Article pubs.acs.org/IECR
Passage of a Single Bubble through a Liquid−Liquid Interface K.K. Singh†,‡ and Hans-Jörg Bart*,† †
Chair of Separation Science and Technology, TU Kaiserslautern, Kaiserslautern 67663, Germany Chemical Engineering Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India
‡
ABSTRACT: The phenomenon of passage of a single bubble through the interface between two immiscible and quiescent liquids is studied by using numerical simulations. The liquids considered are viscous and Newtonian. The volume of fluid method has been used for tracking the interface between the three fluids. Validation is carried out by using the experimental data reported in the literature. The validated numerical model is used to perform parametric analysis to understand the effects of bubble diameter and relevant physical properties (interfacial tension between the two liquids, viscosity ratio of the two liquids, and the density difference between the two liquids). Specifically, dependence of the retention time and retention height of the bubble on bubble size and physical properties is studied. Correlations for the retention time and retention height are proposed.
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INTRODUCTION “Bubbles, drops, and particles are ubiquitous,” cited from Clift et al.,1 is especially true with industrial processes and equipment dealing with multiphase flows. Specifically, gas− liquid−liquid flows are central to several unit operations such as liquid−liquid extraction2−6 and direct contact heat transfer7 in which gas agitation is used to generate a liquid−liquid dispersion to enhance the interfacial area for mass/heat transfer and film coefficients. There are many reactions in which gas− liquid−liquid flows are important.8,9 Gas−liquid−liquid flows are also prevalent in metallurgical processes10,11 and oil and gas industry.12 Due to their immense industrial relevance, fundamental understanding of the phenomena involving gas−liquid−liquid flows is indispensable. This study, focused on studying passage of a bubble through a liquid−liquid interface, is aimed at understanding one of the phenomena relevant to gas−liquid− liquid flows. The study is basically focused on the behavior of a fluid particle at a fluid−fluid interface. A literature survey reveals that there are many studies on this topic.13−20 But among such studies, the studies in which the fluid state of the matter of the particle is different from the two fluids at the fluid−fluid interface, as is the case with gas−liquid−liquid systems, are very few. Uemura et al.21 identified different stages of the interfacial phenomenon observed when a bubble passes through the interface between two immiscible quiescent liquids (water and silicone oil). In the first stage, the bubble rising through the interface pulls up the heavier liquid retaining a film of the heavier liquid around it. In the second stage, this film of the heavier liquid ruptures at the front of the bubble. In the third stage, the ripples generated due to rupture of the film of the heavier liquid propagate on the bubble surface, and the resulting instability leads to generation of numerous microdroplets of the heavier liquid. In the fourth stage, the heavier liquid thrusts into the bubble, and in the last stage, it penetrates the bubble completely. Dietrich et al.22 studied the phenomenon of passage of a bubble through the interface between two quiescent immiscible viscous liquids using high speed imaging and PIV (particle imaging velocimetry) measurements. The © 2015 American Chemical Society
effects of bubble size and the viscosity of the lighter liquid on the retention time of the bubble at the interface and the height to which the heavier liquid is entrained behind the bubble were studied. The velocity field near the liquid−liquid interface during passage of the bubble through the interface was studied using PIV. In a recent study, the generation of nanoemulsions due to bursting of the bubble at the interface between a thin layer of an oil and surfactant laden water is reported.23 There are some studies in which numerical simulations of the phenomenon of passage of drop (or bubble) through a liquid− liquid interface are reported. Manga and Stone24 reported a numerical model based on the Stokes equation for low Reynolds number (Re ≪ 1) motion of a single drop (or bubble) through a fluid−fluid interface. They used the numerical model to study the effects of physical properties on the evolution of the interfaces, specifically the shape of the drop (bubble) as it passes through the liquid−liquid interface. Their numerical model is, however, applicable only prior to film drainage, i.e., until the drop (or bubble) is still completely surrounded by the heavier liquid entrained by it. In addition to the numerical model, images of a single bubble and two vertically aligned bubbles passing through the liquid−liquid interface between two quiescent immiscible liquids were reported and qualitatively described. Shopov and Minev25 reported a numerical model based on Navier−Stokes equations for the phenomenon of passage of a single bubble through the liquid−liquid interface. However, like the model of Manga and Stone,24 their model too is applicable only prior to film drainage. Kemiha et al.26 studied the phenomenon of the passage of a single bubble through the interface between two viscous quiescent liquids. In the first set of the experiments, both lighter liquid (silicone oil) and heavier liquid (viscous solution of a lubricant in demineralized water) were Newtonian. In the second set of experiments, lighter liquid (silicone oil) was Received: Revised: Accepted: Published: 9478
July 9, 2015 September 2, 2015 September 8, 2015 September 8, 2015 DOI: 10.1021/acs.iecr.5b02488 Ind. Eng. Chem. Res. 2015, 54, 9478−9493
Article
Industrial & Engineering Chemistry Research
where v⃗ is the fluid velocity vector, ρ and μ are the effective fluid density and viscosity, p is the static pressure, τ is the viscous stress tensor, g⃗ is acceleration due to gravity, and F⃗ is the volumetric body force due to surface or interfacial tension. The viscous stress tensor is defined as
Newtonian, but the heavier liquid (solution of poly(acrylamide) in demineralized water) was a shear-thinning liquid. In both sets of experiments, effect of bubble diameter on the variation of the bubble position with time was observed. CFD (computational fluid dynamics) simulations using VOF method were also carried out for the first set of experiments. Qualitative match between the experimental results and the results obtained from CFD simulations was reported. In the most comprehensive study on the topic so far, Bonhomme et al.27 reported experimental and computational VOF studies on the phenomenon of the passage of a bubble through the interface between two quiescent Newtonian liquids. Size of the bubble, viscosities, and densities of the liquids were varied over a wide range to obtain bubbles of different initial shapes. Water and its solution with glycerin were used as the heavier liquid. Silicone oils of different grades were used as the lighter liquid. For each case evolution of the interfaces between the three fluids with time was analyzed. Comparison of the experimental and computational results was reported to be reliable in most cases, but the need for improvements in computational model to capture the phenomenon of film drainage was emphasized. In this study we employ the widely used28−33 VOF method to analyze a relatively less explored phenomenon of the passage of a bubble through the interface between two quiescent liquids. We are specifically interested in quantifying the effect of the bubble diameter and relevant physical properties on the retention time and the retention height of the bubble at the liquid−liquid interface. This information is not yet reported in the literature. The liquids considered are viscous, Newtonian liquids, and the flow regime considered is thus laminar. The numerical model is first validated using the experimental data reported in the literature and then used for parametric analysis in which we try to understand the effects of bubble diameter, interfacial tension between the two liquids, viscosity ratio of the two liquids, and density difference between the two liquids on the phenomenon of passage of the bubble through the liquid− liquid interface. In reality the liquids are seldom quiescent; still, the phenomenon of passage of the bubble through the interface between two quiescent liquids is of fundamental interest. The findings of this study will serve as the benchmark to understand and explain the more realistic cases in which liquids are not quiescent but flowing. The objective of this study is to explain the effect of physical properties on two fundamental quantities (retention time and retention height) which define the extent of interaction of the bubble with the liquid−liquid interface. The knowledge of these quantities can be useful in deciding limits of the operating parameters. For example, the knowledge of the retention time of a bubble can help in deciding the minimum flow rate of the air required to generate an air agitated liquid−liquid dispersion.
τ = μ{(∇⃗v ⃗) + (∇⃗v ⃗)T }
The effective density and viscosity are the weighted average of the densities and viscosities of the individual phases. Weighted average is done by using volume fractions of fluids as given by eqs 4 and 5. 2
ρ=
(2)
∑ μq αq (5)
q=0
where q refers to the phase index and α represents the volume fraction. The heavier liquid is considered as the primary phase (q = 0), the lighter liquid is considered as the secondary phase (q = 1), and air is considered as the tertiary phase (q = 2). The VOF method is used for tracking the interfaces between the three fluids. For this continuity equation for volume fraction, given by eq 6, is solved for the secondary and tertiary phases. ∂αq ∂t
+ ∇⃗ ·(v ⃗αq) = 0 ∀ q = 1, 2
(6)
The volume fraction of the primary phase is computed by using the constraint that the sum of the volume fractions of the three phases should be equal to 1. This constraint is mathematically given by eq 7. 2
α0 = 1 −
∑ αq (7)
q=1
Geometric reconstruction scheme is used for obtaining the interface between the fluids. The construction of the interface is based on the information about the volume fraction and its derivative in a partially filled cell. The interface is assumed to be linear in each cell. Thus, a smooth interface in reality is approximated by a piece-wise linear interface. For a sufficiently small grid size, the piece-wise linear interface is usually a good approximation of the smooth interface. F⃗, the volumetric body force due to surface or interfacial tension, is estimated using the gradient of volume fraction following the CSF (continuum surface force) model proposed by Brackbill et al.34 The expression of F⃗ generalized for a multiphase system is given by eq 8.
COMPUTATIONAL APPROACH Governing Equations. The numerical model is implemented in the commercial software ANSYS Fluent. The mass and momentum conservation equations solved are given by eq 1 and eq 2. The fluids are considered incompressible.
∂(ρv ⃗) + ∇⃗ ·(ρvv⃗ ⃗) = −∇⃗p + ∇⃗ ·τ + ρg ⃗ + F ⃗ ∂t
(4)
2
μ=
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(1)
∑ ρq αq q=0
F⃗ =
∇⃗ ·v ⃗ = 0
(3)
∑ σpq p
