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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

1 D

exp [ -

(ln D - ln Dgm)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 =

Ntot (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 average geometric 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 T et al 2006, Schuster GL 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 E 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 LW 1977, Espenscheid WF et al 1964). The generalized log-normal distribution of order q is expressed as follows:

nq(D) = Cq D q exp [ -

(ln D - ln Dgm, q)2 2 Var(ln D)

]

(4)

Cq =

Ntot (2π)1/2 SD(ln D) Dgm, 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 M 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 M 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 Peer-reviewed: 03-Jan-2008

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