Lattice Boltzmann model with generalized wall boundary conditions for arbitrary catalytic reactivity

A lattice Boltzmann model for multispecies flows with catalytic reactions is developed, which is valid from very low to very high surface Damköhler numbers (${\text{Da}}_{\text{s}}$). The previously proposed model for catalytic reactions [S. Arcidiacono, J. Mantzaras, and I. V. Karlin, Phys. Rev. E 78, 046711 (2008)], which is applicable for low-to-moderate ${\text{Da}}_{\text{s}}$ and encompasses part of the mixed kinetics and transport-controlled regime, is revisited and extended for the simulation of arbitrary kinetics-to-transport rate ratios, including strongly transport-controlled conditions (${\text{Da}}_{\text{s}}\to \infty$). The catalytic boundary condition is modified by bringing nonlocal information on the wall reactive nodes, allowing accurate evaluation of chemical rates even when the concentration of the deficient reactant at the wall becomes vanishingly small. The developed model is validated against a finite volume Navier-Stokes CFD (Computational Fluid Dynamics) solver for the total oxidation of methane in an isothermal channel-flow configuration. CFD simulations and lattice Boltzmann simulations with the old and new catalytic reaction models are compared against each other. The new model demonstrates a second order accuracy in space and time and provides accurate results at very high ${\text{Da}}_{\text{s}}$ ($\sim {\text{10}}^{\text{9}}$) where the old model fails. Moreover, to achieve the same accuracy at moderate-to-high ${\text{Da}}_{\text{s}}$ of $O\left(1\right)$, the new model requires $\sim 2^d\times N$ coarser grid than the original model, where $d$ is the spatial dimension and $N$ the number of species.

Figure 1

The full-way reactive boundary condition steps.

Figure 2

The effect of Damköhler number on transverse mass fraction profiles at a given cross section ($x=0.2\mathrm{mm}$) of a 2D channel ($0.5\mathrm{mm}\times 5\text{mm}$) obtained from CFD simulations: $U_{\text{in}}=7.2\text{m/s}, X_{\text{in},{\text{CH}}_4}=0.1, X_{\text{in},{\text{O}}_2}=0.9$. $T_{\text{in}}=T_{\text{wall}}=\text{const}=1200\text{K}, P=1$ bar. $Y_F$ indicates the mass fraction of the deficient species (methane).

Figure 3

Normalized deficient species wall gradient of mass fraction and wall mass fraction as a function of the catalytic Damköhler number ${\text{Da}}_{\text{s}}$ for the conditions of Fig. 2.

Figure 4

Schematic of the channel geometry with reactive walls and boundary conditions.

Figure 5

The present model's order of convergence.

Figure 6

Comparisons of mole fraction transverse profiles obtained from the old and present catalytic LB models as well as CFD simulations for a kinetically controlled reaction (${\text{Da}}_{\text{s}}=0.03$) at $x=2\mathrm{mm}$.

Figure 7

Comparison of mole fraction axial profiles obtained from the old and present LB models as well as CFD results for kinetically controlled reaction (${\text{Da}}_{\text{s}}=0.03$) at the channel midplane.

Figure 8

Comparison of mole fraction transverse profiles obtained from the old and present LB models as well as CFD results for ${\text{Da}}_{\text{s}}=2$ at $x=2\mathrm{mm}$.

Figure 9

Comparison of mole fraction axial profiles obtained from the old and present LB models as well as CFD results for ${\text{Da}}_{\text{s}}=2$ at the channel midplane.

Figure 10

The impact of Damköhler number on consumption and production of ${\text{CH}}_4$ and ${\text{CO}}_2$: ${\text{Da}}_{\text{s}}=0.0055,0.55,5.5,555$.

Figure 11

Comparison of mole fraction transverse profiles obtained from the present catalytic LB model and CFD results at high Damköhler numbers ($x=0.5\mathrm{mm}$) (a) ${\text{Da}}_{\text{s}}={10}^3$, (b) ${\text{Da}}_{\text{s}}={10}^6$, and (c) ${\text{Da}}_{\text{s}}={10}^9$.

Figure 12

The error in the ${\text{CH}}_4$ mole fraction obtained by the old and the present catalytic models in transient simulation at two axial locations on the channel wall.