International Journal on Engineering Performance-Base Fire Coes, Volume 4, Number 3, p.95-3, A SIMPLE ENGINEERING MOEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PROCTS V. Novozhilov School of Mechanical an Prouction Engineering Nanyang Technological niversity, 5 Nanyang Avenue, Singapore 639798 (Receive February ; Accepte September ) ABSTRACT Analytical moel is elope to preict smoe layer temperatures after activation of sprinler or water mist system. The governing equations are solve to etermine water roplet trajectory, temperature history an aporation rate. As a result, a rate of heat absorption by water spray an temperature history of the hot smoe layer can be preicte. A goo agreement is emonstrate between the analytical moel an the results of full CF simulations.. INTROCTION The problem of preicting smoe layer temperature is typically arising in many fire safety applications. A part of this general problem is a preiction of smoe layer behaviour upon activation of fire suppression systems. Typical examples of such systems are water sprinlers an water mist nozzles. Heat absorption by water spray generally leas to the rop of the overall temperature insie the compartment. Quantitative preiction of this effect is quite important for estimation of attainable conitions insie builings uring fire ents. Primary ifferences between water sprinler an water mist systems are in water ischarge rates an roplet size istribution. Water spray for conventional sprinler is relatively coarse, with significant amount of roplets larger than mm. In contrast, mist spray is efine by NFPA as the spray containing 99% of its volume in the roplets smaller than mm. Recent interest in water mist application is mostly riven by phasing out of halons, an the nee to fin a replacement which can act as volumetric suppression agent. ue to very fine ispersion, mist is capable of flooing the room, although it oes not have as goo flooing properties as gaseous agents. ue to high surface area of the roplet phase, water mist aporates very effectively, an therefore water ischarge quantities may be ept much smaller, compare to conventional sprinlers. Water mist spray oes not generally penetrate to the surface of burning material (ue to low momentum of iniviual roplets), an suppress flame in gaseous phase. In contrast, sprinler sprays are most effective for unshiele fuel surfaces, as roplets easily penetrate to burning material, cool the surface, an suppress pyrolysis process. Computational Flui ynamics (CF) moelling of water sprinler sprays is well-establishe [-3]. Hower, this technique is relatively expensive. Typically, tens of thousans of trajectories are trace to represent water spray statistically. For real fire engineering applications, this may not be necessary in all situations. Basic unerstaning of water spray behaviour can be achie using simple estimations of roplet ynamics, similar to consierations applie in the pioneering wor of Chow an Tang [4]. The moel escribe in the present paper provies a more accurate estimation of roplet interaction with the hot layer of combustion proucts. In contrast to the paper [4], heat transfer problem for the moving roplet is solve in aition to the ynamic problem. The major avancement of the moel is the analytical solution for the roplet motion, obtaine for the case of real rag coefficient correlations, applicable for spherical particles moving in a turbulent gas stream. The equations an the metho to obtain analytical solutions are iscusse first. From the solutions, the roplet trajectories an heat absorption rates for iniviual roplets can be preicte. Such preictions are illustrate for various roplet sizes an ifferent temperatures of the hot layer. Finally, the moel is utilize to mae preictions of temperature histories in compartments after activation of water sprinlers. The results from the present analytical moel are compare to the CF simulations of the similar problem. 95
International Journal on Engineering Performance-Base Fire Coes. RESLTS AN ISCSSION. Sprinler Spray ynamics within the Smoe Layer Consier an interaction of a sprinler spray with the hot layer of certain thicness an temperature T (Fig. ). It is assume that absorption of heat by the spray within the layer is uniform. This assumption is reasonably accurate if the with of the spray is comparable with the length scale of the fire ceiling jet, as has been emonstrate in [7]. The roplets injecte into the hot smoe layer unergo two major stages: () heat-up stage an () aporation stage. The overall heat absorption by a single roplet epens on its resience time within the layer. If the resience time is large, then the roplet may be completely aporate. roplet ynamics an heat transfer equations must be solve simultaneously to obtain roplet trajectory an temperature history. In orer to elop a simplifie analytical solution, let us consier the stages () an () separately... Heat-up stage roplet ynamics: uring the heat-up stage, the roplet is consiere as having constant mass, i.e. there is no aporation. This assumption can be shown to be quite accurate [5]. The general equation of the roplet motion through a gas layer (neglecting forces other than rag an gravity) may be written as the following set of two ifferential equations for the velocity components [6]: m t V m t π ρc 8 π ρc 8 + V + V V + mg () For a typical conventional sprinler the Volumetric Mean iameter is of the orer of to mm. Taing the characteristic roplet size as mm, Reynols particle number for the roplets exiting orifice at velocities of to ms - may be estimate to be Re ~ 7 to 5. In this range the rag coefficient for spherical particles is a wea function of the particle Reynols number [6] an varies between.47 an.6. Therefore, it may be represente by a constant average value with a reasonable accuracy. It is obvious that en with the constant rag coefficient, the equations () are still couple. Hower, since roplets hit the sprinler eflector, their vertical velocity is usually small compare to the tangential velocity. Since the rag coefficient is assume to be constant, then in the case V <<, equations () become ecouple an may be solve separately. The system is simplifie as follows: m t V m t π ρc 8 π ρc 8 V + mg () Sprinler hea roplet trajectories Smoe layer Opening Fig. : Schematic view of smoe layer an roplet trajectories in a compartment 96
International Journal on Engineering Performance-Base Fire Coes The first equation contains only tangential component, an can be easily solve by separating variables. The solution has the following form: π = C t + ρ 8m (3) Substitution of this expression to the secon (V component) equation yiels: V π π m ρc C t V + mg t 8 + ρ 8m (4) To fin solution of (4) one nees to solve homogeneous equation first (which is one by separating variables). Then a metho of variation of parameters may be applie to fin solution of the non-homogeneous equation. The final solution of equation (4) can be written as: g κ V κ g V = + ( t + κ) (5) t + κ where 4 ρ p κ = 3C ρ (6) If the roplet leaves the smoe layer uring its heat-up stage, then the following equation may be easily erive to estimate the roplet exit time t * from the layer: * ( t + κ) * g t g * κ V κ ln + + t = (7) κ 4 Heat absorption: The heat transfer equation for the roplet heat-up stage may be written as: 4 3 T Nu π ( / ) ρpcp = π (Tsm T ) (8) 3 t The correlation for the Nusselt number in the case of spherical roplets moving through the air is taen in the form [6]: / / 3 Nu =.+.6 Re Pr (9) It is seen from this expression that the Nusselt number is a rather wea function of the particle Reynols number (Nu ~ Re / ) an may be represente with sufficient accuracy by its constant average value. roplet temperature at any moment uring the heat-up stage is obtaine by straightforwar integration of equation (8): 6Nu ρ C () t T = exp t (T T ) T sm p p () Expression () allows an estimation of the egree of a single roplet heat-up to be mae. The total heat absorption by the spray can be calculate provie the roplet size istribution in the spray is nown. Taing into account the temperature change along the roplet trajectory (), the heat loss rate to the spray from the layer, which has the temperature T sm is given by: 6Nu Q& * spr = ψ& c f (s) p exp t pc p ρ sm () () s s ( T T ) where ψ& is the water ischarge rate, f(s) is the probability ensity function for the roplet size istribution. The time t * (s) correspons to the en of the heat-up stage for a roplet of a given size. At this time, either roplet aporation begins, or the roplet exits from the smoe layer... Evaporation stage roplet ynamics: Generally, water roplets aporate as they move through the hot gas. Once the roplet is injecte into the hot layer, it will experience a perio of unsteay aporation, followe by a steay-state aporation regime, characterise by approximately constant surface roplet temperature ( wet bulb temperature ) an constant surface regression rate [5]. At high environment temperatures, wet bulb temperature approaches boiling temperature, so it may be reasonably assume that the steay-state aporation occurs at the surface boiling temperature. Equations of motion () apply for both heat-up an aporation stages. They can be re-written in the equivalent form: 3ρC t 4ρ p V 3ρC V + g t 4ρ p () 97
International Journal on Engineering Performance-Base Fire Coes uring the heat-up phase, the roplet iameter oes not change, an the solution is the same as for the set of equations (), that is given by the formulae (3),(5) an (6). uring the steay-state aporation, hower, the equations of motion () must be solve with the roplet iameter varying as a function of time. This function is taen as the theoretically an experimentally observe - law, which implies (t) = (t) t (3) The parameter is the so-calle aporation constant, an may be taen in the form [5]: ( T T ) 8λ g C g sm w = + ln (4) ρ pcg Hfg With the iameter regression rate (4), the equations () can still be integrate analytically. The solution can be written in the following form: 3ρC = ρp V = t + g α t 3 α t t + K where the parameters K an α are given by: K = α g + 3 ρ p α = + 3ρC ( α ) V (5) (6) ( an V are the roplet velocity components at the en of the heat-up stage, an is the initial roplet iameter). Heat absorption: For calculation of the total heat loss to the spray we note that the heat loss ue to aporation of a single roplet is: Q p ( T T ) + H fg m w = m C (7) where the roplet mass loss at a given time is calculate as: ( ) t 3/ 3 m = πρp (8) 6 The total heat loss rate to the spray may then be obtaine as: Q& spr = ψ& C p ( ) ** T T + H t (s) f (s) s w..3 Test calculations fg s 3 / (9) For the purpose of calculations, the roplet size istribution function of the spray is iscretize, an the roplets, therefore, are ivie between finite number of groups. Each group contains roplets of the same iameter. Calculation proceure is esigne in such a way that the equations for both heat-up an aporation stages are solve consecutively along roplet trajectory for one representative roplet from each of the groups. In general, each roplet unergoes both (heat-up an aporation) stages, an the total heat absorption by the roplet epens on its resience time in the smoe layer. The resience time for any particular roplet is etermine using the following equation: * t V ** t () s s V () s s = + () where V (s) an V (s) are taen from the solutions (5) an (5), respectively, is the smoe layer thicness. The resience time t res is equal to: t res = t * + t ** () where t * an t ** are the times for the heat-up an aporation stages, respectively. Equation () oes not have solution if roplet aporates completely in the layer. In that case, the resience time is taen as t * + t where t is the time for complete aporation of the roplet: t = () The total heat absorption rate is obtaine by multiplying the heat absorption rate for a single roplet by a number of roplets in the respective group, an summing up over all groups in the iscretize spray istribution. 98
International Journal on Engineering Performance-Base Fire Coes The typical calculate roplet trajectories are presente in Figs. to 4. The ifference in the roplet behavior can be observe for the ifferent temperatures of the smoe layer, ranging from 3 to 8 K. The trajectories are presente for ifferent representative roplet iameters an ifferent initial velocities. The smoe layer thicness is taen as m in Figs. to 4. Generally, the roplets below 5 µm are affecte by the increase temperature of the smoe. Compare to the ambient conitions (3 K, Fig. 4), fine roplets of to µm in iameter aporate quite quicly in the smoe layer with the temperatures above 5 K. This may be seen from Figs. an 3. A more important parameter for fire engineering applications is the rate of heat absorption by the whole spray. In orer to valiate the performance of the analytical moel, the results are compare to the CF calculations, which were performe solving exact equations for the two-phase (ropletgas) flow [7]. The results are presente for the AM4 spray nozzle. The roplet size istribution for such a nozzle [8] is shown in Fig. 5. In this case, the meium roplet iameter in the spray is of the orer of 3 µm. Heat absorption rate is plotte in Fig. 6 as a function of the hot layer temperature. It appears that the agreement between the CF an the simplifie moel is quite goo. Note that the heat absorption rate approaches the plateau at high smoe layer temperatures. This plateau correspons to the complete aporation of the spray. Also shown in Fig. 6 is a convective part of the total absorption rate. As expecte for water mist spray, major contribution to heat absorption is from vaporization process. X / m Y / m roplet iameters an initial conitions: : = 3 µm; = ms - ; V = ms - : = 5 µm; = ms - ; V = ms - : = µm; = ms - ; V = ms - : = µm; = ms - ; V = 4 ms - Fig. : Selecte roplet trajectories in a 8 K smoe layer (layer epth is m) Y / m X / m roplet iameters an initial conitions: : = 3 µm; = ms - ; V = ms - : = 5 µm; = ms - ; V = ms - : = µm; = ms - ; V = ms - : = µm; = ms - ; V = 4 ms - Fig. 3: Selecte roplet trajectories in a 5 K smoe layer (layer epth is m) 99
International Journal on Engineering Performance-Base Fire Coes X / m Y / m roplet iameters an initial conitions: : = 3 µm; = ms - ; V = ms - : = 5 µm; = ms - ; V = ms - : = µm; = ms - ; V = ms - : = µm; = ms - ; V = 4 ms - Fig. 4: Selecte roplet trajectories in a 3 K smoe layer (layer epth is m). Number fraction of roplets..8.6.4..9.8.7.36.45.54 roplet iameter /mm Fig. 5: roplet size istribution for the AM4 nozzle [8] Total absorption rate / W 4 3 3 5 7 9 Smoe layer temperature /K present analytical moel _ CF preictions [7]. convective absorption rate, as preicte by analytical moel Fig. 6: Heat absorption rate preictions for the AM4 nozzle (layer epth is m; water ischarge rate is equal to ψ& = 8 liters/min)
International Journal on Engineering Performance-Base Fire Coes The presente moel can also be use to preict temperature history of the smoe layer in compartments. This is accomplishe solving the simple energy conservation equation for the smoe layer: M sm c pg T t sm spr ( T ) Q& (3) sm where M sm is the mass of the smoe layer. For the solution of equation (3), the rate of heat absorption Q& spr ( T sm ) must be upate as the solution progresses using the escribe calculation proceure (equations () to ()). Typical results are presente in Figs. 7 an 8. The first of these figures plots smoe layer temperature versus time for ifferent water ischarge rates. The water ischarge rate of ψ& = 8 liters/min correspons to the nominal operating conitions for the AM4 nozzle. The other curves in Fig. 7 correspon to the reuce water ischarge rates. At the initial stage after sprinler activation, the temperature ecrease rate is roughly linearly proportional to the water ischarge rate. All curves in Fig. 7 approach the ambient temperature lel (3 K) at infinitely long times. The effect of compartment size on temperature ecrease rate can be estimate from Fig. 8. Sample results are presente for the three ifferent floor areas of the room. The water nozzle operates at its nominal lel of ψ& = 8 liters/min. The rooms with smaller floor area emonstrate higher temperature rop rate, until the temperature lels off to the ambient lel. Typically, the thermal time constant of the smoe layer is of the orer of min. Finally, typical roplet heat-up times (time require for the roplet to heat to boiling temperature) are presente in Fig. 9. The heat-up times are presente for ifferent smoe layer temperatures, an for seral roplet sizes, which contain most of the AM4 spray volume. 3. CONCLSIONS A simple analytical solution has been presente for water roplet motion an aporation rates in a hot smoe layer. The moel allows calculation of the heat absorption rates for water sprays an smoe layer temperature histories to be performe. As a calculation test, the results are presente for iniviual representative roplet trajectories in hot layers of ifferent temperatures. The calculations of the total heat absorption rate by the spray is also presente, an a goo agreement is foun between the simplifie moel an the results from comprehensive CF simulations. The agreement between the analytical an CF moeling emonstrates that the present moel can be applie for fire engineering calculations as a quic an economical alternative to full CF simulations. Smoe layer temperature /K 8 6 4 : ψ& = 8 liters/min : ψ& = liters/min 3 : ψ& = 4 liters/min 4 : ψ& = 8 liters/min.. 4. 6. 8... Time / s Fig. 7: Smoe layer temperature histories for ifferent water ischarge rates (roplet size istribution correspons to the AM4 nozzle; layer epth is m)
International Journal on Engineering Performance-Base Fire Coes Smoe layer temperature /K 8 6 4.. 4. 6. 8... Time / s : 5 m x 5 m compartment : 4 m x 4 m compartment 3 : 3 m x 3 m compartment Fig. 8: Effect of the compartment floor area on the smoe temperature rop rate (roplet size istribution correspons to the AM4 nozzle; layer epth is m; water ischarge rate is equal to ψ& = 8 liters/min ) Heat-up time / s.4. roplet iameters:. : = 3 µm.8 : = 5 µm.6 : = 36 µm.4.. 4 4 44 46 48 5 5 Smoe layer temperature / K Fig. 9: Heat-up time for various roplets NOMENCLATRE c p specific heat C rag coefficient roplet iameter f (s) pf for roplet size istribution H fg latent heat of water vaporisation m roplet mass Nu Nusselt number Q & rate of heat absorption Re Reynols number t time T temperature tangential velocity component of the roplet V vertical velocity component of the roplet y coorinate normal to the ceiling Gree symbols ϑ ρ ψ& smoe layer thicness imensionless temperature ensity water ischarge rate Subscripts initial g gas w wet bulb temperature p particle spr spray sm smoe
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