Dispersion parameters in Gaussian models

As we noted in Section 17.7, the variances of the mean plume dimensions can be expressed in terms of the motion of single particles released from the source. (At a particular instant the plume outline is defined by the statistics of the trajectories of two particles released simultaneously at the source. We have not considered the two-particle problem here.) In an effort to overcome the practical difficulties associated with using (17.90) to obtain results for (Ty and Pasquill (1971) suggested an alternate definition that retained the essential features of Taylor s statistical theory but that is more amenable to parametrization in terms of readily measured Eulerian quantities. As adopted by Draxler (1976), American Meteorological Society (1977), and Irwin (1979), the Pasquill representation leads to 

The variables on which Fy and F are assumed to depend are the friction velocity , the Monin-Obukhov length F, the Coriolis parameter /, the mixed layer depth c,. the convective velocity scale ic, the surface roughness zo. 2nd the height of pollutant release above the ground h  

The variances a and a are therefore treated as empirical dispersion coefficients, the functional forms of which are determined by matching the Gaussian solution to data. In that way, Oy and a- actually compensate for deviations from stationary, homogeneous conditions that are inherent in the assumed Gaussian distribution. 

Of the two standard deviations, Oy and a, more is known about o-y. First, most of the experiments from which Oy and a- values are inferred involve ground-level measurements. 

Such measurements provide an adequate indication of Oy, whereas vertical concentration distributions are needed to determine cr.. Also, the Gaussian expression for vertical concentration distribution is known not to be obeyed for ground-level releases, so the fitting of a measured vertical distribution to a Gaussian form is considerably more difficult than that for the horizontal distribution where lateral symmetry and an approximate Gaussian form are good assumptions. 

3 Summary of Gaussian Point Source Diffusion Formulas 

The various point source diffusion formulas we have derived are summarized in Table 18.2. 

We have derived several Gaussian-based models for estimating the mean concentration resulting from point source releases of material. We have noted that the conditions under 

TABLE 18.2 Point Source Gaussian Diffusion Formulas 

In Section V we derived several Gaussian-based models for estimating the mean concentration resulting from point source releases of material. We have seen that the conditions under which the equation is valid are highly idealized and therefore that it should not be expected to be applicable to very many actual ambient situations. Because of its simplicity. 

If a plume is photographed or measured by a device having an averaging time of the order of a fraction of a second, the appearance will have a sinuous form. The sinuosities appear to increase in amplitude and characteristic wavelength as one observes at greater distances from the source. Generally, eddies of aU scales can be expected to be present in the atmosphere but not necessarily to an equal degree. Near the source, the diffu- 

The form of the instantaneous cross-sectional distribution will be a function of the initial conditions and, again, the averaging time associated with the instantaneous observation. Close to the source, the measure- 

Based on the manner of derivation of the Gaussian equations in Section III, we see that the dispersion parameters a-y and are originally defined for an instantaneous release and are functions of travel time from release. Since the puff equations depend on the travel time of individual puffs or releases, the dispersion coefficients depend on this time, i.e., these coefficients describe the growth of each puff about its own center. This is basically a Lagrangian formulation. 

In determining how the dispersion coefficients depend on travel time one may employ atmospheric diffusion theory or the results of experiments. Because of the difficulty of performing puff experiments, however, the coefficients are usually inferred not from instantaneous releases but from continuous releases. Thus, the dispersion coefficients derived from such experiments are essentially a measure of the size of the plume envelope formed by sampling a real meandering plume emitted from a 

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Weber, A. H. (1976). Atmospheric Dispersion Parameters in Gaussian Plume Modeling, EPA-600/4-76-030A. U.S. Environ. Prot. Agency, Washington, D.C. 

The aim of dispersion models is the prediction of atmospheric dilution of pollutants in order to prevent or avoid nuisance. Established dispersion models, designed for the large scale of industrial air pollution have to be modified to the small scale of agricultural pollutions. An experimental setup is described to measure atmospheric dilution of tracer gas under agricultural conditions. The experimental results deliver the data base to identify the parameters of the models, For undisturbed airflow modified Gaussian models are applicable. For the consideration of obstacles more sophisticated models are necessary,... 

It is apparent from equations 3.2.4-3.2.7 that the determination of the concentration field is dependent on the values of the Gaussian dispersion parameters a, (or Oy in the fully coupled puff model). Drawing on the fundamental result provided by Taylor (1923), it would be expected that these parameters would relate directly to the statistics of the components of the fluctuating element of the flow velocity. In a neutral atmosphere, the factors affecting these components can be explored by considering the fundamental equations of fluid motion in an incompressible fluid (for airflows less than 70% of the speed of sound, airflows can reasonably be modeled as incompressible) when the temperature of the atmosphere varies with elevation, the fluid must be modeled as compressible (in other words, the density is treated as a variable). The set of equations governing the flow of an incompressible Newtonian fluid at any point at any instant is as follows ... 

The retention time tRjin and the second moment for the Gaussian profile (Eq. 6.61) have been replaced by variables indexed with g . These parameters tg and og must be optimized by curve fitting. Equation 6.143 is only suitable for symmetric peaks. Analytical solutions of, for example, the transport dispersive model (which describes asymmetric band broadening only for a very low number of stages) are not suited to describing the asymmetry often encountered in practical chromatograms. Thus, many different, mostly empirical functions have been developed for peak modeling. A recent extensive review by Marco and Bombi (2001) lists over 90 of them. 

A further parameter, which is of relevance beside dh, is the width of size distribution. This parameter, referenced as polydispersity index (PDI), is calculated from the deviation between the autocorrelation function and values actually measured. For narrowly dispersed samples PDI is below 0.1. PDI values above 0.3 point at widely size dispersed samples, which should be fitted by a more complex model. Several algorithms have been developed, which all base on a sum of autocorrelation functions. As each autocorrelation function results in one Gaussian distribution, the obtained size distribution turns out to be a sum of optionally superimposed curves. Thus, these complex algorithms reproduce reality better but are also more affected by measurement errors. 

Calculations of model spectra (distribution functions/(6jj) of chemical shifts of protons) using Gaussian functions and parameters of dispersion of peaks from the experimental NMR spectra (or theoretical estimations) as described in Chapter 10. Such approach allows us to calculate appropriate NMR spectra of large systems. This information can be used for more reliable and detailed analysis of the experimental NMR spectra. 

In the case of point sources, available dispersion models are typically based on assumption of a Gaussian distribution. The relative complexity and reliability of Gaussian distribution models are affected by parameters such as turbulence although modifications can be made to account for complex atmospherics. Reliability of the Gaussian approach is relatively satisfactory for distances up to about 100 km from the stationary source. Seinfeld and Pandis (1998) have published a comprehensive treatment of Gaussian... 

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