ARTICLE IN PRESS

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Fire Safety Journal 42 (2007) 283–294

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A modeling basis for predicting the initial sprinkler spray

Di Wua, Delphine Guilleminb, Andre W. Marshalla,�

aDepartment of Fire Protection Engineering, University of Maryland, 3104 J.M. Patterson Building, College Park, MD 20742, USAbEcole Nationale Superieure de Mecanique et d’ Aerotechnique (ENSMA), 86960 Futuroscope Chasseneuil Cedex, France

Received 11 October 2006; received in revised form 17 October 2006; accepted 14 November 2006

Available online 12 February 2007

Abstract

The performance of water-based fire suppression systems is governed by the dispersion of the water drops in the spray.

Characterization of the spray is essential for predicting and evaluating the performance of these suppression systems. The dispersion of

the spray is typically modeled using particle tracking methods. The accuracy of the spray characterization using this approach is quite

sensitive to the initial spray specification. A physics-based atomization model has been developed for prediction of the initial spray.

Inputs to this model include injector geometry, injection pressure, ambient environment, and suppressant fluid properties. This

atomization model also accounts for the stochastic behavior of the physical processes governing spray formation and provides

probability distributions of initial drop sizes and locations for the initial spray. This modeling approach can be integrated with drop

dispersion models and CFD models to characterize spray dispersion in quiescent environments or evaluate suppression performance in

fire environments. The drop size predictions using the proposed atomization model have demonstrated favorable agreement with actual

sprinkler spray measurements over a range of operating conditions.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Sprinklers; Atomization; Suppression; Sprays

1. Introduction

Characterization of the water spray is critically impor-tant in evaluating the performance of water-based suppres-sion systems. A recent comprehensive overview of water-based fire suppression is provided in Grant et al. [1]. Theperformance of these suppression systems is primarilyevaluated through full-scale spray dispersion tests andactual fire suppression tests. It is difficult to extrapolate thespray dispersion test performance to real fire scenariosbecause of the potentially strong coupling between the fireand the spray. Alternatively, actual full-scale suppressiontests are expensive making it difficult to generate sufficienttest statistics for proper evaluation of the test results.Predictive models are needed to evaluate spray character-istics or to couple with fire models to predict suppressionperformance. In fact, the atomization model is a criticalmissing link in the modeling of suppressed fires. Sophisti-

e front matter r 2007 Elsevier Ltd. All rights reserved.

esaf.2006.11.007

ing author. Tel.: +1301 405 8507; fax: +1 301 405 9383.

ess: awmarsh@eng.umd.edu (A.W. Marshall).

cated gas phase models are in place for predicting the firedynamics like Large Eddy Simulation (LES). Furthermore,drop dispersion models are well defined for tracking thedrops after the atomization process is complete [2].However, a general model has yet to be provided forpredicting the initial spray properties for sprinklers. Theatomization model developed in this study is a first step inaddressing this deficiency.Some simple correlations have been developed for

estimating characteristic drop sizes based on a fewexperiments [3–5]. These correlations can be used asprimitive predictive models; however, they have a limitedrange of validity and are insensitive to many effects that areknown to influence the initial spray behavior. The data inthese correlations are obtained under quiescent ‘cool’conditions. However, the elevated velocities and tempera-tures in real fires are expected to influence the atomizationprocess. A robust physics-based approach capable ofhandling this coupling has been used to develop theatomization model in this study. The present workprovides the modeling basis to support the design of new

ARTICLE IN PRESSD. Wu et al. / Fire Safety Journal 42 (2007) 283–294284

suppression devices, characterization of spray details, orevaluation of the resulting suppression performance in thepresence of a real fire.

Some experimental work has been conducted to char-acterize the details of the sprinkler spray. The results fromthese experimental investigations provide sprinkler designguidance and provide valuable information for the devel-opment of atomization and spray models. Dundas [4]provides drop size measurements for several sprinklergeometries along with a review of drop size data obtainedin a variety of injectors. The data are correlated based onan expression first proposed by Heskestad [6], dv50/Dorif ¼ C We�1/3, where dv50 is the volumetric mediandiameter, Dorif is the injection orifice diameter, and theWeber number, We ¼ rlU

2Dorif/s, is based on the liquidproperties. The drop size data compiled by Dundas fromvarious injectors demonstrates that the coefficient ofproportionality, C, depends on the sprinkler geometry [4].You’s data reveal more insight into the dependency of thecoefficient, C. His data clearly show that C increases withincreasing injection orifice diameter for upright sprinklers[3]. Prahl and Wendt [7] measured flow patterns from anaxisymmetric laboratory sprinkler and developed modelsto predict these flow patterns. Correlations along withassumed Rosin-Rammler distributions were used toestimate drop size. Initial drop locations were approxi-mated based on wave instability concepts, and droptrajectories were determined from particle tracking. Ad-justments were made to the modeling constants to matchthe predicted and measured flow patterns. More recentlyWidmann [8], Widmann et al. [9], Putorti et al. [10], andSheppard [5] have characterized velocities and drop sizesfrom sprinklers using advanced diagnostics. Widmann usedPhase Doppler Interferometry (PDI) to measure drop sizesand velocities from actual sprinklers having K-factorsranging from 7.2� 10�5m3 s�1 kPa�1/2 (3.0 galmin�1 psi�1/2)to 1.35� 10�4m3 s�1 kPa�1/2 (5.6 galmin�1 psi�1/2). Thismeasurement technique provides detailed information atone point within the spray. Characterizing the overall spraywith this technique is prohibitive because of the number ofpoint measurements required to map out the spraydistribution. Nevertheless, the drop size and velocitymeasurements were taken at a number of locations at agiven plane to determine the mass flux distribution usingthe PDI technique. The mass flux obtained from these PDImeasurements at specified locations compared favorablywith mass flux measurements taken with collection tubes.Widmann also noted deviation from the p�1/3 scaling lawfor drop size at low pressures (around 69 kPa), butobtained better agreement at higher pressures. Putortimeasured drop size and velocity simultaneously using atwo-color fluorescence technique in an axisymmetricsprinkler configuration. Putorti’s measurements providedrop size/velocity correlations, drop size distributions, anddrop trajectories. Sheppard measured velocities very closeto the sprinkler (�0.2m) to characterize the initial sprayvelocity using Particle Image Velocimetry (PIV). This

technique allows for visualization of a cross-section ofthe spray. He presented these measurements in a sphericalcoordinate system having the origin located on thesprinkler centerline at a specified position betweenthe orifice and the deflector plate. Sheppard showed thevariation of radial velocity with polar angle (measuredfrom the sprinkler centerline) at various azimuthal angles(measured from the sprinkler yoke arms). He comparedhis velocity measurements with PDI measurements notingdiscrepancies due to differences in experimental configu-ration and biasing issues related to the differing mea-surement approaches used in the respective diagnostictechniques.Predicting spray characteristics has proven to be

challenging because of the complexity and stochasticbehavior of the breakup process. In fact, it is common tosimply characterize the sprinkler spray using correlations,or curve fits, of available experimental data. Theseexperimental data are often obtained at conditions welloutside of the operating conditions of interest. However,Dombrowski and Johns [11] developed an actual atomiza-tion model based on wave dispersion theory to predict dropsize. This atomization model was developed using fan typeinjectors. Dombrowski described the atomization processin terms of the growth of waves on an infinite unstablesheet. He simplified the wave dispersion equations andintegrated them to quantify the sheet breakup character-istics and then related the sheet disintegration to initialdrop characteristics. This wave dispersion model has beensuccessfully used by Rizk for various types of fuel injectionsystems [12] and is applied to sprinklers in the currentstudy. Marshall and di Marzo [13] have developed acomplete atomization model for sprinklers by integrating afilm formation sub-model proposed by Watson [14] with asheet disintegration sub-model proposed by Dombrowskiand Johns [11]. Furthermore, these models have beenimplemented with a modified stochastic formulationoriginally proposed by Rizk and Mongia [12]. The currentstudy provides the details for this atomization modelingapproach. Results from this atomization model arepresented and comparisons are made with actual sprinklermeasurements and correlations.

2. Model description

2.1. Atomization physics

A spray is formed by breaking up a volume of liquid intosmall drops. This process is referred to as atomization.Sprinklers use atomization to facilitate the dispersion ofwater over a large area to protect commodities not yetinvolved in the fire. The spray also delivers water toburning materials and decreases the burning rate byreducing heat feedback to the fuel surface. Moreover,atomization greatly increases the surface area of theinjected volume of water. In the case of finely atomizedwater mist sprays, this increased surface area results in

ARTICLE IN PRESS

Fig. 2. Sprinkler jet forming a viscous film as it impinges against the

deflector. The effect of the wall on the film can be treated analytically by

considering the dynamics in four separate regions.

D. Wu et al. / Fire Safety Journal 42 (2007) 283–294 285

enhanced evaporative cooling of the hot smoke from thefire and displaces air with inert water vapor. These effectsresult in abatement or even extinguishment of the fire.

For sprinkler sprays, the atomization consists of threedistinct stages. These stages are clearly illustrated in Fig. 1.First, the jet formed at the exit of the injection orificeimpinges on a deflector plate to form a thin film that flowsalong the plate. This film travels beyond the surface of theplate to form an unconfined expanding sheet. This sheetbreaks up more readily than the relatively large-diameterjet formed at the orifice exit. Next, aerodynamic waves areestablished on the liquid sheet, resulting from the inevitablesmall disturbances within the flow. These aerodynamicwaves are unstable and grow to a critical amplitude whichcauses the sheet to break into ring-like ligaments. Theseligaments are also subject to disturbances and the forma-tion of aerodynamic waves. Finally, the waves on theseligaments grow to a critical amplitude and break theligaments into small fragments which contract to formspherical droplets.

2.2. Deterministic model

The atomization model in this study consists of threesub-models based on the previously discussed sheetformation, sheet breakup, and ligament breakup processesgenerated by a jet impinging on an axisymmetric disc. Thiscanonical or ‘ideal’ configuration is characterized as a first-step toward establishing an atomization model for theinitial sprinkler spray. The ‘ideal’ geometry, althoughmissing notable tine, frame arm, and boss features revealsmany important trends related to the effect of changes inambient conditions, injection conditions, and even geome-try. Detailed descriptions of the sub-models for the ‘ideal’sprinkler are provided in the following sections along witha discussion of the suitability and limitations of thesecanonical sub-models for application to actual sprinklers.

Growth of Waves

Ligament → Drop

SheetFormation

Jet

Deflector

Sheet → Ligament

a

Fig. 1. (a) Illustration of the atomization process in convention

2.2.1. Sheet formation

The purpose of the sheet formation model is to predictthe sheet thickness and velocity known to be importantfactors governing the atomization process as will bedemonstrated in the following section. This sheet thicknessand velocity is affected by viscous interaction with thedeflector. For the ‘ideal’ configuration, free surfaceimpinging jet theory is used to characterize this viscousinteraction to determine the liquid film thickness andvelocity at the edge of the deflector.A water jet that impinges on a horizontal plate has been

studied by Watson [14] using boundary-layer theory.Watson describes the radial spread of a liquid jet over ahorizontal plane by four distinct flow regions. Fig. 2 showsthese four regions.

Region I: The stagnation region (rstag ¼ O[ro], where ro isthe radius of the jet). The speed outside the boundary layer

Ddef = 25mm

b

al sprinklers [2]. (b) Photograph of the atomization process.

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rises rapidly from zero at the stagnation point to Uo, thespeed with which the jet strikes the deflector. The effect ofthe wall is contained in a very thin boundary layer, which issmall compared to the film thickness.

Region II: The boundary layer region with Blasiussimilarity solution. The speed outside the boundarylayer is unaffected by the layer and remains almostconstant and equal to Uo. In this region, the boundarylayer grows until the wall influences the entire thickness ofthe film.

Region III: The transition region. The whole flow is ofboundary layer type with velocity profile given by theBlasius solution. The free surface is perturbed by theviscous stresses. The velocity profile changes as r increases;however, the velocity at the free surface remains nearlyequal to Uo.

Region IV: In this region, the speed of the free sur-face decays more quickly with r. Velocity profiles in thisregion can be described by a non-Blasius similaritysolution.

Watson’s theory provides region specific expressions forthe layer thickness based on the radial location for laminarand turbulent flows. The initial thickness of the sheet isgiven by the layer thickness at the edge of the deflector. Thedeflector diameter is thus an important parameter govern-ing the atomization process. Typically, only Region I andRegion II have to be considered in sprinkler flowconfigurations. However, the film has persisted beyondRegion II in a few of the cases studied.

Assuming the motion in the layer is turbulent, for adeflector diameter corresponding to a radial locationwithin Region II where rdorl (as shown in Fig. 2), thesheet thickness is given by

hd ¼ro

2rdþ C1 �

7nlUo

� �1=5

r4=5d , (1)

where rl is the radial location where the boundary layerreaches the surface (boundary of Region II), ro is thehydraulic radius of the jet, nl is the liquid kinematicviscosity, rd is the radius of the deflector plate, Uo is theinitial speed of the jet, and Cl ¼ 1.659� 10�2, which is acoefficient determined from the similarity analysis per-formed by Watson. It should be noted that this coefficientwill change with sprinkler injection geometry. Othercoefficients from the similarity analysis are included insubsequent equations and are denoted as Ci.

For deflector diameter corresponding to a radial locationbeyond Region II, the sheet thickness is given by

hd ¼ C2 �nlQ

� �1=4 r9=4d þ l9=4

rd, (2)

where Q is the flow rate of the jet, C2 ¼ 0.0211, l is anarbitrary constant length, which has to be determined bythe conditions where the boundary layer reaches the freesurface (r ¼ rl). The expression for l is obtained by

matching the sheet velocity at r ¼ rl and is given by

l ¼ C3 � roQ

nl ro

� �1=9

, (3)

where C3 ¼ 4.126.The turbulent flow assumption is not always valid over

the full range of operating conditions. A stability criterionhas been provided by Watson to determine whether theflow is laminar or turbulent. From similarity analysis,Watson derived a critical jet Reynolds number, Re ¼

Q/nl ro ¼ 25,700, above which the flow is turbulent. In thecases presented in the current study, the jet Reynoldsnumber always exceeds this critical Re for injectionpressures above 5 kPa. Therefore, for most cases theregion-specific expressions for turbulent flow are appro-priate for calculating the thickness and velocity of theliquid sheet at the exit of the deflector. The details of thelaminar film formulation can be found in Watson [14].For a sprinkler, Uo can be calculated based on

Bernoulli’s equation assuming inviscid flow, so that

Uo ¼2Dp

rl

� �1=2

, (4)

where Dp is given by the difference between the totalinjection pressure and the environmental pressure and rl isthe density of the liquid. The hydraulic radius of the jet canbe expressed in terms of the sprinkler properties by thedimensional equation

ro ¼K

p

� �1=2 rl2

� �1=4, (5)

where K is the sprinkler K-factor describing the flowcharacteristics of the sprinkler. The K-factor is typicallyexpressed in units galmin�1 psi�1/2 or m3 s�1 kPa�1/2.The average speed of the sheet when it leaves the

deflector plate can be calculated by mass conservation, sothat

U ¼Q

2prdhd¼

K Dp1=2

2prdhd. (6)

The speed of the sheet is assumed to be constant andequal to U throughout the breakup process. The change invelocity due to gravitational acceleration has beenneglected because the breakup time is typically less than10ms, providing little time for gravitational acceleration.After the film leaves the deflector plate, the thickness of thesheet decreases continuously as it expands radially. Thethickness of the sheet is given by

h ¼rdhd

r, (7)

where h is the thickness of the sheet along its radial extentgiven by the radial location, r.Watson’s free surface boundary layer model captures

viscous interaction between the film and the deflector in the‘ideal’ disc configuration quite well. As previously dis-cussed, for typical sprinkler sizes and operating conditions,

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viscous interaction between the film and the deflector islimited. The boundary layer typically does not progressbeyond Region II before reaching the edge of the deflectorresulting in film thicknesses and associated velocitiestypically within 20% of inviscid values.

For an actual sprinkler, the film formation may deviatefrom that predicted by Watson’s model resulting fromcomplex three-dimensional effects. The combined effect ofthe sprinkler boss and tines may direct flow through gapsin the deflector causing it to deviate from the radialtrajectory assumed for the ‘ideal’ configuration and causingviscous interactions with the geometry not accounted forby Watson’s axisymmetric model. No provisions have beenmade to account for the aforementioned geometric effectsin actual sprinkler configurations. These effects willundoubtedly cause profound differences in the initial flowdirection. However with regard to the initial drop size, itshould be noted that Watson’s axisymmetric modelpredicts a relatively small viscous sheet thickening effect(important for atomization). Without measurements it isdifficult to verify if the viscous interaction with thedeflector in actual sprinklers is also small; however, thisresult would be consistent with the short length scalesassociated with the deflector geometry. Comparisonsbetween drop size predicted with ‘ideal’ sprinklers andactual sprinklers in Section 3.1 should shed some light onthe importance of viscous effects in actual sprinklergeometries.

2.2.2. Sheet breakup

The central mechanism for atomization in sprinklers isthe breakup of the liquid sheet formed by the injector intoligaments. This process is easily observed in Fig. 1, whichshows the sheet breakup process in the ‘ideal’ sprinklergeometry. Although this configuration closely resemblesthat of a sprinkler, it should be noted that the ‘ideal’configuration does not include tines which are present inactual sprinklers. Nevertheless, even in a tined configura-tion, a radially expanding sheet or radially expanding sheetfragments are expected. To describe the liquid sheetbreakup process, a wave instability concept is used, whichassumes that the disintegration of a liquid sheet or a jetoccurs when the waves imposed by the surroundingatmosphere reach a critical amplitude. The sheet breaksforming ligaments and drops are produced as the ligamentsdisintegrate. Using this concept, Dombrowski and Johns[11] developed equations describing one-dimensional wavegrowth in an infinite viscous liquid sheet and successfullyapplied them to describe the disintegration of radiallyexpanding sheets generated with fan nozzles. The centralmechanism for atomization is treated in this simplifiedtheory and following Dombrowski the theory is applied tomore complex configurations of practical interest (i.e. the‘ideal’ sprinkler configuration or even actual sprinklers).The suitability of neglecting multi-dimensional wavedispersion effects that may impact wave growth in morecomplex configurations will be determined from the quality

of model comparisons with actual sprinkler data presentedin Section 3.1.In this model, sinusoidal waves are assumed to travel on

the surface of the liquid sheet. A force balance is performedon the undulating sheet considering inertial, pressure,viscous, and surface tension forces. After considerablereformulation and simplification, the force balance can beexpressed in terms of the growth rate of the waves presenton the liquid sheet [11]:

qf

qt

� �2

þmlrl

n2 qf

qt

� ��

2ðranU2 � sn2Þ

rlT¼ 0, (8)

where f is the dimensionless total growth of the wave, s thesurface tension, n the wavenumber of the disturbanceimposed on the liquid stream (n ¼ 2p/l), l the wavelength,ra the air density, rl the liquid density, T the thickness ofthe liquid sheet, and ml the liquid viscosity.To simplify the analysis of the atomization process,

inviscid fluid is first considered with ml ¼ 0, so that Eq. (8)becomes

qf

qt

� �2

�2ðranU2 � sn2Þ

rlT¼ 0. (9)

For a specified n, the wave growth rate increases as thesheet velocity increases, which leads to a shorter sheetbreakup time. Similarly, decreased air density or increasedliquid surface tension results in longer sheet breakup time.Because the wave with the maximum growth leads to the

breakup of the sheet, the corresponding critical wavenum-ber is of interest. Taking the derivative of f with respect to n

and equating to zero yields the critical wavenumber withthe maximum growth in the inviscid formulation:

ðninvÞcrit ¼raU

2

2s. (10)

Since the wavelength is inversely proportional to thewavenumber, the critical wavelength which leads to thebreakup of the sheet increases as the liquid surface tensionincreases but decreases as the air density or sheet velocityincreases.After substituting Eq. (10) into Eq. (9), the sheet

breakup time can be determined by integrating Eq. (9)with respect to time. Assuming that the sheet velocity U

will remain constant until breakup, the breakup radius, rsh,can be determined from calculating the time taken to reacha critical dimensionless amplitude, fcrit, sh. This dimension-less amplitude, f, describes the ratio of the growingdisturbance amplitude to the amplitude of infinitesimalinitial disturbances at the initiation of the sheet. Thecritical value, fcrit, sh ¼ 12, can be determined experimen-tally and has been found not to depend on operatingconditions for plain jets and fan nozzle sheets [11,15].However, in the more complex impinging jet configuration,fcrit, sh may depend on the details of the injector configura-tion and possibly even the injection pressure, because ofpossible sensitivities of initial disturbance amplitudes tothese parameters. Nevertheless, for this initial study

ARTICLE IN PRESSD. Wu et al. / Fire Safety Journal 42 (2007) 283–294288

following previous researchers, a constant value,fcrit, sh ¼ 12, is applied in this model.

Although the inviscid flow assumption simplifies theproblem significantly and provides some insight intothe governing parameters, it is not realistic. For wavegrowth on liquid films with finite viscosity, Eq. (8) issolved for qf/qt which is then integrated to determinethe time to reach breakup (fcrit, sh ¼ 12). Critical breakuptimes are determined over a range of wavenumbers. Thebreakup time, tcrit, sh, is minimized with respect to thewavenumber, ncrit, sh, corresponding to the most unstablewave leading to sheet breakup. The viscous sheet breakupmodel is expected to provide higher fidelity than theinviscid model; however, the effect of flow turbulence onthe wave growth still needs to be considered in futurerefinements.

The sheet is assumed to break up into ring-like ligamentshaving an inner radius equal to the breakup radius,rbu, sh ¼ rd+Utbu, sh, a radial width given by lcrit, sh/2, anda thickness given by the sheet thickness at breakup, andhbu, sh determined from the liquid mass flow. The mass ofthe ligament, mlig, is thus given by

mlig ¼ prlhbu; sh½ðrbu; sh þ p=ncrit; shÞ2� r2bu; sh�. (11)

An equivalent diameter for the ligament can bedetermined from

mlig ¼ p2rld2lig

2rbu; sh þ

d lig

2

� �. (12)

The goal of the sheet breakup analysis is to find thecritical wavenumber with maximum growth which causesthe sheet breakup. The ligament diameter can then beobtained from the mass of the sheet fragment afterbreakup. The ligament diameter is governed by the criticalsheet breakup wavelength, sheet thickness at breakup, andthe sheet breakup location. The ligament diameter is mostsensitive to the change of the critical sheet breakupwavelength, and it increases as the critical sheet wavelengthincreases.

2.2.3. Ligament breakup

The ligaments formed from the sheet breakup are alsounstable and subject to the growth of dilatational wavesalong the axis of the ligament. This wave growth ultimatelyleads to fragmentation of the ligament into drops. Thisbreakup mode can be predicted from a stability analysis ona cylindrical column considering only surface tension andviscous forces.

Aerodynamic forces are neglected because this force isnormal to the axis of the ligament and will not contributeto the growth of dilatational waves along its axis. Theseassumptions for the treatment of ligament breakup arecurrently applied to the ‘ideal’ sprinkler geometry; how-ever, a similar analysis would be equally applicable forligaments formed in the actual sprinklers. The stabilityanalysis provides a simple relationship [15] for the critical

ligament wavelength for breakup, lcrit, lig, given by

lcrit; lig ¼ pffiffiffi2p

d lig. (13)

This fragment will contract into a drop. Conservingfragment mass, the characteristic droplet diameter, ddrop, is

ddrop ¼ d lig3lcrit; lig

2

� �1=3

. (14)

The number of drops that are formed after ligamentbreakup can be expressed as

N ¼6M lig

rlpd3drop

, (15)

determined by conserving mass between the ligament andthe drops. Weber [15] also provides an expression for thebreakup time as

tbu; lig ¼ 242rls

� �1=2d lig

2

� �3=2

. (16)

The distance that it takes for the ligaments todisintegrate into drops is easily calculated from theligament velocity, U, and tbu, lig. The initial drop location,which is the total distance the liquid travels until drops areformed, is given by

rdrop ¼ rd þUðtbu; sh þ tbu; ligÞ. (17)

These atomization relationships provide characteristicinitial spray conditions for a given sprinkler geometry andinjection pressure, fire condition, and liquid suppressant.The initial spray velocity, U, initial spray drop size, ddrop,and initial spray location, rdrop, are completely defined byEqs. (6), (14), and (17), respectively. These quantities aredetermined from the sprinkler geometry (K, rd), injectionpressure drop (Dp), surrounding flow gas phase fireconditions (ra, ma), and liquid properties (s, r1, ml). Itshould be noted that for the current formulation thevelocity of the gas in the vicinity of the sheet was assumedto be zero; however, the sprinkler entrainment velocity andthe fire-induced velocity may significantly change therelative velocity of the sheet. The relative velocity may bemore appropriate for use in Eq. (6) depending on thesymmetry of the gas phase flow.

2.3. Stochastic model

In real sprays, a multitude of drops with different sizesare created. In order to model this behavior, a stochasticanalysis is introduced [12]. In the stochastic atomizationformulation, random behavior with a physical basis isadded into the drop formation model to obtain thedistributed drop characteristics. This physics-based techni-que provides an alternative to specifying a standarddistribution about a calculated characteristic drop size.The liquid sheet critical breakup amplitude, the liquid sheetbreakup wavelength, and the ligament breakup wavelength

ARTICLE IN PRESS

Fig. 3. Predicted initial drop conditions of a sprinkler spray as a function

of injection pressure and ambient temperature, Ddef ¼ 25mm and

K ¼ 3.2 galmin�1 psi�1/2.

D. Wu et al. / Fire Safety Journal 42 (2007) 283–294 289

are treated stochastically. The stochastic model ultimatelyprovides distributions for initial drop size and location.

In the deterministic model, the critical dimensionlessbreakup amplitude is assumed to be a constant. However,this critical condition may vary with the largely unknowndistribution of initial disturbance amplitudes. In thestochastic model, the dimensionless critical sheet breakupamplitude is treated as a discrete random variable f[m]defined over an m-element space to account for theassumed distribution of initial disturbances. This ampli-tude ratio f[m] satisfies a normal distribution specified bythe mean critical sheet breakup amplitude, f ¼ 12 and thefluctuation intensity, If. This fluctuation intensity is amodeling parameter defined as

I f ¼sff, (18)

where sf is the standard deviation of f[m]. The randomvariable f[m] is used in the wave dispersion model resultingin m different critical sheet breakup wavelengths, sheetbreakup times, and sheet breakup locations. Thesedistributed parameters will influence the subsequent liga-ment formation and breakup analysis.

In the sheet breakup model, the sheet is assumed tobreakup into ring-like structures having radial width ofone-half wavelength. These ring-like structures rapidlycontract into torroidal ligaments, which in turn break upinto drops. The sheet, of course, does not always break upinto one-half wavelength fragments. In the stochasticmodel, the radial width of the ligament fragments,Drlig[m, n], is treated as a discrete random variable basedon a chi-square distribution defined for each of the m sheetbreakup realizations over an n-element space. The chi-square distribution prevents the occurrence of negativefragment widths at high fluctuation intensities. Thespecified sheet breakup fluctuation intensity is defined as

I lig ¼slig½m�

Drlig½m�. (19)

This intensity and the mean ligament fragment widthDrlig½m� ¼ lcrit; lig½m�=2� �

calculated from the model areused to determine the standard deviation of sheet fragmentwidths, slig[m]. These quantities are used to define the chi-square distribution and the resulting ligament fragmentwidths, Drlig[m, n].

Similarly, the ligament fragment widths, Dwdrop[m, n, p]are given by a chi-square distribution. The specifiedfluctuation intensity,

Idrop ¼sdrop½m; n�

Dwdrop½m; n�, (20)

and the mean ligament fragment width,Dwdrop½m; n� ¼ lcrit; lig½m; n�, defines this distribution. In allm� n� p drop sizes are obtained in the stochastic modeltogether with the number of drops at each of the possibledrop sizes. In the current study, m, n, and p are specified as

1000, 50, and 50, respectively in order to obtain sufficientstatistics for a smooth drop size distribution.Guidance for If, Ilig, Idrop values have yet to be established

from measurements. These parameters are expected to beinfluenced by the injector geometry and the injectionpressure. Currently, these values can only be estimated untildata or models are available to provide guidance on valuesfor these parameters. Careful measurements are currentlybeing conducted from experiments over a range of config-urations and operating conditions to support continueddevelopment of the atomization model. Predictions havebeen shown to agree with measurements using parameterranges 0.05oIfo0.25, 0.1oIligo0.3, and 0.1oIdropo0.3.

3. Modeling results

3.1. Deterministic analysis

The deterministic atomization model can be used toevaluate the sensitivity of the spray to changes in fluidproperties or sprinkler geometry. Fig. 3 shows initial dropsize and location predictions as a function of ambienttemperature and injection pressure for a sprinkler havingK ¼ 7.7� 10�5m3 s�1 kPa�1/2 (3.2 galmin�1 psi�1/2) andDdef ¼ 25mm. Injection pressures up to 483 kPa (70 psi)are presented, but the model can be used at much higherpressures. If the sheet Re exceeds approximately 105, anadditional shear instability breakup mode (not included inthe model) producing very small drops may becomeimportant. However, this condition is not realized insprinklers even with large orifices operating at extreme

ARTICLE IN PRESS

Fig. 4. Drop size and initial drop location predictions of a sprinkler spray

at standard atmospheric conditions and Dp ¼ 138kPa (20 psi) while

varying the diameter of the deflector and the sprinkler K-factor.

D. Wu et al. / Fire Safety Journal 42 (2007) 283–294290

injection pressures. These modeling results demonstrate thestrong coupling between the ambient temperature and theatomization process. The model suggests that the initialspray at the time of sprinkler activation (sensor tempera-tures typically 330–370K and gas temperatures muchhigher) and at various stages of the fire will differ fromthose measured under room temperature conditions. Asthe ambient temperature increases, the ddrop and rbu, shincrease. Increases in ambient temperatures result in lowerra. These lower densities reduce the imbalance betweencritical aerodynamic pressure and surface tension forces,which is the driving force for wave amplification. Thereduction in this driving force results in a slower wavegrowth rate and a corresponding longer rbu, sh. Althoughthese modeling results were obtained with the deterministicviscous model, the simpler inviscid wave growth equationscan be used to better understand this behavior. In fact,combining Eqs. (9) and (10) reveals that the wave growthrate varies linearly with ra. Furthermore, evaluation ofEqs. (11)–(14) reveals a double impact of ambienttemperature on the drop size. Decreasing ra results inboth longer rbu, sh (and associated thinner hbu, sh) andlonger lcrit, sh. These effects have opposing effects on ddropresulting in relatively small increases in ddrop with increas-ing ambient temperature.

Fig. 3 also illustrates the effect of injection pressure, Dp,on the spray. Increases in Dp and associated increases in U

will increase the imbalance between aerodynamic pressureand surface tension forces resulting in shorter breakuptimes. However, the increased U at higher Dp will have anopposing effect on rbu, sh resulting in only moderatereductions in rbu, sh with increases in Dp. This moderatereduction in rbu, sh (and associated thicker h) is over-whelmed by the substantial reduction in lcrit, sh withincreased U as described in Eq. (10) resulting in significantreductions in drop size when Dp is increased.

The initial drop locations predicted in Fig. 3 aresurprisingly long. Even at relatively high pressures, initialdrops are not completely formed until 250mm. These longdrop formation distances are somewhat misleading. Itshould be noted that the atomization process initiates withsheet breakup, which occurs significantly upstream of thedrop formation region as illustrated in Fig. 1(b). In fact,the sheet breakup times and ligament breakup times, whichsum to the drop formation time are of the same order.

The deterministic atomization model can also be used toevaluate the impact of changing the sprinkler geometry.Fig. 4 shows the effects of changing the K-factor, which is ameasure of the effective sprinkler orifice size, and theeffects of changing the deflector size. The drop size andbreakup length are significantly increased with increasingK-factor. These increases are due to the larger h, requiredby mass conservation. Increasing h increases the sheetstiffness resulting in a slower wave growth rate (lessamplification) for a given driving force for wave growth(pressure/surface tension force imbalance). The slowerwave growth rate results in longer rbu, sh resulting in

increased sheet thinning; however, the larger h resultingfrom the larger K-factor has a dominant effect on the dropsize causing lcrit, sh and corresponding ddrop to increase. Onthe other hand, Fig. 4 shows that ddrop and rbu, sh arerelatively insensitive to changes in the deflector size, rd.Although increasing rd results in reduced U and increasedhd owing to increased viscous interaction between the filmand the deflector, as discussed previously in Section 2.2.1and as demonstrated in Fig. 4, the effect of this viscousinteraction on ddrop is relatively small.

3.2. Stochastic analysis

The stochastic model provides a more realistic view ofthe spray where distributions of the initial drop sizes,locations, and velocities are predicted. These distributionsare important when dispersion and vaporization calcula-tions are required such as in suppression modeling. Figs.5–7 show distributions for initial drop size, velocity, andlocation for a sprinkler having Dp ¼ 138 kPa, (20 psi),K ¼ 3.2 galmin�1 psi�1/2 (7.7� 10�5m3 s�1kPa�1/2), andDdef ¼ 25mm. Fluctuation intensities for the spray arealso specified for the breakup process describing thechaotic behavior of the initial amplitude distribution(If ¼ 0.2), the sheet fragmentation (Ish ¼ 0.3), and theligament fragmentation (Ilig ¼ 0.3). Fig. 5 shows that thedrop size distribution at these conditions is well representedby a log-normal distribution. The median drop size is0.92mm with minimum drop size of 0.2mm and maximumdrop size of 2.8mm compared with a characteristic dropsize of 0.75mm predicted by the deterministic model.Initial locations for ligaments and drops are provided inFig. 6. It is apparent from this figure that drops do not

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initiate from one point, but result from a spatiallydistributed process. The photograph included as Fig. 1(b)also demonstrates this behavior where drops and ligamentsare observed in the same radial span. Fig. 6 shows that thesheet breaks up into ligaments between approximately 0.08and 0.14m. The sheet or ligaments may be present in thisregion. Ligaments begin to break up into drops at 0.18mand continue to form drops until 0.40m.

It is also useful to correlate the stochastic sprayproperties with drop size for specification of the initial

Fig. 5. Probability density function of initial drop size determined from

stochastic model; Dp ¼ 138kPa (20 psi), K ¼ 3.2 galmin�1 psi�1/2,

Ddef ¼ 25mm, If ¼ 0.2, Ish ¼ 0.3, Ilig ¼ 0.3.

Fig. 6. Probability density function of initial drop breakup location determine

Ddef ¼ 25mm, If ¼ 0.2, Ish ¼ 0.3, Ilig ¼ 0.3.

spray in CFD modeling. In this approach, a range ofcharacteristic drop sizes is defined representing the entirespray. The initial location, velocity, and mass fraction arethen specified for each characteristic drop size in thisdistribution. This method allows for the entire spray to bespecified and tracked using a relatively small number ofdrops. Figs. 7–9 show the initial spray properties based ondrop size. Fig. 7 shows that the smallest drop sizes areformed at the earliest times. Although the model specifies auniform velocity distribution for the initial drops, the

d from stochastic model; Dp ¼ 138kPa (20 psi), K ¼ 3.2 galmin�1 psi�1/2,

Fig. 7. Initial drop locations for characteristic drop sizes predicted with

the stochastic model; Dp ¼ 138 kPa (20 psi), K ¼ 3.2 galmin�1 psi�1/2,

Ddef ¼ 25mm, If ¼ 0.2, Ish ¼ 0.3, Ilig ¼ 0.3.

ARTICLE IN PRESS

Fig. 8. Velocities for characteristic drop sizes predicted with the stochastic

model and evaluated at a radial location of 0.65m; Dp ¼ 138 kPa (20 psi),

K ¼ 3.2 galmin�1 psi�1/2, Ddef ¼ 25mm, If ¼ 0.2, Ish ¼ 0.3, Ilig ¼ 0.3.

Fig. 9. Mass/volume fraction for characteristic drop sizes predicted with

the stochastic model; Dp ¼ 138 kPa (20 psi), K ¼ 3.2 galmin�1 psi�1/2,

Ddef ¼ 25mm, If ¼ 0.2, Ish ¼ 0.3, Ilig ¼ 0.3. —— Predicted cumulative

volume fraction; - - - - Rosin-Rammler log-normal curve fit of prediction,

dv50 ¼ 0.92, q ¼ 3.3.

D. Wu et al. / Fire Safety Journal 42 (2007) 283–294292

positive location/size correlation results in a positivevelocity/size correlation if the drops are tracked (allowingfor drag forces) and viewed at a common downstreamradial location. This positive velocity/size correlationevaluated at a radial location of 0.65m and shown inFig. 8 is consistent with measurements in sprinkler sprays[10]. Fig. 9 shows that the predicted drop size distributionis relatively narrow with very small drops (as well as the

very large drops) containing only a small fraction of theoverall mass of the spray.The drop size distribution is often provided in terms of

the cumulative volume fraction. The cumulative volumefraction provides the percentage of the total spray volumecontained in drop sizes smaller than a specified dropdiameter. The predicted cumulative volume fraction isprovided in Fig. 9. The Rosin-Rammler/log-normaldistribution has been found to represent the cumulativevolume fraction for sprinkler sprays. This distribution isgiven by

CVF ¼ð2pÞ�1=2

R dCVF

0 ðq0d 0Þ�1e�½lnðd0=dv50 Þ�

2

2q02 dd 0 ðdCVFpdv50Þ

1� e�0:693ðdCVF=dv50Þq

ðdv50odCVFÞ

8<: ,

(21)

where CVF is the cumulative volume fraction of drops ofdiameter less than dCVF, dv50 is the median volumediameter, q is a correlation coefficient, and q0 ¼ 2((2p)1/2(ln2)q)�1 ¼ 1.15/q. The predicted cumulative volumefraction data are curve fit with the Rosin-Rammler/log-normal expression to determine if the predicted spraybehaves like a typical spray and to determine the curve-fitting coefficients q for the distribution. The Rosin-Rammler/log-normal distribution having dv50 ¼ 0.92 andq ¼ 3.3 compares well with the predicted spray. Thisagreement shows that the atomization model is capable ofpredicting drop size distributions, which are at leastqualitatively consistent with those expected from realsprinkler sprays. However, it should be noted that thevalue for q is higher than typical values for sprinklers(q�2.5) indicating a narrow predicted distribution. Betteragreement may be possible by adjusting the I parameters;however, tuning of the model is reserved until moredetailed data is available.

3.3. Comparison with experiments

A more quantitative evaluation of the model wasobtained by comparing model predictions with actualsprinkler data. As mentioned previously, no attempt hasbeen made to adjust the model for the effect of tines;however, comparisons between the modeled ‘ideal’ config-uration and actual sprinklers should shed some light on theability of the model to capture the essential physics forsprinkler atomization.Assuming negligible secondary atomization, it is reason-

able to compare initial spray predictions with overall spraymeasurements. This is a reasonable assumption becausesecondary atomization does not occur for droplet Wep12[16]. And according to our calculations droplet We

exceeding 12 rarely occurs in sprinkler sprays even forlarge sprinklers operated at high pressures. Althoughnearfield data is not available, the invariance of Yu’smeasurements further downstream support this assump-tion [3]. Yu demonstrated that secondary atomization and

ARTICLE IN PRESS

Fig. 10. Comparison between stochastic model predictions with sprinkler

data using If ¼ 0.2, Ish ¼ 0.3, Ilig ¼ 0.3, Dorif ¼ 12.7mm, Ddef ¼ 31mm, n

Kroesser’s data 1969, J Dundas’ data 1973, ’ stochastic model

predictions, and —— dv50/Dorif ¼ 3.10We�1/3.

D. Wu et al. / Fire Safety Journal 42 (2007) 283–294 293

coalescence is negligible in the farfield of his large sprinklerspray between 3 and 6m. Fig. 10 shows the modelingpredictions compared with data provided by Dundas [4].Dundas showed that data from many sprinklers could becorrelated by

dv50

Dorif¼ C We�1=3, (22)

where 50% of the spray volume is contained in dropssmaller than dv50 (volume median diameter) and the We isbased on the orifice diameter. In Dundas’ research, thecoefficient for 1/2 SSU Duraspeed, Reliable, and Grimesupright sprinklers was determined to be C ¼ 3.1. Modeldrop size predictions for sprinklers with similar geometry(Dorif ¼ 12.7mm and DdefE31mm or rdE15.5mm) werecompared to the experimental data and the correspondingcorrelation at various injection pressures. These modelpredictions agree very well with the experimental data andthe correlation curve. The good agreement between dropsizes predicted by the model and produced by severalactual sprinklers over a range of injection pressuressupports the validity of our modeling assumptions.Although, it should be noted that the model predictionused stochastic modeling parameters of If ¼ 0.2, Ish ¼ 0.3,Ilig ¼ 0.3 for all We. These fluctuation intensities were setsomewhat arbitrarily; even so, they appear to have rationalvalues. These modeling parameters primarily affect thedrop size distribution, but also to a lesser extent modify thecharacteristic drop size. Better guidance for these valueswill be obtained as more detailed data in the breakupregion of the spray is obtained. More actual sprinklermeasurements should also reveal to what extent unmodeled

geometric details are contained within these parameters.The authors are currently conducting spray measurementsin this region to build models for determining thesefluctuation intensities.

4. Conclusions

An atomization modeling basis for sprinklers has beendeveloped using free surface boundary layer and wavedispersion theories. The sprinkler is modeled as anaxisymmetric impinging jet. The effect of the frame armsand tines are not currently incorporated into the model,but will be included in future refinements. The atomizationmodel provides initial drop conditions that can be used totrack an array of characteristic initial drop sizes thatrepresent the entire spray. The atomization model predictsan initial spray having a realistic drop size distribution,which closely matches the Rosin-Rammler/log-normalexpression. Despite the simplified configuration used todevelop the model, median volume diameters, dv50, calcu-lated from predicted distributions show excellent agree-ment with actual drop size measurements from sprinklers.

Acknowledgments

This work is supported by the National Fire SprinklerAssociation (NFSA) and DuPont de Nemours (France).The authors would like to thank the program managersMr. Russell Fleming of NFSA and Dr. Martial Pabon ofDuPont for their support. Special appreciation is given toMessrs. Jonathan Perricone, Andrew Blum, and Ning Renfor their assistance conducting experiments and analysis insupport of the model development.

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