Skip to main content
SpringerLink
Log in

Entropy criterion of random states for granular material in a mixing process

  • Original Paper
  • Published:
Chemical Papers Aims and scope Submit manuscript

Abstract

Mathematical assessment of homogenisation progress of the granular material mixing process is presented. The mixing process was realised using a vessel in the form of two partly penetrating horizontal cylinders equipped with two multi-ribbon agitators. The experimental system consisted of three sets of particles of different colour. Random states of the mixed granular material were characterised by the sampling procedure at different moments of the mixing process. Informational entropy as well as the flow of quantum of information were applied to describe the progress of the homogenisation process. Analysis of this process was based on experimental investigations in the form of informational entropy patterns and described by means of the average informational entropy or the quantum of information.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Akiyama, T., Iguchi, T., Aoki, K., & Nishimoto, K. (1998). A fractal analysis of solids mixing in two-dimensional vibrating particle beds. Powder Technology, 97, 63–71. DOI: S0032-5910(97)03400-1.

    Article  CAS  Google Scholar 

  • Asmar, B. N., Langston, P. A., & Matachett, A. J. (2002). A generalized mixing index in distinct element method simulation of vibrated particulate beds. Granular Matter, 4, 129–138. DOI: 10.1007/s10035-002-0112-8.

    Article  Google Scholar 

  • Austin, L. G. (1971). Introduction to the mathematical description of grinding as a rate process. Powder Technology, 5, 1–17. DOI: 10.1016/0032-5910(71)80064-5.

    Article  Google Scholar 

  • Austin, L. G. (1995). The graphical representation of ash liberation in milled coal. Chemical Engineering Journal, 59, 23–31. DOI: 10.1016/0923-0467(95)03009-3.

    CAS  Google Scholar 

  • Austin, L. G. (1999). A discussion of equations for the analysis of batch grinding data. Powder Technology, 106, 71–77. DOI: 10.1016/S0032-5910(99)00047-9.

    Article  CAS  Google Scholar 

  • Berthiaux, H., Heitzmann, D., & Dodds, J. D. (1996). Validation of a model of a stirred bead mill by comparing results obtained in batch and continuous mode grinding. International Journal of Mineral Processing, 44/45, 653–661. DOI: 10.1016/0301-7516(95)00073-9.

    Article  Google Scholar 

  • Berthiaux, H., & Mizonov, V. (2004). Applications of Markov chains in particulate process engineering: A review. The Canadian Journal of Chemical Engineering, 82, 1143–1168.

    Article  CAS  Google Scholar 

  • Bridgwater, J. (2003). The dynamics of granular materials — towards grasping the fundamentals. Granular Matter, 4, 175–181. DOI: 10.1007/s10035-002-0120-8.

    Article  Google Scholar 

  • Brone, D., Alexander, A., & Muzzio, F. J. (1998). Quantitative characterization of mixing of dry powders in V-blenders. AIChE Journal, 44, 271–278. DOI: 10.1002/aic.690440206.

    Article  CAS  Google Scholar 

  • Chen, S. J., Fan, L. T., & Watson, C. A. (1972). The mixing of solid particles in a motionless mixer — a stochastic approach. AIChE Journal, 18, 984–989. DOI: 10.1002/aic.690180515.

    Article  CAS  Google Scholar 

  • Cho, H., Yildirim, K., & Austin, L. G. (1998). The conversion of sedigraph size distributions to equivalent sub-sieve screen size distributions. Powder Technology, 95, 109–117. DOI: 10.1016/S0032-5910(97)03326-3.

    Article  CAS  Google Scholar 

  • Chou, S. T., Fan, L. T., & Hsu, J. P. (1987). Stochastic analysis of the transient behavior of an msmpr crystallizer; effects of the seed size distribution and size dependent growth, rate, Probability in the Engineering and Informational Sciences, 1, 383–404.

    Article  Google Scholar 

  • Gardner, R. P., & Austin, L. G. (1962). A chemical engineering treatment of batch grinding. In H. Rumpf and D. Behrens (Eds.), Proceedings of the 1st European Symposium on Size Reduction (pp. 217–248). Düsseldorf: Verlag Chemie.

    Google Scholar 

  • Heim, A., Olejnik, T. P., & Pawlak, A. (2005). Using statistical moments to describe grinding in a ball mill for industrial-scale process. Chemical Engineering and Processing, 44, 263–266. DOI: 10.1016/j.cep.2004.02.020.

    Article  CAS  Google Scholar 

  • Henrique, C., Batrouni, G., & Bideau, D. (2001). Diffusion as a mixing mechanism in granular materials. Physical Review E, 63, 011304. DOI: 10.1103/PhysRevE.63.011304.

  • Hoyer, D. I. (1995). Batch grinding simulation — population balance models and self-similar size distributions. Materials Engineering, 8, 1275–1284. DOI: 10.1016/0892-6875(95)00095-8.

    CAS  Google Scholar 

  • Hulburt, H. M., & Katz, S. (1964). Some problems in particle technology: A statistical mechanical formulation. Chemical Engineering Science, 19, 555–574. DOI: 10.1016/0009-2509(64)85047-8.

    Article  CAS  Google Scholar 

  • Khakhar, D. V., McCarthy, J. J., & Ottino, J. M. (1997). Radial segregation of granular mixtures in a rotating cylinder. Physics of Fluids, 9, 3600–3614. DOI: 10.1063/1.869498.

    Article  CAS  Google Scholar 

  • Kruszyński, B. W., & Wójcik, R. (2001). Residual stress in grinding. Journal of Materials Processing Technology, 109, 254–257. DOI: 10.1016/S0924-0136(00)00807-4.

    Article  Google Scholar 

  • Lai, F. S., & Fan, L. T. (1975). Application of a discrete mixing model to the study of mixing of multicomponents solid particles. Industrial & Engineering Chemistry Process Design and Development, 14, 403–411. DOI: 10.1021/i260056a009.

    Article  CAS  Google Scholar 

  • Laurent, B. F. C., & Bridgwater, J. (2002). Performance of single and six-bladed powder mixers. Chemical Engineering Science, 57, 1695–1709. DOI: 10.1016/S0009-2509(02)00052-0.

    Article  CAS  Google Scholar 

  • Laurent, B. F. C., Bridgwater, J., & Parker, D. J. (2000). Motion in a particle bed agitated by a single bed. AIChE Journal, 46, 1723–1734. DOI: 10.1002/aic.690460904.

    Article  CAS  Google Scholar 

  • Makarow, J. I. (1975). Foundations of calculations of powder material blending, Doctor’s Thesis, Moscow: MIKhM.

    Google Scholar 

  • Marchiso, D. L., Vigil, R. D., & Fox R. O. (2003). Implementation of the quadrature method of moments in CFD codes for aggregation-breakage problems. Chemical Engineering Science, 58, 3337–3351. DOI: 10.1016/S0009-2509(03)00211-2S0009-2509(03)00211-2.

    Article  CAS  Google Scholar 

  • Masiuk, S., & Rakoczy, R. (2006). The entropy criterion for the homogenisation process in a multi-ribbon blender. Chemical Engineering and Processing: Process Intensification, 45, 500–506. DOI: 10.1016/j.cep.2005.11.008.

    CAS  Google Scholar 

  • Masiuk, S., & Rakoczy, R. (2008). Kinetic equation of grinding process in mixing of granular material using probability density functions, transient operators and informational entropy. Chemical Engineering and Processing, 47, 200–208. DOI: 10.1016/j.cep.2007.03.001.

    CAS  Google Scholar 

  • McCarthy, J. J., & Ottino, J. M. (1998). Particle dynamics simulation: A hybrid technique applied to granular mixing. Powder Technology, 97, 91–99. DOI: S0032-5910(97)03391-3.

    Article  CAS  Google Scholar 

  • McCarthy, J. J., Khakhar, J. M., & Ottino, J. M. (2000). Computational studies of granular mixing. Powder Technology, 109, 72–82. DOI: S0032-5910(99)00228-4.

    Article  CAS  Google Scholar 

  • Moakher, M., Shinbrot, T., & Muzzio, F. J. (2000). Experimentally validated computations of flow, mixing and segregation of non-cohesive grains in 3D tumbling blenders. Powder Technology, 109, 58–71. DOI: S0032-5910(99)00227-2.

    Article  CAS  Google Scholar 

  • Molina-Boisseau, S., & Le Bolay, N. (2002). The mixing of a polymeric powder and the grinding medium in a shaker bead mill. Powder Technology, 123, 212–220. DOI: S0032-5910(01)00460-0.

    Article  CAS  Google Scholar 

  • Müller, F., Polke, R., & Schäfer, M. (1999). Model-based evaluation of grinding experiments. Powder Technology, 105, 243–249. DOI: 10.1016/S0032-5910(99)00144-8.

    Article  Google Scholar 

  • Niemi, A., Tian, L., & Ylinen, R. (1997). Model predictive control for grinding systems. Control Engineering Practice, 5, 271–278. DOI: 10.1016/S0967-0661(97)00236-0.

    Article  Google Scholar 

  • Ogawa, K., & Inoue, I. (1984). A new definition of quality of mixedness for multicomponents batch mixing. In Proceedings of 8th International Congress CHISA’84, 3–7 September 1984 (Pap. V3.56), Prague: CHISA.

    Google Scholar 

  • Ogawa, K., & Ito, S. (1975). A definition of quality of mixedness. Journal of Chemical Engineering of Japan, 8, 148–151.

    Article  CAS  Google Scholar 

  • Ogawa, K., Ito, S., & Matsumura, Y. (1980). Mixing rate in a stirred vessel. Journal of Chemical Engineering of Japan, 13, 324–326.

    Article  Google Scholar 

  • Pernenkil, L., & Cooney, C. L. (2006). A review on the continuous blending of powders. Chemical Engineering Science, 61, 720–742. DOI: 10.1016/j.ces.2005.06.016.

    Article  CAS  Google Scholar 

  • Ramkrishna, D. (2000). Population balances. Theory and applications to particulate system in engineering, San Diego: Academic Press.

    Google Scholar 

  • Randolph, A. D., & Larson, M. A. (1988). Theory of particulate processes. New York: Academic Press, New York.

    Google Scholar 

  • Runkana, V., Somasundaran, P., & Kaptur, P. C. (2004). Mathematical modeling of polymer-induced flocculation by charge neutralization. Journal of Colloid and Interface Science, 270, 347–358. DOI: 10.1016/j.jcis.2003.08.076.

    Article  CAS  Google Scholar 

  • Sudah, O. S., Coffin-Beach, D., & Muzzio, F. J. (2002). Quantitative characterization of mixing of free-flowing granular material in tote (bin)-blenders. Powder Technology, 126, 191–200. DOI: 10.1016/S0032-5910(02)00006-2.

    Article  CAS  Google Scholar 

  • Unger, D. R., Muzzio, F. J., & Brodkey, R. S. (1998). Experimental and numerical characterization of viscous flow and mixing in an impinging jet contactor. Canadian Journal of Chemical Engineering, 76, 546–555.

    Article  CAS  Google Scholar 

  • Valery, W., & Morrell, S. (1995). The development of a dynamic model for autogenous and semi-autogenous grinding. Materials Engineering, 8, 1285–1297. DOI: 10.1016/0892-6875(95)00096-9.

    CAS  Google Scholar 

  • Yekeler, M., Ozkan, A., & Austin, L. G. (2001). Kinetics of fine wet grinding in a laboratory ball mill. Powder Technology, 114, 224–228. DOI: 10.1016/S0032-5910(00)00326-0.

    Article  CAS  Google Scholar 

  • Yildirim, K., & Austin, L. G. (1998). The abrasive wear of cylindrical grinding media. Wear, 218, 15–20. DOI: 10.1016/S0043-1648(98)00198-7.

    Article  CAS  Google Scholar 

  • Zhukov, V., Mizonov, V., Filitchev, P., & Bernotat, S. (1998). The modelling of grinding process by means of the principle of maximum entropy. Powder Technology, 95, 248–253. DOI: S0032-5910(97)03344-5.

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Chemical Engineering and Environmental Process Protection, Szczecin University of Technology, al. Piastów 42, 71-065, Szczecin, Poland

    Stanisław Masiuk, Rafał M. Rakoczy & Marian Kordas

Authors
  1. Stanisław Masiuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rafał M. Rakoczy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marian Kordas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stanisław Masiuk.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Masiuk, S., Rakoczy, R.M. & Kordas, M. Entropy criterion of random states for granular material in a mixing process. Chem. Pap. 62, 247–254 (2008). https://doi.org/10.2478/s11696-008-0019-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11696-008-0019-x

Keywords

Search

Navigation