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Piloted ignition times, critical heat fluxes and mass loss rates at reduced oxygen atmospheres
ARTICLE IN PRESS
Fire Safety Journal 40 (2005) 197–212 www.elsevier.com/locate/firesaf
Piloted ignition times, critical heat fluxes and mass loss rates at reduced oxygen atmospheres Michael A. Delichatsios FireSERT, University of Ulster, BT37 0QB, UK Received 24 May 2000; accepted 24 November 2004
Abstract Ignition, pyrolysis and burning of materials in reduced oxygen atmospheres occur when recirculating combustion gases are mixed with the air flowing into an enclosure. Still the incoming air can be sufficient for the complete combustion of the pyrolysis gases. Thus, for the prediction of fires in enclosures it is essential to understand the ignition and burning of materials in a reduced oxygen atmosphere even when plenty of oxidizer is available for complete combustion. Previous work employing gaseous fuels has shown that under these conditions, but before extinction, burning of gaseous fuels issuing from a nozzle is complete but radiation from the flames decreases owing to the reduction of their temperature. Complementary to that work, piloted ignition of solids is investigated here at reduced oxygen concentrations by measuring the ignition times and mass loss rates of the solid at ignition. These results were obtained in a cone calorimeter modified to supply air at reduced oxygen concentrations. Two types of plywood, normal and fire retardant 4 mm thick were examined at three imposed heat fluxes 25, 35 and 50 kW/m2 and at oxygen concentrations of 21%, 18% and 15% by volume. Because heating at these heat fluxes and material thickness corresponds to intermediate thermal conditions (i.e. neither thin nor thick), novel analytical solutions are developed to analyze the data and extract the thermal and ignition properties of the material. The same novel solutions can be applied to modeling concurrent or countercurrent flame spread. Moreover, a theory for piloted ignition explains why the ignition times and mass pyrolysis rates are weakly dependent on reduced oxygen concentrations. r 2005 Elsevier Ltd. All rights reserved. Keywords: Ignition; Extinction; Critical mass flux for ignition; Ignition times
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0379-7112/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2004.11.005
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Nomenclature a thermal diffusivity, m2/s c specific heat, J/kg K Ce constant in the surface extinction condition, see Eq. (1) _ ¼ dm=dt mass pyrolysis rate, g/s m F1 function defined in Eq. (3b) F2 function defined in Eq. (4b) k thermal conductivity, W/m K _ 00f m mass pyrolysis per surface area, kg/m2 s MF fuel molecular weight Mo oxidant molecular weight tign ignition time, s Tign ignition temperature, K To initial solid temperature, K q_ 00 imposed heat flux, W/m2 00 critical heat flux, W/m2 q_ cr YF,S mass fraction of fuel at the solid surface, see Fig. 1 Y o;1 mass fraction of oxygen at ambient conditions, see Eq. (1) Greek characters d dth nf no r s wA
material thickness, m thermal depth, Eq. (1), m fuel coefficient for stoichiometric combustion oxygen coefficient for stoichiometric combustion density of solid fuel, kg/m3 Boltzmann radiation constant, 5.67 1011 kW/(m2 K4) combustion efficiency, see Eq. (1)
1. Introduction and background Vitiated burning is common in room fires when the incoming supply air is contaminated with combustion products (e.g. CO2) recirculating from the room of fire origin or other rooms. Thus, the concentration of oxygen in the air can be below normal even though its quantity is still enough to support complete combustion. To determine the growth of the fire and the smoke release, ignition and burning under reduced oxygen atmospheres must be understood. Previous work [1,2] has shown that burning in vitiated air does not change the heat release rate when gaseous fuels burn as non-premixed buoyant flames. Complementary to that work, piloted ignition of solids is examined here at reduced oxygen atmospheres. Prior to this presentation, the physics of piloted ignition, its similarity to flame extinction and a theoretical proposition are introduced.
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NF
199
NF = Non Flammable Fuel Rich Flammability Limit
YF,s
FLAMMABLE REGION Surface extinction line
NF
Fuel Lean Flammability Limit NF YO,∞
Fig. 1. Flammability diagram of fuel vapours issuing from a solid surface plotted in terms of the mass fraction of fuel vapour at the surface, YF,s, against the mass fraction of oxygen in the ambient, YO,N. The diagram delineates the flammable from the non-flammable region by highlighting the rich and lean flammability limits (if fuel and oxygen are completely mixed) and the surface extinction line when extinction of flames occurs near the surface.
The focus is on permanent piloted ignition (corresponding to the fire point) of solids when stable burning is established after ignition of the vapours as the established flame provides sufficient heat feedback to the solid to maintain pyrolysis. Transient ignition (corresponding to the flash point) can occur before permanent ignition, but cannot sustain a flame near the surface [3–5]. Piloted ignition and flame extinction near a solid surface invoke the same critical conditions (expressed for example by the mass pyrolysis rate per unit surface area), which correspond to the first appearance of quasi-steady state pyrolysis and burning of the solid. This observation assumes that the chemical composition of the pyrolysis gases is the same at piloted ignition and at extinction. There is, however, a practical difference. Flame extinction evolves as the flames approach and interact with the surface on their own volition. In contrast, piloted ignition depends on the energy and location of the igniter relative to the surface of the solid. This difference must be considered when analyzing and comparing experimental data on extinction and ignition. As an example, ignition and extinction coincide in opposed flow flame spread because the progressing flame is the cause for both. Some previous results and analysis for extinction are discussed here since they are relevant to piloted ignition as configured by the qualifications of the previous paragraph. Flame extinction near a surface or piloted ignition is usually associated with low straining rates in fires. Therefore, the effects of straining rate (e.g. through the Damkholer number) are generally negligible [3,4,6]. An important diagram is shown in Fig. 1 for extinction at low straining rates deduced from experiments [3]
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and verified numerically [6]. This figure depicts flammability regions in terms of the fuel mass fraction at the surface, YF,s and the oxygen mass fraction in the ambient, YO,N. The straight lines through the origin delineate fuel-rich and fuel-lean flammability limits of a premixed mixture of pyrolysis gases and air. The third straight line corresponds to extinction of flames on the surface of a solid. This line is approximately derived from the condition that extinction occurs at constant mass flux [3]. In this case the extinction conditions are expressed as Y F;s þ wA
nf M F Y O;1 ¼ C e . no M o
(1)
Here wA is the efficiency of combustion near extinction, nf ; no are the stoichiometric fuel and oxygen coefficients and M F ; M o are the molecular weights of fuel and oxygen, respectively. The constant C e is independent of straining rate but it may depend on chemical kinetics of pyrolysis gases burning with air [3,4]. Eq. (1) is valid if the kinetics of the gas combustion are faster than the flow times (high Damko¨hler numbers). Otherwise the critical mass flux at extinction will not be constant but it increases as the Damko¨hler number decreases until no flames can exist below a certain Damko¨hler number [3,4,6]. A direct application of these results, relevant to the present work, is put forward in Fig. 2 which shows that the critical mass flux at ignition becomes larger if nitrogen substitutes oxygen in the oxidizer stream. This happens because at higher nitrogen concentrations the reaction rates and hence the Damko¨hler number decrease. The results and conclusions for flame extinction behaviour (Figs. 1 and 2) assist in the interpretation and support the analysis of the experimental data on ignition times and pyrolysis rates in this work.
⋅ Fuel mass flux at extinction m″, f Rising curve: Extinction controlled by slow kinetics
YN2, Nitrogen concentration in the supply air Fig. 2. Mass flux of fuel at extinction for reduced oxygen (diluted by nitrogen) supply in the combustion air.
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2. Ignition experiments with supply air at reduced oxygen concentrations 2.1. Ignition times Times to ignition through a spark igniter at different levels of external heat flux were measured at reduced oxygen atmospheres in a modified cone calorimeter apparatus [7,8]. In the modified cone, the radiant heaters and the load cell are enclosed in a sealed chamber and the oxidant flows uniformly supplied from the bottom of the chamber [7,8]. The average vertical velocity in the chamber is 0.15 m/s for a 30 lt/s flow rate of the oxidant supply at standard temperature and pressure conditions. Ignition experiments were performed according to the ASTM standard [7,8] that specifies that the samples be conditioned at a temperature of 25 1C and humidity of 50% until their weight is stable. Regular and fire retardant treated plywood were exposed to three different heat fluxes 25, 35 and 50 kW/m2. The supply air for combustion was diluted at three oxygen concentrations of 21%, 18% and 15% by volume by adding nitrogen [7,8]. For oxygen concentrations of 13% or less the samples could not ignite. The sample of plywood was 100 mm 100 mm 4 mm thick insulated at its back surface through 46 mm thick insulating ceramic fibreboard. This arrangement provides excellent thermal insulation up to the time of ignition because during this interval heat losses to the holder or back surface are not significant. The samples were tested in horizontal geometry and ignited by a spark igniter with three repeats for each test condition. An edge-retaining frame resulted in an exposed surface area of 0.0088 m2 [7,8]. For the analysis of the data, it is necessary to identify the thermal behaviour of the plywood during the period preceding ignition, namely whether it is thermally thick, thermally thin or exhibits intermediate behaviour. The actual behaviour can be identified by comparing the sample thickness of 4 mm with an appropriate length scale of the thermal wave propagation at the time of ignition [9] pffiffiffiffiffiffiffiffiffi dth ¼ atign . (2) 2.2. Results for fire retarded plywood Table 1 includes the ignition times for (a) each heat flux imposed on the solid, (b) each concentration of oxygen in the supply air and (c) the three repeats at each test condition. For a nominal thermal diffusivity of plywood 1.2 107 m2/s [5–8], the thermal depth (defined by Eq. (2) and listed in Table 1) is 3.78, 2.6 and 1.62 mm for the three heat fluxes 25, 35, and 50 kW/m2, respectively. Comparing these values with the plywood thickness of 4 mm, it appears that the plywood should behave as thermally thick for the largest heat flux and as thermally intermediate for the lower heat fluxes [9]. Inspection of Table 1 reveals that ignition times are essentially independent of the oxygen concentration, except for the 50 kW/m2 exposure where the variance of the ignition time at 21% and 15% oxygen concentration is about 50% of their average
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Table 1 Ignition times for 4 mm fire retardant plywood at different heat fluxes and oxygen concentrations for three repeats Ignition times (s), thermal depth (mm) and functions F1 and F2 pffiffiffiffiffiffiffiffiffi x ¼ d= atign
F1, Eq. (3b)
F2, Eq. (4b)
1.058
1.148
1.37
2.6 (57 s)
1.529
1.027
1.77
1.62 (22 s)
2.46
1
2.77
Heat flux (kW/m2)
Oxygen 21%
Oxygen 18%
Oxygen 15%
Thermal depth, Eq. (2)
25
124,106,118 Av. ¼ 116 s
113,113,114 Av. ¼ 113 s
126,127,130 Av. ¼ 128 s
3.78 (119 s)
35
50,49,64 Av. ¼ 54 s
62,53,57 Av. ¼ 57 s
70,52,55 s Av. ¼ 59 s
50
19,11,18 s Av. ¼ 16 s
22,18,19 s Av. ¼ 20 s
41,27,20 s Av. ¼ 29 s
value. If the extreme values in the repeat tests for this heat flux are ignored (11 s for 21% oxygen and 41 s for 15% oxygen at 50 kW/m2), the average ignition times are much closer, namely, 18.5, 21 and 23.5 s for 21%, 18% and 15% oxygen by volume, respectively. The average ignition times for each oxygen concentration and heat flux are plotted in Figs. 3a and b against the imposed heat flux using two different ordinates, i.e. pffiffiffiffiffiffiffi 1= tign on the left corresponding to thermally thick conditions and 1=tign on the right corresponding to thermally thin conditions (see Appendix A and discussion in the following paragraphs concerning the choice of coordinates). Fig. 3a includes all data in Table 1 and Fig. 3b includes all data after the extreme values at 50 kW/m2, as noted before, have been ignored. It is noticed that the data in Fig. 3b are more tightly grouped than in Fig. 3a. The focus turns now on a reexamination of the appropriate methods for correlating ignition data. As mentioned, two different ordinates commonly used in the literature represent the ignition times in terms of the imposed heat flux: 1. left ordinate in Figs. 3a and b is the inverse of the square root of the ignition time, pffiffiffiffiffiffiffi 1= tign : This ordinate is appropriate if the solid behaves as thermally thick for which case the ignition data should lie on a straight line [9]. 2. The right ordinate in Figs 3a and b is the inverse of the ignition time 1=tign : This ordinate is appropriate if the solid behaves as thermally thin for which case the ignition data should lie on a straight line [9]. A straight line seems to fit the data in Figs 3a and b for either ordinate, which challenges any decision on which one is the appropriate method. The next two points provide a possible resolution of this dilemma: 1. The intercept with the x-axis (heat flux axis) for the thermally thin representation of the data is a little more than 20 kW/m2. For thermally thin conditions this
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0.025 10 20 30 40 50 Heat flux kW/m2
(b)
0.025 10 20 30 40 50 Heat flux kW/m2
1 / { F1sqrt(tign)}
0.25
0 60
0.05
0.2
y = 0.0054x - 0.0557
0.15
0.04 0.03
0.1
y = 0.0004x - 0.0042
0.05 0 0
(c)
0.075 0.05
0.1 0 0
0 60
y = 0.0052x - 0.0428
0.2
10 20 30 40 50 Heat Flux kW / m2
0.02 0.01 0 60
1/{F2tign}
0
0.1
1 / tign
0.1
1 / sqrt(tign)
0.05
1 / tign
1 / sqrt(tign)
0.075
y = 0.0052x - 0.0415
0.2
0 (a)
0.3
0.1
0.3
203
Fig. 3. (a) Ignition times for fire retardant plywood at different heat fluxes and oxygen concentrations plotted in terms of thermally thick (left) and thermally thin (right) ordinates for all data. (b) Ignition times for fire retardant plywood at different heat fluxes and oxygen concentrations plotted in terms of thermally thick (left) and thermally thin (right) ordinates where extreme data for 50 kW/m2 have been excluded. (c) Ignition times for fire retardant plywood corrected according to Eq. (3a) (left ordinate) and Eq. (4a) (right ordinate). The thermal diffusivity is 1.2E7 m2/s.
intercept would be 1/3 of the critical heat flux for ignition [9], which, therefore, would be 60 kW/m2. (The critical heat flux is the maximum heat flux below which ignition does not occur). This value is exceptionally higher than any critical heat flux for plywood even it if is fire retarded [8,10]. 2. The intercept of the x-axis (heat flux axis) for the thermally thick representation of the data is about 8 kW/m2. This intercept is 64% of the critical heat flux [9], which, therefore, would be 12.5 kW/m2, namely a reasonable value for some types of fire-retarded plywood [8–10]. A more in depth analysis on the most appropriate method of plotting ignition data follows next. Because the imposed heat fluxes and the thickness of the material do not correspond to either purely thick or purely thin thermal conditions, a modification of plotting ignition time data is required in order to determine the correct thermal properties and the critical heat flux of the material (a numerical inversion of the heat transfer equation could be used for this purpose, but such a method would lack an understanding of the physics of the problem and would be difficult to apply). This modification is effected in two different ways starting from the opposite limits of the thermal behaviour of the solid: (a) as a perturbation of the correlation of ignition times for thermally thick and (b) as a perturbation of the
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correlation of ignition times for thermally thin conditions. Appendix A includes the detailed derivation and validation of the modifications, which are the following: 1. of the thermally thick correlation 1 2 q_ 00 q_ 00int pffiffiffiffiffiffiffiffi , pffiffiffiffiffiffiffi ¼ pffiffiffi F 1 tign p ðT ign T o Þ krc pffiffiffiffiffiffiffiffiffi where F1 is the following function of x ¼ d= atign : pffiffiffi F 1 ¼ 1 þ 2 p i erfcðxÞ pffiffiffi 1 2 ¼ 1 þ 2 p pffiffiffi expðx Þ x erfcðxÞ , p
(3a)
ð3bÞ
where erfc (x) is the complementary error function. 2. Modification of the thermally thin correlation 1 q_ 00 q_ 00int ¼ F 2 tign rcdðT ign T o Þ where F2 is the following function of x: x2 2 F 2 ¼ 1 þ 2 expðp2 =x2 Þ. 3 p
(4a)
(4b)
It is shown in Appendix A that the heat flux intercept when employing Eq. (3a) is the same as the heat flux intercept when employing Eq. (4a). Noticeably, numerical solutions in the intermediate thermal regime [9] show that this intercept is equal to the 0.64 fraction of the critical heat flux, q_ 00int ¼ 0:64 q_ 00cr : The proposed methodology (see Eqs. (3) and (4)) is now applied for the interpretation of the ignition data. To plot the ignition data using Eqs. (3a) and (4a) the thermal diffusivity of the material must be known. (For wood based products such as plywood, the thermal diffusivity lies between 0.7 107 and 2 107 m2/s [12]. It should be noticed that the thermal diffusivity (a ¼ k=rc) varies weakly with temperature for most materials [12]). The use of Eqs. (3a) and (4a) for plotting the ignition times enables the determination of the thermal diffusivity of the material by requiring the plotting to conform to the following requirements: 1. The intercepts of heat flux axis in the (linear) plots associated with Eqs. (3a) and (4a) should be the same. 2. The ratio of the slopes of these linear plots, as deduced from Eqs. (3a) and (4a), should be equal to: , 2 1 2 d pffiffiffiffiffiffiffiffi (5) ¼ pffiffiffi pffiffiffi . pffiffiffi rcdðT ign T o Þ p a pðT ign T o Þ krc To satisfy the two previous conditions, the thermal diffusivity was found equal to 1.2E7 as verified in Fig. 3c, where the ordinates are the left-hand sides of Eqs. (3a) and (4a). The average ignition times over all oxygen concentrations were used at
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each heat flux, namely 119, 57 and 22 s at 25, 35 and 50 kW/m2, respectively, as shown also in Table 1. The functions F1 and F2 are included in Table 1, read from the list in Table A1 in Appendix A for various values of their argument, x. The intercepts of the best linear fits in Figs 3c are nearly the same (10.31 and 10.5 kW/m2) and the ratio of their slopes is equal to 13.5s1/2 which compares well with the value 13.02s1/2 from Eq. (5) (where d ¼ 4 mm). The critical heat flux for ignition is equal to 10.4/0.64 ¼ 16.1 kW/m2. Finally, the ignition temperature is ð16:1=sÞ1=4 ¼ ð16:1=56:7Þ1=4 1000 ¼ 730 K assuming that the critical heat flux is equal to the surface reradiation losses. 2.3. Results for regular (untreated) plywood The methodology for plotting time ignition data in the previous section is manifested in Figs. 4a and b for the ignition times for the regular plywood where Fig. 4a presents the standard approach and Fig. 4b uses the modified ordinates defined in Eqs. (3) and (4). These ignition times are listed in Table 2 together with their average value, the thermal depth at ignition and the functions F1 and F2 needed for the modification of ignition times according to Eqs. (3a) and (4a). Table 2 indicates that ignition times are essentially independent of the oxygen concentration, except for the 50 kW/m2 exposure for which the variance in the ignition times is the greatest. The average ignition time for this exposure is 30 s, which, unexpectedly, is larger than the same value, 22 s, for the fire retardant plywood (see Table 1). For this reason, the suggestion is that this time should be closer to the measured time for the 18% oxygen concentration, namely 24 s, thus neglecting the higher values of ignition times for the 50 kW/m2 exposure. All ignition time data are included in Fig. 4a whereas in Fig. 4b, only the average ignition times 102 and 53 s for 25 and 35 kW/m2 and the value of 24 s for the 50 kW/ m2 exposure are used. The soundness and validation of this approach is checked by noting that
0.25
0.08
0.2
0.15
0.06
0.1
0.04
0.05
0.02
0
0 0
(a)
10
20
30
40
50
Heat Flux kW / m2
0.05 y = 0.0046x - 0.0265
0.1
y = 0.00035x - 0.002
0
(b)
0.02 0.01
0.05 0
60
0.04 0.03
0.15
20 40 Heat Flux kW/m2
1/{F2tign}
0.1
0.2
1/sqrt(tign)F1
0.25
1 / tign
1 / sqrt (tign)
1. The thermal diffusivity was found to be the same as that for the fire retarded plywood: 1.2E7 m2/s.
0 60
Fig. 4. (a) Ignition times for regular plywood at different heat fluxes and oxygen concentrations plotted in terms of thermally thick (left) and thermally thin (right) ordinates. (b) Ignition times for regular plywood corrected according to Eq. (3a) (left ordinate) and Eq. (4a) (right ordinate). The thermal diffusivity is 1.2E7 m2/s.
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Table 2 Ignition times for 4 mm regular plywood at different heat fluxes and oxygen concentrations for three repeats Ignition times (s), thermal depth (mm), and functions F1 and F2 Heat flux (kW/m2)
Oxygen 21%
Oxygen 18%
Oxygen 15%
Thermal depth
x
F1
F2
25
93,146,98 Av. ¼ 112
80,94,97 Av. ¼ 90
100,100,112 Av. ¼ 103
4.3 (102 s)
1.14
1.113
1.43
35
51,45,40 Av. ¼ 49
41,49 Av. ¼ 45
64,68,64 Av. ¼ 65
3.1 (53 s)
1.59
1.022
1.53
50
24,30,37 Av. ¼ 30
23,25 s Av. ¼ 24
38,30,42 s Av. ¼ 37
2.3 (30 s) 24 s
2.35
1
2.65
2. The intercepts of the best linear fits in Figs 4b are nearly the same (5.76 and 5.71 kW/m2) and 3. The ratio of their slopes is equal to 13.14s1/2, which compares well with the value 13.02s1/2 from Eq. (5) (where d ¼ 4 mm). From these results, the critical heat flux for ignition is calculated equal to 5.74/ 0.64 ¼ 8.96 kW/m2 and the ignition temperature is ð8:96=sÞ1=4 ¼ 1=4 ð8:96=56:7Þ 1000 ¼ 630 K assuming that the critical heat flux is equal to the surface reradiation losses. In conclusion, it is possible and effective to have a consistent determination of the (equivalent) thermal properties of a material by utilizing the properties of the two methods for plotting the ignition data as illustrated in Figs. 3c and 4b. These properties are determined from the thermal diffusivity (a ¼ k=rc ¼ 1:2 107 m2 =s; for both materials), the slopes of the lines in Figs. 3c and 4b, the density of the material (r ¼ 550 kg=m3 ) and the ignition temperature (deduced from the critical heat flux 730 and 630 K, respectively). Thus, the material properties are: 1. For the fire-retarded plywood: k ¼ 0:166 W=mK; c ¼ 2:51 kJ=kgK given the thermal diffusivity, the density and krc ¼ 0:23 lsðkW=m2 Þ2 obtained from the slope of the curve corresponding to the left ordinate of Fig. 3c and Eq. (3a). 2. For the regular plywood: k ¼ 0:256 W=mK; c ¼ 3:88 kJ=kgK given the thermal diffusivity, the density and krc ¼ 0:546 sðkW=m2 Þ2 obtained from the slope of the curve corresponding to the left ordinate of Fig. 4b and Eq. (4a). The influence of the oxygen concentration in the supply air is finally appraised in the next section.
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3. Critical mass flux at ignition As previously discussed, the experiments have shown (see Tables 1 and 2) that piloted ignition times are weakly dependent on the oxygen concentration of the supply air. In this section, it is shown through the Figs. 5a, b and 6 that the mass flux at ignition is also weakly dependent on oxygen concentration. Figs. 5a and b plot the fuel mass fluxes at the time of ignition for fireretarded plywood and normal plywood, respectively, at three heat fluxes as function of oxygen concentrations. The fuel mass flux is obtained by differentiation of the mass loss histories divided by the exposed area of the sample, 0.0088 m2 [7,8]. Each point represents the average of three repeats at the same test conditions. Because of the small values of the mass loss near ignition, the error of the fuel mass flux at ignition can be significant owing to uncertainties in the exact definition of the ignition time especially at the highest heat flux of these experiments, 50 kW/m2. The magnitude of this error is depicted in Fig. 6 by plotting the mass fluxes at ignition against oxygen concentration for three or two repeats at the same test conditions. Consequently, the following observations are advanced:
8 Fire Retarded Plywood 6 4 kW/m2 kW/m2
50 35 25 kW/m2
2 0 10
(a)
Critical mass flux at ignition (g /m2s)
Critical mass flux at ignition (g/m2 s)
(a) For the two lower heat fluxes 25and 35 kW/m2, the fuel mass flux at ignition is nearly the same and independent of the oxygen concentrations, about 3.4 g/m2 s. (b) For the largest heat flux 50 kW/m2 the fuel mass flux at ignition appears to be higher than 3.4 g/m2 s and also weakly dependent on the oxygen concentration. As seen in Fig. 6, this behaviour is attributed to the uncertainty of exactly defining the ignition times at the largest imposed heat flux, which in turn will affect the determination of the fuel mass flux. Similar variability was observed regarding the measured ignition times at 50 kW/m2 in Tables 1 and 2.
14 18 Volume concentration % O2
22
8 6
Regular plywood
4 50 kW/m2 35 kW/m2 25 kW/m2
2 0 10
(b)
14 18 Volume concentration % O2
22
Fig. 5. (a) Critical fuel mass flux at ignition for fire retarded plywood at different heat fluxes and oxygen concentrations. (b) Critical fuel mass flux at ignition for regular plywood at different heat fluxes and oxygen concentrations.
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10
Critical mass flux at ignition ( g / m2 s)
50 kW / m2 Fire Retarded Plywood 8
6
4
2
0 12
14
16 18 Volume concentration % O2
20
22
Fig. 6. Example of the variability of fuel mass fluxes at ignition for two or three repeats at the same test conditions.
This weak dependence of critical fuel mass flux at ignition on the oxygen concentration (varied by nitrogen addition or reduction) explains and is reinforced by the observed independence of ignition times, critical heat flux and ignition times on the oxygen concentration (see Figs. 3 and 4 and Tables 1 and 2). It is explained also by the behaviour shown in Fig. 2 if it is assumed that the concentration of nitrogen is not high enough to alter significantly the Damko¨hler number. A conclusion is that the thermal and ignition parameters can be determined from experiments at ambient oxygen concentrations. Of course, this conclusion is valid for conditions where the chemistry of the gas phase ignition is fast so that the mass loss rate at ignition is constant and independent of the Damko¨hler number as the horizontal part of the curve in Fig. 2 shows. Given the importance of the Damko¨hler number, a reliable method to measure it at different ignition conditions (velocity, oxidizer chemistry) is needed. Finally, the present results are partly consistent with ignition experiments in wood at forced ventilation conditions (compared to buoyant conditions in this paper) using a pilot igniter (compared to a spark igniter here) [5]. Comparison is made for the low flow velocity, 0.1 m/s (more closely corresponding to buoyant conditions), and oxygen concentrations in the range 13.5–21% [5]. The results of Fig. 10 in [5], considering the experimental scatter, imply that oxygen concentrations do not considerably influence the ignition times or the critical heat flux for ignition. In addition, the observations about the influence of oxygen concentration on the mass
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flux when piloted ignition or extinction occurs are consistent with similar earlier experiments [13–15].
4. Conclusions The conclusions apply for piloted ignition of materials and are based on experimental results for two types of plywood and for reduction of oxygen concentration by adding nitrogen in the combustion air. The main results from this work are: 1. Reduced oxygen atmospheres do not affect the time to ignition and the critical heat flux, as long as the fuel mass flux at ignition is nearly independent of oxygen concentration. 2. The measurement of fuel mass flux at ignition has large uncertainties in current experimental methods such as in the cone calorimeter. Because of its importance, a new method has to be developed for its measurement at different flow and ignition conditions (i.e. Damko¨hler numbers). 3. Thermal properties, critical heat flux and ignition temperature can be determined using a novel method for plotting the measured ignition times (see Figs. 3c and 4b and Eqs. (3a) and (4a)) for intermediate thermal conditions. These results can also be used in calculation of flame spread rates. 4. It follows that, in some cases such as for plywood, measurements at normal oxygen concentrations can yield all quantities related to piloted ignition applicable also for burning in reduced oxygen environments.
Appendix A A.1. Analytic expressions for the ignition time for intermediate thermal conditions We consider the one-dimensional heating of a material of thickness d insulated at its back surface and exposed to a fixed external heat flux q_ 00 : The thermal properties of the material are constant and surface heat losses will be introduced later. The history of the surface temperature can be found in all heat transfer books (e.g. [11,12]), expressed as a series expansion. The material will behave in extreme cases as thermally thin or thermally thick, depending on the ratio of its thickness to the thermal depth at any time of reference. In these extreme cases, there are simple analytic expressions for the history of the surface temperature. These analytic expressions are the first terms in the series expansions obtained as perturbations of the solutions for the purely thermally thick and the purely thermally thin case. The analytic histories for the extreme cases have been used in the literature to correlate experimental data for ignition times and deduce from this correlation the thermal properties and critical heat flux for ignition of the material, as
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discussed in the main text. However, in many cases the experimental data cannot be categorized to represent either purely thermally thick or purely thermally thin conditions. In these cases, numerical inversion of the heat transfer equation or, as proposed here, a modification of the analytic solution for the extreme cases should be applied. The starting point for the sought after modification is the exploitation of the wellknown series expansions of the histories for the surface temperature deduced as perturbations of the solutions of the thermally thin and thermally thick cases, respectively [11]. The first terms of these series expansions, relevant for this work, are: 1. thin case: ðT s T o Þk 1 þ x2 =3 p22 x2 expðp2 =x2 Þ F 2 ¼
2 dq_ 00 x2 x
(A.1a)
which for the purely thermals thin case (x ! 0) is the well known formula: ðT s T o Þk 1 ¼ 2. dq_ 00 x
(A.1b)
Here x is defined in Eq. (A.3). 2. Thermally thick case: pffiffiffi ðT s T o Þk 2 2F 1 ¼ pffiffiffi ð1 þ 2 p i erfcðxÞÞ pffiffiffi 00 dq_ x p x p
(A.2a)
which for the purely thermal thick case (x ! 1) is the well known formula ðT s T o Þk 2 ¼ pffiffiffi . dq_ 00 x p
(A.2b)
Here, d x ¼ pffiffiffiffiffiffiffiffiffi atign
(A.3)
1 i erfcðxÞ ¼ pffiffiffi expðx2 Þ x erfcðxÞ. p
(A.4)
and
An interesting and remarkable observation is that the right-hand sides of Eqs. (A.1a) and (A.2a) take nearly the same values for intermediate thermal conditions in the range of 0:8oxo2 as Table A1 proves. The conclusion is that either of these equations represents the surface temperature histories in the intermediate thermal case. This temperature is also close to the precise surface temperature [11]. Eqs. (A.1a) and (A.2a) (and Eq. (A.3)) together with the aforementioned observation enable the derivation of Eqs. (3a) and (4a) of the main text for the ignition times assuming that ignition occurs when the surface temperature reaches a
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Table A1 List of the exact arithmetic values of the functions F1 (Eq. (3b)) and F2(Eq. (4b)) x
ierfc(x)
F1
F2
pffiffiffi F 1 =x p (Eq. (A.1a))
F2/x2 (Eq. (A.2a))
0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0.2465 0.2409 0.2354 0.23 0.2247 0.2195 0.2144 0.2094 0.2045 0.1996 0.1902 0.1811 0.1724 0.164 0.1559 0.1482 0.1402 0.1335 0.1267 0.1201 0.1138 0.1077 0.102 0.0965 0.0912 0.0861 0.0813 0.0767 0.0724 0.0682 0.0642 0.0605 0.0569 0.0535 0.0503 0.0365 0.026 0.0183 0.0127 0.0086 0.0058 0.0038 0.0025 0.0016 0.001
1.873598 1.853752 1.83426 1.815122 1.796339 1.77791 1.759836 1.742115 1.72475 1.707384 1.674071 1.64182 1.610987 1.581217 1.552511 1.525222 1.49687 1.473125 1.449026 1.425635 1.403308 1.38169 1.361489 1.341997 1.323214 1.305139 1.288128 1.271825 1.256586 1.241701 1.227525 1.214413 1.201654 1.189604 1.178264 1.129356 1.092144 1.064855 1.045009 1.030478 1.020555 1.013467 1.00886 1.00567 1.003544
1.056033333 1.0588 1.061633333 1.064533333 1.0675 1.070533333 1.073633333 1.0768 1.080033333 1.083333333 1.090133333 1.0972 1.104533333 1.112133333 1.12 1.128133333 1.136533333 1.1452 1.154133333 1.163333333 1.172799999 1.182533332 1.192533329 1.202799989 1.213333307 1.224133276 1.23519988 1.246533093 1.258132876 1.269999162 1.282131855 1.294530811 1.30719583 1.320126635 1.333322857 1.403263035 1.479692143 1.56233743 1.650751156 1.744327295 1.842354808 1.944084813 2.048791839 2.15581867 2.264602565
5.157733 4.981597 4.814583 4.656069 4.50549 4.362327 4.226107 4.096396 3.972796 3.85414 3.633596 3.431611 3.246911 3.077017 2.92045 2.776564 2.639796 2.519196 2.405102 2.298671 2.199819 2.107391 2.021933 1.941884 1.866836 1.796426 1.730796 1.66915 1.611669 1.557188 1.505944 1.458158 1.41278 1.37007 1.329869 1.15879 1.027224 0.924514 0.842477 0.775379 0.719918 0.672864 0.632593 0.597404 0.566334
6.282173 6.002268 5.741662 5.498623 5.271605 5.059231 4.860269 4.673611 4.498265 4.333333 4.031558 3.762689 3.522109 3.305985 3.111111 2.93479 2.77474 2.629017 2.495963 2.37415 2.262346 2.159484 2.064635 1.976989 1.895833 1.820543 1.750567 1.685415 1.624655 1.5679 1.514806 1.465064 1.418398 1.374559 1.333323 1.159722 1.027564 0.92446 0.84222 0.775257 0.71967 0.672694 0.632343 0.59718 0.566151
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fixed value, T ignn : In addition, the externally imposed heat flux introduced in Eqs. (3a) and (4a) is reduced by a term, q_ 00int ; due to heat losses from the surface. The equality of the right-hand sides of Eqs. (A.1a) and (A.2a) warrant that the value of q_ 00int is the same in Eqs. (3a) and (4b). Numerical solutions show that this value is 64% of the critical heat flux [9]. References [1] Santo G, Delichatsios MA. Effects of vitiated air on radiation and completeness of combustion in propane pool fires. Fire Safety J 1984;7:159–64. [2] Santo G, Tamanini F. Influence of oxygen depletion on the radiative properties of PMMA. 18th (Int.) Symposium on Combustion, The Combustion Institute, 1980. [3] Delichatsios MA, Delichatsios MM. Critical mass pyrolysis rates for extinction of fires over solid materials. FIRE SAFETY SCIENCE–Fifth International Symposium IAFSS 1997:153–64. [4] Gummalla M, Vlachos DG, Delichatsios MA. Bifurcations and structure of surface interacting methane—air diffusion flames. Combust Flame 2000;120:333–45. [5] Atreya A, Abu-Zaid M. Fire safety science. Third international symposium IAFSS. 1991. p. 177–186. [6] Gummalla M, Vlachos DG, Delichatsios MA. Conditions for extinction of solid fuel combustion: Part I simple model. Combust Flame 2005, submitted for publication. [7] Dowling VP, Leonard J, Bowditch PA. Use of a controlled atmosphere cone calorimeter to assess building materials. Proc Interflam, 99 1999:989–97. [8] Leonard J, Bowditch PA, Dowling VP. The development of a controlled atmosphere cone calorimeter. Technical report, Fire Science and Technology Lab., CSIRO, DBCE Doc. 98/038, 1998. [9] Delichatsios MA. Ignition times for thermally thick and intermediate conditions in flat and cylindrical geometries. Fire safety science—sixth international symposium, IAFSS 1999. p. 233–244. [10] Brescianini CP, Delichatsios MA, Webb AK. 1999 Australian symposium on combustion and sixth Australian flame days, Newcastle Australia, 1999. p. 180. Combust Sci Technol 2003;175:319–31. [11] Luikov AV. Analytical heat diffusion theory. New York: Academic Press; 1968. [12] Roshenow WM, Hartnett JP, Cho YI., editors. Handbook of heat transfer. 3rd ed. 1998. p. 2.64. [13] Bamford CH, Crank J, Malan DH. The burning of wood. Proceedings of the Cambridge Philosophical Society 1946;42:166–82. [14] Thomson HE, Drysdale DD. Critical mass flowrate at the firepoint of plastics. Fire safety science— second international symposium IAFSS, 1988. p. 67–76. [15] Tewarson A. Generation of heat and chemical compounds in fires. In: Di Nenno PJ, et al., editors. SFPE handbook of fire protection engineering. 2nd ed. Bostan: SFPE; 1995. p. 3.53–3.124.
Fire Safety Journal 40 (2005) 197–212 www.elsevier.com/locate/firesaf
Piloted ignition times, critical heat fluxes and mass loss rates at reduced oxygen atmospheres Michael A. Delichatsios FireSERT, University of Ulster, BT37 0QB, UK Received 24 May 2000; accepted 24 November 2004
Abstract Ignition, pyrolysis and burning of materials in reduced oxygen atmospheres occur when recirculating combustion gases are mixed with the air flowing into an enclosure. Still the incoming air can be sufficient for the complete combustion of the pyrolysis gases. Thus, for the prediction of fires in enclosures it is essential to understand the ignition and burning of materials in a reduced oxygen atmosphere even when plenty of oxidizer is available for complete combustion. Previous work employing gaseous fuels has shown that under these conditions, but before extinction, burning of gaseous fuels issuing from a nozzle is complete but radiation from the flames decreases owing to the reduction of their temperature. Complementary to that work, piloted ignition of solids is investigated here at reduced oxygen concentrations by measuring the ignition times and mass loss rates of the solid at ignition. These results were obtained in a cone calorimeter modified to supply air at reduced oxygen concentrations. Two types of plywood, normal and fire retardant 4 mm thick were examined at three imposed heat fluxes 25, 35 and 50 kW/m2 and at oxygen concentrations of 21%, 18% and 15% by volume. Because heating at these heat fluxes and material thickness corresponds to intermediate thermal conditions (i.e. neither thin nor thick), novel analytical solutions are developed to analyze the data and extract the thermal and ignition properties of the material. The same novel solutions can be applied to modeling concurrent or countercurrent flame spread. Moreover, a theory for piloted ignition explains why the ignition times and mass pyrolysis rates are weakly dependent on reduced oxygen concentrations. r 2005 Elsevier Ltd. All rights reserved. Keywords: Ignition; Extinction; Critical mass flux for ignition; Ignition times
Tel.: +61 2 9490 5426; fax: +61 2 9490 5777.
0379-7112/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2004.11.005
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Nomenclature a thermal diffusivity, m2/s c specific heat, J/kg K Ce constant in the surface extinction condition, see Eq. (1) _ ¼ dm=dt mass pyrolysis rate, g/s m F1 function defined in Eq. (3b) F2 function defined in Eq. (4b) k thermal conductivity, W/m K _ 00f m mass pyrolysis per surface area, kg/m2 s MF fuel molecular weight Mo oxidant molecular weight tign ignition time, s Tign ignition temperature, K To initial solid temperature, K q_ 00 imposed heat flux, W/m2 00 critical heat flux, W/m2 q_ cr YF,S mass fraction of fuel at the solid surface, see Fig. 1 Y o;1 mass fraction of oxygen at ambient conditions, see Eq. (1) Greek characters d dth nf no r s wA
material thickness, m thermal depth, Eq. (1), m fuel coefficient for stoichiometric combustion oxygen coefficient for stoichiometric combustion density of solid fuel, kg/m3 Boltzmann radiation constant, 5.67 1011 kW/(m2 K4) combustion efficiency, see Eq. (1)
1. Introduction and background Vitiated burning is common in room fires when the incoming supply air is contaminated with combustion products (e.g. CO2) recirculating from the room of fire origin or other rooms. Thus, the concentration of oxygen in the air can be below normal even though its quantity is still enough to support complete combustion. To determine the growth of the fire and the smoke release, ignition and burning under reduced oxygen atmospheres must be understood. Previous work [1,2] has shown that burning in vitiated air does not change the heat release rate when gaseous fuels burn as non-premixed buoyant flames. Complementary to that work, piloted ignition of solids is examined here at reduced oxygen atmospheres. Prior to this presentation, the physics of piloted ignition, its similarity to flame extinction and a theoretical proposition are introduced.
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NF
199
NF = Non Flammable Fuel Rich Flammability Limit
YF,s
FLAMMABLE REGION Surface extinction line
NF
Fuel Lean Flammability Limit NF YO,∞
Fig. 1. Flammability diagram of fuel vapours issuing from a solid surface plotted in terms of the mass fraction of fuel vapour at the surface, YF,s, against the mass fraction of oxygen in the ambient, YO,N. The diagram delineates the flammable from the non-flammable region by highlighting the rich and lean flammability limits (if fuel and oxygen are completely mixed) and the surface extinction line when extinction of flames occurs near the surface.
The focus is on permanent piloted ignition (corresponding to the fire point) of solids when stable burning is established after ignition of the vapours as the established flame provides sufficient heat feedback to the solid to maintain pyrolysis. Transient ignition (corresponding to the flash point) can occur before permanent ignition, but cannot sustain a flame near the surface [3–5]. Piloted ignition and flame extinction near a solid surface invoke the same critical conditions (expressed for example by the mass pyrolysis rate per unit surface area), which correspond to the first appearance of quasi-steady state pyrolysis and burning of the solid. This observation assumes that the chemical composition of the pyrolysis gases is the same at piloted ignition and at extinction. There is, however, a practical difference. Flame extinction evolves as the flames approach and interact with the surface on their own volition. In contrast, piloted ignition depends on the energy and location of the igniter relative to the surface of the solid. This difference must be considered when analyzing and comparing experimental data on extinction and ignition. As an example, ignition and extinction coincide in opposed flow flame spread because the progressing flame is the cause for both. Some previous results and analysis for extinction are discussed here since they are relevant to piloted ignition as configured by the qualifications of the previous paragraph. Flame extinction near a surface or piloted ignition is usually associated with low straining rates in fires. Therefore, the effects of straining rate (e.g. through the Damkholer number) are generally negligible [3,4,6]. An important diagram is shown in Fig. 1 for extinction at low straining rates deduced from experiments [3]
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and verified numerically [6]. This figure depicts flammability regions in terms of the fuel mass fraction at the surface, YF,s and the oxygen mass fraction in the ambient, YO,N. The straight lines through the origin delineate fuel-rich and fuel-lean flammability limits of a premixed mixture of pyrolysis gases and air. The third straight line corresponds to extinction of flames on the surface of a solid. This line is approximately derived from the condition that extinction occurs at constant mass flux [3]. In this case the extinction conditions are expressed as Y F;s þ wA
nf M F Y O;1 ¼ C e . no M o
(1)
Here wA is the efficiency of combustion near extinction, nf ; no are the stoichiometric fuel and oxygen coefficients and M F ; M o are the molecular weights of fuel and oxygen, respectively. The constant C e is independent of straining rate but it may depend on chemical kinetics of pyrolysis gases burning with air [3,4]. Eq. (1) is valid if the kinetics of the gas combustion are faster than the flow times (high Damko¨hler numbers). Otherwise the critical mass flux at extinction will not be constant but it increases as the Damko¨hler number decreases until no flames can exist below a certain Damko¨hler number [3,4,6]. A direct application of these results, relevant to the present work, is put forward in Fig. 2 which shows that the critical mass flux at ignition becomes larger if nitrogen substitutes oxygen in the oxidizer stream. This happens because at higher nitrogen concentrations the reaction rates and hence the Damko¨hler number decrease. The results and conclusions for flame extinction behaviour (Figs. 1 and 2) assist in the interpretation and support the analysis of the experimental data on ignition times and pyrolysis rates in this work.
⋅ Fuel mass flux at extinction m″, f Rising curve: Extinction controlled by slow kinetics
YN2, Nitrogen concentration in the supply air Fig. 2. Mass flux of fuel at extinction for reduced oxygen (diluted by nitrogen) supply in the combustion air.
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2. Ignition experiments with supply air at reduced oxygen concentrations 2.1. Ignition times Times to ignition through a spark igniter at different levels of external heat flux were measured at reduced oxygen atmospheres in a modified cone calorimeter apparatus [7,8]. In the modified cone, the radiant heaters and the load cell are enclosed in a sealed chamber and the oxidant flows uniformly supplied from the bottom of the chamber [7,8]. The average vertical velocity in the chamber is 0.15 m/s for a 30 lt/s flow rate of the oxidant supply at standard temperature and pressure conditions. Ignition experiments were performed according to the ASTM standard [7,8] that specifies that the samples be conditioned at a temperature of 25 1C and humidity of 50% until their weight is stable. Regular and fire retardant treated plywood were exposed to three different heat fluxes 25, 35 and 50 kW/m2. The supply air for combustion was diluted at three oxygen concentrations of 21%, 18% and 15% by volume by adding nitrogen [7,8]. For oxygen concentrations of 13% or less the samples could not ignite. The sample of plywood was 100 mm 100 mm 4 mm thick insulated at its back surface through 46 mm thick insulating ceramic fibreboard. This arrangement provides excellent thermal insulation up to the time of ignition because during this interval heat losses to the holder or back surface are not significant. The samples were tested in horizontal geometry and ignited by a spark igniter with three repeats for each test condition. An edge-retaining frame resulted in an exposed surface area of 0.0088 m2 [7,8]. For the analysis of the data, it is necessary to identify the thermal behaviour of the plywood during the period preceding ignition, namely whether it is thermally thick, thermally thin or exhibits intermediate behaviour. The actual behaviour can be identified by comparing the sample thickness of 4 mm with an appropriate length scale of the thermal wave propagation at the time of ignition [9] pffiffiffiffiffiffiffiffiffi dth ¼ atign . (2) 2.2. Results for fire retarded plywood Table 1 includes the ignition times for (a) each heat flux imposed on the solid, (b) each concentration of oxygen in the supply air and (c) the three repeats at each test condition. For a nominal thermal diffusivity of plywood 1.2 107 m2/s [5–8], the thermal depth (defined by Eq. (2) and listed in Table 1) is 3.78, 2.6 and 1.62 mm for the three heat fluxes 25, 35, and 50 kW/m2, respectively. Comparing these values with the plywood thickness of 4 mm, it appears that the plywood should behave as thermally thick for the largest heat flux and as thermally intermediate for the lower heat fluxes [9]. Inspection of Table 1 reveals that ignition times are essentially independent of the oxygen concentration, except for the 50 kW/m2 exposure where the variance of the ignition time at 21% and 15% oxygen concentration is about 50% of their average
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Table 1 Ignition times for 4 mm fire retardant plywood at different heat fluxes and oxygen concentrations for three repeats Ignition times (s), thermal depth (mm) and functions F1 and F2 pffiffiffiffiffiffiffiffiffi x ¼ d= atign
F1, Eq. (3b)
F2, Eq. (4b)
1.058
1.148
1.37
2.6 (57 s)
1.529
1.027
1.77
1.62 (22 s)
2.46
1
2.77
Heat flux (kW/m2)
Oxygen 21%
Oxygen 18%
Oxygen 15%
Thermal depth, Eq. (2)
25
124,106,118 Av. ¼ 116 s
113,113,114 Av. ¼ 113 s
126,127,130 Av. ¼ 128 s
3.78 (119 s)
35
50,49,64 Av. ¼ 54 s
62,53,57 Av. ¼ 57 s
70,52,55 s Av. ¼ 59 s
50
19,11,18 s Av. ¼ 16 s
22,18,19 s Av. ¼ 20 s
41,27,20 s Av. ¼ 29 s
value. If the extreme values in the repeat tests for this heat flux are ignored (11 s for 21% oxygen and 41 s for 15% oxygen at 50 kW/m2), the average ignition times are much closer, namely, 18.5, 21 and 23.5 s for 21%, 18% and 15% oxygen by volume, respectively. The average ignition times for each oxygen concentration and heat flux are plotted in Figs. 3a and b against the imposed heat flux using two different ordinates, i.e. pffiffiffiffiffiffiffi 1= tign on the left corresponding to thermally thick conditions and 1=tign on the right corresponding to thermally thin conditions (see Appendix A and discussion in the following paragraphs concerning the choice of coordinates). Fig. 3a includes all data in Table 1 and Fig. 3b includes all data after the extreme values at 50 kW/m2, as noted before, have been ignored. It is noticed that the data in Fig. 3b are more tightly grouped than in Fig. 3a. The focus turns now on a reexamination of the appropriate methods for correlating ignition data. As mentioned, two different ordinates commonly used in the literature represent the ignition times in terms of the imposed heat flux: 1. left ordinate in Figs. 3a and b is the inverse of the square root of the ignition time, pffiffiffiffiffiffiffi 1= tign : This ordinate is appropriate if the solid behaves as thermally thick for which case the ignition data should lie on a straight line [9]. 2. The right ordinate in Figs 3a and b is the inverse of the ignition time 1=tign : This ordinate is appropriate if the solid behaves as thermally thin for which case the ignition data should lie on a straight line [9]. A straight line seems to fit the data in Figs 3a and b for either ordinate, which challenges any decision on which one is the appropriate method. The next two points provide a possible resolution of this dilemma: 1. The intercept with the x-axis (heat flux axis) for the thermally thin representation of the data is a little more than 20 kW/m2. For thermally thin conditions this
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0.025 10 20 30 40 50 Heat flux kW/m2
(b)
0.025 10 20 30 40 50 Heat flux kW/m2
1 / { F1sqrt(tign)}
0.25
0 60
0.05
0.2
y = 0.0054x - 0.0557
0.15
0.04 0.03
0.1
y = 0.0004x - 0.0042
0.05 0 0
(c)
0.075 0.05
0.1 0 0
0 60
y = 0.0052x - 0.0428
0.2
10 20 30 40 50 Heat Flux kW / m2
0.02 0.01 0 60
1/{F2tign}
0
0.1
1 / tign
0.1
1 / sqrt(tign)
0.05
1 / tign
1 / sqrt(tign)
0.075
y = 0.0052x - 0.0415
0.2
0 (a)
0.3
0.1
0.3
203
Fig. 3. (a) Ignition times for fire retardant plywood at different heat fluxes and oxygen concentrations plotted in terms of thermally thick (left) and thermally thin (right) ordinates for all data. (b) Ignition times for fire retardant plywood at different heat fluxes and oxygen concentrations plotted in terms of thermally thick (left) and thermally thin (right) ordinates where extreme data for 50 kW/m2 have been excluded. (c) Ignition times for fire retardant plywood corrected according to Eq. (3a) (left ordinate) and Eq. (4a) (right ordinate). The thermal diffusivity is 1.2E7 m2/s.
intercept would be 1/3 of the critical heat flux for ignition [9], which, therefore, would be 60 kW/m2. (The critical heat flux is the maximum heat flux below which ignition does not occur). This value is exceptionally higher than any critical heat flux for plywood even it if is fire retarded [8,10]. 2. The intercept of the x-axis (heat flux axis) for the thermally thick representation of the data is about 8 kW/m2. This intercept is 64% of the critical heat flux [9], which, therefore, would be 12.5 kW/m2, namely a reasonable value for some types of fire-retarded plywood [8–10]. A more in depth analysis on the most appropriate method of plotting ignition data follows next. Because the imposed heat fluxes and the thickness of the material do not correspond to either purely thick or purely thin thermal conditions, a modification of plotting ignition time data is required in order to determine the correct thermal properties and the critical heat flux of the material (a numerical inversion of the heat transfer equation could be used for this purpose, but such a method would lack an understanding of the physics of the problem and would be difficult to apply). This modification is effected in two different ways starting from the opposite limits of the thermal behaviour of the solid: (a) as a perturbation of the correlation of ignition times for thermally thick and (b) as a perturbation of the
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correlation of ignition times for thermally thin conditions. Appendix A includes the detailed derivation and validation of the modifications, which are the following: 1. of the thermally thick correlation 1 2 q_ 00 q_ 00int pffiffiffiffiffiffiffiffi , pffiffiffiffiffiffiffi ¼ pffiffiffi F 1 tign p ðT ign T o Þ krc pffiffiffiffiffiffiffiffiffi where F1 is the following function of x ¼ d= atign : pffiffiffi F 1 ¼ 1 þ 2 p i erfcðxÞ pffiffiffi 1 2 ¼ 1 þ 2 p pffiffiffi expðx Þ x erfcðxÞ , p
(3a)
ð3bÞ
where erfc (x) is the complementary error function. 2. Modification of the thermally thin correlation 1 q_ 00 q_ 00int ¼ F 2 tign rcdðT ign T o Þ where F2 is the following function of x: x2 2 F 2 ¼ 1 þ 2 expðp2 =x2 Þ. 3 p
(4a)
(4b)
It is shown in Appendix A that the heat flux intercept when employing Eq. (3a) is the same as the heat flux intercept when employing Eq. (4a). Noticeably, numerical solutions in the intermediate thermal regime [9] show that this intercept is equal to the 0.64 fraction of the critical heat flux, q_ 00int ¼ 0:64 q_ 00cr : The proposed methodology (see Eqs. (3) and (4)) is now applied for the interpretation of the ignition data. To plot the ignition data using Eqs. (3a) and (4a) the thermal diffusivity of the material must be known. (For wood based products such as plywood, the thermal diffusivity lies between 0.7 107 and 2 107 m2/s [12]. It should be noticed that the thermal diffusivity (a ¼ k=rc) varies weakly with temperature for most materials [12]). The use of Eqs. (3a) and (4a) for plotting the ignition times enables the determination of the thermal diffusivity of the material by requiring the plotting to conform to the following requirements: 1. The intercepts of heat flux axis in the (linear) plots associated with Eqs. (3a) and (4a) should be the same. 2. The ratio of the slopes of these linear plots, as deduced from Eqs. (3a) and (4a), should be equal to: , 2 1 2 d pffiffiffiffiffiffiffiffi (5) ¼ pffiffiffi pffiffiffi . pffiffiffi rcdðT ign T o Þ p a pðT ign T o Þ krc To satisfy the two previous conditions, the thermal diffusivity was found equal to 1.2E7 as verified in Fig. 3c, where the ordinates are the left-hand sides of Eqs. (3a) and (4a). The average ignition times over all oxygen concentrations were used at
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each heat flux, namely 119, 57 and 22 s at 25, 35 and 50 kW/m2, respectively, as shown also in Table 1. The functions F1 and F2 are included in Table 1, read from the list in Table A1 in Appendix A for various values of their argument, x. The intercepts of the best linear fits in Figs 3c are nearly the same (10.31 and 10.5 kW/m2) and the ratio of their slopes is equal to 13.5s1/2 which compares well with the value 13.02s1/2 from Eq. (5) (where d ¼ 4 mm). The critical heat flux for ignition is equal to 10.4/0.64 ¼ 16.1 kW/m2. Finally, the ignition temperature is ð16:1=sÞ1=4 ¼ ð16:1=56:7Þ1=4 1000 ¼ 730 K assuming that the critical heat flux is equal to the surface reradiation losses. 2.3. Results for regular (untreated) plywood The methodology for plotting time ignition data in the previous section is manifested in Figs. 4a and b for the ignition times for the regular plywood where Fig. 4a presents the standard approach and Fig. 4b uses the modified ordinates defined in Eqs. (3) and (4). These ignition times are listed in Table 2 together with their average value, the thermal depth at ignition and the functions F1 and F2 needed for the modification of ignition times according to Eqs. (3a) and (4a). Table 2 indicates that ignition times are essentially independent of the oxygen concentration, except for the 50 kW/m2 exposure for which the variance in the ignition times is the greatest. The average ignition time for this exposure is 30 s, which, unexpectedly, is larger than the same value, 22 s, for the fire retardant plywood (see Table 1). For this reason, the suggestion is that this time should be closer to the measured time for the 18% oxygen concentration, namely 24 s, thus neglecting the higher values of ignition times for the 50 kW/m2 exposure. All ignition time data are included in Fig. 4a whereas in Fig. 4b, only the average ignition times 102 and 53 s for 25 and 35 kW/m2 and the value of 24 s for the 50 kW/ m2 exposure are used. The soundness and validation of this approach is checked by noting that
0.25
0.08
0.2
0.15
0.06
0.1
0.04
0.05
0.02
0
0 0
(a)
10
20
30
40
50
Heat Flux kW / m2
0.05 y = 0.0046x - 0.0265
0.1
y = 0.00035x - 0.002
0
(b)
0.02 0.01
0.05 0
60
0.04 0.03
0.15
20 40 Heat Flux kW/m2
1/{F2tign}
0.1
0.2
1/sqrt(tign)F1
0.25
1 / tign
1 / sqrt (tign)
1. The thermal diffusivity was found to be the same as that for the fire retarded plywood: 1.2E7 m2/s.
0 60
Fig. 4. (a) Ignition times for regular plywood at different heat fluxes and oxygen concentrations plotted in terms of thermally thick (left) and thermally thin (right) ordinates. (b) Ignition times for regular plywood corrected according to Eq. (3a) (left ordinate) and Eq. (4a) (right ordinate). The thermal diffusivity is 1.2E7 m2/s.
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Table 2 Ignition times for 4 mm regular plywood at different heat fluxes and oxygen concentrations for three repeats Ignition times (s), thermal depth (mm), and functions F1 and F2 Heat flux (kW/m2)
Oxygen 21%
Oxygen 18%
Oxygen 15%
Thermal depth
x
F1
F2
25
93,146,98 Av. ¼ 112
80,94,97 Av. ¼ 90
100,100,112 Av. ¼ 103
4.3 (102 s)
1.14
1.113
1.43
35
51,45,40 Av. ¼ 49
41,49 Av. ¼ 45
64,68,64 Av. ¼ 65
3.1 (53 s)
1.59
1.022
1.53
50
24,30,37 Av. ¼ 30
23,25 s Av. ¼ 24
38,30,42 s Av. ¼ 37
2.3 (30 s) 24 s
2.35
1
2.65
2. The intercepts of the best linear fits in Figs 4b are nearly the same (5.76 and 5.71 kW/m2) and 3. The ratio of their slopes is equal to 13.14s1/2, which compares well with the value 13.02s1/2 from Eq. (5) (where d ¼ 4 mm). From these results, the critical heat flux for ignition is calculated equal to 5.74/ 0.64 ¼ 8.96 kW/m2 and the ignition temperature is ð8:96=sÞ1=4 ¼ 1=4 ð8:96=56:7Þ 1000 ¼ 630 K assuming that the critical heat flux is equal to the surface reradiation losses. In conclusion, it is possible and effective to have a consistent determination of the (equivalent) thermal properties of a material by utilizing the properties of the two methods for plotting the ignition data as illustrated in Figs. 3c and 4b. These properties are determined from the thermal diffusivity (a ¼ k=rc ¼ 1:2 107 m2 =s; for both materials), the slopes of the lines in Figs. 3c and 4b, the density of the material (r ¼ 550 kg=m3 ) and the ignition temperature (deduced from the critical heat flux 730 and 630 K, respectively). Thus, the material properties are: 1. For the fire-retarded plywood: k ¼ 0:166 W=mK; c ¼ 2:51 kJ=kgK given the thermal diffusivity, the density and krc ¼ 0:23 lsðkW=m2 Þ2 obtained from the slope of the curve corresponding to the left ordinate of Fig. 3c and Eq. (3a). 2. For the regular plywood: k ¼ 0:256 W=mK; c ¼ 3:88 kJ=kgK given the thermal diffusivity, the density and krc ¼ 0:546 sðkW=m2 Þ2 obtained from the slope of the curve corresponding to the left ordinate of Fig. 4b and Eq. (4a). The influence of the oxygen concentration in the supply air is finally appraised in the next section.
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3. Critical mass flux at ignition As previously discussed, the experiments have shown (see Tables 1 and 2) that piloted ignition times are weakly dependent on the oxygen concentration of the supply air. In this section, it is shown through the Figs. 5a, b and 6 that the mass flux at ignition is also weakly dependent on oxygen concentration. Figs. 5a and b plot the fuel mass fluxes at the time of ignition for fireretarded plywood and normal plywood, respectively, at three heat fluxes as function of oxygen concentrations. The fuel mass flux is obtained by differentiation of the mass loss histories divided by the exposed area of the sample, 0.0088 m2 [7,8]. Each point represents the average of three repeats at the same test conditions. Because of the small values of the mass loss near ignition, the error of the fuel mass flux at ignition can be significant owing to uncertainties in the exact definition of the ignition time especially at the highest heat flux of these experiments, 50 kW/m2. The magnitude of this error is depicted in Fig. 6 by plotting the mass fluxes at ignition against oxygen concentration for three or two repeats at the same test conditions. Consequently, the following observations are advanced:
8 Fire Retarded Plywood 6 4 kW/m2 kW/m2
50 35 25 kW/m2
2 0 10
(a)
Critical mass flux at ignition (g /m2s)
Critical mass flux at ignition (g/m2 s)
(a) For the two lower heat fluxes 25and 35 kW/m2, the fuel mass flux at ignition is nearly the same and independent of the oxygen concentrations, about 3.4 g/m2 s. (b) For the largest heat flux 50 kW/m2 the fuel mass flux at ignition appears to be higher than 3.4 g/m2 s and also weakly dependent on the oxygen concentration. As seen in Fig. 6, this behaviour is attributed to the uncertainty of exactly defining the ignition times at the largest imposed heat flux, which in turn will affect the determination of the fuel mass flux. Similar variability was observed regarding the measured ignition times at 50 kW/m2 in Tables 1 and 2.
14 18 Volume concentration % O2
22
8 6
Regular plywood
4 50 kW/m2 35 kW/m2 25 kW/m2
2 0 10
(b)
14 18 Volume concentration % O2
22
Fig. 5. (a) Critical fuel mass flux at ignition for fire retarded plywood at different heat fluxes and oxygen concentrations. (b) Critical fuel mass flux at ignition for regular plywood at different heat fluxes and oxygen concentrations.
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10
Critical mass flux at ignition ( g / m2 s)
50 kW / m2 Fire Retarded Plywood 8
6
4
2
0 12
14
16 18 Volume concentration % O2
20
22
Fig. 6. Example of the variability of fuel mass fluxes at ignition for two or three repeats at the same test conditions.
This weak dependence of critical fuel mass flux at ignition on the oxygen concentration (varied by nitrogen addition or reduction) explains and is reinforced by the observed independence of ignition times, critical heat flux and ignition times on the oxygen concentration (see Figs. 3 and 4 and Tables 1 and 2). It is explained also by the behaviour shown in Fig. 2 if it is assumed that the concentration of nitrogen is not high enough to alter significantly the Damko¨hler number. A conclusion is that the thermal and ignition parameters can be determined from experiments at ambient oxygen concentrations. Of course, this conclusion is valid for conditions where the chemistry of the gas phase ignition is fast so that the mass loss rate at ignition is constant and independent of the Damko¨hler number as the horizontal part of the curve in Fig. 2 shows. Given the importance of the Damko¨hler number, a reliable method to measure it at different ignition conditions (velocity, oxidizer chemistry) is needed. Finally, the present results are partly consistent with ignition experiments in wood at forced ventilation conditions (compared to buoyant conditions in this paper) using a pilot igniter (compared to a spark igniter here) [5]. Comparison is made for the low flow velocity, 0.1 m/s (more closely corresponding to buoyant conditions), and oxygen concentrations in the range 13.5–21% [5]. The results of Fig. 10 in [5], considering the experimental scatter, imply that oxygen concentrations do not considerably influence the ignition times or the critical heat flux for ignition. In addition, the observations about the influence of oxygen concentration on the mass
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flux when piloted ignition or extinction occurs are consistent with similar earlier experiments [13–15].
4. Conclusions The conclusions apply for piloted ignition of materials and are based on experimental results for two types of plywood and for reduction of oxygen concentration by adding nitrogen in the combustion air. The main results from this work are: 1. Reduced oxygen atmospheres do not affect the time to ignition and the critical heat flux, as long as the fuel mass flux at ignition is nearly independent of oxygen concentration. 2. The measurement of fuel mass flux at ignition has large uncertainties in current experimental methods such as in the cone calorimeter. Because of its importance, a new method has to be developed for its measurement at different flow and ignition conditions (i.e. Damko¨hler numbers). 3. Thermal properties, critical heat flux and ignition temperature can be determined using a novel method for plotting the measured ignition times (see Figs. 3c and 4b and Eqs. (3a) and (4a)) for intermediate thermal conditions. These results can also be used in calculation of flame spread rates. 4. It follows that, in some cases such as for plywood, measurements at normal oxygen concentrations can yield all quantities related to piloted ignition applicable also for burning in reduced oxygen environments.
Appendix A A.1. Analytic expressions for the ignition time for intermediate thermal conditions We consider the one-dimensional heating of a material of thickness d insulated at its back surface and exposed to a fixed external heat flux q_ 00 : The thermal properties of the material are constant and surface heat losses will be introduced later. The history of the surface temperature can be found in all heat transfer books (e.g. [11,12]), expressed as a series expansion. The material will behave in extreme cases as thermally thin or thermally thick, depending on the ratio of its thickness to the thermal depth at any time of reference. In these extreme cases, there are simple analytic expressions for the history of the surface temperature. These analytic expressions are the first terms in the series expansions obtained as perturbations of the solutions for the purely thermally thick and the purely thermally thin case. The analytic histories for the extreme cases have been used in the literature to correlate experimental data for ignition times and deduce from this correlation the thermal properties and critical heat flux for ignition of the material, as
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discussed in the main text. However, in many cases the experimental data cannot be categorized to represent either purely thermally thick or purely thermally thin conditions. In these cases, numerical inversion of the heat transfer equation or, as proposed here, a modification of the analytic solution for the extreme cases should be applied. The starting point for the sought after modification is the exploitation of the wellknown series expansions of the histories for the surface temperature deduced as perturbations of the solutions of the thermally thin and thermally thick cases, respectively [11]. The first terms of these series expansions, relevant for this work, are: 1. thin case: ðT s T o Þk 1 þ x2 =3 p22 x2 expðp2 =x2 Þ F 2 ¼
2 dq_ 00 x2 x
(A.1a)
which for the purely thermals thin case (x ! 0) is the well known formula: ðT s T o Þk 1 ¼ 2. dq_ 00 x
(A.1b)
Here x is defined in Eq. (A.3). 2. Thermally thick case: pffiffiffi ðT s T o Þk 2 2F 1 ¼ pffiffiffi ð1 þ 2 p i erfcðxÞÞ pffiffiffi 00 dq_ x p x p
(A.2a)
which for the purely thermal thick case (x ! 1) is the well known formula ðT s T o Þk 2 ¼ pffiffiffi . dq_ 00 x p
(A.2b)
Here, d x ¼ pffiffiffiffiffiffiffiffiffi atign
(A.3)
1 i erfcðxÞ ¼ pffiffiffi expðx2 Þ x erfcðxÞ. p
(A.4)
and
An interesting and remarkable observation is that the right-hand sides of Eqs. (A.1a) and (A.2a) take nearly the same values for intermediate thermal conditions in the range of 0:8oxo2 as Table A1 proves. The conclusion is that either of these equations represents the surface temperature histories in the intermediate thermal case. This temperature is also close to the precise surface temperature [11]. Eqs. (A.1a) and (A.2a) (and Eq. (A.3)) together with the aforementioned observation enable the derivation of Eqs. (3a) and (4a) of the main text for the ignition times assuming that ignition occurs when the surface temperature reaches a
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Table A1 List of the exact arithmetic values of the functions F1 (Eq. (3b)) and F2(Eq. (4b)) x
ierfc(x)
F1
F2
pffiffiffi F 1 =x p (Eq. (A.1a))
F2/x2 (Eq. (A.2a))
0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0.2465 0.2409 0.2354 0.23 0.2247 0.2195 0.2144 0.2094 0.2045 0.1996 0.1902 0.1811 0.1724 0.164 0.1559 0.1482 0.1402 0.1335 0.1267 0.1201 0.1138 0.1077 0.102 0.0965 0.0912 0.0861 0.0813 0.0767 0.0724 0.0682 0.0642 0.0605 0.0569 0.0535 0.0503 0.0365 0.026 0.0183 0.0127 0.0086 0.0058 0.0038 0.0025 0.0016 0.001
1.873598 1.853752 1.83426 1.815122 1.796339 1.77791 1.759836 1.742115 1.72475 1.707384 1.674071 1.64182 1.610987 1.581217 1.552511 1.525222 1.49687 1.473125 1.449026 1.425635 1.403308 1.38169 1.361489 1.341997 1.323214 1.305139 1.288128 1.271825 1.256586 1.241701 1.227525 1.214413 1.201654 1.189604 1.178264 1.129356 1.092144 1.064855 1.045009 1.030478 1.020555 1.013467 1.00886 1.00567 1.003544
1.056033333 1.0588 1.061633333 1.064533333 1.0675 1.070533333 1.073633333 1.0768 1.080033333 1.083333333 1.090133333 1.0972 1.104533333 1.112133333 1.12 1.128133333 1.136533333 1.1452 1.154133333 1.163333333 1.172799999 1.182533332 1.192533329 1.202799989 1.213333307 1.224133276 1.23519988 1.246533093 1.258132876 1.269999162 1.282131855 1.294530811 1.30719583 1.320126635 1.333322857 1.403263035 1.479692143 1.56233743 1.650751156 1.744327295 1.842354808 1.944084813 2.048791839 2.15581867 2.264602565
5.157733 4.981597 4.814583 4.656069 4.50549 4.362327 4.226107 4.096396 3.972796 3.85414 3.633596 3.431611 3.246911 3.077017 2.92045 2.776564 2.639796 2.519196 2.405102 2.298671 2.199819 2.107391 2.021933 1.941884 1.866836 1.796426 1.730796 1.66915 1.611669 1.557188 1.505944 1.458158 1.41278 1.37007 1.329869 1.15879 1.027224 0.924514 0.842477 0.775379 0.719918 0.672864 0.632593 0.597404 0.566334
6.282173 6.002268 5.741662 5.498623 5.271605 5.059231 4.860269 4.673611 4.498265 4.333333 4.031558 3.762689 3.522109 3.305985 3.111111 2.93479 2.77474 2.629017 2.495963 2.37415 2.262346 2.159484 2.064635 1.976989 1.895833 1.820543 1.750567 1.685415 1.624655 1.5679 1.514806 1.465064 1.418398 1.374559 1.333323 1.159722 1.027564 0.92446 0.84222 0.775257 0.71967 0.672694 0.632343 0.59718 0.566151
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fixed value, T ignn : In addition, the externally imposed heat flux introduced in Eqs. (3a) and (4a) is reduced by a term, q_ 00int ; due to heat losses from the surface. The equality of the right-hand sides of Eqs. (A.1a) and (A.2a) warrant that the value of q_ 00int is the same in Eqs. (3a) and (4b). Numerical solutions show that this value is 64% of the critical heat flux [9]. References [1] Santo G, Delichatsios MA. Effects of vitiated air on radiation and completeness of combustion in propane pool fires. Fire Safety J 1984;7:159–64. [2] Santo G, Tamanini F. Influence of oxygen depletion on the radiative properties of PMMA. 18th (Int.) Symposium on Combustion, The Combustion Institute, 1980. [3] Delichatsios MA, Delichatsios MM. Critical mass pyrolysis rates for extinction of fires over solid materials. FIRE SAFETY SCIENCE–Fifth International Symposium IAFSS 1997:153–64. [4] Gummalla M, Vlachos DG, Delichatsios MA. Bifurcations and structure of surface interacting methane—air diffusion flames. Combust Flame 2000;120:333–45. [5] Atreya A, Abu-Zaid M. Fire safety science. Third international symposium IAFSS. 1991. p. 177–186. [6] Gummalla M, Vlachos DG, Delichatsios MA. Conditions for extinction of solid fuel combustion: Part I simple model. Combust Flame 2005, submitted for publication. [7] Dowling VP, Leonard J, Bowditch PA. Use of a controlled atmosphere cone calorimeter to assess building materials. Proc Interflam, 99 1999:989–97. [8] Leonard J, Bowditch PA, Dowling VP. The development of a controlled atmosphere cone calorimeter. Technical report, Fire Science and Technology Lab., CSIRO, DBCE Doc. 98/038, 1998. [9] Delichatsios MA. Ignition times for thermally thick and intermediate conditions in flat and cylindrical geometries. Fire safety science—sixth international symposium, IAFSS 1999. p. 233–244. [10] Brescianini CP, Delichatsios MA, Webb AK. 1999 Australian symposium on combustion and sixth Australian flame days, Newcastle Australia, 1999. p. 180. Combust Sci Technol 2003;175:319–31. [11] Luikov AV. Analytical heat diffusion theory. New York: Academic Press; 1968. [12] Roshenow WM, Hartnett JP, Cho YI., editors. Handbook of heat transfer. 3rd ed. 1998. p. 2.64. [13] Bamford CH, Crank J, Malan DH. The burning of wood. Proceedings of the Cambridge Philosophical Society 1946;42:166–82. [14] Thomson HE, Drysdale DD. Critical mass flowrate at the firepoint of plastics. Fire safety science— second international symposium IAFSS, 1988. p. 67–76. [15] Tewarson A. Generation of heat and chemical compounds in fires. In: Di Nenno PJ, et al., editors. SFPE handbook of fire protection engineering. 2nd ed. Bostan: SFPE; 1995. p. 3.53–3.124.