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Statistics of Size distributions
The “moments” will come in when you do area, volume distributions
We also define “effective areal” diameter and “effective volume” diameter
2/1
11
1
BN
iiii
t
DDNN
DMean Diameter
Standard Deviation
Geometric Mean
nth moment
Histogram Discrete distribution Continuous dist.
2/1
11
1
BN
iiiii
t
DDDnN
D dDDDnN
Dt
)(1
BN
iiii
tD DDDN
N 1
22/11
1 dDDnDDN t
D )()(1 2
BN
iiii
tg DDN
ND
11ln
2
11exp
dDDnD
ND
tg )()ln(
1exp
2/
11
1nN
iiii
t
nB
DDNN
D
dDDnDN
D n
t
n )(1
Effective DiametersConsider Nt aerosol particles, each with Diameter D. This is a
“monodisperse” distribution.
The Total surface area and volume will be:
Now consider the total surface area and volume of a polydisperse aerosol population
Substituting our definition for moments, we have
3/13
2
DD
DD
v
a
33
22
6)(
6
)(
DNdDDnDV
DNdDDnDS
T
T
3
2
6DNV
DNS
T
T
dDDnDV
dDDnDS
)(6
)(
3
2
Effective Diameters
Substituting our definition for moments, we have
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.
So…
33
22
DD
DD
v
a
3
2
6DNV
DNS
T
T
3
2
6 vT
aT
DNV
DNS
Board Illustration: Consider a population of aerosols where 900 cm-3 are 0.1 m, and 100 cm-3 are 1.0 m. Compute Da, Dv.
Converting size distributions
)()(11
DdNdDDnDnNi
i
i
i
D
D
D
D
iii
Concentration:
Rule of thumb: Always use concentration, not number distribution, when converting from one type of size distribution to another
DdDnDdDndDDnDdN eo ln)(lnlog)(log)()(
)(ln)(log)( DnDnDn eo
NOT
Converting size distributionsExample:
Let n(D) = C = constant. What are the log-diameter and ln-diameter distributions?
CdDdDDnDdN )()(
Problem:Let n(D) = C = constant. What is the volumetric number distribution,
dN/dv?
Even though n(D) is constant w/ diameter, the log distributions are functions of diameter.
CDCD 303.210ln
DdDne ln)(ln
CDDd
dDCDne
ln)(ln
Dd
DdDnDn e
log
ln)(ln)(log0
The Power-Law (Junge) Size Distribution
n(D)=1000 cm-3 m2 D-
Linear-linear plot
Log-Log plot slope = -3
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.
Need lower+upper bound Diameters to constrain integral properties
Only accurate > 300 nm or so.
Linear in log-log space….
n(D)=C D-
The log-normal Size Distribution
2
2
2
2
lnln
2
1exp
2)(
lnln
2
1exp
2)(ln
g
g
g
t
g
g
g
te
DD
D
NDn
DDNDn
Note that power-law is simply linear in log-log space, and was unbounded
CDn
CDn
bmxy
lnlnln
Let’s make a distribution that is quadratic in log-log space (curvature down)
2
lnlnexp
22
lnexp
2
ln2lnexp
lnln2
lnln
2
2
22
2
2
2
gDDC
DC
DDCn
CDD
n
bmxnxy
The Log-Normal Size Distribution
Linear-linear plot
Log-Log 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 Log-normal
3 parameters: Nt, Dg and g
No need for upper/lower bound constraints goes to zero both ways
Usually need multiple modes.
Nt=1000 cm-3, Dg = 1 m, g = 1.0
2
2lnln
2
1exp
2)(
g
g
g
tDD
D
NDn
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).
2
2
2
2
lnln
2
1exp
2)(
lnln
2
1exp
2)(ln
g
g
g
t
g
g
g
te
DD
D
NDn
DDNDn
)(lnDne4/)(Dn
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 AerodynamicsS+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 #
pDKn
2 = mean free path of air molecule
Dp = particle diameter
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
BBN
2
1
Gas molecule self-collision cross-section
Gas # concentration
Free MolecularRegime
ContinuumRegime
TransitionRegime