TSI Instruments SMPS 3938 - Page 192

B-10 Electrostatic Classifier Model 3082 and SMPS Spectrometer Model 3938
The formulas used to calculate Table B-1 are shown below. They are taken
from Wiedensohler [1988]. The mean free path is taken from Kim et al.
[2005]. To calculate the fraction of particles carrying zero, one or two
charges, use equation B-2 which is an approximation of the Fuchs model.
Equation B-2 is valid for size ranges:
1 nm  D
p
 1000 nm for N = -1, 0, 1
20 nm  D
p
 1000 nm for N = -2, 2
 





















5
0i
i
nm
p
D
logN
i
a
10)N(f
Equation B-2
Table B-2
Coefficients for Equation B-2 used for Radioactive Neutralizers (Models 3077 and 3077A).
a
i
(N)
N=-2
N=-1
N=0
N=1
N=2
a
0
-26.3328
-2.3197
-0.0003
-2.3484
-44.4756
a
1
35.9044
0.6175
-0.1014
0.6044
79.3772
a
2
-21.4608
0.6201
0.3073
0.4800
-62.8900
a
3
7.0867
-0.1105
-0.3372
0.0013
26.4492
a
4
-1.3088
-0.1260
0.1023
-0.1553
-5.7480
a
5
0.1051
0.0297
-0.0105
0.0320
0.5049
For the fraction of particles carrying three or more charges, use equation
B-3 which is based on a derivation by Gunn from 1956.
2
p0
2
i
i
2
p0
p0
2
e
kTD2
2
Z
Z
ln
e
kTD2
N
exp
kTD4
e
)N(f























Equation B-3
where:
e = elementary charge
= 1.60217733E-19 A·s
0

= dielectric constant
= 8.854187817E-12 A·s/(V·m) (for air)
D
p
= particle diameter [m]
k = Boltzmann’s constant
= 1.380658E-23 J/K (for air)
T = Temperature [K]
N = number of elementary charge units
Z
i+
/Z
i-
= ion mobility ratio
= 0.875 (Wiedensohler) for Neutralizers 3077 and 3077A
= 0.975 (Knobel et al.) for Neutralizer 3088