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Research Papers

The Coefficient of Restitution of Spheroid Particles Impacting on a Wall—Part I: Experiments

[+] Author and Article Information
Ming Hu

Department of Aeronautics and Astronautics,
Fudan University,
Shanghai 200433, China

Yrjö Jun Huang

Department of Aeronautics and Astronautics,
Fudan University,
Shanghai 200433, China
e-mail: jun_huang@fudan.edu.cn

Fei Wang

College of Physics and
Electromechanical Engineering,
Hexi University,
Zhangye 734000, Gansu, China;
Department of Aeronautics and Astronautics,
Fudan University,
Shanghai 200433, China
e-mail: wangfeiwd@hxu.edu.cn

Martin Smedstad Foss

Petroleum Technology,
Institute for Energy Technology,
Kjeller NO-2027, Norway

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 27, 2017; final manuscript received January 4, 2018; published online February 2, 2018. Assoc. Editor: Weinong Chen.

J. Appl. Mech 85(4), 041006 (Feb 02, 2018) (10 pages) Paper No: JAM-17-1654; doi: 10.1115/1.4038920 History: Received November 27, 2017; Revised January 04, 2018

Coefficients of restitution (CoR) is used to scale the kinetic energy dissipation, which is a necessary parameter for discrete element modeling simulations of granular flow. Differences from the collision of spherical particles, CoRs of spheroid particle are affected not only by materials, particle size, and impacting velocity, but also by the contact inclination angle of the particle. This article presents our experimental investigation to measure the velocities of translation and rotation using high-speed camera and calculate the CoR in normal direction of prolate spheroid particles impacting flat targets. The results show that this CoR of a prolate spheroid particle is composed of two parts, translation and rotation. The effect from the contact inclination angle is not obvious for a given velocity. When the contact point is close to a pole, the first part plays a major role. On the contrary, the second part dominates the CoR, when the contact point is close to the equator. A dimensionless number, e*, is defined to scale the proportion of velocity due to rotation in the total rebound velocity at the contact point. The relationship between the contact inclination angle, ϕ, and e* for 25 deg < ϕ < 90 deg is given in this article.

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Figures

Grahic Jump Location
Fig. 1

A sketch of a binary collision

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Fig. 2

A sketch of the experimental system

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Fig. 3

The spherical and spheroid particles used in the experiments

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Fig. 4

The radii of curvatures for a prolate spheroid in the form of x2+y2/a2+z2/c2=1. APB̂ is the meridian arc between two poles A and B; r′ is the radius of curvature for this curve at point P; GPĤ is normal to APB̂ at point P, and r″ is the radius of curvature for arc GPĤ at point P. r′ and r″ are the principal radii of curvature at point P.

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Fig. 5

The sketch of the velocity calculation. (a)–(c) are frames obtained from the same animation with time gaps of 0.0052 s (N = 4), and (d) shows the profiles, positions, velocities and trace.

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Fig. 6

The experimental results and the data fitting of spherical particles: (a) for particle #1, (b) for particle #2, (c) for particle #3, and (d) for particle #4. Brilliantov et al.'s model [7], Schwager's model [20], and Bridges' model [17] are presented using dash-dotted lines, dash lines, and solid lines, respectively.

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Fig. 7

CoR, e, of spheroid particle #5 with different contact inclination angles, ϕ, and data fitting using three models [7,17,18,20]. Brilliantov et al.'s model [7], Schwager's model [20], and Bridges' model (power law model) [17,18] are presented using dash-dotted lines, dash lines and solid lines, respectively.

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Fig. 8

CoR, e, of spheroid particle #6 with different contact inclination angles, ϕ, and data fitting using three models [7,17,18,20]. The linetypes have the same meaning as those in Fig. 7.

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Fig. 9

CoR, e, of spheroid particle #7 with different contact inclination angles, ϕ, and data fitting using three models [7,17,18,20]. The linetypes have the same meaning as those in Fig. 7.

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Fig. 10

eT of spheroid particle #5 with different contact inclination angles, ϕ, and data fitting using the power law, Eq. (20), where ζ = 0.139, same as that for the power curve in Fig. 7. The symbols, *, △, ∘ and □ are used to present the results of 0 < χ < 25.5 deg, 22.5 deg < χ < 45 deg, 45 deg < χ < 67.5 deg and χ > 67.5 deg, respectively.

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Fig. 11

eT of spheroid particle #6 with different contact inclination angles, ϕ, and data fitting using the power law, Eq. (20), where ζ = 0.243, same as that for the power curve in Fig.8. The symbols have the same meaning as those in Fig. 10.

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Fig. 12

eT of spheroid particle #7 with different contact inclination angles, ϕ, and data fitting using the power law, Eq. (20), where ζ = 0.156, same as that for the power curve in Fig. 10.

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Fig. 13

eω of spheroid particle #5 with different contact inclination angles, ϕ, and data fitting using the power law, Eq. (21), where ζ = 0.139. The symbols have the same meaning as those in Fig. 10.

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Fig. 14

eω of spheroid particle #6 with different contact inclination angles, ϕ, and data fitting using the power law, Eq. (21), where ζ = 0.243. The symbols have the same meaning as those in Fig. 10.

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Fig. 15

eω of spheroid particle #7 with different contact inclination angles, ϕ, and data fitting using the power law, Eq. (21), where ζ = 0.156. The symbols have the same meaning as those in Fig. 10.

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Fig. 16

When the net force passing the center of the spheroid particle, the net torque due to the force F becomes zero and ωr = 0

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Fig. 17

A sketch of a second collision. (a) before the first collision, c is the first collision point; (b) after the first collision; and (c) second collision happens due to a high angular velocity ωr, and c′ is the second collision point.

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Fig. 18

The linear relationship between e* and ϕ in the range of 25 deg < ϕ < 90 deg. Here, the symbols △, □ and * are used for the prolate spheroid particles #5, #6, and #7, respectively.

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