Particle and Aerosol Technology_3_2019 .pdf

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Center for Nano Particle Control

Seoul National U., Mechanical & Aerospace Eng.

6 ) (

3

dp

 (  d

_p²

)

For spheres, once you know the diameter, then you can determine others such as volume , surface are . For non-spheres, particle volume and surface

area are independent.

First, let us consider spherical particles. ⇒ n(d

_p

) Often we need to tell about “Some representative particles”

instead of detailed distributions.

- Mean diameters

① mean (arithmetic average)

 



  

^



0

) (

N dd d d n n

d n N

d d

^p ^p ^p

i i i i

p

② median: diameter for which one half the total number of particles are smaller and

one half are larger

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mode median n(dp)

- or the diameter to cut half the area under the size distribution function curve





^d ^p _p _p

p

f d dd

d

F ( )

0

( )

p

dd

d dF f ( ) 

- or the diameter at which the cumulative distribution function is equal to 0.5

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Center for Nano Particle Control

③ mode: most frequent size: the diameter that have the highest number

) 0

( 

p p

dd d dn

If particle size distribution function is symmetric (like normal distribution), mean= median = mode.

However, we usually have asymmetric distribution with the tail at larger sizes.

mode < median < mean

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In real case most aerosols do not have symmetric distribution

: skewed distribution (if particle formation ends)

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1. If particle size distribution has one mode, uni-modal distribution

2. Well grown single component aerosol two-modes

: bi-modal: formation + growth

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nucleation

mode accumulation mode

dp

n(dp)

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3. three-modes: tri-modal: atmospheric aerosols generated by different mechanisms

① formation ② growth ③ generation by mechanical means

n(dp)

dp

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Multi-modal nature of atmospheric aerosols

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N

g

d d

d  (

₁ ₂

...)

¹

n N n

d

₁ ₂

...)

¹

(

¹ ²



N d d

_g

  n

ⁱ

ln

ⁱ

ln

④ geometric mean

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* Log-normal distribution - normal distribution

 





 



 





₂

2

) exp (

) 2

(   

p p

p

d N d

d n



² ¹²

1 ) (

 





 







   N

d d

n

_i _i _p

: standard deviation

d

N d n

_i _i

 

: mean diameter

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However, in real case most aerosols do not have symmetric distribution

: skewed distribution (if particle formation ends)

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n(dp)

dp

 





 



 





g g p

g p

d N d

d

n  

²



2

ln 2

) ln exp (ln

ln ) 2

(ln

N d d

_g

  n

ⁱ

ln

ⁱ

ln

Log-norml distribution

12 2

1

) ln

ln (ln

 





 







  

N

d d

n

_i _i _g



g

d

g

: geometric mean diameter



g

: geometric standard deviation

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 0



) 1 (ln

)

(

_p

p

n d

d d

n 



 1



g

Monodisperse aerosols should have or

d

p

ln ln d

p

 d ln d

p

 dN

i

 n (ln d

p

) d ln d

p

p p p

p

dd n d dd

d d

n 1 ( )

)

(ln 

 Particle numbers between and

For log-normal distributions

median

g

d

d 

^,

d

g

, 

_g

To completely define the log-normal distributions, you should know N,

(only three parameters), then you know the complete distributions

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* Log-probability graph

) ( d

_p

F : cumulative distribution function







^ln( ⁾

(ln ) ln )

(ln

_p ^d^p ^p

d d

_p

N d d n

F

 





 



 



g g

p

d

erf d

 ln 2

) ln(

2 1 2

1 if log-normal

g g

p g

d d

F

erf ln

ln 2 ln 1

ln 2 2 1

2 ) ( 1

1



 

 

 

 

 



*Revised

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g g

p g

d d

F

erf ln

ln 2 ln 1

ln 2 2 1

2 ) ( 1

1



 

 

 

 

 



g g

p

d

d ) ln 

ln( 

If

841 . 0 2 )

( 1 2

1 2

1  

 erf

F

*Revised

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% 50

%

84

ln

ln ln

ln

ln 

_g

 d

_p

 d

_g

 d  d

% 50

% 84

d d

g





lndp







 

1 2

2) (F 1 erf

CMD(Count median diameter) or Dg ( geometric mean diameter can be determined from this log-probability graph by reading the diameter

corresponding to 50% cumulative farction. GSD( geometric standard deviation) can be also determined directly from this graph by finding the diameter corresponding to 84%

cumulative fraction.

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The cumulative fraction scale ( or percent) is converted to a

probability scale. This probability scale compresses the percent scale near the median ( 50%) and

expands the scale near the ends such that a cumulative plot of a log-normal distribution will yield a straight line. When the particle diameter scale is logarithmic, the graph is called a log-probability plot. When the size scale is linear, the graph will yield a straight line for a normal distribution and is called a probability graph.

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Steepness of the curve on log-probability curve indicates “broadness of the log-normal

size distribution”. Less steep curve represents ⇒ narrower distribution (with small 

_g

)

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Up to this point, we have dealt with number distribution as a function of particle size.

Sometimes, we may need to define mass distribution, surface area distribution as a

function of particle size. For example, catalytic application of particles, surface area is

an important factor, and the surface area distribution is needed to be known.

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) (

)

(

_p _p² _p

s

d d n d

n   ㎛

2

/ cm

²

· ㎛



^



0

n

_s

( d

_p

) dd

_p

S ⇒ total surface area concentration

) 6 (

)

(

_p ³_p _p

v

d d n d

n  

㎛

³

/ cm

³

· ㎛

p p

v

d dd

n

V ( )



0^

 : total volume concentration

⇒ Weighted Distributions

) (

)

(

_p _q _p^q _p

q

d a d n d

n 

Since particle diameters vary over several orders, horizontal axis is generally using log-scale: log

d

p

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Center for Nano Particle Control

p p

q p

p

q

d d d n d dd

n (log ) log  ( )



) ( 303

. log 2

) ( )

(log

_p _q _p

p p p

q p

q

d n d

d d

d dd n

d

n  

For example) if you know can you find n

_m

(m ), n

_m

(log d

_p

) ?

p p

m

m dm n d d d

n ( )  (log ) log

3

6

^p

d

^p

m   

p p

p

d dd

dm 3

²

6 

 

p p

p

d

d dd

d log  2 . 303

) ( 9

. 6 )

(log d mn m

n

_m _p



_m



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If you know n(m) instead of n

_m

(m ) , then

) ( )

( m mn m

n

_m



) ( 9

. 6 )

(log d m

²

n m

n

_m _p





- Moment average

Definition of moments of distribution function



^



0

(

_p

)

_p^v _p

v

n d d dd

M

Zeroth moment M

0

 

0^

n ( d

_p

) dd

_p

 N

First moment

0 0 1

1

n ( d ) d dd : d M M

M  

^ _p _p _p _p



: mean diameter

*Revised

The need to use the moment average for particle statistics arises because aerosol size is frequently measured indirectly. For example, if you have a basket of apples of different sizes, you could determine the average size by measuring each apple with calipers, summing the results and dividing by the total numbers of apples. This is the direct measurement. If, however, each apple were weighted on a balance and the

weights summed and divided by their number, the average mass would be first obtained. Then assuming the apple as spheres, you can indirectly calculate the average diameter of apple corresponding to the average mass.

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Center for Nano Particle Control

S M

₂





  

^



0

2 2

2 ⁱ

(

_p

)

_p _p

s

n d d dd

N N

d  d 





^



0

2

n ( d

_p

) d

_p

dd

_p

2nd moment M

0 2

2

M M

d

s

 : 2nd moment average

d

s

: diameter of average surface area

N M d

p p m

3

6

3

6

 

  



^



0

3

n ( d

_p

) d

_p

dd

_p

3rd moment M

0 3

3

M M

d

_m

 : 3rd moment average

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p P i i

p

N

d d n

1

 





 



  

* Weighted mean diameters - arithmetic mean (count mean)

0 1

M M N

n d

_p

  d

ⁱ ⁱ



- surface area mean diameter

2 3 2

2

M M d

n

d d d n

i i

i i i

sm

 

 



1

: sauter diameter (mean volume-surface diameter) d

sm

d

3

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- mass mean diameter

3 4 3

3

6 6

M M d

n

d d n d

d

i i p

i i i p

mm

 



 



S d M

p

sm

6 )

( 



S/M: specific surface area can be measured by nitrogen gas adsorption, so called, BET method then you can measure d

_sm

.

- If you have log-normal distribution or you force or approximate the distribution

as log-normal, then you can easily obtain other mean values from one mean value.

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Weighted Distributions

If number distribution is log-normal, which distributions do weighted distributions have ?

since

) (

)

(

_p _p² _p

s

d d n d

n   ( )

) 6

( d

_p

d

_p³

n d

_p

n 





) ( )

(

_p _q _p^q _p

q

d a d n d

n 

 

 





 



 







g g p

g p

q p q

p q

d d

d Nd d a

n  

²



2

ln 2

ln exp ln

ln 2

) (



_p



q

p

q d

d  exp ln

 

 

 



 

  

 



 



 



^q _g _g ^p ^g ^g

p q

q d

q d d

N q d a

n 

 





²

2 2 2

2

ln 2

ln ln

exp ln 2 ln

ln exp

)

(

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is also log-normal with geometric standard deviation, and median diameter,

) (dp n

_q

 

_g

g median g

g

d q

d ln ln

²



ln  

mass

Same slop i.e. same σ_g lnd_p

MMD CMD

50%

count

Log-probability graph

count mass

lndp

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① : median diameter of weighted distribution e.g. mass median diameter: q=3

② d

_qm

: mean diameter of qth moment distribution e.g. surface area mean: q=2

mass mean: q=3

) ln

exp( q

² _g

CMD

qMD   q

q q

qm

M

d  M

^¹





 



 



 

  



_g

qm

q

CMD

d ln

²



2

exp 1

(35)

③

e.g. diameter of average volume p=3

mode

  



 



 



_p _q

p

CMD p d

d

^p

ln

²



exp 2

1

 

 



 

_g

p

CMD

d ln

²



2 exp 3



g



CMD

d ˆ  exp  ln

²



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Center for Nano Particle Control

Log-normal distribution : different

representative diameters

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Different weighted distributions of

atmospheric aerosols