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In the case of the log-normal PSD defined by Eq. 1 and Eq. 2 of Log-normal particle size distributions, the parameters of the PSD can, in principle, be calculated, and the PSD can be reconstructed by knowing the first 3 moments of the PSD, M0 = Ntot, i.e. the total concentration of particles, M1, and M2. Indeed, the moments, Mr of the log-normal distribution are expressed as follows (for example, Thomas JC 1986):
| Mr = Ntot exp[ r lnDgm + r 2 Var(lnD) / 2 ] | (1) |
where Dgm = exp[Avg(lnD)], where Avg(x) is the average of x and Var(x) is the variance of x. In contrast to the original formulation in Thomas JC (1986), Ntot converts the moments of the probability distribution to the moments of the particle size distribution.
It can be now seen that the first three moments define the three parameters of the log-normal distribution, i.e. Ntot, lnDgm, and Var(lnD) via the following equations:
| M0 = Ntot | (2) |
| M1 = Ntot exp[ lnDgm + Var(lnD) / 2 ] | (3) |
| M2 = Ntot exp[ 2 lnDgm + 2 Var(lnD) ] | (4) |
However, parameters of the log-normal distributions are best evaluated by fitting a log-normal function to the log-transformed frequency particle size distribution data: ln[n(D)] = f [ln(D)]. This applies especially to particle populations characterized by very wide particle size ranges, exceeding the size range accessible to a particle sizing method(s) used. Indeed, in such a case, parameters such as Ntot can be evaluated only approximately.
| CITATION: Jonasz M. 2006. Moments of a log-normal distribution (www.tpdsci.com/Tpc/PsdLogNmlMom.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 25-Aug-2007 Modified: 04-Jan-2008 Peer-reviewed: 03-Jan2008 |
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