Combustion and Flame 149 (2007) 366–383
www.elsevier.com/locate/combustflame
A reduced mechanism for methane and one-step rate
expressions for fuel-lean catalytic combustion of small
alkanes on noble metals
S.R. Deshmukh, D.G. Vlachos ∗
Department of Chemical Engineering and Center for Catalytic Science and Technology (CCST), University of Delaware,
Newark, DE 19716-3110, USA
Received 19 June 2006; received in revised form 10 February 2007; accepted 22 February 2007
Available online 27 April 2007
Abstract
A reduced mechanism and a one-step rate expression for fuel-lean methane/air catalytic combustion on an Rh
catalyst are proposed. These are developed from a detailed microkinetic model using a computer-aided model reduction strategy that employs reaction path analysis, sensitivity analysis, partial equilibrium analysis, and simple
algebra to deduce the most abundant reaction intermediate and the rate-determining step. The mechanism and the
one-step rate expression are then tested on Pt catalyst. It is found that the reaction proceeds effectively via the
same mechanistic pathway on both noble metals, but the effective reaction orders differ due to the difference in
the adsorption strength of oxygen. Based on the homologous series idea, the rate expression is extended to small
alkanes (ethane and propane; butane is also briefly discussed) and is found to reasonably describe experimental
data. Estimation of the relevant parameters in the rate expression for various fuels and catalysts using the semiempirical bond-order conservation theory, quantum mechanical density functional theory, and/or simple experiments
is discussed. Finally, it is proposed that detailed microkinetic models with coverage-dependent parameters can
assist in rationalizing the apparent discrepancies between experimental data from various research groups.
© 2007 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
Keywords: Catalytic combustion; Rate expressions; Model reduction; Microkinetic modeling; Alkanes; Methane; Ethane;
Propane; Butane; Platinum; Rhodium; Oxidation; Noble metals
1. Introduction
Catalytic combustion has so far found limited
applications. However, the need for distributed and
portable power generation that relies on modularity
and small scales may render catalytic combustion an
appealing technology. The emergence of the hydro* Corresponding author. Fax: +1 (302) 831 1048.
E-mail address: [email protected] (D.G. Vlachos).
gen economy [1] relies on the production of pure hydrogen. To this effect, various routes, such as steam
reforming, partial oxidation, and autothermal reforming, are being explored. Among these, steam reforming (typically of natural gas) followed by water-gas
shift (WGS) is currently the most economically viable route [2]:
Cn H2n+2 + nH2 O = nCO + (2n + 1)H2 ,
CO + H2 O = CO2 + H2 .
0010-2180/$ – see front matter © 2007 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
doi:10.1016/j.combustflame.2007.02.006
(1)
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
367
Since the net reaction is endothermic, thermal energy
needs to be supplied for sustained hydrogen production via the combustion of hydrocarbons either homogeneously (gas-phase) or heterogeneously (catalytic):
3n + 1
O2 = nCO2 + (n + 1)H2 O.
Cn H2n+2 +
2
(2)
2. Hierarchy of catalytic reaction models
Currently, a lot of attention is devoted to hydrogen production in microdevices for portable applications [3–9]. Recent studies on gas-phase microburners [10–12] and hydrogen production using homogeneous combustion as an energy supply route [13,14]
have revealed the feasibility of such an operation, on
the one hand, but the susceptibility of gaseous flames
to radical and thermal quenching, on the other, arising
from confining flames in small devices. The high temperatures associated with homogeneous combustion
also limit the choice of materials for device fabrication. Given the small scales of these devices and the
large surface-area-to-volume ratios, catalytic combustion appears to be a promising alternative to gaseous
combustion. Elimination of flames makes integration into compact devices easier. The lower temperatures (compared to those in homogeneous combustion) generated via catalytic combustion can significantly widen the operating window of these microdevices [13]. Fuel-lean catalytic operation can eliminate
NOx and CO formation, without coking of the catalyst.
Even though catalytic combustion has been studied intensively, both experimentally and theoretically
[15–21], accurate and reliable rate expressions for
the combustion of alkanes are not readily available,
but are highly desirable for design and optimization studies of microchemical devices and possible
homogeneous–heterogeneous hybrid systems, e.g.,
thermally stabilized combustion.
In this paper, we briefly review the hierarchy of
catalytic kinetic models [22] to re-emphasize the importance of detailed reaction mechanisms. Given the
substantial computational requirements of large reaction networks in computational fluid dynamics (CFD)
simulations, we carry out model reduction of detailed kinetic models following ideas from Ref. [23].
Specifically, we analyze detailed microkinetic models for fuel-lean catalytic combustion of methane on
rhodium (Rh) and platinum (Pt) catalysts and then
perform systematic computer-aided model reduction
(without any a priori assumptions) to develop reduced
kinetic models and an easy-to-use reduced rate expression. Estimation of the parameters involved in
the reduced rate expression is also discussed. The
above framework is then extended mainly to ethane
and propane; butane/air combustion is also briefly discussed.
Here r is the reaction rate (mol/cm2 /s), kapp is the
apparent (or effective) reaction rate constant, C is
the concentration (mol/cm3 ), α is the reaction order,
app
Aapp is the apparent pre-exponential factor, Ea is
the apparent activation energy (kcal/mol), R is the
ideal gas constant (1.987 cal/mol/K), and T is the
temperature (K). Only the apparent activation energy
and the reaction orders are often estimated via fitting the parameters of Eq. (3) to experimental data.
A summary of such parameters for methane combustion on Pt, gathered from the literature, is given in
Table 1. Such rate expressions provide no insight into
the physics and their regime of applicability is ill defined. As a result, the wide scatter in the parameters
in Table 1 is not surprising, making the usefulness of
power-law rate expressions questionable.
Langmuir–Hinshelwood (LH) type kinetic rate expressions are also commonly used to describe reaction
rates. They are developed by making a priori assumptions about the rate-determining step (RDS), partial
equilibrium (PE) of some reactions, quasi-steady state
(QSS) of some species, and/or the most abundant reactive intermediate (MARI) on the surface. To illustrate this approach, let us consider the fuel (F) and
dissociative oxygen adsorption followed by the oxidation reaction between the adsorbed species,
Surface reaction rates have traditionally been
modeled with one-step rate expressions of a powerlaw form,
αO2
α
Fuel
CO
r = kapp CFuel
F+∗
kF,1
F∗
kF,2
O2 + 2∗
kO,1
F∗ + O∗
kr
kO,2
2
app
αFuel αO2
= Aapp e−Ea /RT CFuel
CO .
2
(3)
(PE),
2O∗
(PE),
Products
(RDS).
(4)
Here ∗ denotes a vacant site or an adsorbed species.
Considering the adsorption/desorption steps in PE
and the surface reaction as the RDS, the reaction rate
is calculated as
kr KF PF KO PO2
,
r = kr θF∗ θO∗ =
(5)
(1 + KF PF + KO PO2 )2
k
k
where KF = kF,1 and KO = kO,1 are the equilibrium
F,2
O,2
constants, k stands for the rate constant, θ is the surface coverage, and Pi is the partial pressure of the ith
species. Examples of LH models for methane combustion on Pt can be found in the work of Trimm
368
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
Table 1
Literature power-law rate expressions (see Eq. (3)) for methane combustion on Pt
app
Investigator
Ea
(kcal/mol)
A
αO 2
αFuel
Molar ratio
(O2 /CH4 )
Yao [57]
–
−0.6
1.0
1–4
–
1.0
1.0
0.6
Ma et al. [24]
21 (wire)
24 (Al2 O3 )
20.6
(>813 K)
44.7
(<813 K)
21
−0.17
0.95
0.86–1.3
Firth and Holland [77]
Aube and Sapoundjiev [56]
47.8
13
0.0
–
1.0
1.0
10.5
–
Song et al. [78]
32.2
0.5
1.0
–
Fullerton et al. [58]
22
–
–
2.33
Aryafar and Zaera [79]
32
−0.1
1.1
3–10
Kuper et al. [80]
36
–
1.0
–
Otto [81]
Niwa et al. [82]
27
29.3
0.0
0.0
1.0
0.9
6
2
Garetto and Apesteguia [53]
Anderson et al. [83]
18
24.6
0.0
–
1.0
–
5
–
Cullis and Willatt [84]
Kolaczkowski and Serbetcioglu [85]
24
31.3
–
0.0
–
0.72
0.4
–
Trimm and Lam [25]
0.75
1.20 × 104
(α +α
)
(mol/m2 h/kPa O2 Fuel )
–
1.35 × 104
(1/s)
1.3 × 1011
(cm2.5 /mol0.5 s)
1 × 108
(ml/s/g)
6600
(1/s/cm2 )
1.78 × 108
(mol/m2 /s)
–
1.08 × 109
(1/min)
–
1.35 × 106
(cm3 /cm3 s)
–
2.84 × 108
(mol/m2 /s)
and co-workers, who proposed Eq. (5) with parameters determined at different temperatures (e.g., kr =
995.94, KF = 0.0112, KO = 1.405 at 663 K; the units
of pressure were not explicitly given but are presumably kPa) [24]. The form
r=
kr KF XCH4 XO2
(1 + KF XCH4 + KO XO2 )2
has also been proposed, where X stands for mole
fraction (e.g., kr KF = 9.36 × 10−8 kmol/kg-cat s,
KF = 7.94 × 10−4 , KO = 4.20 × 10−3 at 853 K for
nonporous Pt/Al2 O3 ) [25]. The last form assumes adsorption of molecular oxygen onto the catalyst surface
(viz., O2 + ∗
kO
kO
O∗2 ), which further undergoes
reaction with the adsorbed fuel molecule.
An advantage of a LH rate expression is that reaction orders could vary from positive (small KP terms)
to negative (large KP terms) as operating conditions
change. The assumptions made in LH models rely
mainly on intuition or at best on limited knowledge
of reaction energetics. Adequate description of experimental data via a LH model is typically considered as
validation of the assumptions made. If the model fails
to describe data, a different set of assumptions is made
and the process is repeated. This methodology does
not necessarily guarantee that the underlying assumptions are correct and that the derived rate expression
captures the physics of the reaction mechanism over
a broad range of conditions, which is typically delimited by the available experimental data. Since the
coverages of surface species vary considerably with
operating conditions and within the reactor itself, so
do the RDS and PE conditions. A good example of
this case is hydrogen combustion on Pt (see Ref. [26]
for examples of rate expressions in different operating regimes) and the partial oxidation of methane
on Rh where combustion occurs near the entrance
of a monolith followed mainly by steam reforming
downstream [27].
Microkinetic models that describe all relevant elementary reaction pathways are needed to overcome
the limitations discussed above and provide insights
into the mechanistic pathways [28]. A number of microkinetic models are available for simple fuels, such
as H2 and CH4 , on noble metal catalysts [28–35]. As
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
emphasized in our previous work, microkinetic models are not free of problems; quite the contrary. Major limitations related to their development have been
outlined in [36–39] and are not repeated here. The
foremost challenge in microkinetic model development is the estimation of kinetic parameters. In a series of recent papers [23,27,29,37,39], we have been
developing a multiscale hierarchical approach that results in thermodynamically consistent [40], comprehensively validated microkinetic models. The development is not the subject of this work.
CFD simulations using microkinetic models for
design and optimization are a CPU-intensive task [23].
Hence, reduced kinetic models and one-step rate expressions for the combustion of small alkanes on
noble metals are desirable. In the next section we outline an approach to accomplishing this task starting
with methane combustion on Rh.
3. Rh microkinetic model validation and
reduction using a posteriori computer-aided
analysis
We have recently proposed a thermodynamically
consistent reaction mechanism for the C1 chemistry
on Rh, comprising 104 reactions (52 reversible reactions) [27]. It contains elementary reaction steps for
H2 oxidation, CO oxidation, CO–H2 coupling (carboxyl (COOH) and formate (HCOO) related pathways as well as CO oxidation via OH, which are
important in the water-gas-shift (WGS) and preferential oxidation (PROX) reactions), CH4 oxidation
and reforming, and those associated with oxygenates
(CH3 OH and CH2 O) decomposition. This model was
extensively validated for H2 oxidation, CO oxidation,
WGS, PROX, catalytic partial oxidation, autothermal
reforming, CO2 reforming, and oxygenate decomposition with relevant experimental data [27,29]. Since
the microkinetic model was not previously validated
for methane-lean combustion, in this section, we first
validate the microkinetic model for the C1 chemistry
369
on Rh [27] under methane-lean combustion conditions. Then we develop a reduced microkinetic model
and a reduced rate expression using a computer-aided
methodology (see Fig. 1 and the following discussion). It is well known that transport phenomena (internal in the catalyst and external of the catalyst effects) can be important and must be accounted for
in modeling (analysis has shown that transport effects are not as important for these experimental
conditions). Model predictions with a simple fixed
bed reactor model and the CHEMKIN and Surface
CHEMKIN framework [41,42] are compared with the
experimental data of Burch et al. [43] for 1% CH4 in
air on a 1% Rh/Al2 O3 catalyst at atmospheric pressure. The steady state governing equations in a plug
flow reactor are as follows (gas-phase species, surface
species, and site conservation, respectively, at each
spatial location z),
aσi Mi
dyi
=
, i = 1, . . . , Ng ,
dz
ρu
σi = 0, i = 1, . . . , Ns − 1,
Ns
θi = 1,
(6)
(7)
(8)
i=1
where ρ is the density (g/cm3 ), u is the velocity
(cm/s), z is the reactor length (cm), Ng and Ns are
the numbers of gas and surface species, respectively,
y is the mass fraction, M is the molecular weight
(g/mol), θ is the surface coverage, σ is the species
consumption or production rate (mol/cm2 /s), and a
is the catalytic area per unit volume (cm2 /cm3 ). Here
(ρu) represents the mass flux, which is constant at
each cross-section of the reactor. The species rate can
account for internal and external mass transfer effects
(not explicitly considered here). The resulting stiff
system of differential–algebraic equations is solved
using the DDASSL solver [44].
As shown below and found also experimentally,
under the fuel-lean catalytic combustion conditions
of interest to this work, H2 O and CO2 are the major
products expected (CO and other minor products are
Fig. 1. Schematic showing the important steps in the computer-aided model reduction methodology.
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S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
Fig. 2. Comparison of (a) full and one-step rate expression
model prediction with experimental data and (b) 104-reaction (full) mechanism and 15-reaction (reduced) mechanism for fuel-lean methane combustion on Rh at atmospheric
pressure (1% CH4 in air, a residence time of τ = 33.3 s at
300 K, assuming a = 3200 cm−1 ) [43].
typically not observed). As a result, the overall stoichiometry of Eq. (2) holds and the methane conversion is the only independent variable that can be used
to test catalyst activity and model predictive ability.
Fig. 2a shows that the microkinetic model (solid line)
is able to capture well the experimental data (symbols) for fuel-lean combustion of methane on Rh.
Having ascertained the validity of this detailed
mechanism under fuel-lean conditions, computeraided model reduction is undertaken to develop a reduced rate expression. The high computational speed
of simulations using microkinetic models within ideal
reactors provides an efficient platform for surface reaction mechanism reduction [23]. Specifically, we
employ the recently introduced computer-aided reduction methodology of [7] to identify key steps and
reaction intermediates in a given mechanism under
a wide range of operating conditions. This methodology is based on reaction path analysis (RPA),
PE analysis (PEA), and sensitivity analysis (SA) of
key responses, coupled with small parameter asymptotics as depicted in Fig. 1. Principal component
analysis (PCA) may also be necessary.
RPA is first performed to identify the subset of
dominant (high-rate) reactions. For this, the contribution of all reactions toward the production and
consumption of a given surface species is determined. The contribution of a reaction to a species
is deemed important and the reaction retained in the
overall mechanism if it contributes more than a certain threshold (taken here to be 10%) to the total
production or consumption rate. This analysis is performed over a wide range of fuel–air equivalence
ratios (Φ < 1.0) and at various temperatures. It is
seen that a small subset of reactions (almost the entire
set is indicated in Fig. 3a) are dominant. In this subset, methane adsorbs dissociatively on vacant sites
to give CH∗3 . Oxidation occurs through a series of
O∗ -assisted H abstractions (OH∗ also plays a role in
the oxidation of CH∗2 to CH∗ ) and the subsequent oxidation to carbon monoxide and carbon dioxide. In the
process, the adsorbed oxygen gets reduced to water
via the H∗ atoms. Based on the RPA predictions, the
original 104-reaction microkinetic model is reduced
to a 24-reaction (12 reversible reactions; 11 of them
are shown in Fig. 3a) network.
Of the eight adsorbed surface species participating
in the oxidation pathway described above, the MARI
is inferred from the coverage of the surface species
along the reactor. Fig. 3b shows that the surface is
saturated with adsorbed oxygen (O∗ ); i.e., O∗ is the
MARI. This is not surprising given the fuel-lean operation and the fact that methane adsorption is activated,
and is in line with previous studies [45–47]. Further
analysis of the oxygen coverage on the catalyst surface is presented later.
The steady-state balances for the surface species
(based on the microkinetic model) can be simplified
using reaction rate information from the RPA. These
reductions of surface balances lead naturally to identification of approximate low-dimensional manifolds.
Detection of reaction pairs in PE is such an example. Alternatively, PEA could also be used to ascertain whether PE holds for a certain reaction pair. PE
for a reversible reaction pair (e.g., adsorption of O2
and desorption of O2 ) holds when the rate of the forward reaction is equal to the rate of the backward
reaction. Therefore, under PE conditions, the ratio of
the rate of forward reaction to the sum of the rates
of forward and backward reactions should be equal
to 0.5. This PE criterion is assessed for the adsorption/desorption of major species (O2 , CO2 , and H2 O)
and the results are shown in Fig. 4a. PE is fulfilled
for the adsorption/desorption of O2 , CO2 , and H2 O
at various temperatures and equivalence ratios. Since
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
371
Fig. 3. Panel (a) shows the mechanistic pathway for fuel-lean methane combustion using reaction path analysis. Panels (b) and (c)
show the coverage of important surface species at the reactor exit on Rh and Pt catalysts, respectively, for different temperatures
and equivalence ratios. O∗ is the MARI on both catalysts. Experimental conditions correspond to those of (1% CH4 in air,
τ = 33.3 s at 300 K, assuming a = 3200 cm−1 ) [43] for Rh and to (Φ = 0.35 and a residence time of τ = 50 ms; a is fitted to
220 cm−1 ) [50] for Pt.
O∗ is the MARI, only the approximation regarding
PE of oxygen will be used to compute its surface coverage as a function of the gaseous concentration (see
below).
The RDS is identified next through a pairwise SA.
In this SA, the pre-exponentials of forward and backward reaction steps are simultaneously perturbed by
the same amount to preserve the equilibrium constant.
The absolute value of the sensitivity coefficient with
respect to the exit conversion in the reactor is shown
in Fig. 5a. Based on the sensitivity coefficients, the
dissociative adsorption of methane on Rh is the RDS.
Other important reactions occurring on the Rh surface
are oxygen adsorption/desorption, H2 O∗ -mediated
CH∗ reduction, hydroxyl dissociation/formation, and
water dissociation/formation. Clearly, methane adsorption and oxygen adsorption/desorption are the
most important steps controlling CH4 conversion.
The 24-reaction subset deduced from RPA can be
further trimmed by performing SA with respect to
consumption of reactants, CH4 and O2 , and formation
of products, CO2 and H2 O. A reduced mechanism
of 15 reactions, presented in Table 2, is deduced that
faithfully captures the predictions of the original 104reaction microkinetic model for methane-lean combustion, as shown in Fig. 2b. In this mechanism only
some key reaction steps are reversible (R), since the
reverses of the rest of the reaction steps play no role
under fuel-lean catalytic combustion conditions.
Based on the RDS, the reaction rate for methane
combustion on Rh can be written as the rate of adsorption of methane,
ads X
θ 2.
r = kCH
4 CH4 ∗
(9)
The MARI implies that
θO + θ∗ = 1.
(10)
The partial equilibrium of the oxygen adsorption/desorption step (O2 + 2∗ ↔ 2O∗ ) implies
ads X θ 2 = 2k des θ 2 .
2kO
O2 ∗
O 2 O∗
2
(11)
372
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
Fig. 4. Validity of the partial equilibrium (PE) assumption for the adsorption–desorption of oxygen, water, and carbon dioxide
as a function of equivalence ratio at various temperatures for (a) Rh and (b) Pt catalysts. The parameters are those of Fig. 3.
Fig. 5. Panels (a) and (b) show sensitivity analysis data on Rh and Pt catalysts, respectively, for only the most important reactions
out of the 52 reversible ones at different equivalence ratios. The reaction numbers correspond to Table 1 of Ref. [27]. The
dissociative adsorption of CH4 is the RDS on both catalysts. Panels (c) and (d) identify “total oxidation” as the overall reaction
stoichiometry for Rh and Pt catalysts, respectively. The parameters are those of Fig. 3.
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
373
Table 2
Reduced microkinetic model for fuel-lean methane catalytic combustion on Pt and Rh
s or A
No.
Reaction
1(R)
O2 + 2∗ → 2O∗
2(R)
2O∗ → O2 + 2∗
3(R)
O∗ + H∗ → OH∗ + ∗
4(R)
OH∗ + ∗ → O∗ + H∗
5
H∗ + OH∗ → H2 O∗ + ∗
6
2OH∗ → H2 O∗ + O∗
7(R)
H 2 O∗ → H 2 O + ∗
8(R)
H 2 O + ∗ → H2 O ∗
9
CH4 + 2∗ → CH∗3 + H∗
10
CH∗3 + O∗ → CH∗2 + OH∗
11
CH∗2 + O∗ → CH∗ + OH∗
12
CH∗ + O∗ → CO∗ + H∗
13
CO∗ + O∗ → CO∗2 + ∗
14(R)
CO∗2 → CO2 + ∗
15(R)
CO2 + ∗ → CO∗2
6.86 × 10−4
5.44 × 10−7
9.04 × 1018
5.29 × 1013
9.73 × 1017
5.24 × 1018
1.62 × 1017
4.25 × 1021
1.63 × 1022
2.42 × 1016
2.69 × 1017
1.55 × 1022
7.87 × 105
8.13 × 1014
1.43 × 10−4
2.53 × 10−5
7.10 × 102
35.5
6.64 × 1019
1.06 × 1018
7.49 × 1019
2.56 × 1018
2.25 × 1019
4.55 × 1015
7.95 × 1018
5.85 × 1011
1.83 × 1011
1.53 × 106
4.69 × 10−2
6.46 × 103
β
Ea
0.766
1.997
1.039
3.034
1.379
0.049
1.487
−0.756
−0.246
1.2898
0.567
−1.244
2.589
−0.645
1.162
1.407
−1.529
−0.884
0.031
0.578
−0.126
0.399
0.300
1.124
0.437
2.384
0.523
2.774
0.250
−1.946
0.0
0.0
49.5 − 32.0 · θO
95.5 − 42.0 · θO
8.6 + 6.9 · θO
14.0 − 4.2 · θO
27.2 − 10.1 · θO
24.3 − 7.8 · θO
13.4 − 22.1 · θO
17.4 − 21.4 · θO
22.6 − 33.2 · θO
23.2 − 30.8 · θO
9.6
10.4
0.0
0.0
9.6
10.3
11.1 + 1.0 · θO
7.4 − 1.6 · θO
−0.3 + 6.2 · θO
19.6 − 2.8 · θO
45.8 − 4.4 · θO
43.2 − 6.3 · θO
20.4 − 4.6 · θO
23.2 − 5.6 · θO
3.1
4.9
0.0
0.0
Note. The parameters in the first row are for Pt and in the second row for Rh. (R) indicates a reversible reaction. Activation
energies are in kcal/mol and the rate constants in mol/cm2 s (CHEMKIN format). Primes indicate parameters in CHEMKIN
format (see text). Note that due to refitting of physical parameters in CHEMKIN format, parameters may appear out of their
usual physical range (see equation following Eq. (15)). In addition, one may have to suppress the option of CHEMKIN that does
not allow for an effective sticking coefficient to be greater than 1.
Equations (10) and (11) allow us to evaluate the coverage of vacancies, θ∗ , as
θ∗ =
1+
1
ads X
kO
O2
2
des
kO
2
(12)
,
which can be used in conjunction with Eq. (9) to obtain a simplified rate expression for the fuel-lean combustion of methane on Rh:
rCH4 =
ads X
kCH
CH4
4
1+
ads X
kO
O2 2
.
(13)
2
des
kO
2
In Eq. (13), X is the mole fraction, and the rate constants and reaction rate are in turnover frequency units
(TOF), i.e., molecules per catalyst site per second.
They are computed using semiempirical techniques
with a modified Arrhenius form for desorption (or reaction) and adsorption, respectively
des
T β
and
Tref
ads
ads
sPtot e−Ea /RT
T β
k ads =
.
√
Tref
Γ 2π MRT
des
k des = Ae−Ea /RT
(14)
Here A is the pre-exponential (1/s), s is the sticking coefficient, Ptot is the total pressure, and β is the
temperature exponent. In our formalism, activation
energies are, in general, coverage- and temperaturedependent. The reference temperature is taken as
Tref = 300 K.
In order to facilitate use via CHEMKIN [42],
we have refitted the parameters with temperatureindependent activation energies (Ea ) and converted
the units of parameters (into mol, cm, and s), namely
374
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
Table 3
Parameters for the one-step rate expression for fuel-lean catalytic combustion given via Eq. (13)
Fuel/catalyst
sFuel
Ades
O2
(1/s)
sO2
Eaads (Fuel)
(kcal/mol)
Eades (O2 )
(kcal/mol)
CH4 /Pt
0.116
8.41 × 1012
0.0542
7.3 + 1.9 · (T /300)
52.8 − 2.3 · (T /300) − 32.0 · θO
ads = 0.154
βFuel
des = −0.796
βO
ads = 0.766
βO
0.4
8.41 × 1012
0.0542
1.7 + 2.2 · (T /300)
52.8 − 2.3 · (T /300) − 32.0 · θO
ads = 0.154
βFuel
des = −0.796
βO
ads = 0.766
βO
0.06
8.41 × 1012
0.0542
0.0
52.8 − 2.3 · (T /300) − 32.0 · θO
ads = 0.154
βFuel
ads = 0.766
βO
2
0.06
des = −0.796
βO
2
8.41 × 1012
0.0542
0.0
52.8 − 2.3 · (T /300) − 32.0 · θO
ads = 0.154
βFuel
des = −0.796
βO
ads = 0.766
βO
0.229
4.31 × 1012
0.0481
8.1 + 1.9 · (T /300)
82.8 − 2.3 · (T /300) − 42.0 · θO
ads = 0.788
βFuel
des = 1.997
βO
ads = 1.199
βO
C2 H6 /Pt
C3 H8 /Pt
C4 H10 /Pt
CH4 /Rh
Fuel/catalyst
sFuel
CH4 /Pt
C2 H6 /Pt
C3 H8 /Pt
C4 H10 /Pt
CH4 /Rh
709.55
1339.0
0.0249
0.0249
35.5
2
2
2
2
ads
βFuel
−1.529
−1.827
0.154
0.154
−0.884
2
2
2
2
AOdes
2
des
βO
sO
9.04 × 1018
9.04 × 1018
9.04 × 1018
9.04 × 1018
5.29 × 1013
1.039
1.039
1.039
1.039
3.034
6.86 × 10−4
6.86 × 10−4
6.86 × 10−4
6.86 × 10−4
5.44 × 10−7
2
2
ads
βO
Ea ads
Fuel
Ea des
O
0.766
0.766
0.766
0.766
1.997
9.6
4.4
0.0
0.0
10.3
49.5 − 32.0 · θO
49.5 − 32.0 · θO
49.5 − 32.0 · θO
49.5 − 32.0 · θO
79.5 − 42.0 · θO
2
2
Note. θO is calculated using Newton’s method (see text). In the top set, the activation energy is coverage- and temperaturedependent and the rate constants are in turnover frequency units. In the bottom set (CHEMKIN format), the activation energy is
coverage-dependent (but temperature-independent) and the rate constants are in mol/cm2 s units. A small increase in activation
energies of higher hydrocarbons results in better prediction of conversion data (see Fig. 10 and text discussion). Note that in
CHEMKIN format, due to division of physical parameters with the reference temperature and density of sites related terms (see
equations following Eq. (15)), parameters may appear out of their usual physical range in the bottom part of the table. In addition,
one may have to suppress the option of CHEMKIN that does not allow for an effective sticking coefficient to be greater than 1.
= A T β e−Ea /RT and
kdes
s
RT β −E /RT
kads = n
T e a
.
Γ
2π M
(15)
Here Γ is the site density (2.49 × 10−9 mol/cm2
for Rh),
A =
A
β
Γ n−1 Tref
,
s =
s
β
,
Tref
and n is the reaction order. The reaction rate is now in
mol/cm2 /s. Using CHEMKIN units, X in Eqs. (12)
and (13) should be replaced with the concentration of
the corresponding species.
Table 3 summarizes the parameters needed to
compute the reaction rate. The values reported correspond to Ea , A, s in our formalism (top part of
the table) and to Ea , A , and s in the CHEMKIN
formalism (bottom part of the table). In computing
the rate, one needs to know the coverage of oxygen because the activation energy of desorption is
coverage-dependent. The oxygen coverage can be obtained from Eqs. (10) and (12), whose combination
leads to solving the following nonlinear equation:
ads X /k des
kO
O2 O 2
2
.
θO =
ads
des
1 + kO XO2 /kO
2
2
The rates of other species can be calculated from
the overall reaction stoichiometry and the methane
consumption rate in Eq. (13). Since many overall reaction stoichiometries, such as steam reforming, partial oxidation, and total oxidation, are plausible, it becomes important to deduce the overall reaction(s) for
any given system. This exercise was recently found
to be important in spatially resolving the combustion
and reforming zones within a partial oxidation reactor [27]. The rates of major stable species, such as O2 ,
H2 , CO2 , CO, CH4 , and H2 O, are evaluated using the
detailed microkinetic model and their ratios over that
of CH4 are shown in Fig. 5c. It is found that under
fuel-lean conditions, the reaction proceeds predominantly via the complete combustion chemistry, viz.,
CH4 + 2O2 → CO2 + 2H2 O.
(16)
Predictions of the developed one-step reduced rate
expression (Eq. (13)) are compared against the exper-
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
imental data of Burch et al. [43] and the microkinetic
model in Fig. 2a. The good agreement between them
highlights the success of our reduced model.
A last comment pertains to the thermodynamic
consistency of Eq. (13) (and similar reduced rate expressions). A systematic procedure for including the
backward rate is entirely possible and easy. This aspect is very important for equilibrium-limited reactions, such as steam reforming and the WGS. For fuellean catalytic combustion of CH4 , the equilibrium
constant (based on mole fractions) under a wide range
of conditions of equivalence ratio, temperatures, and
pressure is very high, so one can neglect the reverse
reaction, as done here.
The underlying assumptions about the MARI, PE,
and the RDS remain practically the same over a broad
range of conditions explored, i.e., for equivalence ratios as high as 0.99 and temperatures up to 1200 ◦ C.
Indeed, the predictions of the microkinetic and the
reduced models have been found to be in good agreement under this relatively broad window of conditions. Even at higher temperatures, the predictions
differ only by about 10–20% (data not shown) despite the reactions being driven toward equilibrium
and the lack of a well-defined RDS. At higher temperatures, gas-phase chemistry may become important, and incorporating gas-phase effects into surface
model reduction is necessary for accurate predictions
(this is beyond the scope of this work). The detailed
and the reduced models are in good agreement even
at higher pressures of 5–20 atm, which are of practical interest [48], as shown in Fig. 6a. The fact that
assumptions do not change under fuel-lean conditions
makes PCA unnecessary. Finally, this rate expression
holds also under ignition conditions, as discussed in
a later section.
4. Mechanism reduction for methane combustion
on Pt
The quest for better and more efficient catalysts
for commercial processes is an ongoing journey for
the chemical industry. Tools such as high-throughput
screening using microreactors are being developed
to this end, but analysis of data from such experiments is challenging. A simple theoretical model with
a few catalyst-based parameters, which can be estimated from first principles calculations or simple
experiments, can be valuable for catalyst screening.
Thus, if the assumptions made in the derivation of
Eq. (13) hold for other catalysts, data on just oxygen adsorption/desorption and methane adsorption on
various catalysts will be sufficient for predicting and
comparing relative catalytic activity. In order to assess the generality of Eq. (13) developed on Rh and its
375
Fig. 6. Comparison of reduced and full model predictions at
higher pressures for (a) Rh and (b) Pt. The parameters are
those of Fig. 3.
potential applicability toward rapid screening of catalysts, fuel-lean methane combustion chemistry is next
investigated on Pt.
Pt is a catalyst commonly used for many processes,
including combustion of fuels. The methodology used
above for methane combustion on Rh is employed
again, and hence, only the important findings are
reported next. A recently proposed detailed microkinetic model of 104 reaction steps (52 reversible reactions) for C1 chemistry on Pt [49] is used as the
starting point. RPA, PEA, and SA are performed on
this microkinetic model to identify the MARI, RDS,
and PE, as well as the oxidation pathways and the
overall reaction stoichiometry.
RPA indicates that the subset of important species
and reactions on Pt is the same as on Rh (see Fig. 3a).
The only mechanistic difference between the two catalysts is that all the H∗ abstractions on Pt (including
the one to form CH∗ from CH∗2 ) are O∗ assisted (as
against mediation via O∗ and OH∗ to form CH∗ in
the case of Rh). Examining the surface coverages over
a wide range of equivalence ratios, O∗ is again found
to be the MARI, as seen in Fig. 3c. The major gasphase species, O2 , CO2 , and H2 O, are again in PE
(see Fig. 4b). The dissociative adsorption of CH4 is
found to be the RDS (via the pairwise SA reported in
376
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
duced model to temperatures higher than 1300 K results in differences from the microkinetic model of
more than 10%. Again, gas-phase reactions may become important at high temperatures, but we have not
investigated whether the reduced rate expression is
adequate for such conditions. Under fuel-rich conditions, the MARI, the PE, and the RDS are different,
and thus, the rate expression will not hold for the entire length of the reactor (it may actually hold near the
entrance). In summary, this model could be applied to
fuel-lean conditions under which gas-phase chemistry
is unimportant.
Fig. 7. Comparison of reduced model prediction with experimental data for fuel-lean methane combustion on Pt [50–54].
The parameters are Φ = 0.35, τ = 50 ms, a is fitted to
220 cm−1 for [50]; Φ = 0.86, τ = 0.125 s at 300 K, a is
adjusted to 2.5 cm−1 for [54]; CH4 :O2 :N2 = 1:4:95, (aτ ) is
fitted to 4.55 s/cm for [51]; CH4 :O2 :He = 4:20:76, (aτ ) is
fitted to 2.64 s/cm for [52]; CH4 :O2 :N2 = 2:9.8:88.2,
(aτ ) is fitted to 14.88 s/cm for [53].
Fig. 5b) and the overall reaction is again consistent
with the total oxidation stoichiometry (see Fig. 5d).
These results are in line with those observed for Rh,
and hence, the reduced mechanism (see Table 2) and
the one-step rate expression Eq. (13), with catalystdependent parameters (see Table 3), have been found
to be adequate (data not shown) for methane-lean
combustion on Pt.
To validate the predictions of the reduced rate
expression for Pt, experimental data from fixed bed
reactors [50–53] as well as a stagnation point reactor [54] are modeled over a wide range of temperatures and equivalence ratios. Due to the lack of information of the catalyst loading, only the active catalyst
area per unit volume, a, is adjusted for one data point
and the rest are predicted using the model. Fig. 7
shows excellent agreement between the predictions
of this reduced model and the experimental data of
[50,51,54]. Fair agreement is found for the data sets
of [52,53] and possible sources of this discrepancy
are discussed in the following section. The reduced
rate expression predictions are also in good agreement
with the microkinetic model at higher pressures, as
shown in Fig. 6b.
To explore the regime of applicability of the rate
expression, the underlying assumptions were evaluated for various equivalence ratios and temperatures.
The MARI, PE and the RDS remain the same for
equivalence ratios as high as 0.99 and temperatures up
to 1000 ◦ C, with the predictions of the microkinetic
and the reduced models being within a few percent
of each other (data not shown). At higher temperatures, the reactions are driven toward equilibrium and
no clear RDS can be defined. Extrapolating the re-
5. Apparent activation energy and reaction
orders
In this section we attempt to rationalize the disparity in the experimentally estimated parameters observed in Table 1 for CH4 combustion on Pt. Equation (13) indicates that the reaction order in methane
is unity; however, an estimate of the reaction order in
oxygen is not
obvious, because calculations indicate
ads X /k des in the denominator of
that the term kO
O2 O 2
2
Eq. (13) is on the order of unity (at sufficiently low
temperatures (e.g., room temperature), this term exceeds 1 and negative-order O2 kinetics is expected).
A fit of the rate predicted by the detailed microkinetic
model to the power-law functional form of Eq. (3) under relevant conditions provides a reaction order in
oxygen that is close to zero. This result seems reasonable given that while O∗ is the MARI, the weaker
binding of O∗ on Pt gives rise to a high fraction of vacant sites (see Fig. 3c). An apparent activation energy
of ∼7 kcal/mol for Pt is determined from this fitting,
which is consistent with the RDS being the dissociative adsorption of methane.
An estimate of the activation energy can also be
obtained through analysis of the experimental data
shown in Fig. 7 using an integral fixed-bed reactor
model [55] (this is a crude approximation for some
of these experiments). The reaction is assumed to be
first-order in CH4 and zero-order in O2 , as obtained
from our analysis. The conversion (Ψ ) can, thus, be
expressed as
Ψ = 1 − e−kapp aτ ,
(17)
app
e−Ea /RT
and τ is the residence
where kapp = Aapp
time (s). From the conversion data, one can obtain
kapp aτ . An Arrhenius plot of ln(kapp aτ ) vs 1/T
app
(with a slope of −Ea /R) allows the determination
of the activation energy from the data despite the uncertainty in the active catalyst loading.
Using this approach, apparent activation energies with moderate values of ∼14 kcal/mol [50] and
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
377
and 8 − 42 × θO kcal/mol on Rh at 300 K. The effective Eades (O2 ) for Rh, assuming a monolayer of O∗ ,
is ∼20 kcal/mol larger than that for Pt, leading to
des that renders k ads X /k des 1, for the
a lower kO
O 2 O2 O 2
2
conditions analyzed. Equation (13) indicates a firstorder kinetics in CH4 and negative first-order kinetics
in O2 . Assuming the above reaction orders, analysis
of the microkinetic model predictions gives an apparent activation energy of ∼24 kcal/mol, whereas a fit
to the experimental data [43] gives an activation energy of ∼20 kcal/mol. These values correspond well
to the activation energy of 113 kJ/mol for CH4 on Rh
reported in Firth and Holland [60].
Fig. 8. Arrhenius plot for the determination of the activation
energy for fuel-lean methane combustion on Pt using integral fixed-bed reactor analysis and assuming first-order in
CH4 and zero-order in O2 kinetics.
∼18 kcal/mol [51,54] and large values of ∼37 kcal/
mol [52] and ∼51 kcal/mol [53] (see Fig. 8) for CH4
combustion on Pt have been extracted herein from the
respective experimental data. The activation energies
reported in Table 1 fall within the range computed
here (e.g., the activation energy reported by Aube
and Sapoundjiev [56] is 13 kcal/mol, while Yao [57],
Fullerton et al. [58], and Garetto and Apesteguia [53]
report an activation energy close to 20 kcal/mol). It
is difficult to unambiguously reconcile the differences
between these values, since details of the experimental setup are not available for all cases. For example,
temperature uniformity and control, residence time,
and feed dilution are very different. The so-called
compensation effect [59], viz., a tradeoff between
pre-exponential factor and activation energy, may also
occur, and this may explain the difference among the
moderate values and also between the moderate values and our rate expression model.
It is clear from Fig. 7 that the microkinetic model
can describe different data sets and rationalize moderate differences in activation energies between various
experiments. The difference between the activation
energies from the one-step rate expression Eq. (13)
and the direct fits to the experimental data can be rationalized by the coverage-dependent desorption activation energy of oxygen and the weak O2 partial
pressure dependence that Eq. (13) shows; these features are missing from a power law model. The fair
agreement with some of the data in Fig. 7 (higher apparent activation energies) indicates that the activity
of the catalyst is probably lower due, for example,
to different activation procedures and the presence of
metallic vs oxide forms of Pt.
A similar analysis has been performed on Rh.
Eaads (CH4 ) remains nearly the same on both the catalysts; 7.3 kcal/mol on Pt and 8.1 kcal/mol on Rh
at 300 K. Eades (O2 ) is 50 − 32 × θO kcal/mol on Pt
6. Combustion of higher alkanes
The rate expression developed above adequately
captures the physics of the fuel-lean combustion of
methane. The fact that the MARI and the RDS do
not change between Pt and Rh indicate that simple
model-based rapid evaluation and screening of catalysts may be possible. Furthermore, the applicability of this rate expression to higher alkanes, such as
ethane and propane, is also appealing given that no
reliable mechanisms exist for these fuels. In doing
that, one tacitly assumes that the RDS and the MARI
remain the same, but this appears reasonable based
on the homologous nature of small alkanes. Differences in adsorption of larger alkanes (e.g., multiple
sites for adsorption and higher sticking coefficients)
are of course expected and may break down the validity of this rate expression, so further work is needed
in this topic.
Six parameters are required to predict the catalytic combustion rate of small alkanes, as indicated
in Eq. (13), namely the activation energies and prefactors of oxygen adsorption and desorption and of fuel
adsorption. The oxygen parameters on Pt and Rh have
already been estimated (see Table 3). For other catalysts, these parameters may be unknown. As for the
fuel, adsorption rate constant parameters need to be
estimated. These parameters can be obtained via various means, namely from simulation, experiment, or
a combination of both.
In our work, we obtain activation energies theoretically for adsorption of fuel and adsorption–
desorption of oxygen using the simple bond-order
conservation (BOC) theory [61,62]. When experimental heats of chemisorption (input to BOC) are
lacking [63,64], we carry out DFT calculations to
obtain those and their coverage dependence. Sticking probabilities can be obtained for nonactivated
processes via molecular dynamics (MD) simulations
[65,66]. Experimentally, temperature-programmed
desorption (TPD) data, if available, are ideal to obtain
378
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
or validate desorption parameters. Previous work on
catalytic ignition shows that it is the sticking coefficients of the fuel and oxidant and the desorption of the
MARI that determine the ignition temperature [67,68]
(see also below). These are relatively straightforward
experiments to conduct. The same parameters appear
in the rate of combustion, Eq. (13). Therefore, ignition or rate data are redundant.
In this work, a hybrid route is employed where
BOC theory with (or without) input from DFT is
used to evaluate the activation energy barriers (e.g.,
Eaads (Fuel)) as well as the strength of the repulsive
adsorbate–adsorbate interactions on the required crystallographic plane, e.g., [69]. Pre-exponentials are
obtained from TST or temperature-programmed desorption (TPD) experimental data. Finally, the sticking
coefficients (e.g., sFuel ) are estimated from ignition
experiments, as discussed next, and the parameters
are validated against combustion rate (or conversion)
data.
Sticking coefficient estimation and model validation
for higher alkanes
The idea of estimating sticking coefficients from
ignition data has previously been used in stagnation
point reactors [67] and boundary layers [68]. Aghalayam and Vlachos [67] assumed a simple chemistry
model of breakdown of the fuel into a surface covered with adsorbates with no coverage dependence of
the activation energies. Adsorption of the fuel (activated) was assumed to be the RDS. No PE for oxygen
was reported (due to lack of interactions; the incorporation of interactions herein makes O∗ desorption
significantly faster) and the desorption activation energy of oxygen was fitted. The rest of the parameters
were calculated via the BOC theory. Trevino and coworkers [68] assumed a completely oxygen-covered
surface (θO = 1). Adsorption of the fuel (nonactivated) was assumed to be the RDS. The parameters
for the reaction steps (activation energies and sticking
coefficients for the fuel and oxygen) were fitted to the
experimental data. Both research groups attempted to
describe the same set of experimental ignition data of
Vesser and Schmidt [46]. A significant difference in
our approach is that (1) a detailed chemistry model is
used to arrive at the RDS (dissociative adsorption of
fuel) and (2) coverage-dependent reaction parameters
are employed. Using the parameters of the microkinetic model and a simple ignition criterion based on
a continuously stirred tank reactor (CSTR) model, the
experimental methane ignition data is first predicted
to demonstrate the validity of this approach (obviously a CSTR is a simple model that allows easy
derivation of an analytical criterion, but more complex models, such as a stagnation geometry, could
also be used; since ignition is mainly kinetically controlled, the transport model exerts a second-order effect).
In a typical ignition experiment, the catalytic surface is resistively heated and its temperature is monitored. The onset of the combustion chemistry takes
place at the ignition point marked by a discontinuous jump in the temperature, as indicated in Fig. 9a.
w
Therefore, at ignition dP
dT = 0, where Pw is the
power supplied by resistive heating. Appendix A details the derivation of an analytical ignition criterion
using the CSTR material and energy balances with the
reaction rate given by Eq. (13). Using algebraic manipulations, the ignition criterion is
ads
kCH
4
Eaads (CH4 )
RT 2
1+
ads
kCH
−
4
−
ads −0.5
βCH
4
T
ads 2
kO
2
des
kO
2
ads β ads −β des −0.5
kO
O2 O2
2
des
T
kO
2
1+
−
Eades (O2 ) RT 2
ads 3
kO
2
des
kO
2
ρCp
=
,
τ (−Hr )a
(18)
where Cp is the specific heat capacity at constant
pressure (erg/g/K) and Hr is the heat of reaction
(erg/mol).
Predictions of the ignition temperature for fuellean methane combustion using Eq. (18) are compared to the experimental data of Vesser and Schmidt
[46] in Fig. 9c. Good agreement is found. Since the
activation energy for oxygen desorption is coveragedependent and O∗ is the MARI, in solving Eq. (18)
the surface coverage of O* at the ignition point is an
unknown. This can be obtained in an iterative manner or using Newton’s method (see also the section
on Rh). To simplify matters, the use of an average
value is suggested. Fig. 9b shows the oxygen coverage at the ignition point using the microkinetic model
vs. equivalence ratio. An average value of 0.675 is obtained, which compares well to the value of 0.6 that
can be deduced from Trevino’s work [45] (θO = 1
was used to derive a criterion and then Eades (O2 ) was
tuned to fit the ignition data; this tuning corresponds
to an O∗ coverage of 0.6). The value of θO = 0.675
is used to estimate the coverage dependent activation
energy of oxygen desorption in the calculation of the
ignition criterion for higher alkanes on Pt without further adjustment. Also, since β was not available for
higher alkanes on Pt, it was assumed to be the same as
that of CH4 ; i.e., Eq. (18) was used. Simulations performed by setting all βs equal to zero (those explicit
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
379
Fig. 9. Panel (a) shows typical ignition curves, whereas panel (b) shows the oxygen coverage at ignition for various equivalence
ratios using the microkinetic model (CH4 on Pt). Panel (c) compares the ignition temperatures predicted using the simple algebraic criterion against the experimental data of Vesser and Schmidt (an inlet flow rate of 3 slpm corresponding to τ ∼ 2 s;
a is adjusted to 0.1 cm−1 ) [46]. The activation energy Ea values reported in panel (c) are the approximate values at the ignition temperature. Panel (d) compares the predicted ignition temperatures for methane ignition on Rh/Al2 O3 against catalytic
microreactor data (an inlet flow rate of 2 slpm corresponding to τ ∼ 15 ms; a is fitted to 8 × 104 cm−1 ).
in Eq. (18) and in the rate constants ks) underpredicted the ignition temperature by about 40 ◦ C (error
of ∼5%) for the case of CH4 ignition on Pt, implying
a slight effect of β on accuracy.
No ignition data for fuel-lean alkane combustion
on Rh were found in the literature. However, fuelrich ignition data could also be used because, prior to
ignition, the catalyst surface is covered with O∗ and
the dissociative adsorption of CH4 is still the RDS irrespective of the fuel–air equivalence ratio. One can
thus extend the derived ignition criterion to fuel-rich
operation. By evaluating coverages (data not shown),
in analogy to Fig. 9b, a value of θO = 0.9 is used to
calculate the activation energy of oxygen desorption
on Rh. Fig. 9d indicates good agreement between the
predictions of the ignition temperature for fuel-rich
ignition of methane on Rh and the experimental data
from a catalytic microreactor, described in Ref. [70].
Thus, having demonstrated the predictive capability
of the analytical ignition criterion, the ignition data
for higher alkanes is used to extract the sticking coefficients.
This methodology is preferred over regression of
all parameters using ignition data, e.g., [68], since
fewer parameters are fitted. Fig. 9c shows the fitting
of the ignition data on Pt of Vesser and Schmidt [71].
The extracted sticking coefficients are within a factor
of 2 of those reported in the literature. The calculated
sticking coefficient of methane is 0.116 (the value
reported by McMaster and Madix [72] on Pt(110)–
(1 × 2) is in the range of 0.02–0.1 and that by Aghalayam and Vlachos [67] is 0.1), that for ethane is 0.4
(the value reported by McMaster and Madix [72] on
Pt(110)–(1 × 2) is in the range of 0.1–0.3 and that
by Aghalayam and Vlachos [67] is 0.14), that for
propane is 0.06 (the value reported by Aghalayam
and Vlachos [67] is 0.03), and that for butane is 0.06
(data not shown) (the value reported by Aghalayam
and Vlachos [67] for butane is 0.01).
Based on the heats of chemisorption of ethane
(7.6 kcal/mol [73]), ethyl (51.4 kcal/mol, a value intermediate to the values of 53.6 [73] and 48.6 kcal/
mol [74]), propane (21.1 kcal/mol [75]), and propyl
(41.1 kcal/mol [74]) and the BOC framework, activation energies for ethane and propane dissociative adsorption are estimated (see Table 3). Using the above
sticking coefficients and BOC calculated activation
energies in Eq. (13), the fuel-lean combustion data for
380
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
details of which will be presented in a forthcoming
publication. The important parameters in Eq. (13),
viz., the activation energies and sticking coefficients,
are tabulated in Table 3.
7. Conclusions
Fig. 10. Prediction using the one-step rate expression vs
experimental data for fuel-lean higher alkane combustion.
Panel (a) is for the ethane combustion data of Gracia
et al. (C2 H6 :O2 :N2 = 0.3:16:83.7) [76]; panel (b) is for
the propane combustion data of Garetto and Apesteguia
(C3 H8 :O2 :N2 = 0.8:9.9:89.3) [75]. The solid lines indicate
predictions with the original activation energies ((aτ ) is fitted to 0.26 s/cm for ethane and to 0.03 s/cm for propane).
The dashed lines show predictions with higher activation energies ((aτ ) is fitted to 6.9 s/cm for ethane and to 1.0 s/cm
for propane).
ethane [76] and propane [75] on Pt have been modeled as shown in Figs. 10a and 10b. Good agreement
between the model predictions (solid lines) and the
experimental data indicates the usefulness of this simple approach. Better agreement with the experimental data can be obtained by increasing the activation
energies of ethane and propane dissociative adsorption by ∼4 kcal/mol (see dashed lines in Figs. 10a
and 10b). This increase in activation energy is within
the uncertainty limits of the BOC predictions. Such
uncertainty arises from uncertainty in thermophysical
data, heats of chemisorption, and the bond order of
the transition state used in the calculation of activation energies via the BOC theory, along with the variability of these quantities with crystallographic plane.
The propane combustion rate expression was also validated with catalytic microreactor experiments of [70]
using a pseudo-two-dimensional reactor model, the
A microkinetic model for fuel-lean combustion
of methane on Rh was analyzed using reaction path
analysis and sensitivity analysis. It was found that
the majority of the catalyst is covered by oxygen,
and the rate-determining step is the dissociative adsorption of the fuel. Total methane oxidation occurred
under these conditions via surface oxygen assisted hydrogen abstraction steps. Using inferences about partial equilibrium, most abundant reaction intermediate
(MARI), rate-determining step (RDS), and zero-order
asymptotics (simple algebra), a reduced microkinetic
model and a one-step rate expression describing the
methane combustion kinetics were developed and validated against experimental data as well as the predictions of the original 104-step detailed microkinetic reaction network. The MARI and the RDS were
found to be unchanged on Pt (albeit apparent reaction orders are quite different on the two catalysts),
leading possibly to a “universal” rate expression for
fuel-lean methane combustion on at least some noble
metals. Furthermore, the validity of this expression
for other higher alkanes (ethane and propane) was
demonstrated. This development relied on the use of
the BOC theory to estimate the activation energies of
key steps. Fuel sticking coefficients of higher alkanes
were fitted to experimental ignition data using a new
algebraic criterion.
The simplicity of the one-step rate expression for
fuel-lean combustion of simple alkanes lies in the fact
that the parameters involved can easily be estimated
using semiempirical techniques and/or simple experiments. This rate expression, thus, allows comparison
of combustion catalysts, without resorting to complex
experiments, and provides insights into reactor optimization and catalyst development. Finally, we have
shown that microkinetic models can be used to reconcile apparent differences between experimental data.
They are able to explain moderate variation in apparent kinetic parameters in some instances and indicate
data sets that are not in line with other measurements
due, possibly, to differences in catalyst preparation,
activation, and/or stability.
Acknowledgments
This work was supported by the Army Research
Office under Contract DAAD19-01-1-0582. Any
S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (2007) 366–383
opinions, findings, and conclusions or recommendations expressed are those of the authors and do not
necessarily reflect the views of the Army Research
Office. The catalytic microreactor ignition experiments (data reported in Fig. 9d) were performed by
Justin A. Federici.
Appendix A. Derivation of an analytical ignition
criterion
The time-dependent species mass balances in
a continuous stirred tank reactor (CSTR) are
(y in − yi ) σi Mi
dyi
= i
−
,
dt
τ
ρ
where t stands for time, y is the mass fraction, and
τ is the residence time. The superscript “in” denotes
inlet conditions. The time-dependent energy balance
in a CSTR can be written as
Pw
rHr
(T in − T )
dT
V −
=
−
,
dt
τ
ρCp
des = AΓ e
kO
2
Pw =
(T in − T )VρCp
τ
At ignition, one has
−V
which gives
ads Eaads (CH4 ) 0.346 − T
kCH
dr
RT 2
4
=a
ads
dT
kO 2
2
1+
des
ρC
ads
dkCH
4
dT
1+
ads 2
kO
2
des
kO
2
−
ads
kCH
4
des k des
kO
O
2
ads
kO
2
1+
2
ads
dkO
dk des
ads O2
2
dT −kO2 dT
des )2
(kO
2
ads 3
kO
,
2
des
kO
2
where
−E ads /RT sCH4 PCH4 e CH4
T 0.154
,
Tref
2π MCH4 RT
T 0.766
ads = sO2 PO2
kO
,
2
2π MO2 RT Tref
ads =
kCH
4
k55
−
kO
ads
kO
2 1.062
des
T
kO
2
1+
2
−
Eades (O2 ) RT 2
ads 3
kO
,
2
des
kO
ads − 0.5 and 1.062 = β ads −
where 0.346 = βCH
O2
4
des − 0.5. Therefore, the ignition criterion gives
βO
2
p
dr =
has dT
τ (−Hr ) . In the absence of gas-phase reactions, one can obtain
T −0.796
.
Tref
2
rHr .
dPw
= 0.
dT
Assuming a constant ρCp and a constant Hr , one
dr
=a
dT
2
The temperature exponents define the β terms (the
values shown above are for CH4 on Pt). In order to
simplify the algebra, temperature-independent activation energies (Eades (O2 ) and Eaads (CH4 )) are considered. The derivatives of the individual rate constants
are then
des
des
dkO
des Ea (O2 ) + −0.796 ,
2
= kO
2
dT
T
RT 2
ads
dkO
ads −1 + 0.766 ,
2
= kO
2
dT
2T
T
ads
ads
dkCH
ads −1 + Ea (CH4 ) + 0.154 ,
4
= kCH
4 2T
dT
T
RT 2
where V is the reactor volume (cm3 ), r is the reaction rate, and the summation is over all reactions and
reflects the total heat generated. Pw is the power exchanged. At steady state, the power exchanged is
des /RT
−EO
381
Eq. (18).
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