VDROP: A comprehensive model for droplet formation of oils and gases in liquids - Incorporation of the interfacial tension and droplet viscosity
Introduction
Study of droplet formation and breakup processes is important for oil spill events in the marine environment. The movement of oil droplets in the water column is greatly affected by the droplet size. For a given oil, large droplets have larger buoyancy than smaller droplets, and thus rise to the water surface rapidly when submerged. Studies by Elliott et al. [1] reported rapid transport of large oil droplets and slow transport of smaller droplets, causing the formation of the comet-shaped oil slick on the water surface. Boufadel et al. [2] evaluated the effect of buoyancy on the transport of oil droplets, and demonstrated that large droplets get advected faster than small droplets due to the larger Stokes drift [3] near the surface, resulting in the comet shape distribution.
The droplet size distribution (DSD) is not only important for the transport of oil but also for its fate, as increasing the portion of small droplets results in an increase in the surface area (per unit mass). A large surface area enhances the dissolution of hydrocarbon components in the water column [4]. In addition, as the biodegradation of low solubility oil components in the droplets occurs at the water–oil interface, an increase in the surface area increases oil biodegradation [5], [6]. Therefore, there is a need to obtain the droplet size distribution and its evolution with time an oil spill occurs at the surface or subsurface of a water body.
Models for the droplet size distribution range from correlations to detailed simulation of DSD. Correlations (or correlation models) rely on dimensionless numbers, such as the Weber number – ratio of destructive forces to resisting forces due to interfacial tension (IFT), and the Reynolds number to obtain characteristic diameters at steady state (i.e., after a long time of mixing) [e.g. [7], [8], [9], [10]]. In situations where viscosity also contributes to the resistance to breakup of droplets, a modified Weber number was used through the incorporation of a viscosity group [7], [11], [12] to obtain the equilibrium (i.e., steady state) characteristic size. Droplet coalescence was accounted for explicitly in these approaches. In addition, since correlation approaches can only provide characteristic diameters, the DSD was assumed to follow analytical functions, such as the normal distribution [13], [14], the lognormal distribution [9], [15], [16], and the Rossin–Ramler distribution [9], [12], [17].
Population balance models rely on solving differential equations numerically for the droplet size distribution (DSD). The models take account of various mechanisms causing breakup and coalescence of droplets in a phenomenological way along with mechanisms resisting breakup [18], [19], [20], [21], [22]. Successful population balance models are those that incorporate the physics of the problem to the maximum extent possible without making the code too complex or dependent on a large number of parameters. In these models, breakup of oil droplets in turbulent flow is viewed as the result of collision of droplets with turbulent eddies and of a breakage efficiency that depends on oil properties and mixing intensity [23], [24]. Coalescence is typically assumed to occur as a result of collision between droplets due to mixing and a collision efficiency that depends on oil properties [23], [25].
A major advantage of numerical population balance models over correlation models is that the prior provide the transient DSD. The steady-state DSD is obtained by simply running the numerical model for a long enough duration (typically less than a few hours), and obtaining the output. In spite of many peer reviewed studies using population balance models, only few studies considered oil viscosity effects in droplet breakage, and these occurred under steady state conditions [11], [26]. Accounting for the role of the droplet viscosity in resisting breakup is not only important for high viscosity oils, but also for situations where surfactants are used reducing the surface tension forces by orders of magnitude, which would render viscosity forces (of even low viscosity oils) important in resisting the breakup of droplets [27], [28], [29].
Therefore, in the context of dealing with oil spills in marine environment, the objective of this paper was to develop a numerical model – VDROP using population balance equation to predict the transient and steady state DSD of oils of various properties. In particular, the model accounts for resistance to breakup of droplets due to viscous forces within the droplet. The VDROP model was extensively validated using experimental data from chemical reactors. The model was then used to simulate the DSD resulting from an oil slick in breaking waves.
Section snippets
Methodology
For a liquid–liquid dispersion system, the population balance equation has been widely used for predictions of droplet formation processes. Most previous studies used different forms of integral–differential equation of the population balance equation [e.g. [24], [25], [26]]. And thus, the following population balance equation is introduced for discrete droplet classes:
Validation of VDROP model
Model validations are conducted using available literature data of stirred-tank experiments. Turbulence inhomogeneity in stirred tanks has been observed in numerous experimental studies [e.g. [24], [45]]. Flow field in the turbulently agitated tank can be divided into two regions – impeller region where it is assumed that the impeller stream has a uniform and high intensity of turbulence; and circulating region with larger volume, but lower turbulence [46], [47], [48]. For simplicity, many
Simulation of droplet formation due to wave effects
It is important to know the dispersion of oil droplets at sea due to waves. Under the action of waves, the coherent oil film floating on the water surface is dispersed into small droplets and get advected and spread in the water column. It is reasonable to assume that the wave breaking and the resulting high mixing energy causes droplet breakup while the effects of non-breaking waves is to advect the droplets and to keep them more or less in suspension just below the water surface until the
Discussion
We have used a function for breakup that is time-dependent to match observation, and we provide arguments in this Section to support such an approach.
The equilibrium of forces on a droplet states that the destructive forces are equal to the stabilizing forces:where Fp is the destructive force (assumed due to the dynamic pressure resulting from turbulence); Fviscous is the resistant force due to the stress tensor in the droplet (due the viscosity within the droplet); and FIFT is
Conclusion
A comprehensive numerical model, VDROP, capable of simulating the transient droplet size distribution in turbulent regimes was presented in this paper. The model accounts for the resistance to droplet breakage due to interfacial tension and due to the viscosity of the dispersed phase. For the latter, a new formulation was introduced to account for the fact that the breakup of high viscosity droplets is time-dependent, and that they need to reside in high energy zones to breakup. The model,
Acknowledgement
This research was, in part, supported by the Department of Fisheries and Ocean Canada (DFO), Contract No. F5211-130060.
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