1. Derivation of the Gaussian plume model Distribution of pollutant concentration c in the flow field (velocity vector u ≡ u x, u y, u z ) in PBL can be generally described by Reynolds equation in the form: Gaussian plume dispersion model The inercial or advection term is non-linear - it contains the product of unknown parameters u and c: The Reynolds equation can be solved only numerically (along with other three equations describing the components of the flow velocity vector). are turbulent diffusion coefficients for each spatial directions

2. Derivation of the Gaussian plume model Gaussian model is simplified - does not include a transport in complex flow fields. The model describes only smoke plume drift by constant velocity along a linear path. Derivation of this model, therefore, does not take into account the nonlinear advection term and based on equation involving only the turbulent dispersion. For constant and isotropic turbulent dispersion is the equation simply in the form: This simplified spherically symmetric diffusion equation has an analytical solution in the form of a concentration function of two variables ( t, r ) Q is mass flow of pollutant (g/s) r is distance from the centre (from source)

3. Derivation of the Gaussian plume model Analytical solutions - the spherically symmetric function of time and distance from the source is expressed by the graph: It is obvious that the concentration is high but rapidly decreasing with increasing distance from the center in a short time t 1 after the pulse emission of pollutant from the source. Concentration after a longer time t 2 is lower with slow decline. To derive the model of plume is necessary to establish transport - a shift from the source in the direction of the x-axis. We can simple define: is constant velocity of drifting flow

4. Derivation of the Gaussian plume model We have the function of pollutant concentration including the shift in the x-axis: t1t1 t2t2 t3t3 t3t3 t2t2 t1t1 Plume model is obtained by integrating over all states in the time

5. Derivation of the Gaussian plume model Resulting formula after integration (and neglect of small terms) expresses the time- independent 3D field of spatial distribution of pollutant concentration corresponding to smoke plume. The anisotropy of turbulent dispersion can be re-introduced for the y and z-axes: The relations for dispersion parameters corresponding to Gaussian standard deviations are finally introduced.

6. Derivation of the Gaussian plume model The resulting formula of Gaussian smoke plume model is: The fraction in the formula expresses the concentration in the plume axis. By dimensions is: The exponential terms represent a lateral dispersion in the y and z- axes, they are dimensionless and can have values ​​ in the interval (0,1)

7. Application of the Gaussian plume model In practical applications, z does not mean the distance from the axis of plume but represents the height difference between the ground level at the source and at the reference point (point where the pollutant concentration is calculated). The reflection of pollutant from the ground at the level of source is also included in the formula. h - is effective height of the source (stack).

8. Application of the Gaussian plume model The second possibility is the inclusion of pollutant reflection from the ground level at the reference point. This approach is appropriate to calculate the concentration of the pollutant in reference points on the plateau..

9. Application of the Gaussian plume model Both approaches for calculating of pollutant reflection from the ground can be included continuously by a factor theta. This factor is calculated as the ratio of vertical surfaces defined at the junction between the source and the reference point as shown. Theta factor can have values ​​ in the interval (0,1) A 2 (rectangle)A1A1 Reference point

10. Application of the Gaussian plume model The dispersion parameters are dependent on the length of the plume x- axis. They are calculated based on the stability classification of the atmosphere by the tabulated parameters a y, a z and b y, b z.