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Deirmendijan (1969) introduced a generalized gamma function (also referred to as a modified gamma function) to model the frequency particle size distribution (PSD) of the atmospheric aerosol:
| n(D) = t D α exp( -βD γ) | (1) |
which reduces to Eq. 1 of Gamma-function PSD, if γ = 1. Here, D is a nondimensional particle diameter that is numerically equal to the actual particle diameter (see also Power-law PSD ), α, β, and γ are constant factors, and the concentration factor, t, is expressed as follows
| t = |
|
(2) |
where Ntot is the total number of particles and Γ(x) is the gamma function (for example, Press WH et al 1988, p. 213). Note that when the particle radius, i.e. D / 2 is used (for example, Risović 1993, Deirmendijan 1969) instead of the particle diameter, D, parameters β and t re-scale to 2γβ and 2α+1t respectively.
The modified gamma-function PSD assumes a maximum at Dp:
| Dp = [ (α / β) ( 2γ / γ )]1 / γ | (3) |
Sample values of the modified gamma-function parameters for atmospheric aerosols (Deirmendijan 1969; Haze M aerosol) are t = 2.13e5, α = 1, β = 12.65, γ = 1/2.
Risović (1993) extended the application of this approximation to dispersions of marine particles. Based on an analysis of over 70 PSDs, he proposed to approximate a marine PSD by a sum of two "standard" modified gamma-functions (components). He suggests the following values of the parameters of these components:
| α1 = 2, β1 = 52 / 2γ1, γ1 = 0.145 to 0.195 (γ1, avg = 0.157) | (4) |
| α2 = 2, β2 = 17 / 2γ2, γ2 = 0.192 to 0.322 (γ2, avg = 0.226) | (5) |
The modified gamma-function parameters shown above have been expressed in the current notation (Eq. 1). The concentration factors, t1, t2, and the actual values of the γ parameters of the two components are to be obtained by fitting the experimental data. Fig. 1 shows a sample decomposition of a marine PSD into such two components.
Jonasz and Fournier (1996) also suggested existence of two "standard" components of marine PSD in their analysis of over 400 experimental PSDs in a particle diameter range of 0.5 to 200 µm in terms of log-normal components. However, their analysis shows that a PSD can contain other log-normal components as well.
Peng F et al (2007) applied the Risović (1993) model to dispersions of particles in inland waters in connection with their study of contribution of mineral particles in inland waters to light scattering.
See also: Variable-slope power-law PSD for alternative method of approximating a PSD with a variable curvature.
| CITATION: Jonasz M. 2006. Modified gamma-function PSD (www.tpdsci.com/Tpc/PsdGmmMod.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 20-Jan-2007 Modified: 16-Aug-2007 Peer-reviewed: PENDING |
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