Distributed Activation Energy Model
of Heterogeneous Coal Ignition
John C.
C. Chen
Chen
John
We present a model that simulates the conventional
used for
conventional tube-furnace experiment .used
f~r ignition
i~ition studies.
studi~s: The
particle-to-partIcle variations
vanatlOns in
III reactivity
reactivIty by
Distributed Activation
Activation Energy Model of Ignition accounts for particle-to-particle
having
having a single
single preexponential factor and a Gaussian distribution of activation
activation energies among
am~ng the
th~ particles.
particles.
III
show that the model captures the key experimental
experimental observations,
observations, namely,
namely, the linear increase
Illcrease in
The results show
ignition frequency
frequency with
with increasing
increasing gas temperature and the variation of the slope of the ignition
ignition frequency
frequen'!'
with oxygen
fit
oxygen concentration. The article also
also shows
shows that adjustments to the model parameters permit a good fit
experimental data.
with experimental
NOMENCLATURE
N
OMENCLATURE
A0
d
E
E
E0
Eo
h
He
k
n
Q
Q
R
S
T
y
Y
preexponential factor in
in Arrhenius
Arrhenius rate
rate
preexponential
constant of
of ignition
ignition (kg m
m--2Z s --l1))
constant
diameter (m)
diameter
energy of
of ignition
ignition (kJ
(kJ mol
mol-I)
activation energy
t)
mean of
of Gaussian
Gaussian distribution
distribution
mean
(kJ mol-1
mol-- I ))
(kJ
heat transfer
transfer coefficient
convective heat
m--2z K)
((W
Wm
heat of
of reaction
reaction (J kg --I)
heat
t)
thermal conductivity
conductivity (W
(W m
m--I
thermal
t K)
order of ignition
ignition with
with respect
respect
reaction order
reaction
to oxygen
heat generated
or heat
heat loss (W)
(W)
heat
generated or
universal gas constant
constant (8.314 X
× 1010 -33 kJ
I
1
molmo1-1 KK -1))
external
external surface area of particle
particle (m Ze))
temperature
temperature (K)
molar ratio of CO
CO to COz
CO 2
Greek
Greek Symbols
Symbols
X0
Xo22
6'
E
(T
orbb
orf
i
particle surface
oxygen mole fraction at particle
emissitivity of coal
Stefan-Boltzmann
S t e f a n - B o l t z m a n n constant
constant (5.67 X
×
4)
1010 _88 W mm -2Z KK -a)
standard
standard deviation of Gaussian distribution (kJ mol-I)
mol- t)
Subscripts
Subscripts
conv convection
g
gas
p
particle
particle.,
rad
radiation
radiation
IINTRODUCTION
NTRODUCTION
Numerous
Numerous experiments
experiments have
have been
been conducted
conducted
over
over the
the past
past three
three decades
decades to
to study
study the
the ignition
of pulverized
pulverized coals under
under conditions
conditions reletion of
vant
vant to utility boilers.
boilers. The
The conventional
conventional experiment
ment is based
based on
on one
one developed
developed by Cassel
Cassel and
and
Liebman
Liebman [1], and
and consists of
of a tube
tube furnace
furnace
containing
0 2 and
containing a heated
heated mixture
mixture of
of Oz
and inert
inert
gas. The
e m p e r a t u r e of
The ttemperature
of the
the furnace, and
and hence
hence
the
the gas, is the
the independent
independent variable in this
arrangement.
arrangement. The
The experiment
experiment is conducted
conducted by
dropping
dropping a small batch
batch of
of presized
presized coal partiparticles into
into the
the hot
hot gas and
and visually observing for
ignition. In such
a flash of light, which signals ignition.
experiment the
the particle
particle concentration
concentration is
an experiment
enough that
that each particle can be
typically low enough
considered to behave independently
independently of all othconsidered
The furnace temperature
temperature is then
then deers. The
creased and
and the experiment
experiment is repeated
repeated to decreased
the minimum
m i n i m u m gas temperature (or the
termine the
which ignition occritical gas temperature) at which
condition is termed
termed critical ignition.
curs. This condition
particle size or the Oz
0 2 concentraconcentraFinally, the particle
changed and, again, the critical ignition
tion is changed
condition is found for the new operating
operating concondition
dition.
THEORY OF HETEROGENEOUS
HETEROGENEOUS
THEORY
IGNITION
IGNITION
Essenhigh et al. [2] describe in detail the theEssenhigh
heterogeneous ignition, that is, ignition
ory of heterogeneous
For a
that occurs at the solid-gas interface. For
coal particle exposed to an oxidizing environdetermined by the balance
ment, ignition is determined
between heat
heat generation
generation at
at and
and heat
heat loss
loss from
between
the particle
particle surface.
surface. The
The heat
heat loss
loss from
from the
the
the
surface of
of aa panicle
particle at
at temperature
temperature Tp
Tp is the
the
surface
sum of
of the
the losses
losses caused
caused by convection and
and
sum
radiation:
radiation:
surface
surface is given by the
the kinetic
kinetic expression:
expression:
S
(1)
Equation 1 assumes
assumes that
that the
the surroundings
surroundings inEquation
radiation exchange
exchange are
are in thermal
thermal
volved in radiation
equilibrium with
with the
the gas.
equilibrium
The radiative
radiative loss term
term is relatively unimunimThe
portant until
until the
the panicle
particle temperature
temperature exceeds
portant
1500 K.
K For
For the
the convective-loss term, we
~- 1500
assume that
that the
the Nusselt
NusseIt number
number equals
equals 2, as is
assume
appropriate for very small particles, which leads
leads
appropriate
= 2k
/d
1
be
•
to hh =
2kg/dp.
Thus,
Eq.
can
be
rewritten,
g
p
on a per-external-surface-area basis, as:
on
Qlo~
S
dQge,
dQgen = dQlos
dQloss~
dJ;,
=
~
(6)
(6)
drp
dJ;,'
It is presumed
presumed for the
the purpose
purpose of
of this study
that
that certain of the
the variables in Eqs. 2 and
and 4 are
are
known a priori (He'
(He, A0,
A o, n, dp,
d p , and
and e), or
or are
are
fixed by the experimental conditions (T
(Tgg and
and
Xo2).
Xo). The
The values of these
these variables used for the
the
base
base case (described below) are
are shown in Table
Tpp and
1. The
The remaining
remaining unknowns,
unknowns, T
and E, can
then
then be determined
determined by the
the simultaneous
simultaneous solution of
of Eqs. 5 and
and 6; the
the relation of these two
parameters to T
Tgg for the base case is shown in
parameters
Tp
manner repreFig. 1. T
p determined in this manner
sents the critical ignition temperature, whereas
E can be interpreted as the critical (or maximum) activation energy that a particle may
mum)
under the given conditions.
have and still ignite under
A detailed expansion of Eqs. 5 and 6 is given in
the Appendix.
The gas thermal
thermal conductivity, kg, in the
The
boundary layer around
around a heated
heated particle is
boundary
given by a linear fit to the conductivity of air
over the temperature
temperature range of 300-2000
300-2000 K:
[T+T]
(3)
(3)
Equation
Equation 3 represents an approximation for
the conductivity evaluated at the mean
mean of the
free-stream and particle-surface temperatures,
temperatures,
and it is noted that the variation of conductivity with temperature may be represented by
higher order representations.
The heat generated by a spherical carbon
particle undergoing oxidation on its external
M O D E L FORMULATION
FORMULATION
MODEL
Figure 2 shows typical data [3] obtained from
an ignition experiment conducted by varying
the gas temperature while holding all other
TABLE
TABLEI
Values
Values of Parameters
Parameters in
in the Base
Base Case
Case of the Model
Model
Variable
Variable
Value
Value
A0
500
500
100
100
120
120
9210
9210
1.0
1.0
16.0
16.0
0.8
0.8
de
E~
Hc
n
~r
e
Units
Units
kgmkgm -22
s -1
S-I
/Lm
/~m
kJ
kJ mol-I
mol- 1
kJ
kJ kg-I
kg- 1
-kJ
kJ mol-I
mol- 1
--
(4)
(5)
(5)
Qgen =OJoss,
2kg
(Tp - Tg) + eCro(Tp4 - Tg4). (2)
dp
W
5 Tp
7.0 X× 10lO_5[
W- •
P + Tg]
kkg== 7.0
g
m K"
g
2
mK
.
Diffusion effects are
are neglected
neglected because
because at
at the
the
relatively low particle
of pulverparticle temperatures
temperatures of
ized-coal ignition,
ignition, the
the oxidation reaction
reaction is kinetically controlled.
At
At the
the critical ignition condition, the
the following two conditions
conditions are
are satisfied [2]:
Q10ss =
= Oloss,conv
Qloss,eonv +
+ QIoss,
Qloss. rad
rad
Qloss
= hS(T e - Tg) + ~o'bS(Tp 4 - rg').
= Hc x°2A°exp
Remarks
Remarks
arbitrarily chosen
chosen to
to illustrate
illustrate model
model
arbitrarily
arbitrarily chosen
chosen to illustrate
illustrate model
model
arbitrarily
arbitrarilychosen
chosen to
to illustrate
illustrate model
model
arbitrarily
for the
the reaction
reaction CC + 1/2
1/202
~ CO
CO
for
O 2 .....
arbitrarilychosen
chosento
to illustrate
illustrate model
model
arbitrarily
arbitrarilychosen
chosen to
to illustrate
illustrate model
model
arbitrarily
arbitrarilychosen
chosento
to illustrate
illustrate model
model
arbitrarily
140
140
1300
1300
Q
1200
~ 1200
::l
~
1100
1100
//
~~ 1000
1000
t-.I¢c:
O 900
900
u
/
// /
120
///
E
~
/
~oo w
100
/~
/
///
70O
700
i'ii
Ij
:¥
1///
80
80
i
800
800
U
60
1200
1200
60O
600
600
6OO
'7~
'0
/
.E9
-~ 800
80o
"~
:.e
//
// /
:~c:e-
"§
// //
//
//
// //
1000
1000
T
T~g (K)
Fig.
temperature (dashed
Fig. 1.
1. Relation
Relation of critical
critical ignition
ignition temperature
(dashed
line)
temline) and critical
critical activation
activation energy
energy (solid
(solid line)
line) to gas
gas tembase case
perature, TTg,
for base
case listed
listed in Table
Table 1.
1.
g , for
parameters constant. The
parameters
The data
data shown was obthe experiment as detained by conducting the
tained
scribed earlier except that, at each temperarepeated
ture, 10 to 20 tests for ignition were repeated
to obtain
obtain a frequency or probability of ignition.
Fig. 2 shows that
that ignition frequency increases
approximately linearly with gas temperature,
temperature,
and
and this is inconsistent with the heterogeneous
ignition theory previously described. If
If all particles of a coal sample
sample used in an experiment
have the same
same reactivity, that
that is, they are de100
100 , - - - . - - - - - - -
N
80
80
".e
~
"O
-0
C1J
60
ct'~
~
C1J
"l
::J
•
---,
w
.~
<Jl
••
•
•
••
•
•
•
•
•
40
4o
<Jl
•
•••
Cl..
20
O
• ---.___--_,_--__I
o -+---~....
700
750
800
850
900
Gas
Gas Temperature
T e m p e r a t u r e (K)
(K)
Fig.
Fig. 2.
2. Typical
Typical data from
from a conventional
conventional ignition
ignition experiexperirelation between
between ignition
ment showing
showing the relation
ignition frequency
frequency (or
probability) and gas temperature for
bituminous coal.
probability)
for a bituminous
coal.
Ref. 3.
Data extracted
extracted from
from Ref.
3.
scribed by a common Arrhenius
Arrhenius rate constant
as in Eq. 4, then
then the data
data would show an
ignition frequency of 0%
0% until the critical gas
temperature, corresponding to that
temperature,
that at the critical ignition condition, is reached. At any gas
temperature above this, the ignition frequency
temperature
be 100%. Note that the observed igniwould be
trend is not an artifact of the
tion frequency trend
experiment; this same behavior is reported
from a variety of ignition experiments [4, 5, 6]
including thermogravimetric analyzers and
laser-based
laser-based studies.
One
One of the reason why ignition frequency
should increase gradually with increasing gas
temperature is somewhat
Within any
temperature
somewhat obvious: Within
sample of coal, there exists a distribution of
reactivity
reactivity among the particles. Thus, in the
conventional ignition experiment, in which a
batch of perhaps
perhaps a few hundred
hundred particles of a
batch
sample is dropped
dropped into the furnace, there is an
increasing probability (or frequency) that at
least one particle has a reactivity that
that meets or
exceeds the critical ignition condition set forth
in Eqs. 5 and 6 as the gas temperature
temperature is
increased. Of
Of course, there exist other
other variations among
among the particles within a sample, such
as particle size and specific heat. Variation
Variation in
size alone could account for the observed
increase in ignition frequency with gas temperature. It cannot account for another experimental observation, however, namely the variamental
tion in the slope of the ignition frequency with
oxygen concentration. (This behavior is described in a later section.) A distribution in
specific heat
heat would only affect the rate at which
a particle attains its equilibrium temperature,
temperature,
but would not change this value or the reactivbut
ity. Perhaps
Perhaps other
other variations could cause the
observed behavior of ignition frequency. It is
our premise
premise that the distribution in reactivity
dominates
dominates all other
other variations, however, and
therefore accounts for the observed behavior.
The Distributed
Distributed Activation Energy Model of
Ignition (DAEMI)
(DAEMI) models the conventional ignition experiment by allowing for the particles
within the coal sample
sample to have a distribution of
We prescribe that
that all the particles
reactivity. We
the same properties, including the preexhave the
ponential factor in the Arrhenius
Arrhenius rate constant
ponential
constant
describing their ignition reactivity, and
and that
their activation energy is distributed according
to the Gaussian (or normal) distribution:
1
f ( E ) = (27rtr2)0. 5 exp
-
(E
-
2tr2
RESULTS AND DISCUSSION
E0) 2
(7)
E 0 is the mean and IT
0" is the standard
where Eo
deviation of the distribution. The expression
E+iJ>.E
jE
fee+ a E!(E)
f ( E ) dE
dE
(8)
describes the frequency or probability that particles within a sample have an activation en+ aE.
A E. Accordingly,
ergy in the range E to E +
the distribution satisfies the condition that
f~_~ !(E)
f ( E ) dE
dE = l.
1.
r~-oo
DAEMI divides a prescribed
prescribed distribuThe DAEMI
AE == 3
tion into discrete energy intervals of aE
ld mol-I,
tool-1, and considers only the energy range
kJ
E 0 --3 t3lT
r to Eo
E 0 ++3 3lT,
o ' , rather than -00
- ~ to
of Eo
+o0.
+
00.
The latter simplification still covers
99.73% of the distribution. The model then
calculates the frequency of being in each of
these intervals by numerically integrating Eq. 8
for each of the intervals.
An ignition experiment is modeled by assuming that 10
1055 particles are in the initial
sample, and that they are distributed among
the various AE
aE intervals according to the calculated frequency for each interval. Each
Each simulation of an experimental run under
under a given set
of conditions is conducted on a batch of 100
randomly selected particles from the sample,
keeping in mind that no particle can be selected more
more than once. Whether
Whether or
or not ignition occurs for a run is determined by the
particle
particle in the
the batch
batch of
of 100 with the lowest
activation energy. If
If this particle's
particle's reactivity
equals
equals or
or exceeds that determined
determined by the
the critical ignition condition (that
(that is, its activation
energy is less than
or equal
equal to
to the
the critical
than or
energy determined
determined by solution of
of Eqs. 5 and
and 6),
the
the batch
batch is defined as ignited. This is consistent
tent with our
our observation
observation [6] that
that single-parsingle-particle ignition is discernible to the
the eye, and
and
certainly to
to a photon
photon detector. This procedure
procedure
is repeated
repeated 20 times at each
each condition, just
just as
in actual
actual experiments,
experiments, to
to determine
determine an
an ignition
frequency at
at this condition. Finally, the
the gas
temperature
temperature is varied
varied several
several times and,
and, each
each
time, 20
20 runs
runs are
are conducted.
conducted.
Figure 3 shows model results of ignition frequency versus gas temperature for one hypothetical sample for which Eo
E 0 = 120 kJ mol-I,
mo1-1,
IT
= 16
and A o0 ==5 500
or=
1 6 kkJ
J mol-I,
mo1-1,and
0 0 k gkgm m--22 S-I.
s -1.
The other parameters of this base case calculation are listed in Table 1. It is obvious that the
DAEMI
DAEMI exhibits the experimental characteristic of increasing ignition frequency with increasing gas temperature. As stated earlier,
this is expected as an increase in gas temperature leads to an increase in the maximum
activation energy that a particle can have and
still be ignitable (see Fig. 1), and therefore to
an increased probability of having at least one
particle within each batch of a simulated run
that is reactive enough to ignite.
Figures 4 and 5 display the effects of varying
E 0 and IT,
or, respectively, on the DAEMI.
DAEMI. An
Eo
increase in Eo,
E0, which shifts the Gaussian distribution to higher energies (Fig. 4a), has the
effect of shifting the ignition-frequency data to
higher gas temperatures (Fig. 4b). This is the
expected behavior as a representative batch
from the higher Eo
E 0 sample contains, on average, particles with higher activation energies,
require a higher temperature
temperature to induce
which require
Note that E
Eo0 has only a slight effect
ignition. Note
on the slope of the data in Fig. 4b.
100
--..
>R..
~
• ••
80
80
v
••
>u
ec
0)
"1
:J
0"
c-
660
0
•
•
O)
u:,-.
I.l_
c
440
0
0
O
¢c
• •
••
:-e
o~
$
._~
20
0
•
i
6600
00
A ~
w w
i
i
800
80O
11000
000
11200
200
T g (K)
Tg
Fig. 3.
3. RResults
from D
Distributed
Activation
Model
Fig.
e s u l t s from
istributed A
c t i v a t i o n EEnergy
nergy M
odel
for bbase
case ((Table
1).
oof
f IIgnition
g n i t i o n ((DAEMI)
D A E M I ) for
a s e case
T a b l e 1).
12000 , . . - - - - - - - - - - - - ,
8000
<Jl
a;
7000
(a)
(a)
<Jl
a; 10000
10000
U 6000
.€
U
°w
'';:;
'n:l
Q..
8000
'-
4000
0 4000
a; 3000
t-~
.D
E
E 2000
'-
6000
'-
'-
Z
Z
Z
Z
n:l
0...
5000
0O
a;
.D
E
E
~
~
I000
1000
2000
0
60
90
150
120
0+----..,.....
0
40
80
180
Activation Energy
Activation
Energy (kJ marl)
mo1-1)
100
1oo
----
•
• •
0~
~
>u
e-c:
a;
~)
-s
~
0-
a;
'u..
"
cc:
0
O
•~-e--
:-ec:
80
80
C
0
0
~
(b)
160
•
>-
•••
80
~
u
Q)
•
~
~
u..
"
0
20
20
•
m
600
tC
.-::
•~cc:
,
i
20
800
1000
0
1200
Tgg (K)
T
the effect
Eo0 on
Fig. 4. DAEMI
D A E M I results showing
s h o w i n g the
effect of
of E
o n ignition frequency.
tion
frequency. (a)
(a) Solid-colored
S o l i d - c o l o r e d distribution
d i s t r i b u t i o n corresponds
corresponds
to EEo0 = 110 kJ mo1-1,
mol-I, and
the distribution
a n d the
d i s t r i b u t i o n shown
s h o w n in
outline
to EEo0 = 120 kJ m
mol-I;
pao u t l i n e corresponds
c o r r e s p o n d s to
o l - ~ ; all other
o t h e r parrameters
a m e t e r s are
are as listed in Table
T a b l e 1. (b) Ignition
I g n i t i o n frequency
frequency
data
Eo0 =
mol- 1 (
Eo0 =
mol-- l1
d a t a for
for E
= 110 kJ mo1-1
( I.) ) and
and E
= 120 kJ tool
(0).
(o).
Figure 5 shows that
that an increase in the standard
dard deviation (a
( o r)-the
) - - t h e spread
spread of the GaussGaussian distribution (Fig. 5a)-has
5 a ) - - h a s two effects: a
shift of the ignition frequency to lower temperatures and
and a slower rise of ignition frequency
temperature (Fig. 5b). These findings
with gas temperature
are somewhat
because the average
somewhat unexpected because
both
activation energy (Eo)
( E 0) is the same
same for both
the use of a
samples, and
and they result from the
small batch
batch (l00
(100 particles) in each run. The
shift to lower temperatures
temperatures is caused
caused by the
fact that, statistically, the most reactive particle
in the larger atr batch will have a lower activa-
"
•
"
"
•
00
0
O0
40
0O
0
••
•
Q)
'-
40
40
00
AA
60
6o
0-
•
(b)
"
"
c:
0
200
--4~__,
c-
60
60
0
100 ,----1oo
",.e
0
120
Activation Energy (kJ
Activation
(k] marl)
tool -I)
-
0
Jk
4000
"
m
i
600
800
800
A
1000
1000
1200
1200
Tgg (K)
T
the effect of
Fig. 5. DAEMI
D A E M I results showing
s h o w i n g the
of u
o- on
on ignition
frequency.
to
frequency. (a) Solid-colored
S o l i d - c o l o r e d distribution
d i s t r i b u t i o n corresponds
c o r r e s p o n d s to
u
mol--~,1, and
the distribution
cr = 20 kJ mol
a n d the
d i s t r i b u t i o n shown
shown in outline
outline
mol
corresponds
to ut7 = 12 kJ m
c o r r e s p o n d s to
o l -~1I;; all other
o t h e r pparameters
a r a m e t e r s are
are
as listed in Table
T a b l e 1. (b) Ignition
I g n i t i o n frequency
f r e q u e n c y data
d a t a for u
tr =
= 20
u~ r == 12
mol--n1 (( A...))..
kkJJ mmolo l 1i (
( I.) )
a nand
d
1 2 kkJ
J tool
than the most reactive particle
tion energy than
batch because of its wider
from the smaller aor batch
spread
spread in distribution. Thus, the higher reactivity allows for ignition to occur at lower temperThe wider spread
spread and small batch
atures. The
batch also
approach to 100% ignition
cause the slower approach
frequency because the probability of having a
batch containing only relatively higher energy
batch
particles is increased, which requires higher
gas temperatures
temperatures until all runs result in igniparameters
tion. Clearly, by adjusting
adjusting the two parameters
of the Gaussian
Eo0 and
Gaussian distribution
distribution (Eq. 7), E
and a,
~r,
the DAEMI
D A E M I can be fitted to ignition data, such
that accurate valas those in Fig. 2, provided that
ues for the parameters shown in Table 1 are
available.
D A E M I in its present form fails
fails to
The DAEMI
capture a second experimental characteristic,
namely, the change in slope of the ignition
frequency data with oxygen concentration, as
shown in Fig. 6. The lines shown are linear
regressions of experimental data from Ref. 3.
(The data points are explained below.) In fact,
DAEMI shows a very weak dependence on
the DAEMI
oxygen concentration (not shown), so the quesparameter has not
tion remains as to what parameter
100
100
50
50
25
-~"
G
~
0
v
A-
0
IO0
100
Q)
~
::J
25
25
Q)
o0
100
100
:.ee'-c
~
~
1
I
1
kJ
C+
+ "20z
z-&-O2 ~
~ CO; H~,eo
H~,co = 9,21O
9,210kg C
kgC
kJ
C+
co 22 = 32,790+ Oz
O2 ~
--* COz;
CO2; H~
H~,co
32,790;gK
,
kg C
C
l
-~
l
y
1
Y H'
1 H''
He = Y
y +
+ 1 H ' c,co
+ Y + 1 Hc,
co2,
c,eo +
e,eo
2,
l
~
75
20o/002
50
50
25
0
100
f..••
10%
75
0 2
50
50
25
25
00
i
600
600
700
700
•
_.ff~O
l
i
8800
00
900
900
(9)
Thus, the value of H
H cc in Eq. 4 is dependent on
the relative amounts of CO and COz
C O 2 formed
during ignition, and is given by the expression:
50010Oz
0 2
50"10
5o
50
0-
u.
u..
e'C
.0
0
100%
Oz
100"10 Oz
.I °°"
75
o"
...
eC
~
4
75
been or is incorrectly accounted for. The obvious candidate is the reaction order, n, which
must vary with X0
Xo22 in order to cause the
observation. We can think of no fundamental
basis for this behavior, however.
A second, less obvious parameter
parameter is the heat
of reaction, H cc,' of ignition. So far in the presentation of the DAEMI, it has been assumed
that the product of coal ignition is CO, as
shown by the value for H
H cc in Table 1. It is well
known that the product of carbon oxidation is
both CO and COz,
CO 2, however, with the relative
amounts dependent on both temperature and
oxygen partial pressure [7, 8]. The significance
of this is that their heats of reaction are vastly
different:
1000
1000
TT8g (K)
Fig. 6. Linear
Linear regressions of
of experimental
experimental data
data from Ref.
Ref.
33 (shown as solid
of free-stream
solid lines), showing the
the effect
effect of
oxygen concentration
on ignition
ignition of
of aa high-volatile bitumibitumiconcentration on
nous
5 - 9 0 /~m.
h e data
nous coal
coal of
of diameter
diameter 775-90
p,m. T
The
data points
points reprerepresent
A E M I , including
sent results
results from
from the
the D
DAEMI,
including the
the modification
modification
to
O and
to account
account for
for the
the production
production of
of both
both C
CO
and CO2
CO 2 and
and
adjustments
0, (r,
u, and
and n,
n, as
as
adjustments to
to the
the base
base case
case values
values for
for EEo,
described
described in
in the
the text.
text.
(10)
(10)
where y =
O / m o l COz'
CO 2. We
= mol C
CO/mol
We have asCO
sumed here that energy released by any CO
that oxidizes as it diffuses away from the particle surface does not affect the ignition.
DAEMI should now
This modification to the DAEMI
Measurements
show the experimental trend. Measurements
that at higher particle
particle temperature
temperature
[7, 8] show that
Tg ), the
the molar
molar ratio
(which results from higher Tg),
CO/COz
increases and
and consequently He,
H c ' the
the
C
O / C O 2 increases
of heat
heat released
released during ignition, deamount of
amount
creases. Therefore,
Therefore, the
the result of
of a set of
of runs
creases.
decreased oxygen level not only
conducted at a decreased
conducted
the ignition frequency data
data to a higher
shifts the
Tg (a
(a direct result of
of the decreased
decreased Xo2)
Xo) but
but
Tg
reduces the slope of
of the
the rise (an
(an indirect
also reduces
of the decreased
decreased He).
H c )'
because of
result because
Direct measurements
measurements of
of the
the C
CO
Direct
O //CO
C O 2z ratio
Du et
et al. [7] in a thermothermohave been
been made
made by Du
have
(TGA) using soot
soot as the
the
gravimetric analyzer (TGA)
carbon material. Measurements
Measurements were
were made
made
carbon
of 667-873
667-873 K
K and
and
over the
the temperature
temperature range
range of
over
-, .g
0
>u
>,
c::
c
(1)
::J
"~
C"
(1)
L..
I,Ju..
c
c::
0o
•~
c::
:-E
tlO
I O0
100
80
60
40
2o
20
0
100
100
80
80
60
40
40
20
0o
100
80
60
40
20
0
•
100% 02
•
/.
oZ
e
.6/_
0 2
6 0 % 02
60%
,
20% 02
0 2
•
••
o ° ~
/
•
•
••
60O
600
800
800
I
I
1000
1000
1200
1200
1400
1400
T
Tg (K)
(K)
g
Fig. 7. DAEMI results, including
including the modification
modification to acacFig.
CO2,
showing
count for the production of both CO and CO
2 , showing
effect of oxygen
oxygen concentration on ignition
ignition frequency.
frequency.
the effect
The solid lines indicate linear regressions of the data
points.
oxygen partial
partial pressures
pressures from 0.1 to 1 atm. The
The
results at an oxygen partial
partial pressure
pressure of
of 0.21
atm
atm are
are correlated by the expression:
mo, CO
[-3214]
mol CO
[ -3214]
- - -2 = 59.95
59.95exp
tool
exp TpCK)
Tp(K) J
CO 2
mol CO
(11)
(11)
This correlation is incorporated
incorporated into
into the
the
D
A E M I and
DAEMI
and the
the model
model results
results are
are shown
shown in
Fig. 7. Notice
Notice that
that the
the model
model now
now clearly possesses
sesses the
the desired
desired characteristic.
characteristic. Furthermore,
Furthermore,
it captures
both the
the decrease
decrease in the
the slope
slope of
of
captures both
the
the ignition frequency with
with decreasing
decreasing oxygen
concentration,
of the
the slope's
slope's
concentration, and
and the
the slow rate
rate of
decrease
decrease until
until low oxygen concentrations,
concentrations,
showing
showing the
the nonlinear
nonlinear behavior
behavior with Xo2
X0 2 displayed in experimental
data (Fig. 6).
experimental data
By adjusting
adjusting the
the mean,
mean, the
the standard
standard deviation
tion and
and the
the reaction
reaction order,
order, n,
n, of
of the
the base
base
case,
A E M I can
of D
DAEMI
can be
be fit
case, this
this current
current version
version of
to
to the
the experimental
experimental data
data shown
shown in
in Fig.
Fig. 6. (The
(The
particle
particle diameter
diameter has
has also
also been
been changed
changed to
to
83.0
JLm to match the mean of the sample used
83.0/xm
in Ref. 3.) The model results using Eo
E 0 == 84.0
1
1
, a
kJ mol, and n =
mo1-1,
o- == 4.0 kJ molmol-1,
= 0.4, are
plotted as data points in Fig. 6 over the regression lines and show that a satisfactory fit
fit is
achieved with minimal effort in parameter adjustment,
justment, despite the uncertainties in the values of other parameters. (Because of the small
value of acr used, the energy intervals into
which the distribution is divided was decreased
from 3 kJ moltool 11 to 1 kJ molm o l - I1 to obtain these
results.) It should be noted that the ignition
parameters reported above represent merely a
rough fit to the experimental data; it is certainly possible that another set of parameters
can also fit the data satisfactorily, especially if
a different value for A o0 were chosen. This
extra effort may be worthwhile as certain parameters have strong theoretical justification
(the reaction order, n, for example) for being
within a particular range. We have begun work
to examine this issue in more detail.
Although
Although we have assumed in this study that
pulverized-coal ignition occurs heterogeheterogeneously without influence from any volatile
matter that may be present,
present, and even though
the results closely fit the
the experimental data,
data, it
the
cannot be said that
that the D
DAEMI
that
cannot
A E M I confirms that
ignition is purely a heterogeneous process. Very
models of homogeneous ignition have been
been
few models
presented, and
and none
none have been
been tested against
against
presented,
the available experimental data
data because
because of the
the
inherent difficulty and
and uncertainty in modeling
modeling
inherent
devolatilization and
and the combined solidsolid- and
and
gas-phase
reactions.
gas-phase
CONCLUSIONS
CONCLUSIONS
The D
DAEMI
has been
been formulated to model
model
The
A E M I has
It acconventional coal-ignition experiments. It
counts for particle-to-particle variations
variations in recounts
within a sample
sample by allowing for a disactivity within
tribution in activation energies
energies among
among the
the
tribution
particles and
and a single preexponential
preexponential factor.
particles
The model
model captures
captures the
the main
main characteristics
characteristics
The
of actual
actual experiments: the
the gradual
gradual increase
increase in
of
increasing gas
gas tempertemperignition frequency with increasing
of the
the slope
slope of
of the
the
ature and
and the
the variation
variation of
ature
concentration. Fiignition frequency with
with 0022 concentration.
ignition
has been
been shown
shown that
that adjustments
adjustments to
to
nally, it has
the model
model parameters
parameters can
can be
be used
used to
to fit experithe
mental data and extract reaction rate constants.
of the U.
U.S.
of Energy
The support of
S. Department of
DE-FG22-94MT94013) for this project is
(Grant DE-FG22-94MT94013)
gratefully acknowledged.
Note that the neglect of the TTp
dependence in
p dependence
kg
k s introduces
introduces a small error in Eq. 13.
Following Eq. 6, we set Eq. 12 equal to Eq.
E/RTp:
13 and solve for the quantity E/R~:
2k
4
2k gs
y
dpT
zrp +
+ 4eubTp
4 e°'b Tr4
E
E
p
=
REFERENCES
REFERENCES
1. Cassel,
Cassel, H. M.,
M., and Liebman,
Liebman, I.I. Combust.
Combust. Flame
Flame
1.
3:467-475 (1959).
(1959).
3:467-475
2. Essenhigh,
Essenhigh, R
R. H.,
H., Mahendra,
Mahendra, K
K. M.,
M., and Shaw,
Shaw, D.
D. W.,
W.,
2.
Combust. Flame
Flame 77:3-30 (1989).
(1989).
Combust.
3. Zhang,
Zhang, D.,
D., Wall,
Wall, T.
T. F.,
F., Harris,
Harris, D.
D. J.,
J., Smith,
Smith, I.I. W.,
W.,
3.
Chen, J., and Stanmore,
Stanmore, B.
B. R,
R., Fuel
Fuel 7:1239-1246
7:1239-1246
Chen,
(1992).
(1992).
4. Tomeczek,
Tomeczek,J., and Wojcik,
Wojcik, J., Twenty-Third
Twenty-ThirdSymposium
Symposium
4.
(International) on
on Combustion,
Combustion, The Combustion
Combustion InstiInsti(International)
Pittsburgh, 1990,
1990, pp. 1163-1167.
tute, Pittsburgh,
Boukara, R,
R., Gadiou,
Gadiou, R,
R., Gilot,
Gilot, P., Delfosse,
Delfosse, L.,
L., and
5. Boukara,
Prado, G.,
G., Twenty-Fourth
Twenty-FourthSymposium
Symposium (International)
(International)on
Prado,
Combustion, The Combustion
Combustion Institute, Pittsburgh,
Pittsburgh,
Combustion,
1993, pp. 1127-1133.
1993,
6. Chen,
Chen, J., Taniguchi,
Taniguchi, M.,
M., Narato, K,
K., and Ito, K,
K., ComCorn6.
bust. Flame
Flame 97:107-117
97:107-117 (1994).
(1994).
bust.
7. Du, Z., Sarofim,
Sarofim,A. E,
F., and Longwell,
Longwell, J. P., Energy
Energyand
7.
Fuels 5:214-221
5:214-221 (1991).
(1991).
Fuels
8. Mitchell,
Mitchell, R
R. E., Kee,
Kee, R
R. J., Glarborg,
Glarborg, P., and Coltrin,
Coltrin,
8.
M. E.,
E., TwentyTwenty-Third
Symposium (International)
(International) on
on
M.
Third Symposium
Combustion,
Combustion, The Combustion
Combustion Institute, Pittsburgh,
Pittsburgh,
1990, pp. 1169-1176.
1990,
H c xd2Ao exp
The
The denominator
denominator is recognized to be Qgen/S
(Eq. 4), which by Eq. 5 is also QlossiS
Qloss/S (Eq. 2).
Thus Eq. 14 can be rewritten as:
2k
4
2ksg
y
drTp
Tp +
+ 4eUb~
4eCrbZp4
E
E
p
RTr
2ks
dr ( Tr - Tg) +
eo'b(Tp4- Tg4)
(15)
(15)
relation for E //RR~
T p is substituted
substituted into the
the
This relation
expression Q
Q10ss
=
0
to
obtain
a
funcQgen
Qtos~
=
obtain
gen
tion, F,
F, which is a function of T
Tp
p only:
F(Tp)
=
=
Received
1995; accepted
Received 30 July
July 1995;
accepted 4 February
February 1996
1996
(14)
(14)
[
Qgen -- Qloss
Q10s s
Qgen
=
He xg,~Ao
Xo 2 A o
= Hc
APPENDIX
APPENDIX
Expansion
Expansion of
of Eqs.
Eqs. 55 and
and 66
In
In order
order to determine
determine the
the critical
critical ignition
ignition temperature
and
critical
activaof the
the particle,
particle, Tr,
and
critical
,
perature of
p
tion
E, Eqs.
Eqs. 5 and
and 6 are
are solved simultasimultation energy,
energy, E,
neously.
neously. Qgen
Qgen and
and Qtos~
Qloss are
are given in
in Eqs.
Eqs. 22 and
and
4, and
and lead
lead to
to the
the following derivatives with
with
respect
respect to
to temperature:
temperature:
dQgen
dQsen
E] (RT/
E)
n
c X~32Ao
exp [ d~ = SH
SHe Xo 2 A o exp RTp
dQ
2k S
dQtoss
2ks
3
~
4eUb STp.
+ 4eOrbSTp
dTr = --gs
dp
+
3.
dTp
dp
(12)
(12)
(13)
(13)
× exp
2ks
- - C - r, - 4 br,
. . . .
-C(rr-
r,)
4]
----~4-
4'
r - r, )
(4
4) = O.
0.
2k(g T
2ks
Tgg ) -- eO'b(Zp
- subp
T 4-- Zg
Tg4)
d ( T pr _- T)
d rp
(16)
(16)
The reasonable
reasonable root
root of
of F(Tp)
F(Tp ) corresponds
corresponds to
to
The
the critical
critical ignition
ignition temperature
temperature of
of the
the partipartithe
and substitution
substitution of
of this
this value
value into
into Eq.
Eq. 15
cle, and
produces the
the critical
critical activation
activation energy
energy at
at the
the
produces
critical ignition
ignition condition.
condition.
critical