Size Distributions
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Many processes and properties depend on particle size
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Fall velocity
Brownian diffusion rate
CCN activity
Light scattering and absorption
Others
There are a number of quantitative ways to represent the size distribution
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Histogram
Number Distribution
Number Distribution function
Volume, area, mass distributions
Cumulative distributions
Statistics of size distributions: Median, mode, averages, moments…
Others
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We will review a number of these next
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Our primary goal is to explain the Number Distribution Function, which is the most
widely used
Size Distributions
The Histogram
Simplest form of distribution – Very instrument-based
Lots of structure at small sizes
Few particles at largest sizes
Di –Di+1
Ni
i  1, 2, . . . N B
NB = Number of size bins
Di = Lower-bound particle diameter for bin i
Di+1 = Upper-bound particle diameter for bin i
Ni = concentration of particles in bin i (cm-3)
Size Distributions
Cumulative Properties from Histogram
NB
Total Concentration
Nt   Ni
i 1
NB
Total Surface Area
Total Volume
Di –Di+1
i 1
NB
Vt  
i 1
Ni
i = density of aerosol substance in bin i
St  Di Di 1 Ni
Total Mass
Note: We don’t 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.

D D 
6
i
NB

i 1
6
Mt  
3/ 2
i 1
Ni
i Di Di 1  N i
1/ N
 N

Ag    Ai 
 i 1 
1 N

 exp  ln Ai 
 N i 1

3/ 2
Size Distributions
Cumulative Distributions
i 1
Cum. Concentration
N ( Di )   N j
j 1
i 1
Cum. Surface Area
S ( Di )   D j D j 1 N j
j 1
i 1
Cum. Volume
i
Di –Di+1
Ni
N ( Di 1 )   N j
V ( Di )  
j 1
j 1
Cum. Mass

D D 
6
i 1

j 1
6
M ( Di )  
3/ 2
j
j 1
Nj
 j D j D j 1  N j
3/ 2
A Cumulative distribution gives the concentration (or some other property) of particles
smaller than diameter Di
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Cumulative values are properties at bin boundaries, not bin centers!
They are monotonically increasing in size
N(DNB+1) = Nt
Ni  N ( Di 1 )  N ( Di )
Different instruments should report the same function, just sampled differently
Size Distributions
The Number Distribution
More uniform way to present instrument data
i
Di –Di+1
Ni
N ( Di 1 )   N j
j 0
ni = aerosol number distribution for bin i
DDi = Di+1–Di is the bin width
Ni = niDDi
ni = has units of
ni
(cm-3
mm-1)
i
N ( Di 1 )   n j DD j
j 0
Area under the curve = total aerosol concentration, N
Size Distributions
The 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
Size Distributions
The Log Number Distribution
ni
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)
ni0 
Ni
Ni

D log Di logDi 1 / Di 
Size Distributions
The 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
n( D p )  lim
Di1  Di
Ni
Di 1  Di
This looks a lot like the definition of the derivative.
If we use the cumulative distribution, we get…
ni
The log-diameter distribution is the derivative of
the cumulative distribution with log of diameter
n0 ( Dp ) 
dN ( D) 
dN( D) 
  2.303D

d log D 
dD 
n( D ) 
N Di 1   N Di 
dN( D)
 lim
Di1  Di
dD
Di 1  Di
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
Size Distributions
Other Distribution Functions
n( D ) 
Number Distribution
dN
dD
Surface Distribution
nS ( D ) 
dS
 D 2 n( D)
dD
Volume Distribution
nV ( D) 
dV  3
 D n( D )
dD 6
Mass Distribution nM ( D) 
dM 
  ( D ) D 3 n( D )
dD 6
Aerosol distributions span orders of magnitude in size, and are often best shown as a function
of log-diameter. We must use the identity nN0 (logD)  2.303DnN ( D)
This lowers the power of Dn in the functions above.
Note the “shifting of the peaks” from number  area  volume
Statistics of Size distributions
Histogram
Mean Diameter
Standard Deviation
1
D
Nt
Discrete distribution
1/ 2
NB
 N D D 
i 1
D 
1
Nt
Geometric Mean
1
Dg  exp
 Nt
nth moment
1
Dn 
Nt
NB
i
1
D
Nt
i 1
i
 N D D 
NB
i 1
1/ 2
i
i 1
i
D
1/ 2
NB
 n DD D D 
i 1
Continuous dist.
i
i
i
D
i 1

2

1


N
ln
D
D

i
i i 1 
2
i 1

NB
n/2
 N i Di Di 1 
i 1
The “moments” will come in when you do area, volume distributions
D 
1
Dn( D)dD
Nt 
1
( D  D ) 2 n( D)dD

Nt
1

Dg  exp  ln(D)n( D)dD
 Nt

Dn 
1
D n n( D)dD

Nt
More Statistics
In-class…
• Power-law distributions
• Log-normal distributions
• Properties of each