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Dependence of flashover on assumed value of the discharge coefficient
Fire Safety Journal 36 (2001) 25}36
Dependence of #ashover on assumed value of the discharge coe$cient Alan N Beard* Department of Civil & Owshore Engineering, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland, UK Received 19 August 1999; received in revised form 18 July 2000; accepted 14 August 2000
Abstract A non-linear model of #ashover, FLASHOVER A1, has been used to explore the dependence of the critical "re size necessary for the onset of #ashover on the assumed value for the discharge coe$cient, C . It has been found that, overall, there is a signi"cant dependence on C . Experiments should be conducted in order to determine empirically the values of discharge coe$cients for di!erent cases. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Flashover; Non-linear; Instability; Model
1. Introduction Over the last two decades there has been a great increase in the mathematical modelling of "re development within buildings. In particular, a number of deterministic models [1,2] have been developed to characterize the development of "re growth in compartments and their use will certainly increase. Models of this kind have a twofold importance: (a) to aid understanding of the fundamental processes of "re development and (b) in direct practical terms such as assessment of a speci"c design. In parallel with these developments there has been a burgeoning of activity in the theory of non-linear dynamical systems as may be seen in the books by Thompson and Stewart [3] and Strogatz [4] which indicate the breadth of applications. The fundamentally non-linear nature of many systems is being realized and instabilities associated with phenomena such as jumps, oscillatory behaviour and deterministic
* Tel.: #44-0131-449-5111; fax: #44-0131-451-5078. 0379-7112/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 9 - 7 1 1 2 ( 0 0 ) 0 0 0 4 8 - 5
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A.N. Beard / Fire Safety Journal 36 (2001) 25}36
Nomenclature A A A A A C C C C C ,C Conex D G H H H G H H h h
area of the ceiling critical "re area, at instability "re area total internal surface area, excluding vent area of the upper surface discharge coe$cient for #ow from the vent speci"c heat at constant volume of the hot layer gases speci"c heat at constant pressure of ambient air speci"c heat of the upper surface material constants relating to Eq. (8) dimensionless constant e!ective conductive coe$cient rate of gain of energy of the upper layer height of the compartment heat of combustion of the fuel enthalpy #ow rate into the plume from the lower space gases and the fuel minimum height of the neutral plane heat of vaporization of the fuel convective heat transfer coe$cient from the #ame to the "re base convective heat transfer coe$cient from upper space gases to the upper surface k thermal conductivity of the material of the upper surface ¸ rate of loss of energy of the upper layer ¸ , ¸ sides of the compartment ¸ distance from the ceiling to the "re surface ¸ mean beam length through the "re volume k extinction coe$cient for the #ame volume m rate of entrainment of air into the plume m fuel volatilization rate N fraction of the heat released going into the upper layer p pressure within the upper layer QQ rate of heat conduction through the upper surface material QQ rate of heat release D QQ rate of energy going into the upper layer from the "re G q net heat #ux to the surface of the fuel R radius of the "re S stoichiometric air/fuel mass ratio ¹ temperature of the upper layer (K) ¹ temperature of the "re base (K) ¹ temperature of the upper surface (K) ¹ ambient temperature (K) t time
A.N. Beard / Fire Safety Journal 36 (2001) 25}36
t ;Q = w Z
27
characteristic "re time rate of change of internal energy of the upper layer width of the ventilation opening rate of work done associated with the upper layer depth of the upper layer
Greek letters a d d e e e o s
thermal di!usivity of the upper surface material physical thickness of the upper surface material e!ective thickness of the heated layer emissivity of the upper surface emissivity of the lower surface emissivity of the "re base density of the upper surface material combustion e$ciency con"guration factor from the underside of the smoke layer to the "re base. con"guration factor from the upper surface to the "re base
chaos are being explored. Application of the concepts of non-linear dynamics to "re was conducted by Thomas et al. [5] in the early 1980 s. In more recent years Beard et al. [6}8] have created three models which associate #ashover with an instability within the system. Work applying non-linear concepts to #ashover has also been carried out by Graham and Makhviladze [9]. Of the three models described in Refs. [6}8] the "rst was a single state-variable model, the second a two state-variable model and the third a three state-variable model. The third model, called FLASHOVER A1, is the most developed and has been described in detail in Ref. [8]. Speci"cally, the model of Ref. [8] predicts the critical "re area necessary to create an instability in the system. This "re size is then associated with the onset of #ashover. The model may also be used to predict the critical heat release rate and the critical temperature for the onset of #ashover (for recent experiments see [18]). FLASHOVER A1 has already been used to explore the e!ect of compartment size [10]. In this work, it has been used to consider the e!ect of the assumed value for the discharge coe$cient. Of course, the onset of #ashover would be expected to depend on other factors as well.
2. Summary of the model A summary of the model is given here and further details are given in Appendix A. A full description has already been published and is given in Ref. [8]. A zonal formulation has been assumed, as illustrated in Fig. 1, which makes the following basic assumptions:
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A.N. Beard / Fire Safety Journal 36 (2001) 25}36
Fig. 1. The compartment "re assumed.
(1) A rectangular cuboidal compartment has been assumed of sides ¸ by ¸ and height, H. A "re has its base at a distance ¸ below the ceiling. (2) There is a single ventilation opening of rectangular shape and width, =, which extends from #oor to ceiling. (3) A "re of circular perimeter and radius, R, entrains air and produces smoke which rises to form a hot layer of depth, Z, and uniform temperature, ¹. Smoke leaves the enclosure through the ventilation opening. (4) The lower layer consists of air at ambient temperature. (5) The three state-variables for the model are: (1) Temperature of the upper layer, ¹; (2) Radius of the "re, 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 #ux to the surface of the fuel to rate of change of "re radius. Three coupled ordinary di!erential equations have been constructed for the temporal rates of change of ¹, Z and R. 2.1. Conservation of energy Applying the principle of conservation of energy to the upper layer results in the equation G";Q #¸#w ,
(1)
where G is the rate of gain of energy of the upper layer, ¸ the rate of loss of energy of the upper layer, w the rate of work done associated with the upper layer and ;Q the rate of change of internal energy of the upper layer. Each of the terms in Eq. (1) has been further de-composed: G consists of convective and radiative energy associated with the combustion together with enthalpy from the lower space gases and the fuel. ¸ consists of energy lost from the layer through the upper surfaces and the layer base together with enthalpy associated with the #ow of hot gases out of the compartment. Work done is associated with the temporal rate of change of the upper layer depth, given by w "Ap dZ/dt,
(2)
A.N. Beard / Fire Safety Journal 36 (2001) 25}36
29
where A is the area of the ceiling, p the pressure within the upper layer and Z the depth of the upper layer. The rate of change of internal energy is given by ;Q "AC d(o¹Z)/dt, (3) where C is the speci"c heat at constant volume of the smoke layer gases, o the density of the upper layer, ¹ the temperature of the upper layer and t the time. The heat release rate, QQ , is assumed to be given by QQ "sm H if m /m 'S (4) "s(m /S )H if m /m (S , (5) where s is the combustion e$ciency, m the fuel volatilization rate, m the rate of entrainment of air, H the heat of combustion and S the Stoichiometric air/fuel mass ratio. The fuel volatilization rate has been assumed to be governed by the net heat #ux to the surface of the pyrolysing fuel, q , via m "A q /H , (6) where A is the surface area of the pyrolysing fuel and nR; R the radius of the circular perimeter of the "re, q the net heat #ux to the surface of the fuel and H the heat of vaporization of the fuel. The term q 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 #ame to the fuel. 2.2. Conservation of mass Applying the principle of conservation of mass to the upper layer results in the equation m #m "m #A d(oZ)/dt, (7) where m is the mass #ow rate of air entrained from the lower layer and m the mass #ow rate from the smoke layer out of the vent. The mass #ow rate out of the vent is assumed to be given by the expression derived by Rockett [11]. The mass #ow 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 in#ow during the latter stages. The general form for the rate of change of "re radius has been taken from the work of Mitler [14] dR/dt"!C log (1!X), (8) where C is a constant and X"q /(p¹), where p is the Stefan's constant and ¹ the #ame temperature.
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A.N. Beard / Fire Safety Journal 36 (2001) 25}36
In order to avoid problems with the logarithmic term as X approaches 1, the "rst three terms in the logarithmic expansion have been taken. 3. Di4erential equations for the system Manipulation of the above equations enables three coupled ordinary di!erential equations to be derived with the general form d¹/dt"F (¹, Z, R), dZ/dt"F (¹, Z, R), dR/dt"F (¹, Z, R). These equations de"ne a system having the three state-variables ¹, Z and R.
(9) (10) (11)
4. Instability and 6ashover The di!erential equations (9)}(11) above may be solved to "nd the conditions which give rise to instability. The ideas of non-linear dynamical systems theory have been described very brie#y in Ref. [7]. 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 "eld associated with a dynamical system may be used to gauge the stability of the state [3,7]. A system becomes unstable if the real part of an eigenvalue, j, becomes positive. As the "re grows a radius may be reached at which the system becomes unstable. After that the "re would be expected to increase rapidly. If there is su$cient fuel, that is the maximum radius is large enough, to allow a ventilation-controlled (VC) "re to exist then that is the state which would be expected to result. For a domestic-sized room, of typical contents and ventilation opening, that would correspond to a post-#ashover "re. If there is insu$cient fuel present to allow ventilation control then a serious "re 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 "rst, 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 "re 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. Simulations conducted Simulations have been carried out to investigate the e!ect on the calculated critical heat release rate of changing the assumed value of the discharge coe$cient. Full details of input are given in Appendix B.
A.N. Beard / Fire Safety Journal 36 (2001) 25}36
31
A compartment of dimensions ¸ "5 m, ¸ "5 m;3 m high with a single vent, extending from #oor to ceiling, of width 0.4 m has been assumed as a &base case'; the boundary has been assumed to be of brick. The fuel is #exible polyurethane foam and the "re base has been assumed to be on the #oor. The aspect ratio has been de"ned as c"¸ /¸ . This ratio has been varied whilst keeping the total internal surface area, A , constant. The reason for this was to eliminate any possibility that change in the predicted values was due to change in the internal surface area rather than aspect ratio or C per se. The total internal surface area, which does not include the vent, is given by A "2(¸ ¸ )#H(¸ #2¸ )#H(¸ !=). (12) This assumes the vent to be in the side of length ¸ . That is, the general form of the compartment is as shown in Fig. 2. A for this compartment comes out to be 108.8 m. A general problem exists in that when conducting simulations for enclosures a value for the discharge coe$cient needs to be assigned. Work by Steckler et al. [15], using a square compartment, found experimental values for #ow coe$cients to generally be about 0.7. As there will be uncertainty about the value of C in a real-world case, a likely range may be taken with 0.7 as a central point; given the results of Steckler et al. In this study three assumed values have been taken for C , ie 0.5, 0.7 and 0.9, in order to gauge the e!ect of assuming di!erent values for discharge coe$cient. Fig. 3 shows variation in calculated critical heat release rate, Q , with discharge coe$cient, C , for four values of aspect ratio, c. Overall it is seen that the critical heat release rate rises sharply with assumed discharge coe$cient, especially for the largest value of c considered, ie 4.26. As an illustration, graphs of "re radius, R, upper layer temperature, ¹ and upper layer depth, Z, are given in Fig. 4 for the case in which ¸1"8.0 m and ¸2"2.82 m, giving an aspect ratio of 2.84; at C "0.5. The variation in the least negative eigenvalue with time is also shown in Fig. 4.
Fig. 2. The compartment assumed.
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A.N. Beard / Fire Safety Journal 36 (2001) 25}36
Fig. 3. Variation in critical rate of heat release, Q , with discharge coe$cient, C . (c"aspect ratio; constant total internal surface area. Crosses indicate computed points.)
A.N. Beard / Fire Safety Journal 36 (2001) 25}36
33
Fig. 4. Illustrative results for the case with c"2.84 and C "0.5: (a) Upper layer temperature, ¹: (b) Upper layer depth, Z; (c) Fire radius, R; (d) Eigenvalue, j.
6. Discussion The model FLASHOVER A1 has been used to explore the dependence of the predicted value of the critical heat release rate to produce #ashover on the assumed value of the discharge coe$cient associated with #ow through the ventilation opening. With aspect ratio de"ned as c"¸ /¸ and the vent in the side of length ¸ then it has been found that variation in aspect ratio, at given C , does not have a very considerable e!ect on the "re size necessary for #ashover except for assumed values of the discharge coe$cient which are relatively high. This is at a total internal surface area of 108.8 m. At smaller values of A , however, a marked dependence on aspect ratio has been found; ie for values below about 95 m [16]. Overall, it has been found that the critical "re size for #ashover rises signi"cantly with increase in the assumed value of the discharge coe$cient. It is clearly desirable that discharge coe$cients for di!erent cases be estimated as well as possible. To conclude: overall, simulations using the non-linear model FLASHOVER A1 indicate that the critical "re size for #ashover has a marked dependence on the assumed value of the discharge coe$cient for the ventilation opening. Also, it has been found that, at "xed C , the critical heat release rate does not depend dramatically on the aspect ratio of the compartment, except for relatively high assumed values of the
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A.N. Beard / Fire Safety Journal 36 (2001) 25}36
C . This is at the particular value of total internal surface area considered; ie 108.8 m. From earlier work [16], however, it has been predicted that below about 95 m then there is a marked dependence on aspect ratio, at "xed assumed value of C . There is a need for experimental work to test the inferences drawn. In particular, the importance of future experimental work aimed at estimating the discharge coe$cients for di!erent con"gurations is apparent.
Appendix A. Further details of the model More details of the model are given here. A full description has been given in Ref. [8]. The term G of Eq. (1) has been assumed to be given by G"QQ #HQ (A.1) G G where QQ is the rate of energy going into the upper layer from the "re; ie convected in G via the plume together with radiation from the #ame and HQ the enthalpy #ow rate G into the plume from the lower space gases and the fuel. QQ has been assumed to be given by G QQ "N QQ , (A.2) G D where QQ is the rate of energy production via combustion, N that fraction of D QQ which goes into the upper layer and N has been assumed to be constant in this D work. The enthalpy #ow rate into the plume, HQ , has been assumed to be given by G HQ "C m ¹ #C m ¹ , (A.3) G where C is the speci"c heat at constant pressure of ambient air. The speci"c heat at constant pressure of the volatiles is assumed to be C . For the #ow rate of entrained gases, m , a switching function has been assumed from the Zukoski plume equation in the early stages to the vent in#ow equation of Prahl and Emmons in the latter stages: m "m EXFUN#m (1!EXFUN), (A.4) N where m is given by the Zukoski equation [12], m is given by the expression of Prahl N and Emmons [13]. EXFUN"0
if Z"H
if Z((H!H ) "exp+Conex[H!Z!H ]/(H!Z)], H (Conex"a dimensionless constant). "1
if
(H!H
)(Z(
The minimum height of the neutral plane, H , is assumed to be a "xed ratio, H , of the vent height; ie H "H H.
A.N. Beard / Fire Safety Journal 36 (2001) 25}36
35
The rate of heat conduction through the upper surface has been assumed to be given by QQ "A D (¹ !¹ ), (A.5) where A is the area of the upper surface; ie adjacent to the smoke layer, D the e!ective conductive coe$cient for upper surface material, ¹ the temperature of the upper surface and ¹ the ambient temperature. The temperature of the upper surface, ¹ , 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 e!ective conductive coe$cient, D , is given by k /d, where d"d is the physical wall thickness if the characteristic "re time, t , is greater than the thermal penetration time of the wall and d the e!ective thickness if the characteristic "re time is less than the thermal penetration time. d "(at ) , a"thermal di!usivity"k /o C , where o is the density of the upper surface material, k the thermal conductivity of the upper surface material and C the speci"c heat of the upper surface material. The upper surface material forms, essentially, the wall/ceiling. (d ) Thermal penetration time [17]" . 4a In the equation for rate of change of "re radius (see [14]) C "C R/(¸ K ), where R is the "re radius, ¸ the mean beam length to the "re base for the #ame volume, assumed to be a cylinder, giving ¸ "1.42R (see Ref. [19]), K the extinc tion coe$cient for the #ame volume, C the exponent in the equation R"C exp(C t), derived from experimental tests conducted by Kawagoe et al. on #ame spread over polyurethane [20].
Appendix B. Input for the simulations The following gives the numerical input for the simulations, outwith values of parameters speci"cally stated in the main text. See Ref. [8] for further explanation relating to input parameters. C : as given in main text; C : 1.04 kJ/kg, see [17]; C : 0.84 kJ/kg, see [17]; C : 0.021/s, see [20]; Conex : 3, found by experience to give a plausible change-over DENC : 0.5, see [8]; h : 0.02 kW/m K, see [17]; h : 0.01 kW/m K, see [21]; H : 3 m; H : 2190 kJ/kg, see [22]; H : 2.87(10)kJ/kg, see [22]; H : 0.45, see [23]; K , K : both 1.3/m, see [22]; k : 0.00069 kW/m K, see [17]; ¸ , ¸ : as speci"ed in main text; ¸ : 3 m; N : 0.7, see [24]; S : 9.78 (poly urethane foam), see [22]; ¹ : 293 K; ¹ : 600 K, see [22]; ¹ : 1400 K, see [17]; t : 300 s; = : 0.4 m; d : 0.06 m, approximate thickness of a brick; e : 1; e : 0.5; e : 0.5;
: 0.95; : 0.95; s : 0.85; o : 1.1 kg/m, see [17]; o : 1600 kg/m, brick, see [17].
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A.N. Beard / Fire Safety Journal 36 (2001) 25}36
References [1] Beard AN. Evaluation of "re models: report 11, overview. Unit of Fire Safety Engineering, University of Edinburgh, October 1990 (see also reports 1}10). [2] Beard AN. Limitations of computer models. Fire Safety J 1992;18:375}91. [3] Thompson JMT, Stewart HB. Nonlinear dynamics and chaos. Chichester: Wiley, 1986. [4] Strogatz SH. Nonlinear dynamics and chaos. Reading, MA: Addison-Wesley, 1994. [5] Thomas PH, Bullen ML, Quintiere JG, McCa!rey BJ. Flashover and instabilities in "re behaviour. Combustion Flame 1980;38:159}71. [6] Beard AN, Drysdale DD, Holborn PG, Bishop SR. A non-linear model of #ashover. Fire Sci Technol 1992;12:11}27. [7] Bishop SR, Holborn PG, Beard AN, Drysdale DD. Nonlinear dynamics of #ashover in compartment "res. Fire Safety J 1993;21:11}45. [8] Beard AN, Drysdale DD, Holborn PG, Bishop SR. A model of instability and #ashover. J Appl Fire Sci 1994}95;4:3}16. [9] Graham TL, Makhviladze GM, Roberts JP. On the theory of #ashover development. Fire Safety J 1995;25:229}59. [10] Beard AN, Drysdale DD, Holborn PG, Bishop SR. FLASHOVER A1: a model for predicting the conditions for #ashover. Conference on Fire Safety by Design, University of Sunderland, 10}12th July, 1995. [11] Rockett JA. Fire induced gas #ow in an enclosure. Combustion Sci Technol 1976;12:165}75. [12] Zukoski EE, Kubota T, Cetegen E. Entrainment in "re plumes. Fire Safety J 1980/81;3:107}121. [13] Prahl J, Emmons HW. Fire induced #ow through an opening. Combustion Flame 1975;25:369}85. [14] Mitler HE. The physical basis for the Harvard computer "re code. Home "re project technical report No. 34, Division of Applied Sciences, Harvard University, October 1978. [15] Steckler KD, Baum HR, Quintiere JG. Fire induced #ow through room openings-#ow coe$cients. 20th Symposium (International) on Combustion, The Combustion Institute, 1984. p. 1591}1600. [16] Beard AN. Dependence of #ashover on total internal surface area. Proceedings of the Third International Seminar on Fire & Explosion Hazards, Lake Windermere, UK, April 10}14th, 2000. [17] Drysdale DD. Introduction to "re dynamics. Chichester: Wiley, 1999. [18] Luo M, Beck V. A study of non-#ashover and #ashover "res in a full-scale multi-room building. Fire Safety J 1996;26:191}219. [19] Tien CL, Lee KY, Stretton AJ, Radiation heat transfer. SFPE handbook of "re protection engineering. Boston: Society of Fire Protection Engineers, 1988. p. 1}92}1}106. [20] MizunoT, Sasagawa F, Horiuchi S, Kawagoe K.Burning behaviour of urethane foam mattresses. Fire Sci Technol 1981;1:33}44. [21] Atkinson GT, Drysdale DD. Convective heat transfer from "re gases. Fire Safety J 1992;19:217}45. [22] Alpert RL. In#uence of enclosures on "re growth: vol 1; test data, test 7. Report No. OAOR2, Factory Mutual Research Corporation, Norwood, NJ, USA, 1977. [23] Babrauskas V, Williamson RB. Post-#ashover compartment "res: basis of a theoretical model. Fire Mater 1978;2:39}53. [24] Heskestad G. Engineering relations for "re plumes. Fire Safety J 1984;7:25}32.
Dependence of #ashover on assumed value of the discharge coe$cient Alan N Beard* Department of Civil & Owshore Engineering, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland, UK Received 19 August 1999; received in revised form 18 July 2000; accepted 14 August 2000
Abstract A non-linear model of #ashover, FLASHOVER A1, has been used to explore the dependence of the critical "re size necessary for the onset of #ashover on the assumed value for the discharge coe$cient, C . It has been found that, overall, there is a signi"cant dependence on C . Experiments should be conducted in order to determine empirically the values of discharge coe$cients for di!erent cases. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Flashover; Non-linear; Instability; Model
1. Introduction Over the last two decades there has been a great increase in the mathematical modelling of "re development within buildings. In particular, a number of deterministic models [1,2] have been developed to characterize the development of "re growth in compartments and their use will certainly increase. Models of this kind have a twofold importance: (a) to aid understanding of the fundamental processes of "re development and (b) in direct practical terms such as assessment of a speci"c design. In parallel with these developments there has been a burgeoning of activity in the theory of non-linear dynamical systems as may be seen in the books by Thompson and Stewart [3] and Strogatz [4] which indicate the breadth of applications. The fundamentally non-linear nature of many systems is being realized and instabilities associated with phenomena such as jumps, oscillatory behaviour and deterministic
* Tel.: #44-0131-449-5111; fax: #44-0131-451-5078. 0379-7112/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 9 - 7 1 1 2 ( 0 0 ) 0 0 0 4 8 - 5
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A.N. Beard / Fire Safety Journal 36 (2001) 25}36
Nomenclature A A A A A C C C C C ,C Conex D G H H H G H H h h
area of the ceiling critical "re area, at instability "re area total internal surface area, excluding vent area of the upper surface discharge coe$cient for #ow from the vent speci"c heat at constant volume of the hot layer gases speci"c heat at constant pressure of ambient air speci"c heat of the upper surface material constants relating to Eq. (8) dimensionless constant e!ective conductive coe$cient rate of gain of energy of the upper layer height of the compartment heat of combustion of the fuel enthalpy #ow rate into the plume from the lower space gases and the fuel minimum height of the neutral plane heat of vaporization of the fuel convective heat transfer coe$cient from the #ame to the "re base convective heat transfer coe$cient from upper space gases to the upper surface k thermal conductivity of the material of the upper surface ¸ rate of loss of energy of the upper layer ¸ , ¸ sides of the compartment ¸ distance from the ceiling to the "re surface ¸ mean beam length through the "re volume k extinction coe$cient for the #ame volume m rate of entrainment of air into the plume m fuel volatilization rate N fraction of the heat released going into the upper layer p pressure within the upper layer QQ rate of heat conduction through the upper surface material QQ rate of heat release D QQ rate of energy going into the upper layer from the "re G q net heat #ux to the surface of the fuel R radius of the "re S stoichiometric air/fuel mass ratio ¹ temperature of the upper layer (K) ¹ temperature of the "re base (K) ¹ temperature of the upper surface (K) ¹ ambient temperature (K) t time
A.N. Beard / Fire Safety Journal 36 (2001) 25}36
t ;Q = w Z
27
characteristic "re time rate of change of internal energy of the upper layer width of the ventilation opening rate of work done associated with the upper layer depth of the upper layer
Greek letters a d d e e e o s
thermal di!usivity of the upper surface material physical thickness of the upper surface material e!ective thickness of the heated layer emissivity of the upper surface emissivity of the lower surface emissivity of the "re base density of the upper surface material combustion e$ciency con"guration factor from the underside of the smoke layer to the "re base. con"guration factor from the upper surface to the "re base
chaos are being explored. Application of the concepts of non-linear dynamics to "re was conducted by Thomas et al. [5] in the early 1980 s. In more recent years Beard et al. [6}8] have created three models which associate #ashover with an instability within the system. Work applying non-linear concepts to #ashover has also been carried out by Graham and Makhviladze [9]. Of the three models described in Refs. [6}8] the "rst was a single state-variable model, the second a two state-variable model and the third a three state-variable model. The third model, called FLASHOVER A1, is the most developed and has been described in detail in Ref. [8]. Speci"cally, the model of Ref. [8] predicts the critical "re area necessary to create an instability in the system. This "re size is then associated with the onset of #ashover. The model may also be used to predict the critical heat release rate and the critical temperature for the onset of #ashover (for recent experiments see [18]). FLASHOVER A1 has already been used to explore the e!ect of compartment size [10]. In this work, it has been used to consider the e!ect of the assumed value for the discharge coe$cient. Of course, the onset of #ashover would be expected to depend on other factors as well.
2. Summary of the model A summary of the model is given here and further details are given in Appendix A. A full description has already been published and is given in Ref. [8]. A zonal formulation has been assumed, as illustrated in Fig. 1, which makes the following basic assumptions:
28
A.N. Beard / Fire Safety Journal 36 (2001) 25}36
Fig. 1. The compartment "re assumed.
(1) A rectangular cuboidal compartment has been assumed of sides ¸ by ¸ and height, H. A "re has its base at a distance ¸ below the ceiling. (2) There is a single ventilation opening of rectangular shape and width, =, which extends from #oor to ceiling. (3) A "re of circular perimeter and radius, R, entrains air and produces smoke which rises to form a hot layer of depth, Z, and uniform temperature, ¹. Smoke leaves the enclosure through the ventilation opening. (4) The lower layer consists of air at ambient temperature. (5) The three state-variables for the model are: (1) Temperature of the upper layer, ¹; (2) Radius of the "re, 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 #ux to the surface of the fuel to rate of change of "re radius. Three coupled ordinary di!erential equations have been constructed for the temporal rates of change of ¹, Z and R. 2.1. Conservation of energy Applying the principle of conservation of energy to the upper layer results in the equation G";Q #¸#w ,
(1)
where G is the rate of gain of energy of the upper layer, ¸ the rate of loss of energy of the upper layer, w the rate of work done associated with the upper layer and ;Q the rate of change of internal energy of the upper layer. Each of the terms in Eq. (1) has been further de-composed: G consists of convective and radiative energy associated with the combustion together with enthalpy from the lower space gases and the fuel. ¸ consists of energy lost from the layer through the upper surfaces and the layer base together with enthalpy associated with the #ow of hot gases out of the compartment. Work done is associated with the temporal rate of change of the upper layer depth, given by w "Ap dZ/dt,
(2)
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where A is the area of the ceiling, p the pressure within the upper layer and Z the depth of the upper layer. The rate of change of internal energy is given by ;Q "AC d(o¹Z)/dt, (3) where C is the speci"c heat at constant volume of the smoke layer gases, o the density of the upper layer, ¹ the temperature of the upper layer and t the time. The heat release rate, QQ , is assumed to be given by QQ "sm H if m /m 'S (4) "s(m /S )H if m /m (S , (5) where s is the combustion e$ciency, m the fuel volatilization rate, m the rate of entrainment of air, H the heat of combustion and S the Stoichiometric air/fuel mass ratio. The fuel volatilization rate has been assumed to be governed by the net heat #ux to the surface of the pyrolysing fuel, q , via m "A q /H , (6) where A is the surface area of the pyrolysing fuel and nR; R the radius of the circular perimeter of the "re, q the net heat #ux to the surface of the fuel and H the heat of vaporization of the fuel. The term q 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 #ame to the fuel. 2.2. Conservation of mass Applying the principle of conservation of mass to the upper layer results in the equation m #m "m #A d(oZ)/dt, (7) where m is the mass #ow rate of air entrained from the lower layer and m the mass #ow rate from the smoke layer out of the vent. The mass #ow rate out of the vent is assumed to be given by the expression derived by Rockett [11]. The mass #ow 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 in#ow during the latter stages. The general form for the rate of change of "re radius has been taken from the work of Mitler [14] dR/dt"!C log (1!X), (8) where C is a constant and X"q /(p¹), where p is the Stefan's constant and ¹ the #ame temperature.
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In order to avoid problems with the logarithmic term as X approaches 1, the "rst three terms in the logarithmic expansion have been taken. 3. Di4erential equations for the system Manipulation of the above equations enables three coupled ordinary di!erential equations to be derived with the general form d¹/dt"F (¹, Z, R), dZ/dt"F (¹, Z, R), dR/dt"F (¹, Z, R). These equations de"ne a system having the three state-variables ¹, Z and R.
(9) (10) (11)
4. Instability and 6ashover The di!erential equations (9)}(11) above may be solved to "nd the conditions which give rise to instability. The ideas of non-linear dynamical systems theory have been described very brie#y in Ref. [7]. 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 "eld associated with a dynamical system may be used to gauge the stability of the state [3,7]. A system becomes unstable if the real part of an eigenvalue, j, becomes positive. As the "re grows a radius may be reached at which the system becomes unstable. After that the "re would be expected to increase rapidly. If there is su$cient fuel, that is the maximum radius is large enough, to allow a ventilation-controlled (VC) "re to exist then that is the state which would be expected to result. For a domestic-sized room, of typical contents and ventilation opening, that would correspond to a post-#ashover "re. If there is insu$cient fuel present to allow ventilation control then a serious "re 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 "rst, 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 "re 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. Simulations conducted Simulations have been carried out to investigate the e!ect on the calculated critical heat release rate of changing the assumed value of the discharge coe$cient. Full details of input are given in Appendix B.
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A compartment of dimensions ¸ "5 m, ¸ "5 m;3 m high with a single vent, extending from #oor to ceiling, of width 0.4 m has been assumed as a &base case'; the boundary has been assumed to be of brick. The fuel is #exible polyurethane foam and the "re base has been assumed to be on the #oor. The aspect ratio has been de"ned as c"¸ /¸ . This ratio has been varied whilst keeping the total internal surface area, A , constant. The reason for this was to eliminate any possibility that change in the predicted values was due to change in the internal surface area rather than aspect ratio or C per se. The total internal surface area, which does not include the vent, is given by A "2(¸ ¸ )#H(¸ #2¸ )#H(¸ !=). (12) This assumes the vent to be in the side of length ¸ . That is, the general form of the compartment is as shown in Fig. 2. A for this compartment comes out to be 108.8 m. A general problem exists in that when conducting simulations for enclosures a value for the discharge coe$cient needs to be assigned. Work by Steckler et al. [15], using a square compartment, found experimental values for #ow coe$cients to generally be about 0.7. As there will be uncertainty about the value of C in a real-world case, a likely range may be taken with 0.7 as a central point; given the results of Steckler et al. In this study three assumed values have been taken for C , ie 0.5, 0.7 and 0.9, in order to gauge the e!ect of assuming di!erent values for discharge coe$cient. Fig. 3 shows variation in calculated critical heat release rate, Q , with discharge coe$cient, C , for four values of aspect ratio, c. Overall it is seen that the critical heat release rate rises sharply with assumed discharge coe$cient, especially for the largest value of c considered, ie 4.26. As an illustration, graphs of "re radius, R, upper layer temperature, ¹ and upper layer depth, Z, are given in Fig. 4 for the case in which ¸1"8.0 m and ¸2"2.82 m, giving an aspect ratio of 2.84; at C "0.5. The variation in the least negative eigenvalue with time is also shown in Fig. 4.
Fig. 2. The compartment assumed.
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Fig. 3. Variation in critical rate of heat release, Q , with discharge coe$cient, C . (c"aspect ratio; constant total internal surface area. Crosses indicate computed points.)
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Fig. 4. Illustrative results for the case with c"2.84 and C "0.5: (a) Upper layer temperature, ¹: (b) Upper layer depth, Z; (c) Fire radius, R; (d) Eigenvalue, j.
6. Discussion The model FLASHOVER A1 has been used to explore the dependence of the predicted value of the critical heat release rate to produce #ashover on the assumed value of the discharge coe$cient associated with #ow through the ventilation opening. With aspect ratio de"ned as c"¸ /¸ and the vent in the side of length ¸ then it has been found that variation in aspect ratio, at given C , does not have a very considerable e!ect on the "re size necessary for #ashover except for assumed values of the discharge coe$cient which are relatively high. This is at a total internal surface area of 108.8 m. At smaller values of A , however, a marked dependence on aspect ratio has been found; ie for values below about 95 m [16]. Overall, it has been found that the critical "re size for #ashover rises signi"cantly with increase in the assumed value of the discharge coe$cient. It is clearly desirable that discharge coe$cients for di!erent cases be estimated as well as possible. To conclude: overall, simulations using the non-linear model FLASHOVER A1 indicate that the critical "re size for #ashover has a marked dependence on the assumed value of the discharge coe$cient for the ventilation opening. Also, it has been found that, at "xed C , the critical heat release rate does not depend dramatically on the aspect ratio of the compartment, except for relatively high assumed values of the
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C . This is at the particular value of total internal surface area considered; ie 108.8 m. From earlier work [16], however, it has been predicted that below about 95 m then there is a marked dependence on aspect ratio, at "xed assumed value of C . There is a need for experimental work to test the inferences drawn. In particular, the importance of future experimental work aimed at estimating the discharge coe$cients for di!erent con"gurations is apparent.
Appendix A. Further details of the model More details of the model are given here. A full description has been given in Ref. [8]. The term G of Eq. (1) has been assumed to be given by G"QQ #HQ (A.1) G G where QQ is the rate of energy going into the upper layer from the "re; ie convected in G via the plume together with radiation from the #ame and HQ the enthalpy #ow rate G into the plume from the lower space gases and the fuel. QQ has been assumed to be given by G QQ "N QQ , (A.2) G D where QQ is the rate of energy production via combustion, N that fraction of D QQ which goes into the upper layer and N has been assumed to be constant in this D work. The enthalpy #ow rate into the plume, HQ , has been assumed to be given by G HQ "C m ¹ #C m ¹ , (A.3) G where C is the speci"c heat at constant pressure of ambient air. The speci"c heat at constant pressure of the volatiles is assumed to be C . For the #ow rate of entrained gases, m , a switching function has been assumed from the Zukoski plume equation in the early stages to the vent in#ow equation of Prahl and Emmons in the latter stages: m "m EXFUN#m (1!EXFUN), (A.4) N where m is given by the Zukoski equation [12], m is given by the expression of Prahl N and Emmons [13]. EXFUN"0
if Z"H
if Z((H!H ) "exp+Conex[H!Z!H ]/(H!Z)], H (Conex"a dimensionless constant). "1
if
(H!H
)(Z(
The minimum height of the neutral plane, H , is assumed to be a "xed ratio, H , of the vent height; ie H "H H.
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The rate of heat conduction through the upper surface has been assumed to be given by QQ "A D (¹ !¹ ), (A.5) where A is the area of the upper surface; ie adjacent to the smoke layer, D the e!ective conductive coe$cient for upper surface material, ¹ the temperature of the upper surface and ¹ the ambient temperature. The temperature of the upper surface, ¹ , 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 e!ective conductive coe$cient, D , is given by k /d, where d"d is the physical wall thickness if the characteristic "re time, t , is greater than the thermal penetration time of the wall and d the e!ective thickness if the characteristic "re time is less than the thermal penetration time. d "(at ) , a"thermal di!usivity"k /o C , where o is the density of the upper surface material, k the thermal conductivity of the upper surface material and C the speci"c heat of the upper surface material. The upper surface material forms, essentially, the wall/ceiling. (d ) Thermal penetration time [17]" . 4a In the equation for rate of change of "re radius (see [14]) C "C R/(¸ K ), where R is the "re radius, ¸ the mean beam length to the "re base for the #ame volume, assumed to be a cylinder, giving ¸ "1.42R (see Ref. [19]), K the extinc tion coe$cient for the #ame volume, C the exponent in the equation R"C exp(C t), derived from experimental tests conducted by Kawagoe et al. on #ame spread over polyurethane [20].
Appendix B. Input for the simulations The following gives the numerical input for the simulations, outwith values of parameters speci"cally stated in the main text. See Ref. [8] for further explanation relating to input parameters. C : as given in main text; C : 1.04 kJ/kg, see [17]; C : 0.84 kJ/kg, see [17]; C : 0.021/s, see [20]; Conex : 3, found by experience to give a plausible change-over DENC : 0.5, see [8]; h : 0.02 kW/m K, see [17]; h : 0.01 kW/m K, see [21]; H : 3 m; H : 2190 kJ/kg, see [22]; H : 2.87(10)kJ/kg, see [22]; H : 0.45, see [23]; K , K : both 1.3/m, see [22]; k : 0.00069 kW/m K, see [17]; ¸ , ¸ : as speci"ed in main text; ¸ : 3 m; N : 0.7, see [24]; S : 9.78 (poly urethane foam), see [22]; ¹ : 293 K; ¹ : 600 K, see [22]; ¹ : 1400 K, see [17]; t : 300 s; = : 0.4 m; d : 0.06 m, approximate thickness of a brick; e : 1; e : 0.5; e : 0.5;
: 0.95; : 0.95; s : 0.85; o : 1.1 kg/m, see [17]; o : 1600 kg/m, brick, see [17].
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References [1] Beard AN. Evaluation of "re models: report 11, overview. Unit of Fire Safety Engineering, University of Edinburgh, October 1990 (see also reports 1}10). [2] Beard AN. Limitations of computer models. Fire Safety J 1992;18:375}91. [3] Thompson JMT, Stewart HB. Nonlinear dynamics and chaos. Chichester: Wiley, 1986. [4] Strogatz SH. Nonlinear dynamics and chaos. Reading, MA: Addison-Wesley, 1994. [5] Thomas PH, Bullen ML, Quintiere JG, McCa!rey BJ. Flashover and instabilities in "re behaviour. Combustion Flame 1980;38:159}71. [6] Beard AN, Drysdale DD, Holborn PG, Bishop SR. A non-linear model of #ashover. Fire Sci Technol 1992;12:11}27. [7] Bishop SR, Holborn PG, Beard AN, Drysdale DD. Nonlinear dynamics of #ashover in compartment "res. Fire Safety J 1993;21:11}45. [8] Beard AN, Drysdale DD, Holborn PG, Bishop SR. A model of instability and #ashover. J Appl Fire Sci 1994}95;4:3}16. [9] Graham TL, Makhviladze GM, Roberts JP. On the theory of #ashover development. Fire Safety J 1995;25:229}59. [10] Beard AN, Drysdale DD, Holborn PG, Bishop SR. FLASHOVER A1: a model for predicting the conditions for #ashover. Conference on Fire Safety by Design, University of Sunderland, 10}12th July, 1995. [11] Rockett JA. Fire induced gas #ow in an enclosure. Combustion Sci Technol 1976;12:165}75. [12] Zukoski EE, Kubota T, Cetegen E. Entrainment in "re plumes. Fire Safety J 1980/81;3:107}121. [13] Prahl J, Emmons HW. Fire induced #ow through an opening. Combustion Flame 1975;25:369}85. [14] Mitler HE. The physical basis for the Harvard computer "re code. Home "re project technical report No. 34, Division of Applied Sciences, Harvard University, October 1978. [15] Steckler KD, Baum HR, Quintiere JG. Fire induced #ow through room openings-#ow coe$cients. 20th Symposium (International) on Combustion, The Combustion Institute, 1984. p. 1591}1600. [16] Beard AN. Dependence of #ashover on total internal surface area. Proceedings of the Third International Seminar on Fire & Explosion Hazards, Lake Windermere, UK, April 10}14th, 2000. [17] Drysdale DD. Introduction to "re dynamics. Chichester: Wiley, 1999. [18] Luo M, Beck V. A study of non-#ashover and #ashover "res in a full-scale multi-room building. Fire Safety J 1996;26:191}219. [19] Tien CL, Lee KY, Stretton AJ, Radiation heat transfer. SFPE handbook of "re protection engineering. Boston: Society of Fire Protection Engineers, 1988. p. 1}92}1}106. [20] MizunoT, Sasagawa F, Horiuchi S, Kawagoe K.Burning behaviour of urethane foam mattresses. Fire Sci Technol 1981;1:33}44. [21] Atkinson GT, Drysdale DD. Convective heat transfer from "re gases. Fire Safety J 1992;19:217}45. [22] Alpert RL. In#uence of enclosures on "re growth: vol 1; test data, test 7. Report No. OAOR2, Factory Mutual Research Corporation, Norwood, NJ, USA, 1977. [23] Babrauskas V, Williamson RB. Post-#ashover compartment "res: basis of a theoretical model. Fire Mater 1978;2:39}53. [24] Heskestad G. Engineering relations for "re plumes. Fire Safety J 1984;7:25}32.