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Computation of the behavior of fire in an enclosure
Corrlbustion and Flame
131
Computation of the Behavior of Fire hi an Enclosure* Yoshio Tsuchiya and Kikuo S u m i Fire Research Section. Division of Building Research, National Researdh Council of Canada. Ottawa, Ontario. Canada
An improved method is presented for the computation of the behavior o£ fire in an enclosure. The method takes into account the composition and geometrical shape of fuel and the change of equilibrium gas composition with temperature, It is applied to fires in a single compartmem during "developed "+and *'decay" periods. Temperatures of gas and building elements, I~*~P. of combustion, rates of various heat losses, and composition of fire atmosphere are calculated. Uncertain factors ~hat should bc invesligated in future studies to improve the present eomputatior~ include the oxygen deficiency factor in the developed period of fire, the rate of penetration of fire into various fuels, :and ~:hecombustion behavior ¢~fchar.
Introduction lnform:ation on the behavior of fire in an enclosure is important in the design of buildings from the safety point of view. Many us~'ful data, such as rate of heat evolution and temperatures of gas and building elements, can now be obtained by computer simulation of compartment fires without resorting to a large number of costly, full-scale experiments. Several investigators have used this approach [1-5]. Kawagoe and Sekine studied the behavior of fire in a compartment with various sizes of window openings. The influence of the opening factor [1] and the properties of buildin~ materials [4] on the temperature of fire were investigated. Calculations were simplified by assuming that the rate of burning and the composition of th,~ fire atmosphere are constants; the two factors are actually functions of fire temperature. Odeen [5] considered the geometrical shape of the fuel and presented a method for the calculation of temperature * Published with the approval of the Director of the Division of Building Research, National Research Council of Canada.
during the decay period whe n the limited supply of fuel is gradually diminished by the fire. He did not calculate the rate of combustion during the "developed" period of fire but assumed various values for this parameter. In the method of computation described in this paper the rate of combustion and the composition of fire atmosphere are considered to be functions of temperature and fuel composition. In this way the computation can be applied to various types of carbonaceous fuel. For the calculation of fire atmosphere a~ equilibrium between CO + H 2 0 and CO2 + H2 was considered; the inclusion of H2, which has been neglected by previous workers, increases the accuracy of the calculations of the heat evolved in a fire and the heat loss with outflow gas. The temperature during the decay period was computed by a method similar to that used by Odeen. Rate of combustion was calculated in two different ways: the controlling factor of one was air supply, and that of the other was the surface area of fuel. The rate of combustion, based on each factor, was computed at every reference time, and the smaller Combustion & Flame. 16~ 131-139(1971) Copyright © 1971 by The Combustion Institute Published by,American Elsevier Pug[ishingCompany. Inc.
YoshioTsuchiyaand Kikuo Sumi
132 of the two rates was adopted. A decrease in the surface area of fuel during a fire brings on the decay period. In the present study the following values, which describe the behavior ~of fire in an enclosure, were computed: fire temperature, temperature profile of the building elements, composition of the fire atmosphere, rate of combustion, rate of heat evolution, and rates Of various heat losses during the developed period and the decay period. The given values for the computations were dimensions of the enclosure, dimensions of the openings, physical properties of the building element, and composition, amount, and geometrical shape of the fuel. Method of Computation
The heat evolved by the combustion of fuel is equal to the sum of the heat losses, that is, radiation and conduction to the ,enclosing material, Q~ and Qc; radiation through a window opening, Qo; and heat content of outflow gas, Q~ : Qr = QR + Qc + Qo + Qo
(1)
Q~, the Amount of Heat Evolved in a Fire in Unit Time
Qr = qoR
where the numerator on the right-hand side is the volume of inflow air in unit time at standard temperature and pressure (STP); L, the volume of air necessary for the combustion of unit weight of fuel at STP; ~/, the coefficient of contraction of the opening, h~, the height of the neutral plane; W, the width of opening; To, the outside temperature; and v, the mean velocity of inflow air. Also, v = 2 (2g,hi ~ o
~ ) '/z
(41
and h
h, = ~ ~ T
(5)
where Po and p, are the densities of inflow air and outflow gas, respectively, and V, is the volume ratio of outflow gas and inflow air. The values p, and V, are functions ef both fire temperature and composition of fuel, and L is a function of fuel composition. Heat Evolved from Unit Weight of Fuel The amount of heat evolved from a unit weight of fuel, qa, may be calculated from the composition of the fire atmosphere: qa = qc
100- w. 100
w~
Ct ~
-- C.,(CO) - c3(H2)
(2)
(6)
where qo is the heat evolved by the combustion of a unit weight of fuel, and R is the rate of combustion of fuel. The rate R is expressed as R~ waen the controlling factor for combustion is air supply, and as R/when the controlling factor is fuel surface. The smaller value is always adopted for R.
where q, is the heat of combustion of unit weight of dry fuel, W~ is the weight per cent of water in the fuel, and (CO) and (H 2) are the number of moles of CO and H z in outflow gas evolved from unit weight of fuel. Factors cl, c2, and c3 are heat of vaporization of water, heat of combustion of CO, and heat of combustion of hydrogen, respectively.
Ra~e of Combustion In the developed stage of fire, where air supply is the controlling factor, the rate of combustion depends on the rate of airflow into the room due to the buoyancy effect :
Composition of Fire Atmosphere The combustion reaction of fuel must be considered in calculating the values L, Pt, and V~. The reaction may be written as follows:
R~, = vn.h~W 273/TO L
C~HmO. + wH20 + k¢(1 - x) 02 = (3)
bCO + cCO 2 + dH z + eH20
(7)
Behaviorof Fire in an Enclosure
133
where k, is the number of moles of oxygen necessary for complete combustion of I mole of fuel, w is the number of moles of water in 1 mole o f fuel, a n d x is an oxygen-deficiency factor. Then k, = l + m/4 - n/2
(8)
From material balance, l=b+c
(9)
2w + m = 2d + 2e
(10)
and w + n + 2k¢(1 - x ) = b + 2 c + e
(11)
where g is the number of moles of nitrogen : g = (79/21)k,(1 - x)
(17)
Qa, Qc, Qo, Heat Losses Dae to Radiation and Conduction to the Building Elements and to Radiation through an Opening F o r given temperatures of the fire atmosphere heat losses can be calculated by using the Stefan-Boltzmann law and the temperature profile of the enclosing element, which is obtained by the usual numerical method for the calculation of one-dimensional heat flow [3, 5].
In the calculation of the composition of fire atmosphere at the temperature of fire the equilibrium reaction
Heat Loss Owing to Outflow Gas The heat removed by the outflow gas in unit time, Qo, is
C O ' + H 2 0 = CO, + H ,
Q6 = R i ~ m i H i )
(12)
is considered. The equilibrium relation for this reaction is K = cd/be
(i 3)
Tlae equilibrium constant K was calculated as a function of temperature from the usual chemical thermodynamic relations. Equations 9, 10, 11, and 13 are solved simultaneously to determine the composition of the fire atmosphere. One may calculate the values L, V,, and p,, mentioned in the previous paragraph, a n d determine R a, the rate of combustion when air supply is the controlling factor, and Q r , the heat evolved during the developed period of fire. Thus, L =
V,
Pf
(lO0/21)kJl - x ) x 22.4 1 2 1 + m + 16n + 18w (b+c+d+e+g)T (l,O0/21)k~(l- x)To
(14)
(15)
28(b+g)+44c+ 18e + 2d 1 273 b+c+d+e+g x 2--2~.4x - T (16)
(18)
where m i is the number of moles of ith component, and H~ is the enthalpy increase of 1 mole of ith component from initial temperature, To, to the temperature of fire. Generally, H =
Cp d T
(19)
To
and Cp, the molar heat capacity at constant pressure, can be expressed as follows: Cp = ~t + ~ T + 7 T 2 + 6 / T 2
(20)
The coefficients ct, t , ~,, and 6 are given in the literature [6]. Temperature of Fire The temperature of fire atmosphere that satisfies Eq. 1 was determined in the computer program by successive approximations. As shown in Fig. 1, terms other than QR are obtained from an assumed value of T; QR is determined by difference. From QR a new value of T is calculated. Using the new value, the calculation is repeated and another new value of T is obtained. This iteration is continued until the variation between two successive ealCombustion &Flame, 16, 131-139 0971)
134
Yoshio Tsuchiya and Kikuo Sumi
I so,
T; 1
!1
r'j_, = ,ltTi_,} I I •=
"" ~
1
Comp* =: f 2 ( T j ) H = f3(l:omp, QG
= f4 (HI
Ra
~ fs(Tj)
Tj)
El F = f 6 ( c o m p , 0 0
place. This is the start of the decay period of fire. Odeen I'5] calculated the fire temperature of the decay period by considering the rate of penetration of the fire into the fuel and the ratio of volume to surface area of fuel. The fire temperature for the decay period has now been calculated using an approach similar to that of Odeen. The surface-area-controlled rate of combustion of fuel, Rl , may be writlten as follows:
Ro)
Rs = _ _ _ ~ , F_.~f
Si(&/.r.o)~lJ'~
= f7(Tj)
Oc : f s ( t j ,
t ' i . 1)
QR
= QF-QO-QO-Oc
Tj
=
fg(T'j -1,
QR)
* Comp : Composition of f l u e gas Figure I. Calculation of temperature of fire in the computer program,
eulated values is within preselected limits• These calculations are carried out at each reference time at 'intervals of At. The computation terminates when the fire temperature decreases to a preselected level.
(21 )
where vp~. is the rate of penetration of the fire into the fuel perpendicular to the surface, F o and F, are the weight of fuel at times zero and t, Sy is the smallest dimension of the origiraal fuel, and fo is a dimensionless geometrical factor• The value fo is related to the ratio of the surface area to the volume of the fuel, which has values for a board, cylinder, and sphere or cube of 2, 4, and 6, respectively. The weight of fuel at time t is calculated as follows : F, = F o - ~ R dt
(22)
where R is the rate of combustion (in the developed period, R~, and in the decay period,
Rf).
Decay Period of Fire
Examples of Computation
The rate of combustion of fuel in a room with a small opening depends mainly on the supply of air. If progressively larger openings are considered, at some critical size the rate of decomposition (or production of combustible gases and vapors) becomes the controlling factor. The former condition prevails during the de'~eloped period of fire in most practical situations. At a certain stage, however, as the fire gradually consumes the fuel, the transition from the air-controlled to the fuel-controlled (or surface-area-controlled) condition takes
The results of the computations are shown in Figs. 2-6. The following parameters have not been defined or described and are now introduced : fire load, F L , the weight of fuel per unit floor area, and opening factor, Os:
0~-= (h)I/2(A~/A,) where h is the height of the window, Aw is the area of the window, and A r is the total area of the inside surface of the room, including the opening area.
135
Behavior of Fire in an Enclosure
Behavior of Fire in a Room Consider, as an example, a fire in a compartment 3 m by 6 m by 3 m high, with a concrete wall thickness of 20 cm. The window opening is 3 m wide by 0.9 m high. The fuel is wood in the form of 10-cm cubes and has a water
content of 13~o by weight. The t,~re load is 50 kg/m 2. The temperatures of the gas and the surface of the building element, the weight of fuel, and the rate of combustion, all calculated, are plotted against time measured from the start
1200
[ [tOOO [
=:
'
'
'
'
'
'
F,'RE~OAO'SO'O,M'2 ' OPENING FACTOR 0.0285 M½ FUEL WOOD LOCM CUBES
-
~o
~,~MPERATURE OF GAS 8O0
~
600
~:o~
400
2O0
V wE,O~TO,.~DEL'~". "~TEOFCOMO~ST,ON 0
20
40
60
I00
80
120
140
160
TIME. MINUTES
Figure 2. Behavior of fire.
t200 ~
~
-
-
~
1000 ] l-
.'-~. .'" ~,. OAS~"~°"
°
//
.
FIRE LOAD SO gOl M 2 ~ OPENING FACTOR 0.0285 M FUEL WOOD IOCM CUBES
~
"
ALL SURFACE
""
DEPTH ICM
~
600
~ 400
~ 4CM
,-
200
60 80 100 120 140 160 rIME, MINUTES Figure 3. Temperature profile of gas and building element at different depths. 0
20
40
Combustion & Flame.
16.131-139 (1971)
136
Yoshio Tsuchiya and Kikuo Sumi
of the developed period in Fig. 2. The rate of combh:;tion decret, ses sharply at a certain stage where '.~he transition of the controlling mechanism ~'rom air supply to surface area of fuel takes place. The temperature of the gas starts to fail soon after the transition point, and the 12OO
I
I
[
decay period of fire occurs. The temperature of the gas falls more rapidly than that of the building element, so that at a certain stage the heat flow reverses, with the heat transferred from the building element to the gas.
I
~
[
i
1
J
I
I
FIRE LOAD 50KGIM 2 OPENING FACTOR O,0285M½ V pen 0. Q3 M IHR WOOO CUBES iCM~ISCM
lOO0
800 ¢: =
~:
600
=
c,.
~ 4110
200
I
=
20
I
i
40
I
=
l
I
80
60
I
I
lOO
I
I
r
120
t
E40
160
TIME, MINUTES
Figure 4. EITcct o f fucl size. 1200
I J l FIR'i LOAD SO KGIM 2 FUEL SCM DIA. WOOD STICKS T OPEr~ING FACTOR 0.0055--0.0540 M~
lOOO
8OO
.~ 600
400
200 0.0540 r
I 40
t
r 8D
O. 0285 r
I i20
i
I
10O
i
i 200
i
l 240
r
I ZSO
r
I 320
TIME, MINUT£S Figure 5. Temperature versus time for different opening factors.
i 360
Behavior of Fire in an Enclosure
137
"Iem~rature Profile of the Building Element During a Fire The temperature of the gas and the temperature profile of the building element, as calculated for the fire in the example, are presented in Fig. 3. Effect of Size of Feel on Fire Temperature The same comp~rtment as specified in the example was used. The fuel, however, was wood in the form of cubes varying in size from 1 to 15 cm. The gas temperature was calculated during the fire and is presented in Fig. 4. The smaller fuels yielded higher maximum temperatures, and the larger fuels produced longer decay periods. Effect of Opening Factor The dimensions of the room and the width of the window were the same as in the example, but the height of the window was varied from 0.3 to 1.6 m. The relation between gas temperature and time is shown in Fig. 5, and that between maximum gas temperature and opening factor in Fig. 6. Discussion
The present computation, which describes the behavior of fire in an enclosure, includes a FUEL WOOD 5CM DbA. STICKS FINE LOAD 3O-II0 NGIl~ i'
. ~--*50
./.// ,,~ 800
~600
.ofJ ~400
0. O0t
I 0.0!
0. I
OPENING FAI3TOR, J'~ AWIA¥ I.~} Figure 6. M a x i m u m gas temperature versus opening ['actor.
number of improvement~ over the computa, tions of previous workers. It will be further refined w h e n more data from experimental fires become available. One uncertain factor in the present calculation is oxygen deficiency; which was assumed to be 0.1. This value is within the range found in experimental fires. When it was used in calculations, the rate of combustion in the developed period was about 360 (h)L'~2Aw kg/hr, wl~Lich agrees with that found by Thomas [7]. Actually, however, oxygen deficiency depends on both the rate of air supply and the rate of gaseous fuel supply, or the rate of decomposition of fuel..Tn the developed stage of a fire the rate of gaseous fuel supply increases as the fire temperature rises, while the rate of air supply is affected by two countering factors: in,:rcasing buoyancy force and increasing volume of product gas. From these considerations the oxygen-deficiency factor is expected to increase daring the developed period of a fire. This factor reaches a maximum because the increasing oxygen deficiency reduces the heat evolution and the rate of decomposition in the fire. The variation of oxygen deficiency is far greater when fuel is a volatile liquid than when it is a solid. The rate of vaporization of fuel increases with temperature, and the gaseous fuel produced prevents inflow of air so that combustion takes place largely outside the opening. The authors suggest that the oxygen-deficiency factor should be determined from experimental fires. Another uncertain factor is the rate of penetration of ~fire into fuel. Schafferi i[8] determined this value for three different kinds of wood with different water contents and .different specific gravities at various irradiation levels. These data should be obtained for different types of fuel. In the present study the rate of penetration for wood was assumed to be 0.03 m/hr. Fortunately, the variation of the rate of penetration has a relatively small effect on fire behavior. Combusrlml & Flame.
16. 131-139(197t)
138 The cooling curve in the decay period calculated in the present paper varies to a large extent with the size of fuel and to a lesser extent with the opening factor and thermal conductivity of the building dement. In the example shown in Fig. 6, where the fuel consists of wood cubes varying in size from 1 to 15 era, the calculated cooling rates were from 7.5 to 62°C/min. These values probably cover too wide a range and are in most eases faster than the experimental values of 7 and 10°C/min found by Kawagoe and Sekine [2]. A possible explanation for this disagreement is that the evolution of heat by the glowing of char was not considered in the present study. The combustion of char is another field that deserves further study. Kawagoe and Sekine [ I ] used a value of 3558 kcai/kg for the complete combustion of wood. Values ranging from 3990 to 4422 kcal/kg for the complete combustion of wood having a water content of 11.8-13.3~ are found in the literature [9]. When a value in this range is applied to Kawagoe and Sekine's method, the calculated temperature is generally much higher than that found in experimental fires, especially when the opening factor is large, because temperature increases linearly with increase of opening factor on a semilogarithmie scale. In the present computation the temperature levels offat large opening factors, as shown in Fig. 6; this agrees with the findings of Thomas [7] and of Gross and Robertson [10].
Conclusion A method is presented for the computation of the behavior of fire in "~n enclosure. The composition and the geometrical shape of fuel are considered in the calculation of temperature of fire, composition of fire atmosphere, rate of combustion, and rates of various heat losses during the developed and decay periods. The computations show that computer results describe the behavior of fire very well. Uncertain
YoshioTsuchiyaand Kikuo $umi factors that should be considered or experimentally determined to improve the computation described in this paper include the oxygen deficiency factor in the developed period of fire, the rate of penetration of fire into various fuels, and the combustion behavior of char. The computer program was written in FORTRAN IV. The listing of the program may be obtained from the authors.
Nomenclature
Af Aw Af ct c2 ca
C. .fi fo F FL gr h h~
H, H~ k~
K L m~
O; q. q~
of' floor, m z area of window, m 2 area of inside surface, including area of window, m 2 heat of vaporization of water; cl = 583.9 kcal/kg at 25°C heat of combustion of carbon monoxide; c2 = 67.636 kcal/mole at 25°C heat of combustion of hydrogen; c a = 57.798 kcal/mol¢ at 25°C molar heat capacity at constant pressure, kcal/mole °C function geometrical lhctor of fuel weight of fuel, kg fire load, kg/m 2 acceleration of gravity, m/hr 2 height of window, m height of neutral plane, m heat content, kcal/mole number of moles of oxygen necessary for complete combustion of I mole of fuel equilibrium constant volume of air needed for combustion of I kg of fuel, ma/kg number of moles of a component in flue gas evolved from 1 kg of fuel opening factor, m I'z heat released from 1 kg fuel in fire, kcal/kg heat of combustion of dry fuel, kcal/kg area
Behavior of Fire in an Enclosure
Qr QR
Qc Qo Q6 R~ Rr S.r t, tj To T
T~, j v t,o~. V, w W~ W
I0"
heat gvolved from the combustion of fuel in a fire, kca!/hr radiation to the wall, keal/hr convection to the wall, kcal/hr radiation to outsid,: through window, kcal/hr heat carriedby outflow gas, kcal/hr rate of combustion of fuel, air supply controlled, kg/hr rate of combustion of fuel, surface area of fuel controlled, kg/hr size of fuel, m finite short time, hr temperature of outside, °K temperature of fire, °K temperature of wall at ith reference point at jth reference time mean velocity of inflow air, m/hr rate of penetration of fire into fuel, m/hr volume ratio of outflow gas and inflow air number of moles of water in 1 mole of fuel water content of fuel, wt % width of window, m
139 x ~t, fl, 7, 6 Po Pt r/
factor of oxygen deficiency coefficient of heat capacity density of air at T o, kg/m a density of flue gas at T, kg/m 3 coefficient of contraction of window
References 1. KAWAGOE, K., and SEKINE, T., Japan Building Res. l,~zst. Occasional Rept. I I (June 1963). 2. KAWAGOE, K., and SEKINE. T.. Japan Build~,lg Res. I,nst. Occasional Rept. 17 (March 1964). 3. KAWAOOE,K., Japan Building Res. Inst. Res. Paper 29 (October 1967). 4, KAWAGOE,K., Japan Bull. Fire Prevention Sac,, 13. 29 (1965). 5. ODEEN.K,, Sweden. Roy. last. TecbnoL Bull. 10 (1963). 6. PERRY,J. H., Chemical Engineers' Handbook. 4th ¢d.. F'. 3-116. McGraw-Hill: New York (1963). 7. THOMAS,P. H.. Rescarch. 13, 69 0960). 8. SCHAI:EER,E. L., U.S. Forest Serv. Res. Paper FPL 69 (1967). 9. LANGE.N. D.. Handbook ofCht, mistry, 9th ed., p. 1565 0956). 10. GROSS, D., and ROBERTSON,A. F., Tentb Synlposium (International) on Combustion. pp. 931-942. The Combustion Institute: Pittsburgh 0965).
(Received August, 1970: revised October, 1970)
Combustion & Flame. 16, ]31-139 (19711
131
Computation of the Behavior of Fire hi an Enclosure* Yoshio Tsuchiya and Kikuo S u m i Fire Research Section. Division of Building Research, National Researdh Council of Canada. Ottawa, Ontario. Canada
An improved method is presented for the computation of the behavior o£ fire in an enclosure. The method takes into account the composition and geometrical shape of fuel and the change of equilibrium gas composition with temperature, It is applied to fires in a single compartmem during "developed "+and *'decay" periods. Temperatures of gas and building elements, I~*~P. of combustion, rates of various heat losses, and composition of fire atmosphere are calculated. Uncertain factors ~hat should bc invesligated in future studies to improve the present eomputatior~ include the oxygen deficiency factor in the developed period of fire, the rate of penetration of fire into various fuels, :and ~:hecombustion behavior ¢~fchar.
Introduction lnform:ation on the behavior of fire in an enclosure is important in the design of buildings from the safety point of view. Many us~'ful data, such as rate of heat evolution and temperatures of gas and building elements, can now be obtained by computer simulation of compartment fires without resorting to a large number of costly, full-scale experiments. Several investigators have used this approach [1-5]. Kawagoe and Sekine studied the behavior of fire in a compartment with various sizes of window openings. The influence of the opening factor [1] and the properties of buildin~ materials [4] on the temperature of fire were investigated. Calculations were simplified by assuming that the rate of burning and the composition of th,~ fire atmosphere are constants; the two factors are actually functions of fire temperature. Odeen [5] considered the geometrical shape of the fuel and presented a method for the calculation of temperature * Published with the approval of the Director of the Division of Building Research, National Research Council of Canada.
during the decay period whe n the limited supply of fuel is gradually diminished by the fire. He did not calculate the rate of combustion during the "developed" period of fire but assumed various values for this parameter. In the method of computation described in this paper the rate of combustion and the composition of fire atmosphere are considered to be functions of temperature and fuel composition. In this way the computation can be applied to various types of carbonaceous fuel. For the calculation of fire atmosphere a~ equilibrium between CO + H 2 0 and CO2 + H2 was considered; the inclusion of H2, which has been neglected by previous workers, increases the accuracy of the calculations of the heat evolved in a fire and the heat loss with outflow gas. The temperature during the decay period was computed by a method similar to that used by Odeen. Rate of combustion was calculated in two different ways: the controlling factor of one was air supply, and that of the other was the surface area of fuel. The rate of combustion, based on each factor, was computed at every reference time, and the smaller Combustion & Flame. 16~ 131-139(1971) Copyright © 1971 by The Combustion Institute Published by,American Elsevier Pug[ishingCompany. Inc.
YoshioTsuchiyaand Kikuo Sumi
132 of the two rates was adopted. A decrease in the surface area of fuel during a fire brings on the decay period. In the present study the following values, which describe the behavior ~of fire in an enclosure, were computed: fire temperature, temperature profile of the building elements, composition of the fire atmosphere, rate of combustion, rate of heat evolution, and rates Of various heat losses during the developed period and the decay period. The given values for the computations were dimensions of the enclosure, dimensions of the openings, physical properties of the building element, and composition, amount, and geometrical shape of the fuel. Method of Computation
The heat evolved by the combustion of fuel is equal to the sum of the heat losses, that is, radiation and conduction to the ,enclosing material, Q~ and Qc; radiation through a window opening, Qo; and heat content of outflow gas, Q~ : Qr = QR + Qc + Qo + Qo
(1)
Q~, the Amount of Heat Evolved in a Fire in Unit Time
Qr = qoR
where the numerator on the right-hand side is the volume of inflow air in unit time at standard temperature and pressure (STP); L, the volume of air necessary for the combustion of unit weight of fuel at STP; ~/, the coefficient of contraction of the opening, h~, the height of the neutral plane; W, the width of opening; To, the outside temperature; and v, the mean velocity of inflow air. Also, v = 2 (2g,hi ~ o
~ ) '/z
(41
and h
h, = ~ ~ T
(5)
where Po and p, are the densities of inflow air and outflow gas, respectively, and V, is the volume ratio of outflow gas and inflow air. The values p, and V, are functions ef both fire temperature and composition of fuel, and L is a function of fuel composition. Heat Evolved from Unit Weight of Fuel The amount of heat evolved from a unit weight of fuel, qa, may be calculated from the composition of the fire atmosphere: qa = qc
100- w. 100
w~
Ct ~
-- C.,(CO) - c3(H2)
(2)
(6)
where qo is the heat evolved by the combustion of a unit weight of fuel, and R is the rate of combustion of fuel. The rate R is expressed as R~ waen the controlling factor for combustion is air supply, and as R/when the controlling factor is fuel surface. The smaller value is always adopted for R.
where q, is the heat of combustion of unit weight of dry fuel, W~ is the weight per cent of water in the fuel, and (CO) and (H 2) are the number of moles of CO and H z in outflow gas evolved from unit weight of fuel. Factors cl, c2, and c3 are heat of vaporization of water, heat of combustion of CO, and heat of combustion of hydrogen, respectively.
Ra~e of Combustion In the developed stage of fire, where air supply is the controlling factor, the rate of combustion depends on the rate of airflow into the room due to the buoyancy effect :
Composition of Fire Atmosphere The combustion reaction of fuel must be considered in calculating the values L, Pt, and V~. The reaction may be written as follows:
R~, = vn.h~W 273/TO L
C~HmO. + wH20 + k¢(1 - x) 02 = (3)
bCO + cCO 2 + dH z + eH20
(7)
Behaviorof Fire in an Enclosure
133
where k, is the number of moles of oxygen necessary for complete combustion of I mole of fuel, w is the number of moles of water in 1 mole o f fuel, a n d x is an oxygen-deficiency factor. Then k, = l + m/4 - n/2
(8)
From material balance, l=b+c
(9)
2w + m = 2d + 2e
(10)
and w + n + 2k¢(1 - x ) = b + 2 c + e
(11)
where g is the number of moles of nitrogen : g = (79/21)k,(1 - x)
(17)
Qa, Qc, Qo, Heat Losses Dae to Radiation and Conduction to the Building Elements and to Radiation through an Opening F o r given temperatures of the fire atmosphere heat losses can be calculated by using the Stefan-Boltzmann law and the temperature profile of the enclosing element, which is obtained by the usual numerical method for the calculation of one-dimensional heat flow [3, 5].
In the calculation of the composition of fire atmosphere at the temperature of fire the equilibrium reaction
Heat Loss Owing to Outflow Gas The heat removed by the outflow gas in unit time, Qo, is
C O ' + H 2 0 = CO, + H ,
Q6 = R i ~ m i H i )
(12)
is considered. The equilibrium relation for this reaction is K = cd/be
(i 3)
Tlae equilibrium constant K was calculated as a function of temperature from the usual chemical thermodynamic relations. Equations 9, 10, 11, and 13 are solved simultaneously to determine the composition of the fire atmosphere. One may calculate the values L, V,, and p,, mentioned in the previous paragraph, a n d determine R a, the rate of combustion when air supply is the controlling factor, and Q r , the heat evolved during the developed period of fire. Thus, L =
V,
Pf
(lO0/21)kJl - x ) x 22.4 1 2 1 + m + 16n + 18w (b+c+d+e+g)T (l,O0/21)k~(l- x)To
(14)
(15)
28(b+g)+44c+ 18e + 2d 1 273 b+c+d+e+g x 2--2~.4x - T (16)
(18)
where m i is the number of moles of ith component, and H~ is the enthalpy increase of 1 mole of ith component from initial temperature, To, to the temperature of fire. Generally, H =
Cp d T
(19)
To
and Cp, the molar heat capacity at constant pressure, can be expressed as follows: Cp = ~t + ~ T + 7 T 2 + 6 / T 2
(20)
The coefficients ct, t , ~,, and 6 are given in the literature [6]. Temperature of Fire The temperature of fire atmosphere that satisfies Eq. 1 was determined in the computer program by successive approximations. As shown in Fig. 1, terms other than QR are obtained from an assumed value of T; QR is determined by difference. From QR a new value of T is calculated. Using the new value, the calculation is repeated and another new value of T is obtained. This iteration is continued until the variation between two successive ealCombustion &Flame, 16, 131-139 0971)
134
Yoshio Tsuchiya and Kikuo Sumi
I so,
T; 1
!1
r'j_, = ,ltTi_,} I I •=
"" ~
1
Comp* =: f 2 ( T j ) H = f3(l:omp, QG
= f4 (HI
Ra
~ fs(Tj)
Tj)
El F = f 6 ( c o m p , 0 0
place. This is the start of the decay period of fire. Odeen I'5] calculated the fire temperature of the decay period by considering the rate of penetration of the fire into the fuel and the ratio of volume to surface area of fuel. The fire temperature for the decay period has now been calculated using an approach similar to that of Odeen. The surface-area-controlled rate of combustion of fuel, Rl , may be writlten as follows:
Ro)
Rs = _ _ _ ~ , F_.~f
Si(&/.r.o)~lJ'~
= f7(Tj)
Oc : f s ( t j ,
t ' i . 1)
QR
= QF-QO-QO-Oc
Tj
=
fg(T'j -1,
QR)
* Comp : Composition of f l u e gas Figure I. Calculation of temperature of fire in the computer program,
eulated values is within preselected limits• These calculations are carried out at each reference time at 'intervals of At. The computation terminates when the fire temperature decreases to a preselected level.
(21 )
where vp~. is the rate of penetration of the fire into the fuel perpendicular to the surface, F o and F, are the weight of fuel at times zero and t, Sy is the smallest dimension of the origiraal fuel, and fo is a dimensionless geometrical factor• The value fo is related to the ratio of the surface area to the volume of the fuel, which has values for a board, cylinder, and sphere or cube of 2, 4, and 6, respectively. The weight of fuel at time t is calculated as follows : F, = F o - ~ R dt
(22)
where R is the rate of combustion (in the developed period, R~, and in the decay period,
Rf).
Decay Period of Fire
Examples of Computation
The rate of combustion of fuel in a room with a small opening depends mainly on the supply of air. If progressively larger openings are considered, at some critical size the rate of decomposition (or production of combustible gases and vapors) becomes the controlling factor. The former condition prevails during the de'~eloped period of fire in most practical situations. At a certain stage, however, as the fire gradually consumes the fuel, the transition from the air-controlled to the fuel-controlled (or surface-area-controlled) condition takes
The results of the computations are shown in Figs. 2-6. The following parameters have not been defined or described and are now introduced : fire load, F L , the weight of fuel per unit floor area, and opening factor, Os:
0~-= (h)I/2(A~/A,) where h is the height of the window, Aw is the area of the window, and A r is the total area of the inside surface of the room, including the opening area.
135
Behavior of Fire in an Enclosure
Behavior of Fire in a Room Consider, as an example, a fire in a compartment 3 m by 6 m by 3 m high, with a concrete wall thickness of 20 cm. The window opening is 3 m wide by 0.9 m high. The fuel is wood in the form of 10-cm cubes and has a water
content of 13~o by weight. The t,~re load is 50 kg/m 2. The temperatures of the gas and the surface of the building element, the weight of fuel, and the rate of combustion, all calculated, are plotted against time measured from the start
1200
[ [tOOO [
=:
'
'
'
'
'
'
F,'RE~OAO'SO'O,M'2 ' OPENING FACTOR 0.0285 M½ FUEL WOOD LOCM CUBES
-
~o
~,~MPERATURE OF GAS 8O0
~
600
~:o~
400
2O0
V wE,O~TO,.~DEL'~". "~TEOFCOMO~ST,ON 0
20
40
60
I00
80
120
140
160
TIME. MINUTES
Figure 2. Behavior of fire.
t200 ~
~
-
-
~
1000 ] l-
.'-~. .'" ~,. OAS~"~°"
°
//
.
FIRE LOAD SO gOl M 2 ~ OPENING FACTOR 0.0285 M FUEL WOOD IOCM CUBES
~
"
ALL SURFACE
""
DEPTH ICM
~
600
~ 400
~ 4CM
,-
200
60 80 100 120 140 160 rIME, MINUTES Figure 3. Temperature profile of gas and building element at different depths. 0
20
40
Combustion & Flame.
16.131-139 (1971)
136
Yoshio Tsuchiya and Kikuo Sumi
of the developed period in Fig. 2. The rate of combh:;tion decret, ses sharply at a certain stage where '.~he transition of the controlling mechanism ~'rom air supply to surface area of fuel takes place. The temperature of the gas starts to fail soon after the transition point, and the 12OO
I
I
[
decay period of fire occurs. The temperature of the gas falls more rapidly than that of the building element, so that at a certain stage the heat flow reverses, with the heat transferred from the building element to the gas.
I
~
[
i
1
J
I
I
FIRE LOAD 50KGIM 2 OPENING FACTOR O,0285M½ V pen 0. Q3 M IHR WOOO CUBES iCM~ISCM
lOO0
800 ¢: =
~:
600
=
c,.
~ 4110
200
I
=
20
I
i
40
I
=
l
I
80
60
I
I
lOO
I
I
r
120
t
E40
160
TIME, MINUTES
Figure 4. EITcct o f fucl size. 1200
I J l FIR'i LOAD SO KGIM 2 FUEL SCM DIA. WOOD STICKS T OPEr~ING FACTOR 0.0055--0.0540 M~
lOOO
8OO
.~ 600
400
200 0.0540 r
I 40
t
r 8D
O. 0285 r
I i20
i
I
10O
i
i 200
i
l 240
r
I ZSO
r
I 320
TIME, MINUT£S Figure 5. Temperature versus time for different opening factors.
i 360
Behavior of Fire in an Enclosure
137
"Iem~rature Profile of the Building Element During a Fire The temperature of the gas and the temperature profile of the building element, as calculated for the fire in the example, are presented in Fig. 3. Effect of Size of Feel on Fire Temperature The same comp~rtment as specified in the example was used. The fuel, however, was wood in the form of cubes varying in size from 1 to 15 cm. The gas temperature was calculated during the fire and is presented in Fig. 4. The smaller fuels yielded higher maximum temperatures, and the larger fuels produced longer decay periods. Effect of Opening Factor The dimensions of the room and the width of the window were the same as in the example, but the height of the window was varied from 0.3 to 1.6 m. The relation between gas temperature and time is shown in Fig. 5, and that between maximum gas temperature and opening factor in Fig. 6. Discussion
The present computation, which describes the behavior of fire in an enclosure, includes a FUEL WOOD 5CM DbA. STICKS FINE LOAD 3O-II0 NGIl~ i'
. ~--*50
./.// ,,~ 800
~600
.ofJ ~400
0. O0t
I 0.0!
0. I
OPENING FAI3TOR, J'~ AWIA¥ I.~} Figure 6. M a x i m u m gas temperature versus opening ['actor.
number of improvement~ over the computa, tions of previous workers. It will be further refined w h e n more data from experimental fires become available. One uncertain factor in the present calculation is oxygen deficiency; which was assumed to be 0.1. This value is within the range found in experimental fires. When it was used in calculations, the rate of combustion in the developed period was about 360 (h)L'~2Aw kg/hr, wl~Lich agrees with that found by Thomas [7]. Actually, however, oxygen deficiency depends on both the rate of air supply and the rate of gaseous fuel supply, or the rate of decomposition of fuel..Tn the developed stage of a fire the rate of gaseous fuel supply increases as the fire temperature rises, while the rate of air supply is affected by two countering factors: in,:rcasing buoyancy force and increasing volume of product gas. From these considerations the oxygen-deficiency factor is expected to increase daring the developed period of a fire. This factor reaches a maximum because the increasing oxygen deficiency reduces the heat evolution and the rate of decomposition in the fire. The variation of oxygen deficiency is far greater when fuel is a volatile liquid than when it is a solid. The rate of vaporization of fuel increases with temperature, and the gaseous fuel produced prevents inflow of air so that combustion takes place largely outside the opening. The authors suggest that the oxygen-deficiency factor should be determined from experimental fires. Another uncertain factor is the rate of penetration of ~fire into fuel. Schafferi i[8] determined this value for three different kinds of wood with different water contents and .different specific gravities at various irradiation levels. These data should be obtained for different types of fuel. In the present study the rate of penetration for wood was assumed to be 0.03 m/hr. Fortunately, the variation of the rate of penetration has a relatively small effect on fire behavior. Combusrlml & Flame.
16. 131-139(197t)
138 The cooling curve in the decay period calculated in the present paper varies to a large extent with the size of fuel and to a lesser extent with the opening factor and thermal conductivity of the building dement. In the example shown in Fig. 6, where the fuel consists of wood cubes varying in size from 1 to 15 era, the calculated cooling rates were from 7.5 to 62°C/min. These values probably cover too wide a range and are in most eases faster than the experimental values of 7 and 10°C/min found by Kawagoe and Sekine [2]. A possible explanation for this disagreement is that the evolution of heat by the glowing of char was not considered in the present study. The combustion of char is another field that deserves further study. Kawagoe and Sekine [ I ] used a value of 3558 kcai/kg for the complete combustion of wood. Values ranging from 3990 to 4422 kcal/kg for the complete combustion of wood having a water content of 11.8-13.3~ are found in the literature [9]. When a value in this range is applied to Kawagoe and Sekine's method, the calculated temperature is generally much higher than that found in experimental fires, especially when the opening factor is large, because temperature increases linearly with increase of opening factor on a semilogarithmie scale. In the present computation the temperature levels offat large opening factors, as shown in Fig. 6; this agrees with the findings of Thomas [7] and of Gross and Robertson [10].
Conclusion A method is presented for the computation of the behavior of fire in "~n enclosure. The composition and the geometrical shape of fuel are considered in the calculation of temperature of fire, composition of fire atmosphere, rate of combustion, and rates of various heat losses during the developed and decay periods. The computations show that computer results describe the behavior of fire very well. Uncertain
YoshioTsuchiyaand Kikuo $umi factors that should be considered or experimentally determined to improve the computation described in this paper include the oxygen deficiency factor in the developed period of fire, the rate of penetration of fire into various fuels, and the combustion behavior of char. The computer program was written in FORTRAN IV. The listing of the program may be obtained from the authors.
Nomenclature
Af Aw Af ct c2 ca
C. .fi fo F FL gr h h~
H, H~ k~
K L m~
O; q. q~
of' floor, m z area of window, m 2 area of inside surface, including area of window, m 2 heat of vaporization of water; cl = 583.9 kcal/kg at 25°C heat of combustion of carbon monoxide; c2 = 67.636 kcal/mole at 25°C heat of combustion of hydrogen; c a = 57.798 kcal/mol¢ at 25°C molar heat capacity at constant pressure, kcal/mole °C function geometrical lhctor of fuel weight of fuel, kg fire load, kg/m 2 acceleration of gravity, m/hr 2 height of window, m height of neutral plane, m heat content, kcal/mole number of moles of oxygen necessary for complete combustion of I mole of fuel equilibrium constant volume of air needed for combustion of I kg of fuel, ma/kg number of moles of a component in flue gas evolved from 1 kg of fuel opening factor, m I'z heat released from 1 kg fuel in fire, kcal/kg heat of combustion of dry fuel, kcal/kg area
Behavior of Fire in an Enclosure
Qr QR
Qc Qo Q6 R~ Rr S.r t, tj To T
T~, j v t,o~. V, w W~ W
I0"
heat gvolved from the combustion of fuel in a fire, kca!/hr radiation to the wall, keal/hr convection to the wall, kcal/hr radiation to outsid,: through window, kcal/hr heat carriedby outflow gas, kcal/hr rate of combustion of fuel, air supply controlled, kg/hr rate of combustion of fuel, surface area of fuel controlled, kg/hr size of fuel, m finite short time, hr temperature of outside, °K temperature of fire, °K temperature of wall at ith reference point at jth reference time mean velocity of inflow air, m/hr rate of penetration of fire into fuel, m/hr volume ratio of outflow gas and inflow air number of moles of water in 1 mole of fuel water content of fuel, wt % width of window, m
139 x ~t, fl, 7, 6 Po Pt r/
factor of oxygen deficiency coefficient of heat capacity density of air at T o, kg/m a density of flue gas at T, kg/m 3 coefficient of contraction of window
References 1. KAWAGOE, K., and SEKINE, T., Japan Building Res. l,~zst. Occasional Rept. I I (June 1963). 2. KAWAGOE, K., and SEKINE. T.. Japan Build~,lg Res. I,nst. Occasional Rept. 17 (March 1964). 3. KAWAOOE,K., Japan Building Res. Inst. Res. Paper 29 (October 1967). 4, KAWAGOE,K., Japan Bull. Fire Prevention Sac,, 13. 29 (1965). 5. ODEEN.K,, Sweden. Roy. last. TecbnoL Bull. 10 (1963). 6. PERRY,J. H., Chemical Engineers' Handbook. 4th ¢d.. F'. 3-116. McGraw-Hill: New York (1963). 7. THOMAS,P. H.. Rescarch. 13, 69 0960). 8. SCHAI:EER,E. L., U.S. Forest Serv. Res. Paper FPL 69 (1967). 9. LANGE.N. D.. Handbook ofCht, mistry, 9th ed., p. 1565 0956). 10. GROSS, D., and ROBERTSON,A. F., Tentb Synlposium (International) on Combustion. pp. 931-942. The Combustion Institute: Pittsburgh 0965).
(Received August, 1970: revised October, 1970)
Combustion & Flame. 16, ]31-139 (19711