367
JOURNAL
OF COLLOID AND INTERFACE SCIENCE 23,
367‑378 (1967)
On
the Theory of Charging of Aerosol Particles by Unipolar
Ions
in the Absence of an Applied Electric Field
BENJAMIN Y. H. LIU,
KENNETH T. WHITBY, AND HENRY H. S. YU
Particle Technology Laboratory, Department
of ‑Mechanical Engineering, University of Minnesota,
Minneapolis, Minnesota, 55455
Received
October 28, 1966; revised December 1, 1966
SUMMARY
The charging of aerosol
particles by unipolar ions in a gaseous medium and in the absence of an applied
electric field has been analyzed from the standpoint of the kinetic theory and
compared with theories based upon the macroscopic diffusion of ions and
experimental data. It is shown that the quasi‑steady charging rate can be
expressed by the simple equation, (dg/dt = e-g where g is a dimensionless particle charge and t is a
dimensionless particle charging time. The particle charging rate is shown to be
unaffected by the mean free path of the ions, and hence by pressure. The
theoretical equation shows good agreement with the experimental data when the
mean thermal speed of the ions is taken as 1.18x104cm/sec, which
corresponds to a molecular weight of 460 for the corona ions used in the
charging experiments. It is also shown that the theory based upon the solution
of the continuous macroscopic diffusion equation for ions cannot be applied to
the electrical charging of aerosol particles in the micron and submicron size
range because of the low concentration of ions normally encountered and the essentially
discontinuous nature of the charging process.
INTRODUCTION and the electrogasdynamic power generator
Aerosol particles in a gaseous medium
(2), although the exact charging theory
containing ions will he
charged as a result applicable to these devices is complicated
of the thermal motion of the
ions and the owing to the presence of an applied electric
collision between ions and
particles. If the field. It is also of considerable
importance
medium contains both positive
and negative in the ionic equilibrium of the atmosphere,
ions, the particle charge
will also be bipolar since the mechanism of particle
charging
with sonic particles being
charged posi‑ affects both the removal from the
atmos-
tively and some negatively.
If ions of only phere of small ions produced by
ionizing
one sign are present, then
the resulting par‑ radiation and the resulting statistical
distri-
ticle charge will also be
unipolar. The par‑ bution of electrical charge on aerosol
par-
ticle charge depends
primarily on the size ticles. Finally, the recently developed
of the particle, the
concentration of small method (3) of aerosol size measurement
by
ions in the ‑surrounding
medium, the length electrical charging and mobility ‑analysis
in
of time to which the
particles tire exposed to the size range from 0.01 p to 1.0 u has
also
ions, and to somewhat lesser
extents the made the detailed understanding of the
nature and characteristics
of the ions. charging process a practical and
important
The charging of aerosol particles by small problem.
ions is an important factor
in the design and Traditionally, the particle charging
process
operation
of the electrostatic precipitator (I) has been considered as a macroscopic
368 LIU, WHITBY, AND YU
diffusion
process in which ions are assumed to diffuse continuously in a quasi‑steady
state toward the particle under the action of a concentration gradient. The
rate of capture of ions by the particle is assumed to be equal to the quasi‑steady
ionic flux which can be obtained by solving the steady‑state diffusion
equation with appropriate boundary conditions. Various expressions have been
obtained for the steady‑state ionic flux by various investigators using
different boundary conditions. Examples are the equations of Arendt and
Kallmann (4), Gunn (5), Bricard (6), Natanson (7), and others However, in order
for the results of these analyses to apply to the actual charging process
occurring in a gaseous medium containing aerosol particles, it is essential
that the concentration of small ions in the gaseous medium be of such a
magnitude that the charging process may be considered essentially as a continuous
process. However, the maximum concentration of small ions in all practical
particle charging devices, because of the limitation by space charge or
recombination, seldom exceeds 108 ions/c.c., which corresponds to a
partial pressure of 10-11 atmosphere or 10-8 torr for the
ions. In the free atmosphere the natural concentration of small ions will be
even lower by several orders of magnitude. Under such conditions the capture of
ions by an aerosol particle in the micron and submicron size range is essentially
discontinuous, and the results based upon the solution of the continuous
macroscopic diffusion equation can not be expected to apply to the actual
charging process. This may be shown by an order of magnitude calculation .
......Calculation based on Kinetic
Theory of Particles Charging Process and Diffusion Theory of Particles Charging
Process......Comparison of theories and experimental data ......Results
.....Discussion.....
376
CONCLUSION
The
charging of
aerosol
particles by unipolar
ions in a gaseous
medium
fluid in the absence of an applied electric‑ field has
been
analyzed from the
stand point of the
kinetic theory. The result has
been compared and found to show good
agreement with the experimental data, when the
mean thermal speed of the
corona
ions used
in the charging experiments as 1.18x104
cm/sec. This is equivalent to
assuming a value of
460
for the
molecular
weight of the ions. The molecular weight thus determined
and the measured mobility of the ions of 1.1
cm2/
volt.sec., at
atmospheric pressure both tend to support the
view that the
corona ions used in the charging
experiments are molecular
clusters formed by the interaction of the charge and dipole moment of the
molecules in the cluster. However, the exact nature of
these molecular clusters could not be
determined from the measurements made.
The
experimental data show that the charging process is unaffected by the mean free
path of the ions and hence
by pressure. This is in
agreement with the result of the theoretical analysis based upon the method of
kinetic theory of gases and is in variance with
theories based upon the macroscopic diffusion of ions. It
has also been shown that the predicted increase in the particle chargingrate
in Murphy's analysis is entirely the result of neglecting the curvature of the
free path of ions in the vicinity of a charged particle. The correct equation,
taking into account the curvature of the ion free path, agrees with an equation
first derived by White.
The
failure of the theories based upon the solution of the continuous steady‑state
diffusion equation to
account for the experimental
facts has been shown to be due to the low concentration of ions normally
encountered in any aerosol charging process and the essentially discontinuous
nature of the charging process .
______________________
378
NOMENCLATURE
a particle radius, cm.
C speed of an ion, cm./sec
"Electrical Aerosol Particle Counting
`c mean thermal speed of ions, cm./sec.
cm, minimum speed
of an ion to reach the particle surface, cm./sec.
D diffusion coefficient of ions, cm2/sec.
Dp, particle
diameter, cm.
e
base of natural logarithm).
f distribution function for ionic speed.
I ionic flux to particle, number/sec.
k Boltzmann's constant.
m mass of an ion.
N local concentration of
ions, number /c.c.
N0 concentration
of ions at great distances fromthe
particle, number c./c.
r distance from surface of particle, cm.
R distance from center of
particle, cm.
S surface area, cm2
T
absolute temperature, °K.
t time during which particles are exposed toions, sec.
V
volume, cm3
x
dimensionless distance = r/a, dimensionless.
Zi
electrical mobility of ions cm2/volt.sec
g dimensionless particle charge, Eq. [161.
d dimensionless quantity in 'Murphy's equation, Eq.[28), and Fig. 3.
D thickness of vacuum shell, cm.
e elementary unit of charge, 4.8.10-10e.s.u.
q angle, radian.
l mean free path of ions, cm.
x dimensionless energy of
ion, Eq. [8].
t dimensionless time, Eq.
[191.
f angle, radian.
y electrostatic potential
energy of ion, erg.
REFERENCES
1.
White, H.J. "Industrial Electrostatic Precipitation." Addison Wesley
Publishing
Co., Inc., Reading, Massachusetts,
1963.
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______________________
Ozone 
Ionization