Particle size statistics

Impactors and Particle Size Distribution (2)

Ju-Hyeong Park, Sc.D., M.P.H., C.I.H.

National Institute for Occupational Safety and Health

Division of Respiratory Disease Studies

Field Studies Branch

Aerosol measurement and size distribution

Diameter

Concentration

1. Concentration

Particle mass, surface area

or number per unit

volume

2. Size distribution

Concentation versus particle size

Increasing

Complexity

Aerosol particle sizing

0.1 0.2 0.4 0.7 0.9 1.0 1.2

Sample

Size particles

Assign to size bin

Bins

Diameter / µm 0.4 µm < d < 0.7 µm

Conversion of a discrete particle size distribution

to a continuous distribution

Particle size bins

Nu m b er concent ra tion 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 1 2 3 4 5 6 Number C oncentation / B in Particle Diameter / Particle Diameter / Continuous Distribution Number concentation is proportional to the

0 0.0 0. 0.1 0. 0.2 0. 0.3 0. 0 1 2 3 4 5 6

Number Concentation / Bin

Particle Diameter / Particle Diameter /

Continuous Distribution

The smooth continuous distribution is obtained by joining the bin mid-points dn /d( d)

Continuous graph

™

Y-axis

‰

Differential particle concentration (or number)

¾

Particle number normalized by range of particle diameter

of the interval (or bin)

¾

d

n

/d

d

™

X-axis

‰

Particle diameter range of the interval

¾

d

d

™

Integrated area under the curve=total # of particle

d

n

d

d

Properties of particle size distribution

™Asymmetrical (or skewed) distribution

‰

Long tail to the right

¾

Large number (fraction) of small particles

¾

Small number (fraction) of large particles

‰

Large range of particle size

¾

Several orders of magnitude in particle diameter

‰

No negative particle size

™

Mode < Median < Mean

™

Geometric mean (

d

_g

or GM)

Log

d

_g

=(

Σ

n

_i

log

d

_i

)/N

Æ

d

_g

=exp{(

Σ

n

_i

log

d

_i

)/N}

Arithmetic mean and GM

™Arithmetic mean

‰

d

_a

=(

Σ

n

_i

d

_i

)/N = (summation of all areas of the

bars) / (total number of particles)

™Geometric mean

0 0.0 0. 0.1 0. 0.2 0. 0.3 0. 0 1 2 3 4 5 6

Number Concentation / Bin

Particle Diameter / Particle Diameter /

Continuous Distribution

The smooth continuous distribution is obtained by joining the bin mid-points

dn

/d(

d)

Asymmetric (skewed) distribution

0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 d n /d ( d ) Diameter / µm

Lognormalormal Size Distribution Normalized distribution (n=1) Area inder curve = 1

CMD

Modal Diameter

Note that when plotted as dn/d(d), the modal and count median diameters of a lognormal distribution are different

Distribution skewed to smaller particle diameters

Mode

Geometric mean (Median)

Arithmetic mean

0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 d n/d( d ) Diameter / µm +σ -σ d p 68% of all particles are between ± σ

Normal Size Distribution Normalized distribution (n=1)

Area inder curve = 1

Distribution characterized

by n, and d p σ

Normal distribution

Mean = Mode = Median

Normal distribution

™

⎯

dp: arithmetic mean particle diameter

™

σ

: standard deviation

™

ddp

: particle diameter interval

™

df

: frequency of occurrence of particles of

diameter

dp

df

=

n

σ

2

π

e

−

(

d

p

−

d

p

)

2

2

σ

2

_dd

p

™

Y-axis

‰

Differential particle concentration (or number)

¾

Particle number normalized by range of log-transformed

particle diameter of the interval (or bin)

¾

d

n

/dlog(

d

)

™

X-axis

‰

Range of log-transformed particle diameter of the

interval

¾

dlog(

d

)

™

Integrated area under the curve=total # of particle

‰

∫

{d

n

/dlog(

d

)}dlog(

d

)

Log transformation of continuous graph

Lognormal distribution with arithmetic scale

0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 d n /d ( d ) Diameter / µm

Lognormalormal Size Distribution Normalized distribution (n=1) Area inder curve = 1

CMD

Modal Diameter

Note that when plotted as dn/d(d), the modal and count median diameters of a lognormal distribution are different

Distribution skewed to smaller particle diameters

Mathematical function of lognormal

distribution

df

=

n

2

π

Log

(

σ

_g

)

e

−

(

Log d

( )

−

Log CMD

(

)

)

2

Log

( )

σ

dLog d

( )

_p

df

=

n

σ

2

π

e

−

(

d

−

d

)

2

σ

_dd

p

Two ways of log-transformed graph

™Transform the original particle size data using

logarithm, and then plot them on normal

arithmetic scale of the graph

‰

To calculate all statistics mathematically

‰

Exponentiate log-transformed statistics

™

Transform x-axis scale of the graph, and then

plot the original particle size data on it

‰

Do not transform the data

0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 d n/d( d ) Diameter / µm +σ -σ d p 68% of all particles are between ± σ

Normal Size Distribution Normalized distribution (n=1)

Area inder curve = 1

Distribution characterized

by n, and d p σ

Log normal distribution (first approach)

Mean = Mode = Median

Lognormal distribution with log scale

0 0.5 1 1.5 0.01 0.1 1 10 100 d n/d L og( d ) Diameter / µm CMD ×σ g CMD/σ g 68% of all particles are between CMD / σ g and CMD × σ g

Lognormal Size Distribution

Normalized distribution (n=1) Area inder curve = 1

Distribution characterized by n, CMD and σ

g

Count Median Diameter (CMD)

16% of all particles are less than CMD/σ_g

84% of all particles are less than CMD*σ_g 50% of all particles are less than CMD

Cumulative size distribution

Area inder curve = 1

CMD 84%

16% 50%

Probability scale of lognormal distribution

™Percent of particles less than a given particle

diameter

‰

<CMD/(2

σ

_g

)- 5%

‰

<CMD/

σ

_g

- 16 %

‰

<CMD (median)- 50%

‰

<CMD*

σ

_g

- 84%

‰

<CMD*(2

σ

_g

)- 95%

™

Any pattern??

‰

Symmetry of probability

Lognormal distribution with log scale

Lognormal Size Distribution

Normalized distribution (n=1) Area inder curve = 1

Distribution characterized by n, CMD and σ

g

Count Median Diameter (CMD)

16% of all particles are less than CMD/σg

84% of all particles are less than CMD*σg

50% of all particles are less than CMD

Cumulative plot

to

log-probability plot

0 0.2 0.4 0.6 0.8 1 0.1 1 10 Fr actio n o f p ar ticles smaller th an d Diameter / µm CMD ×σ g CMD/σ_g Lognormal Size Distribution Normalized distribution (n=1)

Area inder curve = 1

CMD 84%

16% 50%

Log-probability paper

Probability scale (probit)

1 2 5 10 20 30 40 50 60 70 80 90 95 98

Log

scal

e

(

par

ti

cle size:

µ

m)

Log-probability plot of lognormal distribution

0.01 0.1 1 10 .001 .01 .1 1 5 10 20 30 50 70 80 90 95 99 99.9 99.99 99.999 Di am et er / µm

% Distribution less than diameter d (probit)

16% 50% 84% CMD ×σ g CMD/σ g CMD Lognormal Size Distribution

1

2

3

1/2 =

σ

= 2/3

Count median diameter and

σ

_g

™CMD (count median diameter)

‰

50% of all particles are less than CMD

™Geometric standard deviation

‰

CMD*

σ

_g

/CMD or

‰

CMD/(CMD/

σ

_g

)

CMD, SMD, and MMD

Log-probability plots for count, mass, and surface

area lognormal distribution

0.01 0.1 1 10 100 .001 .01 .1 1 5 10 20 30 50 70 80 90 95 99 99.9 99.99 99.999 Number weighted Surface-area weighted Mass weighted Diameter / µm

% Distribution less than diameter d (probit)

16% 50% 84%

Plots are parallel - σ

g

(given by gradient) is the same for each weighting

Cascade impactor data reduction

0

0.70

1.2

2.2

4.0

9.0

d

₅₀

(

µ

m)

0-0.7

0.7-1.2

1.2-2.2

2.2-4.0

4.0-9.0

>9.0

100.0

16.4

Total

3.7

3.7

0.6

79.3

78.7

Back

filter

18.9

15.2

2.5

855.1

852.6

5

56.7

37.8

6.2

853.6

847.4

4

88.4

31.7

5.2

861.0

855.8

3

99.4

11.0

1.8

844.1

842.3

2

100.0

0.6

0.1

850.6

850.5

1

An example using count data

Data

Plotting upper size range vs. fraction/µm

Log scale of particle diameter (lognormal distribution)

Particle size (

µ

m)

Fr

act

ion/

µ

m

Cumulative lognormal plot (cum vs. size)

Particle size (

µ

m)

Cumu

lative frac

tion

(%)

Cumulative fraction in probability scale/

µ

m

1 2 5 10 20 30 40 50 60 70 80 90 95 98

Pa

rticle

s

ize

in

lo

g

sc

ale

(

µ

m)

Log-probability plot of count data

84%

50%

Take-home practice

Stage cut-point (diameter d)

Mass concentration Cumulative mass concentration greater than

diameter d

% Cumulative mass concentration greater than

diameter

% Cumulative mass concentration less than

diameter (µm) (mg/m3_{) (mg/m}3₎ 10 0.4209 5.62 1.3330 3.16 3.3770 1.78 6.4771 1 9.5248 0.5 12.7579 0.285 8.1861 0.158 5.2389 0.05 2.9853 Sum (mg/m3₎

1. Fill out the blank columns.