Mathematical models optimization studies

Empirical energy functions can fulfill the demands required by computational studies of biochemical and biophysical systems. The mathematical equations in empirical energy functions include relatively simple terms to describe the physical interactions that dictate the structure and dynamic properties of biological molecules. In addition, empirical force fields use atomistic models, in which atoms are the smallest particles in the system rather than the electrons and nuclei used in quantum mechanics. These two simplifications allow for the computational speed required to perform the required number of energy calculations on biomolecules in their environments to be attained, and, more important, via the use of properly optimized parameters in the mathematical models the required chemical accuracy can be achieved. The use of empirical energy functions was initially applied to small organic molecules, where it was referred to as molecular mechanics [4], and more recently to biological systems [2,3]. 

Catalytic crackings operations have been simulated by mathematical models, with the aid of computers. The computer programs are the end result of a very extensive research effort in pilot and bench scale units. Many sets of calculations are carried out to optimize design of new units, operation of existing plants, choice of feedstocks, and other variables subject to control. A background knowledge of the correlations used in the "black box" helps to make such studies more effective. 

In this paper we present a meaningful analysis of the operation of a batch polymerization reactor in its final stages (i.e. high conversion levels) where MWD broadening is relatively unimportant. The ultimate objective is to minimize the residual monomer concentration as fast as possible, using the time-optimal problem formulation. Isothermal as well as nonisothermal policies are derived based on a mathematical model that also takes depropagation into account. The effect of initiator concentration, initiator half-life and activation energy on optimum temperature and time is studied. 

Reaction mechanism of azo dyes removal Limited study has focused on the reaction mechanism of azo dyes removal. The research of mechanistic and mathematical models to optimize the integrated process and to characterize the interaction between the reactant and azo dyes should be carried out in the future. 

Another study was performed on a catalytic hydrogenation of 1,3,5-trimethyl-benzene to 1,3,4-trimethylcyclohexane, which is a typical first-order reversible reaction [168]. By optimizing various operating conditions it was possible to achieve a product purity of 96% and a reactant conversion of 0.83 compared to a thermodynamic equilibrium conversion of only 0.4. The results were successfully described with a mathematical model derived by the same authors [169]. Comparison to a real countercurrent moving bed chromatographic reactor yielded very similar results for both types [170]. 

A mathematical model was found for each studied response. From the models, the contoured curves and the response surfaces were plotted, and the optimal points were sought and confirmed. 

The coefficient of correlation values (R2) were quite low for the estimated friability mathematical models. It seems that this response was independent of the studied parameters and their levels and, in our case, friability was not an important response to be optimized. This response could only be an evaluated property. Lindberg and Holmsquist [6] had also obtained low R2 values for this response (R2=0.57 and 0.68). 

Optimization studies then, both because of their inherent difficulty and the fact that they are recurrent, will involve repeated use of the mathematical model. It is for this reason that its adaptation to a computer is important. In spite of the great reduction in calculating cost due to the automatic computer, however, the total cost of a study will usually not be so small that efficient design of the investigation is not important. The problem is quite similar to that encountered in the statistical subject of design of experiments, and it seems apparent that the two fields should have several common aspects. 

Step 3 Algorithmic Development The mathematical model (7.1), which describes all process alternatives embedded in the superstructure, is studied with respect to the theoretical properties that the objective function and constraints may satisfy. Subsequently, efficient algorithms are utilized for its solution which extracts from the superstructure the optimal process structure(s). 

Domain of factors is marked O . The figure clearly shows that intervals of factor variations are part of the domain of factors when the optimization problem is being solved. This is necessary in order to realize movement towards optimum in this domain. The experiment domain is in the same figure marked by letter E". In studies with an objective of approximation or interpolation, that is mathematical modeling, the factor-variation intervals cover the whole of the domain of factors. For a two-factor experiment the upper level of factors X and X2 corresponds to values Xlmax,-and X2max, while the lower levels have values Xlmin, X2min. Domain of factors O is in that case called intcrpolational, and E the domain of extreme experiment. 

The last twenty years of the last millennium are characterized by complex automatization of industrial plants. Complex automatization of industrial plants means a switch to factories, automatons, robots and self adaptive optimization systems. The mentioned processes can be intensified by introducing mathematical methods into all physical and chemical processes. By being acquainted with the mathematical model of a process it is possible to control it, maintain it at an optimal level, provide maximal yield of the product, and obtain the product at a minimal cost. Statistical methods in mathematical modeling of a process should not be opposed to traditional theoretical methods of complete theoretical studies of a phenomenon. The higher the theoretical level of knowledge the more efficient is the application of statistical methods like design of experiment (DOE). 

If the behaviour of complex chemical (in our case catalytic) reactions is known, it will be clear in what way these reactions can be carried out under optimal conditions. The results of studying kinetic models must be used as a basis for the mathematical modelling of chemical reactors to perform processes with probable non trivial kinetic behaviour. It is real systems that can appear to show such behaviour first far from equilibrium, second nonlinear, and third multi dimensional. One can hardly believe that their associated difficulties will be overcome completely, but it is necessary to approach an effective theory accounting for several important problems and first of all provide fundamentals to interpret the dependence between the type of observed kinetic relationships and the mechanism structure. 

It was estimated that, if all the Surfmers contributed to stabilization, the surface coverage would be close to 20% at the end of the process. When Surfmer burial is considered, the minimum surface coverage is in the region of 14.7-15.0 % [35]. The authors have also studied the influence of the addition procedure on the evolution of the Surfmer conversion and concluded that, despite the low reactivity due to the presence of the alkenyl double bond, the incorporation could be increased to 72% from the original 58% obtained with a constant feeding rate. A mathematical model able to describe Surfmer polymerization was used in the optimization process [36]. 

High conversion with an optimal reaction rate [7, 11, 75, 95], increase of the turnover numbers, i.e., the moles of substrate converted per mole of enzyme deactivated [3, 75, 95], and high stereospecificity of the compound of interest are targets of particular interest in the operation of these batch reactors [10,11,48, 77]. The achievement of these goals requires the study of different variables type and concentration of peroxide, substrates and cofactors, enzyme activity and purity, composition of the reaction medium, pH, temperature, or agitation. Such optimization requires a deep knowledge of the system and a mathematical model that represents it satisfactorily. The kinetic model obtained in batch experiments is the... 

Elaboration of toxicokinetic data of chemical warfare agents is essential for designing effective antidotes, improving first aid, and optimizing therapeutic regimen and medical care. It has to be eonsidered that data obtained from in vitro or in vivo animal studies need earefiil extrapolation to humans whieh, at least, requires sophisticated mathematical models to eonsider basie interspecies differenees (Langenberg et ah, 1997 Levy et ah, 2007 Sweeney et al, 2006 Worek et al, 2007). 

In this study, we focused our attention on investigating the adsorption dynamics in column packed with activated carbon fiber. By optimizing the breakthrough curve data with a mathematical model, effective overall mass transfer coefficient was obtained. And it can be given reasonable predictions compared with the experimental data of breakthrough curve. 

Thus, expectedly no rigorous mathematical models are available that can accurately describe the detailed flow behavior of the fluid streams in a membrane separation process or membrane reactor process. Recent advances in computational fluid dynamics (CFD), however, have made this type of problem amenable to detailed simulation studies which will assist in efficient design of optimal membrane filtration equipment and membrane reactors. 

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