Threshold frequency of standing wave at hydrodynamic separation of particles

In this paper, the question of the separation of particles in a standing wave in closed resonator at first resonance frequency is considered. The threshold values of the frequency of the standing wave, as well as the densities and radii of the particle, which allow to separate the particles depending on their radius and density are studied.

To investigate the separation of particles, a 1D resonator with length L filled with air is considered. In a Cartesian coordinate system, equations of particle motion are expressed as 2 2 d , d where x -Cartesian coordinate, t -time, 2 x -particle coordinate, 1 sin v U t   -air velocity,  -air density, 2  -particle density, r -particle radius,  -coefficient of kinematic viscosity of the air.
To study the particle drift the evolutionary equations of a particle are written [1]: where  -period average position of the particle,  -velocity of particle drift, -drag coefficient of the particle with account of additional mass forces. From (1) the local equilibrium velocity of particle drift can be introduced as In the resonator, a standing wave is received, therefore the direction and speed of the drift is determined by the parameter A. Formula to acceleration shows that for any particle there is a threshold frequency c  . At frequencies lower than this frequency c   the acceleration of the particle drift is determined by the Stokes force and is directed to the node of the velocity wave. At frequencies of higher than threshold frequency c   , the acceleration of particle drift is determined by the inertia forces of the carrier medium (i.e. by the forces of the added masses and the dynamic force of Archimedes) and is directed to the antinode of the wave. Each radius corresponds to its own threshold frequency, namely: for radius 1

Separation of particles from the same material
Hz ). A large particle has a larger relaxation time, which corresponds to lower threshold frequency. If the frequency of the standing wave  lays between threshold frequencies (2) (1) cc , then for a larger particle, this frequency will be greater than its threshold value, and the particle will drift into the antinode of the wave. For a smaller particle, the frequency will be less than its threshold value, and the particle will drift to the node. We choose as the frequency of the standing wave value  . Figure 1 shows good agreement between trajectories and evolution for each of the two particles. The smaller particle drifts toward the node, and the larger toward the antinode. In this case, the average drift velocity of particles can be determined from the corresponding angles of inclination. Modulo it is about the same for both particles and has the order 166 m/s . The larger particle drifts toward the antinode of the wave, and the smaller toward the node. In this case, the absolute values of the particle drift velocity are close to each other. Particle with a threshold radius c r =116 m  stays at rest and has zero drift velocity.

Separation of particles of the same size
Consider the drift of two particles of the same radius 100 m r   : one of Styrofoam    . Figure 5 shows good agreement between trajectories and evolution for each of the two particles. The particle of the Styrofoam drifts to the node, and the water -to the antinode.   Figure 4 shows good agreement between trajectories and evolution for each of the two particles. In this case, the absolute values of the particle drift velocity are close to each other. Foam rubber particle with a threshold density

Appendix
From the evolution equation, a formula for dimensionless vibration acceleration of a particle is obtained. The existence of threshold frequencies at which the direction of particle drift changes is determined. The existence of radii and densities corresponding to the threshold frequencies at which the particles practically do not drift is defined.