Mathematical and Numerical Techniques

Correlation methods discussed include basic mathematical and numerical techniques, and approaches based on reference substances, empirical equations, nomographs, group contributions, linear solvation energy relationships, molecular connectivity indexes, and graph theory. Chemical data correlation foundations in classical, molecular, and statistical thermodynamics are introduced. 

Air quality models use mathematical and numerical techniques to simulate the physical and chemical processes that affect air pollutants, namely PM, as they disperse and react in the atmosphere. Based on meteorological and emission data inputs, these models are designed to characterise primary PM emitted directly into the atmosphere and, in some cases, secondary PM formed as a result of complex chemical reactions within the atmosphere. 

The good results obtained for chains with small unit cells make it probable that with further improved mathematical and numerical techniques and applying even larger computers, in the next few years the calculations on the electron structure and properties of biopolymers will reach the same level of sophistication as those for the above mentioned simple chains,... 

The mathematical details outlined here include both analytic and numerical techniques usebil in obtaining solutions to problems. 

Unfortunately, the exponential temperature term exp(- E/RT) is rather troublesome to handle mathematically, both by analytical methods and numerical techniques. In reactor design this means that calculations for reactors which are not operated isothermally tend to become complicated. In a few cases, useful results can be obtained by abandoning the exponential term altogether and substituting a linear variation of reaction rate with temperature, but this approach is quite inadequate unless the temperature range is very small. 

The mathematics involved are somewhat cumbersome, and numerical techniques are needed. The conclusions reached are the following ones ... 

Perre and Turner, The Use of Macroscopic Equations to Simulate Heat and Mass Transfer in Porous Media, in Turner and Mujumdar (eds.), Mathematical Modeling and Numerical Techniques in Drying Technology, Marcel Dekker, New York, 1996, pp. 83-156. 

Turner, I., and Mujumdar, A.S., eds. 1997. Mathematical modeling and numerical techniques in drying technology. New York Marcel Dekker. 

Configurations that Emphasize the Role of Internal Pressure. In Mathematical Modeling and Numerical Techniques in Drying Technology Turner, I., Mujumdar, A.S., Eds. Marcel Dekker New York, 1997 83-156 Ch. 2. 

To optimize this approach the boundaries of the engineering system are necessary in order to apply the mathematical results and numerical techniques of the optimization theory to engineering problems. For purposes of analysis, they serve to isolate the system from its surroundings, because all interactions between the system and its surroundings are assumed fixed/frozen at selected, representative levels. However, since interactions and comphcations always exist, the act of defining the system boundaries is required in the process of approximating the real system. It also requires defining the quantitative criterion on the basis of which candidates will be ranked to determine the best approach. Included will be the selection system variables that will be used to characterize or identify candidates, and to define a model that will express the manner in which the variables are related. 

Hosseinalipour SM, Mujumdar AS. A model for superheated steam drying of particles in an impinging stream dryer. In Turner I, Mujumdar AS, eds. Mathematical Modeling and Numerical Techniques in Drying Technology. New York Marcel Dekker, 1996, pp 537-574. 

Drops of liquids have held researchers interest for many years. As mathematical curiosities for famous early fluid dynamicists, the pendant drop from a capillary provided an interesting and practical challenge. Young (1805) and Laplace (1806) independently developed the theory of surface tension and drop formation while the first analytical solutions to their theory were completed by Gauss in 1830. Much of the early woik on pendant drops involved numerous methods involving the determination of drop volume or shape with experimental techniques and using the available theory to determine surface tension of the liquid-gas interface. These methods are well detailed in the works by Adamson, Padday, and Reed Hah. ITie studies of the early researches developed into the rich and diversified field of interfacial fluid dynamics. The advancement of theory and numerical techniques has steadily increased the ability of researchers to better understand and control interfacial behaviors. 

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