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Theoretical analysis of coal stockpile self-heating
Fire Safety Journal 67 (2014) 107–112
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Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf
Theoretical analysis of coal stockpile self-heating K. Hooman a,n, U. Maas b a b
School of Mechanical and Mining Engineering, The University of Queensland, Australia Institut für Technische Thermodynamik, Karlsruhe Institute of Technology, Karlsruhe, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 10 February 2014 Received in revised form 20 April 2014 Accepted 11 May 2014
This paper theoretically analyses the problem of coal stockpile self-heating under free convection (no wind) conditions. Scale analysis has been used to drive expressions for the stockpile temperature and inflection point (when the stockpile is completely dry) as functions of the key parameters including the coal type, moisture content, and particle diameter as well as the ambient air temperature. Theoretical predictions are then successfully compared with experimental data and CFD simulations results available in the open literature. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Spontaneous combustion Scale analysis Porous medium Heat transfer Moisture
1. Introduction By self-heating we mean the process of temperature rise in a medium due to internal heat generation from chemical or physical processes taking place within the reactive matter. For a wet coal stockpile, the main processes that take place simultaneously are the transfer of (internally-generated) heat and moisture (evaporation) to the ambient. Spontaneous combustion may happen when all of the generated heat within the reactive pile cannot be transferred away to the environment by natural convection. As such, a part of the generated heat is stored in the pile and overheats the coal to its thermal runaway point. The time needed for the temperature to reach a critical value for moist coal is much longer than that required for a dry one [1–3]. Furthermore, the water transfer into coal can also be affected by the relative pressure [4,5]. Including the influence of chemical exothermic reactions on phase change of the moisture during the drying process, Dong [6] used a 1D mathematical model to determine the effect of moisture content on the maximum temperature rise in a coal stockpile. The influence of coal stockpile height, slope angle and moisture was investigated by Akgun and Essenhigh [7]. Their model predicts that the ignition temperature is a function of the bed porosity, pile shape, and the coal type and the time to ignition is usually in excess of half a month or more. Bouddour et al. [8] simplified mathematical models of heat and mass transfer in wet porous media using
n
Corresponding author. Tel.: þ 61 7 33653677; fax: þ 61 7 33654799. E-mail address: [email protected] (K. Hooman).
http://dx.doi.org/10.1016/j.firesaf.2014.05.011 0379-7112/& 2014 Elsevier Ltd. All rights reserved.
asymptotic expansions for periodic structures. A 2D CFD simulation is reported to investigate the wind effect on dry coal self-heating in [9]. There have been few studies that investigate the effects of moisture transport in reactive porous media on the heat generation process and water phase change. The present work offers scale analysis of self-heating within coal stockpiles of typical size and physical properties. Solutions allow the prediction of the coal temperature within the stockpile. The inflection point, the point in time when the coal stockpile water content first goes to zero is also of interest. The solution also gives the temperature at this point and the time at which it occurs. The former is a measure of coal self-heating while the latter gives an indication as to how much time the stockpile has before it poses a safety concern on its environment.
2. Mathematical modeling In this work we consider stockpiles which have a geometry that can be represented by two spatial coordinates. The specific geometry we use for the analysis is depicted in Fig. 1 and translational symmetry is assumed in z-direction. This is representative of the typical shape of a coal stockpile. Using the Boussinesq approximation to model the fluid density variation, the volume-averaged governing equations [10–17] to cover porous domain, can be written as Continuity: ∂u ∂v þ ¼0 ∂x ∂y
ð1Þ
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Nomenclature A C CF CP d D D1 Dm Dw DT E g h k K L _ m M M*
O p Q Q1 R t T (u,v,w)
constant (s 1) concentration (mol kg 1) form drag coefficient (CF ¼0.55) specific heat at constant pressure (J kg 1 K 1) solid particle diameter (m) diffusion coefficient (m2 s 1) characteristic length (m) isothermal mass transfer coefficient (m2 s 1) isothermal total mass transfer coefficient (m2 s 1) non-isothermal mass transfer coefficient (m2 s 1 K 1) activation energy (J mol 1) gravitational acceleration (m s 2) enthalpy (J kg 1) thermal conductivity (W m 1 K 1) permeability of the porous medium (m2) length of the stockpile (m) rate of moisture evaporation (kg s 1) mass of porous media (kg) _ m M (s 1) symbol to show order of magnitude pressure (Pa) volumetric heat generation rate (W m 3) heat of reaction (J kg 1) universal gas constant (J mol 1 K 1) time (s) temperature (K) velocity components (m s)
Momentum:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ∂u ∂u ∂u ∂p ∂2 u ∂2 u υu u2 þ v2 þ u þv ¼ ε þ ν þ C F εu ε þ ∂t ∂x ∂y ∂x K K ∂x2 ∂y2
ð2aÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ∂v ∂v ∂v ∂p ∂2 v ∂2 v υv u2 þ v2 þu þv ¼ ε þν þ C εg β ðT T i Þ þ ε ε v F ∂t ∂x ∂y ∂y K K ∂x2 ∂y2
ð2bÞ Energy: 2 ∂T ∂T ∂T ∂ T ∂2 T ðρC p Þe þ ðρC p Þa u þ v ¼ ke þ þ ð1 εÞQ þ M n ρe hvl ∂t ∂x ∂y ∂x2 ∂y2 ð3Þ The source terms are oxidation (heat generation) and evaporation, respectively. Here, the evaporation rate divided by the porous medium mass is denoted as Mn which changes with time. In order to simplify the mathematical formulation, we assume that the heat release is governed by a simple one-step reaction. This is a
Fig. 1. Schematic representation of the problem.
v* V (x,y,z)
velocity scale (m s 1) volume of the stockpile (m3) Cartesian coordinates (m)
Greek symbols
α β ε θ μ ν ρ ω
thermal diffusivity (m2 s 1) thermal expansion coefficient (K 1) porosity porous medium (stockpile) side angle viscosity (Pa s) fluid kinematic viscosity (m2 s 1) density (kg m 3) moisture content (kg kg 1 of dry porous)
Subscripts a dry e f i l s sat t v vl
air dry stockpile effective fluid phase initial liquid solid saturation total vapor difference between vapor and liquid
rough approximation, because it is well known that the oxidation of coal is governed by numerous processes like formation of volatile compounds, pore diffusion and heterogeneous reaction, and can be locally diffusion or reaction limited [23]. These processes could be accounted for by hierarchical modeling concepts, but this is out of the scope of this work and subject of future research. Therefore the oxidation term is be modeled as: E Q ¼ ρa Q 1 Aexp ð4Þ RT where Q1 is the heat of reaction, A is the pre-exponential factor, R is the universal gas constant and E is the activation energy [10]. Note that the fluid (and thus effective porous) properties in the above equations change with time. As time evolves and water evaporates, the properties change. Although the values of the heat of reaction, the pre-exponential factor, and the activation energy for this global reaction (in contrast to reaction parameters based on elementary reactions [19]) depend on local conditions like moisture content, temperature etc. and may change with time [20–24], we assume, for simplicity, constant values. An extension to moisture-dependent values is possible in the model; subject of future work. Assuming local thermal equilibrium between the solid and fluid phases in the porous medium, a single energy equation model, like Eq. (3), is adopted to study the heat transfer from the stockpile. Numerical simulation of the stockpile based on local thermal non-equilibrium (two-equation model) showed results very close to those obtained from Eq. (3) as reported by Ejlali [18]. Thus, Eq. (3) is preferred to a more complex twoequation model in this work. The conservation equations for moisture in liquid and vapor forms and the energy equation are derived based on Whitaker's theory as detailed by Kallel et al. [12]. The generic forms of the unsteady equations for moisture movement in the solid porous
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109
materials are given by Murugesan et al. [13]. In view of the above, using Darcy's law for capillary liquid mass flux and Fick's law for diffusive vapor flux, following [13], one has the following equations for water in the form of liquid and vapor, respectively, as follows
magnitude as the original term in Eq. (2b), i.e.
2 2 ∂ ωl ∂ ωt ∂ 2 ωt ∂ T ∂2 T ¼ Dml þ þ þ D þ Mn tl ∂t ∂x2 ∂y2 ∂x2 ∂y2
ð5aÞ
2 2 ∂ ωv ∂ ωt ∂ 2 ωt ∂ T ∂2 T ¼ Dmv þ D Mn þ þ tv ∂t ∂x2 ∂y2 ∂x2 ∂y2
ð5bÞ
These are the orders of magnitudes of the transient, convection, pressure, viscous, form drag (Forchheimer), Darcy drag, and buoyancy forces from left to right. A force balance, by balancing two or more terms, in the above equation is what we propose to use instead of actually solving Eq. (2b). The result form such force balance is expected to be accurate within an order of magnitude. One observes that the convective terms scale with Oðvn 2 =D1 Þ and the shear terms scale with Oðυvn= D1 2 ). Most of the coal stockpiles are packed enough to assume very low permeability values in the order of O(K)o10 9 m2. With such low permeability values, the Darcy drag will be the dominant resistance (as the stockpile length scale is D1 O(1m)) to be balanced by the driving force which is the buoyancy term, i.e.
Adding the above equations leads to an equation for the total moisture transport, defined as ωt ¼ ωv þ ωl , as: 2 2 ∂ ωt ∂ ωt ∂2 ωt ∂ T ∂2 T ¼ ðDml þ Dmv Þ þ þD Þ þ þ ðD tv tl ∂t ∂x2 ∂y2 ∂x2 ∂y2
ð5cÞ
Finally, the species flux balance at the interface is given by ∂C ∂C ∂T D ¼ Dmv þ Dtv ∂n ∂n ∂n
ð5dÞ
The values of the diffusion coefficients are taken from [15] to be D ¼0.0256 m2 s 1, Dmv ¼ 10 12 m2 s 1, Dml ¼ 10 8 m2 s 1, and Dtv ¼ Dtl ¼ 10 12 m2 s 1 K 1. Similar to the kinetic parameters these coefficients may depend on the local conditions like temperature, structural changes of the coal… However, following [15] these parameters are assumed to be constant. Furthermore, [20,21] have shown that for large stockpiles, moisture transport can take place to or from the stockpile depending on the outside air humidity. Here, however, we only considered a case where ambient air is dry allowing for transfer of moisture from the stockpile to the air.
3. Theoretical analysis Conducting the scale analysis, we take the whole stockpile as the region of interest. The aim is to find appropriate terms that are of the same order of magnitudes as the terms in the governing equations. For this purpose, we need to find appropriate length, time, temperature, and velocity scales. One can start with the mass continuity equation to assume that velocities scale with (u,v,w) along (x,y,z) so that w u v O ð6Þ O O D1 D1 L Note that the size of the pile in (x,y) direction are assumed to be of the same order of magnitude while, as mentioned before, the extension in z is much longer in such a way that O(L) c O(D1). One can argue that when L-1, then translational symmetry reduces the z-wise derivatives to zero, i.e. ð∂=∂zÞ ¼ 0. This essentially reduces the problem to a 2D one where the mass continuity will also lead to the same order of magnitude for the x- and ycomponents of velocity, which is u v and O(u)¼O(v)c O(w). Now, one can move on to Eq. (2b) and make use of the fact that O(x)¼ O(y)¼ D1 as well as our observation that O(u) ¼O(v). That is, we made the assumption that stockpile is equally extended in (x,y) directions and that velocities, from an order of magnitude point of view, are comparable. With these, one can replace x and y with D1 and u and v with a velocity scale vn magnitude of which is yet unknown. Similarly, we use a time scale t, a pressure scale, and a scale for the temperature difference ΔT which are yet to be expressed in terms of known parameters. Then, we can replace each term in Eq. (2b) with a term which is of the same order of
vn^ vn^ εp ; O v ; O ; O O D1 t D1
ν
vn^ D21
! ; O C F ε2 vn^
rffiffiffiffiffiffiffiffi! vn2 ευvn^ ; O ; Oðεg βΔTÞ K K
ð7Þ
g βΔT
υvn^ K
ð8Þ
This can be rearranged to give the velocity scale as a function of the temperature scale as follows vn^
Kg βΔT
υ
ð9Þ
Now the remaining problem is to find the temperature scale. This is more complex than finding the velocity scale mainly because the heat transfer is linked to mass transfer. More appropriately, the process can be divided into three stages. The first stage is when the moist substance is heated up to a higher temperature; still lower than the saturation temperature. Throughout this stage, there is no or negligible mass transfer and the last term on the right-side of Eq. (3) is negligible. As such, making use of Eqs. (4 and 9), accurate within an order of magnitude, the actual terms in Eq. (3) can be estimated by the following ones, which are showing orders of magnitudes of different terms in the energy equation, as follows ΔT Kg βΔT ΔT ΔT O ðρC p Þe ; OðρC p Þa ; O ke 2 ; t υ D1 D1 E ð10Þ O ð1 εÞρa Q 1 Aexp RT The above are orders of magnitudes of the transient, convection, conduction, and oxidation terms from left to right and we are yet to estimate which term has to balance what other term or terms. According to [10], for coal stockpiles, typical values of Q1 and A are orders of magnitudes higher than the convection and conduction terms. This leaves a heat generation-storage balance as ΔT E O ð1 εÞρa Q 1 Aexp ð11Þ O ðρC p Þe t RT Rearranging the above algebraic equation, the order of magnitude analysis gives the temperature increase, as a function of time, which reads ð1 εÞρa Q 1 A E t ð12aÞ ΔT exp ðρC p Þe RT Note that the time scale in the above equation, and thus for how long this equation is valid, is still unknown. However, the above equation is valid till the stockpile temperature is below the saturation temperature beyond which any additional heat will lead to evaporation of the moisture from the stockpile and thus activating the mass transfer process. The above equation can, implicitly, as the right-hand side also contains T, give the temperature rise, on top of the stockpile initial temperature, Ti, as time
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goes on
ð1 εÞρa Q 1 A E t T Ti exp ðρC p Þe RT
ð12bÞ
The second stage is when the temperature equals the saturation temperature, say after a time scale tsat, and the coal dehumidification process starts. As a result, we assume that the stockpile temperature does not increase (remains constant at Tdry) but the heat released as a result of oxidation is spent on the evaporation of the moisture trapped in the coal particles. This suggests a balance between internal heat generation and heat of evaporation, i.e. E OðM n hvl ρe Þ O ð1 εÞρa Q 1 Aexp ð13Þ RT dry This can be rearranged to give ð1 εÞρa Q 1 A E exp Mn hvl ρe RT dry
ð14Þ
with V being the volume occupied by the stockpile, the above equation can be further simplified to give the evaporation rate ð1 εÞV ρa Q 1 A E _ m ð15Þ exp hvl ρe RT dry Furthermore, one notes that the time it takes from saturation to the coal being dry t dry t sat scales with t dry t sat
ωt
ð16Þ
Mn
i.e. the time needed for drying scales with the moisture content and inversely with the evaporation rate. Combining Eqs. (14) and (16) gives the drying time scale as hvl ρe ωt E exp ð17Þ t dry t sat RT dry ð1 εÞρa Q 1 A On the other hand, from mass transfer point of view, avoiding sophisticated drying models and relying on a very simple mass transfer argument, one can assume that the time required by vapor trapped in the coal to reach the particle surface (diffusion time) can be obtained by balancing the transient term with the diffusion one in Eq. (5c) as
Δωt
t dry t sat
ðDmv þ Dml Þ
Δωt d
2
ð18Þ
Rearranging the above equation, and using d as the average particle diameter and Dw ¼ Dmv þ Dml , one has 2
t dry t sat þ
d Dw
ð19Þ
Interestingly, Stauch and Maas [19] reported a d2 law for the burning rate constant of a spherical particle, like coal, immersed in a gas environment. The above scaling argument, suggests a similar function for the dependence of the diffusion time scale on the particle diameter. Equating Eqs. (17) and (19) gives Tdry as T dry
E 1 R Lnðð1 εÞðρ =ρ Þðd2 =Dw ÞðQ 1 A=ωt hvl ÞÞ a e
ð20Þ
Finally, when the stockpile is completely dry, the mass transfer term, the last one on the right-side of Eq. (5c), has to be deactivated again (similar to stage 1) and the heat generated internally will be stored in the stockpile (which is now dry). This gives a temperature rise (on top of the dry temperature) as ð1 εÞρa Q 1 A E ðt t dry Þ ð21Þ exp T T dry ðρC p Þe RT Note that the effective thermal capacity of the (now dry) stockpile is different from the initial one used with moist coal stockpile. One
also notes that we consider a case when the ambient humidity is well below that of the stockpile. Furthermore, it is assumed that the air temperature is lower than Tdry. This allows for the transfer of heat and water from the stockpile to the surrounding air. The above simple theoretical analysis, based on scaling arguments, leads to very interesting observations. Based on the above formulae, a very simple predictive tool is provided based on which the stockpile temperature scale is given as a function of time. These predictions are expected to be accurate within an order of magnitude of the results from sophisticated numerical simulations or time-consuming and difficult experimental observations. What is yet to be established is the accuracy of these predictions which will be addressed in the forthcoming section.
4. Results The results reported in this paper are general enough to cover all coal types and are expected to be applicable to any open stockpile geometry. However, in order to be able to compare the results with the available experimental [16] and numerical data [17], we focus on a frustum-shaped Lignite coal with A¼ 1000 1/s, Q1 ¼ 22 MJ kg 1, and E ¼55 kJ mol 1 [10]. As described in Section 2 Fig. 1 illustrates a schematic diagram of the problem considered here. Based on this configuration we performed calculations of the self heating process in the stockpile. Parameters are set to be the same as those of the experiments conducted in [16], i.e. d ¼3.4 mm, ε ¼0.2, and ωt ¼ 0.18. Furthermore, similar to [17], the effective properties are linked to those of fluid and solid phases as ðρcp Þe ¼ εðρcp Þf þ ð1 εÞðρcp Þs
ð22aÞ
ke ¼ εkf þ ð1 εÞks
ð22bÞ
Local temperatures in the stockpile have been calculated and compared with experimental results [16] in [17]. Here we focus on results of the temperature at 2 m of the stockpile from the ground, where we show only three of the measurement locations (T11, T14, and T15 of [16], see[16] for details on the measurement locations. Fig. 2 depicts the predicted stockpile temperature versus time for the scale analysis results, for CFD results [17], and for the experimental studies [16]. These results can be seen as representative for the heating process. First of all, we note that there are differences between the experimental results (T11, T14, T15) although the measurement locations were equivalent taking into account the symmetry of the configuration. This underlines the fact that the properties of the coal vary statistically. Nevertheless, the experiments indicate that there are three stages of the self heating process (marked in the figure). During the first stage the moist substance is heated up to a higher temperature; still lower than the saturation temperature. Here, the temperature is predicted using Eq. (12b). The second stage starts when the temperature equals the saturation temperature, and the coal dehumidification process starts. During this phase the stockpile temperature does not increase because the heat released is spent on the evaporation of the moisture trapped in the coal particles. Duration of this stage is then predicted using Eq. (19). Finally, when the stockpile is completely dry, the heat generated internally will again lead to a temperature increase, given by our Eq. (21), on top of Tdry; here predicted by Eq. (20). One also notes that Tdry (or Tsat) marks the start of second stage and is explicitly obtainable from Eq. (20) without the need to the time-history of stockpile. One can substitute this value in Eq. (12b) to solve for the saturation time, tsat, which marks the end of the first stage. 2D CFD results, 2 m above the ground on the stockpile centerline, are generally in good agreement with the theoretical
K. Hooman, U. Maas / Fire Safety Journal 67 (2014) 107–112
60 55 50
Tdry(oC)
predictions especially before the dry coal is heated up (stages 1 and 2). Experimental data are collected at different heights (at each level). However, the average of the thermocouple readings is very close to the theory. While the theory predicts a sharp transition to phase change, the local temperatures are marking the same point in time. CFD data also tend to be very close in prediction while it marks a more gradual transition. This is most likely due to the fact that the temperature can change during the second phase, which is not accounted for in the simple analysis. Eq. (15) is used to obtain Fig. 3 which shows the volumetric hourly rate of water evaporation from the stockpile versus the drying temperature. The drying temperature in that equation is treated as a parameter with all other parameters fixed. As seen, with higher drying temperature the evaporation rate rockets up and, as a results, one would expect the drying process to speed up. One notes that, according to current predictions, the evaporation rate increases linearly with the size (volume) of the stockpile with a fixed drying temperature; which is a function of a number of key parameters according to Eq. (15).
111
45 40 35 30 25
0
0.1
0.2
0.3
0.4
wt Fig. 4. Drying (inflection) temperature versus moisture content.
Finally, Eq. (20) is used to generate Fig. 4 which illustrates the drying temperature versus the moisture content (with all other parameters constant). As seen, the drying temperature enhances as the moisture content increases. That is, the existence of moisture postpones the drying and thus reduces the risk of spontaneous combustion. However, what cannot be verified using the current theory is the addition of water at some intermediate stages and the interaction of additional water with the oxidation reactions. In other words, the oxidation formulation for Q in our thermal energy equation has not been tried for a cyclic stockpile humidification.
Current Theory Experiment (T11; Ozdeniz (2010)) Experiment (T14; Ozdeniz (2010)) CFD; Ejlali et al. (2011) Experiment (T15; Ozdeniz (2010))
70 65 60
T (oC)
55 50 45 40
5. Conclusion Stage 1
35
Stage 3
Stage 2
30 25 0
500
1000
1500
2000
2500
t (hours) Fig. 2. Stockpile temperatures versus time from CFD, experiment, and theory.
1
References
Evaporation rate (kgh-1m-3)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 25
A theoretical model is presented to study the spontaneous combustion of moist coal stockpile subject to cooler and less moist ambient air. Results are then compared to CFD and experimental data available in the literature. Closed form solutions for inflection point and the required time are obtained based on scale analysis. Obviously, the developed model invokes many simplifications, and could be improved by including in more detail the underlying physics. But the model allows to perform parametric studies.
45
65
85
105
Tdry (oC) Fig. 3. Evaporation rate versus dry temperature (parameters as those of Fig. 2).
[1] A. Arisoy, Numerical modelling of coal spontaneous combustion with moisture included, in: Eighth International Mine Ventilation Congress, A.D.S. Gillies (Ed.), 2005, pp. 501–506. [2] S. Krishnaswamy, S. Bhat, R.D. Gunn, P.K. Agarwal, Low-temperature oxidation of coal. 1. A single-particle reaction-diffusion model, Fuel 75 (3) (1996) 333–343. [3] K. Brooks, V. Balakotaiah, D. Luss, Effect of natural convection on spontaneous combustion of coal stockpiles, AlChE J. 34 (3) (1988) 353–365. [4] A.L. McCutcheon, W.A. Barton, M.A. Wilson, Kinetics of water adsorption/ desorption on bituminous coals, Energy Fuels 15 (6) (2001) 1387–1395. [5] D. Charriere, P. Behra, Water sorption on coals. J. Colloid Interface Sci. 344(2): 460–467. [6] C.X. Dong, Effect of drying heat and moisture content on the maximum temperature rise during spontaneous heating of a moist coal pile, Coal Prep. 14 (3–4) (1994) 223–236. [7] F. Akgun, R.H. Essenhigh, Self-ignition characteristics of coal stockpiles: theoretical prediction from a two-dimensional unsteady-state mode, Fuel 80 (2001) 409–415. [8] A. Bouddour, J.L. Auriault, M. Mhamdi-Alaoui, J.F. Bloch, Heat and mass transfer in wet porous media in presence of evaporation-condensation, Int. J. Heat Mass Transfer 41 (15) (1998) 2263–2277. [9] A. Ejlali, S.M. Aminossadati, K. Hooman, B.B. Beamish, A new criterion to design reactive coal stockpiles, Int. Commun. Heat Mass Transfer 36 (7) (2009) 669–673.
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[10] A. Ejlali, K. Hooman, Buoyancy effects on cooling a heat generating porous medium: coal stockpile, Transp. Porous Media 88 (2) (2011) 235–248. [11] B. Moghtaderi, B.Z. Dlugogorski, E.M. Kennedy, Effect of wind flow on selfheating characteristics of coal stockpiles, Trans. IChemE 78 (2000) 445–453. [12] F. Kallel, N. Galanis, B. Perrin, R. Javelas, Effect of moisture on temperature during drying of consolidated porous materials, J. Heat Trans.-Trans. ASME 115 (3) (1993) 724–733. [13] K. Murugesan, D.C. Lo, D.L. Young, C.W. Chen, C.M. Fan, Convective drying analysis of three-dimensional porous solid by mass lumping finite element technique, Heat Mass Transfer 44 (4) (2008) 401–412. [14] G.I. Kelbaliev, M.P. Manafov, Mass transfer in the process of drying of porous materials, J. Eng. Phys. Thermophys. 82 (5) (2009) 991–999. [15] N. Amanifard, A.K. Haghi, A numerical study on drying of porous media, Korean J. Chem. Eng. 25 (2) (2008) 191–198. [16] A.H. Ozdeniz, Determination of spontaneous combustion in industrial scale coal stockpile, Energy Sources 32(a) (2010) 665–673. [17] A. Ejlali, D.J. Mee, K. Hooman, B. Beamish, Numerical modelling of the selfheating process of a wet porous medium, Int. J. Heat Mass Transfer 54 (2011) 5200–5206.
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Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf
Theoretical analysis of coal stockpile self-heating K. Hooman a,n, U. Maas b a b
School of Mechanical and Mining Engineering, The University of Queensland, Australia Institut für Technische Thermodynamik, Karlsruhe Institute of Technology, Karlsruhe, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 10 February 2014 Received in revised form 20 April 2014 Accepted 11 May 2014
This paper theoretically analyses the problem of coal stockpile self-heating under free convection (no wind) conditions. Scale analysis has been used to drive expressions for the stockpile temperature and inflection point (when the stockpile is completely dry) as functions of the key parameters including the coal type, moisture content, and particle diameter as well as the ambient air temperature. Theoretical predictions are then successfully compared with experimental data and CFD simulations results available in the open literature. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Spontaneous combustion Scale analysis Porous medium Heat transfer Moisture
1. Introduction By self-heating we mean the process of temperature rise in a medium due to internal heat generation from chemical or physical processes taking place within the reactive matter. For a wet coal stockpile, the main processes that take place simultaneously are the transfer of (internally-generated) heat and moisture (evaporation) to the ambient. Spontaneous combustion may happen when all of the generated heat within the reactive pile cannot be transferred away to the environment by natural convection. As such, a part of the generated heat is stored in the pile and overheats the coal to its thermal runaway point. The time needed for the temperature to reach a critical value for moist coal is much longer than that required for a dry one [1–3]. Furthermore, the water transfer into coal can also be affected by the relative pressure [4,5]. Including the influence of chemical exothermic reactions on phase change of the moisture during the drying process, Dong [6] used a 1D mathematical model to determine the effect of moisture content on the maximum temperature rise in a coal stockpile. The influence of coal stockpile height, slope angle and moisture was investigated by Akgun and Essenhigh [7]. Their model predicts that the ignition temperature is a function of the bed porosity, pile shape, and the coal type and the time to ignition is usually in excess of half a month or more. Bouddour et al. [8] simplified mathematical models of heat and mass transfer in wet porous media using
n
Corresponding author. Tel.: þ 61 7 33653677; fax: þ 61 7 33654799. E-mail address: [email protected] (K. Hooman).
http://dx.doi.org/10.1016/j.firesaf.2014.05.011 0379-7112/& 2014 Elsevier Ltd. All rights reserved.
asymptotic expansions for periodic structures. A 2D CFD simulation is reported to investigate the wind effect on dry coal self-heating in [9]. There have been few studies that investigate the effects of moisture transport in reactive porous media on the heat generation process and water phase change. The present work offers scale analysis of self-heating within coal stockpiles of typical size and physical properties. Solutions allow the prediction of the coal temperature within the stockpile. The inflection point, the point in time when the coal stockpile water content first goes to zero is also of interest. The solution also gives the temperature at this point and the time at which it occurs. The former is a measure of coal self-heating while the latter gives an indication as to how much time the stockpile has before it poses a safety concern on its environment.
2. Mathematical modeling In this work we consider stockpiles which have a geometry that can be represented by two spatial coordinates. The specific geometry we use for the analysis is depicted in Fig. 1 and translational symmetry is assumed in z-direction. This is representative of the typical shape of a coal stockpile. Using the Boussinesq approximation to model the fluid density variation, the volume-averaged governing equations [10–17] to cover porous domain, can be written as Continuity: ∂u ∂v þ ¼0 ∂x ∂y
ð1Þ
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Nomenclature A C CF CP d D D1 Dm Dw DT E g h k K L _ m M M*
O p Q Q1 R t T (u,v,w)
constant (s 1) concentration (mol kg 1) form drag coefficient (CF ¼0.55) specific heat at constant pressure (J kg 1 K 1) solid particle diameter (m) diffusion coefficient (m2 s 1) characteristic length (m) isothermal mass transfer coefficient (m2 s 1) isothermal total mass transfer coefficient (m2 s 1) non-isothermal mass transfer coefficient (m2 s 1 K 1) activation energy (J mol 1) gravitational acceleration (m s 2) enthalpy (J kg 1) thermal conductivity (W m 1 K 1) permeability of the porous medium (m2) length of the stockpile (m) rate of moisture evaporation (kg s 1) mass of porous media (kg) _ m M (s 1) symbol to show order of magnitude pressure (Pa) volumetric heat generation rate (W m 3) heat of reaction (J kg 1) universal gas constant (J mol 1 K 1) time (s) temperature (K) velocity components (m s)
Momentum:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ∂u ∂u ∂u ∂p ∂2 u ∂2 u υu u2 þ v2 þ u þv ¼ ε þ ν þ C F εu ε þ ∂t ∂x ∂y ∂x K K ∂x2 ∂y2
ð2aÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ∂v ∂v ∂v ∂p ∂2 v ∂2 v υv u2 þ v2 þu þv ¼ ε þν þ C εg β ðT T i Þ þ ε ε v F ∂t ∂x ∂y ∂y K K ∂x2 ∂y2
ð2bÞ Energy: 2 ∂T ∂T ∂T ∂ T ∂2 T ðρC p Þe þ ðρC p Þa u þ v ¼ ke þ þ ð1 εÞQ þ M n ρe hvl ∂t ∂x ∂y ∂x2 ∂y2 ð3Þ The source terms are oxidation (heat generation) and evaporation, respectively. Here, the evaporation rate divided by the porous medium mass is denoted as Mn which changes with time. In order to simplify the mathematical formulation, we assume that the heat release is governed by a simple one-step reaction. This is a
Fig. 1. Schematic representation of the problem.
v* V (x,y,z)
velocity scale (m s 1) volume of the stockpile (m3) Cartesian coordinates (m)
Greek symbols
α β ε θ μ ν ρ ω
thermal diffusivity (m2 s 1) thermal expansion coefficient (K 1) porosity porous medium (stockpile) side angle viscosity (Pa s) fluid kinematic viscosity (m2 s 1) density (kg m 3) moisture content (kg kg 1 of dry porous)
Subscripts a dry e f i l s sat t v vl
air dry stockpile effective fluid phase initial liquid solid saturation total vapor difference between vapor and liquid
rough approximation, because it is well known that the oxidation of coal is governed by numerous processes like formation of volatile compounds, pore diffusion and heterogeneous reaction, and can be locally diffusion or reaction limited [23]. These processes could be accounted for by hierarchical modeling concepts, but this is out of the scope of this work and subject of future research. Therefore the oxidation term is be modeled as: E Q ¼ ρa Q 1 Aexp ð4Þ RT where Q1 is the heat of reaction, A is the pre-exponential factor, R is the universal gas constant and E is the activation energy [10]. Note that the fluid (and thus effective porous) properties in the above equations change with time. As time evolves and water evaporates, the properties change. Although the values of the heat of reaction, the pre-exponential factor, and the activation energy for this global reaction (in contrast to reaction parameters based on elementary reactions [19]) depend on local conditions like moisture content, temperature etc. and may change with time [20–24], we assume, for simplicity, constant values. An extension to moisture-dependent values is possible in the model; subject of future work. Assuming local thermal equilibrium between the solid and fluid phases in the porous medium, a single energy equation model, like Eq. (3), is adopted to study the heat transfer from the stockpile. Numerical simulation of the stockpile based on local thermal non-equilibrium (two-equation model) showed results very close to those obtained from Eq. (3) as reported by Ejlali [18]. Thus, Eq. (3) is preferred to a more complex twoequation model in this work. The conservation equations for moisture in liquid and vapor forms and the energy equation are derived based on Whitaker's theory as detailed by Kallel et al. [12]. The generic forms of the unsteady equations for moisture movement in the solid porous
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materials are given by Murugesan et al. [13]. In view of the above, using Darcy's law for capillary liquid mass flux and Fick's law for diffusive vapor flux, following [13], one has the following equations for water in the form of liquid and vapor, respectively, as follows
magnitude as the original term in Eq. (2b), i.e.
2 2 ∂ ωl ∂ ωt ∂ 2 ωt ∂ T ∂2 T ¼ Dml þ þ þ D þ Mn tl ∂t ∂x2 ∂y2 ∂x2 ∂y2
ð5aÞ
2 2 ∂ ωv ∂ ωt ∂ 2 ωt ∂ T ∂2 T ¼ Dmv þ D Mn þ þ tv ∂t ∂x2 ∂y2 ∂x2 ∂y2
ð5bÞ
These are the orders of magnitudes of the transient, convection, pressure, viscous, form drag (Forchheimer), Darcy drag, and buoyancy forces from left to right. A force balance, by balancing two or more terms, in the above equation is what we propose to use instead of actually solving Eq. (2b). The result form such force balance is expected to be accurate within an order of magnitude. One observes that the convective terms scale with Oðvn 2 =D1 Þ and the shear terms scale with Oðυvn= D1 2 ). Most of the coal stockpiles are packed enough to assume very low permeability values in the order of O(K)o10 9 m2. With such low permeability values, the Darcy drag will be the dominant resistance (as the stockpile length scale is D1 O(1m)) to be balanced by the driving force which is the buoyancy term, i.e.
Adding the above equations leads to an equation for the total moisture transport, defined as ωt ¼ ωv þ ωl , as: 2 2 ∂ ωt ∂ ωt ∂2 ωt ∂ T ∂2 T ¼ ðDml þ Dmv Þ þ þD Þ þ þ ðD tv tl ∂t ∂x2 ∂y2 ∂x2 ∂y2
ð5cÞ
Finally, the species flux balance at the interface is given by ∂C ∂C ∂T D ¼ Dmv þ Dtv ∂n ∂n ∂n
ð5dÞ
The values of the diffusion coefficients are taken from [15] to be D ¼0.0256 m2 s 1, Dmv ¼ 10 12 m2 s 1, Dml ¼ 10 8 m2 s 1, and Dtv ¼ Dtl ¼ 10 12 m2 s 1 K 1. Similar to the kinetic parameters these coefficients may depend on the local conditions like temperature, structural changes of the coal… However, following [15] these parameters are assumed to be constant. Furthermore, [20,21] have shown that for large stockpiles, moisture transport can take place to or from the stockpile depending on the outside air humidity. Here, however, we only considered a case where ambient air is dry allowing for transfer of moisture from the stockpile to the air.
3. Theoretical analysis Conducting the scale analysis, we take the whole stockpile as the region of interest. The aim is to find appropriate terms that are of the same order of magnitudes as the terms in the governing equations. For this purpose, we need to find appropriate length, time, temperature, and velocity scales. One can start with the mass continuity equation to assume that velocities scale with (u,v,w) along (x,y,z) so that w u v O ð6Þ O O D1 D1 L Note that the size of the pile in (x,y) direction are assumed to be of the same order of magnitude while, as mentioned before, the extension in z is much longer in such a way that O(L) c O(D1). One can argue that when L-1, then translational symmetry reduces the z-wise derivatives to zero, i.e. ð∂=∂zÞ ¼ 0. This essentially reduces the problem to a 2D one where the mass continuity will also lead to the same order of magnitude for the x- and ycomponents of velocity, which is u v and O(u)¼O(v)c O(w). Now, one can move on to Eq. (2b) and make use of the fact that O(x)¼ O(y)¼ D1 as well as our observation that O(u) ¼O(v). That is, we made the assumption that stockpile is equally extended in (x,y) directions and that velocities, from an order of magnitude point of view, are comparable. With these, one can replace x and y with D1 and u and v with a velocity scale vn magnitude of which is yet unknown. Similarly, we use a time scale t, a pressure scale, and a scale for the temperature difference ΔT which are yet to be expressed in terms of known parameters. Then, we can replace each term in Eq. (2b) with a term which is of the same order of
vn^ vn^ εp ; O v ; O ; O O D1 t D1
ν
vn^ D21
! ; O C F ε2 vn^
rffiffiffiffiffiffiffiffi! vn2 ευvn^ ; O ; Oðεg βΔTÞ K K
ð7Þ
g βΔT
υvn^ K
ð8Þ
This can be rearranged to give the velocity scale as a function of the temperature scale as follows vn^
Kg βΔT
υ
ð9Þ
Now the remaining problem is to find the temperature scale. This is more complex than finding the velocity scale mainly because the heat transfer is linked to mass transfer. More appropriately, the process can be divided into three stages. The first stage is when the moist substance is heated up to a higher temperature; still lower than the saturation temperature. Throughout this stage, there is no or negligible mass transfer and the last term on the right-side of Eq. (3) is negligible. As such, making use of Eqs. (4 and 9), accurate within an order of magnitude, the actual terms in Eq. (3) can be estimated by the following ones, which are showing orders of magnitudes of different terms in the energy equation, as follows ΔT Kg βΔT ΔT ΔT O ðρC p Þe ; OðρC p Þa ; O ke 2 ; t υ D1 D1 E ð10Þ O ð1 εÞρa Q 1 Aexp RT The above are orders of magnitudes of the transient, convection, conduction, and oxidation terms from left to right and we are yet to estimate which term has to balance what other term or terms. According to [10], for coal stockpiles, typical values of Q1 and A are orders of magnitudes higher than the convection and conduction terms. This leaves a heat generation-storage balance as ΔT E O ð1 εÞρa Q 1 Aexp ð11Þ O ðρC p Þe t RT Rearranging the above algebraic equation, the order of magnitude analysis gives the temperature increase, as a function of time, which reads ð1 εÞρa Q 1 A E t ð12aÞ ΔT exp ðρC p Þe RT Note that the time scale in the above equation, and thus for how long this equation is valid, is still unknown. However, the above equation is valid till the stockpile temperature is below the saturation temperature beyond which any additional heat will lead to evaporation of the moisture from the stockpile and thus activating the mass transfer process. The above equation can, implicitly, as the right-hand side also contains T, give the temperature rise, on top of the stockpile initial temperature, Ti, as time
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goes on
ð1 εÞρa Q 1 A E t T Ti exp ðρC p Þe RT
ð12bÞ
The second stage is when the temperature equals the saturation temperature, say after a time scale tsat, and the coal dehumidification process starts. As a result, we assume that the stockpile temperature does not increase (remains constant at Tdry) but the heat released as a result of oxidation is spent on the evaporation of the moisture trapped in the coal particles. This suggests a balance between internal heat generation and heat of evaporation, i.e. E OðM n hvl ρe Þ O ð1 εÞρa Q 1 Aexp ð13Þ RT dry This can be rearranged to give ð1 εÞρa Q 1 A E exp Mn hvl ρe RT dry
ð14Þ
with V being the volume occupied by the stockpile, the above equation can be further simplified to give the evaporation rate ð1 εÞV ρa Q 1 A E _ m ð15Þ exp hvl ρe RT dry Furthermore, one notes that the time it takes from saturation to the coal being dry t dry t sat scales with t dry t sat
ωt
ð16Þ
Mn
i.e. the time needed for drying scales with the moisture content and inversely with the evaporation rate. Combining Eqs. (14) and (16) gives the drying time scale as hvl ρe ωt E exp ð17Þ t dry t sat RT dry ð1 εÞρa Q 1 A On the other hand, from mass transfer point of view, avoiding sophisticated drying models and relying on a very simple mass transfer argument, one can assume that the time required by vapor trapped in the coal to reach the particle surface (diffusion time) can be obtained by balancing the transient term with the diffusion one in Eq. (5c) as
Δωt
t dry t sat
ðDmv þ Dml Þ
Δωt d
2
ð18Þ
Rearranging the above equation, and using d as the average particle diameter and Dw ¼ Dmv þ Dml , one has 2
t dry t sat þ
d Dw
ð19Þ
Interestingly, Stauch and Maas [19] reported a d2 law for the burning rate constant of a spherical particle, like coal, immersed in a gas environment. The above scaling argument, suggests a similar function for the dependence of the diffusion time scale on the particle diameter. Equating Eqs. (17) and (19) gives Tdry as T dry
E 1 R Lnðð1 εÞðρ =ρ Þðd2 =Dw ÞðQ 1 A=ωt hvl ÞÞ a e
ð20Þ
Finally, when the stockpile is completely dry, the mass transfer term, the last one on the right-side of Eq. (5c), has to be deactivated again (similar to stage 1) and the heat generated internally will be stored in the stockpile (which is now dry). This gives a temperature rise (on top of the dry temperature) as ð1 εÞρa Q 1 A E ðt t dry Þ ð21Þ exp T T dry ðρC p Þe RT Note that the effective thermal capacity of the (now dry) stockpile is different from the initial one used with moist coal stockpile. One
also notes that we consider a case when the ambient humidity is well below that of the stockpile. Furthermore, it is assumed that the air temperature is lower than Tdry. This allows for the transfer of heat and water from the stockpile to the surrounding air. The above simple theoretical analysis, based on scaling arguments, leads to very interesting observations. Based on the above formulae, a very simple predictive tool is provided based on which the stockpile temperature scale is given as a function of time. These predictions are expected to be accurate within an order of magnitude of the results from sophisticated numerical simulations or time-consuming and difficult experimental observations. What is yet to be established is the accuracy of these predictions which will be addressed in the forthcoming section.
4. Results The results reported in this paper are general enough to cover all coal types and are expected to be applicable to any open stockpile geometry. However, in order to be able to compare the results with the available experimental [16] and numerical data [17], we focus on a frustum-shaped Lignite coal with A¼ 1000 1/s, Q1 ¼ 22 MJ kg 1, and E ¼55 kJ mol 1 [10]. As described in Section 2 Fig. 1 illustrates a schematic diagram of the problem considered here. Based on this configuration we performed calculations of the self heating process in the stockpile. Parameters are set to be the same as those of the experiments conducted in [16], i.e. d ¼3.4 mm, ε ¼0.2, and ωt ¼ 0.18. Furthermore, similar to [17], the effective properties are linked to those of fluid and solid phases as ðρcp Þe ¼ εðρcp Þf þ ð1 εÞðρcp Þs
ð22aÞ
ke ¼ εkf þ ð1 εÞks
ð22bÞ
Local temperatures in the stockpile have been calculated and compared with experimental results [16] in [17]. Here we focus on results of the temperature at 2 m of the stockpile from the ground, where we show only three of the measurement locations (T11, T14, and T15 of [16], see[16] for details on the measurement locations. Fig. 2 depicts the predicted stockpile temperature versus time for the scale analysis results, for CFD results [17], and for the experimental studies [16]. These results can be seen as representative for the heating process. First of all, we note that there are differences between the experimental results (T11, T14, T15) although the measurement locations were equivalent taking into account the symmetry of the configuration. This underlines the fact that the properties of the coal vary statistically. Nevertheless, the experiments indicate that there are three stages of the self heating process (marked in the figure). During the first stage the moist substance is heated up to a higher temperature; still lower than the saturation temperature. Here, the temperature is predicted using Eq. (12b). The second stage starts when the temperature equals the saturation temperature, and the coal dehumidification process starts. During this phase the stockpile temperature does not increase because the heat released is spent on the evaporation of the moisture trapped in the coal particles. Duration of this stage is then predicted using Eq. (19). Finally, when the stockpile is completely dry, the heat generated internally will again lead to a temperature increase, given by our Eq. (21), on top of Tdry; here predicted by Eq. (20). One also notes that Tdry (or Tsat) marks the start of second stage and is explicitly obtainable from Eq. (20) without the need to the time-history of stockpile. One can substitute this value in Eq. (12b) to solve for the saturation time, tsat, which marks the end of the first stage. 2D CFD results, 2 m above the ground on the stockpile centerline, are generally in good agreement with the theoretical
K. Hooman, U. Maas / Fire Safety Journal 67 (2014) 107–112
60 55 50
Tdry(oC)
predictions especially before the dry coal is heated up (stages 1 and 2). Experimental data are collected at different heights (at each level). However, the average of the thermocouple readings is very close to the theory. While the theory predicts a sharp transition to phase change, the local temperatures are marking the same point in time. CFD data also tend to be very close in prediction while it marks a more gradual transition. This is most likely due to the fact that the temperature can change during the second phase, which is not accounted for in the simple analysis. Eq. (15) is used to obtain Fig. 3 which shows the volumetric hourly rate of water evaporation from the stockpile versus the drying temperature. The drying temperature in that equation is treated as a parameter with all other parameters fixed. As seen, with higher drying temperature the evaporation rate rockets up and, as a results, one would expect the drying process to speed up. One notes that, according to current predictions, the evaporation rate increases linearly with the size (volume) of the stockpile with a fixed drying temperature; which is a function of a number of key parameters according to Eq. (15).
111
45 40 35 30 25
0
0.1
0.2
0.3
0.4
wt Fig. 4. Drying (inflection) temperature versus moisture content.
Finally, Eq. (20) is used to generate Fig. 4 which illustrates the drying temperature versus the moisture content (with all other parameters constant). As seen, the drying temperature enhances as the moisture content increases. That is, the existence of moisture postpones the drying and thus reduces the risk of spontaneous combustion. However, what cannot be verified using the current theory is the addition of water at some intermediate stages and the interaction of additional water with the oxidation reactions. In other words, the oxidation formulation for Q in our thermal energy equation has not been tried for a cyclic stockpile humidification.
Current Theory Experiment (T11; Ozdeniz (2010)) Experiment (T14; Ozdeniz (2010)) CFD; Ejlali et al. (2011) Experiment (T15; Ozdeniz (2010))
70 65 60
T (oC)
55 50 45 40
5. Conclusion Stage 1
35
Stage 3
Stage 2
30 25 0
500
1000
1500
2000
2500
t (hours) Fig. 2. Stockpile temperatures versus time from CFD, experiment, and theory.
1
References
Evaporation rate (kgh-1m-3)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 25
A theoretical model is presented to study the spontaneous combustion of moist coal stockpile subject to cooler and less moist ambient air. Results are then compared to CFD and experimental data available in the literature. Closed form solutions for inflection point and the required time are obtained based on scale analysis. Obviously, the developed model invokes many simplifications, and could be improved by including in more detail the underlying physics. But the model allows to perform parametric studies.
45
65
85
105
Tdry (oC) Fig. 3. Evaporation rate versus dry temperature (parameters as those of Fig. 2).
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