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Radiation blockage in small scale PMMA combustion
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Proceedings of the
Proceedings of the Combustion Institute 33 (2011) 2657–2664
Combustion Institute www.elsevier.com/locate/proci
Radiation blockage in small scale PMMA combustion Fenghui Jiang a,⇑, J.L. de Ris b, Haiying Qi a, M.M. Khan b a
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, PR China b FM Global Engineering & Research, 1151 Boston-Providence Turnpike, P.O. Box 9102, Norwood, MA 02062, USA Available online 25 September 2010
Abstract Fuel vapors, soot and products of combustion near a burning fuel surface often block much of the radiative heat feedback to the fuel surface of a burning object. The blockage phenomenon was clearly observed in experiments of small 9.5 cm diameter PMMA pool fires subjected to external radiation burning in ambient atmospheres of different oxygen concentrations. The measurements are explained by a one-dimensional model of a diffusion flame, which focuses on the radiant absorption and emission of the soot–gas mixture of the flame. An approximate Band Model was developed and inserted into Radcal to compute gas absorption and emission from MMA vapor, CO2 and H2O. The Radcal results are included in the one-dimensional diffusion flame model to provide a greater understanding of the radiation blockage and burning rates in various ambient atmospheres and externally imposed radiant heat fluxes. A comparison between experimental data and model prediction shows a good agreement. Ó 2010 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Radiation blockage; Gas–soot mixture; PMMA combustion; Gas absorption; Oxygen concentration
1. Introduction Flame heat transfer blockage occurs when hot flame gases encounter a cold surface. The cooling by the surface creates a dense intervening layer of soot and gases that partially block radiation from reaching the surface. The effect is further enhanced by mass transfer of pyrolysis gases leaving a burning fuel surface and convects heat away from the surface. Knowledge of flame heat transfer blockage is of broad importance for improving the under-
⇑ Corresponding author. Address: FM Engineering Consulting (Shanghai) Co., Ltd., Unit 03-09, 3rd Floor, Building One, One Corporate Avenue, 222 Hubin Road, Luwan District, Shanghai 200021, PR China. Fax: +86 21 2329 8100. E-mail address: [email protected] (F. Jiang).
standing and prediction of fire burning rates and heat release rates. The blockage effect needs to be included when calculating flame heat transfer in fire growth models. The blockage of flame heat transfer is of continuing interest in the field of flame radiation research [1–3]. Gritzo and Nicolette modeled the blockage of flame heat transfer by a cool soot layer surrounding a cold objects suddenly immersed into a large flame volume [2]. In principle, radiation blockage can be predicted once one knows the composition and properties of the mixture of soot and gases [4]. In compartment fires, radiation blockage is observed prior to flashover. The relatively cool pyrolysis gases immediately above the fuel surface can block radiation coming from the ceiling itself and the hot flames just beneath the ceiling [5]. In pool fires, the cooler soot laden gases near the fuel surface block the incoming radiation. This
1540-7489/$ - see front matter Ó 2010 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2010.08.007
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F. Jiang et al. / Proceedings of the Combustion Institute 33 (2011) 2657–2664
phenomenon is very pronounced. Indeed, because of blockage, the radiant heat flux back to the fuel surface rarely exceeds 60 kW/m2 for pool fires [6,7]. Depending on their spectral characteristics the fuel vapors immediately above the fuel surface can block considerable radiation [8,9]. In vertical wall fires, de Ris et al. compared measurements of the outward to measurements of inward radiation fed back to the burning fuel surface and found that the inward radiation was significantly less than the outgoing radiation. The blockage of the inward radiation increased from 20% for very lean flames to almost 40% for quite rich flames [3]. There have been numerical studies for radiation blockage and heat transfer in radiating medium between flame and fuel surface [10,11]. Bedir and Tien used Radcal when calculating radiation from laminar PMMA – air diffusion flames and found that the surrounding cooler combustion gases block as much as 40% radiation coming from the hot interior [12]. Further understanding of burning of materials in small scale fires is vital for predicting fire performance of the same materials in large scale fires. The purpose of this study is to (1) experimentally and theoretically extend the conceptual and quantitative understanding of the flame heat transfer blockage phenomenon; and (2) establish a theoretical model for predicting burning and heat release rates based on an understanding of flame heat transfer in the presence of gas–soot mixture above the fuel surface.
ambients, while being subjected to incident radiant fluxes ranging from 0 to 60 kW/m2. Figure 1 focuses on the mixture of vapors and soot near the fuel surface for flames burning in air subjected to 60 kW/m2. The transparency of the flames indicates the absence of soot. Figure 2 compares the mass loss rates for pyrolysis in pure nitrogen to that for burning in air over a range of external heat fluxes, q_ 00ext , up to 60 kW/m2. The pyrolysis data approximately follows a straight line: m_ 00p ¼ k p ðq_ 00ext q_ 00srad Þ
with slope, kp, and intercept q_ 00srad for surface reradiation. The inverse of the slope, 1/kp = DHg agrees with the thermodynamic heat of gasification DHg = 1.6 kJ/g for PMMA [14]. The result suggests that the black PMMA absorbs all the incident radiation. Evidently the thin layer of pyrolysis vapors is almost transparent to the external heat flux. Instead of being heated by a flame, the pyrolysis vapors are rapidly cooled by the nitrogen atmosphere supplied at ambient temperature. This causes the vapors to rapidly condense into a much smaller volume of tiny liquid/ solid droplets of mist immediately upon release from the fuel surface. The white appearance of mist suggests that the droplets are very tiny causing them to scatter rather than absorb the radiation. When the sample burns, the mass transfer rate (m_ 00b ) data similarly follow a straight line; however, with reduced slope: m_ 00b ¼ k b q_ 00ext þ m_ 000
2. Experiment 2.1. Experimental apparatus (FPA)
ð1Þ
ð2Þ m_ 000
device – fire propagation
All experiments were conducted in the wellknown FPA (ASTM Standard E-2058). The unique features of the FPA include: (a) tungsten-quartz external heaters that provide constant radiant heat flux (as high as 65 kW/m2) to the test samples, such that the flux remains approximately uniform across the sample surface; (b) ambient atmospheres of various chemical compositions including oxygen vitiated and/or oxygen enriched mixtures, as well as pure nitrogen; (c) 9.5 cm diameter round horizontal samples and (d) placed in a standard sample holder [13]. The samples for the present study consisted of 25 mm thick Black PMMA (polymethyl methacrylate) with black (Thurmalox) painted top surface.
are the slope and intercept of Here kb and the experimentally obtained linear function. Such a linear dependence is typical of most fuels when the external heat fluxes q_ 00ext are less than 60 kW/ m2. m_ 000 is the “free burn” mass loss rate in the absence of external radiation: m_ 000 ¼ k p q_ 00f net ð3Þ 0
In the presence of external radiation the mass transfer, m_ 00b , is directly proportional to the net heat transfer to the surface:
2.2. Experimental observation and result Black PMMA samples were pyrolyzed in a pure nitrogen atmosphere and also burnt in air as well as 18%, 30% and 40% oxygen–nitrogen
Fig. 1. PMMA burning in air showing vapor and soot mixture inside flame (20.9% oxygen (air) ambient and 60 kW/m2 incident heat flux).
F. Jiang et al. / Proceedings of the Combustion Institute 33 (2011) 2657–2664
Mass Loss Rate (g/m2s)
45 40
Combustion in Air (1st Batch Exp) Combustion in Air (2nd Batch Exp) Pyrolysis in Nitrogen
35 30 25 20 15 10 5 0
Other experimental results are shown in figures in the later section of this paper in comparisons with model predictions.
b p
3. Theoretical model Black PMMA with Thurmalox Paint 0
10
20 30 40 50 Incident Heat Flux (kW/m 2)
60
70
Fig. 2. Mass loss rate of burning in air and pyrolysis in nitrogen (slope of line b, kb, is clearly less than slope of line p, kp).
m_ 00b ¼ k p ½ð1 bext Þq_ 00ext þ q_ 00f net
ð4Þ
Here (1 bext) is the fraction of external radiation that penetrates the flames and reaches the surface. The flames absorb the remaining fraction, bext. The fraction absorbed by the flames heats the flames causing them to emit additional radiation some of which is transferred inward toward the fuel surface. Based on the fuel surface energy balance q_ 00f net includes: (1) the increased radiative heat transfer from the flames, (2) the convective heat transfer which now is reduced because of the increased mass transfer, (3) minus the surface re-radiation, q_ 00srad . Equating Eqs. (2) and (4) while substituting Eq. (3) for m_ 000 , yields: m_ 00b ¼ k p ½ð1 bext Þq_ 00ext þ q_ 00f net ¼ k b q_ 00ext þ k p q_ 00f net 0
ð5Þ Solving Eq. (5) for the ratio of slopes: . k b =k p ¼ 1 bext þ ðq_ 00f net q_ 00f net Þ q_ 00ext 0
ð6Þ
Rearranging Eq. (6) and defining an attenuation factor A as: A¼
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q_ 00f net q_ 00f net j0 1 k b =k p ¼1 bext q_ 00ext bext
ð7Þ
The denominator of the factor A is the absorbed radiant energy from the external heat flux, and the numerator is an additional heat flux reaching the fuel surface as a result of providing the external heat flux. Figure 2 shows the ratio kb/kp is clearly less than unity. The A factor is therefore always less than 1 according to Eq. (7), indicating the absorption is stronger than the additional emission so that the total heat transfer to the fuel surface is attenuated. This attenuation of the incident radiant flux in the presence of flames provides an experimental understanding of the radiation blockage, which has been ignored by previous investigators [7,14–16] leading at times to erroneous (>20%) evaluation of the heats of gasification of solids.
A one-dimensional steady-state model of diffusion flame is developed as illustrated in Fig. 3. The theoretical analysis and modeling focuses on absorption and emission by the gas–soot mixture above the fuel surface as demonstrated in the experiments. 3.1. Model description As shown in Fig. 3, an infinite fuel surface at temperature Ts is located at x = 0. As a result of fuel gasification driven by heat feedback to the fuel surface, fuel vapor is released at a constant mass flux m_ 00 . On the opposite side, the ambient temperature Tamb and oxygen concentration Yo1 are maintained at a screen a distance d away. Oxygen and fuel vapor diffuse toward an infinitely thin flame sheet located between the fuel surface and screen. The flame sheet, having temperature Tf divides the entire space into two zones: fuel zone and oxidant zone. The fuel-rich zone contains a mixture of fuel vapor, nitrogen and combustion products (CO2, H2O, soot) that diffuse but do not react due to lack of oxygen. The chemical reactions are sufficiently fast to consume all oxygen and most of the fuel at the infinitely thin flame sheet. Owing to incomplete combustion, a very small amount of unburned fuel gas bypasses the flame, diffuses through the oxidant zone and is released at the screen. Soot particles are generated at the flame and transported into the oxidant zone where they are partially oxidized. The amount of soot eventually released at the screen is set equal to the experimentally measured smoke yield. For simplicity, it is assumed that the soot retains the same chemical composition of the original fuel.
Fig. 3. One dimensional diffusion flame model.
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In summary, the oxidant zone contains a mixture of oxygen, nitrogen, combustion products (CO2, H2O, soot) and “inert” fuel gas. Oxidization of soot is the only chemical reaction occurring within the oxidant zone. The screen carries away all the heat and combustion products generated by the flame, as well as any unburned “inert” fuel gas and “residual” soot. The mixture of soot and gases gradually absorb the external radiation as the incident heat flux q_ 00ext passes through the flame. The attenuation of the external radiation can simply be described by the Bouguer’s Law, q_ 00e ðxÞ ¼ q_ 00ext efðdxÞ , where fðd xÞ is the local optical depth experienced by a typical ray entering from the ambient. At the surface q_ 00e ð0Þ ¼ q_ 00es ¼ ð1 bext Þq_ 00ext : Radiation from the soot–gas mixture is treated separately from the external radiation. The flames both emit and absorb radiation. The net flame _ 00 radiation q_ 00r ¼ q_ 00þ has both inward, q_ 00þ r q r r 00 and outward, q_ r components. The inward flame 00 00þ radiation incident on the surface is H_ s ¼ q_ r ð0Þ. The outward radiant flux emitted by the fuel surface is q_ 00s ¼ q_ 00 r ð0Þ. The flame radiation released to the ambient is H_ 00d ¼ q_ 00 r ðd Þ. The fuel surface is assumed Lambertian with unit emissivity (es = 1). The highest temperature occurs at the flame. Heat is conducted away from the flame both inward toward the fuel surface and outward toward the ambient, eventually reaching the fuel surface with flux q_ 00cs and the screen with flux q_ 00cd , respectively. Simultaneously the mass transfer coming from the surface convects heat away from the surface toward the ambient. The convection and conduction are coupled to the heating of the flame by radiation, d q_ 00R =dx by the equation for conservation of total enthalpy: dhT d k dhT d q_ 00 m_ 00 ¼ ð8Þ þ R dx dx dx cp dx The total enthalpy varies smoothly across the flame where sensible enthalpy is generated by the release The total enthalpy, Pof chemicalRenthalpy. T ðZÞ hT ¼ Y i ðZÞ½h0i þ T Ref cpi ðT ÞdT , is expressed as the sum of Rthe chemical enthalpy h0i plus sensible T ðZÞ enthalpies T Ref cpi ðT ÞdT summed over all species Yi (Z), that are functions of the mixture fraction Z. The mixture fraction is similarly a conserved property satisfying: dZ d k dZ m_ 00 ¼ ð9Þ dx dx cp dx One can obtain rather accurate temperature profiles by including temperature and species dependent specific heats together with unit Lewis number and equal species diffusivities [17]. With these assumptions, the species mass fractions become simple algebraic functions of mixture fraction Z on both the fuel and oxidant sides of
the diffusion flame. Finally, by using the equation of state, one can express the local temperature and species mass fractions as functions of distance from the surface x. The theoretical model is established in an Excel spreadsheet, instead of a CFD program. A technique has been developed for taking Radcal results into Excel to solve the flame heat transfer blockage problems. A major advantage of using Excel is that all parameters are viewable during solution processes. 3.2. Emission and absorption of radiation by gas and soot mixture It remains to evaluate the net radiative heat flux q_ 00R ðxÞ and its derivative d q_ 00R =dx which is source of heat in Eq. (8). The radiative heat fluxes depend upon profiles of both temperature and absorption/emission coefficients. 3.2.1. Gas emission and absorption Radiation and absorption by gases is particularly important for problems of radiation blockage as demonstrated by the experiments. The gases involved are mainly fuel vapor (MMA vapor), carbon dioxide and water vapor. Carbon monoxide contributes very little radiation and thus is ignored. The modeling of radiation in this study has the following major difficulties: (i) Limited data is available for the fuel vapor. (ii) Radiation within a medium containing products of combustion depends on the temperature and species distributions throughout the entire field. The distributions depend in turn on the combustion and radiative heat transfer. (iii) One must consider the wavelength dependence of the radiation. This study uses the Fortran 90 program Radcal to overcome these difficulties. It is based on the statistical narrow band model for radiation [18,19]. However, importantly one must provide it with the spectral characteristics of the MMA vapor. I. Approximate Band Model – MMA absorption and radiation: Methylmethacrylate (MMA: CH2@ C(CH3)COOCH3) vapor is the major pyrolysis product coming from heated PMMA. It has particularly high absorptivity and emissivity within 8 wavelength bands centered near 2940 cm1 (3.4 lm), 1730 cm1 (5.8 lm), 1440 cm1 (6.9 lm), 1310 cm1 (7.6 lm), 1170 cm1 (8.5 lm), 1035 cm1 (9.7 lm), 950 cm1 (10.5 lm), 820 cm1 (12.2 lm) according to the previous measurements [20]. In general, the absorption (and emission) of radiation, jk (k, T), by a gas depends on both
F. Jiang et al. / Proceedings of the Combustion Institute 33 (2011) 2657–2664
wavelength k and the gas temperature. The Planck mean absorption coefficient jP (T) gives the absorption of blackbody radiation per unit length over very short path-length, ‘ ! 0. It is a function of temperature, defined by: R1 jk ðk; T Þebk ðT Þdk jP ðT Þ ¼ jðT ; ‘Þj‘!0 ¼ 0 ð10Þ eb ðT Þ Here eb (T) = rT4 is the integrated black body radiation and ebk (T) is its spectral distribution. The function jp (T) is known for MMA vapor from the literature [8,20]. In this study, jk (k, T) for MMA vapor is approximated by a single wide band having a form similar to that used by Radcal. The wide band is assumed uniform between the wavelengths k1 and k2 and zero elsewhere. That is jk ðk; T Þ ¼ j0 ðkÞ jðT Þ
ð11Þ
Here j0 (k) is unity for k1 6 k 6 k2 and zero for elsewhere. Constant k1 and k2 are the wavelength to be determined. Thus: Rk k1 k2 ðT Þ½ k12 ebk ðT Þdk j jP ðT Þ ¼ eb ðT Þ k1 k2 ðT ÞðF 0k2 T F 0k1 T Þ ¼j
ð12Þ
Rearranging this equation, the uniform absorption coefficient between k1 and k2 can be obtained: k1 k2 ðT Þ ¼ j
jp ðT Þ F 0k2 T F 0k1 T
ð13Þ
In a homogenous MMA vapor (XMMA = 1.0) of path-length ‘ at the atmospheric pressure (Pamb = 1.0 atm), the Goody statistical model yields the following expression for the absorption coefficient: S d T 0 =T jðT ; ‘Þ ¼ ð14Þ ½1 þ ðS d ‘T 0 =4Gd T Þ1=2 Here Sd (= S/d) is the mean line-strengthto-spacing parameter. Gd = cL (1/d), cL is the spectral line half-width (m) and 1/d is the mean inverse line spacing or the mean strong-line parameter (m1). For a very short path-length ‘ ! 0, the single wide band absorption within constant k1 and k2 can be expressed as follows according to Eq. (14): k1 k2 ðT Þ ¼ S d T 0 =T jk1 k2 ðT ; 0Þ ¼ j
ð15Þ
An expression of Sd can be derived by equating Eqs. (13) and (15): Sd ¼
jp ðT Þ TT0 F 0k2 T F 0k1 T
constant
ð16Þ
With this equation, Sd is approximated as being a constant independent of temperature and wavelengths between constant k1 and k2.
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For given jp (T), Sd, k1 and k2 are solved from Eq. (16) with following restriction: X ½ðS d Þi Sd 2 ¼ Min i ¼ temperature interval i ð17Þ k1 and k2 are adjusted and determined for minimum variation of Sd. For MMA vapor, following solution is obtained based on the jp (T) provided in Ref. [20]: 8 for k < 3:203 lm > <0 S d ¼ 223:6 atm1 m1 for 3:203 lm k 20:165 lm > : 0 for k > 20:165 lm
ð18Þ The maximum variation of Sd is 0.77% for temperature ranging from 600 to 1400 K. The solved k1 = 3.203 lm and k2 = 20.165 lm are comparable with the actual range of the spectral lines from 3.4 to 12.2 lm. The other Goody model parameter, Gd, can be approximated by achieving a good fit with measurements of the emissivity over various path-lengths [20] in the range from 0.004 to 0.100 m. Therefore, following approximate spectral absorptivity is obtained for MMA vapor under the assumption of the approximate band model: 8 for k < 3:203 lm > <0 2Gd ¼ 0:0273 for 3:203 lm k 20:165 lm > : 0 for k > 20:165 lm
ð19Þ Here 2Gd is the line-width to line spacing ratio. In reality, absorption occurs in the neighborhood of many individual spectral lines in response to quantum jumps as radiation interacts with the different energy levels of the MMA molecules. For long path-lengths, individual spectral lines increasingly saturate preventing further absorption with increasing path-length in the neighborhood of the spectral line. This reduces absorption through path-length ‘: This effect of saturated lines is modeled with the coefficient Sd/4Gd. For MMA vapor, the coefficient, in this study, is approximated as constant, independent of wavelength and temperature. For a general gas it depends on both wavelength and temperature. II. Approximate analytical solution of gas absorption: In obtaining analytical solutions for radiance and optical thickness of an inhomogeneous gas volume, the “grey body” assumption is employed for a radiating gas volume with a temperature profile of T (x) and mole fraction distributions Xi (x) of species i. This outward radiance can be numerically evaluated if both the temperature, T (x), and the outward optical thickness, n (x), are known. The
F. Jiang et al. / Proceedings of the Combustion Institute 33 (2011) 2657–2664
0
T ðxÞ
i ¼ fuel; CO2 ; H2 O
S di ðxÞGdi ðxÞX i ðxÞ S g X g R ðxÞ
0.2
0.1
ð22Þ
i
Here Sd and Sg are constants. Xg_R (x) is an overall mole fraction of the radiating gases and is assumed to be linear functions of the mixture fraction Z. Rearranging Eq. (20) with substitutions of Eqs. (21) and (22) and combinations of all the constants, a simpler expression of the outward optical thickness is therefore obtained: R x X ðxÞdx ud 0 g TRðxÞ ng ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð23Þ R x X ðxÞdx 1 þ V d 0 g TRðxÞ Similarly, for the inward direction of x from d to 0, let y = d x, the inward optical thickness, fg (y), as function of y can be derived as: R y X ðyÞdy us 0 g TRðyÞ fg ðyÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð24Þ R y X ðyÞdyffi 1 þ V s 0 g TRðyÞ Here ud, Vd, Xg_R_f, us and Vs are unknown constants. To determine the inward and outward gas optical thicknesses fg (y) and ng (x), Vs and Vd are set to be 0 (weak line limit condition). The gas absorption and optical thickness parameters us, ud and XR_g_f are adjusted and solved by matching the analytical and Radcal radiances and optical thicknesses. Solution of the parameters allows full evaluation of the gas optical thicknesses by Eqs. (23) and (24) (Fig. 4), giving that the temperature profile is solved from the energy conservation. 3.2.2. Soot absorption and radiation Radcal is only used to evaluate absorption and radiation by gases. With the grey body and ideal gas assumptions, soot absorption and emission can be evaluated by the commonly used expression of mean absorption coefficient given in Ref. [5], therefore, the coefficient js is a function of the location x. In computation, data of soot concentrations is based on the measurements in Ref. [21].
Flame
g
Total
g
0.0 0.0
i
Total
PMMA Freeburn in Air
0.2
0.4
0.6
0.8
1.0
x/d
ð20Þ
For simplicity, the following approximations are employed: X S di ðxÞX i ðxÞ Sd X g R ðxÞ ð21Þ X
0.3
and
Curtis–Godson approximation is usually used to extend the Goody statistical model and evaluate the overall optical thickness for inhomogeneous mixtures of gases: P R x ½ i S di ðxÞX i ðxÞdx T0 0 T ðxÞ ng ðxÞ ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ½P S ðxÞX ðxÞdx2 u x i u i di u 0 t1 þ T 0 P T ðxÞ 4 Rx½ S ðxÞGdi ðxÞX i ðxÞdx i di
Optical Thickness
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Fig. 4. Inward, f and outward, n optical thickness profiles.
Similar to the gas optical thicknesses, soot optical thicknesses are also defined for the inward and outward directions respectively. The outward optical thickness of soot is defined as: Z x ns ðxÞ ¼ js ðxÞdx ð25Þ 0
Let y = d x, similarly, the inward optical thickness of soot is: Z y js ðyÞdy ð26Þ fs ðyÞ ¼ 0
For a gas–soot mixture, the total optical thickness is then calculated approximately as a sum of the individual optical thicknesses of gases and soot: Outward direction: nðxÞ ¼ nTotal ðxÞ ¼ ng ðxÞ þ ns ðxÞ
ð27Þ
Inward direction: fðxÞ ¼ fðyÞ ¼ fTotal ðyÞ ¼ fg ðyÞ þ fs ðyÞ
ð28Þ
Figure 4 shows predicted optical thicknesses resulting from gas absorptions (ng and fg) as well as gas and soot absorptions (nTotal and fNotal) as a function of the dimensionless distance above the fuel surface. (1) Compared to soot, the greater contribution of gas absorption indicates heavy importance of gas absorption and radiation in these small scale fires; (2) Absorption coefficients (slope of the optical thickness curves) are larger near the fuel surface than elsewhere due to the presence of fuel vapors. (3) The outward optical thicknesses (nTotal and ng) are significantly greater than the inward optical thicknesses (fNotal and fg), this is a contributing factor to higher outward radiation to the ambient than inward radiation to the fuel surface. Based on Eqs. (24), (26), and (28), at x = 0 (y = d), the inward optical thickness of the entire zone is: fd ¼ fðx ¼ 0Þ ¼ fTotal ðy ¼ dÞ ¼ fg ðdÞ þ fs ðdÞ ð29Þ
F. Jiang et al. / Proceedings of the Combustion Institute 33 (2011) 2657–2664 45
0.30
40
Mass Loss Rate (g/m2 s)
0.35
0.25 d
0.20 0.15
G ases Soot Total
0.10 0.05
Black PMMA
0.00 0.5
1.0
1.5
2.0
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Black PMMA with Thurmalox Paint
35 30 25
Exp. (40%O2) Exp. (30%O2) Exp. (18%O2) Exp. (18%O2) Pred. (40% O2) Pred. (30% O2) Pred. (18% O2)
20 15 10 5 0 0
10
20
30
40
50
60
70
Incident Heat Flux (kW/m2 )
Fig. 5. Optical thickness of the inward directed external radiation showing (1) insensitivity to q_ 00ext and (2) the increasing importance of soot with increasing oxygen concentration.
Figure 5 plots computed inward optical thicknesses, fd, of gases, soot and total versus the ambient oxygen concentration. It shows: (1) soot and total optical thicknesses increase, while gas optical thickness slightly decreases, with increasing oxygen concentration, and (2) for q_ 00ext = 10, 20, 30, 40, 50 and 60 kW/m2, the computed fd values do not vary significantly with q_ 00ext . This implies that soot is increasingly important in radiation blockage as ambient oxygen concentration increases, and the blockage factor, bext, is nearly independent of the external heat flux. These are in agreement with experimental observations. Upon determination of the temperature profile by the energy conservation equation, all the heat fluxes as shown in Fig. 3 can be numerically evaluated by the radiative transfer equations and radiative heat flux equations under various conditions. Therefore, with an assumption of conical shape flame for actual buoyant turbulent diffusion flames in the FPA, the mass loss rate was predicted based on the fuel surface heat balance described by Eqs. (3) and (4) in which q_ 00f net and q_ 00 are equal to F cs H_ 00 q_ 00 þ q_ 00 with and f net
s
0
s
cs
Mass Loss Rate (g/m2 s)
without external heat flux, respectively. Fc-s is a constant conical view factor. H_ 00s and q_ 00cs vary with the external heat flux and ambient oxygen concentration. 45 40 35
First Batch Experiment Second Batch Experiment Ref [14] Ref [15] Ref [16] Model Prediction
30 25 20 15 10 5 0
Black PMMA Burning in Air 0
10
20
30
40
50
60
70
Incident Heat Flux (kW/m2 )
Fig. 6. Predicted and experimental mass loss rates for burning in air.
Fig. 7. Effects of ambient oxygen concentration on measured and predicted mass loss rates.
Figure 6 compares experimental data against our model predictions of PMMA burning in air. The experimental data include both ours and those from previous investigations [14–16] by others using various apparatuses, such as cone calorimeter. Figure 7 extends the comparison to other oxygen ambients, giving different slopes of the straight lines derived experimentally and theoretically. Agreement of our predictions with experiment is quite encouraging. Additionally, Eq. (6) actually provides a “bridge” between theoretical model (left side) and experiments (right side) for comparisons. The ratios of slopes in various ambient oxygen concentrations obtained from our experiments are in reasonable agreement with the combination of heat fluxes computed from the theoretical model. 4. Conclusion 1. Gas absorption and radiation is very important for evaluating flame heat transfer, especially for fuel gases near the surface. It is demonstrated that gas radiation from small fires tends to be more important than soot radiation. The reason for this is that the radiation from the spectral bands for gases saturates more readily, while weaker soot becomes increasingly important as the fire scale increases. Moreover, the saturation of gas spectral bands makes gas radiation more difficult to analyze. The Approximate Band Model established in this study for gas absorption and radiation provides reasonable results. 2. The experimental study and theoretical analysis provide four main reasons for attenuation of heat transfer back to the fuel surface: a. Cold fuel gas near the surface with a large absorption coefficient limits and blocks the inward radiation. b. Significant convective blockage for large B numbers. c. The large oxidant to fuel mass ratio causes larger outward to inward radiation.
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3. 4.
5. 6.
F. Jiang et al. / Proceedings of the Combustion Institute 33 (2011) 2657–2664
d. With increasing external radiation and fuel burning rate, the radiative fraction of flames decreases. Agreement of model predictions of mass loss rates with the experimental data is acceptable. Both experiments and computations illustrate that (a) soot is increasingly important for radiation blockage as the ambient oxygen concentration increases and (b) the blockage factor is nearly independent of the external radiation. Accurate modeling of flame temperature is essential for heat transfer and radiation blockage analysis. Further measurements would be helpful for model verification, and further analysis is needed to develop engineering analysis techniques for characterizing radiation blockage effects.
References [1] J.L. de Ris, Proc. Combust. Inst. 17 (1979) 1003– 1016. [2] L.A. Gritzo, V.F. Nicolette, Fire Sci. 15 (1997) 427– 442. [3] J.L. de Ris, G.H. Markstein, L. Orloff, P.A. Beaulieu, Proc 7th International Symposium for Fire Safety Science, International Association for Fire Safety Science, 2003, pp. 259–270. [4] K.Y. Lee, Z.Y. Zhong, C.L. Tien, Proc. Combust. Inst. 20 (1984) 1629–1636. [5] C.L. Tien, K.Y. Lee, A.J. Stretton, Radiation Heat Transfer, SFPE Handbook of Fire Protection Engineering, The National Fire Protection Association Press, Quincy MA USA, 2002, pp. 1-73–1-89.
[6] A. Hamins, S.J. Fischer, T. Kashiwagi, M.E. Klassen, J.P. Gore, Combust. Sci. Technol. 97 (1994) 37–62. [7] A. Tewarson, S.D. Ogden, Combust. Flame 89 (1992) 237–259. [8] K. Wakatsuki, G.S. Jackson, A. Hamins, M.R. Nyden, Proc. Combust. Inst. 31 (2007) 2573–2580. [9] J.M. Suo-Anttila, T.K. Blanchat, A.J. Ricks, A.L. Brown, Proc. Combust. Inst. 32 (2) (2009) 2567–2574. [10] S.L. Chen, H.K. Ma, D.Y. Chen, Int. Commun. Heat Mass Transfer 20 (1993) 145–157. [11] S.W. Baek, C. Lee, Combust. Flame 75 (1989) 153– 163. [12] H. Bedir, J.S. Tien, Proc. Combust. Inst. 27 (1998) 2821–2828. [13] J.L. de Ris, M.M. Khan, Fire Mater. 24 (2000) 219– 226. [14] R.S. Magee, R.D. Reitz, Extinguishment of Radiation Augmented Plastic Fires by Water Sprays, FMRC Serial No: 22357-1, FM Global Research, 1974. [15] D. Hopkins Jr., J.G. Quintiere, Fire Safety J. 26 (1996) 241–268. [16] P. Beaulieu, N.A. Dembsey, Fire Materials DOI 10 (2007) 1002. [17] P. Chatterjee, J.L. de Ris, C.W. Lautenberger, Proc. 5th International Seminar on Fire and Explosion Hazards, 2007, pp. 404–414. [18] W.L. Grosshandler, RADCAL: A Narrow-Band Model for Radiation Calculations in a Combustion Environment, National Institute of Standards and Technology (NIST) Technical Note 1402, 1993. [19] W.L. Grosshandler, J. Heat Mass Transfer 23 (1980) 1447–1459. [20] S.H. Park, A.J. Stretton, C.L. Tien, Combust. Sci. Technol. 62 (1988) 257–271. [21] R.A. Beier, P.J. Pagni, J. Heat Transfer 105 (1983) 159–165.
Proceedings of the
Proceedings of the Combustion Institute 33 (2011) 2657–2664
Combustion Institute www.elsevier.com/locate/proci
Radiation blockage in small scale PMMA combustion Fenghui Jiang a,⇑, J.L. de Ris b, Haiying Qi a, M.M. Khan b a
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, PR China b FM Global Engineering & Research, 1151 Boston-Providence Turnpike, P.O. Box 9102, Norwood, MA 02062, USA Available online 25 September 2010
Abstract Fuel vapors, soot and products of combustion near a burning fuel surface often block much of the radiative heat feedback to the fuel surface of a burning object. The blockage phenomenon was clearly observed in experiments of small 9.5 cm diameter PMMA pool fires subjected to external radiation burning in ambient atmospheres of different oxygen concentrations. The measurements are explained by a one-dimensional model of a diffusion flame, which focuses on the radiant absorption and emission of the soot–gas mixture of the flame. An approximate Band Model was developed and inserted into Radcal to compute gas absorption and emission from MMA vapor, CO2 and H2O. The Radcal results are included in the one-dimensional diffusion flame model to provide a greater understanding of the radiation blockage and burning rates in various ambient atmospheres and externally imposed radiant heat fluxes. A comparison between experimental data and model prediction shows a good agreement. Ó 2010 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Radiation blockage; Gas–soot mixture; PMMA combustion; Gas absorption; Oxygen concentration
1. Introduction Flame heat transfer blockage occurs when hot flame gases encounter a cold surface. The cooling by the surface creates a dense intervening layer of soot and gases that partially block radiation from reaching the surface. The effect is further enhanced by mass transfer of pyrolysis gases leaving a burning fuel surface and convects heat away from the surface. Knowledge of flame heat transfer blockage is of broad importance for improving the under-
⇑ Corresponding author. Address: FM Engineering Consulting (Shanghai) Co., Ltd., Unit 03-09, 3rd Floor, Building One, One Corporate Avenue, 222 Hubin Road, Luwan District, Shanghai 200021, PR China. Fax: +86 21 2329 8100. E-mail address: [email protected] (F. Jiang).
standing and prediction of fire burning rates and heat release rates. The blockage effect needs to be included when calculating flame heat transfer in fire growth models. The blockage of flame heat transfer is of continuing interest in the field of flame radiation research [1–3]. Gritzo and Nicolette modeled the blockage of flame heat transfer by a cool soot layer surrounding a cold objects suddenly immersed into a large flame volume [2]. In principle, radiation blockage can be predicted once one knows the composition and properties of the mixture of soot and gases [4]. In compartment fires, radiation blockage is observed prior to flashover. The relatively cool pyrolysis gases immediately above the fuel surface can block radiation coming from the ceiling itself and the hot flames just beneath the ceiling [5]. In pool fires, the cooler soot laden gases near the fuel surface block the incoming radiation. This
1540-7489/$ - see front matter Ó 2010 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2010.08.007
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phenomenon is very pronounced. Indeed, because of blockage, the radiant heat flux back to the fuel surface rarely exceeds 60 kW/m2 for pool fires [6,7]. Depending on their spectral characteristics the fuel vapors immediately above the fuel surface can block considerable radiation [8,9]. In vertical wall fires, de Ris et al. compared measurements of the outward to measurements of inward radiation fed back to the burning fuel surface and found that the inward radiation was significantly less than the outgoing radiation. The blockage of the inward radiation increased from 20% for very lean flames to almost 40% for quite rich flames [3]. There have been numerical studies for radiation blockage and heat transfer in radiating medium between flame and fuel surface [10,11]. Bedir and Tien used Radcal when calculating radiation from laminar PMMA – air diffusion flames and found that the surrounding cooler combustion gases block as much as 40% radiation coming from the hot interior [12]. Further understanding of burning of materials in small scale fires is vital for predicting fire performance of the same materials in large scale fires. The purpose of this study is to (1) experimentally and theoretically extend the conceptual and quantitative understanding of the flame heat transfer blockage phenomenon; and (2) establish a theoretical model for predicting burning and heat release rates based on an understanding of flame heat transfer in the presence of gas–soot mixture above the fuel surface.
ambients, while being subjected to incident radiant fluxes ranging from 0 to 60 kW/m2. Figure 1 focuses on the mixture of vapors and soot near the fuel surface for flames burning in air subjected to 60 kW/m2. The transparency of the flames indicates the absence of soot. Figure 2 compares the mass loss rates for pyrolysis in pure nitrogen to that for burning in air over a range of external heat fluxes, q_ 00ext , up to 60 kW/m2. The pyrolysis data approximately follows a straight line: m_ 00p ¼ k p ðq_ 00ext q_ 00srad Þ
with slope, kp, and intercept q_ 00srad for surface reradiation. The inverse of the slope, 1/kp = DHg agrees with the thermodynamic heat of gasification DHg = 1.6 kJ/g for PMMA [14]. The result suggests that the black PMMA absorbs all the incident radiation. Evidently the thin layer of pyrolysis vapors is almost transparent to the external heat flux. Instead of being heated by a flame, the pyrolysis vapors are rapidly cooled by the nitrogen atmosphere supplied at ambient temperature. This causes the vapors to rapidly condense into a much smaller volume of tiny liquid/ solid droplets of mist immediately upon release from the fuel surface. The white appearance of mist suggests that the droplets are very tiny causing them to scatter rather than absorb the radiation. When the sample burns, the mass transfer rate (m_ 00b ) data similarly follow a straight line; however, with reduced slope: m_ 00b ¼ k b q_ 00ext þ m_ 000
2. Experiment 2.1. Experimental apparatus (FPA)
ð1Þ
ð2Þ m_ 000
device – fire propagation
All experiments were conducted in the wellknown FPA (ASTM Standard E-2058). The unique features of the FPA include: (a) tungsten-quartz external heaters that provide constant radiant heat flux (as high as 65 kW/m2) to the test samples, such that the flux remains approximately uniform across the sample surface; (b) ambient atmospheres of various chemical compositions including oxygen vitiated and/or oxygen enriched mixtures, as well as pure nitrogen; (c) 9.5 cm diameter round horizontal samples and (d) placed in a standard sample holder [13]. The samples for the present study consisted of 25 mm thick Black PMMA (polymethyl methacrylate) with black (Thurmalox) painted top surface.
are the slope and intercept of Here kb and the experimentally obtained linear function. Such a linear dependence is typical of most fuels when the external heat fluxes q_ 00ext are less than 60 kW/ m2. m_ 000 is the “free burn” mass loss rate in the absence of external radiation: m_ 000 ¼ k p q_ 00f net ð3Þ 0
In the presence of external radiation the mass transfer, m_ 00b , is directly proportional to the net heat transfer to the surface:
2.2. Experimental observation and result Black PMMA samples were pyrolyzed in a pure nitrogen atmosphere and also burnt in air as well as 18%, 30% and 40% oxygen–nitrogen
Fig. 1. PMMA burning in air showing vapor and soot mixture inside flame (20.9% oxygen (air) ambient and 60 kW/m2 incident heat flux).
F. Jiang et al. / Proceedings of the Combustion Institute 33 (2011) 2657–2664
Mass Loss Rate (g/m2s)
45 40
Combustion in Air (1st Batch Exp) Combustion in Air (2nd Batch Exp) Pyrolysis in Nitrogen
35 30 25 20 15 10 5 0
Other experimental results are shown in figures in the later section of this paper in comparisons with model predictions.
b p
3. Theoretical model Black PMMA with Thurmalox Paint 0
10
20 30 40 50 Incident Heat Flux (kW/m 2)
60
70
Fig. 2. Mass loss rate of burning in air and pyrolysis in nitrogen (slope of line b, kb, is clearly less than slope of line p, kp).
m_ 00b ¼ k p ½ð1 bext Þq_ 00ext þ q_ 00f net
ð4Þ
Here (1 bext) is the fraction of external radiation that penetrates the flames and reaches the surface. The flames absorb the remaining fraction, bext. The fraction absorbed by the flames heats the flames causing them to emit additional radiation some of which is transferred inward toward the fuel surface. Based on the fuel surface energy balance q_ 00f net includes: (1) the increased radiative heat transfer from the flames, (2) the convective heat transfer which now is reduced because of the increased mass transfer, (3) minus the surface re-radiation, q_ 00srad . Equating Eqs. (2) and (4) while substituting Eq. (3) for m_ 000 , yields: m_ 00b ¼ k p ½ð1 bext Þq_ 00ext þ q_ 00f net ¼ k b q_ 00ext þ k p q_ 00f net 0
ð5Þ Solving Eq. (5) for the ratio of slopes: . k b =k p ¼ 1 bext þ ðq_ 00f net q_ 00f net Þ q_ 00ext 0
ð6Þ
Rearranging Eq. (6) and defining an attenuation factor A as: A¼
2659
q_ 00f net q_ 00f net j0 1 k b =k p ¼1 bext q_ 00ext bext
ð7Þ
The denominator of the factor A is the absorbed radiant energy from the external heat flux, and the numerator is an additional heat flux reaching the fuel surface as a result of providing the external heat flux. Figure 2 shows the ratio kb/kp is clearly less than unity. The A factor is therefore always less than 1 according to Eq. (7), indicating the absorption is stronger than the additional emission so that the total heat transfer to the fuel surface is attenuated. This attenuation of the incident radiant flux in the presence of flames provides an experimental understanding of the radiation blockage, which has been ignored by previous investigators [7,14–16] leading at times to erroneous (>20%) evaluation of the heats of gasification of solids.
A one-dimensional steady-state model of diffusion flame is developed as illustrated in Fig. 3. The theoretical analysis and modeling focuses on absorption and emission by the gas–soot mixture above the fuel surface as demonstrated in the experiments. 3.1. Model description As shown in Fig. 3, an infinite fuel surface at temperature Ts is located at x = 0. As a result of fuel gasification driven by heat feedback to the fuel surface, fuel vapor is released at a constant mass flux m_ 00 . On the opposite side, the ambient temperature Tamb and oxygen concentration Yo1 are maintained at a screen a distance d away. Oxygen and fuel vapor diffuse toward an infinitely thin flame sheet located between the fuel surface and screen. The flame sheet, having temperature Tf divides the entire space into two zones: fuel zone and oxidant zone. The fuel-rich zone contains a mixture of fuel vapor, nitrogen and combustion products (CO2, H2O, soot) that diffuse but do not react due to lack of oxygen. The chemical reactions are sufficiently fast to consume all oxygen and most of the fuel at the infinitely thin flame sheet. Owing to incomplete combustion, a very small amount of unburned fuel gas bypasses the flame, diffuses through the oxidant zone and is released at the screen. Soot particles are generated at the flame and transported into the oxidant zone where they are partially oxidized. The amount of soot eventually released at the screen is set equal to the experimentally measured smoke yield. For simplicity, it is assumed that the soot retains the same chemical composition of the original fuel.
Fig. 3. One dimensional diffusion flame model.
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In summary, the oxidant zone contains a mixture of oxygen, nitrogen, combustion products (CO2, H2O, soot) and “inert” fuel gas. Oxidization of soot is the only chemical reaction occurring within the oxidant zone. The screen carries away all the heat and combustion products generated by the flame, as well as any unburned “inert” fuel gas and “residual” soot. The mixture of soot and gases gradually absorb the external radiation as the incident heat flux q_ 00ext passes through the flame. The attenuation of the external radiation can simply be described by the Bouguer’s Law, q_ 00e ðxÞ ¼ q_ 00ext efðdxÞ , where fðd xÞ is the local optical depth experienced by a typical ray entering from the ambient. At the surface q_ 00e ð0Þ ¼ q_ 00es ¼ ð1 bext Þq_ 00ext : Radiation from the soot–gas mixture is treated separately from the external radiation. The flames both emit and absorb radiation. The net flame _ 00 radiation q_ 00r ¼ q_ 00þ has both inward, q_ 00þ r q r r 00 and outward, q_ r components. The inward flame 00 00þ radiation incident on the surface is H_ s ¼ q_ r ð0Þ. The outward radiant flux emitted by the fuel surface is q_ 00s ¼ q_ 00 r ð0Þ. The flame radiation released to the ambient is H_ 00d ¼ q_ 00 r ðd Þ. The fuel surface is assumed Lambertian with unit emissivity (es = 1). The highest temperature occurs at the flame. Heat is conducted away from the flame both inward toward the fuel surface and outward toward the ambient, eventually reaching the fuel surface with flux q_ 00cs and the screen with flux q_ 00cd , respectively. Simultaneously the mass transfer coming from the surface convects heat away from the surface toward the ambient. The convection and conduction are coupled to the heating of the flame by radiation, d q_ 00R =dx by the equation for conservation of total enthalpy: dhT d k dhT d q_ 00 m_ 00 ¼ ð8Þ þ R dx dx dx cp dx The total enthalpy varies smoothly across the flame where sensible enthalpy is generated by the release The total enthalpy, Pof chemicalRenthalpy. T ðZÞ hT ¼ Y i ðZÞ½h0i þ T Ref cpi ðT ÞdT , is expressed as the sum of Rthe chemical enthalpy h0i plus sensible T ðZÞ enthalpies T Ref cpi ðT ÞdT summed over all species Yi (Z), that are functions of the mixture fraction Z. The mixture fraction is similarly a conserved property satisfying: dZ d k dZ m_ 00 ¼ ð9Þ dx dx cp dx One can obtain rather accurate temperature profiles by including temperature and species dependent specific heats together with unit Lewis number and equal species diffusivities [17]. With these assumptions, the species mass fractions become simple algebraic functions of mixture fraction Z on both the fuel and oxidant sides of
the diffusion flame. Finally, by using the equation of state, one can express the local temperature and species mass fractions as functions of distance from the surface x. The theoretical model is established in an Excel spreadsheet, instead of a CFD program. A technique has been developed for taking Radcal results into Excel to solve the flame heat transfer blockage problems. A major advantage of using Excel is that all parameters are viewable during solution processes. 3.2. Emission and absorption of radiation by gas and soot mixture It remains to evaluate the net radiative heat flux q_ 00R ðxÞ and its derivative d q_ 00R =dx which is source of heat in Eq. (8). The radiative heat fluxes depend upon profiles of both temperature and absorption/emission coefficients. 3.2.1. Gas emission and absorption Radiation and absorption by gases is particularly important for problems of radiation blockage as demonstrated by the experiments. The gases involved are mainly fuel vapor (MMA vapor), carbon dioxide and water vapor. Carbon monoxide contributes very little radiation and thus is ignored. The modeling of radiation in this study has the following major difficulties: (i) Limited data is available for the fuel vapor. (ii) Radiation within a medium containing products of combustion depends on the temperature and species distributions throughout the entire field. The distributions depend in turn on the combustion and radiative heat transfer. (iii) One must consider the wavelength dependence of the radiation. This study uses the Fortran 90 program Radcal to overcome these difficulties. It is based on the statistical narrow band model for radiation [18,19]. However, importantly one must provide it with the spectral characteristics of the MMA vapor. I. Approximate Band Model – MMA absorption and radiation: Methylmethacrylate (MMA: CH2@ C(CH3)COOCH3) vapor is the major pyrolysis product coming from heated PMMA. It has particularly high absorptivity and emissivity within 8 wavelength bands centered near 2940 cm1 (3.4 lm), 1730 cm1 (5.8 lm), 1440 cm1 (6.9 lm), 1310 cm1 (7.6 lm), 1170 cm1 (8.5 lm), 1035 cm1 (9.7 lm), 950 cm1 (10.5 lm), 820 cm1 (12.2 lm) according to the previous measurements [20]. In general, the absorption (and emission) of radiation, jk (k, T), by a gas depends on both
F. Jiang et al. / Proceedings of the Combustion Institute 33 (2011) 2657–2664
wavelength k and the gas temperature. The Planck mean absorption coefficient jP (T) gives the absorption of blackbody radiation per unit length over very short path-length, ‘ ! 0. It is a function of temperature, defined by: R1 jk ðk; T Þebk ðT Þdk jP ðT Þ ¼ jðT ; ‘Þj‘!0 ¼ 0 ð10Þ eb ðT Þ Here eb (T) = rT4 is the integrated black body radiation and ebk (T) is its spectral distribution. The function jp (T) is known for MMA vapor from the literature [8,20]. In this study, jk (k, T) for MMA vapor is approximated by a single wide band having a form similar to that used by Radcal. The wide band is assumed uniform between the wavelengths k1 and k2 and zero elsewhere. That is jk ðk; T Þ ¼ j0 ðkÞ jðT Þ
ð11Þ
Here j0 (k) is unity for k1 6 k 6 k2 and zero for elsewhere. Constant k1 and k2 are the wavelength to be determined. Thus: Rk k1 k2 ðT Þ½ k12 ebk ðT Þdk j jP ðT Þ ¼ eb ðT Þ k1 k2 ðT ÞðF 0k2 T F 0k1 T Þ ¼j
ð12Þ
Rearranging this equation, the uniform absorption coefficient between k1 and k2 can be obtained: k1 k2 ðT Þ ¼ j
jp ðT Þ F 0k2 T F 0k1 T
ð13Þ
In a homogenous MMA vapor (XMMA = 1.0) of path-length ‘ at the atmospheric pressure (Pamb = 1.0 atm), the Goody statistical model yields the following expression for the absorption coefficient: S d T 0 =T jðT ; ‘Þ ¼ ð14Þ ½1 þ ðS d ‘T 0 =4Gd T Þ1=2 Here Sd (= S/d) is the mean line-strengthto-spacing parameter. Gd = cL (1/d), cL is the spectral line half-width (m) and 1/d is the mean inverse line spacing or the mean strong-line parameter (m1). For a very short path-length ‘ ! 0, the single wide band absorption within constant k1 and k2 can be expressed as follows according to Eq. (14): k1 k2 ðT Þ ¼ S d T 0 =T jk1 k2 ðT ; 0Þ ¼ j
ð15Þ
An expression of Sd can be derived by equating Eqs. (13) and (15): Sd ¼
jp ðT Þ TT0 F 0k2 T F 0k1 T
constant
ð16Þ
With this equation, Sd is approximated as being a constant independent of temperature and wavelengths between constant k1 and k2.
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For given jp (T), Sd, k1 and k2 are solved from Eq. (16) with following restriction: X ½ðS d Þi Sd 2 ¼ Min i ¼ temperature interval i ð17Þ k1 and k2 are adjusted and determined for minimum variation of Sd. For MMA vapor, following solution is obtained based on the jp (T) provided in Ref. [20]: 8 for k < 3:203 lm > <0 S d ¼ 223:6 atm1 m1 for 3:203 lm k 20:165 lm > : 0 for k > 20:165 lm
ð18Þ The maximum variation of Sd is 0.77% for temperature ranging from 600 to 1400 K. The solved k1 = 3.203 lm and k2 = 20.165 lm are comparable with the actual range of the spectral lines from 3.4 to 12.2 lm. The other Goody model parameter, Gd, can be approximated by achieving a good fit with measurements of the emissivity over various path-lengths [20] in the range from 0.004 to 0.100 m. Therefore, following approximate spectral absorptivity is obtained for MMA vapor under the assumption of the approximate band model: 8 for k < 3:203 lm > <0 2Gd ¼ 0:0273 for 3:203 lm k 20:165 lm > : 0 for k > 20:165 lm
ð19Þ Here 2Gd is the line-width to line spacing ratio. In reality, absorption occurs in the neighborhood of many individual spectral lines in response to quantum jumps as radiation interacts with the different energy levels of the MMA molecules. For long path-lengths, individual spectral lines increasingly saturate preventing further absorption with increasing path-length in the neighborhood of the spectral line. This reduces absorption through path-length ‘: This effect of saturated lines is modeled with the coefficient Sd/4Gd. For MMA vapor, the coefficient, in this study, is approximated as constant, independent of wavelength and temperature. For a general gas it depends on both wavelength and temperature. II. Approximate analytical solution of gas absorption: In obtaining analytical solutions for radiance and optical thickness of an inhomogeneous gas volume, the “grey body” assumption is employed for a radiating gas volume with a temperature profile of T (x) and mole fraction distributions Xi (x) of species i. This outward radiance can be numerically evaluated if both the temperature, T (x), and the outward optical thickness, n (x), are known. The
F. Jiang et al. / Proceedings of the Combustion Institute 33 (2011) 2657–2664
0
T ðxÞ
i ¼ fuel; CO2 ; H2 O
S di ðxÞGdi ðxÞX i ðxÞ S g X g R ðxÞ
0.2
0.1
ð22Þ
i
Here Sd and Sg are constants. Xg_R (x) is an overall mole fraction of the radiating gases and is assumed to be linear functions of the mixture fraction Z. Rearranging Eq. (20) with substitutions of Eqs. (21) and (22) and combinations of all the constants, a simpler expression of the outward optical thickness is therefore obtained: R x X ðxÞdx ud 0 g TRðxÞ ng ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð23Þ R x X ðxÞdx 1 þ V d 0 g TRðxÞ Similarly, for the inward direction of x from d to 0, let y = d x, the inward optical thickness, fg (y), as function of y can be derived as: R y X ðyÞdy us 0 g TRðyÞ fg ðyÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð24Þ R y X ðyÞdyffi 1 þ V s 0 g TRðyÞ Here ud, Vd, Xg_R_f, us and Vs are unknown constants. To determine the inward and outward gas optical thicknesses fg (y) and ng (x), Vs and Vd are set to be 0 (weak line limit condition). The gas absorption and optical thickness parameters us, ud and XR_g_f are adjusted and solved by matching the analytical and Radcal radiances and optical thicknesses. Solution of the parameters allows full evaluation of the gas optical thicknesses by Eqs. (23) and (24) (Fig. 4), giving that the temperature profile is solved from the energy conservation. 3.2.2. Soot absorption and radiation Radcal is only used to evaluate absorption and radiation by gases. With the grey body and ideal gas assumptions, soot absorption and emission can be evaluated by the commonly used expression of mean absorption coefficient given in Ref. [5], therefore, the coefficient js is a function of the location x. In computation, data of soot concentrations is based on the measurements in Ref. [21].
Flame
g
Total
g
0.0 0.0
i
Total
PMMA Freeburn in Air
0.2
0.4
0.6
0.8
1.0
x/d
ð20Þ
For simplicity, the following approximations are employed: X S di ðxÞX i ðxÞ Sd X g R ðxÞ ð21Þ X
0.3
and
Curtis–Godson approximation is usually used to extend the Goody statistical model and evaluate the overall optical thickness for inhomogeneous mixtures of gases: P R x ½ i S di ðxÞX i ðxÞdx T0 0 T ðxÞ ng ðxÞ ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ½P S ðxÞX ðxÞdx2 u x i u i di u 0 t1 þ T 0 P T ðxÞ 4 Rx½ S ðxÞGdi ðxÞX i ðxÞdx i di
Optical Thickness
2662
Fig. 4. Inward, f and outward, n optical thickness profiles.
Similar to the gas optical thicknesses, soot optical thicknesses are also defined for the inward and outward directions respectively. The outward optical thickness of soot is defined as: Z x ns ðxÞ ¼ js ðxÞdx ð25Þ 0
Let y = d x, similarly, the inward optical thickness of soot is: Z y js ðyÞdy ð26Þ fs ðyÞ ¼ 0
For a gas–soot mixture, the total optical thickness is then calculated approximately as a sum of the individual optical thicknesses of gases and soot: Outward direction: nðxÞ ¼ nTotal ðxÞ ¼ ng ðxÞ þ ns ðxÞ
ð27Þ
Inward direction: fðxÞ ¼ fðyÞ ¼ fTotal ðyÞ ¼ fg ðyÞ þ fs ðyÞ
ð28Þ
Figure 4 shows predicted optical thicknesses resulting from gas absorptions (ng and fg) as well as gas and soot absorptions (nTotal and fNotal) as a function of the dimensionless distance above the fuel surface. (1) Compared to soot, the greater contribution of gas absorption indicates heavy importance of gas absorption and radiation in these small scale fires; (2) Absorption coefficients (slope of the optical thickness curves) are larger near the fuel surface than elsewhere due to the presence of fuel vapors. (3) The outward optical thicknesses (nTotal and ng) are significantly greater than the inward optical thicknesses (fNotal and fg), this is a contributing factor to higher outward radiation to the ambient than inward radiation to the fuel surface. Based on Eqs. (24), (26), and (28), at x = 0 (y = d), the inward optical thickness of the entire zone is: fd ¼ fðx ¼ 0Þ ¼ fTotal ðy ¼ dÞ ¼ fg ðdÞ þ fs ðdÞ ð29Þ
F. Jiang et al. / Proceedings of the Combustion Institute 33 (2011) 2657–2664 45
0.30
40
Mass Loss Rate (g/m2 s)
0.35
0.25 d
0.20 0.15
G ases Soot Total
0.10 0.05
Black PMMA
0.00 0.5
1.0
1.5
2.0
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Black PMMA with Thurmalox Paint
35 30 25
Exp. (40%O2) Exp. (30%O2) Exp. (18%O2) Exp. (18%O2) Pred. (40% O2) Pred. (30% O2) Pred. (18% O2)
20 15 10 5 0 0
10
20
30
40
50
60
70
Incident Heat Flux (kW/m2 )
Fig. 5. Optical thickness of the inward directed external radiation showing (1) insensitivity to q_ 00ext and (2) the increasing importance of soot with increasing oxygen concentration.
Figure 5 plots computed inward optical thicknesses, fd, of gases, soot and total versus the ambient oxygen concentration. It shows: (1) soot and total optical thicknesses increase, while gas optical thickness slightly decreases, with increasing oxygen concentration, and (2) for q_ 00ext = 10, 20, 30, 40, 50 and 60 kW/m2, the computed fd values do not vary significantly with q_ 00ext . This implies that soot is increasingly important in radiation blockage as ambient oxygen concentration increases, and the blockage factor, bext, is nearly independent of the external heat flux. These are in agreement with experimental observations. Upon determination of the temperature profile by the energy conservation equation, all the heat fluxes as shown in Fig. 3 can be numerically evaluated by the radiative transfer equations and radiative heat flux equations under various conditions. Therefore, with an assumption of conical shape flame for actual buoyant turbulent diffusion flames in the FPA, the mass loss rate was predicted based on the fuel surface heat balance described by Eqs. (3) and (4) in which q_ 00f net and q_ 00 are equal to F cs H_ 00 q_ 00 þ q_ 00 with and f net
s
0
s
cs
Mass Loss Rate (g/m2 s)
without external heat flux, respectively. Fc-s is a constant conical view factor. H_ 00s and q_ 00cs vary with the external heat flux and ambient oxygen concentration. 45 40 35
First Batch Experiment Second Batch Experiment Ref [14] Ref [15] Ref [16] Model Prediction
30 25 20 15 10 5 0
Black PMMA Burning in Air 0
10
20
30
40
50
60
70
Incident Heat Flux (kW/m2 )
Fig. 6. Predicted and experimental mass loss rates for burning in air.
Fig. 7. Effects of ambient oxygen concentration on measured and predicted mass loss rates.
Figure 6 compares experimental data against our model predictions of PMMA burning in air. The experimental data include both ours and those from previous investigations [14–16] by others using various apparatuses, such as cone calorimeter. Figure 7 extends the comparison to other oxygen ambients, giving different slopes of the straight lines derived experimentally and theoretically. Agreement of our predictions with experiment is quite encouraging. Additionally, Eq. (6) actually provides a “bridge” between theoretical model (left side) and experiments (right side) for comparisons. The ratios of slopes in various ambient oxygen concentrations obtained from our experiments are in reasonable agreement with the combination of heat fluxes computed from the theoretical model. 4. Conclusion 1. Gas absorption and radiation is very important for evaluating flame heat transfer, especially for fuel gases near the surface. It is demonstrated that gas radiation from small fires tends to be more important than soot radiation. The reason for this is that the radiation from the spectral bands for gases saturates more readily, while weaker soot becomes increasingly important as the fire scale increases. Moreover, the saturation of gas spectral bands makes gas radiation more difficult to analyze. The Approximate Band Model established in this study for gas absorption and radiation provides reasonable results. 2. The experimental study and theoretical analysis provide four main reasons for attenuation of heat transfer back to the fuel surface: a. Cold fuel gas near the surface with a large absorption coefficient limits and blocks the inward radiation. b. Significant convective blockage for large B numbers. c. The large oxidant to fuel mass ratio causes larger outward to inward radiation.
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d. With increasing external radiation and fuel burning rate, the radiative fraction of flames decreases. Agreement of model predictions of mass loss rates with the experimental data is acceptable. Both experiments and computations illustrate that (a) soot is increasingly important for radiation blockage as the ambient oxygen concentration increases and (b) the blockage factor is nearly independent of the external radiation. Accurate modeling of flame temperature is essential for heat transfer and radiation blockage analysis. Further measurements would be helpful for model verification, and further analysis is needed to develop engineering analysis techniques for characterizing radiation blockage effects.
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