TPDSci PSD Power law Alternate forms

Power-law PSD: Alternate forms

The power-law form of PSD (see Power-law PSD) is preserved if the latter is expressed at the logarithmic scale, i.e. as n_ln(D). The concentration factor, t, remains the same, but the slope changes from s to s_ln = s - 1. Indeed, by using Eq. 4 in Particle size distribution, we have:

n_ln(D)	= n(D) D
	= tD^-s D
	= tD^{-(s - 1)}	(2)

If the frequency size distribution, n(D), follows a power law with a slope s, then the cumulative size distribution (see Particle size distribution), N(D), also follows a power law but with a slope S = s -1, and a concentration constant, T = t / (s - 1). Indeed, we have:

N(D)	= ∫_D^∞ n(x) dx
	= ∫_D^∞ tx^-s dx
	= t / ( -s + 1) x^{-s + 1} \|_D^∞
	= t / (s -1) D^{-(s - 1)}
	= T D^-S	(3)

Note that s must exceed 1 for the power-law approximation to the cumulative PSD to exist. This condition is easily met for natural PSDs (for example, Jonasz 1983). Also note that a power-law PSD with a slope, s > 1 must, by definition, apply to a range of the particle size that is finite at the small-size end, otherwise the total number of particles in a sample would be infinite.

CITATION:
Jonasz M. 2006. Power-law PSD: Alternate forms (www.tpdsci.com/Tpc/PsdPwLwAlt.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY:
Published: 6-Jan-2006
Modified: 18-Jul-2006
Reviewed: PENDING

Topics in Particle and Dispersion Science