Question

Give 5 ways of minimizing the environmental impacts from stack emissions using Gaussian model.

Answer 1

Gaussian model:

The Gaussian model is perhaps the oldest (about 1936) and the most commonly used model type. It assumes that the air pollutant dispersion has a Gaussian distribution, meaning that the pollutant distribution has a normal probability distribution. Gaussian models are most often used for predicting the dispersion of continuous, buoyant air pollution plumes originating from ground-level or elevated sources. Gaussian models may also be used for predicting the dispersion of non-continuous air pollution plumes (called puff models). The primary algorithm used in Gaussian modelling is the Generalized Dispersion Equation For A Continuous Point-Source Plume.The Gaussian models, calculate the concentration of the pollutants by using the equation of the Gaussian dispersion given as equation (a). The value of concentration in output is directly proportional to flow of pollutants (Qp), to height of emission sources (H) but inversely proportional to the wind speed (v) and to the deviation line standards values ( ?y and ?z) which depend on the transport time (t = x/v). The semplificative hypotheses initially mentioned are following: atmosphere stationariety and homogeneity ,flat surface of the ground, absence of vertical flows. •Axis x directed as the average of the wind speed , it is considered that the emission happens in a point to height H=hs+ ?h (plume elevation)

Air pollution control – Gaussian model:

In order to describe the Gaussian model mathematical equations we consider a coordinate system where the origin is at ground level. The X and Y-axis extend horizontally and are perpendicular to each other. The Z-axis extends vertically perpendicular with the X and Y-axis.Let a and bbe the x and y coordinates, respectively, of the pollution emission position. The stack pollution emission occurs at some height h above the ground (z=0)

  

Assuming that the plume spread has a Gaussian distribution1, the concentration, C (gm-3), of gas or aerosols (particles less than about 20 ?m diameter) at position x, y, and z from a continuous source with an effective emission height, H, is given by equation 1 (present in the attatchment).where Q (gs-1) is the uniform emission rate of pollutants, U(ms-1) is the mean wind speed affecting the plume and ?y(m) and ?z (m) are the standard deviations of plume concentration distributed in the horizontal and vertical planes, respectively. Y is given by equation 2 (in attatchment),here ? (rad) is the mean wind direction (0???2?).

In Eq. (1) the variable x does not appear explicitly in the formula, but the ?y and ?z standard deviations depend on the X variable given by equation 3 (in attatchment).Eqs. (2), (3) provide a change of coordinates of the pollution emission point in the mean wind direction.The second exponential term in eq (1) accounts for the pollutant reflection on the ground.

The Gaussian parameters ?y and ?z depend on the distance from the source and takes into account the atmospheric turbulence. The most common tabulated data is due to Pasquill and Gifford .The effective emission height, H, is the sum of the physical stack height, h (m), and the plume rise, ?H (m).

There are several equations to model the plume rise ?H. The plume rise is directly proportional to the gas exit velocity (how fast the plume is coming out) and the plume buoyancy (in relation with the gas out temperature). It is also inversely proportional to the wind speed and to the stability class parameters (the more stable it is, the lower the plume will be). The most used formula for determining the plume rising height in stable atmosphere is due to Briggs , given by equation 4 (in attatchemt).where F is the buoyancy flux parameter and s is the parameter of atmospheric stability.The buoyancy parameter F is determined by equation 5 (in attatchemnt).

where d (m) is the internal stack diameter, Vo (ms-1) is the stack gas exit velocity, g is the acceleration of gravity (9.806 ms?2), To (K) is the gas out temperature and T_e (K) is the environment temperature.

The parameter s of atmospheric stability is determined as follows in equation 6 (in attatchment).

where d?/dz is the potential temperature’s gradient. As a default approximation, for a stable atmosphere, d?/dz is taken as 0.020 K m?1.

To proceed with a multiple source scenario we need to further extend our notation. Let Ci, i=1,…,n, be the contribution of source i for the total pollutant concentration, where n is the number of pollution sources distributed in a given region. The use of the i subscript also extends to the ith source specific parameters (ai, bi, hi, di, Qi, (Vo)i, ?Hi and Hi).

By further assuming the pollutant as chemical inert, its concentration can be computed by superposition of the npollution sources. The pollutant concentration at a point (x,y,z) can be computed by summing up individual source contribution ?i=lnCi(x,y,z,Hi).

With the optimal air pollution control in mind we can devise the following mathematical programming problems.

In the planning phase, the computation of the minimal stack height, while keeping the pollution level below some threshold C₀, in a given area R, at ground level, can be formulated as follows in equation 7 ( in attatchment) .

where fi, i=1,…,n, are construction cost functions, associated with the stack height. The simple bound constrainthlb?h?hub is to be understood as componentwise and it allows to consider legal and technical bound on the stacks, respectively.

For a given region R, with fixed pollution sources the maximum air pollution concentration (l?) can be estimated by solving the following mathematical programming problem given in equation 8.

The points (x?,y?)?R where ?i=lnCi(x?,y?,0,Hi)=l? are global maximizers (where the maximum pollution concentration is attained) that make the constraint active and are the positions where the sampling stations should be placed.

Lagrangian model:

A Lagrangian dispersion model mathematically follows pollution parcels (also called particles) as the parcels move in the atmosphere and they model the motion of the parcels as a random walk process. The Lagrangian model then calculates the air pollution dispersion by computing the statistics of the trajectories of a large number of the pollution plume parcels. A Lagrangian model uses a moving frame of reference as the parcels move from their initial location. The gaussian puff model is also a lagrangian model.

Eulerian model:

The most important difference between the Lagrangian and Eulerian models is that the Eulerian model uses a fixed three-dimensional Cartesian grid as a frame of reference rather than a moving frame of reference When using a conventional model, we will need to obtain at least one (preferably two or more) year's data on the meteorology of the area, with high data recovery and verifiable data accuracy. In the simplest situations, the data may be limited to that necessary to provide reliable hourly average estimates, at a representative site of wind speed,wind direction,air temperature ,mixing height ,atmospheric stability.

Box model:

The box model is the simplest of the model types. It assumes the airshed (i.e., a given volume of atmospheric air in a geographical region) is in the shape of a box. It also assumes that the air pollutants inside the box are homogeneously distributed and uses that assumption to estimate the average pollutant concentrations anywhere within the airshed. Although useful, this model is very limited in its ability to accurately predict dispersion of air pollutants over an airshed because the assumption of homogeneous pollutant distribution is very often unrealistic.