TPDSci PSD Log-normal

Log-normal particle size distributions

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The log-normal (frequency or differential) particle size distribution (PSD) is expressed as follows:

n(D) = C

exp [ -

(ln D - ln D_gm)²

2 Var(ln D)

]

(1)

where D_gm is here the median particle size, Var denotes variance, and the concentration factor, C, is expressed as follows

C =

N_tot

(2π)^1/2 SD(ln D)

(2)

where N_tot is the total number concentration of particles and D_gm = exp[Avg(lnD)], where Avg(x) denotes the average of x. D_gm has the meaning of the geometric mean diameter, i.e. D_gm = (D₁D₂ ... D_N)^1/N, where N is the number of particles.

The function expressed by Eq. 1 and Eq. 2 is a distribution function of the particle diameter, D, whose natural logarithm, ln D is normally distributed. Indeed, in that case the usual normal (probability) distribution, P(x), applies to x = ln D. Here, P(x) should be interpreted as [1 / N_tot ] dN / d lnD, where N is the cumulative particle size distribution and N_tot ≡ N(0). Hence,

n(D)	= - dN / dD
	= - (dN / d lnD) (d lnD / dD)
	= - (dN / d lnD) (1 / D)	(3)

which explains the presence of D in the denominator of Eq. 1. Note that Eq. 3 implies that while D_gm is the mediam diameter of the log-normal approximation to the particle size distribution n(D) (Eq. 1), lnD_gm is the modal argument of n(lnD).

The log-normal distribution is frequently used to approximate the particle size distribution of aerosol (for example, Nousiainen et al 2006, Schuster et al 2006, Whitby KT 1978, Davies CN 1974). In particular, Davies CN 1974 discusses a simple algorithm for fitting multi-mode log-normal approximations for aerosol particle size distributions. Limpert et al (2001) have recently reviewed applications of the log-normal distribution across sciences.

Note that another definition of the log-normal PSD has also been used. This latter PSD is the 0-th order case of the generalized log-normal distribution (Casperson 1977, Espenscheid WF et al 1964). The generalized log-normal distribution of order q is expressed as follows:

n_q(D) = C_q D^q exp [ -

(ln D - ln D_{gm, q})²

2 Var(ln D)

]

(4)

C_q =

N_tot

(2π)^1/2 SD(ln D) D_{gm, q} ^q+1 exp[(q+1)² SD(ln D) / 2]

(5)

where q is the distribution order. With this notation, the PSD expressed by Eq.1 and Eq. 2 is the -1-st order generalized log-normal distribution.

Espenscheid WF et al 1965 used the 0-th order distribution to model the PSD of aerosol particles. Jonasz and Fournier (1996) fitted multimode 0-th order log-normal approximations to PSD's of aquatic particles (see also Jonasz M and Fournier 2007, p. 422). The fitting algorithm described by Jonasz and Fournier is implemented in a computer program.

CITATION:
Jonasz M. 2006. Log-normal particle size distributions (www.tpdsci.com/Tpc/PsdLogNmlP.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY:
Published: 25-May-2006
Modified: 04-Jan-2008
Reviewed: 03-Jan-2008

Topics in Particle and Dispersion Science