A simple Gaussian model is used to estimate dispersion from an elevated continuous source. The model makes the following assumptions:

constant wind speed

no wind shear

flat topography

The equation for an elevated continuous point source (non fumigation) release is:

where,
 

C(x,y,z)	concentration in air at (x,y,z)	gm / m3
Q	emission rate from the stack	gm / sec
Us	wind speed at source height	m / sec
	horizontal dispersion coefficient	m
	vertical dispersion coefficient	m
y	cross - wind distance	m
z	vertical distance	m
h	effective stack height	m

Normally the plume is positively buoyant and the source height must be adjusted so as to incorporate the plume rise since this will have a significant effect on ground level concentration distribution.

There are two possible ways to incorporate a plume rise:

the effective source height method, and

the variable plume model method.

The "Effective Source Height" method is independent of downwind distance x, while the variable plume model takes into account the tilt of the plume. The following figure shows the source arrangement for effective source height and variable plume model.

The effective source height 'h' is computed as:

where,

hs = physical chimney height

ht = maximum terrain height between the release point and the point for which the calculation is made ( ht >= 0 ).

The above equation may be simplified for ground level concentrations by setting z = 0

The above equation may be simplified for centerline ground level concentrations by setting y = 0

The equation for a ground source (non fumigation) may be written as:

The above equation does not produce realistic results near the source (i.e. for small downwind distances). The following correction is suggested:

where,

q_o is the initial volume flux (m³ / sec) 

---

 The maximum concentration is given by the following equation for a passive, continuous plume:

The equation assumes that  = 

The location of maximum concentration is at:

Ground level concentration under fumigation cases is:

The following figure shows limited mixing condition using effective stack height method.

Puff

For a puff plume the equation for ground level concentration is:

The Gaussian distribution used in Gaussian Models is given by:

where,
 

	is any real number
	is any real number > 0

Plume Dispersion Parameters

The use of the Gaussian model requires the specification of plume spread rates in the horizontal direction (sigma y) and in the vertical direction (sigma z). The numerical value of the sigmas may be obtained by different methods.

Experimental data.

Modified experimental curves such as the use of intensity of turbulence by Csanady (1973) and the displacement method by Weil (1974).

Use of the Lagrangian auto correlation function (see Draxler 1976).

Use of the moment-concentration method (see Smith 1972 and Kumar 1977).

Use of Taylor's statistical theory (see AMS recommendations).

In this course, we will limit ourselves to experimental data and the recommendations made by the American Meteorological Society.

The most popular sigma curves currently in use are based on experiments performed in open field studies (e.g., Pasquill, 1961, and Sutton, 1974, TVA (Carpenter et al., 1971), and Brookhaven National Laboratory (BNL) (see Smith and Singer, 1965). The main advantages of these curves are that they are easy to use and widely accepted by government agencies. Case must be taken in comparing these results, and the following points may by helpful:

Nature of Release: Real life releases may be continuous or instantaneous and be idealized as weather point or line sources. For example, an instantaneous point source situation occurs in cases such as an explosion.

Sampling Time: During experimental studies the values of stigmas depend heavily on sampling time for the same results for a 10 minute average and a two hour average plume can differ by as much as a factor of 2. For an instantaneous plume, boundaries are quite irregular while a 10 minute average plume will have a more regular and much wider boundary (a two hour plume is still wider). Away from the source, large turbulent wind eddies play a dominant role in diffusing the cloud, and the effect if smaller eddies diminishes gradually.

Release Height: Releases in practical applications are usually made above ground. Due to the generally prevailing emissions control regulations, elevated releases are much more common.

Terrain Features: Experiments are conducted under widely varying terrain conditions such as open fields, flat deserts, low/rough hills, low woods, rolling terrain and dense scrub.

Velocity Field: One interesting point which may be helpful to note is that in most field experiments on dispersion, conditions such as uniform sky cover and constant wind direction during the sampling time are favored.

A family of sigma curves was suggested by Pasquill (1961) based on available experimental data and theoretical expectations. Later studies indicated that the Pasquill curves fit the experimental data for the Prairie Grass experiments (see Barad, 1968). The curves are based on smoke plume elevation H_sp (visible portion) and angular spread q using the relations:

s_z=H_sp/2.14

s_y=q_x/4.28

The numerical coefficient 2.14 is just the 10% ordinate of the normal error curve.

Several versions of the Pasquill curves appear in the literature. Some of the comments on these curves are (Beychok 1979):

Turner's (1970) graphs of sigmas are more faithful representatives of the Pasquill (1961) estimates than any other.

Two sets of plots (p. 102-103 and p. 408-409) given by Slade (1968) are neither consistent with the original work of Pasquill, nor with other. The U.S. NRC Regulatory Guide 1.45 uses the curves given on page 102-103.

Gifford's (1961) plots are not consistent with the Pasquill estimates.

BNL s-curves are plotted for distances ranging from 10 m to 60 km using wind gustiness as a measure of atmospheric stability based on 15 years of field experiments. Most of these studies involve the emission of very small oil fog droplets from a source 108 m above ground. The s_y data are based on actual measurements while s_z is derived using:

s_z=Q*e^-Fg/C*s_y*U*p

where, Fg is an adjustment term for the Gaussian equation and C is measured concentration. U has invariably been assumed constant (however it should increase with height).

The points which should be mentioned about BNL curves are:

values of s's are based on the one hour averages from 108 m releases.

for large downwind distances (>10km) experimental curves are extrapolated.

The difference between the Pasquill curve and the BNL curve is explained by Singer and Smith (1965) as:

in stable cases sigma z is a very gentle function of distance, and only with precise data may one determine the difference between the straight line (BNL curve) and a curved plot (Pasquill curve).

U is held constant with distance.

TVA Dispersion Coefficients

Carpenter et al (1971) presented dispersion coefficients for use in dispersion models (Coning dispersion, Inversion Break up dispersion and Trapping) using 20 years of comprehensive field surveillance and documentation of emissions from TVA power plants. The study included a varied range or unit sizes, stack heights and meteorological conditions. The average potential temperature gradient with height was used as an indicator for atmospheric stability.

Sigmas are calculated by employing the relation

s_p=Area/[C_peak*(2*p)^0.5]

where the area is equal to the base times the average height of concentration profile along the axis and C_peak is the maximum concentrations in that profile.

In a number of cases, s_z is calculated using

C_max=Q/[2*U*s_y*s_z*p]

and thus, the distribution is considered Gaussian i.e.,

C=C_max exp[-0.5*(x_g/s)²]

TVA results for unstable cases are not available. In fact these cases are lumped with neutral cases. Also, TVA curves represent nearly instantaneous situations.

Since the TVA plumes are from hot industrial stacks, the initial large values may have been caused by strong vertical mixing due to the self generated turbulence on these plumes. During stable conditions, TVA s_z values show relatively little increase with distance which reflects very small values of vertical turbulence. TVA curves are 3-5 minute averages for 100m stacks.

Sutton Curves

Sutton (1947) used vertical diffusion coefficients (C_y and C_z) to obtain sigmas.

where u is the kinematic viscosity of the air and v', w' are the eddy velocities across wind and in the vertical respectively.

and 2s_i²=C_i²*x²ⁿ                   i = y or z

Here n relates to the turbulence and is normally determined by the vertical transfer of momentum as indicated by the shear of the wind near the surface. The value of n lines between 0 and 1 (very turbulent to low turbulence conditions).

Sutton's equations for sigmas have not been confirmed for non-neutral stability. Sutton curves are based on 3 minute averages from ground level releases and then extrapolated for a 100m release.

AMS Recommendations On Sigmas

The dispersion coefficients are dependent on stability, release height, surface roughness, height of the mixing layer, as well as other factors. A detailed review workshop was held by the American Meteorological Society in 1977, and they recommend an equation for sigma y based on Taylor's statistical theory of one-dimensional and homogenous turbulent dispersion. For sigma z, no recommendation was made for the best technique comparable to the sigma theta method for sigma y. Suggestions to account for wind shear and surface roughness were included in the report.

Empirical Formulas

Several authors have used power law formulas and different type of polynomials obtained using curve fitting to experimental date for use in diffusion equations. A summary of these parameters is given in the following tables.