Mathematical model, behavior

Test of the Mathematical Model. In the mathematical model, behavior of the polymer monolayer was related to two parameters the degree of compression, (1-A/A ), and the ratio of the relaxation rate to the compression (expansion) rate, q. The results from three typical hysteresis experiments with three different polymers were chosen for comparison to the theory. 

The idea of using mathematical modeling for describing materials behavior under loads is well known. Some physical phenomena, which can be observed in materials during testing, have time dependent quantitative characteristics. It gives a possibility to consider them as time series and use well developed models for their analysis [1, 2]. Usually applied... 

In general, the desorptive behavior of contaminated soils and soHds is so variable that the requited thermal treatment conditions are difficult to specify without experimental measurements. Experiments are most easily performed in bench- and pilot-scale faciUties. Full-scale behavior can then be predicted using mathematical models of heat transfer, mass transfer, and chemical kinetics. 

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. 

Once fundamental data have been obtained, the goal is to develop a mathematical model of the process and to utilize it to explore such possibilities as produc t selectivity, start-up and shut-down behavior, vessel configuration, temperature, pressure, and conversion profiles, and so on. 

It is not easy to see why the authors believe that the success of orbital calculations should lead one to think that the most profound characterization of the properties of atoms implies such an importance to quantum numbers as they are claiming. As is well known in quantum chemistry, successful mathematical modeling may be achieved via any number of types of basis functions such as plane waves. Similarly, it would be a mistake to infer that the terms characterizing such plane wave expansions are of crucial importance in characterizing the behavior of atoms. 

This requirement also makes good sense. A calibration is nothing more than a mathematical model that relates the behavior of the measureable data to the behavior of that which we wish to predict. We construct a calibration by finding the best representation of the fit between the measured data and the predicted parameters. It is not surprising that the performance of a calibration can deteriorate rapidly if we use the calibration to extrapolate predictions for... 

Since electrochemical processes involve coupled complex phenomena, their behavior is complex. Mathematical modeling of such processes improves our scientific understanding of them and provides a basis for design scale-up and optimization. The validity and utility of such large-scale models is expected to improve as physically correct descriptions of elementary processes are used. 

Atmospheric aerosols have a direct impact on earth s radiation balance, fog formation and cloud physics, and visibility degradation as well as human health effect[l]. Both natural and anthropogenic sources contribute to the formation of ambient aerosol, which are composed mostly of sulfates, nitrates and ammoniums in either pure or mixed forms[2]. These inorganic salt aerosols are hygroscopic by nature and exhibit the properties of deliquescence and efflorescence in humid air. That is, relative humidity(RH) history and chemical composition determine whether atmospheric aerosols are liquid or solid. Aerosol physical state affects climate and environmental phenomena such as radiative transfer, visibility, and heterogeneous chemistry. Here we present a mathematical model that considers the relative humidity history and chemical composition dependence of deliquescence and efflorescence for describing the dynamic and transport behavior of ambient aerosols[3]. 

In experiments with lowland rice Oiyzci saliva L.) it was found that roots quickly exhausted available sources of P and sub.sequently exploited the acid-soluble pool with small amounts deriving from the alkaline soluble pool (18). More recalcitrant forms of P were not utilized. The zone of net P depletion was 4-6 mm wide and showed accumulation in some P pools giving rather complex concentration profiles in the rhizosphere. Several mechanisms for P solubilization could be invoked in a conceptual model to describe this behavior. However using a mathematical model with independently measured parameters (19), it was shown that it could be accounted for solely by root-induced acidification. The acidification resulted from H" produced during the oxidation of Fe by Oi released from roots into the anaerobic rhizosphere as well as from cation/anion imbalances in ion uptake (18). Rice was shown to depend on root-induced acidification for more than 80% of its P uptake. 

The experimentalist often formulates a mathematical model in order to describe the observed behavior. In general, the model consists of a set of equations based on the principles of chemistry, physics, thermodynamics, kinetics and transport phenomena and attempts to predict the variables, y, that are being measured. In general, the measured variables y are a function of x. Thus, the model has the following form... 

Parameter estimation and identification are an essential step in the development of mathematical models that describe the behavior of physical processes (Seinfeld and Lapidus, 1974 Aris, 1994). The reader is strongly advised to consult the above references for discussions on what is a model, types of models, model formulation and evaluation. The paper by Plackett that presents the history on the discovery of the least squares method is also recommended (Plackett, 1972). 

The nature of the mathematical model that describes a physical system may dictate a range of acceptable values for the unknown parameters. Furthermore, repeated computations of the response variables for various values of the parameters and subsequent plotting of the results provides valuable experience to the analyst about the behavior of the model and its dependency on the parameters. As a result of this exercise, we often come up with fairly good initial guesses for the parameters. The only disadvantage of this approach is that it could be time consuming. This counterbalanced by the fact that one learns a lot about the structure and behavior of the model at hand. 

Based on alternative assumptions about the mechanism of the process under investigation, one often comes up with a set of alternative mathematical models that could potentially describe the behavior of the system. Of course, it is expected that only one of them is the correct model and the rest should prove to be inadequate under certain operating conditions. Let us assume that we have conducted a set of preliminary experiments and we have fitted several rival models that could potentially describe the system. The problem we are attempting to solve is ... 

Two classes of mathematical models have been developed those which are specific and attempt to describe the transport and degradation of a chemical in a particular situation and those which are general or "evaluative" and attempt to generally classify the behavior of chemicals in a hypothetical environment. The type of modeling discussed here, equilibrium partitioning models, fall into the latter category. Such models attempt, with a minimum of information, to predict expected environmental distribution patterns of a compound and thereby identify which environmental compartments will be of primary concern. 

A number of mathematical models have been developed in recent years which attempt to predict the behavior of organic water pollutants. >2>3 Models assume that compounds will partition into various compartments in the environment such as air, water, biota, suspended solids and sediment. The input to the models includes the affinity of the compound for each of the compartments, the rate of transfer between the compartments, and the rates of various degradation processes in the various compartments. There is a growing body of data, however, which indicates that the models to date may have overlooked a small but significant interaction. A number of authors have suggested that a portion of the compounds in the aqueous phase may be bound to dissolved humic materials and are not therefore truly dissolved. 

In order to understand the nature and mechanisms of foam flow in the reservoir, some investigators have examined the generation of foam in glass bead packs (12). Porous micromodels have also been used to represent actual porous rock in which the flow behavior of bubble-films or lamellae have been observed (13,14). Furthermore, since foaming agents often exhibit pseudo-plastic behavior in a flow situation, the flow of non-Newtonian fluid in porous media has been examined from a mathematical standpoint. However, representation of such flow in mathematical models has been reported to be still inadequate (15). Theoretical approaches, with the goal of computing the mobility of foam in a porous medium modelled by a bead or sand pack, have been attempted as well (16,17). 

The quantitative water air sediment interaction (Qwasi) model was developed in 1983 in order to perform a mathematical model which describes the behavior of the contaminants in the water. Since there are many situations in which chemical substances (such as PCBs, pesticides, mercury, etc.) are discharged into a river or a lake resulting in contamination of water, sediment and biota, it is interesting to implement a model to assess the fate of these substances in the aquatic compartment [34]. 

The design q>roblem can be approached at various levels of sophistication using different mathematical models of the packed bed. In cases of industrial interest, it is not possible to obtain closed form analytical solutions for any but the simplest of models under isothermal operating conditions. However, numerical procedures can be employed to predict effluent compositions on the basis of the various models. In the subsections that follow, we shall consider first the fundamental equations that must be obeyed by all packed bed reactors under various energy transfer constraints, and then discuss some of the simplest models of reactor behavior. These discussions are limited to pseudo steady-state operating conditions (i.e., the catalyst activity is presumed to be essentially constant for times that are long compared to the fluid residence time in the reactor). 

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