Modeling natural systems

Although this section has emphasized possible errors in the observed data, the above considerations should not be used as a crutch to support an invalid or inaccurate model. In truth, in most circumstances the observed data are our best real indication of system behavior. Combining model simulations with an informed skepticism of both model and the observed data can lead to a better overall understanding of modeling natural systems. 

Quite complicated porphyrin systems have been constructed with the object of modeling natural systems with biological functions. Some have holes, pockets, and protection for the metal from oxidation by 02. 

James, R. 0., and Mealy, T. W. Adsorption of hydrolyzable metal ions at the oxide/water interface II. Charge reversal of Si02 and Ti02 colloids by adsorbed Co(II), La(III), and Th(IV) as model systems, J. Colloid Interface Sci. 40, 53-64 (1972b). Benjamin, Mark, Adsorption of cadmium, copper, zinc, and lead on oxide surfaces in model natural systems, Ph.D. Thesis, Stanford Univ. (in press). 

Some researchers use plug-flow reactors (PFRs), also known as packed bed reactors or column reactors (if run vertically) to model natural systems. In an ideal plug-flow or column reactor, fluid is pumped or drained through a packed bed of mineral grains and every fluid packet is assumed to have the same residence or contact time (Hill, 1977). The residence time equals the ratio of the pore volume of the reactor (Vo) divided by flow rate Q. With no volume change in the reaction, radial flow, or pooling of fluid in the reactor (Laidler, 1987), the outlet concentration varies from the inlet concentration according to ... 

As a result of these tests I in the pellet form was chosen for additional investigation because its release characteristics were slower. The rest of the release experiments were performed in a synthetic water to better model natural systems. 

In other words, because thermodynamics only applies to equilibrium states, our geochemical models apply only to areas of local equilibrium, and therefore we can only successfully model natural systems which have areas of local equilibrium. But it is in fact very difficult to determine whether natural systems do have such areas of local equilibrium, and on what scale. This problem will be discussed in more detail at the end of this chapter ( 3.11). 

The ion-hydration approach for describing electrolyte chemistry has been described here by Wolery and Jackson. This alternative approach is based on some of the pioneering work done by Stokes and Robinson (43) and attempts to examine electrolyte speciation in terms of the degree of solvation. Since this approach and concept is relatively new, it remains a question whether these methods can be applied to modeling natural systems. 

It is important to note that most of the diffusion data summarized in Table 2 (see Appendix) were obtained in order to quantify either the transport rate of the element of interest (e g. O, C) or the solid state properties of the crystalline phase, chiefly, the nature of defects. Because most of these studies used isotopically labeled compounds, we assume that the rates of isotopic exchange can be adequately represented by these diffusivities. Therefore, the utility of diffusion data in modeling natural systems depends on selection of the appropriate D and its quality. What constitutes a successful (ideal) diffusion experiment ... 

We have now completed our survey of the thermodynamic principles required to model natural systems. It only remains to gain practice in formulating problems involving natural systems in thermodynamic terms. Quite often, that is the hardest part. Once the problem is set up in terms of relevant reactions and components, the equations can be solved by anyone who has absorbed the previous chapters. However, choosing the appropriate components and setting up the relevant balanced reactions only comes from experience. In this chapter we explore a few situations that have been investigated by thermodynamic methods. 

Although ligand binding and metal properties are mainly controlled by the first coordination sphere in simple models, natural systems are submitted to much more complex influences by the supramolecular enviromnent. Such phenomena are observed in supramolecular models when metal binding and cavity are sufficiently interdependent. 

Mass Transfer and Kinetics in Rotary Kilns. The rates of mass transfer of gases and vapors to and from the sohds iu any thermal treatment process are critical to determining how long the waste must be treated. Oxygen must be transferred to the sohds. However, mass transfer occurs iu the context of a number of other processes as well. The complexity of the processes and the parallel nature of steps 2, 3, 4, and 5 of Figure 2, require that the parameters necessary for modeling the system be determined empirically. In this discussion the focus is on rotary kilns. 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

LGs can also serve as powerful alternatives to PDEs themselves in modeling physical systems. The distinction is an important one. It must be remembered, however, that not all PDEs (and perhaps not all physical systems see chapter 12) are amenable to a LG simulation. Moreover, even if a candidate PDE is selected for simulation by a LG. there is no currently known cookbook recipe allowing a researcher to go from the PDE to a LG description (or vice versa). Nonetheless, by their very nature, LGs lend themselves to modeling any partial differential equation (PDE) for which the underlying physical basis for its construction involves a large number of particles with local interactions [wolf86c]. 

Why Do We Need to Know Ihis Material Chemical kinetics provides us with tools that we can use to study the rates of chemical reactions on both the macroscopic and the atomic levels. At the atomic level, chemical kinetics is a source of insight into the nature and mechanisms of chemical reactions. At the macroscopic level, information from chemical kinetics allows us to model complex systems, such as the processes taking place in the human body and the atmosphere. The development of catalysts, which are substances that speed up chemical reactions, is a branch of chemical kinetics crucial to the chemical industry, to the solution of major problems such as world hunger, and to the development of new fuels. 

While these calculations provide information about the ultimate equilibrium conditions, redox reactions are often slow on human time scales, and sometimes even on geological time scales. Furthermore, the reactions in natural systems are complex and may be catalyzed or inhibited by the solids or trace constituents present. There is a dearth of information on the kinetics of redox reactions in such systems, but it is clear that many chemical species commonly found in environmental samples would not be present if equilibrium were attained. Furthermore, the conditions at equilibrium depend on the concentration of other species in the system, many of which are difficult or impossible to determine analytically. Morgan and Stone (1985) reviewed the kinetics of many environmentally important reactions and pointed out that determination of whether an equilibrium model is appropriate in a given situation depends on the relative time constants of the chemical reactions of interest and the physical processes governing the movement of material through the system. This point is discussed in some detail in Section 15.3.8. In the absence of detailed information with which to evaluate these time constants, chemical analysis for metals in each of their oxidation states, rather than equilibrium calculations, must be conducted to evaluate the current state of a system and the biological or geochemical importance of the metals it contains. 

Special considerations are required in estimating paraimeters from experimental measurements when the relationship between output responses, input variables and paraimeters is given by a Monte Carlo simulation. These considerations, discussed in our first paper 1), relate to the stochastic nature of the solution and to the fact that the Monte Carlo solution is numerical rather than functional. The motivation for using Monte Carlo methods to model polymer systems stems from the fact that often the solution... 

Note that the models aim to define general descriptive bulk parameters for an aquifer. Values for bulk adsorption or weathering will reflect the weighted range for a variety of phases present in complex, natural systems, and provide an overall measure of the behavior of the aquifer. In contrast, individual phases can be examined in the laboratory to determine the specific processes involved. 

In the past few years a variety of workshops and symposia have been held on the subjects of model verification, field validation, field testing, etc. of mathematical models for the fate and transport of chemicals in various environmental media. Following a decade of extensive model development in this area, the emphasis has clearly shifted to answering the questions "How good are these models ", "How well do they represent natural systems ", and "Can they be used for management and regulatory decision-making "... 

Verification is the complement of calibration model predictions are compared to field observations that were not used in calibration or fidelity testing. This is usually the second half of split-sample testing procedures, where the universe of data is divided (either in space or time), with a portion of the data used for calibration/fidelity check and the remainder used for verification. In essence, verification is an independent test of how well the model (with its calibrated parameters) is representing the important processes occurring in the natural system. Although field and environmental conditions are often different during the verification step, parameters determined during calibration are not adjusted for verification. 

The process of field validation and testing of models was presented at the Pellston conference as a systematic analysis of errors (6. In any model calibration, verification or validation effort, the model user is continually faced with the need to analyze and explain differences (i.e., errors, in this discussion) between observed data and model predictions. This requires assessments of the accuracy and validity of observed model input data, parameter values, system representation, and observed output data. Figure 2 schematically compares the model and the natural system with regard to inputs, outputs, and sources of error. Clearly there are possible errors associated with each of the categories noted above, i.e., input, parameters, system representation, output. Differences in each of these categories can have dramatic impacts on the conclusions of the model validation process. 

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. 

Figure 2. Model vs. natural system inputs, outputs, and errors. (Reproduced with permission from Ref. 2.)...

System Representation Errors. System representation errors refer to differences in the processes and the time and space scales represented in the model, versus those that determine the response of the natural system. In essence, these errors are the major ones of concern when one asks "How good is the model ". Whenever comparing model output with observed data in an attempt to evaluate model capabilities, the analyst must have an understanding of the major natural processes, and human impacts, that influence the observed data. Differences between model output and observed data can then be analyzed in light of the limitations of the model algorithm used to represent a particularly critical process, and to insure that all such critical processes are modeled to some appropriate level of detail. For example, a... 

Output Errors. Output errors are analogous to input errors they can lead to biased parameter values or erroneous conclusions on the ability of the model to represent the natural system. As noted earlier, whenever a measurement is made, the possibility of an error is introduced. For example, published U.S.G.S. stream-flow data often used in hydrologic models can be 5 to 15% or more in error this, in effect, provides a tolerance range within which simulated values can be judged to be representative of the observed data. It can also provide a guide for terminating calibration efforts. 

Tables I and II present the results of the Work Group discussions for the screening and site-specific level models, respectively. The assessment in these tables is based on a ranking scale between 0 and 100 0 indicates situations where no testing has been attempted and 100 identifies areas where extensive testing has been completed with sufficient post-audits to validate the predictive capability of relevant models. The scores can also be interpreted to mean the extent to which additional field testing would improve our understanding of how well the models represent natural systems. It is important to note that the scores do not indicate model accuracy per se they show the degree to which current field testing has been able to identify or estimate model accuracy.

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