TT 65 61965
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U.S.S.R. LITERATURE ON AIR POLLUTION AND
RELATED OCCUPATIONAL DISEASES
Volume 11
AERODYNAMIC PRINCIPLES OF INERTIA SEPARATION
By
A.I. Pirumov
Candidate of Technical Sciences
Translated and Arranged
By
B.S. Levine, Ph.D.
Introduction to translated edition by
Knowlton J. Caplan, MS ChE
Senior Research Engineer
Hart-Carter Co.
Minneapolis, Minn.
This survey was supported by
PHS Research Grant AP-00176
Awarded by the Division of Air Pollution
of the Public Health Service
Department of Health, Education and Welfare
Distributed by
CLEARINGHOUSE FDR FEDERAL SCIENTIFIC AND TECHNICAL INFORMATION
Springfield, Virginia, 22151
-i-
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STANDARD
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March 8, 1965
INTRODUCTION TO ENGLISH TRANSLATION
Volume 11 of this series of translations by Dr. B. S, Levine is
essentially an attempt to analyze cyclone dust collector performance by
mathematic derivation from various fundamental physical phenomena.
Each section of the conventional cyclone design is considered separately,
and a few types of axial flow and peripheral discharge cyclones are con-
side red.
The author usually begins with fundamentals of physics, and
makes some necessary simplifying assumptions, yet soon gets involved
in partial differential equations, seemingly without productive result.
The author arrives at the conclusion that present theory is not very
useful in providing scientific basis for design of cyclones. Some old
irrational theories are dispelled, and an experiment is reported which
shows no significant difference in performance of four different types of
well-designed cyclones. The treatment is primarily theoretical mathe-
matics, and there is little empirical or application information, and
what there is has been covered extensively in the American literature.
The primary value of this work is its demonstration of inadequate
present theory, and its possible use in selecting areas for further
theoretical study.
Hart-Carter Co. Knowlton J. Caplan, MS ChE
Minneapolis, Minn. Senior Research Engineer
-v-
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Table of Contents
Page
Chapter 1 - , • ,' " •
INTRODUCTION . " , -I
Present Day Technique of Industrial Discharge
Purification from Dust ' .1
Chapter II ' , '
CALCULATION OF CYCLONE SEPARATION CAPACITY ' 10
The Gravitational Theory of Cyclone Separation , -. 10
'.. The P. N. Smukhin and P. A. Koiizov Centrifugal-Theory - 12
i. The Muhlrad-Davies Formula - '-18
L Estimating Dust Catchers Efficiency . •' .22
Chapter III
MECHANICAL ANALYSIS OF THE INTERACTION BETWEEN A
PARTICLE AND THE MEDIUM IN A CURVILINEAR STREAM 27'
Aerosol Particle Inertia Resistance to the Carry-Away
Effect of a Curvilinear Stream • " - 27
!. Differential equations of Dust Particles Movement in a
Curvilinear Flow - 30
i. Solving the Differential Equation of a Particle Movement . / 35
"-. Separation of Particles at the Initial Movement Section .'- " 38
>. Separation of Particles Beyond the Initial Section - 44
Chapter IV
AERODYNAMIC EFFECT OF TRANSVERSE VELOCITY GRADIENT
FLOWS ON BODIES CARRIED ALONG-BY THEM . 47
. Applicability of Stoke s Formula to the Condition of Ambient
Curvilinear Flow Around Suspended Bodies 47
-vi-
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Table of Contents (Cont'd)
- - Page
Chapter IV (Cont'd)
Z. The_Taylor Theorem 49
3. Experimental studies of movements of S olid Bodies
in a Rotating System 55
4. The Taylor Effect 57
5. A Case of Potential Medium Movement.
Effect of Particle Rotation on the Flow - 6t
6. Resistance of a Rotating Particle 67
7. Adjoined Vortex of a Rotating Particle ' 68
8. Dust Particle Movement in a Bordering Layer 71
Chapter V
SOME AERODYNAMIC CHARACTERISTIC
OF CYCLONE APPARATUSES 76
1. Pressure Distribution in Cyclone Apparatuses 76
Z. Distribution of Rotation Velocities at the Bottom of a
Cyclone Apparatus - Radial Flow 81
3. Effect of Radial Cyclones Flows on the Dust S eparation
Efficiency 85
4. Effect of Turbulence on Cyclone Separation Efficiency 90
5. Effect of other Hydrodynamic Factors on Cyclone Separation
Efficiency 94
CONCLUSION 100
-vii-
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Chape r 1
INTRODUCTION
" ' 1. Fresent day technique bf^ihdus trial"
discharge purification from dust
Solid materials reduced to a state of minute particles acquire new
and qualitatively different properties. For instance, practically inert
cement clinkers reduced to powder acquire the well known binding property.
Reducing some materials to a state of high dispersion facilitates certain
technological processes. It is in this c onnection-that cam-b-u-siion of coal-in-
the form of fine powder is now widely used in combustion chambers of
special construction, and powdered ore is used in the processes of enrich-
ment or concentration. There are many other technological processes
based on the principles of complete or partial conversion of the material
to hydrosol or aerosol of different dispersion degrees followed by collation
of the particles. Dispersion of materials and their conversion to the
aerosol state is not only desirable but actually essential in many combus-
tion processes and in their mechanical and thermic processing. In this
connection complete particle collection is of great importance to the cost of
technological processes. For example, as late as 1941, 25% of the material
of cement producing plants found its way into the atmospheric air because
of the inefficient operation of the dust catching installations. For similar
reasons economic losses in the non-ferrous industry reach tremendous
proportions. Atmospheric air is also being polluted by discharges coming
from ventilation systems, pneumatic materials carriers, gases coming
from industrial furnaces and .other installations, containing ash, highly
dispersed metallic sublimates and liquid aerosols.
Dust discharged into the atmospheric air may persist for compara-
tively long periods of time in a state of suspension by the effect of different
air currents, ultimately coagulating into agglomerates followed by gradual
settling upon the surface of the earth and upon other surfaces. Air sus-
pended dust particles absorb ultra violet rays, which are essential to the
normal vital activities of living organisms. Condensation of water vapor
upon air-suspended dust particles leads to the formation of local fogs. The
unfavorable effect which suspended dust has on the respiratory passages has
been well established. Records indicate that attempts had been made in
the early middle ages to control or limit pollution of urban atmospheric
air. Legal measures adopted under public pressure for a long time had
been of palliative character. During the dawn of the industrial era, i.e.
at the beginning of the XIX century, the heavy clouds of smoke coming from
smoke stacks and enveloping a city in a dark smoky cloud had been looked
upon as symbols or indexes of production and reflected the prosperity of
the land. The rapid and vigorous development of industry and planless
distribution of industrial production establishments created unbearable
-1-
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living conditions to local populations as early as the middle of the past
century. In time appropriate legislation established limits of dust concen-
trations discharged into the ,atmosphe ric air. However, with the rapid
increase in industrial production the absolute amount of discharged dis-
persed matter was also constantly increasing. This condition and the com'-.
plexity of control over the actual or absolute density of the discharged
particulate matter into the atmospheric air contributed to the high dusti-
ness of the atmospheric air. This was further worsened by the inefficiently.
operating dust catching installations.
The larger aerosol fractions and the aggregates which formed as the '
result of coagulation of the finer fractions fell out of suspension-and settled '
in close proximity to their original discharge into the atmosphere. It has
been estimated that up to 900 g of dust settled annually per 1 m3 of city ,
territory. For example, the annual amount of settled dust in Leeds exceeded
1 kg/m3, and the daylight intensity in this city was only 50% of that found in
nearby villages (4). The amount of settled dust in territories located close
to extensive production plants was within the annual range of 2 - 7 kg/m2
(5, 6). Settled dust caused considerable damage to city and suburban park
and garden vegetation. Industrially discharged dusts frequently contained
substances which dissolved in atmospheric precipitation and formed chemi-
cally active acid or alkaline compounds; these became adsorbed by the air
suspended dust particles and, upon settling on architectural structures,
monuments, statues and other art and decorative objects, damaged or des- .
troyed them. For example, fuel coal and shale dust contained high concen-
trations of sulfur compounds. Data presented in Table 1 show that the total "
amount of such substances expressed as SO3 could constitute 10% of the ash ,"
Table 1 weight (7). In '- '
individual cases -
corrosion proces-
ses effected by
active dusts were
sufficiently potent
to damage build-
ings and other
constructions to
a point of irrep-
ability. Thus,
corrosive destruction was noted of the steel components of a building hous-
ing a sinter roasting plant. In the course of 4 years the corrosion processes
weakened the strength of the supporting steel elements by 30% or more. A
careful and thorough investigation proved that the corrosion of the steel
supporting elements was caused by the action of dust which discharged dur-
ing the processing of ore containing phosphorus, arsenic and chlorine com-
pounds. Damage was noted which was caused to roofs of buildings housing .
electrolytic departments of aluminum production plants, the discharged
dust of which contained fluorine compounds. The damage -was caused by
SAMPLE
Ho.
1
, 2
3
4
5
(1
TYPE OF FUEL
KASHPIRSK si ALE
PERCEIIT BY
OF SOj IN
2,t
2.7
3,"i
4,4
GO
in.r,
WEIGHT
THE ASH
-z-
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hydrofluoric acid formed when the dust adsorbed moisture. Aluminum pro-
ducing plants discharge considerable quantities of highly dispersed aluminum
dust; upon agglomeration the latter settled down and adsorbed moisture,
forming alkaline solutions which penetrated into the bricks and cement blocks
of which the" pFant blaiFdihg was~ constructed" and caused: considerable-daTnag-e~ "
by gradually and continuously ^weakening them. During the summer hot days
and freezing winter days the moisture evaporated increasing the concentra-
tions of the penetrating alkaline solutions, thereby intensifying the damaging
and destructive effects. Similar damage can be caused by easily soluble
neutral salts. Solutions of the latter easily penetrate into the pores of
structural material, and, as the moisture gradually evaporates, the salt
-solutions become concentrated to the point of crystallization, which in turn
causes damage to the structural materials. As an example of such physico-
mechanical damage or of material destruction, mention can be made of the
effect of chemically neutral dust of soda producing plants.
Damage to building materials can also be caused by the combined
action of chemical and mechanical dust effects. As an example, such dam-
age or destruction caused by alkaline aluminate or alkaline sulfate dust solu-
tions formed by dissolved dust discharges coming from one Ural aluminum
producing plant seriously damaged the cement-asbestos roofing material,
the iron reinforced cement constructions and the outside facing of brick and
other wall types. The damage in some instances was so great that within
1-2 years parts of the constructions had to be replaced (8). In this connec-
tion it must be added that in many instances aerosol dust particles acted as
nuclei for the condensation of active fumes, which cause considerable damage
to building constructions. In some cases such damage may reach catastro-
phic proportions. This happened in London in 1952 when a "still fog" was
created by the discharges coming from numerous industrial enterprises and
which persisted for four and one-half days; according to official reports the
smog caused over 4, 000 deaths (9).
The toxic effect of such fog, or smog, is due, to a large degree, to
its gaseous components. It must be noted, however, that during this fog,
or smog, which lasted from 5 to 9 of December the particulate aerosol con-
centration was 20 times as great as the concentrations of similar aerosols
in the air during days preceding the smog, reaching a concentration of
5 mg/m3 (10). The 1952 London smog was not an exceptional phenomenon.
A similar srnog occurred in the small town of Donora in the USA in 1948
which seriously affected 40% of the town's population.
Industrial enterprises, and especially production and processing
plants, are located according to specially developed plans in the USSR, and
smog phenomena, simiiar to the ones above described, rarely, if ever,
occur in the USSR. The problem of atmospheric air dust prevention in the
USSR arid in many foreign countries compelled the attention of sanitary
authorities, and recently roused health department authorities to intense
-3-
-------
preventive and corrective activity; as a resul^air sanitization, especially"
in the direction of dust abatement, has been notably advanced."
~*£y * , '
- The first paten-t^Cox-a-dus-t-caich-ej^ of the-cyclone,' type- was-is-s-oed-i-n— ~~
1880, and the first electrostatic dust filter was built in 1906. Many dust
catchers built on different operating principles are in use at present, _such
as gravitational, inertia, 'electrostatic, thermic, ultrasonic, and Ventiiri
rapid coagulators. There are several varieties of wet type dust catchers
operating on the principles of "gas washing, " or particle absorption. In
many instances gases are freed of dust by passing them through layers of
porous materials, such as sand, metallic screens, textile material, paper
and artificial fiber filters, and many other filters. The need for such a
variety of dust catchers arises from the different conditions surrounding or
accompanying the sources and manner of dust creation. Thus, high temper-
ature of the gases to be purified precludes the possibility of using textile or
paper filters, and purification of wet gases and trapping pf wet aerosol"
particles can be attained only with the aid of wet dust catchers. Fiber paper
filters can be used only where the initial dust concentration is low," and dry
filters become rapidly clogged by lump-forming fibrous dust, etc.
The type of gas purifying installation is determined in many cases by -
the volume of gas being purified and by technical and cost considerations.
As a rule, fabric and electrostatic filters are used in cases where the '•
trapped dust contained enough valuable substances to pay for the cost of the
purifying equipment installation, operation and maintenance. Such gas puri-
fying installations generally occupy considerable space horizontally and
vertically, so that it frequently becomes necessary to install them in special
annexes.
Distribution of some dust filters is determined to a large extent by
their compact construction, initial cost and installation and simplicity ,of
operation and maintenance. Inertia dust catchers, and basically dust
catchers of the cyclone type, meet the above specifications or qualifica-
tions; because of that inertia dust catchers have been in wide use, and there
is every probability that they will remain in wide use for years to come.
This is especially believed to be the case with the .cyclone-type of inertia
dust separators. The inertia separation phenomenon operates in all dust
catching equipment to a greater or lesser degree; it is also an important
factor in the technique of grading loose materials; it is important, there-
fore, that practical sanitarians, technicians and technologists have a basic
knowledge and clear understanding not only of the principles which underly
the operation of inertia dust separators, but also of details of their con-
struction and operation.
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2. Inertia separation
A dust particle moving with a gas current tends to maintain its original
direction and velocity force. Any external force which exerts its influence on
the particle changes its velocity in accordance with the following law of Newton:
dv
dt ' .
j
in which rn — ', is the particle mass;
v — ; is its velocity;
> l_ — i is the time, and
._F_—i is the external force.
The aerosol particle velocity can be changed by the force of gravity, electro-
static and radiometric forces, by forces related to aerodynamic particle inter-
action, by sound pressure force and by the force of aerodynamic gas effect.
The particle mass is the primary measure of its inertia; the greater the par-
ticle mass, all other factors being equal, the less is its velocity and the
greater is the difference between the velocities of the gas and of the particle.
The difference between the vector velocity of a moving particle and of the
ambient gas represents the vector velocity of the moving particle in relation
to the gas, i.e. the velocity of the particle separation from the gas. Each
particle transposition in relation to a viscous or fluid medium is accompanied
by the emergence of viscous resistance forces which tend to equalize the speed
of the aerosol particle with the velocity of the medium stream. Under practi-
cal conditions equalization of velocities can be attained only at sufficiently pro-
longed particle movement which can be conveniently evaluated as the ratio of
a given particle time migration to its size, as shown in the following formula:
r-
' = T~_"
in which
-jI~P:i (sec). <
?i—j is the mass density of the aerosol particle;
V ! is the gas viscosity coefficient, and
T ~ is the particle radius (11).
N.A. Fuchs had shown that this magnitude determined the character of
irregular movement in all cases, and, therefore, can be designated as the
"relaxation time" of a moving particle. In fact, if the value of T is used,
then the solution of the equation representing the dust particle movement,
which is at rest at the moment when t =O and which slowly settles down
under the influence of the force of gravity,
and,
-5-
-------
assumes the form of
v = vs (\-e~~), I - - ,
in \vhich vs=^g— and represents terminal stationery particle sedimenta-
tion velocity. Similarly, for a particle -with initial velocity of v0 , and in the
absence of external forces, with the exception of the aerodynamic force, the
value of (v) could be expressed as follows: " ' ~ .
When t = T the particle velocity will amount to — part of the final velocity in .
the first case and ^ part of the initial velocity in the second case.
The absolute velocity and trajectory of an aerosol particle are-deter-
mined by the current velocity proper and by the aerosol particle velocity in
relation to the velocity of the current carrying it. Under certain conditions
the moving particle may become separated from the current. Under practi-
cal conditions the aerosol particle may lose momentum, i.e. its velocity,
may become reduced as the result of friction or because it became insepar-
ably attached to an immovable surface, or because it became trapped in the .
sedimentation chamber into which part of the general current has been
diverted after the concentration of the aerosol particles in that part of the
current had become considerably enhanced. Thus, dust trapping by packed
or fabric filters is the result of dust particles becoming adsorbed to the
walls of sinuous channels formed by the packing material, or by the fabric
fibers.
Dust particle coagulation in the field of sound waves can be explained
largely by their inertia. Accordingly, dust catchers, in which the aero-.
dynamic effect of a carrying gas current is the dominant external force of
primary consideration, can be classed as inertia separators, the most
typical of which are the cyclone type of dust catchers, the general construc-
tion of which is schematically represented in Figure 1. The raw gas enters
the upper cylindrical part of the cyclone (see arrow) tangentially and in a
whirling movement descends from the cylindrical part of the apparatus into
the lower conical part, and, retaining its whirling movement, exits from the
exhaust opening of the conical part.
Ascending and descending whirling currents are thus formed, the •
velocity and direction of which undergo continuous changes, as a result of
which the velocity of any particle carried by the current differed from the
current velocity at any given moment. Aerodynamic forces, the emergence
of which is conditioned by.the arising differences in the velocities of the •_
-6-
-------
Figure 1
i i
fn T
(K'TSIBE VORTEX
INSICE VORTEX
Fis. I. SCHEMATIC
CYClC'iE CRAWINS.
gas current and the carried dust particles,
distort or deflect the particle trajectories.
Dust particles of considerable mass reach
the -walls of the cyclone and become separated
from the current or gas medium. At this
point, the force of gravity pulls such particles
down into the conical part of the cyclone and
from there into the bunker-collector. The
role played by the force of gravity will be
further discussed later. As stated previously,
cyclones are now in wide use, and it is fair
to conjecture that the present reasonably
satisfactory state of city atmospheric air may
be due to the high efficiency of the cyclone
type of inertia separators. According to
opinions expressed in the literature inertia
forces of a separating particle are related
only to centrifugal forces which fails to de-
fine the exact nature of a particle's inertia
force. This question will be discussed in
detail in Chapter ILL.
Dust catchers of the cyclone type are being used on a progressively
greater scale in many technological processes. For instance, cyclones and
hydrocyclones are now being used by the coal and ore mining and ceramic
industries in the concentration and grading processes. Cyclone apparatuses
are now used by the power or energy industry as steam dehydrators and as
settling installations in the process of hydrosol separation. Favorable
results have also been obtained by the use of the cyclone process in cyclone
combustion chambers used in burning coal dust (12, 13, 14). Dust separa-
tion by other types of inertia separators is controlled by the same basic
principles as described above for cyclone separators. Spontaneous dust
aerosol separation from air currents, which get into contact (friction) -with
surfaces of obstacles and thereby develop cross gradient velocities, is
merely a special case of inertia separation. Schematic drawings in Figure
2 illustrate five types of cyclones built on the same principle; they are now
in -.vide use. The drawings have been made on the same scale, and all these
types were characterized by the same degree of efficiency. The variety of
tvpe; illustrated in Figure 2 reflects the lack of opinion unanimity regard-
ing the most convenient and efficient cyclone design, which in turn reflects
the lack of knowledge regarding the mechanics of most efficient cyclones.
Inertia separation results from the interaction of forces many of
which are difficult to account for due to their complexity and unstability.
For this reason the theoretical analysis of these phenomena involves the
introduction of simplifying assumptions. Also, since aerosol particles are
formed under different conditions and from different basic substances or
-7-
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Figure 2
I - CYCLONE 9-k
2 - COIIICAL CYCLOUt
3 - CYCLOKE SIOT
4 - CYCLOKE LIOT
5 - CYCLOUE Nil OS »z
Fia. 2. CYCLONE TYPES IN PRESENT USE.
materials, the form, sizes and surface configurations of aerosol particles
are highly varied. To simplify the theoretical analysis it is assumed that
aerosol particles exist in the form of spheres. The radii of such assumed
spheres are defined as the average of the greatest and smallest particle
measurements. In some instances the concept of particle radius is based
on its volume equivalent, or on its resistance force to the general gas flow.
A single dispersion system may consist of particle sizes ranging from 10"
to 10~ m. Sizes of aerosol fractions may be commensurate with the gas
molecules and may be in a state of Brownian movement. As the result of
impact against other similar or larger particles such fine sol fractions
easily coagulate into aggregates, thereby substantially affecting the state of
the dispersion system in the course of time. The relative velocity of
particles the sizes of which are commensurate with the sizes of the gas
molecules is practically that of zero, while the velocity of the larger
particles can be as great as tens of meters per second. This creates a
condition whereby particles of the same dispersion system may impede or
accelerate the velocity of other particles with which they may come in con-
tact, or they may alter their original course. Accordingly, it becomes
impossible to account for all the complex phenomena, and the theory of
inertia separation is abstracted from the system availability and examines
the movement of isolated particles. Resistance of the gaseous medium to
the relative movement of particles depends to a considerable degree upon
the size of particles and their rate of movement. Thus, for highly dispersed
ff
aerosols with r less than 10-' m the rate of particle movement has no
effect on the velocity movement of the medium molecules creating no second-
ary gas flows. Accordingly, the gas resistance is determined solely by the
circumstance prevailing when a moving particle received more frontal than
lateral impacts. Correspondingly, the resistance is, as a rule, propor-
tional to the cross section surface, i.e. to the square of the particle
radius.
-8-
-------
In a more general case particle movements generate in the medium
hydrodynamic flows which determine the medium resistance, which is
expressed in aerodynamics by formula (l.l)
j
fc
(I.I)
in which VG - is the rate of relative particle movement;
d - is the particle diameter;
p - is the medium density;
ty " - "is the resistance coefficient, which is a univalent
function of the Reynolds number.
Re =--
in which y - is the kinematic coefficient of the gas viscosity.
For low velocity movements and particle sizes, when 0 < Re < 1,
the resistance coefficient can be assumed equal to o = —rl_ and formula
(I.I) takes the form of Stokes formula:
r
F = 3 -y
(1.2)
Consequently, depending upon the size of the particles, their movement
resistance in any aerodynamic system is conditioned by different laws. In
addition, the aerodynamic frontal body resistance can change within wide
limits with changes in the degree of turbulence of the surrounding medium
flow. It has not been satisfactorily explained to what extent this condition
extended over bodies the size of which was smaller than body volumes, the
random transposition of which produced current turbulence. It must be
remembered that the basic problem of the theory of inertia separation is to
find the lowest particle size, the relative velocity of v.hich is at the level
of perceptibility threshold. The Re numbers corresponding to the conditions
of this problem cannot be great; the same is true of the permissible error re-
sulting from the application of Stokes' law. Because of the above considera-
tions it has been agreed that the medium resistance could be determined with
the aid of formulas (I. 2). To further simplify the analytical solution of inertia
separation problems it has been generally agreed to regard as negligible the
electrostatic forces conditioned by aerosol particle charges.
-9-
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Chapter II ,
CALCULATION OF CYCLONE SEPARATION CAPACITY,
I. The Gravitational Theory of Cyclone Separation . -->-•-
Regardless of the separation process efficiency which takes place inside
the cyclone apparatus and, in particular, in its descending current, the size
of particles which may be carried out from the apparatus through the exhaust
pipe is determined by the velocity of the gases leaving the cyclone. Prior to
1920-1930, i.e. when early studies of the nature of air currents had been
first reported in the literature, it was generally believed that particle sepera—-
tion was caused basically by the force of gravity acting on the suspended'
particles of the ascending gas flow. According to this concept, the range.of
aerosol particles separation from the gas flow was determined by the balance
between the force of gravity and the force of the particle carrying gas flow,
as expressed in the form of the following generalization'
in which w - represents the gas or air flow-rate in the exhaust pipe, 'and
y - represents the specific weight of the particle.
a
It follows from equation (II. l) that the velocity at which a particle of given
size (r) would remain suspended in the ascending axial flow would be as
presented in the following equation:
; «'; = ~7T ' ~JT "•"• ; , ' (II-2/
*
and, accordingly, the radius of such particles should be as presented in.
equation:
r = 2,12 y nz^-. • (n.-3)
i °
However, the gravitational settling of particles from the ascending flow
plays a secondary role. This follows from the fact that separation takes
place with cyclones not only in the vertical, but also in the horizontal and
in the upturned positions. The incorrectness of the early cyclone separa-
tion concepts is also verified by the corollary which follows from equation,?!
(II. 3), namely that as the velocity in the exhaust pipe diminished the separa-
tion efficiency of the cyclone increased. At the same time the force of
gravity determined the upper size limit of particles carried out of the
cyclone.
-------
The gravitation theory fails to take into consideration velocity changes
which constantly occur in the exhaust part of the cyclone. In actuality the
ascending gas or air flow is of the vortex nucleus nature, the velocity of
which decreases centrifugally, i.e. from the periphery towards the center; it
equals zero along the vortex axis. The rotation vetocrty retained" rts-o~rigirtart
rate in the exhaust part of the cyclone; accordingly, the kinetic energy of the
axial part of the flow is less than at the periphery. It frequently happens in
practice that the total energy of the axial layers is less than the energy in
the air space of the gas exits. This results in a condition in which the in-
side void created by the outgoing gas becomes filled with outside air, creat-
ing a reverse vortex, as shown in the schematic illustration in Figure 3.
Flows directed tow-ard^ the inside of the cyclone
exhaust part had been noted by P. N. Smukhin and
P.A. Kouzov (15), by D. N. Lyakhovskii (16) and
others. If the outgoing flow is directed immed-
iately into the atmosphere, then the gas particles
disperse along straight rays tangentially to their
original trajectories the moment the limiting
effect of the exhaust chamber walls ceases to
exist. These tangential straight rays form angles
with the cyclone axis, the tangets of which equal
to the ratio between the tangential rotation veloc-
ity in the outgoing cross section of the cyclone
and the composite axis velocity at the same
section:
Figure 3
P '» •'//
Vu ! ' i« ,„(/
Vh'-V
\\ :•' 'I
U/
a*
Fll. ?. * tl AC *.!!"! Of CtC'1'"? i.RY
CYCLC-.E FLOWS.
tga = —.
l&r
It follows from the above that the gas outflow from
the cyclone occurs not along the entire cross
section of the cyclone exhaust part, but along its
extreme outer circular part. Accordingly, actual
outflow velocities may be considerably in excess
of average velocity rates in some places, a fact
which explains the carry-out of relatively large
particles from the cyclone; it should be added in
this connection that the higher the velocities of
the inflowing gases into the cyclone, the more
pronounced will be the above phenomenon (18).
-11-
-------
2. The P. N. Smukhin and P. A.Koiizov Centrifugal Theory
P.N. Smukhin and P. A.Koiizov were the first among the investigators
of the principles of cyclone operation. Their studies had been made primar-
ily with deep cyclone exhaust chambers, such as are used in the LIOT cyclone.
Aerodynamics has shown that in cases where the flow movements are in con-
centric circles and the fluid particles do not revolve (potential flow), the dis-
tribution of velocities takes place according to the following equation:
^,/? = k = const.
. 4)
Aquation (II. 4} indicates that rates of a potential circular flaw velocity are
determined by the "law of surfaces, " i.e. the flow velocity was inversely
proportional to the distance from the revolving axis. Velocity distribution
in revolving flows of true viscous gases deviates somewhat from the "law of
surfaces. " The extent of deviation is determined by the number of the flow
Re and by their boundary conditions. Some investigators (2) found that
the distribution of descending flow velocities in cyclones of different con-
struction accorded with the following equation: wRx = const. In some
instances (x) ranged from 0.5 to 0.7. In the cyclone flow nucleus and in its
descending flow x = - 1, i. e. -i- = const. By analogy with the revolving move-
ment of solid particles constant k in this case can be designated as the angle
velocity of the rotating flow, and from here on \vill be denoted as oj. In a
rotating flow %vhere •£ = const, the flow assumes the character of a vortex,
since each current particle revolved around its own axis. P.N. Smukhin and
P.A. Koiizov had found that distribution of velocities through the cross sec-
tion of the circular space formed between the outside cylinder of the cyclone
apparatus and its exhaust part was also of vortex type at some considerable
length. Other investigators also noted similar cyclone currents (21). Ex-
perimental curves of velocity changes obtained in making such studies are
illustrated in Figure 4. The curves show that changes expressed by equation
(II. 4), which accorded with the law of potential flow and were noted at cross
sections 1, 2, etc., were replaced by changes characteristic of vortex cur-
rents. Such observations constituted the basis of the cyclone separation
theory of P.N. Smukhin and
P. A. Koiizov and represented
the first experimental demon-
stration of the fact that
inertia separation operated
in accordance with certain
laws.
The theory is based on a
simplifying assumption that
3. particle inertia force was
equal to the centrifugal force
determined by its participa-
tion in the revolving flow.
Figure 4
SEC.
0-20
FlS. 4. EXP£HIME»T«L CURVES OF TA*S£»TML VELOCITY
CHAWS IN CtCLONE LIOT.
-12-
-------
In addition, it is assumed that the particle velocity differed from the gas
velocity only by the presence of a radial component. The tangential particle
flow velocity was regarded at any particular moment as equal to the velocity
of the gas flow at any given point. It -will be shown below that the supposition
re-gardtng the-equality of particles' and gas tangential velocities— expressed a —
condition which did not exist along the entire path traveled by an aerosol
particle in the cyclone apparatus. The centrifugal force is directed along the
rotation radius, so that on the basis of the discussed condition it follows:
in— which -(v—)--represents- the- parrtrcle-radiarlr velocity, i. er the- separation- —
velocity, From equation (II. 5) it follows that separation velocity is equal to:
"f'^R' (H.6)
0)
in which m = R which is the angle velocity of the current rotation. The
angular revolving velocity of individual layers of the rotating flow is variable
in general cases, abating in the direction from the center towards the
periphery, with the exception of gas flow in cyclones having exhaust pipe of
relatively deep insets with w = const. In such cases, based on equation
(II. 6), it follows that:
If in equation (II. l) 'R^ denotes the radius of the cyclone exhaust pipe and R3
denotes the radius of the outer cyclone cylinder, then the equation deter-
mines the time it takes a particle which entered under most unfavorable
conditions to reach the wall of the outer cylinder. The duration of particle's
remaining in a descending flow can be determined approximately with the aid
of formula (II. 8):
2-'/ftTn' Vrt'
in which R-av - (^i + ^s)/^- ~ represents the average flow radius, and
n - represents the number of flow rotations in
the cyclone - cylinder-
It is difficult to determine the true magnitude of the latter. The gas flow
revolving inside the cyclone becomes wider, and, in the course of its revolv-
ing, becomes diluted, so to speak, by secondary flows. To explain the
cause of appearance of secondary flows, it is necessary, first, to refer to
the well investigated phenomena which accompany flow direction changes at
deflection points. Equation (II. 4) indicates that the gas velocity decreased
from the axis towards the periphery. Eiler's equation '\p~+ $, u'* =TcorisI
-13-
-------
Figure 5
ALONG A -
Flfi. 5. FCF.MATIOK OF A DUAL
VORTEX IK SMOOTH CHAHCI
FLOW COURSE.
applies to the entire gas flow; accordingly, as the
»
flow moves away from the axis its pressure in-
creases, reaching a maximal value in the air
layers abutting the outer deflection wall. The
increased pressure causes the air to flow along
the outer walls in the direction of low pressures
forming vortices schematically illustrated in Fig. 5.
From a kinematic point of view the cyclone repre-
sents an extended flow deflection. Unlike the case
under discussion, the upper part of the secondary
cyclone vortex tends to flow along the shortest path
towards the opening of the outflow pipe, and the
lower part, dispersing through the vertical cylinder
element, ultimately becomes the ascending cyclone
flow (see Fig. 3). Figure 6 is a schematic drawing
of secondary cyclone flows as conceived by A. Ter-
Linden and based on measurements made of flow
velocity components. The formation of flow vortices
in the cyclone had been noted also by Professor Van-
Tongeren (23). Secondary flows considerably
affected the cyclone apparatus efficiency, in partic-
ular the upper branch of the vortex, which consti-
tuted the shortest particle carry-out path through
the exhaust pipe. However, this unfavorable flow effect can be considerably
counteracted by increasing the depth of the exhaust pipe inset, which is a
construction characteristic of cyclone separator
LIOT. In this connection it was anticipated that if
the increased depth of the exhaust pipe inset alone
could not completely eliminate the overflow phenom-
- 1 i I _ enon, then the dust density in the gas overflow might
,s-\ \ \ \(?f I become reduced by the effect of the centrifugal
forces coming into play when the gas descended in
the form of a spiral-shaped flow. As an overall
consideration, the existence of secondary flows in
a cyclone having an extended exhaust pipe will
introduce no substantial distortion in the effect of
the physical model used as the basis in developing
a method of calculating the separation degree in the
cyclone cylindrical part. It should be borne in mind,
here, that the concept of the number of flow turns or
deflections is a conditional one. Under practical
conditions (n) is evaluated on the basis of air flow
observations in the cyclone models built of trans-
parent material. It has been generally accepted
that n = 1/3 turns depending upon the height of the
cyclone cylindrical part.
Figure 6
Fie. 6. SKETCH OF SECONDARY
FLOWS, ACCORSINS TO
-14-
-------
By eliminating the time factor from equations (II. 7) and (II. 8)-, the
following can be arrived at: .- - . -
k~2r./i»
and then arrive at
Aquation (11.9) represents the P. N. Smukhin and P. A . Koiizov formula for
the determination of the diameter of the smallest particle inflowing from
layers farthest from the outer cyclone wall, which have reached the wall
before they had been carried beyond the limits of the cyclone cylinder.
Such diameters are frequently referred to as the "critical dimension" or
"critical size." The assumption was that all particles which reached the
\valls lost their velocities through impact against the walls, which caused
them to fall out of the flow and to slide downward along the wall into the
conical cyclone part. If the gas volume (Q) flowing through the cyclone
per unit time is known, then the average dimension of the vertical compon-
ent velocity within the limits of the circular space between the outer and
inner cyclone cylinders can be computed with the aid of the following
euation: ___________ _.
Using equation (II. 10) it is possible to arrive at the time a particle remained
in a descending flow with the aid of formula (ll.ll)
in which Hcyl represents the height of the circular cylinder part. From
equation (II. 7) derive equation_(ll. 1Z).
By quick gross simplification
ra = 2 -«,/?, "
in which n£ represents the number of flow turns per sec. , and accordingly;
."""' in**- :
-Jr.'n'T '"/A ' I
-15-
-------
then with the aid of equation (II. 11) arrive at equation (II. 13) (21). Experi- .
mental data indicate that average tangential rotation velocity in-the circular
chamber of cyclone LIOT was equal to:
j "a v~ (1,6-1,7) ' |-
accordingly, the number of flow turns per second should be:
solution of equation (II. 5) can be presented as follows:
__
Integration constant (C) can be determined by remembe ring that the moment
the observed particle entered the cyclone, i.e. when t = O, the distance be-
tween the particle and rotation axis R = R2-S, where S is the distance from
the outer cyclone wall. Under such conditions
which makes
For particl-es which reached the outer wall, R ~ R2, and the time
determined on the basis of this expression must be equal to or coincide with
the time of their presence in the cyclone cylinder descending flow, i.e.
—L_ln ^ = 2"/1. t
-16-
-------
from which it follows that:
-hr *'
A-j-5
r= 1,532 Y £ Injr. (n. 14)
' - n PJ w /?,—i> x '
Derived equation (II. 14) enables to determine not only the "critical
dimension" of particles, but, as will be shown Later, the cyclone fraction
efficiency as well. This equation, as equation (II. 12) proposed by
S.E. Butakov, is a modified form of the P.N. Smukhin and P. A. Koilzov
equation (II. 9); when S - R2- R. it assumes the same form of expression.
Assuming that the tangential velocity had not changed in the radial
direction, then the following can be derived from equation (II. 2):
~ i x v j — i
2-u-o
where uj-, represents initial velocity of the air inflow into the cyclone.
after which determine that:
For particles which move through distance S in radial direction in the time
determined by equation (II. 4), R~ can be determined by the following equa-
tion:
, ^
R, - 2 -a 5 - — - - - (/?, - 5)=,
from which derive - by the equation shown below:
On the quick and gross assumption that' 5? = 1, derive the Rosin-Ramler
Re*
-17-
-------
formula (24).
(11.15)
More accurate results can be obtained with the aid of the following equation:
(11.16)
3. The Muhlrad-Davies Formula
Gas rotation characterized by a constant angular velocity, as for
instance in the LIOT cyclone, represents a special case seemingly asso-
ciated -with the effect of friction against the surface of the inner tube. It
is possible that the effect of such friction becomes enhanced by specific
construction features of such cyclones. Velocity distribution in cyclones
having short exhaust pipes differs from the above described and, according
to experimental data, approaches the hyperbolic distribution, i.e. velocity
distribution noted in potential flows.
Figure 7 presents a projection diagram of tangential velocities in such
cyclones according to data found in the above mentioned references (22).
The sketch shows an ascending flow depicted as
a vortex field, the velocities of which decreased
towards the periphery. According to formula
(II. 4) the irrotational flow in the descending cur-
rent persisted in the conical cyclone part, and,
in particular, in cyclones having deeply set in
exhaust pipes. Cyclone LIOT is only one of many
cyclones of similar construction in present use;
since the conical cyclone part has a notable effect
on the separation efficiency, it must be assumed
that any flow which closely approximated the
potential constituted a more characteristic case
of cyclone flow.
Assuming that velocity distribution followed the
law of potential flow (II. 4) and integrating equa-
tion (II. 6), derive the following:
Figure 7
~~
*,
., /\
-^*-
i/N
(/• ->v~,
t —
Fie. 7. A BIASIUH or TANGENTIAL
(v*T> AN* 9 ASIA I (wt) VELOCITIES
ACCO»OIH TO TER-LINDER.
-18-
-------
From equations (II. 4 and II. 8) derive the following equation:
Eliminate the time factor from the last two equations and derive:
r = 0,75 jx £ l^ '/• (II.18)
Follow the procedure used in deriving equation (II. 14) and derive the follow-
ing equa-tio_nJ_
Now, express equation (11.18) in the following easy to analyze form:
i
= 0,75 YrRf [/?J-<*«-«'].- (II. 20)
In the well known work of Muhlrad (25, 26) the minimal size of separat-
ing particles was determined as follows: in particles which reached the
outer cyclone cylinder wall, the distance the particles traversed radially
could be determined using the following formula:
Assume further Rm as the variable radius R (v/hich is not so in actuality)
and determine time (t) using formula (II. 21) shown below:
O ^ « / 1 \
(11.21)
R,-S
Frotn formula (II. 19) derive formula (II.-22):
Equate the right members of equations (II. 21 and II. 22) perform some con-
versions and derive the Muhlrad formula:
• / 1- R42-(/?-S)4
r = 0.75 I/ -— ^ ---- 1
V - nk P» _.
Rs + _. s>
11.23
'
-19-
-------
A more nearly true integral computation in equation (II. 21} can be
arrived at by the following equation: ' ' . _ ,
Sk
R,-S
Davies (27) substituted for a^o the velocity of the linear gas flow as it
entered the cyclone, and on the basis of experimental data equated (t) with
the ratio of the cyclone height to such velocity and arrived at the following
formula for the determination of separation efficiency:
As seen, this formula differs from formula (II. 20) basically in the assump-
tions made for the purpose of its derivation. It was shown above that accord-
ing to experimental data changes in the tangential velocities were more in
accord with the following equation:
, wRl = const. (11.25)
When x =0.5 (22, 26, 28) r can be computed according to the following
equation:
The assumption regarding the constancy of angular velocity in the Smukhin
and Kolizov theory cor re spo.nded to the value of the exponent in equation
(II. 25) x = 1, i. e. w/R = const. "When x = O, the equation becomes
w = const. , which agrees with the assumption made by Rosin and others
in arriving at their generalizations. Finally, using x = 1, and taking the
above mentioned provisos into account, derive the V. Muhlrad formula.
Note in this connection that results of some investigations of vortex pipes
(energy spacers) pointed to the possible existence of all three distribution
types. The schematic illustration of vortex pipe is shown in Figure 8. Com-
pressed air is fed into the pipe tangentially through a snail-type device
(helix) (1), and under the effect of diaphragm (2) flo'vs in a rotating manner
to the right, as in cyclones. The right end of-the tube is equipped with a
throttle, and the left extends out into the atmosphere. The gas flow in the
helix is accompanied by the Rank effect, which can be described as follows:
-20-
-------
Figure 8
TWISTINS NEIIX
DUPHR/kSM
THROTTLE
COOLEI AIR EXIT
WASMEI »IR EXIT
Fi«.
SCHEME OF VOSTICM.
:;r.-1 cuiio'i C'; urVEPAn'HE.
the temperature of the air enter-
ing through the throttle is con-
siderably above the temperature
of the air flowing out through the
diaphragm, - i.e. , it is lowenr
than the initial air temperature.
The boundaries of this interest-
ing effect can be comprehended
by examining the curves shown
in Figure 9. The experimental
curves illustrate the functional
relation-ship between- tUe. attain-
able temperature transitions and
the weight of particles and also
the cold air at different initial
pressures; in the experiments
under consideration the pressure
ranged between 1. 5 and 10 atrnos-
O
pheres (30).
Temperatures of the hot air
flow are indicated on the upper
part of the graph, transitions
and temperatures of the cold
air flows on the lower part
of the graph. The temperature
transitions can exceed the
80 - 100° range (31). The temp-
erature separation effect can be
explained as follows: the distri-
bution of tangential velocities in
the initial section of the rotating
gas flow in the vortex pipe follows
The law of surfaces. Due to the
high velocity of the compressed
air flow the tangential velocities
of the axial gas portion can attain
great values. The .internal fric-
tion of the gas layers which rotate
rt different velocities develops a
elocity of the gas layers. At the same
- Continues to increase, and the
-21-
-------
12 R
lo AT CROSS-SECTION OFF AT 3050;
2. DITTO *T 10.7 D; 3. DITTO AT 21.5 D;
4. DITTO AT 53.5 D0
FIG. 10. RADIAL CHANGES IN RADIAL VELOCITIES.
layers. As a consequence, and in the presence of resistance at the end of
the tube, the peripheral layers overcome the resistance, whereas the flow,.
nucleus becomes separated and exits via the diaphragm.
Figtrre- 10 Figure 10 presents experimental
curves of angular velocity changes
at different cross sections along the
tube length. Measurement determi-
nations were made in air under
initial pressure of 3 atmospheres
and 20° with LL =0.2, using special
sounds (31). The graph shows that
the angular velocity of the peripheral
gas layers reached more or less
constant values beginning with cross
section 2 at a distance of approxi-
mately 10 tube diameters from the
first cross section. The angular
velocity abated towards the axial
part of this cross section. It can be
assumed that in the space between
cross sections 1 and 2 there existed
a cross section of prevailing con-
stant tangential velocity. Accordingly, the 3 values of x hypothecated in the
previously discussed separation theories could be realized in actuality.
Studies of the vortex pipes explained the cyclone operation to some degree,
since the high temperature peripheral layers in the vortex pipe corresponded
to cyclone flow layers rich in sol particles. In particular, these investiga-
tions confirmed the existence of component radial velocities. At the same
time it was established that no axial ascending flow existed in the tail section
of the tube, indicating that its formation was completed before it reached
that tube section.
4. Estimating Dust Catchers Efficiency
Industrial dusts are aggregates of particles varying from the size
of molecules to hundreds of ix in diameter. For this reason estimation of
particle separation efficiency based on trapping particles of smallest diam-
eter ("critical size") does not reflect the true efficiency of cyclone operation,
i.e. the ratio between the trapped particle weight and the initial dust content
of the raw air or gas. Apparently dust particles of diameters smaller than
those determined by the above presented formulas reached the outer cyclone
walls; this may be particularly true of particles which entered the apparatus
under a set of conditions more favorable for their separation, as for example,
particles not far removed from the outer walls of the apparatus. The assump-
tions which formed the basis for the derivation of formulas presented in pre-
ceding paragraphs were to a degree conditional. Therefore, these formulas
-22-
-------
must be checked against one another and with results of practical deter-
minations. It appears evident that formulas of the type of (11.14), (II. 15),
and (II. 20) represent an analytical connection between minimal particle
sizes and the distance to the outer wall at the point of their entrance into
the cyclone, i.e. functional relationship.
r=f(S).
At a. given width (a) of the cyclone outflow attachment, the above expression
of functional relationship assumes the following form in any cor responding
scale change.
r = /(4)' (U.Zb)
The sol particles are evenly distributed over the flow cross section
?_t the moment of their entry into the cyclone, so that the ratio (S/a) deter-
mines the portion of particles of a given size contained in an S wide flow
section. Solve equation (II. 26) with reference to the argument (independent
variable) and obtain the functional efficiency expression.
Figure 11
Each functional value thus obtained determines the portion
of particles of a given size (r) which reach the inner cy-
clone surface in the course of their separation movement;
accordingly, it follows that an area delineated by the
function curve (11.27) and by the coordinate axes, ex-
presses the total of different size particles which reached
the outer surface. Contrary to the accepted concept per-
taining to the separation mechanism, such particles
should be regarded as
Figure 12 caught (or trapped).
Fig. 12 present frac-
tion efficiency curves
of a cyclone type sche-
matically illustrated
in Fig. 11, at 20° air
temperature and
15 m/sec. velocity.
The dust was of
cement origin of
;, 2, 600 kg/in3 specific
gravity. The disper-
sion dust composition
is presented in Fig.
13 in form of curves.
In such cases the
number of rotations
I - ACCi'^SllS TO P.S'.SiitlKHIN ASS P.A.KOU2CVJ
2- Acec-ins TO ROSIN, RAMLEH *«» IHGLEKAH;
3 - ACCOBBINS TO HULR*».
Fu.
C«?V£S Of FHACTICNS EFFICIENCY.
-23-
-------
is usually taken as 2.
Determination of fraction efficiency with the aid of formula (II. 14)
requires that angular velocity (a) be known. On the basis of experimental
data the flow velocity in the proximity of the outer cyclone wall is assumed
to be the same and can be expressed as follows:
Use the fraction denominator as 1. 7 and determine the angular rotation
velocity for the example under discussion as follows:
U) = V>—
T5"
1,7-0,5
= 17,6 secT
-i
Substitute the value of
and obtain:
in formula (II. 15)
/• = 0,132-10
-5
III
0,5 _
0,5—5
I - ACCORDING TO PoNo Sr.UKIIIH Af!8 P0A(,
Kci'zov; 2- ACCORDING TO Rosi'i, HAULER
ADD I f.'CLEMA'IJ 3- ACCORD INS TOUULRAOo
F|S0 I30 NOr!06RAM FOR THE D£T£RM| NAT IOH
OF CYCLO'iE EFFICIENCYo
Curve 1 of Figure 12 represents fraction
cyclone efficiency computed by the above
presented procedure. A condition in which
0) = const, can be attained by lowering the
inner tube to the level of the conical cyclone
part beginning.
Curve 2 (<*> = const. ) of Figure 12 represents
the fraction efficiency of the same cyclone
computed according to formula (11.15). The curve is at a higher level than
curve 1, which is according to expectation. Curve 3 (WR = const.) occupies
an intermediate level and is a basis of functional relationships formulated by
generalization (11.20). The above discussion leads to the conclusion that the
law of flow velocity changes cross sectioiially can affect the cyclone effi-
ciency separation substantially. The difference in the computed value of the
minimal diameter in the case under discussion can be within the range of
20 - 75%, depending upon the assumed conditions.
If the dispersion composition of dusts suspended in the polluted air is
known and can be expressed by the function p = F (r), then the dust separa-
tion efficiency, in percent of the initial dust content, can be expressed by
the following equation:
/TT
(r)dr
-24-
-------
Since the integrals in the numerator and denomenator of this formula repre-
sent the dust concentration in the purified gas (numerator) and in the gas
entering the cyclone (denominator). Dispersion or fraction composition of
a true dust, with few exceptions, are difficult to formulate analytically.
Where such a graph is available, the subintegral numerator formulation
(II. 28) can be found by the method of graphic multiplication of the component
functions. (For other, more obvious, method of graphic cyclone efficiency
determination see references 7 and 15). Obviously, the area of the curve
obtained by multiplying the ordinates of the fraction composition and fraction
efficiency curves represents numerically the weight of the particles. Fig.
13 contains such tracings of the above analytical example. Curves I, 2, and
3 correspond to the graphs of fraction efficiency presented in-Ftg. 12. The
ratio between their areas and the area of the curve of initial dispersion
composition, computed according to formula (II. 14) amounted to 72.4%, by
formula (II. 15) to 85.8%, and by formula (II. 20) to 72.5%. All examined
formulas established the particles' "critical size." Depending upon the '
degrees of correctness and completeness with which the hydrodynamic
phenomena, which determine the separation of particles, had been taken
into account, the computed results may differ somewhat. At the same time,
the existence of a boundary which divides the non-separable or incompletely
separable small particles from the larger separable ones has been indicated
by all known methods of inertia separation computation. Actually there is no
such boundary, since it has been known that not even the larger fractions
become completely separated, and slip through of larger particles has been
observed frequently.
Figure 14
W _ 60 &0 103'
I - IIIOI VID.A1. HICH CAPACITY CYCLC'US;
2 - HIGH £FrKJ£V:c SATTcRV CYC'.O'JcS.
FIG. 14. OPERATSGSAL CURVES of CYCLOSIES'
FRACTIONAL EFFICIE'.'CY.
Figure 14 shows operational carves of
fraction efficiency of large size single
cyclones (curve l), and small diameter
high efficiency cyclones used in group
installations (curve 2). These curves
have been constructed from data repre-
senting averages of many years of obser-
vation, and constitute an objective
evaluation of the efficiency of correctly
constructed and operated dust catchers
(35). The graph demonstrates that
there are no particle sizes in practice,
all fractions of v/hich could anderqo
complete settling or precipitation.
Actually the cyclone efficiency gradually
increased with the increase in the size
of the dust particles. Generally, cyclone
efficiency is much below the computed,
and the above discussed formulas can be
used only for the qualitative evaluation of the intensity of the separating
process (36). It must be noted, however, that, even when used for purely
-25-
-------
qualitative evaluation, the formulas still have their shortcomings. For
example, according to the discussed formulas, the size of the separated
particles appeared inversly proportional to the flow velocity, and, in
particular, to the velocity of inflow into the apparatus. -Actually, with
the increase in flow velocity the cyclone efficiency at first increases and
later decreases. A greater operative regularity was noted in cyclones
of decreased diameters. This is also true only up to a certain limit,
beyond which the cyclone efficiency begins to decrease, unless other
cyclone components have been reduced in size accordingly. It was noted
that the effect of decreased cyclone diameters was more pronounced in
the case of coarsely dispersed dust than in the case of finely dispersed
dust. A comparative study of the curves shown in Figure 15-brings out
the above mentioned effect more vividly. Curve 1 demonstrates the
change in the separation efficiency of relatively coarse fly-ash in which
particles of 50 p, or over amounted to 48%, and curve 2 demonstrates the
separation efficiency of finely dispersed dust. Sizes of the exhaust tubes
which were taken as the basis of measurement of geometrically similar
cyclones were marked on the abscissa.
Figure 15
95
I •= COARSELY DISPERSED DOS?.
2 - OUST OF PtHE
0
100
203 300 WO dV>\
Fie. 15* EFFECT OF CHAHSES in
CYCLOQE OtHEGSOOUS On SEPARATION
EFFICSEIJCV.
-26-
-------
Chapter III
MECHANICAL ANALYSIS OF THE INTERACTION BETWEEN A
PARTICLE AND. THE MEDIUM IN A CURVILINEAR STREAM
1. Aerosol particle inertia resistance to the carry-away
effect of a curvilinear stream.
Visualize that an aerosol particle velocity consists of two vectors, one
of which (w) is equal by its magnitude and direction to the medium velocity
at-the- pmnt occupied by the particle center. Then, the second vector (vc)
Figure 16
16.
A- PARTICLE VELOCITY EXCEEDS MED IUM VELOCITY;
B = PARTICLE VELOCITY IS LOWER THAN MEDIUM VELOCITY
DUSRAIUTIC PRESENTATION CF PARTICLES' AKC MEOIL'H I liTER (VCTI OK 111 A CURYILIKEAH FLOW,
will represent the particle movement velocity in relation to the medium, i.e.
its separation velocity; it is the vector of the sought magnitude. In this way,
the problem of finding the absolute movement acceleration becomes reduced
to the determination of the composite movement acceleration *( y = w + vc, of
which w can be regarded as a known magnitude. According to the Conolis
theorem acceleration in a composite particle movement can be represented
27-
-------
by equation (ill.l)*,
dv di\ , - - 4^ _
dT--rf7-(ulH)#l-r7r/?a-2u^c, ' (III-. 1)
where u; - represents the angle rotation velocity of the flow
layers surrounding the axis;
R - represents the distance from the center of the
flow rotation.
The angular acceleration can be determined by formula (III. 2):
,
"at ^~dT> "^dJ ~ ™y~dj • ' (III; 2)
Assuming that the velocities were distributed radially, as in the rotation
of a solid body, then ^ = const. ; accordingly, when the movement becomes
stabilized dx/dt = O, and the absolute movement acceleration of equal
diameter particles is equal to the geometric sum of the separation move-
ment velocities, the even medium rotation movement velocities and the
deflection of the Coriolis acceleration. In such a case the acceleration
modulus is equal to:
(1II.3)
Since the radial movement of a particle is equal to
dR
•°* = -dt>
then by determining
obtain
^Origins of the movable coordinates coincide with the origins of the immov-
able, i.e. with the flow rotation_center; it is assumed that the coordinate
system of (x, y) is of the rectangular type and rotates at angle velocity u>.
-28-
-------
In a potential movement,' i.e. when ojR = const. , the angular velocity of
layer rotation is
; ft_
or by differentiation with reference to
where i is the unit vector of the radius vector. Correspondingly, the
third member of the right part of formula (lll.l), which expresses the
particle velocity, becomes equal to
Formula (ill. 4) represents a second supplementary particle acceleration
according to the hyperbolic law of velocity change. The vector of this
acceleration is directed opposite to the tangential velocity vector, and,
like the latter, is perpendicular to the particle radius-vector. Therefore,
for the determination of this vector's direction it is only necessary to
deflect the tangential vector velocity 180° in the direction opposite to the
course of rotation. The value of the numerical coefficient in formula
(III. 4) varies with the character of velocity distribution. Thus, at distri-
bution (o/?0>5=1 const, frequently noted in true revolving streams, this
coefficient is equal to 1. 5. By taking into account formula (ill. 4) accel^
e ration of a particle in a potential rotating flow equals to:
(m.5)
and the product of mass times acceleration is equal to:
m -^- = in ^ -j- """ |"> /?] + m 2<" •z'c — ni • 2 «> VR . (III0 6)
Each member of the right part of this equation represents the force
of particle action on the medium which carries it along. In particular,
vector m-'2«i'JR represents the effect of a particle, which in its relative
motion, passes through layers of progressively reduced velocity, on the
-29-
-------
medium braking influence. In a rectangular system of coordinates ( XOY)
the abscissae axis of which at a given moment passes through the particle
centers and coincides with the axis of the flow rotation, the radial particle
velocity represents the projection of the particle vortex velocity along
axis X. - '" - ' " "" ^ - -
i VK = ^cv - ', •
Therefore,
and the sum of the Coriolos deflection and tangential inertia forces is
equal to:
— rn-2~»\vc— ~VR\ = -m 2«>i>cv. (m> 7)
The vector direction of this force is determined by vector deflection vcv- by
90° in the direction of the flow rotation. It follows from the above deter-
mination that the supplementary inertia force (III. 7) was directed radially.
Depending upon the ratio between the flow and particle velocities this force
may coincide with the centrifugal force, or it can be directed in the oppo-
site side.
2. Differential equations of dust particles movement
in a curvilinear flow
According to the above computations and the Stokes lav/, the equation
of particle movements in a potentially resisting medium can assume the
following form (ill. 8) • . - ' '
0. " , (IK. 8)
Reduce equation (ill. 8) with reference to (m) and find that by projecting
coordinates XOY unto the axis of equation (ill.8) assumes the following
form:
t.9)
^-U-L-;, = r>
at
-30-
-------
The following ratios will prevail in a general case:
dx
correspondingly,
t'r r r ,
-- = -.-, and
at ~ the solid lines point to the components of
the particle's inertia force, F is the force of the medium viscosity resist-
ance. According to the schematic illustration forces F are directed
towards the rotation axis in both cases, i.e. they augment the mutual
direction approximation of vectors w and v". The medium viscosity resist-
ance forces lowered the particle movement velocity in the first case and
enhanced it in the second case. Therefore, force F tends to equalize the
velocities of the medium and of the particles. The separation movement
velocity (vc) in both cases abates as the re_sult of the medium resistance,
and as a consequence acceleration vector ^t>i is always _diLrected towards
the side opposite to that of ^ and the inertia force • 'dva is directed
towards the side opposite to that of the medium resistance-fprce. The
action of this force is always towards the periphery of the medium flow.
The Coriolis (deflection) force acts in the direction opposite to that
of the flow rotation velocity. Since this force forms a right angle with the
-31-
-------
direction of vector velocity of the separation movement its direction will
be towards the rotating periphery in the first case, and towards the rota'-'
tion center in the second case. The tangential inertia force acts in the
direction of the medium movement. The vector of the resultant of the-,- „"
latter two components acts towards the periphery in the first case and
coincides with the centrifugal force; it acts in the opposite direction, i.e.
in direction of the rotation center, in the second case. A comparison of
the two equations with equation (II. 5) makes possible to conclude that
equation (III. ll) can be derived from equation (III. 10) by discounting the
first and third members, so that it will represent only a first approxima-
tion. Relaxation time (T) is very short for very small particles. There-
fore, I/T » 2 u>, and the supplementary force can be disregarded-. However,
for particles of, say, 100 u, size the relaxation time is as follows:
/ 2 ' -
KG/SEC.' onn
= 260 —-, * = 20
In the example analyzed in Chapter II the derived angular velocity was
uj = 17.7 rad/sec. Under such conditions (l/j) is of the same order of mag-~
nitude as to , and the radial component of the Conolis force may be com-
mensurable with the centrifugal force. The supplementary inertia force
(ill. 7) comes into play only when the particle's tangential velocity differed
from the medium tangential velocity. As the flow enters the rotation point
of individual layer velocities change and become functions of the distance
from the rotation axis, and in comparison with the initial velocity the
peripheral layer velocities become reduced, while velocities of the cen-
tral layers increase. At point M (Fig. 17) the flow velocity is equal to the
initial velocity; as a result, the velocities of the particles which upon
entering are deflected to the right of point M temporarily exceed the gas
flow velocity, while to the left of point M the velocities of the particles
will be below the gas velocity. If the length and duration of the gas flow
are considerable the tangential velocities of the particles and of the gas
will become equalized, creating a condition of a "quasistationary" move-
ment. It follows from equation (III. 11) that:
Jilti. = _ dt.
; fey
which by integration becomes
(III. 12)
-32-
-------
The integration constant is a function of (x) and can be determined under
following conditions: the viscosity force effect appears with some delay;
therefore, t - O at the moment the flow entered the deflection rotation,
and the particle's movement velocity remains the same as that of the gas
flow a tr±lre time rtrerrters the cyclone apparatus, i.e. v = Wo . Atr that
moment the gas flow velocity has become a function of the radius a; = T (R),
even though In actuality its direction coincided v.ith that of the particle
velocity. Under such conditions
= In we>fl = In [=•,-«•
so that
(111.13)
Formula (III. 13) represents the difference between movement velocities of
the flow and the particle, i.e. the velocity of the particle's ambient medium
flow.
Figure 17
C|G. 17. ~K:TC,-: f- v.TE-Ti.M. ?!.;-' •, ;L-::I- i £•; i :: k c. - v ILIKE^ FLC,-.
the rotation center at which the
particle became deflected, the right sice of equation (ill. -13) may be negative
or positive. The projection of the absolute particle velocity in the direction
of the tangential sas velocity is equal to:
-33-
-------
It follows from this formula that, in the course of time, the tangential par-
ticle velocity will tend to approach the tangential gas velocity which will
ultimately lead to a "quasi-stationary state" the shorter will be the relaxa-
tion time, i.e. other conditions being equal, the shorter the diameter the
sooner will the "quasi-stationary state" be reached. The part of the particle^
trajectory within the limits of which
i t_
~ > T
essentially differ from O represents a section of nonstationery movement of
the particle and can be designated as the initial section, the length of which
is proportional to the relaxation time, and, therefore, differed with the'
particle size. - . -
Equation (III. 12) indicates that value vcy can be regarded as practically
equal to O where values of
t_
T"
are very small. The value of 01 — o> (R) can be of the order of 10; therefore,
in practical computations it can be assumed that:
_t
T
= 0 when e
V 3
or
t > 7'-.
Table 2 presents (T) values for particles of
time (t) from the moment the particle entered the
Table 2
in u,
T in sec.
t in sec.
different sizesx 'and also
rotation deflection at the
end of which the move -
ment can be regarded as
practically "quasi-
stationary. " Computa-
tion results indicate that
beginning with particles
of 5 jj, diameter and above
the length of the initial
section is commensurate
with the general length
of the moving particle
path in the cyclone appar-
atus. When particle
diameters exceed 100 u, .
the supplementary
forces act throughout the
entire period of being
inside the cyclone.
Values of T have been computed for particles of 2.5 g/cm3 specific
gravity weighed in the air at 20° C.
1
5
iO
20
30
40
r,o
60
70
80
-------
3. Solving the differential equation of a particle movement
t_
Substitute v = Ce and o> = (k/x2) in equation (ill. 10) and derive"
the following equation:
*±.L--.JLce- =0
(It* • -. dt x* x»
or
Equation (III. 14) can not be solved with the desired accuracy; however, by
factorizing the desired function into a power series an approximate solution
can be obtained. It is convenient to change the independent variable (t) by
using t _ i_e"7, then | J l=~—i In (1 — I) ', now, by diffe rentiating with respect
to (Z, ) obtain
accordingly
\ dt ~~
and the second (x) derivative will be
1 rf ,_,_
T rf;
after differentiation with respect to (§ ), derive
Substitute the derivative values in (III. 14) and introduce designations
2£cT2 = A and - 2A:2 = B, derive the following final equation (ill. 15).
xt(\-i)*%£-Ax(l-i)-B = Q. (III. 15)
Factorization of function x = f (c) into a Taylor series in the vicinity of
point § = O, which corresponds to to = O, yields the following equation:
x = a + ait+*.?+-!!i.l' + -gv+... (in. 16)
-35- '
-------
Constant coefficient (a) represents the distance between the particle
and the rotation axis at t = O, i.e. as it enters the rotation threshold, or
a = f (t ) =x0- Coefficient cl—j' (,t0) determines the value of velocity
(dx/dt) and at the same initial moment t = O. Evidently, at this moment
particle velocities, which, already differed from the gas velocity a4 points ------
occupied by their centers, directionally still differed from the gas velocity,
as the results of which
"= «.-=<>
Individual factorisation members can be determined by the method of
indeterminate coefficients. To do this it is necessary to find members of '
equation (ill. 15) by means of equation (III. 16). Since
(III. 17)
(HI. 18)
then, by equating coefficients of the same degree with respect to (?) , derive
the follov/ing:
4- f • j -f 4 a' «-• ("3-2 « ,)
Solve these equations and find value of factorization coefficients as follows:
A . B
-36-
-------
' ' " A
a, = 2o, — ^r
-
«''
Thus, solution of equation (ill. 15) appears in the form of the following
infinite series:
+ ( **? _ _' ' I *"^£_ 1_ , I . __'_.' ~ \ { \ - r--,-
\~Z ^n 1 T"o 3" ^ Y. T IOA ..3 / I / I •"' /j-jj -^g\
where x - represents abscissa values of particles in a moving system
of coordinates which changed positions with the medium. Ordinate values
of particles belonging to the coordinate system can be derived by means of
differential equation (ill. 11). Correlation (ill. 12) was derived previously,
By integration derive
(HI. 2o)
when t = O, y = O and c.,^ - (wa - w) 7- ; substitute the value of c^ in equation
(III. 20) and derive finally
, y~(v.'0--v) (\-e~~)-. (III. 21)
Series (ill. 19) represents an equation of particle movements in rotation
field in a plane. Together with expression (ill. 21) this series determines the
distance of the particles from the center of flow rotation and the path traveled
by the particles during a given time. With a change in the time factor, the
variable members of factorization which contained component (c) can be of
positive or negative values, depending upon the ratio between (x^-) and (xg-XjJ.
-37-
-------
Equation (ill. 19) offers a physical explanation of the inertia separation
process at the initial section. A particle which entered a smooth curve, "
tends to continue in a straight line direction, and, as a consequence, changes
its position in relation to axis (x). Viscosity resistance forces exerted upon -
a particle impart to its trajectory a curvilinear character; in the course of
time the magnitude of relative (x) displacement increases. It should be
noted in this connection that differentiation with respect to (t) (ill. 19) makes
possible the determination of the particles' radial separation velocity.
Series (ill. 19) and its derivative series (dx/dt) are divergent series at
.large (t) values (t -. °= , x -» °° ). For this reason, it becomes impossible
to make use of the obtained functional relationships in the investigation of
particle movements at high (t) values. At the same time, there exist inter-
vals of (t), ( T), and (XQ) values when the magnitudes of the series' members
tend to diminish rapidly. Under such conditions it becomes possible to use
only the initial members of a series in order to obtain a satisfactorily
accurate conception of the particles' movements.
4. Separation of particles at the initial movement section -
To make the obtained functional relationships applicable in practice
it is necessary first to establish the exact distribution of velocities in the
flow as it entered the deflection threshold. It was noted previously that the
flow picture in such instances is a highly complex one. On the one hand,
there exist vortex flows of secondary circulation, \\hile, on the other hand,
the flow widens so that its boundaries become indeterminate,. as, for
instance in a cyclone. Under such conditions it becomes difficult to arrive
at accurate mathematical expressions for velocity fields. Usually it is
assumed that the flow cross section is a constant one. However, this
assumption is partly contrary to reality, when applied to the entire course
of the descending cyclone flow. According to some authors this assumption
can be applied to the gas flow at its entry into the cyclone (39).
Examine a flat horizontal flow in a curvilinear channel of a simple
depth formed by two concentric surfaces, as, for example, of the cyclone
wall and of the exhaust tube. Consider, as above, that the medium is of -
-38-
-------
viscous nature only in the immediate particle ambiency; assume also that
the %-elocities at the channel cross section had been distributed according to
the hyperbolic law. Use the system illustrated in Fig. 17. The distribution
of velocities along axis (x) can be expressed by the following law: w x =
= const. , and the ratTo of Velocity fw)~at any point Eo veTocity (w^ at rtre
extreme point (R^) can be expressed as follows: (w/w2) = (x2/x). The
volume flowing through the channel can be expressed as follows:
1 Q = (' wdx = K',.<2 In ~ .
I X,
Since Z/ (xg - x) =w , where w is the average velocity in the conveying
channel, then
(111.23)
from \vhich derive the law of velocity distribution in the channel under exami-
nation as expressed in equation (ill. 24).
-^. (111.24)
x In -£>.
Formula (ill. 24) makes possible the determination of constant value of (c),
v-hich is a component of equation (ill. 22). Taking into consideration the
equivalent values of (x) in equation (III. 24) and of (x ) in equation (ill. 22),
find that
c=WHl'l--*^$-}. (111.25)
The same formula (ill. 24) indicates that the constant in the hyperbolic law of
velocity changes can be expressed as follows:
(III. 26)
The functional relationships thus established facilitate the analyses of
particles' separation movement in a curvilinear channel. Much of the above
discussion and analytical procedures is based on assumptions, and the final
results can be only approximate . On the other hand, the number of factors
taken into consideration in this discussion was considerably greater than
in the immediately preceding discussion; accordingly, the results will more
nearly approach actuality.
-39-
-------
Now, analyze a concrete example: gas or air at 20° contained parti-
cles weighing 2. 5 g/cm3; the flow velocity is 15 m/sec. in the direction of
a curvilinear channel having an outer curvature, the radius of whiclvis 0.5.
In this case the constant will be:
'.,_ 15(0.5 — 0,05) ^ocp a2/sec.
A. — ;r~? ~* •" * '
0,05
Note: For each XQ entry there is a correspondingly defined value for
c = w0 =w (R).
Values of (c) and of factorization coefficients can be computed for
particles of 20 p, size, as shown in Table 3.
Table 3
1 . - - -
! , in0 m
i 0,45
0.4
0,35
0,3
d,?5
0,2
0.15
0.1
' in
m/sec.
8.4
7,r,5
h.6
5,25
3.3
0.45
- 4,5
—14,25
a±w.
in m
K.l
1.94
2,12
3,07
4
5,3
(..15
ii.l
-"' 10'
in m
0,1 )!>
0,85
1,1
1.51
2,18
3.42
6
13,.%
-^-10"
in m
0,11
0,53
0.7J
0.97
1.49
2,47
1.77
13,35
- ' 10*
in m
0 2f>
0,35
0.47
0,7
1.0')
1.97
3.92
12.17
Data in Table 3 show that the magnitude of the first factorization coefficient •
becomes reduced for higher (C) values. This is due to the fact that the par-
ticles lag behind the gas medium in the region where the gas flow has become
rearranged with a consequent velocity rise. Hence, a correction factor has
to be introduced in the calculation of the particle trajectory on the assump-
tion that the tangential velocities of the medium and of the particles were
equal. The magnitude of such a correction factor is of considerable signifi-
cance during the initial period of the flow. Actually each of the adduced £
coefficients is multiplied according to equation (ill. 22) by £ =_ 1 - e T
to a degree which, equals the numerical index of coefficient (a). At low
vaj-ues of (t) and ~. the value (5 n) rapidly-diminishes as (n) increases.,
(see Table 4), and the particle position can be determined by the value of
coefficient (aa), since values of the succeeding members become vamshmgly
small (approach O). For particles entering the deflection threshold in the
proximity of the inner flow boundary, the value of (x) in the moving coordi-
nate system at first appears less than (xo). In the course of time the posi-
tive members of the series gradually increase and (x) values begin to rise.
Data in Table 4 also indicate that in the region of w(R) < w absolute
values of the series' initial coefficients become reduced, and in the region'
of w (R) > w che values increase. At t = <» and in practice at (t/j) > 7,
|=1 and (x) tends asymptomatically towards infinity, . i. e. series (ill. 22)
-40- -
-------
Table 4
t
•
0.01
0,05
0,1
0,2
0,3
0.4
0,5
1
2
3
4
5
6
7
i
i t -r '
(1,00'iH
0,0 |H7
0,09."j2
O.INI.'J
O''5'l '
o'..i_«ir,
0,3935
0.(.321
0.81)47
0.9.-.0-2
0,9847-
0,9933
0,9975
0,9991
^
~0««~.io~
0/21 10-'
0,91 10'2
0.33-10 '
0.67- 10 -1
0.11
0.154
0.4
0,75
0.9
0,94
0,99
0,994
0,993
C>
0.96 10-6
0.12 10-3
O.S2 10 -^
O..'i0-10~-
0.17-10- '
0.3v 10"'
O.'il-lO"1
0,25
0.1x5
O.Sfi
0^4
0.93
0,991
0,997
e<
0.92-10"8
0,57 10"5
OS> 10" ^
o'.n-io-2
0,45 -1Q-2
O.l'2-lO-1
0,24 10- '
0.16
0 ~>F>
o!si
0.93
0,97
0988
0.994
i
i
:.
*
[
0903-10
O.l'75-lO
0.7* -!0
0.19 10
0,12 -;o
0.35
diverges at all m (XQ) values. By taking into account an adequately- great
number of members of the series, it becomes possible to apply polynome
(111.19) for the construction of particles' trajectories. It should be added
that the greater is the selected time (t) or (t/r) and the larger are the par-
ticles, the greater should be the number of elements of the series taken into
account.
Table 5 presents increment magnitudes (£x) for some values of (t/r)
at different distances (XQ) of particles' entering the deflection.
Table 5
0.3
0.8
1
450
400
350
300
250
200
150
100
0.015
0,018
0,023
0.023
0,038
0,51
0,062
0,021
0,12
0.147
0,1 EG'
0.237
0,314
0.423
0,549
0,3-36
:
0.256
0315
0394
0,?C>6
o.ccc
0
1.9u
2.46
3.15
4.17
5,74
S.3'5
13.1-
22.85
2,56
3.17
4 OS
5.42
7.54
10,26
17,7
""
Thus, it becomes possible to compute the movement of the larser
particles, however, to avoid gross errors as particle sizes increase, it is
necessary to reduce the time interval. Ordinates of the particles can be
computed by formula (III. 21). Magnitudes for similar particles of 20
diameters are listed in Table 6.
Examine the movement of particles in a mobile coordinate system,
the rotation velocity of which equals the velocity of the gas flow. Here, for
-41-
-------
Table 6
X0 III HH
450
400
350
300
250
200
150
100
•*
III SEC.
14,75
1R.G5
24.45
.33.2
47,8
74,5
132,5
•282
-1-o.r
f
O'lG'
0°20'
0"28'
(K'.'jfj'
0°"iO'
Ifl20'
2n20'
5°10'
Y
ism
1 <>
2,2
1,9
1,5
0.9
0
-1,3
-4,1
-L-o.3 i -L.o.5
• i t
V*
0"48'
10
1024'
jnjg'
2"30'
•p
7"
153'
Y i
1 M MM i *
t
C.7
6,1
5.1
u
0,3
— 3.5
1°2'
1"40'
2°20'
3°
4"10'
G"40'
11 "40'
-I1,3J25(150'
Y
IN ^a1
,
10
9.1
8
(3,3
4.
0.5
— 5.4
-17.7
-i.,o*
?''
Y
IX IW
1
2"
2"40'
3°30'
6"50'
Ifr^O'
18MO*
410
1-1.2
ii.i
K S.r,
5 (>
0.8
— 7.0
h moment (t) or its ratio (t/7 ) there will be a correspondingly determined
ition in the plane of the coordinate axes. This position is characterized
the axes' rotation angle -o^i^t, where uj = (k/x|). Values of deflection
les corresponding to (t/r ) and (XQ) are also listed in Table 6. Values of
express the advance in the case of positive values, and delay or lag in
case of negative values of the particle movement in relation to the gas
stream. Data in the
—' • — Table indicate that
(y) values increased
in the course of time
having an asympto-
matic tendency at
t -» °= in the direc-
tion of a defined
final value. Form-
ula (III. 21) shows
that this value is
equal to:
Particle trajectories
are illustrated in
Figure 18. The same
figure illustrates the
rectilinear velocities
of the flow section
just before the deflec-
tion threshold and
velocities of the hyper-
bolic section of the
rotating flow com-
puted by formula
(UI.24).
• ISO
fit. 18, PARTICLES' MOYEHEXT T«*JECTORIES IR A
flow.
-42-
-------
The trajectories are illustrated as per scale. Fig"/18 also illustrates tra- i
jectories of larger particles with T = o. 0275 (60ui) and T = 0. 1715 (150ui). Posi- [^
tion of particles was determined as follows : axis of abscissae was deflected
in the direction of the flow at angle (y) , after which a determination was made
of-dT.s~tance~xr = x,-, -F ATTT " AT perp-endrcular v-a's~ e-r^cted~atrt±rs -point aTTd~varltre-s-
of (y) were marked on it, the positive values in the upward direction and the
negative values in the downward direction. Attention should be called to the
fact that all trajectories deflected from the rectilinear course of the initial
particle movement in the direction of the flow deflection, i.e. the particles
were "blown off" their course by the deflective flow. The trajectory curva-
tures progressively increased in the course of time, the angle between the
particle trajectories and the c or res-po-nding flow line became smaller,
approaching, but never reaching zero.
Figure 19 presents velocity curves of ZO ^ size particles computed for
certain XD values using polynome (ill. ZZ). The curves in Fig. 19 show that
Figure 19
M/SE6.
FiGo 19° CURVES OF SEPARATION VELOCITY CHANGES OF PARTICLES i
?0 IH DIAMETERo
the velocities rise at first reached a maximum at (t/r) ^ 1, then began to
abate. Velocity curves of 60 u, particles are of a similar shape, although
here the maximum moved slightly in the direction of the coordinates'
origin. In the w < ^o region the particle trajectories are less distorted
than in region ^ > w0 The larger the particle the less distorted are its
trajectories. Dotted lines in Figure 18 establish the mutual distribution of
150 (j, particles at time t = 0.075 T. Using polynome (111.19) it is possible
to investigate the movement in limited time intervals only, which become
shorter as the particles increase and as (XQ) becomes smaller. The move-
ment of particles can be followed farther out to the left than to the right.
Duration of particle movement, shown in Figure 18 is determined by the
values of (t/-), which for 20 p, particles equal 3 to the right and up to 1 to the
left; for 60 p, particles the corresponding values are 1 to 0.5, and for 150 [j,
particles 0. Z to 0.05. Small errors resulting from the use of limited
number of order elements are indicated by the solid line curves.
-43-
-------
5. Separation of particles beyond the initial section
vcy practically equals zero beyond the initial section; accordingly,
the third member of (ill. 10) and both members of the left part of equation
(III. 11) become zero. The tangential particle velocity at any given moment
is practically equal to the flow velocity. However, the direction of tangen-
tial flow velocity continuously changes, while the particles tend to follow a
rectilinear course. For this reason the absolute particle velocity always
differs from the flow velocity by a magnitude equal to its radial component
vcx, i.e. its radial separation velocity. Acceleration vector direction in
this case coincides with the direction of vector F, so that the differential
equation of'the "quasi-stationary" movement can be expressed by generali-
zation (LLI. 27).
^ + 4^-^ = 0. (III. 27)
In the case of a flow \vith -jj = const, equation (ill. 27) can be easily
squared (32) to assume the following form:
c1 and cs - represent integration constants;
Xj^ and >s - represent roots of a characteristic equation, and
which are determined by formula
" 2- — 2 - v
Assuming that .,= -(— + '« -j,
equation (III. 27) will then be as follows:
- c—
x==C1c-'-'+C.«! ' . (III. 28)
Integration constants can be determined by assuming t =O and x =XQ,
i.e. at the deflection threshold separation velocity will equal (dx/dt) =O,
from which it follows that: Cl + Cj = ATO,
-44-
-------
and accordingly, the integration constants can be expressed as follows:
'1 +
For really small particles, O2f <£ '' and d «_A-p, C2 ~ 0. Substituting values of
the constants in equation (ill. 28), derive the following:
= x0ea •', from which it follows that
or
/•==.
The last equation duplicates expression (ill. 7) which determines the separa-
tion time by disregarding the movement separation force of inertia and the
Coriolis deflection force. It has been generally considered that in the case
of highly dispersed particles, when values of tu^ T^ are very small, the
allowable error was of no significant magnitude. Recent methods for the
computation of the separation movement of sol particles in curvilinear flows,
as indicated in Chapter I I, are based on type (II. 7) functions. In this connec
tion it is interesting to compare particle trajectories obtained by using such
functions with trajectories computed according to equation (III. 14).
Formula (II. 7) can be expressed also as follows;
i
at i = 0, x = x<>, c~ 4
or ( *'— fo from \vhich it follows that:
i .x'-.x40-|-4-2/e: -'" ' (HI. 29)
Using this equation it is possible to construct particle trajectories for
the above analyzed flow. Since v , - O, then y = O, and the position of
particles will be determined by (x) values by means of equation (ill. Z9) and
-45-
-------
by (to) values, listed in Table 6. The computed trajectories are presented in
Fig. 18 and can be compared with the more accurately illustrated or depicted
particle movements shown in the same Figure 18. Trajectories were com-
puted for ZO, 60 and 150 p, particles. Particles of each size were assigned
one- (XQ)- value, e.g.: for 20^, — xo =0. 15-rrt; for 60 u. — XQ = 0. 25 m, and- fa*- -
150 ^ — XQ = 0. 35 m. Arrows indicate positions along movement trajec-
tories occupied by particles at the same moment, computed according to
more accurate formulas. A comparison of the two sets of values brings out
a characteristic property of trajectories constructed with the aid of a simpli-
fied formula. Observation indicated that whereas true trajectories deflected
from the original straight line movement along the entire stretch in the direc-
tion of the flow, the simplified trajectories tended to deflect in the opposite
direction. Such a movement is entirely unnatural and and cannot be explained
on the basis of some sort of physical factors. It must have been caused by
the fact that the total value of the differential equation members which were
disregarded in the process of the formulas' simplification, were commen-
surate with the total value of the remaining members, i.e. it was significant
enough to introduce qualitative as well as quantitative errors into the final
calculation and^as a result, the constructed trajectories proved to be false.
Angles formed by tangents to the flow lines and to the movement
trajectories were, in fact, smaller than the angles similarly formed on the
basis of the simplified formulas. In this connection the actual particle radial
movement time is greater, and the settling efficiency may be considerably
lower than indicated by these formulas.
A comparison of trajectories of different particle sizes might lead to
the conclusion that the error becomes smaller with the reduction in particle
sizes, and that such errors may be disregarded where the particles were
very minute. However, it must be remembered that even a small error in
angle measuring can lead to considerable errors in determining the position
of a particle, especially where value (t/T ) is great. Thus, it must be con-
sidered that methods for the calculation of inertia separation of a body in
curvilinear flows based on the assumption of a uniform movement (dvc/dt =
= O) lead to results contrary to actual physical phenomena. Calculation must
be conducted only by using complete differential equations.
-46-
-------
Chapter IV
AERODYNAMIC EFFECT OF TRANSVERSE VELOCITY GRADIENT
FLOWS ON BODIES CARRIED ALONG- BY THEM
1. Applicability of Stoked formula to the condition of
ambient curvilinear flow around suspended bodies
Following established practices, it was assumed in the preceding dis-
cussion that outer forces applied to a particle acted according to the Stokes
formula. Figure 20 presents lines of a liquid flowing around an immovable
spherical body at low-Re numbers.
Figure ZO The lines are completely symetncal
in relation to a plane passing through
the sphere center perpendicular to the
direction of the flow. It had been
proven in theoretical hydromechanics
(42) that under conditions of such flows
over a spherical body the force exerted
by the flow consisted of two components.
One represented the resultant of the
normal forces, i.e. the medium pres-
sure on the sphere and is equal to
F|60 20. FLOW LINES OF A RECTILIHEAS POTENTIAL
FLOW AROUND A SPHERE AT RE < 1.
rv(
the other is equal to the resultant of the tangential tensions, i.e. the force
of medium friction against the surface of the sphere, and is equal to -innrV;
accordingly, the fluid resistance (F) to the sphere movement can be expressed
in the form of the following equation, in the presence of small Re numbers;
F is made up as follows: 1/3 of it is caused by the pressure of the sphere -
surrounding fluid and 2/3 is caused by the fluid friction against the sphere
surface.
Figure 21 illustrates a curve which depicts the functional relationship
existing between the coefficient of resistance to the spherical particle move
ment according to Stokes formula, and the Re number:
At high Re numbers the greater part of the resistance results from the fluid
pressure on the sphere surface, the friction against the surface being only a
minor contributor. Actual resistance changes proportionally with h" , in
-47-
-------
which (n) continuously increases approaching
2. In its general form the functional relation-
ship between the aerodynamic force exerted
on the spherical particle 'andTthe velocity "of
its relative movement can be expressed by
formula (1.2). In the range in which the
Stokes formula is applicable ^ = (24/Rc),
(Fig. 21); in its logarithmic form the plot
of the functional relationship assumes the
form of a straight line (l). The actual or
absolute functional relationship plot is in the
form of a curve (2). This curve was plotted
on the basis of average data of several experi-
mental tests, collected and processed by N.A.
Fuchs (11). A comparison of curve (l) with ,
. curve (2) shows that the curves coincided in
the region of small Re values. Where Re
values are great, the actual resistance
' exceeds the value determined by the Stokes
' formula. Thus, at Re = 1000 ,i,2/^;i = 19.5.
determined by the Stokes formula and the actual
-I 0 I Z 3
FIG, 2l<> CURVES FOR ^ = $ Re.
I-RESISTAHCE COEFFICIENT ACCORDING
TO STOKES; '
2-OITTO ACCORDING TO EXPERIMENTAL DATA;
3-CUHVE OF ERRORS
The ratio between velocities
velocities is as follows:
This indicates that the error in the determination of velocities with the aid
of Stokes' formula is considerably smaller than in the resistance calculation.
For example: at Re = 1000 the ratio between the velocities is (vCl/vc2^ =
= 4.416.
The functional relationship existing between the magnitude of this
error and the Re number is illustrated by curve 3 in Figure 21. Calculations
described in the preceding paragraph were based on Stokes formula. Maxi- .
mal v velocities in the case of XQ = 0.45 m were as follows: for 20u. ,
particle's - 0. 4 m/sec. , for 60(j, particles -3.5 m/sec. , and for 150(j, parti-
cles - 5 m/sec. The Reynolds numbers were correspondingly 0.5, 14', arid'
50, while the maximal relative errors were - 4, 28, and 80%.
As the value of XQ diminishes, the absolute value of the allowable error
in velocity calculation increases. The abscissa increments (AX) are directly
proportional to v . For this reason curvatures of particle trajectories are
actually greater than shown in Figure 18, which is essentially a first approxi-
mation The situation is somewhat different in the case of ordmate trajectory
determinations. In this case maximal Re values (atx0 = 0.1 m) are 19.1 for
20u. particles, 57.3 for 60^ particles,and 141 for 150^ particle s. Correspond-
ingly the errors in calculating vcy values are: 144, 186, and 232%. The
tangential component of separation velocity (vcy) diminishes in the course
-48-
-------
t_
of time in accordance with the exponential lav/(e r), i.e. very rapidly.
Accordingly, larger errors occur only in the course of the first movement
period. The greater the examined time interval the less will be the
general calculation error. Analyzing the calculation results arrived at
above and accounting for the manifeTsted~errors, it is possible ten each the—
conclusion that the trajectories of moving particles did not differ from the
corresponding flow lines as much as it might api>ear from the results
obtained by the use of simple formulas presented in Chapter 11. It is
undoubtedly more appropriate to use formula (I. 1) in the place of the Stokes
formula; but this would require extremely complicated solution procedures.
Generally speaking^ the-Stokes formula-can hardly produce desirably
accurate results. The magnitude of allowable error can be established by
the method of successive approximations, which offers the possibility of
introducing corresponding corrections. At the same time it is well known
that the Stokes formula was derived primarily for and proved applicable to
cases of spheres submerged by rectilinear incoming flows free from the
predominance of gradient velocities. The symmetry of lines of a curvilinear
flow surrounding a stationery sphere is disturbed; in consequence this very
fact disturbs and even invalidates the basic assumptions on which the Stokes
formula rests. As a result, the unqualified application, of Stokes formula in
cases of curvilinear flows becomes conditional; precise analysis indicates
that this formula is inapplicable even to a case of curvilinear movement of
particles which settle out of a potential horizontal flow (43). This condition
embraces also regularities expressed in (I.I) «nen the magnitude of the
resistance coefficient (••), which constituted a part of that formula, was
determined experimentally under conditions of a rectilinear flow.
No reference was found to studies for the determination of errors aris-
ing from the application of Stokes' or of formula (I. 1) to the calculation of
resistance of bodies to an ambient flow around then-,, and no mathematical
theory has been advanced relative to movements of solid bodies in a fluid
medium characterized by the presence of vortices. Movements of bodies
in regularly rotating fluid v.-nere x = const, are excepted. G. Taylor pub-
lished a series of studies during 1916-1923 or. the nature of solid bodies'
movement in rotating flows. Results of Taylor's investigations can be of
considerable help in the more precise determination of farces acting on sol
particles in the course of their relative movement in rotating flows.
2. The Taylor Theorem
Consider at first a two-dimensional flow originating in a quiescent non-
viscous fluid within which an infinitely long solid cylinder is moving. Let
,1, 1 stand for the function of such movement. By definition flow functions con-
stituting the fluid velocities in axes systems XOY (Fig. 22) relate to j-1 as
-49-
-------
indicated by the following two equations:
Figure 22
Component velocities'will be'as follows:
I
dt
~dtf
In addition, the normal velocity component at
any point on the cylinder surface should be:
.X
' {
(IV. 1)
FIG. 22. WITH REFERENCE TO THE
TAYLOR Th£OB=n DERIVATION.
where S and n — are correspondingly the
tangent and the normal with respect to the
surface.
Equations representing ideal fluid movement in the absence of body
forces can be represented as follows:
dt
(IV. 2)
Assume further that the entire system had been caused to rotate around
an axis parallel to the cylinder axis and that its angle velocity (tu) was con-
stant in relation to time and space. The movement can be analyzed as one
belonging to a system of moving coordinates (XOY) which rotate at the same
angle velocity (o>). At time (t), following the movement initiation the axis of
abscissae of the system forms an angle (o>t) with axis (OX). The flow
function (>''3)- of the resulting movement is equal to the sum of functions of
the superimposed flows; therefore,
(IV. 3)
where R
represents the distance from the rotation axis.
Components of the velocity resulting from the movement are as
follo\vs:
-50-
-------
and
(IV. 4)
(IV. 5)
An equation expressing the acceleration of a sol particle undergoing a
movement analogous to the one under present consideration was derived as
described in the preceding chapter. The equation is as follows:
dw.
Projections of components constituting (composing) complete accelera-
tion of fluid particles are expressed by the following two equations:
"v, " V
^r ui1 ' dv -*
Introduce into the equations the flow functions and obtain the following equa-
tions :
_
dl dt
and
(IV. 6)
(IV. 7)
Equations representing the movement resultant can be expressed as
follows:
di
dx
\_dpi
dt
(IV. 8)
i
From equations (IV. Z) and (IV. 8) derive the following two equations:
. .
- (Pa -Pi)=
and with the aid of expressions (IV. 6), (IV. 7) and (IV. 3) derive the following:
-51-
-------
Ox
from which derive the following:
' (IV. 9)
Equation (IV. 9) is the basic Taylor formula; it determines the pressure
in a two-dimensional curvilinear flow surrounding the cylinder. With the aid
of this formula it is possible to obtain the magnitude of the force characteriz-
ing the effect of the flow on the cylinder. Denote by (F) the aerodynamic
force acting per unit of cylinder height as it moves in a stationary medium,
and by (T) the force of a surrounding curvilinear flow acting upon the cylinder.
According to equation (IV. 9) these forces must be of different magnitudes; the '
difference between the two magnitudes can be determined by formulas in which-
the selected curvilinear integral was taken along contour (L) of the cylinder
cross section. (Fig. 22) ,/ . - .
'.r — r» = P Y (P: — />i) COS X <&, N
(i)
•-»
ry — ^y = P 9' (/'s — Pi) sm^ds.
U)
and
(IV. 10) .
Use formula (IV. 9) and substituting cosf.ds = dy. sinx^=*'—dx, obtain the
following:
and
The first integral of the right side of equation (iV.ll) can be expressed as
follows:
ti) «•)
Integrate and derive:
y--ds. (iv.iz)
(i)
-52-
-------
If flow function ( ,1,-J is computed .on the basis of the overflowing contour,
then
*! =0
and consequently
y~dT- ' (IV. 13)
(.L) L)
Denote coordinates of the cylinder cross section center by XQ and y ,
then coordinates at any point can be expressed as follows:
x =
In a general case of a free cylinder movement the following prevails:
- ^- = «'-» = ("cv-2rj) cos y. + (i;Cy + Q;) sin x. (iv.14)
where n - represents the cylinder angle rotation velocity around its own
axis for a general case of movement fi / O.
v and v - represent velocity components at the cylinder cross
ex cy t
section center.
Substitute expression (IV. 14) in equation (IV. 13) and obtain the following:
C-} (L)
= (j) [(.Vo + T)) (v^ - QTJ) dr, - (yt + rt) (vcy+Q<) d*<\ . (IV. 15)
(D
Using Green's formula express the curvilinear integral (IV. 15) with respect to
contour (L.) of the cylinder cross section through the integral with respect to
the surface of this cross section ( A):
= j j {— \ (y.
+
-53-
-------
Differentiate expressions within the square brackets and obtain:
«,,; . (IV 16)
(L) (i) I - .
where ^ = TT r2 ; r = cylinder radius. In an analogous manner Taylor
found:
vcz. I (IV. 17)
U> ,. J
Calculation of the second integral in the right members of equations (IV. 11)
can be made as follows:
, r " (IV. 18)
.
-Zlv,,. , '(IV. 19)
Now, transform equations (IV. 11) into the following:
.
), . (IV. 20)
i
i
The right part of expression (IV. 21) represents aerodynamic force com-
ponents, supplementary to those which are generated by the movement of a
cylinder in a stationary medium. The first force component is directed-
towards the center of the flow rotation, and the second is directed at right
angles towards the flow rotation.
Prudman investigated a s'phere movement in a similarly rotating
fluid (44). The complex mathematical solution of this problem likewise
leads to the determination of two supplemental components of aerodynamic
forces: one centrifugal, and the other directed along the normal of the rela-
tive movement. Calculations made on the basis of a sphere produced the
following:
t
Note that in both cases the magnitude of the components is equal to the
centrifugal force of the replaced fluid volume. This force must have been
-54-
-------
brought about by the existance of static pressure gradients acting along the
radius of the flow rotation; accordingly, it is designated as the force of
hydrostatic pressure. With regard to the other members, the following
should be considered: cofactor u>r2 is a velocity gyration or circulation, and
the^member of equation (IV, .2\Y 2- r-p t> v~c = p~Foi determines the buoyant force
per unit of cylinder which moved within a rotating fluid. It can be assumed
that the second member of the right part of equation (IV. 22) represented the
magnitude of the buoyant force, which in the case of a sphere cannot be
determined directly with the aid of the N.E. Zhukovskn theorem. Taylor's
was with reference to a non-viscous fluid. However, in the following years
W.P. Dean (45) had shown that Taylor's reasoning was equally true even
with reference to a viscous fluid. On the basLs- of- this-r it cart be— re-a-s.o-n.e-d~
that if a resistance force (F) developed by a rectilinear flow surrounding a
sphere in the presence of small Re can be expressed by the Stokes formula,
then the total value of aerodynamic force (T) must be equal to:
r.^SKrpi'c-f m,u[«fl) +-TrpTvc (IV. Z3)
Equation (IV. 23) is applicable to movement of aerosol particles in a
curvilinear flow rotation of constant velocity.
3. Experimental studies of movements of solid bodies
in a rotating system
According to the principle of mechanics a solid body moving recti-
linearly in a stationary system as the result of exterral forces can continue
its straight course \vhen the system undergoes rotation only apon the appli-
cation of supplemental forces, the intensity of \vhich must equal the combined
centrifugal and Coriolis resistance forces, and their action must be_in the
opposite direction. It will be necessary to apply a force of 2McwVf(Mc=K r2pa),
per unit of cylinder length, normal in relation to the direction of velocity VG
and of force Ao2/?, acting in the direction of the rotation center.
It can be concluded from equation (IV. 21) that when the moving cylinder
density equals the medium, the medium pressare forces equal the forces
needed to apply to the cylinder to cause it to maintain the rectilinear course
against the rotation of the entire system. Thus, if the cylinder length equals
the extension of the medium alone its rotation axis, then theoretically the
rectilinearity of its mo\en-.ent, if it had been such before the rotation onset,
should remain undisturbed in the course of the system's uniform rotation.
In the case of a sphere movement the^supplemental forces should be equal to
2/Vfii»t' with respect to normal lj\j —jL-r3?,] and /yf~2/j_ acting in the
direction of the rotation center.
-55-
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Equation (IV. 22) shows that the force of fluid pressure on a sphere is not
adequate to c ompe nsa te for the first component. As a c onsequenc e , the
sphere should not move rec U line a r 1\ under the influence of the same exter-
nal forces in a revolving system. It can be anticipated that in a counterclock-
wise revolving system the sphere would undergo deflection to the right.
This difference in the movements of a cylinder and a sphere served the
basis of Taylor's experimental check on his theorem (47). Fig. 23 is a sche-
matic illustration ot laylor's experiment. A glass cylinder reservoir filled
with saline solution, was placed into a pan having a transparent bottom; the"
reservoir was forced to rotate by a water jet corning from a nozzle at a rela-
tively great velocity ta ngent ia llv to the outer glass cylinder surface, The
"lass cylinder rotation velocity was kept under
Figure 23
I - C r L I '< 3: •> I C A L i 01 Y ; ?
T it: A i": " - ,.•• i o i >K • i N6'
4 - HA^BLE f R THREAS TI i H
'. - CELL U LOU IRItfet; '• _
L • VI L .
Figure 24
< ontrol within a wide1 range. A 20 mm diameter
brass cylinder 150 mm long was prepared, both
end.-> of which were sealed; a hollow glass ball
served as the spherical body. '1 he ratio between
the specific gravities ot the saline and the cylin-
(ierical and spherical bodies were such that upon
immersion the latter remained in a state of
s t, 111 ona r \' suspension, Changes in the positions
o: the bodies were brought about by means of thin
pliable filament-, (2 ;:• Fig. 3). The photographic
picture in Fig. 2-J illustrates one moment in the
i ylmder movement. 'I he picture shows that the
i \T i ndi• r m ox e el rei• 1111 :iea rl \ , since the di rec tion
ol the lower brandies coincide with the- dire, c tion
ol the upper steering ^lament. (Due to acciden-
tal causes tiii cylinder has assumed a lopsided
position, whu h in lnrn c'aused the tilaments to
: o r m ,i n a c ut e a n g i e ).
1 lie sphere illustrated
in Fig. 25 was sharply
de fleeted to the left
(counterclockwise)
because the reservoir
was moving clockwise,
as shown by the rela-
tive positions of the
pliable filaments. In-
crease in the reservoir
movement increased
the sphere deflection.
FlS. 2* PKO'OSRJPH SHCVIN6
CYLIKIER PCSITICK.
FlS, 25, PXOTOSRJPH .7 HOW IN*
SPH'-: FCSITION.
-56-
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The high degree of precision to which the Taylor theoretical calcula-
tions had been verified is regarded exceptional by theoretical hydrodynamics
to this day. (48,49). Therefore the Taylor theorem can be regarded as a
dependable basis for the evaluation of resistances created by curvilinear
{lows surrounding s oliaT bodies. - - -
4. The Taylor effect
The density of the medium, in this case a gas, constitutes a factor in
the second and third members of the right part of equation (IV. 23). As a.
result of this, and on the basis of their order and the order of the first, or
Stokes, member, it may be concluded that they can be dropped, i.e. T ^ F,
From this it follows that at low Re the resistance of a sphere in a curvilinear
flow can be determined with a satisfactory degree of accuracy by the lav/ of
Stokes. Hov. ever, an isolated case of sphere movement in a rotating flow
can be conceived in v.hich this assumed condition might be entirely erroneous..
as will be shown later. Here we have in mind a slow moving sphere, the rate
of which is either constant or only slightly variable; this is very characteris-
tic for the separation movement of aerosols, if a two-dimensional cylinder
of infinite length elicits a two-dimensional movement of the fluid, then the
movement of a similar tri-dimensional spherical body must simultaneously
elicit a tri-dimensional fluid flow.
J. Proudman iri his investigation (44) arrived at the following hydro-
dynamic equations for "low velocity'1 motion: by the term ''low velocity"
motion Proudman meant velocities so low that squares or multiples of such
velocities could be disregarded.
OP
dt "~~r, dr
^r, , ^ dP
dt -*~*~^*l — c)y <
fo:> d P
dt dz
n P 1 / - i i\ 7
in which ' — "? T~ 1A~"T>";(n •
(IV. 24)
It follows from equations (IV. 24) that where the low motion velocity is also
uniform, dP/dz = O, i.e. fluid flows generated by uniform slow motion must
likewise be two-dimensional in cases of tri-dimensional body movement.
Some years later Taylor (52) confirmed and restated the Proudman postulate
as follows: the circulation over an arbitrary contour (L) in a fluid the flow
of which is determined by the movement of a tri-dimensional body and simul-
taneously by rotation around its axis (OZ), is equal to:
-57-
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dx + ^'y, dy -\-'&Il dz)
The first of the two integrals represent the usual generalization which
expresses the velocity circulation in a flow. The second integral is equal to
where
A= -
v.-hich is the area of the contour projection upon a plane perpendicular to the
rotation axis. If the fluid viscosity is disregarded, then the circulation will
not change even with time, and the circulation of the motion under study can
be regarded as constant and equal to
r = Tj + 2mA = const, i
Evidently, if the relative body motion velocity is low, then the change in the
magnitude of A v/ill be small. Constancy of the projection area of the fluid
contour can be expressed by equation
Apply the Stokes reconversion to the latter integral and obtain the following
gene ralization:
d'""', aaV _ n I -7—' ' -L'"- )1 rfs = °. i
-,--" •'"' o: \ "A ' "y J\ (IV. 26)
(51
in which s - denotes the contour delimited area;
Li, 1, m,n - denotes the directing cosines of the external normal
v.dth respect to area ds.
Generalization (IV. 26) for any contour can be satisfied by the following
equations: - •
-58-
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f dtr,, da>, _ dxr, , _dtr.v,_ = Q
"oT~ dz '" U> dx 1 rfy
Since fluids are noncompressible, then it follows, from what was stated
above, that
The above conditions point to the fact that flow velocity is independent
of (z) indicating that the flow retains a piano - parallel character. Stated
otherwise, if a moderate tri-dimensional disturbance is imparted to a rotat-
ing fluid, then the resulting motion will be one in which two fluid particles,
existing in the same line parallel to the rotation axis prior to the tri-dimen-
sional disturbance, will remain in the same line at the same distance from
one another after the introduction of the tri-dimensional fluid disturbance.
As a matter of practical consideration it follows from the above that trans-
position of a tri-dimensional body will elicit the movement of a fluid cylinder,
the base of which will be equal to the diametric c ross-section of the body and the
height will be equal to the fluid depth in the direction of the rotation axis or,
in any case, to that part of it in which conditions of uniform fluid rotation pre-
vail. These conclusions of Taylor were brilliantly confirmed in a series of
experiments. Figure 26 is a schematic illustration of one of Taylor's experi-
ments. A rectangular hermetically closed vat of 200 x 300 x 100 mm was
filled with water and uniformly
rotated using an electric motor. Figure 26_
The top of the vat consisted of a J B g 'P
heavy glass plate. The transpo- '
sition of bodies was brought about,
with the aid of a long thin worm |
gear (F) operated by a stnall
electric motor (L,) fastened out-
side to the tank bottom as shown.
By this arrangement it was pos-
sible to impart to the body a slow '
and uniform movement. Flow t
lines generated by the fluid flow-
ing around the moving bodies were FIG. 26. SCHEMA OF TAYLOR'S SECOND SERIES OF EXPERIMENTS.
observed by introducing a dye
through a metallic tube (M). In order that the dye-containing apparatus
function at the proper time, specifically after the fluid flow velocity in the
reservoir attained a uniform rate, the bodies under investigation were imple-
mented so as to serve simultaneously as dye containing reservoirs. Tube (M)
was in contact with the lower part of the investigated body, while the upper
part remained air-filled. Compressed air was forced into reservoir (N)
through valve (O). The pressure increase caused the water to partially
penetrate into chamber (E); reduction in the reservoir pressure caused the
-59-
-------
r--
air to expand, thereby forcing some dye into the reservoir \'ia tube (M); the
compressed air was released from the resen'oir by opening valve (P).
Uniform and continuous release of compressed air through valve (P) was
attained by means of a capilliary tube (Q). Fig. 27 illustrates the results
of such an experiment. A cylinder 25 0111"1 high and 30 mOl lon~ was trans-
posed along the bottom of a reser\"oir.
The point from which the dye solution th read emL','gl:d is indicated by
(A) in Fig. 27. Arrow (B) points to the origin of tl:e w,d, r thrl:ad splitting}
(A) part of the tinted water thread streanH"cl ;tf'oll!1d till' right side of the solid
obstacle and formed an enveloping film (C) \vl;;ch 1)<" ,."j"t " ,d the surface of
the connected cy linde r. The othL: r part of the dye solutiC,t) thread streamed
around the left side of the solid obstacle and tl,l'n movl:d "way from the sur-
face forming vortices (D). Fig. 27 is a top-view photogr;jph; it shows beyond
any doubt that the cylinderical water volume above the solid body was not
tinted. To ascertain that this liquid FJ.gure 28
cylinde r actually moved with the body --
the open tip of tube (M) was brought to
a point above the uppe r base of the
solid body. A photograph of this ex-
pe riment is presented in Fig. 28.
Point (A) indicates the place of dye
discharge. Accumulation of tinted
fluid inside the water cylinder is
shown at (D). In this experiment the
dye was released at the very edge of
the reservoir as shown on the right
side of the illustration wl:ich also
A dye solution was released 30 0101
above the upper base of the cylinder at
a distance of 25 0101 from the c ylinde r.
In the case of rectilinear flow around
the cylinder the dye solution thread in
all probability should pass above the
cylinder. In the case under considera-
tion the dye streamlet moved more or
less rectilinearly only up to the point
corresponding with a vertical projec-
tion of the frontal critical cylinder
point, after which it split as if it had
enc ounte red a solid objec t. The
illustration indicates that the obst;:tcle
was in the form of a cylinder, the base
of which was equal to the base of the
solid cylinder.
FiQure 2.
~
F ! G. ?""7.
-: !." .
,\" L r-" I: Iti T r '. :11,..,'-.
j q "~IL: T ~('I.UM[.
,I:',)
r A
" .
,.,
..
FI(.~. :~
At)r;Ut~~ 0F
P:-:r;,T0~:' ~.,! '~.~
u: ~.;'l~ I E'!i ;:\,,0,,;"
~ J~I '!t.D
,L !~~E.
-60-
-------
shows that the released dye was transposed together with the solid body to a
distance equivalent to half the cylinder length; at the same time not even a
trace of it penetrated beyond the limits of the vertical fluid cylinder. Taylor
reported identical results of experiments in which the spherical body moved
at low velocity. He" noted" thTafTthe diameter of the Ve rficaf ^disturbance cylin-
der equalled the sphere diameter (46). Taylor had also noted that during the
transposition of the body which created the disturbance cylinder, the latter
was under the effect of ambient mediu-n flow. This medium flow around the
liquid cylinder becomes similar to that, of a medium flow around the solid
cylinder. Furthermore, in front of each cylinder section there formed a
critical point, and resistance to the solid cylinder movement was conditioned
by the height of the entire indueed- cyttnde r, i.e. by the depth of the-fluid in
the tank. There is reason to believe that creation of the Taylor effect was
conditioned by the piano-parallel character of the medium rotation. In actual
curvilinear flows, and in particular in cyclones, the flat character of the
flow can be disturbed to a greater extent than in Taylor experiments. Never-
theless, there is no doubt that ranges existed within the limits of which
velocity changes were still so slight that the disturbance in the ambient flow
surrounding a particle retained its two-dimensional character. Where the
height of the formed fluid figures measured in millimeters or millimeter
fractions, the actual resistance to the motion of a particle will differ con-
siderably from the resistance determined by the Stokes formula. Corres-
pondingly, velocities of particles separation may be considerably below those
determined by Stokes formula.
The concomitant movement of a sphere and liquid cylinder can be regarded
as the sum of several movements, one of which is in the nature of a rotation
around its own axis at velocity yj . Apparently there existed some connection
between this rotation movement and the buoyancy force; it should be remem-
bered that the Taylor effect did not apply to the movement of highly dispersed
sols, since the latter created no flows in the liquid. Sol systems include par-
ticles of different sizes, and the influence of this effect on the different
particles manifested itself at different intensities. At the same time, and
due to the interaction between the particles, there occurred an averaging
of this effect.
5. A case of potential medium movement.
Effect of particle rotation on the flow
Taylor's studies pertain to the one of two basic rotation instances
which is most characteristic for liquids in which u> = const. No exact
analytical solution of the problem related to body movement in potential
medium flows appeared thus far. At the same time Taylor's study can serve
as a basis for some concepts regarding body movement in potential flows
presented by external turbulence or vorticity fields.
-61-
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Absence of rotation movement in particles of a flow medium is a basic
condition which determined the existence of potential flow velocities. In this
connection the term rotation refers to the deflection of the liquid elements
around their own central axes. In contradistinction, a body within the limits
of the flow nucleus rotates in relation to the original position of its symmetry"
axis at a velocity equal to the angular vorticity velocity as shown in Figure 29.
Apparently the particle undergoes a rotation motion in relation to its medium
in the first case.
Figure 29
FlC. 29. Tl« INSTANCES OF CURVILINEAR
FLOW ROTATION.
Assume that one of the spherical fluid parti-
cles of a small radius (r) has suddenly solid-
ified; the transformation of the liquid parti-
cle into a solid could not effect any change in
its motion, i.e. it will continue to rotate in
relation to the liquid particles surrounding
it. Now, assume that this particle acquired
a density ^ different from the density of the
surrounding medium; under such conditions
the body thus formed will be essentially a
spherical aerosol particle. The tangential
air velocities in the proximity of the diamet-
rically juxtaposed points A and B of this sphere
are different, so that formula (IV. 27) in which
(IV. 27)
k denotes a constant determined by the law of surfaces, and R^ and Rg repre-
sent distances of points A and B from the axis of the flow rotation. Obviously
the particle acquires a rotatory movement around its axis perpendicular to
the rotation plane. Had the sphere movement developed without the rotation
around its axis, i.e. had the movement been the same as of a solid body
rigidly fixed by radius R, then velocities of points A and B would have been
determined by the angular velocity of the rotation transfer, and, accordingly,
velocities of points A and B would have been equal:
R'
** = *
R
To make velocities of points A and B coordinate with formula (IV. 27) it is
necessary to impose upon the "rigid" sphere rotation a secondary rotation
movement around its own axis. Particle rotation is elicited by the field
of nonuniform tangential medium velocities; the magnitude of the rotation
velocity is conditioned by the particle form, its mass (inertia), and other
factors.
-62-
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Under the effect of molecular cohision force the gas layer directly
abutting the sphere -surface adheres to it, as it were. Particle velocities
of this gas layer approximate the body velocity; as a result the rotating sol
particle imparts to the medium new velocity components. Thus, Figure 17
shows that particles of the medium in the proximity oFpoi~nts CT and" O1"'of'
the sphere acquire a velocity directed at right angles to the flow. It can be
naturally expected that the rotating sphere should create in the surrounding
medium a movement which will cause the trajectories of its particles to
form circuits distributed along plains perpendicular to the axis of the sphere
rotation having their centers on the rotation axis. Where Re is of small
value, the distribution of velocities elicited in a medium by a rotating par-
ticle can be studied math:enTatrca.ily (53, 34). D-iffere-rrtral eq-uatiron- oi-N-av'e-
Stokes and continuity equations of flows elicited by the motion of a viscous
noncompres sible fluid in cylinde rical coordinates f,
-------
^ , 9 , cp (see Fig. 31). 1 = X sin 9; under such conditions equation (IV. 32)
becomes as shown below: (IV. 33) '
Figure 31
Fis0 3!,, SPHERICAL COOROI HATES.
= 0.
(IV. 33} -
The Laplace operation can be expressed as
follows: " ' "
~d'ti
'sm-'O
in which the last member of the right part of the
equation becomes equal to O, since:
U>9
Substitute equation (IV. 33) and obtain the differential equation for the
velocity imparted to the medium by a rotating particle. (IV. 34)
a'i i 2 da'= i 1 d'ttie , clgO da>o w? „
~F" H —T )V T7 "I " " =1—a~T = 0.
(IV. 34)
As the distance from the rotating sphere increases, velocities elicited by it
abate; at a sufficiently great distance the effect of the rotating sphere com-
pletely disappears; it is, therefore, possible to assume that when
>. = co, w, — .
From the adhesion, condition it follows that when \ = r
= Q rsmO.
Solution of equation (IV. 34) can be found in expression (IV. 35)
(IV. 35)
in which f (\) is an unknown function of the distance from the particle center.
Introduce expression (IV. 35) into equation (IV. 34) and obtain
< ?L 4-1 d!- — M 4- -°-'B f-(^- — j(x) = 0 ''
-64-
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Now, transform and substitute the partial derivitives by complete derivitives
and obtain a linear Eiler type uniform differential equation of the second order.
It is now easy to establish that this equation can be satisfied by values of
f (A) = X and f (>) ^=l/>2. On the basis of well known properties of linear
differential equations solve equation (IV. 36) and obtain the following function
/(>) = *iM--r. (iv. 37)
Constant of equation (IV. 37) can be determined on the basis of above estab-
lished limiting conditions. When /. = co and & = 0
since it follows from equation (IV. 35) that i(™)=0, and the value of c2 ls
constant.
Over the surface of a sphere ). — r, therefore,
substitute these values into equation (IV. 37) and find
using equation (IV. 3 /) find velocity values as shown in (IV. 38). Now using
i^=--'^£-sinfJ. (IV. 38)
the^last equation examine the velocity distribution elicited in the medium by
the rotating sphere. Then, examine velocity changes in diametrical cross-
section plane perpendicular to the rotation axis. Since in this case sin 9 = 1,
it follows that (IV. 39} in \vhich u denotes the per.pheral velocity along the
surface of the diametrical particle c ross-sec tion. Thus, it is seen that
velocity changes in this plane deviate from the law of flat surfaces and
assumes the form of (IV. 40).
-U?f^ = r12 = i/r!. (IV. 39)
U.V-1 = const. (Iv. 40)
-65-
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Now, find the geometric loci of points in which the medium velocity value
becomes constant, as illustrated below:
^ ~ it- ~ "' i ' :
-- = ^V =.const, ' ^
in \vhich n > 1. Substitute expression above into equation (IV. 38) and obtain:
/,- = rir- sin
since
B '
in which x denotes the distance of the point under observation from the axis
of rotation; x =
, A- = r \'' n sin2').
(IV. 41)
Figure 32
Figure 3Z illustrates isochor velocities elicited in a gas by particle
rotation. These isochors are drawn for values of n = 3, 6, 9, ... The graph
also shows that the medium rotation
velocity rapidly abated with the increase
in the value of n. The above arguments
apply to a case in which the liquid moves
in a manner similar to a viscous medium
in every particular respect; such a fluid
movement is characteristic of a highly
, viscous fluid but not of gases, the vis-
l cosity of which is practically insignifi-
cant. In actuality such a gas movement
prevails only in the region of Re < 1.
:160 32« I30CHCPC3 OF ROTATING 6AS0
Where the Re number is great, the picture sharply changes. With an
increase in the Re number the low viscosity of a gas completely disappears.
Accordingly, the closest approximation could be obtained by completely dis-
regarding the viscosity forces. It should be noted in this connection that the
Eiler equation obtained under such conditions failed to satisfy the limiting
conditions of adhesion to particle surfaces. At the same time, such condi-
tions persist v.here the motion is characterized by high Re numbers. In
actuality \vnere the Re numbers were great, v iscosity forces were manifest
mainly in the layer immediately adjacent to the surface of the body, i.e. in
the limiting layer. It can be assumed that outside of this layer the motion
\\ill be that of an ideal fluid. In this way, and in distinction from the first
instance of high Re numbers, abatement of surface velocity of a sphere in a
stationary medium to O, or to a potential flow velocity, will occur in the
limiting layer of the last or end thickness. The thickness of the limiting
-66-
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sphere layer can be determined from the motion equation established by
L. Prandtle (55). In the case of laminar flow the thickness of the limiting
layer is equal to:
d
, Re'
in which d - denotes the particle diameter. Under practical conditions
the thickness of the limiting layer of a rapidly rotating particle equals tenths
and hundredeths of the particle diameter.
6. Resistance of a rotating particle
Gas suspended particles are under the effect of the ambient medium
flow during the process of their inertia separation. Examine first the nature
of such gas ambience at high Re numbers; consider the advancing current as
potential. The radius of the flow deflection is inmeasurably greater than the
size of the particles, so that the path of the advancing medium flow can be
regarded as rectilinear. Assume that the sphere illustrated in Fig. 20 begins
to rotate in the direction indicated by the arrows in Fig. 33. The sphere rota-
tion will elicit a rotating gas flow supe rimposed upon the main flow, as the
result of which the flow velocities in the proximity of the lower portion of the
sphere will decrease, while the flow lin-es
Figure 33 in the meridian section will appear approxi-
mately as shown in Fig. 33. According to
the Bernoulli law an increase in the velocity
at the upper part of the flow is connected
with a reduced pressure; likewise, as a re-
sult of deceleration in the lower streamlets
the pressure within them will increase; as a
result, the rotating body will experience an
upward fluid pressure.
Figure 34 presents photographically the flow
lines formed around a rotating cylinder (55).
The pressure upon the surface is greater in
the flow region underneath the cylinder than
the pressure in the advancing stream. It
follows from the symmetry of the flow lines
that the resultant pressure force acts in the
direction of the normal towards the ambient
flow and represents the buoyancy force which
Flf.. ~~. F(.~'W LINES OF A POTENTIAL
AMIIENT FL< • •: »n ; •; > ; TSTP :--_ri
Figure 34
is I = o vc per unit of cylinder length.
The N.E. Zhukovskii theorem applies to a
case of cylinderical turbulence overflow by
a rectilinear potential flow, and, as such,
is of no value in computing the force of a
sphere buoyancy.
-67-
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The case of a sphere rotation at low Re values accords with the Zhukovskii
theory even to a lesser degree. The flow resulting from the super position of
a viscous rotation movement over an advancing flow cannot be regarded as
turbulent free. However, as was shown by the work of 1\. Ya. Fabrikant (56),
a buoyancy force arises alsQ in the case of three-dimensional ambient flows.
In the case presented in Fig.' 34 the cylinder surface rotation velocity exceeded
the ambient flow velocity. At lower velocities the character of ambient flows
changes, Thus, lowering the rotation velocity to 1/3 brings into evidence two
critical points, as shown in Fig. 35. Further decrease in rotation velocity as
shown in Fig. 36 ma~:D;- '.
at
u
w
o
approaches the forward point of the cylinder along the path of the ambient
flow, and the buoyancy force tends to approach zero. The second resistance
force component, the fon:e of the fluid friction against the rotating surface,
likewise un de rgoes conside rable change during the body rotation. In fact, due
to the symmetry disturballce resultants of these forces at high rotation veloci-
ties constitute a pair of forces which ret;1t'd the sphere rotation; accordingly,
the medium resistance to the relative mo\'ement of the a':rosol particles
cannot be evaluated with the aid of Stokes' formula in this case as well.
7,
Adjoined vortex of a rotating particle
The hydrodynamic factors which dete rmine the rotation velocity
of a dust particle cannot undergo rapid changes in the case of slow moving dust
particles; therefore, it can be assumed that such velocity changes only slightly
with time and to some degree retains its constancy at different time intervals.
Likewise it can be justifiably assumed that the Proudman statement regarding
two-dimensional turbulences elicited by "slight" body movements in rotating
flows is applicable als 0 to flows elicited by the separation move ment of rotat-
ing aerosol particles. It follows from the above that turbulence elicited by
flows around rotating sol particles can also be of a two-dimensional character,
-68-
't,'\. ' I
1\'
-------
i.e. it extends in the liquid along the rotation axis on both sides of the parti-
cles to a height within the limits of which the flow movement as such can be
regarded as of a piano-parallel character. In contradistinction to the first
case~, a-cy1^ nd'j r i'j-aJ— y^lu.enp--wViirh follow^ the motion of a sphere rotates with
it with reference to the ambient medium. Thus, this fluid volume \vill repre-
sent nothing else but a vortex elicited by the particle rotation, i.e. an adjoin-
ing vortex. Naturally, such an affirmation must be verified experimentally;
experiments of Taylor cannot be regarded as exhaustive checks because they
had been conducted under somewhat different flow conditions. Nevertheless,
assume the existence of an adjoined vortex as an adequatelv reliable working
hypothesis. Evidently the axis of such a vortex coincide with the axis of the
rotating particle. Vortices are formed in many instances in flows* ambient
with regard to rotating and immovable bodies of corresponding configurations.
However, the vortex and body axes coincide only within the limits of the flow
and of the surrounded body length. The vortex termini deflect from the body
borders, deflect and extend along the flow into infinity. The system of termi-
nal vortices in this case determines the buoyancy force abatement and the
induction resistance appearance (48). Thus, the buoyancy force of rotating
bodies becomes fully apparent only in instances of two-dimensional body sur-
rounding flows. It should be noted that the buoyancy force of a rotating sphere
can attain high values. For example, this force can distort flight trajectories
of heavy tennis balls and of artillery projectiles (55). Taking into account the
tri-dimensional character of a flow, presented by isochores (Fig. 32), it can
be assumed that the configuration of vortex following a rotating sphere v/ill be
as illustrated in Figure 38. In a limitless fluid the extension of a coexisting
vortex might be infinite; this is also true of its buoyancy force. Actually,
space occupied by a fluid is always limited; therefore, the
Figure 38 buoyancy force of adjoined vortices is also functionally limited,
although it can at cimes attain great values, as shown by the
/" ' intensive sucking in power of atmospheric storms and whirl-
pools.
Fie. 39.
VQBTEX OF A
ItiS PARTICL
BOTAT-
A vortex circulation increases as it flows around long objects;
therefore, long logs or boards are more easily and more
rapidly sucked into a water whirlpool than short and round
objects. Thus, E. Poe in his book "Descent in a Maelstrom"
describes a case in which a iLsherman's long boat was approach-
ing a whirlpool center faster than smaller objects, such as
empty kegs. Similarly, atmospheric storms suck in relatively
large and heavy objects, the rotation of which can be of high
velocity. Upon reaching the storm nucleus the objects cease~
to rotate and rise to the surface with the ascending storm
current. As the body caught by the storm from the surface of
ground continues to move its velocity rapidly approaches the
velocity of the air current in the storm nucleus and the bodyr
assumes a "quasistationcry" state. In such cases the turbu-
lent effect of the body on the current will be very slight, and
-69-
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the resistance, according to the Taylor theory, can be very great. As a
consequence, vc values must remain small, and as the body moves upward
it approaches the storm nucleus limits. At times the objects caught by the
wind spout are carried into the upper part of the cyclone and are carried off
tens of kilometers from the point of origin. The greater part of the objects
carried off by the wind spout emerges from the storm nucleus and enter^the
limits of the, so-called, normal field of the whirlpool dispersion (57).
In the initial stages of the process the velocity of the body picked up by
the storm from the ground is below the air velocity. Under such conditions
the buoyancy force aided by the direction of the ambient medium flow acts in
the direction of the spout center. In the final stage of the process the picture
changes. At the time when the body emerges from the storm nucleus into
its external field, and thereafter, the tangential body velocity is greater than
the air velocity, and the buoyancy force acts in the opposite direction, i.e.
from the center towards the periphery thereby facilitating separation. In this
connection the little known experiments of C.Z. Weyher (58) of the Paris
observatory are of considerable interest. In Figure 39 a sphere S fixed on
axis AB was brought into rotation. The sphere rotation elicited air movement
perceived by the observers. The sphere was made of metallic platelets to en-
hance its effect on. the ambient air. An air-filled rubber balloon M was placed
in the equatorial plane E-E'. The
Figure 39 following was observed: the balloon
~ enc ountered by the air current began
\ ,, .x to rapidly rotate around the sphere
/ \ \ approaching it, even when it was
[ ^ * } / originally placed at some distance
from the sphere. A protecting belt F
was installed to prevent the balloon
from coming in actual contact with the
sphere. In the course of time the
rubber balloon rotation velocity m-
FIGO 3?0 SCHEMATIC i tLUSTRATi on OF BAYER'S
EXPERIMENT.
creased, and it began to move away
from the sphere. Another experiment
produced more indicative results.
Gilded spangles were introduced into
the sphere surroundings. At first all
the spangles moved toward this sphere,
forming a solid envelope rotating
around the sphere; this soon broke up into separate individual gilded particles
which intermittently approached towards and then moved away from the sphere.
Some spangles succeeded in making a half turn around the sphere, others came
close to it without coming in direct contact with the metallic platelets, while
some penetrated into the sphere through the openings between the platelets,
only to emerge from them again. It appeared as though particles, the tangen-
tial velocity of which was below the velocity of the ambient medium behaved as
if they had been under the effect of vortical centrifugal forces, while particles
-70-
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which moved faster than the ambient medium had been under the influence of
centrifugal forces. Unfortunately, the phenomenon of a conjoined vortex for-
mation has been investigated only slightly. There are no quantitative data on
the basis of which an adequate evaluation could be made of the hydraulic effect
of~the sFrhuITaheous vorfices. More than ttratj as Taylor had"mdtcaLt;d, tire
theoretical basis of this phenomenon had likewise beer, insufficiently investi-
gated. Tne reasoning advanced regarding vortices arising as a result of par-
ticle movement is expressed essentially in the form of an hypothesis which
needs further and more profound theoretical and experimental elaboration.
At the same time, it has been known that in some instances, particle rotation
itself exerted a substantial effect on their movement trajectories. This has
reference to the mo.v.ement in the bordering layers of true fluid flows.
8. Dust particle movement in a bordering layer
Rotation of particles can be observed not only in rotating flows, but in
the border regions of rectilinear flows. Layers of such flows adjoining
immovable surfaces a-re characterized by the property of velocity gradience.
Due to adnesions the velocity of particles in a flow is ecual to zero at its
stationery border, while beyond the boundary layer it can reach considerable
magnitudes. The thickness of the border layer depends upon the nature of the
flow; in the case of laminated flow, as shown in Figure 40, velocity chances
from zero to maximal (for instance, the axial velocity of a flow through pipes)
occur gradually, and the velocity gradient is of comparatively small value.
Due to the intensive c ross-sec tional inter-mixture of the fluid the velocity
profile in turbulent flow is as shov, n in Figure 41. In this case the velocity
change occurs primarily in a very thin laminar layer v.hich persists close to
Figure 40.
FLOW
FlCo '-I. Sr-ITCH 'J.~ Tl'nDilLEKT FLOW
to the s-irfaces which limit the flow. - or this reason the velocity gradient
within the limits of a laminary surface layer of a turbulent flow reaches high
values. Correspondingly, the angular velocity of particle rotation in a border
layer car. also become high. Gastershtadt (60) observed the movement of wheat
grain _n pneumatic conduits and noted that the wheat grain angle rotation velocity
could be as hign as 60. 000 rpm. P. de Felice (61) photographed the "bouncing"
movement of sand particles in a rectilinear conduit and found that angular
-71-
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velocity was of the order of 33,000 rpm. To obtain a visual picture of how
such a high angle rotation velocity can arise, examine a particle which settled
upon the bottom of an air conduit as a result of its own force of gravity. The
thickness of the border layer in the order to its magnitude is commensurate
with the particle measurement. Accordingly, at the time the lowest surface
point of a particle, which came in contact with the bottom of the conduit, finds
itself in the zone of immovable gas, the upper particle surface is under the
effect of the carrying-away force of the flow, the value of which differs con-
siderably from zero. Only minute particles commensurate with the medium
molecules can remain immovable by adhering to the conduit surfaces in a
manner similar to the gas molecules. The basic particle mass is carried in
the direction of the air current along the conduit bottom. Depending upon the
form of particles, their movement can be also a sliding one; however, due to
the continuous action of the tilting momentum the rolling movement predomi-
nates. Taking into account the ratio of flow velocity to particle sizes, it is
possible to conclude that angular rotation velocity can reach values noted by
Gas te rshtadt and de Felice. Now, assume that a particle rolling along the
bottom of the conduit became transposed beyond the limit of the thin laminar
border layer while it continued to rotate. Beyond the limits of the laminar
layer the rotating particle will be under the influence of conditions closely
approximating those implied in the N.E. Zhukovskii theorem and will require
a buoyancy force. The magnitude of this force for a cylinderical particle of
height and radius r, and without accounting for the effect of the limited cylinder
height will bejsqual to:"
It can' be easily seen that the buoyancy force is greater than the particle's
force of gravity beginning with the moment at which the following equation
prevails: __
Accordingly, it is only necessary that the particle velocity in the de Felice
experiment differ from the air velocity by 0.04-0.1 m/sec. in order that the
particle buoyancy force exceed the gravity force. The buoyancy force causes
the particle to move upward until it ceases to rotate, after which it begins to
move horizontally for a short time and comes to rest upon the conduit bottom.
The upward particle movement in turn engenders a buoyancy force directed
along the flow path, i.e. it contributes to the equalization of the particle
velocity with that of the flow. The velocity abating effect of this force is no
more able to support the particle wnich begins to descend. The descent of
the still rotating particle apparently may again elicit a buoyancy force, in this
instance, in the direction opposite to the medium flow. The retardation effect
of this force, in turn, is reflected in the return of the buoyancy force, which
imparts a slanted character to the descending trajectory branch. By means of
the N.E. Zhukovskii theorem it is possible to follow the particle movement
beyond the limits of the laminar border layer where the movement can be
-72-
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regarded as potential. According to N.A. Fuchs the cause of particles emerg-
ing from this layer can be the resilient collision against small wall surface
elevations. At the same time, the previously mentioned conclusions, drawn
from the investigation of N. Ya Fabrikant, indicate that a rotating particle
can acquire a Buoyancy force even within an ambfent laminar flow. Such are
the present day approximate concepts regarding the mechanics of a particle
separation from the walls of a rectilinear conduit. The theory applicable to
such phenomena in the random movement of an infinite cylinder is a corollary
of the Taylor theory, and is amply described in his previously mentioned
work (46). In fact, the velocity of a laminar flow along a flat surface wall
can be approximately presented by the following functional expression within
the-framework of axes (XOY) so constructed that the ordinates are in a posi-
tion perpendicular to the flow direction and to the wall.
uu = a Y
x
in which Q; denotes the proportionality coefficient.
On the basis of generally accepted concepts of mechanics, a gradient
flow can be regarded as a flow rotation around instantaneous rotation centers
continuously transposed along a straight line. On the basis of such a concept
Q; represents something different from the angular flow rotation velocity
around instantaneous centers. In both instances a also equals the angular
rotation velocity of all particles in the flow, including those of the suspended
sol particles. Taylor proved that forces which acted on a cyclotomic cylinder,
moving in a gradient rectilinear flow, differed from forces which acted on the
same cylinder in a flow free from gradient velocity from component
2n r3 p Qf v directed perpendicular to the cylinder movement towards the
negative ordinates, and from component nrpc^ yo, directed toward the
positive coordinates (y is the ordinate of the cylinder c ross-section center
of gravity). It can be easily seen that these forces are analogous to the
previously derived supplemental aerodynamic foi'ce components.
Properties of a gradient flow are analogous to those of a fluid rotation
movement (46); therefore, the concept of a two-dimensional flow elicited by
a slow body transposition, i.e. the engendering of adjoined vortices, can be
extended to include a gradient flow. Characteristically, the centrifugal
component can be greater than the buoyancy force, especially in instances
of small v and relatively great y , which means in larmnary flow of fluid.
Prandtle observed that small objects floating along a river shore are carried
toward the center in the course of time. Similar movement phenomena had
been observed also by P. Bass (64). The observed phenomenon also applies
to particles moving along the walls of a cyclone. The velocity gradient per-
sists also outside of a laminared layer; therefore, the rotation velocity of a
particle abated more gradually, and it might be expected that in their centri-
figual movement individual particles might finally lodge within the limits of
the ascending cyclone flow. This conclusion is confirmed by the studies made
• -73- -
-------
FIG
EXPERIMENTAL T, DAKIELS CYCLONE.
by T.C. Daniels with an experimental uniflow cyclone (67); the schematic
drawing of the cyclone is shown in Fig. 42. The dust-polluted air entered the
cyclone through a paddle-type rotating device (l). The dust-laden peripheral
layers of the twisted flow passed through the ring-shaped slit (3) and entered
the dus-t accumulator (5)-. The
Figure 42 purified air left the apparatus re-
taining its original flow-direction
via the conical apparatus section
(4)7 In the course of the experi-
ments the length of the straight
section and dimensions of the exit
openings were being changed,
whenever it was necessary, by
installing appropriate insertions.
The diameter of the cylinderical
part of the apparatus (2) measured
50 mm. Best total efficiency was
obtained with cyclones 240 mm
long; further increase in the cyl-
inder length resulted in a slight efficiency decrease; increase in the width of
the slit opening increased T] consistently. The diameter of the conical exit
part of the apparatus was 40 mm.
Tests had been conducted with air passing through the cyclone at rates
of 85 and 130 m3/hr. Results are presented in the form of curves in Fig. 43.
Examination of the fractional efficiency curve (1) pointed to a reduction in the
separation efficiency as particle diameters increased; this was made evident
by the fact that the slope of the curve, from left to right, gradually abated.
This phenomenon was also observed by P. A. Figure 43
Koiizov (18). At operational capacity of 130 m3/hr. ,\ _*Ti_% .
i.e. at higher velocity of gas inflow into the cylin- ';
der and a correspondingly increased angular veloc-
ity (a) the drop in the effectiveness of large partic-
les separation becomes pronounced. Curve 2 illus - ^
trates results of experiments conducted under simi-
60
50
30 >_4
lar conditions; its course indicates that beginning
\vith particles lOOii in diameter the separation ~
efficiency sharply dropped, as shown by the curve
(2) of Fig. 43, which dropped sharply in the region
of d > lOOii.
From the viewpoint of the mechanical sepa-
ration theory this efficiency reduction in the
fractional separation of large particles appeared
as an irregular and unexpected phenomenon which
Daniels explains by assuming that the particles |>2_ cu(ms Of FHACTPOtlAL EFFICIEHCY
ricocheted from the cyclone wall, he also believed AT 85 AMD 135 M3,Am CAPACITY 3- BIS-
that the intensity at which the particles recoiled
w
30
20
>0
0 20
*fO SO 80 WO 120 HO 160 't
d U ;
43. T. DANIELS CYCLONE EFFICIEHCY
OF
OUST.
-74-
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from the cyclone wall varied in-direct proportion to the initial gas velocity
and, accordingly, with the rate at which the suspended particles were carried
along. An analysis of this phenomenon on the basis of hydrodynamic factors
offers a more accurate explanation. With increase in gas velocity, the rota-
tiem- momentum impartcd-by- fchc gas to the particte-alrS-eHmcrearsed, -as~a—
result of its impact against the cyclone wall. Because of this and of high gas
flow velocity the initial particle rotation velocity also increased simultaneously
with the increase in its buoyancy force which in turn similarly affected the
particle's radial transposition. This effect is more pronounced in the case of
large particles the diameter of which is greater than the depth of the laminar
border layer. In the case of large diameter cyclones, the separation of large
particles is generally not as pronounced. This maybe due to the fact that the
buoyancy force effect is of short duration, so that in large diameter cyclones
they fail to carry the particles beyond the limits of the descending flow. In
addition the cyclone construction actually facilitates the carrying off of the
rebounding particles. The overall experimental cyclone efficiency at 85 m3/hr.
operational capacity amounted to 71.2%. In the next experiment the cyclone
walls were moistened with water which entered the cyclone tangcntially close
to the paddle-type rotating device, so that the cyclone walls had been always
covered by a thin layer of water. Experiments had been conducted at the same
85 m3/hr. operational capacity. Results are presented in the form of curves
in Fig. 44. The curves show that moistening the cyclone walls with water re-
sulted in a considerable increase of cyclone operation efficiency, amounting to
an average of 91.5%. Daniels correctly explained this phenomenon by the fact
that the water layer formed over the cyclone wall
Figure 44 agglomerated the moistened particles and pre-
._ __ — — vented them from ricocheting from the wall.
It should be noted that Taylor's study of a flow
having a constant velocity gradient can be regarded
as a special case of gradient flow approximating a
case of hyperbolic type of velocity change in a
cyclone, therefore, Taylor's conclusions can be
regarded as applicable to the explanation of phe-
nomena occurring in a descending cyclone flow.
Changes in the cross-section of the conical cy-
clone part constantly disturbed the flat character
of the gas flow. Apparently, this condition found
its reflection in the enhanced separation efficiency.
brought about by the development and introduction
of the conical part of the cyclone apparatus. It has
been established of late that enhanced efficiency
was brought about not only by the conical narrow-
ing of the cyclone apparatus, but also by widening
its diameter (68).
(I 20 1.0 60 Sil tuO 120 tlO ISO
d \i,
Fie. ?-*,. EFFECT OF WALL I-ETTIKS on
T.DA'MELS CYCLOIIE EFFICIENCY.
I - Fn/CTIOt.1 EFFICIENCY CU=. VE AT
6*5 PP/hF.. CnPACITY 2 - FR ACT I OK
CC. rOC.iTION OF ORIGINAL OUST
3- SiTTO Or TfllPP-0 DUST,
-75-
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Chapter V
SOME AERODYNAMIC CHARACTERISTICS OF CYCLONE APPARATUSES
1. Pres scrre distribution in cyclone apparatus-e-s--
It was previously noted that the carrying-off of particles by curvilinear
flows was effected by the force of medium viscosity resistance to the par-
ticles relative movement, and by the force of hycLrodynamic pressure. In
this connection it might be apropros to establish the pressure distribution
in the cross-section plane of a cyclone. Changes in the gas flow velocity are
'always coupled with changes in the static pressure existing in the flow. In a
curvilinear flow the pressure change occurs also perpendicular to the lines of
flow. This change is conditioned by centrifugal forces, and the static pres-
sure gradient is determined in such cases by the following formula.
' dp = ?£dR. (V.I)
Pressure (p) can be determined with the aid of this formula at any given point
of a curvilinear flow. In a special case, where distribution of flow velocities
is determined by the hyperbolic law : ^ = -— .-
' f\
The integration constant can be determined from the fact that at the external
cyclone wall, where R = R2, the pressure reaches maximal value of p = p2
Under such conditions
and
(V.3)
According to the last formula (V. 3) the pressure continually abates as
the radius diminishes with the approach to the vortex nucleus. A study of
pressure changes in the region of R < R» occupied by the nucleus of a flat
vortex can be made with the aid of the Euler formula for ideal fluid movement
(60). If the body force be disregarded, then for movement within planes
parallel to (xy) these equations assume the following form:
J^'-c _L ..dv<- = L . -d!L • \
* Ox "• . -y dy ?
-------
The linear velocity of fluid elements in a nucleus is equal to v = t«R. Since
R2 = x-j + y2, the components of linear velocity along the coordinates equal to
..... . I _Vr_r= «> /?COS L^-VL= " :"»>'
•vy —. m /?sln \vy\ = ID A".
from which it follows that
(iv '
and equation (V. 4) assumes the form of
' -o,^ = --L.^" '
•f ttx •
: - 2 ^^- (V. 5)
Integrate the last equation (V. 5) and obtain
(V.6)
At the border of the vortex nucleus, i.e. where R = Rg , determine the
pressure with the aid of expression (V.3). Accordingly, the integration
constant will be
now substitute this value in equation (V. 6) and derive
(V.7)
This generalization indicates that the pressure continues to abate also inside
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the vortex'nucleus as the value of (R) diminishes. On the rotation axis
R =O, and the pressure equals: •
/»„ = /7.,-f ?~-? n?
(V.8)
The last formula yields
the cyclone apparatus.
Figure 45
Fi6« 45 - CURVES OF
VELOCITY AND STATIC PRES-
SURE CHANGES (PRESSURE
Of0?) 111 THE CYCLONE,,
the value of minimal excess pressure developed in
Figure 45 is a schematic presentation of pres-
sure changes according to formulas (V. 3) and
(V.8), and Figure 46 presents curves of experi-
mental pressure change in the conical part of
cyclone LIOT (Zl). The cyclone was tested while
operating under pressure. Figure 45 also illus-
trates approximate distribution of tangential
velocities (curve 1). Now, using formulas (V. 3)
and (V. 7) determine the pressure force exerted
upon a spherically shaped particle in a curvilinear
flow at distance (R) from the rotation center which
is conditioned by pressure gradient of the flow.
To do this compute the force exerted on an ele-
ment part of a sphere surface close to point M, as
shown in Figure 47. The area of this element,
which is within the rectilinear coordinates, is
equal to:
1 ~ ~ ~ —— - — -
=^ dx dy.
Figure 46
BO K
MM
FIGO 46 - EXPERIMENTAL CURVES
OF PRESSURE DROP (STATIC PRES-
SURE CHANGES) IN CYCLOME LIOT.
(SEE Fi60 4)
Convert to the system of spherical coordinates
r, 9, 'o, and obtain Fig. 31. (p. 64).
, -c -}- / = r2, dx dy ~ r, dr^ d ? .
Substitute the latter expression in the preceding
and obtain :
Since = r sin 6 and dr1 = r cos 6, then
"t//=rrsin9(/erf'f.
(V. 9)
The pressure force exerted on selected surface
element is equal to ~dN = padf, and the projection
of this force in the direction of the rotation radius
-78-
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Fioure.47
is:
0
Fie. 47. RELATIVE TO THE DET£=«I;JATIO« OF
M£SI3" P>iESSt.'°E FOFCES 00 SbSP-NOEB AEtCSCL
PARTICLES.
^•V>, = -/>„/•-sin 6 cos 9 cos(9 — a). Angle a is
very small. For practical purposes tangent values of this angle for aerosols
are of significance only within the limits of 10"6 < tg"
-------
The overall pressure force exerted upon the particle can be determined by
quadruplicate integral of generalization (V.12), as shown in (V.13).
O 0
The first integral can be converted to a tabulated form by the following substi-
tutions :
a — cos 6; du = — sinOdB
and by the corresponding change in the integration limits (53), as shown in
A —
i
in
extend the logarithm into a dual member series and discard values of second
or higher degree of stnallness and obtain
Substitute the value of I, into equation. (V. 13) and perform second integration
to obtain the final equation
i
in which rru is the mass of gas volume replaced by the particle. The differ-
ence in pressure at points M and M1 for a particle in the ascending flow is
determined by expression (V. 7) as follows:
P'a - P» = P i/ - Rl] = — 2 R r^ p cos 0,
d\! — _ 2 /Pr1 01% sin H cos- QdSd-f.
The overall pressure force is as follows:
r/2-,2
rt \
A'=-
The first integration yields: • r? "'
/, = — [ sin 8 cos2 9 d 0 = ~ .
-80-
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The second integration leads to equation (V.14)
»-- -
Thus, the pressure force exerted upon a particle is determined in all cases
by the centrifugal force of the medium replaced by the particle, which is a
variety of the Archimedes force. By juxtaposing the obtained value with
formula (IV. 23) it can be seen that the computation resulted in the determina-
tion of one component of the aerodynamic force effect of a curvilinear flow on
a particle.
2. Distribution of rotation velocities at the
bottom of a cyclone apparatus
Radial Flow
Rotation of liquid layers elicits the rotation of adjacent liquid layers.
Therefore, the rotation movement can come to a stop only when it reaches
the fluid boundaries, provided that it does not stop as a result of friction
against the container walls. In the cyclone apparatus its bottom constitutes
one of the liquid boundaries. Some properties of a flow adjacent to the bottom
fluid boundary are of considerable practical interest. Currents generated by
a rotating fluid in the proximity of an immovable base are examined by the
theory of boundary layers (55, 70) and are characterized by the following
property. In fluid particles located at some distance from an immovable base
the radial pressure gradient is counterbalanced by a centrifugal force. In
closer proximity to the immovable base the tangential particle velocity abates
due to friction; this, in turn, reduces the centrifugal force. The radial
gradient of a static pressure remains the same as at a considerable distance
from the wall; the balance between the static pressure force and the centri-
fugal force becomes disturbed, giving rise to a radial flow in the direction
of the rotation axis, i.e. radial and actual flow. As a consequence of its
continuity this flow elicits an axial flow directed along the external normal
towards the base.
U. Bedevadt (70) investigated a case of fluid rotation at a constant
angular velocity. He used the Navie r-Stokes equations considerably simpli-
fied by the symmetry of the flo-ws generated in relation to fhe rotation axis,
and also the equations of flow continuity. In integrating the equations distance
(z) from the base was replaced by the dimensionless value C = z}' "'- ln which
v is the coefficient of kynematic viscosity, and the component velocities are
presented by formulas (V.15).
or,), K-, = r' v.., w(c). (v.is)
-81-
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'-"*
Figure 48
The system of equations thus obtained was solved with the aid of expansion
of functions F, G, H, into series of power £. Computed values of these
functions are presented graphically by curves in Figure 48. The maximal
radial velocity value is attained when z^]' v , and is equal approximately
to one half the velocity value attained aL - -
some distance from the base. The value
of function F is independent of the radius;
therefore, as can been seen from formula
(V.15), the radial velocity abates in the
direction from the periphery towards the
center, while the axial velocity remains
unchanged. With the aid of these formulas
it is possible to establish the tangent of
angle a. formed by the flow lines with the
plane of the immovable base:
~-
Kf-C)
Computations show that in cyclones the
thickness of the boundary layer, within
the limits of \vhich there occurs an inten-
sive superflow (O^C<13), measures in ji.
In atmospheric dust storms, where (k)
values are great and (a:) values are small,
the thickness of such a layer can be con-
Fie. 48. CHANGES in ROTATING GAS VELOCITY
IN THE PROXIMITY OF THE STATIONARY OASE
(BOTTOM),
siderably greater, and the flow velocity can be as high as hundreds of meters
per second. The sucking in effect of a tornado eddy can be explained by a
partial vacuum creation, i.e. by the effect of Archimedes forces, which can
be determined by formula (V. 14). Therefore, it is of interest to compare
force N acting on a particle with the radial flow force which draws it towards
the rotation center by a radial flow engendered at the earth's surface. The
intensity of this force can be expressed by the following formula:
By comparing this expression with expression (V.14) it is possible to estab-
lish the ratio between the values of the tangential and radial velocities, which
are equally capable of manifesting particle c arry away effects .
'«'«=V-T-?T- (y-16)
Tangential rotation velocities as high as 200 m/sec. have been observed in
atmospheric storm eddies. The exte rnal , nucleus radius ranged between
25 to 200 meters (57). By substituting < w = 200 m/sec. and R - m into expres-
sion (V. 16) it can be found that Wr>, for example for a particle of ,'r =50 m at
-82-
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air temperature of 20°, it can be determined that:
2 25 KXi-1,16'1-4-101 n A~ /
( "'R = 9.1U10.9.8I.1.M-50 ~°'03 m/sec.
In this way the drawing in effect created by the pressure forces developed •
in a powerful storm is equivalent to the carry-away effect of a snow drift
blowing at a rate of 0. 03 m/sec. from which it follows that the sucking in
effect cannot be explained on the basis of the Archimedes forces alone.
As soon as the object under observation comes into motion, it acquires
a radial and simultaneous-tangential velocity. Curves in Fig. 48 shows, that
the tangential velocity of the whirlpool is always greater than the radial flow
velocity in the ground or snow drift. The mechanism of particle acceleration
in both instances is the same; therefore, it can be assumed that at any given
moment the tangential particle velocity is greater than its radial velocity.
The particle rotation movement is followed almost immediately by the appear-
ance of a centrifugal force. It can be easily conceived that the intensity of
this force will exceed the Archimedes force the very moment the peripheral
particle velocity will reach a value at which it1.,—1 ^- xp, , or only thousandths
of the tangential whirlpool velocity. A further comparison of the centrifugal
force with the aerodynamic force, with which the radial flow carries off the
particles, shows that the ratio of the two forces is equal to ,T(0 -- . This
ratio indicates that the radial flow can carry to the center of rotation small
size particles (T ~">> 1) which can be easily observed under actual conditions.
Characteristically, the property of rotation flows to transpose small
particles along an immovable base in the direction of the rotation center,
which is diametrically contrary to the principle of cyclone action has also
found its practical application. Thus, in the Geiger and other types of sand
catchers the water which carries the sand particles is run through a special
deflection chamber. As the sand moves over the immovable bottom it
concentrates in the mid-section of the silo from which it is removed by a
special pump (64). In the cyclone apparatuses, especially in those with flat
bottoms, the lowest or bottom currents can reach velocities of considerable
magnitudes. Upon uniting with the secondary flows, as described in Chapter
2, these bottom currents control the character of the entire secondary circu-
lation generated in the cyclones. The radial components of this circulation,
directed towards the apparatus axis, impede the centrifugal separation move-
ment of the particles; as a result this flow constitutes one of the factors which
negatively affect the cyclone apparatus efficiency. Dotted lines in Fig. 7
(p. 18) indicate the magnitude which such radial components can reach. The
existence of radial compoents has been confirmed also by results of other
recent investigations. (16, 72, 73)
E. Fcifel (74) (75) vas the first to conjecture the existence of radial
flows. Figure 49 schematically illustrates the Feifel cyclone apparatus
-83-
-------
employed in an attempt to utilize the radial flow. The raw gas was run in
through a tangential slit in the central-part of the apparatus, and came out
through a circular opening in the apparatus top lid. Much of the dust con-
centrate was removed through circular slits distributed over the perimeter
of the ttppe-r and lower flat lids. Feifel believed that the gas flow in cyclones
might have resulted from superposition of a whirling flow over a plane flow.
The field of plane flow velocities can be determined by the following expres-
sion:
--".
-------
By utilizing this function's properties it is possible to derive an equation for
the family of flow lines of the resultant flow, as shown below:
/• = ce
(V.18)
in which (c) represents an arbitrary constant. In this way flow lines gener-
ated by the superposition of an overflow over a vortex are in the form of
spiral curves. In accordance with the method of gas supply into the cyclone
and with the space character of the flow, the flow lines in the cyclone differ
considerably from the ideal system indicated by equation (V.18). In particu-
lar, in cyclones of the generally known type, the radial flow is engendered
by the secondary flows, the general scheme of which was analyzed in preced-
ing paragraphs.
A secondary circulation flow is established in a cyclone apparatus
presented in Fig. 49, simultaneously with the appearance of a vortex move-
ment connected with the formation of a static pressure gradient. At the mid -
part of the inflowing slit the velocity is higher. This circumstance imparts
to the secondary flows some symmetry, and the air flow is from the central
slits towards both end walls. As in the previously examined case the upper
branch of the secondary flow streamed toward the outflow section, while the
lower branch of the secondary flows formed an ascending flow. In the course
of their movements the boundary air layers encounter the circular slit where
the pressure is equal to that of the atmosphere, and exit through it. Since the
boundary layers are relatively heavily dust laden it carries some of the dust
out with them. According to recorded data (76) batteries of miniature Feifel
cyclones operated at high efficiency. Such battery cyclones had at one time
found wide application in Germany, replacing electrostatic filters, since
their efficiency was almost as high and their construction did not require the
use of scarce nonferrous metals. However, the gas inflow slit openings of
such cyclones often became clogged with dust necessitating frequent clean-
ing; therefore, the Feifel type of battery clone is used rarely.
3. Effect of radial cyclones flows on the
dust separation efficiency
W. Bart (79) suggested that the radial flow be taken into account, there -
by introducing an additional required condition for the separation of particles,
namely, that the centrifugal force of particles at the boundary of the basic
flow must be equal to the force of the radial flow, as shown by the following
equation,
m -r,-"- =6- rtiK' ,
Kn A '
in which /? B ic> B represent correspondingly the radius of the ascending vortex
flow and the velocity over its outside surface. In this connection it is assumed
that particles, the centrifugal force of which exceeded the carry-away force
-85-
-------
remained in the descending flow, while smaller particles were carried by
the ascending flow out of the cyclone before they became trapped by separa-
tion. Under the above conditions the minimal diameter (r) of particles
remaining in the descending flow was: . • = .
• = 2,12]/
^R_
.„*
(V.19)
Assuming further that the radius of ascending vortex flow in a conical
cyclone is equal to the radius of the exit opening along its entire height,
W. Bart proposed that the radial velocity be regarded as constant along the
entire apparatus height and equal to :
in which H represented the cyclone height, and
R^ represented the radius of the lower exhaust pipe cros s-section.
Substitute the value of u)-^ in equation (V.19) and obtain:
(V.20)
Figiire 51 Figure 51 is a schematic presentation of a
W. Bart cyclone. The dotted lines indicate the
surface of the ascending flow according to Bart's
computation; actually the radial velocities change
along the entire cyclone height.
In the lower part of the cyclone conus, i.e. at
the deflection point of the extreme flow lines, the
c ros s - section of the ascending flow is of minimal
size. The volume of the ascending flow increases
further on due to the adjoining mass of secondary
flows. On the assumption that the vertical velocity
of this flow does not change along its length, it
might be concluded that the cross-section of the
ascending flow must increase with its height, so
that the flow becomes conical-shaped (Z6). Such an
assumption was confirmed by the observations of
'. A. Ter-Landen and by photographs taken of the axial
flow in water models (2Z). The straight line char-
acter of the axial flow lines formed in the cyclone conus can be distorted at
points where the secondary flows possessed maximal kynetic energy. This
may explain the local narrowing of the axial flow cross-section at transition
FlQo 51o SCnEHATOC
o? tits OoBatn
-86-
-------
points from the cylinderical to the conical cyclone part noted in the sche-
matically illustrated secondary Hows.
Figure 52 illustrates graphically that the size of minimal particle
diameters slightly inc reases, with the depth (H)_to which the cyclone exhaust
tube nas been submerged at u>0 = 15 m/sec. , as shown in Fie. 11 (p 23) 'A
mechanical analysis of the movement of particles in a curvilinear flow was
presented in the preceding sections.
Nov.-, examine also the effect of radial
flow on the movement of particles. Examine
first a particle which enters a curvilinear
flow at distances xrQ from -the rotation axis;
assume that the flow velocity is constant
along the entire cyclone height. The aero-
dynamic force exerted upon a particle at
distance (x) from the cyclone axis should be
-,= b-/-(i —,
were
Figure 52
V
e ? e a 10 n i? 13 ;* is
rt«. 52. CrFECT Or IlfSST 5£PT8 0.- THE
•XFAL'ST TC3£ 0=3 CYCLO'ic c-?ICIE^CY.
Under the effect of this force the particles acquire a degree of supplemental
radial acceleration. By taking the radial flow into account equation (W 14)
becomes as follows: "' '
-------
:"' : - (V.23)
* he right member of this equation can be expressed as follows:
'-!-... -(V.24)
Xov/, examine the movement of particles which are under the flow effect in
t..e case of cyclone schematically presented in Fig. 11 as shown on page 23.
Tc be able to compare the obtained results with the results shown in Fig. 20
at any future time, assume a cyclone flow in which k = 2.92 m2/sec. The
cvclone efficiency will be
Q = \ li'dx,
and since
£,= -*- then Q = Arln^- ~ ~
x x\
Substitute into this expression values of y^ = 0. 5 and xx = 0. 3 and find that
Q = 2.92 In 1.66 = 1.48 m3/sec; which accords with the initial inflow
velocity of
MS _ ,, o „
Disreard the exhaust tube immersion and assume that H = 2. 26 then
£caa.te coefficients of equal degrees and obtain values of expansion coefficients.
A -i.'tJL
Q,1 ' fl3
-88-
-------
Radial shifts of any x* particle up to size Z0\i computed under above condi-
tions are listed in Table 7 below. For comparative purposes values are also
given for x computed without taking radial flow into account. The radial flow
"' Ta'b'le' T' ' - ' ' ' ------ .:- - _ -
400
.350
300
T
0
0.018
0.023
0.028
i
0.014
0,018
0,024
0.3
;
0,147
0,1 8G
0.237
0.111
0,145
0.192
0
0,315
0.391
0.5
4.17
3
1.48
I.H.3
0.25H
3.17
4.08
5,42
2rtl
0..32H
effect is seen more clearly in Figure 19 as shown on page 43, in which the
dotted lines are loci of velocity values computed by taking radial flow into
account. Curves in Figure 19 show that the velocity of centrifugal particle
movement abated markedly under the effect of the axial flow. It can be
easily imagined that in the course of time the velocity of this particle's
movement will assume a negative direction, i.e. under the effect of radial
flow the particle begins to be carried towards the cyclone axis. Formula
(V. ZO) was derived for the computation of "minimal diameters" by taking
the radial flow into account. Now, apply this formula to the example under
examination and obtain the following:
x = 0,3 H Wr,-o.3 = ~7pr- = 9,73 rn/sec.
0.3
and
>.v
1,83 10-" 0,104
2500-'l,73~
= 2 p,
Thus, by simplifying the concept of the inertia separation mechanism it is
possible to arrive at the conclusion that the radial flow affected only the
separation of particles less than 5y,-in diameter.- In fact,-as indicated by
the examination and figures presented above, velocities
ex
are greater
than velocities v for particles ZOu, in diameter beginning with _L ~ 2>'
in cases of greater J_ values, the radial flow affect extends over considerably
larger particles. In the examined example the flow affect on particles up to
50p, 111 diameter was actually observed. Similar observations had been
recorded in the use of large cyclones under practical conditions. It was
demonstrated that the flow effect rapidly abated in small diameter cyclones
with the reduction in the initial velocity. Thus, the results of the investiga-
tion lead to the practical conclusion that incomplete separation of small
particles can be explained mainly by the radial flow phenomenon. At the
-89-
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same time, it is clear that the slip-through of large particles cannot be
explained on the basis of the radial flow alone. The cause of this is a more
complex one, and it must be assumed that the previously discussed buoyancy
forces played an important part in this phenomenon.
4. Sffect of turbulence on cyclone separation efficiency
It has been known that the aerodynamic resistance of cyclone appara-
tuses decreased with increase in dust density, and that within the limits of
some gravimetric concentration, the aerodynamic cyclone resistance
increased with increase in the size of the particles (ll). These phenomena
are to a large extent connected with the fact that particles moving in rela-
tion to the medium reduced the flow viscosity. This is primarily true of
the laminar layer. Loss of pressure through friction in the laminar type of'
.low, as for example, in the tubes, can be determined by formula (V.25).
(V.25)
J// — "' Re
whe re
\vhere 1 - denotes the tube length, and
d - denotes the tube diameter.
The jumping movement of the rotating particles and of their co-vortices
create a turbulence in the laminar flow of this layer, i.e. they elicit an
action equivalent to the increase in the Re (Reynolds) number. In addition,
if the a\erage cross -section velocity wcp remains constant, then, according
to formula (V.25), the pressure loss must continually abate. This functional
relationship operates to some extent in the field of basic turbulent flows, for
which
in v.-hich '" /~' T ',
'*: = P ~a and ' /-
is the friction coefficient, which likewise diminishes within certain limits
with increase in the Reynolds number. It has been known that by creating
an artificial flow turbulence the resistance of the sphere can be reduced by
50-/0 (69). A similar phenomenon apparently occurred in the case under con
sideration. It was previously noted that the movement of small particles
did not elicit the co-occurrence of vortices; consequently, the resistance
abating effect of large particles can be easily understood. The complex
-90-
-------
picture of the aerosol particles' effect on the aerodynamic structure of dust
containing flows does not fully explain the turbulence of the laminar boundary
layer of a cyclone flow, nor can it fully explain the effect of the aerodynamic
structure on the course of particle separation. We are dealing here pri-
marily with turbulent aerosol flows.
One of the basic characteristics distinguishing turbulent flows is the un-
organized movement of gas particles in all directions, resulting in continuous
gas intermixing. The presence of suspended particles, the density of which
differed from the flow density, affected the intermixing intensity. Aerosol
particles carried away by the mass intermixing movement impeded the gas
flow by their force of inertia, while-th-e-simultane-eusty- occurring-f rtct-ion-
transformed the particles' kinetic energy into heat (81). Under certain condi-
tions intermixing may cease completely, and the flow will assume a stabilized
laminar or stream-like flow. Such an effect has been noted frequently in liquid
suspensions, as, for example, in argillaceous suspensions, when an increase
in the concentration of hydrosol particles was followed by an increase in the
critical value of Re.
The flow structure within cyclone apparatuses has not been investigated.
Nevertheless, there is reason to assume, first, that the flows are of a turbu-
lent character, and second, that the turbulence degree, which depended not
only on the Re number but also on the aerosol concentration, v/as subject to
change within wide limits. The effect of turbulence on aerosol particles'
separation effected by external forces, is connected primarily with inherent
turbulent flow velocity components, which are normally directed towards the
main flow, i.e. the transverse pulsating velocity.*
Imagine an isolated horizontal small area in a horizontal plane flow. In
the presence of the imaginary small area of transverse pulsating velocities,
volumes of fluid will be transposed from the upper layers to the lower, and
vice versa. It can be taken for granted that the volume'of uncornpressible
liquid transposed in both directions will be absolutely the same. The trans-
posed gas volumes will carry with them aerosol particles at a rate which will
depend upon the mass of particles drawn into the current and upon their pre-
viously acquired velocities. Now, assume that the particles immediately ac-
quire a velocity equal to the velocity of the gas volumes; on the basis of the
Prandtl hypothesis assume also that the velocity of the moving mass remained
constant between impacts in the course of transposition from one layer into
*The transverse and longitudinal pulsating movements of the flow partic-
les can be regarded as the result of superposition of vortex movements of
different magnitude which naturally arise in the flow, on the basic flow. These
extensive vortex movements usually arise as the result of sudden flow disturb-
ances. Pulsations under present observation are of a more regular character
and smaller magnitudes.
-91-
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another. The following two types of turbulence have been generally recog-
nized: the isotropic and the anisotropic. The isotropic turbulence is defined
as a flow in which the average values of squared impulse velocities" are the
same in all directions,
in which u)' is the pulsation velocity. It has been established in viscous fluid
dynamics that the magnitude of transverse pulsating velocity is proportional
-------
Assume conditionally that there was a well defined boundary between
the laminar and turbulent parts of the flow, as schematically illustrated in
Figure 53. Assume further that the turbulent part of the flow is in the form
of an infinite number of vortices of infinite
•Figure 53 - dimensions whieh-are^ moving-eha-o4iea-ily=-iH—
" j - - - - different directions but which are for the
^^ ^^^ J moment in positions indicated in Fig. 53. As
(^^ ) Cv CJ vortices move about they carry off flow-sus-
pended sol particles and impart to their
velocities supplemental components. The
time during which the particles remain in
each of the moving vortices is very brief,
and since the particles are also endowed with
FISO 53» GENERAL SCHEMATIC ILLUSTRATION <- c • .- • . • , , . .
OF THE MAUWER OF DUST PANICLES SETTLiHB a force of ln£rtia' lt is reasonable to expect
on THE WALLS OF A TURBULENT FLOW, that the velocities of transposition and rotation
I - DIFFUSING GAS VORTICES; 2- COURSE movements acquired by them substantially
OF AEROSOL PARTICLES: 3- HARO Fl OU i • rr i r i i i
BOUHDARYJ 4- LAM^ LAYEP Bomm. differed from analogous velocities possessed
by the gas masses. Particles originally
located at the boundary of the laminar layer which move by their own force
of inertia as a result of external impulses towards the flow boundary encoun-
ter no impulses from other whirling masses capable of changing their move-
ment. Such particles will be within the limits of th-e laminer layer if their
dimensions are small as compared with the layer, or they will come in con-
tact with a channel wall.
Examine the first case. Depending upon the velocity of the particles
at the moment they were caught by the laminar flow, their movement will be
along one of the following paths: uhere the velocity of the particles moving
in the direction of the flow is less than in the laminar layer (vc < O) then,
under the influence of buoyancy forces, the particles will again, immerge
from the laminar layer, will be picked up by the radial velocity and carried
into the flow depth, as shown by projectory 1 in Fig. 53 above; where the
velocity of the particles moving in the same direction is greater than in the
laminer layer (vc > O), the particles will move towards the channel wall
until they come in contact with it; thereafter, their velocities will abate, and
the particles will assume a jumpy movement, as illustrated by trajectory II,
Fig. 53. In the second case, the upper part of the particles will feel the
effects of high velocities of the turbulent flow, while the lower part will be
in the zone of zero velocities. In such a case the particle will break away
from the wall and \vill reenter the flow. In both cases minute particles may
remain attached to the wall by the molecular adhesion force.
S.K. Friedlander and H. Johnstone (83) conducted experiments devoted
to the study of turbulent flow effect on the deposition of particles on walls.
Artificial monodispursed aerosols were run through vertical glass and copper
tubes 0. 54 -2.5 cm in diameter at different rates. The number of deposited
particles per section of the tube was determined by a special device. The
aerosol structure v.as kept under strict control throughout the experiment.
-93-
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The experiments resulted in a number of interesting conclusions.
At the first part of the dispersion section (Rer=8-10~<) no deposition
of particles on the wall occurs as a general rule. Apparently these low Re
values correspond to the section where the boundary layer is stilt thin and^ "V
the velocities are practically constant. By dispersion section is meant the
initial part of a flow close to the channel entrance. Within the limit of this
dispersion flow turbulent boundary layers are being formed. The dimension-
less length of a dispersion section is denoted by
in which x - denotes the distance from the point of flow entrance into,
the channel up to the section under study;
R - denotes the channel radius.
At Re values within the interval of1 8-10"1 < Re r< 2 • 1CT3 the number of
particles settling upon the walls rapidly increases until a limiting value is
reached which remains unchanged throughout the length of the tube. This
experimental series was conducted at values of Re - 12,600 and 14,900. On
the whole, the number of particles which separated from the flow and settled
upon the tube walls increased substantially and regularly with increase in th'e
flow velocity in complete accordance with the'previously discussed theoretical
considerations. It was also noted in the course of the experiments that some
particles were forced to break away from the walls; this was eliminated by
coating the walls of a control section with an adhesive substance. There is
reason to believe that an experimental investigation of the turbulent structure
of curvilinear flows might open up new means for the managed operation of
inertia separation proces s .
5. Effect of other hydrodynamic factors on
cyclone separation efficiency
It was previously noted that the upper branch of a secondary vortex
movement formed in cyclones, augmented the process of sol particles re-
moval at the mouth of the exhaust tube. Observations made with the aid of
transparent experimental laboratory cyclone models indicated that aerosols
accumulated in the upper part of the apparatuses immediately below the lid,
forming rotating dust rings. The position of the dust-rings was not a stable
one; their rotating movement encountered obstacles in the form of streams
flowing out of the cyclone entrance opening. This created impacts which at
times knocked the dust-ring down increasing the aerosol concentrations at
the exhaust opening (77). Such aerosol accumulation could be diverted by
making a slit in the upper part of the cylinder. This device was installed
in some of the Feifel elements. The negative effect of the upper branch of
the secondary vortex flow was reduced in the Van-Tangeren cyclone by a
-94-
-------
device which caused part of the aerosol to become diverted through a slit
opening from the region of high pressure into the lower cyclone part. Re-
ports in the literature (22, 23, 77) indicated that the Van-Tongeren cyclone
operated at a relatively high efficiency. The effect of aerosol particles carry-
out can be completely averted by increasing the depth of the exhaust pipe set-
in. Figure 56 presents curves of plotted results obtained in studying the
effect of depth of the exhaust pipe set-in on the cyclone efficiency. Curve 1
is based on data obtained by Ter-Linden,
and Curve 2 is based on data obtained by
A. M. Gervasev (78). An initial sharp
rise in the performance efficiency
appeared in both instances, probably
due to the elimination of the aerosol
carry-out effect. Later the efficiency
be gan to drop, probably due to inc rease
in radial velocities, which accords with
the principle illustrated in graph 52 on
page 82.
The conical part of the cyclone
greatly affected the cyclone separation
efficiency. It has been established that
the cyclone separation efficiency increased with the reduction in the angle of
cone tapering. Curves in Figure 57 represent plots of data obtained in the
experimental study of hydrocyclones and illustrate the general character of
this functional relationship. Now, examine the mechanism of action in the
conical part of a cyclone apparatus with a flat bottom, such as the cyclone
Figure 54
FIG. 54. ;.. T,.J(}..STAGE IJA~-TONGERErI
CYCLUfH I ~".H~ LA T ION.
Figure 56
'I %
".... I
/
f -
./ ,/ '2
V
/
95
90
85
80
75
70
65
60
o
0,5
1.0
1,0
1,5
2,5
F I '. '.
~ ':' ~ E i: T ~lf
':X~"'~T TUE INSET OE~TH.
:1
,!
Figure 55
c)
.. '(
,
"'
J-l
-
-,
,) .
J
s
1- ',. I ;FLC'.!; ;.'... .J-.;r: I F I ~ J4D
~~TF~JW; 3- JAL~U~IE SCR~tN: -
. '. I T - LI :., c.'
. T' " ~ '
. ,.
FJ". 55.
SCH
-------
Figure 57
9
ICO
$0
BO
TO
SO
SO
«?
30
20
w
0
% . .
(a OESREES)
— 1 —
-
^
^
rOv-,
!
y
-
x
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L.
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to
30
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60 W
ris, 57. EFFECT OF COME TAPER ins AIISLE.
heating chamber investigated by Lyakhovskii,
shown by the solid lines in Fig. 3, page 11.
The aerosol, in this case pulverized coal, is
introduced into the heating part through an
opening at the bottom of the apparatus. In '
the described experiment the opening was
covered up which caused the flow to become
completely identical with the flow in the
usual cyclones. The schematic drawing
shows that, regardless of the complete aero-
sols' separation their settling down could not
have taken place in this-case because the
settled down aerosol would have been picked
up from the flat bottom by radial air flows
and would have been carried away by the
ascending flow. Figure 58 schematically
presents the construction of an original cyclone dust-catcher with a flat
oortom built by the Danlabo firm (85). The flat bottom was equipped with
slit-like openings (5) protected by deflectors, or •
baffle plates, which separated the lower radial ' Figure 58
part of the flow containing the highest aerosol
concentration. The layers thus separated entered
the lower part of the chamber where the dust
settled by gravitation, and the partially purified
gas reentered the apparatus through an axial
opening.
Thus, it appears that the lengthening of the
cylinderical part of the apparatus can substan-
tially change the dust precipitation condition. As
the flow velocity abated, slowest moving individual
layers streamed towards the exit opening. Con-
siderable lengthening of the cylinder tube should
induce the formation of a zone free from radial and
axial component velocities. By further lengthening
the cylinder tube, it is also possible to attenuate
tne tangential flow velocity components by the
increased air friction against the cyclone walls,
and the settling of the aerosol in the formed
stagnation zone v. ould proceed unimpeded.
Fie. 53. SCHEMATIC ILLUSTRATION
OF OANLAOO FLAT BOTTOM CYCLONE,,
I- BAOOLE-SHAPED BEFLECTOR; 2-
CYCLOUE BODY; 3- FLAT BOTTOM;
A- CIRCULAR OPENtllS; 5- OREHIIIS
FOR THE REMOVAL OF THE OUST COM-
CENTHATE; 6- 6AS EXHAUST TUDE.
-96-
-------
Figure 59 is a schematic presentation of a Pratt-Daniel battery cyclone
element, the construction and mechanism of which is practically analogous
with the Venturi tubes (vortex forming tubes). The open end of the tube is
connected with a general dust collecting chamber; the increased pressure pre-
. vailingj.n this chamber plays the part of a throttle (Fig. 8, p. 21). The periph-
eral part of the dust-laden air overcomes the back pressure and passes into
the chamber, while the axial part of energy reduced air forms
an axial flow. Conical narrowing of the apparatus creates
radial flows and simultaneously slows down the entire flow.
The flow system in such a case can be pictured as shown by
the dotted lines in Fig. 3 on page 11.
Figure 59
\J V
Fie. 59. PPATT-
DAHIELS CYCLONE
ELEMENT,,
The space in the lower section of the apparatus repre-senti-
a dead zone, the dimensions of which increase with the de-
crease in the conicity and enhances the precipitation conditions.
The separation efficiency in the Danlabo tube increased follow-
ing the removal of part of the dust concentrate from the pre-
cipitation chamber through tube (6) shown in Fig. 58. This
phenomenon deserves attention; this was also noted during
tests made with many other apparatuses. Now, examine the
precipitation process of individual dust particles in the Danlabo
tube; assume that the separation process in the separating part
of the apparatus "proceeded at an efficiency approximating
100%, and that all individual particles passed into the precipi-
tation chamber. It is clear that as the aerosol concentrate flows through
this chamber, the following will take place: all particles of considerable
weight and particles which had undergone aggregation as they passed Through
the cyclone will become precipitated, while the finely dispersed particles
will be carried out of the precipitating chamber through axial opening.
The separation process extends also to the ascending rotating cyclone
flow. The efficiency of this process is determined by inertia forces many
times greater than the gravity forces; therefore, it is greater than the
efficiency of the chamber gravitational precipitation. As a consequence,
part of the particles which had not precipitated in the chamber reenter the
descending flow increasing the number of small particles not susceptible to
gravitational precipitation. Accumulation of such particles can substantially
change the separation conditions in the course of time. As a matter of fact,
the separation movement of large particles will begin to encounter more and
more resistance in the form, of impact against slow-moving or practically
stationary small particles. In other words, the accumulation of highly dis-
persed particles can elicit an effect equivalent to an increase in medium
viscosity, i..e. an increase in viscosity resistance to movement ot particles.
Such increase in "viscosity" can be brought about in hydrocyclones artifi-
cially by the introduction ot heavy suspensions. By such means it is possible
to attain considerable enhancement in the separation of pov/ders of similar
granulometric composition but of different volumetric or gravimetric compo-
sition (86, 87). The effect of fine aerosol particles accumulation becomes
-97-
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particularly manifest in instances where the construction of the separating
part of the dust catcher does not favor particle coagulation. For instance,
G.M. Kharchenko observed dust exhausts which periodically reappeared in
experiments with counter current rotating dust catchers (88). It can be
a«5«med that at-times, such exhausts coincided with the moment of critical
'•. ccumulation of small particles.
By partially sucking off gas from the dust precipitating part of the
;::aratus a condition can be induced at which the number of highly dispersed
rticles removed with the gas will equal the number of similar particles
'.'i.:h reappeared in the precipit?.tion chamber, with the exception of the
.;. ~. 7ulated particles. At the sarne time, this condition prevents the accumu-
; - •/.on of such particles .inside the apparatus, and the medium resistance then
. . am.es a function of the gas viscosity only. It can be assumed that the
• ct of fine particles accumulation appeared in all inertia dust catchers.
In this connection, the effect of the apparent rise in viscosity should increase
" 1 . ihe increase in the disproportion between the inertia separation effic-
i- _/ :;: the separation part of the apparatus and the gravitational sedimen-
la.-on efficiency of the precipitating part of the apparatus, i.e. the greater
is die inertia separation efficiency as compared with the gravitational
efficiency.
Iqure 60
i he Danlabo cyclone was designed for the purification of small volumes
of combustion gas used in gas driven turbine engines. Large gas volumes are
usually purified by means of battery aggragates consisting of many individual
cyclones. Figure 60 is a schematic illustration of a multicyclone trade-named
"TU31X'! manufactured by Pratt-Daniel in France, and Figure 61 is a sche-
matic presentation of battery cyclones presently manufactured in Holland, the
characteristic feature of which is a horizontal
arrangement of curved cyclone elements. The gas
rotation is brought about in such apparatuses by the
tangential inflow of the raw gas into the cyclone ele-
ments. Industrial aerosols frequently contain large
size particles which
cause abrasion of the
inner cyclone surfaces,
and v.-hich clog the gas
intake and dust exit
openings. Such effects
h"'••:•: been eliminated
in "resent day battery
cyclones by retaining
the original gravita-
tional precipitation
capacity of the appa-
ratus to precipitate
large suspended par-
ticles within the limits
Figure 61
!ATTI*T CYCLONE
.ius"
FlS. 61. SCHEHATIC IU8STIUTION Of
IATTESY CYCLONE ACCOSilNS TO TEM-
LlffSEN.
I.-COARSE BUST; II - FINE IBST
-98-
-------
of the intake part. Accordingly, the dust collecting bunkers consist of two
sections: one for the collection of the coarse, and one for the collection of-
the fine dust particles. The cyclone elements are united by common distri-
buting chambers and collectors of the purified gas. The dust-removing
O-peningS- cif.-tKe_r y^l TmP plprnpnt-c 3 rp Tjkev.'ise c"rmpcted to a commnn rtiidf- .
collecting chamber. Smaller sized cyclone apparatuses separate the dust
from the gas more efficiently. Laboratory tests made with small individual
cyclone elements yielded high purification coefficients, fully comparable
with those of electrostatic filters, and the recent trend has been to use
small-cyclone battery aggregates. However, practice indicated that the
general efficiency of such aggregates was lov/er than the efficiency expected
according to tests made with individual small battery cyclone elements.
Experimental results and practical experience with cyclones established
the fact that reduced pressure developed in the lov/er section of the
apparatus; faulty sealing of the dust carrying ducts created an inten-
sive air inleakage which reduced the dust separation effect to zero.
Operation of the dust collecting bunker under absolutely leak-proof
conditions can be affected by the simultaneous action of cyclone elements
so as to create a pressure condition which will prevent the sucking in of
air through the dust removing opening as effectively as a tightly closing
shut-off.
Now, assume that for some reason the lowered pressure in the cyclone
elements become unequal; under such conditions a negative pressure and
sucking in of the gas will appear in the elements where the pressure fell
below the pressure drop in the bunkers, and the gas loss in the bunker will
be compensated by gas flowing via the other cyclone elements; under these
conditions the gas circulation within the aggregate boundries will be from
the inflowing part through the dust collecting bunker into the collector of the
purified gas. Such a gas circulation has been noted frequently in battery
cyclones. The origin of such circulation is connected primarily with the
unequal distribution of flow velocity and pressures in the distribution section
of battery cyclones. In fact, the problem of bringing about absolutely identi-
cal conditions throughout the entire distribution chamber, crossed by exhaust
tubes of individual elements, is a very complex one. Attempts have been
made recently to solve this problem by way of introducing special gas dis-
tributing elements. Circulation currents can arise also as a consequence of
diffei'ences 111 the dimensions of cyclone elements occurring due to faulty
manufacture, or to the abrasive aerosol particles, or to the formation of
nodular excrescence by the finer fractions.
Looking into a conical cyclone through its open dust vent, it may appear
as if the reverse suction effect extended over the entire vent cross-section,
thereby blocking the exit of the separated particles from the cyclone. Dust
particles separated in the proximity of the apparatus walls assume a rotation
movement in the lower part of the cyclone, while friction of such particles
against the wall causes rapid erosion and abrasion of the conical part of the
cyclone. This phenomenon can be observed in cyclones the dust outflow vents
-99-
-------
of which had been left open. W. Muhlrad noted such erosive and abrasive
effect in battery cyclones (26). As the result of the separation process,
separated particles accumulate in the cyclone. Such a phenomenon was
noted in experiments conducted with battery cyclones (7). Under such cir-
cumstances the rate ofseparation movement becomes more and more the
function of impact of particles and not of the gas viscosity; such is also the
case in the Danlabo cyclone. It can be assumed that the accumulation of
particles may reach in time enormous proportions and that the process of
particles separation may completely stop; in such an event sol particles
carried out from the bunker by the circulating'flow, and particles which had
been carried from the distribution chamber into the defective element will
penetrate into the purified gas collector. Removal by suction of some of the
gas from the dust collecting bunker will impede the process of such accumu-
lation and will considerably enhance the apparatus efficiency in a manner
similar to the case in the Danlabo cyclone. Increased efficiency resulting
from partial gas removal from the dust bunker was a special suction device
nr.tcd in all types of battery cyclones. The effect of finely dispersed partic-
les accumulated in the dust collecting bunker is less in evidence in battery
cyclones consisting of conical elements. This may be a corollary effect of
a lowered circulation intensity and of the favorable conicity effect on the
coagulation of finely disbursed particles.
CONCLUSION
The course of separation process taking place in curvilinear flows,
especially in cyclone apparatuses, can be pictured as follows: as the dust
particles enter the cyclone their force of inertia causes them to move along
the initial rectilinear projectories. The carry-away force of the rotating^
"gas flow gradually but progres sively deflects the trajectory of the particles
in reverse proportion to the particle mass; trajectory deflections of all par-
ticles are in direct proportion to the distance of their inflow point from the
outer cyclone wall. This initial particle movement section is characterized
by a high energy separation of the largest particles from the peripheral part
of the flow. The extension of the initial section depends upon the size of the
particles, and in the case of large particles includes the entire path of their
movement inside the cyclone apparatus. Thereafter, the tangential particle
velocity tends to approach the gas flow velocity, i.e. there appears a period
of "quasi-stationary movement. " At this point the separation of particles is
conditioned by continuous changes in the vector of the tangential gas velocity
and as a consequence by the permanent disturbance of the established move-
ment manifested in the form of a radial centrifugal component of the particle
velocity in relation to the medium.
Theoretically particles of all sizes could be separated from the gas
flow if the flow time had been of adequate duration; however, there are other
hindering hydrodynamic factors, such as the radial flow, especially in
-100-
-------
cyclones with flow deflection, and buoyancy forces. The radial flow effect
manifests itself in carrying with it the suspended particles towards the rota-
tion center. This carrying away effect acts first on the finest particles and
on particles suspended in the innermost part of the flow. In this case the
e&Rcep-E-s— oj-^fine'-' a-nd of "crerscr-s-er1'- particles arre-used cunditroTralryT the~arbs-o^-
lute size of particles under discussion is determined by hydrodynamic factors,
The radial flow constitutes the basic factor which effects the separation of
fine fractions. This factor determines essentially only one efficiency limit
of certain cyclone types, namely, the efficiency of fine fractions separation,
The action of buoyancy forces may appear also as a counter action to the
centrifugal movement of particles. Buoyancy forces may cause the suspended
particles to move toward the rotation center in some instances. This is
particularly true of particles which are carried into the limiting layer of the
flow. In isolated short moments a combination ot conditions may arise which
v.ill favor the appearance of the Taylor effect. Under such conditions, all
other particles, the flow velocity of which is less than the flow velocity of the
gas, can move toward the center, or, in any case, can change the centrifugal
flo\v velocity through the effect of buoyancy forces. Particles in their imme-
diate proximity may be affected similarly. The separation of particles the
velocity of which is greater than the gas velocity can also be accelerated in
the presence of prevailing Taylor effect. In such a case a buoyancy force is
imparted to all but the finely dispersed particles. As a general rule, however,
this afiects mostly the large-size particles. For this reason buoyancy forces
apparently determined the second cyclone construction efficiency limit,
namely, the separation of large fractions.
Through the action of inertia forces and centrifugal vortex forces,
peripheral layers of the cyclone flow become continually enriched with aero-
sol particles including the, so-called, "jumping" particles. Increase in the
initial concentration of particles in the untreated gas, increases their density
in this layer, and raises the probability of particle impacts. As a result of
such impacts the impulse-like movement of isolated "jumping" particles be-
comes absorbed to some degree by particles still moving toward the cyclone
walls. Accordingly, the separation efficiency should increase with the
increase in the initial concentration. This has been actually observed under
practical conditions. It should be emphasized here that an increase in
efficiency undei' the above conditions is connected also with a general intensi-
fication of kynematic aerosol coagulation, which appears with an increase in
aerosol density. Actually the number of impacts increases not only in the
limiting layer but in the entire flow; therefore, the impact of large and
rapidly moving particles against small ones and formation of aggregates must
occur at greater intensity. As the c ross-section of the conical part of the
apparatus gradually decreases the density concentration in the limiting layer
may become so great as to prevent the occurrence of particles' "jumping"
movements. The film of concentrated aerosols acquires the property of a
heavy dust-cloud and moves as a unit being less and less under the influence
of the gas flow.
-101-
-------
The movement of the film particles in contact with the cyclone walls
is predominantly of a sliding character. As a result of friction the film
rotation velocity gradually abates. In cyclones the conical part of which is
of considerable height and in the presence of a dead zone at the end of the
cone; the~frlrm~ strops rrroving~and~t:he particle's" precipitate.- After the
part of the cone becomes filled with the precipitated dust to a level at which
the radial flow velocity components begin to erode the accumulated precipi-
tate, further particle settling ceases. Therefore, the cyclones must be
equipped with hermetic dust collectors of considerable capacity to allow
uninterrupted cyclone operation.
No accurate or adequate inertia separation theory has been advanced;
this makes rational improvement in the development of cyclone apparatuses
difficult; proposed improvements are almost entirely based on empiricism.
Examination of the construction of cyclone apparatuses disclosed that there
was a prevailing tendency to base their operational control on individual
hydromechamcal phenomena, for instance, in the case of cyclones designed
by P. N. Smukhnin and P. A. Koiizov the deeply set-in exhaust pipe moderates
the radial flow effect almost along the entire height of the conical part, creat-
ing conditions favorable to the separation of aerosol particles at the cyclone
walls. At the same time the radial flow velocities below the mouth of the
exhaust tube is greater in these cyclones than in cyclones of other construc-
tion, lowering to some degree the above described advantage.
The screw type lid of cyclones, as of cyclones NIIOGaz, can moderate
the negative effect of the upper branch of a secondary vortex to some extent.
An attempt was made to utilize such an upper flow in the Van-Tangeren cy-
clones, and in the Feifel cyclones the secondary flow was artificially enhanced
to bring about a more efficient separation. In the SIOT, the Bart and other cy-
clones the conical form was designed for the maintenance of a more or less
constant radial flow velocity. In this way it was possible to prevent the
appearance in isolated flow sections of higher velocities capable of carrying
out large particles. At the same time the
continuous c ross -section narrowing facili-
tates the displacement to the axial part of
the gas containing an even greater concen-
tration of sol particles. In the barrel-
shaped cyclones such as the Walter cyclone
type the above described unfavorable effect
was obviated by inverting the conical part
of cyclones. This reduced the radial flows
in the cyclone upper part and increased the
length of the particles' separation path.
Among the cyclones of latest construction,
the one built by Arno Andreas deserves
special attention. Figure 62 is a schematic
presentation of such a dust catcher; it is
essentially a polysectional uniflow cyclone.
Figure 62
7
FIG, 62, SCHEMATIC ILLUSTRATION OF SINGLE-
STASE A, ANDREAS CYCLONE,
-102-
-------
The gas to be purified enters the apparatus tangentially through pipe conduits
(1) in the form of individual portions into each of the elements (2); the dust
particles become separated and se'ttle in chambers (3), and the-purified gas
enters into the next conical element. Under such conditions of cyclone con-
fhi-nngh f'np ppr^pViPral part of the cyclone. elejr __ .
ments, and the mid-part of these elements becomes filled \vith the purified
air or gas according to the original description of the apparatus (89), the
introduction of clean air through conduit (4) into the first element has not
been provided for. - -
Similar solutions were proposed by Czechoslovakian specialists Ya.
Ions and A. Oleksa (90). Their apparatus is essentially a single-step uni-
Figure 63
.-1C. r-J. o:j£".i TIC I LLPST3 ATI 0" Or LJtTCST
:V;LC>' •;. 1 - '- i---7;"!;;;« si CY;LG>;£;
Z - Kcis CYCL:•'!•:; 3 - IVISYMJIIT'ICAL c*s
I'jfic. crci.-:,;:: ^ - CYCLON; fillOG^z.
flow cyclone. The gas to be purified is
admitted along the entire peripheral part
of the cyclone through a rotating paddle -
shaped screen. A safety layer can be
created ir. the central part of the apparatus
by -admitting air through a second rotating
screen installed coaxially with the first.
G. i\-iczeck of Czechos lovakia (17) made a
comparative study of several cyclone types,
such as are schematically presented in
Figure 63 v.hich included the Van-Tangeren
tvpe oi cyclones having an internal chute
for the oassage of gas from the upper cy-
clone oart into the lower to prevent the
negative effect of the upper vortex branch
of the above analyzed secondary circula-
tion; the Corso cyclone equipped with a
tar.oential feed-pipe and rotating paddle
apparatus: the cyclone equipped with an
axiaiiv symmetrical gas feed pipe with a
rotating apparatus; and cyclone NIIOGaz
with a slightly shortened cylmderical
part. All these cyclone types are of
recent design; they had been described in
the literature severally as apparatuses
which yielded favorable purification re-
sults under conditions at which other types
of cvclor.e dust catchers proved of low
efficiency. The cylinderical parts of all
the tested cyclones were 307 mm in diam-
eters. All cyclones were equipped with
settling bunkers and had been tested under
identical conditions. Artificial dust sus-
pensions were created with ground brown
coal ash; 90% of the suspended ash partic-
les on the weight basis consisted of
-103-
-------
particles less than 40p, in diameter. The air was pre-heated to 50° for the
purpose of preventing possible particle coagulation.
Results of the investigation established that at identical air dust density
pe-r unit-area of- the-apparaturs- c ros-& —s-eetion the- du^fe-e-ateh^n-g-effixiency-of—
the four types of cyclone models was identical; the same was true under simi-
lar conditions regarding the loss of cyclone pressure head. The Miczeck
comparative study offered a basis on which to obtain a correct picture regard-
ing the comparative advantages of the four types of tested cyclones. Of par-
ticular value was the convincing evidence presented by Miczeck's study
regarding the useless effort, wasted in many lands, in making empirical
attempts to solve problems which in fact could only be solved on the basis of
strictly scientific principles.
Assumptions made in the field of dust technique studies often have no
experimental or theoretical justification, and many attempts have been made .
to solve problems connected with inertia separation v/ithout due consideration
for the underlying basic scientific principles. Progress in the development
of dust-catching techniques depends upon future profound studies, first, of
such basic problems as hydrodynamics of tridimensional turbulent rotation
fluid movements, of the Taylor effect, and of the interaction between partic-
les in a sol system. Only such studies will offer cyclone designing engineers
scientific bases for the solution of the wide range of problems connected with
inertia separation and will narrow or possibly eliminate the gap between dust-
catching techniques and other branches of modern technology.
In conclusion, the author wishes to express his appreciation and thanks
to Professor N. Ya. Fabrikant, Editor of this monograph, for his valuable
suggestions, and to P. Ya. Kochina and N.A. Fuchs, Doctor of Physico-
Che mical Sciences, for their assistance and suggestions in composing this
monograph, and to V. V. Bazhanova for her valuable assistance in the
preparation of the manuscript.
-104-
-------
JJHTEPATVPA
I A r T e K > p T, O u e T e K K a p e .a, MeraJuoitepaMtmecKHe TpH.
f HX HarOTOB.ieHHe, CDOiicrsa n npHMeHCHHe. CvanpoBma, 1959.
2 A^bnepOQHq M. A, V a o B B. H., yxcn/iyarauHoaHaa npoeepaa
& 10 (1950).
5 AHjpHaHOB A PI. VKJOO B H, TexnHxa O«JHCTKH cepHncroro
rasa or orapKosofi nu^H, Bonpocu no-iyjeHHa cepHHcroro rasa H3 it
iaiia H cepu. TocxHHHajiaT. 1957.
6 BarypHH B B. BeHrH^aima HaiuHiiocrpoHTeJibHhiJt saaoaoa,
1353
7. EeJiaesH A, CX5 OMRCTKe rajos n miKjiOHax, XMM npoM . J& 8 (1949)
8 Bonpocu ra3004HCTKH Ha s-ieKipM^ctKHx cranuHax H npoMnpejnpHsnHflx (tsa-
TepHa-ibJ HayMHO-TexHHtecKoro coeemanHH 1949 r ).
1951.
9 Bonpocu OWHCTKH eoaayxa or nu.iu, cCopHHK noa pta. J\. C.
naaar, 1940.
10 TaaooMHCTHTe-'ibHue ycrpoficraa H us ane.npe!ine iia s-ieRTpimecKHx
H npoMuui.TeHHbix npeanpHSTHsx C6op«HK MarepHajioQ HayJHO-Te
CKoro cosemaHHsi 1951 r, TocsMepromaaT, 1953.
II TepaacbeB A M, Fluj;ey^oBrtTe,iH CHOT, FlpoipKaaaT. 1954
12 PopaoH A M, A/iflAxta.iOB H A. raaoomJCTKa
(JjH.ibTpaMH, MeraJi.iyprMiaaT, 1956.
13 PopaoH T. .M. rieftcaioD H JI . nu^ey^as^HBaHHe D
ueTa^i.TyprHH, MeTa^J!ypriJ3flaT, 1958.
14 TyaeuMyK B A, Cp2BHiiTpj7>,Hbie HcnuTaHun miK.ioHHUii annaparoa
paa^MMHMX KOHcrpyhUHH, HsoecTiiu BTH, /<"» 9 (1947).
15 Zlepra^es H , OincrKa fluMoejx raaon B ueHTpofi?K, Moupue sojioyflODHre^iH CHcreMu BTH, PocaHepro
I13.H3T, 1950.
17 Eropoe H H, Ox.na«aenne rasa B CKpy66epan, Pocxnun3aaT. 1954
18. /K e 6 p o D c K n A C PI, 3.ncKTpo:pH;]bTpu, Poc3iieproH34aT, 1950
19 >i< e H H Hi e tt H. H., PoTzwioHHbie nu^eoT^e^KreJin, Poccrpo Anwar.
1958.
20 3a.iorHHH P.IllyxepC M, OvHcrsa ^UUOBHX raaon, PooRepro-
nsaar, 1954.
21 K a .1 H M y uj K H n M P( , O6ecnu.iHB2tom.He ycraHoenn, Has MiiHncrep
craa Ko\iMyHa.ibHoro xoaauciDa PC'tCP, 1957.
22 KataiorH H,I 5itKT,no()H,Ti!TpH, liK^'io.'jb-1 :: CaTsp^fiaae UHK^OCH u^cra «Fa
ToofHCTKa» X? 1 — 28, Pocxnwusaar, 1950.
'23 KHpnHMes E "S", Hotire Tnmu 6aTap«(iHtis unejjouoo, UeKeur, J6 S
(12SO)
24 Koyaon PI. A., UHK^OHU noHcrpyumni J1HOT, 1134. ^IHOT BUCTtC
1939.
25 K o y 3 o o Pi. A, UHK^OHU J1HOT c Doannoft n^ieHKofi, H3«. /1HOT
IS52.
26. Ko^HuaiUDU^H A B, UlrpouSepr 3 A, OvHcrita rasoa
npH Btin.icEke i})eppociiyicBon o 3^eKrpone43x, Plpo^HjjWT, 1954
27 KyxaKoe H. E, PaDC'.c^ xoaaScrto ueTa.ijiyprnsecKHz lasoaoB, Me-
Ta/i.iyprnanaT, 1951.
28. £( y H a n o B H E., PasoHTiie raaoaoro soarflcTBa wera^/iyprniecKHx
33BOAOB. Mera-'i-iyprH^aaT, Id59.
29 K y i e p y K B B, OMHCTIO or nt?.iH BCMTM-TSUHOHHUX H npowuiu-ieHHUx
BiaOpocoB. PoccrpoiiHSAaT, IS55.
JO Ky'JepyK B B. KpacH^os P. M, Hoaue cnoco6y O-IHCTHK sca-
or nu^;i, AUnurH3, 1&50.
31 H H u K e B ti M E. A , HepHau xiera^/iyprHa KanHia^HCTHiccKHx crpaH,
aT, 1960.
32 OjucTKa atiwoBUx ra3OB OT /leryMefl 3O-Tbi H cepu, ccopHHK noa pea.
M C. UJxpooa, Pcc3HeproH3jaT, 1934.
33 OjHcrxa npoijbtai.ieHHbix Bu6pocos a aTMOCtpepy, coopHHK noa pea.
T E Bo^abipeaa, Mearwa. 1953.
34 Ooicrxa npoMbim^enpux raaoa, coopHHK noa pea B. PI Be^ioaa, H H Ero-
poaa H B. H V/xooa, PocxHUHsaar, 1947.
-105-
-------
35. flojoineBHHKOB 5. , TapraKOBcrtHfi B. H., O uero-
flxxe pac^era CatuHH JLIS 3syKOBOH Kcary.isuHH a3pO3OJ»efi c xuijKofl JIH--
cnepCHoii 4>a3oA. XHU. npcw , A» 6 (1959).
35. fl o 3 H H M E.. M y x * e H o a H. PI , T a p a T 3. 3.. FleHBafi
cmicofi- OHJKnKn- taawx OT IIUJIH, jjuua-JL-Tyuatta,- rocx-H1'U«&3*v"l951. ---
37 n o 3 H H .M E.. MyxjieHoa H. n . T a p a T 3. fl.. FleHHue au-
Jiey.iOBH7e-TH. ra3oosHCTKie^H H Ten-'K<-5wenHHKH, PocxHHHiiaT, 1959.
38. ripefHCTeHCKHfi C. A, UeHTpnc^rHpOBaiiHe aspcjaieu B
ATOMH3AOT, ISSO.
39. PetjeCHHKOB H. H. C n o p e u n n .1 iO. A.
yctpOHCTB3 iia >r.ieOpnKftHbjx tpa^DHxaz, Yr-TerexHaiar, 1952.
40 PvieHKoK P., 'Aapro^HHB. A.AjiHTpueBCKaaH. M.
.Moxpue 30.ioy.iOBHTe.in H nu^ey.TOByTe.iH. yr-nerextiXjaT. 1953.
II. P>K05o;stune jKajaHH"J no npoexTHposanmo, Haroroa-ieHHio, HOHtaxiy H
3xcn.i\3TautiH UHK.TOHOD HHHOTA3, focxHMHajiaT, 1S5S.
42. P) \03ojamne yK332H/ neiwaryx 3O-Toy.iOBHTe.'1en H CaiapeAHUx UHK.IOHOS Ha
TpocraHUHsx. FooneproHsaaT, 1957.
44 PbiCHH C A, Flukes. TO5HT?..-iH H (Jirf.lbTpH, roCCTpOfiH3J3T. 1941.
45. Gai-HTapHUe Hopuu npoeMHpoaaHHS npouutu-ieHHUX npejnpMaraS, HI01— W
MejrH3. 1958.
46. C M > x H K H n. H, Koysoa n A., UeHTpo6ex5HUe nu.ieaTi«JSTeJjn-
UHK.ioHbi, OHTH, 1935' \
\1 CHWOHOBHM B C, HopHcrue BO3a>uiHue (fjH.ibTpu. PoccrpoSioaaT.
1953.
48 CarapHH B. M . flep.iH C B, ZiBH&eHHe H o6ecntwino3HHe rasoa
B ueuewiHOM npoMjBoz.rrBp, r\i 6 (1S5^.
S3 C K a H a. o /-i. O., V n o D B. H., Cccp^Kscctie Tcxnn^acnD
fcpbJfci c :iarprE3He-iH£ti aTttcotspKOfo coiny^a. Popo^caos
MCCKEU, .\i 8 (1255).
54 Co.^oDbeo H. S, Epunaoo n H. Crpeataqyn H. A.
OCHOEU T2XHi:!in 6£joncci'j5CTi; u nporj!EorroixapHi>3 Tesnans o antini?-
CkOn npdead^'cn'nOvTa, foC&buliiafiT, 13£0.
35 Tpyj.u CoDsmaHaa no oMRrrixe npcKjuiii-^HcaJi roses, AAera^jjyprai^aT, 1641.
56 Tpyiu HHHOrsj, cun 1. POCZHMCSJUT. 1957.
57. V -/s o o B. H , OSoiy^HOJHHe v"<«Tpc^aJjtTpO3, rotinMswuisT, 1544.
58 y n o D B. H., O t,oab,x rr.nax oTeserrteiiHux w;«KTpojrperffro3 £jia nora-
HKsi s,WKTpc4>^'Ii>Tpoa, 3.'er.Tpiii«cT K>, Ni 8 (1951).
59 y f\ o o B. H.. CoHnrapdcn oxpana ariroc&epiKwo cos^yxa, A-teiras. ISSS.
60 V « o D B. H . OHKCTka orxoAnoai npoaHiU£«jHux rasoo.
1959.
61. y M o D 3. H., CaHtrraf.1 o-TeiHcpjecnan oust'sa ctscren
npuMeaseuui Ha -ren-'o-sux weKipocraMUnaa CXXP, Pur:3«:!S D
,\i 3 (1953)
62. y is o o B. H , Chncrna npoubin^euFux ra»3 s.iet£rpc^>tu]t,Tpnna. Poc-
63 y i: o D B. H., XouyToaHncoo FI. C. HOBWJ OTeqecr&Kiiitce an-
napaTti A.IS TOHmyJ CRUCIKH j.oaeHHO."o rjija, CTJUIS. Ai 2 (1952).
64. UjKecpcoH B Jl , 3vieKTp;mecKan o=:i:cnca rosos,
I960
65. II! H e e p c o s B. /J , 35-".cxTp;:<;ccss3 c=;5CTKa raioa a ssiJc^s
uuoi.'ierfHDCfn, PccK^anj^iT, 1S3S.
65. U]neepcoH 5. .T., E r o p o o H. H.. SweOTpti'MCusa o=jncrna rasoa.
Pocxctinjji^T, 1933.
67. UJ y r o n 8 Zl , /Uxattawcirne nty3ey;x>Di!Taan, nsa ParTpotieK^HTa. 1&30.
63. 1O p b e o H. B.. O od6op« oiuwyajiHas yc.ioanS pagoru euopocrutjE Dfc!°
-ley^iODHTeTeft, U&eTHue ucTa.i^ttf .v-b 12 (19SG).
69. American Air Fillers Co. GUI A. (rtpccnetrr-Bara.Tor. 1935—1207),
70. Ait Pollution Handbook, QUA, 1955 (cnpn^ontinn)t.
71 Ttve South African Ind Chem.. 10. W 2. 31 (I25S).
-106-
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