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Flashover control under fire suppression conditions
Fire Safety Journal 36 (2001) 641–660
Flashover control under fire suppression conditions V. Novozhilov* School of Mechanical and Production Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore Received 30 May 2000; received in revised form 12 February 2001; accepted 22 March 2001
Abstract Possibility of flashover development under fire suppression conditions is investigated using a zone model approach. Analytical models are developed to model the effect of water sprinklers and water mist systems interactions with the ceiling smoke layer, and to assess the effects of such interactions on flashover development. Different regimes of heat transfer between water spray and smoke layer are investigated, and the limits of these regimes are identified. Critical conditions for flashover development during suppression are formulated. The derived conditions allow the critical water application rates required for flashover prevention to be found as functions of spray and fire characteristics. r 2001 Elsevier Science Ltd. All rights reserved. Keywords: Flashover; Fire suppression; Zone model
1. Introduction Flashover is a stage in compartment fire development which can be described as a rapid transition from a slowly growing to a fully developed fire. The underlying mechanism in this phenomenon is essentially a positive feedback from fire environment to the burning fuel. Formation of a hot ceiling layer at the early stages of fire leads to radiative feedback to the fuel, which, in turn, results in increase in the burning rate and the temperature of the smoke layer. If heat losses from the compartment are insufficient, a sharp increase in the fire’s power (i.e. flashover) will eventually occur. *Tel.: +65-790-4335; fax: +65-791-1859. E-mail address: [email protected] (V. Novozhilov). 0379-7112/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 9 - 7 1 1 2 ( 0 1 ) 0 0 0 1 9 - 4
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Nomenclature a1;2 A Af cp CD d D F f G’ h H HV Hf g L’ LC m m’ out Nu Q’ 0 q00 Re t T U V WC WV z
model coefficients surface area fire burning area specific heat drag coefficient droplet diameter fractional height of the inter-zone boundary droplet size distribution density of droplet size distribution rate of heat gains into smoke layer convective heat transfer coefficients height of compartment height of vent latent heat of water vaporisation rate of heat losses from smoke layer length of compartment droplet mass mass outflow rate from compartment Nusselt number heat release rate of fire heat flux to fuel from fire Reynolds number time temperature tangential velocity component of the droplet vertical velocity component of the droplet width of compartment width of vent coordinate normal to ceiling
Greek symbols a emissivity D smoke layer thickness Dhc heat of combustion Dhvap heat of vaporisation of fuel w efficiency of combustion W dimensionless temperature r density s Stefan-Boltzman constant t dimensionless time ’ C water discharge rate
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Subscripts 0 initial w wet bulb temperature cr critical p particle spr spray u upper zone v vents Theoretically, flashover has been studied by both non-linear and computational fluid dynamic (CFD) approaches. Luo et al. [1,2] demonstrated that detailed CFD modeling can predict flashover development in a complicated multi-room geometry. The application of field modeling to flashover is, however, still a difficult task because of strict computational requirements on the accuracy of radiation modeling and the lack of reliable models of flame spread over various fuels. Disadvantage of CFD approach is that each solution is only applicable to a particular physical configuration. On the other hand, flashover can be rather effectively analysed using zone modeling of fire and methods of non-linear dynamics. Early attempts to investigate thermal instabilities, which may be interpreted as flashover, were made by Thomas [3,4]. Further investigations of flashover by methods of non-linear dynamics have been conducted in a number of studies [5–10]. Essentially, the non-linear models have been using one [5–7], two [8,9] or three [10] state variables to describe fire development. From the non-linear dynamics point of view, flashover may be interpreted as bifurcation in a space of appropriate controlling parameters. It has been shown that in some models such bifurcations can be described as cusp catastrophes of controlling surfaces, representing fire evolution in a compartment. Early in fire development the combustion products are usually segregated in a wellstirred ceiling layer with roughly homogeneous properties. For this reason zone models are able to achieve reasonably good qualitative and quantitative agreement with experiments, as has been demonstrated in [9]. In some cases, they can also provide valuable analytical solutions which clearly identify the important physical effects and have general meaning, in contrast to case-oriented field model results. The mechanism of positive non-linear feedback makes flashover phenomenon similar in many respects to the classical thermal explosion theory. In a most consistent way, this analogy has been used in [5,6]. The zone model developed in [5,6] allows prediction of the critical conditions for flashover to be made, and the influence of different physical or geometrical parameters to be investigated. However, the problem of flashover development in the presence of fire suppression systems has not been addressed in the previous studies. Since it is well known that fire suppression systems actually often fail to suppress fire, fire evolution under suppression conditions should be predicted. If fire is not suppressed, the question of whether or not flashover will happen still needs to be answered.
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The present study is concerned with the influence of sprinkler and water mist systems on possible flashover development in compartment fires. It is clear that even if the water spray cannot suppress fire completely, it can restrict fire growth and prevent flashover due to cooling of the hot smoke layer. Therefore, the effect of sprinkler operation on flashover is of significant interest for fire safety. Preliminary considerations of this effect by means of zone modeling have been made by Novozhilov and Kent [11]. In the present study, a more sophisticated zone model, which covers a wider range of operating conditions, is developed to predict effect of water-based fire suppression systems on flashover development. The model provides more accurate estimations of critical water application rates and allows different regimes of water spray interaction with smoke layer to be considered. Critical conditions for flashover are obtained as correlations between fire characteristics and sprinkler discharge rate. The results are generalised for arbitrary droplet size distributions.
2. Flashover zone modeling and the thermal explosion theory The considerations of the present study are based on the zone model [5], which expands classical thermal explosion theory to flashover phenomenon and represents general and convenient way to investigate flashover conditions. We briefly overview here the results necessary for the present study. Principal assumptions and results of the model [5] are as follows. In an enclosure with one opening (Fig. 1), flashover is principally described by four stages. The hot buoyant plume develops at the first stage following ignition, and then reaches the ceiling and spreads as a ceiling jet (second stage). During the third and fourth stages, the hot layer expands and deepens, and establishment of the flow through the opening takes place.
Fig. 1. Diagram of the compartment.
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Fig. 2. Schematic of the zone model.
At the second stage a well-stirred layer of combustion products is formed, and a zone approach may be applied. Under this approach, the compartment may be divided into two layers which are represented by average temperatures T and T0 (Fig. 2). Flashover is assumed to happen during the initial, fuel-controlled stage of fire, so that the temperature of the lower layer may be assigned an ambient value. The thickness of the smoke layer, D; is considered as an input parameter and does not change with time. Other assumptions are listed in [5]. Application of zone modeling involves the consideration of the heat balance equation for the upper hot smoke layer which is: dT mcp ¼ G’ L’ ; ð1Þ dt where t is time, m; cp and T are mass, specific heat and temperature of the smoke layer, respectively. The functions G’ and L’ on the right-hand side describe rates of heat gains and losses for the layer. In the absence of water spray, the right-hand side should generally include the terms corresponding to the rates of heat gain for the smoke layer and heat losses due to outflow through the opening and due to convective and radiation losses to compartment boundaries and fuel. In the present study, we only consider the case of large thermal inertia of compartment walls, which implies that the time scale for the wall heat-up is much larger than that for flashover. This assumption is not restrictive as it retains all the major effects in the system behaviour while removing unnecessary mathematical complications. The cases with low and intermediate thermal inertia may also be considered [5,6]. The Eq. (1) is non-dimensionalised based on the ambient temperature T0, heat release rate of fire Q’ 0 : ð2Þ Q’ 0 ¼ wAf ðq00 =Dhvap ÞDhc and the characteristic heating time of the upper layer: t ¼ mcp T0 =Q’ 0 : *
ð3Þ
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In dimensionless variables the Eq. (1) for the case of large thermal inertia of compartment walls is reduced to [5]: dW ¼ 1 þ a1 ðW4 1Þ a2 ðW 1Þ: dt
ð4Þ
The model coefficients a1;2 are determined by both fire physical parameters and dimensions of the compartment (Fig. 1): a1 ¼ w
Dhc aU sAf T04 =Q’ 0 ag s½Ac þ ð1 DÞAV Af T04 =Q’ 0 Dhvap
ag sAf T04 =Q’ 0 ag s½AU ð1 DÞAV T04 =Q’ 0 ; a2 ¼ ½AU ð1 DÞAV hc T0 =Q’ 0 þ ð1 DÞAV hc T0 =Q’ 0 þ m’ out ð1 DÞAV cp T0 =Q’ 0
ð5Þ
and have clear physical meaning. The first coefficient describes the rate of net heat gains into the smoke layer due to radiation. Parameter a2 characterises the net convective heat losses from the hot layer which occur due to outflow through the opening and the heat exchange between the smoke and cold surroundings. The problem of flashover resembles the classical thermal explosion theory because in both cases the system behaviour is essentially determined by competition between heat gains and losses. As has been described in [5], the Eq. (4) can be investigated by similar techniques which are well developed in the thermal explosion theory. The quantitative analysis is provided by a diagram shown in Fig. 3. In general, there can be three solutions of the heat balance Eq. (1). The lower and upper points of intersection between heat gain and heat loss curves are stable and represent fuelcontrolled and ventilation-controlled fires, respectively. The intermediate solution is unstable, and small perturbations around this point will result in a large change of temperature. The critical conditions for flashover existence are determined by both
Fig. 3. Semenov’s diagram for flashover. Adapted from [5]. PFsteady-state fire conditions; P Fflashover point; W Fcritical temperature; G’ Frate of heat gain into the smoke layer; L’ Frate of heat loss from the smoke layer.
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Fig. 4. Flashover critical boundary [5] and the effect of water discharge rate.
the equations G’ ðW * Þ ¼ L’ ðW * Þ and dG’ dL’ ðW * Þ ¼ ðW Þ: ð6Þ dW dW * It easily follows from Eq. (4) that the critical conditions are written in the following form: a1 ðW4 1Þ a2 ðW * 1Þ þ 1 ¼ 0; *
4a1 W3 a2 ¼ 0:
ð7Þ
*
These two equations determine critical boundary in the plane ða1 ; a2 Þ in the parametric form: a1 ¼ ð3W4 4W3 þ 1Þ1 ; *
*
a2 ¼ 4W3 ð3W4 4W3 þ 1Þ1 ; *
*
*
ð8Þ
where the critical temperature W serves as a parameter. The critical curve separates regions of parameters representing flashover and low intensity fire (Fig. 4).
3. Sprinkler interaction with the smoke layer Activation of a fire suppression system will have effect not only on fire suppression, but also on the possibility of flashover in the case where fire is not fully suppressed. In the case of sprinkler/water mist system, primary effect on flashover development will be cooling of the smoke layer. Such cooling will lead to subsequent reduction in the radiative feedback to fuel, which can be significant as radiation intensity depends on the smoke temperature in strongly nonlinear way. We consider a situation where the sprinkler/water mist system fails to suppress fire directly (e.g. due to obstruction to direct action of the spray on fuel or low
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momentum of the spray), and therefore its only influence on fire development is through absorption of heat from combustion products. The interaction between water spray and fire source is not considered here. The most interesting case which can lead to such a scenario is an application of a very fine water mist. There are well realised difficulties with delivering fine water droplets into the fire region. However, even if the spray is completely evaporated in the ceiling jet, its presence is likely to have a profound fire restriction effect. The model described by Eq. (1) conveniently provides the temperature evolution in the compartment as a solution of an ordinary differential equation. To include the effect of fire suppression system, an additional heat loss associated with water spray needs to be included into Eq. (1). The net heat loss to the spray would be determined by heat-up and mass loss rates of single droplets and their residence times within the hot layer. We consider the interaction of a sprinkler spray with the hot layer of thickness D and temperature T (Fig. 2) and assume uniform absorption of heat by the spray within the layer. This assumption is reasonably accurate if the width of the spray is comparable with the length scale of the fire ceiling jet, or if a number of water suppression systems are operating simultaneously over a large fire area. Activation temperatures for suppression systems are usually low compared to typical critical flashover temperatures, so activation times are assumed to be zero in the present study. In the presence of water spray, there will be an additional heat loss from the smoke layer. The regimes of heat transfer may be different depending on the spray and smoke layer characteristics. The two limiting cases are the purely convective heat loss to the spray without evaporation, and the complete evaporation of water mist. The intermediate case involves both convective and evaporative heat transfer. The above three regimes are analysed using analytical approaches, and the boundaries of applicability of each of these regimes are investigated.
4. Results and discussion 4.1. Coarse spray (non-evaporation) limit In many cases involving conventional sprinklers, heat transfer between smoke layer and water spray is predominantly convective [12]. This assumption becomes increasingly accurate for larger droplets and thinner smoke layers. For such conditions droplets would only experience heat-up to a certain temperature, and evaporation rate may be considered negligible [15]. The total heat absorption by a single droplet of diameter d in the absence of evaporation can be represented as follows: Qdrop ¼ 43pðd=2Þ3 rp cp ðTD T0 Þ;
ð9Þ
where rp is the density and cp is the specific heat of water. The temperature T0 is the initial droplet temperature; TD is the temperature of the droplet at the exit from the
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smoke layer. It is seen from here that the only parameter which should be obtained from the model is the droplet exit temperature TD : To calculate the heat loss rate to water spray Q’ spr ; it is necessary to solve the equations of motion and heat transfer for a single droplet. Since it is assumed that no evaporation takes place, the mass of the droplet does not change and the dynamic and heat-up problems for the droplet can be solved separately. The analytical solution for the droplet motion has been obtained in [11] using the laminar expression for the drag coefficient. Such approach underestimates the drag in the range of Reynolds numbers which are relevant for the sprinkler spray. More accurate approach is developed in the present study as demonstrated below. The general equation of the droplet motion through a quiescent environment (neglecting forces other than drag) may be written as a following set of the two equations for the velocity components [13]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dU p m ¼ d 2 rCD U 2 þ V 2 U; dt 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dV p 2 ¼ d rCD U 2 þ V 2 V þ mg: ð10Þ m dt 8 For a typical conventional sprinkler the volumetric mean diameter is of the order of 1–2 mm. Taking the characteristic droplet size as 1 mm, Reynolds particle number for the droplets exiting orifice at velocities of 10–20 m/s may be estimated to be Re B700–1500. In this range drag coefficient for spherical particles is a weak function of particle Reynolds number [13] and varies between 0.47 and 0.6. Therefore, it may be represented by a constant value with reasonable accuracy. Obviously, even with the constant drag coefficient the above equations are still coupled with each other. However, since droplets hit sprinkler deflector, their vertical velocity may be assumed to be small compared to the tangential velocity. Since drag coefficient is assumed to be constant, then in the case V5U Eq. (10) become decoupled and may be solved separately. The system is simplified as follows: dU p m ¼ d 2 rCD U 2 ; dt 8 dV p 2 ¼ d rCD UV þ mg: m dt 8
ð11Þ
The first equation contains only tangential component, and can be easily solved by separating variables. The solution has the following form: 1 1 p 2 U¼ d rCD t þ : ð12Þ U0 8m Substitution of this expression to the second (V-component) equation yields: 1 dV p 1 p ¼ drCD m þ d 2 rCD t V þ mg: ð13Þ dt 8 U0 8m
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To find solution of (13) one needs to solve homogeneous equation first (which is done by separating variables). Then a method of variation of parameters may be applied to find solution of non-homogeneous equation. The final solution of the Eq. (13) can be written as V¼
kðV0 ðg=2ÞkÞ g þ ðt þ kÞ; tþk 2
ð14Þ
4 rp d : 3CD r U0
ð15Þ
where k¼
Since the droplet distance from the ceiling, z is determined by Z t VðsÞ ds z¼
ð16Þ
0
(Fig. 2), then the following equation is easily derived to estimate droplet exit time t * from the smoke layer: g t* g ð17Þ k V0 k ln 1 þ þ t * ðt * þ 2kÞ ¼ D: 2 4 k The thickness of the smoke layer is expressed through the fractional height of the thermal discontinuity plane, D; and the height of compartment, H : D ¼ ð1 DÞH: The heat transfer equation for the droplet heat-up may be written as dTd 3 4 3pðd=2Þ rp cp dt
¼
kNu 2 pd ðT Td Þ: d
ð18Þ
The correlation for the Nusselt number in case of spherical droplets moving through the air is provided in the form [13]: Nu ¼ 2:0 þ Re1=2 Pr1=3 :
ð19Þ
It is seen from this expression that Nusselt number is a rather weak function of the particle Reynolds number (NuBRe1/2) and may be represented with sufficient accuracy by its constant average value. Droplet heat-up at the moment when the droplet exits smoke layer is obtained by straightforward integration of the Eq. (18): " " ## 6kNu TD T0 ¼ 1 exp t * ðT T0 Þ: ð20Þ rp c p d 2 Expressions (17) and (20) allow the calculation of heat absorption by the spray according to (9). The total heat loss rate to the polydisperse spray can be written in terms of droplet size distribution in the spray: Z N ’ cp Q’ spr ¼ C f ðsÞðTD T0 Þ ds; ð21Þ 0
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’ is a sprinkler discharge rate, f ðsÞ ¼ dFðsÞ=ds is the density of the where C distribution. The distribution function F is defined here in such a way that FðsÞ is a mass fraction of droplets having diameters dos: Taking into account the temperature change along the droplet trajectory (20) and the expression (21), the heat loss rate to the spray from the layer having the temperature T is given by " " ## Z N 6kNu ’ cp Q’ spr ¼ C f ðsÞ 1 exp t * ds ðT T0 Þ: ð22Þ rp c p d 2 0 Once the formula for the additional heat loss rate is obtained, the critical conditions for flashover can be analysed using the Eq. (4) and conditions (7). The form of the relationship (22) implies that heat loss rate to water spray is proportional to the temperature difference between the smoke layer and environment. This means that such loss can be accounted for by modification of the convective heat transfer coefficient, a2. In the presence of sprinkler, the general temperature history Eq. (4) has the same form, but with the modified convective heat loss coefficient: dW ¼ 1 þ a1 ðW4 1Þ A2 ðW 1Þ: dt
ð23Þ
The expression for the modified convective heat transfer coefficient follows from (22) immediately: " " ## ’ cp T0 Z N C 6kNu 1 exp t * ðsÞ f ðsÞ ds: ð24Þ A 2 ¼ a2 þ rp cp s2 Q’ 0 0 The transition from flashover to low intensity fire due to water discharge by sprinkler may be interpreted in the plane of governing parameters ða1 ; a2 Þ; as shown in Fig. 4. Suppose the coefficient a1 is fixed and the system is initially in the flashover area (Fig. 4). As water discharge rate (and the coefficient a2 correspondingly) increases, the point ða1 ; a2 Þ moves to the right until it crosses the flashover boundary at some critical value of a2 : The critical water discharge rate will depend on the distance between the point ða1 ; a2 Þ representing initial parameters and the critical curve which we denote as a2 ¼ jða1 Þ: The critical sprinkler discharge rate to prevent flashover is therefore given by "Z " " ## #1 N 6kNu Q’ 0 ’ * Ccr ¼ 1 exp t ðsÞ f ðsÞ ds ðjða1 Þ a2 Þ: ð25Þ rp cp s2 cp T0 0 The expressions (5) for the coefficients a1 ; a2 allow the critical water application rate to be determined as a function of any geometrical or physical parameters of the fire. For a monodisperse spray, critical water application rate is plotted in Fig. 5 for different smoke layer thickness. It is apparent that the dependence on the hot layer thickness is very weak, especially for large droplets. The assumption of nonevaporation is not true below some minimum diameter. Due to this reason, the
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Fig. 5. Critical water application rates for purely convective heat transfer regime. Different smoke layer thickness: (1) D ¼ 0:2 m; (2) D ¼ 0:4 m; (3) D ¼ 0:6 m; (4) D ¼ 0:8 m.
curves in Fig. 5 cover different ranges of droplet diameters. Exact boundaries for applicability of different regimes of the spray/hot layer interaction will be given below. 4.2. Fine spray (water mist) limit Very fine water mist may be expected to evaporate completely in the hot smoke layer. Generally, the results of CFD simulations [14] show that it is generally the case for the droplets with diameters under 100 mm. In this case the opposite limit (complete evaporation) is achieved. In fact, water mist systems often fail to deliver mist into the burning region and suppress fire. However, they may still be able to prevent flashover. Assuming total evaporation, the temperature history equation takes the form: dW ’; ¼ 1 þ a1 ðW4 1Þ a2 ðW 1Þ gC dt
ð26Þ
where the additional heat sink is proportional to the water application rate and the parameter g, defined as g¼
cp ðTw T0 Þ þ Hf g : Q’ 0
ð27Þ
The Eqs. (26) and (27) are obtained upon assuming total evaporation of the spray, ’ ðcp ðTw T0 Þ þ Hf g Þ; and non-dimensionalising which delivers a heat loss rate of C this expression using the heat release rate Q’ 0 :
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Fig. 6. Critical water application rates for complete evaporation regime. Different values of the heat loss coefficient a2 : (1) a2 ¼ 1:0; (2) a2 ¼ 1:5; (3) a2 ¼ 2:0; (4) a2 ¼ 2:5:
The critical conditions (7) are written in this case as follows: ’; a1 ðW4 1Þ a2 ðW * 1Þ þ 1 ¼ gC * 4a1 W3 *
a2 ¼ 0
ð28Þ
from which the following expression for the critical water application rate follows: " # 4=3 ! 1=3 ! a a 2 2 1 ’ cr ¼ g C a1 1 a2 1 þ 1 : ð29Þ 4a1 4a1 Critical water application rate increases as a function of a1 (Fig. 6), as higher values of this parameter imply higher heat gain rates. The curves starting points (Fig. 6) correspond to the points of the critical curve a2 ¼ jða1 Þ: In a more general sense, the critical condition (29) may be interpreted in the space ’ ; a1 ; a2 ), rather than on the plane as it is the of the three independent parameters (C case for freely burning fire [5,6]. The Eq. (29) determines the critical surface in that space shown in Fig. 7. The flashover area is below the surface (between the critical surface and the plane ða1 ; a2 Þ). Above the surface the flashover is impossible. The critical surface intersects with the ða1 ; a2 Þ plane exactly by the critical curve a2 ¼ jða1 Þ: 4.3. General case Generally, water droplets evaporate as they move through the hot gas. Once droplet is injected into the hot layer, it will experience a period of unsteady evaporation, followed by a steady-state evaporation regime, characterised by constant surface droplet temperature (‘‘wet bulb temperature’’) and constant surface regression rate [15].
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Fig. 7. Critical surface for flashover in the case of water mist limit.
In the first approximation, it is sufficiently accurate to assume that the droplet evolution can be split into two stages: 1. heat-up to wet bulb temperature without evaporation, and 2. steady-state evaporation conditions. At high environment temperatures, wet bulb temperature approaches boiling temperature, so we assume that the steady-state evaporation occurs at surface boiling temperature. During both stages the equations of motion (11) apply, which can be written in the equivalent form: dU 3rCD ¼ dt 4rp dV 3rCD ¼ dt 4rp
1 2 U ; d 1 UV þ g: d
ð30Þ
During the heat-up phase, the droplet diameter does not change, and the solution is the same as for the set of Eqs. (11), that is given by the formulae (12) and (14). During the steady-state evaporation, the equations of motion (30) must be solved with the droplet diameter changing as a function of time. This function is given by the theoretically and experimentally observed ‘‘d 2 -law’’, which implies d 2 ðtÞ ¼ d02 ðtÞ kev t:
ð31Þ
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The parameter kev is a so-called evaporation constant, and may be taken in the form [15]: 8lg Cg ðT Tw Þ kev ¼ ln 1 þ : ð32Þ Hf g rp Cg With the diameter regression rate (31), the Eqs. (30) can still be integrated analytically. The solution can be written in the following form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# #1 " 3rCD d0 kev 1 1 1 2t þ ; U¼ U1 2rp kev d0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " 2d02 kev a 1 kev g 1 2t 1 2t þK 2 3 kev d0 d0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V¼ ; kev a 1 2t d0 "
ð33Þ
where the parameters K and a are given by d02 ga þ ða 1Þ V1 ; kev 2rp kev 1 : a¼1þ 3rCD d0 U1
K¼
ð34Þ
(U1 ; V1 are the droplet velocity components at the end of the heat-up stage). Heat loss to the spray for the general case is calculated using the residence time of the droplet in the smoke layer. The criterion for the general case is Z
t1
ð35Þ
VðsÞ dsoD; 0
where t1 is a droplet heat-up time to the boiling temperature. The residence time is determined using the following equation for the evaporation period time t * * : Z
t1 1
V ðsÞ ds þ 0
Z
t* *
V 2 ðsÞ ds ¼ D;
ð36Þ
0
where V 1 ðsÞ and V 2 ðsÞ refer to the solutions (14) and (33), respectively. The residence time is obtained as t2 ¼ t1 þ t * * :
ð37Þ
The Eq. (36) does not have solution if a droplet evaporates completely in the layer. In that case, the residence time is taken as t1 þ tev where tev is the droplet
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evaporation time tev ¼
d02 : kev
ð38Þ
For calculation of the total heat loss rate to the spray we note that the heat loss due to a single droplet is Qd0 ¼ md0 cp ðTw T0 Þ þ Hf g Dmd0 ;
ð39Þ
where the droplet mass loss is calculated as Dmd0 ¼ 16prp ½d03 ðd02 kev tÞ3=2 : The total heat loss rate to the spray is then obtained as " 3=2 ## Z N" k ev ’ cp ðTw T0 Þ þ Hf g 1 1 2 t * * ðsÞ f ðsÞ ds: Q’ spr ¼ C s 0
ð40Þ
ð41Þ
4.4. Boundaries for the limiting cases The critical conditions (25) and (29) are obviously different and depend on the regime of heat loss from the spray (predominantly convective or predominantly evaporative). It is important to estimate the regions of applicability of the each of these regimes and corresponding critical conditions. In this section, we give the boundaries of the two limiting regimes, that is nonevaporation and complete evaporation regimes. For any given droplet diameter d; temperature of the smoke layer T and the layer thickness D the boundary for non-evaporation regime corresponds to conditions where the droplet exit temperature is exactly Tw : This condition is given by rp cp t* T T0 ln ¼ ; ð42Þ d 2 6kNu T Tw which determines minimum diameter of droplets which do not undergo evaporation. The droplet exit time should be calculated from (17). The boundary for the opposite limiting case, complete evaporation, is calculated using the solutions (33) and (34) for a droplet evaporating in a smoke layer. This boundary would correspond to the maximum diameter of droplets which can be completely evaporated by a layer of a given temperature and thickness. The calculated boundaries are presented in Fig. 8 for the two different gas temperatures as functions of the smoke layer thickness. The boundaries depend rather weakly on the thickness D: The gap in droplet diameters between the two regimes is of the order of 200–300 mm. For the droplets in this gap neither of the limits apply, and the spray influence on flashover is determined by the heat loss calculated according to the general case solution (41).
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Fig. 8. Limits of applicability of different heat transfer regimes FFF convective heat transfer - - - - - - - - complete evaporation. Smoke layer temperatures: (1) T ¼ 700 K; (2) T ¼ 500 K.
4.5. Example of calculation of the critical flashover conditions In order to estimate the influence of a sprinkler, consider the example given in [5] for the fire with the heat release rate of 155 kW/m2 and radius of burning area of 0.15 m. Compartment dimensions for this case are 0.4 m 0.4 m 0.4 m. Fractional height of the thermal discontinuity plane is taken as 0.5. The other relevant parameters may be found in [5]. Controlling parameters can be calculated as a1 ¼ 0:0018; a2 ¼ 0:45: Eq. (4) gives the flashover time ([5]) of 14.6 s. In the case of non-evaporation limit, the critical value of a2 is required. This value has been found in [5] to be A 2 ¼ 0:49: The critical flashover temperature for these conditions may be found from (8) to be 1172 K. Non-evaporative regime for such critical temperature occurs for all droplets with the diameters higher than 480 mm, which can be demonstrated as explained above, using the condition (42). Therefore, if as an example we consider a monodisperse spray with the droplets of 500 mm in diameter, purely convective heat transfer conditions will apply for all temperatures up to the critical. The minimum water flow rate required to prevent ’ cr ¼ 2:99 g=s: flashover can be calculated from (25) to be C General dependencies of the critical application rates on droplet diameter (for a monodisperse spray) are presented in Fig. 9 for different parameters a1 ; a2 in the flashover region. For small droplets, full evaporation limit applies, and the critical flow rate is constant up to a certain droplet diameter (region A in Fig. 9). In the region C, Fig. 9 (non-evaporation limit), the critical application rate is determined by the relationship (25). The boundaries of the limiting regions are determined in the way, similar to Fig. 8. There is an intermediate region between the two limiting regions. In this region, critical water requirements should be increasing with the diameter, which is schematically shown by dotted
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Fig. 9. Water application rates required for flashover prevention as a function of droplet diameter in a monodisperse spray. A: region of complete evaporation (mist limit), B: intermediate region, C: region of convective heat transfer (non-evaporation limit), – – – – boundaries of the limiting regimes. Fire parameters: Q’ 0 ¼ 100 kW; D ¼ 0:5 m; a2=0.5. (1) a1 ¼ 0:011; (2) a1 ¼ 0:059; (3) a1 ¼ 0:018:
lines in Fig. 9. The calculation of the exact dependence is, however, more complicated. The critical application rate should be found from the following set of equations: ’ cr ; a1 ðW4 1Þ a2 ðW * 1Þ þ 1 ¼ g* C *
4a1 W3 a2 ¼ *
d*g ’ cr ; ðW ÞC dW *
where the parameter g* is given by " 3=2 ## Z N" 1 kev * * g* ¼ cp ðTw T0 Þ þ Hf g 1 1 2 t ðsÞ f ðsÞ ds: s Q’ 0 0
ð43Þ
ð44Þ
Note, that the conditions (43) and (44) also describe a full evaporation regime as a limiting case. In such case t * * ðsÞ ¼ tev : As expected, water mist (if evaporated uniformly through the smoke layer) is most effective in the prevention of flashover. Non-evaporation case provides the upper limit for the critical water application rate. For most real sprays, the critical rate will lie between the two limiting cases and may be calculated using the conditions for the intermediate case (43). In both limits the critical discharge rate is much less than used in real operations of sprinklers and water mist systems, which suggests that fire control is achievable with much more economic water supply rates. Finally, we give the most general equations which are applicable for a polydisperse spray. For such (most practically important) sprays, different droplets would be in the different regimes of heat transfer. Most fine droplets are expected to evaporate completely, while for the largest droplets evaporation may be negligible. For each value of the non-dimensional temperature W the droplet size distribution is divided
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by some droplet diameter d * ðWÞ into the ‘‘fine’’ (dpd * ðWÞ) and the ‘‘coarse’’ (d > dðWÞ) tails. Conditions (43) are applicable for the ‘‘fine’’ part, and nonevaporation conditions (24) are applicable for the ‘‘coarse’’ part. These considerations lead to the following general formulation for the critical flashover conditions: ’ cr ; a1 ðW4 1Þ A2 ðW ÞðW 1Þ þ 1 ¼ g* C *
*
*
dA2 d*g ’ cr ; ðW * ÞðW * 1Þ A2 ðW * Þ ¼ ðW * ÞC 4a1 W3 * dW dW
ð45Þ
" " ## ’ cr cp T0 Z N C 6kNu A2 ðW * Þ ¼ a2 þ 1 exp t * ðsÞ f ðsÞ ds r p cp s 2 Q’ 0 d * ðW Þ *
ð46Þ
where
and 1 g* ðW * Þ ¼ Q’ 0
Z
d * ðW Þ *
0
"
" cp ðTw T0 Þ þ Hf g
3=2 ## kev 1 1 2 t * * ðsÞ f ðsÞ ds: s ð47Þ
The function d * ðW * Þ is determined as a boundary for the non-evaporation regime (similarly to upper curves, Fig. 8). Using this function, the critical application rate ’ cr may be determined from (45). C
5. Conclusions Flashover development of fire under suppression conditions has been investigated by expanding the existing flashover zone modeling to the case of water sprinkler/ water mist system activation. Analytical models have been developed to consider heat transfer between water spray and hot smoke layer in different regimes. It has been shown that the equations of droplet motion and heat transfer can be solved analytically with sufficient accuracy in the cases of convective heat loss and steady-state evaporation. As limiting cases, these models include cases of purely convective and complete evaporation regimes. This analytical approach can be successfully used in the further development of fire zone models involving water sprinkler applications. Critical conditions for flashover during application of water-based fire suppression systems has been formulated, and critical water requirements to prevent flashover have been estimated. The derived critical conditions allow the minimum water flow rate, required to prevent flashover, to be found as a function of droplet size distribution in the spray and fire geometrical and physical parameters. Critical water discharge rate in the case of water mist system is less than that for conventional sprinkler, and therefore such systems are more effective in the prevention of flashover. The present estimations show that despite problems with
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delivery of the suppressant into the fire region, water mist systems can still be very effective in fire control because of their ability to prevent flashover. For either water mist or conventional sprinklers, the amount of water required to prevent flashover is significantly less than delivered under regular operating conditions by commercial fire suppression systems. Limitations of the current model may arise from the following most important sources: (i) actual non-uniformity of heat absorption by the spray, and (ii) air motion (entrainment into the spray), which is currently neglected, but can change evaporation dynamics at high water discharge rates. As critical water rates reported in the present paper are low, air motion may be neglected. It may, however, play significant role in heat exchange for large compartments and high water release rates. In order to increase the accuracy of predictions and remove simplifying assumptions, it would be beneficial to investigate the interaction between hot smoke layer and water spray by means of numerical CFD simulations. Such simulations can provide more accurate distributions of the heat loss rate Q’ spr under various conditions. The degree of uniformity of heat absorption within the smoke layer may also be examined in this way.
References [1] Luo M, Beck V. A study of non-flashover and flashover fires in a full-scale multi-room building. Fire Saf J 1996;26:191–219. [2] Luo M, He Y, Beck V. Application of field model and two-zone model to flashover fires in a full-scale multi-room single level building. Fire Saf J 1997;29:1–25. [3] Thomas PH. Fires and flashover in roomsFa simplified theory. Fire Saf J 1980/81;3:67–76. [4] Thomas PH. The growth of fireFignition till full involvement. In: Cox G, editor. Combustion Fundamentals of Fire. New York: Academic Press, 1995. p. 273–328. [5] Graham TL, Makhviladze GM, Roberts JP. On the theory of flashover development. Fire Saf J 1995;25:229–59. [6] Graham TL, Makhviladze GM, Roberts JP. The effects of the thermal inertia of the walls upon flashover development. Fire Saf J 1999;32:35–60. [7] Beard AN, Drysdale DD, Holborn PG, Bishop SR. A non-linear model of flashover. Fire Sci Technol 1992;12:11–27. [8] Bishop SR, Holborn PG, Beard AN, Drysdale DD. Nonlinear dynamics of flashover in compartment fires. Fire Saf J 1993;21:11–45. [9] Holborn PG, Bishop SR, Drysdale DD, Beard AN. Experimental and theoretical models of flashover. Fire Saf J 1993;21:257–66. [10] Beard AN, Drysdale DD, Holborn PG, Bishop SR. A model of instability and flashover. J Appl Fire Sci 1994/1995;4:3–16. [11] Novozhilov V, Kent JH. Effect of Sprinkler Operation on Flashover Development of Compartment Fires. 1999 Australian Symposium on Combustion and the Sixth Australian Flame Days, The University of Newcastle. pp. 175–79. [12] Kung HC. Cooling of room fires by sprinkler spray. ASME Trans J Heat Transfer 1977;99:353. [13] Faeth GM. Evaporation and combustion of sprays. Prog Energy Combust Sci 1983;9:1–76. [14] Novozhilov V, Moghtaderi B, Fletcher DF, Kent JH. Numerical simulation of enclosed gas fire extinguishment by a water spray. J Appl Fire Sci 1995–96;5(2):135–46. [15] Lefebvre AH. Atomization and Sprays. New York: Hemisphere, 1989.
Flashover control under fire suppression conditions V. Novozhilov* School of Mechanical and Production Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore Received 30 May 2000; received in revised form 12 February 2001; accepted 22 March 2001
Abstract Possibility of flashover development under fire suppression conditions is investigated using a zone model approach. Analytical models are developed to model the effect of water sprinklers and water mist systems interactions with the ceiling smoke layer, and to assess the effects of such interactions on flashover development. Different regimes of heat transfer between water spray and smoke layer are investigated, and the limits of these regimes are identified. Critical conditions for flashover development during suppression are formulated. The derived conditions allow the critical water application rates required for flashover prevention to be found as functions of spray and fire characteristics. r 2001 Elsevier Science Ltd. All rights reserved. Keywords: Flashover; Fire suppression; Zone model
1. Introduction Flashover is a stage in compartment fire development which can be described as a rapid transition from a slowly growing to a fully developed fire. The underlying mechanism in this phenomenon is essentially a positive feedback from fire environment to the burning fuel. Formation of a hot ceiling layer at the early stages of fire leads to radiative feedback to the fuel, which, in turn, results in increase in the burning rate and the temperature of the smoke layer. If heat losses from the compartment are insufficient, a sharp increase in the fire’s power (i.e. flashover) will eventually occur. *Tel.: +65-790-4335; fax: +65-791-1859. E-mail address: [email protected] (V. Novozhilov). 0379-7112/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 9 - 7 1 1 2 ( 0 1 ) 0 0 0 1 9 - 4
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Nomenclature a1;2 A Af cp CD d D F f G’ h H HV Hf g L’ LC m m’ out Nu Q’ 0 q00 Re t T U V WC WV z
model coefficients surface area fire burning area specific heat drag coefficient droplet diameter fractional height of the inter-zone boundary droplet size distribution density of droplet size distribution rate of heat gains into smoke layer convective heat transfer coefficients height of compartment height of vent latent heat of water vaporisation rate of heat losses from smoke layer length of compartment droplet mass mass outflow rate from compartment Nusselt number heat release rate of fire heat flux to fuel from fire Reynolds number time temperature tangential velocity component of the droplet vertical velocity component of the droplet width of compartment width of vent coordinate normal to ceiling
Greek symbols a emissivity D smoke layer thickness Dhc heat of combustion Dhvap heat of vaporisation of fuel w efficiency of combustion W dimensionless temperature r density s Stefan-Boltzman constant t dimensionless time ’ C water discharge rate
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Subscripts 0 initial w wet bulb temperature cr critical p particle spr spray u upper zone v vents Theoretically, flashover has been studied by both non-linear and computational fluid dynamic (CFD) approaches. Luo et al. [1,2] demonstrated that detailed CFD modeling can predict flashover development in a complicated multi-room geometry. The application of field modeling to flashover is, however, still a difficult task because of strict computational requirements on the accuracy of radiation modeling and the lack of reliable models of flame spread over various fuels. Disadvantage of CFD approach is that each solution is only applicable to a particular physical configuration. On the other hand, flashover can be rather effectively analysed using zone modeling of fire and methods of non-linear dynamics. Early attempts to investigate thermal instabilities, which may be interpreted as flashover, were made by Thomas [3,4]. Further investigations of flashover by methods of non-linear dynamics have been conducted in a number of studies [5–10]. Essentially, the non-linear models have been using one [5–7], two [8,9] or three [10] state variables to describe fire development. From the non-linear dynamics point of view, flashover may be interpreted as bifurcation in a space of appropriate controlling parameters. It has been shown that in some models such bifurcations can be described as cusp catastrophes of controlling surfaces, representing fire evolution in a compartment. Early in fire development the combustion products are usually segregated in a wellstirred ceiling layer with roughly homogeneous properties. For this reason zone models are able to achieve reasonably good qualitative and quantitative agreement with experiments, as has been demonstrated in [9]. In some cases, they can also provide valuable analytical solutions which clearly identify the important physical effects and have general meaning, in contrast to case-oriented field model results. The mechanism of positive non-linear feedback makes flashover phenomenon similar in many respects to the classical thermal explosion theory. In a most consistent way, this analogy has been used in [5,6]. The zone model developed in [5,6] allows prediction of the critical conditions for flashover to be made, and the influence of different physical or geometrical parameters to be investigated. However, the problem of flashover development in the presence of fire suppression systems has not been addressed in the previous studies. Since it is well known that fire suppression systems actually often fail to suppress fire, fire evolution under suppression conditions should be predicted. If fire is not suppressed, the question of whether or not flashover will happen still needs to be answered.
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The present study is concerned with the influence of sprinkler and water mist systems on possible flashover development in compartment fires. It is clear that even if the water spray cannot suppress fire completely, it can restrict fire growth and prevent flashover due to cooling of the hot smoke layer. Therefore, the effect of sprinkler operation on flashover is of significant interest for fire safety. Preliminary considerations of this effect by means of zone modeling have been made by Novozhilov and Kent [11]. In the present study, a more sophisticated zone model, which covers a wider range of operating conditions, is developed to predict effect of water-based fire suppression systems on flashover development. The model provides more accurate estimations of critical water application rates and allows different regimes of water spray interaction with smoke layer to be considered. Critical conditions for flashover are obtained as correlations between fire characteristics and sprinkler discharge rate. The results are generalised for arbitrary droplet size distributions.
2. Flashover zone modeling and the thermal explosion theory The considerations of the present study are based on the zone model [5], which expands classical thermal explosion theory to flashover phenomenon and represents general and convenient way to investigate flashover conditions. We briefly overview here the results necessary for the present study. Principal assumptions and results of the model [5] are as follows. In an enclosure with one opening (Fig. 1), flashover is principally described by four stages. The hot buoyant plume develops at the first stage following ignition, and then reaches the ceiling and spreads as a ceiling jet (second stage). During the third and fourth stages, the hot layer expands and deepens, and establishment of the flow through the opening takes place.
Fig. 1. Diagram of the compartment.
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Fig. 2. Schematic of the zone model.
At the second stage a well-stirred layer of combustion products is formed, and a zone approach may be applied. Under this approach, the compartment may be divided into two layers which are represented by average temperatures T and T0 (Fig. 2). Flashover is assumed to happen during the initial, fuel-controlled stage of fire, so that the temperature of the lower layer may be assigned an ambient value. The thickness of the smoke layer, D; is considered as an input parameter and does not change with time. Other assumptions are listed in [5]. Application of zone modeling involves the consideration of the heat balance equation for the upper hot smoke layer which is: dT mcp ¼ G’ L’ ; ð1Þ dt where t is time, m; cp and T are mass, specific heat and temperature of the smoke layer, respectively. The functions G’ and L’ on the right-hand side describe rates of heat gains and losses for the layer. In the absence of water spray, the right-hand side should generally include the terms corresponding to the rates of heat gain for the smoke layer and heat losses due to outflow through the opening and due to convective and radiation losses to compartment boundaries and fuel. In the present study, we only consider the case of large thermal inertia of compartment walls, which implies that the time scale for the wall heat-up is much larger than that for flashover. This assumption is not restrictive as it retains all the major effects in the system behaviour while removing unnecessary mathematical complications. The cases with low and intermediate thermal inertia may also be considered [5,6]. The Eq. (1) is non-dimensionalised based on the ambient temperature T0, heat release rate of fire Q’ 0 : ð2Þ Q’ 0 ¼ wAf ðq00 =Dhvap ÞDhc and the characteristic heating time of the upper layer: t ¼ mcp T0 =Q’ 0 : *
ð3Þ
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In dimensionless variables the Eq. (1) for the case of large thermal inertia of compartment walls is reduced to [5]: dW ¼ 1 þ a1 ðW4 1Þ a2 ðW 1Þ: dt
ð4Þ
The model coefficients a1;2 are determined by both fire physical parameters and dimensions of the compartment (Fig. 1): a1 ¼ w
Dhc aU sAf T04 =Q’ 0 ag s½Ac þ ð1 DÞAV Af T04 =Q’ 0 Dhvap
ag sAf T04 =Q’ 0 ag s½AU ð1 DÞAV T04 =Q’ 0 ; a2 ¼ ½AU ð1 DÞAV hc T0 =Q’ 0 þ ð1 DÞAV hc T0 =Q’ 0 þ m’ out ð1 DÞAV cp T0 =Q’ 0
ð5Þ
and have clear physical meaning. The first coefficient describes the rate of net heat gains into the smoke layer due to radiation. Parameter a2 characterises the net convective heat losses from the hot layer which occur due to outflow through the opening and the heat exchange between the smoke and cold surroundings. The problem of flashover resembles the classical thermal explosion theory because in both cases the system behaviour is essentially determined by competition between heat gains and losses. As has been described in [5], the Eq. (4) can be investigated by similar techniques which are well developed in the thermal explosion theory. The quantitative analysis is provided by a diagram shown in Fig. 3. In general, there can be three solutions of the heat balance Eq. (1). The lower and upper points of intersection between heat gain and heat loss curves are stable and represent fuelcontrolled and ventilation-controlled fires, respectively. The intermediate solution is unstable, and small perturbations around this point will result in a large change of temperature. The critical conditions for flashover existence are determined by both
Fig. 3. Semenov’s diagram for flashover. Adapted from [5]. PFsteady-state fire conditions; P Fflashover point; W Fcritical temperature; G’ Frate of heat gain into the smoke layer; L’ Frate of heat loss from the smoke layer.
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Fig. 4. Flashover critical boundary [5] and the effect of water discharge rate.
the equations G’ ðW * Þ ¼ L’ ðW * Þ and dG’ dL’ ðW * Þ ¼ ðW Þ: ð6Þ dW dW * It easily follows from Eq. (4) that the critical conditions are written in the following form: a1 ðW4 1Þ a2 ðW * 1Þ þ 1 ¼ 0; *
4a1 W3 a2 ¼ 0:
ð7Þ
*
These two equations determine critical boundary in the plane ða1 ; a2 Þ in the parametric form: a1 ¼ ð3W4 4W3 þ 1Þ1 ; *
*
a2 ¼ 4W3 ð3W4 4W3 þ 1Þ1 ; *
*
*
ð8Þ
where the critical temperature W serves as a parameter. The critical curve separates regions of parameters representing flashover and low intensity fire (Fig. 4).
3. Sprinkler interaction with the smoke layer Activation of a fire suppression system will have effect not only on fire suppression, but also on the possibility of flashover in the case where fire is not fully suppressed. In the case of sprinkler/water mist system, primary effect on flashover development will be cooling of the smoke layer. Such cooling will lead to subsequent reduction in the radiative feedback to fuel, which can be significant as radiation intensity depends on the smoke temperature in strongly nonlinear way. We consider a situation where the sprinkler/water mist system fails to suppress fire directly (e.g. due to obstruction to direct action of the spray on fuel or low
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momentum of the spray), and therefore its only influence on fire development is through absorption of heat from combustion products. The interaction between water spray and fire source is not considered here. The most interesting case which can lead to such a scenario is an application of a very fine water mist. There are well realised difficulties with delivering fine water droplets into the fire region. However, even if the spray is completely evaporated in the ceiling jet, its presence is likely to have a profound fire restriction effect. The model described by Eq. (1) conveniently provides the temperature evolution in the compartment as a solution of an ordinary differential equation. To include the effect of fire suppression system, an additional heat loss associated with water spray needs to be included into Eq. (1). The net heat loss to the spray would be determined by heat-up and mass loss rates of single droplets and their residence times within the hot layer. We consider the interaction of a sprinkler spray with the hot layer of thickness D and temperature T (Fig. 2) and assume uniform absorption of heat by the spray within the layer. This assumption is reasonably accurate if the width of the spray is comparable with the length scale of the fire ceiling jet, or if a number of water suppression systems are operating simultaneously over a large fire area. Activation temperatures for suppression systems are usually low compared to typical critical flashover temperatures, so activation times are assumed to be zero in the present study. In the presence of water spray, there will be an additional heat loss from the smoke layer. The regimes of heat transfer may be different depending on the spray and smoke layer characteristics. The two limiting cases are the purely convective heat loss to the spray without evaporation, and the complete evaporation of water mist. The intermediate case involves both convective and evaporative heat transfer. The above three regimes are analysed using analytical approaches, and the boundaries of applicability of each of these regimes are investigated.
4. Results and discussion 4.1. Coarse spray (non-evaporation) limit In many cases involving conventional sprinklers, heat transfer between smoke layer and water spray is predominantly convective [12]. This assumption becomes increasingly accurate for larger droplets and thinner smoke layers. For such conditions droplets would only experience heat-up to a certain temperature, and evaporation rate may be considered negligible [15]. The total heat absorption by a single droplet of diameter d in the absence of evaporation can be represented as follows: Qdrop ¼ 43pðd=2Þ3 rp cp ðTD T0 Þ;
ð9Þ
where rp is the density and cp is the specific heat of water. The temperature T0 is the initial droplet temperature; TD is the temperature of the droplet at the exit from the
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smoke layer. It is seen from here that the only parameter which should be obtained from the model is the droplet exit temperature TD : To calculate the heat loss rate to water spray Q’ spr ; it is necessary to solve the equations of motion and heat transfer for a single droplet. Since it is assumed that no evaporation takes place, the mass of the droplet does not change and the dynamic and heat-up problems for the droplet can be solved separately. The analytical solution for the droplet motion has been obtained in [11] using the laminar expression for the drag coefficient. Such approach underestimates the drag in the range of Reynolds numbers which are relevant for the sprinkler spray. More accurate approach is developed in the present study as demonstrated below. The general equation of the droplet motion through a quiescent environment (neglecting forces other than drag) may be written as a following set of the two equations for the velocity components [13]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dU p m ¼ d 2 rCD U 2 þ V 2 U; dt 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dV p 2 ¼ d rCD U 2 þ V 2 V þ mg: ð10Þ m dt 8 For a typical conventional sprinkler the volumetric mean diameter is of the order of 1–2 mm. Taking the characteristic droplet size as 1 mm, Reynolds particle number for the droplets exiting orifice at velocities of 10–20 m/s may be estimated to be Re B700–1500. In this range drag coefficient for spherical particles is a weak function of particle Reynolds number [13] and varies between 0.47 and 0.6. Therefore, it may be represented by a constant value with reasonable accuracy. Obviously, even with the constant drag coefficient the above equations are still coupled with each other. However, since droplets hit sprinkler deflector, their vertical velocity may be assumed to be small compared to the tangential velocity. Since drag coefficient is assumed to be constant, then in the case V5U Eq. (10) become decoupled and may be solved separately. The system is simplified as follows: dU p m ¼ d 2 rCD U 2 ; dt 8 dV p 2 ¼ d rCD UV þ mg: m dt 8
ð11Þ
The first equation contains only tangential component, and can be easily solved by separating variables. The solution has the following form: 1 1 p 2 U¼ d rCD t þ : ð12Þ U0 8m Substitution of this expression to the second (V-component) equation yields: 1 dV p 1 p ¼ drCD m þ d 2 rCD t V þ mg: ð13Þ dt 8 U0 8m
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To find solution of (13) one needs to solve homogeneous equation first (which is done by separating variables). Then a method of variation of parameters may be applied to find solution of non-homogeneous equation. The final solution of the Eq. (13) can be written as V¼
kðV0 ðg=2ÞkÞ g þ ðt þ kÞ; tþk 2
ð14Þ
4 rp d : 3CD r U0
ð15Þ
where k¼
Since the droplet distance from the ceiling, z is determined by Z t VðsÞ ds z¼
ð16Þ
0
(Fig. 2), then the following equation is easily derived to estimate droplet exit time t * from the smoke layer: g t* g ð17Þ k V0 k ln 1 þ þ t * ðt * þ 2kÞ ¼ D: 2 4 k The thickness of the smoke layer is expressed through the fractional height of the thermal discontinuity plane, D; and the height of compartment, H : D ¼ ð1 DÞH: The heat transfer equation for the droplet heat-up may be written as dTd 3 4 3pðd=2Þ rp cp dt
¼
kNu 2 pd ðT Td Þ: d
ð18Þ
The correlation for the Nusselt number in case of spherical droplets moving through the air is provided in the form [13]: Nu ¼ 2:0 þ Re1=2 Pr1=3 :
ð19Þ
It is seen from this expression that Nusselt number is a rather weak function of the particle Reynolds number (NuBRe1/2) and may be represented with sufficient accuracy by its constant average value. Droplet heat-up at the moment when the droplet exits smoke layer is obtained by straightforward integration of the Eq. (18): " " ## 6kNu TD T0 ¼ 1 exp t * ðT T0 Þ: ð20Þ rp c p d 2 Expressions (17) and (20) allow the calculation of heat absorption by the spray according to (9). The total heat loss rate to the polydisperse spray can be written in terms of droplet size distribution in the spray: Z N ’ cp Q’ spr ¼ C f ðsÞðTD T0 Þ ds; ð21Þ 0
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’ is a sprinkler discharge rate, f ðsÞ ¼ dFðsÞ=ds is the density of the where C distribution. The distribution function F is defined here in such a way that FðsÞ is a mass fraction of droplets having diameters dos: Taking into account the temperature change along the droplet trajectory (20) and the expression (21), the heat loss rate to the spray from the layer having the temperature T is given by " " ## Z N 6kNu ’ cp Q’ spr ¼ C f ðsÞ 1 exp t * ds ðT T0 Þ: ð22Þ rp c p d 2 0 Once the formula for the additional heat loss rate is obtained, the critical conditions for flashover can be analysed using the Eq. (4) and conditions (7). The form of the relationship (22) implies that heat loss rate to water spray is proportional to the temperature difference between the smoke layer and environment. This means that such loss can be accounted for by modification of the convective heat transfer coefficient, a2. In the presence of sprinkler, the general temperature history Eq. (4) has the same form, but with the modified convective heat loss coefficient: dW ¼ 1 þ a1 ðW4 1Þ A2 ðW 1Þ: dt
ð23Þ
The expression for the modified convective heat transfer coefficient follows from (22) immediately: " " ## ’ cp T0 Z N C 6kNu 1 exp t * ðsÞ f ðsÞ ds: ð24Þ A 2 ¼ a2 þ rp cp s2 Q’ 0 0 The transition from flashover to low intensity fire due to water discharge by sprinkler may be interpreted in the plane of governing parameters ða1 ; a2 Þ; as shown in Fig. 4. Suppose the coefficient a1 is fixed and the system is initially in the flashover area (Fig. 4). As water discharge rate (and the coefficient a2 correspondingly) increases, the point ða1 ; a2 Þ moves to the right until it crosses the flashover boundary at some critical value of a2 : The critical water discharge rate will depend on the distance between the point ða1 ; a2 Þ representing initial parameters and the critical curve which we denote as a2 ¼ jða1 Þ: The critical sprinkler discharge rate to prevent flashover is therefore given by "Z " " ## #1 N 6kNu Q’ 0 ’ * Ccr ¼ 1 exp t ðsÞ f ðsÞ ds ðjða1 Þ a2 Þ: ð25Þ rp cp s2 cp T0 0 The expressions (5) for the coefficients a1 ; a2 allow the critical water application rate to be determined as a function of any geometrical or physical parameters of the fire. For a monodisperse spray, critical water application rate is plotted in Fig. 5 for different smoke layer thickness. It is apparent that the dependence on the hot layer thickness is very weak, especially for large droplets. The assumption of nonevaporation is not true below some minimum diameter. Due to this reason, the
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Fig. 5. Critical water application rates for purely convective heat transfer regime. Different smoke layer thickness: (1) D ¼ 0:2 m; (2) D ¼ 0:4 m; (3) D ¼ 0:6 m; (4) D ¼ 0:8 m.
curves in Fig. 5 cover different ranges of droplet diameters. Exact boundaries for applicability of different regimes of the spray/hot layer interaction will be given below. 4.2. Fine spray (water mist) limit Very fine water mist may be expected to evaporate completely in the hot smoke layer. Generally, the results of CFD simulations [14] show that it is generally the case for the droplets with diameters under 100 mm. In this case the opposite limit (complete evaporation) is achieved. In fact, water mist systems often fail to deliver mist into the burning region and suppress fire. However, they may still be able to prevent flashover. Assuming total evaporation, the temperature history equation takes the form: dW ’; ¼ 1 þ a1 ðW4 1Þ a2 ðW 1Þ gC dt
ð26Þ
where the additional heat sink is proportional to the water application rate and the parameter g, defined as g¼
cp ðTw T0 Þ þ Hf g : Q’ 0
ð27Þ
The Eqs. (26) and (27) are obtained upon assuming total evaporation of the spray, ’ ðcp ðTw T0 Þ þ Hf g Þ; and non-dimensionalising which delivers a heat loss rate of C this expression using the heat release rate Q’ 0 :
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Fig. 6. Critical water application rates for complete evaporation regime. Different values of the heat loss coefficient a2 : (1) a2 ¼ 1:0; (2) a2 ¼ 1:5; (3) a2 ¼ 2:0; (4) a2 ¼ 2:5:
The critical conditions (7) are written in this case as follows: ’; a1 ðW4 1Þ a2 ðW * 1Þ þ 1 ¼ gC * 4a1 W3 *
a2 ¼ 0
ð28Þ
from which the following expression for the critical water application rate follows: " # 4=3 ! 1=3 ! a a 2 2 1 ’ cr ¼ g C a1 1 a2 1 þ 1 : ð29Þ 4a1 4a1 Critical water application rate increases as a function of a1 (Fig. 6), as higher values of this parameter imply higher heat gain rates. The curves starting points (Fig. 6) correspond to the points of the critical curve a2 ¼ jða1 Þ: In a more general sense, the critical condition (29) may be interpreted in the space ’ ; a1 ; a2 ), rather than on the plane as it is the of the three independent parameters (C case for freely burning fire [5,6]. The Eq. (29) determines the critical surface in that space shown in Fig. 7. The flashover area is below the surface (between the critical surface and the plane ða1 ; a2 Þ). Above the surface the flashover is impossible. The critical surface intersects with the ða1 ; a2 Þ plane exactly by the critical curve a2 ¼ jða1 Þ: 4.3. General case Generally, water droplets evaporate as they move through the hot gas. Once droplet is injected into the hot layer, it will experience a period of unsteady evaporation, followed by a steady-state evaporation regime, characterised by constant surface droplet temperature (‘‘wet bulb temperature’’) and constant surface regression rate [15].
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Fig. 7. Critical surface for flashover in the case of water mist limit.
In the first approximation, it is sufficiently accurate to assume that the droplet evolution can be split into two stages: 1. heat-up to wet bulb temperature without evaporation, and 2. steady-state evaporation conditions. At high environment temperatures, wet bulb temperature approaches boiling temperature, so we assume that the steady-state evaporation occurs at surface boiling temperature. During both stages the equations of motion (11) apply, which can be written in the equivalent form: dU 3rCD ¼ dt 4rp dV 3rCD ¼ dt 4rp
1 2 U ; d 1 UV þ g: d
ð30Þ
During the heat-up phase, the droplet diameter does not change, and the solution is the same as for the set of Eqs. (11), that is given by the formulae (12) and (14). During the steady-state evaporation, the equations of motion (30) must be solved with the droplet diameter changing as a function of time. This function is given by the theoretically and experimentally observed ‘‘d 2 -law’’, which implies d 2 ðtÞ ¼ d02 ðtÞ kev t:
ð31Þ
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The parameter kev is a so-called evaporation constant, and may be taken in the form [15]: 8lg Cg ðT Tw Þ kev ¼ ln 1 þ : ð32Þ Hf g rp Cg With the diameter regression rate (31), the Eqs. (30) can still be integrated analytically. The solution can be written in the following form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# #1 " 3rCD d0 kev 1 1 1 2t þ ; U¼ U1 2rp kev d0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " 2d02 kev a 1 kev g 1 2t 1 2t þK 2 3 kev d0 d0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V¼ ; kev a 1 2t d0 "
ð33Þ
where the parameters K and a are given by d02 ga þ ða 1Þ V1 ; kev 2rp kev 1 : a¼1þ 3rCD d0 U1
K¼
ð34Þ
(U1 ; V1 are the droplet velocity components at the end of the heat-up stage). Heat loss to the spray for the general case is calculated using the residence time of the droplet in the smoke layer. The criterion for the general case is Z
t1
ð35Þ
VðsÞ dsoD; 0
where t1 is a droplet heat-up time to the boiling temperature. The residence time is determined using the following equation for the evaporation period time t * * : Z
t1 1
V ðsÞ ds þ 0
Z
t* *
V 2 ðsÞ ds ¼ D;
ð36Þ
0
where V 1 ðsÞ and V 2 ðsÞ refer to the solutions (14) and (33), respectively. The residence time is obtained as t2 ¼ t1 þ t * * :
ð37Þ
The Eq. (36) does not have solution if a droplet evaporates completely in the layer. In that case, the residence time is taken as t1 þ tev where tev is the droplet
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evaporation time tev ¼
d02 : kev
ð38Þ
For calculation of the total heat loss rate to the spray we note that the heat loss due to a single droplet is Qd0 ¼ md0 cp ðTw T0 Þ þ Hf g Dmd0 ;
ð39Þ
where the droplet mass loss is calculated as Dmd0 ¼ 16prp ½d03 ðd02 kev tÞ3=2 : The total heat loss rate to the spray is then obtained as " 3=2 ## Z N" k ev ’ cp ðTw T0 Þ þ Hf g 1 1 2 t * * ðsÞ f ðsÞ ds: Q’ spr ¼ C s 0
ð40Þ
ð41Þ
4.4. Boundaries for the limiting cases The critical conditions (25) and (29) are obviously different and depend on the regime of heat loss from the spray (predominantly convective or predominantly evaporative). It is important to estimate the regions of applicability of the each of these regimes and corresponding critical conditions. In this section, we give the boundaries of the two limiting regimes, that is nonevaporation and complete evaporation regimes. For any given droplet diameter d; temperature of the smoke layer T and the layer thickness D the boundary for non-evaporation regime corresponds to conditions where the droplet exit temperature is exactly Tw : This condition is given by rp cp t* T T0 ln ¼ ; ð42Þ d 2 6kNu T Tw which determines minimum diameter of droplets which do not undergo evaporation. The droplet exit time should be calculated from (17). The boundary for the opposite limiting case, complete evaporation, is calculated using the solutions (33) and (34) for a droplet evaporating in a smoke layer. This boundary would correspond to the maximum diameter of droplets which can be completely evaporated by a layer of a given temperature and thickness. The calculated boundaries are presented in Fig. 8 for the two different gas temperatures as functions of the smoke layer thickness. The boundaries depend rather weakly on the thickness D: The gap in droplet diameters between the two regimes is of the order of 200–300 mm. For the droplets in this gap neither of the limits apply, and the spray influence on flashover is determined by the heat loss calculated according to the general case solution (41).
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Fig. 8. Limits of applicability of different heat transfer regimes FFF convective heat transfer - - - - - - - - complete evaporation. Smoke layer temperatures: (1) T ¼ 700 K; (2) T ¼ 500 K.
4.5. Example of calculation of the critical flashover conditions In order to estimate the influence of a sprinkler, consider the example given in [5] for the fire with the heat release rate of 155 kW/m2 and radius of burning area of 0.15 m. Compartment dimensions for this case are 0.4 m 0.4 m 0.4 m. Fractional height of the thermal discontinuity plane is taken as 0.5. The other relevant parameters may be found in [5]. Controlling parameters can be calculated as a1 ¼ 0:0018; a2 ¼ 0:45: Eq. (4) gives the flashover time ([5]) of 14.6 s. In the case of non-evaporation limit, the critical value of a2 is required. This value has been found in [5] to be A 2 ¼ 0:49: The critical flashover temperature for these conditions may be found from (8) to be 1172 K. Non-evaporative regime for such critical temperature occurs for all droplets with the diameters higher than 480 mm, which can be demonstrated as explained above, using the condition (42). Therefore, if as an example we consider a monodisperse spray with the droplets of 500 mm in diameter, purely convective heat transfer conditions will apply for all temperatures up to the critical. The minimum water flow rate required to prevent ’ cr ¼ 2:99 g=s: flashover can be calculated from (25) to be C General dependencies of the critical application rates on droplet diameter (for a monodisperse spray) are presented in Fig. 9 for different parameters a1 ; a2 in the flashover region. For small droplets, full evaporation limit applies, and the critical flow rate is constant up to a certain droplet diameter (region A in Fig. 9). In the region C, Fig. 9 (non-evaporation limit), the critical application rate is determined by the relationship (25). The boundaries of the limiting regions are determined in the way, similar to Fig. 8. There is an intermediate region between the two limiting regions. In this region, critical water requirements should be increasing with the diameter, which is schematically shown by dotted
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Fig. 9. Water application rates required for flashover prevention as a function of droplet diameter in a monodisperse spray. A: region of complete evaporation (mist limit), B: intermediate region, C: region of convective heat transfer (non-evaporation limit), – – – – boundaries of the limiting regimes. Fire parameters: Q’ 0 ¼ 100 kW; D ¼ 0:5 m; a2=0.5. (1) a1 ¼ 0:011; (2) a1 ¼ 0:059; (3) a1 ¼ 0:018:
lines in Fig. 9. The calculation of the exact dependence is, however, more complicated. The critical application rate should be found from the following set of equations: ’ cr ; a1 ðW4 1Þ a2 ðW * 1Þ þ 1 ¼ g* C *
4a1 W3 a2 ¼ *
d*g ’ cr ; ðW ÞC dW *
where the parameter g* is given by " 3=2 ## Z N" 1 kev * * g* ¼ cp ðTw T0 Þ þ Hf g 1 1 2 t ðsÞ f ðsÞ ds: s Q’ 0 0
ð43Þ
ð44Þ
Note, that the conditions (43) and (44) also describe a full evaporation regime as a limiting case. In such case t * * ðsÞ ¼ tev : As expected, water mist (if evaporated uniformly through the smoke layer) is most effective in the prevention of flashover. Non-evaporation case provides the upper limit for the critical water application rate. For most real sprays, the critical rate will lie between the two limiting cases and may be calculated using the conditions for the intermediate case (43). In both limits the critical discharge rate is much less than used in real operations of sprinklers and water mist systems, which suggests that fire control is achievable with much more economic water supply rates. Finally, we give the most general equations which are applicable for a polydisperse spray. For such (most practically important) sprays, different droplets would be in the different regimes of heat transfer. Most fine droplets are expected to evaporate completely, while for the largest droplets evaporation may be negligible. For each value of the non-dimensional temperature W the droplet size distribution is divided
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by some droplet diameter d * ðWÞ into the ‘‘fine’’ (dpd * ðWÞ) and the ‘‘coarse’’ (d > dðWÞ) tails. Conditions (43) are applicable for the ‘‘fine’’ part, and nonevaporation conditions (24) are applicable for the ‘‘coarse’’ part. These considerations lead to the following general formulation for the critical flashover conditions: ’ cr ; a1 ðW4 1Þ A2 ðW ÞðW 1Þ þ 1 ¼ g* C *
*
*
dA2 d*g ’ cr ; ðW * ÞðW * 1Þ A2 ðW * Þ ¼ ðW * ÞC 4a1 W3 * dW dW
ð45Þ
" " ## ’ cr cp T0 Z N C 6kNu A2 ðW * Þ ¼ a2 þ 1 exp t * ðsÞ f ðsÞ ds r p cp s 2 Q’ 0 d * ðW Þ *
ð46Þ
where
and 1 g* ðW * Þ ¼ Q’ 0
Z
d * ðW Þ *
0
"
" cp ðTw T0 Þ þ Hf g
3=2 ## kev 1 1 2 t * * ðsÞ f ðsÞ ds: s ð47Þ
The function d * ðW * Þ is determined as a boundary for the non-evaporation regime (similarly to upper curves, Fig. 8). Using this function, the critical application rate ’ cr may be determined from (45). C
5. Conclusions Flashover development of fire under suppression conditions has been investigated by expanding the existing flashover zone modeling to the case of water sprinkler/ water mist system activation. Analytical models have been developed to consider heat transfer between water spray and hot smoke layer in different regimes. It has been shown that the equations of droplet motion and heat transfer can be solved analytically with sufficient accuracy in the cases of convective heat loss and steady-state evaporation. As limiting cases, these models include cases of purely convective and complete evaporation regimes. This analytical approach can be successfully used in the further development of fire zone models involving water sprinkler applications. Critical conditions for flashover during application of water-based fire suppression systems has been formulated, and critical water requirements to prevent flashover have been estimated. The derived critical conditions allow the minimum water flow rate, required to prevent flashover, to be found as a function of droplet size distribution in the spray and fire geometrical and physical parameters. Critical water discharge rate in the case of water mist system is less than that for conventional sprinkler, and therefore such systems are more effective in the prevention of flashover. The present estimations show that despite problems with
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delivery of the suppressant into the fire region, water mist systems can still be very effective in fire control because of their ability to prevent flashover. For either water mist or conventional sprinklers, the amount of water required to prevent flashover is significantly less than delivered under regular operating conditions by commercial fire suppression systems. Limitations of the current model may arise from the following most important sources: (i) actual non-uniformity of heat absorption by the spray, and (ii) air motion (entrainment into the spray), which is currently neglected, but can change evaporation dynamics at high water discharge rates. As critical water rates reported in the present paper are low, air motion may be neglected. It may, however, play significant role in heat exchange for large compartments and high water release rates. In order to increase the accuracy of predictions and remove simplifying assumptions, it would be beneficial to investigate the interaction between hot smoke layer and water spray by means of numerical CFD simulations. Such simulations can provide more accurate distributions of the heat loss rate Q’ spr under various conditions. The degree of uniformity of heat absorption within the smoke layer may also be examined in this way.
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