The dynamics of gas bubbles in liquids has been studied using a variety of optical and acoustic techniques for industrial applications and for basic physics research. High-speed photography (See, e.g., T. G. Leighton, The Acoustic Bubble. (Academic Press, London, 1994), pp. 119–207; H. C. Pumphrey and A. J. Walton, “Experimental study of the sound emitted by water drops impacting on a water surface,” European Journal of Physics, 9(3), 225–231 (1988); and P. Di Marco et al., “Experimental Study on Terminal Velocity of Nitrogen Bubbles” in FC-72, Proc. Experimental Heat Transfer, Fluid Mechanics and Thermodynamics 2001, ed. by G. P. Celata et al., GR, Sep. 24–28, 2001, ETS, Pisa, pp. 1349–1359.) is the most widely used technique, but requires complex image processing to extract quantitative information about bubble behavior. Laser Doppler anemometry (See, e.g., R. Mahalingam et al., “Velocity measurements in Two-Phase Bubble-Flow Regime with Laser-Doppler Anemometry,” J. Am. Inst. Chem. Eng. 22, 1152–1155 (1976).) has been used to study bubble terminal velocity, and the laser-Schlieren (See, e.g., D. S. Hacker and F. D. Hussein, “The Application of a Laser-Schlieren Technique to the Study of Single Bubble Dynamics,” Ind. Eng. Chem. Fund. 17(4), 277–283 (1978).) technique has been used to study bubble shape and terminal velocities. Optical interferometry (See, e.g., A. Gelmetti et al., “An optical interferometer for gas bubble measurements,” Rev. Sci. Instrum. 67(10), 3564–3566 (1996); and L. Rovati et al., 64 (6), 1463–1467 (1993).) has found use in the study of bubble oscillations in a sound field. These optical techniques require both a transparent liquid and window access to the liquid through the container. Radio-frequency probes (See, e.g., N. Abuaf et al., “Radio-frequency probe for bubble size and velocity measurements,” Rev. Sci. Instrum. 50(10), 1260–1263 (1979).) have also been used to investigate bubble size and terminal velocity. Passive listening (See, e.g., T. G. Leighton and A. J. Walton, “An experimental study of the sound emitted from gas bubbles in a liquid”, Euro. J. Phys. 8, 98–104 (1987).) at acoustic frequencies is typically used to study bubble resonance. Ultrasonic pulsed Doppler procedures have been used for bubble detection (See, e.g., R. Y. Nishi, “Ultrasonic detection of bubbles with Doppler flow transducers,” Ultrasonics, 10, 173–179 (1972).) and terminal velocity measurements (See, e.g., H. Kellerman et al., “Dynamic modeling of gas-hold-up in different electrolyte systems,” J. Appl. Electrochem. 28, 311–319 (1998).). Typically, the above-mentioned techniques are used to study only one or two aspects of the behavior of bubbles.
There are three principal stages to the evolution of a gas bubble: (1) formation and growth at the tip of a nozzle located in a liquid; (2) detachment and resonance; and (3) ascent to terminal velocity.
In the first stage of evolution, the bubble grows to a specific size at the opening of the nozzle, the radius of the nozzle opening and the properties of the surrounding liquid determining the ultimate size of the bubble (See, e.g. M. S. Longuet-Higgins, B. R. Kerman, and K. Lunde, “The release of air bubbles from an underwater nozzle,” J. Fluid Mech. 230 (1991) p365–390.) As the bubble pinches off and detaches from the nozzle, it resonates (breathing-mode) briefly at a natural frequency determined primarily by its radius and the liquid density. The frequency f0 of this resonance oscillation was first calculated by M. Minnaert in “On Musical Air-Bubbles and the Sounds of Running Water,” Phil. Mag. 16, 235–248 (1933) to be:                                           f            0                    =                                    1                              2                ⁢                π                ⁢                                                                  ⁢                                  R                  0                                                      ⁢                                                            3                  ⁢                  γ                  ⁢                                                                          ⁢                                      p                    0                                                  ρ                                                    ,                            (        1        )            where R0 is the radius of the bubble, γ is the ratio of specific heat at constant pressure to the specific heat at constant volume of the gas, ρ0 is the hydrostatic pressure of surrounding liquid, and ρ is the liquid density. This equation is reasonably accurate for the mm-sized bubbles. For significantly smaller bubbles, Equ. 1 must be modified to account for the effects of surface tension (See, T. G. Leighton, supra). The bubble resonance can be detected and quantified using a hollow cylindrical piezoelectric transducer surrounding the bubble.
After detachment from the nozzle, the bubble accelerates to its terminal velocity which depends on the size of the bubble. For low viscosity fluids, such as water, the behavior of the rising bubble falls within several regions. Small bubbles (less than 0.035 cm radius) are spherical and rise substantially vertically at a speed determined by Stokes' Law. Larger bubbles (0.035 cm to 0.07 cm), have internal air circulation, which reduces shear stresses at the interface leading to a velocity higher than predicted by Stokes' Law. Between 0.07 cm and 0.3 cm, bubbles are elliptical and follow a spiral or zigzag path. Drag increases due to vortex formation in the bubble wake. Bubbles greater than 0.3 cm form spherical cap shapes (See, e.g., L.-S. Fan and K. Tsuchiya, Bubble Wake Dynamics in Liquids and Liquid-Solid Suspensions (Butterworth-Heinemann, Boston, 1990), pp. 36–43.).
The terminal velocity U0 depends on the buoyant and drag forces on the bubble (See, H. Kellerman et al., supra):                                           U            0                    =                                                    8                3                            ⁢                                                gR                  0                                                  C                  D                                                                    ,                            (        2        )            where g is the acceleration due to gravity, R0 is the radius of the bubble and CD is the drag coefficient. The drag coefficient depends on physical properties of the liquid and the size of the bubble. A theory by G. Bozzano and M. Dente, “Shape and terminal velocity of single bubble motion: a novel approach,” Computers & Chemical Engineering. 25 (2001) 571–576, is useful for calculating the drag coefficient because it covers a wide range of bubble sizes and liquid properties. The drag coefficient is calculated using Reynolds, Eotvos, and Morton numbers, which depend on the surface tension, density, and viscosity of the liquid and the bubble size. An equation for terminal velocity applicable for air bubbles between 0.07 cm and 0.3 cm is given by (See L.-S Fan and K. Tsuchiya, supra):                                           U            0                    =                                                    c                ⁢                                                                  ⁢                σ                                                              R                  0                                ⁢                                                                  ⁢                ρ                                                    ,                            (        3        )            where c=1.8 for a single component liquid (c is between 1.0 and 1.4 for mixtures). The presence of contaminants (e.g., surfactants, suspended particles) has a significant effect on the rise of the bubble due to the Marangoni effect and the immobilization of the air-liquid interface (See, e.g., G. Liger-Belair et al., “On the Velocity of Expanding Spherical Gas Bubbles Rising in Line in Supersaturated Hydroalcoholic Solutions: Application to Bubble Trains in Carbonated Beverages,” Langmuir 16, 1889–1895 (2000).).
The path of the rising bubbles is largely determined by the Reynolds number, NR. For low Reynolds numbers (NR<130), the bubble travels substantially vertically. For higher Reynolds numbers (130<NR<400), the tip of the wake behind the bubble becomes unstable and oscillates at a low frequency, leading to a zigzag path. For even higher Reynolds numbers (400<Re<350,000), vortices are periodically shed from alternate sides of the bubble on a plane that slowly revolves around the bubble, leading to a spiral path (See T. G. Leighton, supra).
As the mm-sized bubble rises, it also undergoes shape oscillations (P. Di Marco et al., supra). The frequency of these oscillations (See T. G. Leighton, supra) is given by:                                           f            n                    =                                    1                              2                ⁢                π                                      ⁢                                                            (                                      n                    -                    1                                    )                                ⁢                                  (                                      n                    +                    1                                    )                                ⁢                                  (                                      n                    +                    2                                    )                                ⁢                                  σ                                      ρ                    ⁢                                                                                  ⁢                                          R                      0                      3                                                                                                          ,                            (        4        )            where fn is the frequency of oscillations, n is the mode number, and σ is the surface tension.
Both terminal velocity and shape oscillations can be monitored by observing the Doppler frequency shift of sound reflected from the bubble. The speed, U, of the bubble is related to the speed of sound, the frequency of the sound source used to interrogate the bubble, and the frequency received by the detector utilized for the measurement according to:                               U          =                      v            ⁢                                          (                                                      f                    r                                    -                                      f                    s                                                  )                                            (                                                      f                    r                                    +                                      f                    s                                                  )                                                    ,                            (        5        )            where ν is the liquid sound speed, fr is the received frequency, and fs is the source frequency (See, e.g., D. G. H. Andrews, “An experiment to demonstrate the principles and processes involved in medical Doppler ultrasound,” Phys. Educ. 35(5), 350–353 (2000)).
Accordingly, it is an object of the present invention to provide an apparatus and method for measuring liquid characteristics from the properties of bubbles formed therein.
Additional objects, advantages and novel features of the invention will be set forth, in part, in the description that follows, and, in part, will become apparent to those skilled in the art upon examination of the following or may be learned by practice of the invention. The objects and advantages of the invention may be realized and attained by means of the instrumentalities and combinations particularly pointed out in the appended claims.