Size Distributions

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Size Distributions

• Many processes and properties depend on particle
  size
• Fall velocity
• Brownian diffusion rate
• CCN activity
• Light scattering and absorption
• Others
• There are a number of quantitative ways to
  represent the size distribution
• Histogram
• Number Distribution
• Number Distribution function
• Volume, area, mass distributions
• Cumulative distributions
• Statistics of size distributions Median, mode,
  averages, moments
• Others
• We will review a number of these next
• Our primary goal is to explain the Number
  Distribution Function, which is the most widely
  used

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Size DistributionsThe Histogram
Simplest form of distribution Very
instrument-based
Lots of structure at small sizes Few particles at
largest sizes
Ni
Di Di1
NB Number of size bins Di Lower-bound
particle diameter for bin i Di1 Upper-bound
particle diameter for bin i Ni concentration
of particles in bin i (cm-3)
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Size DistributionsCumulative Properties from
Histogram
Total Concentration Total Surface Area Total
Volume Total Mass
Ni
Di Di1
ri density of aerosol substance in bin i
Note We dont have an average diameter for the
bin only the bin boundaries. Above I use the
geometric mean. Sometimes it makes sense to
estimate where the particles are within the bin
based on the concentrations of neighboring bins,
and then calculate the effective mean diameter.
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Size DistributionsCumulative Distributions
Cum. Concentration Cum. Surface Area Cum.
Volume Cum. Mass
Ni
Di Di1

• A Cumulative distribution gives the concentration
  (or some other property) of particles smaller
  than diameter Di
• Cumulative values are properties at bin
  boundaries, not bin centers!
• They are monotonically increasing in size
• N(DNB1) Nt
• Different instruments should report the same
  function, just sampled differently

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Size DistributionsThe Number Distribution
More uniform way to present instrument data
Ni
Di Di1
ni aerosol number distribution for bin i DDi
Di1Di is the bin width Ni niDDi ni has
units of (cm-3 mm-1)
Area under the curve total aerosol
concentration, N
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Size DistributionsThe Number Distribution
Histogram
Number distribution
Small bin width at small sizes leads to
amplification of concentrations here relative to
histogram
Instruments with different Di would produce very
different histograms, but similar number
distributions
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Size DistributionsThe Log Number Distribution
Aerosol distributions span orders of magnitude in
size, and are often best shown as a function of
log-diameter. Now, the area under curve is NOT
equal to total concentration. To remedy this, we
can create a log number distribution (not shown
above)
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Size DistributionsThe Number Distribution
Function
Distributions are often represented in models or
analytically, as continuous functions of
diameter. This is as if we had an number
distribution with perfectly precise resolution
This looks a lot like the definition of the
derivative. If we use the cumulative
distribution, we get
We think of the number distribution function as
the derivative with diameter of the cumulative
distribution When n(D) is plotted vs. D (NOT
logD), then the area under the curve total
concentration
The log-diameter distribution is the derivative
of the cumulative distribution with log of
diameter
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Size DistributionsOther Distribution Functions
Number Distribution Surface Distribution Volume
Distribution Mass Distribution
Aerosol distributions span orders of magnitude in
size, and are often best shown as a function of
log-diameter. We must use the identity This
lowers the power of Dn in the functions above.
Note the shifting of the peaks from number ?
area ? volume
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Statistics of Size distributions
Histogram
Continuous dist.
Discrete distribution
Mean Diameter Standard Deviation Geometric
Mean nth moment
The moments will come in when you do area,
volume distributions
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More Statistics
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In-class

• Power-law distributions
• Log-normal distributions
• Properties of each