Entrainment and mixing properties of a simple bubble plume 1

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1 Entrainent and ixing properties of a siple ule plue 1 C. Bergann, D.-G. Seol, T. Bhauik & S.A. Socolofsky Coastal & Ocean Engineering Division, Departent of Civil Engineering, Texas A&M University, College Station, Texas, USA 1 INTRODUCTION Multiphase plues occur in a wide range of environental applications, including ule ABSTRACT: Two-fluid and ixed-fluid integral odels of ule plues are copared to experiental easureents of full-field velocity in laoratory-scale unstratified air-ule plues. Two-fluid odels treat the uoyancy of the ules and the entrained fluid separately; ixed-fluid odels treat the uoyancy of the ules and entrained fluid as a ixture. Laoratory data are otained using particle tracking velocietry (PTV) for the ules and particle iage velocietry (PIV) for the entrained fluid. The tie-averaged velocity profiles of the entrained fluid atch Gaussian profiles; tie-averaged velocity profiles of the ules resele top-hat profiles. The laoratory data provide a direct easureent of the entrainent coefficient, which lies etween.1 at the plue ase and.6 far fro the source. Bias in the ixedfluid odel toward a higher flow rate and a narrower plue give poor coparison with the entrained fluid velocity and oentu flux. By contrast, the two-fluid odel atches all state space variales well. plues for reservoir aeration and destratification (McDougall 1978, Asaeda & Ierger 1993, Wüest 1993), oil and gas plues in accidental oil-well lowouts (Socolofsky & Adas 3,, Yapa & Zheng 1999), and direct ocean caron sequestration (Crounse ) to nae a few. In each application, it is desirale to apply a siple odel to predict the interaction of the ultiphase plue with its surroundings. Integral plue odels are an exaple of such odels and have een developed extensively in past studies. However, detailed easureents of the full-field ixing properties have not een availale to test odel forulations and to validate odel coefficients. As a first step to fill this need, we present full-field velocity easureents in an unstratified, laoratory-scale ule plue to test the applicaility of ixed-fluid and twofluid integral odels. The ixed-fluid and two-fluid integral odels are derived in Section along with an analysis of the ajor differences etween the odels. Section 3 descries the experiental data and presents results for the tie-averaged velocity profiles of the entrained fluid and ule rise velocity. The coparison etween the odel and the experiental data is presented in Section 4. INTEGRAL MODEL EQUATIONS The integral odels presented here are derived fro the self-siilarity assuption lateral profiles have the sae shape at different plue heights. Although ultiphase plues are known to violate self siilarity even in the unstratified case, odels ased on the self-siilarity assup- 1 Pulished as: Bergann, C., Seol, D.-G., Bhauik, T., and Socolofsky, S. A., Entrainent and ixing properties of a siple ule plue, in Environental Hydraulics and Sustainale Water Manageent, Proc. IAHR 4 th Int. Syp. Environ. Hydr., Hong Kong, China, Deceer 15-18, Lee and La, eds., A.A. Balkea Pulishers, London, Vol. 1, pp , 4.

2 tion have een successfully applied y any authors (e.g. McDougall 1978, Asaeda & Ierger 1993, Wüest et al. 1993). For the odels derived here, the lateral variation of a state variale is assued to e Gaussian, and is given y the equation r X( z, r) = X ( z)exp (1) λ z ( ) where X is the state variale of interest, X is the centerline value of the state variale, r is the radial coordinate, is the characteristic profile width, and λ is a scaling factor that allows different state variales to have a characteristic width proportional to the ase width. Because the functional variation in the radial direction is known, state variales can e integrated in the lateral direction to otain integral fluxes that are functions of height only. This reduces the prole to one diension. Thus, integral plue odels consist of a syste of conservation equations in the for of first-order ordinary differential equations. To derive the plue equations, we ake two ain assuptions. First, a turulence closure odel is needed. The entrainent hypothesis states that the induced entrainent velocity at the oundary of a turulent plue is proportional to a characteristic upward velocity within the plue (Morton et al. 1956). For a ultiphase plue, we take the centerline velocity U of the continuous phase as the characteristic plue velocity. Second, we ake the dilute plue assuption: the dispersed phase void fraction C is sall so that ( 1 C) 1. Under these approxiations, the conservation of ass equation gives the classic result d π ( U) = π α U () where α is an entrainent coefficient and is the characteristic width of the continuous phase velocity profile. The driving forces in a ultiphase plue are due to uoyancy. The dispersed phase (ules, droplets or particles) drives the plue either against gravity in the case of a dispersed phase lighter than the aient fluid or with gravity in the case of a heavier dispersed phase. If the aient environent is also staly stratified, then the entrained continuous phase also contriutes to the plue forcing and always acts in a direction opposite to the dispersed phase. There are two coon ethods to treat these opposing forces. First, in two-fluid odels, the uoyancy of the dispersed phase and the uoyancy of the continuous phase are tracked separately. Second, in ixed-fluid odels, the uoyancy of the ixture is treated together y taking the average density of the dispersed and continuous phase ixture as a single variale. The differences etween these two approaches are seen clearly in the conservation of oentu equation. The oentu of the dispersed phase is negligile; thus, the net oentu is M = π U for oth odels. The applied force of the dispersed phase is not negligile, however, and two forcing ters appear in the two-fluid odel, whereas, a single forcing ter appears in the ixed-fluid odel. For the two-fluid odel, the conservation of oentu gives d π π U C g g γ ( λ + λ 1 ) = w where γ is a oentu aplification factor that accounts for the added oentu of the turulence aove that of the ean flow (first proposed y Milgra 1983), λ 1 is the spreading ratio etween the ule and velocity profile (generally less than 1), and λ is the spreading ratio etween the continuous phase uoyancy and velocity profile (generally greater than 1). The uoyancy of the dispersed phase is given y the reduced gravity g of the continuous phase (suscript for ules), and the uoyancy of the continuous phase is given y the reduced gravity of the entrained fluid (suscript w for water). C is the void fraction of the continuous phase, which accounts for the fact that the whole plue width is not occupied y ules. By contrast, the conservation of oentu for the ixed-fluid odel has a single forcing due to the ixture, given siply y the reduced gravity g of the ixture, resulting in the equation (3)

3 d π U π = λ γ g where λ is the spreading ratio etween the uoyancy and velocity profiles. In each case, g is the difference etween the aient density ρ a at the height z and the constituent density (dispersed phase, entrained phase, or ixture) noralized y a reference density ρ r and ultiplied y the gravity constant g. To close the set of equations, conservation equations for the uoyancy flux in the plue are needed. The two-fluid odel requires two equations, one each for the uoyancy of the dispersed and continuous phases. For the continuous phase, uoyancy changes due to the entrainent of stratified aient fluid. Tracking the conservation of continuous phase uoyancy results in the classic plue uoyancy equation for a stratified aient (e.g. Morton et al. 1954) d πλ U g w πu N = 1+ λ. The uoyancy conservation of the dispersed phase depends on its physical and cheical properties. For inert, incopressile particles or droplets, the uoyancy would e invariant. A coon application of two-phase plue odels is for reservoir restoration. There, the fluid is air and is assued to oey the ideal gas law. In this case, the conservation of dispersed phase uoyancy is given y where N is the Brunt-Väisälä uoyancy frequency g ρ( dρ ) ( + ) d πλ U U C g 1 d gq H A = 1+ λ H z 1 T where Q is the air flow rate at standard teperature and pressure (STP), H A is the atospheric pressure head, H T is the total pressure head at the release point, and z is the height aove the injection. Because ules rise faster than the entrained fluid, the uoyancy flux of ules is adjusted y the factor U = (1 + λ 1 )u s, where the slip velocity u s is the terinal rise velocity of a single ule in a quiescent aient. Equations (5) and (6) close the syste of equations for the two-fluid odel. To derive the uoyancy conservation equation for a ixed-fluid odel, McDougall (1978) coined equations (5) and (6) to otain a single equation. Assuing an unstratified aient, g = C g. (7) Then, ultiplying the ters inside the derivative on oth sides of equation (6) y U /(U + U ) and taking λ 1 = λ = λ, the left-hand-side of equation (6) ecoes identical to that of equation (5). Since oth equations (5) and (6) account for different effects, they are treated as parts of a total derivative and coined to otain d πλ U g d gqhu A = πu N + (8) 1+ λ ( H z T )( U + U ) which is the equation derived in McDougall (1978). Hence, only equation (8) is needed to close the syste of equations for the ixed-fluid odel. The integral odel equations presented aove are solved using a fourth-order Runga-Kutta algorith. The state-space variales in the two-fluid odel are the flux variales given y Q = π U J = π U πλ F = w Ug w πλ U C g 1 F = 1+ λ 1+ λ1 and for the ixed-fluid odel, the state-space variales are given y (4) (5) (6) (9)

4 Tale 1. Model flux equations for the two types of integral odels derived in Section. Conservation Two-fluid odel flux equations Mixed-fluid odel flux equations equation dq dq Mass: = α πj = α Moentu: ( 1+ λ ) ( ) QF 1+ λ 1 dj = + γ J + QU J QF w πj ( 1+ λ ) dj QF = γ J df df w = QN = QN + Buoyancy: df d gq H d gq H J A A = ( H z) T ( H z)( J + QU ) T Q = π U J = π U πλ U g F = 1+ λ Tale 1 presents the equations for the two integral odels in ters of the flux variales. (1) 3 EXPERIMENTAL METHODS To copare the ehavior of siple ule plues to the equations derived aove, detailed experients were conducted in a 4 c y 4 c cross-section y 7 c deep Plexiglas tank under unifor aient conditions at the Hydroechanics Laoratory of the Coastal and Ocean Engineering Progra at Texas A&M University. Full-field velocity easureents of the dispersed phase were otained using particle tracking velocietry (PTV), and easureents of the continuous phase were y particle iage velocietry (PIV). Air ule plues were created for three air flow rates (.5, 1. and 1.5 l/in at STP) using a standard aquariu air stone (average ule size 3 in diaeter). PTV easureents of the ules were otained in the asence of seeding particles using a continuous Ar-Ion laser and a high speed Phanto caera (5 fps at 51 y 51 pixels and 8 it color depth). Velocity easureents of the continuous phase were otained using an Nd:YAG laser (5 J/pulse) and a high-resolution Flow Master caera (4 doule fraes per second at 18 y 14 pixels and 1 it color depth). The ules could e filtered fro the iages ecause they were righter than the seeding particles; hence, the analysis could e applied to the ules and entrained fluid separately for the highresolution iages. The ule velocities otained fro each caera were siilar. Tie-average velocity fields (1 fraes for the PIV and 5 fraes for the PTV) were otained for the fluid and ules after accounting for plue wandering (the tendency for the plue centerline to eander slowly during an experient). Figure 1 shows typical velocity profiles otained for the entrained fluid (a) and the ules (). The Gaussian curve fit to the entrained fluid shows good agreeent with the easured data. The ule velocity profile ore closely reseles a top-hat profile, and the curve fit to the data in the figure is the reverse of a shallow water wake profile suggested y Monkewitz (1988) and given y U( z, r) = U ( β 1+ sinh ( r/ ) ) where β is a paraeter that adjusts the flatness of the central part of the profile. Taking β = 1 gives a profile very siilar to the Gaussian profile. The profile shown in the figure is for β = 3. (11)

5 (a) () U/U.6 U/U r/ r/ Figure 1. Measured and fitted velocity profiles for the entrained fluid (a) and ules (). The profiles are noralized y the centerline velocity of the fitted curve for each phase. The Gaussian profiles fitted to the easured entrained fluid velocity provide direct easureent of the variation of U and with height, which in turn define the variation of Q and J with height. Fro these easured values and equation (), the effective entrainent coefficient α is also otained and was found for all three experients to have the functional relationship H T α = (1) 55z This scaling is not intended to apply to field-scale plues; scaling with the ule Froude nuer proposed y Milgra (1983) did not provide a collapse of the data. The entrainent coefficient varied etween.1 near the plue source and.6 far fro the source. These values are coparale to the pure-plue value of RESULTS AND DISCUSSION The integral odels derived aove were run for each of the three flow rates conducted in the experients using the entrainent coefficient value otained fro equation (1). This coparison is useful to validate the forulation of the ule expansion ter ut does not validate the effects of stratification since the experients were unstratified (stratified experients are currently planned). Figure presents the results for the two-fluid and ixed-fluid odels copared to the easured data at a gas flow rate of 1. l/in at STP. Both odels appear to agree well for predictions of the flow rate and plue width, however, the variation of the velocity and, consequently, the oentu do not atch well for the ixed-fluid odel. This is ecause the ixed-fluid odel consistently over-predicts the flow rate Q and under-predicts the plue width. For oth odels, the spreading ratio λ was taken as.8, and the odels were nearly insensitive to λ. The only reaining free paraeter is the oentu aplification factor. The data in the figure are for a est-fit value of γ = 1.5. The ixed-fluid odel result does not significantly iprove until an unrealistic value for γ of 8 or greater is reached. Trials were also attepted with the right-hand-side of equation (8) ultiplied y a factor ½ to account for effectively adding equations (5) and (6) to otain equation (8). This did not iprove the agreeent. Coparisons with the other flow rates in the experients give siilar trends, and the twofluid odel tracks the variations due to different ule flow rates well. These coparisons indicate that the two-fluid odel includes ultiphase plue physics consistent with the experiental data. The poor agreeent for the ixed-fluid odel for the plue oentu flux has iportant consequences in stratified flows, where the plue will e arrested as the oentu approaches zero due to the negative uoyancy of the entrained fluid. Experients at other scales are necessary to evaluate the variation of α at the field scale.

6 .4 (a) 5 ().3 4 Q/(Bz 5 ) 1/3. U /(B/z) 1/ (c).8 (d).3.6 /z. M/(Bz ) / Figure. Coparison of the two-fluid (solid line) and ixed fluid (dashed line) integral odels to the easured experiental data (dotted line). 5 SUMMARY AND CONCLUSIONS Mixed-fluid and two-fluid integral odels of unstratified ule plues are copared to detailed experiental easureents of full-field plue velocity. The experiental data were otained using PTV for the ules and PIV for the entrained fluid. Tie average profiles allow the direct easureent of the entrainent coefficient, which lies etween.1 near the plue source and.6 in the upper regions of the plue. The two-fluid odel tracks all of the state space variales well (entrained volue flux, centerline velocity, plue width, and oentu flux) and its trend with ule flow rate also agrees with the experiental data. The ixed-fluid odel over-predicts the flow rate and under-predicts the plue width so that the entrained fluid velocity and oentu flux do not track the experiental results. This has iportant consequences in stratified flows, where the plue arrests as the oentu flux approaches zero due to the entrainent of aient water. REFERENCES Asaeda, T & Ierger, J Structure of ule plues in linearly stratified environents. J. Fluid Mech. 49: Crounse, B.C.. Modeling uoyant droplet plues in a stratified environent. MS thesis, Dept. of Civil & Envirn. Eng., MIT, Caridge, MA. McDougall, T.J Bule plues in stratified environents. J. Fluid Mech. 85(4): Milgra, J.H Mean flow in round ule plues. J. Fluid Mech. 133: Monkewitz, P.A The asolute and convective nature of instaility in two-diensional wakes at low Reynolds nuer. Phys. Fluids 31(5): Morton, B.R., Taylor, G.I. & Turner J.S Turulent gravitational convection fro aintained and instantaneous sources. Proc. Royal Soc. 34A: 1-3.

7 Socolofsky, S.A. & Adas, E.E.. Multi-phase plues in unifor and stratified crossflow. J. Hydraul. Res. 4(6): Socolofsky, S.A. & Adas, E.E. 3. Liquid volue fluxes in stratified ultiphase plues. J. Hydraul. Eng. 19(11): Wüest, A., Brooks, N.H. & Ioden, D.M Bule plue odeling for lake restoration. Wat. Resourc. Res. 8(1): Yapa, P.D. & Zheng, L Modeling underwater oil/gas jets and plues. J. Hydraul. Eng. 15(5):