Equations, mathematical dispersion interaction

The adsorption of smaller polar molecules, such as water and carbon dioxide, was more complex, and Dubinin (1975) concluded that the overall pore filling process could be expressed as a two-term equation, each term having the mathematical form of Equation (11.1). In the low-filling region, the interaction with the cationic sites was considered to be the most important contribution, with the normal dispersion interactions becoming more important at higher loadings. 

Meteorology plays an important role in determining the height to which pollutants rise and disperse. Wind speed, wind shear and turbulent eddy currents influence the interaction between the plume and surroimding atmosphere. Ambient temperatures affect the buoyancy of a plume. However, in order to make equations of a mathematical model solvable, the plume rise is assumed to be only a function of the emission conditions of release, and many other effects are considered insignificant. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

The value of the constants, a, b, c, etc., can either be determined experimentally or theoretically (107). For a colloidal dispersion of rigid spherical particles the value of a is 2.5. Therefore, Equation 13 equals the Einstein equation (Eq. 12) at low-volume fractions. A rigorous theoretical treatment of the interactions between pairs of droplets has established that b = 6.2 for rigid spherical particles. Experiments have shown that Equation 13 can be used up to particle concentrations of about 10% with a = 2.5 and b = 6.2 for colloidal dispersions in the absence of long-range colloidal interactions (107). It is difficult to theoretically determine the value of higher order terms in Equation 13 because of the mathematical complexities involved in describing interactions between three or more particles. 

For the first time, there was a mathematical expression for the impedance dispersion corresponding to the circular arc found experimentally. The equation introduced a new parameter the somewhat enigmatic constant a. He interpreted a as a measure of molecular interactions, with no interactions a = 1 (ideal capacitor). Comparison was made with the impedance of a semiconductor diode junction (selenium barrier layer photocell). 

At r = 300 K, fcr 3 X lO J, which is an order of magnitude less that the dispersion contribution. The actual difference between the two terms (dispersion and electrostatic) will be reduced by mathematical cancellations in the second (dispersion) term in Equation (4.47), but only rarely will the electrostatic contribution constitute the dominant factor in the total interaction. The presence of imaginary frequencies in the second term may cause some problems in terms of physical concepts of the processes involved however, their use is actually a result of mathematical manipulations (i.e., tricks) that disappear as one works through the complete calculation. 

The theory of light-matter interaction on which Cauchy based this equation was later found to be incorrect. In particular, the equation is only valid for regions of normal dispersion in the visible wavelength region. In the infi-ared, the equation becomes inaccurate, and it cannot represent regions of anomalous dispersion. Despite this, its mathematical simplicity makes it useful in some applications. 

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