This is a repository copy of The effects of gas-phase and in-depth radiation absorption on ignition and steady burning rate of PMMA.

Transcription

1 This is a repository copy of The effects of as-phase and in-depth radiation absorption on inition and steady burnin rate of PMMA. White ose esearch Online UL for this paper: Article: Stas, J (214) The effects of as-phase and in-depth radiation absorption on inition and steady burnin rate of PMMA. Combustion and Flame, 161 (12) ISSN This manuscript version is made available under the CC-BY-NC-ND 4. license euse Unless indicated otherwise, fulltext items are protected by copyriht with all rihts reserved. The copyriht exception in section 29 of the Copyriht, Desins and Patents Act 1988 allows the makin of a sinle copy solely for the purpose of non-commercial research or private study within the limits of fair dealin. The publisher or other rihts-holder may allow further reproduction and re-use of this version - refer to the White ose esearch Online record for this item. Where records identify the publisher as the copyriht holder, users can verify any specific terms of use on the publisher s website. Takedown If you consider content in White ose esearch Online to be in breach of UK law, please notify us by in eprints@whiterose.ac.uk includin the UL of the record and the reason for the withdrawal request. eprints@whiterose.ac.uk

2 The Effects of Gas-Phase and In-Depth adiation Absorption on Inition and Steady Burnin ate of PMMA J. E. J. Stas Enery esearch Institute University of Leeds Leeds LS2 9JT United Kindom Keywords: PMMA; in-depth absorption; inition; pyrolysis. Abstract Althouh well studied, there are still interestin features about the inition and steady burnin rate behaviour of PMMA when heated by thermal radiation. In this contribution, the dependence on external heat flux of inition delay time and steady mass flux of PMMA are investiated numerically. Inition is modelled by the critical mass flux criterion. The heat transfer model includes effects of heat lost by out assin, chane of volume durin deradation and absorption of radiation in both condensed and aseous phases. Model results are compared to experimental data for both inition delay time and quasi-steady mass flux across a rane of heat fluxes from 2 to 21 kwm -2 and the importance of both as-phase and condensed-phase radiation absorption effects are discussed. Nomenclature oman Symbols c Specific heat capacity / Jk -1 K -1 D Thermal diffusivity / m 2 s -1 h Convection heat transfer coefficient / Wm -2 K -1 H eaction heat / Jk -1 J Natural lo of pre-exponential factor - 1 -

3 k Arrhenius reaction rate / s -1 l Sample thickness / m m Mass flux / ks -1 m -2 m Mass production rate per unit volume / ks -1 m -3 q Heat flux / Wm -2 s t T y z Location of exposed surface of sample / m Time / s Temperature / K Distance into sample normal to exposed surface / m Distance above sample normal to exposed surface / m Greek Symbols Absorptivity Characteristic thermal penetration depth / m Emissivity Absorption coefficient Thermal conductivity / Wm -1 K -1 Gas-phase absorption parameter / sm 2 k -1 Density / km -3 Stefan-Boltzmann constant / Wm -2 K -4 Important Subscripts a Ambient conditions i A quantity at inition A quantity at the initial location of the exposed surface (y = ) s A quantity at the movin exposed surface (y = s) A steady-state value 1. Introduction For hih heatin rates, heat loss mechanisms are insinificant for short exposures and so one miht expect that inition behaviour of simple solids under such conditions would be dominated by external heat flux. If the often-used enineerin - 2 -

4 simplification of inition at a critical surface temperature is adopted, it follows that the reciprocal of the square root of inition time should be proportional to external heat flux. Interestinly, this turns out not to be the case [1] - [5]. After a thorouh analysis of a one-dimensional mathematical model for inition (includin an alternative criterion for inition) Bal & ein [5] concluded that the observed variation of inition time from the inverse square root correlation at hih external heat flux could not be explained by: o Takin heat losses into account at the exposed surface. o Usin a critical mass flux rather than a critical surface temperature inition criterion. o Usin a more detailed reaction scheme for condensed phase deradation. o Incorporation of temperature dependent thermal properties. They went on to demonstrate that the deviation of inition delay time from the inverse square root correlation could be explained by in-depth absorption of radiation in the condensed phase. Measurements of the transmissivity of black PMMA (Polycast) [4] suest that the absorption coefficient is approximately 96m -1, indicatin a characteristic absorption depth of ~1mm. If the external heat flux is q at the exposed surface, T is temperature and is thermal conductivity then inorin in-depth absorption, the characteristic thermal penetration depth assumin conduction heat transfer is ~ T / q, where T is a physically sinificant temperature interval. For inition it is sensible to take T as the characteristic kinetic temperature rane [6], which corresponds approximately to the temperature rane over which the material pyrolyses at constant heatin rate. Thus it seems reasonable that in-depth absorption of radiation becomes important when 1, i.e. when q T with as above, =.2 Wm -1 K -1 and approximately 3 kwm -2 accounted for at hih radiation heat flux.. For PMMA T ~ 15 K (see Fiure 4 below), T is and so it is likely that in-depth absorption should be - 3 -

5 In-depth absorption of radiation in the condensed phase is only part of the problem, however. Kashiwai s observations of inition of PMMA by CO 2 laser [1] show that there is sinificant absorption of radiation by aseous MMA in the plume above the deradin sample and the role of as-phase absorption in inition and steady burnin rate is unclear. Measurements of quasi-steady mass flux (the approximately steady mass loss rate per unit surface area due to out-assin of volatiles for an initially thick sample, also referred to as burnin rate ) as external heat flux is varied indicate a nonlinear trend at hih heat flux [2] - [4]. This is unexpected from simple ablation theory, which predicts that steady mass flux should be proportional to the ratio of net heat flux at the exposed surface to heat of asification [7] - [11]. If radiation absorption in the condensed phase provides an explanation for the observed variation of inition delay time, then does it also account for the observed trend in mass flux? In this paper, a mathematical model of inition and pyrolysis of PMMA is carefully investiated. Naturally in-depth absorption of radiation is included, but other features not considered in Bal & ein s model [5] are also included such as heat lost by outassin and chane of volume durin deradation. Chane of volume occurs as the sample pyrolyses. It is unlikely that this will have much effect on inition behaviour, especially at hih heat flux, but an early paper by the author [12] demonstrates that it can have a sinificant effect on mass flux at moderate heat flux. Bal & ein accounted for as-phase absorption usin the device of a as-phase absorption coefficient, assumin that a constant fraction of the incident radiation is absorbed by the pyrolyzate above the exposed surface. In this approach a more involved model of as-phase absorption is developed, where the amount of absorption depends on the mass flux of pyrolyzate. We bein with a brief discussion of the expected, order of manitude, effect of condensed phase absorption on inition. We then o on to consider a more detailed model of pyrolysis and inition toether with a simplified model for as phase absorption of radiation. Where possible parameter values for the model are chosen based on either literature values or experimental data and, prior to the main analysis, results are compared with experimental cone calorimeter data for mass flux. The roles of condensed- and as-phase absorption of radiation in inition and steady burnin rate are then investiated numerically

6 2. Inition at a Critical Temperature 2.1 The General Problem The simplest possible model of inition is the critical surface temperature approximation, where inition is assumed to occur as soon as the exposed surface reaches a characteristic temperature T i. Bal & ein [5] concluded that inition behaviour at hih heat flux was not reatly affected by the choice of inition criterion (critical temperature or critical mass flux) and so for convenience we shall adopt the simpler approach for this order of manitude analysis. Coincident with this approximation of inition is the additional assumption that the solid remains inert for temperatures below T i. When in-depth absorption of radiation occurs in a 1-D translucent solid, the total heat flux q consists of two components: q T / y q. Here the first part is conducted heat flux, where y is distance below the exposed surface, and q is transmitted radiation flux. Thus for an inert solid with constant thermal properties, conservation of enery implies that T T c q. (1) t y y Here is density and c is specific heat capacity. Usually, the in-depth radiation term is modelled by Beer s law: q 1 q exp( y) coefficient already defined above.. Here is the absorption At the exposed surface ( y ) of the solid there are three possibilities for the fate of incident radiation: it can be absorbed, reflected or transmitted. For simplicity it shall be assumed that there is no reflected radiation. The unexposed surface ( y l ) is assumed to be well insulated and opaque, so the net heat flux there is zero. Under these conditions, the boundary conditions are - 5 -

7 T q h Ta a T at y, (2) y 4 4 T T T q at y l. (3) y Here is absorptivity, q is incident radiation, h is the convection heat transfer coefficient, T a is ambient temperature, is emissivity and is the Stefan-Boltzmann constant. When Kirchoff's law applies, the emissivity and absorptivity are equal, however there is no uarantee that this is always true. Applyin the boundary conditions at the exposed and unexposed surfaces, it follows that the steady-state temperature distribution T (y) is iven by y 1 T ( y) T q dy, (4) where. 4 4 T (the temperature at y = ) satisfies q ht T T T a a The solution of the transient inert problem presents no particular difficulties. Delichatsios and Zhan present an approximate solution in [13]. However for what follows below it is worth considerin the hih heat flux limit in some detail. 2.2 Hih Heat Flux Limit Here we note that if the external heat flux is sufficiently hih, so that q h T 4 4 T T T i a i a, where T i is the temperature at inition, then heat losses are neliible 1 up to inition. This is illustrated in Fiure 1, where the reions 1 Note that this is equivalent to the external heat flux bein very much reater than the critical heat flux (see next section)

8 above the curves correspond to hih external heat fluxes for different values of the 3 radiation parameter T i / h T i / h 1 3 T i / h T i / h.5 q ht i 1. 3 T i / h.25 3 T i / h T a / T i Fiure 1. Hih heat flux limits. When this is the case, settin T T T 1 a y q q y y 1 t 1 q dy, (5) lc 2l and substitutin this expression into (1) it follows that T 1 must satisfy the standard diffusion equation 2 2 T1 / t D T1 / y, where D / c is the thermal diffusivity. Furthermore, application of the boundary conditions (2) and (3) without surface heat losses implies that T / y at both boundin surfaces. Since the initial 1 temperature is assumed to be ambient throuhout, the initial temperature distribution for T 1 must be - 7 -

9 y q y y 1 T y q dy 1(, ) 1. (6) 2l If Beer's law is assumed, it is immediately apparent from Eqn. (5) that for inition at a critical temperature T i, the inition time t i will be of the form t i l D Ti f Ta lq T 2, a, l,. (7) Alternatively, in the hih heat flux limit (HHFL) for a thick solid, the temperature profile may be approximated by T ( t) Ta T ( t) T ~ Ta, y ( t), y/ ( t), y ( t), (8) where T ( ) is aain the temperature at the exposed surface y = and (t) is the t thermal penetration depth (now a function of time). Interatin the temperature equation (1) over the thermal penetration depth, we find that c T / t dy q 1 1 e. Now usin the facts that d dt Tdy T d / dt T / t dy follows that a, Tdy T ~ 2 T a and d / dt d / dt d / d it c d T 2 dt dt d s Ta ~ q 1 1 e. (9) The boundary condition on the exposed surface means that T T / and so a q q T i T a /, it when the external heat flux is sufficiently hih, such that 2 follows that will always be small. Therefore d / dt ~ 2D, in the HHFL and so ~ 2Dt. Hence inition time t i and inition temperature will be approximately - 8 -

10 related by T i q 2Dt / and consequently in terms of the parameter ~ Ta i roupins iven in Eqn. (7) above, t i 2 l 1 T ~ D T i a 1 2 Ta. lq 2. (1) In particular, it follows that t i q 2D ~ T T 1 i a (11) and so we see that 1/ ti will reduce in direct proportion to. 3. Inition at a Critical Mass Flux: eactive Solid 3.1 Enery Equation Consider for simplicity a solid that on heatin underoes a sinle-step first-order deradation reaction of the form dm / dt k( T) m, where m is the ratio of sample A mass to initial mass, k( T) exp( J T / T), J is the natural loarithm of the preexponential factor and T A is the activation temperature. Bal & ein [5] showed that inition behaviour was not reatly affected by inclusion of a more sophisticated pyrolysis model, so only this simple first-order sinle-step approach will be investiated. Also, as above, heat transfer in one direction only (normal to the exposed surface) is modelled. Now consider a thick sample of solid where temperature radients are not neliible. As the solid is heated as is enerated, which we assume escapes as soon as it forms (so that the local density remains constant). This implies that there will be a reduction in sample volume as asification proceeds (see Fiure 2)

11 Total mass of as liberated from entire solid per unit surface area = ds y = s(t) y = s(t + t) ds Control Volume m ( y, t) m ( y y, t) y y + y Sample at time t Sample at time t + t Fiure 2. Definition of control volume. The deradation reaction implies that the eneration rate of as per unit sample volume will be iven by m k. Let the location of the exposed surface be iven by y s(t), where s ( ). Now consider a control volume fixed in space, located between y and y y, and let m ( y, t) denote the mass flux of solid that flows past point y at time t due to the reduction of volume. Conservation of mass then implies that m / y m and interatin this, we find that the local mass flux is iven by l m kdy. Since the remainin solid mass per y unit area is s, it follows that the location of the exposed surface will be iven by the solution of the intero-differential equation ds dt l s kdy, s ( ). (12) Applyin similar considerations to conservation of enery, it is possible to show that - 1 -

12 ct m ct q m c T H t y, (13) where, as above, the total heat flux q T / y q, H is the heat of the asification reaction and m c T is the rate of outflow of heat per unit sample volume due to out-assin, c bein the specific heat capacity of the as. Usin the expressions for m and convenient form m, the enery equation may be rewritten in more l T c ct H q T T c c kdy k. (14) t y y y y Note that this equation must be solved throuhout the volume of remainin solid s( t) y l. 3.2 Gas-Phase Attenuation of Incident adiation It is possible that radiation incident on the exposed surface may be attenuated by asphase absorption above the sample. Beaulieu [2], [3], was unable to estimate attenuation for a pyrolysin PMMA sample but instead used a propylene burner with an embedded heat flux aue. She first measured the heat flux from the propylene flame only and then varied the output from the apparatus' external source and noted the new heat flux readins. She concluded that there was no attenuation by either the flame or the propylene as. Earlier, Kashiwai [1] conducted an eleant series of experiments on PMMA (and red oak) usin a CO 2 laser as the radiation source. In his experiments, a small hole was cut in the samples and a heat flux aue was placed directly beneath them so that he could measure attenuation throuh the pyrolyzate prior to inition. He concluded that there was sinificant attenuation prior to inition and that the ratio of incident heat flux to initial heat flux, i.e. 1 - attenuation, decreased approximately exponentially with time. Note that there is no contradiction

13 between these observations: Kashiwai measured attenuation of the radiation by aseous MMA and Beaulieu measured combined attenuation by propylene and a flame. There is no reason why these effects should be the same. To estimate the effect of as-phase attenuation, consider a column of pyrolyzate above the sample surface bein advected upwards with speed v. In an instrument such as an oxyen consumption calorimeter, the mass flow rate of air is always much reater than mass flow rate of pyrolyzate, so to leadin order we may assume that v is constant. Furthermore, consider a layer at heiht z above the sample surface at time t. The layer has taken time z/ v to reach heiht z, and so the mass flux of pyrolyzate at the sample surface when the layer first formed was m t z/ v be no mixin between layers in the column, in time step. Assumin there to t it follows that the thickness of the layer will be z vt and the mass of pyrolyzate in the layer will be m Am t z/ vt, where A c is the cross-sectional area of the column. It is p c reasonable to suppose that attenuation of radiation throuh the layer will be directly proportional to the mass of pyrolyzate in the layer, since each particle has the potential to absorb radiation. q ( z, t) q ( z z, t) / q ( z, t) m ( t z/ v) t i i i In other words, we would expect that, where q i ( z, t) is the manitude of radiation at heiht z and time t. It therefore follows that q i ( z, t) satisfies q i q im t z/ v, (15) z v where is a constant of proportionality with dimensions m 2 k -1 that will depend on the absorbin properties of the pyrolyzate. Now, if the heiht of the column of pyrolyzate is z max and the radiation at z = z max has manitude q ext, then the manitude at the sample surface will be q s, where zmax / v q s q ext exp m t d. (16)

14 Physically z max / v is the time taken for the pyrolyzate to advect throuh the column and if this is small compared to the timescale on which out-assin from the sample occurs, then q q s ext t exp m, (17) where z max / v is an alternative constant with dimensions of sm 2 k -1, which we shall refer to as the as-phase absorption parameter. Physically, if one were able to measure the radiation flux at the exposed surface durin pyrolysis, then would correspond to the radient of the raph of q / escapin pyrolyzate. ln plotted aainst the mass flux of ext q s It must be appreciated that the model set out above will be a ross oversimplification of the actual physical situation and serves only as a first-order approximation to quantify the effect of as phase absorption. The flow field above the sample surface durin out-assin of volatile species will be complex and it is likely that this will have a lare effect of the local density of pyrolyzate and hence attenuation. Furthermore, it should be noted that the nature of the experimental setup, includin its physical dimensions, the wavelenth of incident radiation and the total flow rate of ases, will also affect the value of. 3.3 Post Inition Behaviour and Numerical Solution In order to model post-inition behaviour, an inition criterion is required toether with an additional term in the boundary condition at the exposed surface to account for heat flux from a flame. Here the device of critical mass flux [14] is used to determine inition and followin earlier observations [9], [1] it is assumed that the heat flux from a flame is constant. From the discussion above, we take T y 4 4 m t ht T T T q exp. (18) ext a a Here

15 q, m m in, q ext (19) q, q f m m in, at the exposed surface, where m in is the critical mass flux at inition and f q is the flame heat flux. We shall assume that Beer's law applies and so the in-depth radiation flux is now 1 q exp y s q. ext Due to the nonlinearity of the temperature equation, recourse to numerical methods is inevitable. The method adopted here is a fully-implicit finite difference scheme with a non-uniform rid. In the numerical solution, the node density is reatest at the exposed surface so that temperature variation and the in-depth radiation term q are adequately resolved 2. The node density used in the calculations is shown in Fiure 3. 1 Number of Nodes per mm y / l Fiure 3. Node density (in number of nodes per mm) used in numerical calculations. 2 The in-depth radiation flux at the exposed surface varies on a lenth scale of 1/, so the node spacin y must be sufficiently small so that y << 1/, or equivalently the node density 1/y >>

16 4. Parameter Values For convenience it will be assumed that Kirchoff's law is valid, i.e.. The model parameters used in the numerical simulations, with the exception of the asphase absorption parameter, are iven in Table 1 and are justified below

17 Parameter Symbol Value Ambient temperature T a 298 K In-depth radiation lenth scale Critical mass flux for inition Convection heat transfer coefficient Absorptivity Flame heat flux 1 / 1 mm m 2.5 s -1 m -2 in h 25 Wm -2 K q 23 kwm -2 f Thermal conductivity.2 Wm -1 K -1 Specific heat capacity (solid) Specific heat capacity (as) c 1466 Jk -1 K -1 c 196 Jk -1 K -1 Density 12 km -3 Activation temperature Natural lo of preexponential factor Heat of reaction (asification) T A K J 2.76 ln(s -1 ) H.84 MJk -1 Table 1. Model parameter values. The in-depth lenth scale for radiation, thermal conductivity, critical inition mass flux and density were taken in-line with other values used in the literature [5], [14]. The convection heat transfer coefficient was taken as a representative value from a recent detailed experimental investiation [15]. The flame heat flux was taken from other studies [9], [1] and the heat of the asification reaction was calculated from averae bond eneries [16]. The specific heat capacity of the solid was taken as a

18 representative value for PMMA and the specific heat capacity of the as as a representative value for MMA (both obtained from WWW-based data tables). The Arrhenius parameters A and T A for the reaction rate are obtained from constant heatin rate thermoravimetric (TG) experiments for standard PMMA samples in nitroen, previously published in an earlier study [17]. It was felt important to estimate these parameters from an experiment in an inert atmosphere since the oxyen concentration at the exposed surface is likely to be sinificantly below normal atmospheric levels due to out-assin of MMA. There is a problem with usin TG data at standard heatin rates such as 1-5 K/min in that in practice, the heatin rates at the exposed surface of the sample will reatly exceed these values. For example, at 5 KWm -2, the inition time of black PMMA is ~3 s, indicatin a heatin rate at the exposed surface of ~6 K/min. At 1 kwm -2, the correspondin heatin rate is ~18 K/min. Without specialised equipment it is impossible to investiate thermal deradation at such hih heatin rates and so data from our analyser's maximum heatin rate of 5 K/min was used. Naturally, if one were concerned with a detailed examination of the kinetic mechanism then this heatin rate would be too hih. However, in this application heatin rates are so hih that it is likely that many of the individual nuances of the deradation mechanism will be smeared toether. This perhaps may account for the reason why more detailed kinetic mechanisms do not explain the observed inition trend at hih heat flux as noted by Bal & ein [5]. In order to obtain estimates of the Arrhenius parameters, model fits were compared to both experimental mass loss and mass derivative curves. esiduals for the fits to each curve were calculated and the sum of residuals was minimised in order to obtain the best parameter estimates. The resultin best fits are shown in Fiure

19 1..2 Mass / Initial Mass Experimental Mass Fraction Fit to Mass Fraction Experimental Mass Derivative Fit to Mass Derivative.1. Mass Fraction Derivative / K Temperature / K Fiure 4. Constant heatin rate TG data and optimal fits for PMMA at 5 K/min. The absorptivity and absorption coefficient were estimated by comparin model predictions for mass flux to experimental cone calorimeter results at an external heat flux of 5 kwm -2, usin a PMMA sample of initial thickness 12 mm. Fiure 5 shows a contour plot of the residual, defined by t t stop 1 esidual = m m stop 1/ 2 2 exp model dt, (2) where is the experimental mass flux from the cone calorimeter and m model is the m exp predicted mass flux from the model

20 5 esidual 4. Absorption Lenth (1 / ) / mm Absorptivity 18. Fiure 5. Contour plot of mass flux residual. It is apparent from Fiure 5 that the residual has an approximately flat basin (shown by the dashed curve) in the neihbourhood of the minimum. This unfortunately means that there is no clearly defined optimum and any values within the dashed reion will produce ood fits to the experimental data. Therefore, broadly in line with other observations [4], [5] it was decided to take 1 / as 1 mm and seek the optimum that minimised the residual. This process ives the best value of as.56, as Fiure 6 confirms. Fiure 7 shows the optimum fit toether with the experimental data for comparison

21 11 1 / = 1 mm 1 9 esidual / s -1 m Optimum Fiure 6. Optimum for 1 / = 1 mm

22 4 Model with =.56 3 Mass Flux / s -1 m Cone Calorimeter Data at 5 kwm Time / s Fiure 7. Comparison of model with cone calorimeter results for standard parameter values shown in Table 1. Unfortunately there is a lack of information to quantify the as-phase absorption parameter. Kashiwai s earlier work [1] suests that there was sinificant as-phase absorption in his experiments. However, the apparatus used in his work and the spectral characteristics of the heat source are sinificantly different to an oxyen consumption calorimeter and the discussion in 3.2 suests that there is every reason to suppose that the as phase absorption parameter is sensitively dependent on experimental apparatus. Given that inition occurs at a relatively low mass flux, in a well-ventilated experiment such as a cone calorimeter it is reasonable to suppose that as-phase absorption plays little part in inition. However, in the quasi-steady reion where mass flux is sinificantly hiher than at inition, it is possible that as-phase absorption is important

23 5. esults Beaulieu [2], [3], presented empirical inition results that are representative of a broader super-set discussed by Bal & ein [5]. Beaulieu's results were obtained usin 25 mm thick specimens in the AFM apparatus where an infra-red lamp is used as the heat source. Neither Beaulieu or Bal & ein make mention of the earlier work of Kashiwai [1], where a CO 2 laser was the primary heat source used and delivered similar levels of heat flux at the exposed surface. Fiure 8 shows results from the model with the parameter set iven in Table 1 with the experimental data of Beaulieu [2], [3], Kashiwai [1] and the coated PMMA data of Jian et al. [4]. Also shown in the fiure are model results obtained with slihtly different values for and, correspondin to the upper reion of the dashed area shown in Fiure 5. As suested above, it was found that the as-phase absorption parameter had little discernible effect on inition behaviour for values in the rane 1 m 2 sk -1 and so no results are explicitly included in Fiure 8. 1 t i t i Model: = External Heat Flux / kwm -2 Model: Standard Parameter Set Beaulieu Kashiwai Jian et al. Model: =.52, 1/ mm

24 Fiure 8. Comparison between model and experimental data for inition time (in seconds). The obvious point to note is the model confirms that in-depth absorption plays an important role in inition at hih heat flux. Also, the scalin iven by Eqn. (11) is preserved, i.e. 1 / ti reduces approximately in direct proportion to at hih heat flux. Naturally, an optimum choice for could be found by computin the best fit of model predictions directly with the experimental data in Fiure 8. However, the author felt that a stroner case for the validity of the model could be promoted if a value for was found at a sinle heat flux and then used for model predictions across the heat flux spectrum. The raph in Fiure 9 shows the effect of varyin the as-phase absorption parameter without inition when the external heat flux is set to 1 kwm -2. The dependent axis corresponds to the ratio of heat flux at the sample surface to q ext, i.e. 1 attenuation, and the dependent axis is time. The raph in Fiure 1 shows the same heat flux ratio, but with fixed at 5. m 2 sk -1 for the three heat fluxes that Kashiwai used in his experiments [1]. Althouh the manitude of attenuation at this value of is much less than Kashiwai reported, the qualitative behaviour is strikinly similar

25 Heat Flux at Exposed Surface / Initial Value m 2 sk m 2 sk m 2 sk m 2 sk m 2 sk Time / s Fiure 9. Effect of as-phase absorption parameter on attenuation Heat Flux at Exposed Surface / Initial Value kwm kwm kwm Time / s Fiure 1. Effect of external heat flux on attenuation. Beaulieu [3] reported interestin behaviour for the steady-state sample mass flux at hih external heat flux. As heat flux increased, her measurements showed a nonlinear

26 variation in steady mass flux with external heat flux at odds with the prediction from simple ablation theory that steady mass flux is proportional to the ratio of external heat flux to heat of asification. esults from the model proposed here suest that this variation can be explained, at least in part, by as-phase absorption of radiation. The raph in Fiure 11 shows Beaulieu s data for steady mass flux toether with model predictions for various values of the as-phase absorption parameter. We note that for external heat fluxes below 7 kwm -2, the empirical values aree with model predictions for zero as-phase absorption ( = ). However for heat fluxes above this value areement only improves when is increased. Analysis of the total sum-ofsquares residual 3 (Fiure 12) indicates that ~ 6. 1m 2 sk -1 provides the best overall fit. 7 = m 2 sk -1 = 2.5 m 2 sk -1 = 5. m 2 sk -1 6 = 7.5 m 2 sk -1 = 1. m 2 sk -1 5 Mass Flux / s -1 m Beaulieu Model External Heat Flux / kwm -2 Fiure 11. Comparison between model results for various values of the as-phase absorption parameter and Beaulieu's AFM data for mass flux [3]. 3 Defined analaously to Eqn. (2)

27 25 2 esidual / s -1 m y =.4631x x / m 2 sk -1 Fiure 12. Goodness of fit of numerical results to observed mass flux. Of course, a major assumption in the model is that flame heat flux is invariant to chanes in external heat flux. In all likelihood this assumption is faulty, but is probably not responsible for the observed behaviour for the followin reason. As external heat flux increases mass flux will also increase, which implies that if combustion conditions remain broadly similar in the flame, then more heat should be released. In other words, if anythin we should expect the flame heat flux to increase with external heat flux, which would have the effect of increasin mass flux above the expected value. In reality, as Fiure 11 shows, the observed mass flux increases with external heat flux more slowly than expected. It is also unlikely that the kinetic mechanism can explain the deviation as increasin external heat flux serves only to increase the temperature in the deradin solid. Consequently since most deradation reaction rates are known to increase with temperature, this is also unlikely to be the cause

28 6. Discussion and Conclusion The analysis above lends further support to the suestion that in-depth absorption of radiation plays an increasinly important role in the inition behaviour of PMMA as external heat flux increases. The effect of in-depth absorption is to extend the inition delay time above what miht be expected when it is inored. Furthermore, it is also likely that the quasi-steady mass flux (or burnin rate) is stronly influenced by absorption of external radiation by the pyrolyzate in the as phase above the sample at hih external heat flux. Experimental results from external heatin by CO 2 laser confirm that absorption by the pyrolyzate is sinificant (althouh there are no simultaneous measurements of mass flux). Experiments usin cone calorimeter type apparatus show that burnin rate is lower than expected at hih heat flux (althouh there are no simultaneous measurements of as-phase attenuation). A model of asphase absorption was developed that predicted radiation attenuation to vary exponentially with mass flux. esults from the model suest that as phase absorption has little effect on inition, but supports the hypothesis that there may be a sinificant effect on mass flux. Unfortunately it is not yet possible to make a firm statement about this effect due to the lack of experimental data correlatin attenuation with mass flux. A further complication that is difficult to address fully is the amount of enery that is actually absorbed by a solid or as from a radiant heat source. The quantity of enery absorbed is stronly dictated by the wavelenth of the incident radiation. The solid or as absorbs radiation mainly in discrete bands of the infra-red spectrum, implyin that the absorption coefficients will depend on wavelenth. In other words, if the wavelenth of the incident radiation chanes, then the relative amount of enery absorbed is likely to chane. It is for this reason that Kashiwai chose to use a CO 2 laser for his study [1] - it is, to a ood deree of approximation, essentially monochromatic. The emmision spectra of other heat sources used in the inition studies cited in this paper will be different, and may well vary with heat flux, and this miht play a sinificant part in some of the discrepancies noted above in empirical observations. In particular, the observation reported by Beaulieu [2] that there

29 appeared to be little as-phase absorption of radiation for propene is at odds with Kashiwai's observations of attenuation by PMMA pyrolyzate usin the CO 2 laser and wavelenth dependence of absorption could well be at the heart of this inconsistency. The model used in the analysis above makes no allowance for wavelenth dependent effects and it may be for this reason that the areement between numerical results and inition delay time is better for the monochromatic CO 2 laser source than the heat source used in the AFM apparatus used by Beaulieu. Another consideration to keep in mind when weihin the results reported here is the relative importance of the thermal deradation mechanism of the solid fuel. PMMA is typically chosen for study because it underoes depolymerisation on heatin. As soon as radical framents form by random cleavae, they rapidly unzip to produce MMA monomer. If the unzippin process is much faster than the random cleavae process (which is typically the case) then the rate-determinin step for monomer production will be determined by the rate of random cleavae. A more detailed study of modellin the deradation mechanism of PMMA was undertaken by the author usin population balance methods [17]. In that work the author was able to demonstrate that for an initial population of molecules with hih deree of polymerisation, when unzippin is much faster than initiation by random scission, it is possible to show that the production rate of monomer is well modelled by a process that is essentially similar to a one-step Arrhenius process. The model above uses a simple one-step deradation process, which is a ross oversimplification of the true mechanism, but at hih heatin rates may not be too limitin for PMMA. The validity of this simplification for other polymers is hihly dubious however

30 eferences [1] Kashiwai T. Experimental observation of radiative inition mechanisms. Combustion & Flame 1979; 34: [2] Beaulieu P. A. Flammability Characteristics at Heat Flux Levels up to 2 kw/m 2 and the Effect of Oxyen on Flame Heat Flux, PhD Dissertation, Worcester Polytechnic Institute, 25. < (accessed 3rd July 212). [3] Beaulieu P. A., Dembsey N. Flammability characteristics at applied heat flux levels up to 2 kw/m 2. Fire & Materials 28; 32: [4] Fenhui Jian, de is J. L. & Khan M. M. Absorption of thermal enery in PMMA by in-depth radiation. Fire Safety Journal 29; 44: [5] Bal N., ein G. Numerical investiation of the inition delay time of a translucent solid at hih radiant heat fluxes. Combustion & Flame 211; 158: [6] Stas J. E. J. Thermal deradation of solids usin sinle-step first-order kinetics. Fire Safety Journal 1999; 32: [7] Kindelan M. & Williams F. A. Theory for endothermic asification of a solid by a constant enery flux. Combustion Science and Technoloy 1975; 1: [8] Tewarson A. & Pion. F. Flammability of plastics I- Burnin intensity. Combustion & Flame 1977; 26: [9] B. T. hodes & J. G. Quintiere. Burnin rate and flame heat flux for PMMA in a cone calorimeter. Fire Safety Journal 1996; 26: [1] Hopkins Jr. & J. G. Quintiere. Material fire properties and predictions for thermoplastics. Fire Safety Journal 1996; 26: [11] Stas J. E. J. A Theory For Quasi-Steady Sinle-Step Thermal Deradation of Polymers. Fire and Materials 1998; 22: [12] Stas J. E. J. A theoretical investiation into modellin thermal deradation of solids incorporatin finite-rate kinetics. Comb. Sci. & Tech. 1997; 123: 261 [13] Delichatsios M. A., Zhan J. An alternative way for the inition times for solids with radiation absorption in-depth by simple asymptotic solutions. Fire & Materials 212; 36:

31 [14] Stas J. E. J., Nelson M. I. A critical mass flux model for the flammability of thermoplastics. Combustion Theory & Modellin 21; 5: [15] Stas J. E. J. A eappraisal of Convection Heat Transfer in the Cone Calorimeter. Fire Safety Journal 211; 46: [16] Stas J. E. J. The heat of asification of polymers. Fire Safety Journal 24; 39: [17] Stas J. E. J. Population balance models for the thermal deradation of PMMA. Polymer 27; 48: