Numerical modeling for compartment fire environment under a solid-cone water spray

Applied Mathematical Modelling 30 (2006) 1571–1586 www.elsevier.com/locate/apm

Numerical modeling for compartment fire environment under a solid-cone water spray B. Yao a, W.K. Chow

b,*

a

b

State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei, Anhui, China Research Centre for Fire Engineering, Department of Building Services Engineering, The Hong Kong Polytechnic University, Hong Kong, China Received 1 August 2001; received in revised form 1 June 2005; accepted 2 August 2005 Available online 19 September 2005

Abstract A mathematical model was developed for simulating the fire environment of a compartment under the action of a solidcone water spray such as those discharged from a water mist fire suppression system. A smoke layer was induced by a fire in the compartment. The solid-cone water spray was discharged to act on the smoke layer, but not on the burning object. Under this condition of having a stable smoke layer, the compartment was divided into three regions. Region 1 is the upper hot smoke layer, Region 2 is the lower cool air layer and Region 3 is the solid-cone spray. The effects on the smoke layer development due to spray-induced flow were considered on the basis of mass, momentum and heat conservation. Water droplets of the solid-cone spray were divided into four typical classes based on the droplet distribution function. The parameters including the smoke layer interface height, smoke temperature and air temperature, smoke flow rate through the opening and oxygen concentration in the air layer were investigated under various heat release rates, water application rates and volume mean diameters of the solid-cone spray. Effective hot gas entrainment and water vapor production suggested that the water spray should contain a variety of droplet size. In this way, a compartment fire can be controlled effectively through indirect interaction such as oxygen concentration depletion.  2005 Elsevier Inc. All rights reserved. Keywords: Solid-cone water spray; Compartment fire; Numerical modeling; Smoke layer

1. Introduction A good understanding of the fire environment is important for providing scientific data for designing workable fire services system. Water mist fire suppression systems (WMFSS) are now commonly used in fire protection engineering with the objective to protect the environment by substituting the use of halon. WMFSS takes the advantages of high efficiency, low water damage and environmental safety [1–5]. Though water sprays are widely used for fire suppression and control, injecting them into a hot smoke layer would change *

Corresponding author. Fax: +852 2765 7198. E-mail address: [email protected] (W.K. Chow).

0307-904X/$ - see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2005.08.003

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Nomenclature A BD CD Cp Cxe Cxi De Di g H HD Hi Hu JGe le L Lc Lvi ml mlO2 mu muO2 m_ ent m_ evap m_ in m_ ml m_ out m_ sg m_ w nd Ni Nui qe Q* Q_ t Ta Td Tg Tl Tu Ude Udi UGc UGe V Vl Vs

floor area of the compartment width of the door discharge coefficient of the open door specific heat of gas phase at constant pressure drag coefficient for droplets on the spray envelope drag coefficient for the ith class drop size diameter of droplets on the spray envelope droplet diameter for the ith class drop size gravity acceleration height of the compartment height of the door height of the layer interface to the floor height of upper smoke layer momentum flow rate of gas phase inside the spray envelope spray trajectory length length of the compartment proportional factor of the heat loss due to radiation and conduction volumetric flow rate for the ith class drop size mass of the lower air layer oxygen mass in the lower air layer mass of the upper smoke layer oxygen mass in the upper smoke layer mass flow rate entrained from the lower air layer by fire evaporation rate of the water droplets inside the spray envelope incoming mass flow rate through the open door mass flow rate entrained from the lower air layer by water spray outgoing mass flow rate through the open door terminal mass flow rate of the gas inside the spray envelope mass flow rate entrained from the upper smoke layer by water spray total class number of drop size droplet number per second for the ith class drop size Nusselt number for the ith class drop size latent heat of vaporization of water normalized total heat release rate of the fire total heat release rate of the fire time from ignition ambient temperature droplet temperature gas temperature inside the envelope temperature of the lower air layer temperature of the upper smoke layer velocity of droplets on the envelope droplet velocity for the ith class drop size gas velocity to enter the envelope perpendicularly average velocity of the gas inside the spray envelope volume of the compartment volume of the lower air layer volume of the spray envelope

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Vs,t Vs,t+Dt Vu W xe Zf ZSl ZSu

1573

volume of the spray envelope at time t volume of the spray envelope at time t + Dt volume of the upper smoke layer width of the compartment spray envelope radius height of layer interface to the fire source position of the layer interface on the z-axis position of the terminal spray on the z-axis

Greek symbols Dt time step for calculation b angle between the normal to the envelope and the vertical k thermal conductivity of gas phase kr radiation loss fraction of the total heat release rate h0 initial injecting angle of water spray qa ambient gas density qG gas density inside spray envelope ql gas density in the lower air layer qL liquid water density qu gas density in the upper smoke layer

the smoke characteristics. Smoke might even be brought down to lower level to give adverse effects to occupants. A poorly designed WMFSS will not control a fire, but cause damage. Therefore, studies on the interaction of water spray with smoke will help designing workable WMFSS. There had been lots of works on hollow-cone spray with large droplet discharged by sprinkler system. Other than experimental studies, the interaction of water spray with hot smoke layer has also been simulated using field models [6–9] and zone models [10–15]. Relatively, there are not so many works on solid-cone water sprays with small droplet discharged from WMFSS, apart from recent numerical simulations using zone models on both types of water sprays [13–15]. The approach of zone models is simple and based on total mass and heat conservation between the divided zones. They are capable of simulating smoke characteristics for compartment fires with satisfactory results and have been widely used for fire safety engineering [16–20]. A major problem in using this approach is whether a clear two-layer structure is formed. Most of the available zone models have been validated for simulating compartment fire environment without a water spray. However, under the action of a water spray, if the ratio of the drag force D due to the spray while traveling through the smoke layer to hot gas buoyancy force B, D/B ratio, is greater than unity, the smoke layer might not be stable, resulting in unclear smoke layer interface [21– 23]. Therefore, there will be problems in using zone models to simulate the sprinkled-fire environment in the compartment when the stability of smoke layer is lost. Interaction of the spray-induced flow with the fireinduced flow should be studied, say with computational fluid dynamics (CFD), or through full-scale burning tests. However, for a solid-cone water spray of relatively small droplet discharged from a WMFSS, with the smaller spray dimension, say 0.5 m, comparing to the compartment dimension, say 5.0 m, the spray envelope and interaction area are both relatively small, resulting in smaller entrained flow from the smoke layer comparing to the total smoke layer volume. Assuming the smoke temperature is 600 K with a layer of 2.0 m thick, the entrained gas flow plus the produced water vapor due to the spray entrainment was calculated in literature [13] to be about 1.1 m3 s1, while the total smoke layer volume is up to 50 m3 and the fire-induced flow rate is about 1.0 kg s1 (1.6 m3 s1 at 600 K) if the interface is 1.5 m high from the 500 kW fire source [24]. The major problem in using zone models in a compartment is whether a clear layer structure can be formed and so the compartment dimension is usually bigger than the fire source dimension in using zone model. In this way, the layer structure will not be affected by the fire plume (convective flow above the fire source)

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significantly. Similarly in this paper with water application, convection is assumed to affect only the fire plume and water spray, but not the other regions. Only some regions of the smoke under the spray nozzle might be affected. Smoke layer in regions other than the spray envelope is assumed to be unaffected by the droplet. A hot and thick enough stable smoke layer is still formed. The region of water spray has been carefully considered in the present model as a separate zone. Heat and mass transfer in other regions are calculated. In this way, for a bigger compartment as in this paper, ÔmicroscopicÕ interaction of water spray and fire plume need not be included. The concept of zone model is applicable to regions other than the fire plume and water spray (where the convection currents are important). A good understanding of the compartment fire environment under solid-cone water spray application is important for designing workable WMFSS. The interesting variables include the oxygen concentration in the lower layer and the smoke flow rate through the opening to the adjacent space. For the compartment fire environment, if the spray characteristics and the effects on the surrounding gas can be reasonably estimated, the smoke characteristics and other parameters of the compartment fire environment can be predicted based on the heat and mass conservation among the two layers and spray region. Therefore, interaction of a solidcone water spray on the development of compartment fire environment should be studied first. A one-dimensional model for simulating the interaction of a water spray with smoke layer was developed by Chow and Yao [13]. In this model, the hot smoke and cool air were taken as two quasi-steady layers, and water droplets of the spray were divided into four typical classes based on the droplet distribution function. The global evolution of the spray was modeled using a Lagrangian approach on the basis of macroscopic balances for single droplets. The equation system can be solved using a space marching Runge–Kutta scheme of the fourth order. Spray characteristics such as droplet diameter and velocity, shape of the spray envelope, gas temperature and flow rate inside the spray envelope were calculated for various smoke layer conditions and reasonable results were obtained. This model is now further developed to study the interaction of a fireinduced smoke layer with a solid-cone water spray in a compartment. 2. The model The physical picture of the compartment fire environment is shown in Fig. 1. A smoke layer was induced by a fire. A solid-cone spray was discharged at some distance away from the burning object. Under this condition, suppression of the burning object can be avoided to simplify the model. The gas flow through the opening and the fire plume would not be affected by the water spray. The compartment can be divided into three regions: • Region 1: upper hot smoke layer, • Region 2: lower cool air layer and, • Region 3: solid-cone spray.

Fig. 1. Physical picture of the compartment fire environment.

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The water mist will evaporate in the entrained hot gas and the mixed gas released in the lower layer smoke will reduce the oxygen. The fire would be suppressed when entraining air of low oxygen concentration from the compartment environment. This is the major difference between sprinkler spray and water mist for fire suppression, especially for the case when the water spray could not reach the fire source directly. Taken as a solid-cone, the spray can be assumed to be one-dimensional, i.e., uniform at the cross-section. Since the characteristic changing time of the smoke layer is much larger than that of the water spray, the characteristics of the water spray can be taken as steady during each time step for calculating the smoke layer characteristics. Other main assumptions made on the model are listed as follows: • The compartment fire environment is at the stage of pre-flashover, and the pressure in the compartment is near the ambient pressure. All gases are taken as ideal gases. • The droplet size distribution is described by a finite number of size classes based on the droplet distribution function [13]. • The water droplets are spherical and evaporate in hot gas through convective heat transfer. The radiant heat attenuation by water droplets and steam is neglected. The interaction between droplets is neglected. • The density difference due to temperature change is included in the momentum equation, but neglected for droplet evaporation. • The shape of the spray envelope is determined by the trajectories of water droplets on the envelope. All gas/ droplet interactions are taken as occurring inside the spray envelope. If the droplets on the envelope thoroughly evaporate, a new envelope will be formed by the outer layer of droplets inside the envelope [13]. • The surrounding gas travels into the spray envelope in a perpendicular direction. The gas inside the spray will be well mixed with the gas in the lower layer if the water spray travels through the interface, and even reaches the floor. Otherwise, the gas will be well mixed with the gas in the upper layer, e.g., all water droplets evaporate in the hot smoke layer.

3. Key equations for Region 1: upper smoke layer For the upper hot smoke layer, the mass conservation equation can be expressed as dmu ¼ m_ ent  m_ w  m_ out ; ð1Þ dt where mu is the mass of the upper smoke layer, m_ ent is the mass flow rate entrained from the lower air layer, m_ out is the outgoing mass flow rate through the opening, and m_ w is the mass flow rate entrained by the water spray from the upper layer. Using the ideal gas assumption, Eq. (1) can be rewritten as qa T a dV u qa T a V u dT u  ¼ m_ ent  m_ w  m_ out ; T u dt dt T 2u

ð2Þ

where Tu and Vu are the temperature and the volume of the smoke layer, Ta is the ambient temperature, and qa is the ambient gas density. The mass flow rate entrained from the lower layer can be estimated by engineering correlations, and one of them is [24]: m_ ent ¼ 0:21ql ðgZ f Þ1=2 Z 2f Q ðZ f ; tÞ1=3 ;

ð3Þ

where Q ðZ f ; tÞ ¼

ð1  kr ÞQ_ ql C p T l ðgZ f Þ

1=2

Z 2f

;

ð4Þ

where Zf is the height from layer interface to the fire source, g is the gravity acceleration, ql is the gas density in the lower layer, Tl is the gas temperature in the lower layer, Cp is the specific heat capacity of the gas, kr is the radiation loss fraction of the total heat release rate, and Q_ is the time-varying total heat release rate of the fire.

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The outgoing mass flow through the opening m_ out can be estimated from Bernoulli conservation equation [25]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 2 q T T a a ; ð5Þ 2g 1  m_ out ¼ C D BD ðH D  H i Þ3=2 a 3 Tu Tu where CD is the discharge coefficient (typical value is 0.7), HD and BD are the height and width of the opening, respectively, and Hi is the interface height to the floor. If the layer interface is above the upper edge of the opening, m_ out will be taken as 0. To simply the calculation process of the heat loss from the fire to the compartment due to radiation and conduction, the heat loss is considered to be proportional to the total heat release rate. The remaining combustion heat is transferred to the hot smoke layer. The energy conservation equation in the upper layer can be expressed as dðmu C p T u Þ dV u _ ¼ C p qa T a ¼ m_ ent C p T l  ðm_ w þ m_ out ÞC p T u þ ð1  Lc ÞQ; dt dt

ð6Þ

where Tl is the temperature of the lower layer and Lc is the proportional factor. From Eqs. (2) and (6), the following equation can be obtained: qa T a dT u _ ¼ m_ ent C p ðT l  T u Þ þ ð1  Lc ÞQ. V uCp Tu dt

ð7Þ

4. Key equations for Region 2: lower air layer For the lower air layer, the mass conservation equation can be expressed as dml ¼ m_ in þ m_ sg  m_ ml  m_ ent ; dt

ð8Þ

where ml is the mass of the lower layer, m_ in is the incoming mass flow rate through the opening, m_ ml is the mass flow rate entrained by the water spray from the lower layer, and m_ sg is the terminal mass flow rate of the gas inside the spray envelope. As assumed earlier, the terminal mass flow of the gas inside the envelope will be well mixed with the gas in the lower layer. Similar to deducing Eq. (2), Eq. (8) can also be rewritten as qa T a dV l qa T a V l dT l  ¼ m_ in þ m_ sg  m_ ml  m_ ent ; T l dt dt T 2l

ð9Þ

where Vl is the volume of the lower layer. The added volume of the upper layer, the lower layer and the water spray is just the total compartment volume V: V ¼ V u þ V l þ V s;

ð10Þ

where Vs is the volume of the water spray. Note that V can be expressed in terms of length L, width W, height H and floor area A of the compartment: V ¼ AH ;

ð10aÞ

A ¼ LW .

ð10bÞ

Time derivative of Eq. (10) can be written as dV dV u dV l V s;tþDt  V s;t ¼ þ þ ¼ 0; dt dt dt Dt

ð11Þ

where Dt is the time step, Vs,t+Dt and Vs,t are the volume of water spray at time t + Dt and t, respectively. The incoming mass flow rate through the opening m_ in is constrained by Eq. (11).

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Energy conservation gives dðml C p T l Þ dV l ¼ qa T a C p ¼ m_ in C p T a þ m_ sg C p T g  m_ ent C p T l  m_ ml C p T l ; dt dt

ð12Þ

where Tg is the gas temperature inside the spray envelope. Similarly, from Eqs. (9) and (12): qa T a V l dT l ¼ m_ in ðT a  T l Þ þ m_ sg ðT g  T l Þ. Tl dt

ð13Þ

By assuming that burning 1 kg of oxygen would give 13.1 MJ of heat, oxygen mass conservation equation for both layers can be expressed as dmuO2 mlO2 muO2 Q_ ; ¼ m_ ent  ðm_ w þ m_ out Þ  13; 100 kJ dt ml mu dmlO2 mlO2 muO2 ¼ 0:23m_ in  m_ ent þ m_ w ; dt ml mu

ð14Þ ð15Þ

where muO2 and mlO2 are the oxygen mass in the upper layer and lower layer, respectively. 5. Key equations for Region 3: solid-cone spray 5.1. Mass conservation equations As shown in Fig. 1, the gas inside the spray envelope consists of the gas entrained from two layers and the water vapor evaporated from the water droplets. A mass conservation equation for quasi-steady conditions can be obtained by neglecting their differences in velocities: m_ w þ m_ ml þ m_ evap ¼ m_ sg ;

ð16Þ

where m_ sg is the terminal mass flow rate of the gas inside the spray envelope, which will be released to the lower or upper layer at the end of the spray envelope; m_ evap is the vaporization rate of water droplets, and can be expressed as Z Z Sl X m_ evap ¼  qL dLvi . ð17Þ 0

nd

Similarly, other variables can be obtained: m_ sg ¼ qG px2e U Ge ; Z Z Su qu xe U Gc dle ; m_ w ¼ 2p Z0 Z Sl ql xe U Gc dle ; m_ ml ¼ 2p Z Su Z Z Sl x2e dz; Vs ¼p

ð18Þ ð19Þ ð20Þ ð21Þ

0

where qG is the gas density inside the spray envelope, qu is the gas density in the upper layer, qL is the liquid water density, UGc is the gas velocity entering the envelope perpendicularly, UGe is the average velocity of gas inside the envelope, nd is the total class number of drop size, Lvi is the volumetric flow rate for the ith class drop size, xe is the envelope radius, le is the spray trajectory length, ZSu and ZSl are the position of the layer interface and the terminal spray on the z-axis. If the spray reaches the floor, ZSl is the total height of the compartment. The momentum flow rate of gas phase JGe can be defined as J Ge ¼ pqG x2e U 2Ge .

ð22Þ

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Therefore, Eq. (16) can be rewritten as " # U Ge T u dxe 1 dJ Ge xe U Ge T u dT g qL X dD i þ  þ U Gc ¼ sin b N i D2i ; 2T g dz 4pqu U Ge xe dz dz 4qu xe nd dz 4T 2g

ð23Þ

where Ni and Di are the droplet number per second and droplet diameter for the ith class size, Tg is the gas temperature inside the spray envelope, and b is the angle between the normal to the envelope and the vertical: sin b ¼

dz . dle

ð24Þ

5.2. Momentum conservation equations A total momentum equation can be written as [13]:  X X Lvi dJ Ge dLvi dU di ¼ ðqL  qG Þg þ Lvi  qL U di . dz U di dz dz nd nd

ð25Þ

For the droplets inside the spray envelope, force balancing of the droplet for the ith class size would give:    dU di 1 qG 3 qG C xi ¼ 1 ð26Þ g jU di  U Ge jðU di  U Ge Þ ; U di 4 qL Di dz qL where Udi and Cxi are the velocity and drag coefficient for the ith class drop size. Since the droplet velocity is assumed to be perpendicular to the entering gas velocity, the momentum equation for the droplets on the envelope can also be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 2 dU de qG g 3 qG C xe U de þ U Gc ; ð27Þ ¼ 1  sin b dz qL U de 4 qL De where Ude, De and Cxe are the velocity, diameter and drag coefficient of the droplets on the envelope, respectively. Balancing the forces normal to envelope, the trajectory of the drops on the envelope can be expressed as [13]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi   d2 xe 3 qG C xe U Gc 1 U Gc qG g cos b  1 1þ ¼ . ð28Þ 4 qL De U de sin3 b dz2 U de qL U 2de sin3 b 5.3. Heat and mass transfer between droplet and gas inside envelope The convective heat from the hot gas to the droplets is assumed to evaporate the water droplets, and the evaporation model for water droplet in hot gas under forced convection can be used: qL Di

  dDi dDi ¼ qL U di Di ¼ 2kNui T d  T g =qe ; dt dz

ð29Þ

where k is the thermal conductivity of gas phase, qe is the latent heat of water vaporization, and Nui is Nusselt number for the ith size class. The drop temperature Td and the gas temperature (between 373 K and 1273 K) can be correlated by [11]: T d ¼ 266  ð1 þ 3:23  104  T g Þ.

ð30Þ

The gas temperature inside the spray envelope can be calculated by the following heat balance equation: m_ w C p T u þ m_ ml C p T l  m_ evap qe ¼ m_ sg C p T g .

ð31Þ

B. Yao, W.K. Chow / Applied Mathematical Modelling 30 (2006) 1571–1586

Combining this equation with Eq. (16) gives:  X qe D2 dDi 1 dT g ¼ qG x2e U Ge þ 2qu xe U Gc ðT u  T g Þ . þ Tg Ni i qL sin b Cp 2 dz dz nd

1579

ð32Þ

6. Numerical experiments The set of Eqs. (6), (7), (12), (13), (14) and (15) for the upper smoke layer and lower air layer is a complete system of coupled ordinary differential equations of the first order. The time-dependent variables are Vu, Vl, Tu, Tl, muO2 and mlO2 , and these equations could be solved using a time marching Runge–Kutta scheme of the fourth order. Similarly, under the quasi-steady conditions, the set of Eqs. (25)–(29) and (32) for the spray envelope is also a complete system of coupled ordinary differential equations of the first order for the distance from the ceiling. The distance-dependent variables are JGe, Udi, Ude, dxdze , Di and Tg, and these equations could also be solved using a space marching Runge–Kutta scheme of the fourth order to get the gas and droplet characteristics inside the spray envelope at a certain time, assuming other conditions for the two layers are fixed. The associated variables such as entrained gas flow rate and droplet evaporation rate due to the spray application can be calculated and used for the numerical solution of the two layers. Other important variables such as smoke flow rate through the opening can be calculated using the solved characteristics of the smoke layer. A compartment of 8.0 m long, 5.0 m wide and 3.5 m high was considered. There was an opening of 2.0 m high and 2.0 m wide, with the sill at the floor level. The nozzle was located downward at 10 cm below the ceiling, relatively far from both the opening and the fire source, and the spray radius was relatively small comparing to the compartment dimension. A computer program has been developed using Visual Digital Fortran software, and some typical cases were selected for numerical simulation. Various heat release rates and spray characteristics were selected for the present study. For calculating the spray characteristics, 1 mm was selected as the space step. For predicting the smoke layer development, 0.1 s was selected as the time step. The fire was taken as a ultrafast growth t-square fire and was located on the floor. The maximum heat release rate was selected as 0.5 MW and 1.0 MW. After reaching the maximum value, the heat release rate kept constant. In this paper, the heat release rate of fire was assumed to be the value of the design fire and unaffected by oxygen concentration decrease in the air layer due to water evaporation, and this oxygen effect on the heat release will be further studied. The heat loss fraction Lc due to radiation and conduction to compartment was assumed to be 0.65, and the radiation loss fraction of the total heat release rate kr was assumed to be 0.35. The initial velocity and spray angle at the nozzle were 20 ms1 and 35, respectively. The volume mean diameter (VMD) of the water spray was selected as 600 lm and 300 lm. For the spray of 600 lm, the diameters of the four classes were: 965, 697, 400 and 132 lm; for the spray of 300 lm, they were: 480, 350, 200 and 65 lm. The water application rate ranged from 0.1 l s1 to 1.0 l s1. The water spray was activated at 20 s after the ignition of the fire, and the total calculating time is 300 s. 7. Results The spray characteristics such as droplet velocity, diameter, spray envelope radius, and gas flow rate while traveling through the stable hot smoke layer have been numerically studied in literature [13]. For a time-varying smoke layer in this paper, these characteristics affected the smoke layer development significantly. The calculated droplet velocity and diameter inside the spray envelope at the time of 300 s after ignition are plotted against the distance from ceiling in Figs. 2 and 3, respectively. The water application rate was 0.5 l s1. Smaller droplets were affected by the smoke layer more significantly than the larger ones, because of the larger heat transfer and drag effects. The larger the droplet, the slower rate of decrease of velocity and diameter. For larger heat release rate, the smoke layer was hotter, and the droplet velocity decreased slower while the droplet diameter decreased quicker.

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Droplet velocity / ms-1

25 1.0 MW, 965 μm 1.0 MW, 697 μm 1.0 MW, 400 μm 1.0 MW, 132 μm

0.5 MW, 965 μm 0.5 MW, 697 μm 0.5 MW, 400 μm 0.5 MW, 132 μm

20

15

10

5

0 0.0

0.6

1.2

1.8

2.4

3.0

3.6

Distance from ceiling / m Fig. 2. Droplet velocity inside the spray envelope (time: 300 s, water application rate: 0.5 l s1).

1000

Droplet diameter / μm

0.5 (1.0) MW, 965 μm

800

0.5 (1.0) MW, 697 μm

600 0.5 MW, 400 μm 1.0 MW, 400 μm

400 0.5 MW, 132 μm 1.0 MW, 132 μm

200 0 0.0

0.6

1.2

1.8

2.4

3.0

3.6

Distance from ceiling / m Fig. 3. Droplet diameter inside the spray envelope (time: 300 s, water application rate: 0.5 l s1).

The gas flow rate inside the spray envelope affected the mass and heat transfer among the two layers and spray envelope significantly. The terminal gas flow rate inside the spray envelope is shown in Fig. 4. The water spray of larger VMD entrained more gas from the surrounding. The larger the heat release rate, the hotter the smoke layer and so the higher the evaporation rate, as shown in Fig. 5. The smoke layer interface height, the smoke layer temperature, and the smoke flow rate through the opening against time are shown in Figs. 6–8, respectively. Without water spray application, the higher the fire heat release rate, the lower the layer interface and the higher the smoke temperature. The smoke flow rate through the opening to the adjacent space increased with the heat release rate as well. Under water spray application, the interface height increased with the entrained gas flow rate by the water spray. This entrainment effect due to water spray was similar to a vent in transferring heat and mass. The smoke flow rate through the opening decreased under water spray application, therefore the smoke damage to the adjacent space decreased as well. However, quite different from the vent, the water spray affected the lower air layer significantly, for the hot gas including water vapor was brought to the lower layer by the spray flow. The temperature and the oxygen concentration in the lower layer are shown in Figs. 9 and 10. For the water spray of VMD 300 lm, the temperature of the lower layer was almost unaffected by the spray, because of the lower entrained gas flow rate; while for the spray of 600 lm, the temperature increased significantly. The higher the heat release rate, the

Gas flow rate in side spray envelope/ kgs-1

B. Yao, W.K. Chow / Applied Mathematical Modelling 30 (2006) 1571–1586

1.0 0.8 0.5 MW, 600 μm 0.5 MW, 300 μm 1.0 MW, 600 μm 1.0 MW, 300 μm

0.6 0.4 0.2 0.0 0

50

100

150

200

250

300

Time from ignition / s

Mass flow rate of liquid water / kgs-1

Fig. 4. Terminal gas flow rate inside the spray envelope (water application rate: 0.5 l s1).

0.52 0.5 MW, 600 μm 0.5 MW, 300 μm 1.0 MW, 600 μm 1.0 MW, 300 μm

0.50

0.48

0.46

0.44 0

50

100

150

200

250

300

Time from ignition / s Fig. 5. Mass flow rate of the liquid water inside the spray envelope (position: floor, water application rate: 0.5 l s1).

Smoke layer interface height / m

3.5 0.5 MW, No water 0.5 MW, 600 μm 0.5 MW, 300 μm 1.0 MW, No water 1.0 MW, 600 μm 1.0 MW, 300 μm

3.0 2.5 2.0 1.5 1.0 0

50

100

150

200

250

300

Time from ignition / s Fig. 6. Smoke layer interface height (water application rate: 0.5 l s1).

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600 Smoke layer temperature / K

560 520 480 440 0.5 MW, No water 0.5 MW, 600 μm 0.5 MW, 300 μm 1.0 MW, No water 1.0 MW, 600 μm 1.0 MW, 300 μm

400 360 320 280 0

50

100 150 200 Time from ignition / s

250

300

Smoke flow rate through the opening / kgs-1

Fig. 7. Smoke layer temperature (water application rate: 0.5 l s1).

1.6

1.2

0.8

0.4 1.0 MW, No water 1.0 MW, 600 μm 1.0 MW, 300 μm

0.5 MW, No water 0.5 MW, 600 μm 0.5 MW, 300 μm

0.0 0

50

100 150 200 Time from ignition / s

250

300

Fig. 8. Smoke flow rate through the opening (water application rate: 0.5 l s1).

380 Lower layer temperature / K

1582

0.5 MW, 600 μm 0.5 MW, 300 μm 1.0 MW, 600 μm 1.0 MW, 300 μm

360 340 320 300 280 0

50

100

150

200

250

300

Time from ignition / s Fig. 9. Lower air layer temperature (water application rate: 0.5 l s1).

Oxygen concentration in lower layer

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0.22 0.20 0.18 0.5 MW, No water 0.5 MW, 600 μm 0.5 MW, 300 μm 1.0 MW, No water 1.0 MW, 600 μm 1.0 MW, 300 μm

0.16 0.14 0.12 0

50

100 150 200 Time from ignition / s

250

300

Fig. 10. Oxygen concentration in the lower air layer (water application rate: 0.5 l s1).

1.70

500

1.65

450

1.60

400 1.55

350 1.50 Interface height

1.40 0.0

300

Upper layer temperature Lower layer temperature

1.45

0.2

0.4

0.6

0.8

Two layer temperature /K

Smoke layer interface height / m

higher the temperature due to more heat brought to the lower layer. However, if the temperature of the lower layer is high enough comparing to the temperature of the upper smoke layer, the layer interface might not be so clear and the zone concept might not be available anymore. The oxygen concentration in the lower layer decreased quickly under water spray application. For larger fire and larger entrainment and evaporation, the oxygen concentration even decreased down to 13%. The water application rate affects the mass, momentum and heat transfer between the water droplet and hot gas, resulting in various spray envelopes and layer development. The smoke layer temperature, the interface height, the smoke flow rate through the opening, and the oxygen in the lower layer at the time of 300 s from ignition against water application rate are shown in Figs. 11 and 12. The VMD of the water spray was 600 lm and the maximum heat release rate was 0.5 MW. The temperature of both layers decreased with the application rate. The height of the smoke layer interface increased with the water application rate and then decreased at a critical water application rate (0.7 l s1). The smoke flow rate and oxygen concentration both decreased with the water application rate and then increased at the same critical value. This implies that too much droplet in the unit volume would reduce the gas entrainment and heat transfer. Therefore, to reduce smoke damage more effectively, using only single nozzle to increase the water application rate is not a good choice, instead, using several nozzles in the compartment might be better. This point will be numerically studied for further research.

250 1.0

Water application rate / ls-1 Fig. 11. Smoke layer interface height and two layer temperature (heat release rate: 0.5 MW, VMD: 600 lm).

0.21

1.8 Smoke flow rate

1.6

Oxygen concentration

1.4

0.20 0.19

1.2 0.18 1.0 0.17

0.8

0.16

0.6 0.4 0.0

0.2

0.4 0.6 0.8 Water application rate / ls-1

0.15 1.0

Oxygen concentration in lower layer

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Smoke flow rate through the opening / kgs-1

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Fig. 12. Smoke flow rate through the opening and oxygen concentration in the lower air layer (heat release rate: 0.5 MW, VMD: 600 lm).

8. Discussion on using the model The model is developed to predict the compartment fire environment under a solid-cone water spray, and the most important variables include the smoke flow rate through the opening to adjacent space and the oxygen concentration in the air layer. First of all, the compartment was divided into three regions on the basis of stability analysis for the smoke layer under the action of a solid-cone water spray. The stable condition in this paper is that the spray dimension is relatively smaller comparing to the compartment dimension and the smoke layer can remain hot and thick enough to keep thermally stable under suitable heat release rate. The water spray is used for reducing the smoke flow rate to the adjacent space and depleting the oxygen concentration to suppress the fire through indirect interaction, and not by direct interaction. However, if the water spray is discharged from a total flooding system, the smoke layer might disappear and the compartment would reach a well-mixed steady state with the temperature value between 50 and 70 C quickly after the water mist activation [26]. Under this condition, the compartment gas can be considered to have uniform characteristics and the time varying temperature can be employed for the whole compartment. But for a zoned or local application system, the smoke layer might be formed under the ceiling and the present model can be used if the smoke layer stability has been identified first. Due to water evaporation, the gas inside the spray envelope consists of large amount of water vapor. Since the gas was released in the lower layer under the numerical conditions in this paper, the incoming gas from the ambient would decrease to a lower value to keep the pressure balance in the compartment. Therefore, the incoming mass flow rate and the movement velocity would be both low, and the assumption of well mixing is reasonable for this model. However, if all droplets evaporated in the upper smoke layer, the gas inside the envelope would be released to the smoke layer, resulting in decreased smoke temperature and increased mass flow rate through the opening. The lower air layer would be almost unaffected. The effects on smoke including concentration reduction of toxic gases (especially dissolvable toxic gas) and particles were not included in the present model, and the smoke characteristics under this condition will be further studied. In this paper, the heat release rate of fire was assumed to be a fixed value after reaching the design heat release rate. Actually, the heat release rate is reduced significantly by the decreased oxygen concentration in the entrained air, and the fire cannot be sustained and will extinguish if the oxygen concentration falls below a critical value. However, the oxygen effect on the heat release rate is quite complicated and will be considered in further research. As seen from the simulated results for various droplets, for a given fire scenario, the larger droplets would transfer more momentum to the surrounding gas and more smoke would be entrained into the water spray envelope; while the smaller droplets would be evaporated to absorb more heat from the entrained hot gas. The more the produced water vapor, the lower the oxygen concentration in the lower layer. As discussed earlier, if the spray momentum is small and the entrained gas and water vapor produced inside the envelope could

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not be released to the lower layer, the oxygen concentration would not be depleted. The high spray momentum and large quantity of water vapor produced are both important for fire control through indirect interaction. Therefore, under the conditions in this paper, the water spray containing droplets of suitable size distribution is more effective for fire protection than the water spray containing only uniform droplets. Assessed from the smoke flow rate through the opening and the oxygen depletion, the spray of VMD 600 lm performs better than the spray of VMD 300 lm due to larger hot smoke entrainment and larger heat transfer. In this paper, the mass flow rate entrained from the lower layer by the fire was estimated directly by engineering correlations (e.g., Eq. (3)). However, these engineering correlations were developed for ambient air condition, therefore, if the temperature of the lower layer is high enough, they should be further verified. Fortunately, recent experimental results show that the sprinkler spray has a minimal effect on the amount of air entrained into a fire plume if the heat release rate is fixed [12]. 9. Conclusions The compartment fire environment under a solid-cone water spray was studied numerically using the present mathematical model developed from the one-dimensional model for simulating the spray envelope. Based on stability analysis for the smoke layer under the condition in this paper, the compartment was divided into three quasi-steady regions, namely: upper smoke layer, lower air layer and one-dimensional spray. The key equations for each region can be obtained on the basis of mass, momentum and heat conservation and solved using a marching Runge–Kutta scheme of the fourth order. This model can be used to predict the compartment fire environment and the smoke flow to the adjacent space. The action of the solid-cone water spray affects the compartment fire environment significantly, and the characteristics of the water spray such as drop size and velocity distribution, and water application rate play the dominating role. Due to the effective hot gas entrainment and water vapor production, it is reasonable to employ a water spray containing a variety of droplet size for controlling a compartment fire effectively through indirect interaction such as oxygen concentration depletion. Acknowledgement This work was jointly supported by the Research Grants Council of Hong Kong (Account No. B-Q408); and the China NKBRSF (Project No. 2001CB409609). References [1] A. Jones, P.F. Nolan, Discussion on the use of fine water sprays or mists for fire suppression, J. Loss Prevent. Process Ind. 8 (1) (1995) 17–22. [2] J.R. Mawhinney, J.K. Richardson, A review of water mist fire suppression research and development—1996, Fire Technol. 33 (1) (1997) 54–90. [3] G. Grant, J. Brenton, D. Drysdale, Fire suppression by water sprays, Prog. Energy Combust. Sci. 26 (2) (2000) 79–130. [4] W.H. Yang, T. Parker, H.D. Ladouceur, R.J. Kee, The interaction of thermal radiation and water mist in fire suppression, Fire Saf. J. 39 (1) (2004) 41–66. [5] Z.G. Liu, A.K. Kim, D. Carpenter, J.M. Canabus, Extinguishment of cooking oil fires by water mist fire suppression systems, Fire Technol. 40 (4) (2004) 309–333. [6] W.K. Chow, N.K. Fong, Application of field modeling technique to simulate interaction of sprinkler and fire induced smoke layer, Combust. Sci. Technol. 89 (2) (1993) 101–151. [7] G.M. Makhviladze, J.P. Roberts, V.I. Melikhov, O.I. Melikhov, Numerical modeling and simulation of compartment fire extinction by a sprinkler spray, J. Appl. Fire Sci. 8 (2) (1998–1999) 93–115. [8] S. Nam, Development of a computational model simulating the interaction between a fire plume and a sprinkler spray, Fire Saf. J. 26 (1) (1996) 1–33. [9] S.C. Kim, H.S. Ryou, An experimental and numerical study on fire suppression using a water mist in an enclosure, Build. Environ. 38 (11) (2003) 1309–1316. [10] G. Heskestad, Sprinkler/hot layer interaction, NIST-GCR-91-590, National Institute of Standards and Technology, Gaithersburg, USA, 1991. [11] L.Y. Cooper, Interaction of an isolated sprinkler spray and a two-layer compartment fire environment, NISTIR 4587, National Institute of Standards and Technology, Gaithersburg, USA, 1991.

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