Encyclopedia of Environmental Science and Engineering, Volume I and II

where D



p and^ s^ are, respectively, the mean and standard devi-

ation of the distribution. The mean diameter D



p^ is defined by

DDfDDp∫∞ ppp()

∞

d (5)

and the standard deviation, indicating the dispersion of the

distribution, is given by

s^2

2

∫∞()DDfDDp p ()pp

∞

d. (6)

In the practical measurement of particle sizes, D



p^ and^ s^ are

determined by

D

nD

N

nD D

N

p

ipi

ipi p







∑

()

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

s



(^212) (7)
where n i is the number of particles with diameter D pi and N
is the total particle number measured.
TABLE 2
Names and defining equations for various average diameters
Defining equations
General case In the case of log-normal distribution
number mean diam. D 1 nD
N
p ln D^1 ^ A^
0.5C^ ^ B^  2.5C
length mean diam. D 2 

nD
nD
p
p
(^2) ln D
2 ^ A^
1.5C^ ^ B^  1.5C
surface mean, Sauter or mean
volume-surface diam. D 3



nD 
nD
sD
S
p
p
p
3
2
ln D 3  A
2.5C  B  0.5C
volume or mass mean diam. D 4 


nD 
nD
mD
M
p
p
p
4
3
ln D 4  A
3.5C  B
0.5C
diam. of average surface Ds nD
N
p
(^2) ln D
s^ ^ A^
1.0C^ ^ B^  2.0C
diam. of average volume or mass Dv nD
N
p
3
3 ln Dv^ ^ A^
1.5C^ ^ B^  1.5C
harmonic mean diam. Dh N
(/ )Dp
ln Dh  A  0.5C  B  3.5C
number median diam. or geometric
mean diam. NMD exp
nDln
N
⎡ p
⎣
⎢
⎤
⎦
⎥
NMD
volume or mass median diam. MMD
exp
 ln

nD D
nD
pp
p
3
3
⎡
⎣
⎢
⎤
⎦
⎥
ln MMD  A
3 C


exp
mDln
M
⎡ p
⎣
⎢
A  ln NMD, B  ln MMD, C  (ln sg)^2
N(total number)  n, S(total surface)  s, M(total mass)  m
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