PB 180 523T
U.S.S.R. LITERATURE ON AIR POLLUTION AND RE-
LATED OCCUPATIONAL DISEASES - VOLUME 18 -
ACOUSTICAL COAGULATION AND PRECIPITATION OF
AEROSOLS
«-
V
B. S. Levine
Washington, D. C.
•>•
1968
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I
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U.S.S.R. LITERATURE ON AIR POLLUTION
AND RELATED OCCUPATIONAL
DISEASES
\
Volume ig
A SURVEY
by
B. S. Levine, Ph. D.
ACOUSTICAL COAGULATION
AND
PRECIPITATION OF AEROSOLS
Processed by
CLEARINGHOUSE FOR FEDERAL SCIENTIFIC AND
TECHNICAL INFORMATION
. U. S. DEPARTMENT OF COMMERCE
Springfield, Virginia
-------
The present English edition is a part of a survey conducted by
B. S. Levine, Ph. D.
of 3312 Northampton St. , N. W.
Washington, D. C. 20015
supported by PHS Research Grant AP - 00176,
awarded by the Division of Air Pollution
of the U. S. P. H. Service
-------
ACADEMY OF SCIENCES USSR
INSTITUTE OF FUEL RESOURCES
Ye. P. Mednikov
ACOUSTICAL COAGULATION
'., AND
PRECIPITiATION OF AEROSOLS
IZD-VO AN USSR
MOSCOW
-------
C ONTENTS
Conducted by Editor In Chief
L. V. Radushkevich . 1
Foreword 1
Chapter 1
Introduction.
1. Special Properties of a High-Intensity Sound Field 4
2. Sources of High-Intensity Sound 21
3. A Brief Outline of the Development of Aerosol
Acoustical Coagulation and Precipitation Problems 31
Chapter 2
Motion of Aerosol Particles in a Sound Field
4. Preliminary Information 41
5. Vibratory Motion of Aerosol Particles 45
• 6. Drift of Aerosol Particles 52
7. Circulating and Fluctuating Aerosol Particle Motions 65
8. Scattering and Absorption of Acoustical Energy
by Aerosol Particles 68
Chapter 3 :
Aerosol Particle Interaction in a Sound Field
9. Preliminary Information 77
10. Orthokinetic Interaction of Aerosol Particles . 83
11. Parakinetic Interaction of Aerosol Particles • 90
12. Attractional Interaction of Aerosol Particles 102
13. Pulsation Interaction of Aerosol Particles 111
Chapter 4
Mechanism and General Principles of Acoustical Aerosol
C oagulation
14. The Course Followed by Acoustical Aerosol Coagulation 118
15. Kinetic Equation of the Process 124
16. Effect of Aerosol and Acoustical Characteristics on the
Kinetics of the Process 128
17. Some Earlier Hypotheses of Acoustical Aerosol Coagulation 136
Chapter 5
Practice of Acoustical Coagulation and Aerosols Precipitation
18. Acoustical Dispersion of Natural and Artificial Fogs 141
- i -
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19- Acousto-Gravitational Precipitation of Industrial Dusts,
Smokes, and Fogs 149
20. Acousto-Inertial Precipitation of Industrial Dusts,
Smokes, and Fogs 156
21. Acoustical Aerosol Separation and Filtration 169
Appendix
1. Evaporation and Condensation Droplet Growth During
Fog Sonication 174
2. Sound Effect on the Processes of Combustion and
Degasification of Fuel Droplets 181
3. Behavior of Precipitated Solid Particles in a Sound
Field of an "Acoustical Dryer" 183
- ii -
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This monograph deals with the physical fundamentals of coalescence
and precipitation of aerosols in the acoustic field and with the basic problems
of practical use of these processes as applied to purification of industrial
smoke, dust, and vapors.
Brief consideration is given also to problems of evaporation, conden-
sation, combustion;, and degasification of vapor drops and desiccation of pre-
cipitated solids in the acoustical field.
The monograph is intended for scientific and engineering personnel
who come in contact with problems associated with precipitation and utiliza-
tion of aerosols.
Editor in Chief
Doctor of Chemical Sciences
Professor L. V. Radushkevich
FOREWORD
Practically all natjural and artificial gases which man encounters in
the course of his activity, contain amounts of suspended admixtures of solid
or liquid materials; which constitute aero-dispersed systems or aerosols.
The atmospheric air man breathes normally contains negligible gravimetric
amounts of suspended admixtures although their content in dense fogs and
clouds, and in mines at times attains several grams and more per m3 of air.
Considerable quantities of suspended admixtures are contained in natural
gases, frequently kilogfams/m3 of the gas. A considerable amount of sus-
pended admixtures is contained also in artificial gases, particularly in fuel
gases. To this category belong the following: coke gas, blast furnace gas,
producer gas, etc., smoke from industrial furnaces and ovens, exhaust gases
of carbon black and similar plants, the ventilating air used in mining and
fuel processing industries. Acid smogs, radioactive aerosols, camouflage
smokes, etc., also contain particulate admixtures.
The presence of suspended admixtures in most of the above enume-
rated gases is undesirable, and it frequently becomes necessary to precipi-
tate and remove same from the dispersed phase, to change it qualitatively,
or to remove it partially or completely. One of the ways by which the first
two problems can be solved is artificial coagulation of aerosols, i.e., aggre-
gation by adhesion or coalescence of aerosol particles by physical means.
Most effective among the latter is high-intensity sonic and ultrasonic, oscil-
-------
latiori which presents possibilities for application in many branches of science
and technology. Consolidation of suspended particles occurring in the course
of acoustic aerosol coagulation induces substantial changes in" the physical
characteristics of aerosols. The reduction of the total surface of a dispersed
phase reduces light scattering. This in turn enhances the system transpa-
rency frequently to full visibility, while increases in the particle size and
mass enhance the particulates1 increased tendency to precipitate from the
gaseous medium under the effect of natural gravity or the induced inertia,
electric attraction, etc. All this is of great value and use to the technology
of dust- and fine droplet-trapping in modern air and gas purification techno-
logy. Modern trapping devices, such as settling chambers, dust extractors,
scrubbers, electrpfilters, basically depend upon size of particulates and the
effectiveness of their precipitation. The lower is the particulate dispersion
the higher is the effectiveness of precipitation. The highest dispersed par-
ticles, several microns or less in diameter which are present in most indus-
trial smokes, dusts, and vapors, precipitate particularly poorly. Prelimi-
nary acoustic coagulation of such particles offers means of substantially
increasing the effectiveness of their precipitation in the dust-droplets trapping
devices.
V
The acoustic coagulation principle was first applied in practice in the
late 40's when powerful acoustic oscillations, acoustic sirens, had been de-
veloped. However, the new attempts failed to attain adequate acoustical
intensification of gas purification. Therefore, more detailed study of the
mechanism of the new process, its possibilities, development of more power-
ful sound generators was undertaken, often inopportunely, in the direction of
introducing the new principle into different industries, urgently in need of
improving their gas purification operations. Many technical' and popular-
science journals carried during those yearsj advertised . articles which hailed
the acoustical method of aerosol coagulation as the panacea for all mishaps
associated with atmospheric pollution. Despite that, the acoustical method
of aerosol coagulation failed to become universal, neither were other physical
methods used in the dust- and droplet-trapping technology.
Acoustical coagulation is a special method. The extent of its appli-
cation is comparatively narrow; yet within its area of application it is
capable of attaining results unattainable by any other method. This became
increasingly apparent during recent years. At the same time, it became
clear that acoustical oscillations could be used to intensify not only the coagu-
lation of suspended particles, but also the process of their direct precipi-
tation; this appeared to provide in some instances a more effective and
economical solution of the intensive gas purification problems. This refers
in particular to the recently discovered method of acoustical filtration of
aerosols which seems to make possible the reduction of residual particle
concentration to fractions of mg/m3 of a gas at a moderate specific energy
consumption for the Bonification and extrusion of the gases. If future tests
- 2 -
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will support the early experimental conclusions, then the application of
acoustical oscillation in the dust- and droplet-trapping technology will become
widespread.
. In considering processes of aerosol coagulation and precipitation in
an acoustical field, the pertinent physical and physico-chemical processes
should be considered. It has been established that accelerated condensation
and degassing of droplets occurred during the Bonification of vapors, while
during the Bonification of humid smokes and dusts and their precipitation pro-
ducts, there occurred drying of the solid dispersion phase. The above pro-
cesses exhibit a distinct effect on the kinetics of aerosol coagulation and
precipitation, which, in the case of coarsely dispersed systems, are sup-
pressed. The above is of a highly practical value from the standpoint of a
successful solution of the following important problems: dispersion of natu-
ral fogs, separation of natural gas condensates, extraction of incidental
gases from crude oil, intensified combustion of atomized liquid and solid
fuels, drying of thermosensitive powdered products, etc.
There exists at the present a considerable volume of theoretical and
experimental material on acoustical aerosol coagulation and associated prob-
lems. However, such data appear extremely fragmentary and contradictpry.
This is associated, to a greater degree, with the uniqueness of the new
method which calls for the knowledge of relatively independent scientific
fields, such as hydrodynamics, acoustics, aerosol mechanics, physical
chemistry, chemical engineering, and includes their lesser developed
branches. Furthermore, it could be associated with exceptional experimental
difficulties in a given field due to lack of essential acoustical equipment on
the one hand and, on the other hand, an unusual.motion of aerosols in the
sound field •which impedes direct observation of the primary events. It is
not surprising that, under these conditions, numerous investigators arrived
at erroneous conclusions concerning the mechanism of the processes under
consideration, their effectiveness, areas of application, etc.
An attempt was made in the present monograph to systematize and
generalize the accumulated materials on acoustical coagulation and precipi-
tation of aerosols. Moreover, particular attention has been given to the
physical aspects of the processes and to their practical applications in tech-
nology. Additional brief consideration has been given to the processes of
evaporation, condensation, combustion, and decontamination of a dispersed
liquid phase and the drying of the precipitated solid-state products in the
sound field.
The author considers it his pleasant duty to express gratitude to
Doctor of Chemical Sciences, B. B. Kudryavtsev, Doctors of Physico-
Mathematical Sciences, V. F. Nozdrev and S. N. Rzhevkin, and Candidate
of Chemical Sciences, I. F. Bogdanov, for their moral support and useful
-3-
-------
advice. The author is grateful to Doctor of Chemical Sciences, L. V. Radus-
hkevich, for reading and editing the manuscript. The author also expresses
his thanks to Z. V. Grigorleva and A. M. Timoshenko for their help in the
graphical illustrations.
CHAPTER 1
INTRODUCTION
1. SPECIAL PROPERTIES OF A HIGH-INTENSITY SOUND FIELD.
The coagulating action which sound can exert on aerosols and other
physical and biological effects of sonification in gaseous media can be ob-
served only in connection with very high-intensity sound, the power level
of which exceeded considerably the pain threshold of the human ear. Such
high-intensity acoustical vibrations are characterized by a number of spe-
cific properties which distinguish them from acoustical vibrations at the
normal level. Acoustical vibrations, or simply sound in the physical sense,
implies compressional and rarefactional displacements of the elastic medium,
in the present case a gas, propagating in a. wave-like motion from the sound
source. The sound source is usually a vibrating or pulsating solid, liquid,
or gaseous body which is excited by aerohydrodynamic or electromechanical
means. The vibrations and pulsations of this body cause alternating com-
pression and rareficatipn of the adjoining gas layer and this, in turn, induces
advancing and regressing displacement of the adjoining gas particles. The
latter causes pressure to change its sign in the layer, resulting in an oscil-
latory displacement of particles in the subsequent gas layer; in this manner,
vibrations propagate from one layer to the next, farther and farther from the
sound source with a velocity, called sound velocity. If the origin (t=0) corre-
sponds to an instant at which the medium assumes an equilibrium and its
particles begin to oscillate according to the wave motion at an observation
point, assuming a sinusoidal sound source, the particle displacement Xg,
vibrations velocity Ug, and sound (i.e., excess) pressure pg vary with time
according to the following equations:
x£ = ;4g sin (•>/, (1.1)
(1.2).
(1.3)
where Ag, UB, and Pg represent displacement amplitude, velocity, and pres-
sure, respectively; CD is the angular velocity equal to 2n£ (where f is the
oscillation frequency in H, *).
* 1 Hertz (HJi= 1 vibration per second; 1,000 Hg =1 kilohertz (kH,).
- 4 -
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The sound velocity and pressure are, thus, shifted in time with re-
spect to the particle displacement by 1/4T, where T is the period of vibra-
tions. .
• •
When the sound wave meets an obstacle
of dimensions greater than acoustical wave-
length X, it is first reflected from the obstacle
according to the laws of geometric optics and,
subsequently, interacts with the direct wave.
As a result of interference, the vibration am-
plitudes increase at some points and decrease
at others. If the obstacle consists of a plane
surface perpendicular to the direction of the
sound wave propagation and is located at a
distance from the sound source equal to the
multiples of X/2, a. so-called standing wave is
formed as a result of interference of the direct and reflected waves (Fig. 1).
In this case, changes in Xg, u,, and pg are defined by the following equations:
Fig. I. Distribution of sound prss-
surs «f oscil Istory velocity •no die-
location in static sound MVSS.
s - b At tiM tj b - c At tise t +•
AT
xg = 1At sin kx0 sin
-------
(1.8)
Pst ia the static pressure of the medium, pg is the gas density, .and y is the
ratio of specific heats, Cp /Cv). The wave equation is derived from the mo-
tion equation of an ideal fluid combined with the continuate equation and equa-
tion of state of the gas, all equations being linearized, i. e., only the first
order terms are considered, and the second order terms are disregarded
[105]. This can be done if the sound pressure is neglibily small in compa-
rison -with the static pressure of the medium, and, therefore,, the difference
in the medium density in the compression and rarefaction regions may not
be accounted for. The assumption is equivalent to saying that, 1) the vibration
velocity is negligibly small in comparison with the sound velocity (the
difference in the propagation velocity of disturbances in the compression
and rarefaction regions can be neglected), or 2) the particle displacement
is negligible in comparison with wavelength, i.e., the displacement is di-
rectly proportional to the sound pressure as in Hook's Law.
co/aoc • r, bar •
iooooo Figure 2 shows amplitude of vibration
. velocity Ug and the corresponding amplitudes
soooa of SOUnd pressure Pg for a high-intensity
30000 sound J* in the following gases, the acoustic
impedances pgcg of which differ sharply: hy-
10000 drogen, air, and compressed methane ( the
basic component of natural gas).
Vibration velocity amplitudes were cal-
culated according to a well known formula for
1000 a plane wave [7]:
0.01
M»-B.
/
y-2
Fig. 2. Aoplttudo valuos «f oscillatory
velocity and sound pressure for sooa
gaees at high sound intensity, I -,
Hydrogen at 760 •• and 20° C. (p c - \\.-ac-
cc -'-«); 2 - Air at 760 a» and i>0° C.
<:*-'):*-Betnano at 50 aT»"
and '-'0° C
* The sound force, or intensity J (or I) is defined as the flow of sound energy
through a 1 cm3 area of the sound wave front per second. In the CGS system,
the unit for the sound intensity, is 1 erg/cm2/sec; in practice, the unit is
1 watt/cm2, equal to 107 erg /cm3 /sec.
** The sound pressure is measured in acoustical bars (1 bar=dyne/cms); it
is related to the sound intensity by the following expression J=pg2/pgcg, in the
experimental measurement of the sound field by means of acoustic probes
(subscript g will not be shown subsequently).
- 6 -
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The effective values of vibration velocity
p, are obtained by dividing quantities Ug and Pg
and the sound pressure
It can be seen from Figure 2 that in the range of sound intensity
J = 0.1 - 1.0 watt/cm2, where a rapid coagulation of aerosols is observed,
the amplitude of vibration velocity Ug attained values of 2.2 - 7.0 m/sec
in air, which amounts to 0.65 - 2.1% of the sound velocity; the sound pres-
sure amplitude attains values of 9,000 - 29,000 bar, which amounts to 0. 9 -
3.0% of the atmospheric pressure.
Figure 3 shows that the magnitude of particle displacement A, also
attained extraordinarily high values for a given sound intensity. The values
of A, can be calculated from the following known relationship [?]:
10
(i.ii)
In the 1, 000 ..- 5, 000 range, which is very often used for particle co-
agulation by acoustical means, A, for J = 0.1 - 1.0 watt/cma lies in the
70 - 1,100|J, range. The latter considerably exceeds aerosol dimensions and
is of the same order of magnitude as the average distance between particles
(see # 9). .
10000
Higher- order terms cannot be neg-
lected for the indicated values of sound pres-
sure, particle velocity, and displacement,
and their presence sharply intensified some
existing acoustical effects, inducing a series
of new effects, which include the following:
1) strong heterogeneity of the sound
field - presence of high sound pressure and
particle velocity gradients, especially near
the sound source;
2) distortion of the sound waveform
during propagation and the resultant appear-
ance of harmonics;
3) increased sound absorption;
4) intensification of the radiation
pressure, up to the threshold of feeling;
5) appearance of a strong incident
flow (or a series of closed vortexes) called
the acoustical ("sonic") wind or current;
6) medium turbulence.
Examine briefly the enumerated
acoustical effects.
0.1
1,0
J, Im/CH*
Fig. 3. Amplitude* of sir pcrticl*
oscillations «t high sound intensity
- 7 -
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1) The heterogeneity of the sound field. In the analysis of many
acoustical problems the sonic field is idealized, namely, the wavefront is
assumed to be an absolute plane. The assumption simplifies the solution of
problems, but in most cases such solutions are unreal.
It was established in
[72, 105] that even in the case
of assumed ideally flat radia-
tors, the near field is exep-
tionally heterogeneous and is
unrelated to the common plane
wave. As an illustration, Fig-
ure 4 shows the distribution
of sound pressure in the vici-
nity of a piston-type radiator
calculated by Shtentsel'. Using
Huygen's principle, he summed
up the spherical waves radiated
at each point in space from
each point of the piston sur-
face. As can be seen, the
near field was heterogeneous
in the transverse and longi-
tudinal directions, and the
heterogeneity increased with
increasing acoustical fre-
quency. Admittedly, the
average pressure in the trans-
verse direction was equal for
all cross - sections, illus-
trating the possibility of
standing waves formation.
F»8. I*. Sound pr«*suro distribution in th« proximity of
* picton •r«di«tor, according to t>ht«nt«el.
a -
- ; «-D/X = -
Moving away from the
sound source, the sound wave-
front gradually smoothes out.
However, it remains partly heterogeneous, and can be considered plain only
at one small central point. As an illustration of this Figure 5 shows the so-
called radiation patterns of piston-type radiators indicating the sound pres-
sure distribution in the case of radiation into free space. Figure 5 also
shows that the sonic beam narrows with an increase in frequency (at a con-
stant radiator diameter D), although the sound wavefront heterogeneity fails
to decrease. Its heterogeneity is more pronounced when the wavefront pro-
pagated in a confined space, as in a tube. In-such a case, the plane sound
wave arises only on rare occasions, when the tube diameter D is small in
comparison with the wavelength X. Otherwise, transverse modes appear.
The following inequality serves as a criterion for the transverse mode for-
- 8 - . .
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1,84-cg
IT~D'
Fig. 5. Direction of piston
•ound transmitter* at differ-,
•nt D/X ratio* ' :
s-to' •
S678SW
PaccmonHue am c/neitKu, en
Ciepxy CHUJy (1,7 M)
Distribution of sound pressure
in th« coagulation chamber.
• - At 3.6 k Hartz frequency,
b - At 6.0 k.
mation [105]:
Figure 6 shows the dis-
(1.12) tribution of sound pressure
along the transverse cross-
section of the coagu-
lation chamber with
a diameter D=20 cm
[85]. A highly pro-
nounced sound pres-
sure peak is seen at
the center of the
chamber which
slightly decreases
with the distance from
the sound source.
In view of the oblique
sound wave reflec-
Fig. 6~. "Distribution of sound pressure tion from the tube
walls and their inter-
ference with direct
waves, sound pres-
sure decreases along the tube, and subsequently increases arid decreases
periodically [97,126, 298]. All above phenomena occur at any sound inten-
sity; however, the sound pressure gradients and, consequently, the particle
velocity, attain enormous values at high intensity levels affecting the hydro-
dynamic stability of the total sound field.
2) Distortion of the sound wavefront. In the process of its propa-
gation, a sound wave generated by a sinusoidal source transformed gradually
from the sinusoidal into a sawtooth "shock" wave (Figure 7a). This may be ,
due to the fact that in the case of large sound pressure amplitudes, different
portions of a wave propagate at different velocities. Regions of increased
pressure (compression), where the medium particles move in the direction
of wave propagation, propagate at an increased velocity. Conversely, re-
gions of decreased pressure (rarefaction), where particles move in the oppo-
site direction, propagate at a decreased velocity, causing the wave profile
in the interval between the greatest compression and rarefaction regions to
become increasingly steeper; as a result the wave"overflows, " generating
an explosion [46, 124, 133]. The onset of this event is retarded to some ex-
tent by the inevitable decrease in the medium particle velocity. The non-
sinusoidal sawtooth wave may be represented as a Fourier series [ll] com-
prising a sinusoidal wave with a fundamental frequency c% (first harmonic)
arid higher order harmonics with frequencies of 2 u)0, 3 u)0, etc. Figure 7b
is a graphical representation of a wave distorted by the second harmonic.
Thus it may be concluded that the sound wave distortion is accom-
panied by the appearance of significant harmonics.
- 9 -
-------
According to Riemann, the relative magnitude of pressure of the
second harmonic Pa/P1; which represents the degree of wave distortion,
may be expressed by the following formula [46, 124, 6]:
(1.13)
Fi
AA
usas
AA
ISCH
g. 8.
Ch
AA
AAJ
AN
15166 rtSJS l6ldS
AA
AA
AN
20 en SO CM 100 en
anges in the sound nave
It may be concluded from this equation that the degree of waveform distortion
increased linearly with frequency u)0, sound pressure at a fundamental fre-
quency PI, and distance x traveled by the wave. .
Figure 8 pre-
sents oscillograms
•which illustrate vari-
ations in the wave-
form (fo=15 kHz) in
air as (a) functions of
sound intensity level
for a fixed distance
from the SOUnd Source Fi9-7- Sound .eve fort, distortion
, . _ en the process of extension.
and(b) functions of dis-a _ General plan; b -Breaking
, r j-- j j doon of the distorted *aves into
tance for a fixed sound
intensity level equal
to 156 db* (time sweep)
[205]. . ; •_' __' '
Figure 9 shows sound pressure levels
of the first six harmonics of a 15 kHz sound
wave as a function of source intensity level
[205]. Nomogram in Figure 9 and equation
(1.13) can be used in finding pressure levels
of any of the first six harmonics at an arbi-
trary pressure level and frequency of and
distance from the sound source.
(/„= 15 ';
£i — st high level intensity sound origin
i and at constant distance of 50 coj -
b - at receding souna source ana sound
intensity of 156 db.
SO
no
Sound
The condition for the existence of a dis-
continuity in air is, according to A. A. Goldberg
[25] , the inequality pbar ^ 2TT/(khl) which is al-
ways satisfied for gases. The distance from
source to discontinuity is determined by the fol-
130 135 HO US fjff fJJ ISO ffS
intensity^eyel^of the trans.it- lowing expression [ 25, 88] I
rtg. v. Sound pressure levels ot tne
first 6 hantonios in relation to sound
level' intensity. (^o= 15
..(1.14)
* Sound intensitylevel expressed in decibels (db) is related to intensity by
the following relationship: L = 10 Iga J/J0 (or I/IQ, where J0 is the standard
sound intensity equal to 10-l8 watts/cm3.
- 10 -
-------
The distorted waveform of an acoustical standing wave varies with
time [133] and, therefore, the sound pressure had a given finite value at
antinodes . The number which indicates by how much the above quantity was
smaller at the nodes is called the standing wave ratio.
3) Increased medium sound absorption. If a sound wave possessed a
small initial oscillation amplitude A,e, or pressure P0> it decreased exponen-
tially with propagation along the x axis [7, 72]:
- *«*
' (1.15)
where ag is the so-called sound absorption coefficient of a gas in cm 1 (if
the absorption coefficient is expressed in db/m), the number expressing «g
,in cm"1 must be multiplied by 868. 6, if conversion is desired.
The sound intensity decreased with distance according to the following
equation:
The above equation is based on an assumption that a portion of the input
energy lost in a layer was proportional to its thickness: dJ/J = a'gdx. The
sound absorption coefficient a 'g represents the portion of acoustical energy
lost in the interval x = 1. Clearly, ty.\ - 2&g.
The absorption of acoustical energy by the gaseous medium may be
explained, under small amplitude conditions, by the internal friction or the
viscosity and thermal conductivity of the medium (heat transfer existed be-
tween the compressed and rarefied portions of the medium). According to
the sonic theory this assumption leads to the following expression for the
abosrption coefficient [7, 46, 233]:
where
b --\-^~, 1'
3 <-„
(T) is the dynamic viscosity, Xg is the coefficient of thermal conductivity, and
T)1 is the second viscosity). The second term in (1.17) is equal to 0.554 for
air and, therefore b«*2T) +7]1. For all practical purposes Bp»2.57] for air
[233]. According to equation (1.16), the absorption coefficient for dry air is
negligibly small under normal conditions; in the 3-10 kHz range otso - 10"3
-rlO^db/m. Measurements yield considerably greater values under atmos-
pheric conditions due to the presence of water vapor in the air [6, 7, 46, 267].
Knudsen checked this for frequencies of 1.5, 3, 6, and 10 kHz and estab-
lished that the sound absorption coefficient was lowest in dry air; subse-
quently, it rises abruptly attaining a maximum at a relative humidity of 12-20%,
- 1 1 -
-------
after which it decreases again. The maximum value of the sound adsorption
coefficient in the 3-10 kHz range is # =0.17 -0.56 db/m. ;The presence o
boundaries causes additional viscous and thermal losses in acoustic energy.
In the case of a tube with radius R, the additional absorption coeffi-
cient according to Kirchkoff [6, 7] is:
It is thus seen that as the port dimensions decrease, sound absorption in-
creases rapidly. The absorption coefficient in a high-intensity acoustic
field is greater than ago by several orders of magnitude. This can be basi-
cally explained by the appearance of harmonics which, according to (1.16),
are absorbed much more readily than the fundamental frequency. In accord-
ance with the theory formulated by Z. A. Goldberg [25], the local sound
absorption coefficient for a sawtooth wave is expressed by the following for-
mula:
where Px is the pressure amplitude at point x. Thus, at high intensities,
the absorption coefficient is directly proportional to the vibration frequency
if (1.16) is included and to the local sound pressure. In the case of air, equa
tion (1.19) acquires the following form:
, (1.19')
which in the 1-10 kHz range yields:
2g = 0.23 - 2.3db/m for J (or I) =0.1w/cmai
ttg = 0. 73 - 7.3db/m for J (or I) = 1.0 w/cm8.
The linear dependence of ag on p was confirmed experimentally in
[97]; the result of that work is illustrated graphically in Figure 10 (f = 13 kHz,
R = 6 cm). The experimental data can be formulated by the following linear
equation:
aj-o^o-l-mp,, (1.20)
where ago is the experimental value of the sound absorption coefficient at
zero sound pressure and includes losses at the vessel walls in addition to
losses in gas; m is the proportionality factor.
However, the absolute value of mp, is 6- to 7-fold lower than theo-
retical values of ag obtained from (1.19); this cannot be explained by poor
measurements alone. In view of sound absorption, coefficient dependence
on sound pressure, sound attenuation with distance is no longer exponential,
and can be determined from the following formula .[98]:
- 12 -
-------
P — Po ~~ all*
fio
a,. «•««*-flip. (1-«"«•*> '
(1.21)
applicable to plane acoustical waves only. In conclusion, it is to be noted
that sound attenuation in free space with distance from the source depends
not only on the acoustical energy absorption by the medium, but also on cer-
tain geometric factors, such as divergence of the acoustical beam (see Fig. 5),
the intensity of which decreased with its cross section area.
4) The acoustical radiation pressure.
When acoustical waves meet an obstacle, a defi-
nite force is exerted on the latter. The force of
radiation pressure is directed along the wave
propagation in the case of a traveling wave and
from a node to antinode (where the acoustical
pressure approaches zero) in the case of a
standing wave. This phenomenon cannot be
described graphically. V.A. Krasilnikov
offered the following explanation in his popular
monograph ''Sonic and Ultrasonic Waves" [46]:
Ctf.iS/N
10
IS
Fig. 10. Functional relation be-
tween sound «ttenu«tion coeffi-
cient end eound pressure (f - 13 Hz.)
"Particle velocity in compression regions is in the direction
of wave propagation, while in rarefaction regions, it is in the
opposite direction. Thus, resistance to particles' motion is
smaller when the particles travel from the compression zone
to the rarefaction zone than when the converse occurs. Since
the pressure is determined by the product of particle velocity
and the resistivity of the medium, as shown in (1.10), pressure
along the wave propagation direction is slightly greater than
that in the opposite direction. In this manner, a constant pres-
sure is generated in the direction of sound propagation. "
The solution of a wave equation including quadratic terms (second
order approximation) shows that sound pressure at the obstacle is not sinu-
soidal. Its average temporal value is not equal to zero as it is assumed in
small amplitude acoustics, the • proof of which was presented in detail in
[72, 205]. Radiation pressure is the average pressure developed by the
acoustic wave at the obstacle, accounting for the interaction between the
sound field and the unperturbed medium. Magnitude of radiation pressure
is obtained readily using the reasoning of Pol1 [100], who writes:
"Let two straight lines mark the boundaries of a parallel
beam. The particles in the beam vibrate sinusoidally in
the direction of double arrows with maximum velocity
of Ug (the designations here and below are the author's -
Ye. M.). Consequently, the static pressure Pflt inside
- 13 -
-------
the beam decreased according to Bernoulli's equation by
a quantity equal to 1/2 pgU g. This causes outside air to
penetrate the beam which strikes the right-hand side
boundary. Moreover, its velocity decreases to zero
and the pressure increases by an amount equal to p =
1/2 p8Usg . This is exactly the radiation pressure. The
quantity 1/2 pgUag is at the same time equal to the ratio
Kinetic energy in sound field volume V,
sound field volume V
i. e., it is equal to density E of the acoustic energy. For
the same reason, sound pressure pr equals volume density
E of the same radiated energy:"
(1.22)
Similarly, the magnitude of radiation pressure at the totally absorbing
obstacle, dimensions of which considerably exceed the wavelength, likewise
equals the above magnitude. In the case of total sound reflection, the radia-
tion pressure is twice as great:
pR=2£ = ^-'. (1.22')
In a general case of an obstacle with a reflection coefficient J3, the
radiation pressure magnitude is determined by the following equation:
= < 1-4-82)7-" ='(1-f'!*)—-• (1-23)
ce
The force on the obstacle generated by the radiation pressure depends,
on the one hand, on the kinetic energy density of vibrational motion averaged
over the time and space, and, on the other hand, on the shape, dimensions,
and the reflectivity of the obstacle. In a field of high-intensity sound having
a considerable energy density (at J (or I) =0.1-1.0 w/cm8 in the air E = 30 -
300 erg /cm3), forces generated by radiation pressure on normal size objects
acquire appreciable magnitudes. This is evidenced by the "levitation" of
glass balls, coins, and similar objects observable in a high-intensity field
of a standing wave [137, 179]. '
5) The acoustical wind. The vibrational motion in a high-intensity
sound is accompanied by a progressive (non-periodic) acoustical mass flow.
The gas flow in a traveling wave is directed away from the sound
source in the beam center, and in the opposite direction on the beam circum-
ference (see Fig. Ha). A series of closed (toroidally closed in a cylinder)
vortexes may be observed in a standing wave bounded by the longitudinal
- 14 -
-------
walls. These are directed from antinodes to
nodes near the walls, as can be seen in Fig.
lib.
It is generally accepted that acousti-
cal winds are the result of a drop in radiation
pressure caused by absorption of acoustical
waves by a medium. It should be noted that
acoustical wind phenomena were initially
discovered with the aid of aerosols by Kundt
while he was observing dust patterns (Dvorak,
1887). Shortly thereafter, Rayleigh also
noted this phenomenon and, by including
medium viscosity, derived the following
equations for non-periodic components in
a standing wave [107]:
for velocity parallel to a plane sur-
face (axes x) .
m
Ui
,,.,,,,.,,.,1
fea
f ™ ^\
u
w
/Mr
/«•
/w
^
"•--
1
(1.24)
Fig. II. Accoueticsl Horn (sound • gnd)
•hick arises in the sound field.
• - b in travelling oaves} b - c static
•ave - | - sound transmitter} 2 - sound
absorbing substance (reflector).
Arrow Barked section equals X /%.
for velocity perpendicular to a plane
surface,
=_ «/»«.*, r i_;y_ 0^)11 (1.
16c« «- y\ J
where x and y are distances from the antinodes measured respectively paral-
lel and perpendicular to the surface; yi is the distance measured from the
wall to the symmetry plane.
The acoustical vortex streamlines may be calculated by means of the
following stream function [73]:
1
— sin 2kx —
(1.26)
Velocity curves and acoustical vortex streamlines given by equations
(1.24), (1.25), and (1.26) can be found in [73]. It should be noted, however,
that a considerable disagreement existed in a high-intensity sound between
the theory and experiment. Experimentally obtained streamlines are less
angular, i.e., they have a more continuous rounded shape; moreover, the
greater the curvature the higher is the intensity [140]. According to Eckart
[187], direct acoustical flow velocity along the axis of a cylindrical vessel
for the case of a traveling wave is:
("*Xi.i« "- *" -"'- (2 Jr —\
-------
which is a function of the ratio of acoustical beam radius R and the tube radius
Eckart's and Rayleigh's equations are applicable only in the case of infinite-
simal sound absorption when the sound absorption coefficient is independent
of intensity. The equations are also inapplicable at low acoustical frequencies,
since they were derived on the assumption that the wavelength was of consider-
ably lesser dimensions than the vessel. In this connection, new theoretical
studies of the acoustical wind were conducted recently. Among these the
work of Westervelt deserves special notice [287]. A. I. Ivanovskii [36]
recently conducted a detailed theoretical study of the problem of acoustical
flow; he found a general solution which gave velocity values for any sound
absorption coefficient fxs and vessel length L with an accuracy up to the factor
$(Rd).
For a8L<*:l, the acoustical wind velocity is:
"2" ^-^^(^np)- I1-28)
The above expression coincides with Eckart's formula (1.27) providing it in-
cluded the value of ago from (1.16), and (1.17), in which b«j 271+T]'.
ForQfgL,;«l, the acoustical wind velocity is:
Ut = 2/o .L- '-*- R*-=— £*f«><*np). (1.30)
The above equations indicate that the acoustical wind velocity was pro-
portional to that portion of acoustical energy which remained in the vessel
after its absorption by the medium (in the case of aerosols, it is necessary to
include the absorption of acoustical energy by the aerosols, as it is discussed
in Paragraph 8 below). Experimental verification of the above formulas ap-
plied to gases, is at present unavailable. Moreover, .there is a complete
lack of concrete data on the magnitude of the acoustical wind velocities in
gases. An exception to this is paper [233], although experiments reported
in it had been conducted unfortunately at a very high frequency (f = 185 khz).
Furthermore, no absolute measurements of sound intensity were taken. How-
* should read a8L »1 [B.S.L.]
- 16 -
-------
ever, it is known that the acoustical wind velocity in gases is considerably
greater than in fluids, and in the very high-intensity fields it attains few
meters/sec [137].
6) Acoustical turbulence. Observation of aerosol behavior under in-
tense sonic ation leads to the conclusion that a gaseous medium is in the state
of turbulence. This is clearly manifested when a narrow stream of smoke is
injected into the sonicated gaseous medium. As it penetrates the regular
acoustical stream, it immediately spreads in all directions. The turbulent
acoustical flow may be the result of superposition of the continuous spectrum
of pulsations of different magnitudes and directions on the advancing -regres-
sing medium flow. Acoustical turbulence formation is connected with the a-
bove described unusual heterogeneity of the near field [10, 46] which, under
high acoustical intensities, generated vortexes carried further by the acous-
tical wind. In addition, interaction takes place between the direct and retro-
gressive streams [36] in traveling waves, on the boundaries of which, as
Kwick has shown [219], rotational motion is generated. In the case of standing
waves, this is assisted by 1), the circulatory motion in the regions between
the nodes and antinodes, and 2), the aerodynamic mixing discovered by Mic-
kelson and Baldwin [236]. All above effects are sustained by the medium vi-
brational energy. Therefore, Reynolds number, which reflected the process
of sound propagation in a gaseous medium, may be expressed according to
V. A. Krasil'nikov [46] as follows:
Re" :=
(1.31)
The effective oscillatory velocity ug stands for average flow velocity, the
wavelength X — scale of motion, and the constant b [see (1.17)] divided by
the medium density pg — kinematic viscosity. By considering relationship
X=cg/f and equation (1.11), the expression for Reao may be represented in
the following form:
&Ag, (1.31')
whence, Reynolds number is directly proportional to the acoustical impedance
and the medium vibration amplitude and indirectly proportional to the medium
viscosity. If A, is expressed in terms of parameters J (or I) and f , the ex-
pression (1.31) can be rewritten as follows:
D at (Pecg)1''' /'•
Re =__-: (|3r>
No experimental data were found on Reynolds critical number Re", for
which the acoustical flow loses its hydrodynamic stability and becomes turbu-
lent. The only work conducted in this direction by Kastner and Shih [209] is
applicable to low frequencies (up to 30 hz), and cannot be used to determine
- 17 -
-------
the behavior of Re" at higher frequencies. Supported by his experience with
fluid sonication, V. A. Krasil'nikov [46] assumed that Re£° was approximately
one order of magnitude lower than that in the tube flow (where Reor = 2300);
likewise, no information was found on the internal structure of acoustical
turbulence. Without such information, it is impossible to evaluate the effect
of this phenomenon on the physico-chemical processes enhanced by sounds.
This author is of the belief [83] that if acoustical turbulence is developed suf-
ficiently, the knowledge about its microstructure at a considerable distance
from the sound source and boundary surfaces may be obtained from the iso-
tropic turbulence theory developed by A. N. Kolmogorov, according to which,
energy of primary large-scale vortex es was gradually transformed into energy
of small-scale eddys (according to the 2/3 power law). Upon reaching the so-
called l!internal" range IQ , in which the viscous forces predominated (Re
-------
It can be seen that the above equation and (1. 32) are identical, indi-
cating that the initial characteristics of acoustical turbulence were properly
selected and the dimensional analysis was applicable, if factor I = i/°2~ (Y +1),
which equals 6. 8 for air, was included in the derived equation (1; 32).
A comparison of theoretical and experimental energy dissipation values
such as was done in [99] and shown in Figure 10, leads to the conclusion that
a better agreement can be obtained for very small values of |, which is evi-
dently due to the insufficient development of acoustical turbulence and, con-
sequently, its incomplete isotropy for similar acoustical parameters.
Nevertheless, for the purposes of subsequent calculations for estab-
lishing the order of-magnitudes of the local acoustical turbulence character-
istics, the above considerations are not essential, since quantity £ has a
fractional power in all subsequent formulas. As a result, the effect of the
magnitude of § is well concealed. According to equation (1.34), the following
values of the energy dissipation are obtained in the 1 - 10 khz frequency range:
at J (or I) =0.1 w/cm8, e =7.3X10B -7.3xlOe,
at J= 1.0 w/cm8, e =2.3xl07- 2.3xl08.
Therefore, it may be concluded that acoustical turbulence was not character-
ized by an exceptionally high intensity; rather, it was equal to the standard
turbulence occurring in the tube with a diameter D = X/§ at fl6w velocities
equal to the oscillatory velocity of a gas (ti)r = us). According to dimensional
analysis [62], acoustical turbulence "subrange" was equivalent to the following
expression:
The above indicated that the dependence of 10 on the acoustical intensity J
(or I) and especially on the frequency of oscillation f was very weak. In the
f = 1 - 10 khz range, at J = O.lw/cm8 we have the following: 10«180 - 100(Jt, and
at J = 1.0 w/cm8, 10«75 — 45p,. The relative oscillatory velocity of gas parti-
cles at distance 1 apart within the limits of the turbulence subrange (1<10) is:
^..(^S-iMl (,.36,
\t>/pg) W'P^*
and outside the subrange (1>10) it is:
where ax and aa are coefficients of proportionality, usually taken as'l/,/!"^
^ and 1, respectively.
- 19 -
-------
The rate of relative oscillatory particle velocity is independent of
the wavelike flow character, since it is assumed that 1«X, and accordingly,
both medium particles are practically in the same oscillatory velocity phase.
The maximum oscillatory velocity within the 1<10 subrange is found. At
1 =10, J = 0.1 w/cma, and f = 1 -10 khz, it lies in the range of v j. = 5. 8 -10.4
cm/sec, which constitutes 3.7 —6. 6% of the oscillatory gas velocity. The
lower frequency limit of turbulent pulsations which originate inside the
acoustical flow, is given by the following relationship:
U l',',f
&>x~-f-~^-, (1.38)
* PSce
The order of magnitude of the upper limit of the turbulent pulsation
frequency spectrum which lies within the 10 "subrange" is:
b/Pg
if i/ -i/
(1.39)
rs e
9
At a normal acoustical intensity, J (or I) =0.1w/cm , the frequency spectrum
of turbulent pulsations is as follows:
at f = 1 khz, ^ =1. 5 -60 hz,
at f = 10 khz, fj = 15 - 190 hz.
At increased acoustical intensity, J (or I) = 1 w/cma, the frequency spectrum
is somewhat broader:
at f = 1 khz, fx = 5.0 - 340 hz,
at £ = 10 khz, ^ = 50 - 1050 hz.
Thus, it may be stated that the frequency spectrum of turbulent pulsations is
comparatively narrow and always lies in the low frequency range below the
fundamental frequency. In conclusion, some data on the nature of motion in
the boundary layer of large obstacles and walls are introduced here.
Assuming that A, « L (where L is the surface length), the boundary
layer thickness in the case of the medium oscillatory motion is equal to the
"depth of penetration" determined with the aid of the following expression [62]:
.y <1-40>
which in the 1-10 khz frequency range is 6 = 70 - 22p,. The energy dissi-
pation in this layer is a few orders of magnitude greater than in the remaining
volume and, according to [62], may be calculated using the following formula:
i ,
e — ' ,./ /* /1 A 1 \
'norm — ^~ 0>U« . I 1.11 I
• 4 • ^ '
- 20 -
-------
Table I
Effect
Threshold of human ear pain aenaa-
tion (1 10)
Beginning 'of aerosol coagulation
effect by sound
(IS6)
Beginning of sound intensification
•ffect on the dry ng process of fibrous
granular, and poeder substances (it)
Rapid aerosol coagulation (itiH); en-
hanced , rat* of fibrous and pondered sub-
• tance drying (a)} foam disintegration
(157); aerosol filtration (30.1 )
Gradual •arming of sound absorbing ma-
terials, such as cotton and the like
(137, 253)7
Burning sensation bet *een touching
finders. Death of animals and in-
sects.
(157, 252, 253)
Strong acoustic*! winds of several
m/see veloeity (137)
Seimning under pressure of sjlass sound
sphere*, coin* and other object present
in the field ef a static nave (137, 179)
Rapid burning of sound absorb i no ma-
terial* far several second* •( 137)
Maximum sound intensity reached accord-
ing to calculations in the throat of
sound siren (208)
Required sound J
intensity , vi/c»)2
0,00001—0,001
0,003—0,01 •
0,01—0,03
0,1-1,0
4 A
1,1)
1-3
3-10
3-30*
10—30
300—1000
Corresponding
-.force level of
souod L , db
110-130
135— MO
140—145
150—160
IfiA
• 1 W
160—165
165—170
165—175
170-175
185— llto
In the fi«.ld of static sound «ava the concept of "intensity sound"
it* force, since under such conditions the current of acoustical energy i»
totally absent. Uncer such conditions it is nor* appropriate to operate *ith
sound intensity values. Honever, under practical conditions units measuring
sound intensity are frequently used conditionally -
however, in contrast to the acoustical turbulence which occurs in the entire
sound field, the phenomenon is limited to the eddy motion only in the boundary
.ayer. The pulsations cannot be isotropic, since the thickness of the boundary
ayer is of the same order of magnitude as the turbulence subrange deter-
mined from (1.35) for e =e boundary. The above specific properties of high-
ntensity acoustical vibrations gave rise to a. whole series of physical and
Jiolbgical effects observed in sonicated gaseous media. The most significant
>f these are shown in Table 1 above.
5. SOURCES OF HIGH-INTENSITY SOUND.
Special sound sources are used to generate high-intensity acoustic
/ibrations in gaseous media. Pneumatic sound radiators, such as sirens,
lave found applications in industrial installations. In laboratory practice,
vhistles, such as the electrodynamic and, occasionally, magnetostrictive
sound radiators, have also been used. Two types of sirens are known: dy-
lamic (rotary) and static (whistle -type). Mainly dynamic sirens have been
- 21 -
-------
~
utilized in industrial practice, heretofore. They cover a wide frequency
range and offer a continuous coverage, have a high factor of energy conver-
sion of compres sed air (gas, steam) into acoustical vibrations (i. e.. acous-
tical efficiency) with practically unlimited acoustical power radiation.
The dynamic siren consists of a circular stator (main body), with
ports located on its circumference, and a rotor with teeth in the form of a
perforated disk or cylinder. The compressed air (steam, gas) is introduced
into the stator housing from a special compres sor. During rotor rotation
the teeth interrupt periodically the flow of compressed air which escapes
from the stator po:':'ts. This gives rise to pulsations of air pres sure which
cause the gaseous medium to oscillate.
t1
f,
"
:
: L
~..;
.
o
<..>
Two types of dynamic sirl'ns of dif-
, ferent construction are known: axial and
radial. The axial siren (see Fig. 12a) ports
through which the pulsating air stream
flows are axially distributed, and the sound
is directed to the object by special horns
(usually exponential). The radial siren
(Fig. 12b) ports are radially distributed
and the sound is directed to the object by
means of a special, usually parabolic re-
flector. Axial sirens are easily designed
when they are to be small: the radial, on
the other hand, are easy to design when
they are to be large. The fundamental fre-
quency of the radiated sonic vibrations is:
Ilt!!!ll~
~
5
I!!!!
ZIIC
(0 = 60 I
(2.1 )
FIll. 12. Schellat Ic ...r.a.,tat ion of aound
..ittinw dyna.ic air.na.
a - exia' ty.,e (I - atator .ith perfora-
tiona el, 2 - rotor .ith perforatiuna 02'
3 - Mouth~i.c.), b - radial ty~e (I-atltor
. i th f unne' -ahaped p.rf orat ion . 02; 2 - Ro-
tor .ith perforationa 01' 3 - Ref lector.)
where z is the number of teeth in the rotor,
and I10 is the rotor rpm. In addition to this
frequency, dynamic sirens radiate (unless
special measures are taken) moderately
strong harmonics. Methods for their cal-
culations are cited in a paper by In.one [37].
generated by a dynamic siren may be calculated by the
The acoustical power
following formula:
W (1 IQ", Lall.
OK = :Ie + h)' '1:1K KfJln,
102
(2.2)
where 0. is the theoretical discharge rate of air through the siren ports in
m/ see; (Xo is the constriction factor for the air stream blocked by a tooth:
() is a coefficient which recorded that portion of air which escaped through
- 22 -
. ,
1
-------
the gap between the stator and rotor; the product «„ (1 + 6) lies in the range
between 0. 7 -1.4 [247]; Lad is the theoretical work performed by the adia-
batic expansion of 1m3 of compressed air (in kg/m3); T]ao is the siren acous-
tical efficiency. Quantities C^and Lftd are calculated by the following formulas:
Qm=(e*S0)w. (2.3)
LU = > w\ (2.4)
where iS^ is the total cross section of siren ports (in m8); eo is the overlap
factor for teeth and ports (equal approximately to 0. 5); yk is the specific
weight of air at pressure equal to pk (in kg/ma); cu is the velocity of air dis-
charge determined by the following formula:
(2.5)
[pa is the absolute pressure of air compressed in the stator body (in kg/ma);
Pk is the absolute pressure of the surrounding medium (in kg/m8); yc *s *he
specific weight of compressed air (in kg/m3); y is the adiabatic curve index
Cp/Cy which is equal to 1.4 for air, and 1.3 for natural gas and steam. ] When
the difference Ap = p<, - px is small in comparison with pk, the velocity is
(UftrfA/Zg Ap/Yo and Ijaa« Ap. The acoustical efficiency of a siren T|ao depends
on the following: 1) geometry and ratio of stator ports and rotor teeth sizes
which determine the form of the radiated pressure pulses; 2) the compressed
air pressure; 3) frequency of oscillations; 4) the presence or absence of a
horn and its shape; and 5) siren design factors (clearance bet-ween the stator
and rotor at the points where the ports are located, the rotor play, etc.).
The method for determining the efficiency of the horn-equipped sirens was
presented by Jones [208] as an electromechanical model. His method was
given a brief treatment in a book by Heuter and Bolt [205]. In the case of
rectangular pressure pulses generated when the ports and teeth are rectan-
gular and when the tooth width is slightly greater than the port width, the
acoustical siren efficiency is equal approximately to [208, 205]:
=1., (2.6)
where y is a parameter approximately equal to 4Ap/YPx • -^ different formula
is proposed in [205]:
1-1- 0,5 (&J)» + 0.42 3
-------
&p
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
V
0
0,286
0,572
0,858
1,144
1,430
.1,714
2,000
2,288
f ((/) | AP
1,000
0,888 •
0,813
0,7f>8
0,712
0,675
0,642
0,618
0,595
0,90
1,00
1,25
1,50
1,75
2,00
3,00
4,00
5,00
y
2,570
2,860
3,570
4,285
5,000
5,720
8,575
'11,420
14,300
0,577
0,556
0,518
0,488
0,463
0,442
0,38U
0,340.
0,309
(T)ao^ 75%) can be achieved only under low air pressures (pc ^0.3 gage atm),
the exception being only sirens with ports in the form of [de] Lavale [17]
nozzles.
Formula (2. 6') Tabla 2
shows that the acous-
tical efficiency of si-
rens also decreases
with the increase in
radiated sound frequ-
ency. Moreover, the
fundamental, frequency
contained no more than
81% of the radiated
power. When the ra-
diated pressure pulsa-
tion is sinusoidal, .such as is the case (96% of all cases) when ports are round
and teeth are rectangular and the widths of both are identical, the siren acous-
tical efficiency is equal to only one half the value calculated by formulas (2.6)
or (2. 6'), and as shown in Table 2. Therefore, the maximum efficiency of
sirens with sinusoidal pressure modulation never exceeded 50%. In practice
the attainable value was 7]ao = 35 - 40%, since the use of pressures below 0.2-
0. 3 gage atm was related directly to design difficulties. Most widely used
sirens are equipped with circular rotor and stator ports. When the port dia-
meters are equal, the fundamental frequency radiates approximately 82% of
the acoustical energy.
It should be stated that the above cited figures for acoustical efficiency
pertain to dynamic sirens equipped exponential horns (conical horn design is
difficult), i.e., horns, the cross sectional area of which increased according
to the following expression [124]:
SJt = S0epjt, (2.7)
•where SQ is the initial cross sectional area of the horn, x is the distance be-
tween a given plane and the initial plane, and (3 is the horn flare factor.
In the absence of an exponential horn, the siren acoustical efficiency
falls abruptly, especially at low frequencies [137]. The theory and design of
hornless sirens, used for signaling, is presented in [91, 92] by M. I. Kar-
novskii. The acoustical efficiency of these sirens fails to exceed 1 - 2% as a
result of failure to match between sound source and sonicated medium.
Every exponential horn has its critical frequency below which the
sound is cut off. This frequency depends on the horn flare factor F124]:
(2.8)
- 24 -
-------
This fact may be useful when the sound is to be cut off without discontinuing
the delivery of the compressed air to the siren. In this case it is only neces-
sary to lower ^ below n, [sic]. *
The rotor in dynamic sirens is rotated by an electric motor or a
pneumatic turbine. The power consumption of the drive mechanism is small
and approaches 100 -400 watts for experimental sirens and 1-2 kwatt for
industrial (the lower values refer to low frequencies and higher values to
higher frequencies). In comparison with air compressor power, this power
is negligible.
Up to now, only the Ultrasonics Corporation of the USA was engaged
in mass production of dynamic siren models. According to specification
sheets [6, 7, 47, 184, Z40, 254, 272, 291],'the U-I, U-2, U-3, U-4, and
other models had acoustical efficiencies up to 50 -70%, which appears as an
overrated evaluation. . • •
Dynamic sirens designed for laboratory use
and experimental - industrial investigation have been
described in the foreign literature [1, 6, 7, 92, 137,
227, 229, 246, 252, 253, 259, 274, 291]. Dynamic
sirens had been developed also in the Soviet Union.
The first working models were the LIOT and MEI
sirens [73, 74, 75]. Later,, sirens were developed
at the IGI AN SSSR [60, 75], UNITI [17, 18], Gint-
svetmet [126], Promenergo[299], Institute of Applied
Geophysics An SSSR [20], and other institutions. In
recent years, "static sirens" became the subject of
extensive research. Static sirens are characterized
by their simplicity and lack of rotating parts and
special drive mechanisms. Static siren consists of
series of whistles distributed radially in a single
annular resonant chamber equipped with an exponen-
tial horn at its output (if one whistle is sufficient,
the annular chamber may be replaced by a parabolic
reflector). Five types of whistles are known capable
of generating high acoustical power (Fig. 13):**
* Should read DO,. . B.S.L..
** The terminology below differs from the one used
in the literature. The latter frequently gives insuf-
ficiently indicative nomenclature. For example,
the Hartman shock wave whistle is called the "gas-
jet-type'1 sonic radiator; this may stand also for the
vibrating jet whistle which, in foreign literature, is
called the "police" or "mouth" whistle [190].
- 25 -
F»g 13- Schematic presenta-
tion of gas floeing (•h.
sound transmitters.
• - the Hartaan •hi*tle t/v«
(I - nozzle; 2 - resonator)} b -
the Kurkin ehistle type (l -
nozzle; 2 - Resonator; 3 - con-
ical device)} c - rod-snaped
•histle (I - nozzlei 2 - resona-
torj 3 - rod—shapeo device)]
d — Mo* vibrating •hi»tl« (I —
slit-vhaped nozzle} 2 - cylindri-
cal or toroidal resonator *ith a
blade-shaped contrivance 3)i e -
the vortex type whistle (l -
cylindrical chamber; 2 - tangen-
tial infloo type) 3—out— f lo«
cylinder).
-------
a) the Hartman shock wave whistle [199, 7] in which the sonic vibra-
tions are generated by shock -waves set up by the impact of a supersonic gas
jet onto a cylindrical resonator; moreover, regular Shockwaves are gener-
ated in the jet stream;
b) the shock wave whistle designed by Kurkin [53, 55, 56] differs from
the preceding whistle by the presence of a conical pin which, inKurkin's
opinion, gives rise to oblique shock waves;
c) the stem-jet whistle [221] differs from the Hartman whistle by the
presence of a concentric rod which eliminates an ineffective central region
of the gas jet;
d) the vibrating jet whistle [190] in which sonic vibrations are gener-
ated by means of directing a plane gas jet onto the edge of a cylindrical (or
toroidal) resonator wall. The feedback from the resonator cavity acting on
the gas jet causes the jet to vibrate, if the jet is sufficiently thin;
e) the vortex -whistle [282, 47] in which the sonic vibrations are gen-
erated by the moving vortex spiral, creating a rarefaction in the center of
the chamber. This is periodically disturbed by the backward gas thrust as
a result of which an obliquely-pulsating gas stream is established at its out-
put.
The acoustical and geometric parameters of the Hartman shock wave
whistles activated by the compressed air can be determined by eight empiri-
cal formulas which follow (d0 is the nozzle diameter in cm, pc is the excess
pressure in gage atm).
I. Maximum attainable frequency
rofff\
>/,. (2.9)
II. Relative range of frequency variation
«M3KC _ «MHH
n * I) '0
iMaxc
'0
100 = 8,85(pc—0,93)»0. ,''(2.10)
III. Distance between the nozzle and the edge of the first instability
region (where the resonator is placed)
a, = [ 1 + 0,04 (PC - 0,93)*] £ — 1,86 CM. (2.12)
V. Total acoustical power (including harmonics)
s
«7« - (295 yPe - 0,93 d{ em. (2.13)
- 26 -
-------
r
VI.
Power necessary to maintain the air stream
--- i \l7e ~<5250(fJc + 1,033)(pc +I,033tDI-I.OI)~ 8m. (2.14)
VII.
Efficiency of a single whistle
Wax
'1 = - 100 0,.
aK we I .
(2.15)
VIII.
V olume of air consumed by the whistle
Q ~ 0,852 (Pc -I- 1,033) tfc M'/ AtUH.
(2.16)
Table 3 shows
the experimental data
for shock wave whistles
of different sizes with
the following relation-
ship: ~= ~= 1 (1 is the
resonator depth). It
may be concluded from
data shown in Table 3
that the acoustical efficiency. of shock wave whistles was low (4 -5%), and
that the radiated acoustical power was relatively low, especially at higher
frequencies (this is not due to frequency increase; it is due to a decrease
in the nozzle cross section, and a proportional decrease in the air flow).
Because of the above reasons, utilization of whistles without resonant cham-
bers and horns is confined principally to laboratory setups. It should be
noted that because the gas jet expanded beyond the nozzle, it has been cur-
rently recommended that the resonator diameter d be slightly larger than
the nozzle diam~ter ~, ncune1y ~ = (1. 25 -1. 33) ~ [158, 172J. Moreover, to
avoid rapid wear of the resonator rims it was recommended to increase dp
up to 1. 65 ~. The whistle efficiency, thus, increases by a factor of three
and highe r .
4. ul
Pc, 41"'"
1
3
4
5
6
2,61
2,61
2,74
3,~1,
3,t6
Figure 14 shows a static siren
designed at the Scientific Research
Institute for Gas Purification in Indus-
try and Sanitation (NIIOGAZ), It is
equipped with Hartman whistles. It
differs from an analogous Boucher
whistle [148, 149, 150, 151, 158, 172J
with respect to the presence of a
movable threaded end section, which
permitted adjustments to be made in
the depth and volume of the resonant
chamber, and an annular aperture
Tab I, 3
f --~'_:._J_~-~~~--'-
11";. .'"
'"" -" .'" I '.Il.
....' %
it,7
18,2
21,,6
32,7
37,0
''0 ..
0.0,"
335
6!Ji
1380
2!1~
3570
13.-'.
38,2
73,1
103.0
145,0
3,9!1
5 ,'.0
5,29
3,52
4,00
16,3
13,4
10,0
8,9
1
Fig. I~. Static HIIOGAI airen .ith an -accouatic..
collector
I - Sir~ body .lth an outsid. d088-ahaped horn,BK.
2 - inaid. do..-ahap.d horn .quipped .ith an inflow
pip. 3. A 80vable butt. 4 - Circular r.aonatlng
cha.b.r. 5 - Ho~zle and 6 - r.aonator (a.ve~al
o08pI...a' 7 - R on9 ahaped al.t f or air auc t .on In9
off
- 27 -
-------
for air exhaust [52]. The acoustical efficiency of this siren at a frequency
of 4.5 khz is approximately 10%. Data on Kurkin - designed shock wave
whistles, whidi are more efficient than the Hartman whistles, are very
scarce. In contrast to the Hartman whistles, Kurkin-type whistles operated
with relatively small excess pressures of the order of 1 gage atm and lower.
The resonator diameter is equal to or greater than the nozzle diameter.
For equal diameters the frequency radiated by the Kurkin whistles is notice-
ably lower than the frequency of the Hartman whistles. The whistle is
mounted at the throat of a special horn with an elliptical or circular cross
section and an exponential taper.
In this siren, as in the static NIIOGAZ type, a vacuum accumulator
is provided to expel air from the siren after its kinetic energy is transformed
into acoustical vibrations. Results obtained by the Kurkin shock wave whistle
with nozzle diameter equal to 10 mm are shown in Table 4.
. . Table H
Angle of rod
opening
Zp.'degreea
Without the
rod
20
40
44
44,5
47,0
51,0
80,6
Pressure p
in et».u.
3,8
1,0
1,0
1,0
1,0
1,0
1,0
1,0
Rate of air
consumption
0 in B3/hr.
255
177
177
177
177
177
177
177
Sound fre-
quency t ,
• in khz
5,3
3,5
3,4
3,5
4,0
4,0
4,0
3,8
Acoustical
force
1,410
1,380
0,473
1,005
0,670
0,734
0,667
0,915
Acoust ical
ef f . coef .
^ ac i« %
10,4
38,9
13,3
28,2
18,8
20,7
18,7
25,7
Rod type
whistle sirens were
first tried out by
Sevori [47]. They
were used as high
intensity sound
sources o n 1 y r '&-"
cently as a result
of research con-
ducted by the Dem-
ister Co. of Sweden
[221]. These whis-
tles work effectively at low gas pressures, down to 0. 3 - 0. 5 gage atm, so
that the velocity of the exhaust gas jet is below the speed of sound. The re-
sonator diameter is much larger than the nozzle diameter, dp = 2. 5 d, . The
effective frequency range is from 6 to 15 khz. At low frequencies the amount
of air used is approximately one third that used by the Hartman whistles. The
Swedish whistles are easily manufactured and adjusted, since they are less
sensitive with respect to accuracy of distance between the nozzle and resona-
tor.
Vibrating jet whistles had been known for a long time; they present
essentially an extension of the Galton whistle design, although their theory
has been developed only recently [190]. The fundamental frequency of the vi-
brating jet whistle is determined by a formula derived from a well-known
formula by Helmholtz for natural frequency of a resonator, as derived by
Rayleigh [107]:
where b is the resonator port length measured along the cylinder axis; K is a
constant which depends on the width 6 of the above port (when 6=7.3 mm,
- 28 -
-------
K = 0.713); V is the volume of the resonator. SinceV/b is the resonator
cross flection, the whistle frequency is dnterminnd by the croon Mr.r.tlon n..rna.
According to Gavreau [190], the acoustical efficiency of the vibrating
jet whistle is determined by the following expression:
• '_ Pi a«6»6
TjaK _ __ .. _
where a is the jet width in m, q,, is the air discharge rate per second in nm3/
sec, and pe is the air pressure in kg/ma. The principle of the vibrating jet
whistle is incorporated in the multiwhistle Gavreau siren [190, 150], the toroi-
dal Lavavasseur siren [224, 7], and Jahn's siren [206, 150, 151].
Vortex sirens have been described by Vonnegut [282]. Greguss [28,
196] showed that to work effectively, the vortex whistle chamber radius R^,
the nozzle radius r8, and the cylinder radius rt must operate on the basis of
the following relationship:
/?K = -^-+rc, (2.19)
'i*
where o/Q is the internal radius of the quasi-rigid cylinder produced by the gas
motion (rH«sr - 0. 07 cm) and z is a constant which depended on the coefficient
of friction between the gas and the cylinder wall and the kinematic gas vis-
cosity (z~ 0..1). The basic frequency of the vortex whistle is determined by
the following formula [282, 235, 47]:
<«»
where 0^P is a constant accounting for the reduction in the vortex rotational
velocity caused by friction against the resonator wall. In contrast to all
previously discussed sound sources, the vortex -whistle radiation pattern has
two lobes; radiation is practically nonexistent along the whistle axis. Acous-
tical efficiency of static whistles attained same values as were cited above for
the dynamic sirens with sinusoidal pressure modulation, i.e., the acoustical
efficiency is equal to one half of function cp(y) shown in Table 2. Thus, on the
one hand, the maximum acoustical efficiency of sirens equipped with Hartman
shock wave whistles, which exhibit stable operation only at air pressures of
the order of 2. 5 - 3 gage atm, is equal to f|ao =19 - 20%. On the other hand, the
maximum efficiency sirens, equipped with Kurkin whistles, -with stable opera-
tion under pressures of 1 gage atm and lower, was greater than T]ao = 28 - 29%.
The maximum efficiency of sirens based on the vibrating jet or vortex whis-
tles, which also work at low pressures, can attain 35 - 38%. Therefore, it
can be concluded that, in comparison with high-efficiency dynamic sirens, the
acoustical efficiency of static sirens was lower by a factor between 2 and 3.
Other deficiencies of static sirens are the following: 1) need of adjustment;
2) lack of frequency regulation; and 3) intensified wear of working parts. Thus,
application of static sirens can be justified when excess-pressure gas or steam
- 29 -
-------
•was available, and a small-size setup was desired, etc. A common deficien-
cy found in dynamic and static sirens up to now has been the presence of air
flowing from the siren to the sonicated medium, i.e., unless a special mem-
brane isolating the air flow and simultaneously reducing the sound intensity,
was used. However, a method has been found recently which eliminates this
deficiency by providing a suction system which draws off the air directly at
the jet exhaust. This method was proposed and patented by Schaufler [260]
and Lavavasseur [225]. In the Soviet Union this principle was incorporated in
double-chamber sirens made at the UNITI [18] and in static sirens with a
vacuum pump designed at the NIIOGAZ as shown in Figure 14.
In laboratory practice, aside from the sirens, applications were found
for the high-power electrodynamic loud speakers and special electrodynamic
radiators with resonating cores developed by St. Clair [178, 47]. The former
provide low acoustical power at frequencies less than 1 - 2 khz; conversely,
the latter exhibit high acoustical efficiencies (20 - 30%) and higher power out-
put only at frequencies of 10 khz and up. Both differ from acoustical sirens
with respect to high fidelity and stability of the tone essential to laboratory
research. The, so-called, electro-pneumatic sound radiators were also de-
veloped [157] in which the sound vibrations are generated with the aid of elect-
rically-modulated air stream. An ER 6786 electropneumatic radiator yielded
2. 5 khz and 1 kwatt at 1 khz (with pe = 1. 75 gage atm, and Q = 4. 3 nm3/min)
with a negligible consumption of electrical power for 20 watt vibration modu-
lation.
In conclusion, acoustical measurement methods will be described.
The fundamental frequency of acoustical vibrations radiated by a dynamic
siren has been usually determined by measuring the siren motor rpm. This
was done with the aid of a stationary or portable tachometer with subsequent
conversion of the obtained value into frequency, using formula (2.1). The fre-
quency of vibrations radiated by a static siren was determined by means of an
acoustical analyzer, e.g., a harmonic analyzer, such as the AN-1-50, or the
AS-3 or AS Ch Kh - 1 spectrum analyzers [113], Equipment of this type is
also necessary for the determination of audio harmonics of dynamic sirens.
The I Ch type frequency meters are not suitable for these measurements. To
determine the audio spectrum, an oscilloscope of type EO-7 and others may
also be used. In addition to the test vibrations, an audio generator, such as
3G-12 and others, reference output is fed to the scope input. The analysis is
carried out by means of Lissajou figures. Acoustical pressure is measured
by calibrated microphone receivers coupled to a vacuum tube amplifier. The
sound intensity is calculated from the formula on page 6.
i
The condenser microphones, with a titanium diaphragm and piezo-
electrical sound receivers, are used at sound intensity levels above 130 -140
db. The miniature ceramic barium titanate receivers, built in the form of
small hollow cylinders or spheres, are especially convenient. In this case
- 30 -
-------
the amplifier is not needed frequently, and the electrical oscillations are
applied directly to the input of a vacuum tube millivoltmeter of type MVL-2
and others. It should be noted, however, that the sensitivity of these recei-
vers strongly depended on the ambient temperature. Therefore, in order to
measure acoustical pressure in a high-temperature environment, it is neces-
sary to use a specially-cooled or ventilated probe. Recently, piezoelectric
receivers were made from a special ceramic material of type TsTS and
others. These are useful at high temperatures of the order of 200° C and
above. Tiie receiver calibration consists of determining its sensitivity ((iv/
bar) at different frequencies. The most reliable method was described by
N. N. Pisarevskii and T. V. Smyshlyayeva [19].
Since piezoelectrical receivers are highly independent of frequency,
and since strong harmonics are present in the high-intensity acoustical fields,
causing distortion of acoustical pressure measurements, their utilization has
been favored over the condenser-type and other receivers. More detailed
data on the acoustical measurement techniques may be found in a monograph
by Beranek [6] and in selective articles published in the Acoustical Journal
of the Academy of Sciences, USSR.
3. A BRIEF OUTLINE OF THE DEVELOPMENT OF AEROSOL ACOUSTICAL
COAGULATION AND PRECIPITATION PROBLEMS
In 1926 -1927 the famous physicist R. Wood and a rich patron of arts,
Loomis, conducted experiments on the properties of powerful ultrasonic vi-
brations in liquids. They discovered several curious physical and biological
effects which evoked a considerable interest in the world of science. Those
experiments stimulated similar studies in gaseous media. Shortly thereafter,
Patterson and Cawood [249] established that, when aerodispersed systems
were exposed to ultrasonic vibrations, aggregation of suspended particles and
their local accumulation had been observed at nodes of standing waves, simi-
lar to dust figures in the famous Kundt experiments [218]. This phenomenon,
called the acoustical or sonic coagulation of aerosols, attracted the attention
of investigators and, in the following years, special experiments had been
conducted in England, Germany, and the Soviet Union.
Andrade conducted an experimental study of suspended particles' be-
havior in sound fields and established that the smallest particles actively
participated in the vibrating motion of the gaseous medium [141]. Conversely,
gas flowing around larger particles became turbulence centers of the medium,
spiralled and zigzagged in the coagulation vessel. At the same time, aerosol
particifces participated in the circulating motion of the gas medium in the node-
antinode region. Moreover, local particle concentration was observed not
only at the nodes near the vessel walls, but also at the antinodes where the
particles concentrated in the form of special discs [142]. The author con-
cluded that the prime force behind the acoustical coagulation of aerosols was
- 31 -
-------
the hydrodynamic attraction of particles, which Konig [215] previously investi-
gated theoretically, and which Rayleigh [107] noted as one of the reasons for
the dust "ridges" formation in Kundt's tube. Using Konig's attraction force
equations, derived on the assumption of the potential flow of acoustical cur-
rent around a sphere, and on the experimental study of the kinetics of acous-
tical coagulation of magnesium oxide aerosols at high ultrasonic frequencies
carried out by Parker [90] t Andrade [2] postulated a mathematical theory
of the process of acoustical aerosols coagulation. However, in so doing he
made serious basic errors, which caused his theory to be rejected immedi-
ately after its appearance [34].
The most thorough investigations were made by Brandt, Freund, and
Hiedemann [1, 9, 163, 167, 168, 170, 200]. These authors studied experi-
mentally coagulation of aerosols (tobacco smoke, ammonium chloride, paraf-
fin oil vapor) under static and dynamic conditions at different acoustical fre-
quencies. They established that coagulation of these occurred more readily
at audio frequencies than at high ultrasonic frequencies, which Andrade and
Parker used in their experiments.
The authors confirmed the relationship between the amplitude of vi-
brating particles and their size by means of a movie camera, and theoreti-
cally derived an approximate equation for the degree of particle entrainment
in the medium vibration, having established by calculation that the flow around
aerosols in the sound field was predominantly viscous.
Being more descriptive than the corresponding Konig's equation [214],
the new equation revealed the dependence of the particle vibration amplitude
and phase displacement on their size and density and on the medium viscosity
and vibration frequency. This enabled the authors to show another effect
which enhanced aerosol coagulation, namely, the collision of small, actively
vibrating particles with the larger but slower ones (orthokinetic effect). The
authors assumed that precisely this effect played the leading role in the pro-
cess, and introduced their theory of orthokinetic coagulation of aerosols in
the sound field [1, 167].
In his dissertation [161], one of the authors, Brandt, demonstrated
that the rate of acoustical aerosol coagulation was a linear function of the
calculated particle concentration, thus advancing a strong argument in favor
of the orthokinetic theory. Soviet scientists S. V. Garbachev and A. V. Se-
vernyi [26, 195] conducted original studies on the elementary process of hy-
drodynamic interaction of water droplets suspended on glass threads in a
sound field. The authors established that ponderomotive forces of attraction
or repulsion developed between the droplets under the effect of sound which
were similar to those arising in the flow. The ponderomotive effect of sound
waves on large bodies in liquids was previously investigated by P. N. Lebedev
[64]. A. B. Severnyi [109] tried to calculate ponderomotive interaction of
droplets in sound fields on the basis of the hydrodynamic theory advanced by
- 32 -
-------
Bjerkness [146], which allowed for their virtual pulsations. However, Se-
vernyi failed to account for all parameters of the problem. As a result, his
equation, derived for the time required for the drops to coalesce, proved to
be theoretically and practically worthless.
Following experiments conducted in the 30's, which contributed con-
siderably to the theory of acoustical coagulation of aerosols and which demon-
strated the high effectiveness of the process, interest towards theoretical
studies abated and attention of investigators turned to the clarification of
practical possibilities for the utilization of discovered phenomenon. The
first possibility, to utilize the coagulating action of sound in the dispersion
of natural water mists, was demonstrated by Amy in 1931, and patented in
the USA in 1934 [139]. The following years, the second possibility - utili-
zation of the coagulating action of sound for the intensified purification of
industrial smokes, dust, and fogs [164, 165, 166, 169, 177, 204], was se-
cured by the corresponding patents.
In 1938, Gies (Lurgi Co., Germany) conducted unsuccessful experi-
ments on the purification of dust-containing industrial gases [193], using
whistles and a magnetostrictive projector as the sound sources. The specific
power consumption used in the aerosol sonication was a hundred times as
great as the power consumed in purifying the gases by other known methods.
The main reason for the low efficiency of the process rested in the extremely
low acoustical efficiency of sound sources used in the experiments (several
percent). In the meantime, St. Clair (U.S. Bureau of Mines) while investi-
gating acoustical coagulation of aerosols, mainly ammonium chloride [179,
181], developed a new electrodynamic acoustical generator (with a resonating
core) with which efficiencies of the order of 20 - 30% could be obtained at
high frequencies [178]. However, his acoustical generator was of little use
for industrial purposes, since its effectiveness at lower frequencies was
lowered sharply; other shortcomings confined its use to the laboratory only.
During the following years, which coincided with World War II, vigor-
ous attempts had been made in the USA to build a powerful, highly efficient
siren-type sound generator for special purposes specified by the U.S. Navy
for the dispersion of fog on airfields. Using the electromechanical analog
method, Jones developed the dynamic siren theory [208], which had the possi-
bility of attaining high acoustical efficiencies. Unfortunately, the value of
Jones' work was not fully appreciated in other countries and, therefore, many
siren models built during subsequent years were frequently beyond criticism
with respect to their efficiency.
Based on Jones1 theory and calculations, Chrysler Corporation built
the "Victory" sirens, having a sound power which attained 50 hp at acoustical
efficiency of 70 - 90% [208]. With these sirens Lamer and Sinclair [266, 156*]
conducted experiments in 1943 on the acoustical dispersion of natural fog at
the Lank en airport in Cincinnati, Ohio. Positive results were obtained under
- 33 -
-------
calm weather conditions when the runways were sonicated for approximately
one minute. The same investigators conducted previously a series of experi-
ments on the dispersion of artificial water vapor. Experiments conducted on
laboratory and larger scales brought positive and negative results, the latter
due to the inappropriate selection of sound'frequencies. Loudspeakers and
small sirens made by the Federal Electric Company were used as sound
sources. The same year Lamer, Sinclair, and Brescie [156]* conducted ex-
periments oil the acoustical dispersion of sea fog in Sandburg, California at
high wind velocities (.5 -6 m/sec). However, the acoustical power of eight
"Victory" sirens was insufficient to attain the desired visibility under such
unfavorable conditions.
During post-war years, the USA Ultrasonics Corporation introduced
series of siren models with efficiencies of 40 - 60%. This turned investigators
to experimental work on the intensification of industrial gas purification. In
1947 the same firm built the first experimental unit for the recovery of highly
dispersed carbon black [254, 184, 272]. It consisted of an acoustical coagu-
lation chamber and two series-connected inertial precipitators - cyclones.
This kind of combined dust-droplet trapping units -will be referred to hence-
forth as acousto-inertial.
Experimental data on the first acousto-inertial unit proved to be very
encouraging, which enhanced further experiments on recovering'other indus-
trial byproducts such as: sulfuric acid vapors, anhydrous sodium carbonate,
molybdenum sulfite, cement dust, open-hearth furnace dust, flying cinders,
and others [183, 184, 186, 216, 240, 241, 269, 273, 281]. Acoustical para-
meters, such as sound intensity, frequency of vibrations, sonication time,
which determine the effectiveness of acoustical coagulation of different indus-
trial aerosols, had been established during these experiments. In particular,
it became apparent that there existed an optimal vibration frequency for each
aerosol at which the process proceeded most efficiently. A strong dependence
of the effectiveness of acousto-inertial precipitation on the physical charac-
teristics of aerosols, such as degree of dispersion, gravimetric concentra-
tion of particles, etc., was discovered. It became clear that 1) highly dis-
persed aerosols possessed the best acoustical coagulability, even though they
required higher sound intensities in order to effect consolidation of o'riginal
particles in a short time to a size which could be effectively precipitated by
inertial precipitators; 2) weakly concentrated aerosols possessed a lower
acoustical coagulability, despite the fact that a highly intensifying action could
be attained in this case by injecting atomized water ("sprinkling") which was
not always applicable; 3) residual concentration of particles was considerable;
etc.
* Reference [156] does not correspond to these authors [B.S.L. ]
- 34 - ' . = '
-------
These conditions considerably narrowed the field of industrial appli-
cation of the acousto-inertial method of aerosol precipitation. However, no
serioTB attempts had been made to develop the process [203, 204, 264]. This
may have been due to the fact that investigation was limited to practical ap-
plications only. Theoretical studies, without which planned development of
technological processes is unthinkable, were completely neglected. Almost
no experiments had been conducted during that period in the USA on the theory
of acoustical coagulation of aerosols. The one exception was a study by St.
Glair and his co-workers on the drifting of aerosol particles under the effect
of radiational sound pressure [179, 181]. Using King's equation for the radia-
tional pressure on a sphere immersed in a non-viscous medium [212], St.
Clair derived the equation of a sphere motion in the direction of nearest anti-
node. St. Clair concluded that the radiational sound pressure was, indeed,
the decisive factor in the intensive coagulation of aerosols in a sound field
[179]. However, elementary calculations failed to corroborate the case in
question [73]. Moreover, Westervelt's work, which appeared shortly there-
after, and which dealt with non-periodic ("constant") forces acting on particles
suspended in the sound field [286], cast a doubt on St. Clair)'s equation. Wes-
tervelt showed that in addition to the radiational pressure, much more sig-
nificant viscous forces acted upon suspended particles. These depended on
periodic variations in the medium temperature. Also attendant were forces
associated with unavoidable waveform distortion of finite-amplitude waves.
In view of the significant limitations imposed on the application of the
acoiisto-inertial method of aerosol precipitation, which became apparent in
the course of experimental work, the proposed method received industrial
acceptance only in isolated cases, such as sulfuric acid vapor, anhydrous
sodium'ashes, sodium carbonate, molybdenum sulfide, etc. This greatly
reduced the interest among the U.S. industrialists in the organization of ex-
perimental works, and, after 1951 -1952, experiments in this direction were
almost completely discontinued.
However, in other countries, the interest in the new method gradually
increased from year to year. Experiments had been conducted in the Soviet
Union, Hungary, German Federal Republic, Japan, Poland, Austria, France,
and Czechoslovakia. The first study of the acousto-inertial precipitation of
industrial aerosols was conducted in the USSR by P. A. Kouzov at the Lenin-
grad Institute of Labor Protection (LIOT) in 1950 -1951. Experiments had
been conducted on the precipitation of aggregated light ashes trapped by means
of electrofilters of power-plant boilers and emitted into the air by mechanical
atomization. These, and other [193, 73] experiments, had indicated that ag-
gregated aerosols possessed infinitesimal acoustical coagulability. As a
result of this it is suggested that investigation of the coagulating effect of
sound should be conducted with natural aerosols.only. It must not be accepted
that Schnitzler's (Lurgi Co., GFR) clearly negative conclusions about the
method [262], which were based on experiments with the acousto-inertial pre-
- 35 -
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cipitation of aggregated magnesium- and aluminum-oxide aerosols were well
founded. Following the unsuccessful Baker soot coagulation experiments
conducted at the VNIIGAZ (Kh. Grigoryan [29]), the present author obtained
positive experimental results with acousto-inertial precipitation of artificial
oil vapor [73, 74], conducted at the Moscow Institute of Energetics during
1953-1955.
The coagulating effect of sound was tested in Hungary in connection
with the intensification of gravitational precipitation of cement dust directly
in the smoke stack (Tarnoczy and Greguss [275]). Extensive experiments
on the acousto-inertial method of precipitation of industrial aerosols had been
conducted in Japan in 1950 -1954 (Oyama, Inoue, Sawahata, and Okada [246,
38]). Japanese investigators experimented with the acousto-inertial precipi-
tation of zinc, carbon black, coal tar, cracking-gas condensate, and obtained
positive results for the last two aerosols. As a result of their experiments
the authors established some empirical relations characteristic of said pro-
cess. In particular, exponential dependence of final particles' concentration
on the specific power consumption and the sonication time had been established.
The authors participated in the construction of an industrial acousto-inertial
plant for purification of cracking gas in Tokyo in 1952 [259].
Considerable interest -was aroused by a study of the acousto-inertial
precipitation of zinc oxide sublimates, conducted during the same period in
Poland at a metallurgical plant in Szopenice (Maczewski, Rowinski and others
[227, 228, 229, 92]. The experiments yielded good acoustical coagulability
of zinc oxide sublimates. However, they also revealed that, as a result of
the periodic decrease in their concentration in furnace gases, the final aver-
age particle concentration exceeded the allowable limits for toxic aerosols.
Experiments had been conducted in Austria on the acousto-inertial
precipitation of ferric and cobalt oxides and others (Jahn [206].).. Experi-r
ments indicated the need to maintain a given critical precipitators'velocity,
above which an intensive disintegration of dry aggregated particles began.
Subsequently, experiments had been conducted in France on the acoustical
dispersion of artificial water vapors and the acousto-inertial precipitation
of smoke. A positive effect of aerosol sonication by sound waves of different
frequencies (produced by a static siren of the "Multiwhistie" type built by
Boucher [148, 149, 150, 151]) -was demonstrated by experiments in acousto-
inertial precipitation of smoke from carbide furnaces (Boucher [150, 151]).
Flue gases, artificial water vapor, etc., had been subjected to acousto-iner-
tial precipitation in Taraba, Brzica [274], Czechoslovakia.
The above cited experimental studies helped to advance the knowledge
in the field of acoustical coagulability of industrial aerosols; they also in-
creased the uncertainty concerning the commercial capacity of the process
due to low efficiency of the sirens used. Moreover, the same studies had
been conducted for practical purposes in most cases. Nonetheless, the
- 36 -
-------
mechanism of the process of acoustical aerosol coagulation remained unex-
plained as before, and this inhibited further development of the process.
In the course of following years, Soviet scientists attained definite
progress in clarifying that mechanism. During^l954 4955, the present author
conducted a critical analysis of theoretical hypotheses on the mechanism of
aerosol coagulation. The analysis was based on materials available at that
time; it was concluded [73, 74] that none of the available materials offered
a satisfactory explanation of the phenomenon. The orthokinetic interaction
associated with irregular motion of particles was acknowledged as the only
viable coagulation system. This system is still valid today in spite of the
fact that many previous ideas about the behavioral rules governing aerosols
in a sound field lost their validity.
S. V. Pshenai-Severin showed [103] that if aerosol interaction in a
sound field was considered, it became necessary to proceed from the viscous
flow concept described in terms of Oseen's hydrodynamic theory, which par-
tially takes into account the inertial terms of the equation of the medium mo-
tion. In view of this, Bjerkness1 and Konig's formulas, which were intro-
duced into the potential flow assumptions and which were used earlier to de-
scribe the hydrodynamic interaction forces acting on spheres, were found
unsuitable for the calculation of attraction forces between aerosols present
in a sound field. Pshenai-Severin established that a longitudinal attraction
occurred between aerosols in a sound field which was similar to the hydro-
dynamic attraction between cloud droplets [101, 102, 104, 250]. This form of
interaction will be referred to henceforth as the attractional interaction.
Using Oseen's concept of flow around aerosols in a sound field, the
author of the present monograph found that hysteresis came into action in the
case of small obstacles [81], As a result, particles of the vibrating medium
assumed a zigzag motion with respect to particles around which the flow oc-
curred. Moreover, a special acoustical flow occurred around each aerosol.
In another treatise [82], the present author showed that in the case of ortho-
kinetic aerosol interaction in the sound field selfcentering of aerosols took
place, as a result of which trapping probability increased. Parallel to self-
centering, a reverse effect appeared possible for a given aerosol distribution.
This effect will be referred to as the self-dec entering of aerosols. Both phe-
nomena represent a new form of aerosol interaction in a sound field. The
effect, referred to in this book as the par akinetic aerosol interaction depended
on the flow hysteresis of the medium around the particles, and on the particle
inertia. A high-speed microfilm of the aerosol behavior in a sound field was
taken recently by O. K. Eknadiosyants and L. I. Buravyi at the Acoustics In-
stitute of the Soviet Academy of Sciences [12]. The study revealed the for-
mation of spatial aerosol aggregates, which proved implicitly the existence
of parakinetic interaction.
Podoshevnikov's experimental study of acoustical coagulation of highly
- 37 -
-------
dispersed dioctylphthalate is of interest [94]. He established the existence
of ah exponential relationship between the quantitative concentration of drop-
lets and the sound pressure exerted during sonication time. This agreed
withlnoue's conclusions [37] based on his experimental study of acoustical
coagulation of tobacco smoke conducted by Brandt, Freund, and Hiedemann.
Transformation of the dispersed composition of droplets, which occurred
during acoustical coagulation of a fog, was investigated [95], In collaboration
with V. A. Gudemchuk and B. D. Tartakovskii, the author established [30]
the existence of a negative effect of longitudinal partitions in the coagulation
chamber on the rate of acoustical coagulation of the fog, which the authors
associated with the observed attendant decrease in the medium turbulence.
This observation definitely refuted P. N. Kubanskii's concept concerning the
expediency of developing precipitational surfaces in the coagulation chambers
[48, 49]. The same was theoretically disproved earlier in the work of this
author [73]. However, contrary to assertions in [30, 93], there appeared no
doubt that orthokinetic particle interaction played a leading role in the process
of acoustical aerosol coagulation. In view of differences in the phase dis-
placement angles of particle vibration with respect to the medium, instan-
taneous aerosol velocities became discernible even at insignificant differ-
ences in the degree of particle entrainment by the vibrating medium. Since
all aerosols were polydispersed to a degree, ortho- and parakinetic inter-
actions in a sound field were not an exception but a rule.
In an earlier work [83], in which the acoustical turbulence theory was
developed, the present author showed that turbulent pulsations which occurred
in the sound field were capable of rapidly drawing aerosol particles to dis-
tances at which self-centering and, subsequently, orthokinetic effects took
place (a concept that Brownian motion was a leading factor of diffusion in the
aggregated volume of particles, was refuted by this author earlier [76],
For better understanding of aerosol behavior in a sound field, the
work by S. S. Dukhin [35] may be of interest. Dukhin presented a theory of
aerosol drift into the node of a standing wave, which depended on the attendant
asymmetry of vibrating motion of the medium particles. This problem was
approximately solved earlier by A. D. Bagrinovskii [4], who unfoundedly con-
sidered this effect as the leading factor in acoustical aerosol coagulation.
In addition to the above cited theoretical works, many experiments
had been conducted in the Soviet Union during recent years on the acoustical
coagulation and precipitation of industrial aerosols, in which new systems
had been tested. One of these consisted of a combination of an acoustical
coagulator and a fabric filter, which permitted to obtain low residual particle
concentrations at an increased rate of gas filtration. The system was first
tested in the precipitation of borehole dust by P. Sh. Shkol'nikova ("Gipro-
nikel'," Leningrad) [130, 131, 132, 300] and was received favorably by indus-
try.
- 38 -
-------
V. P. Kurkin (NIIOGAZ), while working on acoustical coagulation of
highly-dispersed oil burner soot, investigated the following system: acous-
tical coagulator followed by two cyclones and a glass-fiber filter [54, 57, 298].
Kurkin established that the system was more efficient than the known "Cana-
dian" system. In the course of investigation, the author designed a new gas
jet radiator [53, 55, 56], the acoustical efficiency of -which was higher than
that of similar radiators built earlier. At the present time, installation of
an experimental-industrial unit, of similar design, is being completed at one
of the Soviet carbon black plants. The system had been tested on the "Prom-
energo" equipment built at a bronze-brass plant for the extraction of zinc
oxide sublimates [23, 299].
It is now proposed to classify the acoustical coagulators as a class of
dust catching devices which might also include turbulent, condensating, and
electrical coagulators [59]. Acoustical gas separators for low temperature
separation of natural gas condensate, now being tested experimentally, are
of great interest [79,. 84]. A portion of the natural gas excess pressure is
being used in generating sound, and, therefore, the question of economy is
of no importance. This idea had been tested first at one of the natural gas
deposits in the Volgograd region [86]. The prospect of applying the acous-
tical method in the gas industry is favorable; however, serious efforts are
required to overcome different problems. Interesting possibilities are being
opened by the graduated method of acoustical aerosol coagulation [77, 78].
According to this method, coagulable aerosols are passed through series of
coagulation chambers in which they are exposed to the action of acoustical
vibrations with a gradually diminishing frequency optimal for each stage of
particle coagulation.
Another method, described in [76], is also of great interest. In this
method, aerosols are sonicated in an electric chamber, into which charged
droplets of -water or other liquid have been previously introduced. The drop-
lets convey the dust settled on them toward the precipitating electrodes. The
method, called the electrostatic, combines the coagulation and precipitation
of aerosols.
Original experiments on the application of acoustic coagulation for the
purpose of rain-making had been conducted by the El'brus Combined High-
Altitude Expedition [32], A series of additional experiments on the problem
in question had been conducted in the USSR [13, 14, 15, 33, 89]. Among for-
eign works on the acoustical coagulation of aerosols, two proposals are of
specific practical interest: 1) a patent granted to Smith [268] for a method of
catalytic hydrocarbon pyrolysis in which the recovery of suspended catalysts
was attained by means of acoustical coagulation, and 2) Slavik's proposal [24]
to extract harmful gaseous chemical impurities by means of chemical rea-
gents which first combined with the impurities to form suspended solid or
liquid substances and, subsequently, could be precipitated by acoustical co-
agulation.
- 39 -
-------
Recent attempts to utilize acoustical vibrations for the direct intensi-
fication of aerosol precipitation, are of great practical interest. The first
proposal in this direction was patented by Westervelt and Sieck [290]. The
proponents described a method of aerosol separation by specially profiled
sirens -which generated distorted waveforms, for which the maximum par-
ticle drift velocity was attained. Asklof proposed a sonic droplet trap [301],
in which the process of mist filtration through a porous cap was intensified
by means of sonication. Asklof's proposal opened a new phase in the acous-
tical dust-droplet trapping history, since it enabled to lower the residual
aerosol concentration to a fractional part of mg/m3 of a gas at a reasonable
energy consumption. Boucher [147, 157] proposed to intensify the process of
aerosol precipitation in the existing dust-droplet trapping devices - the Ven-
turis c rubber and the cyclone. Sonication may be applied also to sleeve
filters; Abboud [135, 157] presented a method of cleaning these by means of
acoustical waves.
Interesting ideas appeared abroad also in the field of acoustical dis-
persion of natural mists. Boucher presented a new, so-called, thermo-
acoustical method of fog dispersion from airfields [152]. His proposal con-
sisted of a combined sonication and fog heating. He also proposed an acous-
to-chemical method for fog dispersion, which was a combination of sonication
and atomization of hygroscopic substances in the fog [156]. This method is
now being tested in France, England, Sweden, USA, and other countries.
Both methods represent examples of sound utilization in the intensification of
an evaporation process of a dispersed liquid phase. This possibility had been
demonstrated during 1938 -1942 and was based on the assumption that sound
and turbulent pulsations flowed around the aerosols.
With the above in mind, Horsley and Danser [202] proposed in 1947 a
method of spray drying of wet products, such as soap solution, in a sound
field which, as far as is known, remained unadapted by the industry. However,
stimulated by Greguss (1955), the process of sonicated drying of damp pow-
dered materials had been developed. In this field, Boucher carried out inter-
esting experiments [172, 153, 154, 155, 157], which are being continued by the
Acoustics Institute of the Soviet Academy of Sciences [8], and other domestic
and foreign scientific-research institutions. The special feature of the new
method is the fact that drying was attained by a "cold" process without in-
creasing the material temperature. This is invaluable in a case of drying
heat-sensitive materials.
Up to now, sonicated drying had been tested on the following hard-drying
materials: cellulose, carboxyl-methyl, titanium dioxide, colloidal zirconium
hydroxide, silica gel, heat-sensitive ferments and hormones, cellulose ethyl,
etc. In view of the excellent results obtained with acoustical drying of mate-
rials, this method can be recommended now for industrial application.
Greguss [28, 196] conducted interesting experiments on the combustion
- 40 -
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intensification of atomized liquid fuels in metallurgical furnaces used for
smelting malleable cast iron; he used in the process a combination whistle-.
atomizer of his own design and obtained satisfactory results.
Intensification of other physicochemical processes taking place be-
tween dispersed and gaseous phases had also been tried. Thus, as early as
1947, Richardson [256] proposed the use of ultrasonic vibrations for the in-
tensification of ammonia synthesis in the presence of suspended iron catalyst. '
Carlstrom [7] pointed but the possibility of intensification of the amalgamate
condensation process used in dentistry. Investigations at the Universal Ani-
line Company (USA) proposed a process of forming finely dispersed carbonyl
iron in the course of thermal decomposition of pentacarbonyl iron in the
sound field. However, as far as known, their proposal failed to find practi-
cal applications.
Experiments on fluidization of powdered products (gypsum and others)
[237, 217] and the generation of aerosols [134, 207, 270] by sonic and ultra-
sonic vibrations had been conducted. However, the above processes are
based on entirely different physical effects and, therefore, will be considered
briefly.
For the same reason the practical use of aerosol sonication will not
be discussed. The latter had been used to determine the following: vibration
amplitude of a gaseous medium [226, 173, 140, 141, 143, 263], acoustical wave-
form [191], acoustical currents [140], sound field near an absorbing surface
[234], dispersed aerosol composition [174, 198], aerosol concentration, and
many other parameters.
CHAPTER 2
MOTION OF AEROSOL PARTICLES IN A SOUND FIELD
4. PRELIMINARY INFORMATION
Solid or liquid particles suspended in a gaseous medium are charac-
terized by high mobility. Sufficiently small particles (r
-------
ture, due to the following: 1) in one way or another, particles participate
in oscillatory motion of a gas; 2) they propagate forward and drift under the
influence of secondary effects, such as radiation pressure, asymmetry of the
acoustical waveform, etc.; 3) they are carried off by the advancing circu-
lating motion of the sonicated medium, the so-called acoustical wind, and, at
the same time, participate in its turbulent pulsations; and 4) they perform
complex motions which depend on the hydrodynamic par akinetic and attrac-
tional interaction with the neighboring particles. In addition, the suspended
particles have a rotational motion, although this phenomenon is disregarded
here, because it lies outside the scope of the problem at hand.
The relationships for the first three types of aerosol motion in an
acoustical field -will be analyzed in the succeeding paragraphs of this chapter.
Particle motion caused by their mutual interaction will be examined separ-
ately in paragraphs 10 and 11 of Chapter 3. Only those types of aerosol motion
in the acoustic field will.be examined first which are not constrained by the
presence of other particles. For all practical purposes, this occurred only
in the infinitely rarefied, i.e., loosely concentrated, aerosols.
Before proceeding with the concrete analysis of each type of aerosol
motion in an acoustical field, the following is noted: the aim of the subse-
quent sections of this chapter is to derive and analyze equations which corre-
sponded to a type of aerosol motion and which determined their spatial and
temporal displacement as a function of the physical parameters of the particles
and the medium. In all such cases, as also in the entire field of hydrody-
namics, analysis is based on Newton's second law:
•"p-Sr^'F'+F. (4.1)
where nip, Xp, and t are, correspondingly, mass, absolute displacement, and
time necessary for particle displacement; F1 is the "static'1 force acting on
the particle which exists independently of particle displacement (this force
owes its origin to the presence of pressure gradients in the acoustical field,
the forces due to radiation pressure, etc.); F" is the "kinematic" force as-
sociated with the occurrence of a relative motion between particles and med-
ium, i. e., the leading or lagging particle motion with respect to the medium.
This force depended mainly on the nature of the medium flow around the par-
ticles. The flow, as is known, determined the Reynolds number Re = 2rugp/v,
which described the relationship between the inertial (numerator) and the vis-
cous (denominator) forces (see [100], p. 209). For small Reynolds numbers
(Re
-------
the medium, and the described picture changes; the gas boundary layer be-
comes progressively thinner, until it finally disappears completely. In this
case, the viscous losses and the related effects of turbulence can be dis-
regarded. A, so-called, potential body flow regime sets in. The difference
between the viscous and potential states of a streamline flow around spheri-
cal objects ;is well illustrated in Figure 45 of the well-known N. A. Fuchs
monograph [121],
The state of a streamline flow around aerosols developed in oscil-
latory medium under the normally applicable sound field parameters will be
determined first. The maximum possible velocity of flow around aerosols in
a sound field represented the peak value of the medium's oscillatory velocity.
All other motion forms have velocities known to be smaller, since they have
been produced by secondary effects, "sustained" by oscillatory motion energy.
The amplitude of oscillatory velocity U for a "normal" sound intensity of the
order of 0.1 watt/cm8 is equal to 220 cm/sec. The maximum particle radius
which corresponded to this velocity, for which Re =1, equals r«3. 5|0t for air
(y =0.15). The radius of aerosols which tend to coagulate is usually less than
r = 3 - 5\i,. For the greater portion of the oscillation period this corresponded
to Re
-------
tially on the prior motion of the particle, namely, on the acceleration at
prior times tj. When the velocity varied slowly, i.e., when the acceleration
dugp/dt was small, the integral term may be disregarded. In this case, the
expression for the kinematic force acting on the particle was as follows:
. (4.3)
•which is the well-known Stoke formula derived on the assumption of stationary
flow of the viscous medium around a sphere. When Reynolds number is in-
creased (Re>l), a correction factor by Oseen (l-f"3/l6 Re) must be added to
the formula. This, however, should be avoided, if possible, since it intro-
duces a square velocity term which makes the solution of the differential
equation for the particle motion difficult. The term "slow" variation of the
acting force implied that variation of duration T was longer than the particle
relaxation time T, given by the following expression:
•''"-if". .(«)
This parameter, widely used in aerosol mechanics, characterized the rate
at which equilibrium in the system "particle -medium" was reestablished, and
it determined the particle sensitivity to variations in the forces acting on the
particle. This is not surprising, since the given expression was the result
of dividing the momentum acquired by the particle mj,ugp by the Stokes1 force
Fit = 6nTlrugp.
Relaxation time of coagulating aerosols is very short. For a unit
density (pp = 1) in the atmospheric air (T]= 1. 85 X 1CT*) it takes on the following
values:
= 0.1n T =1.2xlO~7 sec
= 1.0p, T =1.2x 10"5 sec
forr=10^ T =1.2x 10"3 sec
It follows that the shorter was the particle relaxation time, the
faster the particle acquired the new velocity value in relation to the medium.
The new value corresponds to the instantaneous magnitude of the force acting
on the particle.
If the relaxation time was small as compared with the medium velo-
city period, the particle barely departed from instantaneous medium velocity
values which corresponded to the time-varying acting force. In hydrodyna-
mics, this motion is referred to as quasistationary.
A general solution of the integro-differential equation for the non-
uniform motion of spherical particles in an irregularly moving medium, ob-
tained by substituting the expressions for F1 and F" into (4.1), has been
derived by Tchen [175, 122*] only recently.
* Reference does not correspond to Author [B.S.L.]
- 44.- -
-------
5. VIBRATORY MOTION OF. AEROSOL PARTICLES
Particles suspended in a vibratory gaseous medium are carried
away by the medium's motion, if they are sufficiently small. The degree of
each particle entrainment, i.e., the ratio between the particle and medium
displacements (or velocities), depended to a great degree, on the physical
parameters of the particle and the medium. The exact theoretical solution
of this problem was offered by Konig [214]. Later, this problem was given
a brief treatment by Sewell in [265]. The same problem was also examined
byS. M. Rytov, V. V. Vladimirskii, and M, D. Galanin [108]. The problem
applicable to real aerosols coagulated in a sound field was reanalyzed by
Brandt, Freund, and Hiedemann [167]. A thorough analysis of the problem
of vibratory aerosol motion in a sound field was presented by N. A. Fuchs
[121] in a monograph with supplements and correction presented in a subse-
quent review [122]. The corresponding equations for aerosol motion will now
be formulated.
A "static" force acting on a particle in a given case is caused by
the presence of a medium pressure gradient. It can be easily understood
that the magnitude of this force was entirely independent of the particle den-
sity. Assume that the latter was exactly equal to the medium density. In
this case, the particle with the velocity Ug must move together with the med-
ium. The force capable of bringing about this type of particle motion is
equal to the product of the particle mass and its acceleration. The mass of
this particle is equal to the mass of the displaced medium rrig, and its accel-
eration is equal to the acceleration of the medium du,/dt. The corresponding
"static" force is
F'~mi~ (s.i)
The "kinematic" force, which varies1 periodically with frequency
-------
Assume:
M = mp+9-mtb (M-"reference mass"),
j ^~m&wb(l +b) = ~mgo)b-\- 6.nt)/- (B - "reference mobility"). Substituting these
notations and disregarding the term _Lmt—Z , we obtain:
du I/O / Q ^ \ vU
t A ^^^P I ^ ** -~- * J • i » » • I If _ * - »* \ K /f*J*
AI — — mM>o(\ 4- o) u., -I- I — TnJo -\ mt \ — . (o.4i
tU B 4 . \ 4 2 J dl
If the medium vibratory motion is expressed by the following equation:
«, = £/, sin «rf. (5.5)
Then, after some transformations, equation (5.4) becomes as follows:
M -* -f- — = - mta>Ut i / 1+3& +-&« +-&» + -&« sin (G)/4-8),
(5.6)
where ,
e,= arctg^±l-. (5.7)
The solution of this equation is as follows [39]:
— t JL 9
~i 2 *~ y A
uP = Ut\/ — sin .{at/ — (cp — 6)],
a» + 3afr 4- - 6» + - (.» + - b*
i i 4
(5.8)
•
where
3 nig 3 3' p«'
For sin [out - (cp - 9) = 1, the particle velocity attains its peak value Up = Up;
whence, the degree of particle entrainment is determined by the following
formula:
jr~t 5 ^
U. -ml 2 2 4
^-|/r+^i,+i>+i>'.">
- 46 -
-------
The phase shift between the particle and medium vibrations is
equal to: 8
H
— 6)= - ' - - - . (5-10)
' / ova a a a x
(3 \ 3 0 00
l+T6)+2'6+T6i + "2".6>^T
It is, thus, seen that expressions obtained by the exact solution of
the aerosol motion problem become complex and involved; as a result, the
relationship between the degree of particle entrainment and the phase shift
of vibrations, and the physical parameters of the particle and medium, re-
main unclear. Thus, the approximate solution of the problem given by
Brandt, Freund, and Hiedemann appears to be very useful and, at the same
time, it provided a degree of accuracy sufficient for all practical purposes.
The approximate solution of the problem needs to include only the viscous
forces given by Stokes' equation (4.3):
....... . ..... .........
m,-£- =
Include equations (5.5) and (4.4) and obtain the following equation:
due
*~ + up = UgSina>l. (5.12)
The general solution of this equation has the following "form [36]:
U =
U
where the phase shift cp is determined by equation;
The phase shift angle between tHe medium vibrations and the en-
trained particle can be explained by the fact that each particle had inertia
which caused it to be entrained into the medium's motion with a certain de-
lay characterized by angle cp. The second non-periodic term in equation (5.13)
corresponded to the initial transient" stage of , vibration which rapidly ap-
proached zero. Therefore, the vibratory particle motion was governed by
the f©110wisg equation* ........ _______ .__• ....... - -- .....
— ep). (515)
Whence, it can be seen that the degree (or the coefficient) of par-
ticle entrainment into the vibratory mediumjflow ^d^hefoj^oj^in^g^expjression*:
* (5.16)
'
* According to the authors, this expression and (5.19) may also be derived
from Konig's (5.9), provided some terms were disregarded. Thus, all terms
in the equation may be disregarded in relation to b4, so long as magnitude of
^ was close to unity. Whennp
-------
In this case, the equation of the vibratory particle motion had the following
form:
-------
particle density p, = 1 gm/
cm8). It is, thus, seen that'
at every frequency there
existed small particles :
which followed the medium
vibrations almost exactly
(jUpsal), and large particles
which remained almost com-
pletely unentraitted by the
vibrating medium (/Hp -* 0).
The complete entrainment
of suspended particles into
the vibratory medium mo-
tion was impossible, since
the particles become en-
trained mainly by action of
the viscous forces which
Fig.15. Degree of U_ involveoent A doyree of Ug f |»« •round teroaol
particle* in the air «t different frequencies (pp = 1 9/f**)
arise only in the presence of the relative medium motion.
Normally, no large disagreement appeared between the degree of
particle entrainment calculated from the simplified formula (5.19), and re-
sults obtained by using the exact equation (5. 9). This is illustrated in Fig-
ure 16, copied from the monograph by N. A. Fuchs [121], in which the solid
line 1 expressed the functional dependence of the degree of entrainment/i p on
the ratio T/T, obtained from the exact formula (5. 9), and the broken line -
from the simplified formula (5.19). The graph also shows the same functional
dependence for the degree of streamline flow jig (see below). Values of r^cu
which correspond to T/T are given at the bottom. In both cases, the maximum
range deviation T/T = 0.2 - 2 did not exceed 4%.
"
0,001 0,01 0,1
0.9
a,e
0.1
0.6
s.s
o.t>
0.3
no
T~
1,0 Vg
US
0.1
0.1
Of
V
a.* '-
0.001 ojsaz ojamunnafi e.oz notions0.1 41 a,*of i
Figure 17 shows the functional
dependence of phase shift angle (delay)
of particle vibrations on the vibrations
of air at identical frequencies. Par-
ticles totally entrained by the vibra-
tory medium motion had a zero phase
angle, and the remaining particles -
TT/2.
Since there is a phase shift be-
tween the particle and medium vibra-
tions, the streamline velocity of the
Fig, 16. Function al relation between particles invol-
ved end degree of aerosol particles f lo» around in the medium around the particles U-. at
air in the folle»ing equasiont . . ,
> =t afcjf) any time instant t is expressed as
I - Degree of particle involvement; 2 - Degree of ;
particle floo-around
° OWS'
Sin O)/ —
(u/ — >-
(5-21)
- 49 -
-------
Include (5.18) and perform simple transformations and get:
, (5.22)
where ^ is the streamline factor of the medium's flow around the particle
which equals:
|if = sin9 = -™==-. ' (5.23)
The amplitude of the streamline velocity equals:
(5.24)
Values of jig are indicated by
dotted lines in Figure 15 for different
frequencies. Attention should be drawn
to the fact that jug was appreciable even
for the smallest particles, for which
the entrainment factor /np-» 1. When
UJT « 1, /LJg equals:
=; on.
(5.25)
Fig,
It can b^s-hown that the large
'particles, for which CUT » 1, were not
fully at rest and were drawn into the
vibratory medium motion as follows:
~ HP^(OT)-'. (5-26)
17 Shift angle of aerosol particles cp in the air
at different frequencies (pp = 1 g/oi»). A'simple relationship existed be-
tween jip and Ug:
(5.27)
(5.28)
The above is conveniently
illustrated in Figure 18. The bold-
face line represents the sinusoidal
velocity of the vibrating air (motion
of the smallest particles, for which
jip = 1, coincided with this line). The
fine line represents the vibratory
motion of a suspended droplet having
a degree of entrainment equal to
f4p = 0. 8. It lags the vibratory med-
ium motion by an angle cp = Arccos
/Llpfts34.4°. The dotted curve repre-
sents the relative medium velocity
flowing around the droplet, for which
Fi9. 18. Schematic drawing illuatratina difference
phases of absolute mediuo oaci I let tan rate (\it i of
absolute aerosol part icle osci I 1st ion vel oci ty i^p )
and of the relative oscillation velocities o< par-
ticles in the medium.
- 50 -
-------
IJ1 =.11 -IJ: = o. 6. This velocity leads the particle velocity by an angle of 9cP
so that the phase shift angle relative to the vibratory motion of the medium
is cp - 9c:f - 53. 13° . '
. --~ - -_. ---
--- ---
Fi~. 19. Photodrapha illu.tratln~ diff.r.ne.
l'n tho u.gr.. of a.roaol OrtiCI.a incr.a..
dropI.ta of pa~.ffin fo~ in oaoil latj~
..diU8 80y...nt f. 10 kft ; .~po.uro t.
I 250 aeo. . ao}
a - Particl. radi I 0.3 - O.61J. J b-
partiel. radii froa ,0.3 to ".5\..1..
Equations (5. 9) and (5. 19) had be en ve ri-
fied experimentally repeatedly and qualitative
confirmation had been obtained by many tests.
Figure 19 shows photographs taken by Brandt,
Freund, and Hiedemann [9, 168], using ex-
posure time longer than the period of gase-
ous medium vibration. The photographs show
droplets of a paraffin oil vapor sonicated at a
frequency of 10 khz. It can be seen that the
largest particles remained at rest (j.J., = 0)
during sonication, and conversely, the small
particles (r< 4.&) actively participated in the
vibratory medium motion. This is evident
from the longitudinal traces (tracks) the
lengths of which were twice that of the vibra-
tion amplitude.
The first quantitative, although con-
siderably crude, verification of formulas
(5. 9) and (5.19), was attempted by Wagen-
schein [283], who measured the vibration
amplitudes of lycopodium spores (r = 15. 85 IJ, Pp = 1.1) at a frequency of 85 hz
and vibration amplitudes from 0.58 to 2.40 mID. The medium vibrations
were set'up by the Zernov method, i. e., a lightweight hermetically sealed
vessel containing aerosols was fastened to the electromagnetically excited
,tuning fork., The vibration amplitudes of the vessel, gas, and particles were
photographed through ,a microscope. The choice of lycopodium spores as the
object of observation was unfortunate due to the presence of small fins on
their wrfaces whic,h distorted the particle mobility. Because of this, the
initial divergence of 20 to 300/0 between the experimental and the theoretical
, data followed. However, this was reduced to 5% after correcting the particle
mobility by the rate of their fall.
In recent years, experiments 'hk 20> EX{,.r..enUI dft. pn d?gr?e ?f=
tern of vortexes was generated behind some f°9
streamlined obstacles, bars, and wedges in
particular. The vortexes broke away from each
side of the obstacle and, under certain conditions, caused a loss in stability
and appearance of transverse body vibrations. A similar phenomenon, hav-
ing a self-oscillatory character [125], had been observed while blowing past.
spherical obstacles [114] behind which, as assumed, a vortex formed. It was
noted previously in the footnote on pages 47-48 that highly streamed aerosols
executed chaotic motions in the sound field due to transverse velocity compo-
nents. In this connection, the following guestion arises: is this caused by
loss in the stability of particles suspended in the\sound field?
The theory indicates [ill] that the loss in stability in the advancing
flow, which is characterized by the Reynolds number Re, begins to appear
•when: . -
where h is the flow width kg is a non-dimensional coefficient and kg^l.
It will.be shown in Section 9 that aerosol streaming in a sound field
had a quasistationary character, meaning that the inequality (5.29) is appli-
cable here. It can be concluded from (5.29) that the stability of suspended
particles was disturbed only when RexD 107, an occurrence which did not apply
to sound fields . Therefore, it follows that the chaotic aerosol motion can be
caused in a sound field by other phenomena which will be examined appropri-
ately. .
6. DRIFT OF AEROSOL PARTICLES .
\
In the course of their vibratory motion, particles suspended in a
gaseous medium underwent a progressive transfer or "drift, " in thie longitu-
. • •• : .- . ---• ' •• '.. 52"- -. ••'•*.• ->,..';,.•
-------
dinal direction in relation to the medium. There are at least four factors
which enhanced drift of aerosol particles in a. sound field:
1) acoustical radiation pressure exerted on the particles;
2) periodical variation in the vibrating medium viscosity;
3) distortion (asymmetry) of the acoustical waveform;
4) a symmetry'in the oscillatory medium motion in a standing sound
wave. In addition, the aerosol particle drift in an acoustical field was en-
hanced by the presence of other particles with developed boundary layers.
This effect,, however, is not comparable and will not be examined here.
Drift induced by acoustical radiation pressure. This type of particle
drift has been known for a long time. It can be estimated quantitatively from
King's equation for radiation pressure exerted on a stationary spherical ob-
stacle of arbitrary density having a radius considerably smaller than the wave-
length: kr« 1. According to King [21Z], it is:
E, (6.1)
in a traveling -wave, and
FR = 2 (kr) (nr«) G (p£/Pp) £ sin 2kx>, (6.2)
in a standing wave, where k is the wave number, G(pg/pg) is the density factor
given by the following equation:
c(P«/Pp) = fl+—(1 —
L 3
(in the case of aerosol particles with pp > pe, G (pt/pp)^•/•); XD *-s the distance to
the closest standing wave node.
The derivation method for these equations consisted of the exact so-
lution of the streamline problem for the case of a spherical object in a sound
field. Viscosity and the heat transfer capability of the medium were disre-
garded.
A simpler and more convenient method was applied subsequently by
Westervelt [286], and by L. D. Landau-and E. M. Lifshits [62]. The essence
of this method is contained in the following: if an object were placed in the
path of a propagating sound wave, the wave became reflected and scattered in
all directions upon incidence with the object. The scattered wave can be visu-
alized as a wave radiated by the object itself. The impulse imparted by the
wave to the object was greater than the impulse scattered by the object in the
direction of the incident wave propagation. This gave rise to an excess force
in relation to the forces of radiation pressure in a sound wave. The energy
flux equal to TTscgE was lost (scattered) by the incident sound wave. Quantity
TTS was the total effective cross-section defined as the ratio of the total scat-
tered flux to the incident energy flux density which, for the case of particles
entrained into the oscillatory medium flow, was defined by the quantity Egp =
|pgUagp = |ig E (E is the absolute energy flux density equal to J/cg).
- 53 -
-------
The impulse flux in the incident sound wave is TTsEgp, and in the
region bounded by the solid angle d 0 in the scattered wave, it is E^dS =
EgpdTTg. Project the latter in the direction of the propagating incident wave,
and get: ....... .. .
*
FR = Ev (l — cos6)dn, ..(6.3)
where 9 is the angle between the direction of the incident and scattered waves.
According to [62], the effective differential cross -section is:
Substitute this expression. into equation (6.3); integrate it, and ob-
tain the following expression for the force in a traveling sound wave due to
radiational drift of a sphere:*
Maidanik [230] and, subsequently, L. P. Gor'kov [27], solved this problem
for a more general case, including compressibility of a sphere and the re-
fraction of the incident wave. This led to an identical expression for the
case of aerosol particles. A comparison of derived equation (6.4) with
equation (6.1) shows that they differed only by a numerical constant which
was 2. 7 times as great in King's equation. In both cases, the direction of
force F^ was identical, i. e., in the direction of sound wave propagation.
The radiational drift was many orders of magnitude higher in the standing
wave field. To calculate this force, it is necessary to include the impulses
of the direct and reflected -waves, the magnitudes of which depended on the
location of the sphere in the sound field. To sum up, the following expres-
sion is obtained for the radiation pressure force acting on a sphere in a
standing s ound wave:
FK " Tn (T~} '*& sin 2k*o (6.5)
The above expression (without factory,) was first obtained by Westervelt
[286]. The same problem was recently solved for the general case by L. P.
Gor'kov [27] who, however, obtained a 25% higher coefficient.
Comparison of equation (6.5) with King's equation (6.2) showed that
they were identical. The only exception was a factor, 1.6 times greater
than the factor in equation (6.2). This was verified experimentally by Rud-
nick [258], who used cork balls with radii r~l mm, suspended by threads in
the standing wave field at frequencies between 400 and 2800 hz. Expression
* Instead of an expression for the drift force F, in all examined cases West-
ervelt [286] used the corrected specific pressure F/TrraE , which he desig-
nated by an index d and called it the "drag coefficient. " The index was partly
useful in comparative estimates of different types of drift, although it proved
insufficient for practical purposes, as it excluded nt.
- 54 -
-------
(6.5) shown that the magnitude of raHlnUonal pressure force acting on aero-
sol pBfl.hilefl in B fllBwling f»»un»l W»VP w»« n funr.Hon r»f t,hHr r;oorflln«.frB.
At the nodes (x^ = e, X /2, and X) aud aaUtiodeu (*u - / /4 nti>t 'in /'ij, th«-. lorr.*-.
FR = 0. Moreover, the particle was stable at the node a, and unstable at the
antinodes. The maximum value of force FR occurred at the midpoint between
the node and antinode, i.e., at x0 =X/8, 3X/8, 5X/8, and 7X/8.
Find expressions for velocities of an aerosol particle due to radia-
tional drift.. Since the discussion centers on the progressive particle motion,
the period of which is larger than the particle relaxation time, it can be as-
sumed that the radiational pressure force FR, expressed by equations (6.4)
or (6.5), was balanced at every instant by the medium's drag expressed by
Stokes1 equation F = 6nT|rVR. Whence, in the case of a traveling wave:
and in the case of a standing wave:
(6.6)
(6.7)
In the first case, the particle motion is uniform, and in the second
it is nonuniform and is governed by the following equation:
igkx=e
Bi
(6.8)
where
Equation (6. 8), derived initially by St. Glair 179, 181 , may be obtained
readily by integrating equation (6. 7), the initial conditions being x^x,, at
t - l^j . The motion of aerosol particles in the sound field in the presence of
radiational drift acquires singular character, best illustrated in Figure 21.
All above equations are valid
when the following two conditions are
satisfied. Condition 1: medium's
oscillation amplitude was small in
comparison with the radius of the
sphere, otherwise the motion of the
medium loses its potentiality; Con-
dition 2: the viscous losses of the
acoustical energy in the boundary
layer around the sphere are com-
paratively small, otherwise the
sound wave fails to reach the sphere
surface and remained unscattered by
it. According to [61], the acoustical
Fig. 21.
Schoatic presentation of •»ro«pl p«rticl«»
drift phenomenon in e sound field (d)
a _ in • floving »«vej b - in • •tatic ««»e
- 55 -
-------
energy absorbed by the boundary layer of a small sphere (kr « 1) with 1 cma
cross section area is equal to 6 v/cgr. Whence, Condition 2 is satisfied for
the following inequalities:
— (&•)*> 6v/ 6v/cgr (6.96)
in the case of a standing wave.
In air and at i- 10khz, this results in r > 700/i for a traveling wave,
and r > 25/1 for a standing wave. The dimensions are even larger at lower
frequencies. Therefore, the incident energy is not so much scattered on
the small-size primary aerosols, as it is absorbed by the viscous boundary
layer around them. As a result, new types of drift arise and are examined
below. However, the radiational drift occurred only due to highly enlarged
aerosol particles in the standing wave field.
Drift induced by periodic variation in medium viscosity. The adia-
batic compression and rarefaction of the medium in the sound field produce a
periodical increase and decrease in the medium temperature. Using the
adiabatic equation pvY = const, and the equation of state for an ideal gas pv =
Rg T, it can be shown that in the case of sinusoidal sound pressure which
varied according to p = Pat + Pg sin tut at a given point, the absolute tempera-
ture of the medium, within the first approximation, varied as follows:
r«r.(i+!^-i-sin«rt, (a)
CT
where T0 is the absolute temperature of the medium in the surrounding space
at the static pressure P,8t . When sound intensity was J = 0.1-1.0 w/cm8,
corresponding to Pg /Pgt = 0.009-0.030 under normal conditions, the air tem-
perature (y = 1.4) in regions of compression or rarefaction changed by A =
± 0.75 -2.5° C.
This substantial temperature variation of a gaseous medium induced
a corresponding change in its viscosity. The viscosity and temperature of
ideal gases, in the thermodynamic sense, were related as follows:
n-r'7'. (6)
This relationship may be derived readily from the following two equations [116]:
1) = — P«fm»m, (B)
3
(m,, va and 1, are the average mass, velocity, and the free path of the gas
molecules respectively; kg is the Boltzmann constant.
- 56 -
-------
According to (b), an increase in the medium temperature during compres-
sion was accompanied by an increase in its viscosity, and a decrease in tem-
perature during rarefaction was accompanied by a decrease in viscosity. .
The difference in. the medium viscosity during compression and rare-
faction resulted in some difference in the magnitude of forces exerted, ac-
cording to Stoke 's law, on the suspended particles during forward and back-
ward motions. This produced an appearance of an excess force, the drift
force F ', acting on the particle in the direction of the sound source in the
case of a traveling wave, or in the direction of a node in the case of a standing
wave. The drift force was predicted theoretically by Westervelt in [285],
which dealt with the mean static pressure and velocity of gas particles in a
plane sound wave. In this study the author accounted for the variation in the
parameters of the oscillating medium caused by the variation of their coordi-
nates for oscillations with finite amplitude. By means, of a corresponding
transformation of coordinates, the author obtained the following expressions
for the instantaneous sound velocity cg and the velocity of medium particles
u, infixed (Euler's) coordinates: '
(6.11)
where cgo is the velocity of sound in the surrounding medium having a density
of pge); |x and §„ are the first and second derivatives of displacement of the
medium's particle at a point x; §t and §xt are Lagrangian velocities of the
medium particle and its derivative at a point x, respectively. On the basis
of expression (c), where the product of the first three terms, ^pgl,, is inde-
pendent of temperature, and the fourth term, VB is proportional to the velo-
city of sound cg as determined by (6.10), the following expression may apply
to the medium instantaneous viscosity in the first approximation:
where \ is the viscosity of the surrounding medium. The product of instan-
taneous values of u, and 7), which appear in the Stokes formula (4.3), is de-
fined as follows:
The mean value of which over one period of •oscillations i is:
Substitute expression (6.13) into Stokes1 formula and take the degree
of streamlike flow/jg into consideration, and obtain the following expression
for the drift force caused by the periodic variation of the medium:*
* In Westervelt's subsequent work [286] and also in the book by Hueter and
Bolt [205], this force was referred to as the "Stokes-type force."
- 57 -
-------
(6-14)
In the case of a standing wave, the above expression must be multiplied by a
factor sin 2 kx,,. The following equation may be obtained for the drift velo-
city by equating expressions for the drift force (6.14) and the drag force F8t =
r • . ' • - '
tf . •!/_.. T-3_..»F (6.15)
It is implied that the instantaneous particle temperature follows the medium
temperature.
Drift conditioned by waveform distortion (asymmetry). If the sound
wave had a distorted, sawtooth, form similar to the one shown in Figure 7,
it indicates that the medium's particles move non- sinusoid ally (see Figure 8).
In the compression phase, which indicated the forward motion, i.e., in the
direction of wave propagation, medium's particles acquire velocity faster
and lose it slower than in the sinusoidal motion. In the rarefaction phase,
where the motion is reversed, the opposite occurs: medium's particles gain
their velocity faster than they lose it. These factors clearly influence the
motion character of particles suspended in a medium. During slow velocity
increase, accompanied by backward medium motion, the aerosol particle is
entrained into its motion to a greater extent than during "normal" sinusoidal
velocity increase, and as a consequence acquires greater velocity. Having
acquired greater momentum, a particle "slips" by medium's particles some-
what further than normally during the subsequent decrease in medium's velo-
city. This becomes considerably enhanced by a more than usually abrupt
decrease in medium's velocity. In the case of the medium's forward motion,
the opposite occurred: the aerosol particle lags medium's particle motion by
the same amount. As a result, the aerosol particle fails to return to its ini-
tial position at the end of a vibration cycle and moves to a point opposite to
the flow direction.
Now, find the expressions for the drift force due to waveform asym-
metry. Use Stokes' equation, as Westervelt did [286, 205], in the form given
by Oseen (second approximation):
(6.16)
The second term of the correction factor in this equation gives an additional
force AF which, in contrast to the basic Stokes' force Fgt, is related to the
velocity of the streamlike flow around a sphere by a quadratic relationship:
Q _
AF = — nr'pjUgplu^l, (6.17)
where Ugp is the streamline velocity vector, and lugpl , its magnitude. In the
case of a sinusoidal variation of streamline velocity, the mean value of this
force over each half-period is zero, indicating that the sinusoidal variation
failed to upset the periodicity of particle's motion. However, the situation
- 58 - • '
-------
changes in the case of an asymmetrically distorted waveform, which generated
harmonics (see Section 1): averaging of the force AF resulted in a nonperiodic
unidirectional force Fh. If the analysis is limited to the second harmonic,
the vibrational streamlined velocity of a particle may be expressed as follows:
uu> = »»//< [sin orf + A, sin (2), (5.18)
where hg = Ug/Ug is the relative amplitude of the second harmonic, and <|) is
its phase shift angle (equal to -rr/2 in Figure 7b).
Square the above expression and substitute into equation (6.17) and
get: n
AF = —
(6.19)
Average the terms in brackets over the first half- period and get
the following:
— \ 2A,sina>/sin(2ci>/-{-(}>)d((o/) = —- sin tS
W v . «WI
* . •
— [hi sin1 (2orf 4- *) d (
-------
that t;hr> max l.nnwt va.lupn of for*.'*1 P'h ari'l vo1*irl.l:y Vh
of magnitude in the standing and traveling
Drift conditioned by asymmetry of vibrational motion in a standing
wave. It is known that a distinct characteristic of standing sound waves is
that the vibrational velocity, displacement, and sound pressure were functions
not only of time, but also of space [see Figure 1 and equations (1.4) - (1.6)].
In each region between a node and antinode, the displacement amplitude and
vibrational velocity rose sinusoidally with distance from the node. In the
course of vibrational displacement in the direction of the node, medium's
particles fell into a zone of increased vibration amplitudes, and upon return -
into a zone of decreased vibration amplitudes, as compared with the initial
vibration amplitude, which corresponded to the start of each displacement.
In other words, medium's particles accelerated in the course of their vibra-
tional displacement in the node direction, and, in comparison with the "nor-
mal" sinusiodal motion starting from each extremal position, they deceler-
ated upon return. There exists direct evidence to the effect that the asym-
metry of vibrational medium motion was related to the finite vibration ampli-
tudes.
If a particle suspended in the above described vibrating medium pos-
sessed appreciable mass, then, by virtue of its inertia, it digressed to some
extent from the described behavior of medium's particles. That is, in the
course of displacement in direction of the node when the medium was accel-
erated, an aerosol particle lagged medium's particles more markedly than
in the course of its sinusoidal motion. During the return travel, when the
medium became decelerated, the aerosol particle leads medium's particles
more notably than during its sinusoidal motion. As a result, the aerosol
particles came closer to the node than the medium's particles vibrating with
it in both cases. With each vibration cycle the indicated shift of an aerosol
particle increased relative to the medium's particles. This might soon lead
the aerosol particle to the node, if other, directly opposed, forces had not
acted on it simultaneously. The described type of aerosol particle drift had
been predicted theoretically by S. S. Dukhin. A simplified mathematical
solution of the drift problem was offered by A. Barginovskii [4]. Recently,
Dukhin obtained an exact solution of this problem [35], Both these solutions
will be presented and discussed.
In the first approximation, the equation of an aerosol particle motion
in a standing wave field, including (6.2) was as follows:
i sin kx sin <
or
^f- J , (6.22)
r-£- + -rf- = Ulsmkxymut. (6.22')
If the difficulties associated with exact integration of the above equa-
tions are averted by utilizing formulas for approximate integration,
- 60 -
-------
dt A/ * <6» A/»
equation (.6*22') yielded the following expression for the (n+1) -th ordinate:
&»~ A/
With the aid of this equation and the initially assigned conditions for the mo-
tion of an aerosol particle, a graph of its drift, in the absence of other for-
ward motion, may be constructed by the method of successive approximations.
This type of graph is a mirror-image of the one shown in Figure 21b. To
arrive at the exact solution of the aerosol drift problem, examine equation
of the oscillating particle as the second approximation, as was done by Duk-
hin. Considering that the particle drift was anharmonic, avoid the general
expression (4.2) for force F" acting in a viscous medium with an arbitrarily
varying velocity. This leads to the following expression for the motion of an
aerosol particle: , . _ . t .,
where Xg and Xp are distances from the node to the medium and the particle,
respectively, and dxg/dt = Ug sin kx,, sin tut.
Equation (6.24) describes motion in a rapidly oscillating field, which
allows to use P. Kapitsa's method [63] for its solution. Since the amplitude
of oscillations Ag was much smaller than the wavelength, |Xg | on the right-
hand side of the equation may be considered constant during one cycle of os-
cillations, Xg = x,,. In this case, the solution of (6.24) is the following equa-
tion for the harmonic particle oscillation with respect to the point Xp = x,,,
namely:
••** — *• = IVU sin kx0 cos Ital + (
-------
where Xp-x,, is determined by equation (6.25).
Substitute the above series into the expressions for Up and dup/dt in
equation (5.2), and average over one period of oscillations, in which the
periodic motion component was zero, and obtain the following equation of
motion of an aerosol particle:* ' - . - . i - tf
n ( - 0) ( | + 1 b\\ sin 2fcce. (6.28)
Consider relationships among (5.23), (5.18), and (4.4) and also the
fact that the left-hand side of the given expression represented the force
averaged over one period of vibrations, i.e., the drift force Fa , and obtain
(6.29)
Equate the above expression with the one for the drag forces F3t =
67]TrrVa, and get the following expression for drift velocity:
9. (6.30)
Thus, it is seen that the force and velocity of a given particle type
drifted in the node - antinode range, and varied in a manner identical to the
variation of the force and velocity in the radiational drift. There were no
other similarities. However, there was a characteristic detail: a "preferred"
frequency of oscillations at which the drift velocity attained a maximum value
(see Figure 23). corresponding to each particle size.
Table 5 presents a. summary of formulas for the force and velocity
of all the examined types of aerosol particle drift in a sound field. With ref-
erence to formulas for drift velocity, it can be seen that inasmuch as all had
been derived from the expression for the drift force with the aid of Stokes1
formula (4. 3), they are valid only for the range in which the corresponding
Reynolds number R = 2rV/v did not exceed a unity. Otherwise, when parti-
cles were large and the sound intensity increased, the velocity should be cor-
* The equation in Dukhin's article [35] contained an error: rrip was shown
instead of trig. [Author]
- 62 - i
-------
rected by Oseen's factor as follows:
(6.31)
The above equation is easily arrived
at by comparing the right-hand side of
the expression for the drag force of the
the sphere in the normal form given by
Stokes and. in the refined form given by
Oseen.
Below are shown absolute values
of drift velocity of spherical aerosols
with unit density (pp = 1), computed by
the above formulas in atmospheric air
at J (or I) = 0.1 w/cma for a traveling
wave (Figure 22) and an undistorted
standing wave.(Figure 23). Curve
analyses in Figures 22 and 23 yield
the following important conclusions:
1. Drift caused by a periodical
variation in medium's viscosity is
predominated in an undistorted tra-
veling wave, although its velocity was
small (V*** = 0. 55 cm/sec). Drift ve-
locity caused by radiative sound pres-
sure was insignificant and can be dis-
regarded in all cases..
2. Drift caused by waveform
V distortion usually predominated in dis-
torted traveling waves, and its velocity
could attain high values (up to V°"«
10 cm/sec and up) if the distortion degree and asymmetry were sufficiently
large.
3. Drift caused by vibrational motion asymmetry predominated in
undistorted standing waves in the case of small particles. In the case of
medium-size and large particles, the predominating drift was due to periodic
variations of the medium's viscosity and radiative sound pressure, respec-
tively. Drift velocity in the first and second cases was small (Vam"?»0.17
cm/sec; V"* « 0.55 cm/sec, respectively), and in the third case, could
attain higher values (when r = 100/1, V\" = 7-40 cm/sec at f = 5 - 50 khz).
4. Drift caused by asymmetry in vibrational motion of the medium
predominated in highly distorted standing waves when particles were small.
When particles were larger, drift due to sound waveform distortion predom-
inated. Drift velocity in the second case could attain high values (when r >
5/i, VB*X > 10 cm/sec), higher than in the radiational particle drift. Drift
- 63 -
«/ V
Fig. 22. Aerosol partial** drift velocity in •
flowing aound »«ve «t J * O.I b«/e»*
. (p s i 9/»3, Mdiu* - tir)
VR-drift velocity resulting fro* sound rmdi»tion
pressure}1 VTI - drift velocity conditioned by
periodic •ediu* velocity chehgoS) V(, - drift
velocity conditiaied by »«ve. for* di»tortian
-------
direction of this type depended on the
phase shift of its harmonics.
5. Particle size and oscilla-
tion frequency highly affected aerosol
drift velocity. High drift velocities
were usually attained only for larger
particles, or at elevated ultrasonic fre-
quencies.
6. Velocity of all types of
aerosol particle drift was directly pro-
portional to sound energy density and,
consequently, to sound intensity. At
sound intensity increased up to J (or I)
= 1 w/cm9, particle drift velocity in-
creased tenfold, if Reynolds number
did not exceed unity. Otherwise, drift
velocity increased more moderately.
0,0001
«' 6,113
i J S 70? a Jff S0 MOM
Fig. 23. Amplitude values of aerosol particulea
Fig.- 23. Aaplitude values of aerosol parti—
cules1 drift velocities in a non-distorted
static sound »ave at J » 0.lb«/c»2(Pp = 1 f/ca";
^ _ drift velocity conditioned by intensity
o? sound radiation; Vja* _ drift velocity con-
ditioned by asyaetric oscillating aediua aove-
nent drift velocity (values cond i t i oned by_.
periodic nediu* velocity changes V"*", can be
deterained «ith the aid of Fig.22)1]
Table 3
Causa of particle
drift
Force and velocity of drift particles
In ftigratiny nave
In static "»ve
Sound radiation
(intercity)
Periodic changes
in MdiiM viscosity
F.
«etrical
latory •igratien
- 64 -
-------
7. CIRCULATING AND FLUCTUATING AEROSOL PARTICLE MOTIONS
Aerosol particle motion executed under the effect of acoustical wind
is referred to as circulating motion. The period during which acoustical
wind velocity changed is greater than the relaxation time of aerosol particles
In view of that, as also in the case of drift, it is apropos to assume that the
particles achieved velocity of the medium almost instantaneously, and, sub-
sequently, follow it entirely. In order to make it more apparent, an equa-
tion will be considered for the motion of particles under the effect of a force
acting from within the medium. If acoustical wind was wa, and if, for the
sake of simplicity, the second and third terms of equation (4.2) had been dis-
regarded, the equation for the particle motion could be presented in the fol-
lowing form: . - - .
m,-^=6nv(Wa — wp), (7.1)
or .
Assume that in a given time interval particle velocity was constant:
wm = const. In that case, solution of equation (7. 11), which is trivial for t = 0,
is[39]: •" (7.2)
According to this solution, velocity of an aerosol particle differed from the
velocity of acoustical wind by 37% at t = T, by 0. 6% at t = ST, and by less than
0.005% at t = lOr. Therefore, aerosol particles almost fully achieved acous-
tical wind velocity for following time periods: . v, . ,A ,
7 6 . npH r— 0,1 |i <« 10-' cetc
> r«=l ji /«10-4 »
> r=10 |i /«10-* >
» r=100|i f«l »
If the third integral term of equation (4. 2) were taken into account,
the correction factor would not exceed 4%, and the order of magnitude would
be preserved.
It can be shown that an identical situation existed also when aerosol
particles moved under the effect of any other slowly varying force, including
the drift forces, discussed in the preceding paragraph. Aerosol particles
are entrained with the same velocity also in the overall gas flow through the
sonic ation chamber.
Now analyze the behavior of aerosol particles at the turn points of
acoustical flow in a standing sound wave. Here, an additional centrifugal
force acted on each aerosol particle, as a result of which the particle moved
radially with a velocity which could be determined from the following formula
.-
..
where R - radius of curvature at the turn of acoustical flow. The value of
- 65 -
-------
radial displacement at the turn with angle cp (radians) is:
AR = V^., (7.4)
where t^ - time of motion at the turn equal to cpR/wa. Substituting the latter,
and also (7.3) into (7.4) yields:
A/? = a. (7.5)
In the present author's work [73], this problem is discussed, the
velocity curves computed according to Rayleigh's formulas (1.24 - 1.26),
and the acoustical flow trajectories in the field of a standing wave with a fre-
quency of 4.25 khz are presented. It can be seen, therefore, that the flow
velocity at the turns had a reduced value, not exceeding wa = 1 cm/sec at
J (or I) = 1 w/cm . Accordingly, in the order of magnitude, the radial particle
displacement at cp = ir/2 did not exceed the value of relaxation time, which, as
is known, lies within the range T = 1.2" (10~7 - 1CT3) sec for the initial sonicated
aerosol particles (r = 0.1 - lOp,). Consequently, the radial displacement of
aerosol particles at the turns can be disregarded safely if the particles failed
to coagulate initially into larger ones. Thorough experimental investigations
of the aerosol particle motion in a sound field had been conducted by Andrade
[140], who obtained, by means of aerosols, satisfactory "prints" of the acous-
tical wind trajectories in the node - antinode region of a standing sound wave
with A. = 56. 5 and A. = 91. 8 cm. Later, this phenomenon was observed by many
investigators who worked in the field of the acoustical aerosol coagulation.
It would seem apropos to compare the direction of the circulating
aerosol particle motion with the direction of their drift, discussed in the pre-
ceding paragraph. The direction of .circulating motion of aerosol particles in
an undistorted traveling wave, determined by the acoustical wind, is opposite
to the direction of particle drift in the central part of the acoustical beam.
Thus, the resultant particle velocity is an arithmetical difference between the
acoustical wind and drift velocities at a given point. The directions of the
above types of particle motion coincide at the beam periphery. Consequently,
the value of the resultant particle velocity becomes equal to the sum of the
acoustic si wind and particle drift velocities. If the waveform remained sub-
stantially undistorted, the acoustical wind velocity usually predominated over
the drift velocity and, consequently, the particle motion was directed away
from the sound source.
The direction of the resultant velocity of aerosol particles in a
standing wave depended on their size. Small particles, the low-velocity
drift of which was determined mainly by the asymmetry of the vibrating
motion of the medium, moved along the central portion of the acousticalbeam
and on its periphery toward an antinode. This occurred with either a de-
creased or increased (by the value of Va) velocity of acoustical flow at a given
point. Large particles, the drift of which was determined mainly by the ra-
dial sound pressure moved towards an antinode in the central portion, and to-
wards a node or antinode on the periphery, if V_ was great. In the inter-
*^
- 66 -
-------
mediate region of a sound field, where, acoustical flow direction was mainly
perpendicular to the direction of the particle drift, the magnitude and direc-
tion of particle motion was determined mainly by the magnitude and direction
of drift velocity in the longitudinal direction, and by the acoustical flow ve-
locity in the transverse direction. As a result, aerosol particles move at a
certain angle to the direction of the sound wave. While participating in the
vibrating and circulating motions of the medium, aerosol particles simul-
taneously participated in the acoustical flow fluctuations induced by acoustical
turbulence. This has been observed visually by many investigations [167, 76,
30, 156, 231, 232].
The question can now be asked: what is the degree of enlargement of
aerosol particles and turbulent fluctuations of the sonicated medium? Although
this question has not been investigated experimentally thus far, nevertheless,
its concept can be obtained if one proceeds from the theory of acoustical tur-
bulence as developed in Section 1. If turbulent fluctuations are presented as
.harmonic vibrations with angular frequency wt, as was done by N. A. Fuchs
[121], the extent of aerosol particle entrainment in these can be calculated
clearly by using equation (5. 9). Considering that frequency spectrum of tur-
bulent fluctuations was in the low frequency range [see equations (1.38) and
(1.39) I and, therefore, WT« 1, a simplified equation (5.16) can be used. In
accordance with this equation, the degree of streamline flow of turbulent fluc-
tuations with frequency wx around the aerosol particles was:
The mean square velocity of the relative particle and medium motions was:
(7.7>
where vr - the mean square fluctuating velocity of the medium. Considering
'! that the \term WiVj represented acceleration of medium's fluctuations a.lt
• 'equation,(7. 7) could be written also in the following form:
(7.8>
The relative motion of the particle and the medium is possible only in the
case of fluctuations of the order of 1 < 10, with the frequency, as indicated by
(1.39), independent of the order. Fluctuations of the 10 scale which, according
to (1.36), have the highest fluctuating velocity, naturally have the highest ac-
celeration ••
In view of this, an equation for the maximum acceleration of fluctuations can
be presented as follows: • « .
In accordance with (7. 8) and (7.10), the maximum value of relative
fluctuating of particle arid medium velocity is:
- 67 -'I'
.. \ '-'•-"•' •'•-• *
-------
_2_
:135
e''*
WP,)V*
^
135
' 6 vr__L ri (7 I \\
T^7 "T i7~t/. u/_ ' • \' • ' • /
With Yaglom's calculations as the basis, V. G. Levich [68, 69] used
as the coefficient in (7.10) and converted ec[u at ion (7.11) to the following:
lv ) ^lVl,_Pp__JL^_r.. (7.H-)
v r>u***c g ^ (6/pg)v<
However, this equation, derived on the assumption that vpg « v10, leads to
exceedingly high values for relative velocity of the particle and medium, and,
similarly, to highly elevated values of fluctuation velocity. The latter at
times exceeded even the vibration velocity, a highly improbable condition;
this can be shown by equating (7.11) and (7. 7) and by finding the value of vt.
In view of this only equation (7.11) will be used hereafter. This action is re-
inforced by a simple experimental fact: particles suspended in a sound field,
as a rule, trace coincident tracks and not a sine curve. The latter should be
traced if velocity of fluctuations were much higher. Based on equations (7.6)
and (7.11), Table 6 presents values of the extent of streamline flow around
••-••-- aerosol particles and
the maximum relative
fluctuating velocity of
particles and medium in
a sound field with fre-
quencies f = 1 khz and
10 khz at the normal
sound intensity of J (or I)
= 0.1 w/cma with numera-
Table 6
r. »
3-
5'
' T p 80C
iO 4fV-ft
,Z-1U •
1/\Q J/V-A
,V)O'lV •
3{\ 4fY-4
,U»1U*
IO 4IV-*
,4'IV*
•
rad/sec
370
1200
. *
Accord 109
to «x*ct
formula
0,0017
0,0055
0,0151
0,0497
0,0420
0,1366
0,1655.
0,4830
4
As per sin-
pi if ied for-
mula
0,0017
0,0055
0,0151
0,0497
0,0420
0,1380
0,1680
0,5520
Cfi/ A AC
5,8
10,4
Bax
en/sec
0,012
0,067
0,100
0,567
0,282
1,680
•1,130
5,850
tor and denominator of
each "fraction" corres-
ponding to f = 1 khz and
f = 10 khz respectively.
Data in Table 6 show-
that relative fluctuating velocity of particles and medium in the 1-10 khz band
is moderate and did not exceed 0.28 - 1.26 cm/sec for particles with r < 5pi.
This constituted 0.2 - 1.1% of the medium vibrating velocity. The above values
are accurate to within a factor §.
8. SCATTERING AND ABSORPTION OF ACOUSTICAL ENERGY BY AERO-
SOL PARTICLES
Every aerosol particle, no matter how small, represented an ob-
stacle to sound waves. A portion of the vibrating motion energy of a medium
is used up in overcoming this obstacle. This is explained by the fact that the
particle: 1) scattered incident sound waves in the surrounding space; 2) ab-
sorbed the energy of vibrating motions of the medium in its boundary layer;
and 3) served as a source of irreversible heat loss, associated with periodical
changes in its temperature. As a result, decrease in sound intensity during
- 68 -
-------
its passage through an aerosol was noticeably greater than during passage
through an unpolluted gaseous medium. The theory of acoustical scattering
on spherical and cylindrical obstacles was analyzed by Rayleigh [107] and
was subsequently developed in the works of Lamb [61], Rzhevkin [105], and
others. The theory indicated [62] that energy carried off per second by waves
scattered on a spherical particle for which r « X, was:
. (8.1)
In accordance with (1.22), total energy flux transferred per second by a sound
wave through 1 cm of wavefront was:
/-£«•• (8-2)
That portion of input energy lost in the aerosol layer due to scattering was, as
also in the case of sound absorption by the medium, proportional to the layer
thickness: '..'"' .
-^ -<4
-------
disregarded. In such case, the following formulas can be obtained for the co-
efficient of acoustical energy absorption:
u OJTVf/I m
(8-8)
rf.
1 (8-9)
In the case of particles with unit density, suspended in air, the above formula
becomes: «
otj; a* 1.73*-««?/* (8.9')
(ru - particle radius in ^). The derivation of the above formulas is relatively
simple, if the following elementary method suggested by Brandt, Freund and
Heidemann is followed [162, 168]. •
The energy absorbed by the viscous medium during the streamline
flow around aerosol particle is quantitatively equal to the work done by the
reactive forces exerted by the particle against the motion of the medium.
Assuming that instantaneous reactive force was determined by Stokes1 for-
mula (4. 3), and that the relative velocity of the medium varied sinusoidally
(5.22), it can be shown that energy absorbed by a particle in 1 sec was:
Ja=
cos
-*
u
A
3
9
0,8
OS
he
03
tr,*
/ll
v,t
0,1
fffff
n /tf
aflL
0.03
(L82
fffff
nhftx
0005
Anat
ILfffll
sum
HB01
w*
'*A
-A
f
/
t
1 /
f
/
f
f
f
^
/
r
/
f
j
(
/
^
y
f
f
j
^
/
-4
1
f
*•
N
•
? " "
t
Y-
t
L . _
/
|
^
\
A\
\
X
>
_-^
/
J
'/
/
/
\y
f
= :-*,
/
r
t
1
^
s
/
\
/
^
\
.
s
\
1
\
-v
\
*
\
\
\.
Hf OjmBfOf Of I 2 J.tSS ttO
Fij. 2^. Theoretical vtlues of dorivod coeffi-
cient of «ound «b«orbed by ••r»«ols «t different
frequencies
The ratio of absorbed energy to total
energy, which passed through an aerosol
layer with thickness dx, and determined
from (8.2), could be expressed by an,
equation identical to (8.3), so that after
substitution of (8.10), equation (8.8) is
obtained, and subsequently, allowing for
(9.1), equation (8. 9) is derived.
Based on calculations with the aid
of (8.9), Figure 24 illustrates coefficient
values of viscous sound absorption by
unit-density aerosol particles in air as a
function of their radius at k = 1 g/m3; in
the case of particles with r < l\i, Cun-
ningham's correction factor for the me-
dium was taken into consideration. As
can be seen, each vibration frequency
corresponded to a critical particle ra-
dius for which sound absorption was
maximal. The value of the critical ra-
dius could be found readily by setting the
derivative do»p /dr to zero:
- 70 -
-------
'r.-l/rX- (8-11)
*** r wfj.
For air with pp = 1, the formula becomes
- 3,65 ' " • •
At a critical radius pif = ,/2/2. In view of this, and in accordance with (8. 9),
the following equation can be written for the maximal coefficient of viscous
absorption: ....
66/M. (8.12)
In the 0.5 - 50 khz band, («p)mmx amounts to 0.0325 - 3.25 db/m for 1 g of
aerosol particles. Therefore, even at a comparatively low weight concen-
tration of critical size particles, the coefficient of absorption yvv can far ex-
ceed the coefficient of sound absorption by the gaseous medium ag, which
attains enormous values at elevated frequencies. The value of ^p can be de-
creased only by changing the frequency of vibrations away from the critical
frequency determined by the ratio (8.11) or (8.II1). The following expression
was derived by Epstein and Car hart [188, 122] for the coefficient of sound ab-
sorption due to the periodic irreversible heat transfer between the particle
and the medium, which leads to a corresponding increase in the medium's
entropy and a decrease in free energy: ... . _.. ,
«Je=i5£»(T._l)(l+>l/PE)|iJ,. (8.13)
fg \ f 2v' •
v . * •
where x - thermal conductivity of the medium, Xg/pgcg; (i§x - degree of tem-
perature compensation of the particle and medium, determined by an equa-
tion identical to the equation for the degree of streamline flow around particles,
in which b = l/r,/2v7u> is replaced by 9 = 1/r ,/2x/u)
(8.14)
Substitute k for n in (8.13), and, considering that usually r Jw>/2\> « 1, obtain
the following expression:
In the case of particles with unit density, suspended in air (x — 0.2; y = 1-4),
(8.15) becomes
» 6.10s* * Ma.. • /ft 1R'\
aj~— v-gtt OO/M . . 10.10 /
(ru - particle radius inpi). .
In the case of large 9, values of particle and medium temperatures
are nearly identical, the decrease in the free energy is negligible and, there-
fore, M.^ ~ 0. In the case of very small 9 the particle temperature is almost
constant, the temperature drop and the decrease in free energy attain practi-
cally their maximum possible values and, therefore, ngx ~ !• The total coeffi •
cient of sound absorption due to the presence of aerosol particles is:
- 71 -
-------
«, = afp + «p + »? ^«P + «?• (8.16)
All coefficients discussed above are additive up to the particle con-
centrations of n < 10s and provided that (y - l)u>x/cg « 1, which in the case of
air yields u> « 4' 1(>D. In the case of water mist |«p | is approximately 40% of
|ttp|- The total coefficient of sound absorption by aerosols,-taking into account
absorption by the medium, is expressed by the following sum:
a' =
-------
results for the additional sound absorption coefficient Ap for the water mist,
containing droplets with mean radius f - 6.25\i for k = 2 g/m3 F213] are pre-
sented: , .
/=25 50 100 500 04
k Ap=4 3 2 0,2dtf.c«r».
According/to Wei Chun-tseu [21, 22], results yielded by Oswatitsch's
theory have been understated. Nevertheless, it can be assumed that at a
frequency of 0.5 khz, additional attenuation was negligible as compared with
the normal sound attenuation in a fog, computed from formulas for the solid
phase (see Table 7 below). As a rule, higher frequencies had been used in
practice for aerosol coagulation. Therefore, it can be considered that addi-
tional sound attenuation did not take place in coagulating fogs, and it was pos-
sible to use above formulas for aerosol with solid dispersed phases. The
presence of suspended particles and increased sound attenuation induced some
decrease in the sound propagation velocity. Zink and Delsasso [297] derived
the following formula for the decrease in sound velocity Acg, due to the pre-
sence of aerosol particles:
where Mp/Mg is the ratio of the dispersed phase masses and the medium;
Ypg - the ratio of their specific heats Cyp/C,;. Cpg - the gas molar heat capacity
at constant pressure; Rg - the gas constant. The first term in the equation
represents decrease in sound velocity caused by increase in the system's
density. The second term determined decrease due to increase in the system's
heat capacity. These factors have affected the sound velocity only when aero-
sol particles participated in the vibrating motion of the medium (p.p > 0), and
when they absorbed and emitted heat (|J.fx > 0). Therefore, sound absorption
effect and decrease in the sound velocity were complementary.
It should be noted that the first purely qualitative experimental in-
vestigations of sound attenuation in aerosols (natural fogs) described by Tyn-
dall [117] failed to reveal the negative effect of suspended. particles on the
transmission of sound. On the contrary, it was noticed that in a dense fog
the sound propagated much further than in a transparent atmosphere. The
reason for this, seemingly paradoxical, phenomenon was concealed, as was
discovered later (sjee Section 1) by the fact that sound absorption depended on
air humidity which/changed considerably upon fog formation. In general,
experimental sound attenuation measurements in natural fogs were highly un-
reliable, since under such conditions sound weakened due to wavefront expan-
sion and to atmospheric density and humidity heterogeneities of the medium
could not be taken into consideration. The first quantitative sound absorption
measurements of aerosols were made by Altberg and Holtzmann [138]. Their
experiments had been conducted at frequencies of 5 - 22 khz, and tobacco
smoke was used as the aerosol. The size of its particles was not determined;
moreover, tobacco/smoke contained much carbon dioxide, -which greatly in-
creased the absorbing capacity of the medium. The results obtained by these
authors were difficult to compare -with the theory of sound -wave attenuation
i - 73 -
-------
in aerosols presented here.
Laidler and Richardson [220] experimentally investigated magnesium
oxide and stearic acid absorption by smoke, and lycopodium. spores at high
ultrasonic frequencies (42, 98, and 695 khz). The first two aerosols were
highly polydispersed, and the lycopodium spores were sufficiently homogene-
ous; their average radius was r « 2.5\i, and their numerical concentration
attained n = 1. 5'10S. Degree of streamline flow around the particles was |ig
« 1, i.-e., and particles remained practically stationary. The following values
were obtained for sound absorption coefficient #p by the lycopodium spore
aerosols: 0.029 and 0.031 at 42 and 98 khz, respectively. According to
Fuchs1 calculations [121], theoretical values of ap were 0.038 and 0.042, re-
spectively, and discrepancy between the theoretical and experimental values
was 30 - 35%, the causes of which will be discussed later.
The rate of sound attenuation in a resonant chamber filled with water
or oil vapor, obtained by the mechanical atomization of the liquid, and also
with ammonium chloride smoke, was measured experimentally by Kndsen
Wilson, and Anderson [213]. These authors measured the sound attenuation
coefficient [3 (db * sec"1) which was related to &p in the following manner:
In the case of water mist containing droplets (f = 6.25^,; k "= 2 /g/m3), results
obtained are shown in Table 7. In the
case of mineral oil with (rcp = 5|j, ; k = T»t>ie 7
1.17 g/m3) and in ammonium chloride
smoke with (r = 0. 5 - 1. Op, , k = 0. 26
•r 1.03 g/m3), results shown in Table 8
had been obtained.;
In addition to above mentioned
experiments, sound attenuation meas-
urements were also made in water mist
at lower frequencies. -Results of these
experiments were as follows:
JAbsorption coef f ictent , (3 „ db in B<
Oscil 'n. fre-; '"" • '' j ri_ ...
quency in .,, Theorct.cal
KHerz Eaperrant'l, According , Accordin-)
,
0,5
1,0
2,0
4,0
6,0
8,0
to Knudsen j to Epstein
' 5 .
7
0,4
10,1
12,0
13,2
- 1 ,
et *l (213) et •! U^o
10,1
13,8
5,0
5 > 7
16,0 I C.3
, *
18, 2
18,8
»',3
7,5
7.7
Oscillation frequency
27,5
58,0
112,0
150,0
200,0
350,0
Absorption coefficient
db, s«c~'
. 4.8
3,5
; 2.8
S,8
8,7
7,2
A decrease in sound attenuation in the 112 hz region was revealed.
- 74 -
-------
Table 8
Osci 1 l« tion
frequency
in kherz
0,25
0,50
1,0
2,0
4,0
6,0
8,0
Extinction coeff.fi. db.sec
First exptl .
«erita _•
•
_ ."
1,5
3,6
7,7
13,6
15,0
21,2
Second exptl Result
••ri«« averager
„:•> I , ,
0,5
1,8
4,0
7,7
12,8
14,4
20,6
0,5
1,6
3,8
7,7
13,2
14,7
20,9
Calculated
theoretical
' value*
2,3
6,3
11,2
14,4
16,3
17,3
18,3
Tabl* 9
Oaci 1 1 at ion
frequency
0,5
1,0
2,0
4,0
6,0
Extinction coeff.p, db.eec"
|
1
1
0,2
0,5
2,0
7,0
11,0
I:
«g"
r
0,4
0,8
4,6
15,5
25,0
1
3
i
0,3
1,5
6,0
16,0
26,0
|
S
I
0,2
3,0
11,0
30,5
51,0
£
3
• 1
1,0
3,0
12,6
37,0
58,0
Computed
coeff .
at k =
I.039/.
0,7
2,8
10,4
35,8
65,5
Data in Tables 7,
8, and 9 indicate that
the discrepancy between
Knudsen's jst jil theoreti-
cal calculations and
those of the experiment
was significantly great,
which can be explained
by the following: first,
Knudsenjet al in calcu-
lating theoretical values
of the sound attenuation
coefficient, failed to
take into consideration
attenuation resulting
from irreversible heat
losses of the medium
caused by heating and
cooling of aerosol par-
ticles, an occurrence
which was investigated
theoretically later. Sec-
ondly, as Fu'chs [121]
pointed out, the authors failed to account for the strong effect of aerosol
polydispersion on the theoretically computed sound attenuation coefficient.
The authors used the arithmetical mean f as the numerical value of the par-
ticle radius obtained from the relationship!! i^rj = nf, whereas they should
have used the arithmetical mean ratio f to the cubic mean radius to the third
power r/r33 instead of value 1/r8 in (8.9). Since r3 was always greater than
f in the case of polydispersed aerosols, failure to satisfy the above condition
leads to highly elevated results.
Epstein and Carhart conducted theoretical calculations of sound
attenuation coefficients, shown in the last column of Table 7, individually
for the fractions, allowing for thermal conductivity of the medium according
to equation (8.13). However, their results proved too low. This could be
explained by the fact that the effects of evaporation and subsequent vapor con-
densation on the fog droplets were disregarded.
Wei Chun-itseu [22] measured the attenuation coefficient of 1 - f sound
in an artificial water mist containing droplets of radius in the order of 5 - 9M-
in an amount of 0. 5 " 10* - 1. 5 * 10*. Frequency of vibrations was varied in the
25 - 250 khz range. Measurements were conducted by the standing-wave
method using a sound level recorder. Values of the attenuation coefficient
were in the 8. 8 - 20. 8 db * sec"1 range (the highest absorption occurred at
35 - 50 hz). These values are much higher than those obtained using Oswa-
- 75 -
-------
titch's theory. Exceptionally good agreement
between the theoretical and experimental data
on sound attenuation by aerosols at acoustical
frequencies had been obtained by Zink and Del -
sasso [297]. Their method of measurements
•was based on comparison of acoustical pulses
transmitted through aerosol and a pure gas.
The attenuation coefficient was determined
from the ratio of amplitudes and sound velo-
city in an aerosol - from a distance bet-ween
sinusoids. Functional dependence of attenu-
OJ3 -
i
*] '
.-
**
3 _t
2
f
1- ~3
\ 1
a ma voy tua swasffff am
Fig. 25. .Sound absorptian by aeros
of aluminum oxide according to 2ink
and Da Isasso.
I - Theoretical sound absorbtion v
coeff resulting froa total losses (a.);
1 - Theoretical sound absorbtion
ation and sound velocity on vibration frequen- coeff resulted fro» non-reversible heat
transmission fro* particles to mediun aru'
vice versa (a»){ 3 _ Tpt,, sound absprb~
Circles indicate exptl « values.
cy was investigated for air, argon, oxygen
and helium. Powdered aluminum oxide was tion coaff
used as aerosol particles (n = 3 • 104 ) with
radii from 2.5 to 7. 5\±. For the purpose of
theoretical sound attenuation coefficient computation, particles had been
classed into four groups with radii of 2. 5, 3.75, 5, and 7. 5n. Figure 25
shows the theoretical and experimental values of the sound attenuation coef-
ficients in air obtained by the authors. Both data types are nearly coinci-
dental. Identical degree of agreement was obtained for helium. Experimen-
tal data for argon and oxygen were slightly higher than the theoretical values
(5 - 10%). At the same time, satisfactory agreement was obtained between
the experimental data on the velocity of sound in aerosols and the theoretical
calculations of the same, conducted using equation (8.18).
Zink's and Delsasso's experiments prove that the theory of sound
attenuation in aerosols with a solid dispersed phase (smoke and dust) has
been refined presently to a high degree and can serve as a dependable basis
for practical calculations. Among other experimental investigations in the
area of sound aerosol attenuation, the work of B. F. Podoshevnikov and B. D.
Tartakovskii [97] should be mentioned. Podoshevnikov and Tartakovskii
measured sound attenuation coefficients at 13 khz in a highly dispersed mist of
dioctyl phthalate (r = 0.28p,) with particles strongly entrained into the vi-
brating motion of the medium (for details see Section 15). On the average,
the attenuation coefficient -was 2.7 db/m higher in this case than in the case
of pure air. A substantial functional dependence of the results on the con-
centration of droplets could not be noticed, which undoubtedly should be at-
tributed to inaccuracy of measurements.
- 76 -
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CHAPTER 3
AEROSOL PARTICLE INTERACTION IN A SOUND FIELD
9. PRELIMINARY INFORMATION
Rules governing motion of individual aerosol particles exclusive of
their mutual interaction had been examined in the preceding chapter. This
situation is nonexistent under practical conditions. The average distance
between coagulating aerosol particles was such that, any difference in their
vibratory velocities may cause a mutual overtaking or interaction through
the medium, thereby distorting the ideal picture of particle motion presented
above. It has been known that different aerosols exhibited different physical
characteristics; in particular, the dispersion state, density, and weight con-
centration. This created a wide variation in the following geometric para-
meters: number of aerosol particles per unit gas volume, average distance
between particles, specific particle surface area, average particle radius,
dispersed phase volume, etc. The above parameters substantially affected
the kinetics of coagulation processes in general, and the acoustical aerosol
coagulation in particular. Their influence, although less significant, was
nevertheless quite apparent during evaporation, drying, desorption, and com
bustion of dispersed, phase in the sound field.
Examine the geometry of an ideal aero-dispersive system, a mono-
dispersed non-evaporating fog with strictly spherical droplets which upon
coagulation formed spheres -with larger radii of same density. The geo-
metric parameters of poly-dispersed systems were easily deduced if the
relationships derived for this case, and for the case of solid particle aggre-
gates characterized by loosely packed primary particles, were used, and if
the analysis had been conducted by parts.
If the weight concentration k(g/m3), radius r, and density pp of the
mono-dispersed aerosol particles are generally known, the number of par-
ticles per 1 cm3 - "numerical" particle concentration - can be determined by
the following expression: •
In the case of small particle dimensions, numerical particle concen-
trations may attain values of the order of 1010 and higher. However, because
of the primary particle coagulation brought about by intensive Brownian mo-
tion, even in the first few seconds after their appearance, the highly dis-
persed aerosol particles begin to aggregate, and their numerical particle con
centration falls to 107 - 108. For example: assume radius of primary par-
ticles, formed by the condensation or sublimation of the substance, to be
- 77 -
-------
r - 0.02^ = 2 X 10 cm, and let the numerical particle concentration be
k = 10 gm/m3. According to equation (9.1), this corresponded to n0 ~ 3. 5 x
1011. According to Smolunovskii [121], the rate of the Brownian coagulation
of aerosols may be determined by the following quadratic relationship:
dn
at
which, upon integration, becomes:
where Ktr is the coagulation constant, equal to 1. 8 X 10 9 cm3/sec for particles
under study.
Calculations made with the aid of this formula show that the numeri-
cal particle concentration decreased with time as follows:
when t = 1 sec n = 5. 5 XlO8 (i. e., by a factor of 640)
when t = 10 sec n=5.5x!07 (i. e., by a factor of 6, 400)
when t = 100 sec n = 5.5x!06 (i. e., by a factor of 64, 000)
The particle radius also increased to r = 0.17, 0.5, and 1.5(i, respectively
(i.e., by a factor of 8.5, 25, and 75, respectively). It follows that highly
dispersed industrial fogs only rarely contained droplets with radii less than
0.2 - 0.5pi, and a thinly dispersed smoke contained, as a rule, aggregates
the dimensions of which were even greater.
Now, assume that n particles are uniformly distributed in 1 cm3.
Divide this volume into a series of cubes, such that each cube contained one
aerosol particle. In this case f/n" particles will be distributed on each side
and, therefore, the average distance between particles sm will be:
s«-~-. - (5.4)
Substitute expression (9.1) into the above equation and perform a simple trans-
formation, and obtain the following expression for the normalized distance SB
= sD/r [73, 74]: ' _
5m=a:160y/?2-. (9.5)
This is the fundamental equation of aerosol geometry. Figure 26 presents
values of calculated [using equation (9. 5) ] normalized distances sm between
particles as a function of gravimetric concentration k for different densities
pp. Thus, it is seen that the average normalized distance between particles
varied only slightly with changes in the gravimetric concentration and par-
ticle density.
In the range of gravimetric concentrations k = 1 - 100 gm/cm3, the
average normalized distance for pp = 1 lies in the range sm = 160 - 35. Actually,
equation (9.5) applied only to mono-dispersed aerosols. It demonstrates how
the average distance varied as particles became larger during coagulation,
provided they remained mono-dispersed. Factually, pp and k remained con-
stant during the coagulation of ideal aerosols, and as a consequence
J - 78 -
-------
Sm = const (9.6>
Whence, average distance between particles at each particle enlargement
stage was linearly related to running radius r:.
(9.7)
Radii of ideal aerosol droplets changed during their coagulation in the fol-
lowing manner. When two identical droplets with radius r0 merged, the
initial numerical concentration n,, decreased to n. The mass remained con-
served, i.e., 3/4 IT r^ ppD,, = 3/4Trrnppn; therefore
r = i
(9.8)
During total coagulation of rig small droplets with radii r8, and nj large drop-
lets with radii rx, which acted as coagulation centers, the mass conserva-
tion principle postulated 3/4Trr3appna + 3/4Tir31ppn1 = 3/4nr3ppnl; therefore:
'•» .
r = r,
where r^,^,, = T^TI is the "normalized" radius of small droplets. If the gravi-
metric concentration of small droplets kg is greater than the gravimetric con-
centration of large droplets kj, formula (9. 9) turns into formula (9. 8), where
*o = rg.n^ = ng, and n = nj. Solid particles form loose aggregates with lower
density p'p. If these were approximately spherical, equations (9-8) and (9.9)
may be represented in the following form:
r-r.l/*i, <9'8'>
*P
•
Now, consider the theory of viscous streamline flow around spherical
bodies. In the case of viscous streamline flow regime, a gas underwent con-
siderable changes which remained unconstrained by the near field, but were
propagated over distances exceeding the average distance between aerosol
particles. Examine a velocity field developed in the region surrounding a
spherical obstacle with radius r in the forward streamline gas flow, the non-
turbulent relative velocity Ugp of which was directed parallel to x axis (see
Figure 27). Assume that the center of coordinates was located at the center
of the sphere, and that coordinates of the point in question were the polar p
and angle 6 between p and the x axis. Denote the radial component of the
velocity vector by v [sic]* and the tangential component by v . According J:o
Oseen's theory [243, 244, 45, 111], which unlike Stokes1 theory, partially
accounted for the gas flow inertia"incident on the sphere, the streamline flow
velocity components can be determined from the following equations:
Should read v . [B.S.L.]
P - 79 -
-------
3 II r
DI> = - 2E- {I -[1 + mp (1 + cos a)l~p (1-cca I) } + utPcos e, (9.10)
. RIp'. .
VI = ! ",p' sin ee-mp (1--<01 I) - utP sin e,
4 p
(9.11 )
r
r
Un
Fi3. 27. Di.~~a. of co~put~d velocltie.
i~ the field .u,.~ounding . ~.~t;cle.
where m = u.p /2\). The last
terms in these equations
m/JI complement the equations
k,I/,,: ' cited in the literature above,
.\
Fia.26. Ave"838 derjlt8d di.tance bet_en .ero.ol particle. of ' :insofar as the case consi-
different .eilfht concentration. and denaity - d d d' t. 11
, ere was lame rlca y
opposed to the case of a
moving sphere in a stationary viscous medium (see 145J, page 393).
Transform equations (9.10) and (9.11) by making Pnon = p/r ("normal-
ized" curve relative to the particle radius), and m = % Re Pnon by substituting
equations (9.10) and (9.11), which will then assume?thefollowing "normalized, It
dimensionless form, flp =~=~(I-[l +.!Rep~il+COS9)Je-~Repap(1--<01')J+
. ~ ~~ 4 .'
+ cas.e,'
(9.12)
. 1
V. 3 --Ra ,.1- ')
t1~=-- = -sinGe' . .-sin8.
"u 4p.. .
making equations for the velocity components of streamline flow around a
sphere more convenient for theoretical analysis. The resultant "normal-
ized" streamline velocity can then be determined from a simple formula.
onp = V;:np + v:np' (9.14)
Figure 28 shows the velocity field in the medium for the case (simi-
lar to ours) of a moving sphere with Re = 1, as calculated by Piercey and
McHugh f25lJ, who based their solutions on Oseents equation refined by
Goldstein rl94l The solid lines indicated the locus of all points in the medium
with equal normalized velocities, i. e., maximum sphere veloci~ies. The
dotted lines indicate direction of normalized velocity at angles between sphere
velocity a nd direction of its motion at infinity. Coordina.te axes represent the
!'normalized" distances, i. e., distances with respect to the sphere radius,
Important conclusions can be drawn by examining Figure 28. A cursory in-
- 80 -
(9.13),
-------
Fig. 28. Fi«ld of v«loctti«» «urrounding « »igrtting
particl* it 9r««t di«Unc«» «h«n R« « I.
spection of the fig-
ure suggests first
that the velocity
field around the
sphere was charac-
terized by strong
asymmetry in the
direction of mo-
tion; a parabolic
"trail, " in this
case a "stagnation"
region, characterized by a reduced streamline velocity, appears behind the
sphere, in the considered case - in front of it; secondly, the "boundary"
layer around the sphere is unusually developed; the velocity differed per-
ceptibly from that of infinity at distant point locations which exceeded the
sphere radii tenfold, and in the "trail" region hundredfold. Figure 28
clearly indicates that the velocities which are 1% of the sphere velocity oc-
curred at distances of about 20 r in front of and at the sides of the sphere,
and at approximately 150 r behind the sphere in the "trail" region. The third
conclusion following from analysis of the figure and equations (9.12) and (9.13)
is: as Reynolds number decreased the boundary layer around the sphere in-
creased accompanied by a decrease in asymmetry The asymmetry was
apparent even at low Reynolds numbers, down to Re = 0, and at distances
greater than those shown in the Figure.
This can be illustrated by supplementing the evidence with the fol-
lowing calculation: using equations (9-12) and (9.13), compute the normalized
velocity of streamline flow around a sphere at low Re numbers in front of
(9 = 180°), on each side of (9 = 90 and 270°), and behind (9 =0°) the sphere at
a distance of pnorB = 75, which corresponded to the average normalized dis-
tance sm between aerosol particles when k = 10 gm/m3. It should be noted that
for large values of pnorm and fixed angles 9, equations (9.12) and (9.13) can
be considerably simplified. *
3 «• • (9.15)
At 9 = 0°
at 9 = 90 and 270°
"op
1 —
2*,
Of
„
—3
*»np
0;
-f 1 (npH Re-*0).
(9.16)
at 9 = 180°
O'np = + 1. "l
0 =. ~* 1 1 V
DJI__ ^ • • ~^ ~t' 1 . U«_
OP *« * *D
Re«^
^ = 0 (npH Re-»0),
0/nnu D<» I * I *"^ *\^ • \*** * * J
^npn — -t\e j *np [ >^ **/•
p - 4
* In contrast to equations (9.12) and (9.13) the direction of the velocity com
ponent vx is here considered to be coincident with the x axis.
- 81 -
-------
It appears certain that the streamline velocity in front of and behind the
sphere abated as 1 /~Or8 and 1/x2norm, respectively. Calculation results are
summarized in Table 10, which show that even when Re - 0, the dHference
Table 10 in streamline velocities
--- . B-;hi~d -th';--;~.re G.te~.I-t;-U;-;aph;~.fl-;;-tr~..t~fU t~.--;;her. in front of and behind the
Reynold ---;;.-:1'::;;--1--- (1=-\1(\" 270') j:-J-- (8-0") -sphere was only 2% of
nu::er -- ""pr T "Ypr' "}' r \ ""pr i "" r I "Ypr '.he flow velocity.
--~:; ~IU"oo --- 0 0 P [-j 0,99 TJo.~ -- 0
0,75 } 1.00
1,00
The fourth conclu-
sion, brought out be l'X-
amining jointly Figure
28 and equations (9.12)
and (9.13) is: the stream-
line flow in front of and in a narrow region behind the sphere was radial at
large distances from the sphere, so that the tangential components may be
practically disregarded when e - 0 and e - 1800; v ~ 0 and the radial compo-
nents may be computed from simplified equations ~9.15) and (9.17).
Now examine the trajectories of medium's particles moving around
a sphere. This case was examined specifically by Tomotika and Aoi [280J,
who used Oseen's approximate solution, as refined by Goldstein, to derive
the following expression for the normalized streamline function *norll = */r~ulp:
{ 3 ( 1) 111 + 3Re ( . t )
'l'op = - !: Pop - - - ------;-- Pup - - +
. ~ p~ ~ p~
+ 3Re (p~p - ~) cos a} sin' 9.
32 p~
When Re - 0, the above equation reduces to the streamline function of Stokes'
£low, which, in reality is an abstract concept:
'l'ap :::a ~ (1 - ~ + ~) p~ sinl a.
2 2pap 2p~
Figure 29 shows the streamlines for a case of a flow around a sphere at
Re = 1 as calculatedJn [280J. Clearly, the streamlines exhibited apparent
asymmetry even near the sphere. Ascertain
whether relationships derived under the as-
sumption of stationary streamline flow around
bodies were pertinent to the considered case,
i. e., streamline flow of vibrating gas around
particles. There are two special character-
istics of streamline flow around aerosol par-
ticles in a high-intensity sound field. First,
the particle dimensions are amall as com-
pared with thicknes s 6 of the boundary layer
("depth of penetration" of the vortex motion)
determined by equation (1. 40), 1. e., the fol-
lowing c()ndition is satisfied:
(!U8}
(9.19}
:~I~~~J:00j~~Y~l
--~ ~ ,.,:-,....---.- --oj
~-~.~::,'" .~===--~~-:.. j
---=..:.-" -',- I f'~~ - " =.:-:-1
_._--~ '.::: .'(..-'" -.-. I
~~~~fZ&d
F i~. 29. F I o. I j".. in the p,..oai.i ty of
enilulhd apherical particle at Re = I (Cf' =
p'"
O:OO5J 0.05J 0.015 0.25 0.35 O.~J O.~. etc.)
- 82 -
-------
(2r)»(B«v.
Secondly, the Reynolds number is small: Re < l.i.e., and the following con-
dition is satisfied:
The case, as noted in [62], corresponded to low oscillation frequencies, in
which the medium motion around obstacles could be considered stationary at
each instant. In this case, equation (1.40) loses its effectiveness, and the
boundary layer surrounding the sphere can be determined from the stationary
streamline flow equations (9.12) and (9.13). These equations were independent
of frequency of oscillations, a fact which has been overlooked occasionally by
investigators, as in [12]. In the case of a multiple streamline gas flow around
aerosol particles, there occur special new features which will be examined
in Section 11. The difference in the vibratory velocities of aerosol particles,
presence of wide streamline flow fields, and the general turbulence of a soni-
cated medium, all affected particle behavior. A complex hydrodynamic inter-
action develops between the aerosol particles. It may be analyzed only if sub-
divided into elementary types of interaction as follows: 1) orthokinetic inter -
action; 2) parakinetic interaction; 3) attractional interaction; 4) pulsation
interaction.
All four interaction types are analyzed in the following paragraphs.
10. ORTHOKINETIC INTERACTION OF AEROSOL PARTICLES
If two adjacent particles differed in size or density and, consequently,
had different velocities, they either converged on or diverged from one another
in the course of their motion. This type of longitudinal convergence or diver-
gence between particles is called the orthokinetic* particle interaction. It
occurred during the drift and pulsating and vibratory particle motions. The
most interesting is orthokinetic interaction which appeared in the course of
vibratory particle motion. This type of interaction was the limiting case of
all other types of orthokinetic interaction, since the velocity of vibratory
motion was greater than the drift velocity and the velocity of pulsating par-
ticles. Examine the conditions under which two particles may converge on
one another orthokinetic ally prior to actual contact, i.e., particle collision
in the case of vibratory motion. It has been assumed normally [167] that
orthokinetic collision between two vibrating particles was possible when the
smaller particle fell into the "aggregate volume" of the larger particle. The
latter was in the form of a cylinder, the radius of which was equal to the sum
* This term (in Greek, orthos - strait, kineticos - relating to motion) re-
flects the nature of the phenomenon much better than the term "kinematic"
as proposed in [121] (in Greek, kinema - motion). This will be particularly
apparent from the text in Section 11.
- 83 -
-------
of radii of both particles ra = rx + ra, and its height was equal to twice the
difference between the amplitude of vibratory particle displacements h^ =
2(Apg - Api) = 2(|o,pg -|j,p1)Ag, with two hemispherical ends. However, this
model must be made more precise. In the first place it provided no allow-
ance for phase differences in the interacting particle vibrations. Secondly,
it disregarded the medium's streamline flow around the particles, as a re-
sult of which the converging particles deflected from one another.
Now, examine both questions in that order. If the velocity of the
medium's vibratory motion varied according to Ug = Ug sin cut, the absolute
difference in velocities of two spherical particles 1 and 2 having radii rx and
ra, respectively, at time t may be determined as follows:
,\Up - - 11 p< -- uri - n/,t Uf sin (tot -- if,) -— |.ipi {'/« sin (u>i — '(,), (10.1)
where p,pl and |o,pa are the degrees of entrainment of the first and second par-
ticles into the medium's vibratory motion, respectively, which are computed
from (5.16), and cpx and cpa are the phase angles of the same particles calcu-
lated from (5.14). The maximum value (peak amplitude) of the difference in
the vibratory particle velocities AUp was attained at time t and phase angle
%. The time may be computed by taking derivative d(AUp)/dt and equating it
to zero:
tg o)/0 -- tg ( * -r
-------
snp
(10.6)
(10-7)
(10.8)
(10.9)
(10.10)
•»np
At the instant the relative particle velocity attained its peak value, the en-
trainment coefficients become:
•top
(10.11)
(10.12)
f*se/r!tf a
J. 30. Relative Migration dagraa of t»o aerosol partielaa U
• t di.ffarant fraquancia* and atza ratioa.
Analogous expressions
are obtained for the
relative particle dis-
placement, since both
velocity and displace-
ment were proportional,
as shown in equation
(1.11). Accordingly, it
can be argued that the
critical distance between
particles, in the direc-
tion of vibratory flow,
for which the orthokin-
etic collision is pos-
sible, was:
An = (IK Af.
and consequently, height of the aggregate volume was:
(10.13)
(10 14)
Height of the aggregate volume computed from this formula exceeded twice
the difference of the vibratory particle displacement amplitudes, if the fact
- 85 -
-------
that the particles were retarded by the presence of surrounding boundary
layers were disregarded.
Examine the trajectory of a small particle as it converged on a
large particle. If the size of the smaller particle were absolutely and rela-
tively negligible, it "obediently" followed the gas motion along the original
streamline, and the probability that it would be captured by a larger particle
was nil, since all the streamlines bent around the larger particle without
bisecting it, except for the center line. However, the size and
mass of aerosol particles were finite and, consequently, the smaller particle
diverged from the original streamline by virtue of its inertia as shown in
Figure 31. The aggregate vol-
ume in this case had the shape
indicated by the dotted lines.
It had been shown in
[66, 121, 122, 129] that the co-
efficient of particle entrainment
was characterized by the fol-
lowing relationship:
TT , . ., . . ,. Fig. 31. Di«gr»ns of orthokinetic iap«ct between aerosol
Unfortunately, the existing for- p«rt»cies-
mulas for calculating the COef- Dotted line indicates the "»ggrea«te" voluM involved
in c single conttct, or i>p«ct, between ptrticles.
ficient of entrainment, Lang-
muir's formula [120] in particular, can not be applied here. In the first
place, the formula pertained to cases -where particle sizes were small in
relation to the streamed sphere. In the case considered, both particles as
a rule were commensurate with one another and the "engagement" effect [121]
was significant. Secondly, it had been assumed that the particles moved uni-
formly, while in the case under consideration, they moved nonuniformly ac-
cording to a sinusoidal, or a more complex, relationship and, moreover, they
started from different initial positions. Thirdly, it had been assumed that the
boundary layer surrounding the sphere was negligibly small in comparison with
its size. In this case, the thickness of the boundary layer considerably ex-
ceeded the dimensions of interacting particles and, consequently, when a small
particle converged on a large one, its motion became retarded.
The exact mathematical solution of the problem related to the coeffi-
cient of aerosol particle entrainment in the course of the particle's singular
orthokinetic approach in the sound field was difficult. Fortunately, there
appeared no need to surmount these difficulties, since in a sound field, where
multiple orthokinetic convergence and divergence of particles prevailed, the
concept of a coefficient of entrainment loses its conventional meaning, as will
be shown later (Section 11). However, to complete the picture of the aggre-
gate volume the approximate critical values of the "singular" coefficient of
entrainment, which characterize the minimum and maximum values of the
- 86 -
-------
cross-section area of the volume, will be computed;
Inertia! particle divergence from the original streamline occurred
mainly in the case of close approaches to the streamed sphere, where the
curvature of a streamline radius was very small (see Fig. 29). In the case
of more remote portions of a streamline, the magnitude of the inertial par-
ticle displacement during the approach was small, since the streamlines
here were bent considerably less. Therefore, the minimum value of the en-
trainment coefficient G Bln occurred when the particle was situated away from
the sphere at the time when its relative velocity dropped to zero. Moreover,
it can be assumed for rough calculations, that the particle moved along with-
out diverging from its original streamline.
If the normalized particle radius was r8norB, the radius vector at the
instant the particle touched the sphere was pnorm = 1 + Tanorm- For the sake of
simplicity assume that contact occurred when 0 = 90° (actually 9 t& 98°, but
this affected the calculations only slightly). The streamline function through
a point (PnorB = 1 + r^,., 0 = 90?) according to (9.22) is:
To compute ealn using equation (9.18) it is necessary to find the ordinate of a
point along the streamline at a distance of pBOrm = h^ = 2p,laAg. The terms
l/panorB and l/pnorB in equation (9.18) can be disregarded when the distance
from the sphere was pnorB > 10. The streamline function was as follows:
Since the question concerns the streamline near the center line, it can be
assumed that pnorn « x^,,, and sin99 = (ynorB/pnorm)a « Ynorm/Xnorm)8 • In addition,
angle 9 between the radius vector and the center line was small under these
conditions and, therefore, cos 0 sa -1 may be assumed for the left hand side
branch of the streamline where 0-nr, and cos 0 » +1, for the right hand side,
•where 9 -» 0. As a result of these approximations, the streamline function
assumed the following form: . . ,
k *n that case, equations (10.18) and (10.19) can be sim-
plified even.further:
- 87 -
-------
^"p " (— -I rr Re) y*p30 (iipH 0-.. n), (10.18')
4
S'np " — i/npoo ("P" 8-*0). (10.19')
Equate expressions (10.16) and (10.18'), take equation (10.15) into considera-
tion, and get the following:
(10.20)
The above expression has a minimum value when Re -»0. Thus, to compute
ealn , use the following formula:
3-HH - ^SLr- (1.25-5-1. 50) (10.20') -
jnp i
In deriving this formula the particle divergence from the original stream line
due to inertia was disregarded, so that the formula yielded somewhat lower
values. The maximum coefficient of entrainment em ax was obtained in a case
when a small particle was in the proximity of a sphere and reached it with a
maximum relative velocity. The approximate value of eaax can be computed
in the following manner:
Let the magnitude of entrainment coefficient el (excluding
the effect of engagement), as calculated by Langmuir's
formula, be known. According to (10.15) the ordinate of
the initial point of the particule trajectory was
. (10,21)
In the case of a finite particle size, its trajectory partly changed its course
and moved up. By examining figure 29, where the streamlines in the vicinity
of the streamed sphere are shown, it may be concluded that if yt was not too
small (yj :> 0 . 4 - 0.5 or ., ej > 0 . 16 - 0.25), an increase in the ordinate of the
initial particle trajectory was roughly equal to particle radius ranorB . Thus,
the initial particle trajectory ordinate, include the engagement effect, is as
follows: irnp«^p + rInpw/^ + rtnp, (10.22)
and the coefficient of entrainment, accounting for (10.15) is:
3-i««(/3l + rInp)«. (10.23)
Langmuir's formula, which is norrually used to compute the entrainment coef-
ficient without allowing for "engagement" effect £j , has the following form
Y-. (10.24)
np /
*Shishkin's simplified formula [120] used under our conditions (comparative-
ly large K) yielded elevated results.
- 88 -
-------
where k is the inertia coefficient of a small particle: >
; . •*<" v
M*
(10.25)
and Ta are the relative velocity and relaxation time of a small particle,
respectively, and rx is the radius of a large particle); kc r is the critical
coefficient of inertia which, according to L. M. Levin F 66] , can be determin-
ed for the Oseen streaming regime by the following formula:
= 1.49
16— 3Re
24 -f- l2Re
(10.26)
Table 11 shows the computed (using the methods described) minimal and maxi-
mal coefficients of entrainment for the singular orthokinetic convergence of
particles in a. sound field of normal intensity (J (arl)= 0.1 w/cms , Uj = 220
cm/sec) at the optimum frequency for each particle pair. The relative vel-
ocity of a small particle Ugx may be calculated from the formual
v8n or B Ug , where v 8B 0 r B is a correction factor for the retardation in the
boundary layer whfch, according to (9.16), is equal to the normalized velocity
at a point (yp „ „ . = 1 + r a n „ ,„ . 8 = 90° ) as follows:
norm
n o r i
».75 c-r
— 1.
(10.27)
tnp
The Reynolds number Re was determined for the absolute value of the stream-
line flow velocity ug p = n/^ ug , where (J, , T is the coefficient of streamline flow
around a particle at an instant when the unperturbed oscillatory velocity
attained its peak value expressed by the equation (10.11).
Table II
ft.
li
1,0
2,0
3,0
4,0
5,0
rt,
it
0,25
0,50
0,5
1,0
1,5
1.0
1,5
2,0
1,0
2,0
3,0
1,0
2.0
3,0
4,0
tap.
0,25
0,50
0,25
0,50
0,75
0,33
0,50
0,67
0,25
0,50
0,75
0,20
0,40
0,60
0,80
sec,
x.
7,5-10-'
3 -10-*
3 -10-«
1,2 -10-*
2,7 -10-*
|l,2-10-»
2,7 -10-» '
4,8 -10-»
1,2 -10-»
4,8 -10-«
1,08-10-*
1,2 -10-»
4,8 -10-*
1,08-10-*
1,92-10-*
A-
0,9i
0,80
0,94
0,80
0,64
0,90
0,80
0,69
0,94
0,80
0,64
0,96
0,86
0,73
0,61
uir
CM/CIK
207
176
207
176
141
198
176
152
207
176
141
2(1
189
160
134
R*i
0,276
0,235
0,552
0/.70
0,376
0,792
0,705
0,608
1,104
0,910
0,752
1,410
1,260
1,067
0,893
*KP
1,110
1,125
1,020
1,045
1,076
0,951
0,976
1,004
0,874
0,912
0,963
0.800
0.835
0,880
0,925.
Hi,
0,88
0,60
0,88
0,60
0,28
0,80
0,60
0,38
0,88
0,60
0,28
0,92
0,72
0,47
0,22
»«p.
0,496
0,580
0,575
0,650
0,632
0,668
0,705
0,730
0,700
0,753
0,777
0,731
0,778
0,799
0,814
«,.
cM/ceic
90,0
76,5
111,2
85,8
42,6
117,5
93,0
61,0
135,4
99,4
47,9
148,0
123,2
81,6
39,4
*
0,72
2,29
1,67
5,15
5,75
4,70
8,37
9,76
4,06
11,93
12,94
3,55
11,82
22,10
15,12
MX.
0
0,255
0,175
0/.92
0,041
0/.78
0,607
0,637
0,450
0,678
0,690
0,426
0.676
0,778
0,719
nfi*.
«a».
0,09
o,3;»
0,00
0,33
0,72
0,15
0,33
0,58
0,09
0,33
0,72
0,06
0,22
0,47
0,82
aa».
0,09
1,00
0,i5
1,45
0,90
1,05
1,65
2,15
0.85
1,75
2,50
0,75
1,50
2.20
2,70
- 89 -
-------
The following conclusions may be derived from data in Table 11:
e =0.3 -0.7 for particles with radii rx = 0. 5 - l|j, and
e = 0.7 - 1.5 for larger particles. Therefore, the aggregate volume
of a large particle constituted a long cylinder with a cross section at each end
of e mln n r^ , increasing towards the middle and attaining e max tr r\ in the
vicinity of the sphere. The cross section directly at the particle center
line was even larger - (1 + r 3n or n )2 TT rx2 .
Moreover, the magnitude of the aggregate volume in each case was
considerably smaller than the volume of a cylinder with radius rx + r3 , for
which the coefficient of entrainment would be C0 yl - (1 + r Sn 0 r n)2 . The
orthokinetic interaction taking place during particle drift is characterized by
small velocities. Therefore, the coefficient of particle entrainment may be
considered here to be close to SQi n . The latter lies within the range of 0.09
and 0.82.
11. PARAKINETIC INTERACTION OF AEROSOL PARTICLES
If two aerosol particles of different size, or density, occupy positions
in which their central axes did not coincide with the direction of oscillatory
motion, they are subject to a transverse, or actually radially-transverse,
interaction called the parakinetic* interaction. This type of interaction was
first analyzed by the present author in [82] . The development of the inter-
action is the result on the one hand of the Oseen streamline asymmetry aris-
ing during a medium flow past the particles, and, on the other hand, by the
particle inertia which enhanced a degree of particle divergence from a pre-
vious trajectory. Since an aerosol particle streaming in a sound field was of
a multiple advancing-regressing nature, the initial particle divergence was
followed by a second and third divergences - one cycle following another, As
a result, the particles rapidly migrated with respect to one another in a short
period of time. The parakinetic particle motion is a complex geometric pro-
cess. Its understanding requires additional data on the theory of streaming
which pertains to the stationary streamlines at large distances from the
streamed sphere, the size of which was of the same order of magnitude as
the average distance between natural aerosol particles. The simplified ver-
sion (10.15) of the streamline function may be used in this case to-draw the
streamline. For the construction of streamlines passing near the center line
(x-axis), it is possible to use even more simplified equations for the stream-
line function (10.18) and (10.19), provided the near field (x^ < 10) was disre-
garded; when solved for yn they yielded the following equations:
*In Greek, para - beside, in the vicinity of, alongside of; kinetikos - relating
to motion.
- 90 -
-------
^"p y ..t
l+A,-!,3.^
2 + 4 *•"> + 16 /
/__*££_ V' ,
l7^!^
\7 4*»P-/
jt<0.
i/np —
(11.1)
'(11.2)
The second equation is the one of Stokes1 streamline derived from the first
equation by setting Re =,0. The ordinate origin of these, yn a>, may be ob-
tained by setting
_oo
Table 12
*np
''X!'
\ Ayn
/ *np
^ Wp
Re ™> 0 < p
aemjf yop
lA*op
'
— 00
0,447
0
0,381
0
1,415
0
1,205
0
i
-200
0,448
0,001
0,382
0,001
1,418
0,003
1,206
0,001
-100
0,451
0,004
0,383
0,002
1,424
0,009
1,210
0,005
—50
0,454
0,007
0,385
0,004
1,435
0,020
1,216
0,011
-25
0,461
0,014
0,390
6,009
1,455
0,040
1,233
0,028
—15
0,471
0,024
0,395
0,014
1,486
0,071
1,249
0,044
—10
0,485
0,038
0,404
0,023
1,533
0,118
1,278
0,073
•
0
^S"-\J^
^^^^
•
>.
^^^
\^>
^^^
-"""^
+ 10
-
0,485
0,038
1,533
0,118
+15
0,471
0,024
•
1,486
0,071
'.
+25
•
0,461
0,014
1,455
0,040
+60
0,454
0,007
1,435
0,020
+ 100
0,451
0,004
1,424
0,009
+200
0,448
0,001
1,418
0,003
+ 00
0,447
a
•
1.4(5
0
Table 12 shows calculated values of yn and Ayn = yn - yB _m for I|FB =
0.1 and \|tn = 1 at Re = 0 and Re = 1 (using equation 11.1 and 11.2). Figure 32
shows exaggerated streamlines constructed by the use of experimental numer-
ical values; (the increment ( A yu ) scale is 100 times the x-axis scale). These
correspond to Ayn = 0. 381 - 0.447 and yn m = 1.206 - 1.415. Figure 32 also
shows the normalized curvature radii R^ = Ri / rx for streamlines calculated
from the following formula (see, e.g., [11]:
3Re\*/i
"np
(11.3)
In this
equat ioni
(nP = rt)
(v1 and v " are the first and second derivatives, of which the first may be
v j n ' n
disregarded as negligible .
The streamlines removed from the center line may be handled by
equation (10.17). The solution of this for pn yields the streamline equation
in the explicit form; for higher values of ijfn with a high degree of convergence,
the equation is of the following form: } ptap=l2C-i+Vr32^;C~^sin-«e. (11.4)
- 91 -
-------
S
£.
R,.f
v.
Rcff+f
. where G = 16 + 3 Re (1 -
cos e).
Figure 33 shows
K.t'0+i it*tf • the exaggerated stream-
lines (the Ayn increments
scale is expanded 20
times) computed from
equation (11.4) for i|rn =
100 and i|»n = 1000, in the
case of two "cut-off"
streaming regimes: a)
Re = 0, and b) Re = 1
(dotted lines indicate
backward flow stream-
lines). All the stream-
lines are situated at a
considerable distance
from the center line (x-
axis). Their ordinates
at the origin (at infinity),
written above each line
at the left, were derived
ZOO x^ in accordance with (11. 4)
- from the following for-
Stationory current Itnas and their curvature radi t in sphcri-rrllli _ . OIK \ L's
ticios in tho proxiraity of central )pna 19 r * °«' ond '«°) (/„ — ( j
When J_!-!1E! (N.6)
"\inp — ,„•. •• \' • /
~200 -HO -WO* -SO
SS
ijo 32i
cal porticios
. rt •-
Pnp + 2P,.p
PnpPnp
where pn' and p^' are the first and second derivatives of the polar, derived by
differentiating equation (11.4):
Ocos 0 -\- --
fnPr = 36 RC G"' (6ReC-« sins 6 — cos0)
r i
* sin 6 f 2sin-40-sin-» 0
+
, (H.6')
- 92 -
-------
*•«.- •'•'•V
-201 -Kit -Ku
S3 c.V tS3 Ztil tiO x
Si 100 IS3 100 liS f
-2Jt -200 -us -ma -s
X3 -MS -ISO -100
Fig. 33. Static current line* and their curvature radii for spherical particles'at great distances
fro« central line (V ris '00 and 9 ^ - 1000) a at Re -0, b at Re . | (broken line indi-
cates direction of reverse engulftent •
-------
From the above stated it may be concluded that streamlines which
passed near the center line were subjected to a relatively small deformation,
excluding the near field, which may be disregarded in the case of singular
streaming. However, the following is far more important .
a) streamlines originating in the case of Oseen's stream-
ing regime (Re > 0) are obviously asymmetrical; more-
over, the right branch is perceptibly steeper than the
left branch. Thus, in the region x^ = 10 - 50 at Re = 1
an increase in the ordinate A yn is greater at the right
than the left by (0. 003 - 0. 015) rx when \|rn =0.1 and by
(0.009 - 0.045) T! when i)fn = 1 (see, table 12);
b) curvature radius of a streamline, excluding the far re-
gions, is small, and is predominantly found in the
range (103 - 105) rx which even in the case of the "critical"
radius r = 3. 5|J. yielded R! =0.35 -35 cm.
c) asymmetry, slope, streamline, and curvatures increas-
ed rapidly with the approach of streamed sphere, re-
latively speaking, it may be said that streamlines far
from the center line were deformed more than those
near the sphere (see figure 29).
;
The asymmetry of streamlines is very high here. The further from the cen-
ter line the more the incident gas flow bent around the sphere with increases
in retardation and curvature; moreover, after its rise, the streamline did
not descend to its previous level, within the boundaries of the figure, as a
result the right branch sloped more than the left branch. The' curvature ra-
dius of a streamline, excluding the inflexion region, was approximately of
the same order of magnitude as in the first case. This is the picture of a
stationary aerosol particle streaming at large distances.
Assume, now, that a medium moved with an amplitude of vibration
A, and analyze the (Dehavior of the medium's vibrating particle:
1) near the center line at distant approaches to the streamed sphere,
where the stationary flow lines exhibited form which can be described by
equations (11.1) and (11.2) and figure 32.
2) far from the center line, symmetrically, -with respect to the
streamed sphere, where the stationary flow lines exhibited form which can be
described by equation (11.4) and figure 33. These positions are notable be-
cause they illustrate two diametrically opposed tendencies of a viscous for-
ward-backward flow around obstacles, which together manifested one facet of
the par akinetic interaction of aerosol particles in a sound field.
In order to see immediately the essence and the functional state of
: - 94 -
-------
these tendencies, assume first that the vibratory motion of the medium was
subject to the right angle law in which absolute values of the incident and re-
verse gas flow^velocity / u, I and I u§ |, respectively, were equal and con-
stant, i. e. , | u, 1 = I u§ | = const. In this case, analyzing the forward motion
of a medium particle, one can be guided by the stationary flow picture as
characterized in Figs. 32 and 33, and in the case of a reverse flow, the mir-
ror image with respect to the x-axis.
Now follow the behavior of medium's particle initially located at
point 1 near the center line to the far left of
the x-axis (Fig. 34a). If the magnitude of the
vibration velocity and the Reynolds number Re
were finite, the particle during the forward
motion moved along a straight Oseen stream-
line as illustrated in Fig. 32. Moving along
this line, the particle finally reaches point 2.
The backward motion of the particle follows a
line constituting the mirror immage of Stokes
streamline, passing through symmetry point
2. It was mentioned earlier that this line was
steeper than the Oseen line. This is the rea-
son why a particle moved in the direction of
point 3 during its backward motion below point
Fig. 3M. Hysteresis of aerosol p»rticies 1 displaced by a somewhat smaller but finite
enguif..nt in . sound field. distance 6 . It has been known, however,
a - close to central line) b — sway'fro* ' .
central line that the velocity behind a streamed obstacle
: was lower than the velocity in front of it (see
1 #9 and 12). Consequently, instead of reaching
point 3, our particle reaches point 3" located somewhat closer to the center of
the obstacle. During the succeeding vibration cycles the displacement process
is repeated several times. As a result the particle approaches the center of
the obstacle in a zig-zag manner.
When the observed particle of the medium is initially at point 1 far
from the center line and left of the x-axis (see fig. 34b) the particle behaved
differently. During the forward motion the medium's particle follows a
straight Oseen streamline, as illustrated in figure 33b, coming to rest even-
tually at point 2. During its return motion the particle followed a streamline
constituting a mirror image, of Oseen's line through point 2. Toward the
end of the return path, this line guides the particle to a higher point 3 displac-
ed from the starting point by an amount 6y . In the succeeding vibration
cycles, this phenomenon occurred repeatedly; as a result, the observed
particle zig-zags continuously away from the center line. The reason for
this phenomenon [81] is the same as in the case of asymmetrical Oseen's
streaming, namely, the medium's inertia; that the vibrating medium w a s
predisposed to the streaming hysteresis appeared evident from the velocity
- 95 -
-------
field shown in Fig. 28: velocity angle inclination at a point on the right hand
side always differed from the identical angle at the symmetrical point on the
left hand side.
For practical purposes, the sonicated medium vibrational velocity
was not constant, but varies sinusoidally, or in some other complex way. It
can be easily seen, however, that even in those cases, the described phenom-
enon held even though same specific characteristics appeared. In the case of
medium's vibration velocity -where sinusoidal ug = Ug sin out, these character-
istics; are reduced to the following:
9
Assume the particle oscillating near the
center line to be at the same point to the left
of x-axis at the starting moment (Fig. 35).
During the initial displacement, when the vi-
bration velocity and Reynolds number are
negligible, the particle will, naturally, move
along the stationary Stokes line. However,
as the flow velocity and, consequently, the
Reynolds number increased the particle mo-
tion followed increasingly greater slopes as
it crossed into a more sharply defined Oseen's
streamline. Having reached the peak
velocity, at a point where the "peak'1 Oseen
streamline passed, the trajectory slope in-
. ; creased in the reverse direction up to point 2,
Fig. 35. Diagraa illustrating the origin .
of autocentrai ization phenomenon of (a) the medium's particle will glide into a new
and autodeeentr.i iz.tion (b) «f .eroso! stokes streamline. The medium's particle
particles in a sound field. ~
follows the entire new Stokes streamline dur-
ing its reverse motion until it reaches point 3,
located below point 1, the distance between the two being somewhat less than i
in Fig. 34. The equation for the medium's particle trajectory in the above
described "sinusoidally developing" Oseen's streaming of a sphere may be
derived by disregarding the retarding effect of the boundary layer, i. e. , far
from the sphere, ursing equations (11. 1) and (11. 2) and by substituting Re = Re
sin yj t, and assumi-ng that sin to t_= J \ - cos2 cot, where cos U) t = (xx - Ag -
x /A = 1 - X - x ) /A . Here Re is the peak Reynolds number, equal to
2rx Ug /v; A^ is the normalized amplitude of vibration equal to Ag /rlt
Equation (11.1), which corresponds to the forward motion, assumes
the following form:
!4.!1^i4.25!r1_(i_2!tzf2LVli '
2 T 4 '"" •*• 16 I { A. ) J
(11.7)
- 96 -
-------
Equation (11.2). which corresponds to the backward motion, remains
unaltered. r
The particle of a medium vibrating sinusoidally at some distance
from the center line and to the left of the y-axis also begins to move along the
Stokes streamline (Fig. 35). However as the flow velocity and, consequently,
Reynolds number increased, the particle assumed a progressively steeper tra-
jectory as it crossed onto a more sharply defined Oseen's streamline. Then,
having reached the peak velocity, at a point where the "peak" Oseen's stream-
line passed, the slope began to level off in the opposite direction and the par-
ticle glided onto a new Stokes streamline at point 2 to the right of the y-axis.
The equation for the medium's particle trajectory in the above described
"sinusoidally developing" Oseen's streaming of a sphere, if the flow retarda-
tion were disregarded, may be derived from equation (11.7) by substituting,
as before, Re = Re sin out and assuming the following:
Agog
(11-7')
This resulted in the following:
/ pnpCDS 6—Xiao
(
\ 1M V.
(11.8)
At first the particle followed the same Stokes line in the reverse
direction, changing over later to a progressively - defined and steeper re-
verse Oseen's line of the same type as shown by a dotted line in Fig. 33b.
As a result, at the end of the return trip, having passed a new "peak" point,
the particle assumes a higher point 3 through which a new Stokes line passes.
Thus, tendency for a particle to move away from the center line of the sphere
is also evidenced in this case. . .
As a result of the streamline
flow hysteresis around a sphere, a
unique acoustical flow develops
directed near the co-ordinate axe s ,
s h o wn by arrows in Fig. 36
with closed loops drawn arbitrarily.
The flow is unique, because the size
of its eddys was comparable to the
vibration amplitude, while the eddy
direction was directly opposite to
that observed for A^ « r by Andrade
[140] . Andrade established that the
condition for the formation of an
eddy was Re >5. Subseque ntly,
Fig. 36. Oi.8r«of .cou.tic.i fio.J^r^i) Scblichting [ 261] showed that under
around sn ••rosol psrticl* suspended in • 6 L J
f »«id ' such conditions internal reverse
- 97 -
-------
flows must be generated near the obstacle surface as predicted by Carriere
[173] . This hypothesis was verified experimentally by West [284] during
tests conducted at Re > 40. Subsequent theoretical and experimental research
conducted by Andres and Ingard [144], Holtsmark, et al. [201], Westervelt,
et al. [225] , and Lane [222] indicated that, as the transition to a viscous
streaming had taken place, the scale of the internal eddys rapidly increased,
displacing the outer eddys further away and, thereby, reducing them to zero,
since the eddys were subsequently dispersed by the turbulent pulsations of the
medium. The latter occurred, actually, in author's case, A8»r, although
it differed in the following:
a) Scale of eddys was independent of medium's vibration frequency
which followed from the assumption that the flow was quasistationary;
b) eddy configuration depended on medium's vibration amplitude,
which thus far, can be shown only geometrically, since the analytical solution
of the problem of an acoustical flow around small obstacles for Ag » r was
handicapped by serious mathematical difficulties.
Now, assume that in the above field, existing around a streamed
spherical particle, an aerosol particle with radius ra was located at the point
of the medium's particle. For the sake of simplicity, assume at first that
the degree of entrainment into the vibratory motion was nearly unity for this
particle, and, therefore, the particle was deprived of its own streaming
field -which could distort the indicated streaming pattern of a sphere. If the
aerosol particle were negligibly small, its behavior would not differ practic-
ally from the behavior of the medium's particles; it would vibrate and zig-zag
together with the latter, as shown earlier. However, if the particle had a
perceptible inertia, its behavior in the streaming field might take on new
features. Moving along curved portions of a streamline, the particle would
be subject to some centrifugal forces which might displace it from the origin-
al streamline into the region of .convexity. Therefore, in the first case,
Fig. 35, an aerosol particle in a forward streaming would not assume point
2, but rather point 21 below, and this would be repeated with each succeeding
cycle. Thus, it could be concluded that the particle inertia contributed to its
hysteresis-like approach to the center line.
In the second case, shown in Fig. 35b, as a result of the centrifugal
force, a particle would not assume point 2, but rather point 2' below. During
the return motion, a particle would assume point 31 below point 3, and this
would be repeated with each cycle. Thus, it may be concluded that the parti-
cle inertia restrained its motion away from the center line as a result of the
streaming hysteresis. The zig-zag approach of an oscillating aerosol parti-
cle to the center line of a streamed particle caused, on the one hand, by the
particle inertia, has been referred to as self-centering of a particle [ 82] . In
contrast, particle motion away from the center line may have been referred
to as self-decentering of a particle. Theoretically speaking the phenomenon
• - .98 -
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of par akinetic counter-balance was also conceivable where moving particles
assumed a position in which none of the above diametrically opposed tenden-
cies predominated. All these phenomena characterized par akinetic inter-
action of aerosol particles in a sound field.
The question can now be asked:
What form did parakinetic particle interaction assume in each real
case?
The answer to this depended upon the respective positions of aerosol
particles, their size and density, and on the medium's viscosity and the mode
of its Bonification. Unfortunately, a general analytical solution of this problem
necessitated the use of computers. However, velocities of the parakinetically
interacting aerosol particles in a sound field could be estimated readily if ob-
servations were limited to the case of vibratory velocity which varied as
I Ug | = | xig j = const. In the case of sinusoidally varying vibratory velocity, all
results should be correspondingly lower.
The hysteresis component of velocity of a parakinetically moving
aerosol particle may be found from the following formula: Vg = f6g (11. 9);
where 6yg is the extent of hysteresis for the forward and backward stream-
line flow around a streamed particle for one period of vibrations. The iner-
tial component of velocity of parakinetically moving aerosol particle may be
determined from the following formula (7. 3):
-"„•$*_ **Pfr . (,,.10)
*. *,„/'
where Up8 and ra are the relative velocity and relaxation time of a small par-
ticle, respectively, and Rt and Rln are the absolute and normalized radii of
curvature of streamlines, respectively, around the larger particle at a given
point. ,•
In the case of particle self-centering (Fig. 35a) the quantity Vg may
be analytically estimated by the following equation:
which may be derived readily from equations (11.1) and (11.2) by assuming that
xiiX =0.
If Re = 1 and Xgn = -10 we get: for $n = 0.1, 6yl3 = 0.0095 r^
for $B = 1, ayjg = 0.030 rx .
At frequencies of / =1-10 kHz the following velocities are obtained for
particles with "critical" radius of rx = 3. 5 p..
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" AftX* ... — _ , - - -
Vg = 0,03 — 0,30 CM I cex air :tj>np = 0,1, .
-'(\ in i r\n -../—>»..- — •-•- ' « . in.n
- u, lu — j ,uu
The magnitude and sign of V0 are different at each point of the par-
ticle trajectory. In the case of a particle with normalized radius ran = 0. 33
(p-ia =0.8) and r^ == 3.5|j, when Ug = 220 cm/sec (J = 0.1 w/cm*) in the range
Rin = 10a - 105 , the following values for the inertial velocities of particle
divergence are derived:
Ve = 0.015 - 1. 5 cm/sec
In the second c.ase, when particles diverged from one another (Fig.
35b), the magnitude of Vg may be estimated by graphical-a'nalytical methods
using Fig. 35b. If Re = 1 and Agn - 50, we get: for tyn = 100, 6ygn <& I, and
for ij,n = 1000, 6 y g-n ~ 3. In accordance with this, the following maximum
velocities of hysteresis separation of particles are obtained for particles
with the "critical" radius of rx = 3. 5p, at a frequency of / = 1 - 10 khz;
V.-**it: 0,35 — 3,5 cM/ceK .t' $np = 100, /nun)
V7*X'— 1.05— 10,5 cMlceK^ Tj>np-1000.
The order of magnitude of V0 is the same as in the first case, if the radii of
curvature were identical. .
A more graphic description of the parakinetically self-centering
particle velocity calls for the introduction of a "normalized" interaction ve-
locity concept defined as the ratio between the absolute velocity and the dis-
tance covered, i.e.", Vg /y . This parameter shows the number of times the
covered distance may be traversed per second for a given absolute interaction
velocity. The reciprocal quantity, y /Vg , characterizes the times necess-
ary to achieve complete coincidence of the particle center lines for the same
absolute value of interaction velocity. The following formula may be derived
for the normalized velocity of hysteresis-assisted self-centering, using equa-
tions (11.1), (11.2) and (11.5):
Figure 37 shows normalized f~ /~Tl. 3Re\ 1. 3 T\~
o • .. / f i—,—^.\i^ __ _ j^—t i
2 . /
V.
velocity of hysteresis-assist- _i
i - -•/ \2| 163M ^
V ~2~1Xt[9 +~W
f. (11.12)
ed particles self-centering J'-oo
as a function of Re for a con- '
stant x3ll=10.Itmaybe
concluded from the graph that hysteresis-assisted coincidence of the center
lines of both particles was achieved (if Re=^ 0) in tenths or hundredths of a
second (divergence occurred even quicker).
Assuming that parakinetic interaction. occurred at an optimum vibra-
tion frequency for each particle pair, so that p,ia may be determined from
(10.10), the normalized velocity ofjinertial self-centering Ve /y ^ , allowing
for (11.10) and (11.5), may be found .from the following formula:
r+-
-!00 -
-------
Fig. 37« Adjusted velocity, or rate, of
hysteresis autoconcentratiort of aerosol
particles in the sound field «t I a O.lbt/
>2
Fig. 38. Adjusted rate of inertia auto-
concentration of aerosol cartic|ea_in a
sound field at J =0.1 hi/cm*
Figure 38 shows the normalized inertia! self-centering velocity as a.
function of the normalized curvature radius of streamline Rln for different
values of normalized radii of a vibrating particle r8n for a case when Ug =
220 cm/sec (J = 0.1 w/cma), f = 1, and Re = 0. It may be concluded from the
graph that the inertia! coincidence of the center lines occurred even faster,
namely in hundredths and thousandths of a second, even though the absolute
value of velocity Vc;was comparatively small. The aforementioned indicated
that the parakinetic-interaction of aerosol particles was very intense. As a
result, the concept of particle coefficient entrainment in a sound field loses
its ordinary sense and meaning. Due to self-centering, orthokinetic collision
was probable not only for vibrating particles which were initially in the agg-
regate volume of the streamed particle, but also for particles which were
far beyond its boundaries. The value of the entrainment coefficient, if only
this term may be applied to a given case, is primarily determined here by
the duration of sonication; moreover, even for the shortest time, it is great-
er than the critical value of the coefficient of entrainment for the stationary
streaming of objects, such as, the coefficient of entrainment of falling drop-
lets estimated on the basis of Oseen's attractional forces, as done by Piercey
and Hill [ 250] .
No experimental data had been found on the parakinetic interaction of
aerosol particles in a sound field, except for the work of O.K. Eknadiosyants
and L. I. Buranov [12] in which the phenomenon of aerosol particles "counter-
- 101 -
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balance" was recorded by a high-speed camera. The aerosol particles in this
case were an aluminum paint dust with a radius in the range of 1. 5 and 15|i.
The frequency of vibrations was 0. 7 —3.0 khz, and the camera speed was from
300 to 5000 frames per second. The investigators observed the manner in
which spatial complexes consisting of 2, 3, 4, and more vibrating particles
formed and moved together across the field of vision.
12. ATTRACTIONAL INTERACTION OF AEROSOL PARTICLES
By the attractional* interaction of aerosol particles in a sound field
is meant their drawing together,brought about by the action of hydrodynamic
forces created by the reciprocal interference of flow fields around the partic-
les, and which are arbitrarily calle'd forces of attraction. These include,
respectively, the Bernoulli [ 2] , Bjerkness [109], and Oseen';[ 103] forces.
The first two were assumed responsible for the acoustical aerosol coagulation
without sufficient grounds; therefore, they have been merely mentioned at the
end of the paragraph, and only because they have been mentioned in many pre-
vious publications. .
The basis ;of attractional interaction of aerosol particles in a sound
field consist, as S.: V. Pshenay-Severin has shown [103], in the hydrodynamic
forces of Oseen. These forces are generated by asymmetry in the velocity
field which surrounded all interacting particles. ' - •
v - •
Unlike the previously mentioned types of aerosol particle interaction,
attractional interaction required no gradient in particle size density. There-
fore, the physical nature of this form of interaction between two equidimen- x
sional spherical particles will be mentioned for the sake of simplicity without
taking into account differences in the phase oscillations of the interacting
particles, as a rule, greatly complicated the understanding of the basic prin-
ciple of the phenomenon.
In #9 it was shown that velocity fields created around each particle
with a flow around it is asymmetrical in the direction of the acoustical motion.
Back of the particle there was a wide parabola-like "stagnated" zone in which
the velocity of the medium was appreciably lower at every point than at the
symmetrical point of the weakly restrained, "frontal" zone formed in front of
the particle.
Now, assume that two aerosol particles, 1 and 2 (Fig.. 39), were
moving under the effect of an acoustical flow in a way that one of them, namely
particle 2, moved along its path in the stagnated zone of particle 1; the latter,
in this case, was in the slightly stagnated frontal zone of particle 2. It can be
*From the Latin attractio -- attraction. In the literature this type of inter-
action is often termed "hydrodynamic", which seems less appropriate, since
this term is too all-embracing and vague.
•- 102 -
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Fig. 39. Oiagraa iI lustrating »Jp«rposi-
tion of velocity field*; of t«o aerosol
particles
seen that in this relative position of the
particles, they retarded each other's move-
ment. In such cases, the "head" particle
(1) actually slowed its motion considerably
less than "tail" particle (2), so that the
particles converged somewhat, as if a-
ttracted to each other. In a reverse med-
ium motion, the particles changed roles:
the role of the "head" particle becomes
transferred to particle 2, while particle 1
played the part of the "tail". The distance
between them again diminished, and cycle
after cycle of oscillatory medium motion
is repeated.
Taking into account that Oseen's hydrodynamic equation is linear, it
is possible, when studying the attractional interaction of the particles, to pro-
ceed from the assumption of the superposition of the flow fields around the
particles. In this case, the magnitude of the force acting on the investigated
particle in the direction of the acoustical flow can be estimated approximately
by the formula: :/ FM = Gjiijr (uo — AoM). (J2.1)
where, ugp - unperturbed flow velocity of the investigated particle, determin-
ed from equation (5.22), and A v, - perturbation of the flow velocity created
by the presence of the second particle at the location point of this particle
center; it is equal to the difference between the same unperturbed flow. vel-
ocity of particle u, p and flow velocity of the second particle v x at a given
point determined on the basis of equations (9.10) and (9-11) or (9.12) and (9. 13).
Consequently, the expression in parentheses in (12.1) was merely the velocity
of the flow around the second particle vz at the center of the investigated
particle since (Ugp -vx) = Ugp - (Ugp - >fc) = ^« Accordingly, the force acting
along the line of centers on "head" particle (1) can be expressed as follows:
- (12.2)
and on "tail" particle 2 of the same
radius:
(12.3)
where v^e - radical component of the flow velocity around particle 2 at the
location point of particle center 1 determined from-equation (9.10) when
P =Pi 8 anc* & = ®19 ~*~ n • v pi ~ radical component of the flow velocity a-
round particle 1 at the location point of particle 2 center determined from
equation (9.10) when p =p la and 6 = 0i8 .
The difference in the motion velocities of
both particles is numerically equal to the
difference in the flow velocities around
them. Therefore, the following expressed
the attractional convergence velocities of
e*
the particles: i/ _«'»> am — f»ll)
" r«ro — WPI — "PI — \vftaa
.Fig. 1*0. Simplified diagraaatic pre-
sentation of particles attraction in-
teraction separated far apart.
anp
- 103 -
(12.4)
-------
If the particles are positioned one behind the other, so that 6 -»0,'
then at great distarices pla velocities v2pi and v1ps assumed a directional
position radially close to the x-axis, as shown in Fig. 40, and could be de-
termined by expressions (9.15) and (9.17), respectively. In this case:
According to this equation, velocity Vattr, with the particles directly touch-
ing (x^ = 2) at Re = 1, is equal to zero (since condition Vattr < 0, which occur-
red when xa < 2 /Re, signified the absence of any interaction characteristic
of the Stokes flow for rj_ = ra ) .
It is known that in a sound field flow velocity around a particle varied
in magnitude and direction, causing variation in the Reynolds number and with
it, according to (12; 5), also variation in the attractional convergence velocity
of the particles. Compare the second term in parentheses in equation (12. 5)
with the first terrrrat the beginning of the medium's oscillatory motion, when
the Re number is small. Velocity V£ttr is close to or greater than zero; as
the oscillatory velocity increased, Re increased rapidly, and when it attained
the peak value, velocity V|t t r attained its maximum value. .When the product
Re ^ » 2, the second term in parentheses in equation (12. 5) (representing
the velocity vps ) can be disregarded, and the expression for the attractional
convergence velocity of particles becomes simplified:
• v • — J-L (12-&)
Vatrp — 2*^ "*"• .
Moreover, the one-sided hydrodynamic effect of the "head" particle on the
"tail" particle has been thereby, achieved.
Use equation (12. 6) and estimate the order of magnitude of the in-
stantaneous attractional convergence velocity of the particles at the mean and
shortest distances between particles x^ = 100 and ^ =10, respectively, for
which the equation will still hold. Under such conditions:
=100 V*ttr =0.015ugp;
whenx,, =10 Vjt tr = 0.15 u,p .
Thus, the magnitude of the instantaneous attractional particle con-
vergence velocity consisted of comparatively small part of the instantaneous
velocity of their flow. In absolute values the figures are substantial. Thus,
at a flow velocity of 100 cm/sec, the attractional particle convergence velocity
V*ttris 1.5 and 15 cm/sec, respectively.
Now, compute the mean attractional particle convergence velocity
for one half period of oscillation for the case shown in Fig.% 36, i. e. , when
9=0. The use of equation (12. 5), where the variable Re is introduced for
this purpose, leads-, to insurmountable difficulties in integrating the particle
convergence equation due to the quadratic term in x. In order to avoid such
- 104 -
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difficulties, Pshenay-Severin [103] proposed the use of formula (12.6), nssum
ing that particle convergence occurred at some instant ^ when Re attainrd
some sufficiently high value Re0 p .
Up to t = t£ , the flow velocity around both particles varied according
to law (5.22), and for tj. ).. (12.10)
The initial conditions are as follows: V^ttp = 0 and A Xjj = 0 when t^ = tln.
The integration of equation (12.10) yields displacement A X^ for time interval
At = 1/2 T - V, in the following form:
,-(Hpf-H )si
'np
— -Jt f 2Mpiig sin (or /npt + (|»J - jij)cos
-------
f- k-horiz
M
the rate of attractipnal convergence of
aerosol particles greatly depended upon
oscillation frequency. For each particle
size there was an optimal oscillation fre-
quency at which the attractional particle
convergence rate had a maximum value.
The greater -was the particle size, the high-
er was the value of the optimal oscillation
frequency; for particles with radius r = 5 --
15|j,, low audio frequencies of the order of
several hundred Hertz are effective, while
for particles with a smaller radius r < 5p,,
it is several khz. To compare these oscil-
oscillating velocities
'-•at'
, -=-400
t
= 700
> 1000
i/ff
F|9- »»l» Adjuatud rateiof autual attrac-
tion approach^ of ^aeroeoi particles at high lation frequency values with the optimal
frequency values observed in the experi-
ment presented difficulties, due! to t h e
occurrence of industrial aerosols coagula-
tion as a rule, at lower sound intensities
than those chosen by the author.
The existence of an optimum frequency was directly indicated by
equation (12.5) because the attractional particle convergence did not exist
(Vattr = 0) in the following two cases:
a) during complete particle entrainment into the oscillatory motion
of the medium when Ug p = 0 and Re = 0;
b) during the full medium flow around the particle when the partic-
les were stationary.
It follows, accordingly, that particle convergence occurred only in
the intermediate case when they had not been fully entrained, a condition
which achieved only at certain oscillation frequency values.
Two more specifics of the presented curves deserved further
attention: the curves' compression along the abscissa, i.e. , compression
of particle radius values with the increase of the particle size, and optimum
displacement in the direction of the smaller particle radius with increase in
sound intensity. Several limitations of the author's conclusions were pointed
out by N. A. Fuchs in his review [122]. It can be shown easily that the
attractional particle convergence occurred even when 0 -« 90° ; however, here,
the active principle was repulsion, which was of lesser interest. The attrac-
tional particle interaction also occurred when there was a difference in their
sizes, but the particle contact here usually occurred orthokinetically, since
its rate was considerably greater. Of particular interest was the case of
attractional particle convergence •which had full medium flow around it, while
the second was fully entrained by the medium. It is understandable that the
-106-
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second particle, while oscillating, tended to approach the first particle at the
acoustical medium'flow rate described in #11. However, unfortunately this
fact could not be used in determining the path of the eddys, a practice common-
ly used by investigators in a case of large obstacles: r»A, (see [140, 261] and
others). The long particle tracks, comparable to the eddys1 scale, impact
mutually appearing as a solid background in the photograph. The problem of
determining eddy paths in the case r « A, may be solved only by synchroni-
zing the pulsed photography with the sound source.
Let us now proceed with a discussion of S. V. Gorbachev and A. B.
Severny [26, 195] experiments in order to confirm the attractional particle
interaction in a sound field as described above. The authors studied the
elementry interaction process of two water droplets with radius r = 100|jt, each
suspended on glass threads at a normalized distance of pigB = 26 between them
in a low-frequency sound field generated by an "Accord" - type loudspeaker.
The oscillation velocity of the medium was actually very small, so that it could
be maintained with certainty that a viscous flow regime around the droplets
occurred. The authors established that at successive droplet distribution
(6 = 0°).an attraction occurred, whereas, in the case of side by side distribu-
tion (9 = 90°) a repulsion occurred. Such particle interaction in a sound field
was clearly characteristic of attractional interaction defined by Oseen's
asymmetrical particle flow which the investigators and the interpreters did
not perceive. Before concluding the present paragraph the hydrodynamic
forces of Bernoulli and Bjerkness will be discussed at some length. The
Bernoulli forces occurred between two bodies resulting from the constriction
of the flow passing between them, and the attendant loss of a portion of the
static pressure in the constricted space. When the viscous losses in the med-
ium are negligible and this occurred at an eddy-less potential streaming, then,
as is known, the following was equally maintained for the full duration of the
fl°W: •: P + %PgU3g = const. (12.12)
The phenomenon is known as the Bernoulli law or integral, which represented
a mathematical expression of the law of energy conservation for an ideal
liquid.
According to equation (12.12), .:the pressure of the medium was .some-
what lower than the pressure from outside in the space between the particles
where the constriction of the flow pipe and the attendant increase in velocity
occurred. As a result, forces are created between the two bodies which strive
to bring .them together. Based on Bjerkness1 [146] hydrodynamic theory,
Koenig derived the following equation for the force acting in a sound field be-
tween spherical bodies along the line of centers [215]:
), (12J3)
- 107 -
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where, Qia -is the angle between the line of centers and the direction of the
oscillation velocity. *
This equation was satisfactorily confirmed by experiments in the early
works of Thomas [279], who studied the interaction of spheres with a radius
of 2. 8 - 7.8 mm in a very low-frequency sound field generated as in [283]
(see #5) the Zernov method. At the same time, considerable disagreement
with the theoretical calculations has been observed in several experiments. On
the basis of experimental study of spherical bodies streaming in a sound field,
Andrade [141] concluded that the reason for this disagreement was the formation
of eddys around the bodies generated when ugpr>0. 35. Citing Cook's observa-
tions [182], the author noted, that in the case of shorter distances between the
spheres the line of centers, which was perpendicular to the direction of the
acoustical flow, at times repelling instead of attracting forces appeared.
Recently, Stashevskii [271], verified experimentally and .refined this
thesis using -wax and glass spheres of diameters 0.34 - 5.45 mm. Experiments
•were conducted in a glass tube of 35 - 40 mm diameter, at frequencies of 184 -
533 nz, and different sound intensities. Later, Adamchik and Stashevskii [136]
noted a similar phenomenon with the spheres arranged in a linear succession;
the repulsion at certain small distances here, was replaced by an attraction.
Doerr [185], established experimentally, that angular dependence of the attrac-
tion forces was expressed by the factor (6 + 4 s,ina0) rather than by the factor
in Koenig's formula (12.13). He used hollow glass spheres of 2. 75 --4.95mm
diameter, suspended on fine threads at distances of 7 - 10 mm. The oscilla-
tion velocity amplitude could attain 40 cm/sec.
Andrade [,2] tried unsuccessfully to use the Bernoulli forces in ex-
plaining the mechanism of acoustical aerosol coagulation. It had been previ-
ously indicated, that a viscous flow regime was characteristic for aerosol
particles located in a sound field, which is remote from the ideal conditions,
for which equation (12.13) was formulated. Viscous losses decreased and, with
some reservations, may be disregarded only at high Re numbers, occurring at
higher sound intensities and larger particle sizes. The latter may occur only
in the case of highly coarse aerosol particles not required, as a rule, for addi-
tional particle coagulation, or as a result of prolonged finely dispersed particles
coagulation. However, the appearance of Bernoulli attractional forces during
the final aerosol coagulation stage has become of little use, since their magni-
tude in any given case was insignificant. To verify this thesis, Brandt, Freund,
and Hiedemann [167], derived the following formula for the convergence time of
two equidimensional spherical particles before contact at their optimal relative
positions with respect to the acoustical flow (0 = ir/2) at a total streaming
*In [215] expressions are given for forces X and Y representing the projection
of the resultant "radial" force Fp and of the "tangential" force F6 perpendi-
cular to it on axis x and y.
- 108 -
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(12.14)
The authors arrived at this formula by equating the attraction force Fp defined
by formula (12.13) to the force of the medium's motion resistance which, accord-
ing to Stokes, equalled to FB t = 6 TT n r (^ dpia /di) , and integrating the obtain-
ed equation from pia when t = 0 to pia = 2r when t = t converge. Strictly speak-
ing, such a procedure is irregular, for when Re >l,the resistance force, is
not a linear function of the radius, as expressed by Stokes1 formula, but a quad-
ratic or nearly quadratic function. In this connection, determine what results
formula (12.14) will yield, if it is assumed that Re was not too large. The mean
normalized distance between aerosol particles lies within the limits Sm = 35 -
160. Accordingly, at a sound intensity J (or I) = 0.1 w/cm1* (U = 250 cm/sec),
following expression for t8oilT in air for particles with unit density; is obtained:
= 138 -- 2.76 xlO5 sec.
1 en T
Thus, it can be seen that even at high particle concentrations (of the order of
k = 100 g/m3), the convergence time before contact is more than 2 minutes,
while at low concentrations (k = 1 g/m3 and lower), it is tens of hours. This
suggests the conclusion that the existence of Bernoulli forces in aerosols under-
going coagulation in a sound field may be disregarded even during the final con-
solidation stage. Forces of attraction (or repulsion), called Bjerkness forces,
originated between two bodies oscillating, or pulsating, under the effect of an
external force. As Kelvin noted long ago, the oscillation velocity of a medium's
particles behind a body located in a sound field always.had been somewhat less
than in front of it; ' This is governed by the scattering of a portion of the part-
icles energy on the"-frontal surface of the body; it was true only in the case of
an ideal .medium. According to Bernoulli's principle, the pressure behind a
body was somewhat higher than in front, and, consequently, in this case the
body was acted upon by some force tending to make it approach the oscillating
object. Another body which was oscillating or pulsating, or both simultaneous-
ly, can serve as such an object. In the latter case, both attractional and repul-
sive forces may occur between the bodies, depending on the oscillation phase.
The described ponderomotive forces of interaction arising during the
oscillating or pulsating motion of bodies was theoretically studied by Bjerkness
[146]. P. N. Lebedev [64] experimentally studied the nature of the pondero-
motive forces in cases of electromagnetic, hydrodynamic, and acoustical os-
cillations; he came to the conclusion that the laws which govern these forces
were identical, i. e. , the forces were independent of the wave field character.
White studying the interaction arising between two water, droplets in
a sound field (109), A. B. Severngi came to the conclusion by taking Ollivier's
observations into account |]242], that the droplets were virtually in a state of
pulsation; each ellipsoidal droplet, extended with respect to the two axes, be-
came transformed into a compressed ellipsoid symmetrical with respect to the
/ - 109 -
-------
same axes. According to the author, the pulsating droplets acted on each other,
creating attraction or repulsion between the droplets.
On the basis of an equation derived by Bjerkness for the forces of in-
teraction of two radially pulsating spheres
•
fij. and Qs - mean quadratic values of the rate of change in volumes of the first
and second spheres, respectively, and having transformed it to apply to virtual
pulsations, Severngi derived, through complex calculations, the following ex-
pression for the force of interaction bet-ween equidimensional particles located
in the plane of a sound wave (6 = 90°): u>
F~ — K—-, (12.16)
; P|.
where K = - 6.19 ' 103-p- C.~^r « - (3.1 - 6.2) 103C. r4.
p *I2+(kr) B
Here, k = — , where T - period of ellipsoidal and symmetrical pulsations
P _
determined by Ollivier's [242] expression: _ / 3«m ' /to IT\
Tp=rY g^ . ....... \ ' J
- droplet mass, 5 - surface tension which for water at 20° C, is 73 dyn/
cm.) ;
Using equation (12.17) and disregarding the medium resistance,
Severngi derived the following equation for the convergence time of droplets
before contact: _ _ ... .
Pit (1 + Pu) (2 + Pu) ""7 _ 1 -I/ Pump fl9 1RV
< v ~^~' (
For droplets of radius r = lOp, , the ratio -j-. «0.01; hence, at a distance be-
tween the droplets of the order of pls «1 mm, and a sound intensity of J = 0.1
w/cm2 (Ug p K 150 cm/sec), teenv = 0.0014 sec which corresponded to an aver-
age convergence velocity of Veonv = 70 cm/sec. The results are more than
favorable. However, they have been obtained with equations not usable in the
case of real aerosol sonication, where viscous streaming of particles prevail-
ed. Moreover, the adduced equations fail to hold even in the case of the final
prolonged aerosol coagulation stage, when viscous streaming yielded to poten-
tial streaming.
Calculate the natural frequency of droplet pulsations of a real water
fog, using formul (12.17), and verify that it is great as compared with oscilla-
tion frequencies of the sound field used in practice. In these, far-from-
resonance, conditions, it is impossible to "swing" the droplet and cause it to
execute virtual pulsations. A simple experimental fact proves the unsoundness
of the theory: solid aerosol particles, not capable of virtual pulsations, undergo
coagulation as successfully as liquid particles, provided they were not too
- 110 -
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coarse and, thus, -adhered poorly. It follows from all that has been discussed
that the Bjerkness and Bernoulli forces in the coagulating of aerosols in sound
fields may be completely disregarded when studying the mechanism of acous-
tical aerosol coagulation. The same may also be said of the new forces of in-
teraction related to the radiational particle drift forces studied by Embleton
[302].
13. PULSATION INTERACTION OF AEROSOL PARTICLES
It was shown in #7, that medium size aerosol particles participated
in some way, in the turbulent pulsations of a sonicated medium. As a result,
it is possible for them to converge on each other at some finite velocity.
Two mechanisms for the convergence of colloidal particles were sug-
gested by Levich [67, 68, 69] , in his theory of turbulent coagulation of col-
loids:
1) the diffusion mechanism, according to which particles converged
by purely diffusional means, as the result of their participation in different
pulsations of the medium in different directions;
2) the orthokinetic mechanism created between different size
particles resulting from the joint participation in pulsations of an amplitude
not exceeding the internal amplitude of turbulence.
Now, examine both mechanisms as they apply to the conditions aris-
ing in the turbulization of a sound field:
The diffusion mechanism. Both equidimensional and particles of dif-
ferent sizes can thus converge. The convergence of equidimensional particles
in the turbulized sound field was practically the same as in a convergence
studied by Levich [67], in a normal turbulent flow in so far as all particles
participated in completely the same way in the oscillatory motion of the med-
ium. Place an aerosol particle of radius r at the origin of spherical coordin-
ates and examine the diffusion flow to this particle occurring in a turbulent
sound field. The equation of steady-state diffusion to the surface of an absorb-
ing sphere of radius a = R, surrounding a particle of radius r« R is of the
following form: .
where, D8 f f - effective coefficient of particle diffusion equal to the sum of
two coefficients: Deff = Dturb + Dbpownlftn .
The boundary conditions are the following:
n = n0 npH a'-+oo, (13.2)
n = 0 npH a^R.
- Ill -
-------
Turbulent particle diffusion is prevalent when the sphere radius satisfies the
following inequality [ 69] :
For aerosol particles this implies that r>0.hi, -whence it follows that the co-
efficient of Brownian particle diffusion D may be practically always dis-
regarded in comparison with the turbulent particle diffusion coefficient; there-
fore, it may be assumed that De f f « Dt u r b . Moreover, the coefficient of tur-
bulent diffusion of aerosol particles may be assumed to be equal to the coeffic-
ient of the turbulent particle diffusion of the medium. For aerosol particles
completely entrained in the turbulent pulsations, this supposition is self evi-
dent. However, for particles partially entrained in the turbulent pulsations,
this assumption is not so evident, yet Chan [175] convincingly proved that even
in this case the coefficients of the turbulent diffusion of the aerosol particles
and the medium are equal [122] .
On the basis of the general expression Efe: I3 /t ^ vj. 1, (1 - length of
"step", and t - time expired in this step), the coefficient of turbulent diffusion
due to pulsations of amplitude 1, taking expressions (1.36) and (1.37) into con-
sideration, may be expressed as follows:
,
~ Vtl
(e/)%/ npH />/.. I (13.5)
Accordingly, rate of diffusional convergence of two particles rapidly decreased
as the distance between them diminished. For concentration gradients (dn/Ba)
a = R at the surface of the absorbing sphere when t » Ra /Dt „ rb , the theory
yields the following simple expression [ 121, 69]:
(&n/da)a = R = 3n/R (13. 6)
where n - numerical aerosol concentration taken as equal to the mean particle
concentration in 1 cm3 . Substituting this expression and also equation (13.4)
into equation (13.1) and taking into account that 1 = R + r » R, obtain the follow-
ing equation for the specific flow of aerosol particles to the sphere:
i j«3/4 (e/b/p,P Rn (13.7)
The total particle flow on the surface"Of the sphere, if (1.32) and
(9.1) are taken into account, is equal to:
N = 4TTR3 j = 9/4(R/pp)3 k/r ^ J^ /^/b1'2 pi* CW . 1Q-8 (13.8)
At weight concentrations of diffusing particles equal to k = 10 g/m3 , and at
normal acoustical parameters (J = 0.1 w/cma and f = 1 - 10 KHz), the particle
flow of density p =1 precipitating on a particle of the same radius (R = 2r)
Willequal: P i N« 0.18- 0.60-1—
sec
- 112 -
-------
Them/ore, this flow is very small, and can be disregarded. The diffusion in-
teraction of different size particles is characterized by a greater intensity, in
a turbulent sound field. All small particles which fall within the aggregate
volume, shown in Fig. 42a by the dash line, forming a cylinder of height 2A13
rounded at the ends
having.a cross sec-
tion e TT rf , where Axa -
the amplitude of the re-
p II ij ii is I ! lative particle motion,
ii iiiiil!!
li ii H!!!! ft
tfu
ifc). U2. Pulsation interaction between aerosol particle* in
a aound field
• —diffusion •echehias) b - orthokinetic MC
and e - the entrainment
coefficient of small
particles, may come in
contact -with each large
particle. For the greater
part of the cylinder length
the entrainment co-
efficient rn agnituce
was determined by
equation (10.20). There-
fore disregarding
the slight thickening
in the middle and the
roundness at the ends,
the study of the diff-
usion particle f 1 ow
in a normal cylinder of height 2Aia with permeable walls of radius R =v^ *\ ,
can be substantially limited. For greater clarity the diagram of diffusional
interaction can be presented in a way that the large particle oscillated with
amplitude Ala, and the small particles diffusionally drifted into the aggregat-
ing cylinder, while the large particle continuously "emptied" the aggregating
cylinder in front of the small particles which diffused into it.
The process of the small particles diffusion into the aggregating cy-
linder is accomplished in a series of short events, of durationtj), a function
of the diffusing particle ordinate and varied from O to T, where T, the period
of oscillation, equals to 1/f. A study of curve x = F(t), shows that the mean
length of the diffusional event equaled
(tD)
T =
The differential equation .for small diffusion particles in the case of a cylinder
is of the following form: dn „ f&n . 1 da\
(I3'9>
(13.10)
The initial and boundary conditions are:
nla, 0) = 0 H —(0,/) = 0 npH a/?.
- 113 -
-------
The thermal conductivity theory [70] postulates the following solution for an
unbounded cylinder when a 1 -- 2 KHz), only the first member in parentheses may be used as
the first approximation.
The mean * Fo U;(Fo),p =
which, when J = 0.1 w/cm for an air medium, yields:
- 114 -
-------
In the frequency range f = 1 - 10 Khz; (Fo) = 0.148 -- 0.047, and conse-
mean
quently, the second term is 48 - 15% less than the magnitude of the first, and
the third term is 6.1 - 0, 7% less. Thus, in the first rough approximation:
, (13.14)
However, if the range is limited to f = 1 - 10 Khz a finer approximation
(± 15%) can be achieved by the following formula:
(dn/da) = 2/3 ng/R. (13.14')
a—xv
which will be used thereafter.
The diffusional particle flow onto 1 cm2 of cylinder surface per sec-
ond (considering that the mean diffusion time is one half of the total sonica-
tion time) is equal to: . i _
The total flow of particles onto the cylinder surface, if equations (1.32) and
(9-1), are taken into account, equals:
~ ' *-ir-^T-r-— (13.16)
* */^ */„ _»;• i / * * '
p/«
The relationship of this flow to the flow onto a sphere determined by formula
(13.8), taking (10.20') into account, is as follows:
'JT^ ' '»
Usually Aia /rs > 50 and, consequently, N'/N > 5. However, the absolute
value of the flow of small particles diffusing into the aggregate volume of the
large particle is small. Thus, under optimal conditions, when f is defined by
formula (10. 5), (ils by formula (10.10), and e by formula (10. 20'), for a par-
ticle of radius rA = -.3.5p, at normal sound intensity (J = 0.1 w/cma), the folio-w-
ing is obtained:
at r8n =0.2(fopt =5.4KHz; nia =0.92; emln = 1.41) N' = 0.8;
at rfln =0.5(fopt =2. 2 KHz; p,18 =0.60; emln =1.33) N1 = 0.3;
Such a flow ensures only a partial filling of the aggregate volume by
small particles during each diffusion event, since the average particle con-
centration inside a given volume is always considerably lower than the aver-
age particle concentration in the surrounding volume. Complete filling of the
aggregate volume by the small particles is achieved at a much greater diffus-
ion flow value, namely:
OTS ^1U> ** J''' Ml 1«\
(13.18)
- 115 -
-------
According to this formula
at ran = 0.2 N1 = 9- 8, at ran = 0. 5 N1 = 2.4,
which is about one order of magnitude above the corresponding values of N1.
However, lit is evident from equation (13.16) that the diffusional
particle flow increased rapidly with an increase in the radius of the empty-
ing cylinder. Determine what radius Rdi fV will ensure complete filling of
the cylinder volume by the small particles through diffusion. Equating ex-
pressions (13.16) and (13. 18), get:
W 03.19.
.
Under the previous conditions, get the following:
at r8n =0.2 Rdiff lr\ = 0. 81 or Rdlf f /(rj. + ra ) = 0. 68
at raa = 0.5 Rdlf;f /rx = 1. 57 or R4lff /(T! + ra) = 1.05 •
i.e., usually Rd j f f <_ rx + rs .
In these computations, the Brownian diffusion of particles was completely
disregarded; it must be considered only when r8 < 0.1.
The orthokinetic mechanism. Particles, the distance between which
is less than the internal turbulence amplitude (1 < IQ ) , are being flowed around
by each pulsation, and if these particles are not equidimensional, a difference
in velocity develops between them, and they converge on each other, and di-
verge in a reverse motion. Taking (7.11) into consideration, the relative ve-
locity of two particles with rj and rg equals:
.(r!-r:) (13.20)
Since the particles are also simultaneously entrained into the oscillatory
motion of the medium in different degrees, and the frequency of this motion
is much higher than the pulsation frequency, particle convergence occurs
sinus oidally, as shown in Fig. 42b.
The sinusoid "pace" is usually much smaller than the radius of a
small particle: (v18:/4f) « ra . Therefore, it can be assumed that all small
particles located in the circular parallelepiped of height 2A1B .length via sin0 ,
where 8 - the angle between the oscillation directions and the investigated
pulse, (sin 0)B = 2 AT), and thickness 2 e^Bln T± can collide with the large
particle in one second. Accordingly, the flow of small particles precipitat-
ing on the large particle under optimal conditions should be equal to
out
- 116 -
-------
In an air medium the following prevailed:
The ratio of this flow to the flow onto a sphere in a normal turbulent field
equals:
Alt
— . " i ;j. 22 >
rt
usually Aj, 3 /TJ and, consequently, N"/N^20.
The absolute value of flow of small particles entering the aggregate
volume of a large particle orthokinetically is also small. Thus, for a partic-
le of radius TJ = 3. 5(j, at normal sound intensity (in accordance with (13.211)),
the following is true:
at ran = 0.2 N" = 1.5,
at ra,i = 0.5 N" = 0.2.
However, the presence of an orthokinetic flow markedly decreased radius
Rdt 1 1 . at which a. diffusion particle flow, sufficient to completely fill an
aggregate volume of radius Ra in the course of each diffusion event, is en-
sured.
At ran = 0.2 which corresponds td Rm = 0.24 (r^ + ra), Rdlff ~
0.40(r1 +rj,);atran =0.5, which corresponds to RB =0.58(r! + ra), Rdlff
~ 0. 83 (rx + ra).
In both cases R^i tt /Ra . Thus, it follows that at normal sound intensities
there existed a thin "barrier" of thickness A R c^_ Rdt 1 1 ~ **•» on *^e path of
the particle to the aggregate volume; most of the particles cannot overcome
this barrier by melans of pulsation. This particle "barrier" is overcome by
the phenomenon of ;particle self-centering examined in #11. At an elevated
noise level, the need for this phenomenon actually vanishes, since, owing to
the tendency toward an increase in orthokinetic flow [see formula (13.21)],
the condition of Rdiff < R» is achieved.
- 117 -
-------
C HAPTER
MECHANISM AND GENERAL PRINCIPLES OF
ACOUSTICAL AEROSOL COAGULATION
14. THE COURSE FOLLOWED BY ACOUSTICAL AEROSOL
COAGULATION .
It is apparent from the preceding chapters that the. behavior of
aerosol particles in a sound field differed essentially from their behavior in
an unperturbed medium, or in a turbulent stream. Because of that the pro-
cess of acoustic aerosol coagulation proceeded in an completely specific
manner. This can.be noted by simple macro-ocular observation of the pro-
cess. When a sound source is switched on, the aerosol at first acquires a
somewhat "tense" state, which is difficult to describe verbally, but is easily
comprehended if it is remembered that at this instant the aerosol particles
take on a vibrating motion, and begin to drift, and move forward under the
action of an acoustical -wind. A peculiar aerosol "orientation" occurred.
Naturally, the behavior details of individual particles are not clear, since
the initial particles, due to their minuteness, could not be seen with the nak-
ed eye, since the human eye perceives separate objects only when their sizes
are of the order of lOOp,. Subsequently, the aerosol becomes "bleached", and
particles become visible, executing a random displacement, drifting, and cir-
culating with the medium. Liquid particles form droplets, and dry hard
particles form cotton-like and filiform structures which conglomerated with
greater tightness than those in natural, Brownian, coagulation. The latter
is illustrated by Fig. 43, which shows a carbon black aerosol deposited on a
glass slide before and after sonication [ 272] .
In a standing wave, local phenomena begin to appear at the instant of
"bleaching". The "bleaching" proceeds in regularly alternating lateral bands,
which gradually widen and encompass progressively larger areas. Particle
aggregates emerging • at that time accumulated and formed transverse "discs"
or, more exactly, "rings", and sometim.es longitudinal "sleeves" [277] at
the antinodes. From there they speed away in the direction of vibration nodes
near the walls of a vessel and settle in the shape of transverse ridges on the
vessel's perimeter. The higher the vibration frequency, the smaller is the
distance between the adjacent ridges. Occasionally, intermediate ridges can
be distinguished on the walls which corresponded to harmonics of a sound
wave. In a traveling wave, where the local phenomena are absent, the formed
particle aggregates settle by gravity on the bottom of the sonication chamber
and also partly on its walls. After the settling of particle aggregates, the
sonication chamber looks practically aerosol free.
. - 118 -
-------
---- +--_.
- ------ ---
----_._---+--- ---
- - - - -_. -
Fiji. 163.
t.. .oo~ .."0801 b.fo,.. 8Onication (.) ....~ .ft..,. aonicaUon
in the COY".. of ~ .ec. \b) .100
Such is the typical picture of highly dispersed aerosol coagulation,
such as, ammonium chloride, - magnesium oxide, titanium tetrachloride,
carbon black, and others, which were sonicated in a stationary state at mo-
derate sound intensities. Figure 44 shows photographs of four consecutive
coagulation stages of ammonium chloride smoke, in the field of a lO-khz
standing wave [179]. The- first photograph shows an aerosol before sonica-
tion, the following two show aerosol during sonication, and the last one -
shows coagulated and p:r:edpitated aerosoi after sonication has be~n discon-
nected.
The coagulation pr"ocess and subsequent precipitation of coagulated
particles at moderate sound intensities "lasts several second decades. At
very high sound intensities coagulatio.it is accomplished in several' seconds j
however, large flakes fail to form due to the instability of sound fields. In-
stead, whirling, fog-like clusters of smoke particles emerge at definite in-
tervals along the chamber, and ridges* appear and disappear at the walls.
When an aerosol is pumped through the sonication chamber, 1. e., when it is
in a flowin~ state, a spatial, rather than temporal, picture of aerosol "bleach-
ing", and of the formation and random motion of particle aggregates is un-
folded, i, e., along .the course of .aerosol' s travel through the sonication
chamber. The original aerosol can be seen at the inlet to the sonication
chamber, the coagulated "bleachedll aerosol, can be seen at the outlet.
Moreover, all the described local phenomena are absent not only in a travel-
in~ wave, but also in a standing wave. This can be explained, in the first
place by the short time it takes the aerosol to pass through the node-antinode
region, and secondly, by the fact that the particle drift velocity vector and the
* Ridges can be easily "washed off" from. the walls by changing frequency of
vibrations.
- 119 -
. .
-------
Significant possibilities are offered in this re-
spect by motion-picture filming of the proces s. This
kind of investigation had been conducted by Brandt,
Freund and Hiedemann [9, 170J. The speed of filming
was 25 frames per second. Figure 45 shows eight con-
secutive microphotographs of acoustical tobacco smoke
coagulation, selected by the authors from a series of
photographs. The first picture (a) represents an aerosol
before sonication. The second picture (b) was taken im-
mediately after switching on the sound source - a magnet-
ostrictive emitter, excited by a tube-type lO-khz oscil-
lator i it can be seen that the particles just started their
vibrating motion. The next four frames (c, d, e, and f)
represent consecutive stages of acoustical aerosol co-
agulation; it can be seen distinctly that the particles
participate in the circulating motion of the medium, and
furthermore, execute some other undefined motion, in-
dicated by the smudginess of the picture. The particles
became progressively larger from picture to picture and
scanned the increasingt"y greater areas of the field of
vision. The last two pictures (g and h) were taken after
the sound source was disconnected, and reflect the precipitation process of the
coagulated particles under the effect of the gravitational force.
~
..fl; '~...:'j
~V" , ';;.
,~ ',~
:- :,.::..." -.:
r ...1
~.~.-::
Fig. Itlt. 1....oniU8 eho-'.'
ride aarosol at different
eoawuletion stages in e'
litat Ie sQAlnd .a.,.
(f . 10khz)
~I
acoustical flow vector alternately changed their signs
during the transfer from one node-antinode region to a-
nother. A more detailed picture of the behavior of the
particles can be secured by observing an aerosol coagu-
lating in a sound field, through an ultramicroscope. It
will be clearly seen that the particles participated in the
vibrating motion of the medium, although this phenomen-
on is also hard to observe, since the particles rapidly
pa.ss through the field of vision. This difficulty can be
somewhat avoided by observing the aerosol near the an-
tinode, where the particle drift velocity approa.ched zero.
However, even then it is not possible to directly observe
the elementary act of particle convergence and' aggrega-
tion, due to the extreme speed and complexity of particle
motion in a sound field; which the human eye is unable to
follow.
Based on':results of conducted investigations the authors divided the
process of acousti!=al aerosol coagulation into two stages. In the first stage
the particles vibrated under the effect of sound and participated in tre general
circulation of a gas, increasing in size through colliding with each other. In
the second stage, when the Farticles became enlarged so that their vibrations
amplitudes decreased to small fractions of th.e gas vibrations I amplitudes, the
- 120 -
-------
It can be concluded from the above mentioned infor-
mation, that if the aerosols were monodispersed, sonication
could have yielded insignificant effects. Actually, in this case
only two forms of interaction occurred: attractional and dif-
fusional. However, the velocity of attractional interaction in
the immediate vicinity of a particle (s = 2r) e'quals zero and,
consequently, the attractional interaction cannot be the cause
of contact, and hence, the coagulation of particles. On the
other hand, the ve-
locity of diffusional
particles inter-
action of the same
size in a sound field
is of the same order
as the velocity of
diffusional interaction
of partides.in an ~rdinary turbulent flow, 1. e., negligible.
( .
.,
,
r.. -J
r,~ . . ;.
. '\
. "" ....
, ,).~
. .
~ '.' '
~.. ' .'
. .
a
..... . .~
. .
. .
..
,
J
t '
I .
1 ~
. .
, ;
':J
I .
. ~ ,~,
, ~"
, .
. I
lit
particles suddenly ceased to vibrate and begain to move along
irregular, often sharply zig-zag trajectories, in the case of
spherical particles, or spiral trajectories, in the case of par-
ticles with irregular shapes. At this stage, aerosol coagulation
pr~ceeded due to the convergence of non-vibrating particles,
and also due to collisions between the non-vibrating and still
vibrating particles.
- ~ -.. -.
'the author s believe, that their inve sUg ations enabled
them to fix the elementary aerosol coagulation processes in a
sound field. It is difficult to agree with them. b('cause in ('ve-
ry frame there were about 400 vibration' cycles which. natur-
ally, prevented them from studying trajectories of interacting
particles*. Nevertheless, the course of the process of acousti-
c~ coagulation of aerosols can be reproduced with a sufficient
degree of accuracy, by basing it on the information of element-
ary interaction processes between aerosol particles and a
sound field, obtained in the preceding chapter.
Fig. ~~. 'MicrophotograPh. of .ucce..ive .cou8tic.I
..ole. coaguldiOll atag.. (..po.....e ti.. 0.04 ..c)
a - before sonication. b - at the ti.. of aound
a.itching in. c - d - . - f - aucc...ive .t."e.
of increa.. in p8rttcl.a. II - h - fal aing out of
incr...ing particl.. wh.n .onicatiOll ia di8con-
t inu.d
Meanwhile, experience indicated that aerosol particle coagulation pro-
ceeded with greater speed in a sound field than in a turbulent flow, at the same
values of turbulent characteristics, which indicated that the intensifying effect
of sou!ld on the process of aerosol coagulation was condit loned largely by the
fact that 'all aerosols were polydispersed and, consequently, there exi. :;t~o
-~l,'~:~';;rtly.- an attempt has been made at the Acoustic::;) Im~'tjl.....~<,: tJ~ ~,,.<,: f.' ,,':rll'j
of S(.'iences; USSR: ,to obtain high-speed moving picturf'::5 of ~(:oudi.;;,j ""~%";'1
tion of aerosols with speeds up to 5,000 frames per sf'::cond. How1~vl:r, t.f.~ !:r:d,
experiments succe-eded only in photographing the formation of spatial TJi2.rtid-=:
groups. This was 'already mentioned in #11.
, , - 121 -
~
"
"
-------
differences in the size of their particles, and also, differences in the velocity
of vibrating motion. The following should be pointed out: regardless of minute
differences in the size of two dissimilar particles, a difference in the velocities
of their vibrating motions (if the particles are sufficiently entrained into the
vibrating motion of the medium) developed during sonication. The following
example will serve as an illustration: The most isodispersed aerosol, tested
as an object of the Coagulating sound effect, appears to be, the highly-dispersed
vapor of dioctyl phthalate described in [94, 95]; the radius of its droplets
ranged from 0.16 to 0.48|j., the mean radius being 0.28^. The vibration fre-
quency was 13 khz. Under these conditions, the degree of particle entrain-
ment was very high amounting to \ipi - 1.0, p,p3 = 0. 997, and \^ - 0. 975, for
the smallest, medium, and the largest sizes, respectively. The arithmetical
difference in the vibrations amplitudes of these particles did not exceed |J.13 =
2.5%. For this reason, the author of [93] maintained, that there was practic-
ally no difference in the velocities of particle motion, and, consequently, no
orthokinetic particle interaction. Meanwhile, calculation of particles, rela-
tive degree, according to formula (10. 3), 'which took into account the difference
in the phase displacement of particle vibrations, yielded different results:
M.IS =7.7%, p,33 =15%, and M.J.3 = 22. 4% of the vibrational velocity of gas. These
values are much greater, particularly, considering the fact, that the selected
vibration frequency was far off from the optimal frequency values (296, 99 and
17kHz) determined by formula (10.10).
All forms of the aerosol particle interaction in a sound field occurred
in polydispersed aerosols, each playing a specific role. This can be under-
stood better if it is noted that in order to form an aggregate with a radius r =
10 - 20|J, from the initial particles, the latter must travel a distance of 500 -
1000(j,, which follows from formula (9. 5). This is a sizeable distance, and
moreover, it is much larger than the average distance between the initial
particles and the amplitude of the medium's vibrations. The convergence of
particles located at considerable distances from the center of coagulation, can
be accomplished only by the diffusional interaction due to turbulence of the
sonicated medium. However, as the distance between particles decreased,
their diffusional convergence velocity rapidly abated and, when a particle a-
pproached the volume of a large particle, its velocity became negligibly small.
Here, the phenomenon of self-centering of small particles around a large
particle occurred, sharply increasing the approach velocity of small particles
towards the aggregate volume, with the exception of the middle portion of the
aggregate volume, where the particles become first repelled and then drawn
by the diffusion and acoustical flow to the outer regions of the volume. The
self-centering velocity of small particles is such that the particles reach the
aggregate volume -within fractions of a second, especially as this is assisted
by the attractional interaction of particles and the orthokinetic mechanism of
the fluctuating interaction.
When a small particle entered an aggregate volume of large particles,
its motion ends, (in one vibration cycle or less) following in an orthokinetic
- 122 -
-------
impact with a large particle. Evidently, charging of the aggregate volume; was
continuous, and its discharging - periodical.
The described course of acoustical aerosol coagulation as applied to
a bidispersed aerosol model is shown
in Fig. 46. Here, numeral 1 denotes a
large particle, as the coagulation cen-
ter, and the remaining numerals denote
small equivalent particles located at
different individual points of the sur-
rounding space. The latter vibrates
with respect to the large particle of
amplitude A18 . The outline of a large
particle aggregate volume is shown by
a dotted line, and a broken line denotes
an arbitrary outline of the pre-aggre-
gate volume in which the self-centering
phenomenon of the particles occurred,
Rdlff - mean volume radius. Zig-zag
arrows, with index V8e , denote small
particles delivery by self-centering in-
to the aggregate volume; identical ar-
rows with index VB40 denote the phen-
omenon of particles self-decentering.
Straight arrows with index Vat t r signi-
fy attractional delivery of small par-
ticles and the acoustical flow; arrows
with index Vort ho signify delivery of .
particles by the ort.hokinetic fluctuating
interaction mechanism. Arrows with
: index Dturb signify small particles dif-
fusional delivery into the pre-aggregate volume. Such is the course of rapid
aerosol coagulation observed in the optimal frequency range, when the differ-
ence between the velocities of vibrating particles is perceptible*.
Oj«gr««»«tic presentation of the cour»»
of acoustical avrosol c
*Recently, A. I. Gulyaev and V. M. Kuznetsov [31] demonstrated that aerosol
coagulation under the effect of recurrent shock waves, such as single shock
wave,, disclosed by E. Richardson [257], failed to produce any perceptible
effect. Obviously, the aerosol coagulation mechanism in this case was the
same as the one described above. This is witnessed by the fact, that the tur-
bulence pressure amplitude intensity (0. 3 atm), used by the authors, attained
approximately the same values as in a sound field (4-6%) and, consequently,
the effectiveness of coagulation with respect to a purely turbulent mechanism,
described by V. G. Levich [68, 69] , was inadequate to explain the high results
obtained. It is not reasonable to maintain that the presence of oblique shock
waves was a factor of any significance.
- 123 -
-------
If no above-indicated difference exists, and this is unavoidable, in
the sonication of coarsely dispersed and the coagulated finely-dispersed aero-
sols, aggregation of particles follows a different course. In that case, a de-
finite difference is preserved only for the drift velocities of particles which
aggregate with each other due to the orthokinetic mechanism and, at the same
time, precipitate due to their own weight. In a standing wave, local accumula-
tion of particles on the walls at the nodes takes place during the process. This
is conditioned by the fact that a "dead" zone forms at these points between two
opposing acoustical flows and the large particles, which rotate with them, fall
into the indicated zone under the effect of the centrifugal force. Since the par-
ticle drift velocity is negligible in comparison with the vibration velocity, the
described orthokinetic aerosol coagulation velocity is very small. According-
ly, this form of aerosol coagulation in a sound field shall be referred to as the
slow aerosol coagulation. The same term will also be used for the coagulation
of equivalent particles, where the aggregating processes proceeded at a slower
rate.
15. KINETIC EQUATION OF THE PROCESS
Assume the interaction of dissimilar particles as the basic acousti-
cal aerosols coagulation course, and establish that the rate of the process was
a linear function of the particles numerical concentration. This can be illus-
trated by the following example of a bidispersed aerosol.
A bidispersed aerosol model is, indeed, not a complete equivalent
of a polydispersed system; however, it exhibited its main distinguishing
characteristic, namely, different particle size; generally, any aerosol can be
represented as a combination of a similar kind of bidispersed. system. Assum-
ing that, our bidispersed aerosol contains n^, large particles and ngO small
particles per unit volume at the initial moment. Assume also, that ng0 » nlo ,
otherwise any perceptible coagulation of particles was impossible (see #9). In
accordance with what has been said in the preceding paragraph, it has been
assumed that particles of equal sizes did not aggregate with each other, and
that the reduction of numerical concentration during the process of coagulation
has been achieved only due to adherence of small particles to large ones. To
set up a kinetic equation of the acoustical aerosol coagulation, it is enough to
turn to the formal kinetics of homogenous chemical reactions, which postulates
the following [43]: the rate of reaction was proportional to the concentration
product of reacting substances, during which each concentration participated
to a degree, in the simplest cases, equal to the coefficient in front of the for-
mula of a given substance in the reaction equation. .The reaction mechanism
of the substances, as is known, had not been mentioned there. Therefore, it
is possible to extend the quoted rule to the case of the bidispersed aerosol,
under present discussion, which, incidentally, is similar to the molecular
reaction of two gaseous substances. Specific characteristics of the case un-
der discussion, is the fact that concentration of one of the "substances",
- 124 -
-------
namely, the large particles, remained constant during the process: nj = n^ =
const. Taking this into account and also the fact that "coefficients in front of
the formula of each substance" in the case under discussion, equal to 1, the
rate of acoustical aerosol coagulation can be expressed by the following
equation: dn/dt.= . K^n (15-1)
where n - particle concentration; taking into consideration that nj » nj , it
can be assumed that the total number of particles n and the number of small
particles ng were the same; t - sonication time; K, - coefficient of acoustical
aerosol coagulation, which is a function of the physical aerosol characteristics
and of the sound field parameters. Thus, it can be seen that the equation of
the process of acoustical aerosol coagulation agreed formwise with the equa-
tion of monomolecular chemical reaction.
It should be noted, that if the equidimensional particles interaction
were the basis of the acoustical coagulation process, the rate of the process
would not be a linear, but a quadratic function of the flowing particle concen-
tration. The corresponding kinetics equation would then coincide with the
equation of bimolecular chemical reaction, similar to one applicable to the
Brownian aerosol coagulation, [see equation (9. 2)] . This would contradict
all experimental data on the kinetics of the acoustical aerosol coagulation
process, which will be examined next. Before that, equation (15.1) will be
integrated by first assuming that time t = 0 and the numerical concentration
of particles n = n., : „ ,. ,,c i\
c - ^ n = n0 e - Ka t (15.2)
The exponential dependence of numerical particles concentration on the coagu-
lation coefficient and the sonication time established by this equation are in
good agreement with the experimental data on the acoustical aerosols coagu-
lation.
Relationship (15.2) was first demonstrated by Brandt [161] in ex-
periments on the acoustical paraffin oil vapor coagulation, conducted at 10 khz
frequency. The radii of the vapor droplets were in the range of 0. 2 - 1. 9M-
(mainly 0.8 - 0. 9n), and their gravimetric concentration was 15 - 20 g/m^ .
Results of these experiments are shown in Fig. 47. According to the curves,
the following values were obtained for the coagulation coefficient:
at J = 0.0067 w/cm8 K. =0.43,
at J = 0.06 w/cm2 K. =0.92,
at J = 0.11 w/cm2 Ka=1.28. '
In comparing the coefficients of different aerosol coagulations, they
will be henceforth, related to normal sound intensity J = 0.1 w/cm2 ; such
coefficient shall be referred to as "normal" coefficient of aerosol coagulation.
In this case the normal coagulation coefficient was Ka ^_ 1.25.
- 125 -
-------
a i t 3 * f.cf*
Fig. k7 . Functional relation
botoeon rate of accougtical
paraffin snog coagulation and
perticle count (f ~ 10 k-hertz)>
I - at I * 9.0066 bt/ca2j 2 - at
32- 3 - at I = 0.10
I - 0.06 bt/co2
bt/co .
AB a. relationship confirmation (15.2), Inoue
[37] quotes also the results of St. Glair's [179, 181]
experiments conducted -with ammonium chloride
smoke, at a 10 - 20 khz frequency. It appears,
however, that this was done irregularly, because
in the experiments not the size of coagulating par-
ticles, but the change in the dispersed light inten-
sity was controlled by them, which were far from
being equivalent. Relationship (15.2) was demon-
strated in a more conclusive manner by experi-
ments on the acoustical coagulation of dioctylphtha-
late vapor, recently conducted by B. F. Podoshev-
nikov [94, 96] . The dioctylphthalate vapor was
taken from LaMer's aerosol generator. The radius
of vapor droplets was in the 0.16 - 0.48p, range
(mainly 0.28|J.). The gravimetric concentration was
1. 8 - 2.1 g/m3, and the acoustical vibration frequen-
cy, radiated with the St. Clair-type electrodynamic emitter, was -13 khz. The
present author obtained the following experimental relationship between p^-,
(p - sound pressure in kilobars):
{L -a c-o.o»p/.-H>.oooi O*.)1 • (15.3)
»e
The second quadratic term of the exponent is extremely small in comparison
with the first term at normal or even elevated sound intensities (J = 0.1 - 1. 0
w/cm2 ) and at moderate sonication times (t ^15 - 20 sec). Therefore, the
second term can be practically disregarded, and the equation will assume the
form of (15.2). Moreover, the normal coagulation coefficient is Ka = 0.08 p =
0.08 • 6.4 = 0.5.
Changes of the dispersed composition during sonication, are charac-
terized by particle size distribution curves according to their numbers, as
shown in figure 48. It can be seen, that in the coagulation process the maxi-
Jtr _ mum shifted comparatively little, despite
the fact that the large particle portions
grew considerably. The exponential
nature of the equation, which described the
kinetics of acoustical aerosol coagulation
is also supported by the results of many ex-
0,1 0,1 0, S 7 9 r.tt periments at the semi-industrial scale on
*n •*•
Fig. 4b. Changes in conposi t ion of acuousti- the acousto-inertial precipitation of
c.i di.ctyiphth.i.t. ..09 «t «y»0 intent, different industrial aerosols, conducted by
I' • 13 «nj, p - ooftu bar. J
\ Zo^'/H*1 dlstr<""*'»" »f drop sites (w r Oyama, Inoue, Savahata and Okada. De-
1.1*2 a/»Jj 2 - after acoustical exposure for ..,,., . . . ,
t • x.-i aee.t 3 « ditto - t a tails of these experiments are presented
in #20 (see Fig. 68 and table 18).
an
8
.
- ditto t =. |t|.tt o»c.
Acoustic coagulability of cracking
- 126 -
-------
gas vapor was investigated by the authors in great detail. Particle radii of
this aerosol ranged between 0.5 - 5. On, mainly 3.0 - 3.5^.. and the weight
concentration fluctuated in the range of 6 - 15 g/ma . The optimum vibration
frequency was 4 khz. According to the authors' calculations [259] . normal
coefficient of coagulation of a given aerosol in an industrial unit set up by
them was Km =0.35-0.37.
Let us pause briefly on Ka , the physical content of the acoustical
aerosol coagulation coefficient. Formal kinetics, enable us to establish only
that the coefficient was proportional to the numerical concentration (nio) of
large particles, which are the coagulation centers:
K. =Kal nio (15.4)
However, this concept can be supplemented on the basis of the earlier
described behavior process, although it should be noted that due to the motion
complexity of small particles the final analytical solution of the problem still
presents a difficult problem.
It has be6n known that each large particle is a depository of those
little particles which fall into its aggregate volume, outlined in Fig. 46 by a
dotted line. This volume can be roughly visualized as a regular circular cy-
linder of cross section ETT rf, and the height - 2Aia , so that its volume is
Q = ZErr TI A18 . The small particles concentration na , filling the aggregate
volume, is directly proportional to the average concentration of particles in
the surrounding space: n, = |3n , where P - space factor, which thus has re-
mained undetermined analytically ($ < 1) . The total number of small particles
(An), entrained by the large particles during time (At) equals
An = — 2Qapmt, JM = — 23ntort4ilt^UgnM. (15.5)
Whence, the rate of acoustical aerosol coagulation can be expressed by equa-
tion (15.1) if Ka is assumed equal to:
K, = 2epn10 r^18 U, . (15.6)
The foregoing indicates that the acoustical coagulation coefficient of
bidispersed aersols represents a portion of small particles, entrained by all
the large particles in unit time. The process of slow aerosol coagulation,
which depended on different drift velocities, is also determined by equation
(15.1), but in this case the acoustical coagulation coefficient is expressed as
follows: • •** -J2 , -,r IT \ /ic -r\
Ka =Enrf(V1 - Va ) n10 . (15.7)
where Vx and Va - drift velocities of large and small particles, respectively.
It is not difficult to arrive at this equation, assuming that during drift each
large particle entrained small particles in the cylinder, the cross section and
height of which are ETT rf and ( Vj. - Va ), respectively.
- 127 -
-------
16. EFFECT OF AEROSOL AND ACOUSTICAL CHARACTERISTICS ON THE
KINETICS OF THE PROCESS
Examine available experimental data relative to the effect of the
physical characteristics of an aerosol and the parameters of a sound field on
the kinetics of the discussed process, implementing these by the individual
theoretical reasons pertaining to the present problem.
A. ggregate state, density, and structure of aerosol
particles . Solid particles grow somewhat faster than liquid particles ,of
equal dispersion, density, and concentration. This may be due to the fact that
solid particles gathered into more bulky, disorganized friable agglomerations
in the shape of flakes, chains, etc. , than droplets did. The ability of these to
precipitate is higher than that of droplets of equal weight, dispite a sharp drop
in their apparent density. This follows directly from the expression for the
particle relaxation time (4.4), in which the particle sizes are expressed in the
second power, and its density only in the first. However, dry aggregates of
solid particles were less stable in an agitated stream and, consequently, they
may become partially disintegrated upon entering a cyclone, or any other in-
ertial precipitator [206] , where the turbulence intensity is very high (e = 109-
1010 ). This pertains in particular to low-dispersion aerosol particles, which,
according to the theoretical studies of B. V. Deryagin, stick to one another to
a lesser degree than the highly dispersed particles. ,
In the case of high sound intensity where (J > 1. 0 w/cm2), the dry
particle aggregates may break down partially also in the sound field; suscep-
tible to this, however, are only the large conglomerates on the ends of which
there builds up a sufficiently perceptible difference in the magnitude of sound
pressure or turbulent pulsation velocity. However, the above mentioned, had
little effect on the results of subsequent precipitation, since the "fragments"
formed during the, breaking up of an aggregate were still sufficiently large and
precipitated with equal ease. This is confirmed by the fact that with increased
sound intensity, the effectiveness of coagulated aerosols precipitation did not
drop but invariably increased (compare, for example, the results of acousto-
inertial precipitation of carbon black, at the sound intensities of 0.5 and 1.0
w/cma, quoted further in #20).
Apparently, the density of initial particles played no significant role,
although there existed no experimental evidence in support of this fact. Of
course, this can be explained by the fact that a change in density produced
directly opposite phenomena. On the one hand, with the increased density,
there was, according to (11.3), an increase in the rate of inertia! self-center-
ing of particles, due to that, an increase in the coagulation coefficient occurr-
ed; in addition, the optimal frequency value has been displaced toward lower
frequencies at which the sound absorption by the medium was diminished. Oh
• - 128 -
-------
the other hand, increased particle density indicated, according to (9.1), a de-
crease in the numerical particle concentration, which appeared in the kinetic
equation of the process, directly and indirectly, as a multiplier in the expres-
sion for the coagulation coefficient. Likewise, no record existed of any
specific effect of initial particles' shape on the rate of process. Spherical (fog
droplets, carbon black), cubic (magnesium oxide), acircular (zinc oxide), etc.,
particles coagulated in the sound field, with approximately identical ease. It
can be stated in general, that the physical characteristics of aerosol particles
exerted comparatively little effect on the kinetics of the acoustical coagulation
aerosol process.
Dispersion structure and the weight concentration
of aerosol particles. Highly dispersed aerosol coagulated with great-
er intensity, than aerosols of medium or lower dispersion. This follows
directly from the kinetic equation of the process (15.1), examined with regard
to 1) findings on the acoustical coagulation coefficient, presented in the pre-
ceding paragraph, and 2) the relation (9.1) for the numerical particle con-
centrations. The acoustical coagulation coefficient changed comparatively
little with the decrease on particle size, whereas the numerical particle con-
centration sharply increased, and this predetermined the process accelera-
tion. Physically, this can be explained as follows: -with the increased par-
ticle dispersion, the total area of the particle cross sections sharply increas-
ed; experimental evidence of.this is the sharply decreased aerosol transpar-
ence; the volume, "combed" by the vibrating particles, increased sharply.
with it, and, as a result, the probability of particle collisions also increased.
However, to enlarge particles of highly dispersed aerosols to sizes effectively
precipitated by gravitational or inertial methods, an extremely large number
of aggregation events is required. For instance, to enlarge droplets from a
radius ro = O.lp, to r = lOp,, ( 10a7.0.1) = 1,000,000 events are required in accord-
ance with (9. 8). Such an operation necessitated a large sonication time, and,
consequently, a greater power consumption. If the acoustical coagulation
coefficient is known, the required sonication time can be found, with the aid
of (15.2), and the following formulas:
for fogs: ^ = 3/K. In(r/r0) (16.1)
for smokes and dusts (which require correction to decrease the
density of particle aggregates): . .
(!6.2)
It follows from these formulas that an increase in the particle dis-
persion entails a sharp increase in the required sonication time. For example,
enlargement of highly dispersed dioctylphthalate vapor droplets (ro = 2.28p,)
to sizes r = 5 - 6^) which could be trapped in cyclones at a normal sound in-
tensity J = 0.1 w/cma when Km = 0.50, in accordance required t^j c^. 18 - 20 sec
- 129 -
-------
according to (16.1). Sonication time can be reduced by increasing the sound
intensity. Thus, according to (15.6), increase in sound intensity to J = lw/cm2
the increased coagulation coefficient to Ka = 1.6, and reduced the sonication
time to t^j — 6 sec. This value is acceptable to a degree for practical purposes;
however, it increased the required energy consumption for the sonication of
an aerosol unit volume more than three times [see formula (20. 8)] . In con-
nection with the above, optimal aerosols for industrial coagulation are of med-
ium size, particularly the ones with particle radii in the order of r = 2 - 4p,.
Aerosols, which contained no particles with radii less than 5-7|j,, coagulated
poorly; this pertains in particular to solid state aerosols characterized by
weak adherence between the particles. The following can serve as examples:
fly ash, mineral dusts, atomized artificial fogs, and also suspended powders
(which are "aggregated" aerosols).
The gravimetric particle concentration also strongly effected the
course of acoustical aerosol coagulation. The greater the weight concentration
the more intensively the aerosol coagulated, as equations (15.1) and (9.1) in-
dicate. This is determined by the fact that at higher concentrations particles
form aggregates from a smaller aerosol volume i.e. , they overcome small-
er distances to the certer of coagulation.
It was established that the process effectiveness
sharply decreased with concentrations below 2-5 g/m3.
In such cases injection of atomized water into a weakly
concentrated aerosol - aerosol spraying - had a good
effect. Moreover, the effectiveness of coagulation and
subsequent particle precipitation sharply increased, as
can be seen in Fig. 49, where the results of acousto-
inertial carbon black aerosol precipitation in the natural
ff S2 nti nf.n*
-------
cated an increase in its viscosity with all the resulting consequences, i. e,. the
rate of acoustical aerosol coagulation drops sharply, (see equation (b) on page
56). In other respects, the process proceeded normally; therefore, increased
temperatures are not detrimental in using this process for purification of hot
gases.
Cases of successful aerosol coagulation at temperatures of 300-500°
C and higher are not new anymore. However, in such cases it is necessary to
make certain that all the emitted acoustical energy was delivered to the aerosol,
since strong sound waves reflection and dispersion occurred at the mixing in-
terface between the cold air generated by a siren and hot aerosol. Increase in
medium pressure indicated an increase in its density, and, similarly in its
acoustical resistance, decreased the vibrational medium velocity according to
equation (1. 9), •which, as will be shown later was directly proportional to the
acoustical coagulation coefficient. The value of the latter decreased sharply,
and the process rate slowed down with it. To maintain the coagulation coef-
ficient at its former level, it is necessary to increase the sound intensity
approximately by as many times as the pressure medium differed from the
normal pressure. An experimental method of acoustical aerosol coagulation
was tested at pressures of the order of 10 [86] and 50 [80] gage atmospheres.
The viscosity, temperature, and gaseous medium pressure are approximate-
ly identical in the majority of aerosols, namely, close to atmospheric condi-
tions; therefore, these characteristics, as a rule, did not determine the
observed difference in the rate of acoustical aerosol coagulation.
Frequency of vibrations. It has been established experi-
mentally that the vibration frequency had a highly significant effect on the a-
coustical coagulation process. This can be corroborated in the laboratory
and also by large-scale acoustical method tests of fog dispersion, and the
acousto-inertial method of industrial dust, smoke, and vapor precipitation,
described in the next chapter.
Figure 50 shows sound level frequency
dependence required for a 50% "bleaching" of the
artificial water vapor. This dependence has
been established by laboratory tests conducted
by Horsley and Seavey at the Ultrasonics Cor-
poration of USA [203] . Fig. 51 illustrates fre-
: quency dependence of degree of acousto-inertial
precipitation of some industrial aerosols, ob-
tained by Inoue and Oyama, and by Svahata and
Okada[38].
1MB '3000
Frequency Hz.
jcaa
, fij. 50. SMind to tensity. l***l. re-
quired to effect 50> clarification
of artificial eater vapor fog at
different sound effect intervals
and »•»• frequency.
The above graphs indicated that there
existed an optimal vibration frequency for every
aerosol at which acoustical coagulation proceeded most efficiently. Even a
- 131 -
-------
-•3*
C
f "
i.
•+*
c
e
». so
O
•**
! IB
»
a.
SO
*
}
/
/
1^"
\
~s
X
\
^.
•**-.
\2
\
' — ,
-
<
iiz.stsstss
KHs frequency
comparatively small frequency deviation from its
optimum sharply affected the process effectiveness.
Thus, in the first case (fig. 50), a 2-khz frequency
shifted from its optimal value (3.0 - 3.5 khz) nec-
essitated a 4-db increase in the sound level, or the
doubling of the sqnication time. In the second case
(fig. 51), a similar frequency shift entailed a_l% -2-
fold increase in the residual concentration. -It can
be concluded from Fig. 50 that optimal frequency
was independent of sound intensity and sonication
time and, consequently, it was a function charac-
eetmnn teristic of the aerosol alone. Moreover, it should
be pointed out that the gravimetric concentration of
aerosol particles likewise had no effect on the mag-
nitude of optimal vibration frequency. This was
shown by Kawamura f210] , who investigated the
effect of acoustical characteristics on the acoustic-
al coagulation process in tobacco and ammonium
chloride smokes. Unfortunately, the frequency dependence of the process was
determined in this investigation in degree of light dispersion by the aerosol
particles, whereas the proof of the theoretical assumption required direct
knowledge of coagulation rate dn/dt.
. Table 13
Table 13 shows
composited data of the
optimal frequency val-
ues of acoustical co-
agulation of different
of dif-
ent industrial sorosolo-
I - Coke gas tarj (r - 0.5 - 5.9 |jj
2 - Aggregated carbon block (r a
0.5 - 15 M>)
3 — Dilute sulfuric acid fog (r r
2.5 - 50 M,)
Aerosol
Particle
rad i i *r, |J,
Optimal os-
ci 1 lating i
frequency **
f opt. kHz.
Author
I. Laboratory investigations
Tobacco sooke
ABiaon. chloride vapor
Artificial atter vapor fog.
0,2+2,5
0,8
0,3--2,6
1,2
—
2,0
7—8
5—6
5—6
3-3,5
iKanaaura [210]
H. P. Tmerskoy
H15I
aerosols. Examination
of these indicated that
a lowering of the
aerosol size lowered
Gas furnace soot
Aggregated soot
Zinc oxide vapor
Coke ga» fog
Cracking gaa fog
Dilute sulfuric acid vapor
II. Plant investigation*
0,03-0,07
0,5-15,0
0,5^-5,0
2,5
0,5-1-5,0
2,5
0,5-^5.0
3,0
2,5-50
3,5-4,0
3,0
3,0—3,5
3,5—4,0
3,5-*-4,0
1,0—2,0
rtt*
tndicaUs (i»ttir»y radius
prM>if»
Kohrs.i and Svvp the optimal frequency
vibrations. This con-
clusion has something
in common with the
conclusion related to
the degree of aerosol
particle entrainment
into the vibrating mo-
tion of the medium, on
the basis of which
Hueter and Bolt [205]
assumed that the fol-
1156J
Stokes [272]
Inoe et aj_
[38. 246]
••witor
magnitude ol' viL>ria.u>ii
- 132 -
-------
frequency in the atmospheric medium: (16.2) where rm - predominant particle
radius in \j>. This equation was derived by the authors from the generalization
which expressed the degree of particle entrainment (5.19), assuming that \if ~
0. 50, without any valid foundation. To verify this assertion examine the data
on degree of particle entrainment and degree of the streamline particle flow
into the vibrating motion of a gas for different aerosols coagulated in the
sound field. "
Table l"»
Table 14 presents data
on degree of entrain-
ment and of stream-
line flow of natural
arid artificial fog drop-
lets. These were used
as objects for the a-
coustical dispersion
method, examined in
#18. The Table shows
that in the successful
experiments entrain-
ment degree was usual-
ly higher than pj, = 0.65
- 0.85. Boucher [156]
interpreted this as indi-
cating that orthokinetic
interaction was the bas-
is for the process of
acoustical aerosol co-
agulation. Experiments
with the acousto-in-
ertial precipitation of
industrial fogs, exam-
ined in #20 yielded
other values for the
entrainment coefficient and, consequently, of the droplets (Table 15) stream-
line flow coefficient. Thus, the degree of entrainment at optimal vibration
frequency was in these experiments considerably below |j,p =0.50, namely,
lip =0.15 - 0.27. The above indicates that the initial condition, assumed in
the derivation of the proposed formula for the optimal vibration frequency was
arbitrary; therefore, it is not surprising that a considerable discrepancy ex-
isted between the formula and results of the experiments.
A greater discrepancy existed between experimental results and
Wyrzykowski's formula /ept = 7.64/p r2m, derived on the basis of consider-
ably simplified assumptions of acoustical aerosol coagulation [ 295] . Refer-
ence is implied in both cases to deviation from the results of laboratory in-
1
Type of fofl
Artificial static fog i
(vertical flue 0 ISO * 1000 mm)
Artificial moving fog :
(horizontal flue
4,0
4,0
2,0
4,5
7,5
1,0
0,5
4,0
0,5
4,0
6,0
15
5,0
S|S-M
*•—
Succeaa
Failure
SIOK
clearing
Suceeas
K«i lure
Success
Coefficient
U) of en-
trainment PI
85.5
85,5
68,8
73,6
2.8
99.8
8*,7
2.7
98.4
8.7
64,1
1,7
77,0
-" • 1
Coeff icient
U) of en.
ju If sent, p,
52,0
52.0
72.9
67,8
99.9
5,3
52.9
99,9
17,9
99.6
76,7
99.9)
64.0
- 133 -
-------
vestigations of the process. It will be shown later that this was due to the fact
that the optimal vibration frequency for industrial installations was affected not
only by the aerosol dispersion, but also by geometrical factors.
Now, analyze equation (15.6) for the bidispersed acoustical aerosol co-
agulation coefficient: the only frequency-dependent terms of this equation are
degree of the respective particle motion p^ a and, possibly, the space factor p.
If it is assumed that the
latter was remotely con-
Tab!* ii nected with the vibrations
frequency, its optimal
value could be deter-
mined from (10.5).
Aerosol
Sulfuric acid
fog
Crack mg gas
hydrocarbon
fos
Coko gas tar
fog
Pravai 1-
ing drop-
let radi-
us,r, (J,
7,5
3
2,5
Droplet
density
Q/B>3
1,7
1.1
1,2
Gas vi»-
cosity T|
g.ca"'
sec
1,8-10-
1,0.10-'
0,9-10-«
Optimal
osci 1-
1 at ion
fre-
quency t
kHz
1,0
4,0
3,5
. **-
1 — »- 0.
— o « _,
•* > o 3.
» c o wa.
0-"" C
£~» 0 '~
i— o s
15
18
27
i *
«*- *4>
8 —H. c
*^ 3 tf
0 *»• '*
t- o a
99,0
98,0
96,5
/ onr ~ 7 ~\ '*"*>'
derived from the condi-
tion of maximal value
10,! 2 . For air as the
medium, this formula
has the following form:
U^-1—^. (16.3')
However, the practical application of equation (16.3) encounters dif-
ficulties associated, first, -with the fact that real aerosols are polydispersed.
Even in cases in which particle size distribution curve was'available, it be-
comes difficult to decide which figures are to be substituted in the formula for
ra and rs pr, except most appropriate ra < rB , and rx < rmlli . Moreover, it
must be remembered that particle dimensions increased during coagulation
and this, in accordance with (16.3), means a continuous decrease of the opti-
mal vibration frequency. Therefore, the value of optimal vibration frequency
depended not only on the initial, but also on the final particle dimensions. The
experimentally observed optimal vibration frequency corresponded to a near
average value between the instantaneous values at the beginning and the end of
the process. In addition to the foregoing, it should be mentioned that in the
case of industrial installations, geometrical factors, such as coagulation
chamber length, begin to exert serious effects on the optimal vibration fre-
quency. This becomes evident during the coagulation of highly dispersed
aerosols, particularly, of carbon black. The theoretical value of the optimal
vibration frequency at the initial stage of particle enlargement, derived from
formula (16.3), can be as high as tens and even hundreds of khz. However,
such high-frequency vibrations attenuate in gases -with distance from their
origin rapidly, so that" an aerosol flowing through the chamber remained un-
sonicated most of the time. Therefore, in practice, low-frequency vibrations,
of the order of 3 - 4 khz and lower, which attenuated much slower proved the
most effective. Coagulation of the original particles which required much
- 134...
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higher frequencies, was accomplished by harmonics, generated during the
propagation of a sound wave, and partially present in the emission spectrum
of the sound sources used. This opinion was confirmed by the intensifying
effect of high-frequencies vibrations "addition" to the coagulator, noted by
Boucher [150, 151] and other investigators. Geometrical factors are in
strong evidence also in tests of acoustical dispersion of natural fogs, where
the sonicated zone is especially large. Here, the value of optimal vibration
frequency as a rule, is even lower than that in industrial coagulators. It is
difficult to consolidate all above factors into a single function. Therefore,
so far, the only reliable and available method of determining the optimal
vibration frequency is direct experimentation with natural aerosols under
working conditions. It is maintained in (33) that the rate of acoustical aero-
sol coagulation was generally independent of the applied frequency. There-
fore, it is recommended, that the process be carried out at the lowest, and,
consequently, at slow attenuating vibrations [89], However, this is a mis-
understanding, originating from the fact that an atomized powder of zinc
white, in the place of zinc oxide aerosols was used in the tests. Aerosols,
prepared in this way, contained adhesive, coagulated aggregates of original
particles and, due to their coarse size, the aggregation of particles occurr-
ed in them in the form of slow coagulation (see #14), i. e. , by means of
orthokinetic collisions between unequal particles, dependent upon the differ-
ence in their drift velocity. This is confirmed by the function pa1^ = const.
derived by the authors. For any type of drift, the particle velocity V ~ p2
(see table 5 in #6); whence,
(Vx - Va) 1^, = Lla = const (Lia - length
of path of large particles with respect to small particles).
Sound intensity. It has been established that the acoustical
coagulation coefficient was a power function of sound intensity. At moder-
ate sound intensity and sonication time, the following relationship holds:
K.
1/3
(16.4)
Inoue [37] was the first to stress this important relationship, which he
based on the experimental data obtained by Brandt and Freund [161, 163],
and Neumann and Norton [239].
By taking into consideration relations (1.9) and (1.10), the above
relationship can be written thus: ,
K* ~P, ~Ug (16.4')
Fig. 52 illustrates the functional dependence
of the numerical concentration of tobacco smoke
particles on the product Agt^, which is proportional
to J1/2t. and U.tj . The data were taken from the
9to'
** experiments of Brandt and Freund. Fig. 53 shows
.-.,. 52. Functional relation be-x**16 functional relationship between the degree of
t.*,*, expression A3ffl (* - W "*'>titanium chloride particles on consolidation and
derived in the experiments of Neumann and
j_
<
^
/
f
A -^
/
/
0
/
r
i m si tza
Afti,m»ct*
..„ w _
and size increase of tobacco stoke
particles
- 135 -
-------
Norton. Since the graphs were plotted on the logarithmic scale, the relation-
ship between the above quantities is linear.
Existence of the above functional relationships is
further confirmed by the results of tests conducted
by B. F. Podoshevnikov with acoustical dioctylph-
thalate, and presented earlier as formula (15.4).
In accordance with (15.4), the coefficient of acous-
tic coagulation increased progressively slower
with an increase in the sonication exposure pt,., a-
bove the 400 - 500 kiloparsec level. Moreover,
it is known that sound absorption by the medium
increased rapidly at the same time, (see #1). In
view of this, application of very high sound intensi-
ties to industrial aerosol coagulation is economic-
ally unsound. The most frequent used sound inten-
sities are of the order of 0.1 w/cm2., and the most
frequently used sound pressure is of the order of
6400 bar.
Fig. S3- Functional relotion be-
toeen growth rate of titaniuo ti—
taniuo teto-achlor i do particles and
oppression
Structure of the sound field. The consolidation of aero-
sol particles proceeded faster in a standing wave field than in a traveling wave
field. This may be due to the fact, that the vibrating motion amplitude was
higher in a standing than in a traveling wave for the same output of the sound
generator, and that a certain, even insignificant (see #17), increase in the
numerical particle concentration took place. In addition, though not required
in practice, considerable particle consolidation can be achieved in a standing
wave field. The foregoing refers, naturally, only to the case of the acoustic-
al coagulation of motionless aerosols, since (see #14), no local phenomena
occurred otherwise. It can be asserted that where the sonication chamber
•was sufficiently long, better results -were achieved with an aerosol moving
against the sound source than away from it. This follows from the fact that
aerosols are initially subjected to the effect of harmonics of the sound wave,
in reversed motion, i.e. , at higher frequencies. No experimental confir-
mation of this has been presented thus far. An attempt to solve this problem
in [94] failed to produce differences in the results of "forward" and "reverse"
aerosol sonication; this may be explained by the small length of the sonication
chamber.
Geometrical dimensions of the sonication chamber, and of the sound
field affected the rate of the acoustical aerosol coagulation process, not only
when they had been reversed. In fact, the longer was the sonication chamber,
the more harmonics were created in the sound wave, and the stronger was the
attenuation of sound entering the chamber; consequently, the smaller was the
average sound intensity inside the chamber with respect to its nominal inten-
sity related, as a rule, to the exit cross section of the sound source. In
accordance with (16.4), this means that under invariant conditions, the acous-
tical coagulation coefficient decreased with an increase in the length of the
- 136 -
-------
sonication chamber. Therefore, the findings of the acoustical aerosol coagu-
lation coefficient, quoted in #15, should be examined from this point of view.
It follow from the experiments of B. F. Podoshevnikov and other [ 30] that the
coagulation coefficient depended also on the cross section of the sonication
chamber. In those experiments, longitudinal partitions (d = 1-2 cm) were
placed in the sonication chamber and some slowing down of the process rate
had been observed (see fig. 57). However, it is not certain that this situation
might exist in sonication chambers of large cross sections.
17. SOME EARLIER HYPOTHESES OF ACOUSTICAL AEROSOL
COAGULATION
A general review of the acoustical aerosol coagulation presented in
#3, showed that earlier attempts had been made to define the process itself
or its mathematical theory. One need not dwell on the first two attempts to
portray the process of acoustical aerosol coagulation, undertaken by Andrade
[2] and A. B. Severnyy [109]; it was shown in #12, that the authors' theoreti-
cal calculations were based on fictitious effects and, therefore, their concepts
were completely unsound. It shall be added only, that if the basis of the a-
coustical coagulation process rested on some attractional forces which requir-
ed no difference in the particle dimensions, as assumed by individual present
day investigators, the rate of the process would be in the nature of a quadratic
function of the numerical particle concentration, which is contrary to experi-
mental results.
The third attempt to portray the process was made by Brandt.
Freund, and Hiedemann [167], and was based on correct premises; unfortun-
ately the level of theortical knowledge of aerosol particles behavior in a
sound field, available at the time -was inadequate to permit the authors to bring
their work to the point of finality.
As the base of their scheme, shown in Fig.
54, the authors used the orthokinetic interaction
of aerosol particles in conjunction with a given
disordered motion which the large particles ex-
ecuted with respect to the small ones. The brok-
en line shows the contours of the aggregating vol-
ume of a large, practically non-vibrating particle;
clearly, the authors represent it as a cylinder
with a radius rx + ra and height 2Ag . The later-
al component of the random motion velocity of a
large particle was designated there by u)B . The
'total aggregating volume was presented as a roun-
ded off parallelepiped of height equal to double
the amplitude of small particle vibrations, and
the length to the product uomt. The authors' basic
Fig. 5U. Dia^rea of computed ortho
kinetic interaction between aerosol
particles in a sound field according
Brandt, Freund Hyde»an.
- 137 -
-------
concept, that the acoustical aerosol coagulation process hinged on the inter-
action of unequal particles, is realistic and, as can be seen from #14 , it
was used in constructing the general flow diagram of the process.
The fact that the orthokinetic particle interaction was considered
by the authors in conjunction with the relative particle motion in the lateral
direction deserved still greater approval. In the absence of the lateral
motion of large and small particles ((Bn = o), the aggregating volume would
be limited to a cylinder, as shown by the broken lines at the left of Fig. 54,
•which -would be void of small particles, immediately after the first attempt
at their orthokinetic convergence.
However, the diagram shown in Fig. 54, is a very primitive model
appropriate only for visual demonstration of aggregating capabilities of the
orthokinetic interaction of vibrating particles, taken in conjunction with their
lateral motion with respect to each other. Introduction of a random motion
of large particles was not well-founded theoretically, and any reference to
the visual observation of this type of motion is not very convincing, since no
proof -was offered that this motion -was executed by the large particles alone
and not in association -with the small particles. The elongated enlarged par-
ticles' tracks, seen in photographs (see Fig. 45), and fixed by the increased
gravitational and drift forces, were distorted under the effect of turbulent
pulsations which almost prevented the particles from converging, and were
more apparent for the small particles at close distances.
Some investigators used to associate acoustical aerosol coagulation,
partially or completely, with the particle drift in a standing wave. For in-
stance, St. Clair [179, 181], attached great importance to the fact, that due
to the radiational drift (the remaining forms of drift were unknown at that
time) particle concentrations increased in the direction of an antinode where
the vibrating velocities were higher. Using equation (6. 8) to illustrate his
concepts the author plotted the distribution of particle concentration with
r = Ip, in the node-antinode intervals of a sound wave for different periods of
sonication time at a frequency of 10 khz (see Fig. 55). In addition, the author
showed, that due to the occurrence of harmonics
an increase of the particle concentration actually
greater than shown in Fig. 55 should be achieved.
However, the use of equation (6.8) for the cal-
culation of the finest aerosol particles drift was
as shown in #6, absolutely incorrect. In fact,
the author's statement pertained only to the case
of a standing wave in stationary aerosols. Fur-
thermore, the author's graph showed that even
under these conditions the effect indicated by him
r = I M- p. s I co3) played a small part. According to the graph,
.long sectien UK - pfc according to st. th antinode particle concentration doubled in 21
CI«ir(E 30u org/co''} f a |Q hHs).
sec, (which is intolerably long), and trebled in
-138 -
-------
42 sec, which is not excessively long for the process of highly dispersed
aerosol coagulation. Moreover, if the presence of drift due to the asymme-
try of vibrating motion in a standing wave of reverse direction were taken into
consideration, then the concentration increase will be smaller than the one
shown in the graph.
Jahn [206] attempted to substantiate the advantege of using audio fre-
quencies with the aid of local phenomena as follows: "In a 22 khz ultrasonic
field, nodes, at which the dust agglomerated were 0. 87 cm apart. In a soni-
cation chamber with a volume of 1 m3 and a height of 1 m there were 128 nodes
•with a total surface area of 128 m2, over which the dust collected and coagu-
lated. If the aerosol volumetric flow velocity through the chamber with a con-
centration of 8 g/m2 was approximately 3 m3 /sec, then 24 g of dust will be
distributed over a surface of 128 m2 . Under such conditions the resultant
agglomerates will be of small sizes. However, if 1. 5 khz. vibrations were
generated in the same chamber, then the distance between nodes for same
dust concentrations in a gas, is approximately 12 cm. In such a case, 24 g
of dust became distributed over eight nodal planes with a total area of 8 m2 .
As a result, considerably larger dust agglomerates will result." This ex-
planation can not be criticized adversely; since low frequencies were also
more effective in the case of traveling waves, where local phenomena are
absent.
A. D. Bagrinovskii [4] emphasized the orthokinetic interaction of
particles in the drift path due to asymmetric vibration movement of the med-
ium in a standing wave. The soundness of this point of view, can be evidenc-
ed by determing the maximum number of small particles entrained in the
drift path by every large particle. The maximum length of the aggregating
cylinder in a standing wave equals the length of the node - antinode interval
i.e. , X/4. Now, assume that the entrained particles were small so that their
their motion, in the same direction, could be neglected and, consequently,
their average concentration could be considered constant along the entire
drift path of a large particle.
Consider equations (15.1) and (15. 7) for the maximum number of
small particles entrained by every large particle and also equations (10.20)
and (9.1), and derive the following equation:
X (W -T- 2,8) cf A
The minimum time in which a large particle traveled across the
node - antinode interval can be computed from the following formula:
(17-2)
where Vd - resultant drift velocity of a large particle.
- 139 -
-------
The preceding formulas used in conjunction with the graph in Fig. 23,
yields the following values when / = 5000 hz at pp =1 and k = 10 g/m3 :
at r = O.lp, NBax = 1.6 - 1. 9 in tfflln -» »
at r = Ip,' NBaX =0.16 - 0.19 in tnln = 2 min
at r = lOp, NBQ3C = 0. 016 - 0.019 in taln = 6. 7 min
It can be seen that even under more favorable conditions a large
particle fails to entrain a single small particle in a short time. Admittedly,
if it is considered that due to turbulent particle diffusion the entrainment
coefficient E is considerably large than Ealn , the resultant figures are high-
er by approximately one order of magnitude, however, this likewise fails to
offer a satisfactory explanation of the intensive aerosol coagulation in a sound
field.
* * *
Some investigators, particularly the ones who conducted aerosol co-
agulation tests in high-intensity sound fields, favored agitation of the sonicated
gaseous medium. The most emphatic statement on the subject was made by
Matula, who wrote in his dissertation [232]: "There is no doubt that sound
field turbulence played a fundamental role in the aerosol coagulation process. "
Such conclusion was prompted by experimental findings, whereby a rapid co-
agulation of aerosols occurred only after the sound intensity attained a suffi-
ciently high level, when the gaseous medium turbulence offered simultaneous
relief. These findings were also confirmed by Boucher [156], who found
strong similarity in the experimental data on aerosol coagulation in turbulent
and acoustical fields, conducted earlier by E. Richardson [257]. Richard-
son's experiments were conducted in a 1-m3 chamber, filled with ammonium
chloride smoke (r ?» l|j,). Turbulence was generated by a rotary fan, and the
sound vibrations by a loudspeaker. The desired coagulation effect was de-
termined from changes in the degree of light beam absorption when passing
through the aerosol. Experimental results for the turbulent field are shown
in Fig. 56a, where the relative change in the degree of absorption is present-
ed as a function of time for different rates of gas pulsations (smoke concen-
tration by weight - 0.25 g/m3). Similar results are shown in Fig. 56b, for
sound fields of different vibration frequencies (smoke concentration by weight
- 0.50 g/m3, n = 9.2 • 104).
An even greater similarity in the results of the aerosol coagulation
in the turbulent and sound fields was observed in the NIIOGAZ experiments,
conducted for the purpose of investigating the dependence of aerosol coagula-
tion rate in a turbulent gas washer (Venturi scrubber) and in an acoustical
coagulator on the Reynolds number. In the first case, Reynolds numbers
were used which represented progressive and oscillatory streams respective-
ly. It was demonstrated that in both cases the coagulation rate was a linear
- 140 -
-------
it a
function of the corresponding Reynolds number,
and, also of the gas stream velocity. However,
the above similarity studies failed to provide
acceptable grounds in support of the assumption
that turbulence of a sonicated medium was the
fundamental mechanism of acoustical aersol co-
agulation. These studies could only confirm the
fact that in both processes the particle aggrega-
tion was initiated by the common orthokinetic
mechanism. The only distinction, according to
Levich[68, 69] and others [122], was that or-
thokinetic particle collisions in a turbulent field
were due to a difference in the degree of particle
entrainment into the turbulent pulsations of the
medium, and, in the sound field, they were due
to the difference in the degree of their entrain-
ment into the vibrating motion of the mediurrii
which, according to E. Richardson, exemplified
a "monochromatic" turbulence of a medium.
ent pulsation velocities M." » Nevertheless, turbulence of the sonicated
i. LL = o» 2, u, =6 c«/«oo; 3, u medium played an important role also in the a-
= 8 era/secj k, H1 - 12 ae/sec; c ' v •
b - in acoustical field (i -free of-coustical aerosol coagulation process, since it
soundj 2 - at f -0.5 hHi «nd M>8 " offered means for the conveyance of remote par-
0.5 CD/sec} 3 - f = 2.0 kH* ano J f
10, » 2 cB/seci >» - f »7.o w> and tides into the aggregating zone. Any artificial
a = 2"° c?'6ec} ~ suppression of turbulence degree of a sonicated
static
-------
Fig. i7. Effaotivoness of acous-
tical dioctylphthalolo fo'j coagu-
lation following tho installation
of longitudinal obstructions,) ac-
cord ing to B. F. Codoohovntkov
si si*
K - Light intensity rottoo in o
fluohoving longi tutunal obetruc-
t ion si
K1 - Ditto without the obstruct tens
tc - 1.3 g/o3 and k « 2.5 g/a3
CHAPTER 5
PRACTICE OF ACOUSTICAL COAGULATION
AND AEROSOLS PRECIPITATION
18. ACOUSTICAL DISPERSION OF NATURAL AND ARTIFICIAL FOGS
It has been known that natural fogs, formed as a result of spontan-
eous damp air cooling near the earth's surface, greatly decreased atmospher-
ic transparency. This hampered, or completely excluded, visual communi-
cation and human orientation, performance of outdoor work, and general
transportation, and particularly air transportation. Despite the use of radar,
even at present, the landing of modern aircraft at airports in foggy weather
is still risky; civil airports are often closed to incoming aircraft. This cir-
cumstance has forced scientists to seek ways of artificially dispersing fogs.
The following methods of artificially dispersing supercooled fogs has been
tested and used to a limited extent:
1) heating of the surface boundary layer up to a temperature at which
the fog droplets evaporated; this so called thermal method of fog dispersion
was conducted, for example, in the FIDO dispersal system with J-33, J-47,
and other burner types;
- 142 -
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2) fog seeding with hygroscopic substances, such as calcium chloride,
aluminum sulf ate, etc. , which absorbed the air moisture and thus created con-
ditions for fog droplets evaporation; this dispersal system, based on Langdon's
proposal, was practiced at the San Diego airport in California, USA;
3) fog seeding with highly supercooled crystalline substances such
as dry ice, propane, etc. , which produced new condensation nuclei for the
moisture, and, subsequently, water crystallization;
4) introduction into the fog of special smokes, such as silver iodide,
lead iodide, etc., which are ready nuclei for the condensation of moisture and
of water crystallization.
The last three methods are different physical -chemical methods of
fog dispersion tested also on clouds [120]; they have been used in several
countries.
Increase in atmospheric transparency can be achieved using the first
two methods by eliminating the air suspended droplets, while on the other
hand, in the remaining two methods, this can be achieved by consolidating*
them with the aid of moisture condensation and crystallization.
Increase in atmospheric transparency by consolidating the fog drop-
lets is believed to be the result of decrease in light scattering in the system.
In natural fogs, the droplet size is usually considerably greater than the wave
length of the optical spectrum (X.. , = 0.40 -- 0.76 p,), and, therefore, light
is scattered here basically according to the law of geometrical optics. In this
case, the coefficient of light scattering is given theoretically by the following
expression [ 16] :
-
From this formula it follows that the greater the weight concentration of the
droplets and the smaller their radii, the greater will be the light scattering
by the fog.
For more thinly dispersed fogs in which the geometrical scattering
yields at first to diffusions! and later so called, Rayleigh scattering, a factor
is introduced into formula (18.1) which depended on the droplet radius and the
radiation wavelength. In the critical case when r« X,. , . the light scatter-
_ _ _ light
'* In view of this, the definition of artificial fog and cloud dispersion methods
for the elimination of fog droplets as given by many authors (see, for example
BSE, v. 30, pg. 314) is inexact. Indeed, artificial fog dispersion is not in-
tended to eliminate the droplets as such, but only to decrease their scattering
ability, which can be achieved both by eliminating as well as consolidating the
droplets .
- 143 -
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ing coefficient is propotional to the product kra /\4., , i.e. , a direct inverse
light
dependence of light scattering on droplet size is observed. The intensity of the
radiative flux passing through the fog diminished with distance just as the in-
tensity of sound, i. e. , according to the following exponential law;
ir^Te-V. 082)
where k.. n - attenuation coefficient equals to k_, . =oc ,. , . P,, , . a» " ,. v.
light light light light light
O1. ,, -- absorption coefficient whose magnitude in fogs can be ignored.)
O
The increase in atmospheric visability during fog dispersion, recor-
ded in experiments photoelectrically, is characterized by a relative decrease
in the light scattering coefficient A <* lt /« et in %, which is equivalent to a
relative increase in the distance at which the former light intensity A X/X-100%
is maintained. Practice has shown that existing fog scattering methods
possessed significant disadvantages which severely limited their use. Thus,
the thermal method required unusually considerable heat which, to a great de-
gree was due to the fact that while heating the atmospheric surface boundary
layer, strong convective flows were generated which drew in masses of cold
air. The method of fog dispersion by seeding with hygroscopic substances re-
quired very large amounts of these substances. Thus, about 25 g of calcium
chloride was required per cubic meter of fog; moreover, overseeding induced
a reverse phenomenon - a decrease in atmospheric transparency. Other fog
dispersion methods also have shortcomings.
Discovery of sound intensifying effect on aerosol coagulation and al-
so on the process of droplet evaporation raised new hope for the solution of fog
dispersion problems. At first, it was proposed to limit efforts to the simple
sonication of natural fogs for the coagulation of droplets, leading, as in the
case of condensation, to their consolidation and, thereby, to a decrease in the
degree of light scattering by the fog. This method was called the acoustical
fog dispersion method. It was first proposed by Amy in 1931, and was regis-
tered under US Patent No. 1980171 in 1934. The author proposed fog disper-
sing by sonicating them with a huge sound generator moving along an airfield
or other area, and according to the author, instantly, coagulating and cooling
the fog droplets entering the sound beam. The sound generator described in
Amy's patent consisted of an assembly of Hartman ultrasonic whistles mounted
on a common panel which turned in all directions like a military searchlight.
This project, however, was not exploited in practice in the USA,
evidently due to the absence of corresponding laboratory work to support the
proposed principles. However, as Brandt [156] recently reported in a private
talk in Germany, experimental tests in natural fog dispersion has been conduct-
ed in 1940 using Hartman ultrasonic whistles with a central core (a sevory? *
•whistle). A noticeable "bleaching" was observed at a radius of about 10 m a-
*Sp. not certain.
- 144 -
-------
round the sound source. In 1943 — 1944 La Mer and Sinclair [156] of the USA,
conducted detailed acoustical method dispersal tests. The authors first con-
ducted laboratory experiments on dispersion of artificial fogs created with an
atomizer at Columbia University. The average droplet radius was 4^, the con-
centration by weight 14 g/ma which is greater than the droplet concentration in
natural fogs (0.2 -- 1 g/m3). The experiments, conducted in a vertical tube of
150 mm diameter and about 1m long with a loudspeaker operating on 0.5 khz
frequency showed that at a mean sound intensity of 150 db, the sonicated volume
was cleared with a standing wave in less than 15 sec. Experiments conducted
in a horizontal tube of 75 mm diameter and a 3. 6 m length using the same
sound source showed that fog sonic at ion in a state of flow also had a positive
effect. Thus, at a flow fate in the order of 0. 5 m/sec, with a corresponding
sonication time of 70 sec. , visibility in the fog rose by 30%. A similar type
of laboratory experiment was subsequently conducted by the U. S. Ultrasonics
Corporation, Taraba in Czechoslovakia, and others.
Experiments conducted by the U.S. Ultrasonics Corporation in 1948
were conducted in a 7400 cm3 rectangular glass vessel equipped at the bottom
with a U-l type dynamic siren operating in the 1 -- 6 khz range. The siren was
separated from the vessel by a thin rubber diaphragm about 25\i thick. The
mean fog droplet radius was Zjj,, and the weight concentration was about 4. 5 g/
m3 . As in all experiments on fog dispersion, control was accomplished by
varying the brightness of a thin light beam passing through the aerosol and
picked up by a photoelectric cell. Experimental results are shown in Fig. 50.
From the figure it is seen that the optimal oscillation frequency was 3. 5 khz;
moreover, this did not account as Boucher indicated [156], for the fact that the
power radiated by the siren fell at high frequencies. To achieve a 50% "bleach-
ing" of the fog, 1 to 20 sec are needed, depending upon the sound intensity
(137 -- 152 db). .
Taraba [156*, 274] conducted experiments on artificial fog disper-
sion in a horizontal tube of 400 mm diameter and a 4. 5 m length, equipped with
a dynamic siren with a maximal power of about 11 kw. Two dust extractors,
connected in parallel, were mounted aft of the tube to pick up the coagulated
droplets. The fog droplet concentration by weight was 5 -- 10 g/m3 and droplet
size was not controlled. The fog was drawn in at the rate of 0. 9 m/sec.
Oscillation frequency varied between 4 -- 17 khz. It was possible to disperse
and to collect up to 99% of the coagulated fog droplets, within a few seconds.
La Mer and Sinclair extended their experiments on artificial fog
dispersion to large spaces, such as a tunnel and at the Columbia University
swimming pool [156, 266]. The tunnel had a 1. 2 x 2.1 cross section and a
length of about 30 m. The fog was produced by a battery of 12 atomizers. The
mean droplet radius was 4. 5\i and the concentration by weight -3 -- 6 g/m3.
The light source was located at the midpoint of-the tunnel at a height of 0. 6 m
*Does not correspond to author [Tr]
- 145 -
-------
from the floor. Observations were conducted from outside the tunnel through
a glass window in a door at one end. The 120-w HOR-type siren operated at a
frequency of 0.6 -- 0.7 khz and was located between the observer and the light
source. With an intensity of sound near the siren in the order of 137 db, in one
case a 75% improvement in visibility was observed, and in another case an im-
provement of 50%. The experiments in the swimming pool of more than 5600
«
m capacity were fruitless. A dynamic siren of the US Federal Electric Com-
pany with a rated power of 2.4 kw was used in the experiments. At a 0. 7 khz
frequency no signs of fog dispersion were noticed, which is explained by the
large fog dispersiveness (rn ca-lp,); all the fog particles at the chosen frequency
were completely entrained into the oscillatory motion of the medium and did
not coagulate with one another (see #16.)
Experiments on a similar scale were conducted by Boucher in France
(Chatillon, 1956) [156]. The dimensions of the experimental chamber were
12. 8 x 14 x 18 m. The sound source was the author's static "multiwhistle" si-
ren system suspended from the chamber ceiling. Two types of fog underwent
sonication; coarse-particle, similar to natural fog (radius r = 2 -- 10(j,, con-
centration 1 -- 2 g/m3), and fine-particle (radius rB = 0.5p,, concentration
1 -- 2 g/m3). The siren operated on two frequencies: 9. 5 and 32 --33 khz.
The coarse-particle fog dispersed well (2 min) at the low frequency (9. 5 khz);
under high frequency sonication (32 --33 khz), no results had been noted even
at a considerably greater power (600 w). On the other hand, the fine-particle
fog dispersed well at the high frequency (32 -- 33 khz) and showed a negative
effect at the low frequency (9. 5 khz). This was in complete agreement with the
conclusions arrived at regarding the dependence of optimal oscillation frequency
on the degree of particle size as described in #16. Under a 1943 US Air Force
contract, La Mer and Sinclair tested the acoustical fog dispersal rrethod under
field conditions [156, 266]. The best known tests were conducted at Lankin
airfield (Cincinnati, Ohio) in March 1943. The natural fog consisted of drop-
lets with a radius of 4 -- 16(J. (mostly 6g,) of 1 g/m3 concentration. The atmos-
phere was still. Four dynamic "Victory" sirens of the Chrysler-Bell Corpor-
ation, each of 35 kw, were used for the fog sonication. These tests are illustra-
ted diagrammatic ally in Fig. 58. The sirens sonicated in an area more than
120 m long, 23 m wide, and about 15 m high. Results of the tests established
that fog qlearing began within 1 min and increased atmospheric visibility from
65 to 130 m.
In December 1943, La Mer, et al tested the acoustical fog dispersion
method over the open sea near Sandburg, California. Fog droplet radius was
5|j,. Eight Chrysler-Bell Corp. sirens of 35 kw each, were used, placed 30 m
apart along 210 m long line at 10 -- 15° of the prevailing wind direction. The
wind velocity reached 5 -- 6 m/sec. Despite unfavorable conditions, a 50% fog
bleaching was achieved. However, it was felt that in windy weather the radi-
ated acoustical power was insufficient, and the method could barely be used.
The U.S. Ultrasonics Corp. repeated the natural fog dispersion tests in 1946,
- 146 -
-------
Ctem
SO*
using powerful U2 dynamic sirens and other types.
Tests were conducted at first, in an open area near
Arcata, and later near Bracey Grove. However, both
attempts were unsuccessful. The fog had very coarse
particles (mean radius 15p>), and at the frequency used
(3. 5 --4 khz), it did not coagulate, since it was not
entrained in the oscillatory motion of the medium.
Recently, fog dispersion experiments had been con-
ducted in Denmark (Ingeni^ren, 1959, 4).
Combined data of all different acoustical fog
dispersion tests are summarized in Table 16. The
general conclusions were as follows:
1). : Coagulation, and resulting "bleaching"
of artificial and natural fogs required some time.
ti« *'t«t"l£*Eu5 C*tM*"Eve.n in small volumes and at high level sound the
of
3Onication time was at best several seconds; the re
quired sonication time was 1 min or more for large
volumes;
2). Appropriate choice of frequencies was of great importance for
the effectiveness of acoustical fog dispersion. Coarse particle fogs required
low sound frequencies, while fine-particle fogs required higher sound frequen-
cies which usually did not exceed 0.5 -- 0.7 khz for natural fogs;
3). Sound damping in open spaces is considered limiting the dis-
tance of effective natural fog dispersion to 25 ~- 30 m in the direction of the
sound beam. Therefore, many sound sirens are required for large space
sonication;
4). The last conclusion indicates that energy consumption in acous
tical fog dispersion was rather high, although it did not exceed the energy con-
sumed by the thermal fog dispersal method.
The unpleasant impact of sirens on the human ear constituted no im-
pediment to the use of the method, due to the fact that beyond the limits of the
dispersion zone the sound rapidly abated to the usual airfield level. However,
many disadvantages of the acoustical fog dispersion method hampered its use
under airport conditions, and attempts had been made to solve the problem by
the process of intensified droplet evaporation (see Appendix 1). .
- 147 -
-------
ah ere tests Bare conducted
LABORATORY
TESTS"
Vertical flue 0
150 x 1000 mo,
Columbia Univ.,, US A
Horizontal flue 0
75 x 3600 ma,
Columbia Univ., USA
Right-anylo chasber
7400 c«3,
Ultrasonic Corp., USA
Horizontal flue 0
400 x 450 ran. No cyclone.
Czechoslovakia
CHAHfaER TESTS"
Tunnel 1.2x2.1x30 a
Columbia Univ., USA
Soianing pool
1 oca 1 1 on ,
Columbia Univ. USA
Experioental chamber
12.6x14x18 a.
Chati 1 lion, France
Ditto
n
n
NATURAL TESTS
Lunken Aeroport ,
Cincinnati, Ohio, USA
Open sea,
Sandberg, Cal if ., USA
Open space near A cat a and
Bace Grove, USA
Open space at Mining Height,
Czechoslovakia
Elbrus High mountain Conplox
Expedition, AN,
USSR Baxan Kenyan
(Sea Paragr. 19)
Average
radius r,
. . r
4
4
2
_
4.5
1
2-10
(predomina-
ting)
0,5
2-10
(predoaina
ting)
0,5
4-16
(predooin*
ting)
5
IS
5
Gravi^
metric
con 90.
14
14
4.5
5-10
3-6
»
1-2
1-2
1-2
1-2
1-2
1
1
1
1
Characteristics of
purified volune
Nonnigrating mass,
static waves
Continuous f lo* at the
rate of 5 en/sec
Nona i grating (Bass,
static waves
Continuous flow at
0.9 n/sec.
a
94 » j continuous flowj
static waves
3
•5600 n ; nonmi grating aass
^
400 • . continuous flow,
static waves
3
100 *T
2600 M3, static mass,
unstaole waves
2600 a3
5100 (Hj quiet ataospherei
running waves
3
100 000 • 1
wind 5-6 n/sec.
Running and stationary waves
Still air} running waves
Rain clouds} running waves.
•All artificial fogs were obtained by Beans of water spray vapor'n.
- 148 -
-------
Uble 15
Source «nd in-
tensity of sound
Loud speaker
I ISO db)
Ditto
Dynaaic siren
(152-137 db)
Dynaaic siren
(II kvt)
Dynamic si ran
(120 vt)
Dynaaic siren
(2.U kvt)
Static siren
(60-100 vt)
Static siren
(60-100 vt)
Static siren
(60 vt)
Static siren
(400 vt)
Four dynanic
(35 kvt sirens]
Eight dynaBic
35 kvt. sirens)
Dynamic siren
0yflfMiP"1
Twenty— five kvt
dynaaic sirens
Oscilla-
tion fr*
qusncy
kHz
0.5
0.5
3,5
4-17
06,07
0,7
9,5
32-33
9,5
32-33
O.M»
0,44
3,5-0»,0
4-17
<0.3
Specific
force
b.y.3.
gas fore*
-
Several
kvt/B3
-
'.2
-
0,15
4
0,02
0,15
2,35
-
-.
-
—
Dispersion
ti«e
15 sac
15 see
1-20 see
(See Fig.
50)
A few see
99* sirens
1 >in.
(brighten
up 50*)
No results
2 Bin
2 >in
15 Bin
15 Bin
1 Bin
slow
clearing
up to 50}
No - re-
suits
Undeter-
• in«d
-
Studies in the year of
Lattair and Sinclair, I9H3
(156,266)
Ditto
1156)
Horsaley>>n(j Sivi, I94b
(156)
Taraba 1936-1957
(156,274)
LaHair and Sinclair, I9M3
(156,266)
Ditto
Boucher, 1956-1957
(156)
Ditto
n
B
LeHair and Sinclair, I943
(156,266)
LaMair, Sinclair and
Breishiz, I943, IW4
(158)
Ultrasonic Corp., U.S.A.
I9U6, (156)
'araba (KOBSOBO) 'sksya
Pray da) « May 1957
GeoPhlLAN, USSR, I960 (32)
- 149 -
-------
19.
ACQUSTQ-GRAVITATIONAL PRECIPITATION OF INDUSTRIAL DUSTS,
SMOKES, AND FOGS
Some industrial dusts and fogs contain very coarse particles, the
precipitation of which from the air or gaseous medium for economic or sanitary
hygienic reasons was usually attained through their own natural gravitational
falling out process. On the other hand, some industrial smokes and fogs con-
sist of highly dispersed particles, which under the effects of high pressure and
temperature exploded easily, or they became highly adhesive, or highly re-
active. In many cases the use of the highly effective precipitation devices as
electrostatic precipitators or fabric dust catchers must be excluded, and only
the gravitational precipitation of the aerosol particles should be applied. The
gravitational precipitation rate of spherical particles, also referred to as
"particle hove ring II rate, is determined by the following known formula r 111:
VI =,-g = 2/9 g pp/x r2 (19.1)
which can be easily derived by equating the expression for the forces of par-
ticle gravity with the formula expressing the forces of the medium's resistance
determined by the Stokes' formula.
: . J11/0,.,~r"p flMPl/HOK {tt';1t'1tl/~ I'fp{,'!M~/, I'
~ "/ J8 fno 1j[J 2UO lJO JOI
i ~~'f~~fi-Itftl~l.tlt~.1f1-+ :-fr("~-
~ ~~:;tiJt!IP}i-.tJL~~, '".5
'~ t:..~:: ;::t.;.:=L0:-1:. ,~'"t~;-;~,~q J:'-
. 1---- -* 0._.. ..... . . -.. - ...,.-". ~--- .... --. --
! "~'ft--:i::~~~_:' ,?~~-:~.'~' tit
,:.. ;~t[I.F1~~~~>,/~~~~~~'
~ ~"I ;:~;:,~f£'U0~~'~' [ . ;W~1-
i ~~f~i~l}~~:~03f' .:$*j~+~if!~' .~
i (jU~}:/t~%1'!*DY0~111~~.-?:
8 ,.!.t-I;i ,::;:;: .,.:'. ;.;.:- :,~;.t. ' .
~ t~Lwf\1f'ij.Jttf. 'I~' ,; .:.;, ',; ,.~t-=r ~ - el1JQ
~ z.Qd-hNj,I".i.' -'- ,
L . 1,05 f~.£.i:-: /II. ,'''1'. . t:' .'F.
~ ~ to 'ff-ilhl/L J:!,
-------
Table 17
dust particles; the particle radii ranged from 5 -- lOOji and greater (15 -- 25\i
particles predominate). A standard Hartman whistle mounted in a parabolic
reflector was used as the source of sound in these experiments. Experimental
results are listed in Table 17 and show that under sonication, dust precipitaion
in the stove gas ducts
was 1.6--7.1~times
greater than in simple
gravitational deposition.
However, these doubt-
lessly favorable results
could be achieved by
a. simpler and more
economical method,
namely, by installing
standard dust extractors
aft of the gas ducts,
which considerably re-
duced the residual dust
concentration in t h e
discharged gases.
Extensive
exp e r ime nt s had
been conducted in the
Soviet Union on the
Acousto-gravitational
precipitation of boring
dust formed during
mining. P.Sh.
Shkol'nikova [130,
131, 132] performed
this type of experiment
at the Leningrad "Gip-
ronikel1" Institute. Laboratory acousto-gravitational precipitation of boring
experiments with dust were conducted under static conditions in a 0.45 - m
dust chamber. The dust generated by boring copperpyrite ore in the Urals
contained up to 40% free silicon dioxide; it had the following particle composi-
tion: particle radius r = 2.5ji, 56.8%; r = 2.5 -- 5.0^, 19. 8%; r = 5 -- 26p,.
33.4%. Experiments lasted up to 60 min. Microscopic dust particle fraction
concentrations were determined with SN-2 type counter, and the submicroscopic
particles with a counter designed by Ye. A. Vigdorchik. The sound intensity
generated by the Hartman whistle was about 0.1 w/cm2. Experiments estab-
lished that calculated dust particle concentrations decreased to 50 - 35% of the
original due to sonication. The most favorable frequency was 7 khz; other
frequencies --10 and 13. 8 khz - yielded poorer results.
Pleee of particles
sot tl. ing
First gas flow .
Second *> •
n •>
: First » "
» »
Oscil Utiofl
frequency
kHz
4,6
1.2
3,0
3,0
3,0
.
»
' » »
»
» »
D »
•
4,6
Generator
s tented
doaneard)
5,5
-
3,9
4.0
[Generator
v slanted
• _^ ^ M \
Dust col lee-
tion plent
Door No. 3 :
.Lower door
> »
Door Mo. 1
> M 2
» J6 3
tol (acted du*t
Mount
Free froe
sound
effect
4
43
43
2,00
2,75
During
sound .
treat-
•ent _
28,4
70,3
145,0
5,75
5,00
3.50 2.40
b,za
» N» 1
• > M2
> Mr 3
.Ml
> M 2
' » Nt 3
*
» M 1
> Ni2
» J* 3
» M 1
» M 2
> M 3
> M 1
» Ni 2
• M 3
do«n «ard )
2,00
2,75
3.50
8,&>
4,20
1^,10
7,50
12,00
2.30
Increase
in
preciptd.
dust «ent
7,1
1,6
*•*
1,6
21,80 . 2,7
18,00
5,60 12,00
16,50 137.0Q
26,30
3,00
3,80
0,80
7,60
3,00
167,00
7,85
2,75
6,3
10,75
21,35 , 2,8
4,20
3.80 2.90
0.80
7,tW
3,00
3,80 .
0.80
7, BO
<.9Q
12,00
10,30
4,80
4,40 :
1,6
19,50 > 2,6
- 151 -
-------
Laboratory experiments on the acousto-gravitational boring dust
precipitation under dynamic conditions were conducted with a dust of the fol-
lowing particle profile: r = 0. 5 - 2. 5^, 46.9%; r = 2.5 - 26^, 10.5%, and r =
26 - 150p,, 42.7%. The gravimetric dust concentration ranged mainly between
66 to 1189 g/m3 . The air flow velocity varied from 0.004 to 0.016 m3 /sec..
The following results were obtained during sonication:
at \n=66 g/m3 k, ut = 39.1 g/m3 ^P..i, = 99. 29%
at kln = 453 g/m3 kou t = 41. 3 g/m3 n,, 98 4 p = 99- 82%
atkltt = 753 g/m3 kottt = 21. 3 g/m3 n, r; Ol p = 99. 93%
Clearly, the degree of precipitation was high; however, the residue concen-
tration, although about five times lower than without sonication, greatly ex-
ceeded standard sanitary MAC for dust in the air (10 mg/m3). Research con-
ducted under working conditions in the "III International" mine showed a 99.8%
dust precipitation; however, the residual concentration reached kout = 115.4
mg/mj, which was considerably above the adopted standard MAC.
The experiments led investigators to conclude that acousto-inertial
precipitation of boring and other mineral dusts may be used only in conjunction
with subsequent filtration of the gas. After installation of a glass-wool filter
with strands up to 40|j, in diameter at a filter diameter of 190 mm and a height
of 50 mm, pressure drop 40 - 50 mm hg, they succeeded in lowering the resi-
dual concentration to 2 - 6 mg/m3 .
In this case, the dust load on the filter was decreased considerably
which resulted in an increased filtration rate, thereby, compensating for the
cost in energy consumed by the sonication. This type of process combination
represented a transitional stage in the acousto-inertial precipitation of aerosols
examined in the next paragraph. Studies in boring dust precipitation are being
conducted by this method at "Gipronikel1" [300].
It can be concluded from results of the described experiments that
purely acousto-gravitational precipitation of industrial dusts, smokes, and fogs
were acceptable and promising, in cases in which the gas volume to be purified
was several hundred cubic meters per hour or less, or when the precipitated
products were of special economic worth, or, the use of other precipitation
methods were not feasible. Such cases occurred in modern industry, charac-
terized by a great variety of conditions and demands, as for example, in the
production of phthalic anhydride [88] and other similar substances .
If the volume of gas to be processed was not large, then the absolute
magnitude of the energy consumed in the precipitation of products was small
even when low-efficiency pneumatic sound radiators were used, and in the case
of separation, precipitates from the volume of gas being sonicated by a dividing
- 152 -
-------
diaphragm which prevented gas rarefaction by the siren air. Neither a high
gas temperature or pressure, nor explosion danger, or great reactivity of the
dispersed phase in this case, constituted obstacles to the use of the acousto-
gravitational aerosol precipitation method. This is a rare and significant
quality which had no precedent in dust-droplet entrainment technology.
Promising possibilities are ahead for acousto-gravitational and
acousto-inertial aerosol precipitation methods in separating condensates from
by-product and natural gases. These gases have a high surplus pressure
(100 - 200 atm) [112], part of which could be used for the generation of acous-
tical oscillations without loss in production efficiency. Therefore, the pro-
blem of the energy consumed in sonication is not important in this case. In the
gaseous state are found not only light hydrocarbons, such as, methane and
ethane, but even heavy hydrocarbons boiling at temperatures up to 300 - 400° C.
The amount of such hydrocarbons has been estimated as up to 4 - 5% and higher.
S
u
SO
a
71
a
se
u
a
-n't
u JO[tt so u n M saaoia-ite
At*. pr«»s«ir«
. 60.
ef
Due to the drop in gas pressure
and temperature during its travel along
the borehole shaft, the gas contained a
considerable amount of condensed
heavy hydrocarbon and moisture - a
"film" condensate. The entrainment
of this condensate presented no parti-
; cular problem. However, in avoiding
the deposition of condensates and gas-
eous hydrates during subsequent trans-
Condeneata content in tb. 9.8 .» . funcImportation through gas ducts, the gas
pressure at different throttling t*>f*r.tyr.».can b& additionally throttled at the
(Leningrad point of origin, Kr«»nodar«fc r«gion.J '
- . source, thereby sharply lowering the
gas pressure (Joule-Thompson effect)
and an additional condensation of hydro-
carbon and moisture occurred, changing into fog, which immediately precipi-
tated through gravity into residual, so called, "volume separators. "
The condensation isotherms of natural gases, called reverse or re-
trograde condensations, are of the form shown in Fig. 60. The pressure at
which the greatest number of hydrocarbons changed into a condensate has been
referred to as the maximum condensation pressure, and is usually from 55 -
70 at gage. Due to the high degree of super saturation at this pressure, it con-
densed into a fog in some cases up to 100 - 200 g per normal cubic meter of
gas; when calculated in physical cubic meters at a pressure of 55 - 70 at gage,
it yielded a value of the order of 5.5 - 14 kg/m3. Under such conditions the
condensation process, isothermal distillation and hydrocarbon coagulation
occurred very intensively (see Appendix I). However, to obtain a droplet
consolidation up to the dimensions where the gravitational method of precipita-
tion becomes effective (r > 35 to 50^), the gas must remain in the separator
- 153 -
-------
for not less than 15 sec, which is a considerable length of time.
In order to intensify the condensation, distillation, and coagulation
processes of the condensing hydrocarbons and moisture in the gas being thrott-
led and to increase thereby the permeability and the entrainment ability of the
separating unit, it was proposed that acoustical oscillation be used [80J. In
contrast to the existing method, in which the natural gas is throttled by a stan-
dard cylindrical attachment (" sleeve") mounted in front of the separator, it was
suggested [79J that the gas be throttled inside the separator with a special
sleeve-whistle mounted in it (Fig. 61a)
In the presence of a large
pres sure drop and increase in the
gas density, any predetermined
acoustical intensity of up to 10 kw
and higher may be obtained with a
sleeve-whistle, which is also con-
venient, since the compressed
gases had an increased acoustical
resistance PI CI' caused essential-
ly by an increase in gas density PI '
resulting in a decrease in the os-
cillatory gas velocity ~ [see.
-.- equation (1. 9) J. It is possible to
'---=-1I=-7"'. maintain the oscillation velocity
ICondensate at the normal level only by cor-
. outf low. respondingly raising the sound
I intensity, as compared with its
, Fi9.61. Di.gr... illuatrating the plan of natu,...1 use under atmospheric conditions.
9&8 uparaton sonification. T ki int t th all di
a - 'hiatle-type 988 ,e,.rator (I - The bodyJ 2 - Whi.tle- a ng 0 accoun e sm -
'type connectin.9.. .Ieave't.b _(gas .eparator. .ith a~ac:ou~ti- mentions of the space being soni-
cal .iren of reverae ac Ion I - body. 2 - acou.tlcal ..ren
of rev."'. actiorel connet:bng ni~pl.J cated, and the increase in droplet
concentration in the gas separators,
for effective intensification of the
process taking place in them, it
is sufficient to provide a sound intensity of the order of 0.5 - 1. 0 w/cmij or low-
er. Under these conditions, the sound level close to the separator wall reached
125 - 130 db, although, at a distance of 5-10m, it falls to permissible levels. *
*The permissible noise level depends on the frequency of the highest component
of the sound spectrum:fo and according to the adopted norms [110J has the fol-
lowing values:
Purified
9aa
outflow
PurifIed
1 II..
. outf low
r.
I,.
t
" I
.'
N. tural t.
_.'!.~~I:'.':"~I
. i~;:~~ lrl
-----
"--gas.~
inflow
Frequency Permissible sound Frequency Permissible sound
level L, db level L, db
350 90 3000.4000 70
800 85 5000 75
1600 80 6000 80
, 2000 75 10. 000 85
- 154 -
"f~
-------
In the second version of the sound-equipped gas separator (Fig. 6lb,
page 152), gas throttling is done by the usual method, while its self-sonication
is done separately with a static siren mounted at the separator inlet; the gas
discharged from the separator enters into the siren mouth. It can be shown
that with reverse feeding of the siren, radiation of acoustical oscillations oc-
curred in the usual proportion and direction, i. e. , to the side of the widening
of the mouth (the siren's radiation element is a pulsating gas jet which is an
acoustical dipole radiating sound mainly in the direction of the mouth). As the
experiment indicated, sonication of the gas caused the permeability of the gas
separator to increase several times. However, in view of the strong turbina- .
tion of the gas which develops because of this, thereby interferring with the
droplets fall-out, it is more apropos that their precipitation be transferred to
a separate settling chamber, or a direct-flow dust extractor, having lowered ,
at the same time, the frequency of the generated oscillations to the minimum.
Another variant of the sonic ation of gas separators was tested at the
Korobkovskii deposits in the Volgograd oblast1 [86]. A turbodynamic siren
was used as a sound source; it was mounted between the flanges of the separ-
ator inlet pipe. The oscillation frequency was, according to the researchers,
6 khz (according to our calculation this figure was ten times ove r estimated I).
The sound intensity remained constant; it was known only that the pressure
drop used up in the rotation of the siren, the generation of sound, and the
hydrodynamic losses were 1.5 -2.0 atm. Tests were conducted in two separ-
ators for the by-product and for the natural gases, at a pressure of 10 at the
gage. In the first case, the moisture content and condensate at the separator
output were 0. 2 and 0.1 cm3 /nm3 , respectively. Due to the sonic ation the
remaining moisture content decreased by one half and the condensate. by 17
times. In the second case, the amount of condensate entrained in the separa-
tor increased from 19. 5 to 25. 6 cm3/nms , i. e,, by 30%.
Due to moisture in the gas, gaseous hydrate crystals were formed
inside the separator in addition to condensate droplets, and massive gaseous
hydrate growths developed on its walls hindering further normal use of the gas
separator. To prevent the formation of hydrates on the gas separator Avails
and also to pre-stabilize and preheat the entrained condensate, the construc-
tion of a self-heating sound-equipped gas separator was proposed [84] in which
the disadvantages of the separating devices noted above were eliminated (Fig.
62). The vortical whistle-tube 2 differed from the vortical whistle described
in #2 by the fact that its internal chamber BK was somewhat more developed
and equipped at its back end with an additional annular outlet, a regulating
valve PK, as a Rank tube. Due to this, the gas eddy simultaneously excited
the acoustical gas oscillations and separated the gas into cold and hot streams.
The cold sonicated stream discharged from the upper opening OKhG, while the
hot gas stream (10-20%) came from the lower end of the chamber through a re-
duction valve, it flowed under the hood, heated it and the space under it, came
out into the annular space between the hood and wall in the form of a heat
- 155 -
-------
Separated
| gas enit
ICondeneate
* A Ml +
axil
screen TZ, heating the wall and preventing the formation
of gas hydrate growths on it. Rising still higher, the hot
gas, which contained the main body of the condensing hy-
drocarbons, mixed with the sonicated cold stream coming
from the whistle. The condensate droplets and the gas-
eous hydrate crystals precipitated on the heated hood,
where the crystals melted, and the lighter hydrocarbons
dissolved in the condensate, evaporated and left the sep-
arator with the gas.
In the above vertical gas separator, the need for
constructing heating coils and insulation in the boiler de-
vice was eliminated in the above vertical gas separator.
However, the fact could not be ruled out that the turbuli-
zation of the sonicated gas arrested precipitation of drop-
lets also in this case, this being the case, another solu-
tion to the self-heating idea has to be found.
Fig. 62 - Schematic presen-
tation of an automatical ly
heating natural gas separa-
tor.
I - Bodyj 2 = a *hir(e-
type pipe wgustkel 3 — an
uabrella hood} BK - •hirle
chanber of the pipe—type
whistle; OXT - enit of
sonified cold gasj PX - re-
Original experiments in the acousto-gravitational
precipitation of aerosols were recently carried out in
Czechoslovakia by Slavik and Geyd [24]. .The investiga-
tors tried to test the coagulating properties of sound for
combating harmful gaseous impurities. For this purpose
they Pr°P°sed to introduce into the gas corresponding
gaseous reagents, which, after chemically uniting with the
harmful gas impurities, formed suspended solid or liquid
particles. If, for instance, SOg was to be removed from
the atmosphere, ammonia (NH^) could be introduced as a reagent. In the pre-
sence of moisture, SOg might react with NHg in the following way:
- a heated screen.
2NHg =(NH4)8SO3
forming a suspended ammonium sulfate solid.
The above method for the removal of SOa from the atmosphere was
tested in a foundry casting the light alloy Electron. Upon completion of pour-
ing, the mold was covered with a hood on which four Hartman whistles were
mounted in series and were operating at a frequency of 8 - 11 khz. Small
containers with ammonia were placed under the whistles. After several min-
utes the asphyxiating gas was eliminated from the mold and no longer poisoned
the foundry atmosphere.
In conclusion, mention should be made of the experiments using the
coagulating effect of sound for the artificial precipitation of clouds for the
purposes of rainmaking, which are of definite interest to meteorologists and
agricultural workers. The radius of the cloud water droplets varied over a
wide range of 1 - 2 to 70 - 100 p,, with a mean radius of 10 - 20 y, [120]. Ex-
- 156 -
-------
periments with the artificial rain generation by sound were conducted by mem-
bers of the El'brus Alpine Expedition of the Academy of Sciences USSR [32].
Powerful low-frequency acoustical sirens were installed on the slope of the
Baksanskii Pass mountains. The radiated power of each siren was about 25
kw. Each siren was equipped with a mouth having an outlet cross section of
up to 9 m3 . The sirens were directed toward rain clouds hanging above the
pass. Under the effect of the sound the cloud droplets consolidated resulting
in a rain. .
20. ACOUSTO-INERTIAL PRECIPITATION OF INDUSTRIAL DUSTS, SMOKES,
AND FOGS
The acousto-inertial method of aerosol precipitation differs from the
acousto -gravitational method examined in the previous paragraphs; the preci-
pitation process of coagulated particles operates in a separate inertia! preci-
pitator which is incomparably more effective than the sump; this precipitator
is usually a cyclone. The precipitation rate of spherical particles under the
effect of inertial forces developes in a cyclone with an inner curvature radius
R and is determined by formula [121]:
V _
where wt -- tangential velocity of the aerosol motion. The ratio of inertial
particle precipitation rate to the rate of gravitational particle precipitation,
determined by expression (19.1), is equal to: . .
Vf ^ »? (20.2>
Usually wt = 15 - 25 m/sec, and R = 0.1 -- 1.0 m; thus, it is seen that the rate
of inertial particle precipitation is 25-600 times greater than the rate of
gravitational particle precipitation.
A diagram of a typical acousto-inertial industrial dust, smoke and
fog precipitation installation is shown in Fig. 63. The dusty gas and aerosol
first enter coagulation chamber 1; while passing through the chamber, the gas
sustains the acoustical oscillations radiated by sound siren 2 operated by com-
pressed air from compressor 5; the suspended particles consolidate, and a
portion of these is precipitated. The gas then enters inertial precipitator 3,
in which the consolidated particles are separated from the gas. The purified
gas is then discharged into the atmosphere by blower 4, or removed as a pro-
cessed gas. Finally, the entrained dust enters a dust collector. When lower-
ing the gas temperature is required, cooler 6 is mounted in front of the coagu-
lating chamber; when it is necessary to dampen the gas, the chamber is equipp
ed with liquid spray nozzles (7) or steam nozzles. If the siren is activated by
gas instead of air, filter (8) is mounted in front of the compressor.
- 157 -
-------
Oust
Caught
•aterial
Fig. 63. Diagram of main characteristics of « sound--
inertia dust-droplets catcher installation.
I -Coagulation chamber} 2 - acoustical si rent 3 - in-1
ertia procipitators; k - venti(ator| 5 - compressor; •
6 - cooler device} 7 - spray nozzle; tt- filter.
The coagulation chamber,
or simply the coagulator, is a
vertical cylindrical tower lined
with soundproofing material,
such as glass wool, etc. [110].
The sound siren is usually lo-
cated in the upper part of the
chamber. To reflect the inci-
dent waves, the lower part of
the chamber, is flat or conical
with a 45° tangent angle. The
diameter of the coagulating
chamber usually did not exceed
2. 5m. The height of the cylin-
drical chamber part is in the
6-9 m range, sometimes reach-
ing 11 m. The gas motion ve-
locity w is related to the sonication chamber height Ht by the elementary func-
tion w = HB /tg , where tg is the aerosol sonication time. Velocity, w for nor-
mal Hg and t^ ef industrial aerosols, is in the 1.5 .-- 2.0 m/sec. range. This
is considerably higher than the velocities encountered in electrofilters for the
entrainment of fine particles and basically explains the smaller dimensions of
the coagulation installation.
Figure 64a shows a diagram of the
coagulation chamber of the first experiment-
al acousto-inertial installation designed to
trap soot [272]. The dusty gas entered the
// ,/ H i Xi chamber from above, and, after mixing
with air fed from the siren, traveled to the
chamber bottom, a system which may be
called direct flow. The disadvantage of
I this systemis, in the fact that upon enter-
Ing the chamber the aerosol becomes dilut-
: ed by the siren air. The fraction of siren
air is 7 -- 10%, in industrial installations,
while in laboratory-type installations the
amount of air fed from the siren can exceed
'the aerosol amount.
[•?->
4;
!
A
^••^M
^ *
•
f
/s
\
\
*
I
D
4
A
*_*•
A
B*
f\
/
•>
CT
— «•
^M
/
*•—
Fig. 64. Coagulation chaaber types
a - direct flow} b - counter f low; c - com-
bination typej (I - booyj 2 - f'r»n» 3 -
gas inflow} •« - gas outflow).
The counterflow system (Fig. 64b) is more rational. Here, the
aerosol is diluted by the siren air only upon completion of the particle coagula-
tion. This system is normally used under production conditions. The gas
dilution can be sharply decreased by discharging the siren air before mixing
with the gas [73. 260, 225] . A combination system is also possible by mount-
ing a radial siren inside the chamber, as shown in Fig. 64c. Dimentions of
such a chamber are as follows: D. , = 2.4 m; d, = 1.17 m; d_ = 1.3 m; H =9. Om
en 1 2 s
- 158 -
-------
r.
[269J. The gas purification level in an acousto-inertial installation depends on
the degree of particle consolidation achieved in the coagulator, and on the pre-
cipitation effectiveness of the consolidated particles in the inertial precipitator.
It was shown in [119J that the effectiveness of particle precipitation was charac-
terized by the degree of entrainment, or purification, or indirectly by" the
residual weight concentration of particles kO\lt (g/m3) related by the simple
function: . . i
'1 = (1 _.~) 100 %
, ' ia
(20.3)
(~\I - input particle concentration in g 1m3).
Different devices have been tested as precipitators in acousto-iner-
tial installations: simple cyclones, multicyclones, disintegrators. rotary
cyclones, bubble baths, damp wire filters, cloth filters, etc.
Fig. 65 shows a diagram of a
. coagulation chambe,r with multicyclones
made by "Promenergo" for an acousto-
inertial installation to trap zinc oxide'
dust from the discharging gases of a
bronze-brass factory., Fig. 66 pre-
sents fractional efficiencies of several
inertial precipitators' at pP = 1. The
graph shows that to obtain high acousto-
inertial purification factors for indust-
. rial ga.ses. it was necessary to con-
solidate the particles up to r = 5 -'-7 1Jo,
when using multicyclones, and not less
than y = 10--15 1Jo. when using simple
cyclones. Results achieved in the
acousto-inertial precipitation of indust-
rial smokes, dusts, and fogs were
published in the original articles and
also in a series of review articles by
. the Soviet and foreign author s r 7, 58
: 78, 127. 150, 151, 157, 183, 184, 240.
241. 248, 269. 292,]. The most im-
portant of the foreign articles appear-
ed in the collection of translations
ConatruCltion of a _Uicyclone C08iJulation "The Acoustical Coagulation of Aerosols"
c:ha8ber built by -rr....ergo". r lJ recently published by Goskhimizdat. '
I, .." i'l
. Fig. 65.
In view of the above, the author presented only a limited summary of
the acousto-inertial aerosol precipitation. Readers are referred to the origin-
al literature for more detailed information.
Combined data of experiments with the acousto-inertial industrial
" .
- 159 -
~"
-------
Name of
substance
Gas furnace soot
n n n
Ditto, aggregated
Spray soot
Hard cool soot
Sulfuric acid fog
Fog of natural
sulfuric acid
Di lute sulfuric
oci d fog
Artificial oil
fog
Zinc oxide vapors,
generated by
roasting zinc ore
Zinc oxide vapor
generated during
copper melting
Zinc oxide vapor
generated during
brass smelting
Coke gas (tar)
Cracking gas (con—
densate)
n it n
Open hearth smoke
Carbide furnace
smoke
n n n
Aerosol
e
-^ •*
U * -*
--3
W ._ 41
i- -a l.
a it
i. i_
0.03-0.07
0.03-0.07
0.5-15
0.1-0.2
0.5-1 .0
0.5-5.0
0.25-2.5
2.5-60
Predomin-
ated 7.5
0.5-6.0
0.5-5.0
Predomin-
ated 2.5
0.5-40
0.4-0.6
0.5-6.0
0.5-6.0
0.5-5.0
(Predomin-
ated 3.0-
3.5)
2.5 (55$)
0.5-15
Hrvd «jMn—
a ted 0.5)
Ditto
o
L «
•*• *
0 rn
•=ci
> o •?>
<• c.
fi a
o o
.2-12.6
.2-5.1
0,5-2.5
26
0.5-2.4
5-40
1
0.5-1.2
10-40
I-*
0.5-20
10
30-/0
5-?0
6-15
2
0.25-2 .8
0.25 2-a
01
L
3
-**
a
i
o o
a o
e
o> *>
»— -•-
40
40
-
82
80-90
180
50
35-40
40-100
50-350
400
40-60
35
40
150
120
120
i
* — ».
o e '
o - a £
» 0 «r
-------
Table Its
Exptl . chamber
Acoustical
siren
Dynanic
radial
Ditto
Dynaaic
axial
State eith
suction re-
Dover
Dy nao i c
axial
Ditto
Dynaaic
radial
Dynaoic
axial
Ditto
n
Dynaoic
radial
Dynaoic
axial
Ditto
n
n
Dynaaic
Static
n
8*
-•> x
• c£
= l5
•* 9 C
J£~
it
2-4
3
I
4.6
3.6
2.15
2.25
1-2
2 -4
3-3.5
3-9
0.7
4
it
3.5
2.2
7-HO +
26
10.5
*•>
•a — 5.
J«c~
**
c —
0.5-4.0
0.5-1.0
O.I
1.0
0.10-
0.14
O.I
O.I
O.I
0.1-0.2
O.I
0.13
0.6
O.I
O.I
O.I
-
_
§ «
•- «•
..2"
tf *»C
— *-
•— L.
•• 3 0
CO
4.5
1.2
10
' 1
7
*-»
3
4
7
2.4
10
10
2.5
5-S.5
5
6
-
4-6
4-6
Precipitator
Type
2 cyclones 0
secut ively
Ditto
1 or 4 paral-
i lei cyclone*
• i
2 cyclones and
a yla»»—
cloth filter
con ee cut ively
Cyclone 0
0.15-
Paral lei type
•ulti-
cy clone
2 parallel
cyclones
4 parallel
cyclones
Cyclone
t 0.15 •
Cyclone
Cyclone 0 1.35
Cyclone 0 0.15
0.30 • *ith
filters^con-
secutively
inatel led
2 Paral lei
cyclones
Ditto
2 Paral lei
npuluzy"(Not
faBilier to ne
-BSL)
Ty^e • ••*
re tec lone
Parallel
•u 1 1 i cy —
clones
n
4j!a
- ••- «*
& A » f
C26-
40
B-32
68-72
(30).
68-74
(81)**
84
-
69-72
93-95
84-87
70
—
88
76-82
73
45
II
-
• •f C
ea o
'•»••- e
^ 4« K
I5 C*
o.e —
• e c c
• * O *^
83-90
99
95
(99-,98(97)«
87(97)**
99.6-99.9
90
78-12
99.0-99.8
94.98
90-85
99.8
95.5-99 J
95
90.7
94
86
Piece and Year
of investigation
USA, 1947-49 (272,
'273, 26*, 183)
Ditto
Japan, 1950-5''
. 38,246)
USSR. NIISAS iv^y
(54,57)
USSR, IGI, 1959
(8$)
USA, 1949
(218, 241, 243)
USA, New York, |y40
(184)
Japan, 1950-54
(38,246)
USSR, MEI, 1954-5
t73,74 )
Japan, 1950-54,
(38.246)
Poland, Snepenitzy,
1954—5
(i!27,22a)
USSR, Proeenergo,
1958-61, (299)
Japan, 1960-64,
(38,246)
Japan, 1950-54
(38,246)
Japan, Tokio,
1954 (259)
USA, 1961
(281 ,269)
France, 1964
(150,1^1)
- 161 -
-------
aerosol precipitation method conducted in the Soviet Union and abroad, are
listed in Table 18. Data on the following materials, the acousto-inertial preci-
pitation of which was insufficiently covered in the literature are not included
in Table 18. :
1. Soda ash in gases discharged by wasteheat boiler of a paper
factory, being entrained in an industrial acousto-inertial installation of 85, 000
m3 /hr capacity [184] .
2. Molybdenum sulfide, satisfactorily entrained in dry form in an
industrial-size acousto-inertial installation [269].
3. Cement dust from the gases of a cement annealing furnace,
successfully entrained only when water dampened, which caused no dust,
hardening in the precipitator [269].
4. Fly ash of smoke gases from boiler installations [186, 193 216,
254] , which coagulated poorly in dry form, due to the small particle size.
5. Sulfur from gases discharged from a catalyst chamber in which
sulfur dioxide and hydrogen sulfide formed elementary sulfur; the degree of
sulfur entrainment reached 93% [183].
6. Iron oxide from gases formed in oxygen-blown steel converters,
and also cobalt oxide, 90% - entrained in an experimental installation -with a
3000 m3 /hr productivity, at 1 -- 3 khz frequency [206] .
[150,
7.
151].
Antimony oxide showing, according to Bush, even better results
8. Lead oxide, 95 -- 98% - entrained in an industrial installation
of 10, 000 m3 /hr capacity in Czechoslovakia.
9. Different "aggregating" aerosols obtained by mechanical spray-
ing of earlier precipitated powder-form products in a gas which yielded neg-
ative results with aggregated coal dust, ash [73], magnesium and aluminum
oxides [262], etc.
The table contains no information on
materials, the acoustical coagulation of which
was determined not by the degree of inertial
precipitation but by other indices, and also on
a laboratory scale, such as ferromanganese
dust from blast furnace gases [171], cyclo-
hexane oxime [87], flourides [15], etc. Data
presented in Table 18 leads to the following
conclusions:
5-
c
a if.
'« c
101.
SB
M
to
U
e
>. a at
Diaaeter of
particles in |JL
Fij. 66. Fractional efficiency coeffi-
cients of sone inertia precipitators.
I - Loryo disaster cyclone} 2 — saall
dioootor cyclone; 3 - high impact lab -
oratory precipita tor
Highly dispersed gaseous and lamp
black aerosols consolidate sufficiently for
effective precipitation in inertial precipita-
tor a only at an increased sound intensity, or sonication time of the order of
- 162 i-
-------
1 w/cm"1 and higher. Exhaustive precipitation can be achieved only using cloth
filters as the final stage; due to particle consolidation and the decrease in dust
concentration, cloth filters can operate at an increased gas filtration rate.
Less highly dispersed
aerosols, such as sub-
limates of zinc oxide,
molybdenum sulfide,
and others, are suf-
ficiently consolidated
at normal sound inten-
sities (0.1 w/cma) and
attain 90-96% entrain-
ment levels. However,
'*£*-em the residual 20-200
W"* mg/m3 particle COn-
V"'
Fig. 67. Functions! efficiency rela-
tion betaeen acoustical inertia pre-
cipitation of colloidal oil Ifoy)
and initial gravimetric concentre- ,
tion (f = 2.H kHz; I .0.1-4.3 vt cm3)
a - concentration of preeipitetej b -
trapping degree
centration exceeded
the standard MAC for
toxic substances, ne-
cessitating, as in the
Fig. 68. Functional
between ths concentration of re-
sidusl sonified particles of •«•»
industrial aerosols and the pro-
duet of dfto according to Oytaa,
Inoue, Savahata, and Oksda.
i -coke gas tar .ith r -o.s-5.og, previous case, the
k- • 30-70 g/« i 2 - cracking
• AC ^^^
gas condensste «ith r * 0.5-6.0(J,
k. = 5-70 g/-3, 3 - ssgre-
aste soot «ith .r'» 0.5-15^ j kinc
_ 0.5-5.'* 9/*^l ** ~ dilute sul —
furfc scid fog *ith r * 2.5-50,
.0.5-1.2 9/»
supplemental installa-
tion of cloth filters,
justified only for enr
training high-cost
products or for small
gas volumes.
Different polydispersed dusts, such as dust from open-hearths,
blast furnaces, furnaces for annealing cement, etc. , of low weight concentra-
tions of highly dispersed fractions, generally yield a high degree of entrain-
ment only when 1) the gas is water dampened in the coagulating chamber fol-
lowed by precipitation in a damp precipitator which assured a residual con-
centration of 100 mg/m3 , or 2) cloth filters are used as the final stage which
assured a residual concentration of 5--10 mg/m3 or less and increased per-
meability. Natural fogs such as coke gas, cracking gas, etc. , were effect-
ively precipitated by the acousto-inertial method (9B--99. 8%), if their con-
centration and dispersion degrees are moderate, in which case their residual
droplet concentration did not exceed 30--300 mg/m3.
In testing the acousto-inertial industrial aerosol precipitation
method, several empirical relationships were derived. It was established that
the residual particle weight concentration had a linear functional relationship
to the initial particle concentration in the gas, as illustrated in Fig. 67a. Thus,
the dependence of entrainment degree on the initial particle concentration was
expressed by the typical curve shown in Fig. 67b. Thus, it can be concluded
that the entrainment degree fell sharply at the "critical" concentration of
3-5 g/m3 , while at concentration above 10-15 g/m3 , it barely increased. The
- 163 -
-------
latter could be explained [see equation (20.31)] by the fact that with increasing
kj n the ratio k^,,^ /k^B decreased (see Fig. 67a) progressively slower, as ill-
ustrated in Fig. 67a.
It has been established, in the case of acousto-inertial precipitation,
that the residual particle concentration was an exponential function of the spe-
cific air compression energy consumed by the sound siren by the sonication
time §« = deaaPB3rBt0 [246]: k^ = k^e'011'. "' C20^
where m - constant depending on the form of the aerosol. This is illustrated
in Fig. 68 in which are presented results obtained with acousto-inertial pre-
cipitation of different industrial aerosols by Oyama, Inoue, Sawahata, and
Okada [246]. If m is a constant related to constant n - derived by the authors,
through the relationship m = n/27.3, then the following values are obtained:
coke gas -- 0.077; cracking gas -- 0.059: aggregated gas ashes -- 0.033; fog
diluted by sulfric acid -- 0.009. The specific energy consumption is propor-
tional to the product of the sound intensity, and the sonication time deB~rayJt8
[see equation (20.8)],, since §' ~ (yjts )2 . By comparing equation (20.4) with
equations (15.2) and (16.4) get the following relationship:
(20.5>
The above expression implies that weight concentration of coagula-
ted particles decreased more rapidly after their precipitation in an inertial
precipitator than the numerical particle concentration did in the course of
acoustical coagulation. This is theoretically correct, since the heaviest of the
coagulated particles precipitated in the inertial precipitator.
The following expression is obtained for the degree of entrainment,
according to (20.4): ^ _ mn_nhn_. w-*'% (20 6>
je /&, \f.V'\>r
where x imep%iai " efficiency of the inertial precipitator which can be attained
in the purification of a non-sonicated aerosol. The equation holds for consoli-
dated particles up to dimensions at which they become completely entrained by
the precipitator, and for the unbroken consolidated particles therein, a pheno-
menon observed only in fogs.
On the basis of Stokes and Vivian's [273] experimental data, Kidoo
[211] proposed another exponential equation for the gas black entrainment de-
gree: M
^,00-<*"'•%. W>
where M -- coefficient depending on the form of gas black and the effective-
ness of the coagulating and precipitating installations.
-164 -
-------
The question of energy consumption is important for a correct
appraisal of the acousto- inertia! aerosol precipitation method. Consumption
consists of energy lost in the aerosol coagulation and precipitation. The speci
fic energy consumption magnitude for the aerosol precipitation is well known
for all existing precipitator types. For example, for cyclones, it is equal to
an average of 0. 5 Kwh per 1000 m3 of gas; for rotary cyclones, it reaches
several kilowarthours. If the required sound intensity radiated into a coagula-
tion chamber of cross section Soh , is J w/cma and the sonication time is t,
sec, then, taking into account that the total amount of required sound energy
equaled WBoUB 4 = jSoh and the amount of gas being processed equaled Q, =
wSe fc = Hs /ta Sa h , the following expression for the specific energy consump-
tion in aerosol coagulation [74] is derived:
*"""" . (20-8)
-
0.36 TV. Vy V««. 1000-'
where X 8lPaB - the acoustic siren efficiency, X ee* ~ t*ie overall efficiency of
the compressor operating the siren, X oh •„,„ - utilization coefficient of the
acoustical oscillations in the coagulating chamber; one must also take into
consideration losses connected with the non-uniformity of the chamber's
sound field, the absorption of "the sound energy by the aerosol, leakage through
the chamber' openings and walls, the yield from the resonance state, etc. ,
(determined experimentally); H8 -- amplitude of sonication of the coagulating
chamber in meters. Thus, the measure of energy consumption in the process
of aerosol coagulation is the product JtB of the sound intensity and the sonica-
tion time designated as the sonication factor.
Examining equations (15.2) and (20.8) and taking into account rela-
tionship (15. 6), it can be concluded that an increase in J influenced the effect-
iveness of the process less perceptibly than an increase in ts . However, J and
ta affect the specific energy consumption equally, from which it follows that the
sound consumption was less at low values, while the capital expenditures for
equipment are just the opposite, i. e. , greater at small values of t8 since the
permeability of the chamber increased with ts . The optimal values of J and t0
are derived by technical and economic calculations based on experimental data.
Neuman Sodaburg, and Faul (Ultrasonics Corp. , USA) indicated .
[240] the following limits for specific energy consumption: d eBer,-y = 0.65 --
2.0 Kwh per 1000 m3 of gas. Other authors cited even higher values. Thus,
in the course of semi-industrial experiments conducted by Oyama, Inoue,
Sawahata, and Okada [246], the following values were obtained: consolidated
carbon black and zinc oxide -- 4.2 Kwh;* diluted sulfuric acid --5.6 Kwh; coke
gas -- 8.4 Kwh; and cracking gas -- 7.0 Kwh, calculated by the present author,
* In reference [ 89] the value for the specific energy consumption in the soni-
cation of zinc oxide equals only 0.6 Kwh, which, requires further checking.
- 165 -
-------
hence x 8on = 0.65. Energy consumption in the acousto-inertial purification of
cracking gas in an industrial installation designed by the same authors [259],
was only 2. 5 Kwh. The reason for the difference can be found in the fact that in
the experimental installations non-economical sound sirens were used, the
acoustical efficiency of which did not exceed 10 - 15%. For this reason, a value
of 8 Kwh was obtained in the experiments on the acousto-inertial precipitation
of zinc oxide sublimates conducted at the TsIOT(PNR) [227, 228], while in the
precipitation of carbide furnace smoke [150, 151] , values of 3.9 and 7.0 Kwh
•were obtained. In citing his experiments Schnizler [262] indicated a value of
10 Kwh, although he failed to mention the fact that it was obtained in the acousto-
inertial precipitation of aggregates, i.e. , poorly coagulated products. General-
izing on the accumulated experience in the acousto-inertial precipitation of in-
dustrial aerosols, it can be said that, generally the following factors may
account for the increased specific energy consumption in the process.
1) poor acoustical coagulability of an aerosol, due to excessive
particle sizes or their low -weight concentration, etc. ;
2) strong aerosol sound absorptivity due to the "critical" particle
sizes, excessive particle concentration, etc.:
3) oscillations frequency deviation from the optimal value, due to
a) limited sound generator capabilities, b) drive speed variations, c) variations
in the aerosol dispersion composition;
4) low acoustical sound generator efficiency, due to improper design,
manufacture, wear, etc.;
5) non-economical coagulating chamber operating conditions, due to
the system's non-controlled resonance, unsatisfactory chamber height selec-
tion, heavy energy losses through the inlet and outlet ducts, gas temperature
variability, etc. ;
6) considerable aerosoldilution by the siren exhaust, which lowered
aerosol coagulability and increased the energy consumed in thrusting;
7) decreased compressor efficiency, due to considerable deviations
from the rated operating value, air throttling, and other factors;
8) poor choice of inertial precipitator type, or its operation below
optimal conditions: reduced input velocities required increased degree of par-
ticle consolidation, and excessive velocity enhanced destruction of dry particle
aggregates; according to Jahn [206] , the critical velocity in cyclones was 10-
12 m/sec.
Not all above factors can be completely eliminated. Therefore, 2--
4 Kwh per 1000 m3 of gas should be considered) thus far, as the practical total
value of consumption for the acousto-inertial purification of gases. Only when
it is necessary to dry the entrained products can the specific energy consump-
tion reach higher values, as for example, for dust in an open-hearth gas d =
6. 5 Kwh [269].
-166-
-------
Measures making possible to increase the efficiency of acousto-
inertial precipitation of industrial aerosols are listed below:
1) the introduction of water spray or condensed steam [ 189] into dry,
or only slightly dusty gases, and, in the case of fogs, the introduction of en-
training liquid droplets, such as fog recirculation in proper amounts. The in-
troduction into the aerosol of charged water droplets, or some other liquid .
should be considered;
2) lowering the gas temperature to increase the actual particle con-
centration, thereby, stabilizing the condensation temperature of the input steam,
and to decrease the gas volume leading to an increase in the coagulator perme-
ability, or, conversely, the maintenance of a high gas temperature to sustain
the liquid particle state;
3) an increase or reduction in the coagulation chamber height to
achieve the most rational sonication conditions for a given aerosol;
4) the use of high-efficiency sound sirens, such as dynamic sirens
with rectangular pressure pulses, also sirens with internal air circulation;
5) the use of purified gas to operate the sound sirens when it was
under: excessive pressure, as in blast furnace gas, or, if available, low-pres-
sure steam, as exhaust steam from steam engines;
6) acoustical multifrequency aerosol processing, as in Boucher's
[150, 151], demonstrated this as favorable measure;
7) special acoustical filter installation at the gas inlets and outlets
from the coagulating chamber to backfeed acoustical energy into the chamber
[180];
8) the use of more effective dust precipitators, in particular damp
wire filters [184, 239, 240]; the use of electric filters also deserves attention.
A combination of a turbulent coagulator with an electric filter dem-
onstrated experimentally, proved to be extremely useful in purifying blast
furnace gas [ 119] , the residual concentration of which was lowered to 5-10
mg/m3 instead of 20 mg/m3 and the permeability of the electric filter was
increased by 50-100%.
Still better results may be expected from combining an acoustical
coagulator with an electrical filter. The sonication time was fore-shortened,
since electrical filters permit lesser particle consolidation than cyclones, and
the permeability of electrical filters was increased. This reduced the capital
outlay and the consumed sonication energy. An acoustical coagulator also has
the important advantage over a turbulent coagulator in that coagulation can
proceed in a dry form, with the exception of slightly dusty gases, since there
is no danger that the aggregates coagulating in the electrical filters might
disintegrate, since the gas velocity was small.
- 167 -
-------
The acousto-inertial method of dust-droplet entrainment, like other
methods, has its advantages and disadvantages. The advantages are these:
1) high degree of different industrial gas purification, primarily fogs,
but also smokes, for -which damp entrainment is possible;
2) infinitely small dimensions of the entrained particles of fogs,
smokes, and dusts;
3) the independence of the effectiveness of gas purification from the
electrical properties of the particles;
4) applicability to aggressive gases purification, and also gases with
elevated temperature or pressure;
5) explosion and electrical safety of the acousto-inertial equipment;
6) simplicity of the coagulating devices, in particular when using
static sirens and centrifugal compressors;
7) more moderate dimensions and capital expenditures as against
electrical and cloth filters.
The disadvantages of the acousto-inertial method of .dust-droplet
entrainment are these:
1) higher specific energy consumption in the purification of gases,
particularly smokes and dusts; the consumption is always higher than for
electrical filters, but, as a. rule, it is lower than for disintegrators, turbulent
pipes - atomizers, and cloth filters;
2) the need to dampen certain smokes and dusts and to lower the gas
temperature accordingly;
3) unsatisfactory residual particle concentration, not lower than 30
mg/m3 for fogs and 100 mg/m3 for dampened smokes and dusts, where cloth
filters installations are suitable;
4) when using standard sound sirens, certain dilution of the contam-
inated gas by air, by the gas or the siren's steam;
5) the presence of rotary machines - the sound siren, if it is the
dynamic type, and the compressor.
The most important disadvantage is the increase in the specific
energy consumption most evident in the coagulation of highly dispersed aero-
sols, such as gas black, etc. In this connection, the two new systems for
coagulating aerosols in a sound field proposed by the present author are of
interest: the multifrequehcy multistage [77, 78] and the electro-acoustics
[76, 78]. Consider briefly the physical nalture of these systems.
- 168 -
-------
-,-
i
: ;
It was noted in #16, that for every coagulating aerosol there was an
optimal oscillation frequency at which the coagulation process functioned more
effectively. The coarser was the aerosol, 1. e., the greater was its particle
size I the lower was the optimal oscillation frequency value, This follows from
the theoretical investigations reported in #10, 11, and 12 (see Figs. 30 and 41),
and also general experimental investigations (see #16, Fig. 51 and Table B)'.
Aeroaol
infl08
In the
course of aerosol
coagulation a
gradual con-
I 1 t ~i' '. soli d at ion 0 £
r particles oc-
19 f - cur red w h i c h
of, implied that
Ae ro 80 I the oscillation
infl08 - frequency had to
. be decreased if
FIg. 70. Oiagra. of .Iectro-
acouatical ..thod of a.rosol 'it was desired to
c08gulation and pr.cipitation.
I - tlltOtroacouatical Cha8ber. conduct the pro-
II - Sound rec.ptor; III 8Rd IV cess under opti-
- eloctrod... I - a fOIl drop-' 1 d't'
let. 2,3,",5 - Quat partiel.a. ma con llons.
This is a
special requisite
. in the coagulation
of highly dispersed particles requiring huge consolidations. Consequently, the
existing methodology of conducting acoustical aerosol coagulation was a primi-
tive one. In place of the existing system for the acoustical aerosol coagulation
a more efficient system, shown in Fig. 69, was proposed. In this system, the
aerosol passed through a series of chambers with step-down oscillation fre-
quencies with respect to the change in the particle size. However, this did
not imply an increase'in capital expenditures and in energy consumption by the
sirens, since the flow-through velocity in the chamber increased by a corres-
ponding number of times, with three chambers three times, etc.
I
I
r~L~OHI
J II II II I ouUlo.
Ji
Aeroaol
outf 108 '
I
B
Fi9. 69. Oia~ra. of a atep-
""ap.d c089"lati on in.tallation.
I, ", I" - C"9~lation.eh---
b8n11, 2 3 - eorre.pond.n9 a- ).
cQUat.car .inn. (f I> f II> fill:
The nature of the electro-acoustical aerosol coagulation method lies
in the fact that (Fig. 70) previously charged water droplets, or another liquid,
mixed in with the sonicated aerosol as centers of coagulation, and the process
of sonication occurred in an electrical chamber which outwardly appeared as
an electrical filter without a corona. Having become charged, the particles
moved toward the precipitating electrode and in the course of the movement
had undergone a "bombardment" of oscillating polarized dust which penetrated
into the droplets and moved with them to the precipitating electrode. Thus, the
processes of aerosol precipitation and coagulation were combined here in one
piece of equipment. However, the coagulation process proceeded according to
a new method, since the longitudinal oscillat~ry particle motion was achieved
- 169 -
.
, '.
.
..,
Or"';'
'." ~
.
'. ~ ~~ I
, ,
"
-------
acoustically, while the lateral motion was achieved electrically. In the case of
an a-c current, the dust and droplets will undergo only coagulation and not
precipitation on the electrode surface.
21.
ACOUSTICAL AEROSOL SEPARATION AND FILTRATION
Acoustical oscillations intensify the aerosol precipitation by aerosol
coagulation, and also directly. Direct intensification can be obtained primar-
ily with the aid of drift forces, which prevent the aerosol particles from mov-
ing with the gaseous medium, thereby, allowing particles from becoming
separated from the gas flow.. Another possibility is presented by the use of the
orthokinetic effect which makes possible the aerosol particles precipitation on
large, rapidly-settling liquid droplets which then become separated from the
dusty gas. Finally, a third possibility consists in using the gaseous medium
movement generated at the boundary layer of obstacles (see # 1) as a factor
accelerating the turbulent diffusion and the inertial precipitation of aerosol
particles on them.
Examine the proposed methods of excitation and use of the precipi-
tating sound effect. A highly original method of acoustical aerosol separation
was proposed by Westervelt and Sieck [290J. The basis for the method rests
in Westervelt's theoretical work [286J cited in #6, in discussing aerosol par-
ticles drift forces in a sound field; the author described a new form of particle
drfit resulting from a distortion in the sound waveform. It was shown in #6,
that with great waveform distortion, the velocity of this drift type exceeded
the velocity of all other drift types, and in particular, radiation drift. For the
accelrated separation of the suspended particles from the dispersion medium,
Westervelt and Sieck proposed sonicating the medium by means of an artifici-
ally distorted (asymmetric) waveform. This could be achieved in a gaseous
medium by using a specially constructed dynamic siren, having rotor openings
shaped according to one of the variants shown in Fig. 71.
The separation device is a ver-
tical hollow chamber equipped at the
bottom with a radiator, inside of which
traveling sound waves were generated.
The processed gas entered the chamber
: from below and moved upward, while
. the particles suspended in it were re-
tarded by the drift forces, and then fell
out by gravity. The rate at which the
gas was drawn through the separator
was determined by the minimal dimension of the particles expected to undergo
precipitation. It can be concluded from Fig. 72, that precipitation of the smal-
lest particles required a continuous low gas velocity. However, it can be as-
a
I
Fi..71. Profile of dyne.ie airen rotor open-
In~a u.ed in eeroaol aeperetion. (We.ter..lt
end Syke).
- 170 -
-------
(
I
!
i
I
I I
. .
sumed that in the case of low concentration, the par-
. tides will create a suspended filtering layer which
can support smaller particles at increased gas flow-
through velocities. Unfortunately, no experin1ental
data has been found in support of this assumption. It
. appears that the described aerosol separation mcthod
had prospects for use mainly in the removal of dust
from small enclosed spaces which made possible pro-
longed sonication of the dusty gas. This situation
occurred, for example, in the explosive work during
the sinking of drift shafts and in mines.
Ten minutes was the time alloted for ventila-
ting the drifts. Experience had shown that the usual
sonication of mine workings provided for no actual
shortening of the indicated time. For drift sonication,
the use of special sirens with a distorted pressure
F'. 72 . . pulse shape evidently permits considerable shorten-
,~. . Sorllf'ed Venturi acrub- . d . " .
Del" propoaed by tlouch. - lng of rift somcatlon tlme. Power loss was not ex-
~ - iI.. inf loe; 2 - .ate,. 'infto.; Fected to be great in this case, since sonication was
~ - aound recepior; If - 1188 out- . .
flo. eplsodical, and no sound suppresslng measures were
required in the surrounding space, since under the
explosive safety conditions no people were present in
the sonicated area. Boucher [157] proposed that the process of aerosol par-
ticle precipitation be intensified directly in existing dust-droplet entrainment
devices by mounting compact gas-jet sound radiators at certain places in the
devices. This idea has some merit, since its realization required small capi-
tal outlay. particularly when a compressed air supply was available.
,.
Figure 72 shows that sound-equipped Venturi scrubber [149. 157]
proposed by the present author. The liquid in it flowed downward along tre
inclined walls continously directed by diverting baffles through the pipe throat,
and was scattered by the dusty gas moving at a great rate. The gas-jet whistle,
mounted inside the pipe, sonicated the mixed aerosol and, due to increased
orthokinetic interaction and turbulization, there followed a corresponding in-
crease in the number of collisions with water droplets.
It is natural that sonication should have an appreciable effect only
when the mixture remained in the sonication zone an appreciably long time.
For tbis reason, different high-speed gas purifiers were not suitable for soni-
cation, while, on the other hand, ordinary wet scrubbers and gas purifiers, in
which the gas moved, relatively slowly, were suitable. This opinion was con-
firmed by the early experiments of Boucher and Weiner [159]. The authors
studied the effect of an ll-Khz sound on the effectiveness of entraining the weak-
ly concentrated aerosol triethyleneglycol (r. = 25JJ.;~. = 79--136 mg/m3) in a
850-m3/hr gas purifier made by "Smig Industries". * The sound intensity level
* Not sure of spelling.
- 171 -
~,
~
~.
.- .
-------
--- -- --------- _,I_j.lI__II-I," Ii
. I
f
"
I
,
I
S
r
I
I
I
generated by the rod whistles
was 150 db. On the basis of the
conducted experiments the in-
vestigators stated that the soni-
cation of water mist enhanced
.' the efficiency of gas purifiers
. by 10-15%, even with a low con-
centration of aerosol particles.
Figure 73 shows the
sound-equipped conical cyclone
proposed by Boucher [157J.
The sound radiator in the form
of a rod whistle, was installed
in the rarefaction zone located
under the outlet pipe of the cy-
clone. Due to the repelling ac-
tion of the drift forces the sound
. waves created a sound barrier
in the path of the small particles
entrained by the gas. The size
of this rarefaction zone was
comparatively small; therefore,
a high sound intensity was easily
achieved with a moderate energy
consumption. The volume of
compressed air used by a whis-
tle in large cyclones was small
in comparison with the voluITle
of the processed gases. It
seems that a satisfactory effect
could be achieved in a sound-equipped cyclone only when the sound radiator
generated sound waves of sharply asymmetrical waveform, as in the previously
mentioned Westervelt - Sieck separator. The method of acoustical f.og filtration
proposed by Asklof [30 IJ, was based on a totally different physical principle.
Figure 74 presents a schematic diagram of the filter-droplet trap designed by
the author. The processed gas containing fog droplets, or dust mixed with them,
entered the apparatus from below. After the large droplets had been separated
in an internal cyclone, the gas entered through a porous nozzle sonicated from
above by a rod whistle with a reflector. Under the action of the sound, an in-
tensive gas purification from the particles occurred, and the purified gas was
released from the top. -
Ii
.
I
F i~. 73. Son if i ad cy clone: - Fig. 74. D iag,.a. of aeoilati eal
propoaad by B oucha. f i I hr-c:oagu I II tor propoaad by
I - Tangantial outfl08 nOl- by Aakloph08
lla; 2 - ga. outflollJ 3 -
I - Tangential infl08 nipple;
a atrai~ht floll rOQ-ahaped 2 - Inaide cyclone outflo. pipe;
8hiatle; 1+ - IIco~tical 3 - outtloa into ..t IIcrubbel'}
field; 5 - trajectory of 1+ - ring apray.r; 5 - c.ppln~;
particleal .;wa.ant; I) - fit
duat outfloa. 6 - aound gan.rating re ec or.
7 - lIound generator. ~ - 008-
pre.a.d ail' inflo8; ~ - .ano-
.ater; 10 - gall ...it.
In describing this method in [157] Boucher discussed the nature of
acoustical filtration as follows: "It had long ago been noted that during an in-
tense sound radiaiion over a metallic grid, the latter acted as an acoustical
lattice, transmiting and refracting the sound waves over a large area. By
- 172 -
-------
varying the ratio . X /D (ratio of the wavelength to the mean diameter of the grid
openings), it is possible to amplify and concentrate the sound in any space, or
transmit the radiated sound under a large solid angle. Each opening may there-
by be considered as a point of sound source. The interference of sound waves
may also take place.
When operating with a powerful sound generator, the grid transmitted
a considerable amount of acoustical energy. For example, an ultrasonic gas-jet
whistle of 80 w power located 0.3 m above a filtering nozzle of 5.4 cm diameter
can purify the surface layer of tap water located 0. 3 m below the openings. The
radiational pressure is usually so great that the force fields transmitted through
the openings were able to attract the suspended aerosol particles to the grid fila-
ments or to repel them from the nozzle. " Moreover, "when operating with a
liquid aerosol, a high-intensity sound field repelled certain particles, impelled
others to collide, resulting in orthokinetic coagulation, and generated a liquid
film which covered the metallic grid filaments. "
It appears to the present author that the main role in the proposed
acoustical fog filtration process was played not by the radiation pressure, which
was low for small particles, but by the gaseous medium eddy movement in the
boundary layer of the filtering nozzle elements, which enhanced the turbulent
diffusion and inertia! precipitation of the aerosol particles they contained.
Asklof affirmed [301] that by filtering a thin hydrocarbon fog through a 100-
mm thick gauze nozzle at a velocity of 3 m/sec, it was possible to lower the
residual droplet concentration by sonic at ion (f = 8 Khz; J = 0.05 c/cm8) that 10
mg/m3 to 0.5 mg/m3 of gas, i.e. , by twenty times. The author succeeded in
lowering the residual concentration to the same degree in the acoustical filtra-
tion of radioactive aerosol uranium trioxide formed in the evaporation of a 0. 5%
solution of uranium nitrate, and the susequent drying of the droplets [301, 303] .
The nozzle's capacity was 6000 m3 of radioactive gas per hour with a pressure
drop of 25--50 mm Hg O (at a nozzle height of 900 mm). The energy consumed
in sonication (f = 19 Khz; J = 0.1 w/cm2) and throttling of the gas totalled only
0.4 KWH per 1000 m3 of gas; in other cases the energy consumption reached
1.2 KWH [ 157]. The acoustical aerosol filtration method deserves serious
attention; it can attain fine purification of industrial gases, in particular, the
ventilating of factory air at an accelerated filtration rate. The sound waves
for cleaning filter bags of collected dust is of practical interest. According to
the initial plan described in Abboud's patent [ 135] , the sound generator was
mounted inside the filter so as to irradiate the filter bag fibers from the clean
side.
However, it seemed [ 157] that radiation of the dirty side of the fiber
in the schematic shown in Fig. 75 was a more effective and cheaper method of
filter cleaning. It is felt that the cleaning action of sound could be explained by
the fact that the resonating plates inside each bag generated a standing wave,
which induced a perceptible periodic gas pressure variation at nodes, forcing
the fiber to vibrate, provided the oscillation frequency was not high.
- 173 -
-------
-~'-_. ..._i_.~....i.."~.~.-
In conclusion, a method will be briefly des-
cribed, which was proposed by P. N. Kubanskii.
The method was intended for the intensification of
aerosol particle precipitation by a longitudinal
screen mounted inside the coagulating chamber.
The method is bas ed on the author's mistaken c on-
clusion [48, 49J that the eddy-like acoustical flows
. originating in the node-anti~ode intervals of stand-
ing waves supposedly enhanced the aerosol particle
coagulation. Actually, the coagulating action of the
acoustical flows was very slight, since their
whirling velocity was small, and, therefore, the
finer particles did not gravitate into the "dead"
zones formed near the walls at the junction of two
F j" 7<. <'on" f. d I t .. adjacent eddies at each node, as described in # 7.
..' J. 0> .e & eeve- ype f,lter
...r;)f)o"d Oy Abbott :Only small coagulated particle aggregates or
I - Sleeve fi lterl 2 - dust laden 9.S. '
Inf 108. 3 - aound generators; 4 - C08- large prImary particles, such as lycopodium
p"e&~ed air line;, ,n the direction Df .powder Particles [48J having radii exceeding 10..
~ound gene,..tors; 5 - resonance plates. ~ ,
6 - pure 9&& outflo-. 1 - precip'tated could possibly fall out in these zones. This fact
(uttleci do.n) dust . al d b B d d d. d
. was so note y ran t, Freun , an Hln emann
[167J who pointed out that the local aerosol par-
ticle precipitation at the nodes was a secondary e.ffect of sonication which fol-
lowed the process of particle coagulation.
I'
11~~i
_I _L..-----iii""-"'-~ - 1
s
It should be added in a supplemental way, that according to Rayleigh's
equation (1. 25), the acoustical flow velocity in the direction of the wall decreas-
ed linearly as the clearance between the walls decreased. For this reason the
additional baffles failed to increase the amount, and even the coagulated particle
aggregates precipitated at the nodes.
~, i
..
- 174 -
-------
APPENDIX
#1. Evaporation and Condensation Droplet Growth During
Fog S onic ation
In fog sonic at ion, in addition to the acceleration of the coagulation
processes and droplet precipitation, phase transformations characteristic of
these systems have also been observed to accelerate. If the gaseous medium
were not fully saturated with the liquid vapors, then fog sonic ation could pro-
mote acceleration of droplet evaporation occurring in the system. However, if
the gaseous medium were sufficiently supersaturated with liquid vapor, then
the sonication could add to the acceleration of the attendant fog droplets1 con-
densation growth, if the latter were not too great. *
Under certain conditions, the above processes suppressed coagula-
tion and droplet precipitation, and came to the forefront, thus permitting the
use of acoustical oscillation as a means of intensifying natural fog dispersion,
the drying spray solutions, the separation of hydrocarbon gases, etc.
Examine the theoretical principles underlying the intensifying sound
effect on evaporation and condensation growth of fog droplets. It has been
known that the process of droplet evaporation consisted of two elementary pro-
cesses [ 16]: a) separation of the molecules from the liquid surface and for-
mation of a saturated vapor layer, and b) diffusion of molecules from the
saturated layer into the surrounding medium. With these concepts in mind,
Maxwell used the following equation to derive the rate of evaporation from the
droplet surface in a stationary medium [16, 123]:
where D -- coefficient of vapor diffusion [128]: c iat -- concentration of the
saturated substance vapor: c » -- vapor concentration in the surrounding
medium. It can be assumed for a droplet system constituting a fog, with good
approximation that each droplet evaporated at a rate as if it were in a vessel
•with non-absorbing walls of a volume equal to the mean volume of one droplet
in the system. Based on this premise, the theory yields the following expres-
sion for the rate of fog droplet evaporation [ 123]:
3rD 1 (AI.2)
where b -- radius of the vessel, and t -- time.
* Experiments of V. V. Bazilevich and N. P. Tverskii [5] showed that spni-
cation also contributed to the freezing of supercooled fog droplets; however,
since this phenomenon is of no great interest we shall not dwell upon it here.
- 175 -
-------
Due to some "blow-off" of the diffusing boundary layer in a moving
medium, the droplet evaporation rate is usually higher than in a stationary
medium by fn times, where fw is, the so-called, "wind" factor:
Here, (3 -- constant equal to about 0.276; Sc -- Schmidt number, which re-
presented the ratio Y/D> equal to approximately 0.7 for water vapor in air
under ordinary conditions; Re -- Reynolds number characteristic of the flow
around the droplets: Re = 2 rugp/y. In this case, if K is defined as k = (3 Sc|- ,
the rate of droplet evaporation could be determined by the following Fressling
equation: _
— c,o)(l +/CVRe),
This equation leads to the conclusion that the droplet evaporation
rate in a flow increased relatively slowly with increase in the Reynolds num-
ber. Moreover, Kinzer and Hahn's experiments [ 123] showed that when Re <
1, i.e. , in the case of a viscous flow regime around the droplets, the rate of
their evaporation generally did not depend on the rate of the relative motion of
the medium, but on the Re number only. This is due to the fact that the con-
centration and temperature fields around the droplet in a viscous flow were
distorted by the motion of the medium. The circulation inside the droplet
generated by the droplet friction and the gaseous medium which prompted a
temperature equalization in the droplet, also played no significant role at low
Re numbers. In turn, as the Re number increased from 1 to 1. 75, a rapid
twofold increase in the evaporation rate took place, after which the evapora-
tion rate rose more gradually (at Re = 3,' fw =2.5, etc.).
Assume that the accelerating effect of the medium's motion on the
droplet evaporation was apparent beginning with Re ot = 1, then it could be
concluded that the increase in evaporation rate began to appear only at the
following critical value of the rate of the relative motion of the medium:
Uor = v/2, (AI.5)
It follows that the critical motion velocity of the medium rapidly in-
creased with the increase in particle dispersion, and was thereby, generally
speaking, a significant figure. Thus, with particle radius r = 7.5p., the
critical velocity is 100 cm/sec, and for droplets with r = l.Sp,, it is 500 cm/
sec. For comparison, it shall be noted that a three-fold rate of the same
droplets was only 0.66 and 0.026 cm/sec, respectively. It is characteristic
of acoustical oscillations that they aid in creating motion of a relatively high
velocity of the gaseous medium in relation to all the fine particles suspended
in the medium. In the critical case, which occurs at elevated oscillation
frequency values, for which p,g = 1, the flow rate around the droplets becomes
equal to the oscillatory gas velocity.
The minimum water droplet radius is about r = 2p,, which assures a
good flow of air around them ((j, « 0. 8) oscillating with a critical frequency of
3.5 -- 4.0 KHz for large industrial installations.
- 176 -
-------
The theory proves [123] that if the droplets were not too large, their
evaporation under nonsteady- state can be regarded as a quasi- steady- state
process. Hence, it can be concluded that equations (A1.3) and (A1.4)presented
on the previous page, held even in an oscillatory motion of the medium, if the
amplitude of the oscillations was great in comparison with the droplet radius
Ag » r. The critical sound force level at which the intensifying action of
acoustical oscillations began to appear in the process of droplet evaporation
in an air medium is, according to (Al. 5), equal to:
LHP = 1 64—20 log (n, r) d6. ( A1 ' 6)
At such sound levels the gaseous medium could be agitated causing the droplets
to flow around as turbulent pulsations of the medium. Moreover, as demon-
strated in #11, around every particle suspended in a sound field, even at Re <
1, an acousticalflow was generated with an eddy scale comparable to the
oscillation amplitudes .
The question can now be asked: what is the effect of these phenomena
on the droplet, evaporation process in a sound field?
It is possible that the two phenomena had practically no supplemental
effect on the droplet's evaporation process, since the turbulent pulsation rates
and acoustical flows were, as a rule, considerably lower than the critical
velocities determined by equation (Al. 5). R. S. Tyul'panov [118] attempted to
determine, approximately, the variation in droplet radii in the course of their
evaporation in a pulsating flow, assuming the following idealized model of the
process. The author assumed that during the first half of the half -period of
the medium's oscillation, vapors accumulated in the droplet boundary layer
and only partial evaporation from. its outside surface occurred. For the next
half of the oscillation half-period, part of the boundary layer "blows-off", and
a corresponding molar discharge of the vapor accumulated near the droplet
surface. Stating with these assumptions, the author obtained the following
expression for the droplet radius: " , — - - - — - - - — ^- .. .
, r = V 0,328 y- + (rl — 0.328 -) e~ ^
(or -- initial droplet radius). However, only very large particles may be com-
puted with this formula.
It has been known that the droplet size decreased, and the flow rate
decreased, in the course of evaporation resulting in a decrease in the Re
number. As a result, the droplet evaporation rate gradually decreased, and,
with Re less than 1, the intensifying action ceased. Only the parallel particle
coagulation process could retard the onset of this moment; however, droplet
coagulation had a high efficiency only at a sufficiently high particle weight
concentration. In this connection, it seems that coarse-particle fog sonica-
tion was more effective than the sonic ation of less dispersed fogs. Actually,
- 177 -
-------
this is not the case. The total amount of moisture evaporated by all particles,
allowing for (9.1) and (A1.4) equals:
—' (A1.8)
** • cat
where n - computed concentration, and k -- is the droplets1 weight concentra-
tion.
Taking into consideration that the second multiple in parentheses
increased with increasing r more slowly than 1/r2 , it can be seen that the
coarser the fog particles, the lower -was the total amount of evaporated mois-
ture at the same particle weight concentration, and,, under the condition Re >
1, and \i g -* 1 . The intensifying sound action on the evaporation rate was first
proposed for use in solving problems of natural fog dispersion.
It was brought out in #18, that the acoustical method of fog disper-
sion had a limited dispersion range due to its high energy consumption, and
several other disadvantages. In this connection, the idea arose of using soni-
cation in connection with fog heating and, thus, solving the fog scattering
problem by droplet evaporation in a sound field. Rodebush suggested this
first in 1943, in his report on tests with the acoustical fog scattering method.
The new combination fog-dispersion method has been rapidly devel-
oped in recent years by Boucher [ 156] under the name of the ther mo- acoustical
method. It has been previously indicated that the thermal fog dispersion meth-
od necessitated great heat quantities, due to the fact that strong convection air
flows -were generated during the atmosphere heating; the warmer air rose
while the surrounding cooler air, which replaced it, also contained fog drop-
lets. As a result, quantities of fuel were consumed not only in air heating and
in droplets' evaporating in the sector where the fog was being dispersed, but
also in heating the additional surrounding air. It follows that fog droplets
must be evaporated as rapidly as possible to avoid influx of large cold air
masses. This, according to the author, must be done by fog sonication.
The first experiments in thermo-acoustical fog dispersion conducted
in Arcata (USA) in 1950 had been encouraging. A J-33 turbojet engine with an
afterburner (3 • 108 BTU /hr) was used as the air heater, and U-2 and U3A-1 type
dynamic sirens generating about 5 kw acoustical energy at a 3. 5 Khz frequency
were used as the sound source. The fog was scattered from a distance of 300
m and a height of 90--150 m. On the basis of the obtained results Downey and
Smith estimated that ten J-47 type turbojet engines together with sonication
could perform the work done by the FIDO installation (109BTU/hr) in five min-
utes, which is equivalent to a heat output of 50 J-47 type engines. Thus, a five-
fold decrease can be obtained in heat consumption as compared .with the ther-
mal method.
- 178 -
-------
It is highly significant that much acoustical energy was generated as a
side product of the turbojet stream itself. It was assumed that the J-47 engine
radiated from 10 to 25 kw of acoustical energy, the main part of which was in
the low frequency region of the order of 0.3 -- 0. 6 Khz, for small engines --
0.15 Khz, which was useful for the process. This suggested to Boucher the
idea of completely eliminating the sirens from the installation and using the jet
burners with which the air was heated as sound generators. It is calculated
that at least 1% of the jet energy was transformed into acoustical oscillation
energy, however, by installing resonators mounted in the paths of the jets,
this figure can be increased by 5 -- 10%. In this case, the exhaust pipe will be
similar to a Hartman-type shock-jet whistle. In place of the jet burners de-
signed in the above-described manner, vortex whistles described by Greguss
may also be used. Sonication combined with radiant heating of the fog is even
more effective. In fact, the hot gases exhausted by the FIDO heating installa-
tion, similar to those of the jet engines, contain a considerable amount of
water vapor which raises the moisture content of the fog air and thereby
makes fog droplet evaporation more difficult, especially in still weather.
Where heating is alternately turned on and off for economy, it happened that
the content of the water vapors at times, exceeded the initial moisture content
of the atmosphere. This may have occurred, for example, in 1953 in Los
Angeles with the LAX-FIDO installation with which they could not raise the
visability in excess of 180 m, (actually the. heat output here was small).
The use of infrared heat bodies emission to evaporate fog, which is
a rather good absorber of this type of energy, can exclude such unusual cases.
It has been experimentally proven that atmospheric fog mixed with water vapor
absorbed about 50% of the energy radiated by a black body with a temperature
of 3.00 -- 1100° C at a distance of 90 m. However, all assertions require
positive proof, since the noted presence of the critical flow velocity around the
droplets as shown by [equation (Al. 5)], was not taken into account in formula-
ting the propositions. Boucher had also proposed to combine sonic ation with
atomized hygroscopic and other chemical substances indicated at the beginn-
ing of the section. This fog dispersion method which henceforth shall be re-
garded as the acousto-chemical method was described in journal "Interavia"
[152].
The purpose of sonication in the case of atomized hygroscopic sub-
stances is to speed up the -water vapor condensation on the introduced con-
densation nuclei and the evaporation of the fog droplets. The scheme for
carrying out this method for landing strips is shown in Fig. 76. Thus, it is
seen that in this method the sprayers and sound sirens are positioned alter-
nately along both sides of the runway and directed upward vertically. The
hygroscopic nuclei entered the area affected by the sound field following the
convection flow of air. In this case, it is assumed that the height to which the
hygroscopic nuclei were lifted, an operation, which previously required large
amounts of energy, can be reduced. In place of the hygroscopic substances,
- 179 -
-------
',j
"
Fig. 76. DiRgra. of . thero.eou.tical ..thod for .erodro.. "foE liftin9" accord-'
inil to Bouche.
I - Path leading to ticket office; 2 - Ther.ic spraying generator; 3 - acoustical
siren; 4 - co.pressed air conduit; 5 - co.pressor; 6 ~ electrical linl; 7 - visi-
bility recQrdlr; 1:1 - fuel line.
such substances as silver iodide and others may be blown into the fog by the
cold air.
According to reports by the "French Society for the Use of Ultra-
sound", 30 spray nozzles and 30 static sirens of the Boucher system proved
sufficient for a runway 1000 m long. Such an installation was less costly than
the FIDO installation. Each spray nozzle, including the burner and blower,
requires about 7 kw, making a total of 210 kw. The air compressor for power-
ing the sound siren required about 250 kw. This is more economical than the
FIDO thermal fog dispersion system. It has been reported that the acousto-
chemical fog dispersion method, patented by Boucher, had been tried in
several foreign countries: in France, by the Society for the Use of Ultrasound,
in England by the Aerojet Company of London, in Sweden by the Aerojet
Venturi Company of Vekso, and in the USA by Halton Industries Co., of
Matachun, N. J.
In addition to natural fogs, there were many industrial aerosols,
which, when being prepared or used, required accelerated liquid evaporation
from the surface of the dispersion phase. This was the case in spray drying
viscous solutions in the production of soap, dried milk, and other similar
products; in cooling hot gases by water spray in scrubbers and other installa-
tions; in igniting vaporized liquid fuel in internal combustion engines, in
liquid jet engines, and in industrial furnaces, etc. Several of the enumerat-
ed proposals have been patented [202, 207].
Horsley and Danser of the US Ultrasonics Corp., obtained a patent
- 180 -
h.
-------
in 1951 for an acoustical damp products dispersion method [202] of the following
characteristics:
1. The method of processing damp substances included the injection
of different size particles of damp substances into a hot gas to form aerosols
and to bombard the resultant aerosol with high-intensity sound waves with a
sound intensity level of at least 150 db at a frequency of 3. 5 Khz; in consequence,
the sound waves not only promoted the moisture evaporation, but also formed
dry particle aggregates.
2. The method of processing damp substances entailed the injection
of different size particles of the damp substance into a hot dry gas stream and
the generation of sound waves with a sound intensity of at least 150 db at a fre-
quency of about 3. 5 Khz in order to impart to the particles a velocity -which
varied according to the particle dimensions, and then separated the substance
from the gas. This proposal was to be used in drying soap solutions, and
for other similar purposes; however, no published information on its realiza-
tion had been presented. Evidently, the energy consumed in the sonication of
the drying chamber required to obtain good drying results was too great for
such an enterprise as soap manufacturing. However, there were other enter-
prises in which sonication drying by atomization could be successful.
Turning to the intensifying effect of sound on condensation growth of
fog droplets, the following is observed:
The first indication of possible condensation intensification -with
vibrations appeared in the little known work of Karlstrbm (Svensk Tandlak
Tskr, 1950, 43, 285), who proposed to use this effect in the production of
dentistry amalgams. Of great practical interest was an attempt made by the
present author in association with technicians of the Kuban Gas Industry to
intensify droplet condensation and coagulation growth during natural gas se-
paration. The sonicating gas separators proposed for this purpose had been
previously described in #19 [79, 84, 80]. Therefore, here, only a brief
discussion will be presented of the theoretical bases underlying the intensi-
fying action of sound on the condensation growth of hydrocarbon fog droplets
in the described gas separators.
It has been demonstrated [ 123] that the process of condensation
droplet growth proceeded according to the laws of their evaporation. Accord-
ingly, it follows that all equations and arguments previously presented by
the present author also hold for this case. However, conditions in the gas
separators were such that greater success could be expected from fog soni-
cation than from other cases. The hydrocarbon vapor formed in the gas
separators was under excess pressure (p - 50 -- 55 at gage), indicating that
the Reynolds number characterizing the velocity of the gas flow around the
droplet was about p times greater than under atmospheric conditions, since
- 181 -
-------
7) practically did not vary and pg increased p times. At oscillatory velocity of
u cp = 100 cm/sec the Reynolds number for condehsate droplets of radius r =
1 -- 100 p, was in the range Re = 7 -- 700, and the Schmidt number Sc remained
within the usual limits. Accordingly, the wind factor characterizing accelera-
tion of the condensation droplet growth was in the range of f,, = 2 --10, which is
significant.
It should be noted in conclusion that the discussion presented earlier
on the kinetics of droplet evaporation in a sound field also held in relation to
the heat transfer from the droplet to the medium and vice versa. In this case,
when Re < 100 the following equality prevailed [123]:
Nu-2(H-pPr'/.ReH (Al. 9)
where Nu - Nusselt number, Qf/2ir y Xg (T - To), and Pr - Prandtle number;
Y Ag equal in air under normal conditions to about 0.8, Qf -- amount of heat
transferred from the droplet to the medium per unit of time; Xg - heat transfer
coefficient of the medium; Tm - To -- temperature difference between the body
and the medium: (3 ^ 0.3.
00
Studies discussed in [223, 289] are of interest in this connection.
#2. Sound Effect on the Processes of Combustion and Degas
ific at ion of Fuel Droplets
It has been known that liquid fuel effectively ignited only in atomized
form, i.e. , in the aerosol state. Gaseous hydrocarbons were liberated in a
similar form from crude oil solution. The sonication of such aerodispersed
systems induced an actual acceleration of the processes operating in them,
i.e. , the combustion and degasification processes, while the coagulating sound
action had practically no effect on coarsely dispersed droplets. The physico-
chemical relationships between combustion and degasification of liquid fuel
droplets had little in common; therefore, they will be examined separately.
Liquid fuel droplets combustion in a burner induced evaporation of
the individual droplets and the burning out of the vapors near the droplets and
in the space surrounding them, which is according to the law of gaseous fuel
combustion.
It has been known that acoustical oscillations enhanced the process
of droplet evaporation, which offered the first explanation to the intensifying
effect of sound on the process of droplet combustion. In addition, acoustical
oscillation evidently affected even the combustion of liquid fuel vapors, since
a strong turbulization of the gaseous medium had been generated. In any case,
the data presented by Boucher [ 157] on the acoustical intensification of solid
rocket fuel combustion leads to such a conclusion. Experimental research on
• - 182 -
-------
the intensifying effect of sound on the combustion of an atomized liquid fuel was
confined only to the work of Greguss [28, 196J. The author studied the effect of
acoustical oscillation on the combustion of oil in a revolving furnace for the
smelting of cast iron in a casting-mechanical factory in Budapest.
-.. .._-.... -n.
I
. . -....: ',,~ ,~.:v.'
\
-Failure of Tarnoczy and SomhegyiJs
experiments [276J conducted on the inten-
sification of a gas flame combustion by
external sonication, led Greguss to sonicate
the burning mixture from within. For this
purpose, the author constructed a special
"acous tical burner nozzle" out of a standard
vortex whistle which was provided only with
. an additional opening for the injection of
liquid fuel (Fig. 77a). Acoustical oscilla-
. tions were thereby generated directly in
-the burning mixture; in this way the reflec-
tion of the sound waves from the boundary
by the flames, which occurred in the
Tarnoczy - Somhegyi experiments, had
~ been avoided. Moreover I the revolving
. flame stabilized the combustion process.
II
, ,
,
Fi~. 77. ~r89v.. - 8xperi.ent. in the Inten.ifi-
cation of aprayed liquid fu.1 C08,DU.U~ ~n a Figure 77b shows the whistle-burner
furnac. (h.arth) for ...Itlng cok. PlY Iron
a -conatructlon of a .n...I. whi.tl. .p!'ay y~n- arrangement'in a furnace. The oscillation
...ator (I - whi..l. c:ha.b.... 2 - tanw.ntlal air f d b h hi 1 b
inflow; 3 - outflow; It -lu.1 d.liv...tJ D - requency generate y t e w st e- urner
Oi.9ra. of h.arth with bui It-in wh.rl. ty1'8 was 4 Khz while the sound intensity level
.hi.tle type .pray generator. '
reached 150 db. According to the author,
the acoustical efficiency of the whistle was
10 -- 15%. With the aid of sonication, the
temperature required for smelting was reached after 40 - - 50 min. instead of
one hour; fuel consumption dropped 10%. Moreover, metallurgists confirmed
that the casting obtained after this was of a better structure and more suitable
for thermal treatment. Greguss explained the increase in efficiency of atomiz-
ed liquid fuel combustion by the following:
1)
a higher degree of liquid fuel atomization;
2)
intensification of liquid fuel droplet evaporation;
3) increased probability of liquid fuel droplets and moleculcs
collision with the oxidizer molecules, which ensures a better approximation to
stoichiometric ratios;
4) greater uniformity of temperature distribution around the re-
volving flame, ensuring a more homogeneous diffusion factorj
- 183 -
~, .
:~.
-,
- "->
.'
-'.....
~ ;
-------
5) increase in the flame front boundary stability which increased
the completeness of fuel droplet combustion.
The quantitative relationships characterizing sonication of atomized
liquid fuel combustion have not been derived.
There seemed no doubt that acoustical oscillation intensified the
combustion suspended solid fuel particles; however, appropriate data have as
yet to be published.
It should be noted in conclusion, that the possibilities of acoustical
oscillation in the intensification of liquid fuel combustion have not been ex-
hausted in the reports.
Self-oscillating processes often arose in fuel combustion, including
atomized liquid fuels, due to the interaction between flame and air. They
originated from the pulsating character of fuel combustion. This form of com-
bustion, called vibrational, or sometimes pulsating, arises in nearly all heat-
ing and thermal-powered installations, and, in most cases, it played a negative
role in hindering the normal course of combustion; it generated noise, vibra-
tions, and sometimes damaged the furnace and the combustion chamber. In
addition, this form of combustion differed by its increased velocity, and its use
assured considerable gains in increasing the heat load of pulverized coal fur-
naces, of liquid rocket engines, and of industrial furnaces in which the fuel was
ignited in an atomized form. However, the principles underlying vibrational
combustion are not connected with the fuel aerosol state; therefore, an exam-
ination of this process was regarded as beyond the scope of the book. It is
suggested that students interested in this problem familiarize themselves with
reports in [ 106] and [71] and with B. V. Rauchenbach's monograph Vibration-
al Combustion.
The degasification effect of sound on a liquid located in a gaseous
medium is based on the same principle as the vacuum dryer. The latter is
based on the fact that as pressure in the medium decreased, the amount of
liquid being evaporated increased. This follows from Dalton's law which
established that the rate of evaporation was connected with the vapor pressure
in the surrounding medium by the following relationship [ 128]: .
jp = ks Ps - POO/H (Aii.i)
where pfl -- saturated vapor pressure of the liquid temperature; p -- vapor
pressure in the surrounding space; H -- barometric gas pressure; S -- area
of evaporation surface; k -- coefficient dependent on the character of the flow
around the surface of the medium.
During the propagation of the sound wave at the moment of compres-
sion, no changes take place on the surface of the liquid droplet, since the med-
- 184 -
-------
ium is not saturated by the liquid vapors. Then, at the moment of rarefaction,
when the pressure in the medium drops, an additional separation, i. e., desorp-
tion, of the gas from the liquid occurs. A small amount of rarefaction is
thereby generated at the droplet surface which is compensated for by the in-
staneneous repetition of the process. An important experimental confirmation
of the degasification effect of sound on atomized liquids was recently obtained
with a by-product gas from oil at the Korobkovskii deposits in the Volgograd
oblast'. The gaseous by-products are light hydrocarbons which are dissolved
in large amounts in oil and are separated from it by a pressure drop which
occurred in the course of the oil's passage along the borehold shaft and then in
a special separator serving as a trap.
A turbodynamic siren was mounted in the trap inlet pipe for the in-
vestigation of sonic at ion effect on the process of degasification of atomized oil.
The oil-gas mixture was fed through the siren which began to turn it, generated
the sound, and liberated the atomized oil together with the gas into the trap.
Experimental results established that separation of gaseous by-pro-
ducts from oil increased by 13 -- 32% during sonication as compared with
standard means of atomizing oil. However, it has not been ruled out that the
improved oil siren atomization played a role in these experiments. Should
further investigation confirm these results, then sonicating traps will come
into wide use in the oil-gas industry. Naturally, a more reliable, nonrotating
sound generator, like the vortex whistle-burner, should be used instead of .a
turbodynamic siren.
#3. Behavior of Precipitated Solid Particles in a Sound
Field of an "Acoustical Dryer"
Substances passing from the suspended state into a powder generally
are not aerodispersed systems, expecially if their pores are filled with mois-
ture. However, taking into consideration that many powders and pastes are
products of aerosol precipitation and that they contain a coagulated dispersed
phase, it is of interest to examine their behavior in a sound field. Dry powders
reacted to sonication only with a periodically "muddying" of the layer in re-
sonance with the pulsation of the static gas pressure over its surface, or by
forming peaks at the oscillation nodes. This has been observed not only in
parallel sonication of a powder layer, as in a Kundt pipe [218], but also in
perpendicular sonication when transverse oscillation nodes were generated.
The cause of local powder accumulation in both cases was the acoustical flow
generated in the node-antinode intervals. They attracted the precipitated par-
ticles from both sides to the node line, and the lightest particles became sep-
arated from the layer, forming small dust clouds. Damp powders are dried in
a sound field, without any perceptible temperature rise, which drew particular
attention to the acoustical drying method.
- 185 -
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Examine the existing hypotheses on the mechanism of acoustical dry-
ing of materials, It is known that the drying of damp porous materials consisted
of two stages: the evaporation of the liquid from the surface, and its transfer
from the internal pores to the surface. Boucher [154, 155] assumed that acous-
tical oscillations acted only in the first stage, i. e. , they increased the liquid
evaporation rate. He cited Dalton's Law (AII.l), considering that when the
sound waves passed along the damp surface, no variations occurred in the com-
pression areas; whereas in rarefaction areas, an acceleration of the liquid
evaporation occurred similar to the one which took place in the degasification
of droplets, described above. According to Boucher, turbulization of the soni-
cated medium facilitated moisture evaporation.
Greguss, accepting the described mechanism in the initial drying
stage, assumed, on the basis of experiments with the drying of products with
little moisture, that the sound wave also aided in transferring moisture from the
internal pores to the surface [ 197] . It is known that as far as drying was con-
cerned, the rate dropped due to decrease in the material moisture conductivity
medium surface. Acoustical oscillations, according to Greguss, increased the
material moisture conductivity owing to:
a) decrease in the liquid viscosity achieved, according to Altenburg's
data, during sonication, and aiding the acceleration of liquid diffusion from the
pores;
b) the pulsations of bubbles held in the pores and capillaries during
the periodic variations of the medium's pressure and temperature aiding the
liquid separation from the pores and capillaries:
c) the radiational pressure directed into the pores and capillaries
from the liquid into the gas which moved the liquid column outward.
Both investigators, as well as their successors, ruled out the effect
of temperature rise in the surface capillaries on the drying process, which
occurred due to capillaries absorbtion of the incident sound waves energy (see
formula [ 1. 18]); they cited the fact that acoustial drying occurred in a "cold"
method. On the basis of initial drying studies conducted in cooperation with N.
M. Gynkina [8] (see also Acoust. Zh. , 1962, 8, no. 1), Yu. Ya. Borisov
suggested that the deciding effect on drying acceleration was the acoustical flow
generated in the sonicated gaseous medium. His conclusion was based on the
experiments which showed that the beginning of the intensifying action of the
sound nearly coincided with the suspension of the dry powders at the nodes of
standing wave. This, as is known, was due to the effect of the acoustical flows
along the node-antinode intervals. However, this hypothesis, like previous
ones, had not been confirmed.
Experiments conducted in the Ultrasonics Laboratory of the Acous-
- 186 -
-------
tical Institute AS USSR demonstrated that greatest drying acceleration occurred
at the anti-nodes of the oscillatory velocity; where the acoustical flow velocity
was finite, drying has not been accelerated. Sonic pressure was minimal at the
acoustical velocity nodes (theoretically it equals zero), indicating that the
acoustical drying of materials occurred not according to the.principle of vacuum
drying, as Boucher supposed, but according to the principle of convective dry-
ing. Experiments also established that the drying effect in a sound field was
almost equivalent to the effect of convective drying,, achieved at an air flow rate
equal to the oscillatory velocity. This conclusion has .something in common
with the conclusions relating to atomization drying in a sound field (see Appen-
dix I) which is completely natural. The experimentally established fact that the
rate of acoustical drying depended little on the~~emplbyed oscillation frequency
becomes understandable in the light of this observation.
The first experiments with the acoustical drying of materials had been
conducted in 1955 by Greguss, who, with the use of a dynamic siren, obtained
almost a ten-fold acceleration of raw cotton fibers drying at a frequency of 25
Khz. The drying of powdered products was investigated in detail by Boucher.
The following hard-to-dry-products were used: carboxylmethylcellulose,
titanium.dioxide paste, colloidal zirconium hydroxide, and also silica gel,
enzymes, and hormones. Figure 78 shows the results obtained in the drying
of the first three products; it is seen that the sonic at ion ensured a 6 -- 8-fold
drying acceleration.
Silica gel, with a 25% initial moisture Content was fully dried in 15 min.
at a sound intensity of 152 db and at a frequency of 8
Khz; in the vacuum drying method and in thermal dry-
ing by heating up to 92° C with air, only 10 -- 15% of the
moisture content could be removed in such a short time.
Enzymes, which cannot endure heating above 4dP C,
were dried in 14 min. , while the drying rate of thermal
sensitive hormones was increased 3 -- 4 times as
compared with the vacuum method.
Fig. 76. Results of acoustic*I
drying of soo« po.der.d actor- . Boucher also investigated a series of cotton
, - ciMl.STJiitl'SSS- materials, blotting paper, asbestos cartons, and
ioe »t f « |0 kHz; 2 - cj^xy- others [154, 155, 157] ; results of these experiments
aethy I eel lulose at f " •** lc"2> L * ' J r
3 _ tiuniua oeroxid* at f * 9-5 will be discussed at some other time.
SHij l«, 2», 5» sa«e ••tartaU
dri«d natural ly .
Experiments with the drying of ethylcellulose
were conducted at the Acoustical Institute AS USSR
[8]. This material was selected for the first experi-
ments because it oxidized easily and could not be dried at high temperature.
At Stf5 C it dried only in several hours. Due to the high dispersion of ethyl -
cellulose, the material became sonicated in a layer strewn over the bottom of
a small bath tub.
- 187 -
-------
Figure 79 shows the results obtained. Thus, at a sound intensity of
152 db the material completely dried in 60 min. at normal temperature
(20° C), and in 45 min. at 32° C. A change in the sound pressure of 3 db,
equivalent to doubling the sound energy density, accelerated the drying by a
factor of 1. 5.
On the basis of the experimental data,
it can be stated that the following had an effect
on the acoustical drying rate: initial moisture,
the form of moisture adhesion to the material,
X
the thickness and structure of the layer, the
___ . _ presence of material agitation, and sound in -
a a sa M u M se 19 so sg .* • ° .
:.. Tiae in oinutos tensity. Moreover, the threshold sound inten-
'•F.8. 79. Rosuits of art.ficiai dryinB ! sity, below which the drying process failed to
.lathy Icel luloso by tho AcoMetical &nst(- L a/»r-el erateH waa
;.tufco, Acodecy of sciences,, USSR, Decome accelerated, was
- Qt f = 6.7
20% '2 - at f =2.8 hHzj L " 152 <">» It has been established that in the pro-
»^i ° 32»c^f^~- ot f °*2.» ;pagation of sound waves along the surface of the
of Iw7li\l£l*20'£ ;material being dried, the drying process was
only half as fast as when the waves strike di-
rectly. In the present author's opinion, this
could be explained by the folio-wing: when
striking directly,' the sound -wave partially penetrated into the material layer
and easily flowed around its individual elements; the distance between the
elements was slightly greater than their dimensions; this is obvious, even
though the porosity of many materials reached 80 -- 85% and higher. It is
also confirmed that the drying rate depended only partially on the oscillation
frequency, even though at lower frequencies the process was somewhat more
active, evidently due to the -weaker absorption of sound by the medium. This
was contrary to Boucher's initial assertion [172;] , which postulated that the
drying process proceeded smoothly only at frequencies above 10 Khz, and
that 9 Khz was the lower limit at which there was still a noticeable drying
acceleration. He later established that even the lower frequencies enhanced
the acceleration of drying, and that the most effective frequencies were in the
6 -- 10 Khz range [155] .
According to data accumulated by the Acoustical Institute AS USSR
where the drying process had been studied in the 3 -- 15 Khz frequency range,
the results obtained at 2.8 Khz were better than at 6. 7 Khz (see Fig. 79).
The most promising area for the acoustical drying method application,
which is distinguished by its high energy losses, is the drying of easily-oxidi-
zed, low-melting, thermol- sensitive, and explosive materials for which other
methods were not feasible, or which yielded poor results in the final drying
stage. The most expedient use was in the acoustical drying of powder -form
products not in a stationary layer but rather in barrel dryers, or in the
suspended state.
• - 188,'- ' ••-•'•'
------- |