Dispersing Powders in Liquids, Part 1, by Ralph D. Nelson, Jr.
The phrase particle size distribution (PSD) is often used
rather loosely and can mean many things. Most properly it means
the functional relation between the number of particles and some
measure of the particle size. Size is usually taken to mean the
diameter, so the PSD is often presented as a graph of the
logarithm of the total number of particles smaller than particle
diameter d against the diameter itself, d. This plot is based on counting
particles in a series of adjacent size ranges often called
channels.
Sometimes people use the term PSD (incorrectly) when what
they are speaking about is a particle volume distribution
(PVD) or a more complicated dependence on diameter.
Sedimentation, sieve analyses, and light scattering experiments produce
plots or response vs some measure of diameter, but the precise relation
to either diameter or volume is unknown. I prefer to work with the PVD,
since the it is of more practical use. If all the particles in a sample
have similar simple shapes, such as spheres, then a PSD can be converted
to a PVD and vice versa.
The most common way to compare the PVD's of a large number of
slurry samples is to tabulate two measures -- one related to
a characteristic diameter and another related to how broadly the
distribution scatters about that diameter. There are many
different statistical measures that could be used to compute a
characteristic diameter and the scatter -- the ones I use are the
d-fifty-v, written as d50V [m],
and the mid-fifty-breadth-ratio, written as Bmid50V [dimensionless].
What do these mean? Consider the particles
in a sample to be lined up in the order of their diameters.
The d50V is the diameter for which fifty percent of the total volume
of the sample is in larger particles and fifty percent is in smaller particles.
Bmid50V is defined below.
Converting Particle Counts to a PVD
The experimental data usually do not list the diameters of individual particles,
but report the number of particles Ni counted in a series
of n channels with lower diameters dlow,i and upper diameters
dup,i. The lower limit of the bottom channel is nominally taken
as zero (although it really should be taken as the limit of detection)
-- dlow,1 = 0 -- and the upper
size for each channel is the lower size for the next larger,
dlow,i = di-1,up. The di reported in tabulations may
be either the upper bound or the midpoint of the channels. Since
the results of further analysis depend on it, you should find out
which type of d is tabulated.
The formulas for size analysis treat all the particles in a
channel as though they had the same diameter. The channel's
mid-diameter, dmid,i, may be computed either as the
arithmetic mean
dAmid,i = (dup,i + dlow,i) / 2
or as the geometric mean,
dGmid,i = [dup,i dlow,i]0.5.
The boundaries in many electrosensing-zone instruments are set so as to increase in a
geometric progression, where the boundries of each channel are related by dup,i = 21/3 dlow,i.
With this prefactor (21/3 = 1.26) the arithmetic mean is less
than one percent larger than the geometric mean, so the two means
are close enough to be indistinguishable for all practical purposes.
We start the conversion of a set of PSD data -- expressed as a series
of coordinates (di, Ni) --
to a PVD by computing the volume of particles in each channel [m3],
Vi = fVshape,i Ni dmid,i3
The fVshape,i is the volume shape factor relating
particle diameter cubed to particle volume. It is 
/ 6 for a
sphere and 1 for a cube. We normally assume that all the
particles in the sample have the same shape factor, so that
fVshape,i is not a function of particle size. For spheres
and cubes this is true, but for less symmetric shapes or porous
materials it is difficult to relate diameter measurements to
volume. If the axis-length ratios are not the same for all the
particles in a sample, data reduction becomes very difficult and
you should NOT use the following analysis. See Allen (see reference list)
for further discussion.
The net volume of all particles counted and the cumulative percent
of volume in particles smaller than the upper bound of the m-th channel are
Example 2.1 -- Particle Count Data for Ground Quartz
These electric sensing zone data were provided by T. Allen
(private communication). The counts have been corrected for
coincidence. This method provides a good measure of the volume
of nonporous particles regardless of shape.
Channel Upper Count Cumulative
Number Diameter Volume Percent
i dup,i Ni pcumV,i
m %
---- ------ --------- ---------
1 0.79 667,651 1.96
2 1.00 548,691 6.51
3 1.26 427,173 13.64
4 1.59 288,285 23.30
5 2.00 179,310 35.31
6 2.52 101,369 48.85
7 3.19 49,747 62.24
8 4.00 21,980 74.08
9 5.04 8,687 83.36
10 6.35 3,256 90.33
11 8.00 1,094 95.00
12 10.08 320 97.74
13 12.70 96 99.38
14 16.00 14 99.86
15 20.16 2 100.00
Using Parameters to Characterize a Distribution
It is more convenient to describe and compare distributions using a few parameters
that capture the main features of the distribution than to simply list the output
of their multi-channel particle size analyses. In many cases a two-parameter
distribution function works reasonably well. Many distribution functions have
been proposed and used (see Allen's book), but we shall discuss only one here,
just to illustrate the principles. This distribution works best for relatively
narrow distributions of particle size.
Figure 2.3 is a plot of pcumV,m against dup,m
for the powder in Example 2.1. We can interpolate between the points
to determine d25V, d50V, and d75V,
as the diameters at which the plot passes 25, 50, and 75 cumulative
volume percent. The spoken form for d50V is "d-fifty-v".
This value is sometimes incorrectly called the "average particle size",
but no averaging (over number or volume) is involved in its definition.
It is more accurately described as the "sieve diameter for fifty volume
percent passage of spheres", since pcumV,d. Because we often
are concerned about the volume -- or mass, which is directly related to volume
if all the particles are of the same material -- of particles smaller than
or larger than some critical value this is a very useful way to describe
a PVD. Other more complex measures can be readily derived from this distribution
if all the particles have the same composition and the same simple shape
(spheres or cubes).
Fig. 2.3 -- Cumulative Percent of Volume vs Diameter for a Quartz Powder
(dotted lines are at diameters for percent cumulative volumes of 25, 50, and 75%)
There are many ways to characterize the breadth, so be careful to
find out what is meant when you read about or discuss breadth.
The mid-fifty-breadth-ratio, or Bmid50V, is the ratio of
the difference between the two diameter limits that contain 25 % of the volume
on each side of d50V to d50V,
Bmid50V = [d75V-d25V] / d50V
In the simplest case, all particles in the sample have the same
density and the particle mass distribution (PMD) is the same as
the PVD. If the particles are coated with a uniform thickness of
a material which has a different density than the core, or if the
porosity varies with diameter, then the PMD will NOT be the same
as the PVD.
If we have a complete set of dmid,i Ni and know
fAshape,i and 
i, then we can compute the specific
surface area, Asp,comp [m2/kg]. Since many measuring
devices cannot detect (or will underestimate the number of) small
particles present in a sample, and since the equations based on
spheres make no provision for elongated shapes, surface
roughness, or porosity, calculations based on observed volume
distributions are very likely to underestimate the true specific
area. The specific surface area for nonporous spheres may be
computed from the count data by
Since the assumption that the particles are spheres of uniform
density is rarely satisfied, this formula should be used ONLY as
a last resort for estimating the surface area. Experimental
measures of specific surface area are usually easy to obtain and
provide a much more reliable basis for estimating surfactant
demand.
The Rosin-Rammler Volume Distribution
Since the PVD may change dramatically as a slurry passes through
an industrial process no single formula could possibly describe the wide
variety of PVD's observed for real slurries. For example, particles settle
out, are classified, suffer attrition, and agglomerate. These
factors create sharp cut-offs, extremely long tails, or multiple
peaks in the PVD. In spite of these complicating factors, it is
better to use a simple PVD than to assume that all the particles
in a system have the same diameter.
The Rosin-Rammler distribution is convenient because it has only two parameters,
is relatively simple, and can easily be manipulated mathematically.
A useful form for the Rosin-Rammler distribution and the definition of its breadth index
from d50V and d75V are
Allen (see reference list) discusses several other functions
that are used to represent size, volume, and other distributions.
The log-normal distribution, which is often used for pulverized material,
predicts somewhat fewer fines than the Rosin-Rammler distribution.
Since no distribution fits all situations, you should use one that
provides a good fit to the actual distributions of data being compared.
If a Rosin-Rammler distribution fits the data well, then Bmid50V
is related to BRR through
Example 2.1 Linear interpolation of pcumV,i against dup,i
from Example 2.1 or Figure 2.3 gives d50V = 2.58, d25V = 1.65,
and d75V = 4.10 m. From these we calculate
Bmid50V = 0.95 and BRR = 0.668.
Although the Rosin-Rammler function for pcumV,d may be
transformed to a linearized form by taking the logarithm of
(100 - p) / 100 and plotting this against d raised to the power
BRR, this format gives too much visual impact
to the data for the fines, so lines drawn through the points
"by eye" or using simple, unweighted least-squares fits will give
invalid results. A better way to see if the Rosin-Rammler
function fits the data well is to plot both the experimental
pcumV,d points and the Rosin-Rammler function for pcumV,d
against d. If the observed points scatter about the predicted
line with deviations comparable to experimental error, the fit is
acceptable. In Example 2.1 the Rosin-Rammler formula predicts
too many fines and so would be unacceptable for quantitative
simulation.
The Rosin-Rammler distribution may be written to find the d that corresponds
to a particular pcumV,d as
Once we have determined d50V and BRR, we can predict
the number count to be expected in the various channels for the
analysis of a known volume Vsl [m3] of slurry with a
known volume fraction of solids 
s. First, compute
pcumV,d for the upper cutoff of the channels dup,i,
then use
While these NRR,i could be used to compute Asp,
there are so many assumptions involved that the procedure is
useful only for illustrative calculations.
An observed PVD of any shape can be represented as the sum
of several Rosin-Rammler curves with weighting factors fRR,k,
but as the number of contributions increases, the physical meaning
of each new set of parameters -- fRR,k, d50V,k,
BRR,k -- becomes less clear. Little new insight into
the distribution is gained by using five rather than four contributions
to fit a complicated PVD. Since PVD's rarely follow a simple curve, model
calculations using the Rosin-Rammler (or any other theoretical
function) should be done with caution, and the assumptions involved
should be clearly stated. The best approach to simulating an industrial system
is to use a computer program to follow the fate of the particles in each
channel and to use experimental particle volume distributions
based on carefully drawn samples.
Factors Affecting the PVD
The fines end of distribution will decrease and the grit end
increase if the slurry passes through process steps that involve
fouling, agglomeration, filter bleed, slow precipitation, Ostwald
ripening, partial dissolution, or fines separation using an
elutriator or hydroclone. The grit end will decrease and the
fines end increase if the process involves attrition, milling,
sedimentation, or screening.