Continuum Model Simulations

The two-dimensional temperature field of the monolith is presented for a selected case in Fig. 5.9, representative of the continuum model simulations. Mean outlet gas temperatures computed for the individual reactor channels allowed for an estimate of the average monolith exhaust gas temperature. In the case of Fig. 5.9, the computed outlet gas temperature was at 1,089 K, as opposed to a measured value of 1,041 K for the same experimental conditions. 

This figure also illustrates the strong variation of temperature in the radial direction over the axial length covering the rear half of the reactor. Flow field simulations of the single monolith channels for this case revealed that fuel is already consumed up to 99% at half the reactor length, which in turn renders the rest of the reactor volume a heat sink towards the environment. Reduced radial heat transfer (due to the spatially non-uniform thermal properties of the monolith) reduces the 

Karagiannidis S, Mantzaras J, Jackson G, Boulouchos K (2007) Hetero-Zhomogeneous combustion and stability maps in methane-fueled catalytic microreactors. Proc Combust Inst 31 3309-3317 

Siegel R, Howell JR (1981) Thermal radiation heat transfer. Hemisphere, New York, p 271 

Reinke M, Mantzaras J, Schaeren R, Bombach R, Inauen A, Schenker S (2004) High-pressure catalytic combustion of methane over platinum in situ experiments and detailed numerical 

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Sometimes Nature smiles upon us or maybe Nature cannot subvert mathematies (or the other way around). Chromatography is a promising method for the determination of some of the difficult parameters that we have worried about that appear in continuum model simulations of fixed-bed reactors or adsorbers. The approach is not without its difficulties but, properly used, would appear to be very useful. 

TABLE 9.9 Nano-scale Continuum Modeling Simulation for Prediction Vibration Properties of CNTs... 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

The integral equation method is free of the disadvantages of the continuum model and simulation techniques mentioned in the foregoing, and it gives a microscopic picture of the solvent effect within a reasonable computational time. Since details of the RISM-SCF/ MCSCF method are discussed in the following section we here briefly sketch the reference interaction site model (RISM) theory. 

There is a view developing concerning the accomplishments of shock-compression science that the initial questions posed by the pioneers in the field have been answered to a significant degree. Indeed, the progress in technology and description of the process is impressive by any standard. Impressive instrumentation has been developed. Continuum models of materials behavior have been elaborated. Techniques for numerical simulation have been developed in depth. 

Lattice models have the advantage that a number of very clever Monte Carlo moves have been developed for lattice polymers, which do not always carry over to continuum models very easily. For example, Nelson et al. use an algorithm which attempts to move vacancies rather than monomers [120], and thus allows one to simulate the dense cores of micelles very efficiently. This concept cannot be applied to off-lattice models in a straightforward way. On the other hand, a number of problems cannot be treated adequately on a lattice, especially those related to molecular orientations and nematic order. For this reason, chain models in continuous space are attracting growing interest. 

The flat interface model employed by Marcus does not seem to be in agreement with the rough picture obtained from molecular dynamics simulations [19,21,64-66]. Benjamin examined the main assumptions of work terms [Eq. (19)] and the reorganization energy [Eq. (18)] by MD simulations of the water-DCE junction [8,19]. It was found that the electric field induced by both liquids underestimates the effect of water molecules and overestimates the effect of DCE molecules in the case of the continuum approach. However, the total field as a function of the charge of the reactants is consistent in both analyses. In conclusion, the continuum model remains as a good approximation despite the crude description of the liquid-liquid boundary. 

Archontis, G. Simonson, T. Dielectric relaxation in an enzyme active site molecular dynamics simulations intepreted with a macroscopic continuum model, J. Am. Chem. Soc. 2001,123, 11047-11056. 

The methods used for modeling pure granular flow are essentially borrowed from that of a molecular gas. Similarly, there are two main types of models the continuous (Eulerian) models (Dufty, 2000) and discrete particle (Lagrangian) models (Herrmann and Luding, 1998 Luding, 1998 Walton, 2004). The continuum models are developed for large-scale simulations, where the controlling equations resemble the Navier-Stokes equations for an ordinary gas flow. The discrete particle models (DPMs) are typically used in small-scale simulations or... 

The last thirty years have seen a flowering of simulation techniques based on explicit treatments of solvent molecules (some references are given above). Such methods provide new insight into the reasons why continuum methods work or don t work. However they have not and never will replace continuum models. In fact, continuum models are sometimes so strikingly successful that hubris may be the most serious danger facing their practitioners. One of the goals of this present chapter will be to diffuse (but not entirely deflate ) any possible overconfidence. 

Meanwhile, computational methods have reached a level of sophistication that makes them an important complement to experimental work. These methods take into account the inhomogeneities of the bilayer, and present molecular details contrary to the continuum models like the classical solubility-diffusion model. The first solutes for which permeation through (polymeric) membranes was described using MD simulations were small molecules like methane and helium [128]. Soon after this, the passage of biologically more interesting molecules like water and protons [129,130] and sodium and chloride ions [131] over lipid membranes was considered. We will come back to this later in this section. 

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