kind and manner of distributing the devices for loading and emptying the mixer. Significant parameters characterizing the conditions of the process include mass (or volume) shares of particular components, proportion between the mixture’s volume and the working volume of the blender as well as the manner and sequence of passing the elements of the mixture and the mixing intensity [1].

Mixing granular materials is important in a lot of branches of industry such as food and agriculture industries, which is why it is important to get to know the laws governing these processes [1, 13, 30]. The basic problem of mixing granular materials concerns the achievement of dynamically stable states with good parameters of a component’s dispersion. In the case of granular materials, contrary to liquid or gas mixtures, the compositions of the mixture sampled from different places is not always the same. The mixture obtained as a result of mixing a few, or even several, elements often points to lack of uniformity in various places of the layer [1]. Both statistical and non-statistical methods are used for quantity estimation of the mixture’s state. In the case of statistical analysis, the numerical index estimating the uniformity of the mixture is called a degree of mixing, whereas mathmatical expressions that were not built on the basis of statistical values such as variance or standard deviations are called indexes of mixing [1]. One of the cheap and fast ways of mixing big volumes of granular structures is flow mixing. It consists of mixing during the flow from container to container. In industrial conditions, blenders with a cascade system of bottoms placed in the vertical axis of the high silos column are practically used [1]. When the mixed components differ with the density or the size of diameters, the process of segregation takes place apart from the process of homogenization [1, 7, 16]. Unfortunately, while pouring the components of very different densities, the heavier material shows a tendency to concentrate in the core of the container and in its lower parts. It was noticed that materials of greater densities “sink” in those with smaller densities [1, 24, 26]. The use of special inserted strips inside the mixers can significantly affect the variables of the distribution of the mixed components [15, 19, 22]. Mixing the components by the method of normal pouring means that physical properties such as density and dimensions of the diameters are decisive of the results of mixing. It was proved that changing these parameters significantly affects the quality of the obtained mixture [5, 21, 25]. It was also shown in many papers that the quality of the mixture is better with smaller differences between the densities and dimensions of the diameters of the mixed components [1, 4, 5, 6, 8, 14, 15, 17, 18, 19].

Methods of computer image analysis are increasingly more often used to describe the process of mixing in the technology of granular materials. These methods are also successfully used in many fields, e.g. two-phase mixture flows – to estimate the flow structure [27]. Studies by Boss, Krótkiewicz and Tukiendorf [2, 3, 22, 23], based on this method, allowed to classify the typical distributions of components on the surface of selected sections of containers. Variance analysis along the radii of the sections was used to estimate the quality of granular sections. The results made it possible to distinguish and define the ring-like and core state of mixing. In the former case, a ring of grains of the key component occurring in the outer part of the section was noticed. On the other hand, in the case of the core state of mixing, this distribution was characterized by the accumulation of tracer in the middle of the section. The location of grains in the mixer or in the store container can show the properties of even distribution and such placement of grains is the most desired phenomenon. This state of mixing is defined as random and it is characterized by the same or similar distance between the grains [1, 16, 29].

Uniform volume distribution in the container is as important as uniform (random) distribution of the tracer (fig. 1). The tracer can distribute itself uniformly as regards the volume in the whole container; however, on the surface of successive cross-sections it can take different distributions, e.g. ring-like distribution in cross-section Nos. 2 and 10 and a core distribution in cross-section No. 5 (fig. 1). The tracer distributes itself unevenly along the container’s axis [24]. A perfect mixture would be such in which the tracer would be distributed in a random way on the surface of cross-sections and evenly as regards the volume.

Random distribution is difficult to achieve, especially when the mixed elements are considerably different as regards their density, which is caused by the phenomenon of segregation.

As was mentioned earlier, the properties of the mixed media, features of the mixing device and the conditions of conducting the process of mixing are extremely important for the process of homogenization of non-uniform, two-component structures of grains. However, a certain part played by randomness of events taking part in the process should not be neglected since the elements of the mixture are dispersed in the blender by the accidental, chaotic movement of the grains.

The article presents an innovatory way of estimating the randomness of the distribution of a two-component granular mixture during the flow mixing with the use of the geostatic G(y) and Monte Carlo methods. Application of the method is confirmed by the existing views and conclusions from the studies on explaining the mechanism governing the process of mixing non-uniforms granular structures.

METHODS

Funnel-flow method. A granular structure composed of two elements (lead – mustard) was mixed using the method of successive flows. The components differed with the dimension of diameters: lead (4-4.5 mm), mustard (2.5-3 mm), d₁/d₂ = 1.55, and densities (ρ₁/ρ₂ = 8.25). The object of studies was the manner of distributing the tracer (lead) on the surface of particular cross-sections of the blender.

The studies made use of a laboratory model of a flow mixer (fig. 2).

Mixing consisted of pouring a two-component mixture from container to container placed in the construction one upon the other in volume proportions:

Tracer : filler = 1 : 9

The dimensions of containers were the following: the height of the cylindrical part 200 mm, inside diameter 150 mm. The container was built from 10 dismantable rings of the same volume. The tracer (lead) before mixing was begun was placed in the fifth segment. Next, the mixed granular materials were poured from the main container to the lower one; the place of the containers were changed and the following flows took place. Ten successive flows were mixed on the way. It was shown earlier [24] that 10 successive flows is a sufficient number to achieve the state of dynamic balance of the structure.

A computer image analysis. The dismantable construction of the container made it possible to photograph the area od each section of the container. A picture of 10 areas after each of the ten flows was achieved, depending on the number of flows. After 10 flows and for 10 sections, 100 pictures were obtained (fig. 3), which were subjected to further analysis.

Pixelization of the digital picture was performed into 2500 pixels in PATAN ®program. The range corresponding only to the circular section of the mixer’s ring was separated from the pixelized bmp-map of the digital picture. This picture was recorded in format 2⁸ bpp (scale of greyness) and subjected to analysis according to the condition of the arithmetic means (128), accepting all shades below to be dark grains (lead) and all shades above 128 as light grains (mustard) (fig. 5).

In this way the coefficients of grains of the tracer x and y were obtained, which were next in the form of a sheet submitted to analysis in program S-PLUS [12].

One option of this program allows for an analysis of spatial distribution of the given points, defining the statistical randomness of this distribution. Pictures of all photographed sections were analyzed.

STATISTICAL ANALYSIS

The authors’ intention was an attempt to adapt two statistical methods (Ĝ(y) and Monte Carlo method) in order to find out whether the mixing process for a studied structure by a process of components is dependent more on the properties of the mixed media and the manner of mixing or the movement of the grains is a purely random event.

In the content of these needs, two modules of package S-PLUS were used for statistical estimation of distribution of particles on successive areas.

Function Ĝ(y). The analysis of distribution of points in space can be useful in a description of such processes as mixing granular materials [9, 10, 26]. A distribution of points in space is a set of points with the coefficient data irregularly distributed in a limited area of space. The method of the nearest neighbour was used to estimate the randomness of distribution of given points on the surface. This method makes it possible to give objective determination of relations of the distance between the points. Defining the distance to the nearest neighbour d_i as distance from i^th point to the closest point in area A, function G(y) of the empirical distribution of distance d_i can be used to estimate the distribution and randomness of points in space. Function G(y) is described by means of the following formula:

n – number of points in area A [12].

In order to explain interpretation of the graph obtained as a result of using this function, a simple example will be employed.

Figure 6 shows distribution of points in the area. In case of figure 6, the points are distributed evenly in the area. In case of figure 6b, it is possible to observe a clear grouping of points. An example of even distribution is distribution of desert plants, whose regularity is most probably caused by the production of toxic substances that inhibit the growth of plants in the nearest neighbourhood [28]. On the other hand, cluster distribution is most frequently encountered in nature, for example, in the case of animals living in groups.

The relations existing between the points in figure 6 are easily noticeable. However, the relations between grain particles in the volumes of big containers are not so easy to define. Therefore, more precise mathematical models determining the relations occurring between the points in space should be sought. Function G(y) presented in figure 7 can be one of the methods used to describe the spatial distribution of points (granular particles). Graph presented in figure 7 was prepared for the dependencies in the example from figure 6.

In figure 7a we can notice the majority of neighbours (closest points) with high values of distance (cf. fig. 6a). It can be stated on this basis that the points are regularly distributed in the area [9, 10, 12]. In the case of figure 7b, the majority of points with small distances between each other can be seen and such a form of the graph testifies to their clustering.

Monte Carlo method. The second method in the analysis of randomness of points distribution on the surface is Monte Carlo method [12]. An important role in this method is played by drawing lots for the values characterizing the process and the drawing concerns the earlier known distributions [31].

Monte Carlo method allow for example for a simulation of the empirical distributants for the distance of the nearest neighbour with a definite number of simulations. The means from the simulations makes lines of reference – maximum and minimum, which form “an envelope”. Function G(y) will be found inside “the envelope” when the distribution of points is random. When function G(y) “falls outside” the envelope, this testifies to lack of randomness of the distribution.

RESULTS

Function G(y). Successive graphs of the geostatic function G(y) were obtained for the studied distributions of the tracer (lead particles) on the surface of the cross-sections of the blender. Figures 8 and 9 show pictures of these functions for the selected cross-sections and times of mixing here understood as successive steps or flows.

Figures 8a, 8b and 9b clearly show the majority of neighbours with low values of distance to each other (clustering of particles). After a certain time of mixing a significant effect of randomness on the state of the particle distribution was observed on one of the presented sections (for the second ring after nine successive flows) (fig. 9a). This proves the earlier mentioned theory about a certain role of accident in the grain movement and randomness of events. However, another tendency of distributions is observed for distributions in figures 8a, 8b and 9b – the majority of distances with the value of 1. In the case of graph in figure 9a the majority of points with small distances towards each other are not any more so clear as in the other graphs.

Grouping of the points in the area of small distances towards each other testifies to the clustering of particles pointing at lack of randomness in the process of mixing. It should be supposed that differences in the properties of the mixed components exert a greater influence (ρ₁/ρ₂ = 8.25).

Monte Carlo method. Using the Monte Carlo method, the studies performed prediction of particle distribution for all sections after successive steps of mixing (10 sections × 10 steps of mixing = 100 records). Prediction was carried out for 100 records after 20 simulations of coefficients of the location of the tracer. Exemplary graphs are shown in figure 10 and figure 11.

Number 20 of simulations was adopted arbitrarily as a sufficient statistical representation. The examined points of the tracer’s distribution on the surface of the ring were not found inside the “envelope” (fig. 10a, 10b, 11b), which confirms the conclusion drawn on the basis of geostatic function G(y) about the grouping of points, hence lack of randomness of the process. The use of Monte Carlo method also confirmed the conclusion about the role of randomness of the grain movement in the process of mixing (the second section, after nine steps of mixing) (fig. 11a, cf. fig. 9a). It is clear in the picture that in a certain area of the distributant the empirical value will be found inside “the envelope” formed as a result of simulation of the placement of points.

CONCLUSIONS

The method of computer analysis is a good tool in estimating the randomness of a two-component granular mixture.

The use of the geostatic G(y) and Monte Carlo methods proves that the distribution of particle concentration of a two-component granular structure lead – mustard on the surface of particular rings of the container depends on the properties of the mixed media such as relations of diameters and relations of densities. The cause of such course of the process of mixing can be seen in big density, since the relation between the mixed components was ρ₁/ρ₂ = 8.25. However, the role of accidental movement of grains so the randomness of the process should not be ruled out even with big differences of the properties of the mixed components.

REFERENCES

1. Boss J., 1987. Mieszanie materiałów ziarnistych (Mixing of granular materials). PWN, Warszawa – Wrocław [in Polish].

2. Boss J., Krótkiewicz M., Tukiendorf M., 2002. Porównanie metod oceny jakosci stanu mieszaniny ziarnistej podczas mieszania w przesypie [The application of picture analysis as a method to evaluate the quality of granular mixture during funnel-flow mixing]. Inż. Roln. 4 (37), 27-32 [in Polish].

3. Boss J., Krótkiewicz M., Tukiendorf M., 2003. The Application of image analysis as a method of evaluation of granular blend’s quality. Polish J Food Nutr. Sci. 12/53, 2, 22-30.

4. Boss J., Tukiendorf M., 1997a. Mixing of Granular Materials Using the Method of Funnel-Flow, Powder Handling & Processing 9, 4, 341-343.

5. Boss J., Tukiendorf M., 1997b. Mieszanie materiałów ziarnistych metodš wysypu normalnego [Mixing of granular materials using normal discharge]. Zesz. Nauk. Polit. Łódz. Inż. Chem. 22, 30-38.

6. Boss J., Tukiendorf M., 1997c. Mixing of granular materials using the method of funnel-flow. Powder Handling & Processing 9, 4, 341-343.

7. Boss J., Tukiendorf A., 2000: Mixing of Granular Materials Using the Method of Funnel Flow. The best of powder handling & processing bulk solids handling. Trans. Tech. Publications 4, 65-67.

8. Chen S. J., 1972. The Mixing of Particulate Solids in a Motionless Mixer, Bulletin Kenics Co, July.

9. Diggle P.J., 1983. Statistical analysis of spatial point patterns. Academic Press, New York.

10. Gerrard D. J., 1969. Competition quotient: a new measure of the competition affecting individual forest trees. Research Bulletin No. 20, Agricultural Experiment Station, Michigan State University.

11. Horabik J., 2001. Charakterystyka własciwosci fizycznych roslinnych materiałów sypkich istotnych w procesach składowania [The characteristics of significant physical properties of plant granular materials during the storage]. Acta Agrophysica PAN 54, Lublin [in Polish].

12. Kałużny S. P., Vega S. C., Cardoso T. P., Shelley A. A. S+Spatialstats, 1996.

13. Kamiński M., Wirska E., 2000. Modelling of Powder Flow in Silos with Flexible Walls. The Best of powder handling&processing bulk solids handling. Trans Tech Publications, 23-24.

14. Manjunath K. S., 1992. Homogenization of Bulk Powders in Plane Flow Silos, POSTEC 11, 23-24.

15. Reimbert M., Reimbert A., 1976. Silos, Theory and Practice, Trans. Tech. Publications, Germany.

16. Roberts A. W., 1998. Particie technology-reflections and horizons: An engineering perspective. Trans. In. Chem. Eng. 76, A.

17. Sadler J., Johnson T., Mahmound M. H., 1995. Designing Silo for Flow Patterns, ACI Structural Journal, March-April, 219-226.

18. Safarin S. S., Harris E. C., 1995. Desing of Partially Post-Tensioned Circular Concrete Silo Walls, Concrete International, April, 43-46.

19. Schlick H., Gebauer F., Auther A., Gallinat J., 1996. Relationship Between Flow Properties and the Process of Loading In Silos With Central Jones and Plough Fedder Discharge, Bulk Solids Handling 16, February/March, 83-89.

20. S-PLUS 4.0 Release for Windows Copyright © 1988-97 Mathsoft, Inc.

21. Tukiendorf M., 1995. Optymalizacja procesu mieszania przesypowego niejednorodnych układów ziarnistych [Optimization of discharge mixing process of non-homogenous granular systems] Rozpr. dokt., Polit. Łódz. [in Polish].

22. Tukiendorf M., 2002a. Zagadnienie rozkładu koncentracji składników niejednorodnej mieszaniny ziarnistej podczas mieszania w przesypie [The problem of concentration distribution of non-homogenous mixture components during mixing in discharge]. Inż. Aparat. Chem. 4, 139-140 [in Polish].

23. Tukiendorf M., 2002b. Ocena jakosci mieszaniny ziarnistej w przekrojach poziomych silosów podczas mieszania w przesypie [An evaluation of quality status of granular blend in the horizontal profiles of bins during funnel-flow mixing]. Inż. Aparat. Chem. 4, IZACAX 41 (33), 139-140 [in Polish].

24. Tukiendorf M., 2002c. Sposoby oceny jakosci mieszanin ziarnistych w zbiornikach [Ways of evaluation of granular blends in bins]. Acta Sci. Pol., Technica Agraria 1(2), 51-59 [in Polish].

25. Tukiendorf M., 2003a. Wpływ zmiany wymiarów srednic czšstek na wyniki mieszania materiałów ziarnistych podczas wysypu ze zbiorników [Effect of variable particle dimensions on the results of mixing granular materials during their pouring-out from the bins]. Probl. Inż. Roln. 2, 21-26 [in Polish].

26. Tukiendorf M., 2003b. Characteristics of mixing granular materials achieved by using methods of variance analysis and geostatistical functions. EJPAU, Agricultural Engineering, Volume 6, Issue 1. http://www.ejpau.media.pl/series/volume6/issue1/engineering/art-03.html.

27. Ulbrich R., 2002. Rozpoznawanie obrazu w zastosowaniach do badań przepływu mieszanin dwufazowych [Picture recognition In application to research on two-phase flow]. Wyd. Polit. Opol., Pr. zbior., Studia i monografie 138 [in Polish].

28. Ville C. A., Solomon E. P., Berg L. R., Martin D. W., 1996. Biologia [Biology]. Multico, Warszawa [in Polish].

29. Williams J. C., 1976. The segregation of particulate materials. A review Powder Techno. 15, 245-251.

30. Zander L., 2002. Mieszanie – zagadnienie stale aktualne [Mixing – problem constantly current]. Inżynieria i aparatura chemiczna 4, 5-6 [in Polish].

31. Ziliński R., 1970. Metody Monte Carlo [Monte-Carlo methods]. WNT, Warszawa [in Polish].

Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed 'Discussions' and hyperlinked to the article.