Gases mathematical approach

If the linear gas velocity approaches the speed of sound, the simple mathematical model used in equation (4) breaks down. The acceleration term must be taken into account, and the steady-state equation of motion for a straight pipeline with constant diameter may be written (8) ... 

Maxwell spent the next five years at King s College in London. He successfully applied statistical methods to describe the movements of the tiny invisible particles of a gas, an approach adopted a century earlier by the Swiss mathematician Daniel Bernonlli, bnt with less sophisticated mathematics. The Anstrian physicist Lndwig Boltzmann also studied the problem of gas behavior at the same time as Maxwell, and the names of both men are usually associated with the kinetic theory of gases. 

Modem mass spectrometry, photoionization techniques, and ion cyclotron resonance spectroscopy revolutionized gas-phase basidty determination. These techniques have permitted extremely accurate absolute gas-phase basidties to be obtained for a few molecules via thermodynamic cycles using electron affinity and ionization energy data. - Absolute gas-phase basicities for even a few molecules provide valuable references for determination of GB values for bases for which direct GB determination is problematic. The mathematical approach is conceptually straightforward. Consider the general GB equations for Bj and B2 ... 

A Mathematical Approach to Control the Water Content of Sour Gas... 

The Markov method is named after the Russian mathematician Andrei A. Markov (1856-1922) and is a highly mathematical approach that is frequently used for performing various types of safety and reliability analyses in engineering systems. This chapter presents a number of methods and approaches extracted from the published literature, considered useful to perform safety and reliability analyses in the oil and gas industry. 

On the other hand the Thomas-Fermi method, which treats the electrons around the nucleus as a perfectly homogeneous electron gas, yields a mathematical solution that is universal, meaning that it can be solved once and for all. This feature already represents an improvement over the method which seeks to solve Schrodinger equation for every atom separately. This was one of the features that made people go back to the Thomas-Fermi approach in the hope of... 

As our first approach to the model, we considered the controlling step to be the mass transfer from gas to liquid, the mass transfer from liquid to catalyst, or the catalytic surface reaction step. The other steps were eliminated since convective transport with small catalyst particles and high local mixing should offer virtually no resistance to the overall reaction scheme. Mathematical models were constructed for each of these three steps. 

The principal difficulty with these equations arises from the nonlinear term cb. Because of the exponential dependence of cb on temperature, these equations can be solved only by numerical methods. Nachbar has circumvented this difficulty by assuming very fast gas-phase reactions, and has thus obtained preliminary solutions to the mathematical model. He has also examined the implications of the two-temperature approach. Upon careful examination of the equations, he has shown that the model predicts that the slabs having the slowest regression rate will protrude above the material having the faster decomposition rate. The resulting surface then becomes one of alternate hills and valleys. The depth of each valley is then determined by the rate of the fast pyrolysis reaction relative to the slower reaction. 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

Derivation of the Langmuir Equation— Adsorption of a Single Species. The kinetic approach to deriving a mathematical expression for the Langmuir isotherm assumes that the rate of adsorption on the surface is proportional to the product of the partial pressure of the adsorbate in the gas phase and the fraction of the surface that is bare. (Adsorption may occur only when a gas phase molecule strikes an uncovered site.) If the fraction of the surface covered by an adsorbed gas A is denoted by 0Ay the fraction that is bare will be 1 — 0A if no other species are adsorbed. If the partial pressure of A in the gas phase is PA, the rate of adsorption is given by... 

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