1. Introduction

It is difficult to fractionate ultra-fine particles and particles with neutral buoyancy, or uniform electro-magnetically charged surfaces. Existing methods are extremely slow or require prohibitive pressure drops, or extremely high electric or magnetic fields. Hence this exploratory research (cf. [1] and [2]) using acoustic and flow fields was conducted to evaluate the feasibility of fractionating ultra fine suspended particles and at the same time segregating them. In this technology the particle movements due to density and compressibility differences between fluid and particles rather than the particle size are used to fractionate ultra-fine suspended particles. In the following section, the basic derivation for particles in equilibrium in an acoustic field is given, which will be used for the derivation of particle trajectory and concentration equations.

1.1. Acoustic force field

King and Macdonald [3], proposed a mathematical model for the force acting on a spherical particle suspended in a standing acoustic wave field. This analysis was restricted to a rigid particle with a radius much smaller than the wavelength of sound and for a standing wave field created by oppositely travelling, single frequency, and sinusoidal waves. Fig. 1 shows the forces acting on a particle in an acoustic field. These forces are:

1. Primary axial acoustic radiation force (F_PARF)

2. Primary transverse acoustic radiation force (F_PTRF), and

3. Secondary acoustic radiation force (F_S)

Fig. 1. Acoustic forces on a particle.

1.2. Primary axial acoustic force

Almost immediately following the application of an acoustic field, particles experience a time-averaged primary axial acoustic force, F_ac (F_PARF), which is generated by the interaction between particles and the primary wave field. The primary axial radiation force drives dispersed particles toward the velocity antinodes of the resonance field. The magnitude of the force depends on the difference in compressibility and density between the particle and the medium. The primary acoustic force was derived by Yosioka and Kawasima [4] and given below.

(1)

The acoustic energy density, E_ac, is a measure of the energy residing in a wave field. V₀ is the volume of one particle, k = 2π/λ is the wave number of the acoustic radiation, λ is the sound wavelength, x is the axial distance from a pressure node, and k is the unit vector in the axial direction. The response of any solid suspended in a fluid, to a resonant acoustic field, depends on the acoustic contrast factor, G. For any solid particle of size r  λ suspended in a fluid, the acoustic contrast factor is given by:

(2)

where, β_f and β_p are the compressibility of the fluid and particle respectively, and ρ_f and ρ_p are the densities of the fluid and particle respectively.

1.3. Primary and secondary acoustic force

The primary transverse force acts normal to the wave propagation. Of the three forces, the primary axial radiation force generally has the greatest magnitude.