Ideal Ultrasonic Parameters for Delicate Parts Cleaning
(p. 3)
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Sweeping Frequency (continued)
Once a certain amount of ultrasonic energy has been transferred to the fluid medium one must examine how much of that energy is expressed in the form of cavitation. An effective way of representing this is with a mathematical tool known as the acoustic interaction cross-section. Simply, this is the amount of energy subtracted from an acoustic wave by a bubble driven into oscillation. This energy is subsequently re-radiated by the bubble via pulsation or implosion and affects much of the cleaning accomplished by ultrasonics. More precisely the acoustic interaction cross-section is the ratio of the power subtracted from an acoustic wave due to a bubble's presence to the intensity of the incident beam.1 As its name suggests it has the units of area, i.e. square meters.
Figure 3
The acoustic interaction cross-section as a function of bubble radius (mm) for 40kHz ultrasonics. Ω40 is the cross-section at 40kHz, Ω38 is the cross section at 38kHz, and Ω42 at 42kHz. This demonstrates how sweeping the ultrasonic frequency creates a band over which a bubble population is excited.
The cross-section is strongly a function of a bubble's radius, see figure 3, this means that a single frequency picks out it's favorite sized bubble and pumps energy into it preferentially. The resonant bubble radius, at that frequency, is approximately determined by the following equation:
For most aqueous solutions we use κ = 1.3, p0 = 106 dynes/cm2, ρ = 1 gm/cm3. This gives a bubble radius of 0.008 cm at 40 kHz. If we sweep the frequency we are then exciting a range of bubble sizes. For a sweep of plus or minus 2kHz all of the bubbles whose sizes range from 0.0075cm and 0.0083cm are maximally excited. Bubbles whose sizes differ from the resonant size interact less strongly with the incident acoustic field and subsequently absorb less energy for cavitation. This line of thinking would indicate that the larger the sweep bandwidth the better the activity. This is true only to a point. As a transducer is driven off of it's primary resonance, the efficiency with which it converts electrical energy to mechanical energy decreases. It becomes a game of diminishing returns, a larger sweep bandwidth allows you to excite a larger bubble population but with little energy at the ends of the bandwidth. The optimum transducer is designed with a wide bandwidth resonance allowing a significant transfer of ultrasonic energy into the tank over the entire sweep range.
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