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Modelling the effect of solid-phase additives on thermal degradation of solids
Polymer Degradation and Stability 64 (1999) 369±377
Modelling the eect of solid-phase additives on thermal degradation of solids J.E.J. Staggs1 Department of Fuel and Energy, The University of Leeds, Leeds LS2 9JT, UK Accepted 15 March 1998
Abstract We model the eect of a simple solid-phase additive on the thermal degradation of a solid under thermally thick and thermally thin conditions. The mathematical model simulates conditions in bench-scale tests (such as the cone calorimeter test) where a solid sample of typical cross-sectional area 100 cm2 is exposed to a uniform external heat ¯ux and the mass loss rate and eective heat of combustion are determined. We choose a simple model for the decomposition: the additive and solid decompose independently according to ®rst-order kinetics. We assume for simplicity that the thermal conductivity, density and speci®c heat capacity of the two solid phases are identical and hence the discussion centres on the role of the degradation mechanisms of the components in relation to the overall degradation of the solid. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The ®re retardant action of additives which primarily act as heat sinks, such as aluminium oxide trihydrate (ATH) or magnesium hydroxide, involves a number of processes. The endothermic degradation of these ®llers cools surrounding material; the thermal and optical properties of the ®ller may be useful in preventing heat transfer from the exposed surface to subsurface regions as a solid residue builds up at the surface; water vapour liberated as a result of the decomposition of the ®ller (ATH is 35% by weight water) may dilute the ¯ammable vapours produced by the polymer. A striking feature of mass loss rate data for ®lled polymers from cone calorimeter experiments is the presence of (at least) two peaks as shown in Fig. 1. Here we see mass loss rate plotted against time for ethylene-vinyl acetate ®lled with ATH. The heat ¯ux from the cone heater was set to 40 kWmÿ2 and the initial thickness of the test samples was 10 mm. The curves are labelled according to the initial mass fraction of ®ller present in the material. Note that for moderate ®ller loadings (<40% by weight) there are two peaks in the mass loss rate. As the ®ller loading increases, there is a tendency for the second peak to be suppressed, although vestiges remain even for the case of 60% ®ller. 1
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This paper investigates the thermal degradation of an idealised solid comprising a combustible polymer and an inert ®ller. The solid is viewed as a homogeneous mix of ®ller and polymer and we assume that the ®ller and polymer degrade independently of each other according to single-step ®rst-order kinetics. The present work extends earlier theoretical work on the eect of additives in polymer combustion, notably that of Nelson et al. [1], where ignition of ®lled polymers was considered in the thermally thin regime. As in that work, we consider a sample of material degrading under a constant external heat ¯ux, as in the cone calorimeter test [2±5]. The mathematical model of pyrolysis considered here is based on a previous model [6] which includes a transport mechanism in subsurface regions of the solid which accounts for movement of surrounding material as gaps produced by volatilisation are ®lled. Imagine that we are able to label a particular element of material in the solid, and that we can track its progress as degradation occurs. Let the labelled material element be located a distance y below the initial location of the top surface. Let the initial thickness of material be l. In a given time step, some material in the region below the labelled element will volatilise resulting in a net reduction of thickness. We would also observe that the material element at y would itself reduce in thickness, but would also move to a lower location, as material instantaneously ®lls the gaps left by escaping volatiles.
0141-3910/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0141 -3 910(98)00155 -4
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J.E.J. Staggs / Polymer Degradation and Stability 64 (1999) 369±377
Fig. 1. Experimental mass loss rate data for black EVA ®lled with ATH.
P2) degrading according to a single-step ®rst-order process. We characterise the TG curve in terms of a characteristic temperature (CT) and a characteristic temperature range (CTR). The CT is de®ned as the temperature at which the mass fraction equals a ®xed value c, say. Thus the CT at the ®xed mass fraction c where 0 < c < 1 will be denoted by T(c). The CTR is de®ned as follows. Imagine a tangent drawn at the point (T(c), c) on the TG curve, produced so that it meets the T axis at Ta and the line m=1 at Tb. The temperature dierence Ta ÿ Tb is de®ned as the CTR and will be denoted by T(c). For convenience, we will refer to dimensionless versions of these quantities, de®ned by CT ÿ T0 ; T0 CTR dimensionless CTR ; T0 dimensionless CT
In fact, if we track a material element that starts at the top surface, we would be able to follow its progress until ®nally some small portion of it arrives at y=l. This approach is related to that of other authors who have considered in-depth volatilisation using ®nite rate kinetics but have neglected to consider movement of surrounding material [7±12]. 2. Model derivation We consider the solid to be a homogeneous mix of polymer and ®ller, where the two materials are labelled P1 and P2. In order to keep the discussion as general as possible, we do not at this stage specify which of P1 and P2 is the polymer and which is the ®ller. For simplicity we assume that P1 and P2 have the same thermal properties, but have dierent decomposition kinetics and latent heats of vaporisation. Let m1 denote the mass of P1 in a small amount of the mixture and m2 the mass of P2. We assume that P1 and P2 degrade independently of each other via a single step process which is ®rst-order for each species. Hence for an elemental slice of material in a TG experiment, the kinetic equations are ÿdm1 =dt k1 m1 ; ÿdm2 =dt k2 m2 , where k1 A1 exp ÿT1 =T; k2 A2 exp ÿT2 =T. Here t denotes time, A1,2 are pre-exponential factors, T1,2 are activation temperatures and T is temperature. The kinetic equations are characterised by four parameters: two pre-exponential factors and two activation temperatures. Whereas this characterisation is convenient for writing down the equations, it is not particularly useful from a practical standpoint. Some authors [1,13] have adopted alternative descriptions which make use of a characteristic temperature which can be easily related to a constant heating rate TG curve (the temperature of the in¯exion point, or the temperature at a ®xed value of the mass fraction). In this work we adopt an approach that will be familiar to TG experimentalists and so only an outline will be given. Consider a single component of the solid (either P1 or
1
where T0 is ambient temperature. It can be shown that the pre-exponential factor and activation temperature can be easily calculated from the CT and the CTR and the reader is referred to a previous paper [13] for details. The main advantage of the CT±CTR characterisation of decomposition kinetics is the ease with which the two parameters can be determined in practice from a standard constant heating rate TG curve: one simply has to read o the temperature at which the mass fraction is c and the gradient at that point. Furthermore, as we shall see below, the important parameter as far as this work is concerned is the CT ± a direct measure of the decomposition characteristics of the polymer ± which can be easily related to the TG curve (unlike A and TA, where one has to solve the kinetic equation in order to obtain the shape of the curve). Therefore the CT±CTR characterisation also aids in the interpretation of numerical results. Thus, given constant heating rate TG curves for P1 and P2 (labelled C1 and C2, say), a solid containing a mixture of P1 and P2 where the initial mass fraction of P1 in the solid is l0, will have TG curve C=l0C1 + (1 ÿ l0)C2, assuming no interaction between P1 and P2. The shape of C will depend on the CTs and CTRs. If the CTs are markedly dierent and the CTRs are relatively small, C will have two discernible steps. If the CTs are similar, then it may be dicult to distinguish C from a single-step TG curve. For convenience, we recast the kinetic equations in terms of the total mass m=m1 + m2 and the mass-fraction scalar l=m1/m, viz. dl dm l 1 ÿ l k2 ÿ k1 ; ÿmfk1 l k2 1 ÿ lg: dt dt
2
In order to investigate the mass loss behaviour of a larger sample of solid, where temperature gradients may not be neglected, we must link the kinetics to a suitable heat transfer model. We shall speci®cally consider the
J.E.J. Staggs / Polymer Degradation and Stability 64 (1999) 369±377
situation where a horizontal sample is exposed to a constant external heat ¯ux through its top surface. We shall assume that the lower surface of the sample is insulated. Following the approach taken in an earlier paper [6], we adopt a model where volatiles escape from the solid as soon as they are formed and surrounding material instantaneously ®lls in the gaps left by the escaping gases. Under these conditions, the energy equation in the solid takes the form c
DT @2 T 000 k 2 Q_ v ; Dt @y
3
for s(t)4y4l, where l denotes the initial thickness of the solid, y=s(t) denotes the instantaneous location of the top (exposed) surface, rc is the volumetric heat capacity of the solid, k is the thermal conductivity and y an ordinate measuring distance from the initial location of 000 the top surface of the solid. The symbol Q_ v denotes a source term to account for the latent heat of vaporisation of the solid and D/Dt=@/@t + v@/@y is the convective derivative, where v(y,t) is the velocity ®eld imposed by the movement of material as it ®lls in gaps left by escaping volatiles. We apply the kinetic equation for l in a similar way throughout the degrading solid, giving Dl l 1 ÿ l k2 ÿ k1 : Dt
4
Given the kinetic mechanism de®ned above, and following the approach of previous work [6], the source term takes the form 000 Q_ v ÿL1 fk1 l Lk2 1 ÿ lg;
5
where L=L2/L1 and L1,2 are the latent heats of vaporisation of P1 and P2. Furthermore, the velocity ®eld is found by summing the movements of elemental slices of material below a given point y in the solid and hence takes the form l v fk1 l k2 1 ÿ lg dy:
6
y
The energy equation (3) and the mass-fraction equation (4) are combined with the boundary conditions ÿk
@T "q h T0 ÿ T " T40 ÿ T4 @y
@T 0 @y
on y s t; 7
on y l;
8
and the initial conditions T y; 0 T0 ;
l y; 0 l0 y;
371
initial condition) so that we may investigate the eect of ®ller distribution on the mass loss rate (MLR). The other variables are de®ned as follows: " is the emissivity of the top surface, q is the external heat ¯ux, h is the convective heat transfer coecient and is the Stefan± Boltzmann constant. Eq. (3)Eq. (4) and boundary conditions (7) and (8) necessitate a numerical solution and this may be achieved using standard ®nite-dierence techniques [6]. The equations are solved in the dimensionless forms 81 9 1 < = @ @ 1 @2 Da Da E1 dx ÿ 1 ÿ x E1 dx 2 2ÿ E; : ; @x @x @ St x
81 <
0
9 = @l @l Da E1 dx ÿ 1 ÿ x E1 dx : ; @x @ x 0 2 1 ÿ exp ÿ ; Dal 1 ÿ l A exp ÿ 1 1
10
1
11 ÿ
@ @ ÿ on x 0; 0 @x @x
on x 1;
12
where 1 2 A 1 ÿ l exp ÿ ; E1 ; l l exp ÿ 1 1 13 1 2 LA 1 ÿ l exp ÿ ; E ; l l exp ÿ 1 1 14 15 ÿ ÿ Bi ÿ R 14 ÿ 1 and the other variables are de®ned as T ÿ T0 l2 A 1 T1;2 cT0 ; 1;2 ; Da ; St ; T0 T0 L1 "T30 l A2 "ql hl ; ; ; Bi ; R A kT0 k A1 k s yÿs t ; 2 : 1 ÿ ;x l lÿs l
16
In the new variables, the total dimensionless mass loss rate is given by ÿd/d =Da 01E1 dx. 3. Results and discussion
9
where l0(y) is the initial distribution of the mass fraction of P1 in the solid. Note that we shall consider cases where l0 is not constant (hence the generality in the
Unless otherwise stated, the results below were computed with the following dimensionless parameter values: St=1, L=1, =8, Bi=0.33, R=0.035. For a typical polymer which is initially 1 cm thick, with k=0.3
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Wmÿ1Kÿ1, =1000 kgmÿ3, c=3000 Jkgÿ1Kÿ1, so that =10ÿ7 m2sÿ1, =1 corresponds to 1000 s, =8 corresponds to an external heat ¯ux of 72.8 kWmÿ2, Bi=0.33 corresponds to a convective heat transfer coecient of approximately 10 Wmÿ2Kÿ1, R=0.035 corresponds to an emissivity of 0.9 and St=1 corresponds to a latent heat of vaporisation of 0.82 kJgÿ1. The results will be presented mainly in terms of the dimensionless MLR ÿd/d versus dimensionless time . Before we progress to a systematic investigation of the mathematical model outlined above, we ®rst attempt some initial calculations for the case of a diluent ®ller which does not degrade. The goal here is to crudely approximate as best we can, within the limited theoretical framework that our model assumptions provide, conditions similar to those of the experimental results shown in Fig. 1. For the case of EVA ®lled with ATH. Whiteley and Elliot [14] comment that as the ®ller degrades, water is lost leaving a microporous residue of alumina. Of course, we do not model this situation exactly here: . our ®ller either degrades completely without leaving a residue or does not degrade at all, . the thermal properties of the ®ller and polymer are identical, . EVA is a copolymer and it is doubtful that the single-step kinetics assumed in the present work will be adequate to describe the decomposition, . volatiles escape unimpeded and unchanged from the solid as soon as they are formed and do not exchange heat with surrounding material, but it might be expected that similar results may be obtained for the case where the water content of the ATH is low. For the case of a non-degrading ®ller, A is set to 0 in the model equations (13) and (14). Fig. 2 shows the eect of increasing the inert ®ller concentration on the MLR for =5, Bi=0.5, R=0.05, CT1=1.02, CTR1=0.2. The curves are labelled according to ®ller loading. Note the interesting feature of a double peak in the MLR and the fact that the relative size of the second peak reduces with increasing ®ller loading ± in agreement with the experimental data
We ®rst investigate the MLR behaviour as the CT of P2 is varied. The CT of P1 is ®xed at 0.75 (corresponding to a dimensional value of 205 C) and the CTR of P1 and P2 at 0.25 (corresponding to a dimensional value of 68 C). It is assumed that P1 and P2 are uniformly distributed throughout the solid, initially in equal proportions so that l0=0.5. The graph in Fig. 3 plots the total dimensionless mass loss rate against dimensionless time for the parameter values shown in Table 1. In the table CT2 refers to the CT of P2 and Da2 refers to the DamkoÈhler number l2A2/ of P2. Thus the ®rst case CT2=0.75 corresponds to the case where P1 and P2 have identical decomposition kinetics. As CT2 is increased, a characteristic kink begins to form in the MLR curve, with associated multiple local maxima. Note also that the overall mass loss rate is reduced with increasing CT2 (corresponding with increased activation temperature) and that, in particular, the peak MLR reduces. Also, the location of the peak MLR moves from the ®nal maxima to one of the secondary maxima for large CT2. The appearance of the kink may be related to the exhaustion of one of the components in the solid, as the graph in Fig. 4 shows. Here the total MLR is plotted (thick, dark curve) and the parts of the total arising
Fig. 2. Eect of ®ller which does not degrade. CT1=1.1, CTR1=0.25.
Fig. 3. MLR as the CT of P2 is varied. (CT1=0.75, CTR=0.25).
of Fig. 1. The double peak feature arises in the following manner. Initially, the polymer degrades mainly from surface regions, which creates the ®rst peak. There then follows a diusive lag as the temperature in the remaining solid increases. Mass is then ®nally lost from subsurface regions, thereby creating the second peak. Note also that there is little bene®t in adding a ®ller of this nature if ignition resistance is the goal and the polymer has a naturally low MLR per unit area at ignition (or CMLRF for brevity). The numerical results show that the initial MLR is largely unaected by the addition of even large quantities of ®ller. The experimental results shown in Fig. 1 also con®rm this observation. 3.1. CT and CTR of ®ller
J.E.J. Staggs / Polymer Degradation and Stability 64 (1999) 369±377
373
Table 1 Kinetic parameters for Fig. 3 CT2
2
Da2
A
0.75 1.0 1.25 1.5 1.75
32.0049 42.3267 54.0908 67.2975 81.9466
7.0094 108 1.2422 1010 2.2064 1011 3.9252 1012 6.99077 1013
1.0000 17.7219 314.7773 5599.9087 99734.2141
from the degradation of the separate components (labelled P1 and P2 and shown by the thin, light curves). It can be seen from this ®gure that the kink is associated with the exhaustion of P1 and that the ®nal section of the MLR curve is determined almost completely by the degradation of P2. Hence in this case, if P2 is viewed as the combustible polymer and P1 as the incombustible ®ller, we see that sacri®cial degradation of the ®ller could be advantageous in delaying the ignition of P2. In fact, adopting a critical MLR ¯ux criterion for ignition [1], and assuming that ignition will occur at a critical MLR ¯ux (CMLRF) of P2, this ®gure shows that ignition could be delayed by a signi®cant amount (depending of course on the value of the CMLRF). This behaviour is now investigated in greater detail. Let P1 represent the ignitable polymer and let P2 represent the diluent ®ller. The graph in Fig. 5 plots the MLR of P1 versus time for ®llers with various CTs. The CTR for both the ®ller and the polymer is 0.25 and the CT for the polymer is 1.1 (corresponding to approximately 300 C). Note that as the CT of the ®ller reduces, the initial MLR of P1 also reduces. However, the graph shows that there is a cross-over region where the MLR curves intersect, and to the right of this region the MLR of P1 increases as the CT of the ®ller reduces. Hence, if ignition occurred at a critical value of the MLR of P1, an increase in the time to ignition would only be observed if the CMLRF was small. If the CMLRF was to the right of the cross-over region, then addition of the ®ller would actually reduce the time to ignition. This
observation is in agreement with the results of Nelson et al. [1] for thermally thin ignition. Note also that the peak MLR of P1 increases as the CT of the ®ller reduces. Hence in the latter stages of a ®re involving the ®lled material, the price paid for delayed ignition may be increased heat release rate. The graph in Fig. 6 shows the eect of varying the CTR of the ®ller (labelled CTR2) from 41 C to 82 C on the total MLR and the MLR of P1. Note that although there is a large change in the appearance of the total MLR curves (the kink becoming more developed as the CTR of the ®ller reduces), the MLR of P1 is hardly aected (apart from the thermally thin peak in the latter stages) and consequently there will be little variation in the ignition characteristics with the CTR of the ®ller (assuming the CMLRF model for ignition).
Fig. 4. Separate components of the total MLR for the case CT1=0.75, CT2=1.5.
Fig. 6. Eect of CTR of ®ller for CT1=1.1, CT2=0.8, CTR1=0.25, l0=0.5.
Fig. 5. Eect of CT of P2 on degradation of P1 and ignition characteristics of solid. CT1=1.1, CTR1=0.25, l0=0.5.
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J.E.J. Staggs / Polymer Degradation and Stability 64 (1999) 369±377
In summary, although the CT and CTR of the ®ller can change the total MLR curve of the material, the eect on the combustible component of the solid is small. Reducing the CT of the ®ller can have bene®cial eects on the ignition resistance, but can also actually reduce time to ignition if the combustible component has a suciently large CMLRF. Hence it appears that at high loadings (where 50% of the initial mass of the solid is ®ller), assuming that the polymer ignites at a low MLR and that the latent heats of vaporisation of the polymer and ®ller are comparable, the CT and CTR of the ®ller have small eects on the MLR of the combustible component of the solid and the only practical requirement is that the CT of the ®ller is less than (or equal to) the CT of the polymer. 3.2. Filler loading and latent heat of vaporisation We now investigate the in¯uence of the other two important parameters on the degradation of the polymer: the initial mass fraction of ®ller l0 and the ratio of the latent heat of vaporisation of the ®ller to that of the polymer L. Obviously, addition of greater quantities of a ®ller with a lower CT than the polymer will delay degradation of the polymer and this is con®rmed in Fig. 7. Here the MLR of P1 is plotted for various values of the initial ®ller loading up to 50%. The CT of the ®ller is 137 C and that of the polymer is 300 C. Note that there is a signi®cant eect on the initial and midrange MLR, but the peak MLR is unaltered since the ®ller has been exhausted by this time. Note also that from the point of view of ignition resistance, the bene®t of adding the ®ller will increase if the polymer has a naturally high CMLRF. If this is not the case, then the ®ller must be present in relatively large quantities to signi®cantly change the ignition resistance. The graphs in Fig. 8 con®rm the intuitively obvious fact that ideally the ®ller should have a larger latent heat of vaporisation than the polymer. When the ®ller is identical to the polymer in all respects other than the value of its latent heat of vaporisation, the ratio of the
MLR of P1 to the total will be the same as l0 and this is con®rmed in graph (a). In graph (b), the CT of the ®ller is 164 C, compared with 300 C for the polymer. Selecting a ®ller with a much lower CT makes matters worse in the ®nal stages of the degradation where a thermally thin peak develops, but considerably improves the situation in the early stages of degradation where the MLR of P1 is signi®cantly reduced. Therefore, addition of a ®ller with large L and low CT will signi®cantly improve the ignition resistance of a polymer with a naturally low CMLRF. The graph in Fig. 9 shows the eect of changing the external heat ¯ux on the eectiveness of a ®ller with a low CT and a relatively high latent heat of vaporisation. We note that the ®ller becomes more eective as the external heat ¯ux is reduced. For a polymer initially 1 cm
Fig. 7. Eect of mass fraction of ®ller on MLR of P1 for CT1=1.1, CT2=0.5, CTR1= CTR2=0.25.
Fig. 9. Eect of external heat ¯ux for CT1=1.1, CT2=0.6, CTR1= CTR2=0.25, l0=0.5, L=5.
Fig. 8. Eect of ratio of heats of vaporisation L for (a) CT1=1.1, CT2=1.1, CTR1= CTR2=0.25, l0 =0.5, (b) CT1=1.1, CT2=0.6, CTR1= CTR2=0.25, l0=0.5.
J.E.J. Staggs / Polymer Degradation and Stability 64 (1999) 369±377
thick with k=0.3 Wmÿ1Kÿ1, "=0.9, the dimensionless heat ¯ux values shown in Fig. 9 correspond to the following dimensional quantities: 72.8 kWmÿ2 ( =8), 54.68 kWmÿ2 ( =6), 36.48 kWmÿ2 ( =4), 18.28 kWmÿ2 ( =2). Finally, in this section, we investigate the diluent eect of the ®ller on a crude approximation to the eective heat of combustion of the solid as it would be determined in a typical cone calorimeter experiment. This quantity is de®ned as the ratio of the heat release rate to the mass loss rate. Assuming that the polymer ignites at a low CMLRF and that the heat ¯ux from the resulting ¯ame arriving at the surface of the solid is constant (so that it may be viewed as a constant added to in the de®nition of net dimensionless heat ¯ux ÿ), also that the heat release rate from the burning solid is proportional to the MLR of P1, the eective heat of combustion will be proportional to the ratio dm1 =d dm1 : dm dm=d
17
The graph in Fig. 10 plots this quantity as a function of dimensionless time for various initial concentrations of ®ller. The parameter values for the ®ller have been chosen corresponding to the case L=5 in Fig. 7. Of course, when there is no ®ller, dm1/dm=1. When l0 < 1, there is a gradual initial increase in dm1/dm from 0 (since the initial mass loss arises from degradation of the ®ller only) which attains a quasi-steady value of approximately l0. This is followed by a rapid increase (in the thermally thin regime) to a value of 1 as the ®ller becomes exhausted. In summary, we see that the ®ller loading and the latent heat of vaporisation of the ®ller have a greater eect on the ecacy of the ®ller than the kinetic parameters. It appears that the ®ller needs to be present in large quantities to have a noticeable eect if its latent heat of vaporisation is comparable with that of the polymer: the mass loss rate typically reducing in proportion to the polymer content of the solid.
Fig. 10. Approximate dimensionless eective heat of combustion dm1/ dm for L=5, CT1=1.1, CT2=0.6, CTR1= CTR2=0.25.
375
3.3. Initial distribution of ®ller To complete our discussion, the eect of the initial distribution of the ®ller on the mass loss rate is considered. The practical value of this exercise is not clear to the author as it would be to manufacture a material with a ®ller distribution which was anything other than macroscopically uniform. However, it shall be seen below that there are advantages to be gained from nonuniform distribution of ®ller and so include this case for completeness. We shall consider only initial distributions of ®ller which are linear and of the form 1 ÿ l0 x 2 ÿ ax a;
18
where is the overall initial mass fraction of ®ller in the solid and a is a parameter such that 04a42 . Thus when a= the ®ller distribution is uniform. When a < the ®ller concentration increases with depth into the solid and when a > the ®ller concentration reduces with depth. The graph in Fig. 11 plots the MLRs of P1 and the whole solid for the case of a low CT ®ller with comparable latent heat of vaporisation to the polymer (L=1), for the case where 25% of the initial mass of the solid is ®ller. Also shown on this graph is the MLR of P1 for the case of a uniformly distributed ®ller with l0=0.5, so that the surface concentration of ®ller agrees with the linearly distributed case a=0.5, =0.25. Note that as a increases, the MLR of P1 reduces for < 0.25, indicating a bene®cial eect on ignition resistance. Note also the interesting feature that the MLR curve for P1 for the case a=0.5 agrees with the uniformly distributed ®ller for small (<0.07). Hence if the CMLRF of P1 was low enough, this result shows that the same ignition resistance could be achieved using half of the amount of
Fig. 11. Eect of initial distribution of ®ller 1 ÿ l0 x 2 ÿ ax a for 0:25 and CT1=1.1, CT2=0.6, CTR1=CTR2=0.25. The arrows show the direction of increasing a. The curves are plotted at values a=0, 0.125, 0.25, 0.375, 0.5. The total MLRs are shown by the dark curves and the MLRs of P1 are shown by the light curves. The thick light curve shows the MLR for the case of a uniformly distributed ®ller with l0 0:5.
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J.E.J. Staggs / Polymer Degradation and Stability 64 (1999) 369±377
Fig. 12. Consumption of ®ller for case a=0.5.
®ller, provided that the ®ller is distributed throughout the solid in a non-uniform way. Fig. 12 shows the distributions of ®ller in the solid at various stages through the degradation for the case a=0.5. Here the ®ller mass fraction 1 ÿ l is plotted as a function of x, the relative distance into the solid (x=0 corresponds to the top surface and x=1 corresponds to the bottom surface). Note that ®ller is lost rapidly from the surface during the initial stages of degradation and that after approximately =0.32 the ®ller has been exhausted. Increasing the CT of the ®ller would not help ignition resistance unless the CMLRF of P1 was large (as in the case shown in Fig. 5 above), however it would reduce the MLR of P1 during the latter stages of the degradation. 4. Conclusions We have investigated an idealised model of a ®lled polymeric solid decomposing under the action of an external heat ¯ux. One of the main bene®ts of the model is the ability to separate the MLR of the combustible component of the solid from the total MLR, thus enabling inferences about ignition resistance to be made on the basis of a critical mass loss rate ¯ux (CMLRF) criteria. It appears from the numerical results that the main parameters aecting the MLR of the combustible component of the solid are the initial mass fraction of the ®ller l0 and the ratio of the latent heat of vaporisation of the ®ller to the latent heat of vaporisation of the polymer L. The CT and CTR of the ®ller have secondary importance and only become important when L is large. When L is comparable to 1, it appears that the value of the CT of the ®ller is relatively unimportant provided that it is less than that of the polymer. Reducing the CT of the ®ller reduces the initial MLR of P1 but increases it at later times. Thus addition of a low CT ®ller could actually reduce ignition resistance of a polymer which is naturally ignition resistant, i.e. has a high CMLRF. When L is large (5), however, the ignition resistance is improved if the CT of the ®ller is less than that of the polymer. The price to be paid for good ignition resis-
tance is a large peak MLR of the combustible component in the latter stages of degradation. The initial MLR of the polymer reduces approximately in proportion to the initial mass fraction of a low CT ®ller in the solid. Therefore, in order to obtain large improvements in the ¯ammability characteristics of the solid, large concentrations of ®ller must be added. However, we have also shown that the initial distribution of ®ller in the solid can be used to advantage. If the ®ller is distributed such that there is more ®ller at the exposed surface (where it is needed) and decreasingly less ®ller as we move away from the exposed surface, the initial ¯ammability characteristics can be aected with much lower total ®ller content. However, the practicality and economic viability of this approach to ignition resistance is not at all clear at present. In summary, it appears that a solid-phase ®ller with similar thermal properties and latent heat of vaporisation to the polymer works best at low to medium heat ¯uxes and will improve the ignition resistance of a polymer that already has good ignition characteristics. If the polymer does not have naturally good ignition characteristics, then the ®ller must have a high latent heat of vaporisation and a low CT (compared with the polymer) and be present in large quantities. Finally, it should be noted that this model neglects the mechanisms controlling the ¯ow of volatile material through the inert ®ller residue as mass loss progresses. Therefore, we would expect the long-time behaviour predicted by this model, particularly the second MLR peak, to be modi®ed by inclusion of these processes. Furthermore, the results presented herein are valid only when the thermal properties of the ®ller are the same as those of the polymer. A further extension of the work would be to include the facility to model ®llers with dierent thermal properties. Acknowledgements The author gratefully acknowledges the kind cooperation of Mr. P. J. Elliot and Dr. R. H. Whiteley of Raychem Ltd. in providing the experimental data reproduced in this paper. References [1] Nelson MI, Brindley J, McIntosh AC. The eect of heat sink additives on the ignition and heat release properties of thermally thin thermoplastics. Fire Safety Journal 1997;28(1):67. [2] Babrauskas V. Development of the cone calorimeter ± a benchscale heat release rate apparatus based on oxygen consumption. National Bureau of Standards, Washington, DC. NBSIR 1982. p. 82±2611. [3] Babrauskas V. Development of the cone calorimeter ± a benchscale heat release rate apparatus based on the oxygen consumption principle. Fire and Materials 1984;8(2):81±95.
J.E.J. Staggs / Polymer Degradation and Stability 64 (1999) 369±377 [4] Babrauskas V. Construction drawings for the cone calorimeter. Technical report, Center for Fire Research, US National Bureau of Standards, 1985. [5] Standard test method for heat and visible smoke release rates for materials and products using an oxygen consumption calorimeter. ASTM E 1354, American Society for the Testing of Materials, Philadelphia 1992. [6] Staggs JEJ. A theoretical investigation into modelling thermal degradation of solids incorporating ®nite-rate kinetics. Comb. Sci. and Tech. 1997;123(1±6):261. [7] Kindelan M, Williams FA. Theory for endothermic gasi®cation of a solid by a constant energy ¯ux. Comb. Sci. and Tech. 1975;10:1±19. [8] Parker WJ. Ph. D. Thesis. NBSIR 1985. p. 85±3163. [9] Vovelle C, Delfau J, Reuillon, M. Experimental and numerical study of the thermal degradation of PMMA. Comb. Sci and Tech. 1878;53:187.
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[10] Wichman IS, Atreya A. A simpli®ed model for the pyrolysis of charring materials. Comb. and Flame 1987;68:231. [11] Di Blasi C, Wichman IS. Eects of solid-phase properties on ¯ames spreading over composite materials. Comb. and Flame 1995;102:229. [12] Staggs JEJ, Whiteley RH. Modelling the combustion of solidphase fuels in cone calorimeter experiments. Inter¯am `96. Seventh International Fire Science and Eng. Conference Interscience Communications. 1996. p. 103. [13] Staggs JEJ. Thermal degradation of polymers using equivalent ®rst-order single-step kinetics. Submitted to Fire Safety Journal, 1997. [14] Whiteley RH, Elliot PJ. A cone calorimeter study of the eects of alumina trihydrate and magnesium hydroxide on the burning behaviour of an ethylene-vinyl acetate copolymer. Paper presented at the Second Fire and Materials International Conference, September 1993, Crystal City, USA.
Modelling the eect of solid-phase additives on thermal degradation of solids J.E.J. Staggs1 Department of Fuel and Energy, The University of Leeds, Leeds LS2 9JT, UK Accepted 15 March 1998
Abstract We model the eect of a simple solid-phase additive on the thermal degradation of a solid under thermally thick and thermally thin conditions. The mathematical model simulates conditions in bench-scale tests (such as the cone calorimeter test) where a solid sample of typical cross-sectional area 100 cm2 is exposed to a uniform external heat ¯ux and the mass loss rate and eective heat of combustion are determined. We choose a simple model for the decomposition: the additive and solid decompose independently according to ®rst-order kinetics. We assume for simplicity that the thermal conductivity, density and speci®c heat capacity of the two solid phases are identical and hence the discussion centres on the role of the degradation mechanisms of the components in relation to the overall degradation of the solid. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The ®re retardant action of additives which primarily act as heat sinks, such as aluminium oxide trihydrate (ATH) or magnesium hydroxide, involves a number of processes. The endothermic degradation of these ®llers cools surrounding material; the thermal and optical properties of the ®ller may be useful in preventing heat transfer from the exposed surface to subsurface regions as a solid residue builds up at the surface; water vapour liberated as a result of the decomposition of the ®ller (ATH is 35% by weight water) may dilute the ¯ammable vapours produced by the polymer. A striking feature of mass loss rate data for ®lled polymers from cone calorimeter experiments is the presence of (at least) two peaks as shown in Fig. 1. Here we see mass loss rate plotted against time for ethylene-vinyl acetate ®lled with ATH. The heat ¯ux from the cone heater was set to 40 kWmÿ2 and the initial thickness of the test samples was 10 mm. The curves are labelled according to the initial mass fraction of ®ller present in the material. Note that for moderate ®ller loadings (<40% by weight) there are two peaks in the mass loss rate. As the ®ller loading increases, there is a tendency for the second peak to be suppressed, although vestiges remain even for the case of 60% ®ller. 1
Tel.:0113 233 2495; fax: 0113 244 0572; e-mail: [email protected]
This paper investigates the thermal degradation of an idealised solid comprising a combustible polymer and an inert ®ller. The solid is viewed as a homogeneous mix of ®ller and polymer and we assume that the ®ller and polymer degrade independently of each other according to single-step ®rst-order kinetics. The present work extends earlier theoretical work on the eect of additives in polymer combustion, notably that of Nelson et al. [1], where ignition of ®lled polymers was considered in the thermally thin regime. As in that work, we consider a sample of material degrading under a constant external heat ¯ux, as in the cone calorimeter test [2±5]. The mathematical model of pyrolysis considered here is based on a previous model [6] which includes a transport mechanism in subsurface regions of the solid which accounts for movement of surrounding material as gaps produced by volatilisation are ®lled. Imagine that we are able to label a particular element of material in the solid, and that we can track its progress as degradation occurs. Let the labelled material element be located a distance y below the initial location of the top surface. Let the initial thickness of material be l. In a given time step, some material in the region below the labelled element will volatilise resulting in a net reduction of thickness. We would also observe that the material element at y would itself reduce in thickness, but would also move to a lower location, as material instantaneously ®lls the gaps left by escaping volatiles.
0141-3910/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0141 -3 910(98)00155 -4
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Fig. 1. Experimental mass loss rate data for black EVA ®lled with ATH.
P2) degrading according to a single-step ®rst-order process. We characterise the TG curve in terms of a characteristic temperature (CT) and a characteristic temperature range (CTR). The CT is de®ned as the temperature at which the mass fraction equals a ®xed value c, say. Thus the CT at the ®xed mass fraction c where 0 < c < 1 will be denoted by T(c). The CTR is de®ned as follows. Imagine a tangent drawn at the point (T(c), c) on the TG curve, produced so that it meets the T axis at Ta and the line m=1 at Tb. The temperature dierence Ta ÿ Tb is de®ned as the CTR and will be denoted by T(c). For convenience, we will refer to dimensionless versions of these quantities, de®ned by CT ÿ T0 ; T0 CTR dimensionless CTR ; T0 dimensionless CT
In fact, if we track a material element that starts at the top surface, we would be able to follow its progress until ®nally some small portion of it arrives at y=l. This approach is related to that of other authors who have considered in-depth volatilisation using ®nite rate kinetics but have neglected to consider movement of surrounding material [7±12]. 2. Model derivation We consider the solid to be a homogeneous mix of polymer and ®ller, where the two materials are labelled P1 and P2. In order to keep the discussion as general as possible, we do not at this stage specify which of P1 and P2 is the polymer and which is the ®ller. For simplicity we assume that P1 and P2 have the same thermal properties, but have dierent decomposition kinetics and latent heats of vaporisation. Let m1 denote the mass of P1 in a small amount of the mixture and m2 the mass of P2. We assume that P1 and P2 degrade independently of each other via a single step process which is ®rst-order for each species. Hence for an elemental slice of material in a TG experiment, the kinetic equations are ÿdm1 =dt k1 m1 ; ÿdm2 =dt k2 m2 , where k1 A1 exp ÿT1 =T; k2 A2 exp ÿT2 =T. Here t denotes time, A1,2 are pre-exponential factors, T1,2 are activation temperatures and T is temperature. The kinetic equations are characterised by four parameters: two pre-exponential factors and two activation temperatures. Whereas this characterisation is convenient for writing down the equations, it is not particularly useful from a practical standpoint. Some authors [1,13] have adopted alternative descriptions which make use of a characteristic temperature which can be easily related to a constant heating rate TG curve (the temperature of the in¯exion point, or the temperature at a ®xed value of the mass fraction). In this work we adopt an approach that will be familiar to TG experimentalists and so only an outline will be given. Consider a single component of the solid (either P1 or
1
where T0 is ambient temperature. It can be shown that the pre-exponential factor and activation temperature can be easily calculated from the CT and the CTR and the reader is referred to a previous paper [13] for details. The main advantage of the CT±CTR characterisation of decomposition kinetics is the ease with which the two parameters can be determined in practice from a standard constant heating rate TG curve: one simply has to read o the temperature at which the mass fraction is c and the gradient at that point. Furthermore, as we shall see below, the important parameter as far as this work is concerned is the CT ± a direct measure of the decomposition characteristics of the polymer ± which can be easily related to the TG curve (unlike A and TA, where one has to solve the kinetic equation in order to obtain the shape of the curve). Therefore the CT±CTR characterisation also aids in the interpretation of numerical results. Thus, given constant heating rate TG curves for P1 and P2 (labelled C1 and C2, say), a solid containing a mixture of P1 and P2 where the initial mass fraction of P1 in the solid is l0, will have TG curve C=l0C1 + (1 ÿ l0)C2, assuming no interaction between P1 and P2. The shape of C will depend on the CTs and CTRs. If the CTs are markedly dierent and the CTRs are relatively small, C will have two discernible steps. If the CTs are similar, then it may be dicult to distinguish C from a single-step TG curve. For convenience, we recast the kinetic equations in terms of the total mass m=m1 + m2 and the mass-fraction scalar l=m1/m, viz. dl dm l 1 ÿ l k2 ÿ k1 ; ÿmfk1 l k2 1 ÿ lg: dt dt
2
In order to investigate the mass loss behaviour of a larger sample of solid, where temperature gradients may not be neglected, we must link the kinetics to a suitable heat transfer model. We shall speci®cally consider the
J.E.J. Staggs / Polymer Degradation and Stability 64 (1999) 369±377
situation where a horizontal sample is exposed to a constant external heat ¯ux through its top surface. We shall assume that the lower surface of the sample is insulated. Following the approach taken in an earlier paper [6], we adopt a model where volatiles escape from the solid as soon as they are formed and surrounding material instantaneously ®lls in the gaps left by the escaping gases. Under these conditions, the energy equation in the solid takes the form c
DT @2 T 000 k 2 Q_ v ; Dt @y
3
for s(t)4y4l, where l denotes the initial thickness of the solid, y=s(t) denotes the instantaneous location of the top (exposed) surface, rc is the volumetric heat capacity of the solid, k is the thermal conductivity and y an ordinate measuring distance from the initial location of 000 the top surface of the solid. The symbol Q_ v denotes a source term to account for the latent heat of vaporisation of the solid and D/Dt=@/@t + v@/@y is the convective derivative, where v(y,t) is the velocity ®eld imposed by the movement of material as it ®lls in gaps left by escaping volatiles. We apply the kinetic equation for l in a similar way throughout the degrading solid, giving Dl l 1 ÿ l k2 ÿ k1 : Dt
4
Given the kinetic mechanism de®ned above, and following the approach of previous work [6], the source term takes the form 000 Q_ v ÿL1 fk1 l Lk2 1 ÿ lg;
5
where L=L2/L1 and L1,2 are the latent heats of vaporisation of P1 and P2. Furthermore, the velocity ®eld is found by summing the movements of elemental slices of material below a given point y in the solid and hence takes the form l v fk1 l k2 1 ÿ lg dy:
6
y
The energy equation (3) and the mass-fraction equation (4) are combined with the boundary conditions ÿk
@T "q h T0 ÿ T " T40 ÿ T4 @y
@T 0 @y
on y s t; 7
on y l;
8
and the initial conditions T y; 0 T0 ;
l y; 0 l0 y;
371
initial condition) so that we may investigate the eect of ®ller distribution on the mass loss rate (MLR). The other variables are de®ned as follows: " is the emissivity of the top surface, q is the external heat ¯ux, h is the convective heat transfer coecient and is the Stefan± Boltzmann constant. Eq. (3)Eq. (4) and boundary conditions (7) and (8) necessitate a numerical solution and this may be achieved using standard ®nite-dierence techniques [6]. The equations are solved in the dimensionless forms 81 9 1 < = @ @ 1 @2 Da Da E1 dx ÿ 1 ÿ x E1 dx 2 2ÿ E; : ; @x @x @ St x
81 <
0
9 = @l @l Da E1 dx ÿ 1 ÿ x E1 dx : ; @x @ x 0 2 1 ÿ exp ÿ ; Dal 1 ÿ l A exp ÿ 1 1
10
1
11 ÿ
@ @ ÿ on x 0; 0 @x @x
on x 1;
12
where 1 2 A 1 ÿ l exp ÿ ; E1 ; l l exp ÿ 1 1 13 1 2 LA 1 ÿ l exp ÿ ; E ; l l exp ÿ 1 1 14 15 ÿ ÿ Bi ÿ R 14 ÿ 1 and the other variables are de®ned as T ÿ T0 l2 A 1 T1;2 cT0 ; 1;2 ; Da ; St ; T0 T0 L1 "T30 l A2 "ql hl ; ; ; Bi ; R A kT0 k A1 k s yÿs t ; 2 : 1 ÿ ;x l lÿs l
16
In the new variables, the total dimensionless mass loss rate is given by ÿd/d =Da 01E1 dx. 3. Results and discussion
9
where l0(y) is the initial distribution of the mass fraction of P1 in the solid. Note that we shall consider cases where l0 is not constant (hence the generality in the
Unless otherwise stated, the results below were computed with the following dimensionless parameter values: St=1, L=1, =8, Bi=0.33, R=0.035. For a typical polymer which is initially 1 cm thick, with k=0.3
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Wmÿ1Kÿ1, =1000 kgmÿ3, c=3000 Jkgÿ1Kÿ1, so that =10ÿ7 m2sÿ1, =1 corresponds to 1000 s, =8 corresponds to an external heat ¯ux of 72.8 kWmÿ2, Bi=0.33 corresponds to a convective heat transfer coecient of approximately 10 Wmÿ2Kÿ1, R=0.035 corresponds to an emissivity of 0.9 and St=1 corresponds to a latent heat of vaporisation of 0.82 kJgÿ1. The results will be presented mainly in terms of the dimensionless MLR ÿd/d versus dimensionless time . Before we progress to a systematic investigation of the mathematical model outlined above, we ®rst attempt some initial calculations for the case of a diluent ®ller which does not degrade. The goal here is to crudely approximate as best we can, within the limited theoretical framework that our model assumptions provide, conditions similar to those of the experimental results shown in Fig. 1. For the case of EVA ®lled with ATH. Whiteley and Elliot [14] comment that as the ®ller degrades, water is lost leaving a microporous residue of alumina. Of course, we do not model this situation exactly here: . our ®ller either degrades completely without leaving a residue or does not degrade at all, . the thermal properties of the ®ller and polymer are identical, . EVA is a copolymer and it is doubtful that the single-step kinetics assumed in the present work will be adequate to describe the decomposition, . volatiles escape unimpeded and unchanged from the solid as soon as they are formed and do not exchange heat with surrounding material, but it might be expected that similar results may be obtained for the case where the water content of the ATH is low. For the case of a non-degrading ®ller, A is set to 0 in the model equations (13) and (14). Fig. 2 shows the eect of increasing the inert ®ller concentration on the MLR for =5, Bi=0.5, R=0.05, CT1=1.02, CTR1=0.2. The curves are labelled according to ®ller loading. Note the interesting feature of a double peak in the MLR and the fact that the relative size of the second peak reduces with increasing ®ller loading ± in agreement with the experimental data
We ®rst investigate the MLR behaviour as the CT of P2 is varied. The CT of P1 is ®xed at 0.75 (corresponding to a dimensional value of 205 C) and the CTR of P1 and P2 at 0.25 (corresponding to a dimensional value of 68 C). It is assumed that P1 and P2 are uniformly distributed throughout the solid, initially in equal proportions so that l0=0.5. The graph in Fig. 3 plots the total dimensionless mass loss rate against dimensionless time for the parameter values shown in Table 1. In the table CT2 refers to the CT of P2 and Da2 refers to the DamkoÈhler number l2A2/ of P2. Thus the ®rst case CT2=0.75 corresponds to the case where P1 and P2 have identical decomposition kinetics. As CT2 is increased, a characteristic kink begins to form in the MLR curve, with associated multiple local maxima. Note also that the overall mass loss rate is reduced with increasing CT2 (corresponding with increased activation temperature) and that, in particular, the peak MLR reduces. Also, the location of the peak MLR moves from the ®nal maxima to one of the secondary maxima for large CT2. The appearance of the kink may be related to the exhaustion of one of the components in the solid, as the graph in Fig. 4 shows. Here the total MLR is plotted (thick, dark curve) and the parts of the total arising
Fig. 2. Eect of ®ller which does not degrade. CT1=1.1, CTR1=0.25.
Fig. 3. MLR as the CT of P2 is varied. (CT1=0.75, CTR=0.25).
of Fig. 1. The double peak feature arises in the following manner. Initially, the polymer degrades mainly from surface regions, which creates the ®rst peak. There then follows a diusive lag as the temperature in the remaining solid increases. Mass is then ®nally lost from subsurface regions, thereby creating the second peak. Note also that there is little bene®t in adding a ®ller of this nature if ignition resistance is the goal and the polymer has a naturally low MLR per unit area at ignition (or CMLRF for brevity). The numerical results show that the initial MLR is largely unaected by the addition of even large quantities of ®ller. The experimental results shown in Fig. 1 also con®rm this observation. 3.1. CT and CTR of ®ller
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373
Table 1 Kinetic parameters for Fig. 3 CT2
2
Da2
A
0.75 1.0 1.25 1.5 1.75
32.0049 42.3267 54.0908 67.2975 81.9466
7.0094 108 1.2422 1010 2.2064 1011 3.9252 1012 6.99077 1013
1.0000 17.7219 314.7773 5599.9087 99734.2141
from the degradation of the separate components (labelled P1 and P2 and shown by the thin, light curves). It can be seen from this ®gure that the kink is associated with the exhaustion of P1 and that the ®nal section of the MLR curve is determined almost completely by the degradation of P2. Hence in this case, if P2 is viewed as the combustible polymer and P1 as the incombustible ®ller, we see that sacri®cial degradation of the ®ller could be advantageous in delaying the ignition of P2. In fact, adopting a critical MLR ¯ux criterion for ignition [1], and assuming that ignition will occur at a critical MLR ¯ux (CMLRF) of P2, this ®gure shows that ignition could be delayed by a signi®cant amount (depending of course on the value of the CMLRF). This behaviour is now investigated in greater detail. Let P1 represent the ignitable polymer and let P2 represent the diluent ®ller. The graph in Fig. 5 plots the MLR of P1 versus time for ®llers with various CTs. The CTR for both the ®ller and the polymer is 0.25 and the CT for the polymer is 1.1 (corresponding to approximately 300 C). Note that as the CT of the ®ller reduces, the initial MLR of P1 also reduces. However, the graph shows that there is a cross-over region where the MLR curves intersect, and to the right of this region the MLR of P1 increases as the CT of the ®ller reduces. Hence, if ignition occurred at a critical value of the MLR of P1, an increase in the time to ignition would only be observed if the CMLRF was small. If the CMLRF was to the right of the cross-over region, then addition of the ®ller would actually reduce the time to ignition. This
observation is in agreement with the results of Nelson et al. [1] for thermally thin ignition. Note also that the peak MLR of P1 increases as the CT of the ®ller reduces. Hence in the latter stages of a ®re involving the ®lled material, the price paid for delayed ignition may be increased heat release rate. The graph in Fig. 6 shows the eect of varying the CTR of the ®ller (labelled CTR2) from 41 C to 82 C on the total MLR and the MLR of P1. Note that although there is a large change in the appearance of the total MLR curves (the kink becoming more developed as the CTR of the ®ller reduces), the MLR of P1 is hardly aected (apart from the thermally thin peak in the latter stages) and consequently there will be little variation in the ignition characteristics with the CTR of the ®ller (assuming the CMLRF model for ignition).
Fig. 4. Separate components of the total MLR for the case CT1=0.75, CT2=1.5.
Fig. 6. Eect of CTR of ®ller for CT1=1.1, CT2=0.8, CTR1=0.25, l0=0.5.
Fig. 5. Eect of CT of P2 on degradation of P1 and ignition characteristics of solid. CT1=1.1, CTR1=0.25, l0=0.5.
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J.E.J. Staggs / Polymer Degradation and Stability 64 (1999) 369±377
In summary, although the CT and CTR of the ®ller can change the total MLR curve of the material, the eect on the combustible component of the solid is small. Reducing the CT of the ®ller can have bene®cial eects on the ignition resistance, but can also actually reduce time to ignition if the combustible component has a suciently large CMLRF. Hence it appears that at high loadings (where 50% of the initial mass of the solid is ®ller), assuming that the polymer ignites at a low MLR and that the latent heats of vaporisation of the polymer and ®ller are comparable, the CT and CTR of the ®ller have small eects on the MLR of the combustible component of the solid and the only practical requirement is that the CT of the ®ller is less than (or equal to) the CT of the polymer. 3.2. Filler loading and latent heat of vaporisation We now investigate the in¯uence of the other two important parameters on the degradation of the polymer: the initial mass fraction of ®ller l0 and the ratio of the latent heat of vaporisation of the ®ller to that of the polymer L. Obviously, addition of greater quantities of a ®ller with a lower CT than the polymer will delay degradation of the polymer and this is con®rmed in Fig. 7. Here the MLR of P1 is plotted for various values of the initial ®ller loading up to 50%. The CT of the ®ller is 137 C and that of the polymer is 300 C. Note that there is a signi®cant eect on the initial and midrange MLR, but the peak MLR is unaltered since the ®ller has been exhausted by this time. Note also that from the point of view of ignition resistance, the bene®t of adding the ®ller will increase if the polymer has a naturally high CMLRF. If this is not the case, then the ®ller must be present in relatively large quantities to signi®cantly change the ignition resistance. The graphs in Fig. 8 con®rm the intuitively obvious fact that ideally the ®ller should have a larger latent heat of vaporisation than the polymer. When the ®ller is identical to the polymer in all respects other than the value of its latent heat of vaporisation, the ratio of the
MLR of P1 to the total will be the same as l0 and this is con®rmed in graph (a). In graph (b), the CT of the ®ller is 164 C, compared with 300 C for the polymer. Selecting a ®ller with a much lower CT makes matters worse in the ®nal stages of the degradation where a thermally thin peak develops, but considerably improves the situation in the early stages of degradation where the MLR of P1 is signi®cantly reduced. Therefore, addition of a ®ller with large L and low CT will signi®cantly improve the ignition resistance of a polymer with a naturally low CMLRF. The graph in Fig. 9 shows the eect of changing the external heat ¯ux on the eectiveness of a ®ller with a low CT and a relatively high latent heat of vaporisation. We note that the ®ller becomes more eective as the external heat ¯ux is reduced. For a polymer initially 1 cm
Fig. 7. Eect of mass fraction of ®ller on MLR of P1 for CT1=1.1, CT2=0.5, CTR1= CTR2=0.25.
Fig. 9. Eect of external heat ¯ux for CT1=1.1, CT2=0.6, CTR1= CTR2=0.25, l0=0.5, L=5.
Fig. 8. Eect of ratio of heats of vaporisation L for (a) CT1=1.1, CT2=1.1, CTR1= CTR2=0.25, l0 =0.5, (b) CT1=1.1, CT2=0.6, CTR1= CTR2=0.25, l0=0.5.
J.E.J. Staggs / Polymer Degradation and Stability 64 (1999) 369±377
thick with k=0.3 Wmÿ1Kÿ1, "=0.9, the dimensionless heat ¯ux values shown in Fig. 9 correspond to the following dimensional quantities: 72.8 kWmÿ2 ( =8), 54.68 kWmÿ2 ( =6), 36.48 kWmÿ2 ( =4), 18.28 kWmÿ2 ( =2). Finally, in this section, we investigate the diluent eect of the ®ller on a crude approximation to the eective heat of combustion of the solid as it would be determined in a typical cone calorimeter experiment. This quantity is de®ned as the ratio of the heat release rate to the mass loss rate. Assuming that the polymer ignites at a low CMLRF and that the heat ¯ux from the resulting ¯ame arriving at the surface of the solid is constant (so that it may be viewed as a constant added to in the de®nition of net dimensionless heat ¯ux ÿ), also that the heat release rate from the burning solid is proportional to the MLR of P1, the eective heat of combustion will be proportional to the ratio dm1 =d dm1 : dm dm=d
17
The graph in Fig. 10 plots this quantity as a function of dimensionless time for various initial concentrations of ®ller. The parameter values for the ®ller have been chosen corresponding to the case L=5 in Fig. 7. Of course, when there is no ®ller, dm1/dm=1. When l0 < 1, there is a gradual initial increase in dm1/dm from 0 (since the initial mass loss arises from degradation of the ®ller only) which attains a quasi-steady value of approximately l0. This is followed by a rapid increase (in the thermally thin regime) to a value of 1 as the ®ller becomes exhausted. In summary, we see that the ®ller loading and the latent heat of vaporisation of the ®ller have a greater eect on the ecacy of the ®ller than the kinetic parameters. It appears that the ®ller needs to be present in large quantities to have a noticeable eect if its latent heat of vaporisation is comparable with that of the polymer: the mass loss rate typically reducing in proportion to the polymer content of the solid.
Fig. 10. Approximate dimensionless eective heat of combustion dm1/ dm for L=5, CT1=1.1, CT2=0.6, CTR1= CTR2=0.25.
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3.3. Initial distribution of ®ller To complete our discussion, the eect of the initial distribution of the ®ller on the mass loss rate is considered. The practical value of this exercise is not clear to the author as it would be to manufacture a material with a ®ller distribution which was anything other than macroscopically uniform. However, it shall be seen below that there are advantages to be gained from nonuniform distribution of ®ller and so include this case for completeness. We shall consider only initial distributions of ®ller which are linear and of the form 1 ÿ l0 x 2 ÿ ax a;
18
where is the overall initial mass fraction of ®ller in the solid and a is a parameter such that 04a42 . Thus when a= the ®ller distribution is uniform. When a < the ®ller concentration increases with depth into the solid and when a > the ®ller concentration reduces with depth. The graph in Fig. 11 plots the MLRs of P1 and the whole solid for the case of a low CT ®ller with comparable latent heat of vaporisation to the polymer (L=1), for the case where 25% of the initial mass of the solid is ®ller. Also shown on this graph is the MLR of P1 for the case of a uniformly distributed ®ller with l0=0.5, so that the surface concentration of ®ller agrees with the linearly distributed case a=0.5, =0.25. Note that as a increases, the MLR of P1 reduces for < 0.25, indicating a bene®cial eect on ignition resistance. Note also the interesting feature that the MLR curve for P1 for the case a=0.5 agrees with the uniformly distributed ®ller for small (<0.07). Hence if the CMLRF of P1 was low enough, this result shows that the same ignition resistance could be achieved using half of the amount of
Fig. 11. Eect of initial distribution of ®ller 1 ÿ l0 x 2 ÿ ax a for 0:25 and CT1=1.1, CT2=0.6, CTR1=CTR2=0.25. The arrows show the direction of increasing a. The curves are plotted at values a=0, 0.125, 0.25, 0.375, 0.5. The total MLRs are shown by the dark curves and the MLRs of P1 are shown by the light curves. The thick light curve shows the MLR for the case of a uniformly distributed ®ller with l0 0:5.
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Fig. 12. Consumption of ®ller for case a=0.5.
®ller, provided that the ®ller is distributed throughout the solid in a non-uniform way. Fig. 12 shows the distributions of ®ller in the solid at various stages through the degradation for the case a=0.5. Here the ®ller mass fraction 1 ÿ l is plotted as a function of x, the relative distance into the solid (x=0 corresponds to the top surface and x=1 corresponds to the bottom surface). Note that ®ller is lost rapidly from the surface during the initial stages of degradation and that after approximately =0.32 the ®ller has been exhausted. Increasing the CT of the ®ller would not help ignition resistance unless the CMLRF of P1 was large (as in the case shown in Fig. 5 above), however it would reduce the MLR of P1 during the latter stages of the degradation. 4. Conclusions We have investigated an idealised model of a ®lled polymeric solid decomposing under the action of an external heat ¯ux. One of the main bene®ts of the model is the ability to separate the MLR of the combustible component of the solid from the total MLR, thus enabling inferences about ignition resistance to be made on the basis of a critical mass loss rate ¯ux (CMLRF) criteria. It appears from the numerical results that the main parameters aecting the MLR of the combustible component of the solid are the initial mass fraction of the ®ller l0 and the ratio of the latent heat of vaporisation of the ®ller to the latent heat of vaporisation of the polymer L. The CT and CTR of the ®ller have secondary importance and only become important when L is large. When L is comparable to 1, it appears that the value of the CT of the ®ller is relatively unimportant provided that it is less than that of the polymer. Reducing the CT of the ®ller reduces the initial MLR of P1 but increases it at later times. Thus addition of a low CT ®ller could actually reduce ignition resistance of a polymer which is naturally ignition resistant, i.e. has a high CMLRF. When L is large (5), however, the ignition resistance is improved if the CT of the ®ller is less than that of the polymer. The price to be paid for good ignition resis-
tance is a large peak MLR of the combustible component in the latter stages of degradation. The initial MLR of the polymer reduces approximately in proportion to the initial mass fraction of a low CT ®ller in the solid. Therefore, in order to obtain large improvements in the ¯ammability characteristics of the solid, large concentrations of ®ller must be added. However, we have also shown that the initial distribution of ®ller in the solid can be used to advantage. If the ®ller is distributed such that there is more ®ller at the exposed surface (where it is needed) and decreasingly less ®ller as we move away from the exposed surface, the initial ¯ammability characteristics can be aected with much lower total ®ller content. However, the practicality and economic viability of this approach to ignition resistance is not at all clear at present. In summary, it appears that a solid-phase ®ller with similar thermal properties and latent heat of vaporisation to the polymer works best at low to medium heat ¯uxes and will improve the ignition resistance of a polymer that already has good ignition characteristics. If the polymer does not have naturally good ignition characteristics, then the ®ller must have a high latent heat of vaporisation and a low CT (compared with the polymer) and be present in large quantities. Finally, it should be noted that this model neglects the mechanisms controlling the ¯ow of volatile material through the inert ®ller residue as mass loss progresses. Therefore, we would expect the long-time behaviour predicted by this model, particularly the second MLR peak, to be modi®ed by inclusion of these processes. Furthermore, the results presented herein are valid only when the thermal properties of the ®ller are the same as those of the polymer. A further extension of the work would be to include the facility to model ®llers with dierent thermal properties. Acknowledgements The author gratefully acknowledges the kind cooperation of Mr. P. J. Elliot and Dr. R. H. Whiteley of Raychem Ltd. in providing the experimental data reproduced in this paper. References [1] Nelson MI, Brindley J, McIntosh AC. The eect of heat sink additives on the ignition and heat release properties of thermally thin thermoplastics. Fire Safety Journal 1997;28(1):67. [2] Babrauskas V. Development of the cone calorimeter ± a benchscale heat release rate apparatus based on oxygen consumption. National Bureau of Standards, Washington, DC. NBSIR 1982. p. 82±2611. [3] Babrauskas V. Development of the cone calorimeter ± a benchscale heat release rate apparatus based on the oxygen consumption principle. Fire and Materials 1984;8(2):81±95.
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[10] Wichman IS, Atreya A. A simpli®ed model for the pyrolysis of charring materials. Comb. and Flame 1987;68:231. [11] Di Blasi C, Wichman IS. Eects of solid-phase properties on ¯ames spreading over composite materials. Comb. and Flame 1995;102:229. [12] Staggs JEJ, Whiteley RH. Modelling the combustion of solidphase fuels in cone calorimeter experiments. Inter¯am `96. Seventh International Fire Science and Eng. Conference Interscience Communications. 1996. p. 103. [13] Staggs JEJ. Thermal degradation of polymers using equivalent ®rst-order single-step kinetics. Submitted to Fire Safety Journal, 1997. [14] Whiteley RH, Elliot PJ. A cone calorimeter study of the eects of alumina trihydrate and magnesium hydroxide on the burning behaviour of an ethylene-vinyl acetate copolymer. Paper presented at the Second Fire and Materials International Conference, September 1993, Crystal City, USA.