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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                                               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
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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-

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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.
                                      -4-

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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-

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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-

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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-

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      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.

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      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-

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         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-

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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-

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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-

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     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-

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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-

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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-

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 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-

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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-

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      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-

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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-

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                               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-

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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-

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                                    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-

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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-

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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-

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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-

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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-

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  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-

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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-

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                           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-

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                               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-

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                                                   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-

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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-

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                                                        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-

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              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-

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      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-

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      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-

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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-

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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-

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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\'

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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-  -

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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      ?    
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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





                                      -77-

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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" 
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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-

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 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:
                                  --".      
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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-

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 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-

-------
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-

-------
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-

-------
 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
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«?
30
20
w
0
% . .
(a OESREES)
— 1 —
-













^








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y




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to
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                W    £0
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-

-------
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-

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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-

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 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-

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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-

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      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-

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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-

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                                JJHTEPATVPA

   I   A r T e  K > p T,  O u e T e K  K a p e .a,   MeraJuoitepaMtmecKHe TpH.
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        &  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,
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   7.  EeJiaesH   A, CX5 OMRCTKe rajos n miKjiOHax, XMM npoM . J& 8 (1949)
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        TepHa-ibJ  HayMHO-TexHHtecKoro  coeemanHH  1949  r ).
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 II  TepaacbeB   A   M,   Fluj;ey^oBrtTe,iH  CHOT, FlpoipKaaaT.  1954
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 13  PopaoH  T.  .M.  rieftcaioD H  JI .  nu^ey^as^HBaHHe D
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 14  TyaeuMyK  B   A,  Cp2BHiiTpj7>,Hbie HcnuTaHun miK.ioHHUii annaparoa
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                                        -105-

-------
 35.  flojoineBHHKOB  5.  ,  TapraKOBcrtHfi  B.  H.,  O  uero-
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 37  n o 3 H H  .M   E..  MyxjieHoa  H. n .  T a p  a T  3.  fl..  FleHHue au-
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 38.  ripefHCTeHCKHfi  C.  A,  UeHTpnc^rHpOBaiiHe aspcjaieu  B
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 54  Co.^oDbeo  H.   S,   Epunaoo   n  H.  Crpeataqyn H.  A.
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 35  Tpyj.u CoDsmaHaa no oMRrrixe npcKjuiii-^HcaJi roses, AAera^jjyprai^aT, 1641.
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 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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