Models in Space and Time

Due to permanent motion, namely advection and turbulent diffusion, having stochastic characteristics on different time and spatial scales, it is extremely comph-cated to model chemistry and transport (so-called chemistry-transport models, CTM, which are also a basis for climate modeling) in space and time. At the earth-air interface, exchange of matter occurs, emission as well as deposition. 

Computational Methods The implementation of theoretical models within software packages has now become excellent for certain focused problems such as molecular electronic structure or simple Monte Carlo simulations. Very large challenges remain for extending these methods to multiscale modeling in space and time. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

Limited Data First, plant data are limited. Unfortunately, those easiest to obtain are not necessarily the most useful. In many cases, the measurements that are absolutely required for accurate model development are unavailable. For those that are available, the sensitivity of the parameter estimate, model evaluation, and/or subsequent conclusion to a particiilar measurement may be very low. Design or control engineers seldom look at model development as the primaiy reason for placing sensors. Further, because equipment is frequently not operated in the intended region, the sensitive locations in space and time have shifted. Finally, because the cost-effectiveness of measurements can be difficult to justify, many plants are underinstru-mented. 

In particular we would like to treat some essential effects of fluctuations where we assume that, for example, thermal fluctuations exist and are localized in space and time. The effects on large lengths and long times are then of interest where the results are independent of local details of the model assumptions and therefore will have some universal validity. In particular, the development of a rough surface during growth from an initially smooth surface, the so-called effect of kinetic roughening, can be understood on these scales [42,44]. 

Kinetic gelation simulations seek to follow the reaction kinetics of monomers and growing chains in space and time using lattice models [43]. In one example, Bowen and Peppas [155] considered homopolymerization of tetrafunctional monomers, decay of initiator molecules, and motion of monomers in the lattice network. Extensive kinetic simulations such as this can provide information on how the structure of the gel and the conversion of monomer change during the course of gelation. Application of this type of model to polyacrylamide gels and comparison to experimental data has not been reported. 

Input Errors. Errors in model input often constitute one of the most significant causes of discrepancies between observed data and model predictions. As shown in Figure 2, the natural system receives the "true" input (usually as a "driving function") whereas the model receives the "observed" input as detected by some measurement method or device. Whenever a measurement is made possible source of error is introduced. System inputs usually vary continuously both in space and time, whereas measurements are usually point values, or averages of multiple point values, and for a particular time or accumulated over a time period. Although continuous measurement devices are in common use, errors are still possible, and essentially all models require transformation of a continuous record into discrete time and space scales acceptable to the model formulation and structure. 

To address media-specific problems, single-media models for air, surface water, groundwater and soil pollution have been developed and used by different disciplines. Although these models generally provide detailed description of the pollutant distribution in space and time and incorporate mass transfer from other media as boundary conditions, they are not capable of characterizing the total environmental impact of a pollutant release. Multimedia models have been, therefore, developed to predict the concentration of chemicals in multiple environmental media simultaneously with consideration of chemical transport and transformation within and among media [1],... 

In a reactive transport model, the domain of interest is divided into nodal blocks, as shown in Figure 2.11. Fluid enters the domain across one boundary, reacts with the medium, and discharges at another boundary. In many cases, reaction occurs along fronts that migrate through the medium until they either traverse it or assume a steady-state position (Lichtner, 1988). As noted by Lichtner (1988), models of this nature predict that reactions occur in the same sequence in space and time as they do in simple reaction path models. The reactive transport models, however, predict how the positions of reaction fronts migrate through time, provided that reliable input is available about flow rates, the permeability and dispersivity of the medium, and reaction rate constants. 

As will be shown for the CD model, early mixing models used stochastic jump processes to describe turbulent scalar mixing. However, since the mixing model is supposed to mimic molecular diffusion, which is continuous in space and time, jumping in composition space is inherently unphysical. The flame-sheet example (Norris and Pope 1991 Norris and Pope 1995) provides the best illustration of what can go wrong with non-local mixing models. For this example, a one-step reaction is described in terms of a reaction-progress variable Y and the mixture fraction p, and the reaction rate is localized near the stoichiometric point. In Fig. 6.3, the reaction zone is the box below the flame-sheet lines in the upper left-hand corner. In physical space, the points with p = 0 are initially assumed to be separated from the points with p = 1 by a thin flame sheet centered at... 

The reviews by Johnson and by Seinfeld give helpful guidelines in the classification of models by space and time scale. 

On the basis of these estimates, we can identify the flow of nuclear reactions and plot the rivers they follow on the (A, Z) map. By coupling this network of nuclear reactions with models of stars or the Big Bang, which predict temperature and density variations in space and time, we may hope to identify the nature of the elements and isotopes produced, as well as their relative proportions. 

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