Extracting Kinetic Information from Complex GasâSolid Reaction Data

Article pubs.acs.org/IECR

Extracting Kinetic Information from Complex Gas−Solid Reaction Data Christopher L. Muhich,† Kayla C. Weston,† Darwin Arifin,† Anthony H. McDaniel,‡ Charles B. Musgrave,†,§ and Alan W. Weimer*,† †

Department of Chemical and Biological Engineering, University of Colorado, Boulder, Colorado 80309-0596, United States Sandia National Laboratories, Livermore, California 94551, United States § Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309, United States ‡

S Supporting Information *

ABSTRACT: We develop an approach for extracting gas−solid kinetic information from convoluted experimental data and demonstrate it on isothermal carbon dioxide splitting at high-temperature using CoFe2O4/Al2O3 (i.e., a “hercynite” cycle based on Co-doped FeAl2O4) active material. The reaction kinetics equations we derive account for competing side reactions, namely catalytic CO2 splitting on and O2 oxidation of doped hercynite, in addition to CO2 splitting driven by the oxidation of oxygendeficient doped hercynite. The model also accounts for experimental effects, such as detector dead time and gas mixing downstream of the reaction chamber, which obscure the intrinsic chemical processes in the raw signal. A second-order surface reaction model in relation to the extent of unreacted material and a 2.4th-order model in relation to CO2 concentration were found to best describe the CO generation of the doped hercynite. Overall, the CO production capacity was found to increase with increasing reduction temperature and CO2 partial pressure, in accordance with previously predicted behavior. The method outlined in this paper is generally applicable to the analysis of other convoluted gas−solid kinetics experiments.

1. INTRODUCTION High-temperature reduction and/or oxidation reactions are present in many chemical processes such as combustion, carbothermal reduction of metals, NOx remediation, biomass gasification, and solar thermal gas-splitting (STGS).1−5 In the design of reactors where these processes take place and of the processes themselves, it is desirable to know the intrinsic kinetics of the individual reactions in order to confidently apply kinetic models derived from bench-scaled experiments to new reactor and process designs. However, extracting detailed kinetic information can be difficult, especially at the hightemperature conditions required for some of these reactions, and particularly when solids are reacting with gases.6 Many reported studies of high-temperature gas−solid reactions rely either on thermogravimetric analysis (TGA) or modeling of flow-reactor data to extract kinetic information. In TGA experiments, the material is contained in a crucible and heated by a furnace to the desired temperature under various reactive atmospheres. The crucible is connected to a quartz crystal microbalance that measures the differential mass change of the solid during reaction. Kinetic models and their associated parameters are then routinely derived from these data.7−11 In the flow-reactor method, either the solid is passed through the reactor or the gas is passed over the solid and the amount of product is measured. Separately, a computer simulation is developed to model the mass and energy transfer of the system inclusive of kinetic governance. Kinetic parameters are then determined from best fits to the experimental data.12−15 In the solarthermal water-splitting (STWS) research community, solid-state models that describe various rate-limiting phenomena16 are often employed. © XXXX American Chemical Society

The aforementioned approaches suffer from experimental and modeling shortcomings that often lead to erroneous results, such as misidentification of the true rate governing mechanism and its associated energetics. For example, diffusion of the reactant or product gases to and away from the solid in a TGA experiment is not instantaneous, and the rate or extent of mixing is rarely taken into account when conducting the kinetic analysis. This may mislead researchers into incorrectly ascribing the rate-limiting step in the solid-state reaction to a diffusionlike process.17,18 Similar problems plague flow-reactor modeling, even when heat and mass transfer are correctly included in the analysis. The dispersion of the gaseous products in the lines between the exit of the flow reactor and the gas analyzer is seldom accounted for and can result in the erroneous assignment of a diffusion-based kinetic model to the reaction. Even in cases where all gaseous dispersion effects are accurately taken into account,19,20 the individual solid-state reaction models reviewed by Khawam et al.16 are not always sufficient to describe all of the processes that occur when, for example, multiple gas−solid reactions occur simultaneously. These are widely recognized problems, especially with thermal analysis techniques,21 and have recently been addressed using modelfree approaches.22 With this in mind, we will demonstrate the extraction of kinetic data from a complex, high-temperature gassplitting process that occurs in the presence of multiple Special Issue: Scott Fogler Festschrift Received: October 1, 2014 Revised: November 26, 2014 Accepted: December 1, 2014

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dx.doi.org/10.1021/ie503894f | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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itself, as prescribed by Khawam et al.16 However, the equation that represents the oxidation reaction is comprised of a system of equations that represents a parallel reaction network. This approach provides a general roadmap for the extraction of relevant kinetic data from convoluted, high-temperature data sets where multiple gas−solid reactions occur simultaneously.

parasitic gas−solid reactions while preserving the model-based approach. It is well-known that sunlight incident on the earth’s surface has the capacity to supply all of humanity’s energy needs. Unfortunately, the capture and efficient storage of this dispersed resource presents technical and economic challenges that continue to inhibit the widespread implementation of solar-based technologies. High-temperature thermochemical gas-splitting cycles driven by concentrated solar energy, such as STGS, are of interest because these processes directly convert sunlight into CO and/or H2, which can then be stored, transported, used as secondary fuel, or serve as precursors to fungible liquid fuels.23,24 STGS has the potential to be more efficient than indirect methods of gas-splitting, such as photovoltaic-based electrolysis, because it makes use of the entire solar spectrum as process heat.5,25 In two-step solar thermal gas-splitting, a metal oxide is heated using concentrated sunlight to a temperature at which it spontaneously reduces to liberate O2, as shown in eq 1 for an O-vacancy mechanism metal oxide cycle. Subsequently, the reduced oxide MOx‑δ is exposed to CO2 or H2O to reoxidize it to MOx and form CO and/or H2 to complete the redox cycle, as shown in eqs 2a and 2b, respectively. The oxidation step is traditionally performed at a temperature ∼500 °C below the reduction step of the metal oxide.26 MOx → MOx ‐ δ + δ /2O2

2. METHODS 2.1. Kinetic Modeling Methods. An overall reactor model was developed to elucidate the kinetic behavior of a hightemperature, multiple oxidative species gas−solid reaction. Scheffe et al. showed that experimental artifacts such as the finite time required to introduce oxidizing gas into the reacting volume, detector time lag, and the dispersion and mixing of gases downstream of the reaction must be accounted for when developing a kinetic model for gas/solid reactions.20 If not removed, these effects cause the gas production curve, i.e. CO generation rate, to broaden in time, which depresses the peak rate while preserving the total area under the curve. Scheffe et al. developed a rigorous numerical approach that models the dispersive experimental effects in addition to the kinetic processes occurring in the solid, which enables the determination of the intrinsic solid-state kinetics.20 This approach forms the basis for the kinetics extraction method that we developed and describe herein. The numerical model consists of three parts: 1) gas injection dispersion, 2) solid-state reaction, and 3) mixing of the gases downstream from the reaction site but upstream from the detector. The behavior of CO2 injection is modeled by a sigmoidal error function that includes a detector time delay. The oxidation reaction is based on the solid-state models reviewed by Khawam et al. and given generally in eq 3;16 the experiments and modeling presented herein aim to determine the kinetic constants and nature and functional form of the kinetic model f(α)

(1)

MOx − δ + δCO2 → MOx + δCO

(2a)

MOx − δ + δ H 2O → MOx + δ H 2

(2b)

Several materials have been used for these cycles, such as Ceria (CeO2 → CeO2−δ + δ/2 O2),27 zinc oxide (ZnO → Zn(g) + 1/2 O2),28 perovskites (ABO3 → ABO3‑δ + δ/2 O2),19 ferrites (MFe2O4 → MO + 2FeO + 1/2 O2)29,30 and, of particular interest here, doped hercynite materials (CoFe2O4 + 3Al2O3 → 3Co1/3Fe2/3Al2O4 + 1/2 O2).31 The doped hercynite cycle operates by the oxidation and reduction of the iron cations (Fe2+/Fe3+) and is promising because of its relatively low reduction temperature and stability compared to standard ferrites.31−34 Additionally, this material has recently been demonstrated to split water isothermally, meaning that the gas-splitting step can occur at any temperature equal to or less than the thermal reduction temperature.33 While hercynite-like compounds appear to be promising STGS materials, to date no studies examining its oxidative kinetic behavior have been reported. Several steps occur during the oxidation of reduced STGS materials. CO2 or H2O gas adsorbs to the reduced metal oxide surface and dissociates into free O atoms and other surface-bound reactive intermediates that are precursors to the product gas, CO or H2, that desorbs from the surface. The surface O atom exchanges with an O vacancy and diffuses through the reduced metal oxide to oxidize an Fe2+ atom at the reaction front to Fe3+.20 Any of these steps, surface reaction and vacancy exchange, diffusion, or bulk oxidation at the reaction front can be ratelimiting. In this work, we develop a kinetic model of CO2 splitting on doped hercynite and incorporate it into the gas dispersion correction method described by Scheffe et al.20 Here, we develop a gas/solid reaction mechanism that accounts for multiple gas−solid reactions. Solid-state kinetic theory is used to describe the chemical transformation of the redox material

dα = ko[Yoxid(t )]γ f (α) dt

(3)

where ko is the kinetic rate coefficient, Yoxid is the concentration of oxidizing gas fed to the reactor (in this case CO2), γ is the reaction order of the oxidative species, f(α) is the kinetic model under consideration and may be any of those listed in Table 1, and α is the extent of oxidation of the active material. The reaction modes considered here are the shrinking area or volume family (R2 or R3), diffusion family (Dn), and reaction order family (Fn) where n is the order of the reaction. A full development of the new set of reaction modeling equations is given in Section 3.2. Table 1. Solid-State Reaction Models in Differential Form Used in This Work As Developed in Khawam et al.16a model name

description

equation [f(α)]

Fn D1 D2 D3 D4 R2 R3

reaction order of n 1-D diffusion 2-D diffusion 3-D diffusion 4-D diffusion contracting area contracting volume

(1 − α)n 1/(2α) − 1/(ln (1 − α)) (3*(1 − α)2/3)/(2*(1 − (1 − α)1/3)) 3/(2*((1 − α)− 1/3)) 2*(1 − α)1/2 3*(1 − α)1/3

a

Adapted with permission from Khawam et al.16 Copyright 2006, the American Chemical Society.

B



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in the material observed in initial cycles that would otherwise affect the kinetic analysis. Isothermal CO2 splitting experiments were carried out using an idealized stagnation flow reactor (SFR), which has been described elsewhere.20 In each experiment, approximately 150 mg of active material (loosely packed in a shallow bed) was placed in a zirconia boat at the stagnation point in the flow field. This ensured that all redox active surfaces were exposed to the oxidizing stream with minimal interparticle transport resistance. In a given CO2 splitting reaction, the STGS material was heated from room temperature to the desired reduction temperature and allowed to reduce for 30 min. Then, CO2 carried by He was injected into the system for 20 min with an overall flow rate of 500 sccm while maintaining a constant temperature; this causes the material to reoxidize and generates CO. The effluent stream was scrubbed of unreacted CO2 using a liquid nitrogen cooled cryogenic trap before the product gases from the reactor were analyzed by a differentially pumped, modulated effusive beam mass spectrometer (Extrell C50, 500 amu). CO2 splitting reactions were conducted over a 1280 to 1420 °C temperature range and a 310 to 576 Torr CO2 partial pressure range. The total overall pressure was 760 Torr. The specific operating conditions were chosen using a central composite experimental design with k = 2, as shown in Figure 1a.

The downstream dispersion is modeled using a variable number of consecutive continuously stirred tank reactors (CSTR), which are characterized using step changes of an argon tracer gas at each experimental condition. Of the dispersive effects which confound the reaction rate, downstream mixing constitutes the majority of the gas dispersion and is mainly attributable to turbulence in the large volume of the cryogenic CO2 trap. To determine the kinetic rate expression, the parameters of various solid-state reaction models are stochastically fit to the experimental CO generation rates via a minimization of the sum of squares error after the effects of diffusion have been accounted for by the diffusion models. We solved the following set of equations to determine the sum of squares error dαi = ki[YCO2,0,(t − ts)]γCO2,i f (αi) dt

(4)

dαtot = dt

(5)

dYCO, j dt

∑ i

dαi ; αtot ≤ 1 dt

= τj(YCO, j − 1 − YCO, j)

⎛ V ⎞⎛ d α ⎞ YCO,0 = ⎜ CO ⎟⎜ tot ⎟ ⎝ Ftot ⎠⎝ dt ⎠ RSSQ =

2 ̅ (t ) − YCO, j = last(t )] ∑ ωCO[YCO t

(6)

(7)

(8)

where αi is the extent of reaction of species i, Yi,j is the concentration of species i in the jth reactor, τj is the space velocity of the jth CSTR, V is the standard volume of the ith species, Ftot is the total flow rate, ω is the a weighting factor at each time point, and RSSQ is the residual sum of squares error. In this work, we develop a broader set of reaction models which replaces eq 4. This system of equations was implemented in the computational tool Mathematica,35 which numerically solves the system of governing equations (using NDSolve), produces a value for the RSSQ shown in eq 7, and then passes this information to a numerical minimization routine (NMinimize). The fitting routine uses a differential evolution algorithm native to Mathematica that stochastically solves a constrained, mixedinteger optimization problem. Differential weighting is applied to data points surrounding the peak in the H2 signal and, to a lesser extent, the observed tails. In this work, each individual reaction condition was fit independently, and activation energies and pre-exponential factors were determined from an Arrhenius plot; however, eq 4 could be altered to include these kinetic parameters, and eq 8 could be expanded to include the entire set of experimental runs such that the activation energy and pre-exponential factor could be fit by Mathematica35 as well. 2.2. Experimental Methods. Active doped hercynite materials were produced via atomic layer deposition (ALD) on a highly porous, high surface area, polystyrene-divinylbenzene support as described in detail elsewhere.32,36 Layers of Al2O3 were deposited on the polymer support which was subsequently removed leaving nanometer thick walls onto which CoFe2O4 was deposited via alternating CoO and Fe2O3 ALD processes. The material was cycled many times before used in the work described here; this avoids structural changes

Figure 1. a) Central composite experimental design with k = 2 used to determine the operating conditions for CO2 splitting. b) The uncorrected CO and O2 production rate curves observed during the oxidation reaction at 1400 °C and 350 Torr of CO2. c) CO production curve, blank and corrected curves for CO2 splitting at 1400 °C and 350 Torr of CO2. d) O2 production curve, blank and corrected curves for CO2 splitting at 1400 °C and 350 Torr of CO2.

Control runs without the active oxide were conducted at each reaction condition to determine the CO2 splitting attributable to the reactor itself. At each operating condition, the blank CO generation curve was subtracted from the CO generation curve obtained when doped hercynite was present resulting in a CO generation rate that is attributable solely to the active material. This will be referred to as the “corrected CO curve”. A similar process was attempted with the O2 generation curve but resulted in a temporal anomaly that will be described in further detail in the Results and Discussion section. The CO production curves presented here are analyzed with respect to the total doped hercynite mass. C



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3. RESULTS AND DISCUSSION The Results and Discussion section is organized as follows. We first discuss the general findings of isothermal CO2 splitting, which motivated the further development of gas−solid reaction models. We then discuss the expansion of the single solid-state kinetic model before finally discussing the result of applying the expanded solid-state kinetic model to our CO2 splitting data. 3.1. Solar Thermal CO2 Splitting Motivation To Expand the Solid-State Reaction Model. Isothermal based CO2 splitting using the stagnation flow reactor results in significant CO production, as shown in Figure 1b for a CO2 splitting process at 1400 °C and 350 Torr of CO2. In addition to the high peak CO production rate, after roughly 300 s of reaction the CO production continues in an apparently catalytic fashion along with concomitant production of O2. As shown in Figure 1c (blank run), a significant quantity of CO was produced from CO2 reduction on the walls of the SFR, which confounds interpretation of the CO2 splitting behavior of doped hercynite. The CO2 splitting in the absence of reactive material is attributed to catalytic activity on the reactor walls and zirconia boat. At the high experimental temperatures of 1280 to 1420 °C, the alumina walls become catalytic and split the CO2 into CO and O2. The small volume in the SFR and relatively high flow rate make it possible to rapidly quench the gas-phase recombination of CO and O2, otherwise this chemistry would go unnoticed. In order to differentiate between the CO produced from CO2 splitting by the reactor walls and the CO produced from the oxidation of doped hercynite, the respective signals from blank runs were subtracted from the overall CO and O2 signals, as demonstrated in Figure 1c and d. Even after correcting for CO2 splitting caused by the reactor itself, CO production occurs for the duration of the oxidation step, long after the initial CO production peak has subsided. We attribute this to the catalytic splitting of CO2 on the surface of the active material. At the reaction temperatures, the highly redox active surface provided by the material becomes catalytic. The assertion that the continued CO production is due to catalytic behavior is substantiated by the fact that after the oxidation reaction is complete, CO and O2 are generated in a 2:1 ratio, which is expected for CO2 decomposition. The sources and sinks of CO2, CO, and O2 in the SFR are shown schematically in Figure 2a. As mentioned previously, correction of the oxygen signal by subtracting the blank-run data created a temporal anomaly in the product evolution curve. We observed a time delay in the O2 production rate as compared to the blank (Δt in Figure 1d) when hercynite was present in the SFR. Subtracting the blankrun O2 data gives a negative production rate at early times. The delay likely stems from the oxidation of the active material by the O2 that was formed from catalytic CO2 splitting on the reactor walls. The O2 oxidation of the reactive material consumes a portion of the reduced material, thereby decreasing the overall CO production capacity and the rate of CO production. As suggested above, the continued O2 production at a rate higher than that found for the blank is due to catalytic behavior where the CO and O2 are generated in a 2:1 ratio. Because of these complications, new reaction equations that take into account the oxidation of the hercynite material by CO2 and O2, and which include catalytic splitting on the surface of the hercynite material, need to be incorporated into the analysis.

Figure 2. a) Schematic of the sources and sinks of CO2, CO, and O2 in the stagnation flow reactor. b) Schematic of the surface and bulk oxidation reactions occurring during material reoxidation. The Ffamily of models consists of the various reactions on the surface including CO2 and O2 adsorption and dissociation, as well as CO desorption. Ion diffusion, as modeled by the D-family of equations, can be oxygen ion diffusion into the bulk, as shown here, or cation diffusion. The R-family models describe interfacial reactions such as movement of ions from the reoxidized material to the reduced material and phase change across the interface.

3.2. Expanded Solid-State Reaction Model Development. The complicated reaction behavior described above, namely catalytic CO2 splitting and co-oxidation of the active material by O2 and CO2, makes direct use of a single solid-state reaction model inadequate. This is exemplified in Figure 3 where we attempted to fit a single traditional solid-state reaction model (F1, F2, D1, D2, etc.) to the CO production data using the numerical method described above. Here, the predicted CO generation is unable to match either the initial CO generation peak or the catalytic CO production. These shortcomings arise because neither O2 oxidation of the material nor catalytic CO2 splitting on the doped hercynite is included in the reaction model. To account for catalytic CO2 splitting, and O2 uptake by the reduced oxide, additional terms were included in the equations modeling CO generation and the overall reaction progress (α). The new equations for modeling the CO2 splitting behavior that replace eq 4 are developed as follows. The oxidation of hercynite by CO2 and the resulting CO generation is described by eq 9 dα1 d[CO]1 γ1 f (α ) = = k1Y CO 21 dt dt D

(9)



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examined as potential models for f(α 3 ). Because the thermodynamics and kinetics of CO2 interacting with the reactor wall and splitting in the reactor volume are time independent and are related to the CO2 concentration in the reactor, the O2 concentration was related to CO2 by eq 12

YO2 = βYCO2

(12)

where β is a constant which takes into account the thermodynamics, kinetics, surface area of the reactor wall, and reactor volume upstream of the reactive material. Equations 9−12 can be combined into eqs 13 and 14 to model the rate of CO generation and hercynite reoxidation. Equation 13 shows the overall rate of CO production. d[CO]1 d[CO]2 d[CO] γ1 γ2 f (α) + k 2Y CO = + = k1Y CO 21 2 dt dt dt (13)

Combining eqs 9, 11, and 12 results in an overall rate expression for material reoxidation, as shown in eq 14: dα3 dα dα γ3 γ = 1 + = k1Y CO f (α) + k 3β γ3Y CO f (α ) 2 3 21 dt dt dt γ

γ

3 1 = k1Y CO f (α) + k 3′Y CO f (α ) 21 2 3

As described above, the individual reaction steps that occur during doped hercynite reoxidation can be described by the Ffamily of models (surface reactions), the D-family of models (O diffusion), or the R-family of models (reaction at the solid-state oxidation reaction front), as shown schematically in Figure 2b. This does not mean that the simultaneous CO2 and O2 oxidation behaviors are independent of one another and can or should be modeled by different solid-state reactions. Of the three solid-state model behaviors, the D- and Rfamilies have identical reaction rates for both CO2 and O2 oxidation with the caveat that CO2 and O2 oxidations produce different surface O concentrations. The degenerate activation energies and pre-exponential factors arise because the source of O atoms only affects the concentration of surface O atoms, and not the barrier for an O atom in the bulk to hop between vacancies (diffusion limited), or across a reaction front (reaction front limited), nor the frequency with which that O atom vibrates within the lattice, which would affect the preexponential factor for either diffusion or solid-state oxidation. Therefore, diffusion cannot be the rate-limiting step for oxidation by O2 while interfacial reaction is the rate-limiting step for oxidation by CO2 or vice versa. This eliminates the need to consider both a diffusion and interfacial reaction model to describe the behavior of the active materials under simultaneous oxidation by CO2 and O2. If diffusion or the interfacial reaction is the slow step for both O2 and CO2 oxidation, then that reaction step is logically the rate-limiting step for simultaneous oxidation, and eq 14 collapses into eq 15, which involves a single f(α) term to model the reaction:

Figure 3. Stochastically fit CO generation to a single Fn model a) without the inclusion of catalytic CO2 splitting and b) with catalytic CO2 splitting.

where α is the total extent of hercynite reoxidation, α1 is the extent of reoxidation due to oxidative CO2 splitting, (d[CO]1)/ (dt) is the rate of CO generation attributable to oxidative CO2 splitting, t is time, k1 is the kinetic rate coefficient for oxidative CO2 splitting, Y is the CO2 concentration, γ1 is the reaction order with respect to the CO2 concentration, and f1(α) is the gas/solid reaction model. The catalytic portion of the CO2 splitting is modeled by a zeroth-order reaction (with respect to α) as shown in eq 10 d[CO]2 γ2 = k 2Y CO 2 dt

(10)

where (d[CO]2)/(dt) is the rate of CO generation attributable to catalytic CO2 splitting, and k2 is the kinetic rate coefficient of catalytic CO2 splitting. The reoxidation of hercynite by O2 released from the walls of the reactor is accounted for by eq 11 dα3 γ = k 3Y O32f3 (α) dt

(14)

(11)

where α3 is the rate of the doped hercynite oxidation by O2, k3 is the kinetic rate coefficient of O2 oxidation, YO2 is the concentration of O2 in the system, γ3 is the reaction order with respect to the O2 concentration, and f 3(α) is the reaction order model for O2 oxidation. The two unknowns in eq 11 are the rate-limiting step behavior, i.e. f 3(α), and the concentration of O2 at the surface of the active material, i.e. YO2. In choosing an equation to model O2 uptake by the active material, it is important to remember that both O2 and CO2 are oxidizing gases and that the set of potential equations that model CO2 oxidation also necessarily model O2 oxidation. Therefore, the same reaction models described by Khawam et al.16 were

dα 0 γ3 γ )f (α0) = k4(k1Y CO + k 3′Y CO 2 2 dt

(15)

In eq 15, the dependence of the reaction rate on the CO2 and O2 concentrations stems from thermodynamic adsorption considerations, described by k1 and k3′, and the diffusion or interfacial reaction kinetics, such as hopping frequencies or interfacial reaction kinetics, the combined behavior of which is contained in k4. The k1 containing terms can then be used to determine the CO generated from material oxidation. E



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Figure 4. Stochastically fit CO generation curves at 1350 °C and 450 Torr PCO2 including both CO2 and O2 oxidation using an a) Fn model; b) R3 model; c) D1 model; d) D2 model; e) D3 model; and f) D4 model. The dots show experimental data, and the solid lines show the fitted CO generation curves.

If, however, the reaction is limited by surface reactions (Ffamily) for both CO2 and O2 oxidation, the modeling equation does not necessarily collapse into a single f(α) model as occurs for the D- and R-family models. This is because the equations used to model the rate-limiting steps of the CO2 and O2 oxidation reactions could be different [i.e., f1(α) = (1−α)n1 and f 3(α) =(1−α)n2, where n1 ≠ n2]. If the rate-limiting surface process does not distinguish between CO2 and O2 oxidation, for example when the distribution in the initial particle sizes results in particles with different shrinking surface areas available for reaction, and thus different numbers of reaction sites, n1 and n2, are equal. In contrast, if it results from surface processes such as desorption, promotion of electrons from a shrinking supply of, for example, bulk Fe2+ ions, to adsorbing CO2 or O2, or more complex surface pathways where the number of surface sites or electrons required for adsorption and decomposition of O2 and CO2 are different, n1 and n2 will be unequal. Therefore, the f(α) term cannot be factored out of the concentration and rate constant terms as was done to derive eq 15. Modeling surface-limited reactions, then, requires two Ffamily models to characterize simultaneous CO2 and O2 oxidation of the active material, as shown in eq 16:

If the O2 and CO2 oxidation reactions are governed by a combination of surface and bulk based reactions, the overall rate model will still collapse into eq 15 due to unequal surface reaction competition. For example, if the rate-limiting step for CO2 oxidation is diffusion, but the rate-limiting step for O2 oxidation is a surface reaction, then the CO2 surface decomposition reaction must be faster than O2 oxidation because they have the same diffusion rate. Therefore, the CO2 splitting reaction will be the dominant surface reaction and will cover the surface in O atoms and block the O2 surface oxidation reaction, but the CO2 reaction will still be limited by diffusion. This will result in large quantities of CO production, which are modeled by eq 15 where the CO production is diffusion-limited with negligible O2 oxidation (k3′ ≈ 0). A similar, but converse argument can be made with O2 as the main species oxidizing the surface resulting in little or no CO generation, which also leads to the conclusion that eq 15 holds. The same analysis can be performed with an interfacial reaction rather than a diffusion based model. Therefore, we only attempted to fit the doped hercynite material oxidation to a single f(α) model, as shown in eq 15, for the cases of diffusion- and interfacial reaction-limited processes. F



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Figure 5. Stochastically fit CO generation at the material surface using an Fn model including both CO2 and O2 oxidation reactions at 450 Torr PCO2 and a) 1420 °C, b) 1350 °C, and c) 1280 °C and at 1350 °C and d) 576 Torr PCO2, e) 450 Torr PCO2, and f) 310 Torr PCO2. The inserts show the measured (dots) and the model (solid lines) CO production rates that account for reaction and dispersion.

dα 0 γ3 γ1 (1 − α0)n1 + k 3′Y CO (1 − α0)n3 = k1Y CO 2 2 dt

stochastically fit the CO production profile for each of the CO2 splitting conditions examined in this work. As mentioned above, each reaction condition was fit individually, but in future work this method could be expanded to fit all of the conditions simultaneously. The peak CO production rate increases with both increased reaction temperature and increased partial pressure of CO2, as shown in Figure 5. All fitted CO production curves can be found in Supporting Information Figures S.1−S.9. These figures also show that it is necessary to account for downstream diffusion when conducting kinetic analysis; without including dispersion, it would appear that CO2 splitting at 1350 °C and 576 Torr PCO2 is slower than at 1350 °C and 450 Torr PCO2 or even 310 Torr PCO2 due to the increased dispersion when the CO2 partial pressure increases. The increased CO production with increasing temperature is expected because the higher reaction temperatures increase the extent of reduction in the oxide and thereby increases the overall CO production capacity of the material while simultaneously increasing the oxidation driving force and rate of oxidation. Similarly, the increase in CO

(16)

From eq 16, the CO2 and O2 contributions to oxidation of the doped hercynite materials are easily discernible where the terms containing subscripts 1 and 3 represent CO2 and O2 based oxidation, respectively. 3.3. Application of the Expanded Kinetic Model to CO2 Splitting Using Doped Hercynite. The collapsed diffusion and interfacial reaction models, and surface reaction models, as shown in eqs 15 and 16, respectively, were used to fit the CO generation rates found in the SFR experiments. Examples of the resulting CO production described by these models are shown in Figure 4 for the experiment conducted at 1350 °C and 450 Torr CO2. The F-family based model outperforms the D- and R-family of models qualitatively via inspection and quantitatively by the sum of squares error. This result agrees with previous findings for STGS materials, which have determined that the F-family or mixed F- and D-family of models best describes CO2 oxidation behavior.18,19,37−39 Based on these results, the F-family of models was used to G



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Table 2. Stochastically Fit Parameters for CO2 Splitting Modeled Using Eq 16 T

PCO2

Log k1

γ1

n1

Log k2

Log k3

γ2

n2

1420 1400 1400 1350 1350 1350 1300 1300 1280 mean SD

450 550 350 576 450 310 550 350 450

−1.07 −1.39 −1.26 −1.50 −1.50 −1.48 −1.55 −1.60 −1.75

2.4 3.0 2.0 3.0 2.5 2.0 2.0 1.8 3.0 2.4 0.5

1.5 1.5 2.3 2.5 1.7 2.1 1.2 1.7 2.5 1.9 0.5

−3.33 −3.29 −3.21 −3.15 −3.30 −3.22 −3.15 −3.25 −3.34

−1.55 −1.43 −1.33 −1.57 −2.00 −2.00 −1.00 −1.70 −2.71

1.4 2.5 1.5 1.4 1.2 1.2 1.0 1.3 0.9 1.4 0.5

0.8 1.4 2.0 1.5 1.0 1.0 2.0 2.0 1.3 1.5 0.5

production with increased CO2 partial pressure stems from the increased driving force for oxidation, which has been described before,33 as well as increasing the rate of the reaction. The overall kinetic parameters determined in this work are listed in Table 2. It is important to note that while these models offer a reasonably good fit to the experimental data, they are not necessarily unique solutions and therefore interpreting the implicit kinetics of oxidation of the material derived from these fits must be approached with caution. However, the broad findings are useful in illustrating the new gas−solid reaction models, developing a general understanding of the active material behavior, and potentially for use in basic reactor and system modeling. The apparent reaction order with respect to the extent of reduced doped hercynite oxidation (α) was found to be n ∼ 2 and n ∼ 1.5 for CO2 and O2 oxidation, respectively, suggesting that oxidation requires two surface sites for adsorption and reaction. This is in agreement with previous investigations of CO2 splitting on ceria40 and oxidation reactions on ferrite materials.20 The reaction order with respect to the CO2 concentration was found to be ∼2.5. The distribution in the predicted reaction order is likely attributable to the differing initial extents of reduction of the materials, for example materials reduced at 1420 °C are more reduced than materials reduced at 1280 °C and, therefore, have a higher CO production capacity and differing initial reaction driving forces. The reaction rate constants, on the other hand, span a larger range of values. This stems from the Arrhenius-like behavior of reaction rate constants, which vary from one oxidation temperature to the next. The determination of a preexponential factor and activation energy based on the reaction rate constants is complicated by the different reduction extents mentioned above and the simultaneous oxidation by O2 and catalytic CO production. However, as a rough estimate, our model results in a pre-exponential factor and activation energy for the CO2 splitting reaction of 88.2[CO][CO2]−2[O2]−1.5 s−1 and 80.0 kJ/mol, respectively, based on apparent reaction rate constants and an Arrhenius plot, as shown in Figure 6. The activation energy is reasonably close to the activation energy (Ea = 53.9 kJ/mol) of the F2 reaction path for water-splitting on CoFe3O4 on YSZ.20 The extraction of the activation energy from the O2 oxidation reaction is complicated by the fact that we do not know the O2 concentration at the reaction surface because we related the O2 concentration to the CO2 concentration via β, which was then combined with the reaction rate constant to give an effective rate constant. Similarly, it is challenging to determine the reaction order with respect to O2 concentration. Therefore, we leave the analysis of the O2 oxidation for doped hercynite in

Figure 6. Arrhenius plot of the reaction rate constants found in this work.

which O2 is used as the sole oxidizing gas to future work. With the estimates of reaction orders and activation barriers for the CO2 splitting reaction developed in this work, doped hercynite materials can now be simulated in reactor design models.

4. CONCLUSION In this work, we developed a set of gas−solid reaction equations and employed them to extract kinetic behavior from convoluted reaction data sets. We demonstrated the use of these equations by determining the kinetic parameters of isothermal CO2 splitting using doped hercynite materials over the temperature range of 1280−1420 °C and CO2 partial pressure range of 310−576 Torr. The expanded set of gas− solid reaction equations was required to account for CO production from both active material oxidation and catalytic CO2 splitting on the active material’s surface and simultaneous O2 oxidation of the hercynite material. We fit the experimental CO production using the updated models and found that a second-order surface reaction (F-family) model best fit the reaction with respect to the extent of reoxidation, with an apparent activation energy and pre-exponential factor of 80.0 kJ/mol and 88.2[CO][CO2]−2[O2]−1.5 s−1. While these models offer a reasonably good fit, they are not necessarily unique solutions and, therefore, cannot be used to infer implicit kinetics of the material and process but are suitable for reactor design and modeling using doped hercynite. The models and method for developing the models described here are applicable not only to STGS but also to gas−solid reactions in general, particularly to complicated mechanisms involving multiple gas species reacting along various reaction paths. H



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ASSOCIATED CONTENT

S Supporting Information *

Figures of the developed F-based models for all of the experimental conditions explored in this work. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Al Weimer would like to thank Scott Fogler for many inspirational kinetics and reactor engineering discussions during Scott’s consulting visits to Dow Chemical and Adjunct Professor visits to the University of Colorado. Chris Muhich would like the thank Scott Fogler for getting him interested in kinetics and reactor design during his Reaction Engineering Class at the University of Michigan. The authors gratefully acknowledge the National Science Foundation and the U.S. Department of Energy for supporting this research. The work was completed through the National Science Foundation via Grants CBET-0966201 and CBET-1433521 and by the U.S. Department of Energy Fuel Cell Technologies Office. This work was a collaboration between the University of Colorado Boulder and Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.

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