The dissolution of a stationary spherical bubble beneath a flat plate
Introduction
The dissolution of stationary bubbles has been widely investigated in the past for commercial, biological and fundamental interests such as removal of bubbles in glass (Doremus, 1960, Greene and Gaffney, 1959) and polymer melts (Favelukis et al., 1995, Kontopoulou and Vlachopoulos, 1999) and in the determination of the diffusion coefficient of gases in liquids (Houghton et al., 1962, Krieger et al., 1967, Wise and Houghton, 1966). Most recently, lipid, protein or surfactant coated microbubbles are being utilised as ultrasound contrast agents (Christiansen and Lindner, 2005), drug and gene delivery vehicles (Bekeredjian et al., 2005) and blood substitutes (Van Liew and Burkard, 1995), and an understanding of the behaviour of these coated microbubbles is becoming important.
The dissolution of a bubble involves solution of gas into the liquid at the interface, and diffusion of dissolved gas away from the interface into the bulk liquid. It is assumed that the former is fast compared to the latter, hence the dissolution process is assumed to be diffusion controlled. Several authors (Cable and Evans, 1967, Frade, 1983, Duda and Vrentas, 1971) have numerically solved the fundamental problem of an isolated spherical bubble of initial radius with the following assumptions:
- (a)
The bubble contains a uniform and constant gas density. This assumption implies that there is a negligible effect of the changing Laplace pressure on the gas density as the bubble shrinks. This effect is found to be important only when the bubble radius becomes very small and surface tension forces are large.
- (b)
Transfer of gas from the bubble is controlled by Fickian diffusion (dissolution of gas into the liquid is fast) and the concentration of dissolved gas in the liquid at the interface is constant at the equilibrium concentration .
- (c)
The diffusivity, temperature and pressure of the system are constant.
- (d)
The partial specific volumes of solute and solvent are constant but not necessarily equal.
- (e)
The velocity field in the liquid is purely radial, and the bubble is a perfect isolated sphere so that the concentration field is spherically symmetrical.
The governing equations are those of advection–diffusion of solute in the liquid together with a kinematic condition at the moving bubble surface that describes the change in bubble radius in terms of the diffusion flux there. Duda and Vrentas, 1969, Duda and Vrentas, 1971 expressed the problem in a dimensionless form containing the two non-dimensional parametersandThe dimensionless parameter indicates the solubility of the solute in the fluid whereasindicates the importance (relative to diffusion) of advection of solute due to the liquid velocity field which results from the volume change of the system as the bubble dissolves.
By neglecting the advection term, Epstein and Plesset (1950) derived an approximate analytical solution for a stationary bubble which yielded the following equation for the dimensionless bubble radius a in terms of dimensionless time The term in Eq. (4) can be ignored for large dissolution times, thus allowing the equation to be solved with an approximate analytical solution:Others (Bankoff, 1964, Kirkaldy, 1956) discuss a more sophisticated quasi-steady state solution, valid for small values of and long times given byThis equation approaches Eq. (5) for small .
Other studies (Liebermann, 1957, Houghton et al., 1962, Wise and Houghton, 1966; Wigman et al., 2001; Duncan and Needham, 2004) consider the dissolution of a single bubble that is held in place beneath a stationary flat plate, and attempt to account for this by modifying the Epstein–Plesset solution (Eq. (5)) for an isolated bubble. Liebermann (1957) introduced a correction factor of into the Epstein–Plesset solution. This factor was taken from the theory of electrostatics where the capacitance of a conducting sphere drops by a factor of when it is placed next to an ungrounded infinite plane. Thus, in the presence of a flat plate, Eq. (5) becomesWise and Houghton (1968) confirmed this correction factor numerically for a point contact using potential theory assuming quasi-steady diffusion. However, both Manley (1960) and Wise and Houghton, 1966, Wise and Houghton, 1968 comment that the bubble is likely to have a finite area of contact with the plate. This will both reduce the volume of the bubble below that of a perfect sphere and further disrupt the concentration distribution around the bubble. For an air bubble resting below a Plexiglas plate, Wise and Houghton (1966) suggest that the bubble volume be reduced by a factor of 0.947 and the concentration field by a geometry factor of 0.650, giving an overall correction factor of 0.686.
Cable (1967) points out that the ln(2) correction factor will be valid only for low solubilities when the concentration boundary layer is essentially infinite in size. This will only occur for very small values of . As the solubility increases, the shell of liquid around the bubble in which diffusion occurs (i.e., the boundary layer) becomes thinner, and the plate “cuts off” a smaller fraction of the spherical shell (see Fig. 1). In the limit of infinite solubility the diffusion boundary layer is so thin that it does not intersect the plate and the diffusion behaviour becomes that of an isolated bubble. Thus, the correction factor should move from 0.693 towards 1.0 as the solubility increases, provided the effect of liquid velocity remains small.
Takemura et al. (1996) appear to be the only authors to have considered these wall effects through numerical modelling. These authors find that the correction factor does indeed increase as the solubility increases. In the absence of natural convection, they find that the correction factor to the Epstein–Plesset equation moves from 0.72 to around 1.3 as (their ) moves from 0.001 to 1.0. However, the kinematic condition of the bubble interface used in this model is somewhat simplistic and this may have affected the results at the upper end of this range.
It is apparent from the above account that a complete mathematical understanding of the influence of a flat retaining plate on bubble dissolution is lacking in the literature. The aim of this paper is to present a finite difference numerical model of bubble dissolution that explicitly accounts for the presence of the boundary condition on the flat plate, and to compare the model predictions with experimental data and with predictions of existing models.
Section snippets
Numerical model
This model assumes that the bubble makes a point contact with the wall (horizontal plate). Like many studies of isolated bubbles (Epstein and Plesset 1950; Houghton et al., 1962; Krieger et al., 1967; Subramanian and Weinberg, 1981) this work assumes that the effect of liquid velocity can be ignored (), and that the bubble is small enough to remain spherical. However, the concentration distribution is no longer spherically symmetric because of the presence of the wall. Rather than solve for
Experimental techniques
A rectangular cell of dimensions was designed and constructed to study the dissolution rate of a single stationary bubble in an aqueous solution, pictured in Fig. 3. The cell is made of stainless steel material, with rubber seals to prevent any leakage of gas and liquid into and out of the cell. A removable lid allows liquid to be transferred into the cell, this also allows easy cleaning of the cell interior. A stainless steel frame that supports a glass plate is positioned at
Results/discussion
The model equations were solved numerically on a grid in the plane. For the calculations shown the transformation parameter . This gave a suitable non-uniform distribution of computational mesh cells in the physical domain with a high mesh density near the bubble surface where the concentration gradients are highest, and much lower mesh density at large distances from the bubble where the gradients are smaller. Doubling the number of cells in each direction in the
Conclusion
A mathematical model for the dissolution of a stationary spherical bubble below a flat plate has been developed. Analysis of the dissolution curves provided by this model suggests that the correction factor commonly used to correct for the wall boundary condition is truly valid only for a limited range of conditions, namely at long times and low gas solubilities. At short times and for higher gas solubilities the correction factor diverges from this value. When a full numerical solution
Notation
dimensionless bubble radius = the concentration of the dissolved gas in the liquid, the gas concentration in the bulk liquid, gas concentration at saturation, diffusion coefficient of the dissolved gas, dimensionless concentration, dimensionless variable defined by Eq. (1) dimensionless variable defined by Eq. (2) radial co-ordinate dimensionless radial radius of the bubble at time , m initial radius at time zero, m time, s
Acknowledgements
The authors would like to acknowledge Prof. Franz Grieser for his valuable comments and suggestions. Infrastructure support from the Particulate Fluids Processing Centre, a Special Research Centre of the Australian Research Council is also recognised. Judy Lee received a postgraduate stipend from the Department of Chemical and Biomolecular Engineering within the University of Melbourne and this support is also gratefully acknowledged.
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