Diametri e Conversioni

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Zhang, Y. 11/30/2009

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Chapter 3. Particle Size Statistics and Distribution

Statistics and distributions of particle sizes are important means by which we can identify andsolve air quality problems. For example, in a dusty airspace, how much of the total dust fallsinto the respirable particle size range? If the total particle mass in an air stream needs to be

reduced by 90% by passing through a filter, what is the maximum porous size of the filter? Inother cases, one needs to know other particle statistics based on measured data or a knowndistribution function. In this chapter, we will begin to characterize the particle number and massdistribution with respect to particle size. Different particle size averages can be calculatedstatistically with measurement data, or can be calculated with a known size distribution function.Lognormal distribution, one of the most commonly used particle size distributions, is discussedin great detail.

By completing this chapter, the reader will be able to:

Determine particle distributions in count, or in mass, both mathematically and graphically.

Determine other moment averages of particle diameters based on a set of measured data suchas particle size and its corresponding concentration (Table 3-4).

Understand and apply lognormal size distribution function to real life particles. Determine the fraction of particles that are smaller than a particular diameter (or size

classification) for particles with a lognormal size distribution. Apply the log-probability graph to determine the CMD (or MMD), and theg. Apply Hatch-Choate equation to find other moment averages of diameter for particles with a

lognormal size distribution. Be aware of other particle size distributions that can be applied to particular indoor

pollutants.

3.1 Number Distribution

As far as number distribution is concerned, particles are considered to be spherical regardlesstheir shapes and densities. This assumption is necessary for the discussion of particle statistics.There are two categories of spherical particles: monodisperse and polydisperse. The size ofmonodisperse particles is completely defined by a single parameter, the particle diameter,d.Most particles, however, are polydisperse and have particle sizes that range over two or moreorders of magnitude. This wide range of particle sizes even applies to the same particle source.For example, cigarette smoke contains tobacco particles ranging from less than 0.01m to largerthan 1m. Because of the wide range and the fact that the physical properties of particles arestrongly dependent on particle size, it is necessary to accurately characterize these sizedistributions by statistical means.

In measurement of particles, we often encounter grouped particle data. From a statistical pointof view, we consider a total population ofN polydisperse particles, each particle with a diameter,dp. ThisN number of particles can be further divided into ngroups of particles. Each group hasFi particles, or anFi frequency of particle occurrences, with a size range ofdi and mid-pointdiameter, di.


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From the data example in Table 3-1, the total particle populationN is the sum of frequencies forall groups; i.e.,

==

n

i

iFN1

(3-1)

whereFi is the number of particles, or the frequency, in theith group. The fraction for theithgroup of particles, fi, with respect to the total particle population is

N

Ff ii = (3-2)

Table 3-1. Illustration of statistical variables of particle count distribution for grouped data.

Size range*,di-1 di

(m),

di(m)

Mid-sizediameter, di

(m)

Frequency, Fi Fraction, fi,(m-1)

AccumulativeFraction, f

0-3 3 1.5 90 0.041667 0.1253-5 2 4 125 0.086806 0.2995-7 2 6 165 0.114583 0.5287-10 3 8.5 165 0.076389 0.75710-20 10 15 115 0.015972 0.91720-50 30 35 60 0.002778 1.000>50 0Total: N =720

* Upper size range is smaller than, and lower size range is equal to, the size indicated.

Therefore, the sum of the fractions for then-groups of particles, or the total population of theparticles,

===n

ii

n

i

n

i

ii F

NN

Ff 1

1(3-3)

The fraction for theith group particles includes the variable of size range,di, which can bemisleading in some instances becausedi can vary in many magnitudes. A more clearrepresentation of fraction for particle sizedp is standardized as the fraction per micrometer of the

particle size,

N

dFf iii

=

/(3-4)

Obviously, the nominator onthe right side of Eq. 3-4 is the standardized frequency of the totalparticle population,


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i

i

d

FF

= (3-5)

Unless otherwise specified, the frequency and fraction refer to the standardized frequency andstandardized fraction, respectively. They are both in units ofm-1. When the particle fraction isa continuous function, df(dp), the standardized fraction for the size range ofd(dp) can beexpressed as

)()( pp ddddfdf = (3-6)

The accumulative fraction of the total particle population, f, is

1)()( ==== ppii ddddfNF

ff (3-7)

Figure 3-1 shows a graphical representation for data in Table 3-1. The height and width of eachbar in Figure 3-1 are the frequency (Fi) and size range (di) of each particle group, respectively.The curve is the standardized fraction function [f(dp)]. The total area of the bars represents thetotal particle population (N). The total area under the curvef(dp), is unity, which represents theaccumulated fraction as shown in Figure 3-2.

0

40

80

120

160

200

0 10 20 30 40 50 60

Partic le diameter (m)

Frequency

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Fraction/

mFrequency for

grouped particles

Fraction

Figure 3-1. Particle size distribution in terms of frequency in a size range and fraction function.

There are many quantities that can be used to define a size distribution. The most commonlyused quantities for defining the location of a distribution are the mean, mode, median, andgeometric mean. These quantities are described by using the general term of averages, i.e.,they are different types of averages of a given population. The mean diameter, also called


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arithmetic average, pd , is the sum of all the particle sizes divided by the number of particles.

The mean diameter for listed data, for grouped data, and for the frequency function is given by

===

0

)()( pppi

iiii

p dddfd

F

dF

N

dFd (3-8)

wheref(dp) is the fractional function of particles with diameter dp (Figure 3-1).

Thecount median diameter (CMD) is defined as the diameter for which one-half of the particlesare smaller and one-half are larger. It is the diameter corresponding to the 0.5 accumulativefraction in Figure 3-2. Themodeis the most frequent size, or the diameter associated with thehighest point on the fraction function curve. It can be determined by setting the derivative of thefraction function equal to zero and solving for dp. For symmetrical distributions such as thenormal distribution, the mean, median, and mode will have the same value, which is the diameterof the axis of symmetry. For an asymmetrical or skewed distribution, these quantities will havedifferent values. The median is commonly used with skewed distributions because extremevalues in the tail have less effect on the median than on the mean. Most particle sizedistributions for indoor environments are skewed, with a long tail to the right. For such a skeweddistribution,

mode


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0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60

Particle diameter (m)

Cumulativ

efraction

Figure 3-2. Cumulative fraction of the particle population.

For monodisperse particles, the mean is equal to the geometric mean, gcm dd = ; wheredcmrefersto count mean; otherwise, cmg dd < . The geometric diameter is widely used in lognormal

distribution.

Thecount mean diameter, or the arithmetic mean of the number distribution, dcm, for aparticlepopulationN or for grouped particles, is defined as

=

=

=

n

iii

p

cm dFNN

dd

1

1(3-14)

Example 3-1. The following data are obtained using a particle counter within a buildingairspace. What is the most frequent particle diameter? What is the count median and countmean diameter?

Particle diameter range (m) Count/ml0.3-0.5 280.5-0.7 240.7-1.0 261.0-5.0 32

5.0-10 1210-50 3Total N =125

Solution: First the mid-point diameter for each size range is determined. The standardizedfrequency for each size group can be calculated using Eq. 3-5. Other terms related to the CMD

and count mean diameter areFiln(di) andFidi. All calculated variables based on the givendata are summarized in the following table.


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di di Fi F = Fi/di Filn di Fidi

0.2 0.4 28 140 -25.656 11.20.2 0.6 24 120 -12.260 14.40.3 0.85 26 86.7 -4.225 22.14 3 32 8 35.156 965 7.5 12 2.4 24.179 9040 30 3 0.075 10.204 90

Total: N=Fi=125

Filn(dI)=27.4

Fidi=323.7

According to the definition of the count mode, which is the most frequent particle size, the count

mode is 0.4m.

Using Eq. 3-13, the count median, or geometric mean diameter of the particles, is

)(255.1125

4.27exp

1exp m

N

ndFd iig =

=

=

Using Eq.3-14, the count mean diameter, or the arithmetic mean of the number distribution, is

)(59.2125

7.3231

1

mdFN

dn

i

iicm === =

3.2 Mass Distribution

Mass distribution, or distribution of mass with respect to particle size, is the most importantparticle distribution as far as indoor air quality is concerned. Most threshold limit values ofparticulate matter for human exposure are based on mass concentrations as described in Chapter2. Using the same example data set in Table 3-1 and assuming the particle population has adensityp, the mid-size particle mass and the total mass (M) of the particle population can becalculated.

pi

i

dm

6

3

= (3-15)

= ii FmM (3-16)

wheremi is in kg, di is in meters, andp is in kg/m3. The mass fraction, or, to be more exact, the

standardized mass fraction,fim, is inm-1, and the accumulative mass fraction, fm, can be

calculated:


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M

dFmf iiiim

=

/(3-17)

1=== MFm

ff iiimm (3-18)

Figures 3-3 and 3-4 show the standardized mass fraction and accumulative mass fraction withrespect to particle size, respectively. It should be pointed out that in practice, many particlepopulations do not have a smooth normal distribution. It is important to keep in mind that whena distribution function is not available or does not fit a given function, statistical parametersshould be calculated using their definitions. For example, the count mean diameter should becalculated using Eq.3-13.

0

0.005

0.01

0.015

0.02

0.025

0.03

0 20 40 60 80

Partic le diameter (m)

Massfraction/m

Figure 3-3. Mass fraction distribution of a particle population in Table 3-1.


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0

0.2

0.4

0.6

0.8

1

0 20 40 60 80


Cumulativemas

sfraction

Figure 3-4. Cumulative mass fraction of a particle population in Table 3-1.

The commonly used averages for mass distribution include the diameter of average mass, massmean diameter and mass median diameter. The averages of mass distribution are perhaps mostconfusing and deserve careful clarification. Lets start with the example data set in Table 3-2.The average mass of the particle population, m, can be obtained.

N

M

N

Fmm

ii == (3-19)

Table 3-2. Illustration of statistical variables for mass distribution for grouped data, assumingp =1,000 kg/m3,

and all particles are spheres.

Sizerange*,(m),

di(m)

Mid-sizediameter,di,(m)

FrequencyFi

Mid-sizemass,mi, (mg)

Massfraction,fim,,(m

-1)

Accumulativemass fraction,fm

0-3 3 1.5 90 1.7672E-09 3.2498E-05 9.75E-053-5 2 4 125 3.351E-08 0.00128388 0.0026655-7 2 6 165 1.4379E-07 0.00727206 0.0172097-10 3 8.5 165 3.2156E-07 0.01084133 0.04973310-20 10 15 115 1.7672E-06 0.01245761 0.17430920-50 30 35 60 2.2449E-05 0.02752309 1

>50 0 0 1Total: N =720 M =0.00163 1

Experimentally, mcan be obtained by either weighing each size group or by weighing the entireparticle population and dividing by the total particle number N. Assuming that all particles are


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spheres and have a densityp, the diameter related to this average mass, called thediameter ofthe average mass, md , has the following relationship with the average mass:

pm

dm

6

3

= (3-20)

Note that the total mass, M is

=== 333

666 mp

p

p

p

pdNd

dM

(3-21)

31

33

1

3

31

31

6

6

66

=

=

=

=

NdF

dN

N

Mmd

pi

p

p

p

ppm

(3-22)

Themass mean diameter, dmm, for the particle populationN with a total massM, is defined as

+++= N

Nmm d

M

md

M

md

M

md L2

21

1 (3-23)

Or, for the grouped particles, with mid-point diameter di, and mid-point mass, mi,

==

==

3

43

3

1

6

6

1

ii

iiiipi

iip

n

iiiimm

dF

dFddF

dF

dmFM

d

(3-24)

Although it is analogous to the count mean diameter and has an explicit mathematicalexpression, the mass mean diameter has no physical meaning because it averages the diameter(first order of length), yet weighs according to its mass (third order of length), which contributes

to the total mass of the particle population.

The mass median diameter, also called volume median diameter, MMD, is the diameter at whichhalf of the mass has smaller diameters and half of the mass has larger diameters for a particlepopulation. For the grouped particles, the mass median diameter can be calculated using thefollowing equation:


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=

3

3 lnexp

ii

iii

dF

ddFMMD (3-25)

Comparing the fraction distributions of particle count and particle mass for the same particle

population as shown in Table 3-1, the mass fraction distribution is shifted toward the largerparticle size. This is because the few large particles make up a large portion of the total mass.Figure 3-5 reveals that: (1) the averages of mass are larger than their counterpart averages ofcount; for example, the mass mean diameter is larger than the count mean diameter; and (2) thetwo fraction distributions are not necessarily in the same pattern, which indicates that thedistributions may not be the same. Therefore, care should be taken in using distributionfunctions.

0

0.02

0.04

0.06

0.08

0.1

0.12

0 20 40 60 80

Particle diameter ( m)

Fraction/m

Count

Mass

Figure 3-5. Comparison of count and mass fraction distribution for particle population.

Example 3-2. Using the same data in Example 3-1, find the mass median diameter (MMD) andthe mass mean diameter (dmm).

Solution: Using the mid-point diameters (di) and frequencies for each size group (Fi), sums ofterms are calculated and summarized in the following table.

di Fi Fidi3ln di Fidi

3 Fidi4

0.4 28 -1.64 1.79 0.720.6 24 -2.65 5.18 3.110.85 26 -2.59 15.97 13.573 32 949.20 864.00 2592.007.5 12 10200.45 5062.50 37968.7530 3 275496.99 81000.00 2430000.00

N=Fi=125

Fidi3ln di

=286639.76Fidi

3

=86949.44Fidi

4

=2470578.15


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Using Eq. 3-25, the mass median diameter of the particles is

)(02.2744.949,86

34.639,286exp

1exp

3

3

mdF

nddFMMD

ii

iii

=

=

=

Using Eq.3-24, the mass mean diameter is

)(41.2844.949,86

15.578,470,23

4

mdF

dFd

ii

ii

mm ===

3.3 Other Moment Averages and Distributions

The statistics of particles are much more complicated than many other number statistics, such asbirth rate, financial gain, and tourist population, because many particle statistics are derived froma certain power (moment) of the particle size. For example, statistics of particle surface areproportional to the square (second moment) of particle diameter. The statistics of particlevolume or mass are proportional to the cubic of particle diameter (third moment). The particlenumber statistics, which are dimensionless, can be considered the zero power (zero moment) ofthe particle diameter. These characteristics, related to a certain moment of the particle size,make the particle statistics unique compared to other statistics. For example, what does$2orbaby3 mean statistically? For particles,d0 is proportional to the particle count andd3 isproportional to the particle mass. In addition to these physical implications, a more practicalreason for the particle moment expression is that the particle sizes are often measured directly.

The particle size then can be calculated using the moment statistics. For example, particle massdistribution and density can be directly measured using some weighing methods. The particlediameters can then be calculated using the moment statistics.

The averages of particle count (zero moment) and mass (third moment) are mostly used inparticle statistics and indoor air quality, largely due to the available measurement technology andthe threshold limit values. There are two other types of averages: particle length or diameter (d1or first moment), and particle surface area ( d2 or second moment). Of these two, the averages ofparticle surface areas deserve some special consideration, even though the measurement may beextremely difficult. For example, many air cleaning devices, such as filtration and electrostaticprecipitation, depend on the particle surface area. Therefore, averages of particle surfaces could

be particularly useful statistics in the design of air cleaning technologies.

The averages for particle surface areas can be obtained in a similar way to those of theircounterparts for averages of particle mass. Lets consider a particle population as shown inTable 3-1. The average particle surface area, s , can be calculated as

== 2iiii dFNN

sFs

(3-26)


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whereFi is the frequency ofith group, andN is the total number of the particles. Thediameterof average surfaces,

sd , then can be calculated as

21

2

2

1

=

= NdFsd

ii

s

(3-27)

Thesurface mean diameter, dsm, also called Sauter diameter or mean volume-surface diameter,for the particle populationN with a total surface areaS, is defined as

+++= N

Nsm d

S

sd

S

sd

S

sd L2

21

1 (3-28)

Or, for the grouped particles, with mid-point diameter di and mid-point mass, mi,

===

=2

3

2

2

1

1

ii

ii

ii

iiin

i

iiismdF

dF

dF

ddFdsF

Sd

(3-29)

Again, the surface mean diameter has no physical meaning because it averages the diameter (firstorder of length), yet weighs according to its surface area (second order of length), whichcontributes to the total surface area of the particle population.

The surface median diameter, SMD, is the diameter at which half of the total surface has smallerdiameters and half of the total surface area has larger diameters for a particle population. For thegrouped particles, the surface median diameter can be calculated using the following equation:

=

2

2 lnexp

ii

iii

dF

ddFSMD (3-30)

3.4 The Lognormal Distribution

This section discusses the most well-defined and widely used particle distributionlognormaldistribution. The lognormal distribution is a transformation of normal distribution. Manypopulations in nature are normally distributed. Such normal distributions include the heights andweights of a group of grown people, exam grades of a class, and service life of a batch of light

bulbs. The plot of a normal distribution is symmetrical with its arithmetic mean (Figure 3-6). Ina normally distributed particle population, the number fraction function, df, can be uniquelydefined by the geometric mean, dg, and the standard deviation, .

)(2

)(exp

2

12

2

p

gpdd

dddf

=

(3-31)


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2

12

1

)(

=

N

ddF gii (3-32)

Normal distribution has many advantages, especially in its ease of analysis and the simple

relationship among its properties. However, the size distribution of airborne particles is usuallynot normally distributed. It has a skewed (with a long tail at the large particle sizes) distribution.Figure 3-7 shows such a skewed size distribution with the same CMD (15m) as that in Figure3-6. Apparently, the distributions in Figures 3-6 and 3-7 are quite different. Such skeweddistribution is very difficult to analyze and the relationships among the particle properties canonly be calculated individually using the equations described above.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 10 20 30 40 50

Par ticle diameter ( m)

Fraction/

m

Figure 3-6. A particle population with a normal count distribution. The geometric mean diameter dg=15m, andthe standard deviation =4m.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 20 40 60 80 100


Fraction/m

Figure 3-7. A skewed normal distribution with a long tail to the large size side. CMD =15m, andg =2.


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Properties and rules for analyzing the normal distribution can no longer be directly applied onsuch a skewed distribution. However, when the fraction is plotted against the log scale of theparticle size, the fraction distribution is approximately normally distributed. Figure 3-8 showsthe same particle population in Figure 3-7 with the particle size on a logarithmic scale.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5 2


Fraction/lnd

1 10 100

Figure 3-8. When a skewed distribution with a long tail to the large size side is plotted on a log scale, thedistribution appears to be normally distributed.

Since thex-axis is in log scale, the distribution becomes lognormal. In a lognormal distribution,the fraction function can be directly derived from that of the normal distribution by substitutingdg with lnCMD, and with lng.

)(ln)(ln2

)ln(lnexp2ln

1 2

2

p

g

p

g

ddCMDddf

=

(3-33)

2

1

2

1

)ln(lnln

=

N

CMDdF piig (3-34)

whereCMD is the count median diameter, andg is the geometric standard deviation, or GSD.In a lognormal distribution, the geometric mean diameter equals to the count median diameter.Note that either the natural logarithm or the logarithm to the base 10 can be used. The natural

logarithm is used in this book. It also should be pointed out that the termslndp and lnCMD havealready been standardized in Eqs. 3-33 and 3-34, and really should beln(dp/d0) and ln(CMD/d0),respectively, whered0 =1m. The lognormal distribution has been applied in many real worldphenomena. Incomes, populations (for humans, animals and bacteria), stock processes andenvironmental contaminants are some examples. It is particularly widely used in particle sizedistribution, especially for those populations from single sources.


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Eq. 3-33 gives the fraction function of particles having a diameter range of(lndp) and[lndp +d(lndp)]. Sinced(lndp) = d(dp)/dp, Eq. 3-33 can be written in terms of particle sizedp rather thand(lndp):

)()(ln2

)ln(lnexp

2ln

1

2

2

pg

p

gp

ddCMDd

ddf

=

(3-35)

Although Eq. 3-35 is written in terms ofCMD andg, it is important to understand that theparticle surface area and mass distributions are also lognormal, and can be written in terms ofSMD and MMD, respectively, as

)()(ln2

)ln(lnexp

2ln

12

2

p

g

p

gp

ddSMDd

ddf

=

(3-35b)

)()(ln2

)ln(ln

exp2ln

12

2

p

g

p

gpdd

MMDd

ddf

= (3-35c)

For the normal distribution, 95% of the particles fall within a size range defined bydg2. Forthe lognormal distribution, the distribution is normal with respect to lndp, and with a standarddeviation oflng. In analog with the normal distribution, the 95% of the particles with alognormal distribution fall within a size range defined by

[ ]exp1 21nCMD n g (3-36)

This size range is asymmetrical and goes from CMD/g2

to CMDg2

. For g = 20. , 95% of theparticles would have sizes between one-fourth and four times the count median diameter. Forexample, the particle sample in Figure 3-8 with CMD =15m andg =2, 95% of the totalparticles falls in the size range of 3.75 to 60 m.

The cumulative count fraction for the particle population represented in Figure 3-7 is plotted onthe logarithmic scale ofdp and shown in Figure 3-9. The cumulative fraction 0.5 value isintersected with the CMD (15m). Particles smaller thanCMD/g

2 (3.75m) and particleslarger thanCMDg

2 (60m) are less than 5% of the total particle count.


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Figure 3-9. Cumulative fraction of a particle population with a lognormal distribution.

The total fraction between the particle size range of [0,], or the ranges oflndp [-, ], is aknown function.

=

0 2

2

)()(ln2

)ln(lnexp

2ln

1p

g

p

gp

ddCMDd

df

= )(ln

)(ln2

)ln(lnexp

2ln

12

2

p

g

p

g

ddCMDd

1= (3-37)

In many air quality problems, we often need to know the fraction between two particle diametersdp1 anddp2 (0


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g

p

p

CMDdK

ln

lnln = (3-39)

And similar to Eqs. 3-35b and 3-35c, Kp values for particle surface area fractions and massfractions can be written in terms of surface median diameter or mass median diameter,respectively,

g

p

p

SMDdK

ln

lnln = (3-39b)

g

p

p

MMDdK

ln

lnln = (3-39c)

Taking derivatives of both sides of Eq. 3-39 noting thatCMD andg are constants, gives

pgp dKdd ln)(ln = (3-40)

Substituting Eqs. 3-39 and 3-40 into Eq. 3-38 gives

=

2

1 2exp

2

1)(

2

12

p

p

K

Kp

p

pp dKK

ddf

=

12

2

exp

2

1

2

exp

2

122

pp K

p

pK

p

pdK

KdK

K

)()( 12 pp dfdf = (3-41)

The two terms at the right side of Eq. 3-41 represent two integrals of a normal distributionfunction with respect to Kp: f(dp2) is the fraction of particles smaller thandp2, and f(dp1) is thefraction of particles smaller thandp1.

For indoor air quality problems, the particles of concern are usually within the size range of 0.01

to 200m, and theKp values within a range of 3 to 3. Typical values for f(dp) with respect toKpare listed in Table 3-3. More detailed normal distribution values are listed in Appendix 4. If theKp falls between the two adjacent values, an inter-point of the two adjacent f(dp) values can beselected using linear interpolation.

Eq. 3-41 and Table 3-3 combined are very useful for finding the fraction of a given particle sizerange, if the CMD and GSD are given. The fraction of mass can also be obtained by substitutingthe CMD with the mass median diameter, MMD. A variety of types of commercial particle


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instruments measure the CMD and GSD of particles. Thus, the particle count or mass of anyspecified size range can be obtained.

Table 3-3. Typical values for f(dp) with respect toKp, wheref(dp) is the fraction of particles smaller thandp.

Kp f(dp) Kp f(dp) Kp f(dp)

-3.00 0.001350 -1.00 0.1587 1.00 0.8413-2.95 0.001589 -0.95 0.1711 1.05 0.8531-2.90 0.001866 -0.90 0.1841 1.10 0.8643-2.85 0.002186 -0.85 0.1977 1.15 0.8749-2.80 0.002555 -0.80 0.2119 1.20 0.8849-2.75 0.002980 -0.75 0.2266 1.25 0.8944-2.70 0.003467 -0.70 0.2420 1.30 0.9032-2.65 0.004025 -0.65 0.2578 1.35 0.9115-2.60 0.004661 -0.60 0.2743 1.40 0.9192-2.55 0.005386 -0.55 0.2912 1.45 0.9265-2.50 0.006210 -0.50 0.3085 1.50 0.9332-2.45 0.007143 -0.45 0.3264 1.55 0.9394-2.40 0.008198 -0.40 0.3446 1.60 0.9452

-2.35 0.009387 -0.35 0.3632 1.65 0.9505-2.30 0.01072 -0.30 0.3821 1.70 0.9554-2.25 0.01222 -0.25 0.4013 1.75 0.9599-2.20 0.01390 -0.20 0.4207 1.80 0.9641-2.15 0.01578 -0.15 0.4404 1.85 0.9678-2.10 0.01786 -0.10 0.4602 1.90 0.9713-2.05 0.02018 -0.05 0.4801 1.95 0.9744-2.00 0.02275 0 0.5000 2.00 0.9773-1.95 0.02559 0.05 0.5199 2.05 0.9798-1.90 0.02872 0.10 0.5398 2.10 0.9821-1.85 0.03216 0.15 0.5596 2.15 0.9842-1.80 0.03593 0.20 0.5793 2.20 0.9861-1.75 0.04006 0.25 0.5987 2.25 0.9878

-1.70 0.04457 0.30 0.6179 2.30 0.9893-1.65 0.04947 0.35 0.6368 2.35 0.9906-1.60 0.05480 0.40 0.6554 2.40 0.9918-1.55 0.06057 0.45 0.6736 2.45 0.9929-1.50 0.06681 0.50 0.6915 2.50 0.9938-1.45 0.07353 0.55 0.7088 2.55 0.9946-1.40 0.08076 0.60 0.7257 2.60 0.9953-1.35 0.08851 0.65 0.7422 2.65 0.9960-1.30 0.09680 0.70 0.7580 2.70 0.9965-1.25 0.1056 0.75 0.7734 2.75 0.9970-1.20 0.1151 0.80 0.7881 2.80 0.9974-1.15 0.1251 0.85 0.8023 2.85 0.9978-1.10 0.1357 0.90 0.8159 2.90 0.9981

-1.05 0.1469 0.95 0.8289 2.95 0.9984

Example 3-3. The total particle count in a building is 100 particles/ml. The particle sizedistribution is lognormal. The particle analyzer also gives the geometric mean diameter as 1.5

m, and the geometric standard deviation is 4. What is the number concentration of particles

between 0.3 and 10m?


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Solution:

Since dp1 = 0.3m and dp2 = 10m, CMD = 1.5m andg = 4.

161.14ln

5.1ln3.0ln

ln

lnln 1

1

=

=

=g

p

p

CMDdK

368.14ln

5.1ln10ln

ln

lnln 22 =

=

=

g

p

p

CMDdK

From Table 3-3, we find f(dp1) = 0.123, and f(dp2) = 0.9148

The fraction between the size range dp1 and dp2 is

f(dp2 dp1) = f(dp2) f(dp1) = 0.7918

Since the total number concentration CN = 100 particles/ml, the number concentration betweenthe size range dp1 and dp2 is

CN2-1 = f(dp2 dp1) CN = 0.7918100 = 79.2 (particles/ml)

3.5 Log-Probability Graphs

In the previous sections, our particle analysis is based on some known statistics such as CMDand GSD. However, many particle instruments measure only the size and the correspondingcount or mass concentration. If the particle distribution is lognormal, using log-probability

graphs can facilitate much of the practical application of the lognormal size distribution. In alog-probability graph (Figure 3-10), thex-axis is the cumulative particle fraction in probabilityscale. A probability scale is a percentage scale symmetric to the 50% point. It is compressedaround the median (50% of the cumulative value), and expands near the both ends of the axis.They-axis is the particle diameter in log scale. Such graphs are available commercially.

The cumulative particle fraction appears to be a straight line when it is plotted in such a log-probability graph (Figure 3-10). The median diameter can be directly read from the intersectionpoint with 50% cumulative fraction. The slope of the line represents the range of the particlesize. A steep slope indicates a large particle size range, and a shallow slope indicates a smallparticle size range. If the line is horizontal, the particles are monodisperse, i.e., all particle sizes

coincide with the median size. If the distribution is normal and the y-axis is linear, thecumulative fraction is also a straight line, and the graph is called probability graph.


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0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1Percen t less than ind ica ted s ize

Particle

diamerter(

m)

1 2 10 30 50 70 90 98

100

1

10

84% size

50% size =CMD

Figure 3-10. The cumulative fraction of a particle population with a lognormal size distribution is a straight linewhen plotted on a log-probability graph.

In using a log-probability graph, one needs two sets of data: particle size (or size range) andcumulative fraction of the particles. An example of such data sets is listed in Table 3-1, wherecumulative particle count fractions are calculated against size ranges. When particle size rangeis given, the mid-point diameter can be used. However, other weighted mid-diameters can beused to plot against the corresponding cumulative fraction. The weighing methods depend onthe types of statistics such as count distribution or mass distribution. For example, for the size

range of 5-10m, the mid-point diameter is 7.5 m. However, the weighted mid-point diametermay be 7m (less than the geometric midpoint) for particle count cumulative fraction becausemost of the particles in that size range may have smaller diameters than 7.5m. Also, theweighted mid-point diameter may be 8m for particle mass cumulative fraction because most ofthe particle mass in that size range is contributed by particles larger than 7.5m. Computerprograms are available to construct log-probability graphs. These programs can calculate theGSD and medians but the user must provide the weighted data.

For a normal distribution, the standard deviation represents the differences ofdat 84% of itscumulative value (d84%) and thedat 50% of its cumulative value (d50%), or the difference ofdat50% of its accumulated value and thedat 16% of its accumulated value (d16%).

%16%50%50%84 dddd == (3-42)

For a lognormal distribution, which is normal with respect to lndas shown in Figure 3-8, thedifferences are in log scale,


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==

==

%16

%50%16%50

%50

%84%50%84

lnlnln

lnlnlnln

d

ddd

d

dddg

(3-43)

21

%16

%84

%16

%50

%50

%84

=

=

==

d

d

d

d

d

dGSDg (3-44)

Figure 3-11. A log-probability graph. The horizontal axis should be the cumulative fraction of particles, and thevertical axis should be the particle size.


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In Eq. 3-44, the geometric standard deviation of particles with a lognormal distribution is a ratioof two particle diameters; hence, it is dimensionless and must be equal or greater than 1. Wheng =1, the particles are monodisperse, i.e., all particles have the same diameter. Thedimensionless feature of GSD for lognormal distribution has one unique advantage in particle

analysis. Since the GSD is only related to a particle size ratio, not the particle size, all weighted(moment) distributions of any lognormal distribution will be lognormal and have the samegeometric standard deviation. This means they will have the same shape and be parallel to eachother when plotted on a logarithmic scale. Figure 3-12 shows the distribution of count and massplotted on the same logarithmic diameter scale.

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1Percent less than indicated size

Particlediamerter(m)

1 2 10 30 50 70 90 98 99

100

1

10

MMD

CMD

Figure 3-12. Cumulative count and cumulative mass distributions for the same particle population on a log-probability graph.

The mass distribution has the same shape as the count distribution, but it is displaced along theaxis by a constant amount equal toMMD/CMD. The ratio MMD/CMD can be calculatedknowing only the GSD. If the two lines are not parallel, the mid-point diameters are not weighedproperly. Using a computer program, one can adjust the weighed diameters to make the twolines parallel. The relationship among theGSD and average diameters of a particle population

with a lognormal distribution is described by Hatch-Choate Conversion Equations.

3.6 The Hatch-Choate Conversion Equations

It is often necessary to know other characteristics such as the mass median diameter or thesurface median diameter, based on the measurement of one characteristic of particle size, such asthe count median diameter. The statistical parameters of particle size distributions such as the


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medians, means and diameters of average properties can be calculated from a detailed data setcontaining the sizes and the counts or masses of particles using the equations described inSection 3.3. This process is time consuming and may not be accurate if the detailed data areinsufficient. However, if the size distribution is lognormal, any other parameter, dx, can beconverted from the count median diameter (CMD) and the geometric standard deviation (g).

)ln(exp 2 gx bCMDd = (3-45)

Table 3-4Thebvalues in the Hatch-Choate Equation for different types of average diameters.

Type of average diameter Symbol Statistical equation bvalues

Mode d -1

Count median diameter,Geometric mean

CMD, dg

N

dFi lnexp 0

Count mean diameter d N

dFi

0.5

Length median diameter LMD

dF

ddF

i

i lnexp 1

Diameter of average surfacesd

21

2

N

dFi

1

Length mean diameterl

d dF

dF

i

i

2

1.5

Diameter of average massmd

31

3

N

dFi 1.5

Surface median diameter SMD

2

2 lnexp

dF

ddF

i

i 2

Surface mean diameter,Sauter diameter,Mean volume-surface diameter

dsm

2

3

dF

dF

i

i 2.5

Mass median diameter,Volume median diameter

MMD

3

3 lnexp

dF

ddF

i

i 3

Mass mean diameter,Volume mean diameter

dmm

3

4

dF

dF

i

i 3.5

Eq. 3-45 was called Hatch-Choate Equation to commemorate the original study of Hatch andChoate (1929).2 Thebvalue in the Hatch-Choate Equation depends on the types of averages and


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the distribution momentum. Thebvalues for different average diameters are listed in Table 3-4.Detailed descriptions of thebvalues and the derivation of the Hatch-Choate equation can befound from Hinds (1999).1 The corresponding equations to calculate the averages from raw data(total particle population or grouped particles) are also listed in Table 3-4. It is important toremember that those equations can be applied to any type of size distribution, while the Hatch-

Choate Equation is only applicable to the lognormal distribution.

Although Eq. 3-45 is written in terms ofCMD, it can be applied to any other types of averages,i.e., any other parameters can be converted from one type of average and the geometric standarddeviation (g). If one type of average rather thanCMD is known and we need to find anothertype of average, we use the known value to find CMD first, then use theCMD to find theparameter in question.

Example 3-4. The particle concentration in a room is measured using two instruments: aparticle counter and a mass sampler. The particle size distribution is lognormal with ageometric standard deviation of 2.5. In two air samples with the same volume, the particle

counter recorded 6107

particles, the mass sampler collected 0.8 mg of total mass, and thedensity of the particles was 1,000 kg/m3. What is the mass median diameter, MMD?

Solution:

Because the average mass per particle, m,can be calculated from the total number and the totalmass of the particles,

)(10333.1)(106

)(108 147

7

kgparticles

kg

N

Mm

=

==

The diameter of average mass, md , can be obtained3/1

3

143/1

)(1000

)(10333.166

=

=

m

kg

kgmdm

)(94.2)(1094.2 6 mm ==

From Table 3-4, the b value for the diameter of average mass is 1.5. Substituting b = 1.5, md =

2.94m andg = 2.5 into the Hatch-Choate Equation and rearranging gives

)(834.0)5.2ln5.1exp(

)(94.2

)lnexp(

2

2

mm

b

dCMD

g

m

=

=

=


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Applying Hatch-Choate Equation again and noting that b = 3 for mass median diameter, yields

)(4.10)5.2ln3exp(834.0

)lnexp(

2

2

m

bCMDMMD g

==

=

3.7 Other Types of Particle Size Distributions

The most commonly used distribution for characterizing aerosol particle size is the lognormaldistribution described above. Many particle populations, especially particles from a singlesource, are lognormal in size distribution. Several less common distributions, however, havebeen found useful for specific types of aerosols.1 In the following equations,aandbareempirical constants having different values for each distribution.

The Rosin-Rammler Distribution

Rosin and Rammler (1933) developed this distribution, originally based on sizing crushed coal,and it is applicable to coarsely dispersed dusts and sprays.3 It is particularly useful for sizedistributions that are more skewed than the lognormal distribution and for sieve analysis. Themass fraction betweendp anddp+ddp is defined as

pbp

bpm ddadabddf )exp(

1 = (3-46)

Whereadepends on the fineness of the particles andbdepends only on the material. Thisdistribution can be used when cutoff points for the smallest and largest diameters are welldefined. Expressed in particle count distributions, as in Eq. 3-46, the Rosin-Rammlerdistribution has the same form as the Weibull distribution.

pbp

bp ddadabddf )exp(

1 = (3-47)

The Nukiyama-Tanasawa Distribution

Nukiyama and Tanasawa (1939) developed this distribution, which is used for sprays havingextremely broad size ranges.4 The count distribution of particles with diametersbetweendp anddp+ddp is given by

p

p

p ddd

b

addf

= 32

exp (3-48)

Where the empirical constantsaandbare functions of a nozzle constant.

The Power-Law Distribution


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This size distribution function has been applied to the size distribution of atmospheric aerosols.The number of particles with sizes betweendp anddp+ddp is given by

pb

p ddaddf= (3-49)

The Exponential Distribution

The exponential distribution has been applied to powdered materials. The number of particleswith sizes betweendp anddp+ddp is given by

pp ddbdadf )exp(= (3-50)

The total number of particles isa/b.

The Khrgian-Mazin Distribution

The size distributions of cloud droplets are described by the Khrgian-Mazin distribution.5 Thenumber of droplets per unit volume with sizes betweendp anddp+ddp is given by

ppp ddbdaddf )exp(2 = (3-51)

Total number concentration is2a/b3, and the mean diameter is3/b.

Chens Empirical Distribution

This size distribution may be used for particle populations with mixed sources. Chen et al.

(1995) studied dust particles in animal buildings that contained a variety of dust sources such asgrain, fecal materials, danders and hairs.6 The number of droplets per unit volume with sizesbetweendp anddp+ddp is given by

)sin()exp( pp bdaddf = (3-52)

All of the above models are two-parameter models. The empirical constants,aandb, for eachmodel need to be determined before the model can be used. Many methods, such as parameteranalysis, can be used to calculateaandb. However, care must be exercised in obtainingrepresentative data for application of any of these statistical methods.

Discussion Topics

1. Can you name one practical example of a particle surface distribution that may be useful?

2. Why are particle count distribution and mass distribution with respect of particle size morefrequently used?


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3. Why the statistics of particle size is more complicated than other types of statistics such asthe height of adult people, valuation of a stock?

4. Intuitively, mass ir proportional to the volume and density. Why is there no density involvedin those particle average diameters related to mass, such as diameter of average mass or mass

mean diameter?

5. For a particle population, the standard deviation for diameter () has the same unit as thediameter, while the geometric standard deviation (GSD org) is dimensionless. Why?

6. If the particle size distribution is unknown, how do you plan to obtain the statistics of aparticle population? Is a particle counter that gives particle concentrations at different sizesadequate to do the job?

7. There are two methods used to characterize the particle statistics and distributions:calculation and graphical (log-probability graph). What are the limitations of each method?

Problems

1. Use the following measured data to calculate the arithmetic mean (or count mean), geometricmean (or CMD), mass median (MMD) and diameter of average mass. Fi is the frequency oftheith size particles.

d (m) Fi1 33 55 28 1

2. A sample of particle count concentrations in an indoor airspace was measured as in the tablebellow. Assume the particles are spherical with standard density (1000 kg/m3). Calculateand plot the fractions versus particle size for particle count, surface area and mass. Find themodes and median diameters for particle count, surface area and mass.

d(m) Fi1 303 70

5 1008 8010 5012 3014 1516 518 1


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3. Use a log-probability graph to determine theCMD andGSD (g) of the following dust

sample, assuming the dust sample has a lognormal distribution.

d(m) Fi

0.3-0.5 70.5-1 61-2 72-3 53-5 75-10 1210-20 520 - 30 1

Total 50

4. Repeat problem 3, using the following particle diameters instead of the mid-point diameter:a. Lower size of the rangeb. Upper size of the range.

5. A dust sample has a lognormal distribution. The count median diameter is 0.5m and theGSD is 1.8. The dust number concentration is measured at 50 particles/ml. What is the massconcentration in mg/m3? The particles are spherical with a density of 2,000 kg/m3.

6. A dust has a lognormal particle size distribution. It is known that 95% of particle count arewithin the size range of 0.3 and 12m. What is theCMD and GSD of the dust sample?

7. Dust mass concentration in an animal building is measured at 2.8 mg/ m3 using a filtersampler. Assume the dust has a lognormal distribution, with a count median diameter(CMD) of 2.5m and a geometric standard deviation of 2.5. The dust particles are sphericaland have a density of 2,000 kg/m3. Find:a) (a). Dust number concentration.

(b). Mass concentration of particles smaller than 4m.

8. The particle count distribution is lognormal as follows:

)(ln96.0

)223.0(lnexp576.0

2

p

pdd

ddf

+=

(a)What is the particle mass distribution?(b)What is the particle surface area distribution?

9. Dust concentration in a vehicle assembling building is measured at 120 particles/ml using aparticle counter. The particle counter also gives aCMD of 1.5m andGSD of 2.5. The dust


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particles are spherical and have a standard density. What are the number concentration andmass concentration for particles smaller than 10m in diameter?

10.In specifying a lognormal dust distribution in a problem building, a two-stage cascadeparticle impactor. It was found that 10% of the total particle mass are particles smaller than

10m, and 99% are smaller than 100m. Calculate theMMD andGSD of the dustparticles.

11.With the same measurement data in the above problem, use a log-probability graph todetermine theMMD andGSD.

12. Particle concentration in a dust storm is 5 mg/m3. The particle distribution is lognormal withCMD =5m andGSD =2.5. In order to reduce the dust concentration to 0.3 mg/m3indoors, what should the filter porous size be? Assume the particles are spherical with adensity of 1,500 kg/m3, and all particles smaller than the porous size will pass through thefilter.

References

1. Hinds, C. W., Aerosol Technology, John Wiley & Sons, New York, NY, 1999.2. Hatch, T. and Choate, S.P., Statistical description of the size properties of non-uniform

particulate substances,J . Franklin Inst., 207: 369, 1929.3. Rosin, O. G. and Rammler, E., Particle mass distribution for coarse crushed coal,J . Inst.

Fuel, 7: 29, 1933.4. Nukiyama, S. and Tanasawa, Y.,Transactions of Mechanical Engineering (Japan), 5: 63,

1939.5. Pruppacher, H.R. and J.D. Klett, J.D., Microphysics of clouds and precipitation, 2nd ed.,

Kluwer, Dordrecht, The Netherlands, 1997.6. Chen, Y., Y. Zhang Y., and E. M. Barber, E.M. A new mathematical model of particle

size distribution for swine building dust,Transactions of Ame. Soc. Heat. Refrig. Air Cond.Engineers, 101(2): 1169-1178, 1995.

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