Effective continuum

The application of CFD to packed bed reactor modeling has usually involved the replacement of the actual packing structure with an effective continuum (Kvamsdal et al., 1999 Pedernera et al., 2003). Transport processes are then represented by lumped parameters for dispersion and heat transfer (Jakobsen... 

Now, in the statistical limit for which we can treat our final manifold of closely spaced twisted triplets as an effective continuum, the sum in eq. (12-24) can be replaced by a definite integral whose weight function is taken to be the density of vibronic states, p(E). Thus we write... 

The committee acknowledges that there has been substantial research developing biomarkers further along the exposure-effect continuum, including prominent work by Gan et al. (2004) Hecht (2003) Joseph et al. (2005) Kensler et al. (2005) Qian et al. (1994) Rappaport et al. (2005) and Yu et al. (1995). The ultimate objective of the biomonitoring research is to link biomarkers of exposure to biomarkers of effect and susceptibility to understand the public-health implications of exposure to environmental chemicals. 

The above results may be interpreted as follows Figure 3.10 shows the dispersion diagrams co(/0 for the uncoupled and the coupled matter-radiation systems. Thus, the coupling induces, for cK < oj0, a splitting off of the lower state of the effective continuum, repelled to lower energies by its interaction with the matter state K>.126... 

Another relaxation process encountered in isolated molecules is the phenomenon of intramolecular vibrational relaxation. Following excitation of a high-lying vibrational level associated with a particular molecular mode, the excitation energy can rapidly spread to other nuclear modes. This is again a case of an initially prepared single state decaying into an effective continuum. 

A final consideration is how parameters from geometrical models might be incorporated into models to better predict solute dispersion at the macroscopic scale. One approach involves the development of effective continuum models for structured soils and fractured rocks (de Josselin de Jong Way, 1972 Long et al., 1982 Berkowitz et al., 1988). Another approach is the inclusion of dispersion due to porosity variations in Monte Carlo simulations of solute transport in random conductivity fields (Fiori, 1998 Hassan et al., 1998). It is also possible to embed geometrical models within multiregion velocity based models (Gwo et al., 1998). 

State now involves a dissipative irreversible decay into the effective continuum of levels < >/. Questions concerning the precise nature of the prepared state, the contribution from the reversible dephasing processes at intermediate energies, and the role of intramolecular vibrational relaxation at high enough energies are central to the understanding of infrared multiphoton dissociation processes in isolated molecules. ... 

By definition irreversible electronic relaxation processes cannot occur in isolated small and too-many level small (intermediate) case molecules because of the insufficient density of final levels. For long times the molecule senses the presence of a finite number of possible final levels instead of the effective continuum that is required to drive irreversible electron relaxation. When collisional processes are appended, it is clear that the continuous density of states of the colliding pair can provide the necessary driving force for irreversible relaxation. The observed magnitudes of electronic relaxation rates as well as dependencies on the initial state, perturbing molecules, temperature, and so on, are the aspects of the processes that are of central interest. 

Assuming a weak coupling between the chemical species and the transition state which plays the role of an effective continuum, we get the frequency-independent effective Liouvillian... 

In Odegard s study [48], a method has been presented for finking atomistic simulations of nano-structured materials to continuum models of tfie corresponding bulk material. For a polymer composite system reinforced with SWCNTs, the method provides the steps whereby the nanotube, the local polymer near the nanotube, and the nanotube/ polymer interface can be modeled as an effective continuum fiber by using an equivalent-continuum model. The effective fiber retains the local molecular stractuie and bonding information, as defined by MD, and serves as a means for finking tfie eqniv-alent-continuum and micromechanics models. The micromechanics method is then available for the prediction of bulk mechanical properties of SWCNT/polymer com-... 

Computation of Direct Effects - Continuum Solvent Models... 

The photon continuum allows the irreversible radiative decay, the states ip act as an effective continuum that enables irreversible decay into this manifold. The probabiUty of finding the molecule in the ip, manifold at time, complementary to (6.29), is given by... 

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