Faqd ()dd - KOCWcontents.kocw.net/KOCW/document/2016/yeungnam/choiyunhi/... · 2017-01-23 ·...

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학과목 집진공학(集塵工學) 담당교수 장혁상 (810-2547)

단원의 주제 입자상물질의 분포함수(Particle Size Distribution) Page 4

- 입자상분포함수의 이해

* 누적율

F a q d ddp pa

( ) ( )= ò0

dp (mm)

0 10 20 30 40 50

Cum

ula

tive F

ract

ion (%

)

0

10

20

30

40

50

60

70

80

90

100

적색점의 의미는 무엇인가 ?

* 입자분포함수의 형태

dp (mm)

0 10 20 30 40 50

q(d

p) P

robabili

ty D

ensi

ty F

unct

ion

0.00

0.02

0.04

0.06

0.08

0.10

수농도 분포 (Number Distribution)

dp (mm)

0 10 20 30 40 50

Mass

fra

ctio

n/m

m

0.00

0.01

0.02

0.03

0.04

질량농도분포 (Mass Distribution)

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- 입자상분포함수의 Fitting이산분포를 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 loglog

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- 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 갯수vi : 분포구간 i의 입자의 평균부피N : 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 : 분포구간 i의 입자의 중간입경 : 분포전체의 입자질량

.표면적 평균입경(Surface Mean Diameter) : d p, s

m ax m ax

: 분포 구간 i를 차지하는 입자의 표면적 : 분포구간 i의 입자의 중간입경 : 분포전체의 입자표면적

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* 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( d̂ )와 NMD의 관계 exp ln

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● 단원에서의 검토사항

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● 참고문헌

Hinds, W.C. Aerosol Technology: Properties, Bahavior, and Measurement of Air borne Particles,

Chap. 4, Wiley(1982)

입자상 물질의 특성화

Hyuksang Chang, 영남대학교1

Linear-Probability Graph

Cumulative Percent (%)

0.1 1 10 30 50 70 90 99

Parti

cle S

ize in

Diam

ater

(um

or c

m)

10

20

30

40

50

60

70

80

90

100



Log-Probability Graph


0.1 1 10 30 50 70 90 99

Parti

cle S

ize in

Diam

ater

(um

or c

m)

0.1

1

10

100



Linear-Probality Graph


0.1 1 10 30 50 70 90 99

Par

ticle

Siz

e in

Dia

met

er (m m

or cm

)

20

40

60

80

100



Log-Probality Graph


0.1 1 10 30 50 70 90 99

Par

ticle

Siz

e in

Dia

met

er (m m

or cm

)

0.1

1

10

100

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 0.0264-6 160 0.16 16.0 26.4 0.086-8 161 0.161 16.1 42.5 0.08058-9 75 0.075 7.5 50.0 0.0759-10 67 0.067 6.7 56.7 0.06710-14 186 0.186 18.6 75.3 0.46514-16 61 0.61 6.1 81.4 0.030516-20 79 0.79 7.9 89.3 0.019720-35 103 0.103 10.3 99.6 0.003435-50 4 0.004 0.4 100.0 0.0001> 50 0 0 0 100.0 0Total 1000 100.0

Reading: Hinds, Chap 4


Histogram of frequency(count) versus particle size

dpi (m)

0 10 20 30 40 50

Frequency/Count0

50

100

150

200

Q: Which size range has the most particles?

Size Range

(m)

Count

(#)

0-4 1044-6 1606-8 1618-9 759-10 6710-14 18614-16 6116-20 7920-35 10335-50 4> 50 0Total 1000


Frequency/dp (distribution function) vs particle size

dpi (m)

0 10 20 30 40 50

n i(d pi) Size Distribution Function

(frequency/ d p0

20

40

60

80

Q:Total # of particles ?

Size Range

(m)

Count/dpi

(#/m)

0-4 264-6 806-8 80.58-9 759-10 6710-14 46.514-16 30.516-20 19.2520-35 6.8735-50 0.27> 50 0

pi

ii

d

Countn


Standardized frequency/dp vs particle size

dpi (m)

0 10 20 30 40 50

f i(d pi) Probability Density Function

(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 0.0264-6 0.086-8 0.08058-9 0.0759-10 0.06710-14 0.46514-16 0.030516-20 0.019720-35 0.003435-50 0.0001> 50 0 0

N

nf i

i


Continuous Particle Size Distribution

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

continuous PSD.

dp (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 df

d

df

ddp

i

pi p

( )

0


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 q d ddp p

a

( ) ( ) 0

dp (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?


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 dp.

For a symmetrical distribution, the mean, median and

mode have the same value.

dd

N

n d

nd q d ddp

p i pi

i

p p p

( )0


• GEOMETRIC MEAN:

the Nth root of the product of N values

Expressed in terms of ln(dp)

• 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 dp pg

pgp dd

d d d d dpg p

n

p

n

p

nN

pi

n dN

pi 1 2 3

1 11 2 3 ...

/( )

/

lnln

expln

exp( ) ln( )

( )

dn d

N

dn d

N

n d d dd

n d dd

pg

i pi

pg

i pi p p p

p p


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 1m & =1.91g/cm3 10-11 g/cc 1m

1 #/cc 10m 10-9 g/cc 10m

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


dp (m)

0 10 20 30 40 50

Mass fraction/m

0.00

0.01

0.02

0.03

0.04

dp (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?


• Count Mean Diameter: based on number of particles.

• Mass Mean Diameter: based on mass of particles.

dd

N

n d

nd n d ddpn

p i pi

i

p p p

( )0

dm d

md m d ddpm

i pi

i

p p p

( )0

m n n d k n dp p p p p

6

3

1

3Conversion


Moments of the PSD

• Definition: The quantity proportional to particle size

raised to a power; an integral aerosol property

M n d d n d d ddn i pi pi

n

p p

n

p

( ) ( )0

Q: What is Mo?

M n d n d ddo i pi p p

( ) ( )0

Q: What is M1?

Q: What is M1/M0?

Q: What is M2/M0? M3/M0?

Q: Which is larger? M1/M0? (M2/M0)1/2? (M3/M0)

1/3?


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 ndp:

M n n dk i pi pi

k

p p

k

p

( ) ( )0

Q: What is M1/M0

?

p p p p pd d d dd 3 26 2/ /

dN n d

dN n d dd

p p

d p p

( )

( )

(1)

(2)

(3)

n d dd n d ddp p p d p p ( ) / ( )2 2 (4) n d d nd p p p( ) / ( )

2 2


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.

dfd d

dd

n d d

N

p p

p

i p p

1

2 2

1

2

2

2 1 2

exp

/

standard deviation


• 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 dp by ln(dp).

lnln

dn d

Npg

i pi

geometric mean diameter

Why using Lognormal?


ln(ln ln )

g

i pi pgn d d

N

2

1

df

d dd d

g

p pg

g

p

1

2 2

2

2 ln

expln ln

(ln )ln

(1)

(2)

d d dd dp p pln / (3)

df

d

d ddd

p g

p pg

g

p

1

2 2

2

2 ln

expln ln

(ln )(4)

df dg

p pg

g

p

1

3 2 18

2

2

lnexp

ln ( / )

ln (5)

geometric standard deviation

Convert dlndp to ddp


• Features of Lognormal PSD

Q: How much is ln(d84%/d16%)?

ln ln ln

ln( / )

g d d

d d

84% 50%

84% 50%

)/ln(ln2 %50%5.97 ddg

Log-probability graph

For a given distribution, g

remains constant

(nondimensional) for all

weighted distributions.


Moments for lognormally distributed aerosols:

M M kk g

k

g

0

2 29

2 exp ln

ln ln2 0 2

1

2

1

9 g

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.


Hatch-Choate Conversion Eq.

• q: weighted distribution – 0: count

– 1: length

– 2: area

– 3: volume/mass

• p: type of average – 0: median/geometric

– 1: mean

– 2: area

– 3: volume/mass

b = q + p/2

(Table 4.3)

Q: If CMD = 10 m and

g = 2, how much is

MMD? Diameter of

average mass?

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