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Mathematical model


Date: 2015-10-07; view: 441.


In this research fractionation of particles is based on density and compressibility differences of fluid and particles rather than on particle size. Employing the above basic principles of physics of particles in an acoustic field, a mathematical model is developed to calculate trajectories of deflected particles subjected to acoustic standing waves. Table 1 gives the properties of the particles and the fluid that are used in this research.

Table 1.

Properties of particles and suspending medium

Description Solid (SiC) Medium (DI water)
Density (ρ) (kg/m3)
Frequency of sound in medium (f) (kHz)
Viscosity of medium (μ) (N s/μm2) 9.98E−16
Acoustic energy in medium (J/m3)
Power in medium (W/m3) 56 000
Quality factor (Q) of chamber

Rearranging Eqs. Figs. (3) and (5) yields the following equation:

 

(6)

Simplifying Eq. (6) with values in Table 1 yields:

 

x+cx-ksinβx=0, (7)

where

 

x=ν, (8)

where, c = 1.42E6, k = E8, and β = 2.78E−3 are constants representing the physical parameters of Eq. (7). This equation will be constantly used during the mathematical derivation. The notation (′) indicates the derivative, d/dt, and x is the position of the migrating particle in the x-direction between a transducer and a reflector separated by one half wavelength (=λ/2) of the resonant sound at the given frequency. The parameters c, k, and β are all positive constants having the following orders (O) of magnitude: c = O(106), k = O(108), β = O(10−3).

A study of the behavior of the solutions is discussed with an explanation of the available solution techniques. Solutions of Eq. (7) will be used as a basis for concluding some results during the derivation. The above equation is extremely stiff, so most numerical solution methods—even stiff equation solvers—provide little useful information.

Note that Eq. (7) has the form somewhat like a damped nonlinear spring (or pendulum) equation. Several publications cited in the literature assumed instantaneous viscous relaxation where the inertial term dv/dt or x″ was neglected. This type of singular perturbation approximation; namely:

 

cx-ksinβx=0 (9)

has been used to approximate the solution for Eq. (7). It will be shown in this paper that although this approximation produces results that are qualitatively correct, quantitative errors are incurred that can be significant for some applications.


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