Chemical Engineering April 2012 - 43

EXAMPLE OF A SAMPLING PROBLEM, WITH SOLUTION
A
fter several customer complaints, an engineer is assigned the responsibility of
setting up a sampling protocol for a ground product that frequently does not
meet the specification that no more than 5% of the mass, or volume distribution
should be greater than 250 microns (Figure 1). This product is sold in lots consisting of
several tons. This specification should be measured at the 99% confidence level. This
product has a density of 2.5 g/mL. Assuming that correct sampling techniques were
used to obtain a random sample, what is the minimum sample size that needs to be
collected and analyzed?
Solution:
1. Since the mass of the sample is much smaller that the mass of the lot, the equation
for the fundamental error estimation [Equation (1)] can be rearranged as follows to
solve for the minimum sample mass
100
(4)
2. Measure the size distribution on a volume, or mass basis to obtain the diameters
corresponding to the 5th and the 95th percentile (Figure 1)
250 275
Diameter (micron)
3. The 99% confidence level implies that the value of FE is 0.01. The variance of the
fundamental error, Var(FE), is 0.012, or 0.0001. The shape factor (ƒ) can be set at
0.5, assuming that the particles can be approximated by spheres. The particle density (ρ) is 2.5 g/cm3. The fraction of material in
the size class of interest cL is 0.05 (5% > 250 microns). The average diameter in the size class if interest (d1) can be taken as 275
microns (see Figure 1). The granulometric factor (g) is defined as d05/d95 (see Figure 1) is 0.40 for this distribution. Finally, d,
defined as the 95th percentile of the distribution (see Figure 1) is 250 microns. Changing all units to CGS units to obtain the sample
mass (MS) in grams, we obtain the following:
FIGURE 1. Example of size distribution
with information necessary to calculate
minimum sample mass
(5)
Please note that not only a sample of 4.8 g is needed, but an analysis technique that can analyze the whole sample needs to be utilized.
❏
increases. Since the variance is
equal to the square of the fundamental
sampling error, the fundamental
sampling error decreases in
proportion to the square root of the
sample mass
2. The variance of the fundamental
error is a strong function of the

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Chemical Engineering April 2012

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