Atmospheric Dispersion: Gaussian Models

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.

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).

Fig. 6.7

An example of mass continuity. The number of molecules which enter minus the number which exits from the volume equals the number of molecules which remain inside the volume

The balance of the molecule concentration in the cell can be expressed as:

$$ \Delta N\,\Delta x\,\Delta y\,\Delta z = {u_{{1\,}}}{N_1}\,\Delta y\,\Delta z\,\Delta t - {u_{{2\,}}}{N_2}\,\Delta y\,\Delta z\,\Delta t\; $$

(6A.1)

Dividing both sides by Δt and with volume (Δx, Δy, Δz) it results that:

$$ \frac{{\Delta N}}{{\Delta t}} = - \left( {\frac{{{u_2}\,{N_2} - {u_1}\,{N_1}}}{{\Delta x}}} \right) $$

(6A.2)

When $\Delta x \to 0$ and $\Delta t \to 0$ the above equation is expressed as:

$$ \frac{{\partial N}}{{\partial t}} = - \frac{{\partial \left( {u\,N} \right)}}{{\partial x}} $$

(6A.3)

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:

$$ \frac{{\partial N}}{{\partial t}} = - \frac{{\partial \left( {u\,N} \right)}}{{\partial x}} - \frac{{\partial \left( {\upsilon \,N} \right)}}{{\partial y}} - \frac{{\partial \left( {w\,N} \right)}}{{\partial z}} = - \nabla \bullet \left( {v\,N} \right) $$

(6A.4)

Furthermore:

$$ \nabla \bullet \left( {\nu \,N} \right) = N\,\left( {\nabla \bullet \nu } \right) + \nu \left( {\nabla \bullet N} \right) $$

(6A.5)

And with replacement the Eq. (6A.5) to (6A.4) can be written:

$$ \frac{{\partial N}}{{\partial t}} = - N\,\left( {\nabla \bullet \nu } \right) - \nu \,\left( {\nabla \bullet N} \right) $$

(6A.6)

and knowing:

$$ \nu \,\left( {\nabla \bullet N} \right) = \frac{{dN}}{{dt}} - \frac{{\partial N}}{{\partial t}} $$

(6A.7)

it can be concluded that:

$$ \frac{{dN}}{{dt}} = - N\,\left( {\nabla \bullet \nu } \right). $$

(6A.8)

A similar expression can be written for the density ρ_α of air:

$$ \frac{{d{\rho_a}}}{{dt}} = - {\rho_a}\,\left( {\nabla \bullet \nu } \right) $$

(6A.9)

A more general form of the continuity equation, where there are emission sources and chemical reactions is given by the expression:

$$ \frac{{\partial N}}{{\partial t}} = - \nabla \bullet \left( {v\,N} \right) + D\,{\nabla^2}N + \sum\limits_{{n = 1}}^{{{N_{{e,t}}}}} {{R_n}} $$

(6A.10)

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 R_n expresses the variation of the gaseous concentration arising from chemical reactions. The term which is arising from molecular diffusion can be written as:

$$ \eqalign{ D\,\left( {{\nabla^2}N} \right) = D\,\left( {\nabla \bullet \nabla } \right)\,N = D\left[ {\left( {i\frac{\partial }{{\partial x}} + j\frac{\partial }{{\partial y}} + k\frac{\partial }{{\partial z}}} \right) \bullet \left( {i\frac{\partial }{{\partial x}} + j\frac{\partial }{{\partial y}} + k\frac{\partial }{{\partial z}}} \right)} \right]N \hfill \\& = D\left( {\frac{{{\partial^2}N}}{{\partial {x^2}}} + \frac{{{\partial^2}N}}{{\partial {y^2}}} + \frac{{{\partial^2}N}}{{\partial {z^2}}}} \right) \hfill \\}<!endgathered> $$

(6A.11)