Statistics of Size distributions
Histogram
Mean Diameter
Standard Deviation
Geometric Mean
D 
Nt
D 
moment
1/ 2
NB
 N D D 
i
D 
1
Nt
D 
i 1
i
 N  D D 
NB
1
1/ 2
i
Nt
NB

i
1
Nt
i 1
i 1
D
1/ 2
NB
 n  D D D 
i
i
i
Nt
i 1

2
NB

i 1

N i ln  D i D i  1 
2

1
1
D 
Nt
D 
i 1
The “moments” will come in when you do area, volume distributions
We also define “effective areal” diameter and “effective volume” diameter
n
 Dn ( D ) dD
 (D  D )
 1
D g  exp 
 Nt
n/2
N i  D i D i 1 
1
D 
i 1
i 1
 1
D g  exp 
 Nt
n
nth
1
Continuous dist.
Discrete distribution
1
Nt
2
n ( D ) dD

 ln( D ) n ( D ) dD 

D
n
n ( D ) dD
Effective Diameters
Consider Nt aerosol particles, each with Diameter D. This is a
“monodisperse” distribution.
2
S  N T D
The Total surface area and volume will be:
V  NT

D
3
6
Now consider the total surface area and volume of a polydisperse
aerosol population
2
S   D n( D)dD
V 

Da 
D n( D)dD
S    D n( D)dD  NT  D
2
V 
6
 D n( D)dD  NT
3

6
2
D
3
2
 
Dv  D
3
6
Substituting our definition for moments, we have

D
3
1/ 3
Effective Diameters
Substituting our definition for moments, we have
S  NT  D
V  NT

2
D
3
6
We now define an effective areal diamater, Da, and an effective
volumetric diameter, Dv, which are the diameters that would produce
the same surface area and volume if the distribution were
monodisperse.
Da  D
2
Dv  D
3
2
3
So…
Board Illustration: Consider a population of
aerosols where 900 cm-3 are 0.1 mm, and 100
cm-3 are 1.0 mm. Compute Da, Dv.
S  N T Da
2
V  NT

6
3
Dv
Converting size distributions
Concentration:
D i 1
N i  ni  Di 
D i 1
 n ( D ) dD

Di
 dN ( D )
Di
Rule of thumb: Always use concentration, not number distribution, when
converting from one type of size distribution to another
dN ( D )  n ( D ) dD  n (log D ) d log D  n (ln D ) d ln D
o
e
NOT
n ( D )  n (log D )  n (ln D )
o
e
Converting size distributions
Example:
Let n(D) = C = constant. What are the log-diameter and ln-diameter
distributions?
dN ( D )  n ( D ) dD  CdD
 n (ln D ) d ln D
e
dD
n (ln D )  C
e
 CD
Even though n(D) is
constant w/ diameter,
the log distributions
are functions of
diameter.
d ln D
n (log D )  n (ln D )
0
e
d ln D
d log D
 CD ln 10  2 . 303 CD
Problem:
Let n(D) = C = constant. What is the volumetric number distribution,
dN/dv?
The Power-Law (Junge) Size Distribution
n(D)=C D-a
n(D)=1000 cm-3 mm2 D-3
Linear-linear plot
What is the total concenration, Nt?
What is volume distribution, dV/dD?
What is total volume?
What is log-number distribution?
Major Points for Junge Distr.
Log-Log plot
slope = -3
Need lower+upper bound Diameters
to constrain integral properties
Only accurate > 300 nm or so.
Linear in log-log space….
The log-normal Size Distribution
Note that power-law is simply linear in log-log space, and was unbounded
y  mx  b
ln n   a ln D  ln C
n  CD
a
Let’s make a distribution that is quadratic in log-log space (curvature down)
y  nx  mx  b
2
ln n  
ln
D
2
2
 a ln D  ln C
 ln D   2a ln D 
n  C exp  

2



2
 ln D  a
 C exp  
2

2

n (ln D ) 
e
2
a 
 ln D  ln D g 2
 C 2 exp  
2


2 



n(D ) 
Nt
2  g
 1 ln D  ln D g 2
exp  
2
g
 2
2



ln
D

ln
D
Nt
1
g
exp  
2
g
2  g D
 2






The Log-Normal Size Distribution
n(D ) 
Nt=1000 cm-3, Dg = 1 mm, g = 1.0
 1 ln D  ln D g 2
exp  
2
g
2  g D
 2
Nt



What is the total concenration, Nt?
Linear-linear plot
What is volume distribution, dV/dD?
What is total volume?
What is log-number distribution?
Major Points for Log-normal
3 parameters: Nt, Dg and g
Log-Log plot
No need for upper/lower bound
constraints  goes to zero both
ways
Usually need multiple modes.
More Statistics
Medians, modes, moments, and means from
lognormal distributions
Median – Divides population in half. i.e. median
of # distribution is where half the particles
are larger than that diameter. Median of
area distribution means half of the area is
above that size
Mode – peak in the distribution. Depends on
which distribution you’re finding mode of
(e.g. dN/dlogD or dN/dD). Set dn(D)/dD = 0
Secret to S+P 8.7… Get distributions in form
where all dependence on D is in the form
exp(-(lnD – lnDx)2). Complete the squares to
find Dx. Then, the median and mode will be
at Dx due to the symmetry of the distribution
The means and the moments are properties of
the integral of the size distribution. In the
form above, these will appear outside the
exp() term. (i.e. what is leftover after
completing the squares).
n (ln D ) 
Nt
e
n(D ) 
2  g
 1 ln D  ln D g 2
exp  
2
g
 2
 1 ln D  ln D g 2
exp  
2
g
2  g D
 2
Nt
n(D ) / 4
e






n (ln D )
Standard 3-mode distributions
Typical measured/parameterized
urban size distributions
Southern AZ size distributions
Vertical distributions
Often aerosol comes
in layers
Averaged over time,
they form an
exponentially decaying
profile w/ scale height
of ~1 to 2 km.
Particle Aerodynamics
S+P Chap 9.
Need to consider two perspectives
• Brownian diffusion – thermal motion of particle, similar to gas
motions
• Forces on the particle
– Body forces: Gravity, electrostatic
– Surface forces: Pressure, friction
Relevant Scales
•
•
Diameter of particle vs. mean free path in the gas – Knudsen #
Inertial “forces” vs. viscous forces – Reynolds #
Knudsen #
Kn 
2
 = mean free path of air molecule
Dp
Dp = particle diameter
 
Gas molecule selfcollision cross-section
1
2  B N B
Gas # concentration
Quantifies how much an aerosol particle influences its immediate
environment
• Kn Small – Particle is big, and “drags” the air nearby along with it
• Kn Large – Particle is small, and air near particle has properties about the same as the gas
far from the particle
Kn
Free Molecular
Regime
Transition
Regime
Continuum
Regime