Part IV – April 1969 - Papers - Studies in Vacuum Degassing Part I: Fluid Mechanics of Bubble Growth at Reduced Pressures

A formulation is given for describing the rate of expansion of spherical bubbles rising in liquids the freeboard of which is evacuated. The computer solution of the resultant differential equations has shown that, for low freeboard pressures (less than about 5 mm Hg), on approaching the free surface the bubbles expand much less rapidly than predictable from equilibrium considerations. In other words, in this region the pressure inside the bubbles will be significantly larger than the static pressure in the liquid corresponding to the position of the bubble. These theoretical predictions were confirmed by experimental work, using two-dimensional air bubbles rising in mercury. The important consequence of these findings is that the expansion of gas bubbles in vacuum degassing operations will be a great deal less than expected from hydrostatic considerations. This would lead to a significant reduction in the available interfacial area and may explain the apparent poor efficiency of many vacuum degassing units. VACUUM degassing as a treatment for liquid steel has gained widespread popularity in recent years; the number of known installations exceeds several hundred at the present time.' Although much information is available on both the thermodynamics of the system and the overall performance of various industrial units, much less is known about the fundamental aspects of the process kinetics.2-4 The basic physical situation common to virtually all vacuum degassing operations is the interaction between gas bubbles (swarms of bubbles) and the surrounding molten metal, held in a container, the freeborad of which is at a low absolute pressure. Once formed (or introduced from an external source) the bubbles will ascend, due to the buoyancy forces, and, during this ascent, a significant increase in their volume will occur. This progressive increase in the bubble volume is due to two factors: a) the continuous reduction in the static pressure acting on the bubble during its rise; and b) mass transfer into the bubble from the surrounding molten steel. In a recent paper Richardson and Bradshaw developed equations5 for describing mass transfer into gas bubbles from molten metals at reduced pressures. However, in deriving these expressions it was assumed that the pressure inside the bubble was identical to the static pressure in the adjacent liquid. In other words, the volume of the bubble was considered to be in equilibrium with the pressure of the fluid adjacent to it. This assumption, thus their analysis presented, was thought to be reasonably accurate for most of the bubble's ascent; however, it was unlikely to be valid in the immediate vicinity of the free surface, held at a low pressure. It was pointed out in the discussion6 that the region close to the surface may be of considerable importance as both the driving force and the interfacial area available for mass transfer are at their highest value here. The ' 'anomalous" behavior of gas bubbles when approaching a free surface at low pressures was recently confirmed in a preliminary investigation by Szekely and Martins. ?1 Here high-speed motion photography was used to study air bubbles rising in a column of n-tetradecane with a freeboard pressure of 0.1 mm Hg. It was found that significant distortion of the bubbles occurred on approaching the free surface; furthermore, the expansion observed was much less than what one could expect from hydrostatic considerations, i.e., factor a previously discussed. It follows from the foregoing that a detailed study of these phenomena would be justified both from fundamental considerations and because of their potential relevance to technology. The purpose of the paper is to present a more realistic formulation for the expansion of a gas bubble approaching a low-pressure region, together with a comparison of the theoretical prediction with experimental measurement. An inert bubble will be considered in the first instance; it is thought that the understanding of the fluid mechanics is an essential first step toward the realistic formulation of the mass transfer process. This latter problem will be the subject of a subsequent publication. FORMULATION The Physical Model. Consider a spherical bubble, of initial radius Ro, rising in a fluid, having a density pL. Initially let the bubble be at a distance H from the free surface, and at a pressure Pgo, as illustrated in Fig. 1. Pgo, the initial pressure in the bubble, is given by the following equation: pgo = Po = Ptp +pLgH [ la] where Po is the pressure in the liquid corresponding to the initial position of the bubble and Ptp is the pressure at the free surface. The fluid pressure at