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Module 3: Characteristics of Particles - Size Distribution
Interpretation of Raw Data
Size Distribution
Lognormal Size Distribution
Practice
Problems
Objectives
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Determine if a particle size distribution is lognormal.
- Determine the mass median particle diameter and standard deviation of a lognormal particle size distribution.
Particulate emissions from both manmade and natural sources do not consist of particles of any one size. Instead, they are composed of particles over a relatively wide size range. It is often necessary to describe this size range. Particulate matter for size distribution evaluation is measured in a variety of ways. The data must be measured in a manner whereby it can be classified into successive particle diameter size categories.
In the case of cascade impactors, particulate matter is separated into diameter size categories within the impactor head during sampling. The mass of particulate matter contained within each size range is recovered and determined gravimetrically.
Particle count is another method of data interpretation for size
distribution. Figure 1 is a photomicrograph of fly ash particles
collected on a filter. The filter material shown is mixed cellulose
ester (MCE). Fly ash particles tend to have a spherical shape and
their particle diameter is easy to determine. The particles shown here
are in the 1-5 
size range. From the
photomicrograph, the number of particles in a predetermined size range
is tabulated and the distribution plotted.
A histogram (shown in Figure 2) is one of the simplest ways to display a particle size distribution. It is a particle frequency distribution that shows the percentage of particles found in each size range. Frequency can be plotted (on the Y-axis) by number count, surface area, or mass. This lesson focuses primarily on particle distributions based on the mass of the particles. The skewed distribution shown in Figure 2 is typically encountered in the field of air pollution control.
The median, arithmetic mean, and mode help characterize the arithmetic mass distribution (see Figure 2). The median particle size (mass median particle diameter) is the particle diameter that divides the frequency distribution in half; fifty percent of the aerosol mass has particles with a larger diameter, and fifty percent of the aerosol mass has particles with a smaller diameter. The arithmetic mean diameter, usually simply termed the mean diameter, is the arithmetic average particle diameter of the distribution. The value of the arithmetic mean is sensitive to the quantities of particulate matter at the extreme lower and upper ends of the distribution. The mode represents the value that occurs most frequently in a distribution. In particle size distributions, the mode is the particle diameter that occurs most frequently.
When the particle diameters in Figure 2 are plotted on a logarithmic scale against the frequency of occurrence, a bell-shaped curve is generated. As shown in Figure 3 the particle size categories are altered to produce equidistant ranges when plotted on a logarithmic basis. This bell-shaped histogram is called a lognormal curve.
It is not always possible to graph equidistant particle size ranges on a logarithmic scale. In many cases the size ranges are determined by the data collection method (e.g. data obtained from a cascade impactor).
For many manmade sources, the observed particulate matter distribution approximates a lognormal distribution. Therefore, it is often beneficial to work with particle size distributions on a logarithmic basis.
The terms, geometric mean diameter and geometric standard deviation, are substituted for arithmetic mean diameter and standard deviation when incorporating logarithms of numbers. When the frequency of the particle size distribution is based on mass, the more specific term geometric mass mean diameter is used.
The geometric mass mean diameter is the diameter of a particle that has the logarithmic mean for the size distribution. It is the nth root of the product of n terms. The geometric mass mean diameter is expressed in Equation 1 and Equation 2.
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Where:
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Where:
The geometric mass mean diameter is equivalent to the mass median diameter for lognormal graphs due to the bell-shaped symmetrical curve produced. The geometric standard deviation of a lognormal distribution is determined by dividing the mass median particle diameter by the particle size at the 15.78 percent probability or by dividing the particle size at the 84.13 percent probability by the mass median particle diameter.
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Where:
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Where:
The use of logarithmic paper simplifies the determination of the mass median particle diameter and the geometric standard deviation as shown in the following example.
Example Problem 1.
Calculate the Mass Median Particle Diameter and the Geometric
Standard Deviation for a Particle Distribution
Calculate both the mass median particle diameter and the geometric standard deviation for the following lognormal distribution data.
Solution:
Step 1. Plot the distribution data from Table 1 on log-probability paper.
The straight line indicates that the particle size data set is lognormal.
Step 2. Using the graph, determine the approximate particle size at 15.78, 50, and 84.13 percent probability.
Step 3. Determine the mass median particle diameter.
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Step 4. Determine the geometric standard deviation of the particle mass distribution.
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Particle size distributions resulting from complex particle formation mechanisms or several simultaneous formation mechanisms may not be lognormal. As shown in Figure 5, these distributions may exhibit more than one peak (multi-modal).
In these cases, plots of the data on log-probability paper will not yield a straight line. In order to characterize this type of particle size data, it may be necessary to treat the data as two separate lognormal distributions.
Practice Problems
Size Distribution
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Instructions:
- Complete the Practice Problems before proceeding to the next lesson. Click on the button below.