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Chapter Three: Particle Size Statistics

Definitions

Moment of DistributionsLognormal Distribution

A. Definitions: probability density function (pdf) and cumulative distribution

function (cdf)

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Note: the base of distribution, for example, count, surface area, volume or mass

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( )= p pdf f d dd

0 1.0( )

= p pf d dd

( )b

p paabf = f d dd

0( ) ( )= a

p pF a f d dd

( )( ) =

pp

p

dF df d

dd

Definitions:

Mean: mathematical average of all variables

Mode: the values corresponding to the greatest pdf

Median: the values corresponding to cdf = 0.5

Mathematical mean

0( )

= = =

d i ip p p p

i

n dd d f d dd N n

Geometrical mean

1/

1 2 3( )= LN

g Nd d d d d

lnexp

=

i ig

n dd

N

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B. Moment of Distribution

Number mean diameter

1 21 2

= = + +

Li i I

I

n d n n nd d d d N N N N

Mass mean diameter

mass mean diameter =1 2

1 2

= + +

LI

mm I

m m md d d d

M M M

( )

( )

3 4

1

3 3

/ 6

/ 6

= = =

p i i ii i imm

p i i i i

n d dm d n d d

M n d n d

Surface mean diameter

3

1

2

= =

i i ism

i i

s d n dd

S n d

3 3

2 2

( ) ( )

( ) ( )= =m msm

s s

N d dd

N d d

General form:

1/

( ) =

q p

i i iqm qp

i i

p

n d dd

n d

Table2.1 Definitions for Various Average Diameters

Indicated diameter Symbol Definition Description

Mode d0P=-1

dat maximum ni Diameterassociated withthe maximumnumber of

particles in a

distribution.

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Geometric mean dgP=0

( )1log log /i in di n

The nth root ofthe product of all

particle diameters,also for alognormaldistribution themedian diameter.

Arithmetic mean dP=0.5

/i in di n The sum of alldiameters divided

by the totalnumber of

particlesdof average

surfaceds

P=12 /i i in d n

The diameter of ahypothetical

particle havingaverage surface

area.dof average

volume(mass)dv

P=1.5133 /i i in d n

The diameter of ahypothetical

particle havingaverage volume ormass.

Surface mediandiameter

dsmdP=2

( )1 2 2log log /i i i i in d d n d The geometric

mean of theparticle surfaceareas or for alognormal

distribution thearea mediandiameter.

Surface meandiameter ( Sauter

diameter)

dsmP=2.5

3 2/i i i in d n d The averagediameter based onunit surface areaof a particle.

Volume mediandiameter (mass)

dmmdP=3

( )1 3 3log log /i i i i in d d n d The geometric

mean pf particlevolumes (mass) orfor a lognormal

distribution thevolume(mass)median diameter.

Volume meandiameter (mass)

dvmdP=3.5

4 3/i i i in d n d The averagediameter based onthe unit volume(mass) of a

particle.p values assume a lognormal distribution

.

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C. Lognormal Distribution

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Names, Defining Equations and Coefficients a for the Lognormal ConversionEquations for Common Type of Average Diameters

Types of Average b

Distribution(dq)In [ P=0 ]: P=1: P=2: P=3:

Median, Geometric Diameter Area, VTS Volume, Mass

Count(d0) Count median diameter,geometric mean:

Count meandiameter:

Diameter ofaveragesurface:

Diameter ofaveragevolume,diameter ofaverage mass:

lnexp

=

n dCMD

N

=

ndd

N

1/ 22

=

s

ndd

N

1/ 33

=

m

ndd

N

b=0 b=0.5 b=1 b=1.5

Length(d1) Length mediandiameter:

Length meandiameter:

lnexp

=

dLMD

nd

nd2

=lm

ndd

nd

b=1 b=1.5

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Area(d2) Surface mediandiameter:

Surface meandiameter,Sauterdiameter,

Mean volume-surfacediameter:

2

2

nd dlnexp

=

ndSMD

3

2

=sm

ndd

nd

b=2 b=2.5

Volume(d3)

Or mass(d3)

Volume mediandiameter, mass mediandiameter:

Volume meandiameter, massmean diameter:

3

3

lnexp

=

nd dMMD nd

4

3

=mmnd

d nd

b=3 b=3.5