Analysis of a semi-empirical sprinkler spray model

Fire Safety Journal 64 (2014) 1–11

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Analysis of a semi-empirical sprinkler spray model Hamed Aghajani, Siaka Dembele n, Jennifer X. Wen School of Mechanical and Automotive Engineering, Kingston University, London SW15 3DW, UK

art ic l e i nf o

a b s t r a c t

Article history: Received 13 December 2012 Received in revised form 4 October 2013 Accepted 21 January 2014 Available online 7 February 2014

Modelling the atomization process in fire sprinklers has remained a challenge mainly due to the complexity of sprinkler geometry. A review of existing fire sprinkler spray modelling approaches, including film flow and sheet tracking models, showed that they mainly assumed a constant sheet velocity and linear attenuation of the sheet thickness before its disintegration. In the present study, a liquid sheet trajectory sub-model based on the solution of stream-wise conservation equations has been used to predict both sheet thickness and velocity as it radially expands. This will also help to investigate the extent to which a change in the release angle can affect the sheet characteristics. The analysis carried out shows that the proposed approach improves the predictions of mean droplet diameter and initial droplet speed. A semi-empirical approach is further introduced in the study by using experimental volume fraction measurements to characterize sprinkler sprays in the near field. For a given direction predictions have been conducted for droplet volume median diameter, water volume flux and droplet average velocity at different elevation and azimuthal locations. A reasonably good agreement is found for the near field measurements. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Sprinkler Spray Droplets Atomization

1. Introduction Modelling sprinkler atomization is a challenging task, due to the stochastic nature of the breakup process. This complexity is predominantly influenced by the sprinkler geometry. Predictive models are required to evaluate sprinkler spray characteristics for coupling with fire models to predict sprinkler suppression performance. Droplet dispersion models are available [1] for tracking droplets after the initial atomization process is completed, but there is no general atomization model to predict the initial spray characteristics for sprinklers and this is a critical missing link in the modelling of fire suppression with sprinklers. Various experimental studies have been conducted to characterize droplet's size and velocity in sprinkler spray using methods such as photographic techniques [2], laser-light shadow imaging method [3–4], Phase Doppler Interferometer [5] and Particle Image Velocimetry [6]. These studies have provided valuable data for the development of atomization and spray models. One of the first correlations presented from droplet size measurements was suggested by Dundas [2], in the form dV50 =D0 ¼ CWe0  1=3 . Measurements of spray undertaken by Yu [3] showed that any change in the injection orifice diameter and the deflector geometry in an upright sprinkler affects directly the coefficient of proportionality, C. The droplet size distributions of the experimentally studied

n

Corresponding author. Tel.: þ 44 208 417 4720; fax: þ 44 208 417 7992. E-mail address: [email protected] (S. Dembele).

0379-7112/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.firesaf.2014.01.004

sprinklers were expressed as an amalgamated form of log-normal and Rosin–Rammler distributions. Despite the insensitivity of these distributions and correlations to some key parameters relevant to the initial spray, they are still used as primitive predictive models. Yu [3] noticed a deviation from the Δp  1=3 scaling law for droplet size at low pressures (around 69 kPa). The experiments of Sheppard [6] presented the variation of radial velocity with polar angle at various azimuthal angles, as well as a rough approximation of the radial velocity close to the sprinkler ð  200 mmÞ, which is described by 0:6ðΔp=ρf Þ1=2 . Ren et al. [7] and Zhou and Yu [8] conducted series of experiments on simplified and commercial sprinklers, respectively. Both studies investigated the effect of sprinkler geometrical components on the spray formation process and provided more insight to some essential physics of the atomization process. Research was also undertaken by many researchers on the modelling of sprinklers sprays. The underlying simplified atomization physics resulting from a jet impinging on an axisymmetric horizontal disc (deflector plate) are thoroughly discussed in the literature [1,9–10]. At the first stage the jet transforms into a film flow upon impact, moving radially outwards on the deflector surface. The film is transformed into an unconfined sheet as it expands beyond the deflector edge. A sinuous wave grows on the decaying thickness sheet due to existing inertia, surface tension, viscous force and the pressure difference between the sheet upper and lower surfaces. At critical wave amplitudes, the sheet either breaks up into cylindrical strands (ligaments) or disintegrates directly to droplet depending on the jet Weber number [9].

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H. Aghajani et al. / Fire Safety Journal 64 (2014) 1–11

Nomenclature A C Cd Do d dV 50 f g K N N0 n Δp Q q_ Reξ Reζ r Sξ Sζ T t U V

amplitude [m] coefficient of proportionality friction coefficient of the deflector surface sprinkler orifice diameter [m] diameter [m] volume median diameter [m] dimensionless total growth of the wave gravitational acceleration [m/s2] sprinkler K-factor [m3/s kPa  1/2] number of droplets per degree number density [m  3] wave number [m  1] pressure difference at sprinkler orifice between water reservoir and atmosphere volumetric discharge [m3/s] water volume flux [m3/s/m2]

stream-wise Reynolds number: ρg D0
U g  U f
=μg
tangential direction Reynolds number: ρg D0
W g  W f j=μg radial location from sprinkler/radius [m]  stream-wise viscous
forces
in ρg =ð2δÞ 0:79ð1 þ 150δ=  1=4

rÞðReξ Þ ðU g  U f Þ U g  U f tangential-wise viscous
forces ρg
=ð2δÞ½ 0:79ð1 þ 150δ= rÞðReζ Þ  1=4 ð Wg  W f Þ
W g  W f
temperature [K] time [s] velocity [m/s] volume [m3]

As the ligaments expand outwards, aerodynamic forces cause dilatational waves to grow along the ligament. When these dilatational waves reach their critical amplitude, the ligaments break into smaller fragments which contract to form drops due to surface tension. The featured physics in atomization process have been investigated and reported in the literature. The film flow development over a flat plate is predicted by two main models. One of the approaches is an analytical model based on a free-surface similarity boundary layer concept developed by Watson [10]. This model has been a popular choice in numerous applications, hence is used in the present study. Another film model is the integral approach recently formulated by Zhou and Yu [8] where the sheet thickness can be calculated for different degrees of viscous effects. To track the trajectory of the water sheet emerging from a deflector plate, the most widely used and simplest approach assumes an inversely linear decay of sheet thickness and constant velocity of the radially expanding sheet prior to its breakup point known as the Taylor hypothesis [11]. An alternative more rigorous model and detailed sheet tracking approach consists in solving a set of stream-wise continuity equation to resolve the sheet trajectory characteristics as derived by Ibrahim and McKinney [12] and mentioned in [13]. The work presented in the present study builds on this latter model which is referred to as “Detailed Trajectory Model” (DTM) throughout this paper. The aerodynamic instability and disintegration of viscous and inviscid liquid sheets to ligaments and droplets have been studied and reported in the literature [14,15]. The overall goal of the present study is to develop and thoroughly investigate a methodology that is more accurate with further capabilities to predict the near-field initial sprinkler atomization building on and extending some existing models. To achieve this target, two liquid sheet tracking sub-models have

W We z

swirl velocity [m/s] Weber number ρU 2 D=s vertical displacement below fire sprinkler angle [m]

Greek symbols

γ θ δ ν ρ ϕ λ μ ψ s

volume fraction elevation angle (Polar angle) film/sheet thickness [m] kinematic viscosity [m2/s] density [kg/m3] azimuthal angle wave length [m] dynamic viscosity [Pa s] sheet deflection angle surface tension [N/m]

Subscripts 0 bu d Dr f g lig s

ξ ζ

jet break up deflector droplet fluid gas ligament sheet stream-wise coordinate tangential coordinate

been implemented and their accuracy is investigated to shed more light on their capability for the sprinkler initial spray prediction. The present work also further introduces and explores the concept of a semi-empirical approach which is capable of predicting the volume median diameter, average droplet velocity and volume flux of the spray at different elevation and azimuthal locations under the sprinkler with a relatively good degree of fidelity.

2. Mathematical modelling The atomization process relevant to fire sprinklers starts from the moment the liquid jet exits the orifice of the sprinkler and ends when the droplets are formed. This could be categorized as film flow formation, sheet trajectory and growth of instability on sheet's surface toward its breakup, ligament formation and finally droplet formation. Fig. 1 summarises the atomization physics and associated sub-models. Throughout the modelling process, the liquid's temperature and release pressure and the ambient air temperature are required as initial condition as well as the sprinkler's K-factor and its orifice and deflector diameter. The mathematical sub-models are briefly discussed in this section with the modifications and extensions introduced by the present authors. 2.1. Film formation In order to calculate the thickness and speed of the sheet leaving the sprinkler's deflector, tines and slots, the film formation over the deflector should be modelled. Film formation comprises of regions such as stagnation point formation, boundary layer formation and developed boundary layer. The detailed characteristics of each of

H. Aghajani et al. / Fire Safety Journal 64 (2014) 1–11

3

INITIAL CONDITIONS: Liquid properties: , Ambient properties: Sprinkler: K, r , D

BOUNDARY LAYER MODEL (BLM)

FILM FORMATION

INVERSLY LINEAR THICKNESS DECAY (ILTD) SHEET TRAJECTORY DETAILED TRAJECTORY MODEL (DTM) LIGAMENT&DROPLET FORMATION Fig. 1. Atomization physics and modelling approaches.

these stages based on the boundary layer modelling approach of Watson [10] are discussed by Wu et al. [16]. For sprinkler applications the film reaching the edge of the deflector might fall either in the boundary layer formation region, with sheet thickness given by Eq. (1), or in the developed boundary layer region, with sheet thickness obtained from Eq. (2):   7νf 1=5 4=5 r δd ¼ 0 þC 1  rd for r d r r l ð1Þ 2r d U0

δd ¼ C 2 

 1=4

νf Q

9=4

rd þ l rd

Uf r

9=4

for r d 4 r l

ð2Þ

In Eqs. (1) and (2) the constants are C 1 ¼ 1:659  10  2 ; C 2 ¼ 2:11  10  2 , and l ¼ 4:126  r 0 ðQ =νf r 0 Þ1=9 . The criterion for the radial location, measured from the jet/plate stagnation point, where the 1=3 boundary layer affects the main stream is r l ¼ 0:183D0 Re0 . The stagnation point is the centre of deflector or boss in sprinklers. The speed of the sheet leaving the deflector edge can be calculated from the mass conservation principle Q ¼ π D20 U 0 ¼ 2π r d δd U d . In sprinkler applications the volumetric discharge is commonly defined as Q ¼ K Δp1=2 .The velocity at the sprinkler orifice could be determined from the Bernoulli's equation U 0 ¼ ð2Δp=ρf Þ1=2 assuming negligible loss in the discharging process. The hydraulic radius, r0, is then r 0 ¼ D0 =2 ¼ ð4Q = π U 0 Þ1=2 =2. Eqs. (1) and (2) have been employed in the present study for the film/sheet thickness calculations during the formation process.

The sheet characteristics (local thickness and velocity) should be tracked rigorously after it leaves the deflector for a good prediction of the sheet breakup distance (distance that is takes for ligaments to disintegrate into droplets), initial droplet locations and droplet median diameter. A sheet trajectory model is needed for this purpose. A popular hypothesis for predicting the characteristics of an attenuating sheet formed in sprinkler applications is to assume (i) that the sheet velocity, Us, remains unchanged and (ii) that the sheet thickness δ at any arbitrary point is inversely proportional to its radial distance from the stagnation point, r, i.e.:

δd r d r

dU f dδ dr þ U f δ þ rδ ¼0 dξ dξ dξ

ρf U f

dU f sin ψ ¼ ρf g cos ψ þ Sξ  ρf W 2f r dξ

ð5Þ

  Δp g dψ cos ψ 2s cos ψ dψ ¼   ρf W 2f  ρf g sin ψ  r r δ δ dξ dξ

ð6Þ

dW f sin ψ ¼ Sζ þ ρf U f W f r dξ

ð7Þ

ρf U 2f ρf U f

ð4Þ

The system of Eqs. (4)–(7) consists of four equations and five unknowns, W f ; U f ; δ; ψ ; r. An additional equation could be derived from geometrical considerations of the streamline as shown in Fig. 2. dr ¼ sin ψ dξ

ð8Þ

To track the sheet trajectory, its horizontal coordinate z is evaluated (Fig. 2) as

2.2. Sheet trajectory model

δ¼

A more rigorous and detailed sheet trajectory model was presented by Ibrahim and McKinney [12]. This model, formulated in a curvilinear coordinate system, enables calculations of the sheet thickness, speed, deflection angle, and vertical displacement while it is travelling from the deflector edge (Fig. 2). The system of ordinary differential equations for the sheet velocities (Uf and Wf), thickness (δ), deflection angle, ψ, can be expressed in the following simplified forms:

ð3Þ

This hypothesis is referred to as “Inversely Linear Thickness Decay (ILTD)” model in Fig. 1.

dz ¼ cos ψ dξ

ð9Þ

The system of non-linear Eqs. (4)–(9) is solved
with the



boundary conditions u ¼ u , w ¼ wf d , δ
ξ ¼ 0 ¼ δd , f f d f ξ¼0 ξ¼0

ψ ξ ¼ 0 ¼ ψ 0 ¼ 90, rjξ ¼ 0 ¼ r d and zjξ ¼ 0 ¼ 0 using a fourth-order Runge–Kutta method to yield the solutions. This rigorous sheet trajectory model based on the equations from [12] has been implemented in a computer code and is referred to as ‘Detailed Trajectory Model (DTM)’ in Fig. 1. In the context of sprinkler spray applications, this theoretical formulation has been little explored (although it is mentioned in [13]). Beyond that purpose, the DTM will also serve in the present study as a sub-model in the semi-empirical sprinkler approach proposed by the authors. A comparative analysis is also carried out between the DTM and ILTD models.

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H. Aghajani et al. / Fire Safety Journal 64 (2014) 1–11

Fig. 2. Schematic diagram of liquid sheet emerging sprinkler deflector [13].

In the sheet breakup model, the sheet is assumed to breakup into cylindrical strands having radial width of one-half wavelength as assumed by Wu et al. [16]. To get the ligament diameter, the volume of ligaments is calculated by two methods. In the first method the volume is calculated from the multiplication of the ligament's cross   section by length of cylindrical strand V 1 ¼ π r 2lig 2π ðr bu þ r lig Þ , and in the second method the volume is determined by the product of torroid area by sheet thickness ati a h point at which fragmentation happens, V 2 ¼ π ðr 1 þ π =nÞ2  r 21 δ. Equating these two volumes (V1 and V2) in equality and rearranging, the ligament diameter could be obtained from:

  π 3 π π 2 2 2 dlig þ r bu dlig  r bu þ  r bu ¼ 0 ð14Þ 4δ 2δ n 2.4. Droplet formation

2.3. Sheet break-up and ligament formation Due to the existing pressure force between the upper and lower surfaces, surface tension force, inertial force and viscous force, a sinuous wave starts to grow on the attenuating sheet. According to Dombrowski [14], the growth of aerodynamic waves on a liquid sheet could be represented as

ρf δ

 2   ∂f ∂f  2ρg nU 2 þ 2sn2 ¼ 0 þ μf δn2 ∂t ∂t

ð10Þ

Eq. (10) is the balance of the four aforementioned forces after many simplifications, and the dimensionless total growth of the wave, f ¼ lnðA=A0 Þ, is the natural logarithm of the ratio of wave amplitude to amplitude of initial disturbances. It is found that the breakup of the liquid sheets due to the wave growth occurs as the total growth of the wave, f, approaches a constant value of 12 [14]. The solution to Eq. (10) is h i1=2 Z t f ¼ 2=ρf ðρg U s 2 n  sn2 Þ δ  1=2 dt ð11Þ 0

Differentiating Eq. (11) with respect to n and equating to zero, ∂f =∂n ¼ 0, gives the wave with maximum growth. Thus, ninviscid ¼ ρg U s 2 =ð2sÞ and the corresponding values of total growth f, Z t ρg U s 2 =ð2δρl sÞ1=2 dt ð12Þ f¼ 0

For a parallel-sided sheet, Eq. (12) reduces to f ¼ ρg U s 2 t= ð2δρf sÞ1=2 . It is known that the attenuating sheet will disintegrate after some time, which may be called as breakup time, at a location denoted as sheet breakup radius. As the sheet thickness varies with radial location in time and its breakup time value is not known, Eq. (12) is rearranged into t ¼ f ð2ρf sδÞ1=2 = ðρg U s 2 Þ. If the ILTD model, Eq. (3), is employed, and taking into account that the sheet disintegration distance from stagnation point is, r bu ¼ r d þ U s t bu , then the sheet breakup time is found from: qffiffiffiffiffiffiffiffiffiffiffi ð13Þ U s t 3bu þ r d t 2bu  δd r d ðf 2ρf s=ðρU 2s ÞÞ2 ¼ 0 If the detailed sheet trajectory model (DTM) is employed then the sheet disintegration time is calculated using Eq. (12) in the Rt  1=2 dt, by taking into following form f ¼ ρg =ð2ρf sÞ1=2 0 U s 2 δ account the fact that both sheet speed and sheet thickness vary in time. In the present study, the total growth of the wave has been evaluated numerically employing recursive adaptive Simpson quadrature method. The breakup time is the one for which the dimensionless total growth of the wave, f, matches its desired value, here 12.

It has been observed by Dombrowski and Johns [14] that ligaments produced from a liquid sheet will break up through dilatational waves and the critical wave number at which the ligament breakup occurs is obtained from: h i  1=2 1=2 ð15Þ ncr lig dlig ¼ 1=2 þ 3μ=2ðρf sdlig Þ In an inviscid flow, the pffiffiffi critical ligament wave length for break cr up simplifies to λlig ¼ π 2dlig . If it is assumed that the waves grow until they have amplitude equal to the radius of the ligament, then one droplet will be produced per wave length. Thus by mass balance, the relation between droplet size and wave number is, cr 4π =3ðdDr =2Þ3 ¼ π ðdlig =2Þ2 λlig . Combining this with Eq. (15), gives: h i1=6 ð16Þ dDr ¼ ð3π =21=2 Þ1=3 dlig 1 þ 3μ=ðρf sdlig Þ1=2 To calculate the initial droplet location, the same approach of Wu et al. [16] is employed in the present study.

3. Semi-empirical modelling It should be noted that the main output of the modelling approaches summarized in Fig. 1 and described in Section 2 would be a single global characteristic parameter (droplet size, velocity and initial droplet location) for the whole spray for a given set of working conditions. These deterministic approaches hardly mimic the real sprinkler spray pattern. The actual number of droplets generated by a sprinkler is estimated to be 105–108 particles per second. Droplets are generated below and around the sprinkler at different azimuthal locations, ϕi, and elevation (polar) angles, θj, Fig. 2, with uneven characteristic sizes, i.e. different volume median diameters. The difference between the minimum and maximum droplet diameters may reach even up to two orders of magnitude at some operating pressures and ambient temperatures. The main reason to the divergence in characteristic sizes is the sprinkler's complex configuration. The configuration allows multiple mechanisms- combination of jet and sheet and different modes of instability-combination of symmetric and asymmetric waves to play a role in the atomization process. There has been an effort to overcome the shortcomings in deterministic modelling by adding random behaviour to certain parameters in droplet formation model with the use of the stochastic analysis [16–18] to obtain a distribution for some spray characteristics. The stochastic approach can predict a range of global values for the initial droplet diameters, velocities and droplet locations. These stochastic predictions provide a global representation of the spray characteristics as a whole but not the specific detailed local spatial characteristics of the spray. Such detailed local characteristics are important for a better understanding of the atomisation process

H. Aghajani et al. / Fire Safety Journal 64 (2014) 1–11

and also for fire suppression for instance, where the local water volume flux is a key information. This shortcoming is addressed in the present study, where a semi-empirical atomization model is developed by building on a combination of models presented in Section 2 and an empirical input (experimentally evaluated volume fraction, γ). The spray characteristics (diameter, velocity and water fluxes) at various spatial locations could be predicted with the semi-empirical approach. Another rational behind this semi-empirical approach is that there is no existing sprinkler spray model to date (to the best of the authors' knowledge) that includes in its formulation the effects of some important geometrical features of actual sprinklers such as frame arm, boss and tines. The experimentally measured volume fraction, which is the only information needed as input to the semi-empirical model, could be easily obtained and provided by most manufacturers and would take into account the effects of these complex geometrical features. The measurements of volume flux on 16 different types of sprinklers [6] showed that the spray pattern along elevation angles θj is distributed unevenly with at least one high density region. In this study we assumed there is a high probability for formation of large volume median diameters in areas of higher spray densities. This assumption is based on a comparative analysis of the droplet sizes and respective volume fluxes for ranges of sprinklers and working conditions. The sprinklers investigated include one pendant sprinkler and one upright sprinkler in Zhou et al. [4] and the 16 sprinklers experimentally studied by Sheppard [6], consisting of 9 pendant sprinkler and 7 upright sprinklers. Non-uniformity of the spray can be translated into fractional distributions of the flow transported towards the high and low density areas. In the current approach, these fractions are being used to scale up the sheet thicknesses, which imply that the sheet is converted to multiple sheets. To estimate the near field droplet diameter at different local spatial locations of interest, our adopted methodology employs the deterministic approach to calculate the spray characteristics in the direction of favourable spatial location (ϕi,θj). It is assumed that the initial sheet thickness Eqs. (1) or (2) is modified to account for the fractional distribution as

δijd ¼ γ ij  δd

ð17Þ

where γ ij is the volume fraction of the flow from azimuthal position i transported towards elevation position j, as illustrated on the hemisphere of Fig. 2. This technique enables the predictions of spray droplet size, velocities and initial droplet formation radius at any spatial location (combinations of azimuthal and elevations positions). In the authors' opinion the number of sheets (in the current semi-empirical calculation) should be determined through the number of low density and high density regions identifiable in a typical spray. This depicts non uniformity of the spray. From the analysed sprinklers these regions are found to be one of 5–10. In the context of fire suppression, water flux is one of the most important characteristics of the sprinkler spray. The water volume flux defines how much water is transported to each location below the sprinkler, and it varies with both azimuthal location and elevation angle. In spherical coordinates, the differential volumetric flow rate, dq_ , passing through area of, dA can be quantified as π 3 dq_ ðθ; ϕÞ ¼ N 0 d U r ðθ; ϕÞ ð18Þ 6 where N 0 ¼ N=dV is the number density of droplets in a probe volume, dV, and N is the number of droplets in the probe volume. The probe volume in an arbitrary radial location, surrounded between semi-cone with differential cross section, dA ¼ ðr sin θ dϕÞðrdθÞ and the length sweeps the region between initial

5

droplet formation radius, r Dr , and the data collection location, r Meas , with respect to sprinkler position. Therefore the probe volume is calculated as dV ¼ 13 dA ðr Meas  r Dr Þ

ð19Þ

The droplets will slow down due to continuous drag force exerted by surroundings as they approach data collection point until they level at a terminal velocity. In this study, the following Eqs. (20)–(21) are employed to track the droplets and find their speed when passing through any arbitrary measuring surface, where Re o 1 and Re Z 1, respectively [6]: " 18ν ρg  #  f2 t 1 1 2 1 2 ρf d Dr U d ðtÞ ¼ gd ρ þe for Re o 1  gdDr ρf þ 9V 0 νf ρg 9νf ρg 2 Dr f 2 ð20Þ 2

3

3ρg 612νf ∂U d 6 7 qffiffiffiffiffiffiffiffi þ 0:45U 2d ¼ g þ 4 ∂t 8r ρf rU d 1 þ 2rU d

for Re 4 1

ð21Þ

νf

Eq. (21) is solved numerically in the present study. N in Eq. (18) has been estimated through the time it takes for droplets to travel from rDr to rMeas. Knowing this time, the number of sequences that ligaments will break up to droplets has been evaluated. Considering that in each sequence N 1 ¼ ðmlig =mDr Þ=360 droplets will enter the probe volume. The following assumptions have also been made dϕ ¼ 1 3 and dθ ¼ 1 3 .

4. Results and discussion In Section 4.1 the DTM, which has been scarcely applied to fire sprinklers, is investigated for a better understanding of the behaviour of the sheet developed beyond the deflector edge. In Section 4.2, a comparative analysis between the BLM-LTD-Dombrowski approach (Method-1) and the BLM-DTM-Dombrowski approach (Method-2) in predicting the initial droplet size and initial droplet location is conducted. The preliminary verifications study of the proposed semi-empirical approach is undertaken in Section 4.3. For the calculations, the liquid properties are taken as s ¼0.0728 N/m, mf ¼0.798 mPa s, νf ¼ 0.801 mm2/s, the ambient air properties at 300 K are, mg ¼ 19.83 mPa s, νg ¼15.68 mm2/s, and the sprinkler configuration are assumed as K ¼7.7  10  5 m3/s kPa  1/2 and rd ¼12.5 mm. 4.1. Sheet trajectory characteristics In this section the characteristics of the sheet based on detailed sheet trajectory model the DTM, Eqs. (4)–(9), are investigated at different operational pressures and various sheet release angles for the first time, mainly from a mathematical point of view. In order to verify the correct implementation of the model, the same conditions as in [12] are used. The predictions of axial variation of the dimensionless thickness for a non-swirling liquid sheet at four liquid mass flow rates (13.09, 19.13, 25.5 and 31.88 g/s) are shown in Fig. 3. There is a full agreement with the data of Ibrahim and McKinney [12] which is not shown in the graph due to overlapping with the current predictions. Upon verification of the coding in Fig. 3, the velocity and thickness of a radially expanding sheet formed by impingement of jet to a flat deflector are calculated with the DTM and presented in Fig. 4a and b respectively for different pressures. The release pressures of the jet are 69, 207 and 483 kPa and the associated sheet velocities and thicknesses have been non-dimensionalized against their respective values at the edge of the deflector. The release angle of the sheet is assumed to be parallel to the deflector

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H. Aghajani et al. / Fire Safety Journal 64 (2014) 1–11

1 ILTD

0.9

θ=90°

Us / Ud

0.8

θ=60°

0.7

θ=45°

0.6 Expected sprinkler breakup region

0.5 0.4 0.3 0

5

10

15

20

25

30

35

2r/Dd

1

ILTD θ=90°

Fig. 3. Axial variation of dimensionless thickness for a non-swirling liquid sheet.

θ=60°

ILTD

0.9

us/ud

s/ d

1

θ=45° 0.1

p=69 kpa

0.8

p=207 kpa

0.7

p=483 kpa

Expected sprinkler breakup region 0.01

0.6

0

Expected Sprinkler breakup region

0.5

0.3 5

10

10

15

20

25

30

35

2r/Dd

0.4

0

5

15

20

25

30

35

Fig. 5. (a) Radial variation of dimensionless velocity for a liquid sheet leaving the deflector, at three release angles, at P¼ 69 kPa and T ¼ 300 K. (b) Radial variation of dimensionless thickness, for a liquid sheet leaving the deflector, at three release angles, at P ¼69 kPa and T ¼ 300 K and comparison against LTD.

2r/Dd 1

2r/Dd

ILTD p=69 kpa

s/ d

p=207 kpa p=483 kpa 0.1 Expected sprinkler breakup region 0.01 0

5

10

15

20

25

30

35

2r/Dd Fig. 4. (a) Radial variation of dimensionless velocity for a liquid sheet leaving a flat disk parallel to its surface, i.e. release at θ ¼901. (b) Radial variation of dimensionless thickness for a liquid sheet leaving a flat disk parallel to its surface, i.e. release at θ¼ 901 and comparison against LTD.

(θ ¼901) and can be representative of an ideal sheet developed over the tine surface of a sprinkler. For the range of low to high operational pressure, Fig. 4a and b shows that the velocity and thickness always decreases as the sheet moves away from the sprinkler at all pressures in the direction orthogonal to the jet. A smooth and gradual decline in rate and magnitude of velocity is seen in Fig. 4a. Besides that, with an increase in the sprinkler injection pressure, water sheets with greater momentum will be produced and the sheet radius will be amplified with less sheet deflection. The velocity based on the DTM also show the clear limitations of the constant velocity (ILTD) made in some studies

over the sheet breakup length. In Fig. 4b the DTM based predictions of non-linear variation for dimensionless sheet thicknesses have been compared against ILTD predictions. The evaluated trend with ILTD remains the same for the range of studied jet release pressures. The DTM shows sensitivity to the increase in sprinkler pressure after 2r/Dd ¼ 4. In practice (sprinkler applications) the liquid sheet disintegrates into ligament or droplets at shorter radial distances (2r/Dd ¼10–15) than what studied, as shown in Figs. 4 and 5. Existence of the sheet beyond this limit is physically meaningless, due to the pronounced growth of disturbance over the sheet surface. In general the DTM will predict larger sheet thickness at the time of sheet disintegration. In the present model the surrounding gas is assumed to be quiescent. The droplet velocity (terminal velocity) in sprinkler applications can be between 70% and 80% of the jet velocity at far fields, which also implies a sheet breakup prior the limit selected, 2r=Dd ¼ 15. However, extra care must be taken while comparing droplet velocity (at far distance) with the sheet velocity, as droplets may both speed up or decelerate while they are moving downward, until they reach the terminal velocity. The predictions in Fig. 5a and b investigate the effects of a change in the sheet elevation angle on the liquid sheet behaviour using the DTM approach. Studying this change is quite important for real sprinklers as part of the flow affected by the yoke arms, boss, slots and/or tilted tines does not necessarily leave the sprinkler at an angle orthogonal to the initial jet. As shown in Fig. 5a with a decrease in the sheet release angle, the velocity of the sheet will decline faster due to the dominance of capillary force effect, although in constant time duration, sheets with higher elevation angles see greater decrease in their velocities, due to subsided effect of gravity on them. The DTM based calculated rate of change of non-dimensional thickness, δs/δd, Fig. 5b, shows a

H. Aghajani et al. / Fire Safety Journal 64 (2014) 1–11

non-linear decay in radially expanding sheet thickness compared to ILTD. The change in sheet release angle has no effect on estimations made by ILTD as being formulated in Eq. (3). The DTM approach shows that the difference between the predicted sheet thicknesses at different angles is more significant as the elevation angle decreases. The differences between DTM and ILTD predictions are gradually more important as the distance from deflector edge increases, leading to a larger sheet thickness at the time of sheet disintegration, as in Fig. 4b. Sheet modelling based on the DTM, presented in this section is helpful to analyse the ability of this sub-model in sprinkler spray atomization modelling. The effects of pressure and release angle on sheet characteristics have been studied. The results show that the release angle plays a significant role in sprinkler spray atomization and this aspect is well captured by the detailed sheet trajectory model. Also the velocity profiles obtained with the DTM clearly show the non-linearity of the velocity and the limitations of constant velocity assumption used in some studies.

4.2. Influence of the sheet trajectory model on spray modeling The two approaches Method-1 and Method-2, defined above, differ with each other in the sheet trajectory sub-model and the rest of sub-physics are dealt the same way theoretically. It is of interest to better assess the accuracy of Method-2 (which is theoretically more advanced and detailed than Method-1) for sprinkler spray predictions as this has not been reported in the literature to the authors' knowledge. This step is important before the implementation of Method-2 in the semi-empirical approach discussed in Section 4.3. Method-1 is well established for its simplicity and was employed in some studies [1,16], there are more results in the literature based on this method which is therefore employed for comparison purposes with the predictions of Method-2. Our aim is not to claim a superiority of one method over the other (they each have their own advantages/limitations) but to picture the prediction trends of Method-2 which we are interested in. Fig. 6 compares the deterministic predictions of the whole spray initial droplet size and initial droplet location as a function of injection pressure and ambient temperature for a sprinkler spray. The spray characteristics is studied over a range of five injection pressures, 69, 138, 207, 345, 483 kPa and six ambient air temperatures, 300, 400, 500, 700, 900, 1100 K similar to the results of Wu et al. [16]. 1.6 1.4

1100K 900K

700K

Droplet Diameter (mm)

69kPa

500K

1.2

400K 300K

1 138kPa

0.8

207kPa 345kPa

0.6

483kPa

0.4 0.2 0 200

250

300

350

400

450

500

550

600

Initial Droplet Radial Location (mm)

Fig. 6. Predicted initial droplet conditions of sprinkler spray as a function of injection pressure and ambient temperature using Method-1 (solid line) and Method-2 (dashed line).

7

As can be seen in Fig. 6, in general Method-2 predicts smaller droplet diameters and initial droplet locations than Method-1 at most pressures and ambient temperatures. In general the predictions of initial droplet diameters and initial droplet locations from Method-2 are different by roughly 10% and 5%, respectively compared to Method-1. It might seem that as Method-2 is taking into account the air drag and predicts thicker sheet and lower velocity than Method-1, the drop size prediction of Method-2 should be larger than that of Method-1. This trend in Fig. 6 could be explained by taking into account the fact that droplet size is a direct function of ligament diameter and is not directly related to sheet breakup thickness and sheet breakup distance in the formulation. In addition to this, the ligament diameter has been calculated from Eq. (14) in the form of a polynomial where the sheet breakup thickness and sheet breakup distance appeared as part of the coefficients of that polynomial. Hence a clear conclusion could not be drawn, in which Method-2 necessarily led to a larger predicted droplet diameter in all situations. The discrepancies beween Methods 1 and 2 could be explained by the smaller sheet breakup distance and the smaller sheet breakup speed resulting from the implemented DTM. Without claiming the superiority of Method-2 (exact benchmark results which are not available would be needed for a rigorous comparison), the results show the reliability of its predictions and more confidence in employing it for the semi-empirical approach which is one of our main targets. It is worth noting that the preditions in Fig. 6 represent the data for the whole spray, and do not account for the detailed spatial variations. This aspect is addresed among others in the next section. 4.3. Analysis of the proposed semi-empirical approach The semi-empirical approach outlined in Section 3 employs empirically evaluated volume fraction data coupled with Method1 or Method-2 to predict the near-field spatial distributions of the droplet diameter, droplet velocity and spray volume flux. Although water fluxes are important in fire safety, it is worth noting that initial droplet diameters and velocities are also important input parameters to many CFD codes (Lagrangian particles tracking could be employed once these initial parameters are known). One of the rationale of the approach proposed here is that the diameter and velocity are very complex to measure experimentally, in particular in the near field. In contrast volume fractions are relatively easier to measure (estimate) by collecting water at the areas of interest using pans. Such measurements would also directly reflect the impact of the spray geometrical parameters such as tines, boss etc which are not incorporated in any model yet due the many variations available for such parameters. Therefore by using the model proposed, this additional complexity of measuring droplet size and velocity is removed. The water volume fraction obtained from the experimental water fluxes are employed in the model which has its own assumptions and limitations reflected in the predicted data. As such there is no reason to believe that the predicted fluxes would be exactly the same as the experimental values. The comparative water fluxes predictions/experiments are presented for this purpose (to show the model influence on predictions) rather than for a pure validation purpose. However there is a legitimate question of how the model could be used if no experimentally obtained volume fractions are available, this is discussed in Section 4.4 where a uniform distribution could be used as a first approximation. The preliminary verifications of the proposed approach are conducted using the data of Zhou et al. [4] who conducted a series of near field measurements on a pendant fire sprinkler with a K-factor of 205 Lpm/bar1/2, and quantified the spray by water volume flux, droplet size and speed. The spray characteristics were

H. Aghajani et al. / Fire Safety Journal 64 (2014) 1–11

measured at six azimuthal locations ϕ1 ¼90, 107, 123, 140, 157, 180 and for each azimuthal location, the following seven elevation angles are considered θ1 ¼3, 15, 30, 45, 60, 75, 90. In the experiments, the elevation angles, θ ¼ 01 andθ ¼ 901, point respectively to surfaces parallel and right below the sprinkler deflector. The semi-empirical approach requires the spatial distribution of the water volume fractions as input to either of methods (1 and 2). If this information is not readily available, it could be extracted from the experimental spatial volume flux distributions. The total water volume flux analogous to each azimuthal location could be approximated by summing up the partial volume fluxes over available elevation angles: Z θj ¼ 90 q_ ðϕi Þ ¼ dq_ ðθj ; ϕi Þ ð22Þ

1.8

Droplet Size (mm)

8

Method-2

1.2 1 0.8 0.6 0.4

0 0

10

The current approach has been tested on one upright sprinkler, and the corresponding predictions have been reported in [19]. The sprinkler was of type Viking TY5851, with K-factor of 161.4 lpm/ bar1/2. The results have been presented for two pressures (0.76 bar and 1.31 bar). For that sprinkler Method-2 has shown an overall superiority over Method-1. One drawback of the presented semi-empirical model results from the fact that the volume fraction implies the distribution in elevation direction, but does not illustrates the distribution in azimuthal direction. Hence, for instance, for two azimuthal locations, the spray flux may be different; however, if the distribution in elevation direction is the same, γ ij will be the same. Therefore, according to Eq. (23), the same spray characteristics will be predicted for the two azimuthal locations. However, the reality may contradict with this. To give a better picture on the value of evaluated volume fraction, they have been given in Appendix-A (Tables A1 and A2), for the pendant sprinkler working at 3.5 bar. 4.3.1. Median droplet size predictions The initial droplet diameters at various spatial locations have been calculated using both Method-1 and Method-2 in the semiempirical approach. They are compared against the available experimental data in Fig. 7a and b at two azimuthal angles (a) slot flow at ϕ ¼1071 and (b) tine flow at ϕ ¼1231. The sprinkler is activated at 3.5 bar and the deflector diameter is 47 mm. Overall very promising predictions were obtained with Method-2 in particular. Method-1 under-predicts the droplet diameter for all three cases. The average errors from Method-1 for predictions over seven elevation angles at each of azimuthal angles range from 50% to 70%. The maximum error is considerably lower when Method-2 is employed, 20%–40%. The semi-empirical model is able to extend the near-field predictions to elevation angles where experimental measurements are challenging and unavailable due to high water fluxes (901), as shown in Fig. 7a and b. Since the errors of measurements have not been quantified in the experiments, care must be taken in the interpretation of the differences provided here. On the other hand a locally evaluated volume fraction does not always reflect the measured volumetric median diameter, dv50 . One of the contradictions is for elevation angles less than 201. In this particular region, the droplets are not resulting from sheet break in directions pertinent to this region. The droplets are either originating from periodic shedding from sheet surface (mostly due to Kelvin–Helmholtz instability) or the

30

40

50

60

70

80

90

100

Zhou et al. (2010) Method-2

2

Droplet Size (mm)

ð23Þ

20

Elevation Angle

2.5

Having evaluated the corresponding water flux to each azimuthal location, the partial volume fraction at the elevation angles is estimated in the present study as _ θ j ; ϕi Þ dqð _ ϕi Þ qð

Method-1

1.4

0.2

θj ¼ 0

γ ij ¼

Zhou et al. (2010)

1.6

Method-1

1.5

1

0.5

0 0

10

20

30

40

50

60

70

80

90

100

Elevation Angle Fig. 7. (a) Comparison of droplet diameter predictions from two semi-empirical approaches and measurements of Zhou et al. [4] at azimuthal angleϕ¼ 1071. (b) Comparison of droplet diameter predictions from two semi-empirical approaches and measurements of Zhou et al. [4] at azimuthal angle ϕ¼1231.

turbulence driven (ambient air) force dispersed the droplets which are formed at lower elevation levels (θ 4 151). From the modelling point of view, a variation of constant dimensionless total growth, f¼12 in Eq. (11), can affects the droplet size and may also explain the discrepancies to some extent. Further work may be needed to fully investigate the effect of this constant as well as the sheet initial thickness at the deflector edge on predicting the characteristics of real sprinklers. Nevertheless the results show relatively good predictions for Method-2 which is due to its more rigorous sheet thickness evaluation at different sheet angles. The semi-empirical approach also provides the detailed local distributions of the spray characteristics, which is one of its strength.

4.3.2. Droplet velocity Predictions and experimental data for droplet average velocity are presented in Fig. 8a and b respectively for the sprinkler operating pressures of 3.5 bar and 5.2 bar. The predictions obtained from Method-1 and Method-2 are compared against the empirical relations of Sheppard [6] and measurements of Zhou et al. [4]. The errors in average velocity predictions fall within an acceptable range for droplet diameters less than 0.9 mm. The reported sprinkler experimental spray volume median diameter is usually between 0.9 mm to 1.2 mm which means that most of the number densities of the spray have actually a corresponding diameter less than the spray volume median diameter. There are more discrepancies model/experiments for larger droplets which could be explained by both un-quantified experimental errors and simplifications made throughout the model development. Method-1 predicts narrower range of droplet sizes (less than 0.9 mm) and Method-2 predicts broader range (up to 1.6 mm). Droplet velocity predictions with Method-2 show improvement

H. Aghajani et al. / Fire Safety Journal 64 (2014) 1–11

Sheppard (2002) Method-1

25

Method-2 20 15 10 5 0 0

0.5

1

1.5

2

Sheet Breakup Distance (mm)

Average Velocity (m/s)

260

Zhou et. Al (2010)

30

9

240

Method-1

220

Method-2 Upper Limit

200

Lower Limit

180 160 140 120

Droplet Median Diamter (mm)

100 0

Average Velocity (m/s)

35

Zhou et. Al (2010) Sheppard (2002) Method-1 Method-2

30

0.2

0.4 0.6 Pressure (bar)

0.8

1

Fig. 9. Comparison of calculated sheet breakup distance at four pressures for a deflector of 25.4 mm.

25 20

better accuracy, but this technique has not been used in the current research.

15 10 5 0 0

0.5

1

1.5

2

Droplet Median Diamter (mm) Fig. 8. (a) Average velocity predictions using Method-2 and measurements of Zhou et al. [4] at different azimuthal angles for sprinkler operating pressure of 3.5 bar. (b) Average velocity predictions using Method-2 and measurements of Zhou et al. [4] at different azimuthal angles for sprinkler operating pressure of 5.2 bar.

for higher pressure of 5.2 bar (Fig. 8b) compared with the lower pressure of 3.5 bar (Fig. 8a). The correlation presented by Sheppard [6] predicts a constant velocity for all size of droplets which is not in good agreement for droplet with diameters less than one millimetre. The observed discrepancies between predictions and experimental data for large droplets could be explained by noting that in practice the large droplets are usually generated in part of the flow field which is affected by yoke arms. In the present study for the model formulations, no viscous loss was considered for the flow discharge from sprinkler orifice and for the flow which is in contact with boss and yoke arms. The development of a friction coefficient function is required to improve the predictions of velocity at part of the flow affected by yoke arms. The other factors which could contribute to the observed discrepancies (in addition to the unspecified experimental errors) are the uncertainties in theoretical calculation of film velocity and film thickness at the edge of deflector. There is a lack of experimental measurements to verify the implemented film model accuracy. Furthermore, the magnitude of calculated velocity is dependent on the sheet breakup distance and consequently to the value of total growth of wave (f¼ 12 here). In calculating the deceleration of the droplet, the ambient air turbulence has been neglected. The first author of the present study investigated the predictions of sheet breakup in his PhD thesis [20]. Fig. 9 shows how the Methods compare in predicting the sheet breakup distance, where liquid jet impingement with a flat deflector [8] has been used for comparison. In real sprinklers shorter sheet breakup distance is expected. Hence the value of “f” could be tuned accordingly for a

4.3.3. Spray volume flux The predictions of spray volume flux for a pendant sprinkler operating at pressures of 3.5 bar and 5.2 bar with Method 2 are compared to the data of Zhou et al. [4], in Figs. 10a and b and 11. The spray volume flux has been calculated by the method explained in Section 3. The quantified spray at azimuthal angle of 1401, 1571 and 1801 are respectively relevant to slot flow, tine flow and yoke arm flow. To the authors knowledge this is the first attempt to estimate the sprinkler spray volume flux theoretically, and this has been done using the method developed in Section 3. Method-2 is chosen due to the stronger capability shown in predicting droplets diameter, Fig. 7a and b. The volume flux predictions for the range, θ o 151, is largely over-predicted. The reason discussed in 4.3.1 is that the droplets in the dense spray have been carried away into that area from lower regions θ 4151 probably no spray has formed in that region. The volume flux predictions for the middle range elevation angles, 15 r θ r75 is very promising at both pressures and all azimuthal angles. The maximum error model predictions/experiments are below 40%. The predictions are closer to experimental data as the pressure increased from 3.5 bar to 5.2 bar. There is an over-prediction for elevation angle of θ ¼901 at all azimuthal angles. The shortcoming which contributes the most to the discrepancies observed is the uncertainties in sheet breakup distance evaluation due to constant dimensionless total growth, and the subsequent initial droplet formation radius. Additionally the evaluation of film thickness at deflector edge which has a direct effect on sheet thickness and ligament diameter plays a considerable role in the initial droplet diameter estimation and hence affects the volume flux. Furthermore, there is no certainty for the calculated droplet number density entered in the probe volume and this enquires a separate research and validation. The results show the promising potential of the semi-empirical approach presented in the study. 4.4. Analysis of the proposed model – uniform distribution To avoid the dependency of the semi-empirical modelling to the experimentally evaluated volume fractions, a uniform distribution assumption has been investigated with Method-2. In this

10

H. Aghajani et al. / Fire Safety Journal 64 (2014) 1–11

Φ=140º

3

Zhou et al. (2010) Method-2

2.5

1000

Droplet Diameter (mm)

Water Volume Flux (Lpm/m2)

10000

100

10

7-Points estimate =73 ° =90 °

1.5

=107 ° =123 °

1

=140 ° =157 °

0.5

=180°

0

1 0

10000

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20

30 40 50 60 Elevation Angle (°)

70

80

0

90

1000

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40

50

60

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1000 7-Points estimate =73 ° =90 °

100

=107 ° =123 ° =140°

10

=157 °

10

=180 °

1 0

10

0

10

20

30 40 50 60 Elevation Angle (°)

70

80

90

30

40

50

60

70

80

90

Fig. 12. (a) Droplet diameter of pendant sprinkler at 3.5 bar for a uniform volume fraction. (b) Volume flux predictions of pendant sprinkler at 3.5 bar for a uniform volume fraction.

Fig. 10. (a) Volume flux predictions using Method-2 and measurements of Zhou et al. [4] at azimuthal angle ϕ¼ 1401 for sprinkler operating pressure of 3.5 bar. (b) Volume flux predictions using Method-2 and measurements of Zhou et al. [4] at azimuthal angle ϕ ¼1571 for sprinkler operating pressure of 3.5 bar. 10000

20

Elevation Angle

1

Φ=180º

Zhou et al. (2010) Method-2

Water Volume Flux (Lpm/m2)

20

10000

Φ=157º

Zhou et al. (2010) Method-2

10

Elevation Angle

Water Volume Flux (lpm/m2)

Water Volume Flux (Lpm/m2)

2

1000

100

10

angle approaches 901 due to singularity in the solution; hence 881 has been substituted for 901. Considering the fact that no experimentally obtained volume fraction data been have been used in this section, relatively good predictions have been obtained for droplet size and volume flux for the pendant sprinkler using the uniform distribution. However care should be taken not to extrapolate this conclusion unless a wide range of sprinklers are tested experimentally and compared to the semi-empirical model for uniform distribution. A tendency to the formation of smaller droplets has been obtained as the elevation angle increased from 01 ( 2.45 mm) to 881 (  1.8 mm).

5. Conclusions

1 0

10

20

30

40

50

60

70

80

90

Elevationa Angle (°) Fig. 11. Volume flux predictions using Method-2 and measurements of Zhou et al. [4] at azimuthal angle ϕ¼ 1801 for sprinkler operating pressure of 5.2 bar.

scenario the volume fraction γ is taken to be the same for any studied elevation angle. This could be the first approach adopted when no experimental data is available for the volume fractions. The equally distributed spray calculations have been presented in Fig. 12a and b and are compared against experiments at 1 7 azimuthal angles, ϕ ¼73, 90, 107, 123, 140, 157, 1801. The number of studying point along 01–901 elevation angles was taken as 7 (γ ¼1/7) and the angles associated with these points are 01, 151, 301, 451, 601, 751 and 881. Numerical instability occurs as elevation

In the present study a detailed sheet trajectory model has been implemented for sprinkler applications to investigate the sheet radial thickness and velocity at different release angles and operational pressures. The results show the influence of the release angle on sheet characteristics. Combining the detailed sheet trajectory model with other spray formation sub-models resulted in an overall smaller droplet diameter and initial droplet location, compared to available methods in the literature. The proposed model (DTM) offers a more realistic and robust modelling of the sheet trajectory. The results presented in the paper show the good potential of the DTM; however more verification studies based on experimental data are needed for a full quantitative comparison of the DTM with other relatively simpler approaches. In a move towards more realistic spray models, a semiempirical approach has been proposed using volume fractions calculated from water flux measurements as input, and the

H. Aghajani et al. / Fire Safety Journal 64 (2014) 1–11

Table A1 Spray volume flux (Lpm/m2) along the elevation angle in the near field for the K-205 Sprinkler operating at 3.5 bar at seven azimuthal angles [4]. Elevation angle

Appendix A See Tables A1 and A2.

Azimuthal angle 1401

1571

1801

31 151 301 451 601 751 901

1.7 128.6 90.7 138.7 224.5 223 2718.4

2.1 27.6 37.5 53.7 76 119.8 2866.4

12.8 475.8 55.9 45.6 66.2 70.5 2570.7

SUM

3525.6

3183.1

3297.5

Table A2 Volume fractions along the elevation angle in the near field for the K-205 Sprinkler operating at 3.5 bar at seven azimuthal angles. Volume fraction Elevation angle

11

Azimuthal angle 1401

1571

1801

31 151 301 451 601 751 901

0.00048 0.03648 0.02573 0.03934 0.06368 0.06325 0.77105

0.00066 0.008671 0.011781 0.01687 0.023876 0.037636 0.900506

0.00388 0.14429 0.01695 0.01383 0.02008 0.02138 0.77959

SUM

1

1

1

detailed sheet trajectory model. The developed semi-empirical model is capable of predicting the droplets volume median diameter, droplet average velocity and water volume flux for a given direction (at different elevation and azimuthal angles) of pendant type sprinklers with an acceptable agreement. Providing such local detailed information of the initial spray characteristics is a good potential of the proposed semi-empirical approach. The results presented show that the approach is promising and it will be further verified as more experimental data become available in the literature. Acknowledgements The authors gratefully acknowledge the financial support of FM Global and the technical guidance from Drs. Hong Zeng Yu, Yinbin Xin and Xiangyang Zhou of FM Global.

References [1] A.W. Marshall, M. di Marzo, Modeling aspects of sprinkler spray dynamics in fires, Process Saf. Environ. Prot. 82 (2004) 97–104. [2] P.H. Dundas, Technical Report Optimization of Sprinkler Fire Protection the Scaling of Sprinkler Discharge: Prediction of Drop Size. FMRC Serial no. 18792 RC73-T-40. Norwood, MA, Factory Mutual Research Corporation, June 1974. [3] H.Z. Yu, Investigation of spray patterns of selected sprinklers with the FMRC drop size measuring system, in: Proceeding of 1st Intenational Association of Fire Saftey Science, 1986, pp. 1165–1176. [4] X. Zhou, S.P. D'Aniele, H.Z. Yu, Spray pattern measurements of selected fire sprinklers, in: Proceedings of 12th International Interflam Conference, Nottingham, UK, 2010, pp. 177–188. [5] J.F. Widmann, D.T. Sheppard, R.M. Lueptow, Non-intrusive measurements in fire sprinkler sprays, Fire Technol. 37 (2001) 297–315. [6] D.T. Sheppard, Spray Characteristics of Fire Sprinklers (Ph.D. dissertation), North-Western University, Evanston, IL, 2002. [7] N. Ren, A. Blum, C. Do, A.W. Marshall, Atomization and dispersion measurements in fire sprinkler sprays, Atomization Sprays 19 (2009) 1125–1136. [8] X. Zhou, H.Z. Yu, Experimental investigation of spray formation as affected by sprinkler geometry, Fire Saf. J. 46 (2011) 140–150. [9] N. Ren, A.W. Marshall, Characterizing the initial spray from large Weber number impinging jets, Int. J. Multiphase Flows (2012), http://dx.doi.org/ 10.1016/j.ijmultiphaseflow.2012.08.004. [10] E.J. Watson, The radial spread of a liquid jet over a horizontal plane, J. Fluid Mech. 20 (1964) 481–499. [11] G.I. Taylor, The dynamics of thin sheets of fluid: I. Water bells, Proc. Roy. Soc. London Ser. A: Math. Phys. Sci. 253 (1959) 289–295. [12] E.A. Ibrahim, T.R. McKinney, Injection characteristics of non-swirling and swirling annular liquid sheets, in: Proceedings of IMechE, vol. 220, 2006, pp. 203–214. [13] N. Ren, A. Blum, Y. Zheng, C. Do, A.W. Marshall, Quantifying the initial spray from fire sprinklers, in: Fire Safety Science – Proceeding of the Ninth International Symposium, IAFSS, Karlsruhe, Germany, 2008. [14] N. Dombrowski, W.R. Johns, The aerodynamic instability and disintegration of viscous liquid sheets, Chem. Eng. Sci. 18 (1963) 203–214. [15] C.J. Clark, N. Dombrowski, Aerodynamic instability and disintegration of inviscid liquid sheets, Proc. R. Soc. London A 329 (1972) 467–478. [16] D. Wu, D. Guillemin, A.W. Marshall, A modeling basis for predicting the initial sprinkler spray, Fire Saf. J. 42 (2007) 283–294. [17] N.K. Rizk, H.C. Mongia, Model for airblast atomization, J. Propuls. 7 (1991) 305–311. [18] H. Aghajani, S. Dembele, J.X. Wen, An extension of deterministic and stochastic physics based sprinkler spray modeling, in: Proceedings of the of Seventh International Seminar on Fire and Explosion Hazards (ISFEH7), Providence, USA, 2013, pp. 501–510. [19] S. Dembele, H. Aghajani, J.X. Wen, Theoretical modeling of an upright sprinkler spray, in: Proceedings of the 13th International Conference and Exhibition on Fire Science and Engineering, Interflam, 2013, pp. 295–306. [20] H. Aghajani, Modelling the Initial Sprinkler Spray Characteristics of fire Sprinklers (Ph.D. thesis), Kingston University, London, UK, 2013.