# AirWare User Manual

# AERMOD: A Dispersion Model

for Industrial Source Applications

This documentation is extracted from Perry et al. (1998).

## Algorithms for the Convective Boundary Layer

Having defined the general form of the concentration
equation in (1), we now must specify the form of C(*x,y,z*) in that
equation that is appropriate for convective conditions (L < 0) in flat
terrain.
The Monin-Obukhov length, L, (calculated by AERMET) is the height at
which wind-shear (mechanical) induced turbulence is approximately equal to
buoyancy-induced turbulence and is defined as:

where r is air density,
*C*_{p} is the specific heat of air, *T* is the air
temperature, *u*_{*} is the surface friction velocity (also
supplied by AERMET), *k* is the von Karman constant, *g* is the
acceleration due to gravity, and *H* is the surface sensible heat flux.

The dispersion algorithms for the convective boundary layer
(CBL) are based on Gifford's^{0} meandering
plume concept in which a small "instantaneous" plume wanders due to the large
eddies in a turbulent flow. The instantaneous plume is assumed to have
a Gaussian concentration distribution about its randomly varying centerline.
The mean or average concentration is found by summing the concentrations due
to all of the random centerline displacements. The specific model form is a
probability density function (p.d.f.) approach because the probability
distribution of the centerline displacement is computed from *p*_{w}
and *p*_{v}, the p.d.f. of the random vertical (*w*)
and lateral (*v*) velocities in the CBL,
respectively^{0,0,0}.
The total vertical displacement *z*_{c} of the plume centerline
is based on the superposition of the displacements due to the random
*w* and the plume rise^{0,0}.
Thus, the AERMOD approach extends Gifford's model to account for plume rise.
In addition, it includes a skewed distribution of *z*_{c}
because *p*_{w} in the CBL is known to be skewed;
however, the lateral displacement is assumed to be Gaussian.

The average concentration is due to three contributions:

- a "direct" plume which emanates from the stack and
accounts for those plume segments caught in downdrafts sufficiently large
to bring them to the surface,
- an "indirect" plume which accounts for plume segments that first
rise to the CBL top, delay due to buoyancy, and then disperse downwards, and
- a penetrated plume that represents the plume material with sufficient
buoyancy to escape the mixed layer.

The indirect plume is a mathematical device included to satisfy the no-flux
boundary condition at the CBL top, and therefore plays the same role as
the "image" plume in the standard Gaussian plume model.
Where the image plume represents a simple reflection at the mixed layer top,
the indirect plume of AERMOD represents the tendency of the buoyant plume
to hug the mixed layer top before diffusing back toward the surface.
So as a mathematical construct the indirect plume replaces the first image
source term and primarily accounts for the delayed reflection with an
adjustment in its height.
For both the direct and indirect plumes, the interaction with the ground
surface and subsequent interactions with the CBL top are handled
by assuming reflection and using image sources.
That is, only the first (direct) plume interaction with the CBL top
is handled by the "indirect" source or plume.
For material dispersing within a convective layer, the
conceptual picture is a plume embedded within a field of updrafts and
downdrafts that are sufficiently large to displace the plume section within
it.
The p.d.f. of the plume centerline height *z*_{c}
is found from the p.d.f. of *w, p*_{w}, as discussed
in Weil^{0}, and *z*_{c} is
obtained by superposing the plume rise (D *h*)
and the displacement due to the random convective velocity (*w*):

where h_{s} is the stack height, U is the mean wind speed,
x is downwind distance, and D h includes source
momentum and buoyancy effects^{0,0}.

A good approximation to the w p.d.f. in the CBL has
been shown to be given by the superposition of two Gaussian
distributions^{0} such that

where l _{1}
and l _{2} are weighting coefficients for
the two distributions (1=updrafts, 2=downdrafts);
l _{1} + l
_{2} = 1 (e.g. Weil^{0}).
The and s _{i} (i=1,2) are the mean vertical
velocity and standard deviation for each distribution and are assumed to
be proportional to s _{w}.

A simple approach for finding w_{1}, w_{2},

s _{1},
s _{2}, l
_{1}, l _{2} as a function of
s _{w} and the vertical velocity
skewness S = w^{3}/s _{w3}
is given by Weil^{0}.

#### Direct Plume

Following Weil^{0} et al., the
concentration distribution for the direct plume at an arbitrary height, z,
along the plume centerline (y = 0) is given by
where subscript *d* denotes the direct plume,

and

Here, h_{edj} and
s _{zdj} are the effective source height and
vertical dispersion parameter corresponding to each of the two distributions
in Eq. (8); s _{b} is the dispersion due to
buoyancy-induced entrainment with
s _{b} = b
D h/Ö 2 and
b = 0.4; *T*_{L}
is the Lagrangian time scale on the order of
*z*_{i}/s _{w}.

The proportionality coefficients a_{j}
and b_{j} above are defined by:

In obtaining Eq. (9), we use an "image" plume to satisfy the
no-flux condition at the ground, i.e., an image plume from a
source at z = -h_{s}, which results in the exponential terms
containing z + h_{edj} on the right-hand side of Eq. (9).

The image plume at z = -h_{s} results in a
positive flux of material at z = z_{i}.
To satisfy the no-flux condition there, an image source is introduced
at z = 2z_{i} + h_{s}, which then leads to a series of
image sources at z = 2z_{i} - h_{s},
4z_{i} + h_{s}, -4z_{i} - h_{s}, etc.
Thus, the general expression for the concentration distribution along the
plume centerline is

#### Indirect and Penetrated Plumes.

The concentration expression C_{i}(x, 0, z) for the indirect plume
is similar to that in Eq. (14) but with a different emission rate and a
plume height near z_{i}.
The indirect plume is formulated to delay the downward dispersion
of plume mass until decreasing (due to entrainment) plume buoyancy is
overcome by convective turbulent downdrafts.
Expressions for both the direct and indirect plumes account for the
fraction of material that penetrates the elevated inversion due to plume
buoyancy.
The penetrated source concentration expression,
C_{p}(*x,0,z*), is a simple Gaussian form.
For the sake of brevity, we will not detail the formulation of the indirect
plume and the penetrated plume.
The concentration (to be included in Eq. (1) as C(*x,0,z)) *along the
centerline is given by the
sum C_{d} + C_{i} + C_{p}, with "off-centerline"
concentrations obtained by multiplying this value by
exp (-y^{2}/s _{y}^{2}).