Physical processes, mathematical models

Development of Process (Mathematical) Models Constraints in optimization problems arise from physical bounds on the variables, empirical relations, physical laws, and so on. The mathematical relations describing the process also comprise constraints. Two general categories of models exist ... 

The Process Data Warehouse, as described in this subsection, has been implemented using the deductive object-base ConceptBase [204]. The lifecycle model CLiP was realized in ConceptBase (see Subsects. 1.3.4, 2.1.3, and 2.2.3), while the tool environment uses the interfaces offered by the CAPE-OPEN initiative to integrate physical properties, mathematical models, and simulators. Subsequently, it will be shown how this setup is used in an example scenario. In this scenario, the services offered by the PDW Query Assistant are used by the PRIME process integrated environment (cf. Subsect. 3.1.3) for extended situation analysis. 

The ECM is relatively a new powder production technique and as such very few physical or mathematical models are available to simulate this process. Different scenarios of particle formation are illustrated in Fig. 40.2 and will be elaborated... 

Explosion hydrodynamics is related to the investigation of a wide range of unsteady processes developing under pulse loading of liquids, such as wave processes, bubble cavitation and high-rate cumulative jet flows. These phenomena are possible to be analysed in detail only by the combined investigations, both experimental and theoretical, with developing appropriate physical and mathematical models. This approach will be demonstrated below on the examples of some principal results and will refer mainly to the so-called surface effects and the problem of wave field parameters control. 

Production processes using genetically modified cell systems are not so thoroughly understood that they could be modelled comprehensively with physically based mathematical models only. On the other hand there is much heuristical knowledge available fi-om many productions run in industrial practice. This can be described by correlations or at least by simple rules-of-thumb. The main idea behind the model to be described is to exploit all the knowledge and information which is available, may it be provided by means of mathematical models, heuristic knowledge or even by simple... 

There are a number of modeling approaches that can be used with process control systems. Whereas mathematical models based on the chemistry and physics of the system represent one alternative, the typical process control model utilizes an empirical input/output relationship, the so-called black-box model. These models are found by experimental tests of the process. Mathematical models of the control system may include not only the process but also the controller, the final control element, and other electronic components such as measurement devices and transducers. Once these component models have been determined, one can proceed to analyze the overall system dynamics, the effect of different controllers in the operating process configuration, and the stability of the system, as well as obtain other usefid information. 

Empirical models are based on input utput relations that do not take into consideration the description of the processes taking place within the system. This is why it is sometimes referred to as the black-box approach, because the system is considered as a nontransparent box with attention focused only on input and output variables, without much concern about what is going on inside the box. Many procedures exist for determining the appropriate experimental design and the way in which data are converted to equations. It is usually necessary to make some a priori assumptions about the physical structure of the system as well as the mathematical structure of the equations. Empirical models are highly unreliable and can hardly be extrapolated outside the region where the experiments were conducted. However, they are relatively easy and some empirical models are sometimes used as parts of an overall physical (or mathematical) model. Thus, empirical models are unreliable as overall process models, but sometimes they are used as parts of overall mathematical models. 

Chapter 1 provides a general overview and introduction of the principles and techniques of physical and mathematical modeling discussed in the book. It provides the rationale for modeling two-phase flow in gas-agitated reactors of materials processes. Chapter 2 presents the turbulence structure of two-phase jets and the impact on the mixing and chemical reaction rates in materials reactors agitated by... 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

Those based on strictly empirical descriptions Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinefics) are frequently employed in optimization apphcations. These models are conceptually attractive because a gener model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input-output data without any physiochemical analysis of the process. For... 

The aims of the given work ar e investigation of interaction processes of active forius of oxygen with phospholipids under action of natural antioxidant QIO development of chemical model on the basis of physical and chemical behaviour of QIO and corresponding mathematical model. 

In its simplest form, a model requires two types of data inputs information on the source or sources including pollutant emission rate, and meteorological data such as wind velocity and turbulence. The model then simulates mathematically the pollutant s transport and dispersion, and perhaps its chemical and physical transformations and removal processes. The model output is air pollutant concentration for a particular time period, usually at specific receptor locations. 

Once the designer has developed confidence in the analysis techniques pertaining to the various parts of a design concept (whether derived from mathematical models or from physical models), the designer can begin the process of synthesis. Synthesis is basically the combining of the analyses (and any other pertinent information) to... 

They point out that at the heart of technical simulation there must be unreality otherwise, there would not be need for simulation. The essence of the subject linder study may be represented by a model of it that serves a certain purpose, e.g., the use of a wind tunnel to simulate conditions to which an aircraft may be subjected. One uses the Monte Carlo method to study an artificial stochastic model of a physical or mathematical process, e.g., evaluating a definite integral by probability methods (using random numbers) using the graph of the function as an aid. 

Gas-liquid-particle operations are of a comparatively complicated physical nature Three phases are present, the flow patterns are extremely complex, and the number of elementary process steps may be quite large. Exact mathematical models of the fluid flow and the mass and heat transport in these operations probably cannot be developed at the present time. Descriptions of these systems will be based upon simplified concepts. 

Before the advent of modem computer-aided mathematics, most mathematical models of real chemical processes were so idealized that they had severely limited utility— being reduced to one dimerrsion and a few variables, or Unearized, or limited to simplified variability of parameters. The increased availability of supercomputers along with progress in computational mathematics and numerical functional analysis is revolutionizing the way in which chemical engineers approach the theory and engineering of chemical processes. The means are at hand to model process physics and chenustry from the... 

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. 

Chemical and Physical Information Processed in a Mathematical Model... 

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