(25f) Modeling of Diffusion and Catalytic Reactions of Gases in Highly Porous Nanolayers with Dsmc and Openfoam

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 [1],
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 [2]. 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.

[1] 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

[2] 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