# Aerosol - Size Distribution

Size Distribution

For a monodisperse aerosol, a single number - the particle diameter - suffices to describe the size of the particles. However, for a polydisperse aerosol, we describe the size of the aerosol by use of the particle-size distribution. This defines the relative amounts of particles present, sorted according to size. One approach to defining the particle size distribution is to use a list of the size of all particles in a sample. However, this approach is awkward to use so other solutions have been found. Another approach is to split the complete size range into intervals and find the number of particles in each interval. This data can then be visualised using a histogram where the area of each bar represents the total number of particles in that size bin, usually normalised by dividing the number of particles in an interval by the width of the interval and by the total number of particles so that the total area is equal to 1 and the area of each bar is equal to the proportion of all particles in that size range. If the width of the bins tends to zero we get the frequency function:

where

is the diameter of the particles
is the fraction of particles having diameters between and +
is the frequency function

Therefore the area under the frequency curve between two sizes a and b represents the total fraction of the particles in that size range:

It can also be formulated in terms of the total number density N:

If we assume the aerosol particles are spherical, we then find that the aerosol surface area per unit volume (S) is given by the second moment:

and the third moment gives the total volume concentration (V) of the particles:

It can also be useful to approximate the particle size distribution using a mathematical function. The normal distribution is not usually suitable as most aerosols have a skewed distribution with a long tail of larger particles. Also for a quantity that varies over a large range, as many aerosol sizes do, the width of the distribution implies negative particles sizes which is clearly not physically realistic. However, the normal distribution can be suitable for some aerosols, such as test aerosols, certain pollen grains and spores.

A more widely chosen distribution is the log-normal distribution where the number frequency is given as:

where:

is the standard deviation of the size distribution and
is the arithmetic mean diameter.

The log-normal distribution has no negative values, can cover a wide range of values and fits observed size distributions reasonably well.

Other distributions which can be used to characterise particle size are: the Rosin-Rammler distribution, applied to coarsely dispersed dusts and sprays; the Nukiyama-Tanasawa distribution, for sprays having extremely broad size ranges; the power function distribution, which has been applied to atmospheric aerosols; the exponential distribution, applied to powdered materials and for cloud droplets the Khrgian-Mazin distribution.

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