Liquid Loss From Advancing Aqueous Foams With Very Low Water Content

Transcription

1 Naval Research Laboratory Washington, DC NRL/MR/ Liquid Loss From Advancing Aqueous Foams With Very Lo Water Content Michael Conroy Ramagopal Ananth James Fleming Justin Taylor John Farley Navy Technology Center or Saety and Survivability Chemistry Division January 14, 011 Approved or public release; distribution is unlimited.

2 Form Approved REPORT DOCUMENTATION PAGE OMB No Public reporting burden or this collection o inormation is estimated to average 1 hour per response, including the time or revieing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and revieing this collection o inormation. Send comments regarding this burden estimate or any other aspect o this collection o inormation, including suggestions or reducing this burden to Department o Deense, Washington Headquarters Services, Directorate or Inormation Operations and Reports ( ), 115 Jeerson Davis Highay, Suite 104, Arlington, VA Respondents should be aare that notithstanding any other provision o la, no person shall be subject to any penalty or ailing to comply ith a collection o inormation i it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY). REPORT TYPE 3. DATES COVERED (From - To) Memorandum Report 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Liquid Loss From Advancing Aqueous Foams With Very Lo Water Content 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) Michael Conroy, Ramagopal Ananth, James Fleming, Justin Taylor, and John Farley 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Research Laboratory, Code Overlook Avenue, SW Washington, DC d. PROJECT NUMBER 5e. TASK NUMBER 5. WORK UNIT NUMBER 8. PERFORMING ORGANIZATION REPORT NUMBER NRL/MR/ SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) Oice o Naval Research One Liberty Center 875 North Randolph Street Suite 145 Arlington, VA SPONSOR / MONITOR S ACRONYM(S) 11. SPONSOR / MONITOR S REPORT NUMBER(S) 1. DISTRIBUTION / AVAILABILITY STATEMENT Approved or public release; distribution is unlimited. 13. SUPPLEMENTARY NOTES 14. ABSTRACT Applications employing aqueous oams begin ith illing a space ith oam, and the liquid loss (drainage) rom the oam during and ater this process plays a crucial role in its eectiveness. We describe the loss o liquid rom oam and the evolution o its average liquid raction over time. The theoretical model shos a constant drainage rate during the illing process hich decays exponentially ater a static column is ormed. Modeling the advancing oam ront requires a ne time scale, the ill time, hich substantially aects the drainage o liquid. Signiicant eects are also ound by varing bubble size, oam column height, and initial expansion ratio, i.e., the volume ratio o oam to liquid. Foams ith lo ater content are generated experimentally at the bench scale, and the measured loss o liquid is ound to be in good agreement ith the theoretical predictions. 15. SUBJECT TERMS High expansion oam Fire suppression 16. SECURITY CLASSIFICATION OF: a. REPORT Aqueous oams Drainage b. ABSTRACT c. THIS PAGE Unclassiied Unclassiied Unclassiied 17. LIMITATION OF ABSTRACT i 18. NUMBER OF PAGES UL 6 19a. NAME OF RESPONSIBLE PERSON Ramagopal Ananth 19b. TELEPHONE NUMBER (include area code) (0) Standard Form 98 (Rev. 8-98) Prescribed by ANSI Std. Z39.18

3

4 CONTENTS 1.0 INTRODUCTION MODEL EXPERIMENTS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGMENTS REFERENCES APPENDIX A - Calculation o the supericial liquid velocity or a cylindrical channel... 5 APPENDIX B Model parameters or comparison ith Reerence [8] iii

5 NOMENCLATURE A, characteristic drainage constant, m/ s a p, cross-sectional area o channel (Plateau border), B, characteristic drainage constant, C 1, integration constant C, integration constant m / s c v, parameter to account or the mobility o channel alls; deined as the ratio o the average liquid velocity in a given channel ith mobile alls to the average liquid velocity in the same channel i the alls are immobile, dimensionless D b, diameter o a bubble, m Ex, oam expansion ratio, dimensionless G, geometrical constant equal to ( ) / , dimensionless m g, gravitational acceleration, 9.8 m/ s H, height o oam column, m h, height o moving boundary at the top o the oam, m h 0, initial height o the top o the oam, m h, height o the moving boundary at the top o the liquid layer beneath the oam, m h 0, initial height o the top o the liquid layer beneath the oam, m L, channel (Plateau border) length, m M d, mass o liquid drained rom the oam, kg n p, number o channels (Plateau borders) per bubble, # P g, pressure o gas in bubble, Pa iv

6 P l, pressure o liquid in a channel (Plateau border), Pa r, radius o curvature o the interace beteen a gas bubble and a liquid channel, m R, radius o cylindrical oam column, m S, cross-sectional area o oam column, t, time, s t 0, initial time, s t ind t ill, induction time, s m, time required to ill a column ith oam, s V b, volume o gas ithin a bubble, 3 m V d, volume o liquid drained rom a oam, 3 m v, liquid velocity ithin a channel averaged over the cross-section o a channel (Plateau border), m/ s v, liquid velocity ithin a channel averaged over the height o the oam layer, m/ s v inj, injection velocity o oam, m/ s v s, supericial liquid velocity, m/ s z, vertical coordinate along the height o the oam column, m GREEK SYMBOLS α, liquid volume raction o oam, dimensionless α, liquid volume raction o oam averaged over the height o the oam layer, dimensionless α, liquid volume raction o oam injected into oam column, dimensionless α, liquid volume raction at the interace beteen the oam and drained liquid layers, dimensionless γ, surace tension, 0.05 N / m v

7 δ, model constant equal to approximately 0.1, dimensionless δ, model constant equal to 3 π / 0.161, dimensionless a η, reciprocal (multiplicative inverse) o α, dimensionless θ, model constant or comparing oam drainage models, dimensionless µ, viscosity o liquid, kg / ms ξ, permeability coeicient, dimensionless ρ, mass density o liquid, 997 kg / m 3 Φ, generic unction in the standard orm o a linear ordinary dierential equation ψ, generic integration actor used in the analytical solution o a linear ordinary dierential equation Ω, generic unction in the standard orm o linear ordinary dierential equation vi

8 Liquid Loss rom Advancing Aqueous Foams ith Very Lo Water Content 1.0 INTRODUCTION Filling a space ith oam is an essential part o oam applications. One illing method is to apply oam directly onto a surace rom above, here resh oam is added to the top o the advancing oam bed. This method o illing is used, or example, in ire-suppression applications. To suppress ires in contained spaces such as arehouses, hangar bays, and obstructed areas, aqueous oams ith very lo ater content are applied rom oam generators located near the ceiling, quickly illing the space. In addition, aqueous oams ith greater liquid content are sprayed on top o lammable liquids to suppress pool ires and build a oam bed above the pool to prevent re-ignition. Another method o illing a container ith oam is by introducing gas bubbles into a liquid solution containing suractants. The gas bubbles rise to the surace o the liquid solution, orming a oam bed above the liquid. In this case, the resh oam is added to the bottom o the column. Examples o applications employing this method include oam ractionation used by pharmaceutical and ood industries or protein separation, and roth lotation used by the mining industry or mineral separation. The loss o liquid rom oam, also called oam drainage, has a crucial impact on its eectiveness in applications. The dominant orce responsible or drainage is gravity, hich acts on the oam rom the moment it is generated. Hoever, most drainage models describe the drainage o liquid rom oam by starting ith a stationary oam column o ixed height. This approach neglects the eect o gravity during an essential stage o oam applications, the illing process. Signiicant liquid loss could occur over the illing time, hich ould ultimately alter the eectiveness o the oam. Modeling eorts aim to understand oam drainage by considering the lo o liquid through the oam. The liquid lo in oam depends on the geometry o the liquid contained in the interstitial space beteen oam bubbles. The oam bubbles take on shapes similar to polyhedra. In Figure 1, an illustration depicts a bubble shaped as a dodecahedron and the surrounding liquid structures: ilms, channels, and nodes. The ilms are thin sheets o liquid ormed here the suraces o to bubbles meet. The channels are ormed at the intersection o three bubbles: channels are also called Plateau borders in honor o pioneer oam scientist J. A. F. Plateau. The cross section o a channel resembles a triangle ith edges curved toard the interior. The channels can be considered as a netork o pipes that allo liquid to lo through the oam. The junctions here the channels intersect are called nodes; and our channels meet at a single node. The dimensions o ilms, channels, and nodes depend on the bubble size and the liquid content o the oam. The amount o liquid contained in the oam can be quantiied in terms o the oam expansion ratio ( ) Manuscript approved November 8, 010. Ex : the ratio o the oam volume to the volume o liquid 1

9 solution used to orm the oam; or the liquid volume raction ( α ): the ratio o liquid volume to total oam volume (Note that Ex = 1/ α ). Figure 1: Illustration o ilms, channels, and nodes or an idealized dodecahedral bubble. Most o the modeling and experiments on oam drainage have been perormed on stationary oams ith lo expansion ratios. Leonard and Lemlich [1] modeled steady-state drainage rom a oam ractionation column, here oam is continuously ormed at the base o the column. The modeling approach, hich is the basis or most models, involved a description o the liquid lo ithin the channels by considering orces due to gravity, variation in liquid (capillary) pressure along the height o the column, and viscous resistance to liquid lo. The latter varied ith a parameter termed surace viscosity, hich as used to quantiy the mobility o the channel alls. Desai and Kumar [] examined the semi-steady-state liquid raction as a unction o height ithin a column under similar conditions, including the loss o liquid rom the ilms ith time. The model contained channels ith a simpliied cross section, an equilateral triangle. A parameter as introduced to account or the mobility o the channel alls [3]. The eects o viscosity, surace viscosity, and input gas lo rate on the steady-state liquid-raction proile ere examined ith experiments using the suractants sodium lauryl sulate (SDS), lauryl alcohol, and Teepol. The average, steady-state Ex values ranged rom about 10 to 100.

10 Ramani, Kumar, and Gandhi [4] proposed a model or drainage rom a static oam column that accounted or the time-dependent drainage o ilms into nearly vertical and nearly horizontal Plateau borders, here gravity as a driving orce only or the liquid contained in the nearly vertical channels. The model as compared ith experiments using the suractant 4 SDS ith Ex 8 and bubble diameters less than 7 10 m. Podual, Kumar, and Gandhi [5] later changed the bubble geometry rom dodecahedral to tetrakaidecahedral along ith the channel orientation o the model, and the adjustments slightly improved the comparison ith experiment or small bubble sizes. The experiments included oams ith expansion ratios up to and bubble diameters up to 7 10 m. The model o Verbist [6] contained a oam drainage equation hich included nonsteady-state variation in the cross-sectional area o the oam channels, alloing the liquid raction o the oam to change ith both position and time. The ability to calculate a timevarying liquid raction including the changing area o the channels alloed the model to study the ree drainage problem: to describe drainage rom a ixed-height column o oam ith a liquid raction that is initially uniorm. The model vieed the oam channels as a set o independent vertical pipes ith strictly immobile alls. As in most recent models o drainage, the contribution o the ilms as neglected. Channels ere considered in the equations or the conservation o liquid volume and momentum o the model, here an eective viscosity (about 150 times that o ater) as introduced to account or the cross-sectional geometry and random orientation o the channels. The approach o Koehler [7] vieed oam drainage as the lo o liquid through a porous medium, such as a ilter. From this perspective, the proportionality beteen the lo rate o liquid through the oam and the pressure gradient that drives the liquid lo is determined by a permeability unction, hich depends on the local liquid raction. The model calculations ere compared ith experimental results using the suractant SDS and gaseous C F 6 to minimize the diusion o gas beteen bubbles. The expansion ratio studied or ree drainage rom a static oam column as 00 and the bubble diameter as approximately m. The ork suggests, in contrast to the model o Verbist, that the viscous eects could be predominantly acting in the nodes rather than the channels or oams ith highly mobile alls. Magrabi et al [8,9] investigated ree drainage rom AFFF and FFFP compressed-air oams used in ire suppression. The oams initially had an expansion ratio o 0 and a bubble 4 diameter o approximately.6 10 m. The model contained a all mobility parameter similar to that o Desai and Kumar [,3]. The ork suggests that an eect not included in the model, the increase in average bubble size caused by gaseous diusion beteen bubbles, could signiicantly increase the drainage rate in these oams ater a e minutes. A e recent orks have investigated orced and pulsed drainage. These processes involve the continuous and inite addition o liquid to the top o a oam, respectively. Drainage under these conditions has been studied by the non-steady-state models o Verbist [6] and Koehler [7]. In orced drainage, the liquid volume raction becomes uniorm behind a traveling 3

11 liquid ront that moves through the oam at a constant velocity. As shon by the to studies [6,7], the ront velocity displays dierent poer-la dependence on the liquid raction depending on the suractant solution used. Hoever, orced and pulsed drainage models do not apply to the case o illing a container ith oam. Bhakta and Ruckenstein [10] have investigated illing a container ith oam rom the bottom, as in the process o oam ractionation. To account or the illing process, the model included a single moving boundary, the oam ront, and considered the non-steady-state behavior o the liquid content ithin the oam. The experimental data o Germick et al [11] that included bubble diameters o about m to m ere used or comparison. In the study, the eect o oam ill rate on the drainage rate as not investigated. In this ork, e present theoretical analysis and experiments or drainage rom a oam as it ills a container rom above. We introduce to the drainage problem an additional time scale, the time required to ill the column ith oam. This additional time scale changes the eect o drainage on the liquid raction o the oam. To moving boundaries are treated in the problem: the air/oam interace here oam is added at the top and the oam/liquid interace here the oam layer meets the drained liquid that collects beneath it. The height o the oam bed is thereore controlled by both the rate o oam injection and the rate o liquid drainage. We also treat ree drainage as a special case o the model, herein there is a single moving boundary at the oam/liquid interace. We ocus our investigation on one type o aqueous oam used in practice or ire suppression generated ith a suractant designed or achieving high expansion (HiEx). The oams possess greater expansion-ratio values and bubble sizes than those investigated in most previous orks. An important aspect o this ork is that e provide analytical solutions or the liquid drained rom the oam and the average liquid raction o the oam as unctions o time. The analytical solutions oer the ability to quickly predict drainage and to clearly understand ho the various parameters inluence the behavior. Similar to previous orks, the model contains an adjustable parameter called a permeability coeicient that accounts or the geometry o the channels as ell as the mobility o the channel alls. The solutions allo the permeability coeicient or a given oam to be calculated by itting to experimental data. We sho good agreement ith experimental results by treating the permeability coeicient as a constant. We use the model to examine the eect o ill rate, expansion ratio, column height, and bubble size on the drainage behavior..0 MODEL Filling a container rom above involves delivering resh oam to the top o a groing oam bed, and the injection o oam is ceased hen the container is ull. Both during and ater the illing period, liquid drains out o the oam and orms a liquid layer beneath the oam bed. During the process o drainage, bubble ilms empty a portion o their liquid to the surrounding channels. Liquid in the channels moves donard, passing through nodes into loer channels repeatedly as it makes its ay out o the bottom o the oam. As the liquid content o the oam 4

12 diminishes, the ilms reduce in thickness, the nodes shrink in volume, and the cross-sectional area o the channels decreases. There are several physical actors that drive the lo o liquid in oam. For the oams o interest, the orces governing luid transport ithin the oam are not in an equilibrium state hen the oam is generated. For oam ith suicient liquid content, the gravitational orce is initially dominant or the liquid in the channels and nodes, driving liquid donard ithin the oam. Liquid lo can also be driven in a oam hen there is a spatial variation in capillary pressure, the pressure dierence beteen the liquid and gas at the curved surace o a bubble. The capillary pressure is caused by the surace tension at the interace o the to phases, gas and liquid, ithin the oam hich acts to minimize the surace area o the interace. Figure : Surace tension orce at the gas/liquid interace o a bubble. To see ho surace tension gives rise to capillary pressure in a oam, consider the illustration o the curved gas/liquid interace at the surace o a channel in Figure. The surace tension orce is directed along the surace and pulls at each end o a surace length element dl. The horizontal components o the surace tension acting on a surace length element dl 5

13 cancel, but the vertical components do not. The net orce on each curved interace is directed toard the adjacent gas bubble and its magnitude is inversely proportional to the radius o curvature o the surace. The surace tension orce thereore increases the pressure o the gas relative to the surrounding liquid by compressing the bubbles at the curved suraces o the channels. The Young-Laplace la = γ (1) r P g P l provides a relation beteen the gas pressure, P g, and the liquid pressure, P l, that is valid at the gas/liquid interace. Note that the pressure o the liquid in the ilms approaches the pressure o the surrounding gas bubbles because the ilms are very lat, i.e. r is very large. The curvature at the surace o channels can vary throughout dierent regions in a oam. Channels ith more liquid have a larger radius o curvature, and thereore have a smaller capillary pressure, i.e. the liquid pressure gets closer to the gas pressure. In a draining column o oam, the liquid content at the top o the oam becomes smaller than that o the bottom. Hence, a gradient in the capillary pressure is established, hich drives the lo o liquid. For a draining column, the lo due to the gradient in capillary pressure opposes the gravitational lo. Additionally, the lo o liquid through a oam depends on the degree o slip at the channel boundaries. Slip is a term used to describe the lo o liquid relative to its bounding surace; a no-slip condition in a channel means that the liquid velocity at the channel alls is zero. Slo or non-moving liquid at the alls causes a variation in liquid velocity (i.e. stress) across the channel cross section, giving rise to viscous drag that impedes lo ithin the channel. The slip condition depends on the mobility o the gas/liquid interace at the surace o the bubbles. I the surace o the channel is able to reely move along ith the loing liquid, it ould cause plug lo, a uniorm liquid velocity proile across the channel cross section. The slip condition or oam channel lo lies ithin these extrema and can result in dierent drainage rates or the oam as a hole. Evidence suggests that the mobility o the surace and hence the slip condition strongly depends on the suractant. In act, studies have shon that the drainage rate rom a oam can be signiicantly reduced by adding a dierent suractant [1]. A suggestion or the physical origin o the mobility o the channel alls is that the liquid loing nearby could shear suractant molecules rom the gas/liquid surace [7]. I the shearing creates a concentration gradient on the surace, then diusion o suractant on the surace could oppose liquid lo. Hoever, i the suractant in the loing liquid can move to the surace much aster than the rate o shear, then the suractant can quickly replenish the surace ithout establishing an appreciable concentration gradient. An alternative argument involves the idea o viscosity at the surace, hich relects the orces o attraction beteen suractant molecules. The strength o the interaction could also be so large that the channel lo is not strong enough to shear the molecules apart. I one considers the ilms as stationary, then the channel alls ould be immobile because they are anchored to the ilms. Weaker interactions may allo the suraces to shear. 6

14 Further, in a polydisperse oam, bubbles o dierent sizes dier in their gas pressure. Small bubbles exhibit greater pressure and Henry s la states that the solubility o gas in the surrounding liquid ill be greater. Neighboring bubbles ith loer pressure ill have surrounding liquid ith loer gas solubility. Hence, a concentration gradient o dissolved gas is established in the liquid phase, driving the dissolved gas toard the larger bubble(s). This process is called Ostald ripening and it makes an appreciable contribution to coarsening, the process o increasing oam average bubble size over time. The time scale associated ith coarsening can vary signiicantly beteen oams [13]; the study o Magrabi [8] on AFFF oams ormed ith compressed air shos that bubble size begins to increase ithin just a e minutes. Another actor that contributes to coarsening is coalescence, here ilm rupture beteen to bubbles produces a single, larger bubble. In our model, e set up the problem such that the oam is injected at a constant velocity. Further, e consider one-dimensional drainage along the z-direction as depicted in Figure 3. The drainage problem is reduced to a single dimension because the orces that dominate liquid transport are in the vertical direction until the oam approaches equilibrium. Figure 3: Illustration o drainage model including moving boundaries. 7

15 We consider to moving boundaries. The irst boundary, z = h () t, is the air/oam interace that moves toards the top o the oam column at the injection velocity v inj, deined as the oam injection rate per unit cross-sectional area o the container. The second boundary, z = h () t, is the oam/liquid interace, here the solution drained rom the oam pools beneath it. The volume o liquid drained is V = Sh () t, here S is the cross-sectional area o the oam d column. Once oam injection stops, the top o the oam layer is ixed at the height o the column, H, and the oam/liquid interace is the only moving boundary. This situation corresponds to ree drainage o a oam on top o liquid. Our goal or the model is to predict the amount o ater drained rom a oam and the change in the oam expansion ratio over time in bench-scale experiments. Hence, or both illing drainage and ree drainage, the model solves or the height o the moving oam/liquid interace, h () t, and the liquid raction o the oam averaged over the length o the oam layer, α () t. Approximations are made in the model or the oam structure. We neglect the drainage o liquid rom the thin ilms, and e thereore only consider the channels. For large bubbles, the portion o liquid in the ilms is negligible [14]. Further, the shape o bubbles in this ork has been approximated as dodecahedral, here the channels are ormed at the bubble edges. Although the shape leaves a small amount o space unaccounted or, the ork o Podual et al. [5] indicates that the dierence beteen tetrakaidecahedral and dodecahedral geometries in drainage models is negligible or large bubbles. Also, e consider a monodisperse oam, here each gas bubble has the same volume. In practice, oams contain a distribution o bubble sizes. Similar to the approach o Magrabi et al. [8], e assume that the drainage behavior o a polydisperse oam can be captured by modeling the oam as monodisperse, ith a bubble size equal to the average o the bubble-size distribution. We maintain a constant bubble size in the model and thereore neglect the eects o both Ostald ripening and coalescence. For Ostald ripening to occur, there must be a pressure dierence beteen bubbles that yields a concentration gradient in the dissolved gas. For larger bubbles, the radii o curvature ithin the oam ill be larger, causing a decrease in the capillary pressure. I the dispersity in the bubble-size distribution is not large, then the pressure dierences beteen bubbles should be small. Hence, it is likely that the phenomenon that drives Ostald ripening should be less important or large bubbles. By the reasoning above, the gas pressure is treated as a constant in the model. The liquid pressure in the channels is assumed equal to the pressure o the liquid at the bubble/channel interace, given by the Laplace-Young la. As or coalescence in the model, recent ork [13] shos that the onset o coalescence depends primarily on a critical liquid raction and is independent o bubble size. The suractants investigated in Re. [13] ere TTAB and SDBS; each oam studied exhibited a critical liquid raction less than ( Ex = 150). The calculations o our model indicate that this liquid 8

16 raction is approached only near the end o the time or hich e are interested in simulating the drainage process. Assuming that the oams in our study have a similar critical liquid raction or the onset o coalescence, the eect o coalescence should be negligible in our study during the drainage period o interest. The mobility o the channel alls is not explicitly accounted or in the model. We have included an adjustable parameter, the permeability coeicient, hich accounts or the mobility o the channel alls, random orientation o the channels, and corrections to the geometrical approximations o the model. Note that most drainage models include either an adjustable parameter or all mobility or assume that the channel alls are immobile. From the partial dierential equations (PDEs) governing mass and momentum balance o the liquid in a oam, e seek ordinary dierential equations (ODEs) ith time as the independent variable to arrive at analytical solutions that describe the drainage process. By approximation, e solve or the height o the ater layer beneath the oam, h, and the length-averaged liquid raction o the oam, α. A detailed description o the solution is provided belo. We develop the model by considering the partial dierential equations or mass and momentum balance in one dimension. Treating the liquid as incompressible, the liquid volume balance in the oam is given by α + = t z ( αv) 0 () here α = α(,) zt is the liquid raction o the oam and v= vzt (,) is the liquid velocity. The liquid velocity o the model, discussed belo, is an eective vertical velocity o the liquid averaged over the cross-sectional area o a channel. The supericial liquid velocity is vs = αv, hich is the volume o liquid loing through a unit o cross-sectional area o the oam column per unit time. As illustrated in Figure 5, the liquid in the channels travels through the channel cross section at velocity v, hereas the liquid exiting the oam (or a control volume) travels through the oam-column cross section ith the velocity v s. The balance in () is ritten or a control volume o the oam; or the column considered, the control volume corresponds to a disc ith the same cross-sectional area o the oam column, but ith a small thickness. The previous expression states that i the supericial velocity o the liquid coming into the control volume is dierent rom the output supericial velocity, there is a corresponding change in the liquid raction ithin the control volume. 9

17 Figure 4: Illustration o the lo o liquid ithin the oam. 10

18 Figure 5: Illustration o the orces that drive the lo o liquid in oam. Next, e consider some o the geometrical approximations o the model that ill be necessary or the momentum conservation equation. The bubble shape in the model is a regular pentagonal dodecahedron, and this is used to relate the geometrical parameters o the bubble and its corresponding channels to the local liquid raction o the oam. There are n p channels per bubble; or a dodecahedral bubble, the number o channels is equivalent to the number o edges that belong to a single dodecahedron, n p = 10. Each channel has length L and area a p, and the liquid volume contained in the channels per bubble is thereore nal. p p We neglect the additional volume o liquid contained in the nodes, hich should be very small or the high-expansion oams o interest. The volume o gas contained in a dodecahedral 3 bubble is Vb = GL, here G = ( ) / , and the total volume o gas and liquid per bubble is the sum o these quantities. We vie the oam column as a space illed by a collection o bubbles and their respective channels, here the liquid raction o an individual bubble unit is equivalent to the liquid raction o the oam as a hole. We thereore get the expression 11

19 npapl npapl α = α =. (3) n a L + V V 1 α p p b b For a high expansion ratio, α << 1, and e thereore make the approximation npapl = α. (4) V The cross-sectional area o the channel is related to the radius o curvature by a here δ = 3 π / By substituting this relation and a b Vb p = δar, 3 = GL into (4), e have a relation beteen the liquid raction o the oam and geometrical parameters o the bubbles: npδar L r α = = δ 3 GL L (5) here δ 0.1. This approximation as made by Koehler [7] or tetrakaidecahedral bubbles. Note that the relationship beteen cross-sectional area o the channel and local liquid raction is a p G α L n =. (6) p To understand the dependence o the channel area in (6), consider the liquid and gas content o a single bubble. I the volume o the gas bubble remains constant, increasing the liquid raction o the bubble by adding liquid to the channels ill cause the channels to sell, increasing the channel cross-sectional area. Also, or a bubble that, regardless o size, maintains a constant liquid raction, the channel volume is proportional to al but the total volume o the bubble is proportional to 3 L. The liquid volume raction is thereore proportional to raction to be constant as the bubble size increases, p ap a p must also be proportional to / L. For the liquid Hence, the channel cross-sectional area depends on both the liquid raction o the bubble and the square o the channel length. Next, e seek an expression or the momentum balance or a channel that is oriented vertically ithin the oam. For a given liquid and pressure gradient across a channel, the velocity o lo through a channel ill depend on the channel geometry and the slip condition at the channel alls. Deriving an expression or the average liquid velocity in a channel is complicated because the cross-section o the channels has a complex shape and e do not predict or measure the slip condition. Given the diiculty o the problem, e aim to sho, under simpler conditions, that the average velocity should be proportional to the channel area. In Appendix A, e derive an equation or the average velocity through a cylindrical channel ith a circular cross section and a no-slip condition at the boundary. The result is quoted here: L. 1

20 ( 5 ) ( αp ) 1 l ( αpl) v= αl ρ g = ap ρ. g (7) 30πµ α 8πµ α The velocity in (7) is very similar to the ell-knon Hagen-Poiseuille equation or lo in a cylindrical pipe. The same dependence o v on a p is shon in the expression derived by Desai and Kumar [] or general all mobility in channels ith the cross-section o an equilateral triangle ( ) α L ca v p P v l c Pl v = ρg = ρg 0µ 3 z 800µ 3 z. (8) The parameter c v is a velocity coeicient deined as the ratio o the average velocity in a channel to the velocity o the same channel i the alls are strictly immobile. In both cases, the average velocity in the channel can be described by ξα L P v ρg l = (9) µ z here ξ, deined here as a permeability coeicient, accounts or bubble and channel geometry, slip condition, and the orientation o the channels. In this ork, ξ is an adjustable parameter o the model. We reiterate that the pressure in the liquid is ound rom the Young-LaPlace la, = γ (10) r P l P g hich, at a gas-liquid interace ith surace tension γ, relates the pressure o the gas in the bubbles, P g, to the pressure o the surrounding liquid in the channel ith radius o curvature, r. The pressure gradient P / z is ound by substitution o (5) into (10), l Pl γ δ γ δ 1 = Pg =. z z L α L z α (11) An expression or the average velocity in a channel as a unction o the liquid raction then becomes 13

21 L 1 v ξα g γ δ = ρ +. µ L z (1) α Note that, overall, the velocity is proportional to the area o the channel, hich is also proportional to α L. Meanhile, the pressure gradient involves the gradient o 1/r here the radius o curvature r ap L α. The combined liquid-raction dependence in the pressure gradient term should thereore involve the gradient o the radius o curvature (i.e. α ). This is shon by the simpliication 1 α = α z α z (13) Substitution o the previous expression in (1) yields an equation or the average liquid velocity in a channel here e introduce the constants ( ), v= Aα B α (14) z A ξ L ρg = µ (15) and ξlγ δ B =. (16) µ The units or A are ms (velocity) and or B are m s (area/time). For a given oam, (14) indicates that the average liquid velocity in a channel is determined by a dynamic competition beteen the liquid raction and its variation along the height o the column. Equivalently, this is a competition beteen the local channel area and the vertical gradient in the radius o curvature. For regions o oam ith uniorm liquid raction, the average velocity is proportional to the corresponding liquid raction. For cases herein a oam is illed very ast such that the liquid raction is approximately equal to the input, one should expect a constant supericial velocity vs Aα. For sloer illing, the liquid raction ill vary along the height o the column, decreasing at the top and increasing near the oam/liquid interace. For this case, the eects combine in dierent regions. At the end o illing, the liquid raction ill quickly decrease ith column height; the dynamic eects o the liquid raction and its gradient ill oppose one another throughout the height o the oam. Drainage ceases at equilibrium hen these terms are equal in magnitude. 14

22 The average liquid velocity in a channel also depends on multiple parameters. For any case, it is proportional to the permeability coeicient and inversely proportional to the viscosity o the liquid. By deinition, a greater permeability coeicient increases the velocity o the liquid in the channels. The viscosity plays a role because e are considering the case here the liquid at the alls is moving sloer than the rest o the liquid in the channel. A more viscous liquid ill reduce the velocity dierence beteen neighboring liquid particles, and thereore reduce the average velocity ithin the channel, overall. For a given liquid-raction proile, other parameters aect the strength o the terms in (14) on the competition that determines the velocity. The constant A is proportional to L, relecting that the term is proportional to the channel area ap α L. Further, the eect o gravity is contained in A, hich alays acts to increase the liquid velocity loing in a vertical channel. The value o B determines the strength o the eect o liquid raction variation ith height on average channel velocity. In contrast ith A, B increases linearly ith channel length L, hich indicates that A ill be dominant or larger bubbles sizes. An essential parameter in B is the surace tension, γ, hich is responsible both or the existence and strength o the capillarity eect or a given variation in the liquid raction ith height. Boundary conditions must be imposed on α. We treat the case or hich α is constant at the boundaries. During illing, the top o the oam contains the liquid raction o the resh input oam, α. The choice or the liquid raction at the oam/liquid boundary is less obvious. Similar to the approach o Desai and Kumar [3], e assume that the bubbles are close-packed in a ace-centered cubic (cc) arrangement. Further, the shape o the bubbles is approximated as spherical, and the liquid raction at this boundary is assumed equal to the void raction o closepacked spheres in an cc arrangement, α ( z = h ) = 0.6. The boundary conditions are also used during ree drainage or comparison ith the experiments o this ork. Ater illing, the liquid volume raction at the top o the oam signiicantly decreases rom α. Hoever, e ind that reducing the value o the liquid raction at the top boundary belo α has a negligible eect on the predictions o the model. We maintain the same boundary conditions or illing and ree drainage to avoid unnecessary complication o the model parameters. The drainage model in this ork is similar to other drainage models in the literature [6-10], and the equations governing the conservation o mass and momentum ithin the oam can be expressed in the same orm. The general orm is: α 0, t z µ + = = z ξ ( αv) v ρgl [ ] ( ) α γlθ α (17) here 0.46 θ = except as noted belo. The values o the permeability coeicient ξ rom the literature and rom this ork are provided in Table 1. The large values used to model the experiments o Magrabi et al. [8,9] indicate that the oams in their study drain much aster than the oams studied in the other orks listed, hich is assumed to be a result o channel all mobility. The other calculated and empirical values or ξ are or oams ith immobile channel alls. For randomly oriented channels, the reported values or ξ are smaller and vary at most 15

23 by about 0%.; the largest contribution to the dierence beteen these values is the bubble geometry o the model. In this ork, e also calculate ξ or vertically oriented channels ith no slip at the channel alls, hich is the largest physically reasonable value o ξ or this slip condition. Table 1: Permeability coeicients o oam drainage models. Work Calculation method ξ Suractant (type) Bhakta and Ruckenstein (Re. [10])* Koehler et al. (Re. [7]) Magrabi et al. (Re. [8])* Magrabi et al. (Re. [9])* Verbist et al. (Re. [6]) This ork This ork Calculated or randomly oriented channels ith triangular cross section and immobile alls D Calculated or randomly oriented channels ith scalloped-triangular cross section and immobile channel alls T Empirically determined rom drainage theory or randomly oriented channels ith triangular cross section and mobile alls D Empirically determined rom drainage theory or randomly oriented channels ith triangular cross section and mobile alls D Calculated or randomly oriented channels ith scalloped-triangular cross section and immobile alls Empirically determined rom drainage theory or randomly oriented channels ith scallopedtriangular cross section and immobile alls (Re. [7]) D Calculated or vertically oriented cylindrical and equilateral triangular channels ith immobile alls (see Appendix A) D *Mathematical orm o Eq. (17) obtained hen α <<1 θ = 0.41 ξ or initial liquid volume raction 0.05 Equation (6) o this ork used to relate a p to α D Assumed dodecahedral bubble geometry, L=0.816R Model results compared ith experimental data o Re. [11] using Bovine Serum Albumin, BSA (protein) N/A Aqueous ilm-orming oam, AFFF: contains perluoroalkyl sulonate salts (luorocarbon based) Film-orming luoroprotein oam, FFFP: contains hydrolized proteins and luorosuractants (luoroprotein) N/A Buckeye HXF.% High- Expansion Foam Concentrate: contains sodium laureth sulate (hydrocarbon based) Cylindrical: 0.03 Triangular: 0.0 N/A T Assumed tetrakaidecahedral bubble geometry, L=0.7R 16

24 The next step in the development o the model is to obtain ordinary dierential equations rom the partial dierential equations that describe the balance o liquid volume and momentum in the oam. To eliminate the spatial dependence o the equation or liquid volume balance, e integrate () ith respect to the vertical coordinate z rom the height o the liquid beneath the oam, h, to the height at the top o the oam, h : h α dz+ α( h,)( tvh,) t α( h,)( tvh,) t = 0. (18) t h Using Leibniz s rule and the boundary conditions vh (,) t = vh (,) t = 0, this expression yields h d dh dh dz ( h, t) ( h, t) 0. dt α α + α = (19) dt dt h Let us deine α by the expression α = h α (,) h ( h h) z t dz (0) here α represents the length-averaged liquid volume raction. The quantity α is useul or providing an overall description o the liquid quantity in the oam. Substitution o (0) into (19) yields d dh ( ) (, ) dh α h h α h t α ( h, t) dt = dt dt (1) here α ( h,) t is the liquid raction at the top o the oam and α ( h,) t is the liquid raction at the bottom. For the present model, the liquid ractions at the boundaries ill be considered as constants: α = α( h,) t and α = α( h,) t. We consider that liquid drains rom the oam into a liquid layer beneath the oam, and thereore e set α = 1 or the liquid volume balance. The let-hand side o Eq. (1) is the time derivative o the liquid content ithin the oam, hich is controlled by the quantities on the right-hand side. I e consider ree drainage only, dh / dt = 0 and the rate o ater loss can be determined by the rise o the ater layer beneath the oam. Hoever, by considering the case o illing, dh / dt 0, the liquid content in the oam is no determined by the competition beteen illing and drainage. This competition can be controlled by the time scale introduced in the model, the ill time. Integrating the momentum equation (14), e obtain 17

25 h h h v dz = A α dz B( α α ). () h We deine the liquid velocity averaged over the height o the oam layer, v, ith the olloing expression v = h v dz h ( h ). h Inserting deinitions (0) and (3) into () gives the expression (3) ( ) α( ) ( α α ). v h h = A h h B (4) Note that the boundary condition or α in the momentum balance is not the same as that or the liquid volume balance. The next step is to use these length-averaged quantities to determine an expression or the height o the ater layer as a unction o time, h () t. The time derivative o h is the volumetric lo rate o liquid rom the oam layer into the liquid layer per unit area o the oam column. Equivalently, this is the supericial liquid velocity exiting the oam, dh dt = v. (5) s The liquid velocity and the supericial liquid velocity, both space- and time-dependent, are related by vs = αv. I the oam bed ere separated into control volumes along the height o the column, the supericial velocity o the liquid exiting the oam due to drainage ould be determined by the liquid balance o the control volume at the bottom o the oam near the oam/liquid interace. The supericial velocity o the liquid entering this control volume ill depend on the balance over the control volume above, and the same is true or each control volume in the oam. Our approach does not resolve the oam column into control volumes; instead, e seek a description o oam drainage based on the average properties o the oam. We thereore make the approximation on the supericial velocity dh dt = α v (6) and e assume that the error introduced in averaging is small. Solving or v in (4) and substituting in the previous expression yields an equation or the rate at hich the height o the ater layer increases dh dt = A Bα α α α h h (7) 18

26 The expression above is used or both illing and ree drainage conditions. There are to ordinary dierential equations in the model calculations, one or liquid volume balance, Eq. (1), and one or momentum balance, Eq. (7), here h and α are unknon. In the olloing sections, to cases are considered or the model: the general case o illing and the special case o ree drainage. For each, ordinary dierential equations are presented that can be solved numerically. Additionally, approximations are made to obtain analytical solutions, and the conditions or hich they are valid are discussed. FILLING For illing, e integrate Eq. (1) ith respect to time to obtain the expression α( h h ) = α h h + C (8) here the last term is an integration constant representing the initial conditions, C = α( t )( h h ) α h + h. (9) The initial height o the ater layer is set such that h 0 = 0. To prevent a singularity in Eq. (7), the problem is set up at t = 0 such that there is a small amount o oam ith height h 0 = 0.01 m having the same liquid raction α( t0) = α as the oam that ill ill the column. By setting this condition, e sacriice the description o drainage behavior ithin the irst e seconds o oam application, but this is not o interest or this study. Under these initial conditions, C 1 = 0. The liquid balance in (8) states that the liquid content in the oam is equal to the dierence beteen the amount o liquid added to the oam and the amount o liquid that has drained out. Solving or α in (8) yields α h h α = h h. (30) Substitution o this expression in Eq. (7) provides the illing equation dh α h h α h h = A B dt h h h h α α (31) The previous expression is used or the numerical solutions during illing in this ork. We also provide an analytical solution belo. The numerical solutions ere determined by the unction NDSolve (MATHEMATICA, Wolram Research) ith deault settings. NDSolve utilizes a variant o the Livermore Solver or Ordinary Dierential Equations, LSODA, to numerically solve ordinary dierential equations. Both sti and non-sti systems can be detected and solved 19

27 using LSODA, here backards dierentiation ormula (BDF) methods and Adams methods are used, respectively. Approximations are necessary in order to obtain an analytical solution or illing. First, e perorm a Taylor expansion o α about the value α α α + α α α = α α α (3) The value o α must be greater than zero, so the approximation clearly ails i the condition α / α > 0.5 is not satisied. Note that this expansion is equivalent to neglecting the term proportional to ( h / h ) in Eq. (31). Physically, the approximation is valid hen a oam drains much less than 50% o its liquid content as it ills. We substitute Eq. (30) into Eq. (3) to obtain α h h α α α h h (33) and substitute Eq. (33) into Eq. (7) to obtain dh α h h α h h = A dt h h α α B α. α h h (34) To obtain a linear equation, e make the approximation h >> h. The maximum value o h is α h, so the approximation is equivalent to α <<1. This approximation is valid or highexpansion oams, here α is typically 0.01 or less. Using this approximation, Eq. (34) can be expressed as dh dt = A α α α α h α h B. h h h (35) Putting Eq. (35) in standard orm yields dh Aα B Bα + α, α h Aα α α dt h h = (36) h a linear ODE hich can be solved analytically. For mathematical simplicity, e make one more approximation in the linear coeicient; e neglect the term ( B α α / h ). Since 1 h = h + v t is linear ith respect to time, the term Aα h is inversely proportional to t, 0 inj hereas B α α h is inversely proportional to t. Hence, the term 0

28 1 B α α h quickly becomes much smaller than Aα h. The ordinary dierential equation rom hich e obtain an analytical solution is thereore dh dt Eq. (37) can be re-ritten as Aα Bα + h = Aα α α. h h (37) dh h Bα = Aα 1 α α dt α h h (38) Eq. (38) indicates that the drainage rate ill be equal to the dierence beteen the drainage rate o a oam that has α = α (see Eqs. (6) and (4)) and a correction actor that depends on the volume per cross section ratio o the liquid lost to the liquid layer h to the total liquid added to the oam α h. The correction actor accounts or the act that α decreases rom α as liquid is lost rom the oam to the liquid layer beneath. Intuitively, the ratio o the volume o liquid in the liquid layer to the volume o liquid in the oam layer ill depend on ho the rate o drainage compares ith the rate by hich the column is illing ith oam. As shon belo, the competition beteen these rates is captured in the analytical solution. Equation (37) is in the orm dh / dt +Φ () t h =Ω () t. Using the integration actor ( ) ψ = exp Φ( t) dt 1, the solution is given up to a constant by h = ψ Ω() t ψ dt. The integration actor, using dh = vinj dt, is ψ The solution is given by Aα 1 vinj dt = dh = l h = h inj inj exp A α exp A α exp A α n. h v h v = (39) 1

29 Aα Aα v Bα inj vinj h = h Aα α α h dt C h + = h + = h Aα Aα Aα Aα 1 v Aα Bα α α inj vinj vinj v inj h dh h dh Ch v inj v inj Aα vinj Aα Aα + 1 v Aα inj vinj Aα vinjh Bα α v α v inj injh h + Ch vinj Aα + vinj vinj Aα Aα h B α α = + Ch Aα + v A inj Aα vinj Aα vinj Aα 1 B α α = α h v + Ch inj. vinj A + Aα (40) We use the initial condition h(0) = h0 = 0 and h (0) = h 0 at time t = 0, hich yields 1 0 = α + vinj A + Aα Aα B α α vinj h 0 Ch 0 (41) hich can be used to solve or the integration constant C Aα B α α 1 vinj = α h 0 h 0 A v inj + Aα. (4) Substitution o (4) into the inal expression in (40) yields the analytical solution

30 h h h + + Aα Aα Aα Aα 1 B α α B α α 1 vinj vinj = α h + α h 0 0 vinj A A vinj Aα 1 1 B α α vinj = α vinjt+ α h 0 h 0 1. vinj vinj A h 0 vinjt Aα Aα (43) The value o α during illing is ound by substitution o Eq. (43) into Eq. (30). Equation (43) can be simpliied by neglecting h 0. The abbreviated analytical solution or illing is given by h 1 B α α α v t. (44) vinj A + Aα inj For cases here the analytical solution is valid, the abbreviated solution given in Eq. (44) shos very little dierence ith the solution o (43) ater roughly 10 seconds. The irst term in the previous expression quickly becomes dominant during the illing process. This indicates that h has a predominantly linear time dependence, and thereore the drainage rate rom the oam is approximately constant during illing. The irst term in Eq. (44) contains vinjt, hich is the height o oam added rom the start o illing to the time t. The liquid portion o the added oam is α vinjt. The coeicient 1 vinj + Aα (45) is thereore a percentage o the added liquid that is drained (and added to h ) at time t. For a given input liquid raction α, the key parameters that determine this percentage are v inj and A, hich represent the time scales or illing and drainage, respectively. A key point o this study is that an appreciable amount o the added liquid can drain hen the illing and drainage time scales are comparable. Hoever, as one might expect, very little o the added liquid is drained during illing i one ills very ast such that vinj >> Aα. The analytical solution is not valid or very slo oam injection such that Aα >> vinj because the condition or the validity o 3

31 the Taylor expansion o α is not satisied, i.e. α ill not be much less than 50% o α. Hoever, the case can still be described by numerical solution o Eq. (31). FREE DRAINAGE For ree drainage, e set dh / dt = 0 in Eq. (1) and integrate ith respect to time to obtain here e solve or the integration constant ( H h) h C1 α = +, (46) ( ) C1 = α ( t0) H h0 + h0. (47) Note that ree drainage occurs rom a column o ixed height, H. Eqns. (46) and (47) simply state that the liquid content in the oam is equal to the dierence beteen the initial liquid content and the amount o liquid that has drained out, causing the liquid layer to rise by the height h h0. We consider the general case or initial conditions on ree drainage. Solving or α in Eq. (46) yields α h h + α ( t ) [ H h ] =. (48) H h Substituting the previous expression into Eq. (7) yields the ree-drainage equation [ ] dh h h + α ( t ) H h = A dt H h h + α ( )[ ] [ H h ] h t H h B α α. (49) Eq. (49) can be solved numerically. Hoever, e calculate an analytical solution that is nearly identical to the numerical solution or all cases examined in this study. For ree drainage, e change the dependent variable o the ordinary dierential equation to α. Making this change allos the calculation o an analytical solution that is both simple to obtain and valid or a greater range o parameters than the solution obtained using h as the dependent variable. Dierentiating Eq. (46) ith respect to time yields dα dh [ H h ] =. (50) dt dt 4

32 We substitute the previous expression in Eq. (7) to obtain dα Aα Bα = + α. α dt H h (51) [ H h ] As in the illing case, e make the approximation that the height o the ater layer is much less than the height o the oam column, h << H, and divide by α in Eq. (51) to yield 1 dα A B α α 1 = +. α dt H H α (5) Using the relation d 1 1 dt = α α and deining η 1/ α, e can express Eq. (5) as dα dt (53) Eq. (54) is in the orm dη / dt ( t) η ( t) dη B α α A + dt H H η =., +Φ =Ω. Using the integration actor ψ = exp ( Φ( t) dt) 1 the solution is given up to a constant by η = ψ Ω() t ψ dt. The integration actor is B α α B α α t ψ = exp dt = exp (55) H H and the solution is given by (54) B α α t A B α α t η exp exp = dt+ C H H H AH B α α t = + C exp. B α H α Using the initial condition η( t ) 1/ α( t ) = at time t 0, 0 0 (56) 5

33 η ( t ) 1 AH = = + C ( t ) B α α t exp 0 α 0 B α H α 0 (57) yields the integration constant C AH B α α t 0 exp. 1 = α ( t0 ) B α H α (58) The analytical solution or ree drainage is thereore [ ] 1 AH 1 AH B α α t t0 η exp = = + α B α ( t0 ) H α α B α α (59) and the corresponding value o h can be ound by solving or α in Eq. (59) and substituting in Eq. (48). Let us examine a limiting case o Eq. (59). As t approaches ininity, e have η 1 ( ) = α B α ( ) =. AH α (60) The value o α ( ) is the equilibrium liquid raction o the oam as predicted by the model. This expression or α ( ) could also be arrived at rom Eq. (7) by using the condition dh / dt = 0 at equilibrium and the approximation h = H >> h. We note that the average expansion ratio o the oam is given by Ex = 1/ α. We can express Eq. (59) as [ ] B α α t t Ex( t) Ex( ) = ( Ex( t0) Ex( ) ) exp H 0 (61) hich shos that the oam progresses rom its initial expansion ratio to that o equilibrium on an exponential time scale. Note that the exponent controlling the rate at hich the initial expansion ratio progresses to equilibrium strongly depends on the height o the column; the process is sloer or taller columns. 6

34 3.0 EXPERIMENTS To generate oam, e created a liquid oam solution by mixing. parts concentrated liquid oam solution (Buckeye HXF.% High-Expansion Foam Concentrate) ith 97.8 parts distilled ater. The suractant solution is, by eight, 6% ater, 9% hexylene glycol, 38% sodium laureth sulate, and 8% proprietary mix o suractants/stabilizers (reer to MSDS). We poured the solution into a liquid chamber and pressurized the chamber ith nitrogen gas, N. The liquid solution as ed to the oam-generation device illustrated in Figure 6. The liquid solution entered the side o the oam generator and exited donard through a conepatterned nozzle (Model 30-C, SureShotsSprayer.com) onto a copper screen ith a diameter o 6.4 cm and 30x30 mesh cells per inch (40.8% open area) held in place by a rubber gasket. The N pressure as increased in the liquid chamber such that the nozzle sprayed the liquid solution at a ixed volumetric rate o about 60 ml/minute, measured ith a calibrated rotameter (Dyer VFA-33 Visi-Float Flometer). An in-house source provided air lo into the top o the oam generator. A perorated plate and a bed o tightly packed, plastic stras, 6 cm long ith an inner (outer) diameter o 0.5 cm (0.3 cm), conditioned the air lo or even distribution over the copper screen. Air lo rates o 8 L/min and 0 L/min, as determined by a mass-lo controller (Sierra Control Flo-Box Model # 905C-PS-BM-11), ere used to generate oams ith expansion ratios o 100 and 00, respectively. A diagram o the drainage experiment is shon in Figure 6. Foam as generated beneath the oam-generation screen, moves through the oam generator, and exits into a 0- liter, transparent, plastic cylinder ith a diameter o 0.13 m and a height o m. The base o the cylindrical column as ormed ith a sheet o aluminum oil, sloped slightly donard toard the center, here a 1-cm diameter hole alloed the drained liquid to pass out o the cylinder. The liquid passing through the hole entered a unnel, collecting the drained liquid into a 500 ml graduated cylinder. A video camera recorded images o the liquid illing the graduated cylinder. For each oam drainage experiment, the desired air lo as irst set olloed by the liquid. A stopatch as used to record the times o key events. The liquid and air combined on the screen; the liquid oam solution as manually distributed on the screen to start oam generation. The excess liquid, mostly rom the priming process beore oam is ormed, as collected in the passage beneath the screen. Foam exited the generator channel and traveled don the side o the cylinder. The beginning o the ill time as recorded hen the oam moved past the speciied height in the collection cylinder. Once the oam reached the bottom o the cylinder and touched the aluminum oil, a video camera began recording (Pinnacle Studio 1) the drained liquid collecting in a graduated cylinder. Once the oam illed the 0-liter cylinder, the ill time as recorded, the liquid and air los ere turned o, and the oam generator as set aside. A plastic lid as placed on top o the cylinder to minimize oam evaporation. The excess liquid in the channel belo the oam-generation screen as collected and measured in a graduated cylinder. The camera recorded video or about six minutes ater the oam reached the base o the cylinder. The video recording as used to determine the amount o liquid drained versus time. 7

35 Ater six minutes, the drainage rate sloed signiicantly, alloing the accumulating liquid volume in the graduated cylinder to be measured by visual inspection. The volume o drained liquid as recorded at one-minute intervals until the change in volume as less than ml/min. Subsequently, the drained liquid volume as measured in intervals o several minutes until the total drainage time reached about an hour. The oam as let in the cylinder overnight to completely drain, and this liquid volume as used to calculate the expansion ratio o the oam. Additionally, a value o the expansion ratio as also derived rom the net amount o liquid sprayed into the oam during the illing process. Volume o liquid in oam = Liquid lo rate Fill time Excess liquid (6) Total oam volume Ex = (63) Volume o liquid in oam 8

36 Figure 6: Experimental apparatus. Foam bubble sizes ere determined using photographs o the bubbles and a ruler taken at the top o the oam column immediately ater illing. In Figure 7, an example photograph shos the centimeter-size bubbles characteristic o HiEx oams. 9

37 Figure 7: Photograph o oam bubbles or Ex = 90. The ruler markings are in centimeters. The illing process in the experiment contrasts ith the idealized illing process o the model, herein the oam ront has a uniorm height as it travels upard. In the experiment, the oam los into the cylinder and a portion o it slides along the cylinder all as it travels donard. Ater it reaches the cylinder base, the oam takes a e seconds to reach the hole here liquid can drain into the graduated cylinder. The liquid draining rom the oam must travel to the hole in the aluminum oil and accumulate to orm a drop, and each drop must travel don the unnel and into the graduated cylinder. The experiment and model also dier at the bottom boundary. In the model, liquid accumulates beneath the oam. In the experiment, liquid drained rom the oam los into a graduated cylinder beneath the oam column. The dierences beteen the model and experiment are discussed belo. The liquid layer under the oam in the model allos liquid to be transported back into the oam rom the ater layer beneath. Capillary suction is a physical phenomenon that occurs hen the pressure o the liquid in the oam is less than the pressure o the liquid ith hich it comes into contact. The suction o liquid can occur in the model at the beginning o the illing process to some extent, depending on the parameters o the oam. In the experiment, hoever, there is no liquid layer beneath the oam. Under the experimental conditions, the 30