Recovering Boundary Model

Whether the recovering boundary or radiation boundary is applicable depends on a detailed model of the dynamics of the system and must be implemented accordingly. For the purposes of this work, the recovering boundary formalism is simply used to estimate the feasibility of certain approximations and is not implemented in any simulation. 

Single slab. A number of recent calculations of surface electronic structures have shown that the essential electronic and structural features of the bulk material are recovered only a few atomic layers beneath a metal surface. Thus, it is possible to model a surface by a single slab consisting of 5-15 atomic layers with two-dimensional translational symmetry parallel to the surface and vacuum above and below the slab. Using the two-dimensional periodicity of the slab (or thin film), a band-structure approach with two-dimensional periodic boundary conditions can be applied to the surface electronic structure. 

However, in the case of large Kn, the no-slip approximation cannot be applied. This implies that the mean free path of the liquid is on the same length scale as the dimension of the system itself. In such a case, stress and displacement are discontinuous at the interface, so an additional parameter is required to characterize the boundary condition. A simple technique to model this is the one-dimensional slip length, which is the extrapolation length into the wall required to recover the no-slip condition, as shown in Fig. 1. If we consider... 

Fortunately, it is not always necessary to recover the system RTD curve from the impulse response, so the complications alluded to above are often of theoretical rather than practical concern. In addition, the dispersion model is most appropriately used to describe small extents of dispersion, i.e. minor deviations from plug flow. In this case, particularly if the inlet pipe is of small diameter compared with the reactor itself, the vessel can be satisfactorily assumed to possess closed boundaries [62]. An impulse of tracer will enter the system and broaden as it passes along the reactor so that the observed response at the outlet will be an RTD and will be a symmetrical pulse, the width of which is a function of DjuL alone. 

Much like the RISM method, the LD approach is intermediate between a continuum model and an explicit model. In the limit of an infinite dipole density, the uniform continuum model is recovered, but with a density equivalent to, say, the density of water molecules in liquid water, some character of the explicit solvent is present as well, since the magnitude of the dipoles and their polarizability are chosen to mimic the particular solvent (Papazyan and Warshel 1997). Since the QM/MM interaction in this case is purely electrostatic, other non-bonded interaction terms must be included in order to compute, say, solvation free energies. When the same surface-tension approach as that used in many continuum models is adopted (Section 11.3.2), the resulting solvation free energies are as accurate as those from pure continuum models (Florian and Warshel 1997). Unlike atomistic models, however, the use of a fixed grid does not permit any real information about solvent structure to be obtained, and indeed the fixed grid introduces issues of how best to place the solute into the grid, where to draw the solute boundary, etc. These latter limitations have curtailed the application of the LD model. 

The predicted flow rates from the JNC analyses are less than the measured flow. The flow rate for Case 1 is smaller than that of Case 2, which infers that the non-linear anisotropic model predicts a greater rock deformation and thus a greater reduction of the hydraulic conductivity around the drift. The predicted pressure at P4 is drastically changed by excavation in Case 1. The pressure becomes negative and then recovers gradually to the initial state. The effects of the drift construction are predicted to happen two to three days before they are actually witnessed in the data. On the other hand, there is little difference in behavior at P3 for the two cases. This means that the influence of excavation at P3 is very small and the non-linear effect there is negligible. The overall pressure values in P3 are less than the data, which is likely due to the choice of hydrologic boundary conditions. 

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