ABSTRACT Title of Document: Analysis of the Initial Spray from Canonical Fire Suppression Nozzles Ning Ren, Master of Science, 007 Directe By: Professor Anré Marshall, Department of Fire Protection Engineering The performance of a fire suppression spray is governe by injector ischarge characteristics. An atomization moel base on the theoretical evolution of a raially expaning sheet generate by an impinging jet has been establishe in this stuy. The atomization moel preicts characteristic initial rop location, size, an velocity base on injector operating conitions an geometry. These moel preictions have been compare with measure ischarge characteristics from three nozzle configurations of increasing geometrical complexity over a range of operating conitions. Differences between the preicte an measure initial spray are critically evaluate base on the experimentally observe atomization behavior.
Analysis of the Initial Spray from Canonical Fire Suppression Nozzles By Ning Ren Thesis submitte to the Faculty of the Grauate School of the University of Marylan, College Park, in partial fulfillment of the requirements for the egree of Master of Science 007 Avisory Committee: Professor Anré W. Marshall, Chair Professor Arnau Trouvé Professor Peter B. Sunerlan
Copyright by Ning Ren 007
Acknowlegements This work is supporte by FM Global. I woul like to thank the program manager Dr. Burt Yu for his support for this project an my research work. Also I want to thank my avisor, Dr. Marshall for his guiance support an countless time to help me to complete this work. I woul like to thank the rest of my committee members Professor Arnau Trouvé an Professor Sunerlan for their support. I also woul like to thank all my friens especially Xianxu Hu an Wali Al Abulhai in the Department of Fire Protection Engineering for their assistance an encouragement uring my stuy. In aition, I want to thank my partner Anrew Blum for his help in this project. Finally, I woul like to thank my parents who support me all the time. Without their encouragement an support, I cannot achieve my goal an finish my work. ii
Table of Contents Acknowlegements...ii Table of Contents...iii List of Figures...iv Nomenclature...v Chapter 1: Introuction...1 1.1 Motivation...1 1. Literature Review... 1..1 Atomization Theory...3 1.. Spray Measurement...5 1..3 Atomization Moeling...7 1.3 Research Objectives...9 Chapter : Approach...10.1 Atomization Physics...11.1.1 Sheet Formation...1.1. Sheet Trajectory...15.1.3 Sheet Break-up...19.1.4 Ligament Break-up...3. Numerical Moel...4.3 Scaling Law...5 Chapter 3: Results an Analyses...30 3.1 Experiment Description...30 3. Moel Evaluation...31 3..1 Sheet Formation...3 3.. Sheet Trajectory...33 3..3 Sheet Break-up an Drop Formation...38 3.3 Scaling Law Evaluation...43 Chapter 4: Summary an Conclusions...49 Appenices...51 Appenix A: Moel Input an Formulations...51 Appenix B: Description of Stochastic Moel...57 Bibliography...60 iii
List of Figures Figure.1. Description of atomization process... 1 Figure.. Sheet formation from impinging jet... 13 Figure.3. Description of sheet trajectory... 16 Figure.4. Growing wave on the sheet... 0 Figure.5. Description of atomization moel... 5 Figure 3.1. Stanar nozzle configuration... 30 Figure 3.. Dimensionless sheet thickness... 3 Figure 3.3. Sheet trajectory characteristic... 34 Figure 3.4. Sheet trajectory comparison... 35 Figure 3.5. Initial angle...36 Figure 3.6. Inverte PLIF images epicting flow through spaces... 37 Figure 3.7. Snapshots of break-up of liqui sheet... 39 Figure 3.8. Sheet break-up moe photograph... 40 Figure 3.9. Force ynamic at the rim of sheet... 41 Figure 3.10. Sheet formation from tine an boss...43 Figure 3.11. Sheet break-up results... 44 Figure 3.1. Drop size results... 47 Figure 3.13. Drop size in ligament moe... 48 Figure B.1. Initial isturbance on the jet... 58 iv
Nomenclature A A 0 * lig Wave amplitue, m Initial Wave amplitue, m Drop iameter, μm Dimensionless rop iameter Ligament iameter, μm v50 Characteristic rop iameter, μm D boss D D inlet D o Boss iameter, mm Deflector iameter, mm Nozzle inlet iameter, mm Orifice iameter, mm f Dimensionless wave amplitue f cirt, sh Critical sheet break-up imensionless wave amplitue f 0 Critical sheet break-up imensionless wave amplitue g Gravitational acceleration constant, m/s h Measurement elevation, m K K-factor of sprinkler, Lmin -1 bar -1/ l L inlet L jet m lig Characteristic length, m Length of nozzle inlet, mm Length of jet before eflector impact, mm Ligament mass, kg v
n Wave number n crit, sh Sheet break-up critical wave number n critlig, Ligament break-up critical wave number r r Raius, m Deflector raius, m * r Dimensionless eflector raius r, Sheet break-up location, mm bu sh * r bu, sh S T T Dimensionless sheet break-up location Liqui gas friction term Sheet thickness, m Theoretical sheet thickness, m * T * T Dimensionless sheet thickness Dimensionless sheet thickness at the ege of eflector t bu, lig Ligaments break-up time, s U Jet velocity, m/s U sheet Sheet velocity We Weber number, ρ U / σ l D o Wesheet x y z Sheet Weber number, Coorinate Coorinate Coorinate ρ U D / σ l sheet o vi
X sheet Sheet break-up scaling parameter X rop Drop size scaling parameter Greek letters β Thickening factor, T /T0 σ γ λ Surface tension, N/m Rosin-Rammler / log-normal correlation coefficient Wavelength θ boss Angle of eflector boss, º θ space Angle of eflector space, º θ tine Angle of eflector tine, º θ ini Initial sheet angle, º ρ Density, kg/m 3 μ Dynamic viscosity, kg/ms υ Kinetic viscosity, m /s ξ Distance along the sheet, m ω Frequency, 1/s Subcsripts a l Air Liqui (water) vii
Chapter 1: Introuction Fire safety is a worl-wie issue that has challenge mankin for thousans of years. In ancient times, the only way to fight a fire was to use water. In moern times, with the help of technology, people have more choices. Carbon Dioxie an Halons have been introuce as an effective fire suppression alternative. However, recent environmental regulations have banne the use of Halon fire suppression agents because they amage the ozone layer. Currently there is renewe interest in avancing water base fire suppression in the area of injector technology an moeling tools. Sprinklers have been use for more than one hunre years. Compare to other fire suppression systems, sprinklers are cheap, reliable an easy to install, maintain an operate. Several stuies have been conucte focusing on optimizing the rop size an mass flux istribution for optimal suppression performance. Other stuies have focuse on characterizing fire sprinklers by measuring these istributions. However, physical moels for preicting the initial spray from sprinklers have yet to be evelope. This chapter introuces the motivation for this project, reviews the previous work, an presents the objectives for this stuy. 1.1 Motivation The basic mechanisms for water-base fire suppression are clear, which can be summarize as cooling (fire gases an combustible material) an oxygen epletion. 1
These suppression mechanisms are associate with the evaporation of the spray introuce into the fire. The evaporation rate can be increase ramatically by atomizing the spray into small rops, increasing the surface area of the spray through atomizing water into fine rops. As more energy is absorbe by the spray, the temperature of the fire gases an combustible material will be ecrease, effectively reucing the energy release rate. Alternatively, evaporation of water also isplaces the oxygen available for burning, further reucing release rate. Recognizing the importance of the spray characteristics in fire suppression performance, many spray focuse experiments have been conucte. Unfortunately, these experiments are often limite to the measurement of global quantities evaluate in quiescent cool conitions. However, etaile spray characteristics are neee for use as CFD moeling input. Furthermore, the spray characteristics may change in actual operating conitions, which woul inclue a hot turbulent environment an a range of injection pressures. A physics base moel is neee to preict the etails of the initial spray for injector performance analysis an integration with CFD tools. 1. Literature Review Atomization is a complex flow problem, which has been extensively stuie in a number of scientific an engineering isciplines, incluing propulsion, agricultural, chemical processing, an fire protection. Stuy of the atomization process has inclue analysis of internal flow interaction with the surrouning gas an stability analysis. Atomization configurations can be broaly categorize as sheet base or jet base. Due to the complexity of the atomization process, these configurations have
been stuie both theoretically an experimentally. In the following section, previous research is reviewe from both theoretical an experimental perspectives. 1..1 Atomization Theory When a liqui jet is injecte into a gaseous flui, it tens to break up into fragments ue to surface tension an/or aeroynamic forces. Jet atomization was first stuie by Rayleigh [1], who foun that if the ambient gas an liqui viscosity were neglecte, the jet is most susceptible to isturbances having wavelengths 143.7% of the jet circumference. A more sophisticate moel was evelope by Weber [] in 1931, incluing the effect of liqui viscosity an ensity of the ambient gas. Weber s theory has been wiely accepte; however, experiments conucte by Sterling an Sleicher [3] showe poor agreement with this theory. Furthermore, they pointe out that previous researchers ha not fairly teste Weber s theory, because their experiments use relatively long nozzles creating a velocity profile within the jet; an effect not inclue in the theory. However, they i not provie a metho to preict the break-up of the jet consiering the velocity profile effects. On the other han, atomization of liqui sheets was first stuie by Savart in 1833 who observe break-up phenomena of raial expaning sheets prouce by two co-axial colliing jets. It was emonstrate that when thin liqui sheets are generate in the atmosphere, unstable sinuous waves are forme. Squire [4] first solve the linearize equation for parallel liqui sheet instability. Hagerty an Shea [5] foun that uner normal operating conitions, the wavelength is relatively large compare 3
to the sheet thickness an their growth rates are consequently greater than those of the alternative ilatational forms typical of the jet atomization stuie by Rayleigh. Dombrowski [7-11] stuie the effect of ambient ensity on rop formation in sheet base fan-spray nozzle experiments. He foun that the rop size increases with ambient ensity. In his analysis, he assume the relative velocity between sheet an ambient atmosphere is equal to the sheet velocity an that the flow in the sheet is irrotational. Thus, a velocity potential exists. By further assuming that the amplitue of the waves on the sheet is small compare to the sheet thickness an wavelength, a simplifie linear orinary ifferential wave growth equation was formulate. He efine imensionless amplitue in the form of f = ln( A/ A ) 0, an assume the sheet woul break up into ligaments every half wavelength at a critical imensionless amplitue equal to 1. Further, he assume the ligament woul break up into roplet accoring to the simple Rayleigh instability mechanism. Dombrowski etermine the fastest growing wave (most unstable) that cause the sheet to break up. For invisci sheets, he etermine λ = 4πσ / U, where σ is the surface tension of liqui an crit ρ a ρa is the ensity of ambient gas. He also suggeste an approach for etermining λcrit for viscous sheets. Li an Tankin [1-14] also stuie the instability of two-imensional viscous liqui sheets. They foun that the surface tension always opposes the wave growth. In aition, they foun that ilatational isturbances control the instability process for small Weber numbers, while sinusoial isturbances ominate for large Weber numbers. However, they i not provie any ata or criterion about sheet break-up. 4
Huang [15] stuie the break-up of axisymmetric liqui sheets forme by the impingement of two co-axial jets. His results show the break-up istance of the sheet is a function of Weber number which is efine as We = ρ U /σ. He ivie the break-up behavior into three regimes. The first regime occurs from 100 We 500 characterize by a stable liqui sheet, in which the sheet break-up istance increases with We. The secon regime occurs from 500 We,000, which is calle the transition regime. The thir regime is efine from,000 We 30,000, representing an unstable liqui sheet regime. In this regime, the break-up istance ecreases following We -1/3. Clanet an Villermaux [16, 17] stuie the break-up of liqui sheets generate by liqui jets impinging onto a small eflector. They stuie the sheet break-up using water an ethanol an foun result similar to Huang. In aitional to liqui sheet break-up location, they foun that the roplet mean arithmetic iameter follows the / 3 * 1 relationship of / D = ρ We for 1,000 We,000 where ρ * = ρ a / ρ l. 0 1.. Spray Measurement Droplet iameter, velocity an initial location are three essential quantities in the characterization of fire suppression sprays. Theoretically, the mass flux istribution coul be etermine with the knowlege of the istributions of these three quantities. Early spray measurements were conucte using photographic techniques an laser shaowing metho. Dunas [18] use these techniques to test six sprinklers with iameter ranging from 3.1-5.4 mm an with pressures ranging from 5
0.345-5.5 bar. The rop size was measure using a high-spee photographic technique. The photographs were analyze both manually an using an electronic scanner. Dunas s results confirme the correlation first propose by Heskesta [19] that v 50 / D 0 = CWe 1/ 3, where v50 is the volumetric meian iameter, D 0 is the orifice iameter, C is a constant epening on sprinkler geometry. Dunas summarize the C value from ifferent researchers an showe 1.74 < C < 3.1. Yu [0] teste three upright sprinklers using a laser-base imaging technique for rop size measurement. He measure the rop size istribution at elevations of 3 m an 6 m below the sprinkler respectively. Measurements at these two elevations were almost the same, suggesting that these ownstream measurements are useful for characterizing the initial spray in the absence of seconary atomization. Detaile spray measurements have been conucte recently using avance iagnostics such as Phase Doppler Interferometry (PDI) an Particle Image Velocimetry (PIV). The PDI metho provies etaile local rop size an velocity information. Wimann [] measure the rop size an velocity from actual sprinklers with K-factors from 43.5 L min -1 bar -1/ to 81. L min -1 bar -1/. He reporte mean volume iameter, 30, follows We -1/3, except at low pressures (below 0.69 bar). More recently, Sheppar [3] measure the rop velocities using the PIV technique. He presente his result in a spherical coorinate system with the sprinkler at the origin. He showe the variation of raial velocity with polar angle. He also provie a ball-park estimate of the raial velocity close to the sprinkler (~0.m), which coul be expresse as U = l 0.6( P / ρ ) 1/. Furthermore, Sheppar use the PDI technique to characterize the rop size istribution. He measure the rop size at a 6
raial istance of 0.38 m at ifferent azimuthal angles of 0º, 10º, 30º an 60º. However, ue to the limitation of one point measurement, Sheppar i not provie an overall rop size istribution. Sheppar also stuie the roplet trajectory; he calculate terminal velocities assuming spherical roplets, an foun they agree with his experiment. The latest measurement was conucte by Blum [4], who measure the rop size istribution using a Spraytec Particle Analyzer evelope by Malvern Instruments [47]. He measure the local rop size 1 m below the sprinkler at several raial locations. The overall rop size istributions were etermine base on the mass flux an local rop size istribution measurements. Three sprinkler configurations were characterize in these experiments, which inclue a Basis nozzle, a Tine nozzle an a Stanar nozzle. The working pressure for these nozzles range from 0.69 bar to.76 bar. The results show that the rop size behavior is strongly relate to the injector configuration, perhaps suggesting that ifferent atomization moes may occur epening on the geometry. 1..3 Atomization Moeling Atomization moeling is a task full of challenge because of the complexity of the atomization process, which is influence by the nozzle geometry, working pressure, ambient air ensity, ambient temperature, as well as the internal injector flow structure. Two methos are typically use to moel the atomization process. The first approach is to use CFD with first-principle two-phase flow coupling. This approach 7
requires simultaneous simulation of the liqui an gas phase flow with etails. For simulating the two phases uring atomization, Eulerian-Eulerian approach has been establishe [46]. The secon lower fielity approach uses surface stability analysis. In the Eulerian-Eulerian approach, the ifferent phases are treate mathematically as interpenetrating continua. Since the volume of a phase cannot be occupie by the other phase, the concept of phasic volume fraction is introuce. These volume fractions are assume to be continuous functions of space an time an their sum is equal to one. Conservation equations for each phase are erive to obtain a set of equations, which have similar structure for all phases. These equations can be close by proviing constitutive relations that are obtaine from empirical information. Several moels have been evelope for multiphase flow problems. The VOF moel is an Eulerian-Eulerian multiphase moeling approach that shows particular promise for free-surface flow. The VOF moel uses a surface-tracking technique applie to a fixe Eulerian mesh. It is esigne for two or more immiscible fluis where the position of the interface between the fluis is of interest. In the VOF moel, a single set of momentum equations is share by the fluis, an the volume fraction of each of the fluis in each computational cell is tracke through the omain. However, it shoul be note that the sheet can be as thin as 100 microns just before break-up. In orer to moel the sheet break-up using the VOF metho, the gri size must be much smaller than the sheet thickness, which increases the computation time greatly. This resolution requirement prohibits the VOF technique for use as a practical tool for preicting atomization in fire suppression sprays. 8
Although etaile information coul be obtaine from numerical simulation using multi-imensional multi-phase flow moels, this metho is not a practical solution for preicting the initial spray (atomization) for engineering applications, especially if there are numerous sprinklers/nozzles. An alternative metho for CFD applications is to use surface stability theory escribing wave ispersion on the liqui sheet. Starting with a force balance on the liqui sheet an assuming a wavy liquigas interface, the wave ispersion equations can be etermine in terms of a imensionless wave growth rate or imensionless wave amplitue. Several research stuies have evelope wave ispersion equations for various geometries using similar assumptions. These equations are use to characterize the wave growth rate or amplitue preicting break-up or atomization at some critical conitions. The breakup criterion is establishe experimentally an may change with nozzle configurations. 1.3 Research Objectives The current stuy buils on fire suppression spray moeling ieas establishe by Di [6, 45]. A series of moeling moifications have been introuce in the current research to improve the fielity of the moel. Simplifie scaling laws have also been evelope from the moeling theory to facilitate analysis. Discharge characteristics from canonical fire suppression nozzles were also obtaine uring this stuy with Blum [4]. These full scale tests provie a etaile escription of the initial spray in well characterize geometries. Comparisons are mae between the etaile initial spray ata an the refine moel preictions to evaluate the moel an ientify focus areas for improvements. 9
Chapter : Approach In CFD applications, a variety of moels are neee to simulate the physical processes associate with fire suppression. This stuy focuses on specification on the atomization moel which is important. In the following iscussion, the important moels relate to atomization moel will be summarize. In practical analysis, fire suppression is initiate with a nozzle activation moel which controls the start of the spray base on a prescribe activation conition. After activation, a spray atomization moel can be use to efine the characteristics of the initial spray. These characteristics inclue the roplet iameter, initial roplet location, an roplet velocity. The initial spray ata will be use as input to a spray ispersion moel which preicts the trajectory of roplets incluing the thermal, mass, an momentum interaction of the isperse phase (liqui) with the continuous phase (gas). An extinction moel is also necessary to simulate the effects of oxygen epletion on combustion an thermal effects. Although proper specification of the initial spray is essential for CFD analysis of fire suppression, atomization moels have receive limite attention for this application. In CFD software such as FDS, correlations are use instea of atomization moels [48]. These correlations are obtaine from limite full spray test conucte almost ten years ago. The initial spray characteristics are strongly epenent on the sprinkler configuration an the result of limite tests cannot be faithfully applie to all sprinklers. Current correlations are limite for a variety of reasons. For instance, the initial rop iameter, location an velocity correlate strongly; however, current correlations o not capture this coupling. The 10
correlations are also base on cool ambient experiments an o not inclue the effect of the ambient ensity reuction occurring in fires which will likely influence the atomization process an the resulting rop size. Base on essential atomization physics, a moel is uner evelopment to preict the characteristics of the initial spray. This chapter will iscuss the general physics of the atomization process in etail base on previous work an atomization measurements performe by Blum [4] an Ren. A new scaling law evelope in this stuy is also explore..1 Atomization Physics The atomization process can be ivie into four istinct stages as shown in Figure.1 [44]. They are the sheet formation, sheet trajectory, sheet break-up an ligament break-up stages. The moeling of atomization is actually a problem of multiphase flow, which can be solve with multiphase moels. Two istinct of multiphase flow problems arise in fire suppression applications. In the first stage, an immiscible liqui sheet is forme; the interaction of the liqui sheet an ambient air is a problem of free surface flow. After the continuous liqui sheet breaks up into ligaments an roplets, the problem becomes one of a iscrete flui phase in a continuous gas. Although first principle multiphase flow moels have been establishe, they are not suitable for engineering level stuy (see 1..3). The approach use in this chapter is base on stability theory. 11
YJet Jet Growth of waves Defl cor Go thofwaves Deflector Seet Fomati n Sheet Formation Seet igamt Sheet Ligaments Lgamen op Ligaments Drops Drop Formation Figure.1. Description of atomization process (sheet formation, sheet trajectory, sheet break-up, ligament break-up) [6].1.1 Sheet Formation Generally, the liqui sheet is forme by impinging a liqui jet onto a eflector. Different sprinkler configurations are mainly treate by changes in the eflector geometry. Those ifferences coul change the thickness an velocity of the liqui sheet, which will change initial spray characteristics. Three types of sprinklers are analyze in this chapter. The thickness an velocity are etermine from Watson s theory [8] base on free-surface similarity bounary-layer concept. Figure. shows the formation of liqui sheet, which is characterize by four stages. Region I: Impinging region ( r < / ). The irection of water jet is D 0 change in this region from vertically to raially. A raial expaning sheet is forme. There is a stagnation point in the center of the impinging 1
region. In this region, a growing bounary layer has not been forme; the sheet velocity can be approximate as the free jet velocity, U jet. x D U o Stagnation Point U < U o I I I I II I V r r stag r 1 r Figure.. Sheet formation from impinging jet [6] Region II: Bounary-layer eveloping region ( D 0 / < r < rl ). In this region the bounary-layer keeps growing with a Blasius velocity profile. In this region, the bounary-layer thickness is less than the sheet thickness until at the en of this region, where the bounary-layer reaches the free surface ( r = r s ). Region III: In this region ( r > r s ), the free surface velocity is slightly smaller than by Blasius. U 0, but nearly constant an the velocity profile is preicte Region IV: Quickly ecreasing free surface velocity an the bounarylayer affects the whole sheet. The velocity ecreases greatly an the velocity profile is given by a non-blasius similarity solution. 13
In orer to show the effect of viscous interaction with the eflector, a nonimensional sheet thickness is efine as the actual thickness compare to an invisci sheet thickness solution, which is given by T β =, (-1) T 0 where 0 T D / 8r. Typically, the working pressure of a nozzle is above 0.345 bar, 0 = an the jet velocity U0 > 5 m/s. The internal iameter of the nozzle D 0 is larger than 8 mm. The Reynols number of the jet is on the orer of 10 4, inicating the flow is in the turbulent regime. Following Watson [8], in Region II where r < r l, the sheet thickness is given by D0 1/ 5 4 / 5 1/ 5 Region II = 0.045D0 / Re, (-) T ( r) + r 8r where Re is Reynols number. The bounary-layer thickness, δ, is The bounary of Region II is In Region III where δ = 0.303D r 1/ 5 4 / 5 1/ 0 / Re, the sheet thickness is r > r l 5. (-3) r.183 Re 1/ 3 l = 0 D 0. (-4) 0 5 / 4 D r T ( r) Region III = 1.0 + 0.04, (-5) 1/ 4 8r ( D Re) 0 where Q is the flow rate, l is an arbitrary constant length 4 1/ 4 l 9 / = 5.71Re D0 9 / 4. (-6) The non-imensional form of sheet thickness is the sheet thickening factor, which is 14
9 / 5 1/ 5 r 1 + 0.196 Re r < rl D0 β =. (-7) 9 / 4 1/ 4 r 1.0 + 0.179 Re r > rl D0 From conservation of mass, the average velocity of sheet at the ege of eflector is etermine by U 1/ D Q Kp = U 0 = =, (-8) 8r h πr h πr h where K is the K-factor of the sprinkler in unit of Lmin -1 bar -1/. The most important parameter in sheet formation is the sheet thickening factor β an average sheet velocity U. The sheet velocity governs the wave growth rate an the sheet thickness etermines the iameter of the roplet. It is also worth noting that the velocity profile of the sheet will have some influence on the sheet break-up. To simplify the problem, in the following analysis, the average sheet velocity is use instea of using velocity profile [3]..1. Sheet Trajectory After leaving the eflector, the external forces acting on the liqui sheet are only the friction force an gravity force. Distinct from a iscrete object (i.e. roplet), the liqui sheet is a continuous expaning stream, which has a more complex trajectory. Furthermore, the thickness of the sheet changes as the sheet expans raially outwars. Internal forces also affect the trajectory of the sheet especially when the liqui sheet is very thin an the curvature of the trajectory is large. To 15
etermine the trajectory of the sheet, a group of ifferential equations are provie by Ibrahim [5] with consieration of all possible parameters. The moel is base on curvilinear boy-fitte coorinates as shown in Figure.3 where r an z are the raial an vertical coorinate of the cylinrical coorinate system. The variable ξ is the position in curvilinear boy-fitte coorinate, η is the coorinate normal to the sheet, T is the local sheet thickness, U is the local sheet velocity, α is the angle between the meian streamline an axial irection z, an g is the gravitational acceleration constant. The problem simplifie by using the curvilinear boy-fitte coorinates. For the raial expaning sheet, the continuity an momentum equations coul be simplifie as Continuity: ξ ( ρ l UrT ) = 0 (-9) r r ξ g r r + ξ ξ T T T + ξ ξ α η U z Figure.3. Description of sheet trajectory 16
Momentum in stream wise irection: U ρ U = S ξ l + ρ g cosα l (-10) Momentum in irection vertical to the sheet: ρ U l α Pl = ρl g sinα, (-11) ξ η where P l is the pressure ifference between the two sies of the sheet, S is the liqui gas interaction friction term given by S ρ a = [0.79(1 + 150T / r) Re T 1/ 4 ]( U a U ), (-1) where ρ a is the gas ensity, U a is the gas velocity, Re is the Reynols number efine as Re = ρ r U U / μ. The inuce gas velocity shoul be couple a a a with the velocity of the sheet, which introuces extra equations for gas. Actually, the gas inuce velocity is much smaller than the moving sheet. Assuming zero gas velocity is a reasonable simplification. The term P l is the pressure ifference across the sheet given by Pl η ΔP T l = ΔP T g σ cosα α +, (-13) T r ξ where Δ is the gas pressure ifference between the sheet. In this case, ΔP is zero; P g the extra pressure is introuce only by surface tension forces. To close the system, two more equations are neee which are provie by the coorinate transformation from cylinrical to boy-fitte coorinates. In the r irection, it is g 17
r = sinα, (-14) ξ an in the z irection, it is z = cosα. (-15) ξ Now we have five variables, which are U, T, α, r, z an five ifferential equations which are summarize below U r T rt + UT + Ur = 0 ξ ξ ξ U ρ au 1/ 4 ρlu = [0.79(1 + 150T / r) Re ] + ρl g cosα ξ T α σ cosα α ρlu = ρl g sinα. (-16) ξ T r ξ r = sinα ξ z = cosα ξ We also nee five bounary conitions to solve the equation. They are given when ξ is zero, U T α r z ξ = 0 ξ = 0 ξ = 0 ξ = 0 ξ = 0 = U = T = π / θ = r 0 = 0 sheet ini. (-17) The bounary conitions of sheet velocity an thickness are provie by the jet moel. Because there are no goo moels to preict the initial angle, the initial angle is currently etermine experimentally. It is not possible to solve the non-linearly equations analytically, a fourth orer Runge-Kutta metho is use to fin the solution 18
of U, T, θ, r, z. The purpose of the trajectory moel is to preict the local sheet thickness an sheet velocity. In previous atomization stuies, the sheet velocity was regare as a constant by all researchers an the sheet thickness was also treate as a simple function of raius. The trajectory moel provies those values in a more accurate way which will be use in the sheet break-up moel..1.3 Sheet Break-up The raial expaning liqui sheet is inherently unstable an the sheet behavior is typically attribute to a Kelvin-Helmholtz type of instability proucing a flapping sheet. An analysis of the mathematical escription of the flapping sheet provies some insight into the nature of the flag-like instability. First, the wave form on the growing sheet is explore. This exercise will be followe by a stability analysis to etermine the critical conition for sheet break-up. The form of the initial isturbance is y = y exp( i( nx ω )), (-18) 0 t where n = n r + in i an ω = ω r + iωi are wave number an frequency respectively. If the solution of n is a complex number, the wave is spatially unstable where n i is the spatial growth rate, which correspons to a convective instability. If the solution of ω is a complex number, the wave is temporally unstable where ω i is the temporal growth rate, which correspons to an absolute instability. Sheet break-up experiments show that the wave amplitue changes with location an not with time, inicating a spatial instability. The wave equation simplifies to 19
y = y exp( n x)sin( n x ω ), (-19) 0 i r rt Figure.4 shows the waves an ligaments forme from the sheet at ifferent times. Four forces are essential for the evelopment of the flapping sheet instability; they are the pressure force, friction force, surface tension force an viscous force. Subject to these forces, waves will grow on the smooth liqui sheet like a flapping flag. Sinuous waves an ilatational waves are possible. When the amplitue of the waves is large enough, the continuous sheet may break up into a ring like ligament. Figure.4. Growing wave on the sheet; t1; t; t3 The wavelength at break-up governs the ligament iameter. A theory base on linear stability has been evelope by Dombrowski [8] to preict the wave instability for two imensional waves in an invisci gas. In his moel, he assumes the flow is irrotational. A velocity potential is use to escribe the flow motion. By further assuming the wave amplitue is small compare to the wavelength, the free bounary conition can be simplifie. In the viscous force term, there are contributions from 0
normal shear force an sheet thinning. However, Dombrowski estimate that the thinning term is small compare to other forces, an if the wave number is sufficiently large, this term can be neglecte. This simplification results in a straightforwar moel to preict the break-up of the raial expaning sheet. The simplifie equation to escribe the wave growth on the sheet is given by f t μl f + n ρ t l ( ρ nu a ρ T l σn ) = 0, (-0) where t is time, f is a imensionless wave amplitue efine by f = ln( A / A0 ), A is the wave amplitue an A0 is initial wave amplitue, μ l is the liqui viscosity, T is sheet thickness, n is wave number efine by n = π / λ, an λ is wavelength. The sheet velocity, U, an sheet thickness, T, is given by the trajectory moel. Actually, the trajectory an sheet break-up analysis are only weakly couple. Accoring to the linear wave ispersion theory, the wave amplitue is small compare to the wavelength. The effect of waves on the sheet trajectory can be neglecte. However, the sheet thickness an velocity significantly affect the wave growth rate. From the wave ispersion equation, we can see that the viscous force an surface tension force is always playing an opposite role on the wave growth. The pressure force accelerates the wave growth. As the ensity ratio, ρ * = ρ a / ρ l increases, the wave growth rate increases. In real fire scenarios, the gas temperature increases an the ensity ecreases. As a result, the wave growth rate will ecrease. Although the wave number, n, can be any real number, there is only one wave number that makes the wave grow the fastest. This wave number is calle the critical 1
wave number, n crit, which is consiere to be the most unstable wave that will first lea to break-up. The sheet won t break-up until f reaches to critical imensionless wave amplitue. In Dombrowski s theory, is a constant with a value of f crit, sh f crit, sh 1 regarless of working conition. Other researchers also foun that is a f crit, sh constant which is close to 1. It shoul be note that in our experiment, f crit, sh was assume to be a function of nozzle configuration an coul be etermine by experiment. However, it shoul be note that for viscous thinning sheet, it is not available to get the critical wave number analytically, which must be etermine from iterating integral of f. The break-up time is recore when f reaches crit sh. An from the trajectory moel, the corresponing sheet velocity, sheet thickness, an the f, break-up location r crit, sh are foun accoring to the break-up time. The thin flapping sheet is assume to break up into ring-like fragments having a raial extent of half wavelength. From conservation of mass, the fragment of ligament mass is etermine by m lig = πρ T [( r + π / n ) r bu, l bu, sh bu, sh crit, sh sh ]. (-1) Assuming the cross-section of the ligament is circular, the equivalent iameter can be etermine from the Eq. (-) m lig lig lig = π ρ + l rbu, sh. (-) The ligament iameter is not only relate to the ligament mass, but also relate to the sheet break-up location. The sheet break-up analysis reveals that the critical wave number an sheet break-up location are important quantities governing atomization
behavior. In this stuy, the sheet break-up location is carefully measure for evaluation of the atomization moel..1.4 Ligament Break-up The ligaments prouce by the sheet are also unstable. Different from the sheet break-up moel, the surface tension force plays a positive role for the wave growth. Weber [] gave the analysis of ligament break-up, similar to the sheet break-up; there is also a critical wave number which is given by 1/ 1 3μ l n cirt, liglig = + 1/. (-3) ( ρlligσ ) Usually, the iameters of the ligaments are on the orer of 0.1 to 1 mm. For water, the secon term in the brackets is about 0.006 to 0.00, which is negligible compare to 0.5. The unstable waves on the ligament are ilational waves, which will lea to break-up every one wavelength. From conservation of mass, the roplet iameter can be expresse as = / 3 3π lig ncrit, lig 1/ 3. (-4) Combining Eq. (-) an Eq. (-4), the final expression for roplet can be simplifie as = 1. 88 lig The time neee for ligaments to break-up into roplets is given by. (-5) t bu, lig ρl = 4 σ 1/ lig 3 /. (-6) 3
With the knowlege of the sheet break-up location, initial velocity of the ligament, an break-up time of the ligament, the initial roplet location is easily foun. For example, the initial raial rop location is given by = (-7) r rop rbu, sh + U ligtbu, lig Here, the velocity of ligament is consiere as a constant provie by the sheet velocity at break-up. Also any changes to the trajectory of the ligament are neglecte.. Numerical Moel In.1, the important physics an associate governing equations for atomization in the impinging jet configuration were prescribe. The equations escribing the break-up process are too complex to solve analytically. However, a numerical moel consisting of four sub-moels corresponing to the four important physical processes escribe in.1 has been evelope in this stuy. A iagram escribing the moel is inclue in Figure.5 an a etaile summary of equations an moeling parameters are inclue in Appenix A. This numerical moel provies eterministic or stochastic formulations for preicting the initial spray. The Deterministic moel only provies characteristic initial spray quantities. However, fire suppression sprays show strong stochastic behavior. For example, the sheet oes not always break-up at the same istance an the roplets o not have only one iameter. In orer to moel these stochastic behaviors, probability istributions are introuce into the moel to treat the various stages of the break-up process. The sheet critical break-up amplitue, the sheet break-up wavelength, an the ligament 4
Input K, P, D Sheet Formation Moel (Similarity solution) θ, T, U ini sheet Trajectory Moel (Sheet trajectory equation) T,U T, U Output r rop,,u rop Ligament Break-up Moel (Jet instability equation) f, λ sheet t Sheet Break-up Moel (Linear stability solution) Figure.5. Description of atomization moel break-up wavelength are all treate stochastically. This physics-base technique provies an alternative to specifying a stanar istribution about a calculate characteristic rop size. The stochastic moel ultimately provies istributions for initial rop size an location. A etaile iscussion on the stochastic moel is provie by Di [6, 45] an repeate in Appenix B..3 Scaling Law The scaling laws for sheet break-up an roplet iameter are base on the equations in.1 an Appenix B. The scaling laws for sheet break-up use the invisci assumption which greatly simplifies the scaling expressions. This assumption is a reasonable approximation for a low viscosity flui like water ( μ = 0. 0011N/m s). A group of imensionless parameters are use to express the scaling laws. These parameters inclue ρ * = ρ a / ρ l Dimensionless Density 5
We = ρ U D /σ Jet Weber Number l jet 0 r * = r / D Dimensionless Deflector Raius 0 rbu = Dimensionless Sheet Break-Up Distance *, sh rbu, sh / D0 T * = T / D Dimensionless Sheet Thickness at Deflector Ege 0 * = / D0 Dimensionless Droplet Diameter β = T /T 0 Thickening Factor The Weber number base on the sheet velocity is smaller than the jet Weber number, We, ue to viscous eceleration along the eflector an rag forces from the surrouning air, which is We sh We U =. (-8) β The expression for wave number from the viscous ispersion equation is too complex for scale analysis. To simplify the analysis, an invisci moel is use assuming the viscosity of liqui is zero. The wave ispersion Eq. (-0) simplifies to f t ( ρ anu sheet ρ T l σn ) = 0. (-9) Since the wave is growing all the time until break-up, f / t will always be a positive number an can be expresse as f t ( ρ anu = ρlt sheet σn ) 1. (-30) By taking the erivative of Eq. (-30) with respect to n, the critical wave number can be foun when ( f / t) / n = 0, which yiels the maximum growth rate, n crit sh 6
The sheet critical wave number can also be expresse as ρ au sheet n crit, sh =. (-31) σ n * ρ au sheet ρ awe ρ We crit = = σ ρl D0β D0β =. (-3) The imensionless wave growth rate will be f t * = ρ U * = ρ U sheet sheet We D T 0 β 1 rwe D0r T β 1, (-33) From the expression of r = r + U cosθ t, whereθ ini is the sheet initial angle, sheet which is a function of sprinkler configuration. We can fin that r = U cosθ t. An the wave growth equation can be expresse in terms of raius as ini sheet ini f t f = U r sheet cosθ ini * = ρ U sheet rwe D0r T β 1, (-34) f r * ρ = cosθ ini rwe D0r T β 1. (-35) At the en of integral, the imensionless amplitue reaches the critical value. In this stuy, the critical amplitue is a function of nozzle configuration, an in the following analysis, f is use instea of f, where 0 crit f 0 = 3 1 * ρ We 3 / ( r 3 / bu, sh r ini D0r T ). (-36) θ β cos Finally, the imensionless sheet break-up istance can be expresse as 7
r * bu, sh 3 f 0 cosθini = 1 * + * * r r ρ T * β We 1 3. (-37) Eq. (-37) shows the break-up istance is a complex function of Weber number. * 3 f cos 0 θ ini T β However, the equation can be simplifie if 1 * *, which means ρ r We the sheet break-up istance is much larger than the raius of the eflector. Uner this conition, the imensionless sheet break-up istance is given by 1 r * bu, sh r * / 3 3 f 0 cosθ ini * * r ρ * 3 1/ 3 ( T β ) 1/ ( ) = We. (-38) For the same nozzle, the sheet break-up istance woul follow We -1/3. Since we alreay have the wave number an sheet break-up istance, the iameter of the ligament can be foun. Because the sheet break-up istance is much larger than the sheet wavelength, the ligament mass can be simplifie as m lig = πρ T l bu, sh π ρ T l [( r bu, sh r bu, sh bu, sh + π / n / n crit, sh crit, sh ) r bu, sh ]. (-39) The ligament mass is a function of ligament iameter, which is also much smaller than the sheet break-up istance. Thus, the Eq. (-) will be simplifie as, (-40) lig m lig π ρl rbu, sh Combining Eq. (-39) an Eq. (-40), the ligament iameter will be lig T n 1 bu, sh 0 cos D r T β = * crit, sh rbu, shρ We θ ini 1, (-41) From Eq. (-5), the final expression for roplet iameter is given by 8
= 1.88 lig = 3.76 r D0r T cosθ ini = 3.76 * rbu, shρ We 3 / 3 f + 0 ( D r T β cosθ ) * ρ 0 cosθ ini 1 ini 1 D0r T β We 1 1 3 1 * ( ρ We). (-4) The imensionless form of roplet iameter is * * ( T β cosθ ) ini = = 3.76. (-43) 1 D0 1 3 * 1 3 f 0 cosθ ini T β * 1+ ( ρ We) * * ρ r We 1 If the break-up istance is much larger than the raius of eflector, that 1 * 3 f cos 0 θ ini T β is 1 * *, the imensionless roplet iameter is simplifie to ρ r We * 1/ 3 * * * 1/ 3 ( ) 1/ 6 r cos 1/ T β θ ( ) = 3.5 ρ ini We. (-44) f 0 All the possible factors which affect the roplet iameter are inclue in Eq. (- 44). There are two coefficients for the Weber power law provie in Eq. (-44); the first one is the imensionless ensity. Increasing the air ensity will ecrease the roplet iameter. In a real fire, the air ensity is reuce; we can conclue that the roplet iameter will increase. The secon coefficient is the sprinkler configuration. For the same sprinkler, this term is almost a constant. The We is etermine by the jet iameter an working pressure. Increasing the working pressure will ecrease the rop iameter following We -1/3. 9
Chapter 3: Results an Analyses 3.1 Experiment Description Experiments incluing sheet trajectory, sheet break-up an rop size measurements were performe to characterize the initial spray for evaluation of the atomization moel an relate scaling laws. A etaile iscussion of the experimental techniques use for these experiments can be foun in Blum [4]. These experiments were systematically conucte in nozzles of increasing complexity to gain insight into the effect of nozzle geometry on the atomization process. Three nozzle configurations were investigate in this stuy ientifie as the basis, tine, an stanar nozzles. The basis nozzles consiste of a separate injector an eflector isk (three ifferent injectors sizes were use in this stuy). The tine nozzle is use to isolate the effect of the tines an spaces on the eflector. The nozzle was fabricate by removing the boss from a stanar Tyco D3 nozzle creating a flat notche eflector. A conventional Tyco D3 nozzle shown in Figure 3.1 was use for the D inlet Section A-A L inlet 1 D boss A 3 D o A L jet θ tine θ space 4 D ef θ boss Figure 3.1. Stanar nozzle configuration (Tyco D3 fire suppression nozzle); (1) Inlet () Frame Arms (3) Boss (4) Deflector [4] 30
stanar nozzle aing boss effects an representing a practical configuration. Table 1 summarizes the important geometries an flow characteristics for the three nozzles use in this stuy. 3. Moel Evaluation In this section, the preliminary results of atomization moeling will be iscusse. The atomization moel is ivie into several sub-moels, each of which correspons to a physical stage of the atomization process. Each of the sub-moels will be evaluate with comparison to available experimental ata. Table 1. Sprinkler Configuration [4] Initial Angle, θ (º) Basis Nozzle Tine Nozzle Stanar Nozzle ΔP U jet Tine Space Tine Space D (bar) (m/s) o = 3.5 D o = 6.7 D o = 9.7 (0º) (15º) (0º) (15º) 0.69 11.8 9.47 4.00 3.99 4.56 N/A.67 N/A 1.38 16.6 7.87 4.19 3.50 4.03 N/A 4.05 N/A.07 0.4 10.10 5.3 3.49 4.10 N/A 3.41 N/A.76 3.5 9.65 5.53 3.50 4.5 N/A 3.81 N/A Average, θ (º) N/A N/A 9.7 4.74 3.6 4.3 N/A 3.49 N/A 0.69 11.8 49.43 37.58 37.11 30.56 N/A 1.48 N/A Sheet Breakup Distance, 1.38 16.6 39.18 31.91 33.81 9.7 N/A 17.6 N/A r.07 0.4 37.39 30.7 N/A 7.31 N/A 15.30 N/A bu /D o.76 3.5 34.64 8.7 N/A 5.87 N/A 13.46 N/A 0.69 11.8 0.155 0.091 0.085 0.101 0.104 0.087 0.056 Overall Characteristic 1.38 16.6 0.149 0.081 0.076 0.097 0.100 0.073 0.053 Drop Size,.07 0.4 0.143 0.08 0.080 0.094 0.103 0.06 0.048 v50 /D o.76 3.5 0.146 0.081 0.074 0.089 0.101 0.059 0.045 31
3..1 Sheet Formation Due to viscous stress, the average sheet velocity ecrease along the eflector. From conservation of mass, the sheet is thicker than that of an invisci raially expaning film. A imensionless sheet thickness which is efine by Eq. (-8) will be use to escribe the thickening effect of the sheet along the eflector. It shoul be note that the thickness of the sheet may also increase from a hyraulic jump occurring on the shallow liqui layer [8]. However, calculations reveal that hyraulic jumps are not likely to occur on the eflector uner normal operating conitions, but may occur at raial locations well beyon the eflector bounary. Figure 3. provies a general view of the effect of geometry on the sheet thickness. The sheet thickness at the ege of the eflector is provie for a range of eflector geometries, D *, an injection conition, Re. In Region II, the bounary layer Figure 3.. Sheet thickening factor 3
thickness is smaller than the sheet thickness an the sheet will be less affecte by the eflector. However, in Region III, the bounary layer overwhelms the entire sheet. In this stuy, the imensionless eflector iameters are 10.9, 5.7, an 3.9 for the basis nozzles. The large D * configuration (small jets) will prouce substantial sheet thickening at low Re. However, typical sprinkler operating conitions shown in Figure 3. corresponing to moerate Re an relatively small D *. These conitions typically result in small viscous effect with T T ) / T 15%. ( 0 0 r= r < 3.. Sheet Trajectory Because of the forces within the sheet, the sheet trajectory is complex an ifferent from trajectories of free shooting objects. Figure 3.3 shows an example of the sheet trajectory curve with the initial sheet velocity of m/s, initial sheet thickness of 0.5 mm an initial sheet angle of 0º. Initially, the sheet is relatively straight which is similar to the trajectory of a iscrete object (i.e. roplet). As the sheet expans, the sheet thickness becomes smaller. An the surface tension force increases rapily, causing the curvature to increase an the sheet to ben. This kin of behavior can be observe when the working pressure is low. Theoretically, even at high pressures the sheet woul ben similarly. However, it breaks up before reaching this point where the sheet turns back as shown in Figure 3.3. For canonical fire suppression nozzles, the sheet break-up istance is very small, an the trajectory is almost a straight line. Figure 3.4(a) shows the measurement of trajectory for the meium basis nozzle at.07 bar. Figure 3.4(b) shows the comparison of the experiment an simulation. It shoul be note that the initial angle 33
Figure 3.3. Sheet Trajectory Characteristics, U = m/s, T = 0.5 mm, θ init = 0º is etermine experimentally as an input to the trajectory moel as iscusse in the approach. The initial conitions require for the trajectory sub-moel have been presente in Figure.5 an Appenix A. These initial conitions were obtaine from other sub-moels with the exception of the initial angleθ ini init init. Because the eflector is horizontal, θ ini shoul be 0º. Figure 3.4 shows that the sheet initial angle is small. However, θ ini rop size. Although moels to preict θ ini is critical to the ispersion analysis an the initial have not been evelope yet; these angles can be etermine form PLIF tests [4]. Figure 3.5 provies a summary 34
Ege Effect (a) (b) Figure 3.4. (a) Photograph of the sheet trajectory, K = 6.1 Lmin -1 bar -1/, P =.07 bar; (b) Comparison between experimental an moele sheet trajectory, Δ Measurement, Moel Preiction. of measure θ ini for basis nozzles, the tine nozzle at the tine an stanar nozzles at the tine. Unfortunately, the initial angle at the space of the tine an stanar nozzles coul not be measure ue to the limitation associate with the PLIF measurement technique. The initial angle seems to be generally a function of sprinkler configuration, although the scatter at various We cannot be ignore (or explaine). Possible factors governing θ ini 1. Deflector surface effects coul be When the thin sheet leaves the eflector, it experiences a surface tension force attracting it towars the outsie ege of the eflector, bening the sheet ownwar. 35
Figure 3.5. Sheet initial angle: Average; Basis Nozzle Δ D 0 = 3.5 mm, D 0 = 6.7 mm, D 0 = 9.7 mm; Tine Nozzle D 0 = 6.35 mm; Stanar nozzle Ο D 0 = 6.35 mm. Sheet Profile effects The sheet thickness ecreases along the eflector uring its raial expansion proucing a vertical velocity component. Without the vertical support of eflector, the sheet will travel ownwars an the angle introuce here can be estimate from Eq. (3-3). T T r (3-1) r T r T r = (3-) r 36
U T / t T z θ tanθ = = = = r= r = (3-3) U r r / t r r r T r T For the basis nozzle, θ will be approximately 1.5º, however, the contribution of the sheet profile effect seems only about one-thir of the total measure initial angle. The remainer of the eflection may be attribute to the eflector surface forces. In fact the smallest nozzle proucing the thinnest sheet is most easily influence by the eflector surface forces an has an average initial angle of ~9º, over twice that the other nozzles. PLIF images taken in planes orthogonal to a reference tine at raial locations of 1.7,.7 an 6.7 mm are provie in Figure 3.6 an reveal an even more complex view of trajectory behavior in tine an basis nozzles. The trajectory appears to be initially affecte by the spaces in the tine nozzle (.7 mm). But this effect is overwhelme by surface tension forces which keep the sheet intact as the sheet expans (6.7 mm). The stanar nozzle shows a raially expaning sheet similar to Figure 3.6. Inverte PLIF images epicting flow through sprinkler spaces; (a) Top view of measurement locations; (b) Tine nozzle; (c) Stanar nozzle [4] 37
the other nozzles, but the boss irects flow through the spaces, which is apparent at 1.7 an.7 mm. It is unknown how much flui is irecte through the space an if an orthogonal continuous liqui sheet is forme. Improve iagnostics shoul reveal a clearer view of atomization in this region. 3..3 Sheet Break-up an Drop Formation Sheet break-up resulte from a flapping flag-like instability for all nozzle configurations an experimental conitions teste in this stuy. This moe of instability has been observe by other researchers in similar configurations, most notably Dombrowski [8], Huang [15] an Villermaux [17]. A photograph of flapping sheet break-up is shown in Figure 3.7. In the past, Huang s impinging jet experiments inicate two lower Weber number break-up regimes in aition to the flapping break-up regime occurring at We >,000. In the flapping regime, Huang observe clearly visible waves growing on the sheet before break-up. He also observe that the sheet break-up follows a We -1/3 power law unlike the lower Weber number regime which follows a We 1 power law. The sheet break-up power law (We -1/3 ) behavior associate with the flapping regime was also observe in this stuy. However, two break-up moes were observe in the high Weber number flapping regime, which categorize as the rim break-up an ligament break-up moe as shown in Figure 3.8. Rim break-up moe In this moe, the sheet breaks up into roplets as shown in Figure 3.8(a). Waves can be clearly observe on the sheet. The wave amplitue is growing as the sheet expans raially outwar. Previous analysis (.1.3) suggests that sheet break-up 38
Sheet break-up location Wavy sheet Droplet Figure 3.7. Snapshots of break-up of liqui sheet from Villermaux [17] into ligaments occurs when the amplitue reaches a critical value [8]. Although critical amplitue approach is vali, an alternative break-up escription base on We arguments near the ege of the sheet is provie for the current analysis. A force balance at the ege of the sheet is illustrate in Figure 3.9 an given by F F I ρlatλl break up ~, (3-4) σl σ where a is the local maximum acceleration at the break-up location. The inertial force tens to tear the sheet apart, while the surface tension force tens to keep the sheet intact. Villermaux [17] suggests that U * a = ρ 1/3, (3-5) λ yieling F F I σ break * 1/ 3 ρu T ρ Wesheet up ~ =. (3-6) σ 39
(a) (b) (c) Figure 3.8. (a) Rim break-up moe; (b) Holes generate in rim break-up moe; (c) Ligament break-up moe. 40
The sheet Weber number at the break-up location is not intene to provie a critical conition for sheet break-up. This conition is provie by the critical amplitue. On the contrary, the Weber number at the break-up location is intene to escribe the conition at the ege of the sheet which is expecte to be important for escribing the moe of fragmentation an rop formation at the ege of the sheet. Experiment photographs, Figure 3.9, show that rim break-up moe occurs when < 150. We sheet Holes As We increases, holes are generate on the sheet. These holes expan ue to surface tension an may affect sheet break-up an rop formation. Holes can be generate from satellite roplets impacting on the sheet or from extreme large fluctuations in amplitue possibly cause by an upstream event. These holes begin to occur at We sheet > 50 while the flapping sheet is still in the rim break-up moe an continue to be observe in the ligament break-up moe. F I F σ Continuous Sheet Broken Sheet Figure 3.9. Force ynamic at the rim of sheet 41
Ligament break-up moe When We sheet > 150, the sheet breaks up in the ligament break-up moe. In this moe, the sheet will not break-up into roplets irectly. First, it breaks up into ringlike ligaments as shown in Figure 3.8(c) an escribe in..3. These ligaments break up into roplets. The atomization moel in this stuy is formulate base on the ligament break-up moe. The nozzle configuration an operating pressures for fire suppression nozzles are expecte to prouce We sheet > 150. These nozzles shoul prouce flapping sheets that form roplets in the ligament moe. More measurements are require to confirm this assertion, especially in complex geometries. The ae geometrical features of the stanar nozzles prouce an even more complex sheet structure than the basis nozzle. Due to the presence of tines an the boss, the sheet formation moel is no longer vali. The sheet surface interactions are ifferent between the flow through the boss an tine an the flow through the space. The boss irects the flow through the spaces creating istinct raially expaning sheets along the tine, an boss oriente streams in the space as shown in Figure 3.10. The current atomization moeling shortcomings for evaluating actual fire suppression nozzles have been reveale in the previous iscussion. Although the potential for sheet break-up in the rim moel is not likely for typical nozzle operating We sheet conitions an geometries ( > 150), the presence of holes in the sheet an the complicate sheet structure owing to the tines, spaces, an boss will require aitional treatment. The importance of these effects will become evient from analysis of the ata. 4
Sheet from tine Sheet from boss Figure 3.10. Sheet formation from tine an boss for stanar nozzle 3.3 Scaling Law Evaluation The scaling law for sheet break-up location an rop size are base on Dombrowski s analysis [8, 6] presente in.3. Assuming an invisci unstable sheet an a ligament sheet break-up moe. Eq. (-37) shows that the sheet break-up location is not only a function of We, but also relate to the initial sheet thickness, eflector iameter, initial sheet angle, air liqui ensity ratio an critical imensionless amplitue at break-up. A convenient sheet break-up parameter incluing these effects is efine as X sheet 3 f = 0 cosθini * * ρ r 1 * T β. Figure 3.11(a) We shows the break-up ata in terms of the break-up parameter. The imensionless amplitue f 0 has been ajuste to fit the theory for each nozzle accoring to Table. 43
(a) (b) Figure 3.11. Experimental sheet break-up results: (a) Basis Nozzle Δ D 0 = 3.5 mm, Tine Nozzle D 0 = 6.7 mm, Stanar Sprinkler D 0 = 9.7 mm, D 0 = 6.35 mm, Ο D 0 = 6.35 mm; (b) scaling law. For the basis nozzle, this change in f 0 may be relate to changes in the initial isturbance characteristics associate with the ifferent nozzle size. For the stanar nozzle, the flow through the spaces may have reuce * T Table. Critical imensionless wave amplitue. Because this effect is not Nozzle f 0 Basis Nozzle (Small, D0 = 3. 5mm ) 7.13 Basis Nozzle (Meium, D0 = 6. 7mm ) 13.33 Basis Nozzle (Large, D0 = 9. 7mm ) 18.13 Tine Nozzle ( D0 = 6. 35mm ) 10.13 Stanar Nozzle ( D0 = 6. 35mm ) 3.95 44
taken into account, the ajuste f 0 may be set too low having to compensate for the relatively large * T use in the calculations. Nevertheless, the sheet break-up location reuces with increasing We. In fact, for X >> 1, which is the case for most of the measurements, Eq. (-37) simplifies to Eq. (-38), an r * * bu, sh / r ~ We 1/ 3. The We -1/3 power law is clearly observe in Figure 3.11(b). The configuration epenent coefficient effect is also apparent in this figure. Similar to the sheet break-up analysis, rop size scaling law in Eq. (-34) shows that rop size is a strong function of nozzle size an configuration. A characteristic rop size is typically reporte for a spray base on an averaging scheme or base on some measure of a particular feature of the rop size istribution [48]. In this stuy, the v50 or volume meian iameter is use. Fifty-percent of the spray volume is containe in rop sizes smaller (or larger) than the v50. Ieally, the rop size calculate from the measure istribution is expecte to follow the trens etermine from the eterministic analysis. However, the magnitue of the characteristic rop size may iffer from the eterministic values. In fact, in this stuy, the stochastic moel preicts a v 50 1. 3, where is the eterministically calculate rop size. A similar approach of etermining a correlation factor to relate the v50 to eterministic value was also use by Dombrowski [8]. Eq. (-44) can be moifie base on this factor to estimate a scaling law for v50 yieling * * * 1/ 6 1/ r T β 1/ 3 ( ) cos θ ini ( We) X rop * = 4.5 ρ = f. (3-7) 0 Accoring to Eq. (3-7), the imensionless rop size shoul follow a We -1/3 power law. Furthermore the coefficients are given by the ensity ratio, an parameters relate to 1/ 3 45
the nozzle configuration. Figure 3.1(a) shows basis nozzle rop size epenence with We, which is much weaker (~We -1/6 power law) than the theoretical We -1/3 behavior. The ata, however, oes suggest a configuration epenent coefficient relate to this unexpecte weaker power law. Figure 3.1 (b) shows the ata compare irectly against the scaling law in Eq. (3-7). For the basis nozzle, higher We (lower X rop ) ata ten to provie better agreement with the theory. Previously in Figure 3.8, it was suggeste that the sheet break-up moe was etermine by the sheet Weber number at the break-up location an this parameter influence the rop formation behavior. Rim break-up moe an ligament break-up moe were observe in our experiments; however, the scaling laws are base on ligament break-up moe assumptions. The iscrepancy between the scaling law an measure rop sizes suggests that little of the measurements were taken in ligament moe. Figure 3.13(a) shows only ata where We sheet > 150 (ligament break-up moe). The agreement with the scaling law is obvious, although the ata is limite. More experiments, perhaps with bigger nozzles at higher pressures will increase the ata in this moe. Figure 3.13(b) also shows that the scaling law errors become relatively small in ligament break-up moe where the agreement with the scaling law is clear. The error is efine as v50 _ Pr eicte v50 _ Measure error = 100%. (3-8) v50 _ Measure The stanar nozzle isplays a very ifferent rop size behavior from the basis nozzles. A clear We -1/3 power law behavior is observe in Figure 3.1(a). Although the stanar nozzle rop size tren follows the theory, the measure rop sizes are 46
much smaller than those preicte by the scaling laws. This iscrepancy may be relate to the large sheet thickness estimate in the scaling law which oes not account for the mass loss through the spaces an the associate small value of. Both of these iscrepancies will cause the ata to shift away from the theory. More experiments an nozzle configurations are neee to measure the sheet thickness to better apply the theory to actual nozzles. f 0 (a) (b) Figure 3.1. Experimental rop size an scaling law: (a) Villermaux; Basis NozzleΔ D 0 = 3.5 mm, D 0 = 6.7 mm, D 0 = 9.7 mm, D 0 = 6.35 mm, Ο D 0 = 6.35 mm; (b) stochastic moel preiction. 47
(a) (b) Figure 3.13. Dimensionless rop size: (a)drop size in ligament breakup moe, Basis Nozzle Δ D 0 = 3.5 mm, D 0 = 6.7 mm, D 0 = 9.7 mm, Tine Nozzle D 0 = 6.35 mm; () error as a function of sheet Weber number at break-up location. 48
Chapter 4: Summary an Conclusions A physics base atomization moel to preict characteristics of the initial spray has been moifie an evaluate through comparisons with canonical fire suppression nozzles of varying complexity. A trajectory moel has been implemente an experimental observations have been critically examine to evaluate the valiity of the basic moeling assumptions an physics. The moel was evaluate quantitatively by comparing measurements with scaling laws obtaine from moel equations. The main conclusions from this analysis are summarize below. Sheet formation: Analysis shows that the sheet thins as it expans raially; however, viscous interaction with the eflector results in sheets that are thicker an slower than those obtaine from simple invisci analysis. Trajectory: The trajectory moel agree well with measurements (showing a nearly constant trajectory angle), but require specification of the measure starting angle at the ege of the eflector. This starting angle is etermine by the rate of thinning of the sheet an surface tension effects at the eflector ege, not accounte for in the moel. Sheet Break-up: Flapping sheet break-up was observe for all measurements. The sheet break-up location followe the scaling laws base on the moel theory emonstrating a We -1/3 power law behavior. Drop Formation: The sheet breaks up irectly into roplets in rim breakup moe ( < 150), but breaks up into ligaments an then rops in We sheet ligament break-up moe ( > 150), consistent with moeling We sheet 49
assumptions. The moel an associate scaling laws agree with measurement when the sheet breaks up in ligament moe although limite ata at the higher operating pressure an jet iameters associate with this moe. Aitional ligament moe measurement will be performe in future work. Sprinklers: Evaluation of the stanar nozzle with its complex sprinklerlike geometry provies insight into sprinkler behavior. Scaling law trens for the sheet break-up location an rop size following We -1/3 were observe. However, the magnitue of the imensionless rop size was much smaller than that preicte by the scaling laws, perhaps ue to sheet thinning effects owing to flow bias in the spaces not accounte for in the scaling laws. This effect will be explore in future work. 50
Appenices Appenix A: Moel Input an Formulations Sheet Formation Moel The object of sheet formation moel is to etermine the sheet thickness an velocity at the ege of eflector, which is the input of trajectory moel. The input quantities inclue An the output quantities inclue P, ρ l, ρ a, K, D. U, T, r. sheet Table A.1 is the summary of all the quantities an the variable names use in the sheet formation sub-moel Table A.1. Summary of variables in sheet formation moel Symbol Physical meaning Corresponing parameter in coe P Pressure pressure ρ l Density of liqui rol ρ a Density of air roa μ l Kinetic viscosity of the liqui miul K K-factor of the nozzle kfactor D 0 Diameter of eflector D0 r Raius of the eflector ra U Jet velocity ujet U sheet Velocity of the sheet at the usheet eflector ege Q Flow rate of the nozzle(kg/s) qflow r o Raius of the jet rjet r 1 Bounary layer region roturb T Film thickness at the eflector thef ege l An arbitrary constant length lcharturb 51
The following is the list of equations that use in the sheet formation sub-moel U P = ρl 1/, (A-1) r o = K π 1/ ρ l 1/ Q = KP 1/4, (A-), (A-3) T 1/ ro 7ν l + 0.01659 r U o = 1/ 4 9 / 4 ν l r + l 0.011 Q r 5 r 9 / 4 4 / 5 r r < r 1 > r 1, (A-4) Q l = 4.16 r o ν l ro 1/ 9, (A-5) U sheet Q =. (A-6) πr h Sheet Trajectory Moel The object of sheet trajectory moel is to calculate the sheet thickness an velocity as the sheet expans outwars, which will be use in wave ispersion moel. The input quantities inclue U 0, z 0, T, r, θ 0, U a, p g. An the output quantities inclue U, T, r, z, θ, ξ. Table A. is the summary of all the quantities an the variable names use in the trajectory sub-moel 5
Table A.. Summary of variables in trajectory moel Symbol Physical meaning Corresponing parameter in coe U sheet Initial velocity of the sheet usheet Z 0 Initial vertical position of the (assume 0 in the coe) sheet θ 0 Initial angle of the trajectory an U a Velocity of the air (assume 0 in the coe) p g Pressure ifference between the (assume 0 in the coe) sheet U Sheet velocity traj(i).v z Vertical position of the sheet traj(i).z r Raius position of the sheet traj(i).r T Thickness of the sheet traj(i).thick θ Trajectory angle(between the traj(i).angle tangent of the trajectory an horizontal line) ξ Natural axis along the sheet traj(i).s μ a Dynamic viscosity of the air miua Re Reynols number base on the Re iameter of the eflector S Gas-liqui interfacial friction s0 factor g Gravity 9.8m/s The following is the list of equations use in the trajectory sub-moel U r T rt + UT + Ur = 0, (A-7) ξ ξ ξ U ρl U = S + ρl g cosθ, (A-8) ξ ρ U l θ Δp g σ cosθ θ = ρl g sinθ, (A-9) ξ T T r ξ r = sinθ, (A-10) ξ z = cosθ. (A-11) ξ 53
S ρ a = [0.79(1 + 150T / r) Re T 1/ 4 ]( U a U ), (A-1) ρ ar where Re = U U. μ a a Sheet Break-up Moel The object of wave ispersion moel is to calculate the wave growth rate an etermine where the sheet breaks up, which will be use in ligament break-up moel. The input quantities inclue U, T, r, z, θ, ξ. An the output quantities inclue f n crit, sh, crit, sh, λ. Table A.3 is the summary of all the quantities an the variable names use in the wave ispersion sub-moel. Table A.3. Summary of variables in wave ispersion moel Symbol Physical meaning Corresponing parameter in coe f Dimensionless amplitue f0 n Wave number x1, x λ crit,sh Critical wavelength Sheet(i).lamba n crit,sh Critical wave number sheet(i).ncrit lig Ligment iameter lig(i).iameter m lig Ligment mass lig(i).mass r bu,sh Raius location of sheet breakup sheet(i).rbreak, lig(i).r1 T bu,sh Film thickness at break-up sheet(i).thick The following is the list of equations use in the wave ispersion sub-moel f t μl f + n ρ t l ( ρ nu a ρ T l σn ) = 0, (A-13) 54
f t ( ρ anu σn ρ T l ) = 0, (A-14) m lig = πρ h [( r + π / n ) r bu, l bu, sh bu, sh crit, sh sh ], (A-15) m lig lig lig = π ρ + l rbu, sh. (A-16) Ligament break-up moel The object of ligament break-up moel is to etermine the rop size istribution an initial rop location. The input quantities inclue An the output quantities inclue lig, U, r bu,sh., r rop, U rop. Table A.4 is the summary of all the quantities an the variable names use in the ligament break-up. Table A.4. Summary of variables in ligament break-up moel Symbol Physical meaning Corresponing parameter in coe U rop Critical wave number n λ crit,lig Critical wavelength for lig(i).wavelength ligament break-up t bu,lig Ligament break-up time lig(i).time r rop Initial rop loction lig(i).r, rop(i).r1 Drop iameter rop(i).iameter The following is the list of equations use in the ligament break-up sub-moel 1/ 1 3μ l n lig = + 1/, (A-17) ( ρlligσ ) 55
3 1/, 3 / 3 = lig crit lig λ, (A-18) 3 / 1/, 4 = lig l lig bu t σ ρ. (A-19) 56
Appenix B: Description of Stochastic Moel In the eterministic moel, the critical imensionless break-up amplitue f is assume to be a constant. However, this critical conition may vary with the largely unknown istribution of initial isturbance amplitues as shown in Figure B.1. We can see clearly that the surface of jet is not smooth at all. Waves appear on both of the jet an the film on the eflector. In the wave ispersion moel, the imensionless wave amplitue is efine as f = ln( A / A0 ). However, the initial wave amplitue A 0 is unknown an can change the critical imensionless amplitue. In the stochastic moel, the imensionless critical sheet break-up amplitue is treate as a iscrete ranom variable f [m] efine over an m-element space to account for the assume istribution of initial isturbances. This amplitue ratio f [m] satisfies a normal istribution specifie by the mean critical sheet break-up amplitue, f = 1 an the fluctuation intensity, I f. This fluctuation intensity is a moeling parameter efine as I f σ f =. (B-1) f where σ is the stanar eviation of f [m]. The ranom variable f [m] is use in the f wave ispersion moel resulting in m ifferent critical sheet break-up wavelengths, sheet break-up times, an sheet break-up locations. These istribute parameters will influence thee subsequent ligament formation an break-up analysis. 57
Waves growth on the jet Wavy sheet Deflector ege Figure B.1. Initial isturbance on the jet In the sheet break-up moel, the sheet is assume to break up into ring-like structures having raial with of one-half wavelength. These ring-like structures rapily contract into torroial ligaments, which in turn break up into rops. The sheet, of course, oes not always break up into one-half wavelength fragments. In the stochastic moel, the raial with of the ligament fragments, Δ [ m, n], is treate as a iscrete ranom variable base on a chi-square istribution efine for each sheet break-up realizations. The chi-square istribution prevents the occurrence of negative fragment withs at high fluctuation intensities. The specifie sheet break-up fluctuation intensity is efine as r lig I lig lig [ m] = σ. (B-) Δr [ m] lig This intensity an the mean ligament fragment with ( Δ r [ m] =, [ m]/ ) lig λ crit lig calculate from the moel are use to etermine the stanar eviation of sheet 58