- Email: [email protected]
Absorption of thermal energy in PMMA by in-depth radiation
ARTICLE IN PRESS Fire Safety Journal 44 (2009) 106– 112
Contents lists available at ScienceDirect
Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf
Absorption of thermal energy in PMMA by in-depth radiation Fenghui Jiang a,, J.L. de Ris b, M.M. Khan b a b
FM Engineering Consulting (Shanghai) Co. Ltd., Unit 18-08, 18th Floor, Plaza 66, 1266 Nanjing Xi Road, Shanghai 200040, PR China FM Global, Engineering Research, 1151 Boston-Providence Turnpike, P.O. Box 9102, Norwood, MA 02062, USA
a r t i c l e in f o
a b s t r a c t
Article history: Received 6 December 2007 Received in revised form 22 April 2008 Accepted 22 April 2008 Available online 20 June 2008
An experimental technique is developed to quantify the absorption of thermal energy in black PMMA (Polycast) by in-depth radiation in semi-transparent media. In-depth heating occurs when non-reflected incident heat flux enters the solid without first being absorbed at the exposed surface. Transient conduction due to temperature gradients occurs within the solid in response to this in-depth absorption. An analytical model is developed for predicting time to ignition for such in-depth heating situations. Using the measured absorption coefficient, k, the analytical prediction for time to ignition is found to be in excellent agreement with data from experiments of Saito and Delichatsios. & 2008 Elsevier Ltd. All rights reserved.
Keywords: In-depth absorption In-depth radiation Semi-transparent media PMMA Absorption coefficient Time to ignition
1. Introduction It is well understood in the field of fire science that thermal radiation is the dominant mode of heat transfer in large-scale fires. de Ris [1] provides a well-known review and discussion of flame radiation. The study of the ignition phenomenon by absorption of radiation is, therefore, important for fire spread. The objective of the present study is to characterize the absorption of thermal energy in black PMMA by in-depth radiation. There have been a lot of recent research works on ignition aimed at understanding the physical processes determining the time to ignition [2–6], including many experimental and theoretical studies on radiative ignitions of various solid fuels, such as polymers, wood and cellulose materials, etc. Experiments and analysis conducted by Hallman et al. [7,8] in the 1970s reached an important conclusion that times to pilot ignition of polymers are strongly affected by the surface absorption. In that study, average surface absorptances were analytically determined for various radiant energy sources, and the ignition time of a number of polymers was measured and evaluated on the basis of surface absorption. In-depth absorption is quite common, and is also referred to as heat transfer in semi-transparent media [9]. Heat is delivered at significant depth into the interior of the solid instead of effectively
Corresponding author. Tel.: +86 21 6288 1066; fax: +86 21 6288 1069.
E-mail address: [email protected] (F. Jiang). 0379-7112/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2008.04.004
being deposited directly on the surface. This reduces the increase in surface temperature and delays time to ignition. Various theoretical models describing radiative ignition of solid fuels have been established and numerically analyzed [10–12], in which in-depth absorption of the incident radiation by the solid fuel is considered and modeled as an important contributing factor, in addition to a gas phase reaction and absorption. Among these theoretical works, Linan’s study [12] focused on a theoretical determination of conditions necessary for the neglect of the in-depth absorption using an asymptotic analysis. Additionally, many elegant experimental studies were performed to extend the knowledge of pilot ignition by radiation [13–17], which mainly involves time to sustained ignition and the corresponding surface temperature. In 1988, Saito et al. [18,19] at FM Global performed a set of experiments exploring the difference between in-depth absorption and surface absorption by the absence or presence of black carbon powder on the surface. The experiment and results are briefly summarized in Section 5. Based on the available knowledge, the present study focuses on a theoretical explanation of the difference between in-depth and surface absorptances in Saito’s experimental results using a simple analytical approach by properly characterizing the indepth absorption. Furthermore, in-depth absorption becomes increasingly important at high incident heat fluxes for which the reduced depth of heat penetration by conduction becomes comparable to the depth of heat penetration by in-depth radiant absorption. Under these conditions, the time to ignition no longer follows that of a
ARTICLE IN PRESS F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
Nomenclature c hc hR k n q_ 00ext r t tig T T0 Ts
specific heat (J/kg K) surface convective heat transfer coefficient (W/m2 K) surface radiation approximation coefficient (W/m2 K) conductivity (W/m K) surface refractive index external/incident heat flux (kW/m2) surface reflectivity time (s) time to ignition (s) temperature (K) environment/initial temperature (K) surface temperature (K)
thermally thick solid. Recently, Beaulieu [20] observed such delays in ignition times at high incident heat fluxes for many materials, including black PMMA, gray PVC, pine, asphalt shingles, plywood, etc. Delays are also commonly observed for foamed plastics whose low densities result in reduced thermal conductivities and a greater penetration depth of radiant heat. The widespread occurrence of this phenomenon involving high radiant fluxes was therefore a further motivation for the present study. In the present work, the absorption coefficient of black PMMA (Polycast) is experimentally measured. Based on the measurement, the physical processes of in-depth heating by radiation are analyzed by solving the differential governing equations. The preliminary results are then compared with the previous experimental data by Delichatsios and Saito [19] and Beaulieu [20].
2. Experimental approach The experimental apparatus is shown in Fig. 1. It consists of a high-density infrared heater (Research Inc., Model 5208), watercooled copper metal plate and calibrated heat flux gauge (Medtherm Corporation Model No. 48-20-18-21197). Output energy of the infrared heater is concentrated in the near infrared spectrum with typical full voltage peak color temperature output at 2500 K. The peak energy is at the wavelength of 1.15 mm. The heater was operated in this experimental study in the range of 12.5–35.5% of its rated voltage (240 VAC). This approximately corresponds to 42–66% of the rated color temperature (placing it in the range of 1050–1650 K). Thus, the spectral emission of the infrared radiator can be considered reasonably close to that of the flame heat flux from soot emission. A circular hole of 25 mm diameter (D ¼ 25 mm) was milled into black PMMA slabs having a thickness of 12.5 mm leaving a thin membrane of thickness (d) ranging from about 1 to 4 mm (Fig. 1). The bottom surface of the sample is an original surface, while the top surface is carefully made equally as smooth as the original PMMA surface by special fine grinding tools. This provides for proper transmission and reflection of incident fluxes. Each sample of black PMMA of various thicknesses from 1.06 to 3.82 mm is pressed against the water-cooled metal plate beneath the radiator. The heat flux gage is placed beneath the sample to measure the heat flux that is transmitted through the sample. Prior to exposure to the heater, the sample is covered by ceramic paper insulation. This prevents sample heat-up. In order to avoid sample heat-up and emission during the experiment, the sample is well insulated and cooled by a water-
Tig x a ba, bL d dL eout g1 b2 Z k r s y
107
surface temperature at ignition (K) in-depth distance (m) diffusivity ( ¼ k/rc) (m2/s) ratios of heat transfer coefficients sample thickness (mm) ratio of heat transfer coefficients surface emissivity to outside angle of incidence (1) angle of refraction (1) transmissivity absorption coefficient (/m) density (kg/m3) Stefan–Boltzmann constant (W/m2 K4) temperature rise (K), y ¼ TT0
cooled plate. When measuring the transmitted flux, the samples were exposed to the radiative heater for not more than 15 s to minimize the temperature rise in the sample. Readings were taken between 10 and 15 s after the start of exposure. This helps ensure that data readings of the heat flux gage are actually ‘‘transmitted heat flux’’ through the sample, not ‘‘radiant emission’’ from the sample. Exposure of the sample to the radiator is controlled by manually covering the sample with ceramic paper. Data are taken intermittently at intervals of 1 min or 30 s. Final data are obtained by taking an average of more than 8 readings.
3. Experimental results Experimental results are shown in Fig. 2, where the logarithm of the transmittivity, Z, is plotted against the sample thickness. The results show an excellent linear relationship between log of transmittivity and sample thickness. The slope of the straight line in a diagram plotting log10 Z against sample thickness d gives an approximate value of the absorption coefficient k ¼ slope/ (log10 e) ¼ 960.5 m1. The maximum variability of all the experimental data is 714.3%. To explore the effect of wavelength, the temperature of the IR heater elements was changed by varying the voltage. In order to maintain the same heat flux, the distance of the radiative heater from the sample was changed (L in Fig. 1). As shown in Fig. 2, the absorption coefficient remained unchanged. This implies that, within the range of conditions of the tests, black PMMA sample has a good ‘‘gray body’’ behavior within this spectral regime. Experiments were also performed for the black PMMA samples with the top surface painted with flat black Thurmalox paint. For a sample thickness over 1 mm, no transmitted heat flux can be detected. For a very thin plastic sheet (much less than 1 mm) with Thurmalox paint, the maximum transmittivity observed was about 0.04. This measurement of absorption coefficient allows for a quantitative analysis of the effect of in-depth absorption within black PMMA heated by radiation. Below is a preliminary analysis of the in-depth heating based on the measured absorption coefficient. The properties of black PMMA used in the analysis are supplied by the manufacturer [21]. All the coefficients and constants used are listed in Table 1.
4. Theoretical model Using the measured absorption coefficient k, the effect of indepth radiation is analyzed for the following two cases:
ARTICLE IN PRESS 108
F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
Radiator (Lamp)
L D Water cooled plate
Sample
= 1.06, 1.77, 2.66 & 3.82 mm Not to scale
Heat flux gage
Fig. 1. Schematics of experimental apparatus for absorption coefficient measurement.
1.00
η = Iη / I0
Radiator Output = 85 VAC Radiator Output = 40 VAC Radiator Output = 30 VAC
Radiator Output = 65 VAC Radiator Output = 35 VAC Line Fitting
η = 0.4652e-0.9605 δ κ = 960.5 (m-1)
0.10
Black PMMA (Unpainted/Uncoated Surface)
0.01 1.5
1.0
2.0
2.5
3.0
3.5
4.0
Sample thickness, δ (mm) Fig. 2. Measured transmittivity of black PMMA.
Table 1 Material properties (black PMMA) and constants used in the analysis Name
Symbol
Value
Unit
Resources
Conductivity Specific heat Density Refractive index Absorption coefficient Surface reflectivity Surface emissivity Convective heat transfer coef. hR approximation coefficient Ignition temperature Initial temperature
k c r n k r eout ¼ 1r hc hR Tig T0
0.188 1465 1190 1.49 960.5 0.04 0.96 10 20–24 602–701 298.15
W/m K J/kg K kg/m3 – m1 – W/m2 K W/m2 K K K
Manufacturer, Ref. [21] Manufacturer, Ref. [21] Manufacturer, Ref. [21] Manufacturer, Ref. [21] Experimentally measured Calculated from the refractive index Eq. (23) Ref. [7,8,24] Estimated based on previous studies Calculated from Eq. (10) by iteration Calculated from Eq. (26), Ref. [23] Assumed
4.1. In-depth heating without surface heat loss (Case 1) Assumption: All non-reflected incident heat flux enters the solid by radiation. Conduction occurs within the solid due to temperature gradients. There is no surface absorption or surface radiation into the solid. Surface heat loss to the environment is also ignored.
Energy balance equation for solid phase: 2
q y 1 qy ð1 rÞq_ 00ext k kx ¼ e 2 a qt k qx
(1)
In this equation, y ¼ TT0 is the temperature rise, a ¼ k/rc is the thermal diffusivity, r is the surface reflectivity, and k is the
ARTICLE IN PRESS F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
absorption coefficient measured in the experiment described above. Initial and boundary conditions: 8 yðx; 0Þ ¼ 0 > > > < qy ¼0 (2) qxx¼0 > > > : yð1; tÞ ¼ 0 An analytical solution is obtained by Laplace transforms as follows:
y3 ¼
pffiffiffiffiffi ba T 0 ð1 þ bL Þ k2 atþkx x e erfc k at þ pffiffiffiffiffi 2ð1 bL Þ 2 at
y4 ¼
y5 ¼
pffiffiffiffiffi 2 2 ba T 0 x ebL k atþbL kx erfc bL k at þ pffiffiffiffiffi bL ð1 bL Þ 2 at
ba T 0 ð1 þ bL Þ x erfc pffiffiffiffiffi bL 2 at
109
(15)
(16)
(17)
yðx; tÞ ¼ Tðx; tÞ T 0 ¼ y1 þ y2 þ y3 þ y4
(3)
y1 ¼ ba T 0 ekx
(4)
where ba ¼ ð1 rÞq_ 00ext =kkT 0 and bL ¼ (hc+eouthR)/kk are ratios of heat transfer coefficients. ba is a ratio of the non-reflected incident heat flux to the in-depth heat transfer capability; bL is a ratio of the convective and radiative heat loss from the surface to the environment to the in-depth heat transfer capability. 4.3. Surface heating with surface heat loss
y2 ¼
pffiffiffiffiffi ba T 0 k2 atkx x e erfc k at pffiffiffiffiffi 2 2 at
(5)
y3 ¼
pffiffiffiffiffi ba T 0 k2 atþkx x e erfc k at þ pffiffiffiffiffi 2 2 at
(6)
pffiffiffiffiffi x y4 ¼ 2ba T 0 k at ierfcð pffiffiffiffiffiÞ 2 at
(7)
where ba ¼ ð1 rÞq_ 00ext =kkT 0 is a non-dimensional parameter, denoting a ratio of the non-reflected incident heat flux entering the solid to the in-depth heat transfer capability expressed by the product kk in the denominator. This quantity is the heat transfer coefficient formed by conduction for a temperature gradient resulting from the absorption of external radiation within the solid. 4.2. In-depth heating with surface heat loss (Case 2) Assumptions: All non-reflected incident heat flux enters the solid by radiation. Conduction occurs within the solid due to temperature gradient. There is no surface absorption or surface radiation into the solid, but the surface has convective and radiative heat loss to the environment. Energy balance equations for solid phase (same as Eq. (1)): q2 y 1 qy ð1 rÞq_ 00ext k kx ¼ e 2 a qt k qx Initial and boundary conditions: 8 yðx; 0Þ ¼ 0 > > > < qy ¼ hc ðT s T 0 Þ out sðT 4s T 40 Þ k qx > x¼0 > > : yð1; tÞ ¼ 0 Eq. (9) can be simplified by defining R tig sðT 4 T 40 Þ dt hR ¼ R0 tig s 0 ðT s T 0 Þ dt
(8)
(9)
(10)
which introduces the following approximation: sðT 4s T 40 Þ hR ðT s T 0 Þ
(11)
With the hR approximation, an analytical solution of Eqs. (8) and (9) b by Laplace transforms is given as yðx; tÞ ¼ Tðx; tÞ T 0 ¼ y1 þ y2 þ y3 þ y4 þ y5
(12)
y1 ¼ ba T 0 ekx
(13)
For engineering purposes, total surface absorption is usually considered in ignition analysis. It is assumed that a black coated/ painted surface absorbs all non-reflected incident heat flux and heat penetrates into the solid only by conduction. In other words, there is no in-depth absorption or emission. Based on this assumption, a simplified surface heating equation and its solution are given as follows: Energy balance equations for solid phase: rc
qy q2 y ¼k 2 qt qx
Initial and boundary conditions: 8 yðx; 0Þ ¼ 0 > > > < qy 00 _ ¼ ð1 rÞqext hc ðT s T 0 Þ out sðT 4s T 40 Þ k qx x¼0 > > > : yð1; tÞ ¼ 0
(18)
(19)
Using the hR approximation as defined by Eqs. (10) and (11), and letting dL ¼ (hc+eouthR)/kk, an analytical solution of Eqs. (18) and (19) is obtained as [25,26] ð1 rÞq_ 00ext yðx; tÞ ¼ Tðx; tÞ T 0 ¼ kdL pffiffiffiffiffi 2 x x erfc pffiffiffiffiffi edL atþdL x erfc dL at þ pffiffiffiffiffi 2 at 2 at
(20)
Both in-depth absorption Case 1 and Case 2 described by Eqs. (3) and (12), respectively, in the limit as k approaches infinity, reduce to the simple surface absorption case with and without surface heat loss to the environment. Additionally, hRE20 W/m2 K is given in Ref. [22]. In the present study, the approximate value of hR is determined by using the following common iteration technique: Initially giving an assumed hR_assumed, Ts(t) and tig can then be solved from Eqs. (12) and (20) for in-depth heating and surface heating, respectively. Consequently, hR can be numerically evaluated in an Excel spreadsheet according to Eq. (10) with a derivation of a calculated hR_cal. Manually adjusting hR_assumed to obtain (|hR_assumedhR_cal|/hR_cal)o1%, hREhR_assumedEhR_cal. It is found here that hR slightly varies between 20 and 24 W/m2 K and the final result of time to ignition is not sensitive to a slight change of hR.
5. Discussion pffiffiffiffiffi b T0 2 x y2 ¼ a ek atkx erfc k at pffiffiffiffiffi 2 2 at
(14)
The black PMMA sample used in this study is manufactured by Spartech Polycast with the brand name of ‘Polycast’. Some of the properties supplied by the manufacturer are listed in Table 1.
ARTICLE IN PRESS 110
F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
More information is available in Ref. [21]. Other coefficients and constants are tabulated from the relevant literature. A refractive index of the PMMA sample is published by the manufacturer as follows based on Snell’s law [21]: n¼
sin g1 ¼ 1:49 sin g2
(21)
where g1 and g2 are angle of incidence and angle of refraction, respectively. For a given angle of incidence g1, the angle of refraction g2 can be computed as sin g1 g2 ¼ arcsin (22) n Then the surface reflectivity can be obtained as follows: " # 2 1 sin ðg1 g2 Þ tg2 ðg1 g2 Þ þ r¼ 2 sin2 ðg1 þ g2 Þ tg2 ðg1 þ g2 Þ
(23)
For the experiment in this study g1 ¼ 01; hence the reflectivity is calculated to be equal to 0.04. This consequently gives the surface absorptance of 0.96. This value very well matches the surface absorptivity of 0.94–0.96 determined by integrating the monochromatic absorptances over the effective wavelengths of the various radiant energy sources in Hallman’s studies [7,8] for black PMMA (Plexiglas). Under various incident radiant heat fluxes from 0 to 200 kW/ m2, the ignition temperature Tig (the surface temperature at ignition) is not a constant, and is evaluated using Eq. (26) in Ref. [23], in which a simplified model predicting an increase in the ignition temperature with the incident radiant fluxes is developed based on an assumption that a thermal response of a constantproperty solid is governed by the one-dimensional heat conduction equation with internal heat sink (in-depth pyrolysis). In that theoretical model, ignition is assumed to occur when a critical mass flux rate is attained, and the time to piloted ignition uses the critical mass flux as the ignition criterion as the incident radiant flux increases. It is indicated that the pyrolysis reaction is increasingly confined to a thin layer near the surface. This layer must be raised to a higher temperature to achieve the critical mass flux for ignition than at lower radiant flux where the thermal wave penetrates deeper into the solid, involving a larger
volume in the pyrolysis process. The finite depth over which pyrolysis occurs causes increasingly higher surface temperatures at ignition as the incident heat flux is increased [23]. Based on the relevant constants/coefficients given in the model, the calculated ignition temperatures vary from 600 to 700 K (Table 1). Fig. 3 shows inverse square root of time to ignition versus incident radiant heat flux for both the present theoretical and experimental data previously obtained by Saito et al. [18]. The experimental study [18,19] was conducted during 1988 in a former FM Global Research flammability apparatus (not the wellknown FM Global FPA), in which the same quartz heaters applied measured heat fluxes on samples having a thickness 19 mm and diameter 127 mm placed in a circular pan [18]. A quartz plate separated the combustion from the pyrolysis process for a predetermined heat flux being applied to the sample. The apparatus allowed one to measure the radiative properties in the form of a buoyant turbulent diffusion flame. Here we are only concerned with ignition and pyrolysis data. The same brand name black PMMA (Polycast) was used for their experiments as well. The figure includes ignition data for black PMMA, with the exposed surface both uncoated and coated with fine carbon black. Notice that the coated ignition data of Saito et al. do not decrease in slope with increasing incident radiant flux. Apparently, the coating used by Saito et al. managed to block all the incident radiation. The IR lamps used by Saito et al. were the same as those used to measure the absorptivity for the present study. For total surface absorption, as suggested by the classical thermal theory of piloted ignition, Eq. (20) gives a linear correlation in Fig. 3. A comparison with the experimental data demonstrates that some in-depth absorption occurs even in the presence of carbon black on the surface. For in-depth heating, the Case 1 analysis given by Eq. (3) considers no heat loss from the exposed surface to the environment. This leads to a time to ignition which is shorter than for Case 2 (Eq. (12)), where the surface heat loss by convection and radiation is included. With an increase in incident heat flux, more thermal energy penetrates into the solid by in-depth radiation, causing a delay in ignition and causing the curves to bend over. Based on the measured absorption coefficient k, the analytical results are in excellent agreement with the experimental data of Delichatsios and Saito [19]. However, it is noted that for incident
1.0
0.8
Black PMMA coated with carbon black (Experiment, Ref [20, 23]) In-depth heating without surface heat loss (Eq. (3))
0.7
Uncoated Black PMMA (Experiment, Ref [19]) In-depth heating with surface heat loss (Eq (12))
tig, (s-1/2)
Black PMMA coated with carbon black (Experiment, Ref [19]) Surface heating with surface heat loss (Eq (20))
0.6
1/
0.9
0.4
0.5
0.3 0.2 0.1 Black PMMA - In-depth Heating and Surface Heating
0.0 0
50
100 150 . 2 Incident Heat Flux qext ′′ , (kW/m )
Fig. 3. A comparison between analysis and experiments.
200
250
ARTICLE IN PRESS F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
heat fluxes less than 60 kW/m2 the surface heat loss may be slightly overestimated, as shown by Eq. (12) in Fig. 3. It should be noted that there has been no adjustment of properties in this analysis. The conductivity k and specific heat c of black PMMA are assumed to be constant in the present study to make the mathematics simpler. However, these thermal properties are actually functions of temperature. Although exact temperature dependence of these properties is not available, it is understood that they increase as the temperature rises in the range of this study. Qualitatively, with the temperature dependence, increasing conductivity would cause more heat loss from the surface, a slower increase in the rate of rise of surface temperature and an increase in the time to ignition. Increasing the specific heat means more heat storage in the solid for the same temperature rise. The effect would also be to delay the time to ignition. If these temperature dependencies were taken into account, the theoretical curves in Fig. 3 would be expected to deviate even further from linearity. Since the main goal of this study is to explain Saito’s experimental data by properly characterizing the in-depth absorption, temperature dependence of the thermal properties is not given the top priority. However, a further study is necessary to include temperature-dependent thermal properties. This may reproduce the experimental data better. As previously mentioned, Beaulieu [20] measured piloted ignition times for the same black PMMA coated with fine carbon black powder. Beaulieu’s ignition data [20] fell between the two sets of data measured by Saito et al. [18,19], as shown in Fig. 3, suggesting that her black coating transmitted some heat flux. In order to understand and explain the energy loss and delayed ignition time at a high incident flux level, Lautenberger and Fernandez-Pello [23] completed a detailed analysis considering Arrhenius decomposition, but with all the incident heat being deposited on the surface. Although the model predicts an increase in the surface temperature at ignition with the applied heat flux, this variation is not sufficiently strong to explain the experimental observations of Beaulieu [20]. The present analysis has shown that proper accounting of indepth absorption of radiation does reproduce the experimental data well and, therefore, this effect provides a strong explanation for the non-linear behavior of the ignition time data at high incident heat fluxes (above 60 kW/m2). Apparently some in-depth absorption occurs even in the presence of carbon black on the surface. A further study should be conducted to provide analysis for the combinations of surface and in-depth absorption, as well as the spectral absorption of PMMA.
6. Conclusions (a) Both experiments and theoretical analysis show that in-depth radiation is the primary cause of delayed ignition time for black PMMA samples, especially at higher heating rates. To predict the effect of in-depth absorption of radiation, one needs to know the absorption coefficient of the material. (b) The logarithm of the transmittivity of black PMMA follows an excellent linear relationship with the sample thickness (1–4 mm). The absorption coefficient is independent of the intensity of the radiant source (from 3 to 30 kW/m2), suggesting ‘‘gray body’’ behavior, at least over the tested range. (c) The predicted analytical ignition time curves based on the indepth radiation equations and measured absorption coefficient are in excellent agreement with the experimental ignition data of Saito and Delichatsios [18,19]. (d) The experimental and analytical techniques developed in this study can be extended to other polymeric materials for
111
generalizing the understanding of the physical processes of indepth radiation. (e) The value of this study is the understanding of the observed phenomenon of in-depth absorption for high heat fluxes. It is possible now to infer the degree of in-depth absorption from the deviation from linearity of the ignition data at high heat fluxes. For most solids the phenomenon becomes important for heat fluxes exceeding 60 kW/m2. For simple engineering purposes, it is recommended to continue to assume heat absorption only at the surface with its simpler interpretation of ignition data.
Acknowledgments The authors gratefully acknowledge the following contributors: K. Saito for conducting the experiments measuring the ignition times of coated and uncoated black PMMA and permission to publish the data, R.L. Alpert for developing the former FM Research Flammability Apparatus (not the well-known FPA); and P. Beaulieu for providing her experimental data of ignition time measurements. References [1] J.L. de Ris, Fire radiation—a review, in: 17th Symposium (Iint) on Combustion, The Combustion Institute, 1979, pp. 1003–1016. [2] M.M. Khan, J. de Ris, in: Eighth Symposium on Fire Safety Science, The International Association of Fire Safety Science, 2005, pp. 163–174. [3] D. Hopkins, J.G. Quintiere, Material fire properties and predictions for thermoplastics, Fire Safety J. 26 (1996) 241–268. [4] D. Hopkins, Predicting the ignition time and burning rate of thermoplastics in the cone calorimeter, Thesis submitted to the Faculty of the Graduate School of the University of Maryland, 1995, p. 36. [5] R.E. Lyon, Fire and materials 2005, Intersci. Commun. (2005) 1. [6] A. Atreya, Ignition of fires, Philos. Trans. R. Soc. Lond. A 356 (1998) 2787–2813. [7] J.R. Hallman, J.R. Welker, C.M. Sliepcevich, Ignition of polymers (Ignition time depends on incident irradiance, the nature of the radiation source, and the optical properties of the material), SPE J. 28 (1972) 43–47. [8] J.R. Hallman, C.M. Sliepcevich, J.R. Welker, Radiation absorption for polymers: the radiant panel and carbon arcs as radiant heat source, J. Fire Flammability 9 (1978) 353–366. [9] R. Viskanta, E.E. Anderson, Heat transfer in semitransparent solids, Adv. Heat Transfer 11 (1975) 317–441. [10] T. Kashiwagi, A radiative ignition model of a solid fuel, Combust. Sci. Technol. 8 (1974) 225–236. [11] S.H. Park, C.L. Tien, Radiation induced ignition of solid fuels, Int. J. Heat Mass Transfer 33 (1990) 1511–1520. [12] A. Linan, F.A. Williams, Radiant ignition of a reactive solid with in-depth absorption, Combust. Flame 18 (1972) 85–97. [13] H.E. Thomson, D.D. Drysdale, Flammability of plastics I: ignition temperatures, Fire Mater. 11 (1987) 163–172. [14] T. Mutoh, T. Hirano, K. Akita, Experimental study on radiative ignition of polymethylmethacrylate, in: 17th Symposium (Iint) on Combustion, The Combustion Institute, 1978, pp. 1183–1190. [15] D.L. Simms, On the pilot ignition of wood by radiation, Combust. Flame 7 (1963) 253–261. [16] D.L. Simms, Experiments on the Ignition on Cellulosic Materials by Thermal Radiation 5 (1961) 369–375. [17] T. Kashiwagi, Experimental observation of radiative ignition mechanisms, Combust. Flame 34 (1979) 231–244. [18] K. Saito, M.A. Delichatsios, S. Venkatesh, R.L. Alpert, Measurement and evaluation of parameters affecting the preheating and pyrolysis of noncharring materials, Paper Prepared for Presentation at the Fall Technical Meeting, Eastern Section, The Combustion Institute, Clearwater Beach, FL, December 1988. [19] M.A. Delichatsios, K. Saito, Upward fire spread: key flammability properties, Similarity solution and flammability indices, FM Global Research Internal Report, July 1991. [20] P.A. Beaulieu, N. Dembsey, Flammability characteristics at applied heat flux levels up to 200 kW/m2, Fire Mater. (2007). [21] General Catalog of Spartech Polycast Acrylic Sheet, Spartech. /http:// www.spartech.comS. [22] P.A. Beaulieu, Flammability characteristics at heat flux levels up to 200 kW/m2 and the effect of oxygen in flame heat flux, PhD. Dissertation, Worcester Polytechnic Institute, Worcester, MA, 2005. [23] C. Lautenberger, C.A. Fernandez-Pello, in: Fire Safety Science—Proceedings of the 8th International Symposium, The International Association for Fire Safety Science, 2005, pp. 445–456.
ARTICLE IN PRESS 112
F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
[24] L. Orloff, J. de Ris, G.H. Markstein, in: 15th Symposium (Int) on Combustion, The Combustion Institute, 1975, p. 183. [25] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, second ed., Oxford University Press, London, 1959, pp. 482, 494.
[26] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Tenth printing, Applied Mathematics Series 55, National Bureau of Standards, US Department of Commerce, December 1972.
Contents lists available at ScienceDirect
Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf
Absorption of thermal energy in PMMA by in-depth radiation Fenghui Jiang a,, J.L. de Ris b, M.M. Khan b a b
FM Engineering Consulting (Shanghai) Co. Ltd., Unit 18-08, 18th Floor, Plaza 66, 1266 Nanjing Xi Road, Shanghai 200040, PR China FM Global, Engineering Research, 1151 Boston-Providence Turnpike, P.O. Box 9102, Norwood, MA 02062, USA
a r t i c l e in f o
a b s t r a c t
Article history: Received 6 December 2007 Received in revised form 22 April 2008 Accepted 22 April 2008 Available online 20 June 2008
An experimental technique is developed to quantify the absorption of thermal energy in black PMMA (Polycast) by in-depth radiation in semi-transparent media. In-depth heating occurs when non-reflected incident heat flux enters the solid without first being absorbed at the exposed surface. Transient conduction due to temperature gradients occurs within the solid in response to this in-depth absorption. An analytical model is developed for predicting time to ignition for such in-depth heating situations. Using the measured absorption coefficient, k, the analytical prediction for time to ignition is found to be in excellent agreement with data from experiments of Saito and Delichatsios. & 2008 Elsevier Ltd. All rights reserved.
Keywords: In-depth absorption In-depth radiation Semi-transparent media PMMA Absorption coefficient Time to ignition
1. Introduction It is well understood in the field of fire science that thermal radiation is the dominant mode of heat transfer in large-scale fires. de Ris [1] provides a well-known review and discussion of flame radiation. The study of the ignition phenomenon by absorption of radiation is, therefore, important for fire spread. The objective of the present study is to characterize the absorption of thermal energy in black PMMA by in-depth radiation. There have been a lot of recent research works on ignition aimed at understanding the physical processes determining the time to ignition [2–6], including many experimental and theoretical studies on radiative ignitions of various solid fuels, such as polymers, wood and cellulose materials, etc. Experiments and analysis conducted by Hallman et al. [7,8] in the 1970s reached an important conclusion that times to pilot ignition of polymers are strongly affected by the surface absorption. In that study, average surface absorptances were analytically determined for various radiant energy sources, and the ignition time of a number of polymers was measured and evaluated on the basis of surface absorption. In-depth absorption is quite common, and is also referred to as heat transfer in semi-transparent media [9]. Heat is delivered at significant depth into the interior of the solid instead of effectively
Corresponding author. Tel.: +86 21 6288 1066; fax: +86 21 6288 1069.
E-mail address: [email protected] (F. Jiang). 0379-7112/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2008.04.004
being deposited directly on the surface. This reduces the increase in surface temperature and delays time to ignition. Various theoretical models describing radiative ignition of solid fuels have been established and numerically analyzed [10–12], in which in-depth absorption of the incident radiation by the solid fuel is considered and modeled as an important contributing factor, in addition to a gas phase reaction and absorption. Among these theoretical works, Linan’s study [12] focused on a theoretical determination of conditions necessary for the neglect of the in-depth absorption using an asymptotic analysis. Additionally, many elegant experimental studies were performed to extend the knowledge of pilot ignition by radiation [13–17], which mainly involves time to sustained ignition and the corresponding surface temperature. In 1988, Saito et al. [18,19] at FM Global performed a set of experiments exploring the difference between in-depth absorption and surface absorption by the absence or presence of black carbon powder on the surface. The experiment and results are briefly summarized in Section 5. Based on the available knowledge, the present study focuses on a theoretical explanation of the difference between in-depth and surface absorptances in Saito’s experimental results using a simple analytical approach by properly characterizing the indepth absorption. Furthermore, in-depth absorption becomes increasingly important at high incident heat fluxes for which the reduced depth of heat penetration by conduction becomes comparable to the depth of heat penetration by in-depth radiant absorption. Under these conditions, the time to ignition no longer follows that of a
ARTICLE IN PRESS F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
Nomenclature c hc hR k n q_ 00ext r t tig T T0 Ts
specific heat (J/kg K) surface convective heat transfer coefficient (W/m2 K) surface radiation approximation coefficient (W/m2 K) conductivity (W/m K) surface refractive index external/incident heat flux (kW/m2) surface reflectivity time (s) time to ignition (s) temperature (K) environment/initial temperature (K) surface temperature (K)
thermally thick solid. Recently, Beaulieu [20] observed such delays in ignition times at high incident heat fluxes for many materials, including black PMMA, gray PVC, pine, asphalt shingles, plywood, etc. Delays are also commonly observed for foamed plastics whose low densities result in reduced thermal conductivities and a greater penetration depth of radiant heat. The widespread occurrence of this phenomenon involving high radiant fluxes was therefore a further motivation for the present study. In the present work, the absorption coefficient of black PMMA (Polycast) is experimentally measured. Based on the measurement, the physical processes of in-depth heating by radiation are analyzed by solving the differential governing equations. The preliminary results are then compared with the previous experimental data by Delichatsios and Saito [19] and Beaulieu [20].
2. Experimental approach The experimental apparatus is shown in Fig. 1. It consists of a high-density infrared heater (Research Inc., Model 5208), watercooled copper metal plate and calibrated heat flux gauge (Medtherm Corporation Model No. 48-20-18-21197). Output energy of the infrared heater is concentrated in the near infrared spectrum with typical full voltage peak color temperature output at 2500 K. The peak energy is at the wavelength of 1.15 mm. The heater was operated in this experimental study in the range of 12.5–35.5% of its rated voltage (240 VAC). This approximately corresponds to 42–66% of the rated color temperature (placing it in the range of 1050–1650 K). Thus, the spectral emission of the infrared radiator can be considered reasonably close to that of the flame heat flux from soot emission. A circular hole of 25 mm diameter (D ¼ 25 mm) was milled into black PMMA slabs having a thickness of 12.5 mm leaving a thin membrane of thickness (d) ranging from about 1 to 4 mm (Fig. 1). The bottom surface of the sample is an original surface, while the top surface is carefully made equally as smooth as the original PMMA surface by special fine grinding tools. This provides for proper transmission and reflection of incident fluxes. Each sample of black PMMA of various thicknesses from 1.06 to 3.82 mm is pressed against the water-cooled metal plate beneath the radiator. The heat flux gage is placed beneath the sample to measure the heat flux that is transmitted through the sample. Prior to exposure to the heater, the sample is covered by ceramic paper insulation. This prevents sample heat-up. In order to avoid sample heat-up and emission during the experiment, the sample is well insulated and cooled by a water-
Tig x a ba, bL d dL eout g1 b2 Z k r s y
107
surface temperature at ignition (K) in-depth distance (m) diffusivity ( ¼ k/rc) (m2/s) ratios of heat transfer coefficients sample thickness (mm) ratio of heat transfer coefficients surface emissivity to outside angle of incidence (1) angle of refraction (1) transmissivity absorption coefficient (/m) density (kg/m3) Stefan–Boltzmann constant (W/m2 K4) temperature rise (K), y ¼ TT0
cooled plate. When measuring the transmitted flux, the samples were exposed to the radiative heater for not more than 15 s to minimize the temperature rise in the sample. Readings were taken between 10 and 15 s after the start of exposure. This helps ensure that data readings of the heat flux gage are actually ‘‘transmitted heat flux’’ through the sample, not ‘‘radiant emission’’ from the sample. Exposure of the sample to the radiator is controlled by manually covering the sample with ceramic paper. Data are taken intermittently at intervals of 1 min or 30 s. Final data are obtained by taking an average of more than 8 readings.
3. Experimental results Experimental results are shown in Fig. 2, where the logarithm of the transmittivity, Z, is plotted against the sample thickness. The results show an excellent linear relationship between log of transmittivity and sample thickness. The slope of the straight line in a diagram plotting log10 Z against sample thickness d gives an approximate value of the absorption coefficient k ¼ slope/ (log10 e) ¼ 960.5 m1. The maximum variability of all the experimental data is 714.3%. To explore the effect of wavelength, the temperature of the IR heater elements was changed by varying the voltage. In order to maintain the same heat flux, the distance of the radiative heater from the sample was changed (L in Fig. 1). As shown in Fig. 2, the absorption coefficient remained unchanged. This implies that, within the range of conditions of the tests, black PMMA sample has a good ‘‘gray body’’ behavior within this spectral regime. Experiments were also performed for the black PMMA samples with the top surface painted with flat black Thurmalox paint. For a sample thickness over 1 mm, no transmitted heat flux can be detected. For a very thin plastic sheet (much less than 1 mm) with Thurmalox paint, the maximum transmittivity observed was about 0.04. This measurement of absorption coefficient allows for a quantitative analysis of the effect of in-depth absorption within black PMMA heated by radiation. Below is a preliminary analysis of the in-depth heating based on the measured absorption coefficient. The properties of black PMMA used in the analysis are supplied by the manufacturer [21]. All the coefficients and constants used are listed in Table 1.
4. Theoretical model Using the measured absorption coefficient k, the effect of indepth radiation is analyzed for the following two cases:
ARTICLE IN PRESS 108
F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
Radiator (Lamp)
L D Water cooled plate
Sample
= 1.06, 1.77, 2.66 & 3.82 mm Not to scale
Heat flux gage
Fig. 1. Schematics of experimental apparatus for absorption coefficient measurement.
1.00
η = Iη / I0
Radiator Output = 85 VAC Radiator Output = 40 VAC Radiator Output = 30 VAC
Radiator Output = 65 VAC Radiator Output = 35 VAC Line Fitting
η = 0.4652e-0.9605 δ κ = 960.5 (m-1)
0.10
Black PMMA (Unpainted/Uncoated Surface)
0.01 1.5
1.0
2.0
2.5
3.0
3.5
4.0
Sample thickness, δ (mm) Fig. 2. Measured transmittivity of black PMMA.
Table 1 Material properties (black PMMA) and constants used in the analysis Name
Symbol
Value
Unit
Resources
Conductivity Specific heat Density Refractive index Absorption coefficient Surface reflectivity Surface emissivity Convective heat transfer coef. hR approximation coefficient Ignition temperature Initial temperature
k c r n k r eout ¼ 1r hc hR Tig T0
0.188 1465 1190 1.49 960.5 0.04 0.96 10 20–24 602–701 298.15
W/m K J/kg K kg/m3 – m1 – W/m2 K W/m2 K K K
Manufacturer, Ref. [21] Manufacturer, Ref. [21] Manufacturer, Ref. [21] Manufacturer, Ref. [21] Experimentally measured Calculated from the refractive index Eq. (23) Ref. [7,8,24] Estimated based on previous studies Calculated from Eq. (10) by iteration Calculated from Eq. (26), Ref. [23] Assumed
4.1. In-depth heating without surface heat loss (Case 1) Assumption: All non-reflected incident heat flux enters the solid by radiation. Conduction occurs within the solid due to temperature gradients. There is no surface absorption or surface radiation into the solid. Surface heat loss to the environment is also ignored.
Energy balance equation for solid phase: 2
q y 1 qy ð1 rÞq_ 00ext k kx ¼ e 2 a qt k qx
(1)
In this equation, y ¼ TT0 is the temperature rise, a ¼ k/rc is the thermal diffusivity, r is the surface reflectivity, and k is the
ARTICLE IN PRESS F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
absorption coefficient measured in the experiment described above. Initial and boundary conditions: 8 yðx; 0Þ ¼ 0 > > > < qy ¼0 (2) qxx¼0 > > > : yð1; tÞ ¼ 0 An analytical solution is obtained by Laplace transforms as follows:
y3 ¼
pffiffiffiffiffi ba T 0 ð1 þ bL Þ k2 atþkx x e erfc k at þ pffiffiffiffiffi 2ð1 bL Þ 2 at
y4 ¼
y5 ¼
pffiffiffiffiffi 2 2 ba T 0 x ebL k atþbL kx erfc bL k at þ pffiffiffiffiffi bL ð1 bL Þ 2 at
ba T 0 ð1 þ bL Þ x erfc pffiffiffiffiffi bL 2 at
109
(15)
(16)
(17)
yðx; tÞ ¼ Tðx; tÞ T 0 ¼ y1 þ y2 þ y3 þ y4
(3)
y1 ¼ ba T 0 ekx
(4)
where ba ¼ ð1 rÞq_ 00ext =kkT 0 and bL ¼ (hc+eouthR)/kk are ratios of heat transfer coefficients. ba is a ratio of the non-reflected incident heat flux to the in-depth heat transfer capability; bL is a ratio of the convective and radiative heat loss from the surface to the environment to the in-depth heat transfer capability. 4.3. Surface heating with surface heat loss
y2 ¼
pffiffiffiffiffi ba T 0 k2 atkx x e erfc k at pffiffiffiffiffi 2 2 at
(5)
y3 ¼
pffiffiffiffiffi ba T 0 k2 atþkx x e erfc k at þ pffiffiffiffiffi 2 2 at
(6)
pffiffiffiffiffi x y4 ¼ 2ba T 0 k at ierfcð pffiffiffiffiffiÞ 2 at
(7)
where ba ¼ ð1 rÞq_ 00ext =kkT 0 is a non-dimensional parameter, denoting a ratio of the non-reflected incident heat flux entering the solid to the in-depth heat transfer capability expressed by the product kk in the denominator. This quantity is the heat transfer coefficient formed by conduction for a temperature gradient resulting from the absorption of external radiation within the solid. 4.2. In-depth heating with surface heat loss (Case 2) Assumptions: All non-reflected incident heat flux enters the solid by radiation. Conduction occurs within the solid due to temperature gradient. There is no surface absorption or surface radiation into the solid, but the surface has convective and radiative heat loss to the environment. Energy balance equations for solid phase (same as Eq. (1)): q2 y 1 qy ð1 rÞq_ 00ext k kx ¼ e 2 a qt k qx Initial and boundary conditions: 8 yðx; 0Þ ¼ 0 > > > < qy ¼ hc ðT s T 0 Þ out sðT 4s T 40 Þ k qx > x¼0 > > : yð1; tÞ ¼ 0 Eq. (9) can be simplified by defining R tig sðT 4 T 40 Þ dt hR ¼ R0 tig s 0 ðT s T 0 Þ dt
(8)
(9)
(10)
which introduces the following approximation: sðT 4s T 40 Þ hR ðT s T 0 Þ
(11)
With the hR approximation, an analytical solution of Eqs. (8) and (9) b by Laplace transforms is given as yðx; tÞ ¼ Tðx; tÞ T 0 ¼ y1 þ y2 þ y3 þ y4 þ y5
(12)
y1 ¼ ba T 0 ekx
(13)
For engineering purposes, total surface absorption is usually considered in ignition analysis. It is assumed that a black coated/ painted surface absorbs all non-reflected incident heat flux and heat penetrates into the solid only by conduction. In other words, there is no in-depth absorption or emission. Based on this assumption, a simplified surface heating equation and its solution are given as follows: Energy balance equations for solid phase: rc
qy q2 y ¼k 2 qt qx
Initial and boundary conditions: 8 yðx; 0Þ ¼ 0 > > > < qy 00 _ ¼ ð1 rÞqext hc ðT s T 0 Þ out sðT 4s T 40 Þ k qx x¼0 > > > : yð1; tÞ ¼ 0
(18)
(19)
Using the hR approximation as defined by Eqs. (10) and (11), and letting dL ¼ (hc+eouthR)/kk, an analytical solution of Eqs. (18) and (19) is obtained as [25,26] ð1 rÞq_ 00ext yðx; tÞ ¼ Tðx; tÞ T 0 ¼ kdL pffiffiffiffiffi 2 x x erfc pffiffiffiffiffi edL atþdL x erfc dL at þ pffiffiffiffiffi 2 at 2 at
(20)
Both in-depth absorption Case 1 and Case 2 described by Eqs. (3) and (12), respectively, in the limit as k approaches infinity, reduce to the simple surface absorption case with and without surface heat loss to the environment. Additionally, hRE20 W/m2 K is given in Ref. [22]. In the present study, the approximate value of hR is determined by using the following common iteration technique: Initially giving an assumed hR_assumed, Ts(t) and tig can then be solved from Eqs. (12) and (20) for in-depth heating and surface heating, respectively. Consequently, hR can be numerically evaluated in an Excel spreadsheet according to Eq. (10) with a derivation of a calculated hR_cal. Manually adjusting hR_assumed to obtain (|hR_assumedhR_cal|/hR_cal)o1%, hREhR_assumedEhR_cal. It is found here that hR slightly varies between 20 and 24 W/m2 K and the final result of time to ignition is not sensitive to a slight change of hR.
5. Discussion pffiffiffiffiffi b T0 2 x y2 ¼ a ek atkx erfc k at pffiffiffiffiffi 2 2 at
(14)
The black PMMA sample used in this study is manufactured by Spartech Polycast with the brand name of ‘Polycast’. Some of the properties supplied by the manufacturer are listed in Table 1.
ARTICLE IN PRESS 110
F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
More information is available in Ref. [21]. Other coefficients and constants are tabulated from the relevant literature. A refractive index of the PMMA sample is published by the manufacturer as follows based on Snell’s law [21]: n¼
sin g1 ¼ 1:49 sin g2
(21)
where g1 and g2 are angle of incidence and angle of refraction, respectively. For a given angle of incidence g1, the angle of refraction g2 can be computed as sin g1 g2 ¼ arcsin (22) n Then the surface reflectivity can be obtained as follows: " # 2 1 sin ðg1 g2 Þ tg2 ðg1 g2 Þ þ r¼ 2 sin2 ðg1 þ g2 Þ tg2 ðg1 þ g2 Þ
(23)
For the experiment in this study g1 ¼ 01; hence the reflectivity is calculated to be equal to 0.04. This consequently gives the surface absorptance of 0.96. This value very well matches the surface absorptivity of 0.94–0.96 determined by integrating the monochromatic absorptances over the effective wavelengths of the various radiant energy sources in Hallman’s studies [7,8] for black PMMA (Plexiglas). Under various incident radiant heat fluxes from 0 to 200 kW/ m2, the ignition temperature Tig (the surface temperature at ignition) is not a constant, and is evaluated using Eq. (26) in Ref. [23], in which a simplified model predicting an increase in the ignition temperature with the incident radiant fluxes is developed based on an assumption that a thermal response of a constantproperty solid is governed by the one-dimensional heat conduction equation with internal heat sink (in-depth pyrolysis). In that theoretical model, ignition is assumed to occur when a critical mass flux rate is attained, and the time to piloted ignition uses the critical mass flux as the ignition criterion as the incident radiant flux increases. It is indicated that the pyrolysis reaction is increasingly confined to a thin layer near the surface. This layer must be raised to a higher temperature to achieve the critical mass flux for ignition than at lower radiant flux where the thermal wave penetrates deeper into the solid, involving a larger
volume in the pyrolysis process. The finite depth over which pyrolysis occurs causes increasingly higher surface temperatures at ignition as the incident heat flux is increased [23]. Based on the relevant constants/coefficients given in the model, the calculated ignition temperatures vary from 600 to 700 K (Table 1). Fig. 3 shows inverse square root of time to ignition versus incident radiant heat flux for both the present theoretical and experimental data previously obtained by Saito et al. [18]. The experimental study [18,19] was conducted during 1988 in a former FM Global Research flammability apparatus (not the wellknown FM Global FPA), in which the same quartz heaters applied measured heat fluxes on samples having a thickness 19 mm and diameter 127 mm placed in a circular pan [18]. A quartz plate separated the combustion from the pyrolysis process for a predetermined heat flux being applied to the sample. The apparatus allowed one to measure the radiative properties in the form of a buoyant turbulent diffusion flame. Here we are only concerned with ignition and pyrolysis data. The same brand name black PMMA (Polycast) was used for their experiments as well. The figure includes ignition data for black PMMA, with the exposed surface both uncoated and coated with fine carbon black. Notice that the coated ignition data of Saito et al. do not decrease in slope with increasing incident radiant flux. Apparently, the coating used by Saito et al. managed to block all the incident radiation. The IR lamps used by Saito et al. were the same as those used to measure the absorptivity for the present study. For total surface absorption, as suggested by the classical thermal theory of piloted ignition, Eq. (20) gives a linear correlation in Fig. 3. A comparison with the experimental data demonstrates that some in-depth absorption occurs even in the presence of carbon black on the surface. For in-depth heating, the Case 1 analysis given by Eq. (3) considers no heat loss from the exposed surface to the environment. This leads to a time to ignition which is shorter than for Case 2 (Eq. (12)), where the surface heat loss by convection and radiation is included. With an increase in incident heat flux, more thermal energy penetrates into the solid by in-depth radiation, causing a delay in ignition and causing the curves to bend over. Based on the measured absorption coefficient k, the analytical results are in excellent agreement with the experimental data of Delichatsios and Saito [19]. However, it is noted that for incident
1.0
0.8
Black PMMA coated with carbon black (Experiment, Ref [20, 23]) In-depth heating without surface heat loss (Eq. (3))
0.7
Uncoated Black PMMA (Experiment, Ref [19]) In-depth heating with surface heat loss (Eq (12))
tig, (s-1/2)
Black PMMA coated with carbon black (Experiment, Ref [19]) Surface heating with surface heat loss (Eq (20))
0.6
1/
0.9
0.4
0.5
0.3 0.2 0.1 Black PMMA - In-depth Heating and Surface Heating
0.0 0
50
100 150 . 2 Incident Heat Flux qext ′′ , (kW/m )
Fig. 3. A comparison between analysis and experiments.
200
250
ARTICLE IN PRESS F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
heat fluxes less than 60 kW/m2 the surface heat loss may be slightly overestimated, as shown by Eq. (12) in Fig. 3. It should be noted that there has been no adjustment of properties in this analysis. The conductivity k and specific heat c of black PMMA are assumed to be constant in the present study to make the mathematics simpler. However, these thermal properties are actually functions of temperature. Although exact temperature dependence of these properties is not available, it is understood that they increase as the temperature rises in the range of this study. Qualitatively, with the temperature dependence, increasing conductivity would cause more heat loss from the surface, a slower increase in the rate of rise of surface temperature and an increase in the time to ignition. Increasing the specific heat means more heat storage in the solid for the same temperature rise. The effect would also be to delay the time to ignition. If these temperature dependencies were taken into account, the theoretical curves in Fig. 3 would be expected to deviate even further from linearity. Since the main goal of this study is to explain Saito’s experimental data by properly characterizing the in-depth absorption, temperature dependence of the thermal properties is not given the top priority. However, a further study is necessary to include temperature-dependent thermal properties. This may reproduce the experimental data better. As previously mentioned, Beaulieu [20] measured piloted ignition times for the same black PMMA coated with fine carbon black powder. Beaulieu’s ignition data [20] fell between the two sets of data measured by Saito et al. [18,19], as shown in Fig. 3, suggesting that her black coating transmitted some heat flux. In order to understand and explain the energy loss and delayed ignition time at a high incident flux level, Lautenberger and Fernandez-Pello [23] completed a detailed analysis considering Arrhenius decomposition, but with all the incident heat being deposited on the surface. Although the model predicts an increase in the surface temperature at ignition with the applied heat flux, this variation is not sufficiently strong to explain the experimental observations of Beaulieu [20]. The present analysis has shown that proper accounting of indepth absorption of radiation does reproduce the experimental data well and, therefore, this effect provides a strong explanation for the non-linear behavior of the ignition time data at high incident heat fluxes (above 60 kW/m2). Apparently some in-depth absorption occurs even in the presence of carbon black on the surface. A further study should be conducted to provide analysis for the combinations of surface and in-depth absorption, as well as the spectral absorption of PMMA.
6. Conclusions (a) Both experiments and theoretical analysis show that in-depth radiation is the primary cause of delayed ignition time for black PMMA samples, especially at higher heating rates. To predict the effect of in-depth absorption of radiation, one needs to know the absorption coefficient of the material. (b) The logarithm of the transmittivity of black PMMA follows an excellent linear relationship with the sample thickness (1–4 mm). The absorption coefficient is independent of the intensity of the radiant source (from 3 to 30 kW/m2), suggesting ‘‘gray body’’ behavior, at least over the tested range. (c) The predicted analytical ignition time curves based on the indepth radiation equations and measured absorption coefficient are in excellent agreement with the experimental ignition data of Saito and Delichatsios [18,19]. (d) The experimental and analytical techniques developed in this study can be extended to other polymeric materials for
111
generalizing the understanding of the physical processes of indepth radiation. (e) The value of this study is the understanding of the observed phenomenon of in-depth absorption for high heat fluxes. It is possible now to infer the degree of in-depth absorption from the deviation from linearity of the ignition data at high heat fluxes. For most solids the phenomenon becomes important for heat fluxes exceeding 60 kW/m2. For simple engineering purposes, it is recommended to continue to assume heat absorption only at the surface with its simpler interpretation of ignition data.
Acknowledgments The authors gratefully acknowledge the following contributors: K. Saito for conducting the experiments measuring the ignition times of coated and uncoated black PMMA and permission to publish the data, R.L. Alpert for developing the former FM Research Flammability Apparatus (not the well-known FPA); and P. Beaulieu for providing her experimental data of ignition time measurements. References [1] J.L. de Ris, Fire radiation—a review, in: 17th Symposium (Iint) on Combustion, The Combustion Institute, 1979, pp. 1003–1016. [2] M.M. Khan, J. de Ris, in: Eighth Symposium on Fire Safety Science, The International Association of Fire Safety Science, 2005, pp. 163–174. [3] D. Hopkins, J.G. Quintiere, Material fire properties and predictions for thermoplastics, Fire Safety J. 26 (1996) 241–268. [4] D. Hopkins, Predicting the ignition time and burning rate of thermoplastics in the cone calorimeter, Thesis submitted to the Faculty of the Graduate School of the University of Maryland, 1995, p. 36. [5] R.E. Lyon, Fire and materials 2005, Intersci. Commun. (2005) 1. [6] A. Atreya, Ignition of fires, Philos. Trans. R. Soc. Lond. A 356 (1998) 2787–2813. [7] J.R. Hallman, J.R. Welker, C.M. Sliepcevich, Ignition of polymers (Ignition time depends on incident irradiance, the nature of the radiation source, and the optical properties of the material), SPE J. 28 (1972) 43–47. [8] J.R. Hallman, C.M. Sliepcevich, J.R. Welker, Radiation absorption for polymers: the radiant panel and carbon arcs as radiant heat source, J. Fire Flammability 9 (1978) 353–366. [9] R. Viskanta, E.E. Anderson, Heat transfer in semitransparent solids, Adv. Heat Transfer 11 (1975) 317–441. [10] T. Kashiwagi, A radiative ignition model of a solid fuel, Combust. Sci. Technol. 8 (1974) 225–236. [11] S.H. Park, C.L. Tien, Radiation induced ignition of solid fuels, Int. J. Heat Mass Transfer 33 (1990) 1511–1520. [12] A. Linan, F.A. Williams, Radiant ignition of a reactive solid with in-depth absorption, Combust. Flame 18 (1972) 85–97. [13] H.E. Thomson, D.D. Drysdale, Flammability of plastics I: ignition temperatures, Fire Mater. 11 (1987) 163–172. [14] T. Mutoh, T. Hirano, K. Akita, Experimental study on radiative ignition of polymethylmethacrylate, in: 17th Symposium (Iint) on Combustion, The Combustion Institute, 1978, pp. 1183–1190. [15] D.L. Simms, On the pilot ignition of wood by radiation, Combust. Flame 7 (1963) 253–261. [16] D.L. Simms, Experiments on the Ignition on Cellulosic Materials by Thermal Radiation 5 (1961) 369–375. [17] T. Kashiwagi, Experimental observation of radiative ignition mechanisms, Combust. Flame 34 (1979) 231–244. [18] K. Saito, M.A. Delichatsios, S. Venkatesh, R.L. Alpert, Measurement and evaluation of parameters affecting the preheating and pyrolysis of noncharring materials, Paper Prepared for Presentation at the Fall Technical Meeting, Eastern Section, The Combustion Institute, Clearwater Beach, FL, December 1988. [19] M.A. Delichatsios, K. Saito, Upward fire spread: key flammability properties, Similarity solution and flammability indices, FM Global Research Internal Report, July 1991. [20] P.A. Beaulieu, N. Dembsey, Flammability characteristics at applied heat flux levels up to 200 kW/m2, Fire Mater. (2007). [21] General Catalog of Spartech Polycast Acrylic Sheet, Spartech. /http:// www.spartech.comS. [22] P.A. Beaulieu, Flammability characteristics at heat flux levels up to 200 kW/m2 and the effect of oxygen in flame heat flux, PhD. Dissertation, Worcester Polytechnic Institute, Worcester, MA, 2005. [23] C. Lautenberger, C.A. Fernandez-Pello, in: Fire Safety Science—Proceedings of the 8th International Symposium, The International Association for Fire Safety Science, 2005, pp. 445–456.
ARTICLE IN PRESS 112
F. Jiang et al. / Fire Safety Journal 44 (2009) 106–112
[24] L. Orloff, J. de Ris, G.H. Markstein, in: 15th Symposium (Int) on Combustion, The Combustion Institute, 1975, p. 183. [25] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, second ed., Oxford University Press, London, 1959, pp. 482, 494.
[26] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Tenth printing, Applied Mathematics Series 55, National Bureau of Standards, US Department of Commerce, December 1972.