Mathematical consequence modeling

Specialized risk consultants and even insurance risk offices can now offer a variety of software products or services to conduct mathematical consequence modeling of most hydrocarbon adverse events. The primarily advantage of these tools is that some estimate can be provided on the possible effects of an explosion or fire incident where previously these effects were rough guesses or unavailable Although these models are effective in providing estimate they still should be used with caution and consideration of other physical features that may alter the real incident outcome. 

The models you use to portray failures that lead to accidents, and the models you use to propagate their effects, are attempts to approximate reality. Models of accident sequences (although mathematically rigorous) cannot be demonstrated to be exact because you can never precisely identify all of the factors that contribute to an accident of interest. Likewise, most consequence models are at best correlations derived from limited experimental evidence. Even if the models are validated through field experiments for some specific situations, you can never validate them for all possibilities, and the question of model appropriateness will always exist. 

Consequently, modeling of a two-phase flow system is subject to both the constraints of the hydrodynamic equations and the constraint of minimizing N. Such modeling is a nonlinear optimization problem. Numerical solution on a computer of this mathematical system yields the eight parameters ... 

The major difficulty in predicting the viscosity of these systems is due to the interplay between hydrodynamics, the colloid pair interaction energy and the particle microstructure. Whilst predictions for atomic fluids exist for the contribution of the microstructural properties of the system to the rheology, they obviously will not take account of the role of the solvent medium in colloidal systems. Many of these models depend upon the notion that the applied shear field distorts the local microstructure. The mathematical consequence of this is that they rely on the rate of change of the pair distribution function with distance over longer length scales than is the case for the shear modulus. Thus... 

It appears from the above that microcosm and/or mesocosm tests are limited by the constraints of experimentation, in that usually only a limited number of recovery scenarios can be investigated. Consequently, modeling approaches may provide an alternative tool for investigating likely recovery rates under a range of conditions. Generic models, like the logistic growth mode (for example, see Barnthouse 2004) and life history and individual-based (meta)population models, which also may be spatially explicit, provide mathematical frameworks that offer the opportunity to explore the recovery potential of individual populations. For an overview of these life history and individual-based models, see Bartell et al. (2003) and Pastorok et al. (2003). 

Exposure assessment, however, is a highly complex process having different levels of uncertainties, with qualitative and quantitative consequences. Exposure assessors must consider many different types of sources of exposures, the physical, chemical and biological characteristics of substances that influence their fate and transport in the environment and their uptake, individual mobility and behaviours, and different exposure routes and pathways, among others. These complexities make it important to begin with a clear definition of the conceptual model and a focus on how uncertainty and variability play out as one builds from the conceptual model towards the mathematical/statistical model. 

The experimental orders were 0.66 and 1.02, respectively, in excellent agreement with the electrochemical model. Another mathematical consequence of this model is a marked change in the kinetic orders when one of the products of the reaction is added to the initial mixture [238]. Thus if hexacyanofer-rate(II) (commonly called ferrocyanide and abbreviated Feoc) is present at the start of the reaction, the theory forecasts that the order with respect to Feic and I- will increase to 2 and 3, respectively, while the order with respect to Feoc will be — 2. In the event, a plot of lnecat versus ln[Feoc] was a curve that coincided with a line of slope - 2 at the higher concentrations of ferrocyanide. At a constant initial Feoc concentration of 4 x 10 4 mol dm 3,... 

This is very similar in essence to Milner s approach in 1912. It corresponds to a very simple physical model but results in some very complex and well-nigh intractable mathematics. Consequently this approach was dropped, though as will be shown later (Section 10.18), it has been revived in the modem Monte Carlo computer simulation methods of solving the problems of electrolyte solutions. 

Gases are compressible, expand when heated, and take up room proportional to the amount (moles) of gas present. In order to visualize the mathematical consequences of the physical manipulation of gases, we will use as a model system, a cylinder with a moveable piston sketched in Fig. 5-1, which will illustrate changes in the gas properties as temperature, volume, and mass are varied. Exerting force on the piston handle will compress the gas. If the force on the piston remains constant, and the gas is heated, the piston will rise as the gas expands. If the gas is cooled at constant pressure, it will contract and the piston will fall. If we can add gas to the gas chamber, and both the temperature and pressure are maintained constant, the piston will rise to accommodate the increased amount of gas. 

The model produced nothing beyond Kaluza s original and confirmed the periodicity condition observed by Klein - a mathematical consequence of a closed system. The coordinates they used to describe the closed fivedimensional space are strangely reminiscent of the homogeneous coordinates of four-dimensional projective geometry, but without a point at infinity. 

In the last two centuries, a lot of attempts and discussion have been made on the elucidation and development of the various constitutive models of liquids. Some of the theoretical models that can be mentioned here are Boltzmann, Maxwell (UCM, LCM, COM, 1PM), Voight or Kelvin, Jeffrey, Reiner-Rivelin, Newton, Oldroyd, Giesekus, graded fluids, composite fluids, retarded fluids with a strong backbone and fading memory, and so on. Further and deeper knowledge related to the physical and mathematical consequences of the structural models of liquids and of the elasticity of liquids can be found in Ref. [6]. 

Using mathematical models (consequence modeling), which can be at various levels of detail and sophistication... 

Since about 1975, much research has been devoted to consequence modeling. This has involved the modeling, mathematically as well as physically, of the chemical and physical phenomena associated with major industrial hazards (MIH). Such models are used primarily in risk assessments for safety reports and by safety officers. These models may therefore influence very important decisions, such as the design or authorization of chemical plants. Proper attention should therefore be paid to the quality of these models. 

Dense gas mathematical models are widely employed to simulate the dispersion of flammable and toxic dense gas clouds. Early published examples of applications include models used in the demonstration risk assessments for Canvey Island (Health Safety Executive, 1978, 1981) and the Rijnmond Port Area (Rijmnond Public Authority, 1982), and required in the Department of Transport LNG Federal Safety Standards (Department of Transportation, 1980). While most dense gas models currently in use are based on specialist computer codes, equally good and versatile models are publicly available (c.g., DEGADIS, SLAB). The underlying dispersion mechanisms and necessary validation are more complex than any other area of consequence modeling. 

It is clear that the experimental curves, measured for solid-state reactions under thermoanalytical study, cannot be perfectly tied with the conventionally derived kinetic model functions (cf. previous table lO.I.), thus making impossible the full specification of any real process due to the complexity involved. The resultant description based on the so-called apparent kinetic parameters, deviates from the true portrayal and the associated true kinetic values, which is also a trivial mathematical consequence of the straight application of basic kinetic equation. Therefore, it was found useful to introduce a kind of pervasive des-cription by means of a simple empirical function, h(a), containing the smallest possible number of constant. It provides some flexibility, sufficient to match mathematically the real course of a process as closely as possible. In such case, the kinetic model of a heterogeneous reaction is assumed as a distorted case of a simpler (ideal) instance of homogeneous kinetic prototype f(a) (1-a)" [3,523,524]. It is mathematically treated by the introduction of a multiplying function a(a), i.e., h(a) =f(a) a(a), for which we coined the term [523] accommodation function and which is accountable for a certain defect state (imperfection, nonideality, error in the same sense as was treated the role of interface, e.g., during the new phase formation). 

A mathematical model for template polymerization similar to the biological process was elaborated by Simha and co-workers. The purpose of their paper was to explore mathematical consequences of alternative kinetic routes to the formation of polymer chains on polymer templates. However, since then, nobody has tried to use this theory for the description of template processes. 

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