Elsevier

Chemical Engineering Science

Volume 61, Issue 23, December 2006, Pages 7697-7705
Chemical Engineering Science

The dissolution of a stationary spherical bubble beneath a flat plate

https://doi.org/10.1016/j.ces.2006.08.071Get rights and content

Abstract

The dissolution of a single stationary bubble held in place by a horizontal plate is commonly observed experimentally. For several decades the standard approach to the analysis of such dissolution data has been to apply a correction factor of ln(2)=0.69 to the Epstein–Plesset equation for an isolated bubble. In this paper, the transport equations for a stationary bubble touching a plate are solved numerically for the common case where the flow field caused by the change in system volume as the bubble dissolves can be neglected. It is found that the total bubble lifetime is not well characterised by the use of the ln(2) factor. However, in most experimental situations, the initial stages of bubble dissolution are not captured. For low gas solubilities the use of a correction factor of 0.69 to the Epstein–Plesset equation is appropriate once the initial transients have dissipated. The correction factor varies from 0.69 to 0.77 across the full range of situations described in this paper. The mathematical model is validated by comparison to experimental data.

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 R0 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 CE.

  • (c)

    The diffusivity, temperature and pressure of the system are constant.

  • (d)

    The partial specific volumes of solute (v) 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.

Others have developed more sophisticated numerical solutions for isolated bubbles that incorporate the effects of multicomponent solute species (Yung et al. 1989, Cable and Frade, 1987) and surface tension (Cable and Frade, 1988).

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 parametersNa=CE-C0ρg(1-νCE),andNb=ν(CE-C0)1-vCE.The dimensionless parameter Na indicates the solubility of the solute in the fluid whereasε=Na-Nb=Na(1-vρg)indicates 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 t*dadt*=-Na1a+1(πt*)1/2.The 1/(πt*) term in Eq. (4) can be ignored for large dissolution times, thus allowing the equation to be solved with an approximate analytical solution:a2=1-2Nat*.Others (Bankoff, 1964, Kirkaldy, 1956) discuss a more sophisticated quasi-steady state solution, valid for small values of Na and long times given bya2=1-2Nat*ln(1+Nb)Nb.This equation approaches Eq. (5) for small Nb.

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 ln(2)=0.693 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 ln(2) when it is placed next to an ungrounded infinite plane. Thus, in the presence of a flat plate, Eq. (5) becomesa2=1-2ln(2)Nat*.Wise 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 Na. 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 Na (their 1/b) 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 (ε=0), 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 0.04m×0.04m×0.1m 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 120×120 grid in the (η,μ) plane. For the calculations shown the transformation parameter β*=0.5. 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 (η,μ)

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 ln(2) 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

adimensionless bubble radius = R/R0
Cthe concentration of the dissolved gas in the liquid, kg/m3
C0the gas concentration in the bulk liquid, kg/m3
CEgas concentration at saturation, kg/m3
Ddiffusion coefficient of the dissolved gas, m2/s
Fdimensionless concentration, ((CC0)/(CEC0))
Nadimensionless variable defined by Eq. (1)
Nbdimensionless variable defined by Eq. (2)
rradial co-ordinate
r*dimensionless radial co-ordinate=r/R0
Rradius of the bubble at time t, m
R0initial radius at time zero, m
ttime, s
t*

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.

References (30)

  • J.P. Christiansen et al.

    Molecular and cellular imaging with targeted contrast ultrasound

    Proceedings of the IEEE

    (2005)
  • R.H. Doremus

    Diffusion of oxygen from contracting bubbles in molten glass

    Journal of the American Ceramic Society

    (1960)
  • J.L. Duda et al.

    Mathematical analysis of bubble dissolution

    A.I.Ch.E. Journal

    (1969)
  • J.L. Duda et al.

    Heat or mass transfer-controlled dissolution of an isolated sphere

    International Journal of Heat and Mass Transfer

    (1971)
  • P.B. Duncan et al.

    Test of the Epstein–Plesset model for gas microparticle dissolution in aqueous media: effect of surface tension and gas undersaturation in solution

    Langmuir

    (2004)
  • Cited by (21)

    • Dissolution kinetics of trapped air in a spherical void: Modeling the long-term saturation of cementitious materials

      2020, Cement and Concrete Research
      Citation Excerpt :

      Based on boundary conditions, the single void dissolution kinetics (SVDK) model allows for the description of how liquid water continuously enters air voids from the surrounding capillary porosity while maintaining thermodynamic equilibrium at the gaseous air - liquid interface. Results are compared to: validated models of simple systems [16,17] (i.e., a gas bubble suspended in an infinite amount of liquid), micro-fluidic studies [13,16,18,19], and to findings of imaging studies [14] which provide a temporal sense of when near-surface air voids of a given size should become completely saturated. The literature review provided for the formulation of the SVDK models is divided into theoretical and experimental sections.

    • Bubble dissolution in horizontal turbulent bubbly flow in domestic central heating system

      2013, Applied Energy
      Citation Excerpt :

      However, such adaptations require correlations in order to compensate for the imperfect bubble spherical shape and diffusion field. Similar adaptations have been done for the dissolution of microbubbles attached to a wall under flow conditions by Cable [14] and Kentish et al. [15]. Such models have also been adapted in medical science involving the analysis of gas bubble dissolution in whole blood and plasma by Yang et al. [16].

    • Rapid Shrinkage of Lipid-Coated Bubbles in Pulsed Ultrasound

      2013, Ultrasound in Medicine and Biology
      Citation Excerpt :

      The fastest diffusion limited shrinkage rate is obtained by setting the concentration of gas at infinity to zero (f = 0). Although the EP model is for spherical symmetry, it has been shown that the presence of a nearby wall will only slow the rate of shrinkage (in the static case) (Kentish et al. 2006). All quiescent uncoated bubbles shrank more slowly than the theoretical prediction, eqn (5), a result that can be readily understood as being caused by either bubble surface contamination or a reduction in diffusive loss caused by the proximity of the cuvette wall (against which the bubbles rest buoyantly.).

    • Ostwald ripening of aqueous microbubble solutions

      2022, Journal of Chemical Physics
    View all citing articles on Scopus
    View full text