Catalysts are important in every industry: chemical, petrochemical, oil-refining, pharmaceutical, organic synthesis, fuel-energetic industries, environment protection, and biocatalysis to name just a few. The chemical reactions on the surface of a catalyst are a function of the inlet chemical species, temperature conditions, metal formulation, washcoat material dispersion/size, and its age causing an immeasurable number of kinetic possibilities. In order to obtain the chemical species conversion rates, researchers spend a significant amount of experimental time and expense. Often, the use of models helps reduce the amount of experiments needed. In specific, models can help predict costs, material placement, effectiveness, and ageing among other items. Accurate and predictive models are needed as catalysts are being asked to perform longer at higher levels of effectiveness. With respect to modeling the reactions occurring on the surface, there are two pathways a researcher can take: detailed or global kinetics.

Detailed kinetic mechanisms allow the researcher to write the reactions as a function of the catalyst metal on the surface and permit different catalyst formulations. However, they are limited from a feasibility standpoint. One concern is that they still require calibration to experimental data because the measurement of all intermediate species on the surface is not feasible without significant experimental capabilities. Even if this happens, industrial revision of the surface metals and the ageing of the device may alter the kinetic constants. Moreover, in the literature there may not be a consensus as to all of the detailed steps needed for the model. For example, researchers have postulated around twenty different mechanistic steps for the relatively simple reaction of carbon monoxide (CO) with oxygen (O₂) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. In comparison, global kinetics provides a macroscopic viewpoint of the reaction; CO and O₂ enter the catalyst and carbon dioxide (CO₂) leaves. This allows use of a simpler experimental setup when generating data for calibration of reaction rate expressions.

Furthermore, calibration of a detailed mechanism can take longer than a global mechanism because of the increased number of parameters and experimental tests necessary in order to ensure physically based values. Another issue is that catalysts typically consist of a number of different metals on the surface that may require the detailed mechanism to change and additional reactions to account for subsequent interactions (e.g., enhancement of CO oxidation with hydrogen present [7,19,26]). This will necessitate further experimental tests while also increasing the numerical stiffness of the resultant reaction mechanism. Moreover, the influence of catalyst size, shape, and amount make it nearly impossible to determine one set of kinetic constants for one reaction. As a result, creating adaptive global kinetics, that include facets of the detailed mechanism while factoring in metal properties (e.g., dispersion), have the potential to be more predictive without significantly increasing the computational burden.

With respect to the literature, there are relatively few global mechanism efforts that attempt to develop a more analytical set of kinetics that can be applied to different catalyst formulations [27]. Therefore, the goal of this effort is to create a global kinetic mechanism for CO oxidation that predicts the light-off and light-down temperatures as a function of platinum metal dispersion on the surface. This is done in the absence of other species, like H₂, in order to simplify the effort and focus on the fundamentals instead of creating a simulation that acts as a multiple parameter curve-fit. Of importance, CO oxidation is the first step in exhaust aftertreatment devices for internal combustion engines [4], and it is a critical reaction in the production of hydrogen for fuel cell applications [28,29]. Hence, the results generated will be of interest to researchers in both these fields. Moreover, since sintering increases particle size (dispersion decreases) reducing catalyst effectiveness as it ages [30], this model provides a starting point for a more predictive simulation tool suitable for automotive engineers who prefer the global kinetics route over the detailed approach [31]. Furthermore, this effort endeavors to bridge the gap that exists between Mechanical and Chemical Engineers by demonstrating how surface interactions can be incorporated into a global mechanism. Finally, since the model variables are re-calculated before the simulation begins and the global reaction rate expression is the same, no significant increase in computational load occurs. The automotive industry standard one-dimensional monolithic catalyst model utilized by a wide number of researchers provides the basis of the modeling efforts.

A previous publication by the first author describes the history of the monolithic one-dimensional (1-D) catalyst model employed in this paper [31]. For this model, two phases capture the bulk and surface (washcoat/substrate) components in the channel. The bulk gas species equation is written as: (1)

The left hand side of this equation accounts for the advection of species through the channel and the right hand side describes the mass transfer of species from the bulk gas to the surface under the laminar conditions in the channel.

The bulk gas energy equation is written as a function of temperature: (2)

The left hand side of this equation accounts for the propagation of energy through the channel and the right hand side describes the heat transfer between the gas and the surface.

In these equations, calculation of the diffusion and heat transfer coefficients include entrance effects [32,33]: (3)

For these expressions, Heck came to the conclusion that Nusselt and Sherwood numbers should correspond to constant heat flux conditions up until light-off and then correspond to the constant wall temperature condition afterwards [34,35]. This is because up until light-off, the region is kinetically controlled; hence, the heat flux is relatively low and approximately constant [32]. Since the experiments modeled later in this paper consist of both light-off and extinction data in order to explore the hysteresis effects of CO oxidation, the constant heat flux conditions will be employed during light-off as developed by Groppi et al. [32]: (4) with the constant wall temperature conditions employed during the extinction experiments (5) where Gz in the above equation represents the respective mass transfer (Sh→Gz_m) and thermal Graetz (Nu→Gz_t) numbers that are written as a function of distance from the inlet: (6) with (7) As commonly accepted, the Chilton-Colburn analogy equates heat and mass transfer for the laminar flow conditions [36,37].

Simulation of chemistry on the surface of the catalyst happens via a combined treatment of gas concentrations within the washcoat: (8)

This equation is written as an Ordinary Differential Equation and solved until steady state so the user does not have to incorporate an algebraic-differential solver for the non-linear equations. The first term on the right hand side describes species mass transfer between the bulk gas and surface and the second term indicates the reaction rates of the gases on the surface. The difference in the mass transfer source term between this equation and the bulk gas equation relates to the relative volume difference between the bulk and surface as indicated by the void fraction (ε).

Completing the 1-D model requires the inclusion of the monolith energy equation for the combined washcoat and substrate (e.g., one numerical node) in the direction normal to the flow: (9)

The left hand side of this equation represents the storage of energy in this combined structure; whereas, the right hand side terms describe conduction, heat transfer between the gas and the surface, energy generated through catalytic reactions, and heat transfer to the ambient, respectively [38]. Since the experiments simulated from the literature in this paper do not include ambient information, the model assumes adiabaticity as commonly accepted and this term is equal to zero.

The authors chose to reduce the complexity of this initial effort by simulating only the carbon monoxide (CO) oxidation reaction over platinum/alumina: (10)

Choice of this reaction allows the authors to calibrate the global reaction expression to available literature experimental data of varying dispersion, particle diameter, and metal weight with similar BET surface area values. The authors have investigated the history of CO oxidation and determined that most researchers believe that the reaction occurs via the following detailed kinetic steps: (11a) (11b) (11c)

CO adsorption is the first step of the mechanism because of the low initial sticking coefficient of oxygen on platinum as compared to CO [39]. The second step in this mechanism is the dissociative adsorption of oxygen [18,40]. Most researchers believe that the final step is that of adsorbed CO interacting with adsorbed atomic oxygen via a Langmuir-Hinshelwood (L-H) mechanism. This appears to be the most widely assumed prospect, but mechanisms developed using only these conversion reactions do not necessarily predict the most accurate results [8,9,13,14]. The final step would be desorption of CO₂ from platinum that most assume happens instantaneously; hence, Equation (11c) lumps this step into the final detailed reaction. There is still no true consensus as to the Rate-Determining Step (RDS). A number of researchers indicate that both Equation (11b) and Equation (11c) can be the RDS [17,19,41]. It is quite possible that at lower CO concentrations, Equation (11c) is the RDS and at higher CO concentrations, Equation (11b) becomes the RDS. This paper will utilize the RDS equal to Equation (11c) because it has been shown by Wojciechowski and Asprey to accurately fit experimental data [20]. This results in the following global reaction equation: (12) where the adsorption equilibrium constants are equal to: (13) (14)

This is the same global mechanism calculated by Hori and Schmidt in 1975 using molecular oxygen adsorption [5] and Goodman et al. in 1982 for atomic oxygen [12].

At this stage, it is possible to calibrate the pre-exponentials, activation energies, and adsorption heats as is often done by researchers. However, as Wojciechowski and Asprey illustrate [20], there can be two different sets of parameters that match the data captured as indicated in Table 1. One issue with kinetics is that it requires collection of a large amount of experimental data in order to eliminate all potential local minimums during parameter optimization. Moreover, since chemical kinetics are non-linear, it is possible to find a set of parameters that matches the data well but may be non-physically based. Hence, the approach the authors take is to explore from first principles the components of the global rate expression in order to minimize the number of variables to calibrate.

Calibration of the parameters utilized CO oxidation literature data from a 2005 paper by Arnby et al. [23] where the amount of platinum, dispersion, and particle diameter changed between different catalysts as indicated in Table 8. The platinum dispersion and surface area of Arnby et al. were calculated from the amount of adsorbed CO, determined by CO chemisorption measurements. The nomenclature relating to the percentage Al₂O₃ (1, 10, and 100%) indicates their preparations methods where they deposit platinum only on part of the available support material. The choice of this paper over their 2004 paper [22] and others for calibration is because both the platinum to washcoat (ω) and BET surface area remains relatively constant between the different catalysts. Therefore, the difference in the data should ideally relate directly to the impact of dispersion and correspondingly the platinum particle size.

Matlab Simulink provided the platform for model calibration by first finding singular values of a, d, and χ that minimized the least-squares curve-fit (LSQ) value between the experimental and simulated temperatures at the 50% CO conversion points over all light-off and extinction tests. The bounds on these parameters came from the respective literature in this area; specifically, Table 6 for d and Table 2 for χ with E_−b set equal to 200 kJ/mol based on Table 3. This initial calibration to their hysteresis experiments found that the model was unable to match the thermal response from their data. Since they did not include a temperature ramp experiment without chemical reactions, measurement of the amount of heat transfer to the ambient and validation of use of monolith properties from the literature was not possible (e.g., density, thermal conductivity, and specific heat as 1800 kg m⁻³, 1.5 W m⁻¹ K⁻¹, and 1020 J kg⁻¹ K⁻¹ respectively [92]). As a result, the authors included monolith density (ρ_m) as another constant for calibration. Using the initial three calibrated components and a literature value of monolith density, this provided the set of parameters for a final calibration involving all eight parameters (a, b, c, d, χ, ξ, ρ_m, and E_−b) that minimized the LSQ value between temperatures at 50% CO conversion for both light-off and extinction with the final results given in Table 9.

In Figure 2, the model results are presented in comparison to the experimental data. Overall, the model deviates by an average of 5.3K between the 50% conversion points for the light-off and extinction temperatures. In Table 10, a comparison of the Root Mean Squared Error (RMSE) between the simulation and experimental percentages is given. On average, the RMSE indicates a 20% difference in conversion. In order to explore the choice of constants for the last four parameters in Table 5, a sensitivity analysis similar to Ó Conaire [93] was performed by doubling (or halving) the respective value and finding the difference in the 50% light-off and extinction conversion temperatures. Taking the absolute values of the differences, halving S_CO was found to change the temperatures by an average of 6.6 K. When doubling E_b, , and ν_CO, the differences were 20.0, 6.6, and 1.9 K respectively. With respect to the coefficients of Table 9, all model values fall within the respective literature bounds; e.g., at 400K, ∆H_CO = −129 kJ/mol (Table 2), E_−b = 197 kJ/mol (Table 3), and E_CO = 96–111 kJ/mol (Table 6). Overall, it can be inferred that the calibrated model provides reasonable quantitative values in comparison to the experiments.

With respect to a qualitative analysis, the model accurately captures the trend of 50% conversion temperature with dispersion. However, the higher dispersion values do illustrate a delayed light-off and extinction temperature. Part of this difference could be due to the relatively high monolith density value found in Table 9 resulting in a reduced thermal response of the monolith. Moreover, some of the deviation between the model and experiments (50% conversion temperatures, slope, and RMSE values) relates to the choice of a 1-D model that lumps boundary layer phenomenon into the source terms and washcoat diffusion into the reaction rates. This embeds assumptions into the kinetic parameters reducing the physicality of the model. While it is possible to use more advanced and accurate simulation techniques (e.g., 1 + 1-D model by the authors [94]), this added computational effort made it unfeasible for calibration across all three hysteresis experiments.

Further differences between the results are a function of writing only the pre-exponential and activation energy as a function of dispersion. Metal particle size can influence other parameters, such as the sticking coefficient. This is evident in the difference between the light-off and extinction temperatures. The 56% dispersion experiment only differs by a few degrees Kelvin; however, the 44% and 26% cases are dissimilar by over 30 K. As a result, there is likely some additional non-linearity in the kinetic parameters as a function of dispersion. The results of the sensitivity analysis also demonstrate that the parameters held constant during the analysis might play a role. The authors’ endeavored to choose the best values based on the literature in order to reduce the required calibration effort. These may not be the optimum values and may need to be included in the calibration effort. However, the first exploration of a derived dispersion model here warranted the choice of the simplest version in order to explore its practicality. Finally, since the adaptive kinetics only influence the variables in the reaction rate expression as new values, there is no significant addition to the computational burden of the model while increasing its predictive nature with respect to dispersion.

In order to contrast the derived expressions with models available in the literature, the Arnby et al. 2005 experiments were simulated using the four different kinetic options via Wojciechowski and Asprey [20] and Herskowitz and Kenney [42] for the reaction expression of Equation (12) with the coefficients given in Table 1. It is important to note that in order to utilize the pre-exponentials of these authors (k_CO) in the same units as this paper required division by the universal gas constant (given in Table 1) that was then divided by the monolith temperature at the time of computation instead of assuming a singular value at all temperatures. Figure 3 illustrates that the Wojciechowski and Asprey models are close to the highest dispersion rate results but significantly deviate from the lowest. Moreover, the Herskowitz and Kenney results do not accurate capture the experimental conditions. It is important to note that these models provide only singular results and cannot depict the influence of surface chemistry.

In order to validate the model, the Arnby et al. experiments of 2004 were simulated using the coefficients from Table 9. This 2004 paper utilized three different CO concentration levels; e.g., 0.01%, 0.1%, 1%. Moreover, the catalysts in the 2004 paper had varying Pt content, washcoat mass, and BET surface areas as given in Table 8. Investigating Figure 4, this paper’s model accuracy does degrade since it was calibrated to only the 2005 experiments that used 0.1% CO input. In particular, the average deviation between all 50% conversion temperatures increases to 19.3K with the 0.01%, 0.1%, and 1% CO concentration results differing by an average of 14.3K, 10.1K, and 33.5K respectively. Moreover, the RMSE values in Table 10 increase significantly showing a difference of 35, 32, and 55% in conversion percentages. The model does perform best when compared to the 0.1% case since this was the concentration over which the model was calibrated.

Moreover, in the Arnby et al. 2004 experimental data, they found a deviation in the trend for the 41% dispersion case as it incurred a decreased conversion rate over the 48% dispersion case. The authors theorized that this result may be a function of the relative ratio of Pt to washcoat support which is the lowest of all of the 2004 tests. The model as described in Equation (28) does account for this factor; however, it was unable to be captured by the simulation. Another concern is that for the high CO concentration input (1% CO) the RDS may shift to the oxygen adsorption case via Equation (11b) as discussed in Section 3. This would require a different global reaction mechanism for simulation purposes; hence, the authors’ model calibrated to a different RDS would definitely not work. Overall, the model presented here provides the correct trends with dispersion. As dispersion increases, the temperature where light-off and extinction happens, increases. Finally, plotting one Wojciechowski and Asprey value (WA1) set in this figure illustrates that it has the same issue with respect to CO concentration while performing worse at the lowest CO level than the model in this effort. Hence, while the authors’ model does illustrate a significant difference in conversion percentages, its predictability has increased and the methodology for including dispersion in the global rate expression has potential.

While it is indeed possible to recalibrate the model to both the 2004 and 2005 experiments of Arnby et al. using Table 9 as the starting parameters, the fact that these experiments include a significant hysteresis effect results in the calibration of reactor heat transfer coefficients alongside the kinetic expression. A common omission from catalyst reactor papers is a reactor warm-up curve or indication that thermal hysteresis was checked during experimentation [2,22,23,24]. Even with insulated or isothermal reactors, there is still a need to have a temperature rise curve that illustrates the time rate of change of temperature throughout the length of the reactor. This will allow modelers the ability to calibrate the thermal properties (specific heat, density, and thermal conductivity) of the monolith or prove that the kinetic results are not a function of hysteresis. Ideally, just knowing the material of the monolith is sufficient; however, different reactors will have unique thermal conditions and a simple experiment to help modelers eliminate some variability would be significantly helpful. In addition, when experiments contain a large gradient in temperature between the reactor and the ambient, there is a possibility of external heat transfer. A transient heating test of the reactor with the inlet and outlet gas temperatures given as a function of time would help modelers calibrate the associated terms in Equation (9). As a result, spending additional time to calibrate the reaction further was deemed excessive and time was better spent working on setting up a reactor for follow-up experiments.

This paper describes an adaptive global kinetics reaction model of CO oxidation over a platinum/alumina catalyst. This model was developed based on the first principles surrounding detailed kinetic mechanisms. A thorough literature search was accomplished in order to minimize the number of constants to calibrate and the results illustrate that the model provides reasonable to good accuracy while capturing the correct dependency on dispersion.

Future work is planned to create a proper Design of Experiments in order to deduce the dependencies as a function of catalyst weight and dispersion. This will provide for better dependencies along with exploration of the different global CO oxidation reaction expressions. As discussed prior, the mechanisms developed using only these conversion reactions do not necessarily predict the most accurate results [8,9,13,14], again illustrating some deviation between the model and experiments. Hence, additional work that provides a better set of data to calibrate the model will shed insight into the adaptive kinetics phenomena. However, this effort illustrates the capability of such a model and demonstrates improved results in comparison to other models available in the literature.