FUNDAMENTALS OF BUBBLE TRANSPORT IN AN ULTRASONICALLY ASSISTED SEPARATION PROCESS (2008)

The use of ultrasonic fields to induce phase separations of a dispersed solid, immiscible liquid, or gas within a suspending liquid phase has received increasing attention over the past two decades. Dissolved and entrained gases, and thus gas bubbles, have the potential to produce transport inefficiencies in closed-loop flow systems and also have the possibility of causing materials failure. Research conducted in this study investigates the applicability of the use of a resonant ultrasonic wave field to entrap and harvest gas bubbles from a surrounding liquid. By subjecting the bubbles to a standing acoustic field, their migration and coalescence can be managed, allowing for ease of removal. Current techniques for gas-liquid separations include static separators, which can be prone to fouling; vortex separators, which are not suited for low flow; and rotary separators, which tend to be mechanically complex and expensive. This research aims to develop a fundamental understanding of the effect of forces induced by resonant ultrasonic fields on the on the entrapment and coalescence phenomena. Modeling efforts illustrate the relationship between the size of bubbles, their sphericity, acoustic field parameters (frequency and intensity), bubble placement relative to acoustic pressure antinodes, and the bubble coalescence phenomena. Experiments involving small numbers of bubbles (less than ten) are used to test the basic predictions of the model. These results support subsequent studies of the coalescence of swarms of bubbles. Theory and Analysis of the Bubble Distortion and Entrapment Phenomena The initial analytical portion of this research project aims to develop a fundamental understanding of the effect of buoyancy versus acoustic forces on the entrapment and coalescence phenomena. Modeling efforts illustrate the relationship between the size of bubbles, their sphericity, acoustic field parameters (frequency and intensity), bubble placement in regards to acoustic pressure antinodes, and the bubble coalescence phenomena. Analysis of Bubble Sphericity Performing experiments with spherical bubbles is most desirable because it allows for the least sophisticated analytical models. Pressures induced by acoustic fields can distort bubbles from a spherical shape. Therefore knowledge of this practical limit on acoustic field intensity is important. Bubble sphericity was determine based on work by Marston [1, 2], in which he used linear and inviscid theory to approximate the quadrupole projection of the acoustic radiation pressure on a compressible sphere of millimeter size. The pressure calculated in this method can be used to calculate the maximum and minimum possible radius of the deformed bubble. By allowing a maximum aspect ratio (rmax/rmin) of 1.1, the assumption of a spherical bubble can be used. By fixing the aspect ratio, the allowable ranges of parameters for the experimental operating conditions can be found. One of which is the energy density in the fluid within the acoustic chamber. By determining the maximum energy density that is allowed to maintain sphericity for a bubble of a specific size, the intensity (voltage) of the acoustic field needed to fix the position of the bubble can be determined. For example, for a bubble with a 100 μm radius, the maximum average energy density that allowed the sphericity criterion to be satisfied was determined to be 0.406 J/m. Analysis of Bubble Entrapment The physical effects referred to as body forces determine whether bubble entrapment will occur. The body forces consist of gravity, buoyancy, drag, and the primary acoustic force. Assuming the bubbles are widely spaced and there are no bubble-bubble interactions, these are the forces which determine not only the likelihood of bubble entrapment, but the position and velocity of the bubble as well. The primary acoustic force is given by Equation 1 [3], and the buoyancy and gravitational forces are given by Equation 3. The drag force is given by the Hadamard-Rybczynski formula [4] which for the gas of an air bubble in water reduces to Equation 7. 3 1, 4 sin(2 ) ac ac F r E F y π κ κ = (1) 2 2 1 ˆ ˆ 2 6 2 2 ˆ ˆ ˆ [3 ( ) ] ˆ ˆ ( ) [3 ( ) ] r r r F κ σ σ κ κ σ σ ρ σ ρ − = + − (2) 3 4 3 ( ) b l g F r g π ρ ρ = − (3) 0 4 d l F rV πμ = − (4) In Equations 1-4, κ is the wavenumber, r is the radius of the bubble, ac E is the average energy density in the continuous liquid phase, y is the distance of the bubble from a pressure antinode, F is the acoustic contrast factor defined in Equation 5 specifically for a gas bubble (F = -17.7 for the air/water system with a bubble of 100 μm radius), ˆ g