Preparative mathematical models

The Supplement B (reference) contains a description of the process to render an automatic construction of mathematical models with the application of electronic computer. The research work of the Institute of the applied mathematics of The Academy of Sciences ( Ukraine) was assumed as a basis for the Supplement. The prepared mathematical model provides the possibility to spare strength and to save money, usually spent for the development of the mathematical models of each separate enterprise. The model provides the possibility to execute the works standard forms and records for the non-destructive inspection in complete correspondence with the requirements of the Standard. 

Dijkman, H. K. M. and Duvekot, C., Mathematical model of the preparation of polyethylene terephthalate, in Proceedings of the 4th Intema-tionaV6th European Symposium on Chemical Reaction Engineering, Vol. 1, Heidelberg, 1976, pp. 344-352. 

We generally distinguish between two methods when the determination of the composition of the equilibrium phases is taking place. In the first method, known amounts of the pure substances are introduced into the cell, so that the overall composition of the mixture contained in the cell is known. The compositions of the co-existing equilibrium phases may be recalculated by an iterative procedure from the predetermined overall composition, and equilibrium temperature and pressure data It is necessary to know the pressure volume temperature (PVT) behaviour, for all the phases present at the experimental conditions, as a function of the composition in the form of a mathematical model (EOS) with a sufficient accuracy. This is very difficult to achieve when dealing with systems at high pressures. Here, the need arises for additional experimentally determined information. One possibility involves the determination of the bubble- or dew point, either optically or by studying the pressure volume relationships of the system. The main problem associated with this method is the preparation of the mixture of known composition in the cell. 

The Sharpless epoxidation is sensitive to preexisting chirality in selected substrate positions, so epoxidation in the absence or presence of molecular sieves allows easy kinetic resolution of open-chain, flexible allylic alcohols (Scheme 26) (52, 61). The relative rates, kf/ks, range from 16 to 700. The lower side-chain units of prostaglandins can be prepared in high ee and in reasonable yields (62). A doubly allylic alcohol with a meso structure can be converted to highly enantiomerically pure monoepoxy alcohol by using double asymmetric induction in the kinetic resolution (Scheme 26) (63). A mathematical model has been proposed to estimate the degree of the selectivity enhancement. 

Mathematical models that incorporate both flow and evaporative-loss mechanisms have been prepared (158). Non-Newtonian features have been introduced recently to refine the modeling efforts and have resulted in models with quantitative predictive abilities (159). Hence, data comparison between films spun on different spinners, with nominally identical final rotation speeds, may reveal differences in film uniformity. 

An important step in the production process is the preparation of a standard specimen. This specimen is used to qualify principle production parameters such as the long-term stability of the reactive mixture, polymerization cycle, and the performance characteristics of the material obtained. Simultaneous determination of the reaction parameters allows us to use mathematical modelling to optimize the reactive processing regime. 

Mathematical designing of an experiment has been applied to mathematical modeling of solidification and hardness of concrete as a function of three basic factors Xi-cement consumption, kg/m3 X2-percentage of sand in filler mixture, % and X3-water consumption, 1/min. This parameter was measured as response yi-concrete solidification, s. The cement of the same brand and the sand from the same supplier have been used in all design points. A mixture 101 in volume was mixed manually for 3 min and a 7 1 volume for 2.5 min. Samples of 10 x 10 x 10 cm were prepared on a vibration table with amplitude of 0.45-0.50 mm, frequency of 2800 min 1 and under pressure of 80-100 kp/cm2. Concrete solidification was measured 10-15 min after formation of samples by GOST 10181-62. Basic experiment was done by FUFE 23, as shown in Table 2.161. 

A mathematical model has also been proposed for evaluating cellulase preparations. Sattler et al.209 describe a relationship between hydrolysis extent, reaction time, and enzyme concentration. This procedure permits the effectiveness of different enzymes and of different pretreatment methods to be ranked. This method examines cellulose hydrolysis data collected from hyperbolic functions of substrate concentration versus cellulase enzyme concentration at various timed incubations. The model is based on a double reciprocal plot of the relationship... 

The creation of the Institute of Catalysis was Boreskov s life work. He was the director of the Institute from the time of its foundation in 1958. He formulated the main research problems of the Institute the development of scientific bases for foreseeing catalytic action and scientific bases of catalyst preparation and mathematical modeling, combined with solutions of applied problems essential for modem industry. In a fairly short period of time the Institute has gained world-wide recognition. In 1980 it received an International Gold Mercury prize. 

In the frame work of mathematical model by selection of parameters one may obtain solutions interpretive as transition from pathological into healthy state of organism. To have an opportunity for realization of such transitions in modeled system one should know how one or another preparation influence on real parameters. Knowledge of these characteristics allows speaking about statement of problem of optimal control of medical treatment that is the target of application of mathematical models in this region of investigations. 

A solution that is not quite consistent with the observed nature of the problem indicates that the mathematical model used is too crude. In that case, a more realistic model should be prepared by eliminating one or more of the questionable assumptions. This will result in a more complex problem that, of course, is more difficult to solve. Thus any solution to a problem should be interpreted within the context of its formulation. 

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