Maximizing energy transfer in vibrofluidized granular systems

Using discrete particle simulations validated by experimental data acquired using the positron emission particle tracking technique, we study the efficiency of energy transfer from a vibrating wall to a system of discrete, macroscopic particles. We demonstrate that even for a fixed input energy from the wall, energy conveyed to the granular system under excitation may vary significantly dependent on the frequency and amplitude of the driving oscillations. We investigate the manner in which the efficiency with which energy is transferred to the system depends on the system variables and determine the key control parameters governing the optimization of this energy transfer. A mechanism capable of explaining our results is proposed, and the implications of our findings in the research field of granular dynamics as well as their possible utilization in industrial applications are discussed.

Figure 1

Comparison of the variation with the nondimensionalized driving amplitude, $A/d$, of $h^{*}$, the relative increase in a system's center of mass height from its resting position (blue triangles), and $E_T$, the total energy (kinetic and potential) possessed by the system (black circles), as determined from simulations.

Figure 2

Variation of the relative increase in center of mass, $h^{*}$, with frequency and amplitude at a fixed dimensionless input energy $S=1.83$. Data is shown for both experimental (triangles) and simulated (circles) data sets and for two differing bed heights: $N_L=3$ (gray circles and red triangles) and $N_L=6$ (black circles and blue triangles). Panel (a) shows the variation in $h^{*}$ as a function of driving frequency, $f$, while panel (b) plots $h^{*}$ against the dimensionless driving amplitude, $\frac{A}{d}$, where $d$ is the diameter of a particle.

Figure 3

Center of mass position for simulated beds as a function of frequency, $f$, for a variety of differing bed heights, $N_L$, and driving energies, $S$, produced via simulation. The center of mass positions for each $N_L-S$ combination have been arbitrarily normalized such that the positions of the various maxima may be easily compared by visual inspection. Panel (a) shows data for a fixed driving strength $S=1.83$ for beds of depth $N_L=2$ (circles), $N_L=3$ (triangles), $N_L=4$ (squares), and $N_L=6$ (diamonds). Panel (b) gives data for a constant bed height $N_L=3$ driven with input energy $S=1.37$ (diamonds), $S=1.83$ (squares), $S=2.75$ (triangles), and $S=4.13$ (circles).

Figure 4

Mean collision rate (circles) and contact time (diamonds) for particle-base interactions as a function of driving amplitude, $A$, for a fixed base vibrational energy $S=1.83$. Data shown is acquired from DPM simulations and corresponds to the case $N_L=2$. Each data point represents an average taken over $\sim 100000$ individual collisions.

Figure 5

Particle-base collision rate per particle [solid orange (light gray) symbols] and the percentage of these collisions that occur when the base's motion is in the positive vertical (upward) direction [open blue (dark gray) symbols] as a function of the vibrational energy parameter $S$. Data is shown for the case of constant acceleration with $\mathrm{\Gamma }=8$ (circles) and $\mathrm{\Gamma }=13.5$ (triangles) as well as for parameter combinations producing a constant $h^{*}=2.1$ (squares) and $h^{*}=2.8$ (diamonds). In all cases, the resting bed depth is fixed at a value $N_L=3$. The extent in $S$ of the fixed-$h^{*}$ data sets is constrained by the limited overlap between the ranges of center of mass positions achievable for differing $S$-values—for example, the lowest $h^{*}$ achievable in a system driven with $S=7.34$ is still greater than the largest obtainable $h^{*}$ value for an equivalent system with $S=3.26$.

Figure 6

Simulated data showing center of mass position as a function of dimensionless acceleration, $\mathrm{\Gamma }$, for various fixed input energies: $S=1.37$ (diamonds), $S=1.83$ (squares), $S=2.75$ (triangles), and $S=4.13$ (circles). As in Fig. 3, the value of $h^{*}$ has been normalized for ease of comparison.

Figure 7

Number of particle collisions as a function of time for two simulated systems of identical resting depth $N_L=3$ driven with equal dimensionless input energies, $S=3.26$, achieved using frequency-amplitude combinations of (a) $A=12.0,f=5.31 (\mathrm{\Gamma }=1.36)$ and (b) $A=1.0,f=63.7 (\mathrm{\Gamma }=16.3)$. For each point in time, the collision number is shown alongside the instantaneous velocity of the vibrating base-plate, which forms the system's energy source. For ease of comparison, the collision number is normalized such that its maximal value is equal to unity.

Figure 8

Experimentally acquired power spectra for the center of mass motion of a granular bed of depth $N_L=6$ driven with input energy $S=1.83$. This value of $S$ is achieved using three differing combinations of driving frequency, $f$, and dimensionless amplitude, $A/d$: $f=9.55,A/d=1$ (black), $f=11.9,A/d=0.8$ (blue/dark gray), and $f=19.9,A/d=0.48$ (orange/light gray). Vertical dashed lines are used to demarcate the positions of the relevant driving frequencies.

Figure 9

Simulated data showing the variation with bed depth, $N_L$, of ${\mathrm{\Gamma }}_{\text{max}}$, the value of the dimensionless acceleration at which, for a given $N_L$, energy transfer is maximal. Data is shown for input energies $S=1.83$ (circles), $S=4.13$ (triangles), and $S=7.34$ (diamonds).

Figure 10

Simulated data showing relative center of mass position, $h^{*}$, as a function of dimensionless acceleration, $\mathrm{\Gamma }$, for the case $N_L=2$ and $S=4.13$. Data is shown for systems of horizontal dimensions $L_{x,y}/d=4$ (gray crosses), 16 (black circles), 24 (blue triangles), and 48 (red diamonds), corresponding to particle numbers, $N$, of 32, 512, 1152, and 4608, respectively.

Figure 11

Variation in solids fraction, $\eta$, with driving parameters $f$ and $A$ for a fixed energy input $S=3.26$ and bed depth $N_L=3$, as obtained from simulations. Panel (a) shows the variation in the bed's average packing density as the amplitude, $A$, of driving is varied at fixed $S$. Panel (b), meanwhile, shows the one-dimensional vertical packing profiles corresponding to each data point in (a). In this image, lighter shades represent higher $A$ or, equivalently, lower $f$.