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, Cp 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's0 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 pw and pv, the p.d.f. of the random vertical (w) and lateral (v) velocities in the CBL, respectively0,0,0. The total vertical displacement zc of the plume centerline is based on the superposition of the displacements due to the random w and the plume rise0,0. Thus, the AERMOD approach extends Gifford's model to account for plume rise. In addition, it includes a skewed distribution of zc because pw 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 zc is found from the p.d.f. of w, pw, as discussed in Weil0, and zc is obtained by superposing the plume rise (D h) and the displacement due to the random convective velocity (w):

where hs is the stack height, U is the mean wind speed, x is downwind distance, and D h includes source momentum and buoyancy effects0,0.

A good approximation to the w p.d.f. in the CBL has been shown to be given by the superposition of two Gaussian distributions0 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. Weil0). 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 w1, w2,

s 1, s 2, l 1, l 2 as a function of s w and the vertical velocity skewness S = w3/s w3 is given by Weil0.

Direct Plume

Following Weil0 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,


Here, hedj 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; TL is the Lagrangian time scale on the order of zi/s w.

The proportionality coefficients aj and bj 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 = -hs, which results in the exponential terms containing z + hedj on the right-hand side of Eq. (9).

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

Indirect and Penetrated Plumes.

The concentration expression Ci(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 zi. 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, Cp(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 Cd + Ci + Cp, with "off-centerline" concentrations obtained by multiplying this value by exp (-y2/s y2).

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