Electronic Journal of Polish Agricultural Universities (EJPAU) founded by all Polish Agriculture Universities presents original papers and review articles relevant to all aspects of agricultural sciences. It is target for persons working both in science and industry,regulatory agencies or teaching in agricultural sector. Covered by IFIS Publishing (Food Science and Technology Abstracts), ELSEVIER Science - Food Science and Technology Program, CAS USA (Chemical Abstracts), CABI Publishing UK and ALPSP (Association of Learned and Professional Society Publisher - full membership). Presented in the Master List of Thomson ISI.
2004
Volume 7
Issue 2
Topic:
Agricultural Engineering
ELECTRONIC
JOURNAL OF
POLISH
AGRICULTURAL
UNIVERSITIES
Węgrzyn M. 2004. MIXING OF THE HETEROGENEOUS GRANULAR MATERIALS IN A DOUBLE CONE MIXER, EJPAU 7(2), #01.
Available Online: http://www.ejpau.media.pl/volume7/issue2/engineering/art-01.html

MIXING OF THE HETEROGENEOUS GRANULAR MATERIALS IN A DOUBLE CONE MIXER

Marek Węgrzyn

 

ABSTRACT

Mixing of heterogeneous granular materials is accompanied by the segregation phenomenon the result of which is that having obtained the highest value of mixing degree there occurs partial distribution of mixing components up to the equilibrium state. The knowledge of parameters affecting the segregation phenomenon is essential to restrict the disadvantageous phenomenon. The main factors affecting the segregation are dimension differences between tracer grains and continuous phase as well as their density differences.

Key words: mixing, granular materials, segregation, mixer..

INTRODUCTION

Mixing of granular materials is commonly used in many industries. The process is applied in food industry to make various products which are packed or to make intermediate products for further processing or in the case of feeding stuff the product is ready to feeding or storage [9, 10]. Mixing process is particularly important in the production of mixtures where the portion of major component in the finished product is very small. It is the case, for example, while adding drugs to feeding stuff where the dispersion of this active component must be very regular.

Components of granular mixtures vary in terms of grain dimensions density, shape, humidity etc. Those factors have an important influence on the segregation phenomenon concurrent with mixing process [1, 8, 12, 13, 19, 21]. Segregation, unlike mixing is a deterministic process and the more intensive it is, the bigger is heterogeneity of the mixed materials. Thus, even after achieving a maximum degree of mixing there is a partial segregation of the components to the equilibrium state [2, 16]. The differences in grain dimensions and density have the most important influence on the behaviour of the granular material [3, 16].

Researchers focus mainly on the maximum degree of mixing which is very essential to obtain high quality product. However, during further treatment such as storage or transport, the mixture will undergo partial segregation.

The research on equilibrium degree of mixing in a static mixer Kenics 180° for materials of different grain dimensions was conducted by Knapik [11]. Further investigations on the equilibrium state of mixing for grain sets varying in grain dimensions and density were made by Węgrzyn [20]. Tukiendorf [17] carried out research on the equilibrium degree of mixing in the funnel flow mixing. The relationship between tracer grain diameter d1 and grain diameter of continuous phase d2 and the relationship between tracer grain density r1 and grain density of continuous phase r2 was found out by Boss and Węgrzyn [7] for roof static mixer and by Boss and Tukiendorf [5, 6] for funnel-flow mixing.

The results obtained for funnel-flow mixing and static mixer show a little different values which may suggest that the relationship is characteristic only for a granular mixer. The authors goal was to investigate how the equilibrium degree of mixing changes as the heterogeneity of granular material increases as well as to find out the relationship between grain diameter and grain density ratio in the heterogeneous structure during mixing in a double cone mixer.

METHODS AND RESULTS

The investigations on equilibrium degree of mixing were carried out in a laboratory double cone mixer. An additional element applied was a container consisting of ten elements. It allowed the mixture to be divided into ten samples and to analyse their composition. The diagram of the apparatus is presented in Figure 1.

Figure 1. Apparatus diagram:
a) initial position of the continuous phase and the tracer, b) container

The mixer contains a mixing chamber with the capacity of 5000 cm3. Mixing process took place when the mixing chamber was filled in 45%.

Spherical grain materials were used to eliminate the effect of shape factor on the mixing process. Before starting the investigations the materials were fractionated on screens. Grain diameters and their density are shown in Table 1.

Table 1. Particle sizes of material investigated

Dimension, mm

Fraction

white
mustard

rape

red clover

corn

field pea,
pea

Al2O3

agalite

sand

A

              

0.71

B

              

1.22

C

1.73

1.73

1.73

    

1.73

  

1.73

D

2.24

2.24

2.24

    

2.24

2.24

  

E

2.74

  

2.74

    

2.74

2.74

  

F

3.24

  

3.24

  

3.24

3.24

3.24

  

G

      

3.74

3.74

3.74

3.74

  

H

      

4.24

4.24

4.24

4.24

  

I

      

474

4.74

  

4.74

  

J

        

5.24

      

Density
kg∙m-3

970

1100

1200

1230

1360

1680

2400

2400

Every time volume fraction of continuous phase was 90% while tracer volume was 10%. The material of continuous phase was placed at the bottom, then its layer was levelled and tracer material was poured and levelled. Having poured the components the chamber was closed and the rotation of the mixer was turned on for a definite period of time. When the mixing process was completed, the container was put on the outlet at its upper position. After turning the chamber by 180° its contents was poured to the container where the whole mixture was divided into ten samples.

Removing each segment of the container, mixture samples can be obtained. Mixture components obtained from the samples were segregated and the tracer volume in the sample and standard deviation were determined

         (1)

where:
xi – tracer fraction in a sample;
p – probability of the presence of tracer component in the mixture;
n – the number of samples.

Standard deviation served to determine the degree of mixing based on Rose’s formula [15]

                  (2)

where

            (3)

Rose’s definition was chosen from many others as it represents best the character of the changes taking place in the mixed granular system.

The defined values of the mixing degree in the time function for granular materials were presented in the form of diagrams. Examples of changes in the degree of mixing derived from equation (2) as the time function was shown in Figures 2 and 3. The diagrams show that having obtained the maximum degree of mixing, as a result of segregation, a lower value appears and the system tends towards the equilibrium degree of mixing. The value of mixing degree in the equilibrium state Me is definitely lower than the maximum value for a particular grain system.

Figure 2. Degree of mixing for the system: field pea (tracer) – white mustard (continuous phase d2 = 1.73 mm):
a) d1 = 3.74 mm; b) d1 = 4.24 mm; c) d1 = 4.74 mm

Figure 3. Degree of mixing for the system: agalite (tracer) – sand (continuous phase d2 = 1.73 mm):
a) d1 = 3.74 mm; b) d1 = 4.24 mm; c) d1 = 4.74 mm

Figures 2 and 3 show that the mixtures reach the equilibrium state after about 200 s of mixing. To ensure the equilibrium state, the sample composition for three periods of mixing time (210, 300 and 420 s) was analysed making sure that volume fraction of key component in the samples has the same distribution. The equilibrium state was checked on the basis of volume participation of the key component in particular samples for the periods of time given above.

The distributions were compared by verifying the zero hypothesis in the form:

Ho: F1 (x) = F2 (x) = F3 (x),           (4)

where: x stands for the volume fraction of the key component.

An alternative hypothesis is:

H1: F1 (x) ≠ F2 (x) ≠ F3 (x).          (5)

Verification of hypothesis (4) was made by Bartlett test and Kruskal-Wallis sum of ranges test. In all cases investigated relation (4) was fulfilled (hypothesis Ho). This allowed us to determine the value of equilibrium degree of mixing for a particular granular system as an average value out of the three ones obtained for 210, 300 and 420 s.

According to the procedure described the investigations were made to determine equilibrium degree of mixing for all possible combinations of materials presented in Table 1.

The research allowed the values of equilibrium degree of mixing to be determined for different density to diameter ratios of the mixed granular materials. It enabled the following relation

                   (6)

A comparison of the results for pairs of the materials (r1/r2 = const) with changing diameter ratios d1/d2 of the mixed materials is equation:

                        (7)

Figure 4 shows the values of equilibrium degree of mixing for the system: red clover (tracer) – sand (continuous phase) as a function of the d1/d2 diameter ratio.

Figure 4. Equilibrium degree of mixing for the system: red clover (tracer- d1) – sand (continuous phase - d2)

Figure 5 presents another investigated granular system: aluminium oxide (tracer) – field pea (continuous phase). For both granular systems presented in Figures 4 and 5 investigations were made for two different diameters value of the continuous phase.

Figure 5. Equilibrium degree of mixing for the system: Al2O3 (tracer - d1) – field pea (continuous phase - d2)

From the diagrams presented in Figures 4 and 5 it can be clearly seen that there in a maximum of equilibrium degree of mixing at the defined values of diameter d1/d2 and density r1/r2 ratios. In the case of other investigated pairs of granular materials such regularity could also be perceived. For all the granular systems it has been determined for what d1/d2 and r1/r2 values there the maximum value of equilibrium degree of mixing Memax occurs. The maximum values were determined by approximation of the obtained points with polynomial of the second degree and by determining the maximum of the obtained function. Table 2 presents the values of d1/d2 and r1/r2 ratios respectively, where th e value of Memax occurred.

Table 2. Value d1/d2 and r1/r2 for Memax

d1/d2

0.50

0.58

0.69

1

1.43

1.73

2.00

r1/r2

1.92

1.87

1.35

1

0.77

0.66

0.56

The values of d1/d2 and r1/r2 served Węgrzyn [20] for mixing in a roof static mixer and Tukiendorf [17, 18] for funnel flow mixing to present the power function of type:

                   (8)

Since the equilibrium degree of mixing is a random state [4, 12, 14] for homogeneous system in terms of grain size and component density of the mixture it can be expressed as:

for and ,       (9)

thus A = 1, and the proposed function can be expressed as:

(10)

The value of B exponent was calculated by the least square method after reducing the equation (10) to linear form. It is 0.893 for determination coefficient R2 = 0.978. Thus, the function relating density ratios and diameter ratios of mixed component grains can be expressed as:

                   (11)

The knowledge of relation (11) may, by preliminary actions, contribute to the elimination of the disadvantageous segregation phenomenon. Such actions being justified for technological reasons may include preliminary breaking up of one of the components or granulation of the components in order to obtain more favourable conditions of mixing process.

CONCLUSIONS

Mixing process of heterogeneous grain materials in terms of grain dimensions and density is accompanied by segregation phenomenon, which deteriorates quality of the mixture, and decreases the equilibrium degree of mixing compared to maximum value.

For the investigated granular materials the relation, where the segregation phenomenon does not occur or is minimal, can be expressed as:

The relation shows that the value of power exponent depends on the mixing equipment. The knowledge of the determined relation allows us to prepare mixing components in such a way as to eliminate unfavourable effect of segregation.

REFERENCES

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

  2. Boss J., 1991. Czas mieszania materiałów ziarnistych [Time of mixing of granular materials]. Zeszyty Naukowe WSI w Opolu, Studia i monografie 39, Opole [in Polish].

  3. Boss J., Knapik A. T., Węgrzyn M., 1986. Segregation of heterogeneous grain systems during mixing in static mixers. Bulk Solids Handling v. 6, 1, 145-149.

  4. Boss J., Knapik A. T., Węgrzyn M., 1989. The approximation of the transition probability matrix of a mixing process. Powder Handling & Processing 1, No. 1 March, 109-114.

  5. Boss J., Tukiendorf M., 1990. Wpływ niektórych parametrów ziarnistych na stan dynamiczny układu podczas mieszania metod± wysypu ze zbiornika [The influence of some granular parameters on dynamic system during discharge method mixing] Zeszyty Naukowe Politechniki Łódzkiej, Inżynieria Chemiczna 16, 29-36 [in Polish].

  6. Boss J., Tukiendorf M., 1997. Mixing of granular materials using the method of funnel-flow. Powder Handling & Processing 9, No. 4 October/December, 341-343.

  7. Boss J., Węgrzyn M., 1991. Równowagowy stopień zmieszania niejednorodnych materiałów ziarnistych w daszkowym mieszalniku statycznym (The heterogeneous granular materials of equilibrium degree of mixing in the roof static mixer). Inżynieria Chemiczna i Procesowa v. 12, No 3, 473-484.

  8. Gayle J. B., Lacey O. L., Gary J. H., 1958. Mixing of solids. Chi square as a criterion. Ind. Eng. Chem. 50, No. 9, 1279-1282.

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  10. Grochowicz J., 1999. Premiksy i mieszanki skoncentrowane. Składniki, technika produkcji i zastosowanie. [Feed premixes and concentrates: ingredients, manufacturing, technology and their use in feeding]. PAGROS Lublin.

  11. Knapik A., 1985. Analiza procesu mieszania niejednorodnych materiałów ziarnistych w mieszalnikach statycznych. [Analysis of mixing of heterogeneous granular materials in static mixers]. Rozprawa doktorska. Politechnika Wrocławska [in Polish].

  12. Lacey P. M. C., 1954. Developments in the theory of particle mixing, J. Appl. Chem. 4, 257-268.

  13. McCarthy J. J., 1998. Mixing, segregation and flow of granular materials, A Dissertation, Submitted to the Graduate School in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy, Field of Chemical Engineering, Northwestern University, Evanston, Illinois.

  14. Oyama Y., Ayaki K., 1956. Mixing of particulate solids. Kagaku Kogaku 20, 148-155.

  15. Rose H. E., 1959. A suggested equation relating to the mixing of powders and its application to the study of the performance of certain types of machine. Trans. I Chem. Eng. 37, 47-64.

  16. Tanaka T., 1971. Segregation models of solid mixtures composed of different densities and particle size. Ind. Eng. Chem. Process Des. Develop. 10, No 3, 332-340.

  17. Tukiendorf M., 1995. Optymalizacja procesu mieszania przesypowego niejednorodnych układów ziarnistych [Optimization of discharge mixing process of non-homogeneous granular systems] Rozprawa doktorska, Politechnika Łódzka [in Polish].

  18. Tukiendorf M., 2002. Ocena jako¶ci mieszaniny ziarnistej w przekrojach poprzecznych silosów podczas mieszania systemem funnel-flow [The quality evaluation of granular mixture in cross sections of silos during funnel-flow mixing]. Problemy Inżynierii Rolniczej 3 (37), 45-52 [in Polish].

  19. Weindenbaum S. S., Bonilla C. F., 1995. A fundamental study of the mixing of particulate solids. Chem. Eng. Progr. 51, No. 1, 27-36.

  20. Węgrzyn M., Maksymalny równowagowy stopień zmieszania niejednorodnych materiałów ziarnistych w daszkowym mieszalniku statycznym [The heterogeneous granular materials of maximal equilibrium degree of mixing in the roof static mixer] Rozprawa doktorska, Politechnika Wrocławska [in Polish].

  21. Williams J. C., 1976. The segregation of particulate materials. A Review Powder Technol. 15, 245-251.

Marek Węgrzyn
Department of Agriculture and Forestry Engineering
Technical University of Opole, Poland
5 S. Mikołajczyka Street, 45-271 Opole, Poland
Phone (+ 48 77) 400 61 94
Fax (+ 48 77) 400 63 42
e-mail: mweg@po.opole.pl

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