A Burning Model for Charring Materials and Its Application to the Compartment Fire Development Keisuke Himoto 1 and Takeyoshi Tanaka 2

Fire Science and Technorogy Vol.23 No.3(2004) 170-190 170

1 Department of Engineering, Kyoto University, Japan 2 Disaster Prevention Research Institute, Kyoto University, Japan

ABSTRACT A one-dimensional integral model is formulated to describe the transient burning of charring materials. In the model, the sequence of events occurring in a material when exposed to a fire environment is divided into following sequential phases: (I) initial heating phase, (II) pyrolysis phase, (III) char oxidation phase. The calculated mass loss rates are shown to be in reasonable agreement with the existing experimental data. The burning model is then incorporated into a one-layer zone compartment fire model. The fuel inside the compartment is divided into multiple fuel elements, and the compartment fire behaviors are described as the consequence of the burning of each of these fuel elements. The comparison with experimental data in the literature also agreed well.

keywords : Fuel, Wood, Char layer, Burning model, Compartment fire, Zone model

1. INTRODUCTION

There is a substantial risk of urban fires, which involve multiple buildings simultaneously, particularly in the event of a severe earthquake. Although, an urban fire always starts as an individual building fire(s) in the urban area, it easily becomes out of control, once it spreads to the adjacent buildings. The Sakata Fire (1976), or the fire followed the Hanshin-Awaji Eathquake (1995) are the examples. To explore the effective measures for reducing the loss caused by urban fires, it is indispensable to develop a rational model for the urban fire spread. As urban fire is an ensemble of numerous fire-involved buildings, and the building fire itself is nothing but an ensemble of room (compartment) fires, development of a reliable compartment fire model will be a starting point. There are several models available for the compartment fire prediction, which have widely been provided for evaluating fire protection performances of buildings. Above all, zone model, which assumes the compartment as one or two control volumes, has been applied to a wide variety of fire scenarios with success [1~4]. Extending such an experienced model, it is expected that a practical urban fire spread model will be developed effectively. However, accuracy is not the only concern in developing a model for the use of urban fire spread simulation, but there are two additional issues that have to be considered 171 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 172 concurrently. One is that the model and its calculation procedure should be simple. As the urban fire involves numerous buildings, it is crucial to save the computational time for an individual compartment. The other is that the model should predict the transient fire behaviors from ignition till decaying, as buildings at different burning conditions are involved in an urban fire. As the behaviors of a compartment fire depend heavily on the burning behavior of fuel inside, we start with a burning model of the combustibles. Here, a slab of charring solid is assumed as the fuel. Wood, which constitutes a considerable fraction of the fuel load in many building fires, is an example of the charring solid. So, the burning behavior of the fuel when exposed to an external heating from one side is modeled, neglecting the dimensions of the fuel. We then incorporate the burning model into a one-layer zone compartment fire model, for the transient compartment fire prediction.

2. TRANSIENT BURNING MODEL FOR CHARRING MATERIALS

Significant numbers of numerical models are available for the burning of one-dimensional materials. In these models, the rate of pyrolysis is attributed to the temperature rise inside the material. Generally, the temperature rise is calculated by numerically solving the heat conduction equations, which are partial differential equations. This (I) Initial Heating Phase provides a comprehensive description of the burning process [5,6]. However, numbers of approximate solutions for the heat conduction equations have been explored for the purpose of reducing numerical costs. Heat-balanced integral model, which reduces the original equations to the form of ordinary differential equations, have already been applied to the burning of (II) Pyrolysis Phase materials [7~11]. One of the advantages of adopting integral model is that it provides a significant saving in computational time, while it is still able to provide reasonable levels of accuracy for engineering purposes. In this study, we develop an integral model for the transient burning of material with special attention on: (a) modeling the transient burning behaviors from ignition till decaying, including heat generation by (III) Char Oxidation Phase the char oxidation; (b) deriving simple and physically Figure 1 Transition of burning consistent formulae with some assumptions invoked. behaviors of charring material. 171 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 172

2.1 Burning of e Material In the model, the behaviors of the material receiving external heat is divided into the following sequential phases: ( I ) Initial Heating Phase (Figure 1 (I)), ( I I ) Pyrolysis Phase (Figure 1 (II)), (III) Char Oxidation Phase (Figure 1 (III)). In the initial heating phase, the material receives heat flux from external heat sources such as flame or compartment gas. The heat is transferred inwards by conduction and

raises the temperature. It finally ignites when the resulting surface temperature Ts

exceeds the material pyrolysis temperature Tp . In the pyrolysis phase, flammable gas and residual char are produced by the pyrolysis of unreacted virgin material. The flammable gas is released into the gas phase and reacts with oxygen to form a flame above, whereas the residual char accumulates upon the virgin material relaxing the intensity of incoming heat flux. Yet, in the meanwhile, the char oxidizes at the surface and generates heat, though it is moderate compared to that of the flammable gas combustion. These layers are separated with a thin layer

(pyrolysis front) of a constant temperature Tp , at which the pyrolysis takes place at a rate proportional to the net incident heat flux. The char oxidation phase is the phase after all the virgin material has been consumed. In this period, flame upon the material is put out as the flammable gas is not being released anymore. However, char oxidation at the material surface continues and keeps the char temperature high. The char oxidation takes place at the thin boundary between the compartment gas and char layer till all of the char is consumed. 2.1.1 Governing Equations Thermal properties of the material are considered to be constant and uniform for the entire body, and the effect of moisture content is neglected. Then, the heat conduction equations for the virgin layer and the char layer are respectively expressed as,

(1)

(2)

where T is the temperature, α(=k/ρc) is the thermal diffusivity, k is the thermal conductivity, ρ is the density, c is the heat capacity, t is the time, and x is the coordinate whose origin is at the material surface before ignition. The subscripts F and C represent the virgin layer and the char layer, respectively. In the pyrolysis phase, pyrolysis of the material takes place at the boundary of the char layer and the virgin layer, at a rate proportional to the net incident heat flux. So the mass loss rate per unit area of the virgin material is given by, 173 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 174

(3)

in which Lp is the latent heat of pyrolysis. Rate of the char oxidation is considered to depend on its temperature or concentration of the oxygen near the surface. In the case of fire, the material receives considerable heat from the external heat sources such as flame or compartment gas, and the surface temperature TS of the material is assumed to be high enough. Thus is described proportional to the mass fraction of oxygen YO near the material surface as,

(4) where and are the reference char oxidation rate per unit area and the reference mass fraction of oxygen. With the above expression (4), the char oxidation could take place even in the pyrolysis phase. An example of such a case is when no flame is formed upon the material due to the ventilation condition in the compartment fire. 2.1.2 Boundary Conditions

(I) Initial Heating Phase The boundary conditions in the initial heating phase are divided into two parts: before and after the thermal penetration depth δF reaches the material width L. For the former case, the boundary condition takes the form,

(5)

where TS is the surface temperature, T∞ is the ambient temperature, and is the net incident heat flux to the material. Considering that the heat is transferred by radiation and convection, is expressed as,

(6) where is the external heat flux from the flame or compartment gas formed above the material, σ is the Stefan-Boltzmann’s constant, εS and hS are the emissivity and convective heat transfer coefficient of the material surface, respectively.

If the heating continues and the thermal penetration depth δF reaches the material width L before the ignition, the boundary condition takes the form as follows,

(7)

where TL is the backside temperature of the material at which adiabatic condition is assumed. 173 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 174

(II) Pyrolysis Phase In the pyrolysis phase, the material is divided into the virgin layer and the char layer. As for the virgin layer, the boundary conditions are divided further into two parts as with the case of initial heating phase. When the thermal penetration depth is shorter

than the material width, i.e., δ∞+δC+δF

(8)

where δ∞ is the descending width of the surface, δC is the char width, and TP is the pyrolysis temperature. As the char starts to oxidize in the pyrolysis phase, is given by, (9)

where ΔHs is the heat released per unit area of the char while oxidation.

When the penetration depth δ∞ + δC + δF reaches the material width L, then the boundary condition takes the form as,

(10)

As for the char layer, the effect of the thermal penetration depth is negligible and a sole boundary condition is assumed through the pyrolysis phase, which is expressed as,

(11)

The boundary temperature of the char layer and the virgin layer is kept at a constant

pyrolysis temperature TP . (III) Char Oxidation Phase In the char oxidation phase, the boundary condition takes the form as,

(12)

As the char oxidation takes place in the current phase, the incident heat flux to the material surface is expressed with the eqn.(9). The backside of the material is assumed to be adiabatic. 175 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 176

2.2 Integral Model We now derive approximate solutions for the burning of charring materials described in the previous section. As an integral model is derived by solving the conservation equation of energy under the concerning boundary condition, there will be individual solutions for different boundary conditions. 2.2.1 Initial Heating Phase

(I-A) when δF< L

First, we approximate the temperature profile of the virgin layer TF as a quadratic function,

(13) where A, B, and C are the coefficients. These are set so as to satisfy the boundary conditions (5), then the eqn.(13) is expressed as,

(14)

With the eqn.(14), the temperature at an arbitrary depth inside the material is known. Yet at this moment, eqn.(14) does not describe the thermal condition of the material adequately, as it only satisfies the boundary condition. So the conservation equation of energy is imposed for the calculation of δF. Integrating the heat conduction eqn.(1) through the depth of the thermal penetration 0 ≦ x ≦ δF , we obtain the conservation equation of energy as,

(15) where the term on the right hand side represents the heat flux at the material surface. The assumed temperature profile (14) is substituted into the conservation eqn.(15) and the following expression for the time rate change of δF is obtained,

(16)

The penetration depth δF will be described by integrating eqn.(16) in regard to time.

Another unknown parameter in eqn.(14) is the surface temperature TS . This will be calculated from the surface boundary condition, which is,

(17)

Solving the above eqn.(17), TS is obtained as,

(18) 175 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 176

where heff is the effective heat transfer coefficient which is calculated from the following expression,

(19)

The surface temperature TS , which is obtained in the previous computational time step, will be used alternatively in the eqn.(19).

(I-B) when δF= L

When the thermal penetration depth δF reaches the material width L, the boundary condition (7) will be applied to the assumed temperature profile (13), from which we obtain,

(20)

Contrary to the case of the eqn.(14), the unkown parameters in eqn.(20) are the surface

temperature TS and the backside temperature TL . With the similar treatment taken in

deriving dδF /dt, the temperature profile TF is substituted into the conservation equation

of energy, and thus we obtain the time rate change of TL as follows,

(21)

While the surface temperature TS is determined from the boundary condition (7) as,

(22)

2.2.2 Pyrolysis Phase

(II-A) when δ∞+δC+δF

(23)

The conservation equation of energy for the char layer in the pyrolysis phase, takes the form,

(24)

in which the first and second terms on the right hand side of the equation represent the heat fluxes at the boundaries. The third and fourth terms represent the energy change 177 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 178 accompanied by the boundary movement. By substituting eqn.(23) into the conservation eqn.(24), the equation for the surface temperature change is derived as,

(25) in which =0 was assumed. While for the virgin layer, a quadratic function is also assumed for the internal temperature profile, so it is consistent with the initial heating phase. Considering the boundary condition (8), the profile of the virgin layer temperatureT F is obtained as,

(26)

Again, we substitute eqn.(26) for TF into the conservation equation of energy,

(27)

and the following equation for δL is derived,

(28)

With the computed temperature profiles for the char layerT c and for the virgin layer TF , the pyrolysis rate per unit surface area of the material is given from the eqn.(3) as,

(29)

While the pyrolysis, the virgin material is transformed into the char. This accompanies the density change from ρF - ρC of the material. So the descending speed of the pyrolysis front is given by,

(30)

Similarly, the descending rate of the char surface is described with the char oxidation rate as,

(31)

(II-B) when δ∞+δC+δF =L

When the thermal depth δ∞ + δC + δF reaches the material depth L, the boundary condition (10) is imposed for the temperature profile of the virgin material. So the profile is expressed as,

(32) 177 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 178

While for the char layer, the temperature profile (23) is used in succession, as the boundary condition does not change. Substituting eqn.(32) into the conservation equation of energy, we obtain the time rate change of the backside temperature as follows,

(33)

Mass loss rate of the material is derived by substituting eqns.(23) and (32) into eqn.(3), which takes the form as follows,

(34)

2.2.3 Char Oxidation Phase The temperature profile of the material is assumed to be quadratic, in succession to the pyrolysis phase. The coefficients are set according to the boundary condition (12) as,

(35)

The conservation equation of energy for the char layer, which is used for the

determination of the unexposed surface temperature TL , is described as,

(36)

The first term on the right hand side of the equation is the heat flux at the surface, and the second term represents the heat loss due to the surface descent. Substituting

eqn.(35) into eqn.(36), we obtain the time rate change of TL as follows,

(37)

While for the temperature at the exposed surface TS , the boundary condition (12) is applied and determined as follows,

(38)

3. COMPARTMENT FIRE MODEL

Schematic diagram of the compartment fire model is shown in Figure 2. In the present model, the compartment gas is assumed to be black. The gas inside the compartment is heated due to the combustion of flammable gas, which is released by the pyrolysis of solid fuel. At the same time, the oxidation takes place at the char surface and also heats the compartment gas, though moderately. 179 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 180

3.1 Compartment Fire Behavior Referring to Figure 2, conservation equations of mass, energy and chemical species (O=oxygen, F=flammable gas) for the gas filled inside the compartment are described as follows, respectively,

(39)

(40)

(41) where ρ is the density, V is the compartment volume, is the mass flow rate through the opening, CP is the specific heat of air, is the heat release rate of flammable gas combustion. , , are the heat loss rate to the combustible, wall, and outdoor through the opening, respectively (heat loss in conjunction with the transport of mass is not included in ). Y is the mass fraction, and is the production rate of the chemical species. The suffix i and j denote the inside and outside of the compartment, respectively. The line of subscripts ij denotes the direction of mass or heat flow. From the eqns.(39)~(41), we obtain following time diffetentials for temperature T and species concentration Y,

(42)

(43)

Figure 2 Schematic diagram of the compartment fire model. Compartment gas is heated due to the combustion of flammable gas and oxidation of char.

In deriving these equations, the compartment gas was assumed as an ideal gas under the constant atmospheric pressure, thus the relation ρi Ti = ρj Tj was invoked. The compartment fire behaviors are computed by integrating these equations iteratively, in regard to time. 179 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 180

3.2 Burning of Sold Fuel In the compartment model, we assume that the solid fuel is composed of a certain number (N) of elements, and the burning behaviors of each of these elements are reflected to the overall behaviors of the compartment fire. The procedure of burning behavior calculation of the individual fuel pieces is already described above. The compartment gas is assumed as the sole heat source to the solid fuel. The incident heat flux is then calculated as,

(44)

With above , the net incident heat flux is calculated and the net loss from the compartment gas to the solid fuel is obtained as,

(45)

where (k=1~N) is the area of the fuel piece. The difference of the incident heat flux for the fuel elements is expressed by considering the spread of burning area, which takes a form of elementary function. The flammable gas released into the gas phase, accompanied by the pyrolysis of solid fuel, will react with oxygen to generate heat. It is assumed that the speed of chemical reaction is fast that the virtual speed of reaction is determined by the supply rate of flammable gas and oxygen. If all of the supplied flammable gas and oxygen contribute to the combustion, the rate of heat release is expressed as,

(46)

in which the effect of oxygen consumption due to char oxidation is also incorporated.

ΔH0 and ΔHF are the heat of combustion per unit mass consumption of the oxygen and flammable gas, respectively. The rates of pyrolysis and rate of char oxidation are calculated by summing up the reaction rates for all of the considering fuel elements,

(47)

With the heat release rate , the consumption rates of the oxygen and flammable gas are computed as,

(48)

3.3 Mass and Heat Transfer through Compartment Boundaries In the present model, two types of compartment boundary, which differs the mode of mass and heat transfer, are considered. One is the opening, which transfers both mass and heat. The mass flow rate through the opening is calculated with the pressure difference induced by the fire environment. 181 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 182

The compartment pressure is obtained so as to satisfy the conservation equation of energy [12]. Another process, which transfers heat through the opening, is the thermal radiation. The net heat loss from the compartment gas to the outside is given by,

(49) where AD is the area of the opening. The other boundary is the wall, which involves heat transfer by conduction. Thus the net heat loss from the compartment gas to the wall surface is calculated as,

(50)

where TW and AW are the surface temperature and area of the wall, respectively. Here, the wall is assumed as a one-dimensional plate, and TW is calculated by solving the heat conduction equation.

4. EVALUATION OF THE MODELS 4.1 Evaluation of e Fuel Model

4.1.1 Pyrolysis in a Nitrogen Environment The calculation results are compared with the experimental data in which a circular sample of Douglas fir is subjected to external radiation in nitrogen gas [6]. The comparison provides a validation of the pyrolysis model without the effect of the flame. (1) Outline of e Experiment The sample size was 0.1m in diameter and 0.019m in depth, and it was preconditioned at 50◦C in a vacuum oven for at least 7days. The sample was placed on a 0.051m thick substrate insulation, and subjected to the heat fluxes of 20, 40 and 60kW/m2. The apparatus was covered by a cylindrical chamber of 0.61m in diameter and 1.70m in height. Pyrolysis product and ambient gases are removed via an exhaust duct by a constant nitrogen flow of 7.61l/s at 25◦C. (2) Calculation Conditions Parameters used in the calculation are shown in Table 1. As nitrogen gas was being supplied around the wood sample in the experiment, we neglected the effect of flame heat flux and char oxidation.

Note that the heat conductivity of the char kC was varied from 0.4 to 1.2 kW/(m·K), so as the calculated mass loss rate approaches the experimental values. These are larger than the nominal value, which is almost same as the heat conductivity of the virgin wood kF. As a substantial fraction of the incident heat flux is transferred inward by convection or radiation through fissures, the assumed values for kC is so to call virtual 181 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 182

heat conductivity. Correspondingly the density ρC and specific heat cP were also expected to vary in the course of the pyrolysis. These changes are also included in the adjusted

value of kC . (3) Results The results are shown in Figure 3. Similar to the experimental evidence, the mass loss rate rapidly increased to their maximum soon after the ignition. It then dropped as the char layer was formed above the virgin material and increased the thermal resistance between the exposed surface and the pyrolysis front. Later on, the mass loss rate reached more or less steady values.

Among the different heat conductivity conditions, the results for kC =0.8kW/(m·K) showed the best agreements with the experimental data in the steady state of the pyrolysis. However, the peak values in the initial phase were overestimated as the

virtual values of kc changes accompanying the enlargement of fissures.

Table 1 Parameters for the fuel model

-3 -3 kF , kC Heat conductivity 0.15×10 , 0.4~1.2×10 [kW/(m·K)] 3 ρF , ρC Density 500, 100 [kg/m ]

cF , cC Specific heat 1.8, 1.3 [kJ/(kg·K)]

TP Pyrolysis temperature 573[K]

LP Latent heat of pyrosysis 1700[kJ/kg]

ΔHS Heat of reaction of char oxidation 32000[kJ/kg]

ΔHO Heat of combustion of oxygen 13100[kJ/kg]

ΔHF Heat of combustion of gasified wood 12600[kJ/kg] Reference char oxidation rate 0.001[kg/(m2·s)]

The calculation results show that the larger the incident heat flux, the shorter the duration of pyrolysis compared to the experimental values. This was also due to the overestimation of pyrolysis rates in the initial phase. 4.1.2 Pyrolysis in the Atmospheric Environment The calculation results were also compared with the experimental data in which a cuboid sample of Douglas fir was subjected to a radiant heating [11]. Unlike the experiments done by Ritchie et al. [6], these were carried out in the atmospheric environment. 183 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 184

(a) 20 kW/m2 (b) 40 kW/m2

(1) Outline of e Experiment The sample size was 0.096 × 0.096 m2 in exposure surface and 0.05m in depth. Prior to the test, the sample was stored in a desiccator at 50% relative humidity and 20◦C. The sample was placed on a substrate insulation, and exposed to the heat flux of =75 kW/m2 by the cone calorimeter. (2) Calculation Conditions According to the experimental observation, the absorption of the incident heat flux by the flame is not considered [11]. So the overall incident heat flux to the surface is simply given as, (51) where (=35 kW/m2 [11]) is the heat flux from the flame. As the flame covered by the material surface, the effect of char oxidation was neglected for the pyrolysis phase. Whereas for the char oxidation phase, was set to 0. The other conditions are shown in Table 1. The calculated mass loss rate was also compared with the results obtained by the finite difference method. In the model, the pyrolysis of the material was related to the density change, which is expressed as a function of the material temperature as follows (Figure 4),

(52) 183 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 184

where ∆TP is a temperature range at which the pyrolysis occur. The value of ∆TP =50 (K) was adopted for the current calculation. With the expression (52), time rate change of the material is obtained as,

(53)

The temperature rise dT/dt is suppressed during the pyrolysis process. (3) Results The results are shown in Figure 5(a). As with the experiments done in the nitrogen environment, a peak of the mass loss rate was observed just after the ignition. Yet, the calculation overestimated the maximum value. After the peak, the mass loss rate dropped once, and started to increase again gradually at about 20 minutes after the ignition. The mass loss rate continued to increase till it reached the second peak at about 40 minutes. While, in the calculation, it showed almost steady value after the first peak. Same trend was obtained with the finite difference method. Generally, it is perceived that the temperature rise is promoted due to the insulation of the backside surface. However, this is not observed in the calculation results, although the insulation effect was incorporated in the model.

Instead, the effect of latent heat LP was investigated. A value of 255 kJ/kg was adopted from the existing literature [13], which is about one seventh of the value used in the previous calculations. The results are shown in Figure 5(b). Although the agreement of the results are not good, apparent mass loss peaks were obtained in the final stage of the pyrolysis.

Figure 4 Mass loss model for the finite difference method

(a) LP=1700 kJ/kg (b) LP=255 kJ/kg Figure 5 Mass loss rate per unit area in the atmospheric environment. 185 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 186

4.2 Evaluation of e Compartment Fire Model The compartment fire model is evaluated with the results of a reduced-scale experiments carried out by Ohmiya et al. [14]. (1) Outline of e Experiment The experiments were done in a cuboid enclosure (1.8m×1.1m of bed, and 1.1m height) made of perlite board with an opening on one side. The dimensions of the openings are shown in Table 2. As shown in Table 3, cedar plates (W1 and W2), and silver fir sticks (W3) were used as fuels. These samples had been dried in an oven for three days at the temperature of 105℃ , before the burning test. Cedar plates were aligned in a line facing broader sides, and fir sticks were piled up to form a crib. The cedar plates were ignited with a gas burner. Propane gas was supplied at the rate of 50 l/min (rate of heat release is 78.2 kW) until the cedar plates starts self-burning. The fir sticks were ignited with a methanol-soaked textile placed beneath the crib. (2) Calculation Conditions In the calculations of the material burning in the previous sections, the effect of char oxidation was neglected, as no oxygen supply to the material surface was expected in their environment. Yet, in some cases of compartment fire, it is supposed that the hot fluids such as flame or smoke goes up to the upper side of the compartment due to buoyancy. Thus, the fuel, which generally concentrates at the bottom side of the compartment, is more likely to receive oxygen at its surface. However, it is difficult to determine the reference char oxidation rate YO,∞ due to a lack of information. In the current calculation, we determined the value as 0.001[kg/(m2·s)], so that the computed mass loss rate in fully-developed fire was approximated to the experimental results. Assumed properties of the compartment walls are shown in Table 4.

Table 2 Experimental conditions of the opening [14]

Width BD[m] Upper end height HD,T [m] Bottom end height HD,B [m] 0.2, 0.3, 0.4, 0.5, 0.3, 1.0 0.0 0.6, 0.7, 0.8, 0.9 185 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 186

Table 3 Experimental conditions of the fuel [14]

Shape W1 (flat plate) W2 (flat plate) W3 (stick)

Type Cedar Cedar Silver fir

(0.45[m]× (0.45[m]× (0.6[m]× 0.3[m]× 0.3[m]× 0.02[m]× Configuration 0.06[m])× 0.03[m])× 0.02[m])× 4-pieces 8-pieces 70-pieces

Exposed area 1.33[m2] 2.41[m2] 2.62[m2]

Mass 13 [kg] 13 [kg] 8 [kg]

Table 4 Properties of wall (perlite board)

3 ρ w Density 750[kg/m ]

-3 k w Heat conductivity 0.202×10 [kW/(m·K)]

c w Specific heat 1.1[kJ/(kg·K)]

L w Width 0.05[m]

(3) Comparison of Mass Loss Rate in Fully Developed Fires The computed mass loss rate per unit surface area in fully-developed fires was compared with the experimental results in Figure 6. Here, the developed fire is referred to the period when almost steady pyrolysis took place. A parameter (ventilation factor / fuel surface area) was taken for the abscissa axis. The regression curve,

(a) Comparison with the results of W1 tests (b) Comparison with the results of W2 tests 187 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 188

(c) Comparison with the results of W3 tests Figure 6 Comparison of mass loss rate per unit fuel surface area in the developed fires.

(54)

is an equation obtained from the experimental results, including the data of some additional experiments [14]. The calculated results show fairly good agreements with the experimental data for the W1 and W2 tests. In the regime x≦0.07m1/2, the release of heat was restricted as the oxygen inflow was limited (ventilation-controlled fire). Thus, the mass loss rate increased proportionally to the coefficient x (i.e., proportional to the ventilation factor). Whereas in the regime x≧0.07m1/2, the mass loss rates decreased as x increased (fuel-controlled fire). As more oxygen than its maximum oxygen/fuel ratio was supplied, some flammable gas was left unreacted in the enclosure. This caused the dilution of the compartment gas and kept the temperature low compared to the case of ventilation-controlled fires. On the other hand, there were substantial differences for the W3 test. This is due to the particular configuration of the wood crib. Some of the sticks in the crib were covered with the other sticks, to be shielded from the incident heat flux. Consequently, the mass loss rate of the covered sticks was reduced. Whereas in the calculation, it was assumed that the incident heat flux is uniform for every single fuel elements, and thus the mass loss rate was overestimated.

Figure 7 Comparison of the temperature change (W2 test: BD=0.4m, HD,T= 0.3m). 187 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 188

(4) Transition of Temperature inside e Enclosures from Ignition till Decaying

An example of the compartment gas temperature development (W2 : BD =0.4m, HD,T =0.3m) is shown in Figure 7. The calculated gas temperature is compared with the measured temperature at the height of 0.9m in the corner of the enclosure. The gas temperature quickly rose after the initial heating. In the test, and the fuel was fully involved in the flame 7 minutes after the heating had started. The corresponding time in the calculation was about 5 minutes. Before long, the flame ejected from the opening, and this continued till about 18 minutes in the experiment. While in the calculation, the pyrolysis phase continued till 22 minutes. Assuming that the fire was at their most vigorous stage in these periods, the duration of the fully-developed fire was about 11 minutes for the experiment and about 17 minutes for the computation. It is considered that in the experiment, some portion of the fuel was left unburnt. So the duration of vigorous combustion was overestimated in the calculation, in which whole of the combustible was consumed. As for the gas temperature, the computed value agreed well with the experimental value in this period. In the experiment, the gas temperature started to decrease gradually after the flame ejection had ceased. Yet, as some flammable gas combustion still went on, rapid temperature decrease was not observed. On the other hand, there was a notable drop in the calculated temperature after the completion of the material pyrolysis, as the char oxidation was the only heat source.

5. COCLUDING REMARKS

In the present study, burning behaviors of charring materials were formulated according to the following sequential phases: (I) initial heating phase, (II) pyrolysis phase, and (III) char oxidation phase. The approximate solutions were conducted with the integral approach. The calculation results showed reasonable agreement with the existing experimental data when the virtual conductivity of the char layer was adjusted to 0.8×10-3 kW/(m∙K). The burning model for charring materials was then incorporated into a one-layer zone fire model. In the model, the combustible was divided into multiple fuel elements, and the compartment fire is described as a consequence of the burning of each of these fuel elements. The computational results were compared with the existing experimental 2 data. By adjusting the reference char oxidation rate YO,∞ to 0.001[kg/(m ∙s)], a reasonable agreement was obtained when the fuel samples were arrayed in simple configurations. 189 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 190

NOMENCLATURE A : Area [m2]

BD : Width of the opening [m] c : specific heat [kJ/(kg∙K)] ΔH : heat of combustion per unit mass consumption [kJ/kg]

HD,T : upper end height of the opening [m]

HD,B : bottom end height of the opening [m] h : coefficient of convective heat transfer [kW/(m2∙K)] k : heat conductivity [kW/(m∙K)] L : width, depth [m]

LP : latent heat of pyrolysis [kJ/kg] m : mass flow rate through opening [kg/s] 2 mB : mass loss rate per unit area [kg/(m ∙s)] 2 mS : char oxidation rate per unit area[kg/(m ∙s)] 2 mS,∞ : reference char oxidation rate per unit area [kg/(m ∙s)] Q : heat release, or loss [kW] q" : incident heat flux [kW/m2] T : temperature [K] Y : mass fraction [-] V : volume [m3] 2 wF : fuel load density [kg/m ] Γ : production rate of mass of species [kg/s] δ : depth, position [m] ε : emissivity [-] ρ : density [kg/m3] σ : Stefan-Boltzmann's constant (=5.67×10-11) [kW/(m2∙K4)] χ : parameter defined by [m1/2]

SUFFIX B : burning, combustion C : char D : opening F : fuel, virgin layer i : inside compartment j : outside compartment O : oxygen P : pyrolysis S : fuel surface, char layer W : wall ∞ : ambient, initial ( )" : per unit area (∙) : per unit time 189 K.HIMOTO and T.TANAKA A Burning Model for Charring Materials and Its Application to the Compartment Fire Development 190

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