Texts on physics, maths and programming (Несколько текстов для зачёта), страница 32

The approximate solutions for particle trajectories and concentration will be used to compare the experimental observations with expected theoretical values. There is no closed form solution for Eq. (7), and direct application of a Runge–Kutta solver led to rather surprising results that were completely unsatisfactory. Stiff ODE solvers also appeared somewhat inadequate, so a viable approach was to use an approximate solution. If the experimental results can be compared to Fig. 6 and Fig. 7, then it can be stated that the mathematical model derived has solutions that are in good agreement with the actual behavior of the system and can quite possibly be generalized for any type of medium or particle.

6. Experimental procedure and results

The mathematical model given by Eq. (39) for the proposed acoustic field technology was verified by building a flow chamber as shown in Fig. 8 (1). The back electrode of the load transducer was connected to an Anritsu Synthesizer/level signal generator MG 443B through an ENI 1040L power amplifier to power the transducer. The other transducer (acting as a receiver) was connected to a Textronix T912 Oscilloscope.

Fig. 8. Acoustic chamber.

Silicon carbide (SiC), C = 29.97% and Si = 70.03%, 5–20 μm size with a specific gravity of 3.217 was used for the experimental verification. A charged coupled device (CCD) color camera (NEC model NX18A) was fixed to the microscope with a magnification of 10× lens and 10× eyepieces for observation of the particles. A Stocker and Yale Fiber optic light (model number 20) with 150 W intensity was placed behind the chamber to illuminate particles for recording. The CCD camera had two video outputs. One was connected to a TV/VCR for video recording, while the other was connected to a Gateway 2000 P5-120 computer for digital image capturing and analysis card (WinTV 2000).

6.1. SiC particle trajectories in an acoustic field

With large SiC particles, it was possible to track the motion of individual particles during the application of acoustic energy. Individual SiC particles were tracked and their positions were captured using the arrangement discussed earlier.

Movements of particles were tracked at different acoustic energy levels ranging from 0.5 to 5.0 W at equal increments of 0.5 W. Digital capturing at a rate of 30 frames per second was made at all these power levels and each capture was stored in separate AVI files. After converting the AVI files into JPEG frames, Adobe PhotoShop 5 was used to obtain particle displacements for each frame at 1/30 s intervals. Fig. 9 shows photographs of particle movement from four consecutive frames. Using the horizontal and vertical distance calibration charts and pixel positions, the x and y coordinates of tracked particles with time were obtained. Graphs were produced for the experimental axial displacement versus time at different power levels (1).

Fig. 9. Photographs of the displacement of SiC particles in an acoustic field.

7. Comparison between experimental results and mathematical model

In order to predict the motion of particles using the mathematical model, there are four input parameters that should be known. They are the radius of the particle, initial position of the particle from the pressure node, particle compressibility of SiC, and acoustic energy in the fluid. The compressibility of SiC was calculated to be 1.85 using Eq. (2).

Three other variables that were not known were the radius of the particle (r), initial starting position of the particle (x₀) and acoustic energy in the field (E). The equation for the particle trajectory (6) was rewritten and Matlab 5.3 was used for data optimization using a minimization function in the package to obtain the best r, x₀ and E values.

Different combinations of these unknown variables were used as initial guesses for the displacement function and the corresponding optimized values were generated by the fmin search function in the Matlab Optimization software. Any optimization values with radius greater than 6 μm or initial position less than zero or greater than a half wavelength were discarded since the maximum probable size of the SiC particle is less than 12 μm.

The above procedure for optimization was repeated for each experimental data set at different input power values from 0.5 to 5.0 W. Fig. 10 shows the particle trajectories from the experimental results and mathematical model predictions using Eq. Figs. (6) and (43). The experimental data for higher power inputs were comparable to the mathematical model predictions using Eq. (39).

Fig. 10. Particle displacement versus time for 5.0 W power.

8. Summary and conclusions

The fractionation process of suspended particles due to the effect of acoustic standing wave and laminar flow fields has been mathematically modeled through the calculation of particle trajectories and displacement. The particle trajectories move towards the pressure node planes, lying at half wavelength intervals, so that the average distances between the particles are decreased considerably.

Using the mathematical model developed to calculate the trajectories and concentration of the micron size particles suspended in a fluid with acoustic standing waves results in rather accurate qualitative and quantitative results. Because no exact solution exists and there are extremely large parameters in the governing equations, it seemed necessary to supplement standard numerical procedures with somewhat novel approximate asymptotic methods. This approach yields solutions that are in closer qualitative and quantitative agreement with observed results than other methods in the literature. SiC particles were used to capture the individual particle displacements. After using statistical analysis and a data optimization procedure, for input power levels between 3.0 W and 5.0 W, the experimental data matched with the mathematical model prediction.