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Wetting-induced ignition in cellulosic materials
Fire Safety Journal 37 (2002) 465–479
Wetting-induced ignition in cellulosic materials B.F. Graya,*, M.J. Sextona, B. Halliburtonb, C. Macaskilla a
School of Mathematics and Statistics, University of Sydney, Sydney 2006, Australia b School of Chemistry, Macquarie University, Sydney 2109, Australia
Received 17 October 2000; received in revised form 16 April 2001; accepted 10 December 2001
Abstract A mathematical model of the ignition of cellulose is described which includes both the dry oxidation reaction and also the water-mediated exothermic reaction which is probably hydrolysis. It also takes account of endothermic evaporation of water inside the porous material, exothermic condensation of water vapour and transport of water vapour by diffusion. Appropriate boundary conditions are chosen and the results reported here are restricted to one spatial dimension, but are time dependent. Experimental data obtained recently for bagasse (ground extracted sugar cane) are used as input for the model. However, it is quite likely that the results are of general interest for other cellulose-based materials, many of which are of considerable commercial importance and many of which are believed to undergo spontaneous ignition induced by the presence of water or water vapour. A significant number of experimental observations and conjectures made over a long period are shown to be predictable and interpretable by use of the model, and good agreement with experiment occurs where results are available. r 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction There have been many instances of combustible materials being ignited by moisture referred to in the spontaneous combustion literature, mainly involving cellulose [1–3]. This effect has been attributed to the latent heat of condensation (in the case of water vapour condensation from a humid atmosphere), heat of adsorption (wetting) in the case of direct addition of liquid, or indeed both. These effects have been held responsible, without quantitative study, for spontaneous *Corresponding author. Tel.: +61-02-9449-4156; fax: +61-02-9988-3945. E-mail address: [email protected] (B.F. Gray). 0379-7112/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 9 - 7 1 1 2 ( 0 2 ) 0 0 0 0 2 - 4
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Nomenclature Constants E Ew R Q Z Qw rb Zw Ze Zc Lv k DY ; DW h hY ; hW mw rw rb cb cw Ya Wa F
activation energy of dry reaction, 1.08 105 J mol1 activation energy of wet reaction, 6.5 104 J mol1 universal gas constant, 8.31 J mol1 K1 exothermicity of dry reaction, 1.7 107 J kg1 pre-exponential factor of the dry reaction, 2.7 107 m3 mol1 s1 coefficient of the wet reaction, 1.683 108 J m3 s1 mol2 pre-exponential factor of evaporation, 3.41 104 s1 pre-exponential factor of condensation, 4.7 s1 latent heat of vaporization, 42 103 J mol1 thermal conductivity, 0.5 J m1 K1 s1 diffusivities of water vapour and oxygen resp., 2.5 105 m2 s1 heat transfer coefficient, 5 J m2 K1 s1 water vapour and oxygen transfer coefficients resp., 5 m s1 mass of water, 18 103 kg mol1 density of water, 1 103 kg m3 density of dry bagasse, 125 kg m3 heat capacity of dry bagasse, 1.4 103 J kg1 K1 heat capacity of water, 4.19 103 J kg1 K1 ambient water vapour concentration, 1.74 mol m3 oxygen concentration in air, 8.04 mol m3 moles of O2 consumed per kg of bagasse, 33.33 mol kg1
Variables p T t X Y W
% increase in water content temperature, K time, s liquid water, mol m3 water vapour, mol m3 oxygen, mol m3
ignition in coal stacks, grain dust stockpiles, cotton meal stockpiles, cotton bales, linen stacks, dry freshly made chipboard stacks and many other materials. The general concept, as elaborated in an interesting paper by Back [2], has been that the heat liberated will cause sufficient temperature rise within the material to perturb it over the ‘watershed’ separating subcritical from supercritical behaviour, i.e. it has been conceived as a ‘critical initial value’ [4] problem. As pointed out by Gray and Wake [3] this simple but plausible approach, when formulated quantitatively, raises many questions, which are not immediately obvious. For example, it ignores the fact that the wetted material, although hotter as a result of the perturbation, is now a different material, i.e. its thermal properties will all be different. More importantly, the endothermic evaporation of the added
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water will proceed as soon as the perturbation (addition of water) has finished and this reaction has a significant temperature coefficient and hence stabilizing effect. It is not at all clear under what conditions the wetted material will become supercritical, if any exist at all. Some theoretical work has been carried out on this problem by McIntosh et al. [5], where it was shown for an approximate model without spatial variation that a slight change in atmospheric humidity can have a marked effect on the ignition characteristics of a material. Studies on the storage of bagasse (the residue from extracted sugar cane) [6] showed conclusively that the usual high temperatureFsmall sample basket ignition tests were not capable of being extrapolated to the practical low temperatureFoccurs in the large size case. In fact, they predicted critical temperatures for the latter which were of the order of 901C compared with the observed values of around 301C. It became clear in that work that below 1001C at least, a watermediated reaction was taking place. The possibility that this reaction might be of microbial origin was ruled out in follow up work on this project by Dixon [7] who showed in carefully controlled experiments that the progress of self-heating in sterile and normal bagasse plies was identical within experimental error. These results have been confirmed in our own laboratory-scale calorimetric work to be reported elsewhere. In the small-scale tests carried out above 1001C rapid drying out of the sample occurred, ruling out the possibility of obtaining information on the wet reaction. The possible effects of a generic water-mediated exothermic reaction and simultaneous endothermic evaporation in addition to the usual dry high-temperature oxidation were considered by Gray [8] who showed theoretically that an amazing variety of behaviour could arise from such systems including oscillatory behaviour, homoclinic bifurcations and up to five steady states for some ranges of parameter values. This work was generalized by Sisson et al. [9,10] to include spatial variability and this work led to similar general conclusions of Gray [8] in this more general context. Both studies [8,9] assumed an Arrhenius temperature dependence for the watermediated reaction in the absence of more detailed knowledge but more recent studies of the details of this reaction in the case of bagasse [11,12] have shown this not to be the case. In fact, the reaction has a fairly sharp cutoff at around 60–701C and quantitative conclusions cannot be drawn without including this feature, although it turns out that the qualitative predictions made in [8,9] still hold. Detailed modelling of the general type carried out by Sisson et al. has now been performed using the present model and equations, also making direct use of the experimental data for the bagasse wet reaction [13,14] and the general ignition characteristics of wet bagasse in large stockpiles with critical ambient temperatures around 301C have now been obtained satisfactorily using the present model. Here we apply this detailed model, as briefly formulated in the next section, to a case where a small test pile suffered perturbations such as addition of extraneous liquid water to a partially or wholly dry stockpile, increase in outside relative humidity (or decrease in outside ambient temperature resulting in condensation), etc. We also show some results where pure temperature perturbations or isothermal additions of water occur to compare with the general case.
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Although the experimental data used as input to this model are direct measurements on bagasse, we believe that they will be of generic value for similar situations for many cellulosic materials such as hay, straw, some grains, woodchip piles, etc. Recent work by Gong et al. [15], which is a continuation and generalization of the work of Sisson et al. [9,10] addresses the problem of spontaneous ignition in wet lignite. In common with the present paper, they use a multi-variable reactiondiffusion model, but with a quadratic dependence on water concentration and simple Arrhenius dependence on temperature, in contrast with the empirically determined rate law for bagasse used here that exhibits a local maximum at low temperatures and linear dependence on liquid water concentration.
2. The mathematical formulation Consider a one-dimensional pile of thickness w; where the spatial coordinate x is defined as x½w=2; w=2: This is the simplest case that demonstrates the relevant behaviour and the one with which a comparison can be made with the experimental results of Halliburton [11]. The model equations in dimensional form describing the time evolution of temperature T; liquid and vapour concentrations X and Y and oxygen concentrations W are as follows: ðrb cb þ mw Xcw Þ
qT ¼ Qrb ZW expðE=RTÞ þ Qw rb Zw XW expðEw =RTÞ f ðTÞ qt ð1Þ þ Lv Zc Y Ze X expðLv =RTÞ þ kr2 T;
qY ¼ Ze X expðLv =RTÞ Zc Y þ DY r2 Y ; qt
ð2Þ
qX ¼ Ze X expðLv =RTÞ þ Zc Y ; qt
ð3Þ
qW ¼ F rb ZW expðE=RTÞ F rb Zw XW expðEw =RT Þf ðTÞ qt þ Dw r2 W :
ð4Þ
See the nomenclature section for the list of constants. Two reactions, an oxidation reaction and a moisture-dependent reaction, are represented by the first two terms in the energy equation (1) and also in the equation describing the mass balance for the oxygen concentration W : The function f ðTÞ in the wet reaction terms is defined as f ðTÞ ¼ ftanh½0:6ð58 T þ 273Þ þ 1g=2: This form is an empirical fit to the experimental data obtained in [11], and reflects the presence of a maximum in the wet heat release curve that occurs between 501C and 601C (e.g. see Fig. 8).
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Due to the presence of liquid water, the effects of condensation and evaporation must be included, and these appear in the energy equation as the third and fourth terms, respectively, as well as in the mass balance equations for liquid water concentration X and water vapour concentration Y : We allow for the dependence of the total heat capacity of the bagasse pile on liquid water concentration X in the term appearing on the left-hand side of the energy equation. Heat conduction and water vapour and oxygen diffusion appear as the last terms in their respective equations, whereas diffusion of liquid water is assumed to be insignificant. The boundary conditions for this model are as follows: qT k 7hðT Ta Þ ¼ 0 on x ¼ 7w=2; ð5Þ qx DY
qY 7hY ðY Ya Þ ¼ 0 on x ¼ 7w=2; qx
ð6Þ
qW ð7Þ 7hW ðW Wa Þ ¼ 0 on x ¼ 7w=2; qx where the ambient water vapour concentration is defined as Ya ¼ 1:74 mol m3 (Relative humidity). The constants h; hY and hW are the transfer coefficients for heat, vapour and oxygen, respectively, and k, DY and DW are the thermal conductivity and vapour and oxygen diffusivity in the bagasse. To solve the system of PDEs with its associated boundary conditions, we will implement the method of lines where the spatial derivatives are discretized transforming the PDEs into a system of ODEs. Due to the nature of these types of combustion problems, the availability of ODE integrators which can account for stiffness and multiple time scales makes the method of lines an efficient and accurate method of solution. We solve for the interior values Tj ; Yj ; Xj and Wj at the spatial points xj ¼ jDx; j ¼ 2yn 1 using a second order spatial discretization for the heat conduction and vapour and oxygen diffusion terms, as well as the boundary conditions. That is r2 T ¼ ðTj1 22Tj þ Tjþ1 Þ=ðDxÞ2 where Dx ¼ w=ðn 1Þ; and similarly for r2 Y and r2 W : The boundary values for temperature T are found by DW
TB ¼ ð4TB1 TB2 þ 2DxhTa =kÞ=ð3 þ 2Dxh=kÞ; where TB is the boundary point and TB1 ; TB2 are the interior points. The boundary values for Y and W are found in the same manner but with appropriate heat transfer and diffusion coefficients. The steady-state solution to the system of PDEs is found by setting the time derivatives to zero. This implies that the steady-state liquid water level is Zc Xss ¼ expðLv =RTss ÞYa ; Ze which is dependent on the ambient water vapour concentration Ya and the steadystate temperature Tss : To conduct the heat of wetting experiments, the model equations are integrated to their steady-state values, then a liquid water concentration change DX is added with a corresponding temperature perturbation DT: Call this time tw : Note that the value
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of X ð0; t ¼ tw Þ depends on the relative humidity, as well as Tss which is spatially dependent and varies according to other parameters in the model. The temperature perturbation DT is defined as 5p ; DT ¼ 1 þ X ð0; tw Þ where p is the percentage increase in water content. For the calculations reported in later Section 3 we use the mixed boundary conditions, and for the remaining results, we have Dirchlet boundary conditions, i.e. T ¼ Ta ¼ 301C; Y ¼ Ya =(1.74 mol m3) (Relative humidity) and W ¼ Wa ¼ 8:04 mol m3.
3. The bagasse data and simulation In experimental work which will be described in detail elsewhere [11,12] a very large ‘infinite slab’ of bagasse was constructed measuring approximately 8 m 8 m 1.2 m standing in a vertical position. It was instrumented with a large number of thermocouples and moisture sensors enabling the continuous measurement and recording of temperature, humidity and total moisture profiles as functions of time and space. The dimensions of the slab ensured that to a very high degree of approximation the behaviour was one-dimensional, i.e. the only significant spatial variation was across the shortest dimension (1.2 m) of the slab. The material, straight from the mill, contained 50% moisture (on a wet weight basis) and was at a temperature of approximately 501C. Initially, the temperature rose quite quickly (not visible in Fig. 1) to a maximum peak then dropped to a quasi-
Fig. 1. Observed temperature traces in 1.2 m 8.0 m 8.0 m bagasse slab across a section.
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steady value which remained almost constant for many months. This behaviour is shown in Fig. 1. The relatively rapid drop in temperature to slightly above ambient (which was in the region of 301C for most of the test) occurring after some months represents an ‘extinction bifurcation’ caused by a drop in wet reaction rate due to water loss by diffusion and evaporation. The small temperature rise occurring after approximately 400 days coincides with a heavy wetting by a sizeable rainstorm. The large temperature rise back to the original ‘quasi-steady’ state occurring after approximately 450 days was due to extreme wetting by a tropical storm which increased the water content sufficiently to cause transition back to the thermally active state of the slab of material. This temperature rise represents a reversal of the ‘extinction bifurcation’ which arose due to gradual drying out of the pile in the first instance. The fact that the ‘active state’ persisted for a much shorter period probably indicates that the water content was less than the original. If these qualitative interpretations are correct it should be possible to simulate them using the model elaborated in the previous section. This has been done with good agreement with the observed results. Fig. 2 shows the temperature history at the centre of the slab (where the temperature is a maximum) as calculated by the model with a constant ambient temperature of 301C and an initial water content of 50% and an initial temperature for the bagasse of 551C which was assumed to be spatially uniform as was the case experimentally. The system was left to evolve for 500 days by which stage the system had reached a quasi-steady state with centre temperature of 371C and water content (at the centre)
60
o
Centre temperature ( C)
55
50
45
40
35 0
200
400 600 Time (days)
800
1000
Fig. 2. Centre temperature of a bagasse slab. At 500 days, a percentage of the initial liquid water level is added: 10% (dashed) and 70% (solid).
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of 23% w/w. At this time a perturbation of 10% of the initial value of water was added uniformly to the system, with no direct temperature perturbation. The reason for the latter is that the heat of wetting of cellulose is a differential quantity which declines towards zero for initial amounts of water approaching 20% [2]. There is a small temperature rise of a few degrees gradually declining back to the stable steady state. This bears a strong resemblance to the observed perturbation shown in Fig. 1 after 410 days. In Fig. 2 we also show the effect of a much larger (but still isothermal) perturbation of the system by the addition of 70% of the initial value of water uniformly distributed throughout the system, again after around 500 days. In this case the system jumps back to the original quasi-steady state in which it had been for the first 200 days or so (before the extinction bifurcation due to drying out). Subsequently, the drying out process repeats itself and the extinction bifurcation occurs again, as before. These perturbation computer experiments are a rather good representation of the observed behaviour, particularly when Fig. 3 is taken into account. Fig. 3 shows the time-dependent behaviour of the liquid water concentration at the centre of the slab, where in fact it displays a spatial minimum (both experimentally and theoretically). Since the water sampling at the centre of the pile required the taking of core samples, this was only done at a small number of times, but was done for a number of positions across the pile. This showed a minimum at the centre each time but did not provide enough data for a full history plot. The above observations show the transition from an almost inactive stable quasisteady state (with only a few degrees of self-heating above ambient) to a more
Fig. 3. Central liquid water percentage of a bagasse slab. At 500 days, a percentage of the initial liquid water is added: 10% (dashed), 70% (solid).
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reactive, intermediate quasi-steady state showing rather more reaction and selfheating of about 30–351 above ambient temperature. The upward transition occurs as a result of the isothermal addition of water and the downward transition (extinction of the intermediate steady state) as a result of the gradual drying out and consequent quenching of the water mediated reaction. The water loss occurs directly as a result of evaporation, diffusion of water vapour towards the pile edges, followed by loss to the atmosphere. Throughout the simulations the relative humidity was maintained at a constant 70%, very similar to the experimental conditions in Mackay, North Queensland, where the experimental work was carried out. In Fig. 4 we show the effect of isothermal addition of liquid water to a pile under conditions where transition to the ignited state can occur, i.e. thermal runaway. In this case the intermediate state with self-heating of 351 (commonly observed in bagasse piles) is by-passed and the low activity state is converted directly to ignition. In this case we started with a relative humidity of only 5% and a higher ambient temperature of 451C. The pile had a width of 3.75 m and settled into a steady state with 81 self heating. The addition of 33.3% liquid water isothermally resulted in only a transient response, but the addition of 77.8% liquid water isothermally resulted in ignition. This is a conservative estimate of this effect as at the low percentage of liquid water present at the time of the perturbation a significant heat of wetting would also be present. Such effects will be discussed more fully in a later section. Although the conditions of the experimental pile were such that it did not ignite, we believe that our ability to provide a reasonable simulation of what did happen lends weight to our simulation of direct ignition of a pile under different (and not unreasonable) conditions.
70
Centre temperature (o C)
65
60
55
50
45 0
200
400 600 Time (days)
800
1000
Fig. 4. Effect of addition of liquid water for a bagasse slab of 3.75 m and ambient temperature 451C: 33.3% (dashed) and 77.8% (solid).
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4. Ignition by humidity change Back [2] has commented on the possibility of ignition by condensation of water vapour resulting from a relatively rapid increase in relative humidity. He comments specifically on bales of cotton coming from a dry climate igniting when stored in a humid harbour climate. He also refers to the spontaneous ignition of freshly produced (hence warm and dry) insulating board as well as freshly made cardboard and particle board. An increase in humidity can result in energy release by virtue of both the latent heat of condensation and any heat of wetting or absorption. However, in the present model it can also increase the rate of heat release by the water-mediated reaction. The mechanism by which this occurs is basically that evaporation of liquid water is slowed by the increased vapour pressure of water vapour with a resultant increase in water content. We have calculated a series of steady-state curves of centre temperature versus slab width at various external humidities ranging from 0% to 100%, all at an ambient temperature of 300C. These are shown in Fig. 5. This diagram has many interesting features. Firstly, we note that for any given relative humidity there is a maximum pile width above which no low-temperature stable state exists, i.e. ignition will occur. This maximum possible width is represented by the turning point on each of the curves. The lower branches of the curves are stable (solid lines) and the upper branches unstable (dashed lines). The unstable steady states, crudely speaking act as ‘watersheds’ separating ignition from non-ignition for pile temperature. Thus for example, a 5 m wide pile, for any humidity, will ignite if it is built of material hotter 100
Centre temperature ( o C)
90 80 70 60 50
100%
70%
40
50%
20%
10%
5%
30 0
0% 2
4
6
8
10
Width (m)
Fig. 5. Steady-state bifurcation diagrams with ambient temperature of 301C for various relative humidities.
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than 751C but a 2 m wide pile would cool down if constructed of material at this temperature. A 6 m pile, if constructed of material cooler than 551C would cool down to near ambient temperature in external relative humidity of 0%, 5% and 10%. However, for relative humidity of 20% and higher there would be no stable low-temperature steady state for this pile and ignition would occur. Similarly, if a pile were to be constructed in a very low humidity environment it would exist in a stable condition on the lower branch of the curve but an increase of relative humidity at constant temperature to anything greater than 20% would result in ignition. The dashed curve linking the turning points of the bifurcation curves for 0%, 5%, 10% and 20% is a continuous development from the completely dry bifurcation point at nearly 9 m width. In this case there is only the dry oxidation reaction taking place and producing heat. We can think of the wet reaction being a perturbation of this up to 20% relative humidity. However, the dashed curve linking the turning points on the curves for relative humidities 20%, 50%, 70% and 100% is not continuously linked to the former as shown by the discontinuous curves in Fig. 6. For the higher relative humidities the ignition point (the turning point on the curve) represents a qualitatively different phenomenon from dry ignition. The stable steady state which ceases to exist at the bifurcation is the ‘active state’ referred to in an earlier section. It is represented by the flattened part of the curves in Fig. 5 for relative humidities 50–100%. The centre temperature in this case is about 601C, i.e. of the order of 30 degrees above ambient temperature. These figures are not particularly sensitive to input parameters such as density, conductivity, diffusivity, etc. and thus have a generic significance. It can safely be said that for cellulosic materials exhibiting a heat release curve similar to that measured for bagasse, ignition by a ‘dry’ mechanism will be related to a stable subcritical state showing
65
Centre temperature ( o C)
60
55
50
45
40
35 4
5
6
7
8 Width (m)
9
10
11
12
Fig. 6. Turning points of steady-state bifurcation curves shown in Fig. 5.
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only a few degrees of self-heating. On the other hand, ignition by a ‘wet’ mechanism is related to stable subcritical behaviour showing the order of 30–351 self-heating. This has been known for a long time in the sugar industry, but not properly understood or quantified in this way. The ‘rule of thumb’ in North Queensland has long been ‘‘If it gets above 701C it is going to ignite!’’ Ambient temperatures of 30– 351C year round are common so self-heating of 301 has been recognized empirically as stable non-ignition behaviour, in accord with our calculations. The simplest interpretation of humidity-induced ignition is to observe that (in Fig. 5) the bifurcation curve moves to the left as humidity increases and when the turning point passes to a value smaller than the actual width of the pile no stable steady state exists and ignition occurs. The physical mechanisms of condensation, wetting, etc. are built into the bifurcation diagram in the calculation of the position and nature (stable or unstable) of the steady states occurring, and most importantly when they bifurcate and consequently cease to exist.
5. Ignition by direct wetting We can also use this model to study the effects of addition of liquid water directly to the pile, in contrast to condensation from the vapour phase as in the previous section. As an example we will take an ambient temperature of 301C, relative humidities of 20%, 70% and 100%. We consider a pile with an initial temperature of 301C and integrate the differential equations until a stable steady state is reached (obviously, we have to choose the other parameters so that one does exist before the perturbation). Since we are adding liquid water in this case the latent heat is not relevant but any heat of wetting is. Back [2] gives a graphical dependence of this quantity on water content for cotton. His figures show for that case that the heat of wetting drops off quite rapidly with initial water content. We have followed this empirical approach and assumed the relationship DT ¼
5p 1 þ X ð0; tw Þ
between temperature rise and water content at the time of the perturbation. This equation determines the relationship between the two components of the perturbation applied, i.e. the water content increase and the (assumed simultaneous) corresponding temperature increase. For convenience we also assume that these two perturbations are spatially homogeneous, but the results are not at all sensitive to this. For the humidity values mentioned above we have integrated the differential equations to steady state and then applied the perturbation repeatedly until a critical perturbation was determined, i.e. the smallest one which produced ignition. This has been done for a series of pile sizes (widths) and the results are shown in Fig. 7. The apparent irregularities in these curves arise as a direct result of the shapes of the bifurcation curves in Fig. 5. In this figure we have plotted temperature rise as a measure of the perturbation but the relation given above shows that we could
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120
o
Temperature perturbation ( C)
RH = 70% 100
80
60 RH = 0% 40 RH = 0% 20 RH = 70% 0
0
2
RH = 20% 4
6
8
10
Width (m)
Fig. 7. Water-dependent temperature perturbation for relative humidities 0%, 70% and 100%.
equally well have used amount of water added or some arbitrary combination of the two as a suitable norm. The physical interpretation is straightforward, for example the 0% humidity case for an 8 m pile requires only sufficient water addition to raise the temperature by around 101. This case is clearly sensitive to water addition as one would expect. On the other hand, in 70% relative humidity the existence of stable steady states for piles wider than 3.5 m is highly questionable and addition of extra water is not very significant as evidenced by the very steep curve for this case. Since we are using a model which involves a low-temperature exothermic wet reaction it is of some interest to examine how much of the effect shown in Fig. 7 is due to increased reaction rate, due to water concentration and how much of it is due to the temperature increase associated with the wetting. Accordingly, we have repeated the computer experiments with perturbations consisting of a temperature rise only, other conditions being identical with the previous runs. These computations show that the non-thermal component of each perturbation is negligible. We can safely conclude that dropping the wet reaction term would give rather similar results to the above and such ‘wetting-induced ignition’ can occur even in the absence of the wet reaction. Basically, this is a question of sensitivity of the heat release rate to the two independent variables, moisture concentration and temperature. The sensitivity with respect to temperature is much the larger of the two. Fig. 8 shows a number of heat generation curves as functions of reactant temperature for various (constant) water contents corresponding to equilibrium with various values of relative humidity as shown in the figure. Reference to Figs. 5 and 6 reveals the existence of two distinct series of fold bifurcations over a limited relative humidity range. This is easily interpreted in terms of Fig. 8 from which it can be seen that a necessary condition for such a range is the
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1 0.9
Heat generation rate
0.8 0.7 0.6
RH = 100%
0.5 0.4 0.3 0.2 0.1 0 10
RH = 0% 20
30 40 50 o Centre temperature ( C)
60
70
Fig. 8. Dimensionless heat generation rate (dry+wet reaction terms) at equilibrium for relative humidities 0%, 5%, 10%, 20%, 50%, 70% and 100%.
existence of two turning points (local maximum and local minimum) in the heat release curve. These disappear below 18% relative humidity.
6. Conclusions We have formulated and numerically integrated a reaction diffusion model including experimentally measured water-mediated exothermic reaction, dry oxidative exothermic reaction, endothermic water evaporation, exothermic water vapour condensation, water vapour diffusion, heat conduction and appropriate boundary conditions for water vapour and temperature. This model was originally formulated to describe the behaviour of large piles of bagasse and to provide a means of studying the optimization of water loss from the pile in order to increase its calorific value for electricity production. We have shown that this model is capable of describing a wide variety of effects involving addition of liquid water, water vapour or both with corresponding temperature increases arising from condensation and wetting (adsorption). We have shown that addition of liquid water to a pile in a stable steady-state condition can cause ignition. We have shown that increasing relative humidity outside the pile can cause ignition of a pile in a stable steady state already attained by exposure to a lower external relative humidity. Although these calculations use input obtained by direct experiment on bagasse, we believe they may be of generic significance for the possible occurrence of spontaneous ignition induced by interaction of water or water vapour. This may be
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of significance for other commercially important bodies such as cotton bales, woodchip piles, peat stacks, haystacks, chipboard and particle board, brown coal, lignite and other materials.
Acknowledgements The authors wish to thank the Australian Research Council and the Sugar Research Institute, Mackay, Queensland for the award of a SPIRT collaborative research grant and a graduate scholarship to B.H.
References [1] Walker IK. The role of water in combustion. Fire Research Abstracts and Reviews, vol. 9, #1, 1967. p. 5–22 (National Academy of Sciences, National research Council). [2] Back EL. Autoignition in hygroscopic organic materials. Fire Safety J 1981;4:185–96. [3] Gray BF, Wake GC. The ignition of hygroscopic materials by water. Combust Flame 1990;79:2–6. [4] Gray BF, Wake GC. Critical initial conditions for thermal ignition. Math Comput Modelling 1993;18:65–75. [5] McIntosh A, Gray BF, Wake GC. Ignition of a combustible material in the presence of a damp atmosphere. Phys Lett A 1994;191:61–70. [6] Gray BF, Griffiths JG, Hasko SM. Spontaneous ignition hazards in stockpiles of cellulosic materials. J Chem Tech Biotech 1984;34A:453–63. [7] Dixon T. Spontaneous combustion in bagasse stockpiles. Proceedings of the Australian Sugar Cane Technology, Mackay, Queensland, Australia, 1988. p. 53–61. [8] Gray BF. Analysis of chemical kinetic systems over the entire parameter space III, a wet combustion system. Proc Roy Soc A 1990;429:449–58. [9] Sisson RA, Swift A, Wake GC, Gray BF. The self heating of damp cellulosic materials IFhigh thermal conductivity and diffusivity. IMA J Appl Math 1992;49:273–91. [10] Sisson RA, Swift A, Wake GC, Gray BF. The self heating of damp cellulosic materials II, on the steady states of the spatially distributed case. IMA J Appl Math 1993;50:285–306. [11] Halliburton B. Ph.D. thesis, Macquarie University, Sydney, Australia, 2001. [12] Halliburton B. Gray BF, Carras J. in preparation. [13] Macaskill C, Sexton MJ, Gray BF. A reaction diffusion model of stored bagasse. Anziam J 2001;43:13–35. [14] Sexton MJ, Macaskill C, Gray BF. Self heating and drying in two-dimensional bagasse piles. Combus Theory Modelling 2001;5:517–36. [15] Gong R, Burnell JG, Wake GC. Modelling spontaneous combustion in wet lignite. Combust Theory Modelling 1999;3:215–32.
Wetting-induced ignition in cellulosic materials B.F. Graya,*, M.J. Sextona, B. Halliburtonb, C. Macaskilla a
School of Mathematics and Statistics, University of Sydney, Sydney 2006, Australia b School of Chemistry, Macquarie University, Sydney 2109, Australia
Received 17 October 2000; received in revised form 16 April 2001; accepted 10 December 2001
Abstract A mathematical model of the ignition of cellulose is described which includes both the dry oxidation reaction and also the water-mediated exothermic reaction which is probably hydrolysis. It also takes account of endothermic evaporation of water inside the porous material, exothermic condensation of water vapour and transport of water vapour by diffusion. Appropriate boundary conditions are chosen and the results reported here are restricted to one spatial dimension, but are time dependent. Experimental data obtained recently for bagasse (ground extracted sugar cane) are used as input for the model. However, it is quite likely that the results are of general interest for other cellulose-based materials, many of which are of considerable commercial importance and many of which are believed to undergo spontaneous ignition induced by the presence of water or water vapour. A significant number of experimental observations and conjectures made over a long period are shown to be predictable and interpretable by use of the model, and good agreement with experiment occurs where results are available. r 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction There have been many instances of combustible materials being ignited by moisture referred to in the spontaneous combustion literature, mainly involving cellulose [1–3]. This effect has been attributed to the latent heat of condensation (in the case of water vapour condensation from a humid atmosphere), heat of adsorption (wetting) in the case of direct addition of liquid, or indeed both. These effects have been held responsible, without quantitative study, for spontaneous *Corresponding author. Tel.: +61-02-9449-4156; fax: +61-02-9988-3945. E-mail address: [email protected] (B.F. Gray). 0379-7112/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 9 - 7 1 1 2 ( 0 2 ) 0 0 0 0 2 - 4
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Nomenclature Constants E Ew R Q Z Qw rb Zw Ze Zc Lv k DY ; DW h hY ; hW mw rw rb cb cw Ya Wa F
activation energy of dry reaction, 1.08 105 J mol1 activation energy of wet reaction, 6.5 104 J mol1 universal gas constant, 8.31 J mol1 K1 exothermicity of dry reaction, 1.7 107 J kg1 pre-exponential factor of the dry reaction, 2.7 107 m3 mol1 s1 coefficient of the wet reaction, 1.683 108 J m3 s1 mol2 pre-exponential factor of evaporation, 3.41 104 s1 pre-exponential factor of condensation, 4.7 s1 latent heat of vaporization, 42 103 J mol1 thermal conductivity, 0.5 J m1 K1 s1 diffusivities of water vapour and oxygen resp., 2.5 105 m2 s1 heat transfer coefficient, 5 J m2 K1 s1 water vapour and oxygen transfer coefficients resp., 5 m s1 mass of water, 18 103 kg mol1 density of water, 1 103 kg m3 density of dry bagasse, 125 kg m3 heat capacity of dry bagasse, 1.4 103 J kg1 K1 heat capacity of water, 4.19 103 J kg1 K1 ambient water vapour concentration, 1.74 mol m3 oxygen concentration in air, 8.04 mol m3 moles of O2 consumed per kg of bagasse, 33.33 mol kg1
Variables p T t X Y W
% increase in water content temperature, K time, s liquid water, mol m3 water vapour, mol m3 oxygen, mol m3
ignition in coal stacks, grain dust stockpiles, cotton meal stockpiles, cotton bales, linen stacks, dry freshly made chipboard stacks and many other materials. The general concept, as elaborated in an interesting paper by Back [2], has been that the heat liberated will cause sufficient temperature rise within the material to perturb it over the ‘watershed’ separating subcritical from supercritical behaviour, i.e. it has been conceived as a ‘critical initial value’ [4] problem. As pointed out by Gray and Wake [3] this simple but plausible approach, when formulated quantitatively, raises many questions, which are not immediately obvious. For example, it ignores the fact that the wetted material, although hotter as a result of the perturbation, is now a different material, i.e. its thermal properties will all be different. More importantly, the endothermic evaporation of the added
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water will proceed as soon as the perturbation (addition of water) has finished and this reaction has a significant temperature coefficient and hence stabilizing effect. It is not at all clear under what conditions the wetted material will become supercritical, if any exist at all. Some theoretical work has been carried out on this problem by McIntosh et al. [5], where it was shown for an approximate model without spatial variation that a slight change in atmospheric humidity can have a marked effect on the ignition characteristics of a material. Studies on the storage of bagasse (the residue from extracted sugar cane) [6] showed conclusively that the usual high temperatureFsmall sample basket ignition tests were not capable of being extrapolated to the practical low temperatureFoccurs in the large size case. In fact, they predicted critical temperatures for the latter which were of the order of 901C compared with the observed values of around 301C. It became clear in that work that below 1001C at least, a watermediated reaction was taking place. The possibility that this reaction might be of microbial origin was ruled out in follow up work on this project by Dixon [7] who showed in carefully controlled experiments that the progress of self-heating in sterile and normal bagasse plies was identical within experimental error. These results have been confirmed in our own laboratory-scale calorimetric work to be reported elsewhere. In the small-scale tests carried out above 1001C rapid drying out of the sample occurred, ruling out the possibility of obtaining information on the wet reaction. The possible effects of a generic water-mediated exothermic reaction and simultaneous endothermic evaporation in addition to the usual dry high-temperature oxidation were considered by Gray [8] who showed theoretically that an amazing variety of behaviour could arise from such systems including oscillatory behaviour, homoclinic bifurcations and up to five steady states for some ranges of parameter values. This work was generalized by Sisson et al. [9,10] to include spatial variability and this work led to similar general conclusions of Gray [8] in this more general context. Both studies [8,9] assumed an Arrhenius temperature dependence for the watermediated reaction in the absence of more detailed knowledge but more recent studies of the details of this reaction in the case of bagasse [11,12] have shown this not to be the case. In fact, the reaction has a fairly sharp cutoff at around 60–701C and quantitative conclusions cannot be drawn without including this feature, although it turns out that the qualitative predictions made in [8,9] still hold. Detailed modelling of the general type carried out by Sisson et al. has now been performed using the present model and equations, also making direct use of the experimental data for the bagasse wet reaction [13,14] and the general ignition characteristics of wet bagasse in large stockpiles with critical ambient temperatures around 301C have now been obtained satisfactorily using the present model. Here we apply this detailed model, as briefly formulated in the next section, to a case where a small test pile suffered perturbations such as addition of extraneous liquid water to a partially or wholly dry stockpile, increase in outside relative humidity (or decrease in outside ambient temperature resulting in condensation), etc. We also show some results where pure temperature perturbations or isothermal additions of water occur to compare with the general case.
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Although the experimental data used as input to this model are direct measurements on bagasse, we believe that they will be of generic value for similar situations for many cellulosic materials such as hay, straw, some grains, woodchip piles, etc. Recent work by Gong et al. [15], which is a continuation and generalization of the work of Sisson et al. [9,10] addresses the problem of spontaneous ignition in wet lignite. In common with the present paper, they use a multi-variable reactiondiffusion model, but with a quadratic dependence on water concentration and simple Arrhenius dependence on temperature, in contrast with the empirically determined rate law for bagasse used here that exhibits a local maximum at low temperatures and linear dependence on liquid water concentration.
2. The mathematical formulation Consider a one-dimensional pile of thickness w; where the spatial coordinate x is defined as x½w=2; w=2: This is the simplest case that demonstrates the relevant behaviour and the one with which a comparison can be made with the experimental results of Halliburton [11]. The model equations in dimensional form describing the time evolution of temperature T; liquid and vapour concentrations X and Y and oxygen concentrations W are as follows: ðrb cb þ mw Xcw Þ
qT ¼ Qrb ZW expðE=RTÞ þ Qw rb Zw XW expðEw =RTÞ f ðTÞ qt ð1Þ þ Lv Zc Y Ze X expðLv =RTÞ þ kr2 T;
qY ¼ Ze X expðLv =RTÞ Zc Y þ DY r2 Y ; qt
ð2Þ
qX ¼ Ze X expðLv =RTÞ þ Zc Y ; qt
ð3Þ
qW ¼ F rb ZW expðE=RTÞ F rb Zw XW expðEw =RT Þf ðTÞ qt þ Dw r2 W :
ð4Þ
See the nomenclature section for the list of constants. Two reactions, an oxidation reaction and a moisture-dependent reaction, are represented by the first two terms in the energy equation (1) and also in the equation describing the mass balance for the oxygen concentration W : The function f ðTÞ in the wet reaction terms is defined as f ðTÞ ¼ ftanh½0:6ð58 T þ 273Þ þ 1g=2: This form is an empirical fit to the experimental data obtained in [11], and reflects the presence of a maximum in the wet heat release curve that occurs between 501C and 601C (e.g. see Fig. 8).
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Due to the presence of liquid water, the effects of condensation and evaporation must be included, and these appear in the energy equation as the third and fourth terms, respectively, as well as in the mass balance equations for liquid water concentration X and water vapour concentration Y : We allow for the dependence of the total heat capacity of the bagasse pile on liquid water concentration X in the term appearing on the left-hand side of the energy equation. Heat conduction and water vapour and oxygen diffusion appear as the last terms in their respective equations, whereas diffusion of liquid water is assumed to be insignificant. The boundary conditions for this model are as follows: qT k 7hðT Ta Þ ¼ 0 on x ¼ 7w=2; ð5Þ qx DY
qY 7hY ðY Ya Þ ¼ 0 on x ¼ 7w=2; qx
ð6Þ
qW ð7Þ 7hW ðW Wa Þ ¼ 0 on x ¼ 7w=2; qx where the ambient water vapour concentration is defined as Ya ¼ 1:74 mol m3 (Relative humidity). The constants h; hY and hW are the transfer coefficients for heat, vapour and oxygen, respectively, and k, DY and DW are the thermal conductivity and vapour and oxygen diffusivity in the bagasse. To solve the system of PDEs with its associated boundary conditions, we will implement the method of lines where the spatial derivatives are discretized transforming the PDEs into a system of ODEs. Due to the nature of these types of combustion problems, the availability of ODE integrators which can account for stiffness and multiple time scales makes the method of lines an efficient and accurate method of solution. We solve for the interior values Tj ; Yj ; Xj and Wj at the spatial points xj ¼ jDx; j ¼ 2yn 1 using a second order spatial discretization for the heat conduction and vapour and oxygen diffusion terms, as well as the boundary conditions. That is r2 T ¼ ðTj1 22Tj þ Tjþ1 Þ=ðDxÞ2 where Dx ¼ w=ðn 1Þ; and similarly for r2 Y and r2 W : The boundary values for temperature T are found by DW
TB ¼ ð4TB1 TB2 þ 2DxhTa =kÞ=ð3 þ 2Dxh=kÞ; where TB is the boundary point and TB1 ; TB2 are the interior points. The boundary values for Y and W are found in the same manner but with appropriate heat transfer and diffusion coefficients. The steady-state solution to the system of PDEs is found by setting the time derivatives to zero. This implies that the steady-state liquid water level is Zc Xss ¼ expðLv =RTss ÞYa ; Ze which is dependent on the ambient water vapour concentration Ya and the steadystate temperature Tss : To conduct the heat of wetting experiments, the model equations are integrated to their steady-state values, then a liquid water concentration change DX is added with a corresponding temperature perturbation DT: Call this time tw : Note that the value
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of X ð0; t ¼ tw Þ depends on the relative humidity, as well as Tss which is spatially dependent and varies according to other parameters in the model. The temperature perturbation DT is defined as 5p ; DT ¼ 1 þ X ð0; tw Þ where p is the percentage increase in water content. For the calculations reported in later Section 3 we use the mixed boundary conditions, and for the remaining results, we have Dirchlet boundary conditions, i.e. T ¼ Ta ¼ 301C; Y ¼ Ya =(1.74 mol m3) (Relative humidity) and W ¼ Wa ¼ 8:04 mol m3.
3. The bagasse data and simulation In experimental work which will be described in detail elsewhere [11,12] a very large ‘infinite slab’ of bagasse was constructed measuring approximately 8 m 8 m 1.2 m standing in a vertical position. It was instrumented with a large number of thermocouples and moisture sensors enabling the continuous measurement and recording of temperature, humidity and total moisture profiles as functions of time and space. The dimensions of the slab ensured that to a very high degree of approximation the behaviour was one-dimensional, i.e. the only significant spatial variation was across the shortest dimension (1.2 m) of the slab. The material, straight from the mill, contained 50% moisture (on a wet weight basis) and was at a temperature of approximately 501C. Initially, the temperature rose quite quickly (not visible in Fig. 1) to a maximum peak then dropped to a quasi-
Fig. 1. Observed temperature traces in 1.2 m 8.0 m 8.0 m bagasse slab across a section.
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steady value which remained almost constant for many months. This behaviour is shown in Fig. 1. The relatively rapid drop in temperature to slightly above ambient (which was in the region of 301C for most of the test) occurring after some months represents an ‘extinction bifurcation’ caused by a drop in wet reaction rate due to water loss by diffusion and evaporation. The small temperature rise occurring after approximately 400 days coincides with a heavy wetting by a sizeable rainstorm. The large temperature rise back to the original ‘quasi-steady’ state occurring after approximately 450 days was due to extreme wetting by a tropical storm which increased the water content sufficiently to cause transition back to the thermally active state of the slab of material. This temperature rise represents a reversal of the ‘extinction bifurcation’ which arose due to gradual drying out of the pile in the first instance. The fact that the ‘active state’ persisted for a much shorter period probably indicates that the water content was less than the original. If these qualitative interpretations are correct it should be possible to simulate them using the model elaborated in the previous section. This has been done with good agreement with the observed results. Fig. 2 shows the temperature history at the centre of the slab (where the temperature is a maximum) as calculated by the model with a constant ambient temperature of 301C and an initial water content of 50% and an initial temperature for the bagasse of 551C which was assumed to be spatially uniform as was the case experimentally. The system was left to evolve for 500 days by which stage the system had reached a quasi-steady state with centre temperature of 371C and water content (at the centre)
60
o
Centre temperature ( C)
55
50
45
40
35 0
200
400 600 Time (days)
800
1000
Fig. 2. Centre temperature of a bagasse slab. At 500 days, a percentage of the initial liquid water level is added: 10% (dashed) and 70% (solid).
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of 23% w/w. At this time a perturbation of 10% of the initial value of water was added uniformly to the system, with no direct temperature perturbation. The reason for the latter is that the heat of wetting of cellulose is a differential quantity which declines towards zero for initial amounts of water approaching 20% [2]. There is a small temperature rise of a few degrees gradually declining back to the stable steady state. This bears a strong resemblance to the observed perturbation shown in Fig. 1 after 410 days. In Fig. 2 we also show the effect of a much larger (but still isothermal) perturbation of the system by the addition of 70% of the initial value of water uniformly distributed throughout the system, again after around 500 days. In this case the system jumps back to the original quasi-steady state in which it had been for the first 200 days or so (before the extinction bifurcation due to drying out). Subsequently, the drying out process repeats itself and the extinction bifurcation occurs again, as before. These perturbation computer experiments are a rather good representation of the observed behaviour, particularly when Fig. 3 is taken into account. Fig. 3 shows the time-dependent behaviour of the liquid water concentration at the centre of the slab, where in fact it displays a spatial minimum (both experimentally and theoretically). Since the water sampling at the centre of the pile required the taking of core samples, this was only done at a small number of times, but was done for a number of positions across the pile. This showed a minimum at the centre each time but did not provide enough data for a full history plot. The above observations show the transition from an almost inactive stable quasisteady state (with only a few degrees of self-heating above ambient) to a more
Fig. 3. Central liquid water percentage of a bagasse slab. At 500 days, a percentage of the initial liquid water is added: 10% (dashed), 70% (solid).
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reactive, intermediate quasi-steady state showing rather more reaction and selfheating of about 30–351 above ambient temperature. The upward transition occurs as a result of the isothermal addition of water and the downward transition (extinction of the intermediate steady state) as a result of the gradual drying out and consequent quenching of the water mediated reaction. The water loss occurs directly as a result of evaporation, diffusion of water vapour towards the pile edges, followed by loss to the atmosphere. Throughout the simulations the relative humidity was maintained at a constant 70%, very similar to the experimental conditions in Mackay, North Queensland, where the experimental work was carried out. In Fig. 4 we show the effect of isothermal addition of liquid water to a pile under conditions where transition to the ignited state can occur, i.e. thermal runaway. In this case the intermediate state with self-heating of 351 (commonly observed in bagasse piles) is by-passed and the low activity state is converted directly to ignition. In this case we started with a relative humidity of only 5% and a higher ambient temperature of 451C. The pile had a width of 3.75 m and settled into a steady state with 81 self heating. The addition of 33.3% liquid water isothermally resulted in only a transient response, but the addition of 77.8% liquid water isothermally resulted in ignition. This is a conservative estimate of this effect as at the low percentage of liquid water present at the time of the perturbation a significant heat of wetting would also be present. Such effects will be discussed more fully in a later section. Although the conditions of the experimental pile were such that it did not ignite, we believe that our ability to provide a reasonable simulation of what did happen lends weight to our simulation of direct ignition of a pile under different (and not unreasonable) conditions.
70
Centre temperature (o C)
65
60
55
50
45 0
200
400 600 Time (days)
800
1000
Fig. 4. Effect of addition of liquid water for a bagasse slab of 3.75 m and ambient temperature 451C: 33.3% (dashed) and 77.8% (solid).
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4. Ignition by humidity change Back [2] has commented on the possibility of ignition by condensation of water vapour resulting from a relatively rapid increase in relative humidity. He comments specifically on bales of cotton coming from a dry climate igniting when stored in a humid harbour climate. He also refers to the spontaneous ignition of freshly produced (hence warm and dry) insulating board as well as freshly made cardboard and particle board. An increase in humidity can result in energy release by virtue of both the latent heat of condensation and any heat of wetting or absorption. However, in the present model it can also increase the rate of heat release by the water-mediated reaction. The mechanism by which this occurs is basically that evaporation of liquid water is slowed by the increased vapour pressure of water vapour with a resultant increase in water content. We have calculated a series of steady-state curves of centre temperature versus slab width at various external humidities ranging from 0% to 100%, all at an ambient temperature of 300C. These are shown in Fig. 5. This diagram has many interesting features. Firstly, we note that for any given relative humidity there is a maximum pile width above which no low-temperature stable state exists, i.e. ignition will occur. This maximum possible width is represented by the turning point on each of the curves. The lower branches of the curves are stable (solid lines) and the upper branches unstable (dashed lines). The unstable steady states, crudely speaking act as ‘watersheds’ separating ignition from non-ignition for pile temperature. Thus for example, a 5 m wide pile, for any humidity, will ignite if it is built of material hotter 100
Centre temperature ( o C)
90 80 70 60 50
100%
70%
40
50%
20%
10%
5%
30 0
0% 2
4
6
8
10
Width (m)
Fig. 5. Steady-state bifurcation diagrams with ambient temperature of 301C for various relative humidities.
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than 751C but a 2 m wide pile would cool down if constructed of material at this temperature. A 6 m pile, if constructed of material cooler than 551C would cool down to near ambient temperature in external relative humidity of 0%, 5% and 10%. However, for relative humidity of 20% and higher there would be no stable low-temperature steady state for this pile and ignition would occur. Similarly, if a pile were to be constructed in a very low humidity environment it would exist in a stable condition on the lower branch of the curve but an increase of relative humidity at constant temperature to anything greater than 20% would result in ignition. The dashed curve linking the turning points of the bifurcation curves for 0%, 5%, 10% and 20% is a continuous development from the completely dry bifurcation point at nearly 9 m width. In this case there is only the dry oxidation reaction taking place and producing heat. We can think of the wet reaction being a perturbation of this up to 20% relative humidity. However, the dashed curve linking the turning points on the curves for relative humidities 20%, 50%, 70% and 100% is not continuously linked to the former as shown by the discontinuous curves in Fig. 6. For the higher relative humidities the ignition point (the turning point on the curve) represents a qualitatively different phenomenon from dry ignition. The stable steady state which ceases to exist at the bifurcation is the ‘active state’ referred to in an earlier section. It is represented by the flattened part of the curves in Fig. 5 for relative humidities 50–100%. The centre temperature in this case is about 601C, i.e. of the order of 30 degrees above ambient temperature. These figures are not particularly sensitive to input parameters such as density, conductivity, diffusivity, etc. and thus have a generic significance. It can safely be said that for cellulosic materials exhibiting a heat release curve similar to that measured for bagasse, ignition by a ‘dry’ mechanism will be related to a stable subcritical state showing
65
Centre temperature ( o C)
60
55
50
45
40
35 4
5
6
7
8 Width (m)
9
10
11
12
Fig. 6. Turning points of steady-state bifurcation curves shown in Fig. 5.
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only a few degrees of self-heating. On the other hand, ignition by a ‘wet’ mechanism is related to stable subcritical behaviour showing the order of 30–351 self-heating. This has been known for a long time in the sugar industry, but not properly understood or quantified in this way. The ‘rule of thumb’ in North Queensland has long been ‘‘If it gets above 701C it is going to ignite!’’ Ambient temperatures of 30– 351C year round are common so self-heating of 301 has been recognized empirically as stable non-ignition behaviour, in accord with our calculations. The simplest interpretation of humidity-induced ignition is to observe that (in Fig. 5) the bifurcation curve moves to the left as humidity increases and when the turning point passes to a value smaller than the actual width of the pile no stable steady state exists and ignition occurs. The physical mechanisms of condensation, wetting, etc. are built into the bifurcation diagram in the calculation of the position and nature (stable or unstable) of the steady states occurring, and most importantly when they bifurcate and consequently cease to exist.
5. Ignition by direct wetting We can also use this model to study the effects of addition of liquid water directly to the pile, in contrast to condensation from the vapour phase as in the previous section. As an example we will take an ambient temperature of 301C, relative humidities of 20%, 70% and 100%. We consider a pile with an initial temperature of 301C and integrate the differential equations until a stable steady state is reached (obviously, we have to choose the other parameters so that one does exist before the perturbation). Since we are adding liquid water in this case the latent heat is not relevant but any heat of wetting is. Back [2] gives a graphical dependence of this quantity on water content for cotton. His figures show for that case that the heat of wetting drops off quite rapidly with initial water content. We have followed this empirical approach and assumed the relationship DT ¼
5p 1 þ X ð0; tw Þ
between temperature rise and water content at the time of the perturbation. This equation determines the relationship between the two components of the perturbation applied, i.e. the water content increase and the (assumed simultaneous) corresponding temperature increase. For convenience we also assume that these two perturbations are spatially homogeneous, but the results are not at all sensitive to this. For the humidity values mentioned above we have integrated the differential equations to steady state and then applied the perturbation repeatedly until a critical perturbation was determined, i.e. the smallest one which produced ignition. This has been done for a series of pile sizes (widths) and the results are shown in Fig. 7. The apparent irregularities in these curves arise as a direct result of the shapes of the bifurcation curves in Fig. 5. In this figure we have plotted temperature rise as a measure of the perturbation but the relation given above shows that we could
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120
o
Temperature perturbation ( C)
RH = 70% 100
80
60 RH = 0% 40 RH = 0% 20 RH = 70% 0
0
2
RH = 20% 4
6
8
10
Width (m)
Fig. 7. Water-dependent temperature perturbation for relative humidities 0%, 70% and 100%.
equally well have used amount of water added or some arbitrary combination of the two as a suitable norm. The physical interpretation is straightforward, for example the 0% humidity case for an 8 m pile requires only sufficient water addition to raise the temperature by around 101. This case is clearly sensitive to water addition as one would expect. On the other hand, in 70% relative humidity the existence of stable steady states for piles wider than 3.5 m is highly questionable and addition of extra water is not very significant as evidenced by the very steep curve for this case. Since we are using a model which involves a low-temperature exothermic wet reaction it is of some interest to examine how much of the effect shown in Fig. 7 is due to increased reaction rate, due to water concentration and how much of it is due to the temperature increase associated with the wetting. Accordingly, we have repeated the computer experiments with perturbations consisting of a temperature rise only, other conditions being identical with the previous runs. These computations show that the non-thermal component of each perturbation is negligible. We can safely conclude that dropping the wet reaction term would give rather similar results to the above and such ‘wetting-induced ignition’ can occur even in the absence of the wet reaction. Basically, this is a question of sensitivity of the heat release rate to the two independent variables, moisture concentration and temperature. The sensitivity with respect to temperature is much the larger of the two. Fig. 8 shows a number of heat generation curves as functions of reactant temperature for various (constant) water contents corresponding to equilibrium with various values of relative humidity as shown in the figure. Reference to Figs. 5 and 6 reveals the existence of two distinct series of fold bifurcations over a limited relative humidity range. This is easily interpreted in terms of Fig. 8 from which it can be seen that a necessary condition for such a range is the
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1 0.9
Heat generation rate
0.8 0.7 0.6
RH = 100%
0.5 0.4 0.3 0.2 0.1 0 10
RH = 0% 20
30 40 50 o Centre temperature ( C)
60
70
Fig. 8. Dimensionless heat generation rate (dry+wet reaction terms) at equilibrium for relative humidities 0%, 5%, 10%, 20%, 50%, 70% and 100%.
existence of two turning points (local maximum and local minimum) in the heat release curve. These disappear below 18% relative humidity.
6. Conclusions We have formulated and numerically integrated a reaction diffusion model including experimentally measured water-mediated exothermic reaction, dry oxidative exothermic reaction, endothermic water evaporation, exothermic water vapour condensation, water vapour diffusion, heat conduction and appropriate boundary conditions for water vapour and temperature. This model was originally formulated to describe the behaviour of large piles of bagasse and to provide a means of studying the optimization of water loss from the pile in order to increase its calorific value for electricity production. We have shown that this model is capable of describing a wide variety of effects involving addition of liquid water, water vapour or both with corresponding temperature increases arising from condensation and wetting (adsorption). We have shown that addition of liquid water to a pile in a stable steady-state condition can cause ignition. We have shown that increasing relative humidity outside the pile can cause ignition of a pile in a stable steady state already attained by exposure to a lower external relative humidity. Although these calculations use input obtained by direct experiment on bagasse, we believe they may be of generic significance for the possible occurrence of spontaneous ignition induced by interaction of water or water vapour. This may be
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of significance for other commercially important bodies such as cotton bales, woodchip piles, peat stacks, haystacks, chipboard and particle board, brown coal, lignite and other materials.
Acknowledgements The authors wish to thank the Australian Research Council and the Sugar Research Institute, Mackay, Queensland for the award of a SPIRT collaborative research grant and a graduate scholarship to B.H.
References [1] Walker IK. The role of water in combustion. Fire Research Abstracts and Reviews, vol. 9, #1, 1967. p. 5–22 (National Academy of Sciences, National research Council). [2] Back EL. Autoignition in hygroscopic organic materials. Fire Safety J 1981;4:185–96. [3] Gray BF, Wake GC. The ignition of hygroscopic materials by water. Combust Flame 1990;79:2–6. [4] Gray BF, Wake GC. Critical initial conditions for thermal ignition. Math Comput Modelling 1993;18:65–75. [5] McIntosh A, Gray BF, Wake GC. Ignition of a combustible material in the presence of a damp atmosphere. Phys Lett A 1994;191:61–70. [6] Gray BF, Griffiths JG, Hasko SM. Spontaneous ignition hazards in stockpiles of cellulosic materials. J Chem Tech Biotech 1984;34A:453–63. [7] Dixon T. Spontaneous combustion in bagasse stockpiles. Proceedings of the Australian Sugar Cane Technology, Mackay, Queensland, Australia, 1988. p. 53–61. [8] Gray BF. Analysis of chemical kinetic systems over the entire parameter space III, a wet combustion system. Proc Roy Soc A 1990;429:449–58. [9] Sisson RA, Swift A, Wake GC, Gray BF. The self heating of damp cellulosic materials IFhigh thermal conductivity and diffusivity. IMA J Appl Math 1992;49:273–91. [10] Sisson RA, Swift A, Wake GC, Gray BF. The self heating of damp cellulosic materials II, on the steady states of the spatially distributed case. IMA J Appl Math 1993;50:285–306. [11] Halliburton B. Ph.D. thesis, Macquarie University, Sydney, Australia, 2001. [12] Halliburton B. Gray BF, Carras J. in preparation. [13] Macaskill C, Sexton MJ, Gray BF. A reaction diffusion model of stored bagasse. Anziam J 2001;43:13–35. [14] Sexton MJ, Macaskill C, Gray BF. Self heating and drying in two-dimensional bagasse piles. Combus Theory Modelling 2001;5:517–36. [15] Gong R, Burnell JG, Wake GC. Modelling spontaneous combustion in wet lignite. Combust Theory Modelling 1999;3:215–32.