Abstract
A mathematical description of the concentration profile of different chemical compounds in the atmosphere is one of the applications of atmospheric models. The problem which one has to solve is to calculate the spatial and temporal concentration of air pollutants under specific emission conditions.In Chap. 6 the dispersion of air pollutants is examined with the analytical Gaussian approach. The Euler and Lagrange descriptions are used for solution of the continuity equation. Furthermore the limitations of the Gaussian approach are studied together with the calculation of Gaussian dispersion coefficients under different stability conditions. The plume rise from point emission sources is also studied together with the characteristics of plume dispersion.
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- 1.
\( \Gamma (n) = \int_0^{\infty } {{x^{{n - 1}}}\,{e^{{ - x}}}\,dx} \) which has limit for n > 0. In addition \( \Gamma \left( {n + 1} \right) = n\,\Gamma (n) \)
References
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Appendix 6.1The Continuity Equation
Appendix 6.1The Continuity Equation
In a volume cell the mass of air that enters inside the cell, minus the mass which flows out of the cell, equals the final mass remaining inside the cell minus the initial mass. The same relationship occurs for other atmospheric variables such as energy. Fig. 6.7 shows a volume cell with dimensions Δx, Δy and Δz (m). The concentration of air has boundary values N1 and N2 at the surface Δy × Δz (molecules m−3) with corresponding values for the velocity u1 and u2 (m s−1) respectively. The influx and outflux of air from the cell is u1 N1 and u2 N2 respectively (molecules m−3 s−1).
The balance of the molecule concentration in the cell can be expressed as:
Dividing both sides by Δt and with volume (Δx, Δy, Δz) it results that:
When \( \Delta x \to 0 \) and \( \Delta t \to 0 \) the above equation is expressed as:
which is the continuity equation for a gas that is influenced by the velocity in one dimension. At three dimensions in a Cartesian coordination system the above equation can be written as:
Furthermore:
And with replacement the Eq. (6A.5) to (6A.4) can be written:
and knowing:
it can be concluded that:
A similar expression can be written for the density ρα of air:
A more general form of the continuity equation, where there are emission sources and chemical reactions is given by the expression:
In the above equation the coefficient D is the coefficient of molecular diffusion of gas, which expresses the molecular kinetics due to their kinetic energy. The coefficient Rn expresses the variation of the gaseous concentration arising from chemical reactions. The term which is arising from molecular diffusion can be written as:
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Lazaridis, M. (2011). Atmospheric Dispersion: Gaussian Models. In: First Principles of Meteorology and Air Pollution. Environmental Pollution, vol 19. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0162-5_6
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