- 입자상분포함수의 이해

(1)

◌

학과목 집진공학(集塵工學) 담당교수 장혁상 (810-2547)

단원의 주제 입자상물질의 분포함수(Particle Size Distribution) Page 4

- 입자상분포함수의 이해

* 누적율

F a ( ) = ò

₀^a

q d (

_p

) dd

_p

d_p (mm)

0 10 20 30 40 50

Cumulative Fraction (%)

0 10 20 30 40 50 60 70 80 90 100

적색점의 의미는 무엇인가 ?

* 입자분포함수의 형태

d_p (mm)

0 10 20 30 40 50

q(dp) Probability Density Function 0.00 0.02 0.04 0.06 0.08 0.10

수농도 분포 (Number Distribution)

d_p (mm)

0 10 20 30 40 50

Mass fraction/mm

0.00 0.01 0.02 0.03 0.04

질량농도분포 (Mass Distribution)

(2)

◌

- 입자상분포함수의 Fitting

이산분포를 연속분포로 표현하였을 때 연속분포를 수식적으로 가장 잘 표현할 수 있는 함수는 ? 주어진 입자분포를 어떻게 표현할 것인가 ? 대상 입자상물질이 어떤분포함수를 따르는가 ?

.단순정규분포(= Gauss 분포) .대수정규분포

.Rosin-Rammler 분포

* 단순정규분포함수

 _   _   _

_  

 









 _ _^

^

* 대수정규분포함수

 _   _   _

_  

 _ _ 











  _  _^

 _^

* Fitting 함수의 선정

대수-확률지 혹은 정규-확률지 사용에 의한 분포도 평가 .Fitting 되는 종류에 따라 분포결정

.분포용지의 판독으로부터



_

 

_{}

log

_

 log

_{}

 log

_{}



_

  

_{}



_{}

(3)

◌

- 분포함수의 매개변수

* 통계변수

.최빈값 (Mode) / 중앙값 (Median) .평균값(Mean)

▶산술평균: _  



^^^{ } _,

▶기하평균: _{ } 



^___



^

* 입경의 정의 및 상관식

.개수 중앙입경(Number Median Diameter)

주어진 입자분포에서 입자의 갯수누적율이 50%가 되는 점의 입자직경 .질량 중앙입경(Mass Median Diameter)

주어진 입자분포에서 입자의 질량누적율이 50%가 되는 점의 입자직경 .표면적 중앙입경(Surface Median Diameter)

주어진 입자분포에서 입자의 표면적누적율이 50% 가 되는 점의 입자직경

.평균체적 입경(Diameter of the particle with average volume)

__  



  



^^



,  



^^_^^^{ } ^^_^^^{ }^^_^^^{ }^{ }^^{  m ax}_ ^^{  m ax}



n_i : 분포 구간 i를 차지하는 입자의 갯수 v_i : 분포구간 i의 입자의 평균부피 N : 분포전체의 입자갯수

.평균질량 입경(Diameter of the particle with average mass)

  

_

_

  

^^



,  



^^_^ ^{ }^_^ ^{ }^_^   

_{  m ax}



m_i : 분포 구간 i를 차지하는 입자의 질량 N : 분포전체의 입자갯수

ρ_p : 모든 입자의 평균밀도

모든 입자의 밀도가 동일할 경우: d _v = d _m .개수 평균입경(Number Mean Diameter) : d_p_,_n

_{  }



^^_^^^{ } ^^_^^^{ }^^_^^^{ }^{ }^^{  m ax}_ ^^{  m ax}



_ : 분포 구간 i를 차지하는 입자의 갯수

_ : 분포구간 i의 입자의 중간입경

 : 분포전체의 입자갯수

.질량 평균입경(Mass Mean Diameter) : d_p_,_m

_{  }



^

_

_{ } 

_

_{ }

_

_{ } 

_{  m ax}

_{  m ax}



_ : 분포 구간 i를 차지하는 입자의 질량

 : 분포전체의 입자질량

.표면적 평균입경(Surface Mean Diameter) : d_p_,_s

_{  }



^

_

_{ } 

_

_{ }

_

_{ } 

_{  m ax}

_{  m ax}



_ : 분포 구간 i를 차지하는 입자의 표면적

 : 분포전체의 입자표면적

(4)

◌

* Hatch-Choate 변환 방정식(복합 분포 함수) .Median Diameter (중앙입경)

   exp   ln^_ 

 : dimensional weighting factor with respect to _  : 변환대상 입자분포누적중간입경

 : number median diameter ( = 대수정규분포에서 _{ }) _ : 분포의 표준 편차

.Mean Diameter (평균입경)

_   exp

 _{  }



 ln^_



 : dimensional weighting factor with respect to 

_ : 변환대상 입자분포평균입경  : number median diameter  : 분포의 표준 편차

.Hatch-Choate 변환 방정식 적용 예

: Mass Mean Diameter의 경우 입자질량이 d ³에 비례하므로 q=3. 따라서 _{ }  _   exp

 _{  }



 ln^_



: Mass Median Diameter의 경우 입자질량이 d ³에 비례하므로 q=3. 따라서      exp   ln^_ 

: Surface Mean Diameter의 경우 입자표면적이 d ²에 비례하므로 q=3. 따라서 _{ }  _   exp

_{  }



 ln^_



: Surface Median Diameter의 경우 입자표면적이 d ²에 비례하므로 q=2. 따라서    ^  exp ^{ ln}^^ 

: Diameter of particle with average mass와 NMD의 관계    exp  ln^_ 

: Mode(ˆ )와 NMD의 관계 d

   exp  ^{  ln}^^ 

(5)

◌

● 단원에서의 검토사항

(6)

◌

● 참고문헌

Hinds, W.C. Aerosol Technology: Properties, Bahavior, and Measurement of Air borne Particles, Chap. 4, Wiley(1982)

(7)

입자상 물질의 특성화

Hyuksang Chang, 영남대학교 1

Linear-Probability Graph

Cumulative Percent (%)

0.1 1 10 30 50 70 90 99

Particle Size in Diamater (um or cm)

10 20 30 40 50 60 70 80 90 100

 _  

 









 _ _^

^ _

(8)

Log-Probability Graph

Cumulative Percent (%)

0.1 1 10 30 50 70 90 99

Particle Size in Diamater (um or cm)

0.1 1 10 100

 

_

  

 

_



_

 





 

  

_

  

_



^



 

_



^



_

(9)

Linear-Probality Graph

Cumulative Percent (%)

0.1 1 10 30 50 70 90 99

Particle Size in Diameter (

m m

or cm)

20 40 60 80 100

 ^  

 











 

^ ^^

^

^

(10)

Log-Probality Graph

Cumulative Percent (%)

0.1 1 10 30 50 70 90 99

Particle Size in Diameter (

m m

or cm)

0.1 1 10 100

 ^  

 ^ ^















 ^  ^^

 ^^

^

(11)

2016-07-13 Environmental Aerosol Engineering Laboratory 1

Particle Size Distribution

• Monodisperse - All the particles are of the same size

• Polydisperse - Particles are of more than one size (more realistic)

Typical data from measurement

Size Range (m)

Count (#)

Fraction Percent (%) Cumulative Percent (%)

Fraction/size (m

^-1

)

0-4 104 0.104 10.4 10.4 0.026

4-6 160 0.16 16.0 26.4 0.08

6-8 161 0.161 16.1 42.5 0.0805

8-9 75 0.075 7.5 50.0 0.075

9-10 67 0.067 6.7 56.7 0.067

10-14 186 0.186 18.6 75.3 0.465

14-16 61 0.61 6.1 81.4 0.0305

16-20 79 0.79 7.9 89.3 0.0197

20-35 103 0.103 10.3 99.6 0.0034

35-50 4 0.004 0.4 100.0 0.0001

> 50 0 0 0 100.0 0

Total 1000 100.0

Reading: Hinds, Chap 4

(12)

Histogram of frequency(count) versus particle size

d

_p_i

(  m )

0 10 20 30 40 50

Frequency/Count

0

50

100

150

200

Q: Which size range has the most particles?

Size Range

(m) Count (#)

0-4 104

4-6 160

6-8 161

8-9 75

9-10 67

10-14 186

14-16 61

16-20 79

20-35 103

35-50 4

> 50 0

Total 1000

(13)

Frequency/d

_p

(distribution function) vs particle size

d

_p_i

(  m )

0 10 20 30 40 50

n i (d pi ) Size Distribution Funct ion (frequency/  d p 

0 20 40 60 80

Q:Total # of particles ?

Size Range

(m) Count/d

pi

(#/m)

0-4 26

4-6 80

6-8 80.5

8-9 75

9-10 67

10-14 46.5

14-16 30.5

16-20 19.25

20-35 6.87

35-50 0.27

> 50 0

pi i

i

d

Count

n  

(14)

Standardized frequency/d

_p

vs particle size

d

_p_i

(  m )

0 10 20 30 40 50

f i (d pi ) Probability Density F unction (fraction/  d pi )

0.00 0.02 0.04 0.06 0.08

Q: What is the value of the total area?

Size Range

(m) Fraction/size (1/m)

0-4 0.026

4-6 0.08

6-8 0.0805

8-9 0.075

9-10 0.067

10-14 0.465

14-16 0.0305 16-20 0.0197 20-35 0.0034 35-50 0.0001

> 50 0 0

N

f

_i

 n

ⁱ

(15)

Continuous Particle Size Distribution

If the size range is very small, the discrete PSD will approach continuous PSD .

d _p (  m )

0 10 20 30 40 50

q(d p ) Probability Density Function

0.00 0.02 0.04 0.06 0.08 0.10

q d f d

df

p

dd

i

pi p

( )  





 0

(16)

Cumulative Distribution

• Definition:

– The fraction that is less than a specific size

• Why cumulative distribution?

– Can be used to determine some statistical values.

Provide another viewpoint to observe the distribution.

F a ( )  

₀^a

q d (

_p

) dd

_p

d

_p

(  m )

0 10 20 30 40 50

Cumulative Fraction ( %)

0 10 20 30 40 50 60 70 80 90 100

Q: What’s the RED spot?

(17)

 MEAN (arithmetic average):

The sum of all the particles sizes divided by the number of particles

 MEDIAN :

 The diameter for which 50% of the total are smaller and 50% are larger; the diameter corresponds to a

cumulative fraction of 50%

 MODE:

 Most frequent size; setting the derivative of the frequency function to 0 and solving for d _p .

 For a symmetrical distribution, the mean, median and mode have the same value.

d d

N

n d

n d q d dd

p

p i pi

i

p p p

    

  ₀ ^ ⁽ ⁾

(18)

• GEOMETRIC MEAN :

the Nth root of the product of N values

Expressed in terms of ln(d _p )

• For a monodisperse aerosol, otherwise,

• Very commonly used because the an aerosol system typically covers a wide size range from 0.001 to 1000 m

d

_p

 d

_pg

p g

p

d

d 

   

d _pg  d d d _p ⁿ ₁ ¹ _p ⁿ ₂ ² _p ⁿ ₃ ³ ... ¹ ^/ ^N   d _pi ^{n d} ⁽

^pi

⁾ ¹ ^/ ^N

ln ln

exp ln

exp ( ) ln( )

( )

d n d

N

d n d

N

n d d dd n d dd

pg

i pi

pg

i pi p p p

p p

 

  

  

   





 





 





 



(19)

Weighted Distributions

• Why do we need other distributions?

– Aerosols may be measured in different ways, and in indirect ways (e.g. impactors, light scattering)

• What are the other distributions?

– Surface area, mass (volume), volume square ...etc

• Definition: frequency of the property (e.g. mass) contributed by particles of the size interval

• What is the effect?

Ex. A system containing spherical particles (mode size?) Number Concentration: Mass Concentration:

100 #/cc 1m & =1.91g/cm ³ 10 ^-11 g/cc 1m 1 #/cc 10m 10 ^-9 g/cc 10m

Q: How will the PSD on page 5 look like?

(20)

2016-07-13 Environmental Aerosol Engineering Laboratory 10 d_p(m)

0 10 20 30 40 50

Mass fraction/  m

0.00 0.01 0.02 0.03 0.04

d_p(m)

0 10 20 30 40 50

q(d p ) Probability Density Function

0.00 0.02 0.04 0.06 0.08 0.10

Number Distribution Mass Distribution

Q: What is the mode size of the distribution?

(21)

• Count Mean Diameter: based on number of particles.

• Mass Mean Diameter: based on mass of particles.

d d

N

n d

n d n d dd

pn

p i pi

i

p p p

    

  ₀ ^ ⁽ ⁾

d m d

m d m d dd

pm

i pi i

p p p

  

  ₀ ^ ⁽ ⁾

m   _p   n  _p   _p   n  d _p    k n d _p 6

3

1 Conversion 3

(22)

Moments of the PSD

• Definition: The quantity proportional to particle size raised to a power; an integral aerosol property

M

_n

  n d

_i

⁽

_pi

⁾  d

_piⁿ

 

₀^

n d ⁽

_p

⁾  d dd

_pⁿ _p

Q: What is M _o ?

M

_o

  n d

_i

⁽

_pi

⁾  

₀^

n d ⁽

_p

⁾ dd

_p

Q: What is M ₁ ?

Q: What is M ₁ /M ₀ ?

Q: What is M ₂ /M ₀ ? M ₃ /M ₀ ?

Q: Which is larger? M ₁ /M ₀ ? (M ₂ /M ₀ ) ^1/2 ? (M ₃ /M ₀ ) ^1/3 ?

(23)

Volume Moments

• Particle volume, instead of particle diameter, is also used as a variable (i.e. the x-axis is particle volume, not size)

• Definition:

• Conversion of n _ to n _dp :

M

_k

^   n

_i

⁽ 

_pi

⁾  

_pi^k

 

₀^

n ⁽ 

_p

⁾   

_p^k

d

_p

Q: What is M ₁ ^ /M ₀ ^ ?

 _p   d _p ³ / 6  d  _p   d _p ² / 2  dd _p

dN n d

dN n d dd

p p

d p p

 

 (  ) 

( )

(1) (2) (3)

n _ (   _p ) d _p ² / 2  dd _p  n _d ( d _p )  dd _p 

n _d ( d _p )   d _p ² / 2  n _ (  _p ) (4)

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Lognormal PSD

• Various distributions: Power law, Exponential, ...etc. Very limited application in aerosol science

• Normal Distribution: widely used elsewhere, but typically not for aerosol science, because

– most aerosols exhibit a skewed distribution function – if a wide size range is covered, a certain fraction of the

particles may have negative values due to symmetry.

 

df d d

dd

n d d N

p p

p

i p p

   



 





 

 







 





 



1 2 2

1

2

2 1 2

  



exp

/

standard deviation

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• The application of a lognormal distribution has no theoretical basis, but has been found to be

applicable to most single source aerosols

• Useful for particle of a wide range of values (largest/smaller size > 10)

• Its mathematical form is very convenient when handling weighted distributions and moments.

• How to use it? Simply replace d _p by ln(d _p ).

ln ln

d n d

pg

N

i pi

 

geometric mean diameter

Why using Lognormal?

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ln (ln ln )



_g

ⁿ

ⁱ

^d

^pi

^d

^pg

 N 





²

1  

df

d d

g

p pg

g

   

p



 





  1

2 2

2

  _ln ^exp 

2

ln ln

(ln ) ln

(1)

(2)

d ln d

_p

 dd

_p

/ d

_p

(3)

 

df d

d d

dd

p g

p pg

g

   

p



 





  1

2 2

2

 _ln  ^exp 

2

ln ln

(ln ) (4)

df d

g

p pg

g

  

p

  

  1

3 2 18

2

 

2

 

 

ln exp ln ( / )

ln (5)

geometric standard deviation

Convert dlnd _p to dd _p

(27)

• Features of Lognormal PSD

Q: How much is ln(d

_84%

/d

_16%

)?

ln ln ln

ln( / )



_g

d d

 



84% 50%

) /

ln(

ln

2 

_g

 d

₉₇_.₅_%

d

₅₀_%

 Log-probability graph

For a given distribution,  _g remains constant

(nondimensional) for all

weighted distributions.

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Moments for lognormally distributed aerosols:

M _k  M _g ^k  k _g

  

 

0 2 2

9  exp 2 ln 

ln ² ln ⁰ ²

1 2

1  _g 9 ^{M M}

  M

  

 

 _g ^M

M M

 ¹

2 0 3 2

2 1 2

/ /

The statistical variables can be easily determined through the moments!

Ref: Lee, K. W. and Chen, H., Aerosol Sci. Technol., 3, 1984, 327-334.

Lee, K. W., Chen, H. and Gieseke, J. A., Aerosol Sci. Technol., 3, 1984, 53-62.