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Assessment of Overall Rate Expressions and Multiscale, Microkinetic Model Uniqueness via Experimental Data Injection: Ammonia Decomposition on Ru/γ-Al2O3 for Hydrogen Production V. Prasad, A. M. Karim, A. Arya, and D. G. Vlachos* Department of Chemical Engineering and Center for Catalytic Science and Technology UniVersity of Delaware, Newark, Delaware 19716-3110
We introduce a framework for parameter estimation of microkinetic models via injecting data, collected in optimal regions of the “entire” experimental operating space, in models. We demonstrate this framework by combining differential conversion experimental data without transport limitations and models for ammonia decomposition on Ru/γ-Al2O3 for hydrogen production. Experiments indicate that there is significant H2 inhibition even at low ammonia conversions. Statistical model discrimination techniques indicate that multiple microkinetic parameter sets are able to describe quantitatively the experimental data, and some global rate expressions are also adequate. Among the best microkinetic models, nitrogen adsorption/desorption and the NHx dehydrogenation reactions are the kinetically significant reactions. It is found that macroscopic data (conversion) are insufficient for complete model discrimination; microscopic scale data are proposed for further model discrimination. Introduction In the quest for clean and efficient energy production, fuel cells have been identified as a viable candidate. Hydrogen is the primary fuel for low temperature fuel cells, and ammonia is a prime candidate for the production of hydrogen.1 The major advantages of ammonia as a hydrogen storage material are its ability to be stored and transported as a liquid at relatively low pressures, its relatively high energy density and hydrogen storage capacity (∼17 wt %), and the widespread infrastructure for ammonia synthesis. In addition, the hydrogen produced is free of CO and CO2, and there is minimal downstream processing required to remove the unreacted ammonia from the hydrogen stream. In spite of these obvious advantages of hydrogen production by the catalytic decomposition of ammonia, the underlying chemistry is still not well understood. Most previous studies have focused on ammonia synthesis, due to its importance for fertilizers. While research on ammonia decomposition has recently increased,2-23 there is still no consensus on the ratedetermining step (RDS) and the most abundant reactive intermediate (MARI). The associative desorption of N2 and the N-H bond scission have both been proposed to be the RDS in NH3 decomposition.4,19 Also, either elemental hydrogen or nitrogen has been predicted to be the MARI.19,24,25 Different operating conditions and catalyst synthesis methods may explain these differences. In addition, some of the experimental studies have been performed at ultrahigh vacuum conditions on single crystals, and the pressure and material gaps make it difficult to extrapolate these results to conditions that are relevant to fuel cells (high ammonia concentration, atmospheric pressure). While first principle models are emerging as a powerful tool for predicting trends, their quantitative capabilities are still limited. The large number of parameters of complex reactions in conjunction with the inherent uncertainty of the catalyst structure and active site often require the use of free-energy type and semiempirical correlations that introduce further * To whom correspondence should be addressed. E-mail: vlachos@ udel.edu.
parameter uncertainty. Data injection into microkinetic models is therefore necessary to render these models predictive. Parameter refinement is typically cursed with nonuniqueness (multiplicity in values) due to the nonlinear nature of kinetic models. Given the physical and thermodynamic constraints imposed on the parameters of microkinetic models, a natural question raised though is whether such physically based (rather than purely fitting-based) models are unique. In this contribution, we first formulate a framework that combines experiments, high throughput microkinetic modeling, and an informatics-based design of experiments to render the model quantitative in a “global” sense. The approach is illustrated in the catalytic decomposition of ammonia on ruthenium (Ru) to study the chemistry at fuel cell relevant operating conditions and to clarify aspects of the chemistry, such as the RDS and the MARI. In the following sections, we describe the experimental procedure and results, the microkinetic and global rate expression models developed, and their comparison with experimental data and conclude on the uniqueness of microkinetic models and the predicted RDS and MARI. Experimental Details Ru/Al2O3 catalyst was prepared using an incipient wetness technique. γ-Al2O3 was dried at 120 °C overnight prior to impregnation. Ru nitrosyl nitrate solution (Aldrich) (1.7 wt % Ru by weight) was added dropwise to the Al2O3 powder and the powder was shaken to ensure even distribution; then, the powder was dried at 80 °C. The process was repeated until 4 wt % Ru loading was achieved. The catalyst was dried at 80 °C overnight and then calcined at 500 °C for 5 h. The kinetics of ammonia decomposition on Ru/Al2O3 catalyst was measured in a 1/4 in. stainless steel reactor. MKS calibrated digital mass flow controllers were used to control the flow rate of NH3, H2, N2, and He. Figure 1 shows a schematic of the reactor and the locations of the thermocouples. In order to eliminate temperature and concentration gradients, the catalyst powder was diluted with inert Al2O3 of the same particle size (50-75 µm) and, then, packed in the reactor. The catalyst bed
10.1021/ie900144x CCC: $40.75 2009 American Chemical Society Published on Web 04/27/2009
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Figure 1. Schematic of packed bed reactor used for kinetic studies.
length was 1.9-3.8 cm depending on the catalyst loading and dilution. Prior to reactivity measurements, the catalyst was reduced at 450 °C (10 °C/min ramp rate) for 2 h in 10% H2 (balance He) at 200 sccm total flow rate. The reactor products were sampled with a gas-sampling valve, and the composition was monitored with a HP Series 6890 gas chromatograph (GC), using a thermal conductivity detector. Ammonia conversion was calculated using the H2 and N2 mole fractions in the products. Attention has been paid to eliminate transport gradients (see the Appendix). Microkinetic Models Microkinetic models26 are generally superior to global rate expressions in terms of being able to describe data over wider ranges, having a physical basis for most parameters, and being able to identify changes in the important chemistry. Microkinetic models take into account all relevant elementary reactions and do not make a priori assumptions about an RDS, the MARI, or partial equilibrium (PE) conditions. The microkinetic model we use in this work builds on a model previously developed by our group to describe ammonia decomposition on Ru.7,14 The rate constant of elementary reactions is expressed in the modified Arrhenius form
( ) ( )
k i ) A0i
T Tref
βi
Ei , RT
exp -
i ) 1, ..., m
(1)
where m is the number of reactions. The elementary reactions considered for ammonia decomposition are (* denotes vacant sites, and H*, N*, etc. denote adsorbed species): k1f
H2 + 2* {\} 2H*
(R1)
k1b k2f
N2 + 2* {\} 2N*
(R2)
k2b
k3f
NH* + * {\} N* + H*
(R3)
k3b k4f
NH2*+* {\} NH* + H*
(R4)
k4b
k5f
NH3*+* {\} NH2*+H*
(R5)
k5b
k6f
NH3 + * {\} NH3*
(R6)
k6b
There are two major sets of parameters in microkinetic modelssthe activation energies and the pre-exponential factors. We employ the bond-order conservation (BOC) method27 to estimate activation energies, using heats of chemisorption as
inputs (obtained from density functional theory (DFT) calculations or experiments). Surface coverage effects determined from DFT simulations are included in the activation energies, as detailed in ref 28. The pre-exponential factors A0i are estimated (in an order of magnitude sense) using transition state theory (TST). To improve the quantitative capabilities of a mechanism, in previous work, some of the pre-exponential factors were estimated by fitting the model predictions to limited experimental data, subject to thermodynamic consistency.29 The parameters to be estimated were chosen based on sensitivity analysis. The microkinetic model is then combined with a reactor model, and is solved numerically to obtain predictions of gasphase mole fractions, fractional coverage of surface species, conversion, and other relevant quantities. In this work, we model our experimental reactor as a plug flow reactor (PFR) at steady state, given that the Peclet number is larger than 10 (convection significantly faster than diffusion). The method employed to generate relevant experimental data for estimation and refinement of the parameters in the model is described in the next section. Informatics-Based Design of Experiments The computational effort in calculating all parameters of large microkinetic models from first principles is prohibitive. In addition, there is an inherent uncertainty in parameter values associated with estimation methods, even with DFT. An additional uncertainty arises from the complex and often poorly known catalyst structure. In order to render microkinetic models quantitative and predictive, parameter refinement is therefore necessary. To achieve this, we use rational model-based techniques (described in detail in ref 30) in order to refine uncertain parameters and assess the global (i.e., in the entire experimental parameter space) model robustness. We employ informatics based approaches to identify optimal experimental regions; other applications of systems and informatics approaches in catalysis and kinetics may be found in refs 31-33. The overall methodology can be outlined as follows. An exhaustive search of the operating space is conducted to identify operating conditions that are optimal for estimation of parameters of a microkinetic model. This search is performed using a Monte Carlo method and results in a massive set of generated conditions (high throughput modeling). We have developed physics-aided methods (sensitivity, PE, and MARI) and statisticsbased methods (A, D, and E optimal designs) for the design of experiments. In the statistical approaches, we use suitable metrics, obtained from the Fisher information matrix (FIM), constructed from the normalized sensitivity coefficients (NSCs). The massive information is refined using informatics methods, such as principal component analysis (PCA) and clustering, to identify optimal regions for conducting experiments in the following manner: First, operating conditions are classified based on the optimality metric (D optimal metric or NSC value), and points with a high value of the metric are chosen. These points are then grouped into clusters using the partitioning among medoids (PAM) method.34 Conditions that belong to the same cluster share common underlying characteristics. Next, patterns indicating optimal regions are extracted from the clusters by identifying regions that have high probability of finding points that score high on the optimality metric. Performing a relatively small number of experiments in these optimal regions enables us to refine the parameters of a model, and improve its predictive capability and its reliability in identifying important chemistry. Our analysis illustrates that the D optimal and sensitivitybased designs are the most promising methods and generate
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conditions that delineate important chemistry. We have shown that a standard design around the D optimal point may not be useful for (highly nonlinear) kinetics problems, and informatics methods are proposed instead to identify optimal regions of the operating space. We have found that the experiments conducted within these regions have a high probability in providing useful kinetic information. The methodology described above is used to design optimal experiments for parameter estimation in our multiscale model for ammonia decomposition. In this work, we conduct a set of experiments in an optimal region identified using the methods described above. The main characteristics of this optimal region are that the inlet composition space is bounded by the limits (in terms of mole fractions) 0.6 and 0.75 for NH3 and 0 and 0.25 for H2. Reactor temperatures lie between 623 and 723 K, and total flow rates are either 100 or 200 sccm. This data is used to refine parameters of the microkinetic model (called “fitting set” in the sections below). A separate data set is used for model assessment. This data set uses a much broader sampling of the feasible experimental space. For example, the mole fractions of NH3 at the inlet range from 0.1 to 1.0, and the inlet mole fractions of H2 at the inlet range between 0 and 0.4. Total flow rates range between 100 and 400 sccm, and reactor temperatures are in the range 573-723 K. This set will be called the “assessment set” in the following sections. The informatics-based design of experiments provides conditions that are best to estimate parameters. Fuel cell conditions form part of the sampling space. However, the best conditions may not be those of fuel cell operation. The global nature of the models ensures that they apply at fuel cell conditions. Parameter Estimation and Model Discrimination We have employed methods based on numerical optimization for estimating the parameters of the microkinetic modelsthese include gradient-based methods (the “fmincon” function in MATLAB’s optimization toolbox) and simulated annealing.35 Initial guesses for these searches were supplied using models developed earlier in our group and in the literature.14,36,37 It should be noted that none of the literature models provide adequate fits over the entire experimental space studied prior to parameter estimation. An example is shown later, in Figure 9a. The objective function used in the parameter estimation is given below
( ( N
φ ) min θ
∑ i)1
Xmodel - Xexpt Xexpt
)) 2
(2)
where X denotes ammonia conversion, N is the total number of data points in the set used for estimation, and θ is the set of parameters to be estimated. Note that a quadratic objective function is being optimized with respect to the relative difference between model predictions and experimental data. Uncertainty in the experimental conversion scales as a percentage (approximately 10%) of the measured value and, thus, does not affect the objective function for the optimization. The set of parameters to be estimated includes the pre-exponential factors of elementary reactions, binding energies for nitrogen and hydrogen (these are inputs to the BOC calculation for computing activation energies) and their dependence on surface coverage (mainly of atomic nitrogen), and the BOC bond-index parameters for the two most important elementary reactions based on sensitivity analysis (R2: N2 + 2* ) 2N* and R4: NH2* + * ) NH* + H*). Entropic thermodynamic consistency of the overall
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ammonia decomposition mechanism is applied as a constraint in the estimation of pre-exponentials (using thermodynamic constraints and a reaction basis set, as described in ref 29), and enthalpic thermodynamic consistency is ensured by the use of BOC for calculating activation energies and heats of reaction. Since microkinetic models are highly nonlinear, many local minima are found in the optimization. Each of the parameter sets, corresponding to a local optimum, provides a reasonably good fit to the experimental data. To decide on the most appropriate model (parameter set), we use a model discrimination criterion developed in ref 38 on the best models based on goodness of fit. This is a modified Box-Henson criterion that is based on Bayesian statistics. Assuming normally distributed errors in the experiments, the posterior probability of each model being correct is calculated using
( )
p(Mj |Y, σ) ∝ p(Mj)2-pj/2 exp -
Sˆj
2σ2
(3)
Here, p(Mj|Y,σ) is the posterior probability of model j (given data Y with variance σ), p(Mj) is the prior probability, pj is the number of parameters estimated in model j, and Sj is the sum of squared errors for model j. The probability of model j being correct is calculated by normalizing the posterior probability over all models π(Mj |Y, σ) )
p(Mj |Y, σ)
∑ p(M |Y, σ)
(4)
k
k
The model with the highest probability is chosen as the (statistically) most likely model. Results In the following sections, we describe first experimental results. Catalyst characterization results are presented at the outset, and the catalyst surface area estimated experimentally is kept constant in all the models. Next, we describe the effect of temperature and feed composition changes, including the inlet ammonia mole fraction and inhibition by co-feeding the products (H2 and N2). The presentation in terms of one-parameter variation helps establishing major experimental trends that any model should be capable of predicting. For parameter refinement, experiments are conducted in the optimal regions identified using clustering. Details of optimal regions and normalized sensitivity coefficients for the nominal model may be found in ref 30. Model parameters are refined using this data (the fitting/ training set), and model predictions are then compared against experimental data in the entire operating range (training and assessment sets). We include a comparison of the performance of global rate expressions along with those of full microkinetic models against the experimental data. Finally, we discuss model discrimination. Catalyst Characterization. Scanning transmission electron microscopy (STEM) was performed on a JEOL 2010F FASTEM field emission gun scanning transmission electron microscope. Figure 2 shows a typical STEM image of the spent catalyst. Over several images, we measured the particle size of about 200 Ru particles. The corresponding particle size distribution is shown in Figure 3. The Ru number average particle diameter (Navg ) ∑i nidi/∑i ni, di being the particle diameter and ni the number of particles) was 7 nm. The mean surface diameter (Ns ) ∑i nidi3/∑i nidi2) was 8.6 nm, which corresponds to a Ru dispersion of 15%.39
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Figure 4. Effect of NH3 mole fraction in the inlet on the NH3 conversion. Total flow rate ) 200 sccm, balance He. A 96 mg portion of catalyst was diluted with 150 mg inert Al2O3. Figure 2. Typical STEM image of the spent Ru/Al2O3 catalyst.
Figure 3. Ru particle size distribution.
Figure 5. Effect of H2 on ammonia decomposition. Total flow rate ) 200 sccm. Inlet NH3 mole fraction ) 0.3, balance He. A 96 mg portion of catalyst was diluted with 150 mg inert Al2O3.
The total Ru surface area of the fresh catalyst was measured by CO pulse chemisorption using an automated chemisorption analysis instrument from Altamira (AMI-200). A 250 mg portion of the 4 wt % Ru/Al2O3 catalyst was loaded into a 1/4 in. quartz tube. Prior to the pulse CO chemisorption measurement, the catalyst was reduced at 450 °C for 2 h in 10% H2 (balance Ar) at 40 sccm total flow rate and then cooled down to room temperature in He at 30 sccm. The CO uptake was 22 µmol/g, and the corresponding calculated Ru dispersion was 10%, assuming a CO:Ru stoichiometry of 0.6:1.19,40 The Ru surface area for the spent catalyst was also measured, and the calculated Ru dispersion was 8.5%. Microscopy and chemisorption data are in reasonable agreement. The latter is used for the catalyst surface area in all our models. This ensures that the surface area (or the number of active sites) is not an adjustable parameter for the models, and avoids any compensation effects between the surface area and pre-exponential parameters in parameter estimation. Effect of Ammonia Mole Fraction. The points in Figure 4 show the experimentally estimated ammonia conversion as a function of temperature and ammonia mole fraction in the inlet. The lines are refined model predictions and will be discussed later. The total gas flow rate was kept constant at 200 sccm. Data (not shown) was also collected at 400 sccm total flow rate in order to get differential conversions (0-15%). The calculated NH3 reaction order, based on data of Figure 4, is between 0.25-0.45 (with the increasing ammonia mole fraction, the space velocity changes linearly, whereas conversion decreases sublinearly giving rise to a positive reaction order). A transition
from zero order at low temperatures to first order in ammonia at high temperature has been reported in the literature.8,19 The calculated apparent activation energy was 95-100 kJ/ mol (see Figure A2 in the Appendix). The activation energy reported in the literature for supported Ru catalysts ranges between 61 and 138 kJ/mol.4,6,22,23,41 It should be noted that in all the literature studies cited here, Ru chloride was used as the precursor for catalyst preparation, while we used Ru nitrosyl nitrate. Chlorine is known to be a poison for ammonia synthesis and decomposition catalysts.15,42-44 Transport limitations, especially heat transfer, were not checked for and eliminated in some literature studies. Chlorine poisoning and transport limitations could be responsible for the large spread in the apparent activation energy reported in the literature. H2 and N2 Inhibition Effect on Ammonia Decomposition. The effect of flowing H2 and N2 with NH3 in the inlet was studied. H2 was found to significantly inhibit the ammonia decomposition reaction. Figures 5-7 show the effect of H2 inhibition on ammonia decomposition at the same space velocity. It is evident that NH3 decomposition is strongly inhibited by H2 in the inlet, even at inlet mole fractions as low as 10%. This indicates that differential data analysis that leaves product inhibition out will lead to errors. Nitrogen, on the other hand, has no effect on NH3 conversion even with up to 70% N2 mole fraction in the inlet (data not shown). The inhibition by H2 and insensitivity to N2 partial pressure on ammonia decomposition has been reported previously in the literature.8,23 However, as shown in Figures 5-7, the inhibition by H2 persists at all temperatures. This is in contrast to the work by Egawa et
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as with the microkinetic models; also, the catalyst surface area has been fixed at the value obtained from the chemisorption results. The best-fit parameter values provided goodness of fit values that were clearly better (by approximately a factor of 2) than the other local minima. Model 145 is obtained by assuming reaction R2 (N2 adsorption/desorption) to be the RDS and adsorbed nitrogen (N*) to be the MARI. The parameters k2f and k2b represent the kinetic parameters for the forward and backward directions of R2, and K7 represents the equilibrium constant for the composite reaction K7 3 NH3 + * {\} N* + H2 2
Figure 6. Effect of H2 co-feed on ammonia decomposition. Total flow rate ) 200 sccm. Inlet NH3 mole fraction ) 0.3, balance He. A 96 mg portion of catalyst was diluted with 150 mg inert Al2O3.
Figure 7. Effect of H2 co-feed on ammonia decomposition. Total flow rate ) 200 sccm. Inlet NH3 mole fraction ) 0.6, balance He. A 96 mg portion of catalyst was diluted with 150 mg inert Al2O3.
al., where they reported no H2 inhibition at temperatures higher than 327 °C.8 It should be noted that our study was conducted under atmospheric pressure, while the work by Egawa et al. was done at ultrahigh vacuum. Also, Tsai et al. studied the effect of hydrogen inhibition on ammonia decomposition on platinum, and they reported a pressure dependence of the degree of hydrogen inhibition on ammonia decomposition;18 the higher the total pressure, the higher the degree of hydrogen inhibition, and therefore the higher the temperature at which the decomposition of ammonia becomes insensitive to H2 partial pressure. On the other hand, Chellappa et al. reported a zero order reaction rate in H2 on Ni-Pt/Al2O3 at temperatures 520-660 °C,5 which is higher than the temperature range explored in our study. Global Rate Expression Refinement and Assessment. In addition to microkinetic models, we have investigated the possibility of a global rate expression being able to describe the system in the entire experimental space. Most of these global expressions have been derived using an a priori assumption(s) of an elementary reaction being the RDS and a particular surface species being the MARI. Table 1 summarizes the global rate expressions investigated, the assumptions used in their derivation, and the goodness of fit (based on the objective function of eq 2 for each model over all experimental data, including fitting and assessment sets). Table 2 summarizes the parameter values obtained for the best fit with each global expression, and Figure 8 shows parity plots obtained for all the models. Note that the differential reactor assumption is not applied; the global rate expressions are used with a reactor model (PFR) that corresponds to experimental conditions. The parameter estimation has been carried out using the same optimization methods
(R7)
which is assumed to be in PE. The fitted values of activation energies for N2 adsorption and desorption are 40.8 and 20.1 kcal/mol, respectively. The value for desorption is in reasonable agreement with the experimental TPD value (32.7 kcal/mol37) and DFT calculations (27.7 kcal/mol for B5 step sites46), and that for adsorption is much larger than the literature values (7.937 and 9.2 kcal/mol46). The pre-exponential for k2f (assumed to be independent of temperature in the range studied) is estimated to be 2.3 × 104 1/s and that for k2b is 5.6 × 107 1/s. The value for k2f is larger than those reported in the literature (0.241 1/s using TST in ref 46; 56 1/(kPa s) from TPD experiments conducted at atmospheric pressure in ref 37, even if the N2 pressure is assumed to be 1 atm, the maximum possible value). The value of k2b is approximately 5 orders of magnitude lower than that predicted using crude TST and 2 orders of magnitude lower than that estimated by Hinrichsen et al. (2 × 1010 1/s). The equilibrium constant (actually a function of temperature) K7 is not an independent parameter and is herein calculated in a thermodynamically consistent manner (such that the total change in enthalpy and entropy of the system of reactions R2 and R7 matches that for the overall gas-phase ammonia decomposition reaction). As Figure 8 and the goodness of fit given in Table 1 show, model 1 provides good fits to the experimental data and is thermodynamically consistent. However, the estimated pre-exponential for N2 desorption appears physically unrealistic. Model 27 was derived from a full microkinetic model that was previously tested over a small range of operating conditions (no a priori assumption of a RDS was made). On the basis of the results, a posteriori assumptions were made about dominant surface species and reactions being in partial equilibrium (PE). Reactions R1, R5, and R6 were in PE, and H* and N* were the MARI. For parameter estimation, the activation energies for each elementary reaction were fixed at the values reported in ref 7, with N-N adsorbate interactions included. The important parameters in the fit using this model are k2f and k2b. The activation energies in the model are 14.1 and 37.2 kcal/ mol, respectively, which are slightly higher than the literature ones. The pre-exponential factor for k2f is estimated to be 1.4 1/s (in the range reported in refs 37 and 46) and that for k2b is 6.4 × 1013 1/s. Model 2 also provides very good fits to the experimental data. The goodness of fit is insensitive to k2f over a couple of orders of magnitude, while the fit for model 1 deteriorates quickly when k2f is varied. When N-N adsorbate interactions are ignored in computing the activation energies, the fits are not satisfactory, possibly indicating the importance of these interactions for ammonia decomposition as suggested before.14 The estimated parameters of model 3,4 which assumes reactions R2 and R5 to be the RDS and N* to be the MARI, are k5f, k2b, k6f and k6b. The model is not able to capture the
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Table 1. Global Rate Expressions Derived Using RDS and MARI Assumptions model index 1
model
assumptions
-k2bK72([NH3]2 /[H2]3) + k2f[N2] r) K7[NH3] 2 1+ [H2]3/2 -E2f -E2b k2f ) A2f exp , k2b ) A2b exp RT RT
(
)
( )
2
r)
(
ω)
k6f p + k6b NH3
k2f p + k2b N2
k1f p +ω k1b H2
k6f k5f pNH3 k6b
( 1+
4
k5fk6fpNH3 2k2bk6b
k2bK7pNH32
(
r)
pH23 1 + K7 5
)
2
pNH3 pH23/2
)
)
2
2
[( ) ( ) ]
r ) kapp
pNH32 pH23
N2 + 2* ) 2N* is the RDS. N* is the MARI.
45
2.73
NH3 and H2 adsorption/ desorption and NH3* + * ) NH2* + H* are in partial equilibrium. H* and N* are the MARI.
7
2.31
NH3* + * ) NH2* + H* and N2 adsorption/ desorption are both the RDS. N* is the MARI.
4
1.7 × 103
N2 adsorption/desorption is the RDS. N* is the MARI.
4
12.2
N2 adsorption/desorption is the RDS. N* is the MARI.
23, 47, 48
8.5
k1b k4fk5fk6f p p -0.5 k1f 2k2bk5bk6b NH3 H2
3
r)
goodness of fit
( )
-2(k2bω2 - k2fpN2)
1+
source
β
-
pN2
pH23
1-β
Keq2 pNH32
inhibition of NH3 conversion with H2 in the feed and does not provide good fits. Model 4,4 which is similar to model 1 but does not account for the N2 adsorption, provides almost identical fits as model 1. Finally, model 523,47,48 has also been derived using the same assumptions as models 1 and 4 but has been put in the Temkin-Pyzhev form. Keq is the equilibrium constant for the overall ammonia synthesis reaction. The parameters kapp and β are related to the apparent activation energy and reaction order. The apparent activation energy of 28.0 kcal/mol (approximately 117 kJ/mol) is within the range reported in the literature.4,6,22,23,41 The reaction order with respect to NH3 () 2β) is estimated to be 0.54. These values for the apparent activation energy and reaction order with respect to NH3 are in agreement with the experimentally estimated values that we obtained, 95-100 kJ/ mol and 0.25-0.45, respectively (see above and the Appendix). While multiple models may describe the data, it appears that a single overall rate expression (model 2) is physically sound. The goodness of fit of model 2 is relatively insensitive to the pre-exponential factor for N2 adsorption. This may indicate that model 2 has a more robust estimate of parameters than the other models. Microkinetic Model Refinement and Assessment. In this section, we describe the results obtained by refining the microkinetic model described above using the optimal design of experiments. The parameters to be estimated include the pre-
exponential factor of each elementary reaction, binding energies for nitrogen and hydrogen and their dependence on surface coverage, and the BOC bond-index parameters for the elementary reactions R2: N2 + 2* ) 2N* and R4: NH2* + * ) NH* + H* (these were found to be important in the initial model). The fitting and assessment data sets used for the global rate expressions and the microkinetic models are identical. Figure 9a shows the performance of the model with the nominal (literature) parameter set, prior to estimation. There are systematic, large errors in the model predictions, indicating the necessity of optimal experiment design and parameter estimation. Figure 9b shows the performance of the refined model on the fitting set of data, and Figure 9c shows the performance on the assessment set. Figure 9d combines the information from panels b and c. There are multiple sets of parameters that provide good fits to the experimental data, and these results represent the model (parameter set) with the best fits. The model predictions described in Figures 2-5 are also based on this parameter set that is listed in Table 3. Table 3 also lists the range of parameters over all local minima (over 100 in number). Approximate confidence intervals for the parameters of the optimal set were calculated from the FIM and the parameter covariance matrix using the method described in ref 49. A notable feature is that the confidence intervals for the pre-exponential parameters (excluding the sticking coefficients) are of the order of 104 (for
Ind. Eng. Chem. Res., Vol. 48, No. 11, 2009 Table 2. Parameter Values Obtained for the Best Fits with Models Described in Table 1 model 1
k2f ) 2.3 × 10 exp -40.8 RT 4
(
)
k2b ) 5.6 × 10 exp -21.1 RT 7
(
)
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a
model 2
model 3
model 4
model 5
k1f ) 1.1 × 1014 exp -1.9 RT
k5f) 2.6 × 107 exp -18.9 RT
k2b) 3.2 × 109 exp -29.4 RT
kapp ) 6.0 × 108 exp -28.0 RT
(
k1b ) 4.8 × 1011 exp -23.7 RT
(
) )
k2f ) 1.4 × 100 exp -14.1 RT
)
k2b ) 6.4 × 1013 exp -37.2 RT
)
k4f ) 3.6 × 1012 exp -19.1 RT
)
k5f ) 1.5 × 1013 exp -17.5 RT
)
k5b ) 2.9 × 1013 exp -13.2 RT
)
( ( ( ( (
(
)
K7 ) 8.3 × 10-2
k6f ) 5.8 × 10-1 k6b
k2b ) 6.0 × 108 exp -20.0 RT
(
(
)
(
)
β ) 0.27
)
k6f ) 4.4 × 106 (non-activated) k6b ) 1.9 × 1012 exp -18.2 RT
(
a
)
Units of activation energies are kilocalories per mole, and those of pre-exponential factors are inverse seconds.
example, the estimate for A3f is 1.4 × 1013 ( 8.9 × 103). Since the estimated values are typically in the range 1011-1013, with TST estimates placing bounds 2 orders of magnitude in each direction (i.e., between 1011 and 1015 for an estimate of 1013), this is a substantial improvement in the accuracy of the parameter estimates. Microkinetic Model Discrimination. Since many (potentially hundreds) of local minima have been located for the microkinetic models, a model discrimination exercise was conducted using the methodology described in an earlier section (eqs 3 and 4), with a uniform prior being used. In addition to the three best fit models (parameter sets), the global rate expressions described by models 1 and 2 were considered (since they provided the best fits among the global rate expressions).
The parameter sets for the microkinetic models were separated by orders of magnitude for the pre-exponentials/sticking coefficients related to adsorption of H2, N2, and NH3. The model discrimination procedure indicates that the model (parameter set) described in Table 3 is statistically the most probable model of the five considered, with a probability of 40%. However, other models also have significant probabilities in the range 15-25%, indicating that the discrimination measure is not very strong. The parameter sets corresponding to local minima fall roughly into eight families of solutions, of which five have physically reasonable parameters. The parameters of the statistically most probable microkinetic model are within a couple of orders of magnitude of the values of the global rate expression model 2. The main difference is
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Figure 8. Parity plots showing the best fits of each of the global rate expressions listed in Table 1 over the complete set of experimental data (fitting and assessment).
parameters of the microkinetic model are physically reasonable, and this may be considered to be the most plausible model. To exploit the most appropriate model from another perspective, we have calculated the RDS for each of the parameter sets that indicated a local minimum in the optimization (approximately 100 in all). In addition, we calculated which of the reactions were predicted to be in PE for each of these models and also determined the MARI predicted by each model for each operating condition in the fitting and assessment data sets. Some common features of the models (parameter sets) were that either H* or N* were found to be the MARI in every case, and their surface coverage was comparable in many cases. N2 adsorption/desorption and at least one of the NHx dehydrogenation reactions (R3-R5) were always predicted to be out of PE, indicating that they are kinetically significant. This is consistent with the assumptions made in deriving the global rate expression model 2. While different reactions are predicted to be the RDS for different parameter sets at different operating conditions, the most common reactions to be predicted as the RDS from all the possible solutions are reactions R2: N2 + 2* ) 2N*, R3: NH* + * ) N* + H*, and R4: NH2* + * ) NH* + H*. For the microkinetic models chosen from model discrimination, reaction R3 is predicted to be the RDS, and reaction R2 is the next most sensitive reaction. H* is predicted to be the MARI, and reactions R2-R4 (and occasionally R5) are out of PE. It is interesting to note that model refinement changed the order of important reactions; R3′s bond index was not included, and thus, it was modified. However, R3′s preexponential was fine-tuned and its activation energy depends on the elemental heats of adsorption that were estimated from part of the data. As a result, adjustment of the bond index of R3 is not necessary. The fairly large range of parameter variation shown in Table 3 indicates that despite the physical bounds and thermodynamic constraints, multiple minima (parameter sets) still exist. It is clear that macroscopic data injection into multiscale models is insufficient to provide complete model discrimination. Rather, some critical microscopic data is needed. For example, temperature programmed reaction is needed to resolve the MARI on the real catalyst and isotopic labeling experiments will be helpful to elucidate the RDS. Despite this multiplicity, the good news is that all these microkinetic models could be reliable predictors of macroscopic response, such as ammonia conversion, needed for reactor design. The global search and informatics tools ensure that these models are useful in the entire experimental parameter space (globally accurate). Conclusion
Figure 9. Parity plots showing (microkinetic) model predicted conversion against experimental conversion for (a) model before parameter estimation (assessment set), (b) model after estimation (fitting set), (c) model after estimation (assessment set), and (d) model after estimation (fitting and assessment sets). Error bars in a indicate the uncertainty in the experimentally reported conversion.
that the activation energies for nitrogen adsorption and desorption (23.3 and 44.5 kcal/mol, respectively) of the microkinetic model are slightly higher than those in model 2. However, the
First principle microkinetic models lack predictive capabilities. In order to render quantitative models, we introduced a framework for data injection into microkinetic models to refine the model parameters and assess model uniqueness. Our approach was demonstrated for the catalytic decomposition of ammonia on Ru/γ-Al2O3 to produce hydrogen. Experiments were designed to eliminate mass transfer limitations and minimize temperature gradients, while searching and organizing the “entire” experimental parameter space using informatics tools. Traditional data analysis indicates a fractional reaction order with respect to NH3, and strong H2 inhibition on the ammonia conversion, even at low fractions of ammonia. Multiple parameter sets were determined. Bayesian statistics was employed to discriminate the microkinetic models and several global rate expressions. A major conclusion is that some of the best models describing the decomposition of ammonia
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Table 3. Optimal Parameter Set for Microkinetic Model (Second Column) and Range of Parameter Values in Various Parameter Sets (Third Column)a
a
parameter
value
range
QH, binding energy for H (kcal/mol) QN, binding energy for N (kcal/mol) bond index, reaction R2 bond index, reaction R4 S1f (sticking coefficient for H2 adsorption) Ea1f (activation energy for H2 adsorption, kcal/mol) A1b (pre-exponential, H2 desorption), 1/s Ea1b (activation energy for H2 desorption, kcal/mol) S2f (sticking coefficient for N2 adsorption) Ea2f (activation energy for N2 adsorption, kcal/mol) A2b (pre-exponential, N2 desorption), 1/s Ea2b (activation energy for N2 desorption, kcal/mol) A3f (pre-exponential, forward reaction R3), 1/s Ea3f (activation energy, forward reaction R3, kcal/mol) A3b (pre-exponential, backward reaction R3), 1/s Ea3b (activation energy, backward reaction R3, kcal/mol) A4f (pre-exponential, forward reaction R4), 1/s Ea4f (activation energy, forward reaction R4, kcal/mol) A4b (pre-exponential, backward reaction R4), 1/s Ea4b (activation energy, backward reaction R4, kcal/mol) A5f (pre-exponential, forward reaction R5), 1/s Ea5f (activation energy, forward reaction R5, kcal/mol) A5b (pre-exponential, backward reaction R5), 1/s Ea5b (activation energy, backward reaction R5, kcal/mol) S6f (sticking coefficient for NH3 adsorption) Ea6f (activation energy for NH3 adsorption, kcal/mol) A6b (pre-exponential, NH3 desorption), 1/s Ea6b (activation energy for NH3 desorption, kcal/mol) coverage dependence of QH on H-H interactions, kcal/mol coverage dependence of QN on N-N interactions, kcal/mol
66.7 125.2 0.8 0.5 0.08 0.0 2.4 × 1013 25.9 2.2 × 10-7 23.3 1.6 × 1010 44.5 1.4 × 1013 4.3 8.8 × 1012 38.6 7.2 × 1012 19.3 4.6 × 1012 15.6 1.4 × 1013 18.2 8.8 × 1012 10.5 0.63 0.0 4.4 × 1013 14.0 -3.2 -40.0
47.4 - 77.4 105.3 - 153.2 0.19-0.80 0.11-0.60 0.05-1.0 0.0 (3.6-5.0) × 1013 0-47.2 1.0 × 10-7-0.43 0.0-39.6 7.4 × 109-3.6 × 1012 10.8-77.3 1.2 × 1011-1.5 × 1013 1.1-9.2 4.8 × 108-9.4 × 1012 24.9-46.8 1.3 × 1012-2.0 × 1013 3.5-28.2 9.1 × 109-1.3 × 1013 0.9-24.4 1.4 × 1012-1.4 × 1013 13.3-24.9 2.4 × 109-8.8 × 1012 0.1-21.0 6.1 × 10-5-1.0 0.0 3.3 × 1011-7.6 × 1013 9.8-20.9 0.0-(-6.5) (-22.3)-(-47.1)
Activation energies are specified in the zero coverage limit at 623 K.
on Ru/γ-Al2O3 have N2 adsorption/desorption and at least one of the NHx dehydrogenation reactions out of partial equilibrium, and one of these reactions is the rate-determining step (RDS). H* or N* are the most abundant relative species (MARI) in all these models. This finding indicates that the ambiguity in the literature about N2 adsorption/desorption or the reaction NH2* + * ) NH* + H* being the RDS may result from the fact that models based on either premise can adequately describe the system with different parameter sets. It is entirely possible that there is not a single RDS, but instead N2 adsorption/ desorption and some NHx dehydrogenation reactions are all kinetically significant reactions. Our analysis indicates that (macroscopic) kinetic data alone may be insufficient for microkinetic model discrimination; techniques such as temperature programmed reaction, isotopic labeling, and in situ spectroscopy are needed to complement macroscopic data and elucidate the key elementary reactions and the most abundant species and, thus, to enable further model discrimination. From an engineering viewpoint, the fact that multiple models (parameter sets) provide comparable accuracy in predicting experimental data implies that any of these models (with physically reasonable parameters) may be adequate for process design and systems engineering. Acknowledgment We acknowledge financial support from the US Department of Energy (DE-FG02-06ER15795). D.G.V. acknowledges Prof. V. Hatzimanikatis for useful discussions on sampling during D.G.V.’s sabbatical visit at EPFL. Appendix: Investigation of Transport Limitations and Effective Kinetic Parameters High ammonia mole fraction (60-100%) in the feed was chosen to test for transport limitations under conditions of high reaction
Figure A1. External mass transfer limitation test. Both catalyst beds (4 wt % Ru/Al2O3) were diluted with Al2O3 in 1:1.5 ratio by weight. The gas velocity for the 96 mg bed was double that of the 48 mg to fix the space velocity at the same value.
rates and low effective thermal conductivity (thermal conductivity of NH3 is much lower than that of He). In order to check for temperature and concentration gradients in the catalyst bed, several experimental tests were conducted. External mass transfer limitation was investigated by varying the gas velocity while keeping the space velocity constant.50 This was done by varying the catalyst loading and measuring the NH3 decomposition rate. Higher gas velocity leads to higher mass transfer rate between the gas in the bulk and the catalyst pellets. At the same space velocity, a higher mass transfer rate would lead to a higher NH3 conversion, if mass transfer limitations were present. Figure A1 shows that by doubling the gas velocity, the NH3 conversion is not affected. Therefore, we can conclude that the catalyst bed is not affected by external mass transfer limitations. The small catalyst pellets size (50-75 µm) was chosen in order to eliminate internal diffusion limitation. The absence of
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Figure A2. (a) NH3 reaction rate plotted against 1/RT (gmol/J) for 100% NH3 in the inlet. NH3 conversion ranged from 0.9% to 13.5%. (b) Eapp vs NH3 mole fraction in the inlet. A 96 mg portion of catalyst (4 wt % Ru/ Al2O3) was diluted with 150 mg inert Al2O3. The total flow rate ranged from 200 to 400 sccm.
reaction rate at 450 °C, 100% NH3 at the inlet, and 400 sccm flow rate was 0.65 gmol/(kg s). The left-hand side equals 1.2 × 10-3, which satisfies the criterion. The apparent activation energy was calculated at 30%, 60%, and 100% NH3 mole fraction in the feed, and similar values were obtained (95-100 kJ/mol) as shown in Figure A2. The apparent activation energy is expected to be lower under internal mass transfer limitations and increase if the system transitions (as the concentration increases) from diffusion- to reactionlimited conditions. These apparent activation energies are (within experimental error) independent of concentration and close to those reported in the literature for Ru/Al2O3 catalysts (65-140 kJ/mol).4,6,22,23,41 These results support the conclusion that the catalyst bed is free of internal mass transfer limitations. The typical experimental test to check for temperature gradients in the catalyst bed consists of comparing the reaction rate at different bed dilution ratios.50 Figure A3 shows that the catalyst bed is free of temperature gradients up to 400 °C. As shown in Figure A4, higher dilution resulted in about 4-6% increase in NH3 conversion at 450 °C, which is within experimental error. Therefore, it can be concluded that with a bed dilution ratio of 1:1.5, the catalyst bed is free of all transport limitations up to 400 °C. At 450 °C, small temperature gradients could exist in the catalyst bed. However, the effect of temperature gradients on the measured ammonia conversion at 450 °C was within experimental error. Literature Cited
Figure A3. Effect of bed dilution on the NH3 conversion: 48 mg catalyst (4 wt % Ru/Al2O3); flow rate of 100 sccm; 100% NH3 in the inlet. Dilution ratio ) catalyst; inert Al2O3 by weight.
Figure A4. Effect of bed dilution on the NH3 conversion at different NH3 mole fractions in the inlet: 48 mg catalyst; T ) 450 °C; total flow rate ) 100 sccm; balance He. Dilution ratio ) catalyst; inert Al2O3 by weight.
internal diffusion limitations was confirmed using the Weisz-Prater criterion, (rA′ FcRp2)/(DeCAs) , 1, where rA′ is the measured NH3 reaction rate (mol/(kg s)), Fc is the catalyst pellet density (1500 kg/m3), Rp is the pellet radius (30 × 10-6 m), De is the effective diffusivity of NH3 in the catalyst pellet (m2/s), and CAs is the NH3 concentration at the pellet surface (16.7 mol/m3 at 450 °C, assumed to be the same as in the gas-phase). The effective diffusivity is calculated from the equation De ) DNH3-H2φpσ/τ, where DNH3-H2 is the diffusion coefficient of NH3 in H2 at 450 °C (4 × 10-4 m2/s51), φp is the pellet porosity, σ is the constriction factor, and τ is the tortuosity. Typical values for a catalyst pellet are φp ) 0.4, σ ) 0.8, and τ ) 3. The measured
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ReceiVed for reView January 27, 2009 ReVised manuscript receiVed April 1, 2009 Accepted April 7, 2009 IE900144X
