Models modeling approaches, tracing

Heimann M, Kaminski T. 1999. Inverse modelling approaches to infer surface trace gas fluxes from observed atmospheric mixing ratios. In Bouwman AF, ed. Approaches to Scaling of Trace Gas Fluxes in Ecosystems. Amsterdam Elsevier. 

In the spirit of the box model approach, of the several residence times of geophysical interest two now seem to be fairly well established— the mean residence time of a trace substance in the lower stratosphere before entering the troposphere and the mean residence time of C02 in... 

Two general modeling approaches have been employed to assist studies of trace metal impacts in aquatic environments one approach has emphasized biological processes while the other has emphasized chemical controls on metal availability to organisms. 

The first generation of the CPFR process had nine tasks in its generic business model that trace the collaboration process between manufacturers, retailers, and distributors. The nine steps, listed in Table 3.5, provide a flavor for the overall CPFR approach. The first two steps are planning-, the next six are forecasting-, the last step is replenishment. 

It is difficult to trace the historical development of the theory of water. One of the earliest documents attempting an explanation of some anomalous properties of water is Rontgen s (1892) article, which postulated that liquid water consists of two kinds of molecules, one of which is referred to as an ice-molecule. Rontgen himself admitted that his explanation of the properties of water, using the so-called mixture-model approach, had been known in the literature for some time, but he could not point out its originator. An interesting review of the theories of water until 1927 was presented by Chad well (1927). Most of the earlier theories were concerned with association complexes, or polymers of water molecules. There has been little discussion on the structural features of these polymers. 

Let us now consider some general features of the present model. At the very outset, we emphasize that the basic variables of our system are r, P, and i.e., we have a one-component system, and there is no trace of a mixture-model approach. The variables and in (6.32) play the role of convenient intermediary variables. Once we have carried out the summation in (6.32), the dependence of the partition function on disappears. Nevertheless, the nature of the model suggests a new way of looking at this one-component system namely, we decide to refer to a lattice molecule as an L-cule and to an interstitial molecule as an LT-cule (the letters L and H in the present context are chosen to remind us of water in lattice and water in holes. The same notation will be used in Section 6.8 to indicate the type of environment, that of low local density and high local density, respectively.) Once we have made this classifica-... 

Intercalation-induced stresses have been modeled extensively in the Hterature. A one-dimensional model was proposed to estimate stress generation in the lithium insertion process in the spherical particles of a carbon anode [24] and an LiMn204 cathode [23]. In this model, displacement inside a particle is related to species flux by lattice velocity, and total concentration of species is related to the trace of the stress tensor by compressibihty. Species conservation equations and elasticity equations are also included. A two-dimensional porous electrode model was also proposed to predict electrochemicaUy induced stresses [30]. Following the model approach of diffusion-induced stress in metal oxidation and semiconductor doping [31-33], a model based on thermal stress analogy was proposed to simulate intercalation-induced stresses inside three-dimensional eUipsoidal particles [1]. This model was later extended to include the electrochemical kinetics at electrode particle surfaces [2]. This thermal stress analogy model was later adapted to include the effect of surface stress [34]. 

We use computational solution of the steady Navier-Stokes equations in cylindrical coordinates to determine the optimal operating conditions.Fortunately in most CVD processes the active gases that lead to deposition are present in only trace amounts in a carrier gas. Since the active gases are present in such small amounts, their presence has a negligible effect on the flow of the carrier. Thus, for the purposes of determining the effects of buoyancy and confinement, the simulations can model the carrier gas alone (or with simplified chemical reaction models) - an enormous reduction in the problem size. This approach to CVD modeling has been used extensively by Jensen and his coworkers (cf. Houtman, et al.) ... 

First approaches at modeling the viscoelasticity of polymer solutions on the basis of a molecular theory can be traced back to Rouse [33], who derived the so-called bead-spring model for flexible coiled polymers. It is assumed that the macromolecules can be treated as threads consisting of N beads freely jointed by (N-l) springs. Furthermore, it is considered that the solution is ideally dilute, so that intermolecular interactions can be neglected. 

First approaches to approximating the relaxation time on the basis of molecular parameters can be traced back to Rouse [33]. The model is based on a number of boundary assumptions (1) the solution is ideally dilute, i.e. intermolecular interactions are negligible (2) hydrodynamic interactions due to disturbance of the medium velocity by segments of the same chain are negligible and (3) the connector tension F(r) obeys an ideal Hookean force law. 

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