Boundary treatment

Equations (5.46) and (5.47) are referred to as steam methane reforming (SMR) reaction and Water Gas Shift (WGS) reaction respectively. These reactions are homogenous reactions that occur everywhere inside the anode, whereas Equations (5.48) and (5.49) only occur at the active triple phase boundary. Treatment of source terms due to electrochemical oxidation of H2 (Equation 5.49) is already covered in the previous sections and the treatment is similar for electrochemical oxidation of CO (Equation 5.48). Specie source terms due to homogenous reactions in the mechanism are given by ... 

In comparison, the Ewald summation-based methods were developed for the periodic boundary treatment, where the entire system plus its solvent environment are explicitly represented. Thus, Ewald summation-based simulations are particularly suited for studying free energy changes that involve delocalized... 

In this chapter, we will be describing different electro-optic techniques that have been developed specifically for measurement of the flexocoeffi-cients. In these techniques, an electric field is applied to a nematic liquid crystal in a cell with well-defined boundary treatments. The net torque on the medium, which involves dielectric, elastic and flexoelectric components, is set to zero both in the bulk and at the two surfaces in order to And the equilibrium director configuration if a DC field is applied. In AC techniques, the above torques are balanced with the viscous torque. The optical properties of the medium are in turn calculated for the distorted profiles and compared with experimental measurements. The flexoelectric, and in some... 

Figure 1 shows an example where a Cartesian grid is used to discretize the domain including both sides of the boundary. The fluid-fluid boundary or fluid-solid boundary intersects with the grid lines. The governing equations are discretized on the grid using a finite difference method. Near the boundary, special treatment has to be taken in order to incorporate the boundary conditions. Different boundary treatments lead to different variants of the immersed boundary method. 

Mei R, Luo LS, Shyy W (1999) An accurate curved boundary treatment in the lattice Boltzmann method. J StatPhys 155 307-330... 

The halfway bounce-back scheme with interpolation is used to deal with the curved surfaces [22]. This boundary treatment has a superlinear accuracy when the wall surface varies between two adjacent nodes and approximately second-order accuracy if the wall surface is in the middle. This method is easy to implement for complicated boundary conditions without special considerations for the comers. 

Of the three types of boundary treatment, the link atom method is the simplest both conceptually and in practice, and is hence the most widely used. The boundary atom and in particular the frozen orbital methods can potentially achieve higher accuracy but require careful a priori parametrization and bear limitations on transferability (Senn and Thiel 2009). 

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