Size separation of binary mixture under vibration

Size separation of binary mixture under vibration Chuanping Liu, Lige Tong, Shaowu Yin, Peikun Zhang, and Li Wang Citation: AIP Conference Proceedings 1542, 714 (2013); doi: 10.1063/1.4812031 View online: http://dx.doi.org/10.1063/1.4812031 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1542?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phase separation in binary hard‐core mixtures J. Chem. Phys. 101, 3179 (1994); 10.1063/1.468468 Phase separation in binary mixtures with surfactants AIP Conf. Proc. 256, 331 (1992); 10.1063/1.42379 Variational theory of phase separation in binary liquid mixtures J. Chem. Phys. 75, 3594 (1981); 10.1063/1.442469 Vibrational relaxation in binary mixtures J. Acoust. Soc. Am. 64, S60 (1978); 10.1121/1.2004293 The kinetics of phase separation in a liquid binary mixture J. Chem. Phys. 62, 3298 (1975); 10.1063/1.430884

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Size Separation of Binary Mixture under Vibration Chuanping Liu*,+, Lige Tong*, Shaowu Yin*, Peikun Zhang*, Li Wang*,+ *

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School of Mechanical Engineering, University of Science and Technology Beijing, 100083, China Beijing Key Laboratory of Energy Saving and Emission Reduction for Metallurgical Industry, 30 Xueyuan Road, Haidian District, Beijing, China Abstract. By considering as a thermodynamic system, a minimum energy principle is established, in which the granular system changes its size distribution to make itself to be at the lowest energy state as soon as possible on the precondition that it dissipates all of the energy supplied from the vibrating bottom. A model is presented based on this principle to clarify why and how binary mixture separates under vibration. The small particles tend to sink in order to lower the kinetic energy of system, while the heavy particles sink in order to lower the potential energy. The mixture separates finally based on the competition between the two effects. The results of our model qualitatively agree with the previous researches. Keywords: Size separation, granular matter, minimum energy principle PACS: 45.70.-n, 83.80.Fg

INTRODUCTION MINIMUM ENERGY PRINCIPLE Unlike conventional solids, liquids, and gases, binary granular mixtures separate when agitated. By vibrating binary granular mixture, the large particles usually rise to the top of the bed, resulting in the “Brazil Nut” (BN) separation[1,2]. By varying the vibration parameters or the particles’ physical properties, the large particles also can sink to the bottom, resulting in the “Reverse Brazil Nut” (RBN) separation[3]. The similar size separation phenomena are found also in rotating drums[4,5], chute flow[6] and so on. Up to the present, several theories have been proposed to explain the granular size separation, including percolation[1], convection[2], void filling[7,8], condensation[9], etc.. However, because of the strongly energy dissipation and nonlinear characteristics of granular system, the size separation is far from being fully understood, and its mechanism becomes an open question. In this paper, our aim is to contribute to the further understanding of size separation based on a conditional minimum energy principle. Here, we consider the binary mixture as a thermodynamic system, which changes its size distribution to make itself to be at the lowest energy state as soon as possible on the precondition that it dissipates all of the energy supplied from the vibrating bottom. A conditional minimum energy principle is introduced to illuminate why and how the mixture separates under vibration which differs in particle size and/or mass.

By vibrating the particles, the energy is supplied to the granular system from the vibrating bottom. Part of the energy is dissipated as heat due to non-perfect collisions, and the other keeps in the system to change its kinetic or potential energy, as FIGUER 1. JA=ȖAh+dE/dt (1) where, J is the energy flux supplied from the vibrating bottom of unit area, W/m2; Ȗ the energy dissipation rate by the collisions in unit volume, W/m3; E the energy of the system including both the kinetic and potential energy, J; dE/dt the change rate of E, A and h the bed bottom area and the bed height respectively. Energy of system after changing E+dE

JAdt

Energy of vibrating bottom

E

Energy of system

γAhdt

Heat of particles

FIGUER 1 Energy transfer in vibration bed

The entropy production is brought by all of the irreversible processes including the energy transfer from the vibrating bottom to the system, the energy dissipation by collisions and the energy change of the system. (2) ȥ= JAȤJk+ȖAhȤki+EȗE where, ȤJk, Ȥki are the thermodynamic force corresponding to the energy flux JA and energy dissipation rate ȖAh, respectively; ȗE is the coefficient corresponding to the energy change of the system. The ȗE is proportion to the energy of the system E. By

Powders and Grains 2013 AIP Conf. Proc. 1542, 714-717 (2013); doi: 10.1063/1.4812031 © 2013 AIP Publishing LLC 978-0-7354-1166-1/$30.00

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mi/2. The position of the mass center of the all particles when they are rest with mixed (Figure 2(b)) is defined as reference height, as h=0. The particles at the bottom have the largest kinetic energy and the kinetic energy Eki is reduced gradually along the height. However, the reducing rate is related to the size distribution, as it is reduced most quickly when the particles are at the state of “BN” separation by comparing to mixing and “RBN” separation. The energy of the bed includes two parts, the kinetic energy and the potential energy of particles, as E=Ep+Ek, where Ek=™Eki, Ep=™Epi. TABLE 1 shows the comparison of the bed energy between the three types of size distribution, as E(BN)