Dispersion equations

Note that the system (11.55) is valid for small deviations of the interface from Xf when mx f 1 and exp( ix[) 1. Estimations show that the term in the thermal balance equation on the interfece is small in comparison with the term Aca3i and ALa32. Moreover, since Pg.l( g//ilg) 1 and (/Jl/Zilg) it is possible to neglect the second term in the expressions for coefficients an and 01.2,2 and assume that 0 31 = (ml - OgPg.l g), 32 = (ml - ol l)- Then the non-trivial solution of 

The case 6 13 = 0 corresponds to the condition Q = 0 (stationary regime), and we obtain the following dispersion equation for 7 0 

The specific form of the dependence of the complex frequency Q on parameters of the problem found by Eq. (11.58), is presented as follows  

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This is an approximation to the complete dispersion equation [131]. The amplitude of a train of waves originating from an infinitely long linear source decays exponentially with the distance x from the source... 

The index of refraction of synthetic vitreous siUca at 20°C has been fitted to a three-term Sellmeier dispersion equation for wavelengths from 0.2139 to 3.7067 pm (184), where n is the refractive index and X is the corresponding wavelength in micrometers. 

With these two-point boundary conditions the dispersion equation, Eq. (23-50), may be integrated by the shooting method. Numerical solutions for first- and second-order reaciions are plotted in Fig. 23-15. 

The chromatography literature contains a vast amount of dispersion data for all types of chromatography and, in particular, much of the data pertains directly to GC and LC. Unfortunately, almost all the data is unsuitable for validating one particular dispersion equation as opposed to another. There are a number of reasons for this firstly, the necessary supporting data (e.g., diffusivity data for the solutes in the solvents employed as the mobile phase, accurate distribution and/or capacity factor constants (k")) are not available secondly, the accuracy and precision of much of the data are inadequate, largely due to the use of inappropriate apparatus with high extracolumn dispersion. 

Katz et fl/.[l] searched the literature for data that could be used to identify the pertinent dispersion equation for a packed column in liquid chromatography. As a result of the search, no data was found that had been measured with the necessary accuracy and precision and under the sufficiently diverse solute/mobile phase conditions required to meet the second criteria given above. It became obvious that a... 

C. In their first series of experiments, six data sets were obtained for (H) and (u), employing six solvent mixtures, each exhibiting different diffusivities for the two solutes. This served two purposes as not only were there six different data sets with which the dispersion equations could be tested, but the coefficients in those equations supported by the data sets could be subsequently correlated with solute diffusivity. The solvents employed were approximately 5%v/v ethyl acetate in n-pentane, n-hexane, n-heptane, -octane, -nonane and n-decane. The solutes used were benzyl acetate and hexamethylbenzene. The diffusivity of each solute in each solvent mixture was determined in the manner of Katz et al. [3] and the values obtained are included... 

H) was determined for each solute, in each solvent mixture, for 10 different linear velocities ranging from 0.02 cm/sec. to 0.6 cm/sec. Three measurements were taken at each velocity, which produced a minimum of 180 values of (H) for each solute. Each data set was then fitted to each dispersion equation and the respective constants... 

A), (B), (C), etc. calculated. The data were sufficient in both quantity, and quality to allow the most appropriate dispersion equation to be identified. The results obtained are shown in Table 2. 

Table 2. Experimental Values for the Dispersion Equation Coefficients Obtained by a Curve Fitting Procedure...

In summary, it can be said that all the dispersion equations that have been developed will give a good fit to experimental data, but only the Van Deemter equation, the Giddings equation and the Knox equation give positive and real values for the constants in the respective equations. 

Dispersion equations, typically the van Deemter equation (2), have been often applied to the TLC plate. Qualitatively, this use of dispersion equations derived for GC and LC can be useful, but any quantitative relationship between such equations and the actual thin layer plate are likely to be fraught with en or. In general, there will be the three similar dispersion terms representing the main sources of spot dispersion, namely, multipath dispersion, longitudinal diffusion and dispersion due to resistance to mass transfer between the two phases. 

HETP of a TLC plate is taken as the ratio of the distance traveled by the spot to the plate efficiency. The same three processes cause spot dispersion in TLC as do cause band dispersion in GC and LC. Namely, they are multipath dispersion, longitudinal diffusion and resistance to mass transfer between the two phases. Due to the aforementioned solvent frontal analysis, however, neither the capacity ratio, the solute diffusivity or the solvent velocity are constant throughout the elution of the solute along the plate and thus the conventional dispersion equations used in GC and LC have no pertinence to the thin layer plate. 

To confirm the pertinence of a particular dispersion equation, it is necessary to use extremely precise and accurate data. Such data can only be obtained from carefully designed apparatus that provides minimum extra-column dispersion. In addition, it is necessary to employ columns that have intrinsically large peak volumes so that any residual extra-column dispersion that will contribute to the overall variance is not significant. Such conditions were employed by Katz et al. (E. D. Katz, K. L. Ogan and R. P. W. Scott, J. Chromatogr., 270(1983)51) to determine a large quantity of column dispersion data that overall had an accuracy of better than 3%. The data they obtained are as follows and can be used confidently to evaluate other dispersion equations should they appear in the literature. 

EXPERIMENTAL VALUES FOR DISPERSION EQUATION COEFFICIENTS OBTAINED BY CURVE RTTING PROCEDURES TO THE DATA GIVEN IN TABLE II... 

At the effective stack height, the dispersion of the pollutants are assumed to spread out as a Gaussian distribution. The basic dispersion equation considers... 

There are many dispersion equations available, most of them semiempirical. It is not tlie intent of this book to develop each in detail but ratlier to look at tlie one tliat has found tlie greatest applicability today. (In the authors opinion, tlie best atmospheric dispersion workbook published to dale is that by Turner (8))... 

There are inmiy dispersion equations available, most of them semieinpirical. 

The form of the solution of the dispersion equation (11.61) depends on the sign of the determinant D = q + Pl, i.e., on the values of the characteristic parameters g and P. The latter are determined by the physical properties of the liquid and its vapor, as well as the values of the Peclet number. This allows us to use g and P as some general characteristics of the problem considered here. 

Then the dispersion equation for the problem considered here I form ... 

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