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The gamma-function frequency particle size distribution (PSD) is expressed as follows (for example, Risović 1993):
| n(D) = t D α exp( -βD) | (1) |
where D is a nondimensional particle diameter that is numerically equal to the actual particle diameter (see also Power-law PSD ), t, α and β are constant factors. 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) instead of the particle diameter, D, parameters β and t re-scale to 2β and 2α+1t respectively. The gamma-function PSD is closely related to the gamma probability distribution.
If α / β > 0, then the gamma-function PSD assumes a maximum at Dp:
| Dp = α / β | (3) |
Hence, Eq. 1 can be re-writted as follows (for example, Kokhanovsky 2007f).
| n(D) = t D α exp( -αD / Dp) | (4) |
Parameters α and β are related as follows to the average diameter, Davg, and the variance of D as follows:
| α = Davg2 / var( D ) - 1 | (5) |
| β = Davg / var( D ) | (6) |
From Eq. 3, Eq. 5, and Eq. 6 it follows that:
| Davg = Dp ( 1 + 1 / α ) | (7) |
The effective optical particle diameter, Deff (Eq. 1 in Effective optical particle size for a gamma-function PSD can be expressed as follows (for example, Kokhanovsky 2004):
| Deff = Dp ( 1 + 3 / α ) | (8) |
Deirmendijan (1969) gives an expression for the volume concentration, V, of a dispersion with a gamma-function size distribution. That expression can be re-written in the present notation as follows:
| V = (4π / 3) 2 -3β -3 Ntot (α + 1)(α + 2)(α + 3) | (9) |
The gamma function has been used to describe PSDs of populations of bacteria (for example, Ulloa et al 1992, see Fig. 1), phytoplankton (Risović 1993), marine particles (Jonasz and Fournier 2007, see Fig. 2), sludge flocs (for example, Chaignon et al 2002), as well as atmospheric aerosols (for example, Deirmendijan 1964), water droplets in clouds (Mitchell 2000), and snow crystals (Mitchell 1991).
See also Modified gamma-function PSD, Log-normal distributions and Variable-slope power-law PSD for other methods of approximating variable-slope PSDs.
| CITATION: Jonasz M. 2007. Gamma-function PSD (www.tpdsci.com/Tpc/PsdGmm.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 20-Jan-2007 Modified: 20-Feb-2007 Peer-reviewed: PENDING |
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