Article pubs.acs.org/IECR
Effects of Contamination and Shear-Thinning Fluid Viscosity on Drag Behavior of Spherical Bubbles Nanda Kishore,*,† V. S. Nalajala,† and Raj P. Chhabra‡ †
Department of Chemical Engineering, Indian Institute of TechnologyGuwahati, Assam − 781039, India Department of Chemical Engineering, Indian Institute of TechnologyKanpur, Kanpur − 208016, India
‡
ABSTRACT: In this work, the combined effects of contamination and shear-thinning (power-law) viscosity on the free rise of a single bubble have been studied numerically. The influence of insoluble contaminants on the surface of the bubble has been incorporated in the analysis by employing the spherical stagnant cap model which has been employed successfully in Newtonian fluids. The governing differential equations have been solved numerically over a range of conditions: Reynolds number, Re = 10− 200; power-law index, n = 0.6−1; and stagnant cap angle, α = 0°−180°. Finally, the effect of each of the parametersnamely, Re, n, and αon streamline patterns, surface pressure and vorticity distributions, and individual and total drag coefficients is discussed in detail. Briefly, for α > 30° and Re ≥ 50, the recirculation length increases and the separation angle moves forward with the increasing Re; however, mixed trends are observed with respect to the power-law index and the stagnant cap angle. The total drag coefficient increases as the cap angle and/or the power-law index increases and/or the Reynolds number decreases; while mixed trends are observed on the dependence of the ratio of the individual drag coefficients on these parameters.
1. INTRODUCTION
and drag phenomena of contaminated bubbles. Therefore, this work attempts to fill this gap in the current literature.
For a clean spherical bubble freely rising in an unconfined incompressible Newtonian liquid, at low Reynolds number (Re → 0), the drag coefficient is given by (16/Re).1 However, the presence of impurities or surface-active materials is unavoidable in real-life applications. Therefore, the insoluble surfactants can become adsorbed on the surface of bubbles and due to the surface advection caused by the fluid flow, these surfactants move from the front stagnation point toward the rear end. The surfactants accumulate at the rear end of the bubble to form a spherical stagnant cap that is immobile and, thus, inhibits the transmission of stress across the free surface; of course, the rest of the interface remains mobile and acts like a shear-free surface. With the increasing surfactant concentration, the bubble surface is increasingly covered by the surfactants and, in the limiting case of 100% coverage, the bubble behaves as a solid particle.1,2 However, Sadhal and Johnson3 were probably the first to generalize this model to the case of bubbles and drops, and they also reported closed-form expressions for the drag force of contaminated bubbles and drops as a function of the stagnant cap angle (α) in the limit of creeping flow (zero Reynolds number). Subsequently the so-called spherical stagnant cap model has been employed to estimate drag of clean and contaminated bubbles in Newtonian continuous phase at least up to Re = 200.4−19 On the other hand, many high-molecular-weight polymers and their solutions, as well as multiphase mixtures such as slurries, foams, and emulsions encountered in several industrially important applications display shear-thinning characteristics.20,21 Typical applications include degassing of polymeric melts and glasses, use of polymeric solutions in enhanced oil recovery, non-Newtonian fluidized beds, etc. Despite such wide-ranging potential applications, indeed no prior results are available on the effect of surfactants on the flow © 2013 American Chemical Society
2. PREVIOUS WORK The extensive body of knowledge on bubble dynamics in Newtonian liquids has been reviewed among others by Clift et al.,22 Michaelides,23 Rodrigue,24 Kulkarni and Joshi.25 Similarly, the corresponding analytical, numerical, and experimental results for bubble motion and mass transfer, etc., in clean power-law fluids have also been reported in numerous studies, which have been thoroughly reviewed in refs 20 and 26. The next generation of analyses in this field considered the effect of surfactants, shape deformation, and confinement, etc., while the ambient liquid was still Newtonian. At least, in the axisymmetric flow regime, the stagnant cap model has led to reasonable predictions of the drag characteristics (e.g. see refs 1−19). For instance, in the creeping flow limit, the expression for drag force of a contaminated bubble shows that the drag force is almost identical to that of a clean bubble for α ≤ 40°; in addition, it is very close to that of a solid sphere for α ≥ 140°. Between these two limits, the drag of a contaminated bubble strongly increases as the cap angle α increases. However, the analytical expression of Sadhal and Johnson3 is valid only for a very small concentration of surfactants, because of the linear relation for surfactant surface pressure approximation implicit in their work. This led He et al.9 to conclude that the results of Sadhal and Johnson3 underestimated the role of the cap angle on the drag coefficient and they also incorporated weak nonlinear effects. Fdhila and Duineveld10 reported experiments and numerical results on the rise velocity of small bubbles Received: Revised: Accepted: Published: 6049
January 28, 2013 March 24, 2013 April 2, 2013 April 2, 2013 dx.doi.org/10.1021/ie4003188 | Ind. Eng. Chem. Res. 2013, 52, 6049−6056
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(organic and inorganic materials) in quiescent contaminated solutions in the Reynolds number range of Re = 50−200. Their results lend further support to the general validity of the stagnant cap model. The numerical results of Leppinen et al.11 suggest that the effect of contamination on the drag of a fluid sphere (without deformation) is small and the effect of surfactants concentration is found to be significant for a deformable drop. Further support for the validity of the stagnant cap model to mimick the effect of surfactants has been provided by scores of theoretical and experimental studies over wide ranges of conditions that involve the Reynolds number, chemical reactions, etc.12−18 In a recent numerical study, Saboni et al.19 investigated the effect of contamination on the flow and drag behavior of fluid spheres at Re values up to 400, using the spherical stagnant cap model. They proposed a correlation for the drag coefficient of a contaminated fluid sphere as a function of the drag coefficients of a clean fluid sphere and a fully contaminated fluid sphere.3 In contrast, the available literature is very limited on the analogous problem in power-law fluids. For instance, Rodrigue et al.27 experimentally investigated the rising velocity of contaminated bubbles in Carreau model fluids and presented a correlation. Subsequently, they28 employed the standard perturbation technique to obtain an approximate expression for drag of a contaminated bubble in Carreau model fluids in the limit of small Carreau numbers and Reynolds numbers, i.e., for weak shear-thinning and inertial conditions. In a later study, they29 employed a thermodynamic approach (which assumes linear deviation in the surface tension from the equilibrium value) and a physical approximation (based on geometry and boundary conditions) in the limit of small Reynolds numbers. Tzounakos et al.30 experimentally investigated the effect of surfactant concentration on the terminal velocity, shape, and drag coefficient of bubbles in power-law liquids. They found that the bubble shape is independent of the surfactant concentration used in their study; however, the rise velocity and surface mobility of bubbles are strong functions of the concentration of the surfactants. Thus, to the best of our knowledge, no prior numerical results that elucidate the effect of surfactants on the flow and drag phenomena of contaminated bubbles in power-law liquids are available. Therefore, this work is intended to fill this gap in the literature over wide ranges of conditions: Reynolds number, Re = 10− 200; power-law index, n = 0.6−1; and stagnant cap angle, α = 0°−180°.
Figure 1. Schematic representation of uniform flow past a spherical bubble with immobile stagnant cap of surfactants at the rear end.
continuous phase; however, there is no analogous information available for a non-Newtonian continuous phase. Therefore, it is difficult to suggest critical values of the Reynolds number under which the bubble remains spherical in surfactant-laden power-law fluids. Over the range of conditions spanned here, the flow is assumed to be axisymmetric; a sphere-in-sphere computational domain is chosen here to reach the unconfined flow conditions, as shown in Figure 1. Because of axisymmetry, the velocity component in the azimuthal direction (vϕ) and derivatives with respect to ϕ are zero. Assuming the flow to be incompressible, the continuity and momentum equations in their dimensionless forms are written as follows: Continuity equation: 1 ∂ 2 1 ∂ [r vr ] + [vθ sin θ ] = 0 2 r sin θ ∂θ r ∂r
(1)
r-component of the momentum equation: ∂vr v2 ∂ 1 ∂ 1 + 2 [r 2vr 2] + [vrvθ sin θ ] − θ ∂t r r sin θ ∂θ r ∂r εrθ ∂η ⎤ ∂p 2n + 1 ⎡ ∂η =− + + ⎢εrr ⎥ ∂r Re ⎣ ∂r r ∂θ ⎦ ∂v ⎞⎤ 2nη ⎡ 1 ∂ 2 2 ∂ ⎛ 1 ⎜sin θ r ⎟⎥ + (r vr ) + 2 ⎢ ∂θ ⎠⎦ Re ⎣ r 2 ∂r 2 r sin θ ∂θ ⎝
(2)
θ-component of the momentum equation: vv ∂vθ ∂ 2 1 ∂ 1 + 2 [r 2vrvθ ] + [vθ sin θ ] + r θ ∂t r sin θ ∂θ r r ∂r εθθ ∂η ⎤ 1 ∂p 2n + 1 ⎡ ∂η =− + + ⎢εrθ ⎥ r ∂θ Re ⎣ ∂r r ∂θ ⎦ ⎞ 2nη ⎡ 1 ∂ ⎛ 2 ∂vθ ⎞ 1 ∂ ⎛⎜ 1 ∂ ⎟+ ⎜r + [vθ sin θ ]⎟ ⎢ ⎠ Re ⎣ r 2 ∂r ⎝ ∂r ⎠ r 2 ∂θ ⎝ sin θ ∂θ
3. PROBLEM STATEMENT AND MATHEMATICAL FORMULATION Consider a spherical bubble of radius R, steadily translating (with a constant velocity of U0) in an infinite expanse of powerlaw fluid contaminated with surfactants. During the course of bubble translation, as shown in Figure 1, the insoluble surfactants accumulate in the rear of the bubble forming a spherical cap with immobile surface, while the rest of the bubble surface remains mobile. Within the framework of the spherical stagnant cap model, the degree of contamination is expressed by the cap angle α, measured from the rear stagnation point. Because of the presence of insoluble surfactants along the surface of the bubble, a portion of the surface becomes immobile; thus, the bubble retains a spherical shape, even at large Reynolds numbers.2,20,22,23 Furthermore, under these conditions, the Weber number is small enough so that the bubble can retain its spherical shape in a Newtonian
+
2 ∂vr ⎤ ⎥ r 2 ∂θ ⎦
(3)
In eqs 1−3, velocity terms are scaled using the free stream velocity Uo, radial coordinate r using the bubble radius R, pressure using ρUo2, components of rate of deformation tensor by (Uo/R), viscosity by a reference viscosity ηref (ηref = m(Uo/ R)(n−1)), stress components by ηref(Uo/R), and time by (R/Uo). The Reynolds number, Re, appearing in the momentum equation is defined as follows: Re = 6050
ρUo(2 − n)(2R )n m
(4)
dx.doi.org/10.1021/ie4003188 | Ind. Eng. Chem. Res. 2013, 52, 6049−6056
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where ρ is the density of the fluid, m the power-law fluid consistency index, and n the power-law behavior index. For an incompressible liquid, the components of the extra stress tensor are related to the rate of deformation tensor as follows: where i , j = r , θ , ϕ
τij = 2ηεij
Cdf =
2n + 2 Re π ⎧ ⎡⎛ ∂v ⎤⎫ ⎛ ∂v ⎞ v ⎞ ⎨η⎢⎜ θ − θ ⎟sin 2 θ − ⎜ r ⎟sin 2θ ⎥⎬ ⎝ ∂r ⎠ ⎦⎭ r⎠ 0 ⎩ ⎣⎝ ∂r
∫
(5)
dθ r=1
(17)
The power-law viscosity (η) is given by
⎛ Π ⎞(n − 1)/2 η = ⎜ ε⎟ ⎝ 2 ⎠
4. NUMERICAL METHODOLOGY The governing equations (eqs 1−3) subject to the boundary conditions outlined in eqs 7−14 are solved numerically here. A finite difference method based on a simplified marker and cellimplicit algorithm is used on a staggered grid arrangement. This algorithm is a simplified version of the marker and cell method due to Harlow and Welch.32 The modification is introduced here to handle the non-Newtonian viscosity of the fluid effectively and an implicit formulation is used in the present study. The diffusive and non-Newtonian terms of the momentum equations are discretized using a second-order central difference scheme. The convective terms are discretized using the quadratic upstream interpolation for a convective kinematics scheme due to Leonard.33 The final steady-state solution was obtained by using a false-transient time stepping method, and this is why the time-dependent terms are retained in eqs 2 and 3. This procedure involves three steps: in the first step, eq 6 is used to evaluate the dimensionless non-Newtonian viscosity field over the entire flow domain using the previous time-step values of the velocity field. In the next step, the unknown velocity field is predicted using the previous time-step pressure field. Finally, the pressure field is corrected using the predicted velocity field, which is further used to correct the velocity field. For the next time-step calculations, the sum of the predicted and corrected velocity fields can be used as new values of the velocity field. This sequence of time-stepping is continued until the velocity field satisfies the equation of the continuity within a prescribed level. The stopping criterion for simulations is fixed as the maximum difference of any quantity between the two consecutive time steps divided by the time step (Δt) should be 120° it decreases. Irrespective of the Re values and/or of the stagnant cap angle, the drag ratio increases with the power-law index. Before leaving this section, it is worthwhile to revisit the key assumptions inherent in this work. First, the bubbles are assumed to remain spherical over the range of conditions spanned here. In the presence of surfactants, this is a reasonable assumption, because the surfactant-laden bubbles behave more like a rigid particle. However, it is difficult to propose critical values of the Reynolds number, because this information is not available, even for surfactant-laden Newtonian fluids, let alone for surfactant-laden power-law fluids. The second key assumption here is that the effect of the surfactants is taken into account by the simple stagnant cap model. Therefore, it is hoped that the future studies will address these as well as the other pertinent issues, such as convection, diffusion, and adsorption of surfactants on the surface of bubbles rising in power-law fluids as well as other non-Newtonian fluids. Despite these limitations, the results reported herein are believed to be reliable and useful, at least as a first-order approximation for estimating the bubble rise velocity in surfactant-laden powerlaw fluids.
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AUTHOR INFORMATION
Corresponding Author
*Tel.: +91 361 258 2276. Fax: +91 361 258 2291. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The corresponding author (N.K.) gratefully acknowledges the financial support from the Indian Institute of Technology Guwahati (Startup Grant Project − Phase X), Assam − 781039, India.
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6. CONCLUSIONS The effects of surfactants and power-law viscosity on the flow and drag characteristics of isolated bubbles are numerically investigated over wide ranges of pertinent governing parameters. For Re ≤ 20, no flow separation occurred, regardless of the values of the cap angle (α) and the power-law index (n). For Re ≥ 50, no flow separation was observed up to α ≤ 30°, irrespective of the value of power-law index. For α > 30° and Re ≥ 50, the recirculation length increases and the separation angle moves forward (toward the front stagnation point) as the
NOMENCLATURE Cd = total drag coefficient, dimensionless Cdf = friction drag coefficient, dimensionless Cdp = pressure drag coefficient, dimensionless FD = drag force, N LR = recirculation wake length, dimensionless m = power-law consistency index, Pa sn n = power-law behavior index, dimensionless p = pressure, dimensionless ps = surface pressure, dimensionless r = radial distance, dimensionless R = bubble radius, m Re = Reynolds number, dimensionless R∞ = radius of the computational domain, dimensionless Uo = free stream velocity, m/s vr = r-component of velocity, dimensionless vθ = θ-component of velocity, dimensionless vϕ = ϕ-component of velocity, dimensionless
Greek Symbols
α = stagnant cap angle, degree ε = rate of strain tensor, s−1 θ = streamwise direction, degree 6055
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θS = separation point, degree Πε = second invariant of the rate of strain tensor, dimensionless Φ = azimutal direction, degree η = dynamic viscosity of fluid, dimensionless ρ = density of fluid, kg/m3 τ = extra stress, Pa ωs = surface vorticity, dimensionless
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