TPDSci Mie theory Numerical integration of size patterns Power law psd Dim

Mie theory: Integration of particle size-dependent patterns for a power-law PSD. Unit of an integral.

Given that in Eq. 4 (in Mie theory: Integration of particle size-dependent patterns for a power-law PSD), the concentration factor, t, has a dimension of length^-4, ( λ / π )³ has a dimension of length³, and π / λ_n, Σ_c, and x are nondimensional, we have:

[c]	= [ ( π t / 2 ) ( λ / π )³ ( π / λ_n )^s ∫_xmin^xmax Σ_c(x, m) x^-s dx ]
	= [ t ] [ ( λ / π ) ]
	= length^-4 length³
	= length^-1	(1)

where [x] is a shortcut for "dimension of x".

The unit of t is frequently expressed by using two different length units, for example, cm^-3痠^-1, i.e. as the number of particles per unit volume of a dispersion (cm^-3) per unit particle-size interval (痠^-1; see Particle size distribution). One also may need the attenuation coefficient, c, to be expressed in a different length unit, for example, m^-1. In such a case, one must include in Eq. 4 a multiplicative factor, F, to reconcile the various units used. In the example chosen here, we would have:

[c]	= [ t ] [ ( λ / π )³ ]
	= ( cm^-3 痠^-1 ) um³
	= cm^-3 痠²
	= cm^-3 (10^-4 cm)²
	= 10^-8 cm^-1
	= 10^-8 (10^-2 m)^-1
	= 10^-6 m^-1
	= F m^-1	(2)

i.e. F = 10^-6. In that case, and solely for the purpose of numerical calculations, Eq. 4 can be simplified as follows:

c(λ) = F' ( π t_n / 2 ) ( λ_n / π ) ^{3 - s} ∫_xmin^xmax Σ_c(x, m) x^-s dx

(3)

where (in the present example), F' = 10^-6 m^-1, t_n is just a number equal to t expressed in cm^-3痠^-1, and λ_n is also a just a number equal to λ expressed in 痠.

CITATION:
Jonasz M. 2006. Mie theory: Integration of particle size-dependent patterns for a power-law size distribution (www.tpdsci.com/Tpc/MiePtnSzIntgPwLw.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY:
Published: 30-Jun-2006
Modified: 18-Oct-2006
Peer-reviewed: PENDING

Topics in Particle and Dispersion Science