Mathematical Method

The microstructural properties that we address are chain length, number of monomer units of one kind (copolymer), number of branch points, number of unsaturated bonds, and number of reactive monomer units (end groups in polycondensation). Problems may require solution of one or more of these properties simultaneously. Here, we will denote this as the dimensionality of the problem at hand. For instance, growth in addition polymerization can be described by a simple 1D (chain length) reaction equation and population balance. 

Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 2005 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN 3-527-31014-2 

In contrast, growth in polycondensation of a trifunctional monomer A with a bifunctional monomer B requires a 3D description. The 3D distribution Rn,i,k, where subscripts denote chain length, number of A end groups, number of B end groups, respectively, obeys  

Note that this describes a reaction between single end groups of two different molecules end group combinations within one (longer) molecule require a similar approach. 

The discrete Galerkin h-p finite element method (FEM) is a powerful numerical method to solve chain length distributions for a wide set of polymerization prob- 

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Mathematical framework of cone beam 3D reconlruction via the first derivative of the Radon transform, in Herman, Louis, Natterer (eds.). Mathematical Methods in Tomography, Springer, 1991... 

G. Arfken, Mathematical methods for Physicists, 3rd ed., Academic ftess, San Diego, 1985. 

V. I. Arnold, Mathematical methods of classical dynamics. Chap. 7, Springer, New York, 1978,... 

Arnold, V. I. Mathematical Methods of Classical Mechanics. 2nd edition. Springer Verlag, Berlin, Heidelberg, New York, Tokyo (1989)... 

Chemometrics is the discipline which deals wdth the application of statistical and, in a more general sense, of mathematical methods to chemical data. Chemometric methods are used for the extraction of chemical information from chemical data. 

Easy and intuitive data analysis The data analysis process is easy and intuitive, because the pattern recognition only requires the knowledge and intuition of the scientists. DifEcult statistical and mathematical methods are not necessary. 

Fig. l-ll Single and double integrals. (Figure adapted in part from Boas M L, 1983, Mathematical Methods in the Physical Sciences. 2nd Edition. New York, Wiley.)... 

Stephenson G 1973. Mathematical Methods for Science Students. London, Longman. 

One of the most important methods of modem computation is solution by iteration. The method has been known for a very long time but has come into widespread use only with the modem computer. Normally, one uses iterative methods when ordinary analytical mathematical methods fail or are too time-consuming to be... 

Starzak, M. E., 1989. Mathematical Methods in Chemistry and Physics. Plenum, New York. 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

The term theoretical chemistry may be defined as the mathematical description of chemistry. The term computational chemistry is generally used when a mathematical method is sufficiently well developed that it can be automated for implementation on a computer. Note that the words exact and perfect do not appear in these definitions. Very few aspects of chemistry can be computed exactly, but almost every aspect of chemistry has been described in a qualitative or approximately quantitative computational scheme. The biggest mistake a computational chemist can make is to assume that any computed number is exact. However, just as not all spectra are perfectly resolved, often a qualitative or approximate computation can give useful insight into chemistry if the researcher understands what it does and does not predict. 

The most convenient mathematical method of describing pervaporation is to divide the overall separation processes into two steps, as shown in Figure 40. The first is evaporation of the feed Hquid to form a (hypothetical) saturated vapor phase on the feed side of the membrane. The second is permeation of this vapor through the membrane to the low pressure permeate side of the membrane. Although no evaporation actually takes place on the feed side of the membrane during pervaporation, this approach is mathematically simple and is thermodynamically completely equivalent to the physical process. The evaporation step from the feed hquid to the saturated vapor phase produces a separation, which can be defined (eq. 13) as the ratio of... 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

Considerable work has been done on mathematic models of the extmsion process, with particular emphasis on screw design. Good results are claimed for extmsion of styrene-based resins using these mathematical methods (229,232). With the advent of low cost computers, closed-loop control of... 

Any of the mathematical methods discussed previously to calculate P, Z9, and A in a single experiment ate appHcable to flavor permeation. Two... 

The development of combustion theory has led to the appearance of several specialized asymptotic concepts and mathematical methods. An extremely strong temperature dependence for the reaction rate is typical of the theory. This makes direct numerical solution of the equations difficult but at the same time accurate. The basic concept of combustion theory, the idea of a flame moving at a constant velocity independent of the ignition conditions and determined solely by the properties and state of the fuel mixture, is the product of the asymptotic approach (18,19). Theoretical understanding of turbulent combustion involves combining the theory of turbulence and the kinetics of chemical reactions (19—23). 

There are several mathematical methods for producing new values of the variables in this iterative optimization process. The relation between a simulation and an optimization is depicted in Eigure 6. Mathematical methods that provide continual improvement of the objective function in the iterative... 

V. E. Jenson and G. V. Jeffreys, Mathematical Methods in Chemical Engineering, Academic Press, Inc., Orlando, Ela., 1990. 

Construction of Alignment Charts. Of the ways to constmct alignment charts, the bmte force method, which requires some idea of the geometry for the chart, is the easiest method to use. The mathematical method, which uses parametric equations of scale to determine the placement and scale of each axis, is the most accurate, but the most difficult to apply. 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

Amundson, N. R. Mathematical Methods in Chemical Engineeiing, Prentice Hall, Englewood Cliffs, NJ (1966). 

Bender, C. M., and Orszag, S. A., Advanced Mathematical Methods foi Scientists and Engineeis, McGraw-HiU (1978). 

A version of this problem is solved by Jenson and Jeffreys Mathematical Methods in Chemical Kngineeting, Academic Press, 1977). 

References A variety of mathematical methods are proposed to cope with hnear (e.g., material balances based on flows) and nonhnear (e.g., energy balances and equilibrium relations) constraints. Methods have been developed to cope with unknown measurement uncertainties and missing measurements. The reference list provides ample insight into these methods. See, in particular, the works by Mah, Crowe, and Madron. However, the methods all require more information than is tvpicaUy known in a plant setting. Therefore, even when automated methods are available, plant-performance analysts are well advised to perform initial adjustments by hand. 

The paper describes the different chemical sensors and mathematical methods applied and presents the review of electronic tongue application for quantitative analysis (heavy metals and other impurities in river water, uranium in former mines, metal impurities in exhaust gases, ets) and for classification and taste determination of some beverages (coffee, bear, juice, wines), vegetable oil, milk, etc. [1]. 

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