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Module 3: Characteristics of Particles - Size Distribution - Practice Problems

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Instructions:
Work these problems on a sheet of paper and check your answers against those provided below.

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Q icon #1


During sampling, fly ash particles were collected on a filter. A photomicrograph, similar to Figure 1, was taken of the filter for a particle count. A particle count and measurement were performed using a scale at the base of the photomicrograph. The results from the particle count are categorized by size in Table 2.


Table 2

  1. Using the data in Table 2, plot a probability size distribution histogram similar to Figure 2.

    Figure 2

  2. Complete Table 3 by grouping the particles in Table 2 into new size categories. Plot a probability size distribution histogram on a logarithmic basis on the graph shown below. Does the histogram represent a lognormal distribution? Where are the mass median diameter and the geometric mean diameter located?

Table 3

Figure 6

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Answer: i. See graph below (Figure 7).

Solution:
Step 1. Calculate the frequency of particles in each category from Table 2.

Practice Equation 1i(1)

Table 4

Step 2. Graph the histogram.

Figure 7

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Answer: ii. Size distribution is lognormal (see Figure 8). The mass median diameter and the geometric mean diameter are in the 3.79 to 5.92 micrometer range.
Solution:
Step 1. Using the data in Table 2, determine the number of particles and the frequency percent for each size category shown in Table 3.
Practice Equation 1ii(1)

Table 5

Step 2. Graph the histogram.

Figure 8

The mass median diameter and the geometric mean diameter are equivalent because the graph is lognormal. They occur within the 3.79 to 5.92 diameter range.

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Q icon #2
Given the particle size data in Table 6, verify that the distribution is approximately lognormal, and determine the mass median particle diameter and the geometric standard deviation.

Table 6

Link to Appendix C (Blank Log-Probability Graph).

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Answer: The particle size distribution is lognormal.

Equation 2


Solution:
Step 1. Determine the minimum particle size in each of the size categories. Calculate the cumulative percentage of the total particulate catch that is less than each of these minimum sizes.

Table 7

Step 2. Plot the data on log probability paper and confirm that it is a straight line (i.e. a lognormal distribution).

Figure 9

Step 3. Determine the mass median particle diameter and the geometric standard deviation.
Practice Equation 2(3a)

The mass median particle diameter is the 50% probability point on the graph.
The geometric standard deviation can be determined by using either the 15.87 or 84.13 probability values. The equation below uses the 84.13 probability point.
Practice Equation 2(3b)

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Size Distribution
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