Types of models

Models of systems can also be categorized into various types based on the attributes of the system, including  

Each of these categories can then be combined into more descriptive categories, e.g., a nonlinear, time-variant system or a static, time-invariant discrete system. 

Time-variant means that the parameters of the system change over time, such as an autoinduction process that increases a drug s hepatic clearance with repeated administration. Time-invariant or stationary parameters do not change over time. It is typically assumed that a drug s pharmacokinetics are stationary over time so that the principle of superposition1 applies. With a static model, the output depends only on the input and does 

1 Superposition was developed in physics to explain the behavior of waves that pass simultaneously through the same region in space. In pharmacokinetics, superposition states that concentration-time profiles passing through the same relative region in time are additive. For example, if two doses are taken 24 hours apart and the concentration 6 hours and 30 hours after the first dose was 100 and 10 ng/mL, respectively, then the concentration 6 hours after the second dose (which is 30 hours after the first dose) would be equal to 110 ng/mL (100 ng/mL + 10 ng/mL). Thron (1974) presents a comprehensive review of linearity and the meaning of superposition. 

Models are also classified into whether they are deterministic or stochastic. Stochastic (Greek for guess ) systems involve chance or probability, whereas a deterministic system does not. In a deterministic model no randomness is assumed to be present, an assumption that is clearly not realistic. Stochastic models assume random variability and take into account that variability. There is no such thing as a deterministic model—all 

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A multitude of different variants of this model has been investigated using Monte Carlo simulations (see, for example [M])- The studies aim at correlating the phase behaviour with the molecular architecture and revealing the local structure of the aggregates. This type of model has also proven useful for studying rather complex structures (e.g., vesicles or pores in bilayers). 

In this section, we intend to show that for a certain type of models the above imposed restrictions become the ordinary well-known Bohr-Sommerfeld quantization conditions [82]. For this purpose, we consider the following non-adiabatic coupling matrix x ... 

It is important to emphasize that this analysis, although it is supposed to hold for a general three-state case, contradicts the analysis we perfonned of the three-state model in Section V.A.2. The reason is that the general (physieal) case applies to an (arbitrary) aggregation of conical intersections whereas the previous case applies to a special (probably unphysical) situation. The discussion on this subject is extended in Section X. In what follows, the cases for an aggregation of conical intersections will be tenned the breakable situations (the reason for choosing this name will be given later) in contrast to the type of models that were discussed in Sections V.A.2 and V.A.3 and that are termed as the unbreakable situation. 

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

It is worth remarking that the development of both types of model, like so many other aspects of the kinetic theory of gases, relies heavily on ideas of Clerk Maxwell. Some of these were rediscovered by later workers, but there is remarkably little that was not anticipated, at least in outline, by Maxwell. 

More recently, the Wakao-Smith type of model has been generalized by Cunningham and Geankoplis [50] to admit two different sizes of macropore. 

The modeling of solids as a continuum with a given shear strength, and the like is often used for predicting mechanical properties. These are modeled using hnite element or hnite difference techniques. This type of modeling is usually employed by engineers for structural analysis. It will not be discussed further here. 

The relationship of this type of model to a tme differential analysis has been discussed for the case of linear equiHbrium and first-order kinetics (74,75). A minor extension of this work leads to the foUowing relations for a bed section in which dow rates of soHd and Hquid are constant. For the number of theoretical trays,... 

As can be seen from Figure 4, LBVs for these components are not constant across the ranges of composition. An iateraction model has been proposed (60) which assumes that the lack of linearity results from the iateraction of pairs of components. An approach which focuses on the difference between the weighted linear average of the components and the actual octane number of the blend (bonus or debit) has also been developed (61). The iadependent variables ia this type of model are statistical functions (averages, variances, etc) of blend properties such as octane, olefins, aromatics, and sulfur. The general statistical problem has been analyzed (62) and the two approaches have been shown to be theoretically similar though computationally different. 

Fig. 2. Inputs, types of models, and outputs used in air quahty modeling studies.

Many different types of models are used as the foundation for statistical analysis. These models are also referred to as populations. 

A number of statistics have been suggested (39, 40) as measures of model performance. Different types of models and the use of models for different purposes may require different statistics to measure performance. 

Chemical reactions are performed in reactor systems that are derived from one of the following basic types of model reactors ... 

The predicted strain variation is shown in Fig. 2.43(b). The constant strain rates predicted in this diagram are a result of the Maxwell model used in this example to illustrate the use of the superposition principle. Of course superposition is not restricted to this simple model. It can be applied to any type of model or directly to the creep curves. The method also lends itself to a graphical solution as follows. If a stress is applied at zero time, then the creep curve will be the time dependent strain response predicted by equation (2.54). When a second stress, 0 2 is added then the new creep curve will be obtained by adding the creep due to 02 to the anticipated creep if stress a had remained... 

Many different types of models may be produced to aid product development, test theories, experiment with solutions, etc. However, when the design is complete, prototype models representative in all their physical and functional characteristics to the production models may need to be produced. When building prototypes, the same materials, locations, subcontractors, tooling, and processes should be used as will be used in actual production so as to minimize the variation (see also clause 4.4.8.3). 

This type of model derives from the RTD eoneept of maeromixing as deseribed above, whieh is applied on a mieroseopie level using idealized zones and... 

Electron densities, bond densities, and spin densities, as well as particular molecular orbitals may be displayed as graphical surfaces. In addition, the value of the electrostatic potential or the absolute value of a particular molecular orbital may be mapped onto an electron density surface. These maps provide information about the environment around the accessible surface of a molecule. Electrostatic potential maps show overall charge distribution, while orbital maps reveal likely sites for electrophilic and/or nucleophilic attack. Surface displays may be combined with any type of model display. 

To investigate the connection between Ap/p and pn and the various relativistic effects two types of model calculations were performed i) the speed of light c and that way all relativistic effects have been varied artificially, ii) the strength of the spin-orbit coupling has been manipulated separately [8]. 

No written formula is quite as effective as a molecular model to help us visualize molecular shape. Since chemists find that the shape of a molecule strongly influences its chemical behavior, pictures and models of molecules are important aids. A variety of types of models are... 

Takemoto et al.21) synthesized similar types of models, i. e. poly(/3-methacryl-oyloxyethyladenine) 20 (PMAOA), poly(0-methacryloyloxyethyluracil) (PMAOU), and poly(j3-methacryloyloxyethylthymine) (PMAOT), by free-radical polymerizations of /3-methacryloxyethyl compounds of the corresponding base. 

Three types of model study have been performed. The first approach has been to decompose a mixture of two initiators (/.< . one to generate radical A, the other to generate radical B). With this method experimental difficulties arise because the two types of radical may not be generated at the same rate and because homotermination products from cage recombination complicate analysis. 

These types of models were significant improvements of a previously introduced multilayered model17 based on the same principles, as the unfolding models, and taking into consideration the influence of the mesophase layer to the physical behaviour of the composite2A). 

It should be emphasized that whereas the theoretical modelling of An3+ spectra in the condensed phase has reached a high degree of sophistication, the type of modelling of electronic structure of the (IV) and higher-valent actinides discussed here is restricted to very basic interactions and is in an initial state of development. The use of independent experimental methods for establishing the symmetry character of observed transitions is essential to further theoretical interpretation just as it was in the trivalent ion case. 

FIGURE 6.6. The type of model compounds that were used to estimate the electrostatic stabilization in lysozyme (the only hydrogen atom shown, is the one bonded to the oxygen). Such molecules do not show a large rate acceleration due to electrostatic stabilization of the positively charged carbonium transition state. However, the reaction occurs in solution and not in a protein-active site, and the dielectric effect is expected to be very different in the two cases. 

All the models discussed above are based on a deterministic point of view. However, there is another type of model (i.e., a nondeterministic model) that includes the concept of nonequilibrium fluctuation. In the following section, we discuss such a model, i.e., the electrocapillarity breakdown model. 

This chapter focuses on types of models used to describe the functioning of biogeochemical cycles, i.e., reservoir or box models. Certain fundamental concepts are introduced and some examples are given of applications to biogeochemical cycles. Further examples can be found in the chapters devoted to the various cycles. The chapter also contains a brief discussion of the nature and mathematical description of exchange and transport processes that occur in the oceans and in the atmosphere. This chapter assumes familiarity with the definitions and basic concepts listed in Section 1.5 of the introduction such as reservoir, flux, cycle, etc. 

The advent of fast computers and the availability of detailed data on the occurrence of certain chemical species have made it possible to construct meaningful cycle models with a much smaller and faster spatial and temporal resolution. These spatial and time scales correspond to those in weather forecast models, i.e. down to 100 km and 1 h. Transport processes (e.g., for CO2 and sulfur compounds) in the oceans and atmosphere can be explicitly described in such models. These are often referred to as "tracer transport models." This type of model will also be discussed briefly in this chapter. 

Simple three-box models with the atmosphere assumed to be one well-mixed reservoir and the oceans described by a surface layer and a deep-sea reservoir have been used extensively. Keeling (1973) has discussed this type of model in detail. The two-box ocean model is refined by including a second surface box, simulating an "outcropping" (deep-water forming) polar sea (e.g.. Keeling and Bolin, 1967, 1968), and to include a better resolution of the main thermo-cline (e.g., Bjorkstrom, 1979). The terrestrial biota are included in a simple manner (e.g., Bolin and Eriksson, 1959) in some studies Fig. 11-18 shows a model used by Machta (1972) where the role of biota is simulated by one reservoir connected to the atmosphere with a time lag of 20 years. 

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