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