- Email: [email protected]
Flashover and boundary properties
ARTICLE IN PRESS Fire Safety Journal 45 (2010) 116–121
Contents lists available at ScienceDirect
Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf
Flashover and boundary properties Alan N. Beard Civil Engineering Section, School of the Built Environment, Heriot-Watt University, Edinburgh, Scotland, UK
a r t i c l e in f o
a b s t r a c t
Article history: Received 31 July 2009 Received in revised form 28 October 2009 Accepted 7 December 2009
A non-linear model of flashover, FLASHOVER A1, which has been described in earlier work, has been used to explore the dependence of the critical heat release rate (Qfc) for flashover on the properties of the boundary of the enclosure. The compartment is assumed to have a single ventilation opening stretching from floor to ceiling. Specifically, the dependence of Qfc upon the thermal inertia of the boundary has been calculated for several different compartment sizes. The boundary materials considered are: marinite, gypsum, brick and concrete. Variations in Qfc with thermal inertia of the boundary and compartment size have been computed. The findings should be tested by experiment. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Fire Safety Non-linear Flashover Compartment
1. Introduction Over the last three decades there has been a great increase in the mathematical modelling of fire development within buildings. In particular, a number of deterministic models have been developed to characterize the development of fire growth in compartments and their use will certainly increase. Models of this kind have a two-fold importance: (a) to aid understanding of the fundamental processes of fire development and (b) in direct practical terms such as assessment of a specific design. In parallel with these developments there has been a burgeoning of activity in the theory of non-linear dynamical systems. Fire research applying non-linear systems theory has addressed flashover in compartment fires (the subject of this paper) and major fire spread in a tunnel [1]. Application of the concepts of non-linear dynamics to fire was conducted by Thomas, Bullen, Quintiere and McCaffrey in about 1980 [2]. In more recent years Beard, Drysdale, Holborn and Bishop have created three models which associate flashover with an instability within the system. Work applying non-linear concepts to flashover has also been carried out by Graham and Makhviladze [3]. Of the three models created by Beard et al. the first was a single state-variable model [4]; this identified the existence of a swallowtail catastrophe within the compartment fire system. The second was a two state-variable model [5] and the third a three state-variable model. The third model [6], called FLASHOVER A1, is the most developed and has been applied in this work; more details of the model are given later.
E-mail address: [email protected] 0379-7112/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2009.12.002
Specifically, the three state-variable model predicts the critical fire area necessary to create an instability in the system. This fire size is then associated with the onset of flashover. The model may also be used to predict the critical heat release rate and the critical temperature for the onset of flashover. FLASHOVER A1 has already been used to explore the effect of varying the assumed value of the discharge coefficient [7], the length of the lateral dimension for a square compartment [8], total internal surface area [9] and dependence on temperature and aspect ratio [10].
2. Summary of the model A summary of the model is given here and further details are given in Appendix 1. A zonal formulation has been adopted as this is well suited to the application of the concepts of non-linear dynamics. The basic compartment fire assumed is illustrated in Fig. 1. The essential assumptions in the model are: {1} A rectangular cuboidal compartment has been assumed of sides L1 L2 and height, H. A fire has its base at a distance L3 below the ceiling. {2} There is a single ventilation opening of rectangular shape and width, W, which extends from floor to ceiling. {3} A fire of circular perimeter and radius, R, entrains air and produces smoke which rises to form a hot layer of depth, Z, and uniform temperature, T. Smoke leaves the enclosure through the ventilation opening. {4} The lower layer consists of air at ambient temperature.
ARTICLE IN PRESS A.N. Beard / Fire Safety Journal 45 (2010) 116–121
_e m _f m N1
Nomenclature Latin symbols A Af Aus Cv Cp C Cus C1,C2 Conex Dc G H Hc Hi Hnmin Hv k kus L L1,L2 L3 Lfb Kfb
area of the ceiling fire area area of the upper surface specific heat at constant volume of the hot layer gases specific heat at constant pressure of ambient air short-hand for Cus, used in Figs. 2, 3, 4 specific heat of the upper surface material constants relating to Eq. (8) dimensionless constant effective conductive coefficient rate of gain of energy of the upper layer height of the compartment heat of combustion of the fuel enthalpy flow rate into the plume from the lower space gases and the fuel minimum height of the neutral plane heat of vaporization of the fuel short-hand for kus, used in Figs. 2, 3, 4 thermal conductivity of the material of the upper surface rate of loss of energy of the upper layer; compartment side length sides of the compartment distance from the ceiling to the fire surface mean beam length through the fire volume extinction coefficient for the flame volume
{5} The three state-variables for the model are: (1) temperature of the upper layer, T; (2) radius of the fire, R; (3) depth of the upper layer, Z. The principles of conservation of energy and conservation of mass have been used, together with an equation relating net heat flux to the surface of the fuel to rate of change of fire radius. Three coupled ordinary differential equations have been constructed for the temporal rates of change of T, Z and R. 2.1. Conservation of energy Applying the principle of conservation of energy to the upper layer results in the equation: _ G ¼ U_ þ Lþ w
ð1Þ
117
rate of entrainment of air into the plume fuel volatilization rate fraction of the heat released going into the upper layer pressure within the upper layer rate of heat release rate of energy going into the upper layer from the fire net heat flux to the surface of the fuel radius of the fire stoichiometric air/fuel mass ratio temperature of the upper layer (1K) temperature of the upper surface (1K) ambient temperature (1K) time characteristic fire time rate of change of internal energy of the upper layer width of the ventilation opening rate of work done associated with the upper layer depth of the upper layer
p Q_ f Q_ i
q_ 00 R Sr T Tus Ta t tc U_ W _ w Z
Greek symbols
a dus deff
r rus w
thermal diffusivity of the upper surface material physical thickness of the upper surface material effective thickness of the heated layer density of the upper layer; short-hand for rus, used in Figs. 2, 3, 4 density of the upper surface material combustion efficiency
where G is the rate of gain of energy of the upper layer; L the rate _ the rate of work done of loss of energy of the upper layer; w associated with the upper layer; U_ the rate of change of internal energy of the upper layer. Each of the terms in Eq. (1) has been de-composed further: G consists of convective and radiative energy associated with the combustion together with enthalpy from the lower space gases and the fuel. L consists of energy lost from the layer through the upper surfaces and the layer base together with enthalpy associated with the flow of hot gases out of the compartment. Work done is associated with the temporal rate of change of the upper layer depth, given by: _ ¼ Ap dZ=dt w
ð2Þ
where A is the area of the ceiling; p the pressure within the upper layer; Z the depth of the upper layer. The rate of change of internal energy is given by: U_ ¼ ACv dðrTZÞ=dt
ð3Þ
where Cv is the specific heat at constant volume of the smoke layer gases; r the density of the upper layer; T the temperature of the upper layer; t the time The heat release rate, Qf, is assumed to be given by: _ f Hc Q_ f ¼ wm
_ e =m _ f 4 Sr if m
_ e =Sr ÞHc ¼ wðm
Fig. 1. The compartment fire assumed (the fire is shown on the floor although this is not essential for the model).
_ e =m _ f oSr if m
ð4Þ ð5Þ
_ f the fuel volatilization where w is the combustion efficiency; m _ e the rate of entrainment of air; Hc the heat of combustion; rate; m Sr the stoichiometric air/fuel mass ratio.
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The fuel volatilization rate has been assumed to be governed by the net heat flux to the surface of the pyrolysing fuel, q_00 , via: _ f ¼ Af q_ 00 =Hv m
ð6Þ 2
where Af is the surface area of the pyrolysing fuel; = pR ; R the radius of the circular perimeter of the fire; q_00 the net heat flux to the surface of the fuel; Hv the heat of vaporization of the fuel. The term q_00 includes radiative heat transfer from the upper layer and upper surface of the room to the fuel as well as radiative and convective heat transfer from the flame to the fuel. 2.2. Conservation of mass Applying the principle of conservation of mass to the upper layer results in the equation: _ f ¼m _ o þ AdðrZÞ=dt _ e þm m
ð7Þ
_ e is the mass flow rate of air entrained from the lower where m _ o the mass flow rate from the smoke layer out of the vent. layer; m The mass flow rate out of the vent is assumed to be given by the expression derived by Rockett [11]. The mass flow rate of entrained air has been assumed to be given by the expression of Zukoski [12] in the earlier stages and by the expression of Prahl and Emmons [13] for vent inflow during the latter stages. The general form for the rate of change of fire radius has been taken from the work of Mitler [14]: dR=dt ¼ C1 loge ð1OÞ
ð8Þ
Where C1 is a constant.
O ¼ q_ 00 =ðsTf 4 Þ;
s ¼ Stefan0 s constant; Tf ¼ flame temperature
In order to avoid problems with the logarithmic term as O approaches 1, the first three terms in the logarithmic expansion have been taken.
If there is sufficient fuel, i.e., the maximum radius is large enough to allow a ventilation-controlled (VC) fire to exist, then that is the state which would be expected to result. For a domestic-sized room, of typical contents and ventilation opening, that would correspond to a post-flashover fire. If there is insufficient fuel present to allow ventilation control then a serious fire may still result even if the regime is still fuel-controlled (FC). The essential point is that, after instability has occurred, a rapid development to the maximum fuel area available or the maximum sustainable with VC, whichever occurs first, would be expected. For a system with three state-variables there would be three eigenvalues. In order to simplify the eigenvalue calculations it has been assumed that the fire radius is relatively slowly varying by comparison with the upper layer temperature and the layer depth; at least up to the point of instability. This reduces the number of calculated eigenvalues to 2.
5. Numerical results of the simulations Variation in the calculated critical heat release rate, Qfc, with thermal inertia and room size has been found for an aspect ratio, g = L1/L2, of 1. A fixed height of 3 m has been assumed for a compartment and polyurethane foam has been assumed as the fuel. A vent width of 0.4 m has been taken. Given that the vent extends from floor to ceiling, this value of width produces an opening factor approximately the same as that of a typical door; i.e., where the opening factor is W(H)3/2. A fixed value of discharge coefficient (Cd) has been assumed. (It may be noted that Cd would be expected to depend upon g.) Specific input values used are given in Appendix 2. Dependence of flashover upon thermal inertia has already been considered by Graham and Makhviladze [17]. Their consideration was couched in general terms. The work
3. Differential equations for the system Manipulation of the above equations enables three coupled ordinary differential equations to be derived with the general form: dT=dt ¼ F1 ðT; Z; RÞ
ð9Þ
dZ=dt ¼ F2 ðT; Z; RÞ
ð10Þ
dR=dt ¼ F3 ðT; Z; RÞ
ð11Þ
These equations define a system having the three statevariables T, Z and R. The differential Eqs. (9)–(11) above may be solved to find the conditions which give rise to instability. If instability occurs then this is being associated with the point of onset of flashover.
4. Instability and flashover The first order ordinary differential Eqs. (9)–(11) above have been solved simultaneously to find the conditions which give rise to instability. The ideas of non-linear dynamical systems theory have been described very briefly in Ref. [5]. In particular, the concept of eigenvalue has been used in order to determine the conditions which cause the system to lose stability. In general the eigenvalues of the Jacobian matrix of the vector field associated with a dynamical system may be used to gauge the stability of the state [5,15,16]. A system becomes unstable if the real part of an eigenvalue, l, becomes positive. As the fire grows a radius may be reached at which the system becomes unstable. After that the fire would be expected to increase rapidly.
Fig. 2. Critical heat release rate vs. thermal inertia at different room sizes; tc = 600 s.
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119
Fig. 3. Critical heat release rate vs. thermal inertia at different room sizes; tc =1000 s.
Fig. 4. Critical heat release rate vs. thermal inertia at different room sizes; tc = 600 s up to L = 7 m, 1000 s beyond 7 m.
here concerns specific materials, i.e., marinite, gypsum, brick, concrete. Fig. 2 shows the variation in critical heat release rate, Qfc, required to produce instability as a function of thermal inertia, assuming the characteristic time (tc) of the fire to be 600 s. It is seen that while there is a broad increase in Qfc with thermal inertia (krc) the curve exhibits an unexpected maximum at L= 8 m; L= L1 = L2. (Note that in Figs. 2, 3, 4 then kusruscus has been indicated as krc, for clarity.) If a characteristic time of 1000 s is assumed then the maximum at L=8 m is not there, see Fig. 3. However, at tc =1000 s, maxima are seen in other values of L. It may be the case that there exists in the real world a maximum at L=8 m. However, it is also possible that the characteristic time which it is reasonable to assume depends on room size. If it is assumed that a value for tc of 600 s is plausible for values of L up to 7 m and a value of 1000 s is more plausible beyond that then the curves of Fig. 4 are found. It may be noted in Fig. 2 that the value for Qfc at L=4 m (brick) is very slightly higher than the value at L=5 m. This is consistent with previous results found and reported in Refs. [8,9] where a minimum in Qfc is predicted, specifically at around a value for L of 4.3 m. Experimenters are urged to test these results. In any comparison with experiment it needs to be remembered that the values for critical heat release rate predicted by this model identify the moment of onset of instability. That is, the predicted results correspond to the moment of onset of the flashover transition; not a very short time later when flashover may be obvious experimentally.
of a solid, assuming very close proximity of a flame to the surface, has been described in Ref. [18]. That model may be regarded as approximating ignition by flame impingement and may also be seen as a special case of piloted ignition. The model in this present paper is aimed at determining the thermo-physical and geometrical conditions which lead to instability within a compartment fire. Given sufficient fuel and ventilation such instability would be expected to lead to flashover. Specifically, it predicts a critical fire area or heat release rate (Qfc) at which instability would be expected to occur and it would be desirable for the critical fire area or Qfc value not to be reached in a real-world case. This obviously has design implications, for example in terms of fire loading or configuration of room contents. Also, for example, sprinklers might be used to try to ensure that a fire remains below the critical size necessary for instability. Different designs may be compared using the model and the effects of changes to a system estimated. A particular result in this paper is that the critical heat release rate for flashover to occur broadly increases with thermal inertia of the boundaries and with compartment size; assuming that the characteristic fire time increases with room size. It is difficult to know what value of characteristic time should be used in a simulation and it may be that tc depends not only upon room size but upon boundary material as well. (It may be noted that this latter comment would also be expected to apply to other models, including ‘hand-calculation’ models, which employ the concept of characteristic time, such as that of McCaffrey et al. [19].) In the simulations carried out in this paper, though, tc has not been assumed to depend upon boundary material. Experimenters are urged to test these results experimentally.
6. Conclusions The model described here is one application of the concepts of non-linear dynamics to modelling fire development. The scope is wide, however, and the field is open to much more application of such concepts. For example, a simple non-linear model of ignition
Appendix 1. Further details of the model More details of the model are given here.
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The term G of Eq. (1) has been assumed to be given by: _i G ¼ Q_ i þ H
ðA:1Þ
where Q_ i is the rate of energy going into the upper layer from the fire; i.e., convected in via the plume together with radiation from _ i the enthalpy flow rate into the plume from the the flame; H lower space gases and the fuel. Q_ i has been assumed to be given by: Q_ i ¼ N1 Q_ f
ðA:2Þ
where Q_ f is the rate of energy production via combustion; N1 is that fraction of Q_ f which goes into the upper layer; N1 has been assumed to be constant in this work. The enthalpy flow rate into the plume, Hi, has been assumed to be given by: _ i ¼ Cp m _ e Ta þCp m _ f Tf H
ðA:3Þ
where Cp is the specific heat at constant pressure of ambient air. The specific heat at constant pressure of the volatiles is _ e, a assumed to be Cp. For the flow rate of entrained gases, m switching function has been assumed from the Zukoski plume equation in the early stages to the vent inflow equation of Prahl and Emmons in the latter stages: _ p EXFUNþ m _ v ð1EXFUNÞ _ e ¼m m
ðA:4Þ
_ p is given by the Zukoski equation [12]; m _ v is given by where m the expression of Prahl and Emmons [13] EXFUN ¼ 0 ¼1
from experimental tests conducted by Kawagoe et al. on flame spread over polyurethane [22].
Appendix 2. Input data Data used are as in Ref. [6] except: effective thickness of upper surface [20] =(atc)1/2 tc = characteristic time of the fire [20] =600 or 1000 s Four different values of kUS, rUS,CUS have been used. (‘US’= ‘upper surface’) thermal conductivity of marinite [23] = 0.0001154 kW/ kUS = mK rUS = density of marinite [23] =800 kg/m3 CUS = specific heat of marinite [23]= 1.11 kJ/kg K thermal conductivity of gypsum [20] =0.00048 kW/mK kUS = rUS = density of gypsum [20] = 1440 kg/m3 CUS = specific heat of gypsum [20]= 0.84 kJ/kg K thermal conductivity of concrete [20] =0.0011 kW/mK kUS = rUS = density of concrete [20] = 2100 kg/m3 CUS = specific heat of concrete [20] = 0.88 kJ/kg K thermal conductivity of brick [20] =0.00069 kW/mK kUS = rUS = density of brick [20] = 1600 kg/m3 CUS = specific heat of brick [20]= 0.84 kJ/kgK w= combustion efficiency [24] =0.85 3m L3 =
d=
if Z ¼ H if Z oðHHnmin Þ
¼ expfConex½HZHnmin =ðHZÞg
if ðHHnmin Þ o Z oH References
Conex is a dimensionless constant The minimum height of the neutral plane, Hnmin, is assumed to be a fixed ratio, Hnrat, of the vent height; i.e., Hnmin = HnratH. The rate of heat conduction through the upper surface has been assumed to be given by: Q_ cond ¼ Aus Dc ðTus Ta Þ
ðA:5Þ
where Aus is the area of the upper surface; i.e., adjacent to the smoke layer; Dc the effective conductive coefficient for upper surface material; Tus the temperature of the upper surface; Ta the ambient temperature The temperature of the upper surface, Tus, is calculated by assuming the heat conducted through the upper surface to be equal to the rate of heat transfer from the upper layer to the upper surface and solving the resulting quartic equation. The effective conductive coefficient, Dc, is given by kus/d. Where d = dus is the physical wall thickness if the characteristic fire time, tc, is greater than the thermal penetration time of the wall; = deff the effective thickness if the characteristic fire time is less than the thermal penetration time.
deff ¼ ðatc Þ1=2 ;
a ¼ thermal diffusivity ¼ kus =rus Cus
rus is the density of the upper surface material; kus the thermal conductivity of the upper surface material; Cus the specific heat of the upper surface material. The upper surface material forms, essentially, the wall/ceiling. Thermal penetration time [20] ¼ ðdus Þ2 =4a In the equation for rate of change of fire radius, see [14]: C1 ¼ C2 R=ðLfb Kfb Þ where R is the fire radius; Lfb the mean beam length to the fire base for the flame volume, assumed to be a cylinder, giving Lfb = 1.42R see Ref. [21]; Kfb the extinction coefficient for the flame volume; C2 is the exponent in the equation R= C3 exp(C2t), derived
[1] A.N. Beard, A theoretical model of major fire spread in a tunnel, Fire Technology 42 (2006) 303–328. [2] P.H. Thomas, M.L. Bullen, J.G. Quintiere, B.J. McCaffrey, Flashover & instabilities in fire behaviour, Combustion and Flame 38 (1980) 159–171. [3] T.L. Graham, G.M. Makhviladze, J.P. Roberts, On the theory of flashover development, Fire Safety Journal 25 (1995) 229–259. [4] A.N. Beard, D.D. Drysdale, P.G. Holborn, S.R. Bishop, A non-linear model of flashover, Fire Science and Technology 12 (1992) 11–27. [5] S.R. Bishop, P.G. Holborn, A.N. Beard, D.D. Drysdale, Non-linear dynamics of flashover in compartment fires, Fire Safety Journal 21 (1993) 11–45. [6] A.N. Beard, D.D. Drysdale, P.G. Holborn, S.R. Bishop, A model of instability & flashover, Journal of Applied Fire Science 4 (1994–95) 3–16. [7] A.N. Beard, Dependence of flashover on assumed value of the discharge coefficient, Fire Safety Journal 36 (2001) 25–36. [8] A.N. Beard, D.D. Drysdale, P.G. Holborn, S.R. Bishop, FLASHOVER A1: A Model for Predicting the Conditions for Flashover’, Conference on Fire Safety by Design, University of Sunderland, 10–12th July, 1995. [9] Beard, A.N., Dependence of flashover on total internal surface area, in: 3rd International Seminar on Fire & Explosion Hazards, Lake District, UK, 10–14 April 2000. [10] A.N. Beard, Dependence of flashover on temperature and aspect ratio of the compartment, Journal of Fire Sciences 21 (2003) 267–283. [11] J.A. Rockett, Fire induced gas flow in an enclosure, Combustion Science and Technology 12 (1976) 165–175. [12] E.E. Zukoski, T. Kubota, E. Cetegen, Entrainment in fire plumes, Fire Safety Journal 3 (1980/81) 107–121. [13] J. Prahl, H.W. Emmons, Fire induced flow through an opening, Combustion & Flame 25 (1975) 369–385. [14] H.E. Mitler, The physical basis for the harvard computer fire code, Home Fire Project Technical Report no. 34, Division of Applied Sciences, Harvard University, October 1978. [15] J.M.T. Thompson, H.B. Stewart, Nonlinear Dynamics & Chaos, Wiley, Chichester, 1986. [16] S.H. Strogatz, Nonlinear Dynamics & Chaos, Addison-Wesley, Reading, Massachusetts, 1994. [17] T. Graham, G. Makhviladze, J.P. Roberts, The effects of the thermal inertia of the walls upon flashover development, Fire Safety Journal 32 (1999) 35–60. [18] A.N. Beard, Major fire spread in a tunnel: a non-linear model with flame impingement, in: 5th International Conference on Safety in Road and Rail Tunnels (SIRRT5), pp. 511–520, Marseille, 6–10 October, 2003. Organized by University of Dundee and Independent Technical Conferences Ltd.; ISBN: 1-901808-22-X.
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[19] B.J. McCaffrey, J.G. Quintierre, M.F. Harkleroad, Estimating room temperatures and the likelihood of flashover using fire test data correlations, Fire Technology 17 (1981) 98–119. [20] D.D. Drysdale, Introduction to Fire Dynamics, Wiley, Chichester, 1999. [21] C.L. Tien, K.Y. Lee, A.J. Stretton, Radiation Heat Transfer, SFPE Handbook of Fire Protection Engineering, pp. 1–92 to 1–106; Society of Fire Protection Engineers, Boston 1988.
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[22] T. Mizuno, F. Sasagawa, S. Horiuchi, K. Kawagoe, Burning behaviour of urethane foam mattresses, Fire Science and Technology 1 (1981) 33–44. [23] R.L. Alpert, Influence of Enclosures on Fire Growth, vol. 1, Test Data, Test 7, Report no. OAOR2, Factory Mutual Research Corporation, Norwood, New Jersey, 1977. [24] A. Tewarson, Physico-chemical and combustion/pyrolysis properties of polymeric materials, Report NBS-GCR-80-295, BFRL, National Institute of Standards and Technology, Washington, USA, 1980.
Contents lists available at ScienceDirect
Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf
Flashover and boundary properties Alan N. Beard Civil Engineering Section, School of the Built Environment, Heriot-Watt University, Edinburgh, Scotland, UK
a r t i c l e in f o
a b s t r a c t
Article history: Received 31 July 2009 Received in revised form 28 October 2009 Accepted 7 December 2009
A non-linear model of flashover, FLASHOVER A1, which has been described in earlier work, has been used to explore the dependence of the critical heat release rate (Qfc) for flashover on the properties of the boundary of the enclosure. The compartment is assumed to have a single ventilation opening stretching from floor to ceiling. Specifically, the dependence of Qfc upon the thermal inertia of the boundary has been calculated for several different compartment sizes. The boundary materials considered are: marinite, gypsum, brick and concrete. Variations in Qfc with thermal inertia of the boundary and compartment size have been computed. The findings should be tested by experiment. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Fire Safety Non-linear Flashover Compartment
1. Introduction Over the last three decades there has been a great increase in the mathematical modelling of fire development within buildings. In particular, a number of deterministic models have been developed to characterize the development of fire growth in compartments and their use will certainly increase. Models of this kind have a two-fold importance: (a) to aid understanding of the fundamental processes of fire development and (b) in direct practical terms such as assessment of a specific design. In parallel with these developments there has been a burgeoning of activity in the theory of non-linear dynamical systems. Fire research applying non-linear systems theory has addressed flashover in compartment fires (the subject of this paper) and major fire spread in a tunnel [1]. Application of the concepts of non-linear dynamics to fire was conducted by Thomas, Bullen, Quintiere and McCaffrey in about 1980 [2]. In more recent years Beard, Drysdale, Holborn and Bishop have created three models which associate flashover with an instability within the system. Work applying non-linear concepts to flashover has also been carried out by Graham and Makhviladze [3]. Of the three models created by Beard et al. the first was a single state-variable model [4]; this identified the existence of a swallowtail catastrophe within the compartment fire system. The second was a two state-variable model [5] and the third a three state-variable model. The third model [6], called FLASHOVER A1, is the most developed and has been applied in this work; more details of the model are given later.
E-mail address: [email protected] 0379-7112/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2009.12.002
Specifically, the three state-variable model predicts the critical fire area necessary to create an instability in the system. This fire size is then associated with the onset of flashover. The model may also be used to predict the critical heat release rate and the critical temperature for the onset of flashover. FLASHOVER A1 has already been used to explore the effect of varying the assumed value of the discharge coefficient [7], the length of the lateral dimension for a square compartment [8], total internal surface area [9] and dependence on temperature and aspect ratio [10].
2. Summary of the model A summary of the model is given here and further details are given in Appendix 1. A zonal formulation has been adopted as this is well suited to the application of the concepts of non-linear dynamics. The basic compartment fire assumed is illustrated in Fig. 1. The essential assumptions in the model are: {1} A rectangular cuboidal compartment has been assumed of sides L1 L2 and height, H. A fire has its base at a distance L3 below the ceiling. {2} There is a single ventilation opening of rectangular shape and width, W, which extends from floor to ceiling. {3} A fire of circular perimeter and radius, R, entrains air and produces smoke which rises to form a hot layer of depth, Z, and uniform temperature, T. Smoke leaves the enclosure through the ventilation opening. {4} The lower layer consists of air at ambient temperature.
ARTICLE IN PRESS A.N. Beard / Fire Safety Journal 45 (2010) 116–121
_e m _f m N1
Nomenclature Latin symbols A Af Aus Cv Cp C Cus C1,C2 Conex Dc G H Hc Hi Hnmin Hv k kus L L1,L2 L3 Lfb Kfb
area of the ceiling fire area area of the upper surface specific heat at constant volume of the hot layer gases specific heat at constant pressure of ambient air short-hand for Cus, used in Figs. 2, 3, 4 specific heat of the upper surface material constants relating to Eq. (8) dimensionless constant effective conductive coefficient rate of gain of energy of the upper layer height of the compartment heat of combustion of the fuel enthalpy flow rate into the plume from the lower space gases and the fuel minimum height of the neutral plane heat of vaporization of the fuel short-hand for kus, used in Figs. 2, 3, 4 thermal conductivity of the material of the upper surface rate of loss of energy of the upper layer; compartment side length sides of the compartment distance from the ceiling to the fire surface mean beam length through the fire volume extinction coefficient for the flame volume
{5} The three state-variables for the model are: (1) temperature of the upper layer, T; (2) radius of the fire, R; (3) depth of the upper layer, Z. The principles of conservation of energy and conservation of mass have been used, together with an equation relating net heat flux to the surface of the fuel to rate of change of fire radius. Three coupled ordinary differential equations have been constructed for the temporal rates of change of T, Z and R. 2.1. Conservation of energy Applying the principle of conservation of energy to the upper layer results in the equation: _ G ¼ U_ þ Lþ w
ð1Þ
117
rate of entrainment of air into the plume fuel volatilization rate fraction of the heat released going into the upper layer pressure within the upper layer rate of heat release rate of energy going into the upper layer from the fire net heat flux to the surface of the fuel radius of the fire stoichiometric air/fuel mass ratio temperature of the upper layer (1K) temperature of the upper surface (1K) ambient temperature (1K) time characteristic fire time rate of change of internal energy of the upper layer width of the ventilation opening rate of work done associated with the upper layer depth of the upper layer
p Q_ f Q_ i
q_ 00 R Sr T Tus Ta t tc U_ W _ w Z
Greek symbols
a dus deff
r rus w
thermal diffusivity of the upper surface material physical thickness of the upper surface material effective thickness of the heated layer density of the upper layer; short-hand for rus, used in Figs. 2, 3, 4 density of the upper surface material combustion efficiency
where G is the rate of gain of energy of the upper layer; L the rate _ the rate of work done of loss of energy of the upper layer; w associated with the upper layer; U_ the rate of change of internal energy of the upper layer. Each of the terms in Eq. (1) has been de-composed further: G consists of convective and radiative energy associated with the combustion together with enthalpy from the lower space gases and the fuel. L consists of energy lost from the layer through the upper surfaces and the layer base together with enthalpy associated with the flow of hot gases out of the compartment. Work done is associated with the temporal rate of change of the upper layer depth, given by: _ ¼ Ap dZ=dt w
ð2Þ
where A is the area of the ceiling; p the pressure within the upper layer; Z the depth of the upper layer. The rate of change of internal energy is given by: U_ ¼ ACv dðrTZÞ=dt
ð3Þ
where Cv is the specific heat at constant volume of the smoke layer gases; r the density of the upper layer; T the temperature of the upper layer; t the time The heat release rate, Qf, is assumed to be given by: _ f Hc Q_ f ¼ wm
_ e =m _ f 4 Sr if m
_ e =Sr ÞHc ¼ wðm
Fig. 1. The compartment fire assumed (the fire is shown on the floor although this is not essential for the model).
_ e =m _ f oSr if m
ð4Þ ð5Þ
_ f the fuel volatilization where w is the combustion efficiency; m _ e the rate of entrainment of air; Hc the heat of combustion; rate; m Sr the stoichiometric air/fuel mass ratio.
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The fuel volatilization rate has been assumed to be governed by the net heat flux to the surface of the pyrolysing fuel, q_00 , via: _ f ¼ Af q_ 00 =Hv m
ð6Þ 2
where Af is the surface area of the pyrolysing fuel; = pR ; R the radius of the circular perimeter of the fire; q_00 the net heat flux to the surface of the fuel; Hv the heat of vaporization of the fuel. The term q_00 includes radiative heat transfer from the upper layer and upper surface of the room to the fuel as well as radiative and convective heat transfer from the flame to the fuel. 2.2. Conservation of mass Applying the principle of conservation of mass to the upper layer results in the equation: _ f ¼m _ o þ AdðrZÞ=dt _ e þm m
ð7Þ
_ e is the mass flow rate of air entrained from the lower where m _ o the mass flow rate from the smoke layer out of the vent. layer; m The mass flow rate out of the vent is assumed to be given by the expression derived by Rockett [11]. The mass flow rate of entrained air has been assumed to be given by the expression of Zukoski [12] in the earlier stages and by the expression of Prahl and Emmons [13] for vent inflow during the latter stages. The general form for the rate of change of fire radius has been taken from the work of Mitler [14]: dR=dt ¼ C1 loge ð1OÞ
ð8Þ
Where C1 is a constant.
O ¼ q_ 00 =ðsTf 4 Þ;
s ¼ Stefan0 s constant; Tf ¼ flame temperature
In order to avoid problems with the logarithmic term as O approaches 1, the first three terms in the logarithmic expansion have been taken.
If there is sufficient fuel, i.e., the maximum radius is large enough to allow a ventilation-controlled (VC) fire to exist, then that is the state which would be expected to result. For a domestic-sized room, of typical contents and ventilation opening, that would correspond to a post-flashover fire. If there is insufficient fuel present to allow ventilation control then a serious fire may still result even if the regime is still fuel-controlled (FC). The essential point is that, after instability has occurred, a rapid development to the maximum fuel area available or the maximum sustainable with VC, whichever occurs first, would be expected. For a system with three state-variables there would be three eigenvalues. In order to simplify the eigenvalue calculations it has been assumed that the fire radius is relatively slowly varying by comparison with the upper layer temperature and the layer depth; at least up to the point of instability. This reduces the number of calculated eigenvalues to 2.
5. Numerical results of the simulations Variation in the calculated critical heat release rate, Qfc, with thermal inertia and room size has been found for an aspect ratio, g = L1/L2, of 1. A fixed height of 3 m has been assumed for a compartment and polyurethane foam has been assumed as the fuel. A vent width of 0.4 m has been taken. Given that the vent extends from floor to ceiling, this value of width produces an opening factor approximately the same as that of a typical door; i.e., where the opening factor is W(H)3/2. A fixed value of discharge coefficient (Cd) has been assumed. (It may be noted that Cd would be expected to depend upon g.) Specific input values used are given in Appendix 2. Dependence of flashover upon thermal inertia has already been considered by Graham and Makhviladze [17]. Their consideration was couched in general terms. The work
3. Differential equations for the system Manipulation of the above equations enables three coupled ordinary differential equations to be derived with the general form: dT=dt ¼ F1 ðT; Z; RÞ
ð9Þ
dZ=dt ¼ F2 ðT; Z; RÞ
ð10Þ
dR=dt ¼ F3 ðT; Z; RÞ
ð11Þ
These equations define a system having the three statevariables T, Z and R. The differential Eqs. (9)–(11) above may be solved to find the conditions which give rise to instability. If instability occurs then this is being associated with the point of onset of flashover.
4. Instability and flashover The first order ordinary differential Eqs. (9)–(11) above have been solved simultaneously to find the conditions which give rise to instability. The ideas of non-linear dynamical systems theory have been described very briefly in Ref. [5]. In particular, the concept of eigenvalue has been used in order to determine the conditions which cause the system to lose stability. In general the eigenvalues of the Jacobian matrix of the vector field associated with a dynamical system may be used to gauge the stability of the state [5,15,16]. A system becomes unstable if the real part of an eigenvalue, l, becomes positive. As the fire grows a radius may be reached at which the system becomes unstable. After that the fire would be expected to increase rapidly.
Fig. 2. Critical heat release rate vs. thermal inertia at different room sizes; tc = 600 s.
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Fig. 3. Critical heat release rate vs. thermal inertia at different room sizes; tc =1000 s.
Fig. 4. Critical heat release rate vs. thermal inertia at different room sizes; tc = 600 s up to L = 7 m, 1000 s beyond 7 m.
here concerns specific materials, i.e., marinite, gypsum, brick, concrete. Fig. 2 shows the variation in critical heat release rate, Qfc, required to produce instability as a function of thermal inertia, assuming the characteristic time (tc) of the fire to be 600 s. It is seen that while there is a broad increase in Qfc with thermal inertia (krc) the curve exhibits an unexpected maximum at L= 8 m; L= L1 = L2. (Note that in Figs. 2, 3, 4 then kusruscus has been indicated as krc, for clarity.) If a characteristic time of 1000 s is assumed then the maximum at L=8 m is not there, see Fig. 3. However, at tc =1000 s, maxima are seen in other values of L. It may be the case that there exists in the real world a maximum at L=8 m. However, it is also possible that the characteristic time which it is reasonable to assume depends on room size. If it is assumed that a value for tc of 600 s is plausible for values of L up to 7 m and a value of 1000 s is more plausible beyond that then the curves of Fig. 4 are found. It may be noted in Fig. 2 that the value for Qfc at L=4 m (brick) is very slightly higher than the value at L=5 m. This is consistent with previous results found and reported in Refs. [8,9] where a minimum in Qfc is predicted, specifically at around a value for L of 4.3 m. Experimenters are urged to test these results. In any comparison with experiment it needs to be remembered that the values for critical heat release rate predicted by this model identify the moment of onset of instability. That is, the predicted results correspond to the moment of onset of the flashover transition; not a very short time later when flashover may be obvious experimentally.
of a solid, assuming very close proximity of a flame to the surface, has been described in Ref. [18]. That model may be regarded as approximating ignition by flame impingement and may also be seen as a special case of piloted ignition. The model in this present paper is aimed at determining the thermo-physical and geometrical conditions which lead to instability within a compartment fire. Given sufficient fuel and ventilation such instability would be expected to lead to flashover. Specifically, it predicts a critical fire area or heat release rate (Qfc) at which instability would be expected to occur and it would be desirable for the critical fire area or Qfc value not to be reached in a real-world case. This obviously has design implications, for example in terms of fire loading or configuration of room contents. Also, for example, sprinklers might be used to try to ensure that a fire remains below the critical size necessary for instability. Different designs may be compared using the model and the effects of changes to a system estimated. A particular result in this paper is that the critical heat release rate for flashover to occur broadly increases with thermal inertia of the boundaries and with compartment size; assuming that the characteristic fire time increases with room size. It is difficult to know what value of characteristic time should be used in a simulation and it may be that tc depends not only upon room size but upon boundary material as well. (It may be noted that this latter comment would also be expected to apply to other models, including ‘hand-calculation’ models, which employ the concept of characteristic time, such as that of McCaffrey et al. [19].) In the simulations carried out in this paper, though, tc has not been assumed to depend upon boundary material. Experimenters are urged to test these results experimentally.
6. Conclusions The model described here is one application of the concepts of non-linear dynamics to modelling fire development. The scope is wide, however, and the field is open to much more application of such concepts. For example, a simple non-linear model of ignition
Appendix 1. Further details of the model More details of the model are given here.
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The term G of Eq. (1) has been assumed to be given by: _i G ¼ Q_ i þ H
ðA:1Þ
where Q_ i is the rate of energy going into the upper layer from the fire; i.e., convected in via the plume together with radiation from _ i the enthalpy flow rate into the plume from the the flame; H lower space gases and the fuel. Q_ i has been assumed to be given by: Q_ i ¼ N1 Q_ f
ðA:2Þ
where Q_ f is the rate of energy production via combustion; N1 is that fraction of Q_ f which goes into the upper layer; N1 has been assumed to be constant in this work. The enthalpy flow rate into the plume, Hi, has been assumed to be given by: _ i ¼ Cp m _ e Ta þCp m _ f Tf H
ðA:3Þ
where Cp is the specific heat at constant pressure of ambient air. The specific heat at constant pressure of the volatiles is _ e, a assumed to be Cp. For the flow rate of entrained gases, m switching function has been assumed from the Zukoski plume equation in the early stages to the vent inflow equation of Prahl and Emmons in the latter stages: _ p EXFUNþ m _ v ð1EXFUNÞ _ e ¼m m
ðA:4Þ
_ p is given by the Zukoski equation [12]; m _ v is given by where m the expression of Prahl and Emmons [13] EXFUN ¼ 0 ¼1
from experimental tests conducted by Kawagoe et al. on flame spread over polyurethane [22].
Appendix 2. Input data Data used are as in Ref. [6] except: effective thickness of upper surface [20] =(atc)1/2 tc = characteristic time of the fire [20] =600 or 1000 s Four different values of kUS, rUS,CUS have been used. (‘US’= ‘upper surface’) thermal conductivity of marinite [23] = 0.0001154 kW/ kUS = mK rUS = density of marinite [23] =800 kg/m3 CUS = specific heat of marinite [23]= 1.11 kJ/kg K thermal conductivity of gypsum [20] =0.00048 kW/mK kUS = rUS = density of gypsum [20] = 1440 kg/m3 CUS = specific heat of gypsum [20]= 0.84 kJ/kg K thermal conductivity of concrete [20] =0.0011 kW/mK kUS = rUS = density of concrete [20] = 2100 kg/m3 CUS = specific heat of concrete [20] = 0.88 kJ/kg K thermal conductivity of brick [20] =0.00069 kW/mK kUS = rUS = density of brick [20] = 1600 kg/m3 CUS = specific heat of brick [20]= 0.84 kJ/kgK w= combustion efficiency [24] =0.85 3m L3 =
d=
if Z ¼ H if Z oðHHnmin Þ
¼ expfConex½HZHnmin =ðHZÞg
if ðHHnmin Þ o Z oH References
Conex is a dimensionless constant The minimum height of the neutral plane, Hnmin, is assumed to be a fixed ratio, Hnrat, of the vent height; i.e., Hnmin = HnratH. The rate of heat conduction through the upper surface has been assumed to be given by: Q_ cond ¼ Aus Dc ðTus Ta Þ
ðA:5Þ
where Aus is the area of the upper surface; i.e., adjacent to the smoke layer; Dc the effective conductive coefficient for upper surface material; Tus the temperature of the upper surface; Ta the ambient temperature The temperature of the upper surface, Tus, is calculated by assuming the heat conducted through the upper surface to be equal to the rate of heat transfer from the upper layer to the upper surface and solving the resulting quartic equation. The effective conductive coefficient, Dc, is given by kus/d. Where d = dus is the physical wall thickness if the characteristic fire time, tc, is greater than the thermal penetration time of the wall; = deff the effective thickness if the characteristic fire time is less than the thermal penetration time.
deff ¼ ðatc Þ1=2 ;
a ¼ thermal diffusivity ¼ kus =rus Cus
rus is the density of the upper surface material; kus the thermal conductivity of the upper surface material; Cus the specific heat of the upper surface material. The upper surface material forms, essentially, the wall/ceiling. Thermal penetration time [20] ¼ ðdus Þ2 =4a In the equation for rate of change of fire radius, see [14]: C1 ¼ C2 R=ðLfb Kfb Þ where R is the fire radius; Lfb the mean beam length to the fire base for the flame volume, assumed to be a cylinder, giving Lfb = 1.42R see Ref. [21]; Kfb the extinction coefficient for the flame volume; C2 is the exponent in the equation R= C3 exp(C2t), derived
[1] A.N. Beard, A theoretical model of major fire spread in a tunnel, Fire Technology 42 (2006) 303–328. [2] P.H. Thomas, M.L. Bullen, J.G. Quintiere, B.J. McCaffrey, Flashover & instabilities in fire behaviour, Combustion and Flame 38 (1980) 159–171. [3] T.L. Graham, G.M. Makhviladze, J.P. Roberts, On the theory of flashover development, Fire Safety Journal 25 (1995) 229–259. [4] A.N. Beard, D.D. Drysdale, P.G. Holborn, S.R. Bishop, A non-linear model of flashover, Fire Science and Technology 12 (1992) 11–27. [5] S.R. Bishop, P.G. Holborn, A.N. Beard, D.D. Drysdale, Non-linear dynamics of flashover in compartment fires, Fire Safety Journal 21 (1993) 11–45. [6] A.N. Beard, D.D. Drysdale, P.G. Holborn, S.R. Bishop, A model of instability & flashover, Journal of Applied Fire Science 4 (1994–95) 3–16. [7] A.N. Beard, Dependence of flashover on assumed value of the discharge coefficient, Fire Safety Journal 36 (2001) 25–36. [8] A.N. Beard, D.D. Drysdale, P.G. Holborn, S.R. Bishop, FLASHOVER A1: A Model for Predicting the Conditions for Flashover’, Conference on Fire Safety by Design, University of Sunderland, 10–12th July, 1995. [9] Beard, A.N., Dependence of flashover on total internal surface area, in: 3rd International Seminar on Fire & Explosion Hazards, Lake District, UK, 10–14 April 2000. [10] A.N. Beard, Dependence of flashover on temperature and aspect ratio of the compartment, Journal of Fire Sciences 21 (2003) 267–283. [11] J.A. Rockett, Fire induced gas flow in an enclosure, Combustion Science and Technology 12 (1976) 165–175. [12] E.E. Zukoski, T. Kubota, E. Cetegen, Entrainment in fire plumes, Fire Safety Journal 3 (1980/81) 107–121. [13] J. Prahl, H.W. Emmons, Fire induced flow through an opening, Combustion & Flame 25 (1975) 369–385. [14] H.E. Mitler, The physical basis for the harvard computer fire code, Home Fire Project Technical Report no. 34, Division of Applied Sciences, Harvard University, October 1978. [15] J.M.T. Thompson, H.B. Stewart, Nonlinear Dynamics & Chaos, Wiley, Chichester, 1986. [16] S.H. Strogatz, Nonlinear Dynamics & Chaos, Addison-Wesley, Reading, Massachusetts, 1994. [17] T. Graham, G. Makhviladze, J.P. Roberts, The effects of the thermal inertia of the walls upon flashover development, Fire Safety Journal 32 (1999) 35–60. [18] A.N. Beard, Major fire spread in a tunnel: a non-linear model with flame impingement, in: 5th International Conference on Safety in Road and Rail Tunnels (SIRRT5), pp. 511–520, Marseille, 6–10 October, 2003. Organized by University of Dundee and Independent Technical Conferences Ltd.; ISBN: 1-901808-22-X.
ARTICLE IN PRESS A.N. Beard / Fire Safety Journal 45 (2010) 116–121
[19] B.J. McCaffrey, J.G. Quintierre, M.F. Harkleroad, Estimating room temperatures and the likelihood of flashover using fire test data correlations, Fire Technology 17 (1981) 98–119. [20] D.D. Drysdale, Introduction to Fire Dynamics, Wiley, Chichester, 1999. [21] C.L. Tien, K.Y. Lee, A.J. Stretton, Radiation Heat Transfer, SFPE Handbook of Fire Protection Engineering, pp. 1–92 to 1–106; Society of Fire Protection Engineers, Boston 1988.
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[22] T. Mizuno, F. Sasagawa, S. Horiuchi, K. Kawagoe, Burning behaviour of urethane foam mattresses, Fire Science and Technology 1 (1981) 33–44. [23] R.L. Alpert, Influence of Enclosures on Fire Growth, vol. 1, Test Data, Test 7, Report no. OAOR2, Factory Mutual Research Corporation, Norwood, New Jersey, 1977. [24] A. Tewarson, Physico-chemical and combustion/pyrolysis properties of polymeric materials, Report NBS-GCR-80-295, BFRL, National Institute of Standards and Technology, Washington, USA, 1980.