Page 175 - Instrumentation Reference Book 3E

Analysis methods that measure size direcily 159

In a true log-normal distribution, the equation graph. If the percentage over size s is R, it has
becomes been found that
log log( 100/R) = K + n log x
1 (1nx -
1: = __
‘ uG where K is a constant and n a characteristic for
the material.
where now it is [l:d( In x), the area under the The Rosin-Rainrnler distribution is included
curve using a log axis which represents the total for completeness but its use is not generally
quantity, and u now refers to the log distribution recommended.
and is not the same as before. The expression E Sometimes when a distribution covers a wide
is the arithmetic mean of the logarithms of the range of sizes, more than one analysis method has
size so that I is now the geometric mean of the to be used. It is not unusual for a discontinuity to
distribution. On a cumulative percentage dia- occur in the graphs at the change-over point, and
gram 4, the geometric mean particle size is the this can be due to shape or density effects (see
50 percent size and u is found from shape factor, Section 11.5j.
o = InxY4 - Ins = 1n-Y - 1n.xI6
~
= In, .X84 =In- x 11.7 Analysis methods that
x -XI6 measure size directly
11.7.1 Sieving

If x is plotted on base-10 logarithm x probability Sieving is the cheapest, most popular. and prob-
paper, ably the most easily understood method of size
analysis. It also covers a very wide range of sizes,
1 X84 it being possible to buy sieves (screens) ranging in
o = -In 10logl,- mesh size from jpm up to several centimeters.
2 2x1 6
However. sieving of fine materials requires spe-
cial techniques and the British Standard
410:(1962) indicates a lower limit of pm. Sieves
Again X and u define the distribution. are made in a variety of materials from non-
Sometimes o is replaced by In B to show that it metallic (e.g., polyester) to stainless steel. The
is the standard deviation of a log-normal distri- common method of construction is woven wire
bution or fabric but the smallest mesh sizes are electro-
formed holes in plates. The British Standard gives
B = -J-j minimum tolerances on mesh size and wire spa-
cing. etc. American, German. and I.S.O. stand-
Again, if the cumuiative percentage plot is not ards are also applicable with very similar criteria.
truly linear the derivation of the standard devi- The British Standard nomenclature is based on
ation is iiot truly meaningful and the 50 percent the number of wires in the mesh per inch. Thus
size is the:n the median size. However, in practice B.S. sieve number 200 has 200 wires per inch and
such curves are commonly used for comparing with a specified nominal wire diameter of 52 pin
size analyzes and it is sometimes useful for math- has a nominal aperture size of 75pm square. In
ematical treatment to draw an approximate principle all particles less then 75 ym diameter in
straight line. a sample of spherical particles will pass through a
A feature of a log normal distribution is that if B.S. number 200 sieve. The sample is placed in
one method of treatment, for example a mass the uppermost of a stack of sieves covering the
particle size analysis. demonstrates log-normal range of diameters of interest arranged in ascend-
properties, then all the other methods will also be ing order of size from the bottom. The powder is
log-normal. Clearly the values of .? uill be differ- totally enclosed by means of a sealed base and lid.
ent. Log-probability diagrams are particularly use- The stack is agitated until the particles have
fd when the range of particle sizes is large. found their appropriate level and the mass in each
sieve noted. The tolerance on mesh size intro-
duces a measure of uncertainty on the band-
11.6.3 Rosin-Rammler distributions
widths. and clearly irregularly shaped particles
Some distributions are extremely skewed. for with one or more dimensions larger than the
example. ground coal. Rosin and Rammler have nominal size could still pass through. It is cus-
developed a further method for obtaining a tomary therefore to quote particle size when siev-
straight-line cumulative percentage frequency ing in terms of “sieve” diameters.