Shape oscillation of microbubbles

Highlights

• Experimental data on bubble oscillation were collected.

• A simple, easily handled mathematical model of oscillatory motions was setup.

• Expression for the amount of liquid that takes part in the oscillatory motions was derived.

• Frequency and its dependence on the bubble size were evaluated.

Abstract

Microbubbles – gas bubbles of diameter less than 1 mm – became currently of considerable importance for chemical and process engineering applications, mainly because of the recent discovery of an energetically efficient method of their generation with a fluidic oscillator in the gas supply into an aerator. The oscillation should be applied at the microbubble resonant conditions, about which there has been so far known very little. The key problem is the unknown and difficult to evaluate extent of the surrounding liquid that takes part in the microbubble oscillatory motions and represents the inertia term in the governing equation. The oscillation of microbubbles also influences their ascent, which is slow and puts them often into mutual proximity causing their conjunctions. Author evaluated basic data on oscillating microbubbles from high-speed camera frames and used them to setup a simple model, suitable for engineering design purposes.

Introduction

Gas bubbles in a liquid are used in a wide spectrum of processes in chemical engineering, biotechnology, food sciences, and medicine. Typical purpose is to provide an opportunity for diffusion of the gas into the liquid – representative cases being, e.g., supplying O₂ to micro-organisms processing waste water or providing CO₂ to algae growth in a photobioreactor. Bubbles may be also present involuntarily due to decreased gas solubility if the liquid pressure is decreased. This is the case of the well-known problems with cavitation in pumps or hydraulic machinery. This phenomenon has been studied since the pioneering work of Lord Rayleigh in 1917 [1]. It is associated with bubble volume changes, described by Rayleigh’s equation or its later derivatives taking into account additional effects, such as Rayleigh–Plesset equation [2]. Since in these cases bubbles maintain their spherical shape, the equations are ordinary differential equations with bubble radius as only variable. Compression and expansion of the gas inside the bubble needs, to be really significant, an action of quite large variations of pressure.

Easier to generate, even at a constant pressure, are aspherical oscillation modes with the bubble varying its shape. The complexity of the geometry of possible shapes, however, makes much more difficult to solve the equations that govern the aspherical modes. Their theory is, as a consequence, far less developed than that of the volume dynamics. Characteristically, the authoritative reference on bubble oscillation [3], which summarises practically all bubble oscillation research results, devotes only a very small part of its contents (roughly one page out of the total 88) to the shape oscillation. An interesting theoretical analysis of bubble oscillation having several associations with the present problem was recently published by Van der Geld and Kuerten [13]. Practical usefulness of their results is limited by the fact that they analyse a quite different case – that of a bubble near a wall – and also lack any support by experimental data. The paper [13] shows the extreme mathematical complexity of such problems, despite numerous inevitably introduced simplifications. Of particular interest is the derived equation (labelled Eq. (3.15) in [13]) governing the oscillatory deformation of a bubble “near but not very close” to the plane wall. It is essentially the Rayleigh–Plesset equation extended to involve anisotropic (i.e. shape) deformation. It is finally converted to second-order ordinary differential equation describing the oscillation. After introducting very complex expressions for the derivatives it finally contains altogether 21 coefficients. Other know approaches to the bubble shape oscillation problem are even less advanced. They do not attempt evaluation of the oscillation and only limit their attention to study of bubble stability [14], [15], [16], [17]. Yet they encounter serious mathematical difficulties.

In industrial chemical engineering applications, bubbles are mostly introduced into the liquid with the aim to enhance diffusion transport of the gas into the liquid – higher than the transport across the liquid level surface, since the bubbles offer a much larger total transfer surface area. It is obvious that the effect requires the bubbles to be as small as possible. Microbubbles, defined as smaller than 1 mm in diameter, are therefore an attractive possibility. While a topic of discussions for a more than a decade, microbubbles have been so far only seldom used in practical applications – mainly because of energetically inefficient available methods of their generation.

One reason for difficulties associated with generation of microbubbles is the very Laplace–Young law that governs the process. It says that the pressure difference across the bubble surface is inversely proportional to its curvature radius. This mean the total energy spent on generation, from a given gas volume, of many small microbubbles (rather than a few large bubbles) may become quite large in total. In addition, typical mechanisms used for microbubble generation usually wasted much of the input energy. The situation has changed mainly due to the recent research done at the University of Sheffield [18], [19]. The key to the energy efficient mechanism is an oscillation applied to the gas supply flow into the aerator. Particularly advantageous is to use for the purpose a no-moving-part fluidic oscillator. The absence of mechanical (moved or deformed) components results in practically unlimited life, very high reliability, low price, and other advantages. No electric current is needed (which is an advantage in the typically harsh environment such as, e.g., in waste water processing). The hopes associated with this new idea have led even to the promise of generating the so far somewhat enigmatic and elusive nanobubbles [20]. It is obvious that for efficiency of this approach it is mandatory to bring the driving oscillatory action at resonance with the natural frequency of microbubbles.

It is an important and yet not widely known fact that microbubbles may in some respects exhibit unusual behaviour, sometimes even qualitatively different from that of the large (>1 mm) bubbles [21]. In an example discussed in [22] microbubbles demonstrably made possible delivering heat into a vaporisation process without spending (and wasting) energy on heat transfer into the evaporated liquid. Even some mechanical properties of microbubbles and the conditions of their generation may deviate from the usual laws established for the large bubbles. The difference is due to the higher surface energy of microbubbles, again a consequence of the Laplace–Young law. While large bubbles may be relatively easily deformed by acting external forces into a variety of shapes, microbubbles – if departing at all from the spherical shape – resist to all effects much more strongly.