Table Of ContentTOPICS IN CURRENT
AEROSOL RESEARCH
EDITED BY
G. M. HIDY
Science Center, North American Rockwell Corporation,
Thousand Oaks, California 91360
AND
J. R. BROCK
University of Texas, Austin, Texas
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P R E F A CE
DESPITE an ever-widening technological interest in aerocolloidal systems, the fundamental
science of aerosols, on which practical considerations must rely, largely has been set aside
until recently in deference to other problems. However, in the last decade a kind of renais-
sance in aerosol research has taken place in which much of the classical work of the early
twentieth century is being extended. New sophisticated theoretical and experimental
techniques are being developed, and are being applied to understand better the behavior
of aerosol systems.
As scientists in many diverse fields expand their activity in aerosol research, the results
of investigations appear in a wide variety of journals that reach entirely different small
groups of workers. The problems of communication of these scientific results are compli-
cated further by the worldwide character of aerosol science. To help focus attention on the
variety of important aerosol research presently being published, and to open a new channel
for international communications between workers in this field, we have organized a new
series entitled International Reviews in Aerosol Physics and Chemistry. This work will
consist of a collection of monographs of book length, and companion volumes of selected
review articles dealing with several aspects of aerosol science, and its relationship to
the study of the so-called "particulate state of matter." The scope of the International
Reviews will be limited to results which contribute significantly to the state of fundamental
knowledge of aerosol behavior.
Because International Reviews in Aerosol Physics and Chemistry is designed to con-
centrate on the fundamental aspects of aerosol science, it should have considerable
usefulness both to practising scientists and to graduate students in such widely diverse
fields as physics, physical chemistry, meteorology, geophysics, astronomy, chemical
engineering, mechanical engineering, aerospace engineering, environmental sciences, and
medicine.
As the second volume of the series, this book includes two separate articles on aerosol
science, written by well-known investigators. The material presented in these two reviews
represents a viewpoint on different aspects of several problems. Prof. Fuchs and Dr.
Sutugin discuss experiment and theory describing highly dispersed aerosols. Prof. Soo's
article gives a capsule review of his and his colleagues' work devoted to expounding on a
"transport" model for electrically charged aerosols.
In their article, Fuchs and Sutugin describe several properties of very small aerosol
particles, the means of generation of such suspensions, and the dynamical behavior of
such particles, either as single, idealized spheres, or in collections. The article gives some-
what broader coverage to the experimental and theoretical efforts in this subject than in the
first volume of this series. The significance of non-continuum dynamical effects is empha-
sized, as well as the peculiarities of experimental methods required in this subject.
Prof. Soo summarizes in his contribution much of his efforts over the past several years
to construct an aerosol model based on an extension of the theory of transport in multi-
component gas systems. In his model, aerosol particles of a given size class are considered
viii PREFACE
a component in the gas mixture in analogy to a gas molecule. Prof. Soo illustrates his
views with a variety of experimental work, on both uncharged and electrically charged
particle species in gases. It is particularly interesting to note in passing the "success" of
boundary layer treatments of some experimental observations of flowing aerosol suspensions.
Similar results were discussed in a different way in Chapter 7 of Volume 1.
We are very grateful to the authors for their participation in the International Reviews.
It is a great pleasure to acknowledge their extensive efforts and patience in preparation of
the manuscripts for publication.
1970 G. M. HIDY
J. R. BROCK
N O M E N C L A T U RE
a Radius of a particle, or a radius.
a Molecular radius of air.
a
a Radius of a sphere of equivalent volume.
e
a Mean particle radius.
m
A Area.
b A distance or length.
B(S) A function of speed ratio S.
c Specific heat at constant pressure, or a length.
C Specific heat of a particle.
P
Cv Specific heat at constant volume.
(i)
c Mole-fraction of component (q).
C Cunningham correction factor, or a capacitance.
C Drag coefficient of a particle.
D
^Dfm Drag coefficient in free molecule flow.
Skin friction coefficient.
c
Lift coefficient.
L
d Characteristic length for charge transfer.
D Diffusivity.
e Electronic charge (1.6 χ 10~19 coulomb).
Ε Modulus of elasticity, electric field, or energy.
Ε Vectorial electric field.
f Distribution function.
f Force acting on a particle.
f Drag force acting on a particle.
D
fe Force due to electrostatic repulsion.
h Lift force.
Electric drag force.
f Force vector.
fe Force vector due to external field.
F Time constant for momentum transfer from fluid to particle.
p(ip) Time constant for mutual interaction of particles.
F' A drag constant defined after eqn. (4.33).
F* A correction coefficient for non-Stokesian motion.
F
Force per unit mass.
g Gravitational acceleration.
9 Rate of heat generated per particle, including radiation input.
G Time constant for heat transfer between fluid and particle.
Time constant for heat transfer between particle clouds.
Gl Gf
Energy function defined by eqns. (3.29) and (3.30).
65
66 TOPICS IN CURRENT AEROSOL RESEARCH
G Rate of heat generated per unit volume, including radiation.
h Heat transfer coefficient or charge transfer coefficient.
H Inverse relaxation time for charge transfer.
i Electric current.
J Flux of particles or current density.
Heat flux.
k Boltzmann constant,
(l-vî)/^.
k Factor accounting for interfacial velocity.
u
κ Mobility, a constant, temperature coefficient, or coagulation co-
efficient.
Mobility of a particle.
K Effectiveness of momentum transfer from particle to fluid.
m
Particle-fluid interaction length for momentum.
L Particle-fluid interaction length for energy.
T
m Mass of a particle.
m Average molecular mass of air.
a
m, nif Mass of an ion.
2
m Total flow rate.
m Collision rate at the wall.
pw
m* Mass ratio of particulate to fluid phase.
rh* Mass flow ratio of particulate to fluid phase.
M Molecular weight or electric moment.
η Number density.
Number density of particles.
"Ρ
Free electron density.
n, η,· Ion density.
z
"z Number density of particles with Ζ charges.
Ν Number of ions.
N Diffusion convection number.
D
Diffusion response number.
NDF
Ny Drift-diffusion parameter.
D
Ks Electrosurface number.
Κ, Electrothermal number.
Electrothermal number based on external field.
(N,)
EE
Electrothermal number based on particle charge.
N Electroviscous number, or electrodiffusion number.
EB
N Froude number.
FR
N Impact number.
IM
N Knudsen number.
KN
Knudsen number for particle-fluid interaction.
NK
P
N Fluid-particle momentum number.
M
(NJ Fluid-particle momentum transfer number in gravitational field.
E
N Mach number.
M
Ν Nu Nusselt number.
N Peclet number.
PE
N Prandtl number.
PR
NOMENCLATURE 67
N Reynolds number.
Re
(N ) Electric Reynolds number.
Ree
N Shear response number.
s
N Schmidt number.
Sc
N Stanton number.
st
Nsv Space-charge parameter.
N Weber number.
We
Ν Rate of mass transfer in number of moles per unit time.
Ρ Electric dipole moment.
Ρ Static pressure.
q Electric charge.
Q Charge transfer per collision.
Specific charge per unit volume.
Maximum charge for given band structure.
<lo
q\m Charge to mass ratio.
r Radius, radial coordinate.
r* Ratio of the reflected speed to incoming speed of collision.
r Position vector.
R Radius or pipe radius.
R Gas constant based on mass.
S Molecular speed ratio.
t Time.
Τ Absolute temperature.
Τ Wall or surface temperature.
τ Temperature at infinity from particle.
u Velocity, internal energy.
U Velocity.
AU Relative speed.
<υ2>ί/2 Intensity of particle motion.
υ ρ Vectorial fluid velocity.
υ, Vectorial particle velocity.
Speed.
<(Δ*/)2>1/2 RMS relative velocity of particle to fluid.
ν Mean thermal speed of ions.
Volume of a particle.
V Electric potential.
ν Volume.
χ, y Cartesian coordinates.
χ Position vector.
Χι Position coordinate.
Ζ Number of electron holes.
z Saturation charge.
s
ζ Axial coordinate.
α (N) accommodation coefficient, a constant, or Townsend ioniza-
etp9
tion coefficient.
Collision parameter.
68 TOPICS IN CURRENT AEROSOL RESEARCH
ß Empirical constant or diffusion parameter.
y Ratio of specific heat.
r Rate of generation.
Kronecker delta.
Δ Deformation tensor.
δ*,δ*,δ ,θ,θ δ Momentum integrals of boundary layer calculations.
ρ9 ρ9 ρί
ε Permittivity of material.
δ Ion sheath thickness, or boundary layer thickness.
Permittivity of free space.
C Bulk viscosity.
α, J?, η, γ Characteristic groups of pipe flow.
Άοο Fraction of collisions that leads to coalescence.
n Fraction impacted, or collection efficiency, or Kolmogoroff length
scale of turbulence.
θ Azimuthal angle, angle of inclination.
Θ Dilatation tensor.
κ Thermal conductivity of the fluid material.
Knt Thermal conductivity of a suspension.
KK{mq) Thermal conductivity of component (q) in the mixture.
Thermal conductivity of material constituting the particulate phase.
λ Mean free path.
Λ Wavelength, microscale.
λ Debye shielding distance.
Ώ
τ Shear stress, time, dimensionless time, or temperature ratio.
ξ Parameter Fx\ U.
ζ-(2β)μ.
Viscosity of a mixture.
Viscosity of component (q) in the mixture.
μ Viscosity of the fluid material.
Viscosity of the material constituting the particulate phase.
μ
Ρ
ν Poisson ratio of the material.
VJ Stoichiometric coefficient.
Kinematic viscosity of the fluid material.
ν
Density of fluid phase.
Ρ
Density of particulate cloud.
ΡΡ
Ρ Material density of fluid phase.
Density of material.
ΡΡ
σ Surface tension.
5
σ Electrical conductivity.
Φ Volume fraction of particulate phase.
φ Polar angle, electrokinetic potential, or potential of work function.
<Ρο Potential difference of droplets, or surface potential.
Φ Electric flux.
ΦΡ Stream function of particulate phase.
Superscripts
Dimensionless quantities.
NOMENCLATURE 69
(0), (1) ... Order of perturbation.
(AGO Component (q) or (p) of mixture.
Unsubscripted
quantities Fluid phase.
Subscripts
a Molecules
c Contact.
e Electron.
i Ion.
ith, yth, or &th component.
m Mixture.
ο Initial conditions or reference conditions.
Ρ Particulate phase/particle.
pr Reflected particle.
Pi Incoming particle.
w At wall.
I N T R O D U C T I ON
VARIOUS hydrodynamic phenomena of transport, settling, coalescence, and collection of
particulate matter in gaseous suspension are strongly influenced by the fact that they carry
electric charges by nature or by design. Such influences range from being a perturbation
on the hydrodynamic behavior of an aerosol suspension to being an overriding effect.
That aerosol particles are seldomly electrically neutral is substantiated by the following
well-known facts :
Deposition of dust on ceilings.
Collection of dust particles smaller than filter openings.
Residual collection efficiency of an electrostatic precipitator at power-off.
The following chapters delineate recent results on the dynamics of dilute gaseous
suspensions of charged aerosols pertinent to the atmospheric sciences. For general funda-
mentals of hydrodynamics and electrohydrodynamics of multiphase systems, the readers
are referred to a recent monograph/1} The present review emphasizes aspects not included
in that volume and supersedes some of the earlier results, although this review is self-
contained.
While many of the basic relations are applicable to liquid suspensions, our concern in this
part is mainly with gaseous suspensions of solid particles and liquid droplets.
In preparation for treating dynamics of charged suspensions, we shall first consider
charging and transport properties. This is followed by general formulation and examples
of various electrohydrodynamic systems.
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