Page 172 - Instrumentation Reference Book 3E

156 Particlesizing

where X is the mean free path and b depends on be confused with extinction coefficient E. If the
the fluid (e.g., for air at s.t.p. dry, b E 1.7). transmission intensity of a beam of light changes
from IO to It in a path length L
11.4 Optical effects caused by ZT/JO = EL
particles where K is contained within E.
Extinction In Zo1It is the Napierian equivalent
When light passes through a suspension of par- of optical density.
ticles, some is absorbed, some scattered and a Although the value of K has been shown to be
proportion is unaffected, the relative proportions virtually 2, the scattering angle for larger particles
depending on the particle size, the wavelength of ( -30pm) is small and about half the light is
the light, and the refractive indices of the media. forward-scattered. It follows that depending on
The molecules of the fluid also scatter light. the observation distance and the size of the sen-
Some optical size-analysis methods infer size sor, much of the forward-scattered light could be
from measurements of the transmitted, i.e., unaf- received and the effective value of K in the above
fected light, others measure the scattered light. expression could be as low as 1. It will be apparent
Some operate on suspensions of particles, others that the effect of a distribution of particles on
on individual particles. light transmission is not a simple function of the
The theory of light-scattering by particles is projected area.
complicated. Rayleigh’s treatment. which applies Bearing in mind the above limitations on K, it
only to particles whose diameter d << X (the is possible to estimate the transmitted light inten-
wavelength), shows that the intensity of scattered sity through a distribution of particles by sum-
light is proportional to d61X4. It also shows that ming the area concentrations within size bands.
the scattering intensity varies with the observa- In each band of mean diameter d, the effective
tion angle and this also depends on d. As size d area alA is 1.5 KcLlpd, where c is the mass con-
approaches A, however. the more rigorous treat- centration, L is the optical path length, and p is
ment of Mie indicates that the scattering intensity the particle density.
becomes proportional to d2, i.e., particle cross-
sectional area, but that the effective area is differ-
ent from the geometrical area by a factor K, 11.5 Particle shape
known as the scattering coefficient or efficiency
factor, which incorporates d; X and the refractive Although we can attribute to a particle an equiva-
index. Where d (< X the two theories are similar. lent diameter, for example a Fer&t diameter dF,
In the region around d = A, however, K oscillates this does not uniquely define the particle. Two
(typically between about 1.5 and 5) tending particles with the same Feret diameter can have a
towards a mean of 2. Beyond about d = 5X, the very different shape. We can say that the volume
value of K becomes virtually 2, Le., the effective of an equivalent sphere is
cross-sectional area of a particle is twice the geo- 7l
metrical area. As dlX increases the preferred scat-
tering angle reduces and becomes more distinct 6(dF)3
and forward scattering predominates (diffraction). but we must recognize that the actual volume Vis
If the light is not monochromatic, the oscilla- probably very different. Heywood has proposed
tion of K is smoothed to a mean of about 2. the use of shape coefficients. We can assign a
The ratio of the intensity of the transmitted light coefficient CXY,F to a particle such that
IT to the incident light lo is given by the Lambert-
Beer law
-- exp (-K: Thus, if we use another method of size analysis
IT
-
Io which in fact measures particle volume V, know-
where a is the total projected area of the particles ing dF we can calculate QV,F. Similarly, by meas-
uring particle surface area S, we can assign a
in the light beam, A is the area of the beam, and coefficient QS,F so that
again K is the scattering coefficient. This is often
simplified to
optical density D = log,,, Zo/l~ OI~’.F is called the volume shape coefficient (based
= 0.4343K(a/A) on Fer&t diameter) and (YS,F is called the surface
shape coefficient (based on Feret diameter).
The scattering coefficient is sometimes called the Clearly, there are other shape coefficients, and
“particle extinction coefficient.” This should not they can be associated with other diameters.