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Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
1
Modelling oxygen-limited and self-sustained smoldering propagation:
underground coal fires driven by thermal buoyancy
Zeyang Songa,b,*
a College of Safety Science and Engineering, Xi’an University of Science and Technology, Xi’an 710054, P.R.
China; b Shaanxi Key Laboratory of Prevention and Control of Coal Fire, Xi’an University of Science and
Technology, Xi’an 710054, P.R. China
Abstract
Modelling oxygen-limited and self-sustained smoldering propagation is of significance for
prevention of fire hazards and optimization of applied systems. However, two issues remain unsolved
in the conventional models: (a) adjustment of TG-scale kinetic parameters applied to bed-scale
propagation, and (b) decoupling oxidative reaction and oxygen transport in multi-scale porous media.
In this work, an analytic expression of oxidative reaction rates limited by oxygen transport is derived
from the conservation equations of oxygen species transport in gas and solid. Then, both oxidative
reaction rates controlled by the kinetic reaction (Arrhenius equation) and oxygen transport (analytic
expression) are integrated into the conservation equations of mass, energy, and oxygen species
transport. In this model, five-step reaction scheme is considered and their kinetic parameters obtained
from TG experiments are employed without any adjustment. Along with the Darcy air flow driven by
thermal buoyancy, this model is applied to predict oxygen-limited and self-sustained smoldering
propagation of underground coal fires. The proposed model is compared with laboratory experiments
and the conventional model. Results show that the proposed model well predicts the oxygen-limited
and self-sustained smoldering propagations of underground bituminous and anthracite coal fires. The
predictability of the proposed model is better than the conventional model in spite of great effort to
modify kinetic parameters best fitting with experimental data. It is validated that the proposed model
addresses the two puzzled issues in the conventional model with respect to buoyancy-driven, oxygen-
limited, and self-sustained smoldering propagation of underground coal fires. This work may help to
improve models of self-sustained propagation of other smoldering fires and applied smoldering
systems.
* Corresponding author at: College of Safety Science and Engineering, Xi’an University of Science and Technology, Xi’an 710054,
P.R. China. E-mail address: zeyang.song@xust.edu.cn (Z. Song).
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
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Keywords: smoldering propagation, fire spread, underground coal fires, porous media, multiple
scales
Nomenclature
Latin Letters
A pre-exponential factor, s
c discharge coefficient, -
Cp specific heat capacity, J kg-1 K-1
d particle diameter (size), mm
D diffusion coefficient, m2 s-1
Da Damkoeler number, -
E activation energy, kJ mol-1
g acceleration of gravity, m s-2
h boundary heat transfer coefficient, W m-2 K-
1
hsg interfacial heat transfer coefficient, W m-2
K-1
H depth, m
k thermal conductivity, W m-1 K-1
L thickness of coalbed, m
m reaction order of oxygen mass fraction, -
n reaction order of fuel mass fraction, -
Nu Nusselt number, -
P pressure, Pa
Pr Prandtl number, -
q' heat flux of igniter, kW m-2
r chemical reaction rate, kg m-3 s-1
r’ consumption rate of solid species, kg m-3 s-
1
Re Reynolds number, -
RO2 oxygen consumption rate limited by oxygen
transport, kg m-3 s-1
R ideal gas constant, J K-1 mol-1
t time, h
T temperature, K
u air flow velocity, m s-1
U smoldering propagation velocity, cm h-1
x radial distance, m
y vertical distance, m
Y mass fraction, -
Greek Symbols
δ characteristic length of oxygen
concentration gradient, mm
ΔH heat of reaction, MJ kg-1
κ permeability, m2
μ dynamic viscosity, Pa s
υ stoichiometric coefficient, -
ρ density, kg m-3
𝜌 source/sink term of oxygen, kg m-3
σ Stefan-Boltzmann constant, W m-1 K-4
τ dimensionless time, -
φ porosity, -
Subscripts
a ambient
co coal oxidation
e evaporation
g gas
i chemical reaction step
ig ignition
j solid species
p pyrolysis
rad radiation
s solid
T thermal
αo α-char oxidation
βo β-char oxidation
Superscripts
T oxygen transport
R oxidative reaction
Abbreviations
CC Changcun coal sample
CFIPM condensed fuels in inert porous media
DSC differential scanning calorimetry
NAPLs nonaqueous phase liquids
PSFs porous solid fuels
REV representative elementary volume
TG thermogravimetric analysis
XA Xin’an coal sample
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
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1. Introduction
Smoldering propagation naturally occurs in porous solid fuels (PSFs) [1] such as polymer foams
[2, 3], bio-fuels [4-7], oil shales [8, 9], peat [10-15], and coal [16, 17], and recently has been applied
to condensed fuels (including nonaqueous phase liquids (NAPLs) [18-20], sludges [21, 22], faeces
[23-25], digestates [26], and food wastes [27]) in inert porous media (CFIPM) [1]. It is often self-
sustained with limited oxygen either driven by natural buoyancy for smoldering fires or forced by air
compressor for applied smoldering systems. The propagation usually terminates or extinguishes until
the smoldering front reaches the end of fuel beds or the outer cooling boundaries. Such self-sustained
propagations make smoldering fires persistent [17, 28] and applied smoldering system energy-
efficient [1]. It is of significance for modelling of oxygen-limited and self-sustained smoldering
propagation to prevent fire hazards and optimize applied systems.
Smoldering propagation generally represents thermal and chemical waves continuously moving
across porous fuel beds. The substantial processes of smoldering propagation involve multiphase heat
and mass transfer coupled with complex chemical reactions, and cover a wide range of spatial scales
from microscopic pores to macroscopic fuel beds as well as temporal scales from microseconds
(reaction) to minutes (transport). Ohlemiller [29] developed a general model at both scales of particle
and bulk fuel bed to describe the substantial processes of smoldering propagation. Even thought the
solution of the model was not completely addressed, as Ohlemiller [29] pointed out, it has been
considered as a benchmark comparing to existed and being developed models. Since then, modelling
of self-sustained smoldering propagation has become an active research topic [1, 28]. Analytic models
give insight into the structural patterns [30, 31] and benefits to derive a quantitative expression
predicting propagation velocity [2, 32, 33]. Details and valuable insights in terms of analytic models
can be found in the latest review article [1]. Herein, emphasis of literature review is put on numerical
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
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models.
In the light of hypothesis on spatial scale of oxygen concentration gradient, two solution paths have
been explored for numerical models of self-sustained smoldering propagation. Debenest et al. [34-36]
developed microscale models of smoldering propagation. Heat and mass transfer as well as oxidation
reaction with respect to both pores and solid fuels were considered in the conservation equations [34-
36]. Xu et al. [37] employed lattice Boltzmann numerical method to model crude oil smoldering
propagation at pore scale. One of key hypothesis in the micro-/pore-scale models was that the oxygen
concentration gradient at pore scale can not be ignored.
Another path was to use Representative Elementary Volume (REV) method [1]. It has been
considered as an effective and efficient solution for modelling of smoldering propagation. One of
important underneath hypothesis of REV method is that the oxygen concentration gradient at pore
scale is ignorable. Then, oxidation reactions formulated with the Arrhenius equations can be directly
integrated into the REV-based conservation equations. One or multiple oxidation reactions along with
water evaporation and pyrolysis are lumped together to represent the chemical reaction schemes of
smoldering propagation [38-40]. Kinetic parameters (A, E, n, and m) in the Arrhenius equations are
usually obtained from TG-scale experiments [41-43]. This methodology has achieved great successes
in smoldering propagations of both PSFs (polyurethane foam [38, 44], peat [11-13, 45-48], coal [49],
and biomass [50]) and CFIPM (pyrolytic carbon [51], bitumen [52-54], and granular activated carbon
[55-57]). Nevertheless, several puzzles remain in the REV based models.
Firstly, kinetic parameters obtained from TG-scale experiments usually have to be adjusted with
bed-scale propagation experiments. Huang and Rein [40] reported that the fuel consumption rates
(mass loss rate) of numerical computations based on TG-scale kinetic parameters were faster than
experimental observations. Leach et al. [38] and Zanoni et al. [53] revealed that the pre-exponential
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
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factors had to be manually tuned according to bed-scale experiments. Pozzobon et al. [51] adjusted
both pre-exponential factor and activation energy to best fit numerical results with experimental data.
Secondly, it has been well acknowledged that oxidation reaction in smoldering propagation is
controlled by oxygen transport [2, 7, 32, 33, 58-60], however, the Arrhenius equations involved in
the REV conservation equations for smoldering propagation are not in accordance with the
acknowledgement. He et al. [61], Chen et al. [47], Song et al. [62, 63], and Yang et al. [64] derived
analytic correlations of fuel/oxygen consumption rates controlled by oxygen transport and replaced
the Arrhenius equations with the analytic correlations in the conservation equations. Wessling et al.
[49] used the operator splitting method to separate the transport and reaction terms in the oxygen
species transport equation so that the fuel/oxygen consumption rates were technically controlled by
oxygen transport. However, these models were restrained by one-step oxidation reaction and lack of
experimental validations. Moreover, few models are capable to integrate both the kinetic and oxygen-
transport limiting regimes.
Underground coal fires widely spread in almost every continents on the Earth [65]. They emit
enormous greenhouse gases [66, 67] and toxic substance [68], and trigger wildfires [69] and
geohazards such as collapse and subsidence [70]. Smoldering propagation of underground coal fires
has been rarely investigated and not understood well. It is still a big challenge for modelling
buoyancy-driven, oxygen-limited, and self-sustained smoldering propagation of underground coal
fires.
In this work, we attempt to model buoyancy-driven, oxygen-limited, and self-sustained smoldering
propagation of underground coal fires, highlighting how to solve challenging issues in the REV based
models. It is unnecessary for the proposed model to arbitrarily modify the kinetic parameters obtained
from TG-scale experiments. Furthermore, both oxidative reaction and oxygen-transport limiting
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
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processes are integrated into the conservation equations of mass, energy, and oxygen species transport.
Then, smoldering propagation can be flexibly and automatically controlled by either the kinetic
reactions or the oxygen transport. The proposed model is validated by comparisons with experimental
results and the conventional model.
2. Laboratory-scale experiments
Underground coal smoldering fires in the field are large-scale fires with areas of more than 1000
m2 and maximum depth of 100 m. It’s costly to conduct experiments in the field and challenging to
reveal the controlling mechanisms from the field observations disturbed by many geological and
meteorological factors. To address these issues and challenges, a ~1/20 laboratory-scale experimental
framework, as shown in Fig. 1, has been established in our previous work [17, 66, 71]. The primary
fundamental processes of underground coal smoldering fires in the field were considered: downward
transported air flow, upward smoke flow, air flow driven by buoyant smoke, limited oxygen
availability, and smoldering combustion. Air flow velocity, coal burning temperatures, smoke
temperatures and typical gas concentrations (CO, CO2, and O2) were measured. This framework
allowed to experimentally investigate the buoyancy-driven and oxygen-limited underground coal
smoldering fires with cost-efficient, quantitative, and controlled benefits [17, 66, 71]. Experimental
setup and procedures were detailed in our previous work [17, 66, 71].
Two coal samples, Changcun (CC) bituminous coal and Xin’an (XA) anthracite coal, are
considered to examine the proposed model. The ultimate and proximate analyses of two studied coal
samples were presented in our previous work [60]. In practice, underground coal smoldering fires
propagate towards multiple directions. Multiple-dimensional propagation of smoldering fires is very
complicated and remains a challenging topic [14, 29]. Thus, it is usually simplified as one-
dimensional model, which is beneficial to obtain insight on the key roles of heat and mass transfer as
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
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well as combustion chemistry. Two propagation patterns have been proposed [39, 72], that is, forward
propagation and reverse propagation. When the direction of smoldering propagation is the same
direction as air flow, it is called the forward propagation; if the directions of propagation and air flow
are against, it is named the reverse or opposed propagation. Fieldwork observation [68, 73] reported
that most of underground coal smoldering fires belonged to the forward propagation because air and
self-ignition fire source usually coexist at the same location. Hence, the forward propagation is
considered in this work.
Fig. 1. Schematic diagram of laboratory-scale experimental setup for buoyancy-driven, oxygen-
limited, and self-sustained smoldering propagation of underground coal fires.
As shown in Fig. 1, coalbed was piled up by coal particles with mean diameter of 6 mm. The
thickness of coalbed was 0.14 m. The fire depth was 2.6 m. Igniter was placed at the bottom of the
combustion reactor. The heat fluxes of the igniter ranged between 15-25 kW/m
2
. Warm smoke was
produced from coalbed heated by the igniter, which generated thermal buoyance and drew air flow
downward feeding smoldering combustion. Igniter was turned off once a self-sustained smoldering
propagation of coal fire was established. Each experiment was repeated at least twice.
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
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3. Mathematical modelling
Buoyancy-driven, oxygen-limited, and self-sustained smoldering propagation of both CC and XA
coal fires is mathematically modeled in this section. The model is two-dimensional and the studied
domain is limited to the coalbed (see Fig. 2).
3.1 Buoyancy-driven gas flow velocity
Our previous work found that both thermal and solutal (induced by smoke composition and gas
concentration) buoyancy have influences on air flow [71]. However, thermal buoyancy plays a
dominant role [42, 71]. Then, only thermal buoyancy is involved in this work, for the sake of
simplification.
In this work, thermal buoyancy results from the global temperature difference between inlet and
outlet vertical gas column (see air-transport and exhaust pipes in Fig. 1). It is different from several
previous models [11, 12, 45, 46, 48, 49, 74], in which thermal buoyancy was calculated on the basis
of local temperature inside fuel bed. Our previous experimental work [17, 71] concluded that thermal
buoyancy driven gas flow in terms of underground coal smoldering fires was basically subject to the
Darcy’s law. Then, gas flow velocity is written as
g,Ta g,Tg
T()gH L gH
P
uyL
(1)
We assume that smoke is ideal gas. Thus,
g,Ta a g,Tg g
TT
(2)
where g
T denotes the mean temperature of exhaust gas.
Substituting Eq. (2) into Eq. (1), we have
g,Ta a g
()gH L HTT
uL
(3)
In practice, κ could gradually change with thermal degradation [71] and μ is a function of gas
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
9
temperature [55]. Besides, g
T is hard to be quantified, and herein it is replaced by the smoke
temperature ( g
y
L
T) near to the top surface of coalbed. Besides, Eq. (3) neglects the influence of gas
flow friction along the pipe. A discharge coefficient (c) is involved in Eq. (4) to represent these
uncertainties at certain degree [71] and its value is calibrated via experiments. More details associated
with the derivation of the gas flow velocity is given in the Supplementary Martials.
ag
g,Ta () yL
HL HTT
g
uc L
(4)
3.2 Chemical reaction schemes
The chemical reaction mechanism with respect to smoldering combustion has been still not
understood well [1]. The trade-off strategy for self-sustained smoldering propagation is to balance the
complexity of the reaction schemes and research purposes. Detailed chemistry is necessary while
predicting the valuable products from smoldering combustion [75-77]. From the engineering
perspective, it is principal to establish a simple chemical reaction scheme that is essential to minimize
the errors of the tempo-spatial evolution of temperature profiles between numerical and experimental
results, and to rationally inverse the dominant chemical reactions at different reaction zones. At this
point, substantial research progresses on chemical reaction schemes associated with self-sustained
smolder propagations of both PSFs [38-44, 78] and CFIPM [51, 53, 54, 56, 57] have been achieved.
Valuable details and insights can be obtained in review articles [1, 28, 29, 75].
3.2.1 Five-step reaction scheme
Rein et al. [41], Huang and Rein [40] concluded that five-step chemical reaction scheme is a nice
trade-off option for smoldering fires. Hence, in this work, a five-step scheme including water
evaporation, coal pyrolysis and oxidation, and char oxidation is employed, as shown in Eqs. (5)-(10).
e,w e,w e
coal water coal waterνν
H
(5)
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p
,p,gp
coal α - char gasνν
H
(6)
2
o,O 2 o, o,g o
coal O β - char gasνν ν
H
(7)
2
o,O 2 o,a o,g o
α - char O ash gasννν
H
(8)
2
o,O 2 o,a o,g o
β - char O ash gasννν
H
(9)
Chemical reaction (fuel consumption) rates are expressed by the Arrhenius equations:
is
i
2
ijjOi
e
i
nE/RT
m
rYYA
(10)
where i and j denote chemical reactions and solid species, respectively. m equals to 1 and 0 for
oxidation reaction and evaporation/pyrolysis.
Kinetic parameters, stoichiometric coefficients, and reaction heat of studied coal sample are shown
in Table 1. These parameters are obtained from our previous work [60] and the TG-DSC experiments,
except for water evaporation that are referred to the literature [45]. Note that the thermodynamic of
water evaporation and condensation [79, 80] is not considered in this work.
The chemical reaction scheme established in the literature is actually kinetic regime because the
underneath hypothesis of these chemical reaction schemes is that the oxygen concentration is
relatively rich, and oxygen consumption rate is always determined by fuel consumption rate (e.g. Eq.
(10)) and oxygen stochiometric coefficient (υi,O2 in Eqs. (7)-(9)).
3.2.2 Oxygen-transport limiting scheme
It is well acknowledged that oxygen is often lean for smoldering fire propagation because air flow
driven by thermal buoyancy is slow. The condition for underground coal smoldering fires is expected
to be more harsh due to very limited ventilation pathways (e.g., a few centimeter cracks). If the
oxygen is lean, oxidative reactions (Eqs. (7)-(9)) needs to be modified as follows:
222
o,g
o,
2
o,O o,O o,O
1coal O β - char gas
ν
ν
ννν
(11)
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
11
222
o,g
o,a
2
o,O o,O o,O
1α - char O ash gas
ν
ν
ννν
(12)
222
o,g
o,a
2
o,O o,O o,O
1β - char O ash gas
ν
ν
ννν
(13)
Correspondingly, the fuel consumption rates presented in Eq. (10) must be formulated by oxygen
consumption rate (Ri,O2) limited by oxygen transport.
2
2
iiO
iO
1
,
,
rR
ν
(14)
Table 1. Kinetic parameters, stoichiometric coefficients, and reaction heat of five-step chemical
reaction scheme for two studied coal samples.
Parameter Va l ue Unit Ref.
CC coal sample XA coal sample
ne 3 - [45]
Ae 1.318×10
8 s
-1 [45]
Ee 67.800 kJ mol-1 [45]
ΔHe 2.26 MJ kg-1 [45]
ΔHp 0.335 MJ kg-1 Measured
np 9 2 - [60]
Ap 2.592×10
10 1.8960×108 s
-1 [60]
Ep 194.662 164.841 kJ mol-1 [60]
nco 1 2 - [60]
Aco 3.291×10
14 1.430×10
8 s
-1 [60]
Eco 220.612 139.682 kJ mol-1 [60]
nαo 2 1 - [60]
Aαo 1.187×10
10 1.616×10
8 s
-1 [60]
Eαo 166.285 155.030 kJ mol-1 [60]
nβo 1 1 - [60]
Aβo 1.812×10
10 3.460×10
9 s
-1 [60]
Eβo 208.913 170.922 kJ mol-1 [60]
ν
p
,
α
0.731 0.981 - [60]
νo,
β
0.243 0.848 - [60]
ναo,
a
0.203 0.144 - [60]
ν
β
o,
a
0.610 0.167 - [60]
νo,O2 1.796 1.877 - [60]
ναo,O2 2.164 2.108 - [60]
ν
β
o,O2 2.164 2.108 - [60]
ΔHco 23.527 24.583 MJ kg-1 Measured
ΔHαo
(
β
o
)
28.351 27.612 MJ kg-1 Measured
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
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Both kinetic (Eqs. (5)-(10)) and oxygen-transport (Eqs. (11)-(14)) limiting schemes together
consists of a whole comprehensive chemical reaction scheme for smoldering fire propagation. The
approach to determine the kinetic or oxygen-transport limiting scheme and derive the expression of
oxygen consumption rate controlled by oxygen transport are discussed in the Sections 3.4 and 3.5.
3.3 Conservation equations
3.3.1 Mass conservation equations
Five solid species (water, coal, α-char, β-char, and ash) are involved in the chemical reaction
scheme. The mass conservation equations of five solid species are formulated as follows:
water
e
(1 ) r
t
(15)
coal
p
co
(1 ) rr
t
(16)
char
p
,p o
(1 ) rr
t
(17)
char
co, co o
(1 ) rr
t
(18)
ash
αo,a αo o,a o
(1 ) rr
t
(19)
s water coal -char -char ash
(20)
Importantly, note that the oxidative reaction rates ( i, i = co, o, or
) in Eqs. (16)-(19) are
expressed by Eq. (40) derived in the Section 3.5, rather than the Arrhenius equation (Eq. (10)). Solutal
buoyancy is ignored in this work. It indicates that the gas density is only a function of temperature,
but independent on gas species in smoke. Thus, gas density is assumed to be equal to air density.
3.3.2 Energy conservation equations
Zanoni et al. [53, 81] developed one-dimensional thermal non-equilibrium model and proposed a
formula estimating the heat transfer coefficient (hsg) between gas and solid particles in the context of
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
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CFIPM. Importantly, their work proved that gas and solid were not thermal equilibrium for the
smoldering propagation of CFIPM. As for PSFs, the thermal non-equilibrium has been considered in
some models [11, 13, 47, 48, 82], but also not taken account in other several models [12, 39, 45, 46,
50]. In this work, however, the thermal non-equilibrium must be considered, as shown in Eqs. (21)
and (22), because gas temperature (Tg) has to be separated to calculate gas flow velocity (see
discussion in the Section 5.5). Note that the oxidative reaction rates ( i, i = co, o, or
) in Eq. (21)
are expressed as Eq. (40) derived in the Section 3.5, rather than the Arrhenius equation (Eq. (10)).
i=co, o, o
sg
2
s
spseff eff s i i i i s g
i=e,p
6(1 )
(1 )( ) (1 ) ( )
h
T
CkTrHrHTT
td
(21)
gsg
2
gpg pgg g g g s g
6(1 )
()
Th
CCuTkT TT
td
(22)
The heat transfer coefficient (hsg) in Eqs. (21) and (22) is estimated by Eq. (23a) [11, 83], instead
of Eq. (23b) proposed by Zanoni et al. [81]. The reason for this parameter adoption is discussed in
the Section 5.5.
gg 0.6 1/3
sg (2 1.1 )
kNu k
hRePr
dd
(23a)
gg 1.97 1/3
sg 0.001
kNu k
hRePr
dd
(23b)
In Eq. (21), thermal radiation is considered as an additive conduction term formulated as the
Rosseland approximation ( 3
rad s
16 / 3kdT
). It is assumed that coal particles are sphere, and their
surface area per unit volume is 6(1-φ)/d [53]. ps
C and s
k are mean specific heat capacity and
thermal conductivity of five solid species, which are expressed as
ps pj j
CCY (24)
ssjj
kkY (25)
3.3.3 Oxygen species transport equations
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Figure 2 shows a conceptualized diagram of multiple scales (bed, REV, and particle) associated
with oxidative reaction and oxygen transport. The overall porous space consists of voids among
particles and pores inside particles (φ=φ
g
+φ
s
). The overall derivation of oxygen concentration with
respect to time is given by
22 2 2 2
O g O,g g O, O,g O,
gg
(()) ()
ss
tt tt
(26)
The equation of oxygen transport in void can be written as
2
222
O,g 2
g g g O ,g O ,g O ,g
T
Du
t
(27)
where the last term at the right-hand side of Eq. (27) denotes the sink term resulting from oxygen
transport into pore inside the particle.
Fig. 2 Schematic diagram of studied domain and multiple scales (bed, REV, and particle)
associated with oxidative reaction and oxygen transport.
The oxygen transport from the void into pore is described as
2
222
O,s 2
g g s O ,s O ,s O ,s
() ()
R
Du
t
(28)
where the last term at the right side of Eq. (28) is the sink term induced by oxidation reaction (oxygen
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
15
consumption rate). Usually, pores inside raw coal particle are tiny [84]. Oxygen transport in these
tiny pores is dominant by diffusion. However, when pyrolysis of coal particle occurs, pores will
enlarge greatly and the permeability of coal particle will be remarkably improved [85]. Pyrolysis of
single particle-scale (6-9 mm diameter size) coal sample were tested using TG experiments. It was
observed that the raw solid particle was turned into very porous particle with large pores. Similarly,
investigations in the literature revealed that the permeability of coal particle could increase by three
orders of magnitude due to pyrolysis [86, 87]. In the scenario of smoldering forward propagation in
this work, coal particles initially underwent pyrolysis and then oxidation. The permeability of coal
particle could be close to the permeability of coal bed when the oxidation reaction takes place.
Additionally, the influence of ash layer generated by the oxidative reaction of char is not considered
since the ash layer could be sufficiently thin, considering ~6 mm diameter and ~14w.t.% ash mass
fraction of coal particles in this work. Therefore, it is reasonable to hypothesize that u in both Eqs.
(27) and (28) equals to the gas flow velocity in bed scale (see Eq. (4)).
Importantly, according to the mass conservation, the latent correlation between Eqs. (27) and (28)
is that the sink term in void should be equal to the summation of the diffusion and convection terms
in pore (see Eq. (29)), because the oxygen inside the particle comes from the void, as shown in Fig.
2. This correlation provides a different insight into the oxygen mass transfer that have been intuitively
involved in the energy conservation equation in Dosanjh et al. [32], and Torero et al. [2]. Besides, in
Eqs. (28), comparing to the convection, the diffusion is much small so that the diffusion term can be
removed. Therefore,
22 22
2
g s O,s O,s O,s O,g
() T
Duu
(29)
It is hypothesized that the initial oxygen concentration inside particle is zero, that is, the initial
condition of Eq. (28) is ρO2,s|t=0=0. It indicates that the derivation of ρO2,s with respect to time must be
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equal to or larger than 0, as shown in Eq. (30). Otherwise, oxygen inside particle could be negative.
2
22 2 2
O,s
g O,s O,s O,g O,s
() 0
RT R
u
t
(30)
Eq. (30) can be transformed into Eq. (31), which means that the rate of oxygen consumption must
be equal to or slower than the velocity of oxygen transport from the void.
22 2
O,s O,g O,s
RT
u
(31)
The equal and less conditions in Eq. (31)are separately considered in the following.
Firstly, if the rate of oxygen consumption is equal to the velocity of oxygen transport from the void
(22
O,s O,g
RT
), then, 2
O,s 0
t
. Eq. (26) is equal to
22
OO,g
g
tt
(32)
Although the model is two-dimensional, the gas flow for the forward propagation of smoldering
fire is primarily one dimensional, as can be seen in Fig. 2. The convection term in Eqs. (28) and (29)
can be simplified as
22 2
2
O,s O,s O,s
O,s
d
()
d
uu u
x
yy
(33)
Substituting Eq. (33) into Eq. (31), we have
22
222
22
O ,s (surface) O , y + (inside)
O,s O,g O,g
O,s O,s
d0
d()
ys
Ruuu u u
yyy
(34)
Note that δ in Eq. (34) denotes the characteristic length of oxygen concentration gradient and is
approximated as the particle diameter (d) in this work (see Fig. 2). It has been proposed in the previous
literature [49, 62] to characterize the time scale of oxygen transport. The influence of on the self-
sustained smoldering propagation is discussed in the Section 5.5. Substituting Eqs. (32) and (34) into
Eq. (27), we have
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22
22
OO
2
gggOO
Duu
t
(35)
Secondly, if the rate of oxygen consumption is slower than the velocity of oxygen transport from
the void, combining Eqs. (26)-(29), we can obtain
2
222
O2
gg O O O
R
Du
t
(36)
The porosity of pores inside the particles could be ignorable comparing to the voids among
particles (φs<<φg), thus, φg≈φ and φg in Eqs. (35) and (36) can be replaced by φ. This allows to
merge two separated oxygen transport equations (Eqs. (35) and (36)) as one equation. Overall,
combining Eqs. (30), (31), (35), and (36), the oxygen transport equation is summarized as follows
2 2
222
OO
2
OgO O,i
i
( ) min( , ) (i = co, o, o)
Ru
+u D
t
(37)
where the last term at the right-hand side of Eq. (37) denotes a minimization function, and 2
O
R
is
the total oxygen consumption rates resulting from multiple oxidative reactions, which is written as
is
ii
2222
/
i,O i,O j j O iO,i i e
E
RT
nm
RYY Ar
(38)
Table 2. Summary of the kinetic/oxygen-transport limiting regimes, determined condition, fuel and
oxygen consumption rates.
Regimes Determined condition Fuel consumption rate Oxygen consumption rate
Kinetic regime 1Da Eq. (10) is
ii
22
/
i,O j j O ie
E
RT
nm
YY A
oxygen-transport
limiting regime
1Da Eq. (14)
2
O
u
3.4 Determination of kinetic- and oxygen-transport limiting regimes
The kinetic and oxygen-transport limiting schemes presented in the Section 3.2 can be determined
by the minimization function in the Eq. (37). If the oxygen consumption rate equals to the oxygen
supply rate, the chemical reaction is controlled by oxygen transport, that is, the oxygen-transport
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limiting scheme.
The sink term in the oxygen transport equation (Eq. (37)) represents the fact that the oxygen
consumption rate (RO2) must not be larger than oxygen supply rate ( 2
O
u
), which is the root-cause
for smoldering fire spread controlled by oxygen transport. It sheds light on the challenging issue of
determination of kinetic and oxygen-transport limiting regimes. Moreover, the Damkoeler number is
defined in Eq. (39) to determine the kinetic and oxygen-transport limiting regimes. Regimes and
determined conditions are summarized in Table 2.
2
2
is
ii
2
/
i,O j
O
jOi
eERT
nm
u
A
DYY
a
(39)
3.5 Source/sink terms in the mass and energy conservation equations
Table 2 shows that the fuel consumption rates vary with regimes, which further significantly
impact the source/sink terms of the solid-phase mass and energy conservation equations (Eqs. (16)-
(19) and (21)). The relationship between fuel consumption and oxygen consumption is established
with the stochiometric coefficients. Therefore, similar to the sink term of oxygen transport equation
(Eq. (37)), the fuel consumption rates in the mass and energy conservation equations are modified
as
2
is
ii
2
2
O
ijjOi
iO
min( e , ) (i = co, o, o)
E/RT
nm
,
u
rYYA
v
(40)
The minimization function in the Eqs. (37) and (40) can be easily handled using numerical scheme.
Eqs. (16)-(19), (21), and (37) clearly show that the mathematic model developed in this work is very
different from the models [11, 12, 38, 39, 45, 46, 48, 52-54, 56, 57] with chemical related source/sink
terms formulated by the Arrhenius equations.
4. Numerical computation
The mathematical model is numerically computed using a Finite Element Method (FEM) code
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pack, i.e., COMSOL Multiphysics. Recently, COMSOL Multiphysics has been successfully
employed for hill-side underground coal smoldering fires [62, 63] and self-sustained smoldering
propagation of CFIPM [52-54, 56, 57, 81]. In this work, in order to reduce the computation resources,
the computation domain is limited to the right-hand half domain, as shown in Fig. 3. Due to volume
shrinkage, the distance of fire spread was approximate 0.1 m (see discussion in the Section 5.1).
Hence, the height of computation domain is set as 0.1 m. The input parameters for the numerical
computation are shown in Table 3. Kinetic parameters and reaction heat are presented in Table 1.
Fig. 3 Computation domain of underground coal smoldering fire spread.
The initial and boundary conditions for the numerical computation are presented in the following
(Eqs. (41)-(44)).
The initial conditions are
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22
O=0 O
coal =0 w s
water = 0 w s
char/ ash = 0
g=0 a
s=0 a
(1 )
0
t
t
t
t
t
t
Y
Y
TT
TT
(41)
The bottom, top and lateral boundary conditions are presented as follows:
22
O bottom O
g bottom a
b
ottom s a ig
s rad s bottom
bottom s a ig
()()
() ()(>)
TT
qh TTtt
kk T hTTtt
(42)
2
gOtop
ggtop topg a
srad stop topsa
0
()
() ()
D
kT hT T
kk T hTT
(43)
2
gOlateral
g g lateral lateral g a
s rad s lateral lateral s a
0
()
() ()
D
kT h T T
kk T h TT
(44)
Table 3. Input parameters of numerical computation.
Parameter Value Unit Ref.
φ 0.55 (CC coal), 0.45 (XA coal) - Measured
q' 10 (CC coal), 15 (XA coal) kW m-2 Calibrated
κ 1.970×10-7 (CC coal), 7.220×10-8 (XA coal) m2 Measured
ρs 1482 (CC coal), 1453 (XA coal) kg m-3 Measured
Ta 282.30 (CC coal), 293.15 (XA coal) K Measured
c 2.5×10-3 (CC coal), 7×10-3 (XA coal) - Calibrated
ρg 1.185 kg m-3 [62]
ρO2 0.270 kg m-3 [62]
R 8.314 J mol-1 K-1 [62]
Cp,coal 1320 J kg-1 K-1 [11, 45, 48]
Cp,water 4186 J kg-1 K-1 [11, 45, 48]
Cp,char 1260 J kg-1 K-1 [11, 45, 48]
Cp,ash 880 J kg-1 K-1 [11, 45, 48]
kcoal 0.200 W m-1 K-1 [11, 45, 48]
kwater 0.600 W m-1 K-1 [11, 45, 48]
kchar 0.260 W m-1 K-1 [11, 45, 48]
kash 0.800 W m-1 K-1 [11, 45, 48]
krad Rosseland approximation W m-1 K-1 [52, 53, 81]
kg 0.026 W m-1 K-1 [11, 48]
tig 2 h Measured
Dg 2×10-5 m
2 s-1 [62]
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μ 1.850×10-5 Kg m
-1 s-1 [62]
hsg Eq.(23) W m-3 K-1 [11]
hbottom 10 W m-2 K-1 Calibrated
htop
5 W m-2 K-1 Calibrated
hlateral
30 W m-2 K-1 Calibrated
σ 5.670×10-8 W m
-2 K-4 [81]
d (δ) 0.006 m Measured
H 2.600 m Measured
L 0.100 m Measured
Triangle mesh was adopted in this work. Uncertainty resulting from mesh size was evaluated.
Appropriate mesh size was used when computational resource was not very large and mesh quality
was sufficient for the accuracy of computational results.
Models developed in the literature [11, 12, 38, 39, 45, 46, 48, 52-54, 56, 57] have followed the
same mathematic framework with respect to the governing equations, in which the source/sink terms
associated with chemical reactions have been formulated as the Arrhenius equations, although these
models also have had several differences such as the thermal equilibrium\non-equilibrium hypothesis,
heat transfer coefficients, chemical reaction schemes, and numerical analysis methods. By contrast,
in the proposed model, the source/sink terms in terms of chemical reactions in the governing
equations is replaced by a minimization function between the Arrhenius equations and the analytic
derived expression 2
2
O
iO
,
u
v
. It makes a distinct difference from the mathematic models with the
chemical relevant source/sink terms formulated by the Arrhenius equations in the literature [11, 12,
38, 39, 45, 46, 48, 52-54, 56, 57], which, in this sense, are roughly named as the Arrhenius-type
models for the convenience of sematic difference from the proposed model. In order to further
examine the model proposed in this work, the Arrhenius-type model was numerically computed with
the same input parameters, initial and boundary conditions involved in the proposed model, except
for the reaction order (m) of oxygen mass fraction (YO2). In this work, m in the Arrhenius-type model
increased to 2-5, outputting the best fitting with the experimental data.
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5. Results and discussion
5.1 Experimental results
Figure 4 presents temperature history data of seven thermocouples (TC1-TC7) embedded in
coalbed and evolution of exhaust gas concentrations (O2, CO2, and CO) with time. Basically,
experimental data of both bituminous (CC) and anthracite (XA) coal samples share three common
features:
I. The time needed to achieve self-sustained smoldering propagation of underground coal fire is
very long (see the first and second peaks in Fig. 4 (a) and (c)), which almost accounts for half
time burning the whole coalbeds. The long-term duration mainly results from the slow air flow
driven by weak buoyancy. In the beginning, the smoke temperature is low, and then only gentle
thermal buoyancy is generated. Besides, our previous work [71] revealed that solutal buoyancy
induced by heavy gaseous products at this stage is against thermal buoyancy and hinders air
supply to smoldering combustion. Due to the complex chemical reactions and couplings with
air flow, the influence of the solutal buoyancy is ignored in the mathematical model.
II. Secondly, the peak temperatures increase while fire spreads along the fuel bed. It is attributed
to improvement of air supply due to stronger thermal buoyancy (see TC7 at y=0.14 m in Fig.
4 (a) and (c)).
III. Lastly, peaks of TC6 and TC7 (y=0.12 and 0.14 m, see dash lines in Fig. 4 (a) and (c)) are
much lower than others and takes place at almost the same time. As shown in Fig. 4 (b) and
(d), exhaust gas concentrations indicate that coalbed is almost burnt out when the fire front
passes TC5 at y=0.1 m. It indicates that the distance of fire spread is approximately 0.1 m,
which is shorter than the thickness of fuel bed due to the volume shrinkage [71]. This shrinkage
is neglected in the proposed model.
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Fig. 4 (a) and (c)Temperature-time profiles of seven thermocouples embedded in coalbed, and (b)
and (d) evolution of exhaust gas concentrations with time. The left and right columns denote
experimental data of bituminous (CC) and anthracite coal samples (XA), respectively.
There are several distinctions between CC and XA coal smoldering fire propagations. As can be
seen in Fig. 4 (a) and (c), comparing CC coal sample with XA coal sample, the ignition temperature
is lower but the peak temperatures are higher. In addition, due to higher content of fixed carbon of
anthracite coal [60], CO2 concentration is higher but CO is lower, as shown in Fig. 4 (b) and (d).
5.2 Validation of the proposed model
Temperature history data computed by the proposed model is compared to the experimental results
and the Arrhenius-type model, as shown in Fig. 5. Dimensionless time (τ=tUs,avg/l=0.1tUs,avg [cm h-1])
[88] that is more convenient to illustrate the overall performance of smoldering propagation is used
in this work.
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Fig. 5 Temperature-dimensionless time profiles. The left ((a) and (b)) and right ((c) and (d))
columns present bituminous (CC) and anthracite (XA) coal samples, respectively. Shadow zones
denote max-min temperature ranges of replicated experiments. Solid lines (Ts) in the top ((a) and
(c)) and bottom ((b) and (d)) subpanels indicate the proposed and Arrhenius-type models,
respectively.
As can be seen in Fig. 5, after turning off the igniter, self-sustained smoldering propagations are
successfully established in the proposed and Arrhenius-type models. Peak temperatures in either the
proposed model or the Arrhenius-type model are very close to the experimental data. Moreover, both
models capture the feature that the peak temperatures increase while fire propagating from the bottom
to the top of coalbeds. However, fire propagation from the igniter to TC1 is not predicted well by
neither the proposed model nor the Arrhenius-type model. It is mainly attributed to the fact that the
important influence of solutal buoyancy are not considered in the models.
Fig. 5 (a) and (c) shows that the spatiotemporal evolution from TC3 to TC5 of both CC and XA
coal samples is well predicted by the proposed model. By contrast, the performance of the Arrhenius-
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type model is not good, as can be seen in Fig. 5 (b) and (d). Fire propagation predicted by the
Arrhenius-type model is faster for CC coal sample but slower for XA coal sample. Experimental
results show that the duration of fire spread for CC and XA coal samples is around 25 h and 22 h,
respectively (see Fig. 4). However, the duration predicted by the Arrhenius-type model for CC and
XA coal samples is around 8.5 h and 37 h, respectively (see Fig. S1 in the supplementary materials).
In short, the proposed model well predicts the evolution of central temperatures observed in
experiments. The capability of the proposed model to predict buoyancy-driven, oxygen-limited, and
self-sustained smoldering propagation of underground coal fires is better than the Arrhenius-type
model.
5.3 Comparisons of chemical reaction rates in the proposed and Arrhenius-type models
Figure 6 illustrates the comparisons of chemical reaction rates in the proposed and Arrhenius-type
models. The sequential and parallel patterns of endothermic pyrolysis and exothermic oxidation in
the proposed and Arrhenius-type models are the same. For bituminous coal, the magnitudes of five-
step reaction rates are comparable. Nevertheless, as for anthracite coal, coal oxidation and its
successive β-char oxidation are significantly hindered in both models. It indicates that the proposed
model does not change the sequential and parallel patterns of chemical reactions. But, as shown in
Fig. 6, it is evident that all chemical reaction rates become slower in the proposed model because the
exothermic oxidative reaction rates are limited by oxygen transport.
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Fig. 6 Five-step reaction rates in the proposed (solid lines) and the Arrhenius-type (dash lines)
models. (a) and (f), water evaporation; (b) and (g), coal pyrolysis; (c) and (h), coal oxidation; (d)
and (i), α-char oxidation; (e) and (j), β-char oxidation. The left and right columns present
bituminous and anthracite coal samples, respectively. Colorful lines denote chemical reaction rates
at five locations (y=2, 4, 6, 8, and 10 cm) of the central axis.
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Fig. 7 Plots of maximum rates of major oxidative reactions with air flow velocity. (a) and (b): coal,
and α-char oxidation of bituminous coal; (c): α-char oxidation of anthracite coal. Dark green and
orange denote the proposed and Arrhenius-type models, respectively.
As can be seen in Fig. 6, for bituminous coal, magnitudes of coal and α-char oxidation rates are
larger than β-char oxidation; for anthracite coal, α-char oxidation much faster than coal and β-char
oxidation. Herein, these oxidative reactions that paly a major role in the exothermic oxidation of
smoldering fire propagations are called “major oxidation”, and other oxidative reactions are named
“minor oxidation”. In the Arrhenius-type model, oxidation rates at upper coalbed for both bituminous
and anthracite coal samples are significantly larger than those at lower coalbed, as can be seen in Fig.
6 (c)-(e) and (i). The enhanced oxidative reaction at the upper coalbed are attributed to increased air
flow velocity (see Fig. S2 in the supplementary materials). Fig. 7 illustrates the dependence of
maximum rates of major oxidation on the air flow velocity for both models. In the Arrhenius-type
model, they dramatically increases with air flow velocity (exponential functions). By contrast, in the
proposed model, the influence of air flow velocity is much more gentle (linear functions). The linear
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correlation in the proposed model is in accordance with the intuitive derivation of fire spread velocity
on the basis of the oxidative reaction wave (Uavg=ρgYO2u/υO2ρs [59]).
5.4 Two-dimensional spatial distributions of oxygen and solid species
Two-dimensional spatial distributions of oxygen and four solid species (coal, α-char, β-char, and
ash) are illustrated in Figs. 8 (τ=0.5) and 9 (τ=1). Temperature contours with τ=0.5 and 1 are presented
in Figs. S3. Comparing the proposed model to the Arrhenius-type model, several differences can be
observed. The most distinct one lies in the spatial distribution of oxygen specie (see the top row of
Figs. 8 and 9). In the proposed model, the shapes and thickness of oxygen contours reserve almost
the same for either bituminous coal or anthracite coal. However, in the Arrhenius-type model, they
are dependent on coal samples. The oxygen concentration gradients of bituminous coal are thicker
than anthracite coal. Particularly, as can be seen in Fig. 8, in the beginning stage of smoldering fire
propagation of bituminous coal, the oxygen concentration near to the lateral cooling wall and far
downstream fire front is very high. It indicates that the oxygen distribution in the Arrhenius-type
model is sensitive to coal samples and cooling boundaries. This is mainly attributed to the fact that
the Arrhenius equation is highly dependent on kinetic parameters of coal samples and local
temperatures. By contrast, the analytic expression 22
OO
Ru
derived in the proposed model is
independent on kinetic parameters and local temperatures. Nevertheless, the proposed model is tightly
associated with δ. The sensitivity of δ is discussed in the following section. It’s worth mentioning that
the length of oxygen concentration gradient shown in Fig. 9 is at the magnitude of several millimeters,
which is very close to d. This result at certain degree confirms that the approximation δ≈d is
acceptable in this work.
5.5 Uncertainties
Several uncertain factors such as discharge coefficient (c), heat loss, heat transfer coefficient (hsg),
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and characteristic length of oxygen concentration gradient (δ) are involved in the proposed model.
Our previous work [71] revealed that c in Eq. (4) could be technically considered as a discharge
coefficient, similar to natural ventilation of buildings [89]. It can be estimated via experimental data
[71]. However, air flow in this work is too slow to be measured by the anemometer (FMA900R,
OMEGA, USA; range: 0-1.02 m s-1, accuracy: ±2% FS). At this circumstance, a number of
computations in terms of a wide range of c values are conducted (not presented in this paper). It is
practical to adopt a best value when the central temperatures computed from the proposed model were
best matched with experimental data, as shown in Fig. 5. Table 3 shows that c for anthracite coal
sample is two times larger than that for bituminous coal sample. This is in line with practical
conditions in experiments, that is, 0.4 m thick porous foam placed at the entrance of air-transport pipe
(see Fig. 1) for the bituminous coal sample was removed for the anthracite coal sample.
Lateral heat loss has important effect on smoldering propagation [31, 50, 51, 90]. The lateral heat
loss may lead to two different smolder structures, i.e. the convex and concave smoldering fronts [31].
The convex smoldering front is developed by non-uniform reaction from the center to the wall due to
lower temperatures near to the cooling wall [31]. The concave smoldering front is associated with
non-uniform air flow from the center to the wall because air flow resistance at lower temperature
zones is smaller and then air flow near to the cooling wall is enhanced [9, 31, 51, 55, 91, 92]. As
shown in Fig. 10, our experimental observation confirmed that the shape of smoldering front was
convex. This phenomenon was also observed in smoldering peat fires [48]. Thus, the dependence of
air flow resistance and air density on the temperature is not considered in this work. The convective
heat transfer coefficients (htop/lateral/bottom) are obtained using the same strategy as the discharge
coefficient (c). Numerical results show hlateral and hbottom have slight influence on the centerline
temperatures, but htop is tightly relevant to the smoke temperature and air flow velocity (see Eqs. (4)
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and (42)), and then greatly impacts the smoldering propagation of underground coal fires.
Fig. 8 Two-dimensional spatial distributions of oxygen and four solid species (coal, α-char, β-char,
and ash) density (kg m-3) of (a) bituminous coal and (b) anthracite coal with τ=0.5.
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Fig. 9 Two-dimensional spatial distributions of oxygen and four solid species (coal, α-char, β-char,
and ash) with τ=1 for (a) bituminous coal and (b) anthracite coal.
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Fig. 10 Convex structure of smoldering front of underground coal fires observed in laboratory
experiments.
By using the heat transfer coefficient shown in Eq. (23a), as can be seen from Fig. S2 in the
Supplementary Materials, solid and gas temperatures in the coal bed are local thermal equilibrium,
except for the top surface of the coal bed. Remarkable temperature difference between solid and gas
takes place due to the convective heat transfer boundary condition (h
top
). The velocity of air flow
driven by thermal buoyancy were strongly dependent on gas temperature at the top surface of the coal
bed (T
g5
), as shown in Eq. (4). This the main reason that the thermal non-equilibrium model is
considered in this work. It is suggested for modelling of smoldering propagation driven by thermal
buoyancy that the thermal non-equilibrium between solid and gas should be considered even though
solid and gas temperatures in the main body of the porous media are the thermal equilibrium. Hence,
gas temperature at the boundary can be separated from solid temperature to better estimate the
velocity of gas flow driven by thermal buoyancy. The heat transfer coefficient h
sg
(Eq. (23b))
proposed by Zanoni et al. [81] was tested for both the proposed and Arrhenius-type (i.e., conventional
model in the previous version of manuscript) models as well as two coal samples, as shown in Fig.
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S3, with the same input parameters as well as boundary conditions, except for Nu=2+1.1Re0.6Pr1/3
replaced by Nu=0.001Re1.97Pr1/3. As can be seen in Fig. S3, with Nu=0.001Re1.97Pr1/3, temperature
difference between gas and solid is very large (thermal non-equilibrium model). The solid
temperatures increase fast due to the thermal energy from the igniter, but gas temperature slight
increase probably attributed to the low heat exchange from coal particles. Underground smoldering
coal fires driven by thermal buoyancy are very sensitive to the gas temperature. Low gas temperature
barely produce the thermal buoyancy, resulting in extinction of smoldering, as shown in Fig. S3. It
indicates that hsg proposed by Zanoni et al. [81] might not be applicable for thermal buoyancy driven
smoldering propagation of PSFs fires.
Fig. 11 Sensitivities of δ with regarding to (a) average fire spread velocity and (b) average peak
temperatures of smoldering fire propagations of bituminous and anthracite coal samples.
The model in this work proposes an approximate formulation, 22
OO
Ru
, quantifying the
oxygen consumption rates controlled by the oxygen transport. This formulation performs nice
capability addressing the coupling of oxygen transport and oxidation reactions in PSFs. On the other
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34
hand, it brings one uncertain factor, i.e. δ. Figs. 5 and 9 proves that the approximation δ≈d is
acceptable if the fuel bed consists of particles like coalbed in this work. However, it needs further
investigation in the future to determine δ for certain natural fuels (e.g., polyurethane foams and wood
chips) and applied smoldering systems. Dependence of smoldering propagation on δ is examined, as
shown in Fig. 11. For both studied coal samples, Tp,avg coincidently behaves the same. It decreases
with an increase of δ. But, Uavg presents two distinct trends for bituminous and anthracite coal samples.
Moreover, it is observed that self-sustained smoldering propagation can merely be established within
a limited δ-value window. If δ is too large or too small, smoldering combustion will be extinct. With
increasing δ, the oxygen and fuel consumption rates become slower, which consequently results in
drops of Tp,avg and further extinction. However, the oxygen and fuel consumption rates are not only
associated with δ but also ρO2. If δ becomes smaller, the oxygen consumption rate will become faster
and then leads to lower ρO2. However, lower ρO2, on the contrary, restrains the oxygen consumption
rate. This well explains the extinction of smoldering combustion occurred with small δ (≤1 mm).
From the holistic point of view, δ exerts complicated influences on the oxygen and fuel consumption
rates. This may be partially responsible for the opposite trends of Uavg for two different coal samples,
as can be seen in Fig. 11 (a). Interestingly, dependence of Uavg on δ for anthracite coal is similar to
self-sustained smoldering propagation of food wastes [27], which deserves in-depth investigation in
the future. In addition, the derived analytic correlation of 22
OO
Ru
simplify the complicated
oxygen transport inside pores, given the assumption that the influences of pore evolution and ash
production in pore scale on the macroscopic behavior of self-sustained smoldering propagation could
be neglected. Nevertheless, from the perspective of pore scale, pores inside coal particle could expand
during the pyrolysis, and ash is produced and cumulates at the out layer of pores when coal particle
endures oxidation reaction. Both pore size change and ash production are expected to have important
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
35
influence on oxygen transport in particle and pore scales. Yet, these processes are very complicated
and not involved in the proposed model. The proposed model is limited to simulate oxygen transport
through temperature-and-chemical dependent pores inside coal particles.
Fig. 12 Plots of oxidation reaction rates in kinetic/oxygen-transport-limiting regimes with the
dimensionless time at six points (y=0, 2, 4, 6, 8, and 10 cm) of the central axis. Three rows from the
top to the bottom illustrate oxidation rates of coal (rco), α-char (rαo), and β-char (rβo). The left ((a)-
(c)) and right ((d)-(f)) columns denote bituminous (CC) and anthracite (XA) coal samples,
respectively. Dark green circles (○) and orange asterisks (*) present kinetic and oxygen-transport
limiting regimes, respectively.
5.6 Kinetic/oxygen-transport-limiting regimes
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
36
With the determining criteria shown in Table 2, the kinetic/oxygen-transport limiting regimes for
both studied coal samples is plotted in Fig. 12. It is the first time for the proposed model to holistically
identify the kinetic and oxygen-transport limiting regimes for self-sustained smoldering propagation.
The transition from the kinetic regime to the oxygen-transport limiting regime takes place in the major
oxidative reactions (see Fig. 12 (a), (b), and (e)). The minor oxidation is controlled by the kinetic
reactions, as shown in Fig. 12 (c), (d), and (f). As can be seen in Fig. 12 (a), (b), and (e), the critical
points of the kinetic-to-oxygen-transport transition vary with locations. Specifically, the critical
values decreases from O(10-2 kg m-3 s-1) to O(10-8 kg m-3 s-1) as the fire front propagates from the
bottom to the top of coalbed. It indicates that the kinetic reactions play significant role in the initial
stage of ignition, but the oxidative reactions scheme tends to be controlled completely by oxygen
transport once the self-sustained smoldering propagation is established. This result confirms the
hypothesis [1, 93] on the roles of kinetic reaction and oxygen transport in the different stages of
ignition and propagation of smoldering combustion. However, the root cause for the controlling
processes responsible for different stages remains unknown in this work. The underneath mechanism
for the kinetic and oxygen-transport limiting regimes with respect to ignition, propagation and
extinction deserves further investigation in the future.
6. Conclusions
A two-dimensional model for buoyancy-driven, oxygen-limited, and self-sustained smoldering
propagation of underground coal fire was developed. Air flow driven by thermal buoyancy was
formulated as a function of the discharge coefficient and smoke temperature based on the Darcy’s
law. A five-step chemical reaction scheme was established. The conservation equations of solid mass,
gas and solid energy, and oxygen species transport were involved in the model. Oxygen consumption
rate limited by oxygen transport ( 22
OO
Ru
) was analytically derived from the conservation
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
37
equation of oxygen species transport in gas and solid fuel. A minimization function of oxygen
consumption rates in terms of oxidative reaction (the Arrhenius equations) and oxygen transport (the
analytic expression) was introduced to determine the kinetic and oxygen-transport limiting regimes.
It was proved that the proposed model was able to address the challenging issues of adjustment of
kinetic parameters obtained from TG-scale experiments and decoupling of oxidative reaction and
oxygen transport in multi-scale porous media.
It was validated that the proposed model was better than the Arrhenius-type model to predict the
buoyancy-driven, oxygen-limited, and self-sustained smoldering propagation of underground coal
fires. The main reason was attributed to the improvement of the proposed model to handle the
coupling between oxidative reactions and oxygen transport. The derived oxygen consumption rate
reduces exothermic oxidative reaction rates, but had no influence on the sequential and parallel
patterns of the chemical reaction scheme defined by the kinetic parameters from TG-scale
experiments. The analytic expression was in accordance well with the acknowledgement that the
oxidative reaction of smoldering propagation was controlled by oxygen transport. The characteristic
length (δ) of oxygen concentration gradient was an important parameter for the proposed model. Self-
sustained propagation was established within a certain δ-value range (3 mm-13 mm for bituminous
coal and 3-7 mm for anthracite coal in this work). Either too small or too large δ leaded to the
extinction of smoldering propagation. δ can be approximated as particle size d for coalbed, but for
CFIPM and certain PSFs with fibrous and chip solid shapes it should be further investigated. This
work may help to improve modelling of the oxygen-limited and self-sustained smoldering
propagation for other PSFs and CFIPM.
Acknowledgements
This work was funded by National Natural Science Foundation of China (No. 51804168) and the
Combustion and Flame 245 (2022) 112382; https://doi.org/10.1016/j.combustflame.2022.112382
38
Youth Talent Program of Shaanxi Province. Author appreciates three anonymous reviewers for their
constructive comments that are very helpful to improve the quality of this paper.
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1
Supplementary Materials
For
Modelling of oxygen-limited and self-sustained smoldering propagation:
underground coal fires driven by thermal buoyancy
Zeyang Songa,b,*
a College of Safety Science and Engineering, Xi’an University of Science and Technology, Xi’an
710054, P.R. China
b Shaanxi Key Laboratory of Prevention and Control of Coal Fire, Xi’an University of Science and
Technology, Xi’an 710054, P.R. China
* Corresponding author at: College of Safety Science and Engineering, Xi’an University of Science and Technology, Xi’an 710054,
P.R. China. E-mail address: zeyang.song@xust.edu.cn (Z. Song).
2
Velocity of gas flow driven by thermal buoyancy
Simple one-dimensional model of gas flow is shown in Fig. S1(a). Gas flow resistance along the
air-transport pipe is firstly assumed to be ignorable, and will be reconsidered later. Gas flow
pressure at point B and D are
P
B
=ρ
air
g(H+L) (1)
P
D
=ρ
smoke
H (2)
Besides, P
C
≈P
B
=ρ
air
g(H+L) considering the vertical one-dimensional model and ignorable flow
resistance. Then, the pressure drop of the coalbed is
P
CD
= P
C
-P
D
=ρ
air
g(H+L)- ρ
smoke
gH (3)
Given the pressure drop, the velocity of gas flow through the coal bed can be calculated using the
Darcy’s law:
CD air smoke
()PgHLgH
P
uyL L
(4)
Fig. S1 Schematic diagram of one-dimensional gas flow model.
Eq. (4) shown above is an equivalent form of Eq. (1) presented in the manuscript. The derivation
3
with respect to the thermal buoyancy was illustrated in the manuscript and is not repeated here.
The air flow resistance along the air-transport pipe that is ignored before should be considered.
Due to rather uncommon structure, rare model in terms of buoyancy-driven flow could be found in
the literature. Herein, the discharge coefficient (c) that has been widely applied in the natural
ventilation of buildings [1] is used to approximately represent the influence of air flow resistance
along the air-transport pipe on the air flow velocity, as shown in Fig. S1(b).
Combined the pressure drop, the Darcy’s law, and the discharge coefficient (c), the velocity of gas
flow through the coal bed (Eq. (4) in the manuscript) is obtained.
Fig. S2 Temperature- time profiles of the proposed model (top row) and the Arrhenius-type model
(bottom row). The left ((a) and (b)) and right ((c) and (d)) columns present bituminous (CC) and
anthracite (XA) coal samples, respectively.
4
Fig. S3 Gas and solid temperatures at five central locations with hsg=0.001Re1.97Pr1/3: (a) the
proposed model for bituminous coal, (b) the Arrhenius-type model for bituminous coal, (c) the
proposed model for anthracite coal, and (d) the Arrhenius-type model for anthracite coal.
Fig. S4 Variation of air flow velocity induced by thermal buoyancy for (a) bituminous coal and (b)
anthracite coal. The orange solid and dark green dash lines denote the proposed model and the
Arrhenius-type model, respectively.
5
Fig. S5 Solid (Ts) and gas (Tg) temperature contours (℃) of (a)bituminous coal with τ=0.5, (a)
anthracite coal with τ=0.5, (c) bituminous coal with τ=1, and (d) anthracite coal with τ=1.
Reference
[1] P. F. Linden, The fluid mechanics of natural ventilation, Annu. Rev. Fluid Mech. 31 (31) (2003)
309-335, https://doi.org/10.1146/annurev.fluid.31.1.201.
