Modeling of Diffusion and Catalytic Reactions of Gases in Highly Porous Nanolayers with DSMC and OpenFOAM
G. R. Pesch†,‡, N. Riefler†, U. Fritsching†, L. Colombi Ciacchi§,‡, L. Mädler†
† -- Foundation Institute of Materials Science (IWT), Department of Production Engineering, University of Bremen, Germany
‡ -- Chemical Engineering -- Recovery and Recycling (VdW), Department of Production Engineering and Center for Environmental Research and Sustainable Technology (UFT), University of Bremen, Germany
§ -- Hybrid Materials Interfaces Group (HMI), Department of Production Engineering and Bremen Center for Computational Materials Science (BCCMS), University of Bremen, Germany
Gas diffusion in at Knudsen numbers (Kn) above 0.1 requires a mathematical treatment based on the Boltzmann equation. Accepted solutions for the description of diffusion in highly porous geometries are for instance the extended Fickian Model or the Dusty-Gas-Model (DGM). Here collisions between molecules are, due to their vast appearance, only statistically considered. Both models require mean geometry values, such as porosity and tortuosity, to describe the porous structure. This results in a simplification of the exact layer geometry, which hinders an effective and accurate description of diffusion processes inside real inhomogeneous 3 dimensional porous layers, such as gas sensors films or catalysts.
In gas sensors for example, a probe gas diffuses into the porous layer. Due to chemical reactions at the surface (rate of reaction is proportional to probe gas concentration) the porous layer resistance changes, which can be measured by two electrodes and back calculated to the probe gas composition. Fundamental for design and response optimization of sensors of this kind is precise knowledge about the amount and exact position of the reaction fronts within the layer.
The Direct Simulation Monte Carlo (DSMC) method is suitable for the simulation of gas diffusion in such sensors as it uses the exact porous geometry to describe diffusion processes by modeling the tracks and collisions of every single gas molecule inside the layer. Molecular collisions are calculated by employing collision parameters acquired from simplified solutions of the Boltzmann equation. If a model of the porous geometry to investigate is available, the DSMC method is therefore able to accurately describe gas diffusion processes in inhomogeneous layers, without the knowledge of pore diameter, tortuosity, or porosity information. The OpenFOAM implementation of the DSMC code has been extended by the Variable Soft Sphere (VSS) model for binary molecular collisions , which results in a more accurate diffusion rate, compared the Variable Hard Sphere (VHS) model. Further, diffusion of CO into a N2 filled layer has been simulated by the DGM and the DSMC-VSS code. Whereas DSMC and DGM show a good agreement for the diffusion inside isotropic layers, DSMC shows higher accuracy for diffusion inside real gas sensor layers as obtained by an aerosol synthesis method.
To obtain information about the chemical reaction fronts inside the layer, the DSMC method has been extended to describe basic heterogeneous reaction mechanisms, i.e., adsorption, co-adsorption, desorption and reaction of gas species on the surface of the solid . The adsorption is based on the well-known sticking coefficient and implemented by the Kisluik model for precursor mediated adsorption, which describes the chance of a molecule to stick on the surface after hitting the solid as a function of temperature and coverage of the surface. Desorption and surface reaction are modeled through a mean-field rate approximation, i.e., the Polanyi-Wigner equation for desorption and a second-order Arrhenius equation for reaction. With this model we study the catalytic oxidation of carbon monoxide (CO) inside gas sensor layers of 1000 nm thickness in the transition regime (Kn ~ 1) using kinetic parameters taken from the literature for single-crystal Pd(111) surfaces at UHV conditions (Fig 1a).
Investigation of the reaction at different temperatures reveals a clear transition from a kinetic limit at low temperatures (T < 673 K) to a diffusion limit at high temperatures (T > 673 K). In the kinetic limit, the diffusion occurs at much faster rate than the reaction; the latter is therefore the rate-determining step of the overall process. At the diffusion limit, the reaction consumes educts at a much higher rate than their replenishment due to diffusion from the inlet. At this limit, diffusion of educts is thus the rate-determining step and the process is mass-transport limited.
At high temperatures and at steady state (Fig 1b), the layer is separated into three distinct regions. The surface of the layer is poisoned by CO, which is a result of the underlying co-adsorption mechanism. Here, the surface is covered with CO, which is hindering oxygen adsorption (competitive adsorption mechanism). Hence, the reaction rate is low, as only one out of two species (CO) required for the reaction is adsorbed on the surface.
With increasing depth into the layer the CO coverage is decreasing and the oxygen coverage is increasing, which results in an effective reaction area of the layer. The peak reaction rate is reached as both coverages intersect. Going even deeper into the layer, the CO coverage reaches zero, as the effective reaction area consumes all CO. This reflects the mass-transport limitation of the overall reaction. Hence, the reaction rate is again low as only one out of two species needed for the reaction (oxygen) is adsorbed on the surface.
The employed parameters together with the presented reaction system serve to demonstrate the capabilities of the newly developed simulation algorithm. We expect that similar investigations will not only help gaining deeper understanding of the reaction processes inside porous structures but also for the structure optimization of gas sensors and catalysts.
 J.A.H. Dreyer, N. Riefler, G.R. Pesch, M. Karamehmedovic, U. Fritsching, W.Y. Teoh, L. Mädler: Simulation of Gas Diffusion in Highly Porous Nanostructures by Direct Simulation Monte Carlo. Chem. Eng. Sci. (2014), 69-76
 G.R. Pesch, N. Riefler, U. Fritsching, L. Colombi Ciacchi, L. Mädler: Gas-Solid Catalytic Reactions with an Extended DSMC Model. AIChE J. (2015), in press, DOI: 10.1002/aic.14856