A reduced mechanism for methane and one-step rate expressions for fuel-lean catalytic combustion of small alkanes on noble metals

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1 Combustion and Flame 149 (007) 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 , USA Received 19 June 006; received in revised form 10 February 007; accepted February 007 Available online 7 April 007 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. 007 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- 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 []: * Corresponding author. Fax: +1 (30) address: vlachos@udel.edu (D.G. Vlachos). C n H n+ + nh O = nco + (n + 1)H, CO + H O = CO + H. (1) /$ see front matter 007 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi: /j.combustflame

2 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) 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 C n H n+ + O = nco + (n + 1)H O. () Currently, a lot of attention is devoted to hydrogen production in microdevices for portable applications [3 9]. Recent studies on gas-phase microburners [10 1] 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 NO x and CO formation, without coking of the catalyst. Even though catalytic combustion has been studied intensively, both experimentally and theoretically [15 1], 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 [] 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. [3]. 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.. Hierarchy of catalytic reaction models Surface reaction rates have traditionally been modeled with one-step rate expressions of a powerlaw form, r = k app C α Fuel Fuel Cα O O = A app e Eapp a /RT α C Fuel Fuel Cα O O. (3) Here r is the reaction rate (mol/cm /s), k app is the apparent (or effective) reaction rate constant, C is the concentration (mol/cm 3 ), α is the reaction order, A app is the apparent pre-exponential factor, Ea app 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, F + k F,1 F (PE), k F, O + k O,1 O (PE), k O, F + O k r 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 r = k r θ F θ O = k r K F P F KO P O (1 + K F P F + K O P O ), (5) where K F = k F,1 k and K F, O = k O,1 k are the equilibrium O, constants, k stands for the rate constant, θ is the surface coverage, and P i 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

3 368 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) Table 1 Literature power-law rate expressions (see Eq. (3)) for methane combustion on Pt Investigator E app a (kcal/mol) A α O α Fuel Molar ratio (O /CH 4 ) Yao [57] 1 (wire) (Al O 3 ) Trimm and Lam [5] (>813 K) (<813 K) Ma et al. [4] (mol/m h/kpa (α O +α Fuel ) ) Firth and Holland [77] Aube and Sapoundjiev [56] (1/s) Song et al. [78] (cm.5 /mol 0.5 s) Fullerton et al. [58] (ml/s/g) Aryafar and Zaera [79] (1/s/cm ) Kuper et al. [80] (mol/m /s) Otto [81] Niwa et al. [8] (1/min) Garetto and Apesteguia [53] Anderson et al. [83] (cm 3 /cm 3 s) Cullis and Willatt [84] Kolaczkowski and Serbetcioglu [85] (mol/m /s) and co-workers, who proposed Eq. (5) with parameters determined at different temperatures (e.g., k r = , K F = 0.011, K O = at 663 K; the units of pressure were not explicitly given but are presumably kpa) [4]. The form k r K F X CH4 X O r = (1 + K F X CH4 + K O X O ) has also been proposed, where X stands for mole fraction (e.g., k r K F = kmol/kg-cat s, K F = , K O = at 853 K for nonporous Pt/Al O 3 ) [5]. The last form assumes adsorption of molecular oxygen onto the catalyst surface (viz., O + k O O ), which further undergoes k O 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. [6] 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 [7]. Microkinetic models that describe all relevant elementary reaction pathways are needed to overcome the limitations discussed above and provide insights into the mechanistic pathways [8]. A number of microkinetic models are available for simple fuels, such as H and CH 4, on noble metal catalysts [8 35]. As

4 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) 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 [3,7,9,37,39], wehavebeen 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 [3]. 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 (5 reversible reactions) [7]. It contains elementary reaction steps for H oxidation, CO oxidation, CO H 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), CH 4 oxidation and reforming, and those associated with oxygenates (CH 3 OH and CH O) decomposition. This model was extensively validated for H oxidation, CO oxidation, WGS, PROX, catalytic partial oxidation, autothermal reforming, CO reforming, and oxygenate decomposition with relevant experimental data [7,9]. 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 on Rh [7] 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 CHEMKINframework [41,4] are compared with the experimental data of Burch et al. [43] for 1% CH 4 in air on a 1% Rh/Al O 3 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), dy i dz = aσ im i ρu, i = 1,...,N g, σ i = 0, i = 1,...,N s 1, N s θ i = 1, i=1 (6) (7) (8) where ρ is the density (g/cm 3 ), u is the velocity (cm/s), z is the reactor length (cm), N g and N s 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/cm /s), and a is the catalytic area per unit volume (cm /cm 3 ). 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, H OandCO 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.

5 370 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) Fig.. 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% CH 4 in air, a residence time of τ = 33.3 sat 300 K, assuming a = 300 cm 1 ) [43]. typically not observed). As a result, the overall stoichiometry of Eq. () holds and the methane conversion is the only independent variable that can be used to test catalyst activity and model predictive ability. Fig. a 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 [3]. 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 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 4-reaction (1 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 O and desorption of O ) 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 (O,CO,andH O) and the results are shown in Fig. 4a. PE is fulfilled for the adsorption/desorption of O,CO,andH O at various temperatures and equivalence ratios. Since

6 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) 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% CH 4 in air, τ = 33.3 s at 300 K, assuming a = 300 cm 1 ) [43] for Rh and to (Φ = 0.35 and a residence time of τ = 50 ms; a is fitted to 0 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, H O -mediated CH reduction, hydroxyl dissociation/formation, and water dissociation/formation. Clearly, methane adsorption and oxygen adsorption/desorption are the most important steps controlling CH 4 conversion. The 4-reaction subset deduced from RPA can be further trimmed by performing SA with respect to consumption of reactants, CH 4 and O, and formation of products, CO and H O. A reduced mechanism of 15 reactions, presented in Table, is deduced that faithfully captures the predictions of the original 104- reaction microkinetic model for methane-lean combustion, as shown in Fig. b. 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, r = k ads CH 4 X CH4 θ. The MARI implies that θ O + θ = 1. (9) (10) The partial equilibrium of the oxygen adsorption/desorption step (O + O ) implies k ads O X O θ = kdes O θ O. (11)

7 37 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) 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 5 reversible ones at different equivalence ratios. The reaction numbers correspond to Table 1 of Ref. [7]. The dissociative adsorption of CH 4 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.

8 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) Table Reduced microkinetic model for fuel-lean methane catalytic combustion on Pt and Rh No. Reaction s or A β E a 1(R) O + O (R) O O θ O θ O 3(R) O + H OH θ O θ O 4(R) OH + O + H θ O θ O 5 H + OH H O θ O θ O 6 OH H O + O θ O θ O 7(R) H O H O (R) H O + H O CH 4 + CH 3 + H CH 3 + O CH + OH θ O θ O 11 CH + O CH + OH θ O θ O 1 CH + O CO + H θ O θ O 13 CO + O CO θ O θ O 14(R) CO CO (R) CO + CO 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/cm 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 1 θ =, (1) ko ads X O 1 + which can be used in conjunction with Eq. (9) to obtain a simplified rate expression for the fuel-lean combustion of methane on Rh: kch ads X r CH4 = 4 CH4. (13) ( ko ads X O ) 1 + 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 ( ) k des = Ae Edes a /RT T β des and T ref k ads = sp tote Eads a /RT ( ) T β ads (14) Γ. πmrt T ref Here A is the pre-exponential (1/s), s is the sticking coefficient, P tot 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 T ref = 300 K. In order to facilitate use via CHEMKIN [4], we have refitted the parameters with temperatureindependent activation energies (E a ) and converted the units of parameters (into mol, cm, and s), namely

9 374 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) Table 3 Parameters for the one-step rate expression for fuel-lean catalytic combustion given via Eq. (13) Fuel/catalyst s Fuel A des O (1/s) s O Ea ads (Fuel) (kcal/mol) E des a (O ) (kcal/mol) CH 4 /Pt (T /300) (T /300) 3.0 θ O βfuel ads βdes O = βo ads = C H 6 /Pt (T /300) (T /300) 3.0 θ O βfuel ads βdes O = βo ads = C 3 H 8 /Pt (T /300) 3.0 θ O βfuel ads βdes O = βo ads = C 4 H 10 /Pt (T /300) 3.0 θ O βfuel ads βdes O = βo ads = CH 4 /Rh (T /300) (T /300) 4.0 θ O βfuel ads βdes O = βo ads = Fuel/catalyst s Fuel β ads Fuel A des O β des O s O β ads O E ads afuel E des ao CH 4 /Pt θ O C H 6 /Pt θ O C 3 H 8 /Pt θ O C 4 H 10 /Pt θ O CH 4 /Rh θ O Note. θ O 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/cm 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. k des = A T β e E a /RT and k ads = s RT Γ n πm T β e E a /RT. (15) Here Γ is the site density ( mol/cm for Rh), ( A = A Γ n 1 T β ref ) ( ), s s =, T β ref and n is the reaction order. The reaction rate is now in mol/cm /s. Using CHEMKIN units, X in Eqs. (1) 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 E a, A, s in our formalism (top part of the table) and to E a, A,ands 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 (1), whose combination leads to solving the following nonlinear equation: ko ads X O / θ O =. 1 + ko ads X O / 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 [7]. The rates of major stable species, such as O, H,CO,CO,CH 4,andH O, are evaluated using the detailed microkinetic model and their ratios over that of CH 4 are shown in Fig. 5c. It is found that under fuel-lean conditions, the reaction proceeds predominantly via the complete combustion chemistry, viz., CH 4 + O CO + H O. (16) Predictions of the developed one-step reduced rate expression (Eq. (13)) are compared against the exper-

10 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) imental data of Burch et al. [43] and the microkinetic model in Fig. a. 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 CH 4, 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 100 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 0% (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 0 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 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 (5 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 )areo 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, O,CO,andH O, are again in PE (see Fig. 4b). The dissociative adsorption of CH 4 is found to be the RDS (via the pairwise SA reported in

11 376 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) 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 ) 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 [5,53] 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 reduced 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 0 cm 1 for [50]; Φ = 0.86, τ = 0.15 s at 300 K, a is adjusted to.5 cm 1 for [54];CH 4 :O :N = 1:4:95, (aτ )is fittedto4.55s/cm for [51]; CH 4 :O :He = 4:0:76, (aτ )is fitted to.64 s/cm for [5]; CH 4 :O :N = :9.8:88., (aτ ) is fitted to s/cm for [53]. 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 CH 4 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 that the term ko ads X O / in the denominator of Eq. (13) is on the order of unity (at sufficiently low temperatures (e.g., room temperature), this term exceeds 1 and negative-order O 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 CH 4 and zero-order in O, as obtained from our analysis. The conversion (Ψ ) can, thus, be expressed as Ψ = 1 e k appaτ, (17) where k app = A app e Eapp a /RT and τ is the residence time (s). From the conversion data, one can obtain k app aτ. An Arrhenius plot of ln(k app aτ) vs 1/T (with a slope of Ea app /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

12 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) 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 CH 4 and zero-order in O kinetics. 18 kcal/mol [51,54] and large values of 37 kcal/ mol [5] and 51 kcal/mol [53] (see Fig. 8)forCH 4 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 0 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 O 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. Ea ads (CH 4 ) remains nearly the same on both the catalysts; 7.3 kcal/mol onptand8.1kcal/mol on Rh at 300 K. Ea des (O ) is 50 3 θ O kcal/mol on Pt and 8 4 θ O kcal/mol on Rh at 300 K. The effective Ea des (O ) for Rh, assuming a monolayer of O, is 0 kcal/mol larger than that for Pt, leading to alower that renders k ads O X O / 1, for the conditions analyzed. Equation (13) indicates a firstorder kinetics in CH 4 and negative first-order kinetics in O. Assuming the above reaction orders, analysis of the microkinetic model predictions gives an apparent activation energy of 4 kcal/mol, whereas a fit to the experimental data [43] gives an activation energy of 0 kcal/mol. These values correspond well to the activation energy of 113 kj/mol for CH 4 on Rh reported in Firth and Holland [60]. 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 [61,6]. 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

13 378 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) 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., Ea ads (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., s Fuel ) 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 (θ O = 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 () 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. Therefore, at ignition dp w dt = 0, where P w 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 kch ads ( E ads a (CH 4 ) 4 RT βads CH ) T ( 1 + k ads CH 4 = ko ads ) ko ads ( βads O β des O 0.5 T ( 1 + ρc p τ( H r )a, ko ads ) 3 Edes a (O ) ) RT (18) where C p is the specific heat capacity at constant pressure (erg/g/k) and H r 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 is obtained, which compares well to the value of 0.6 that can be deduced from Trevino s work [45] (θ O = 1 wasusedtoderiveacriterionandthenea des (O ) was tuned to fit the ignition data; this tuning corresponds to an O coverage of 0.6). The value of θ O = 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 CH 4 ; i.e., Eq. (18) was used. Simulations performed by setting all βs equal to zero (those explicit

14 S.R. Deshmukh, D.G. Vlachos / Combustion and Flame 149 (007) 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 (CH 4 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 s; a is adjusted to 0.1 cm 1 ) [46]. The activation energy E a 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/Al O 3 against catalytic microreactor data (an inlet flow rate of slpm corresponding to τ 15 ms; a is fitted to 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 CH 4 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 CH 4 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 θ O = 0.9 isusedto 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 of those reported in the literature. The calculated sticking coefficient of methane is (the value reported by McMaster and Madix [7] on Pt(110) (1 ) is in the range of and that by Aghalayam and Vlachos [67] is 0.1), that for ethane is 0.4 (the value reported by McMaster and Madix [7] on Pt(110) (1 ) is in the range of 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 (1.1 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