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Cumulative particle size distribution N(D) is related to the frequency (differential) size distribution, n(D), as follows:
| dN(D) = -n(D) dD | (1) |
where n(D) ≥ 0 by definition (to be consistent with n(D) being a limit of the size-interval normalized histogram of the particle size distribution (for example, Jonasz M and Fournier 2007, p. 272). Hence dN(D) ≤ 0.
From the definition of the derivative and Eq. 1 we have
| dN / dD = [ N(D + dD) - N(D) ] / dD | (2) |
Given that dD > 0, Eq. 1 and Eq. 2 imply that N(D + dD) ≤ N(D). Hence, the cumulative size distribution, N(D), is a monotonically decreasing function of D and must be defined as follows:
| N(D) = ∫D∞ n(x) dx | (3) |
Note that this use of the adjective "cumulative" in this context is contradictory to the usual meaning of that adjective, i.e. accumulation, or growth. However, the above definition of the "cumulative" size distribution is preferred due to the following arguments. First, all methods of particle size analysis can access a limited range of the particle size. Second, many particle populations have size distributions that fall precipitously as the particle size increases (Power-law PSD for natural dispersions). This implies that the inevitable truncation of the size range at a "large" D (approximating D = ∞) imposes in general smaller error on N(D) defined by Eq. 3, than that which the truncation at a "small" D (approximating D = 0) imposes on the "correctly"-defined cumulative size distribution:
| N' (D) = ∫0D n(x) dx | (4) |
| CITATION: Jonasz M. 2006. Particle size distribution: cumulative vs. frequency (www.tpdsci.com/Tpc/PsdDv.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 04-Jan-2008 Modified: 04-Jan-2008 Peer-reviewed: PENDING |
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