Topics in Particle and Dispersion Science

Gamma-function PSD

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The gamma-function frequency particle size distribution (PSD) is expressed as follows (for example, Risović D 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 =

Ntot β α + 1 Γ(α + 1 )

(2)

where N_tot 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 D 1993) instead of the particle diameter, D, parameters β and t re-scale to 2β and 2^α+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 D_p:

Dp = α / β

(3)

Hence, Eq. 1 can be re-writted as follows (for example, Kokhanovsky AA 2007f).

n(D) = t D α exp( -αD / Dp)

(4)

Parameters α and β are related as follows to the average diameter, D_avg, 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, D_eff (Eq. 1 in Effective optical particle size for a gamma-function PSD can be expressed as follows (for example, Kokhanovsky AA 2004):

Deff = Dp ( 1 + 3 / α )

(8)

Deirmendijan D 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 O et al 1992, see Fig. 1), phytoplankton (Risović 1993), marine particles (Jonasz M and Fournier 2007, see Fig. 2), sludge flocs (for example, Chaignon V et al 2002), as well as atmospheric aerosols (for example, Deirmendijan D 1964), water droplets in clouds (Mitchell DL 2000), and snow crystals (Mitchell DL 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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