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Ind. Eng. Chem. Res. 2007, 46, 618-622
GENERAL RESEARCH Mechanism of Turbulent Drag Reduction in Emulsions and Bubbly Suspensions Rajinder Pal* Department of Chemical Engineering, UniVersity of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Emulsions and bubbly suspensions are known to exhibit drag reduction behavior in turbulent flow, that is, the measured friction factors in turbulent flow fall well below the values expected on the basis of laminar flow properties. The mechanism of drag reduction in these two-phase dispersions is not well understood. This study proposes a new mechanism of drag reduction in turbulent flow of emulsions and bubbly suspensions. According to the proposed mechanism, drag reduction in emulsions and bubbly suspensions is caused by a significant reduction in effective viscosity of the dispersion when the flow regime is changed from laminar to turbulent. The experimental friction factor data on unstable and surfactant-stabilized water-in-oil (W/O) and oil-in-water (O/W) emulsions are explained in terms of the proposed mechanism. 1. Introduction A number of experimental studies published in the literature indicate that emulsions (dispersions of oil and water) and bubbly suspensions (dispersions of bubbles in liquids) exhibit significant drag reduction in turbulent flows,1-10 that is, the measured friction factors in turbulent flow fall well below the values expected on the basis of laminar flow properties. As an example, Figure 1A shows the flow behavior of unstable water-in-oil (W/ O) emulsion in smooth pipes. No interfacial additives such as surfactants were present in the emulsion, and the emulsion consisted of water droplets dispersed in continuous oil phase. The friction factors in turbulent flow fall significantly below the well-known Blasius equation for single-phase fluids:
f ) 0.079/NRe0.25
(1)
where f is the friction factor and NRe is the Reynolds number calculated on the basis of laminar flow viscosity. Figure 1B shows another example of drag reduction behavior. Here the friction factor data of unstable oil-in-water (O/W) emulsion in smooth pipes are shown. No interfacial additives were present in the emulsion, and the emulsion consisted of oil droplets dispersed in continuous aqueous phase. The O/W emulsion, like W/O emulsion, also exhibits drag reduction behavior (friction factors in turbulent flow fall below the Blasius equation), but the extent of drag reduction in O/W emulsion is much smaller as compared with the W/O emulsion. It is interesting to note that the drag reduction activity in emulsion tends to diminish (even disappear) when surfactant is present in the system. For example, parts C and D of Figure 1 show the friction factor data of surfactant-stabilized W/O and O/W emulsions. The turbulent flow friction factor data of these surfactant-stabilized emulsions follow the Blasius equation reasonably well. Many experimental studies have been reported on the drag reduction behavior of bubbly suspensions7-10 as well. For the most part, these studies are restricted to turbulent boundary layers on a flat plate. The introduction of bubbles to a liquid
turbulent boundary layer on a flat plate reduces the skin friction significantly. The mechanism of drag reduction in emulsions and bubbly suspensions is not well understood. Nearly all the studies published in the literature tend to attribute drag reduction in these systems to suppression of turbulence of the carrier fluid (continuous-phase) upon introduction of droplets or bubbles. The main objective of this paper is to propose an alternative mechanism of drag reduction in emulsions and bubbly suspensions. 2. New Mechanism of Drag Reduction in Emulsions and Bubbly Suspensions It is proposed that drag reduction in dispersions (emulsions and bubbly suspensions) is caused by a significant reduction in effective viscosity of the dispersion when the flow regime is changed from laminar to turbulent. 2.1. Effect of Droplets and Bubbles on Dispersion Viscosity. The relative viscosity of dispersion (emulsion or bubbly suspension), defined as the ratio of dispersion viscosity to continuous-phase viscosity, is a function of at least three variables, namely viscosity ratio λ, capillary number NCa, and volume fraction of dispersed phase φ:
ηr ) ηr(λ, NCa, φ)
(2)
where ηr is relative viscosity. The viscosity ratio λ is defined as the ratio of dispersed phase viscosity to continuous phase viscosity. The capillary number is defined as the ratio of viscous stress ηcγ˘ (ηc is continuous phase viscosity and γ˘ is shear rate) to interfacial stress σ/R (σ is interfacial tension and R is undeformed particle radius). Pal11 has recently published the following theoretical model for the relative viscosity of concentrated dispersions:
ηr
[
] [
M - P + 32ηr M - P + 32
N-1.25
M + P - 32 M + P - 32ηr
]
N+1.25
exp * Tel.: (519) 888-4567, ext 32985. Fax: (519) 746-4979. E-mail:
[email protected]. 10.1021/ie061020y CCC: $37.00 © 2007 American Chemical Society Published on Web 12/13/2006
)
[ ]
2.5φ (3) φ 1φm
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Figure 1. Pipeline flow behavior of unstable and surfactant stabilized water-in-oil (W/O) and oil-in-water (O/W) emulsions.
where
M ) x(64/NCa2) + 1225λ2 + 1232 (λ/NCa)
(4)
P ) (8/NCa) - 3λ
(5)
N)
(22/NCa) + 43.75λ M
(6)
In eq 3, φm is the maximum packing volume fraction of droplets or bubbles. φm is 0.637 for random close packing of uniform spheres. In the low NCa limit, eq 3 gives the following result:
ηr,o
[
]
2ηr,∞ + 5λ 2 + 5λ
3/2
) exp
[ ] 2.5 φ φ 1φm
(7)
Figure 2. Relative viscosity (ηr) as a function of capillary number (NCa) for emulsion at two different values of viscosity ratio λ.
where ηr,o is the limiting relative viscosity at low NCa. In the high NCa limit, eq 3 reduces to
sponding to random close packing of uniform spheres). On the basis of Figure 2, the following points can be made: (a) At low values of NCa (NCa , 1.0), the relative viscosity of dispersion is constant independent of NCa. At intermediate values of NCa, dispersion viscosity decreases with the increase in NCa. At high values of NCa (NCa . 1), the relative viscosity of dispersion again becomes constant independent of NCa. The decrease in viscosity in the intermediate range of NCa is due to elongation of droplets/bubbles in the direction of flow. (b) When the viscosity ratio λ is greater than unity (O/W emulsion case), the relative viscosity is always greater than unity; and (c) when the viscosity ratio is less than unity (W/O emulsion or bubbly suspension), the relative viscosity is less than unity at high NCa.
ηr,∞
[
]
λ - ηr,∞ λ-1
-2.5
) exp
[ ] 2.5φ φ 1φm
(8)
where ηr,∞ is the limiting relative viscosity at high NCa. Figure 2 shows the plots of dispersion relative viscosity ηr versus capillary number NCa for two different values of viscosity ratio λ. The volume fraction of the dispersed phase is 0.50 (φ ) 0.50). The plots are generated from eq 3 using the maximum packing volume fraction of droplets φm equal to 0.637 (corre-
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Figure 3. Effect of volume fraction of dispersed phase (φ) on the relative viscosity of emulsion when λ < 1.0.
This indicates that the dispersion viscosity can become lower than the viscosity of the continuous phase when NCa is large and λ < 1. Figure 3 shows the effect of volume fraction of dispersed phase (φ) on the relative viscosity of dispersion at a small viscosity ratio (λ) of 0.1. The plots are generated from eq 3. The plots shown in the top portion of Figure 3 are obtained for a φm value of 0.637. While the limiting viscosity at low NCa increases with the increase in φ, the limiting viscosity at high NCa decreases with the increase in φ. The gap between the two limiting viscosities widens with the increase in φ. The bottom portion of Figure 3 shows the effect of φm on model predictions. When NCa f 0, the relative viscosity decreases with the increase in φm at any given φ; when NCa f ∞, the relative viscosity shows the opposite behavior, that is, it increases with the increase in φm. Figure 4 shows the effect of φ on the relative viscosity of dispersion when the viscosity ratio is large, that is, λ ) 5. The plots shown in the top portion of Figure 4 (where φm ) 0.637) indicate that the limiting viscosity at low NCa is higher than the limiting viscosity at high NCa. This behavior is similar to that found in the case of λ ) 0.1. However, both the viscosities (low NCa and high NCa) increase with the increase in φ. With the increase in φm (see the bottom portion of Figure 4), the relative viscosity decreases regardless of the capillary number. 2.2. Laminar Pipeline Flow of Dispersions. In laminar pipeline flow of dilute and moderately concentrated emulsions and bubbly suspensions, the friction factor data can be generally described by the following well-known Newtonian relationship (see Figure 1A-D):
f ) 16/NRe
(9)
Thus emulsions and bubbly suspensions exhibit nearly Newtonian behavior in laminar pipeline flow. This indicates that
Figure 4. Effect of volume fraction of dispersed phase (φ) on the relative viscosity of emulsion when λ > 1.0.
deformation of particles (droplets and bubbles) in pipeline laminar flow is small, and that pipeline laminar flow can be treated as low capillary number flow (NCa , 1.0). To support this argument further, the viscosity data obtained from laminar pipeline flow using eq 9 are presented and compared with the viscosity model predictions at low NCa in Figure 5. The relative viscosities of water-in-oil (W/O) and oil-in-water (O/W) emulsions obtained from laminar pipe flow data are larger than unity, and they increase with the increase in the dispersed phase concentration. The low NCa viscosity equation (eq 7) describes the experimental data reasonably well. 2.3. Turbulent Pipeline Flow of Dispersions. In turbulent pipeline flow of dispersions, the viscous stresses are expected to be much larger than the interfacial stress especially in the wall region where shear rates are high. Indeed Sleicher12 and Collins and Knudsen13 found that in turbulent flow of oil/water dispersions, extensive stretching of droplets occur in the region close to the pipe wall. The lengths of the droplets were found to be as large as four times the diameter. In some instances, the aspect ratio (length to diameter ratio) of droplets was found to be much larger than 4. As an example, Figure 6 shows a photograph of a highly elongated droplet observed by Collins and Knudsen13 in a region close to the pipe wall under turbulent conditions. Thus pipeline turbulent flow can be treated as high capillary number flow (NCa . 1.0) where viscous stresses in the wall region are much larger than the interfacial stress. To support our argument further, the viscosity data obtained from turbulent pipeline flow measurements using Blasius equation (eq 1) are presented in Figure 7. The relative viscosity of water-in-oil (W/O) emulsions (λ < 1) obtained from turbulent pipe flow data is less than unity and it decreases with the increase in the dispersed phase (water) concentration φ. The
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Figure 5. Relative viscosity of unstable water-in-oil (W/O) and oil-inwater (O/W) emulsions obtained from laminar pipeline flow data.
Figure 7. Relative viscosity of unstable water-in-oil (W/O) and oil-inwater (O/W) emulsions obtained from turbulent pipeline flow data.
Figure 6. Photograph of elongated droplet in the pipe wall region, observed by Collins and Knudsen.13
high NCa viscosity equation (eq 8) predicts the same behavior. The values of relative viscosity predicted by the model (eq 8) are somewhat higher than the experimental values. Nevertheless, the model predictions are reasonable and give the right trend. The relative viscosity of oil-in-water (O/W) emulsions (λ > 1) obtained from turbulent pipe flow measurements is larger than unity, and it increases with the increase in the dispersed phase (oil) concentration φ. The viscosity model predicts the same behavior at high NCa. The values of relative viscosity predicted by the model (eq 8) are somewhat lower than the experimental values but they are not unreasonable. It should be noted that the φm value used in the model to describe different sets of viscosity data in Figures 5 and 7 is not the same. This is not unreasonable as φm is sensitive to droplet shape and size distribution. The droplet shape and size distribution are expected to vary with the flow regime (laminar versus turbulent) and type of emulsion (O/W versus W/O). 2.4. Explanation of Drag Reduction. Emulsions and bubbly suspensions exhibit drag reduction behavior in turbulent flows (that is, measured friction factors in turbulent flow fall well below the values expected from the Blasius equation using laminar viscosity in the Reynolds number) because the effective viscosity of these dispersions in turbulent flow (high NCa) is significantly lower than their effective viscosity in laminar flow (low NCa). The reduction in effective viscosity from laminar to turbulent regime occurs due to extensive stretching of droplets
Figure 8. Predictions of friction factor for emulsions (viscosity ratio λ ) 0.1) using the low capillary number viscosity equation (eq 7).
in the direction of flow. To elaborate this point further, Figure 8 shows the predictions of friction factor for emulsions with viscosity ratio λ of 0.10. The volume fraction of the dispersed phase (φ) is varied from 0.15 to 0.53. The data are plotted as friction factor (f) versus Reynolds number (NRe,o) based on the limiting low NCa viscosity. The limiting viscosity at low NCa is calculated from eq 7 with φm ) 0.637. Interestingly, the turbulent flow data fall below the Blasius equation indicating drag reduction behavior. Also, the extent of drag reduction increases with the increase in dispersed-phase volume fraction φ as found in experiments.1 However, it should be noted that the experimental friction factors in turbulent flow of unstable W/O emulsions tend to increase at high NRe (see Figure 1A).
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In other words, the drag reduction activity tends to diminish when turbulence becomes more intensive. This behavior can be explained in terms of the average droplet size. At high NRe, the average droplet size of an unstable emulsion is expected to decrease due to break up of droplets. Small droplets tend to resist deformation as the interfacial stress (σ/R) is large. Consequently, the effective viscosity of the emulsion increases resulting in a decrease in the drag reduction activity. Surfactant-stabilized emulsions exhibit small or negligible drag reduction activity in turbulent flows. The droplets of surfactant-stabilized emulsions are generally very small (of the order of few microns). As explained in the preceding paragraph, small droplets tend to resist deformation due to an increase in the interfacial stress. Furthermore, the presence of surfactant at the surface of the droplets generates Marangoni stresses which oppose deformation of droplets. Thus the droplets of surfactantstabilized emulsions undergo small or negligible deformation in laminar and turbulent regimes. As a consequence, the effective viscosity of such dispersions is nearly the same in laminar and turbulent regimes, and therefore, no drag reduction is observed. 3. Conclusions The laminar pipeline flow of dispersions (emulsions and bubbly suspensions) can be treated as low capillary number flow (NCa , 1.0). The turbulent pipeline flow of dispersions can be treated as high capillary number flow (NCa . 1.0). The relative viscosity calculated from laminar pipeline flow data of dispersion is always greater than unity, and it always increases with the increase in the dispersed phase concentration φ, regardless of the value of viscosity ratio λ. The relative viscosity calculated from turbulent pipeline flow data of dispersion is less than unity when λ < 1, and it decreases with the increase in φ. The relative viscosity calculated from turbulent pipeline flow data of dispersion is greater than unity when λ > 1, and it increases with the increase in φ. Drag reduction in turbulent flow of dispersion (emulsions and bubbly suspensions) is caused by a significant reduction in
effective viscosity of the dispersion when the flow regime is changed from laminar to turbulent. It is possible that the slight discrepancy observed between model predictions and experimental viscosity data (turbulent regime) is due to suppression of turbulence in flow of emulsions and bubbly suspensions. Acknowledgment Financial support from NSERC is appreciated. Literature Cited (1) Pal, R. Pipeline Flow of Unstable and Surfactant-Stabilized Emulsions. AIChE J. 1993, 39, 1754-1764. (2) Pal, R. Emulsions: Pipeline Flow Behavior, Viscosity Equations and Flow Measurement. Ph.D. Thesis, University of Waterloo, 1987. (3) Zakin, J. L.; Pinaire, R.; Borgmeyer, M. E. Transportation of oils as oil-in-water emulsions. J. Fluids Eng. 1979, 101, 100. (4) Angeli, P.; Hewitt, G. F. Pressure gradient in a horizontal liquidliquid flows. Int. J. Multiphase Flow 1998, 24, 1183-1203. (5) Iannou, K.; Nydal, O. J.; Angeli, P. Phase inversion in dispersed liquid-liquid flows. Exp. Therm. Fluid Sci. 2005, 29, 331-339. (6) Lum, J. Y. L.; Al-Wahaibi, T.; Angeli, P. Upward and downward inclination oil-water flows. Int. J. Multiphase Flow 2006, 32, 413-435. (7) Madavan, N. K.; Deutsch, S.; Merkle, C. L. Reduction of turbulent skin friction by microbubbles. Phys. Fluids 1984, 27, 356-363. (8) Skudarnov, P. V.; Lin, C. X. Drag Reduction by Gas Injection into Turbulent Boundary Layer : Density Ratio effect. Int. J. Heat Fluid Flow 2006, 27, 436-444. (9) Si-Young, K.; Cleaver, J. W. The persistence of drag reduction following the injection of microbubbles into a turbulent boundary layer. PreViews Heat Mass Transfer 1995, 21, 300. (10) Deutsch, S.; Fontaine, A. A.; Moeny, M. J.; Petrie, H. L. Combined Polymer and Microbubble Drag Reduction. J. Fluid Mech. 2006, 556, 309. (11) Pal, R. Viscous Behavior of Concentrated Emulsions of Two Immiscible Newtonian Fluids with Interfacial Tension. J. Colloid Interface Sci. 2003, 263, 296-305. (12) Sleicher, C. A. Maximum stable drop size in turbulent flow. AIChE J. 1962, 8, 471- 477. (13) Collins, S. B.; Knudsen, J. G. Drop size distributions produced by turbulent pipe flow of immiscible liquids. AIChE J. 1970, 16, 1072-1080.
ReceiVed for reView August 3, 2006 ReVised manuscript receiVed October 30, 2006 Accepted November 6, 2006 IE061020Y
