EXCERPTS FROM MECHANICS OF AEROSOLS

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• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-01043R001400130002-6
• Release Decision: RIPPUB
• Original Classification: U
• Document Page Count: 144
• Document Creation Date: January 4, 2017
• Document Release Date: December 12, 2012
• Sequence Number: 2
• Publication Date: October 28, 1957
• Content Type: REPORT

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Mekhanika Aerozolev, Moscow, 1955, Chap I, pp 7-25; Chap II, pp 59-67;
Chap IV, pp 107-123; 146-152; Chap lip pp 175-189, 215-219; Chap VI,
pp 220-258.
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EXCERPTS FROM sUECHANICS,OF AEROSOLS"
Mekhanika Aerozoley ORech;tnics of
Aerosols], 1955, Moscow', pages
7-25, 59-67, 107-123, 146-152,
175-189, 215-219, 220-258
/ABLE OF CONTENTS, ?
Chapter I. Classification of Aerosols. Size and
Shape of Particles in Aerosols
1. Classification of aerosols
2. The size of particles in aerosols
Chapter
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N. A. Fuks
Pages
1
1
4
3. Distribution of the size of particles in aerosols 8
4. "Average" size of particles in aerosols 21
14. The Movement of aerosols within a closed space 25
15. Vertical and horizontal electric field methods
and their application 27
A. Determination of the size of elementary
charge and calculation of the law of
resistance of a gaseous medium to the
movement of,small particles
29
B. Measurement of the charge and motion of
particles 31
C. Determining the size of particles 32
D. Determination of apparent density and
the dynamic coefficient of the shapes
of. particles
Chapter IV. Curvilinear Motion of Aerosol Particles
25. General theory of the curvilinear motion
of aerosol particles
,26. Precipitation of aerosols, from a laminar
flow under the action of gravity
27. Precipitation of aerosols from a laminar flow
under the influence of an electric field
33. Slit apparatuses
a
ONO
34
37
37
40
44
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Chapter V. Brownian Movement and Diffusion in Aerosols 64
38. Diffusion precipitation of aerosols in an
immobile medium 64
39. -Diffusion of an aerosol in laminar flow 75
43. Brownian rotation. Orientation of aerosol
particles in an electric field 81
Chapter VI. Convection and Turbulent Diffusion -in Aerosols 86
44. Precipitation of aerosols under conditions
of convection and mixing 86
45. Movement of aerosol particles in a turbulent
stream 98
46. Precipitation of an aerosol in turbulent
flow 107
47. Distribution of aerosols in the atmosphere 117
48. Precipitation of aerosols from the atmosphere 136
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(Pages 7-251
EXCERPTS FROM "MECHANICS OF AEROSOLS"
CHAPTER 1
CLASSIFICATION OF AEROSOLS. SIZE AND SHAPE OF PARTICLES IN AEROSOLS
1. Classification of Aerosols
Aerosols, or aerodispersion systems, are dispersed systems
with a solid or liquid dispersed phase in a gas medium. Up to the
present time therehas been no single universal classification of
aerosols and nd single system of designation of various types of aero-
sols; technical literature in this respect is completely arbitrary.
0
We suggest that a rational classification of aerodispersion systems
must be based on differences between dispersion and condensation aero-
sols on one hand, and between systems with solid and liquid dispersed
phases, on the other. In addition, the designation of individual types
of aerosols must, as far as possible, coincide with the common, non-
technical names given to aerosols (highway dust, natural fog, oven
steam, etc).
Dispersion aerosols are formed through the dispersion (pulverizetien, atomization of solid and liquid bodies and transition of pulverized
bodies into a suspension stated under the action of air currents, vibra-
tions, etc. Condensation aerosols are formed through the volumetric con-
densation of oversaturated vapors and as the resultof gas reactions
leading to the formation of nonairborne products, such as soot. The dif-
ferences between these two classes of aerodiSpersion systems, in addition
to the methdd of origin, are, that dispersion aerosols in most cases are
considerably coarser than condensation aerosols, have a greater degree
of polydispers ion and, in the case of a solid dispersion phase, usually
consist of individual, or slightly agglomerated particles of very irregular
shape ("fragments"). In condensation aerosols the solid particles very
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quickly form loose aggregates consisting of very largo numbers of
primary particles which have regular crystalline or angular shapes.
The differences between aerosols with liquid and solid dis-
persion phases are seen in the fact that in the former the particles
have a regular spherical shape, and in coagulation new spherical in-
dividual particles are formed. Solid particles may have extremely
varied shapes which, upon coagulation form more or less porous ag-
gregates, also with extremely varied shapes, the apparent density of
which may be several times less than the density of the substance of
which they consist.
Based on the foregoing, the designation of various types of
aerosols in this book are the following.
Regardless of the degree of dispersion, both condensation and
dispersion aerosols with liquid particles will be called fogs; in
the Russian language they are referred to be the same word (natural,
i.e., condensation fog, fog formed by the atomization of falling
water, etc). In the given ease the difference between condensation
and dispersion systems is not very great.
Dispersed aerosols of solid-particles, regardless of the degree
of dispersion, will be called dusts. The existing opinion that only
coarse-dispersion systems may be called dusts is incorrect: dusts
with a high degree of dispersion may be formed by artificial separa-
tion, or by natural Separation taking place in the atmosphere.
Finally, condensation aerosols with a solid dispersion phase
will be called smokes. This may not include systems of condensation
origin, containing both solid and liquid particles, the most important
example of which are smokes formed as a result of Incomplete combustion
of fuels, smokes of hygroscopic substances (such as ammonium chloride),
the particles of which may be solid, semi-liquid or liquid, depending
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upon the humidity of the medium, smokes of easily supercooled organic
substances in which liquid particles gradually change into crystalline
particles, etc. It is noted that the demarcation between smokes and
condensation fogs, and even strict adherence to the proposed terminology
sometimes is fairly difficult. However, this demarcation is still
easier than that used by English language authors who combine both
types of aerosols under the same term (smoke).
Furthermore, in practice, aerosols frequently are encountered
which contain particles of both dispersion and condensation origin.
Thus, furnace smoke always contains more or less quantities of salts
mechanically carried away from the furnace grate; the so-called
"atmospheric nuclei of condensation" consist partially of a fine
spray of sea water, and partially of droplets of sulfuric acid formed
by oxidation of sulfuric anhydride of furnace gases. The air of in-
dustrial centers contains large amounts of aggregates of soot, salts,
products of the dry distillation of carbon and atmospheric moisture
in proportions of from tens of microns to tens of millimeters in
length. These aerosols may not be included in any of the exiting
classifications and the term proposed for them is "smog" (smoke + fog).
All the above mentioned types of aerosols may have extremely
varied dispersion, which has a great influence on almost all the
properties of dispersed systems. Because of this it is appropriate
to divide aerosols into high dispersion and coarse dispersion systems
(see Figure 1).
In the theory of aerosols the word "cloud" is used by several
foreign [1] and Soviet [2] authors to indicate all condensation aerosols
with particles of 10-5 cm diameter. In the Russian language this word
has an entirely different meaning: a cloud means a free aerodispersion
system of any type (highway dust, dust ?load, a puff ofsgun smoke, etc)
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having definite size and shape. The word "cloud" will be used in
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. this sense in this book, also.
2. The Size of Particles in Aerosols
In the problem of the lower limit of the size of particles in,
an aerosol it must be taken into account that the size of very fine
particles (approximately 10-7 cm) may be determined by at least two
methods:
(1) measurement of the movement of particles in an electric dield
(see paragraph 27) with the use of an electrometer, and
,
(2) measUrement of the coefficient of diffusion of particles,
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usually through the use of electrometric methods. Thus, in both
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methods onft charged particles may be measured.
Experiments have shown that there are two types of charged
particles in gases, called small (gaseous or light) ions, and large
(heavy or medium) ions. The movement of the former is in the order of
units, and of the latter in the order of 10-3 to 10-4 cm2 per sec-1.
At the present time it has been established that gaseous ions are
molecular aggregates, formed from the charged central molecules (more
properly, ions) and clouds of neutral gas molecules connected to-them
by electrostatic and molecular attraction.
Heavy ions, in distinction from light ions, are formed only in
'gases containing solid or liquid particles in adsinSion,.i.e., comprise
the charged portion high-dispersion'aeros4s, "medium" ions, with mo-
bility of 10'3 to 10'1 cm2 per sec-1 have been discovered, and particles
with 0.2 cola per sec-1 mobility are oOntained in the combustion pro-
ducts of illumination gas and in sodium flames [3,4]. Mobility of the
latter degree also is found gaseous ions of the vapors of several
Organic substances (4], stabil as amyl alcohot.F ' Thus, gaseous ions. cannot
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be dptinguished from.charged particles in aerosOla on the basis of
mobility, but 4hp different behavior of, Ions and particles with respect
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to coagulation may be put to use. In coagulation (recombination),
the gaseous ions which form the neutral molecular complex immediately
decompose, i.e., the ions disintegrate. In the coagulation of aerosol
particles, which is not connected with the presence of charges, much larger
particles are formed. Actually, the mobility of the above mentioned ions
which are formed inflames undergoes a several hundred-fold decrease in
approximately one second (3,4],. Because the probability of the Ore- A
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sence of multiple charges on particles rapidly decreased with a reduc-
tion in the size of the particles (see equation (27.3)) it may be as-
sumod that these ions exhibit a single, elementary charge. In this
case a mobility of 0.2 cm2 per sec-1 corresponds to a radius of
1.5 ? 10-7 cm (see TUble 3). In view of the 'pet that the charged par-
ticle content of an aerosol also decreases rapidly with a decrease in
the size of particles it is quite possible that many very fine particles
are present which cannot be detected or measured. (Particles of the
order 10-7 cm may be detected with condensation nucleus meters, but
their size remains undetermined by this method.) Curiously, this
minimum size of aerosol particles has been obtained in electron micro-
scope investigations because this size conforms to the crucial limits
of magnification of the electron microscope. However; the abundance
of particles with a'redius of r 1.5 ? 10-7 cm (for example, in
silver iodide smoke (,51) forces the conclusion that they contain
-still smaller particles. It is noted that only considerably larger
particles of aerosols may be detected by ultramicroscopic methods.
'This is the status of the experimental side of the problem.
Theoretically it is quite possible that a substance with a very stable
crystalline lattice may give an aerosol with particles 2 or 3 molecules
In diameter. It is true, however, that because of very high speed of
diffusion of such particles they precipitate extremely quickly on much
larger particles (see paragraph 49), on walls, etc.
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Turning to the problem of the upper'limit.of the size of par-
ticles in aerodispersion systems, it is noted that in systems with a
now-mobile medium particles with a diameter of only several hundred
microns precipitate so fast that detection of their suspension state
is difficult. On the other hand, in rapidly rising or turbulent air
currents, such as those found in cumulo-nimbus clouds' from which rain
is falling, the lifting of loose material by the' air, fluidization of
catalysts, during sand- and snow-storms, etc, particles several milli-
meters in diameter may be found in the suspended state. Because of
the current importance of the above mentioned problems, these particles
also must be taken into consideration in the mechanics of aerosols.
Thus, the study of aerosols encompasses systems with a very
wide range of dispersion, from 10-7 to 10-1 cm. It is not surprising
that the transition from the lower to the upper limit is accompanied
not only by quantitative changes of almost all physical properties of
aerosols, but also by changes in the laws which express these changes.
This is especially apparent in the example of the law of resistance of
a gaseous medium to the motion of particles. For very fine particles
(r lee cm), resistance is proportional to the speed and the square
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of the radius of the particle. Between 10 and 10-4 cm there is a.
gradual transition to the law of Stock: resistance remains' proportional
to speed, but the square function of the radius takes on a linear
character. With a greater increase in radius there is an additional
deviation from the law of Stock: at not too low, speeds the propor-
tionality of resistance to speed is disrupted, and at fairly high
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speeds and large diameter of particles the resistance is at. the first
approximation proportional to the square of the radius and the square
of the speed (see equations 10.2, 10.4, and. 10.4). '
Changes in the character of laws which regulate several more
important properties of aerosols are shoWn in Figure lt here, in all
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cases, the transitional region is mostly between 0.5 to 1 ? 10.4 to
10-4 cm. It is not surprising that the properties are combined in
Figure 1 in groups 1 and 2, because these properties are connected
with the ratio of the radius of the particles to the length of the ?
free path of gas molecules composing air ,at an ,atmospheric pressure
of approximately 10- cm, or the average length of a wave of visible
light (A= 0.55 ? 10-4 cm). This coincidence has a circumstantial
character for the remaining properties. The facts cited above enable
the establishment of a natural classification of aerosols according
to dispersion.
The group of high-dispersion aerosols With a radius of particles
primarily less than 0.5 to 1 10-5 cm is characterized by the fact
that the resistance to movement, the speed of evaporation and condensa-
tion of particles is proportional to r2 , the diffusion of light by
particles is proportional to 1,6 , and the coagulation constant is a
function of r . The particles are invisible through ordinary micro-
scopes and may be detected by ultramicroscopy only under very favorable
conditions. The vapor pressure of the dispersed phase of these aero-
sols noticeably exceeds the normal vapor pressure of the substahce.
Consequently, a rapid "eating up" of the smaller particles by the larger
ones may take place in the latter. Finally, Brownian movement of the
particles has a prevailing influence on precipitation due to gravity.
In coarse dispersion aerosols with a radius of particles greater
than 10-4 cm both the size and shape of particles may be determined with
the aid of a microscope: the coagulation constant does not depent on r ;
precipitation has considerable prevalence over Brownian movement, and all
the laws mentioned above alternate with each other,e-as shown in Figure 1.
Finally, systems with a radius of particles of 0.5 - 1 ? 10-5 to
10-4 cm are best isolated in a special group of aaerosols of average
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dispersion and with transitional properties. It must be noted that
this group plays a very great role in research on aerosols because
this size of particles is very favorable for ultramicroscopY, which
is .one of the basic methods of studying aerosols. In addition, in
the formation of condensation aerosols from substances with poor
flexibility of the vapor phase this is the type of system which usually
is obtained in practice.
3. Distribution of the Size of Particles in Aerosols
Most natural and artificially obtained aerosols have a fairly
significant degree of polydispersion. (Relatively monodispersion
aerosols, obtained from the Lamer [6] generator and in the formation
of flower pollen and the spores of several plants: the particles of
clover pollen lie within the limits 24.8 to 26.9p [7].) In view of
the great degree of dependence of the physical properties of aerosols
on their dispersion, as mentioned above, the "average" size of par-
ticles in most cases is inadequate for the classification of aerodis-
persion systems; it is necessary to take into account the distribution
of the size of particles. Before the invention of the electron micro-
scope the ordinary microscope was used for this purpose, which permits
measurement only of the particles of coarse' aerosols (r> 3 to 5 ..10-5 cm)
or the corresponding fraction of high dispersion aerosols. Only the
general number of particles of the finer fractions of the latter were
determined without measurement of their size, or else these fractioni
usually were disregarded: At the present,time the distribution of the
size of particles of aerosols may be determined in systems with a par-
ticle diameter of larger than 1 or 2 ? 10-7 cm.
Tho distribution of particle sizes may be expressed in several
ways. The number of particles df , the radii of which are between
(r, r 4- dr) may be expressed as:
df = f(r)dr
(3.1)
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under the cenditfons
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The curve_expresaing the function ,f(r) is called the dis-.
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tributien density curve of the differential curve of the distribution
.of the size of.partic.les (Figure 2)or, more precisely, the curve of
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the.compt,rtqd diStriblitien,.1V',4d1Stinction from the curve of weight'.
.
dsi;triiati0o#:whAch extire4esSthe weight dg ,pf particles witha
radius (r, r+dr) : dg g (r) dr. . (3r3)'
in. which
co.
g (r)dr
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It must be noted that the area bound by the dlfferential curve of
distribution, the abscissa axis and two verticals to the points r1
andr2 ? .expresses the weight (computed and gravimetric) of particles.
with a radius between r1 and r2 .
In terms of the function g(r) the equation
g (r) fan,1 (r),
may be constructed, in which mr is the mass of particles with radius
; is the proportionality factor, which is easily determined by
the integration:
g(r)dr=1=mrf(r)dr=m,?
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where in is the average (arithmetical) mass of the particles of the
aerosol.
The functions f(r) 'and g(r) may be combined in a simple
equation
g(r) = f ( r)
Thus m = , where 7 is the density and V is the avefage
volur of particles, and at"constant y i.e., in a case in which the
aerosol has uniform composition and 'contains no aggregate's; -the gravi-
metric distribution g(r) is identical with the volumetric distribution
In aerosols with heterogeneous composition (for ex:mple,
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? In technical practice the function GaSr) is used very fre-
quently in studying various industrial dusts. In determining its in-
tegral curve of gravimetric distribution it is called the "characteris-
tic curve" of.a given dust, or the "curve of residues," because it is
determined by the weight of the dust which settles on a given screen
or by the given speed of air movement in an air separator. The curve
b(r) is called the "curve of apertures." It is noted that:
Ey(r) + Pb(r) = (jr) + Ob(r) = 1 (3.8)
By way of clarification of the given example of distribution
of the size of droplets in a stratus cloud [9] determined by micro-
scopic measurement, it is noted that in distinction from a system with
a liquid medium, the direct, continuous curve of distribution of which
may be obtained through sedimentometric methods, in aerosols the weight
of particles in an experiment is determined, the radii of which lie be-
tween end values, i.e., in addition to continuous curves, interrupted
curves, or "histograms" also are determined.
The results of measurement of 100 droplets in the case under
consideration are shown in Table 1.
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TABLE 1
DISTRIBUTION OF SIZE OF DROPLETS IN A STRATUS CLOUD
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1
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Radius intervals of droplets,p,
2.5--4
4--5.5 5.5--7 7--8.5 8.5--10
10-11.5
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Number of droplets
4
6 15 24 24
12
Radius intervals of drciplets,t,
11.5-13
13--14.5 14.5-16 16-17.5
17.5-18 .
Number of droplets
4
4 4 ' 1
2
The histogram shown in Figure 2 was constructed from those data.
It may be used directly for the computation of various average sizes
(see paragraph 4) but in most cases it is desirable to convert the histo-
gram into a continuous curve, smoothing out any irregularities in this
process which are caused by an insufficient number of measurements (for
example, a physically improbable rise in the right region of the histogram).
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This smoothing out of the curie is substantiated in experi-
mental results by the circumstance that the number of measured par-
tides is necessarily limited, and therefore statistical fluctuation
is fairly noticeable. The direct drawing of a smoothed-Out curve of
this histogram is inconvenient. It is better to begin with the con-
struction of experimental points (marked with "xis" in Figure 3) for
the smooth integral curve Fb(r) . Thus it is relatively easy to cor-
rect occasional errors in measurement and fluctuations in the number
of particles in each fraction: Graphic differentiation of the integral
curve results in the differential curve f(r) (Ewe Flgure 2).
In proceeding with the curve of weight' distribution one could
start with construction of a "gravimetric histogram" directly from
experimental data, but in this ease the fluctuations mentioned above
are extremely marked. Therefore, it is better to begin with smoothing
out the f(r) curve of experimental data. Laying out the axis r for
sufficiently narrow intervals r , the average mass of particles
4 3
grr corresponding to the latter is calculated, which gives
3
the curve f(r) and enables determination of the mass mrf(r)Ar
of the dispersed phase in each interval and the average mass in , from
which the desired function of gravimetric distribution may be derived
through formula (3.5). The curve g(r) in Figure 2 is constructed in
this manner. From this, ih turn, is derived the integral curve Gb(r)
(see Figure 3).
In grid or sedimentation analysis of dusts the function G(r)
is determined directly in the experiment. The reverse
pre,aictala of de-
termining the calculated distribution from this function follows the
same method. The construction of quadratic curves and other size dis-
tributions also is similar.
The selection of any method of expression for determination of the
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size of particles in an aerosol depends on the properties which are
to be emphasized. For example, the Computed distribution of the sizes
of particles f(r) (see equations (49.28) through (49.32)) must,be
used for computing the speed of thermal coagulation-of aerosols. The
speed.of evaporation of coarse-dispersion aerosols at any given moment
is determined by a linear distribution of the size of particles rf(r)
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because the speed of evaporation .of particles is proportional to their radius'.
The optical density of coarse-dispersion fogs is determined by ,a quadratic
distribution of particle size r2f(r) , because the reflection and dif-
fusion of light by large droplets is proportional totthe.Square of
their radius. The degree of completeness of precipitation of aerosols
due to the effects of gravity or inertia also is determined by this
distribution.
The examples described above involved differential curves of
distribution. Integral curves are applied primarily in the calcula-
tion of the degree of completeness of separation of the dispersed
phases of an aerosol from the gaseous medium in the various aerosol
generators and in expression of the distribution of particle sizes
through empirical equations (see equation (3.9)). It is noted also
that the gravimetric percentage fraction remaining on'a grid screen of
determined aperture size and passing 'through a grid of another size
usually is used for the classification of industrial powdered sub-
stances, i.e., the size of the function G(r) is given in two deter-
mined values of r .
-It is necessary to mention a type of distribution curve fre-
quently found in literature (Figure 4) in whteh the ordinate axes
intersect at some distance from the initial coordinate. Curves of
this type, which give a definitely 'false representation of the true
character of theAistribution Of particle size, are obtained because of
the terminal?size of the intervals at which the size of the pait*Cles
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are graphed according to measured values. Actually, the curves in
Figure 5 give the actualdstribution of sizes. At sufficiently nar-
row intervals (jr 0.10 , for example .when particles are mea-
sured with an electron microscope, a histogram of the type shown in
Figure 5a is obtained. "Smoothing out" this curve results in a curve
which is very close to the curve of actual distribution. When parti-
cles are measured with an ordinary microscope, however, at best the
smallest interval attainable is r = 0.2p., which gives the histo-
gram shown in Figure 5b, and a curve similar to A in Figure 4.
Finally, at intervals of jr.= 0.514. obtained in ordinary hygienic
procedures, the resulting histogram is of the type shown in Figure Sc,
and the curve is of the type B ? in Figure 4,
Usually, this masking of the true distribution occurs when a
considerable portion of the distribution curve lies within that re-
gion of particle size where the method used for determination of the
size is not favorable because the particles are too small; or in
another case, when because of a very high degree of polydispers ion the
interval between sizes of particles is extended over several orders of
greatness. Assuming that the radius of particles of several aerosols
lies between 0.1 and 200ft. and that the entire length of the abscissa
is equal to 100 mm, then the part of the distribution curve corre-
sponding to the limits 0.1 to lp contains an interval of only 0.5 pm,
and it is impossible to construct a correct distribution .on this graph.
It would seem that under these, conditions a logarithmic scale
could be used for the abscissa axis. Actually, in this condition of
inequality the positien of very fine particles is displaced because
each order of greatness on the graph is not afforded an equal space.
However, another difficulty arises at this point. If, taking a loga-
rithmic scale for the radii, a distribution curve is constructed by
simply plotting the function f(r) on the ordinate axis, then the area
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/ ?
p.
???????????
bound by this curve, the abscissa ax1.s and verticals to the points
T2 ri
and r. is equal eto1
(r) d Ig r (r) dr,
r,
i.e.., this area already is not proportional to the weight orparticles
?
with radius from r1 tor2 , and the curve loses its descriptiveness.
In order to retain the significance of this area rf(r) must be
plotted on the ordinate axis in addition to f(r) , but in aerosols
with a high degree of polydispersion the curve rf(r) practically
coincides with the abscissa axis in the region of a small r value.
Thus one is agaih confronted with the necessity of graphic expression
of the distribution of sizes by means of a curve. In recent times
there has been a tendency in aerosol research to reject the above de-
scribed methods of graphic representation of the distribution of par-
ticle size and to substitute a system of coordinates in which the
distribution is expressed by straight lines. This question is closely
related to another problem which is discussed below.
The use of distribution curves for characterizing.industrial
aerosols, and in the solution of various theoretical and applied
aerosol problems is very inconvenient. These curves are appropriately
represented by formulas with a minimum number of coefficients, the
size of which would characterize a given distribution. In the latter
case itis desirable that the formula be applicable to the greatest
? , ?
_4
possible number of aerodispersion systems, i.e., that only the size
of the coefficient be changed in the transition from one system to
another. With a sufficiently large number of coefficients all distri-
butions eneountered in practice could be represented by a single,
unified formula. However, the selection of coefficients requires a
great deal of work for each case, and in addition, it would be diffi-
cult to describe anygiven physical situation by these coefficients.
BOCali6e of this, multiple-coefficient formulas have not received practical
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V
application. As a rule, only' formulas with two coefficients are used,
because this number of coefficients is in itself a minimum: one co-
efficient characterizes the average size of particles and the other
characterizes the degree of polydispersion of the aerosol.
There still is no theoretical derivation of a dispersion formula '
for very complex and insufficiently studied processes of generation of
both condensation and dispersion aerosols (with one exception; see below)
but there are many dmpirical formulas which are used mostly for a'e:15'osols
produced by the mechanical dispersion of solid and liquid bodies... The
most well known of these are the following:
(1) ,R011er formula [9], of the type described in the foregoing:.
0 (r) = 46r/21 exp(-s/r) (3.9)
(In this formula, at r -4,446 . Gb(r) . Therefore, in this
case the integral curve of distribution must be interrupted at that dis-
tribution of r1 where Gb(ri) = 1.)
which also is used for many industrial pulverized materials with ex-
tremely varied degrees of dispersion.
(2) Razin-Rammler formula [10]
Git,(r) = exp(4r8) (3.10)
which is used for comparatively coarse-dispersion dusts and fogs .pro-
duced by mechanical methods.
A, considerably improved formula for these-fogs has been sug-
gested by NUkiyama and Tanasava ill]
f(r) = a.r2 exp(-brs) (3.11)
where 41. and b are independent, but determined functions s , and
the size of droplets is medium. These functions have been calculated
by the present authors and are presented in table form.
Formula (3.9) may be represented in the form
lg[Gb(r ) / r1/2'
j = lg 4,- 0.434 ?/r .
16 -
? ? -
(3,.12)
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' e
If 1/r is plotted along the abscissa and lg(Gb/r1/2) is
plotted on the ordinate, then as far as is applicable, the experimental
?If
points must be plotted in a straight line, on which $3.? and s may be
easily determined.
Similarly tO foimula (3.10) we obtain
Ig (r) ? 0,434 ars. (3.13)1
In this case a value must be chosen for s at which the ex-
_
perimental points lie in a straight line for the selection of the co-
'ordinates rs and lg G . The experimental data are treated as in
foraula (3.11), which gives the formula
lg Cr) r9 Ig a ? 0,434 bra. (3.14)
Reduction of the formula to a stage in which the particle sizes
are distributed in a straight line greatly eases the task of selection
of coefficients in these formulas, and of smoothing out the curve of
experimental data.
In a very few aerosols, such as that formed by plant spores [12]
the Curve of distribution is symmetrical, very similar to the Gauss
curve, and corresponds to "normal" distribution;
1
1(r) =exp E? (r -?;)2 [49,
(3.15)
where r is the average radius of particles; fe ='(r 7)2 . is the
average quadratic deviation (dispersion) of the size of the radius
from r .
As an auxiliary alternative the following formula.is intro=.
. -
The weight of particles with radius & r1 is equal to
Ts
I , Ts Dri?
I
1 (r)dr = e--(1/2(33 dr .. I e-4.14. (3.17)
13/1.7c o 1177
o 7
? alir
Because according to the function f(r) it is different from 0 only
at r 370 , the lower limit in the integral may be taken as -0:, ,
17 ..
4
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' r ,
.is
If 1/r is plotted along the abscissa and 1g(Gb/r1/2)
plotted on the or4inate, then as far as is applicable, the experimental
points must be plotted in a straight line, on which a- and s may be
easily determined.
Similarly to foi.mula (3.10) we obtain
lg G (r) 0,434 ars. ' (3.13)
In this case a value must be chosen for s at which the ex-
perimental i,oints lie in a straight line for the selection of the co-
'ordinates rs and lg G . The experimental data are treated as in
formula (3.11), which gives the formula
lg (/ (r) r9 = 1g a ? 0,434 1; (3A4)
Reduction of the formula to a stage in which the particle sizes
are distributed in a straight line greatly eases the task of selection
of coefficients in these formulas, and of smoothing out the curve of
experimental data.
In a very few aerosols, such as that formed by plant spores [12]
the burve of distribution is symmetrical, very similar to the Gauss
curve, and corresponds to "normal" distribution:
1
(6 = iirrc exP 1-- ? )7)2 I 2[2], (3A5)
where r is the average radius of particles; le = (r - 7)2 is the
average quadratic deviation (dispersion) of the size of the radius.
from r .
As an auxiliary alternative the following formula is introduced
(3.16)
The weight of particles with radius ri is equal io
r, ? r
? :
(r) dr c--(r-71'12a3 dr et di. (317)1
0 py 2n 0
?
Because according to the function. f(r) it is different from 0 only
at r 3;0 , the lower limit in the integral may be taken as. -cx,. .
17 ..,
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41.?
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? ???
t.
?
4
TI ri. ? . _
/ (r) dr dt = Erl (7-3-3-74)]= Er/E1), (0.18)
where
Er = (4ramp's. functIon) (3.19)
'
2
o ?
but ti. is equal to the'value of k , corresponding to r1 .
.1%
The variable is plotted alowg the ordinate on an-arbitrary
scale (Figure 6), and the corresponding values 0.5[1 + Er1(5)j are
lecated on it, i.e., the weight of particles with the normal distri-
bution under consideration, for which r/.. r +pv-ir t As before,
r is plotted on the abscrssa. It the integral curve Fb is con-
seructed in this "probable" system of coordinates, expressing the
weight of particles with radius less than r , then in the case of a
normal distribution of particle sizes similar to equation (3.16) a
straight line must result, intersecting the abscissa at the point
r = r . The tangent of the angle of intersection of this straight
line with the abscissa is equal to I .
The data of Table 1 for a water fog do not give a probability
grid of a straight line graph, as may be seen in Figure 6 in which
the distribution of the size of droplets is indicated by crosses, in
view of the asymmetry of the differential curve of distribution of
this fog (see Figure 2). It must be mentioned that in the over-
whelming majority of condensation and dispersion aerosols the dis-
tribution curves are asymmetrical with a-greater deviation toward small
values of r . Apparently this is connected with the already mentioned
inequality of very small particles in the selection of the linear size
of-particles as the abscissa value in distribution curves. The dis-
tribution curves are more symmetrical if the logarithm of the radius
Is taken for the abscissa, and they frequently approach the shape of
' the-Gaugs curve. In this case the distribution may be expressed by
18 ?
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S?????
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the formula (logarithmic-normal disiributiOn)
. . .
_ grYdr'="-----i OXD r ( g ig on d la r-
1 P.-- r , .
. . igporgi ? L 2003 r i .0 ... . (3.2,0)'
- g . .
r .L.L....7,
. ,
rg , is the average geo-
metric radius of joarticlls (lg'd )2' = ,(1g r - lg r )?'/,'1..e., is:the
?
-, Vg g?
,:
average quadratic deviation of the logarithm of the iadius. In fereAgn
. _ , " P
? .
literature p usually is called the. "standard geometric deyiation
g, ? ? ? ,? . ,,, ? i ?
? I. I ' t 2 r,.:. ,
As is-seen in Figure 7, the fog .chosen by the present authors as an
_Here lg rg = lg r , and consequently
example (stratps cloud) gives a straight-line curve, 1, in a proba.!
bility-logarithmic grid. (In passing straight lines through experi-.
mental points in probability network's, it must be remembered that points
far removed from the axis have little statistical value because they cor-
respond to a small number of particles.) L. Levin established the
logarithmic-normal distribution of the size of particles in natural
clouds ea the basis of a great deal of material compiled by El'brus
[13]. Recently, this type of distribution has been found in other
aerosols of dispersion and condensation origin; in mineral [14] and
uranium [15] dusts formed during mechanical crushing operations, in
fogs formed by disc pulverization [16], in aerosols of NH4C1 and 112SO4,
formed by mixture of gaseous components [17], etc. In addition to the
other distributions mentioned above, logarithmic-normal distribution
undoubtedly also has theoretical significance [18]. In particular,
as is shown by A. Eolmogorov [1.9], proceeding from simple hypotheses
on ihe character of the-process.of crushing solid particles, it may
' be demonstrated that the distribution of the size of particles follows
an asymptotic path in proportion to the degree the refinement of parti-
cles follows an asymptotic path in 'proportion to the degree the refine-
ment of particles approaches the logarithmic-normal system [20]; It
would be very interesting to clarify the conditions under which this
distribution is obtained in condensation processes.
It is noted further, that in the case of logarithmic-normal
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I.
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: t .
a
-
calculated distribution of sizes (3.20) gravimetric and other derived
'distributions also become logarithmic-normal at the same value of
i.e., all derived distributions are expressed by parallel straight
lines [21].
Actually, in a. distribution corresponding to the 9-th degree
of radius. the folloiving.expr9A8ion is. obtained (a. is the normalizing
*:
t,
factor, containing only constant values);
a esp (2,302 v Ig r)exp [? (1g r ? 1g rg)2 / 2 1/4%1 =
1g pgyr2n
a (Ig r ? Ig rg)2 ? 2,302 v Ig r 2Ig2 Pg
1g pp exp ; 2 1g2 pg.
pg r-- Og 2,2102 v pdrl
_ oxp
1g Pg1r2 a 2 le Pg
The gravimetric distribution of the size of particles in a fog
according to the calculated distribution, represented by straight line
1 in Figure 7, is expressed by the parallel straight line 2.
It is characteristic of all distribution formulas discussed in
this paragraph that points which lie close to one or both regions of
the curve more or less deviate from a straight line. However, this
does not have great practical significance in the characteristics of
aerosols because in integral distribution curves the marginal points
correspond to a small weight of particles. Several authors [9] are
inclined to aver that the formulas proposed by them are correct for
" all ranges, and they explain deviations by experimental errors, pre-
mature precipitation of particles, c"-c. The present author disagrees
with this contention; as already stated, these formulas (with the ex-
ception of (3.20)) cannot be used for the expression of any theoretical
idea. They are more or less lucky equations; purely empirical approxi-
mations or the actual distribution. Furthermore, because of the fact
that each of these formulas has been successfully applied to one .or
another group? Of aerosels it follows that none of them has general'
significance. Nevertheless, the practical value of these formulas .is
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4 ? '
-'beyond doubt because they are used in the determination of all properr
ties of aerosols depending on their dispersion, in the comparison of
-aerosols using only two parameters, etc. The work involved in the
-selection of the fitting formula and in the determination of the size
- of the coefficients usually is considerably less than the work re-
quired by the measurement of a sufficiently large number of particles.
Because of this, for the further development of knowledge of aerosols,
and especially of the theory of the formation of aerosols, it is very
desirable that the indicated processing of measurement data be conducted
on the largest possible number of aerodispersion systems.
Knowledge of the distribution of the size of particles is ex-
tremely important in aerosol research because almost all properties
of aerosols depend to a very great degree on their dispersion. There-
fore, experimental aerosol research must be done on isodispersion sys-
_ tem, but because the production of the latter still has not been com-
pletely solved research must be conducted with polydispersion aerosols,
In this case, more or less promising conclusions may be drawn from the
results of these experiments, noting the distribution of the size of
_particles.
. "Average" Size of Particles in Aerosols
Completely characterization of aerosols requires knowledge of
the distribution of the size of particles. However, in practice the
term "average" frequently is limited to those cases in which, for some
'reason or other, there has been no investigation of the distribution
of the size of particles, but some other property of the aerosol,
which depends upon its dispersion, has been measured, such as the
coefficient of diffusion, expansion of x-ray lines in an x-ray photo-
graph of precipitated Particles, the diameter of the diffraction corona,
etc. Very often the dispersion of aerosols is measured by the calculated
-
gravimetric method, i.e., the gravimetric concentration Ojos of the
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11
,
dispersion phase per unit volume) and. the-calbulated'concentre.tion:-
-
(number of particles per enit,y-olunie): of the aerosol aie measured;
this includes the average mass of particles, and if their density
,
is'%nown, the amerage size:of re the'. particles.. (In ,Soviet literature . f
tl
, ..
term IIparticular,concentratiow' usgq.11y is used to designate the
, .
"V
number of particles per unit aerosol volume. Recause of the ambiguity
:
("particuiar" as opposite of "complete") of the term the present au-
thor will follow the example of G. Romashov [22] and employ the term
"calculated" concentration.)
It must be borne in mind that the "average" particle Sizes de-
termined by various methods 'may be clearly distinguished from each
other. Just as in the case of the calculated, gravimetric, etc,
terms' for the distribution of the size of particles, there are various
average radius values:
(1) average arithmetic radius
CO
(r) dr 2 r?A r? 1 N , (4.1)
where 1?13, is the number of particles per y size interval; r, is
the middle of this interval; N is the total number og particles.
etc.
(2). average quadratic radius (surface average)
?
V. . .q,
ra V = ,r2t (r) dr] [P14.1Al ? I Al ;
v
(4.2).
0
(3) average-cubic radius (average per volume of aYerage weight)
3 co t
rs = 17?P r3 (r) dr] ? I Al (4.3)-
0'
In these averages, derived from-the calculated distribution
of sizes, the large and small particles are equal. Suspension ayerages
derived from the gravimetric distribution g(r) have greater Rractical
value, such as rg (r) dr :?-???.-, I G, , .(4.4)
v
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?-*-
?
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,??????
? ?
A ?
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whpre, ,gv is the yeight.ot.yarticlas:per interval, G' is
, -
-
the total weight of particles,
,
The following also freqUently are used:
'(4) average geometric radius. , determined by the formula
:cot g
Ig rg= lg r r.f r., N; (4.5)
(5) calculated median radius rm , daUFEIned from the condi-,
t-ion F (r/11) = Fb(rm) =0.5 . This implies that half of the particles
have a radius greater than rm and half less than rm ;
(6) gravimetric median radius: rm? determined from the analo-
- gous condition Gc4(rml) = Gb(rm,) = 0.5., i.e., the mass of particles
with radius greater than rm, comprises one-half of the total mass of
_ the aerosol.
It is noted that at normal distribution rm = r , but at the
logarithmic normal distribution rm = rg .
By way of illustration various average particle sizes in the
fog described above are calculated in the following. In this case it
is advantageous to start directly with the experimental data (see Table 1)
without smoothing out the raw data, and the calculation is conducted
according to the formula
7.1. 2 1-,N, I N
V
etc.
(4.6)
Thus we have: r1 = 8.91u,; r2 = 9.4m.; r2 = 9.9pG . From the
curves for Fb and Gb in Figure 3 we have rm = 8.61.4, and rm =
= 11.1FL. ; and from curves 1 6.nd 2 in Figure 7 we obtain the values
rm = 8.6pu and rm, = 11.51a. .
:
Various average values are obtained in the experimental deter-
mination of the average size of.particles, depending upon the method of
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measurement used. Thus, r3 is determined by the calculated-gravimetric
method, r1 is determined bY the "corona"
method (diameter of diffrac-
tion rings), etc.
The rational selection of an average value for the characteriza-
tion of an aerosol, Es?s well as the selection of the distribution curve,
%, ?
is determined by the properties of the aerosol which are to be described.
Therefore, the average quadratic radius r2 must be chosen for expres-
sion of the optical density characteristics of coarse aerosols and the
speed of their precipitation under the influence of a heavily charged
field or under the influence of inertia, r1 is used to characterize
the speed of evaporation of aerosols, etc. In some special instances
more complex average values must be established. Thus, the specific
surface area of an aerosol, i.e., the surface per unit mass or volume
of the dispersion phase, may be characterized by a particle in which
the specific surface area is equal to the aerosol weight. The radius
of this particle rs is determined by the equation
4rrr2/ (r) dr
or
4 4 4
? nr3/ (r) dr = 4ercr3 I ?itr3 = 4nr2 -- nr3
3 2 3 3 a 3
r =:r3/r2.
3 2
(4.7)
In the fog under consideration rs = 11.0f6 . This value is
obtained in investigation of. the problem of the absorption of light
per unit volume of a substance in a coarse aerosol.
It is seen that in the given case the various average valuPs
do not differ greatly. The greater the degree of monodispersion, the
smaller is the difference between average values. The concept of an
average radius for aerosols with particle sizes which are extended
over several orders of greatness loses All physical significance.
The problem of the average radius and distribution of sizes in
aerosols with irregularly Shaped particles will be discussed later.
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[Pages 59"67]
590,67]
4,
CHAPTER II
14. The Movement of Aerosols within a Closed Space
In the case of aerosols contained in a limited space, the motion
of particles involves both movement along with the suspension mediums
" due to convection currents, artificial stirring, etc, and movement in
- relation to the medium. At present we are concerned only with the
latter, and this problem will be studied from the point of view of
the precipitation of particles due to gravity. In the precipitation of
particles of an aerosol which fills a space bounded by walls, the speed
of precipitation is V , and the motion of the suspension medium in the
opposite direction has an average speed of 15V where 45 usually is a
very small volumetric size of the dispersion phase. Whereas in the im-
mediate vicinity of the particles the medium is absorbed by the latter,
? at a greater distance from the particles (i.e., at a distance greater
than the particle radius) the speed in the opposite direction is greater
than V . Thus the speed of precipitation of particles in the -case at
-hand, in distinction from the moveMent of a free cloud, is less than
the speed of isolated particles in an unbounded space by the factor
1 + kl5 where k >1 .
According to Cunningham [46] only one circumstance need be taken
into consideration: in the derivation of the Stock formula one of the
limiting conditions is that the speed of the medium is equal to zero
at an infinite distance from the particle. In the precipitation of a
system of particles in a closed space the speed of the medium must be.
taken as equal to zero at h distance e from the center of a particle
-(dieregarding the opposing motion of the medium), where 2e 7.--n-1/3
is the average distance between precipitating particles. Thus every
particle is exposed to the sails reeistance which it would encounter in
the center of a closed vessel with radius p
? According to the calcu-
lations of Cunningham this resistance in Stock approximations -is equal.
? ?"?
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to 61(116(1 + 1.25f)- According to Oseyen the correction is in-
creasingly smaller at greater values of the Reinholds number Vpr/7 .
All other authors arrive at a correction factor 1 + kfthrough fairly
complicated, though not precise methods, in Which the values of k.are
equal to 5.5 (85]; 7.0(83]; and 4.5 [86]. Obviously, a precise
solution of this problem is very difficult.
T1 difference between the correction factors of type 1 + ktt (I)
and 1 + kl-ef 1 + kfl/3
(II) is quite large, because at ordinary values
of factor / is for practical purposes equal to unity, whereas
factor I/ may be several percent greater than unity. The only re-
search on the small values of , in which we are interested at this
point has been done by Cermak [85], in which he studied the speed of
precipitatien of suspensions of a high degree of monodispersion, con-
sisting of erythrocytes of various animals in water, with radii of
2,4, 3.0, 3.7, and 4.4p. . It was shown that at 154L 0.04 to 0.08
the experimental results agreed with the correction factor 1 + k!f ,
in which the value of k is between 4.8 and 6.9 for various erythro-
cytes. Unfortunately the speed of precipitation of isolated particles
was not measured in this work, but was determined by extrapolation.
Because of the rather scanty data available it may be stated
only that in the precipitation of aerosols in a closed space the re-
sistance of the medium at small values of 15 apparently is equal to
6140V7 (1 + 195) , in which k is approximately 5 or 6. Recently,
during the course of research on the fluidization of dusts (see equa-
tions 58.1 and 58.2) the problem of the speed of precipitation of concen-
trated dispersed systems acquired great importance. In the measurement
of this speed fluidization was accomplished by the transformation of
systems of uniform spheres or bodies with other shapes into a suspended
state in a rising fluid streaV. In this process the concentration of
Spheres is automatically determined because their speed of precipitation
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. ? ?
is equal to the speed of the fluid stream. The results of these ex-
periments may be expressed by the formula
V' = V8(1 -fo)L , (14.1)
where VL is the speed of precipitation of the system of particles,
and VS represents isolated particles. Louis and Hauerman (87] and
Richardson and Zaki (88] obtained the same value, 4.65, for the coef-
ficient CC in the case of the system of spheres. The work of Richard- 4
son and Zaki also included an approximated theoretical calculation of
the speed of precipitation of the system of spheres.. Proceeding from
two models of the distribution of spheres within a space, these authors
obtained two curves (V;4), one of which was approximately 40% higher
and the other 20% lower than the experimental curve.
In conclusion, mention is made of a phenomenon which is fa-
miliar to all who work with aerosols: in precipitating concentrated
aerosols the upper limit of the dispersion phase usually is horizontal
and flat. This phenomenon may be observed both in the laboratory and
in natural fogs. The cause of this is that when the saturation of an
aerosol exceeds that of a medium bordering on the aerosol, hydrostatic'
forces counteract the disturbance of the horizontal upper 14mit of the
aerosol by the action of convection currents. In the given instance
the aerosol behaves like a fluid. It is known that stabilization of
the upper border is observed only when the dispersed phase moves as a
whole with the medium, i.e., at a sufficiently high concentration of
preceding paragraph). Ber!Ape of this, the surface of
aerols charged with chlorine, carbon dioxide, etc, is very stable (89].
the aerosol (see
15. Vertical and Horizontal Electric Field Methods and their Application
The movement of aerosol particles in an electric field is ob-
servable primarily when movement within the field is accompanied 'by
the attraction of gravity. The force acting on a particle in an
'electrical field is equal to qE , where q is the charge of the particle
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?
? ,
4,4i;?-..1 ? .."_?
pOtqhfial-Orttherffeld: The speed
. , .
?
' 3 ? ?
which 'forunila ? (8 . 2 ):*%apii,1 is equal to
? - ?
1\
.??
TE .7: 9AB = 9E:(1. + A I Ori?
of particles, in a.
(15.1).
."': ? ( .
,
.,,,-' , ? The,mpvement of aerosol particles
'--
field:. i.e ---when an 'electrical field is
f'k
e 4 ?? J 1. #, ?
' ? tion41 tield of, tbe earth, is very interesting
? ?
in a vertical electrical
superimposed on the gravita-
? .
v.!
4ppliC4tiOnv'
J, V
ka?
r'
[0] ,Equf
ikcauee of ite Practical
TAO- vertical electral 'tieldlMethdci cleveloPeci by kiln-
?
Ehrenhaft 191] is one of the most ptoductive methods for
.study,ofaerodispersion systems and it plays a very great role in the
'de'VelOpment of ourinowledge 'in this field. The method consists of
aeroiol particles in a test tube, which forms two horizontal conden-
? '
r
,ser/plates and side walls of an insulating material, and is provided
4?
with ports 'for-observation, illumination and overcharging of the par-
.
Observation is through a-horizontal microscope with an ocular
grid, The potential of the field is E = TT/h , where Tr is the po-
tential difference, and h is the distance between the condenser
plates.. The intensity and direction of the electric field may be
.measured at will. The speed of precipitation without an electrical
field Vs
is determined, then the movement of the same particles is
determined under fhe simultaneous action of electrical and gravita-
,
tional fields Vs +VE or. :Vs - VE , depending upon
the electrical field. This
? .
'addit'ion, the charge of the
- .
exactly. counterbalances the.
particles
the direction of
Is where the value VE is involved. In
field- EB s.oinetimes is determined, which
action of gravitational force on the
EB = mg lq = 4 nrliglq.
,designs the_testtube i. provided, with facilities for
,
changing the pressure within the tube within wide limits above or be-
1
low atmospheriC pressure. Tho techniques of the vertical field method
0,
are well Covered in the literature [53, 92, 93], and enable solution
'Of the following series of' tasks.
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,
?
^
- ? ? -
A.-.1.4termination of the Size of Elementary Charge' and Calcula-
tion of the Law of Resistance of a Gaseous Medium to the
Movement of Small Particles
The method first used in the folution of these important tasks
L 1,
-WaS the method cir vertical electrical field [53].
In terms of Vs the
' , ? ,bi
..equation may be expressed,as *40
,
Ve .74 ingn mg(1-1- A )/6nr-ii
or
V. (1 4- A 71.) .
Multiplication cd the square root extracted from (15.4) with (15.1)
?
the followilx equation is obtained:
9
g 7_118 nirE / \ ) 2
=
+A"gst(i+ 2'
(15.5) ? ,
Assuming that the formula of Stock is applicable, the expression
_
contained in brackets, i.e., the size of the charge of particles deter-
mined experimentally, is designated by q In the case of liquid
St
(oil) droplets all dimensional values which are factors of this expres-
sion, such as the viscosity of air, density of particles, field tension,
and the velocities VE and Vs P may be determined experimentally.
When a particle is given an overcharge by x-rays or gamma rays the change
in its charge
is equal to a small positive or negative whole number
of elementary charges E :
Aq = (v ? '2)E, (15.6)
where - 92 usually equals. ?1 , or more rarely, ?2', etc. Simi-
larly,
A:151 ' "1 92) St
finding the common denominator of several values of 6:1St ?
Thus E may be easily determined by
-St
Because from equation (15,5) it follows, that:
then
e,==s1A/(14-dig%
--= (1 + A -9
If the values of
, which have been found for various
droplets,
are plotted for the function Z/r then in the case in
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? \
(15.7)
(15.8)
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?
?.?
",4: -i
which the tnnninghain'foiMula applies these points also must lie on a ' ?
.-
. ..
straight line (15.8). ?The point of intersection of this straight line
, .? , ? , .
with the ordinategives the value of E2/3 , arid the tangent of the
,
.- -
. , ?
angle of inclination with the abscissa axis gives the' value at As,2/3 .
. , - . .
In pra'ctice the-truwradius. r is not determined directly from the'
.experiment, but'?
v. ?
?
(9Vors
rSt 2yg
(15.9)
i.e., the size Of the radius, calculated, assuming the accuracy of the
formula of Stock. However,-at small values of the correction element
AUT, rst differs very, slightly from r , and therefore if r is sub-
stituted for rSt ' - in (15.8) the substitution has little effect on the
results, in'fact the experimental points lie quite within a straight
line. This method resulted in the first accurate determination of the
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.a water water ,vapor fog obtained by condensation of vapor on gaseous ions.
The Water'droplets in.this case have the same elementary charge. Vs
and VE were determined for the movement of the upper limit of a .
cloud, and because of this the value of t was approximately 30%
lower than the true value (06]. This is not surprising, because the
speed of motion of the uppdr boundary always is determined by the
slowest, particles (the smallest particles for Vs measurements and
the largest particles for VB measurements). Because of this, de-
pressed values for the particle charge are obtained, similarly to
equation (15.1).
All the discussion above pertains to individual, spherical par-
ticles. In the transition to non-spherical particles and to aggregates
the following must be taken into consideration: in the case of ag-
gregated particles / indLeates their apparent density. Therefore,
if the true density is substituted for V in (15.5), values somewhat
lower than true may be obtained for q , and thus alsc for t . This
also gave rise to the hypothesis of the "subelectron" [97]. Further-
more, if the particle is not spherical, then r2 /x is substituted
for r2- in (15.4) (see 12.15), and rex is substituted for r (see
12.18) in (15.1). As a result, the expression for q (see formula
- (15.5)) is a factor of x2/3 in the denominator. Because for non-
spherical particles x 1 (see (12.18)), if this. circumstance is
not taken into consideration depressed values again are obtained for
the elementary electrical charge.
Knowing the size of ? and the law of the resistance of the
medium, the following tasks. may be solved by the vertical field method.
B. Measurement of the Charge and Motion of particles
Through repeated measurement of the speed of motion of a particle
in a vertical electric field, directed both up and down, the arithmetical
average of both speeds VE ye; and VE + Vs is taken as equal to VE
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? ?
- Pg.>
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and the proportional number of elementary charges on a particle v.
If the latter is subjected to a several-fold overcharge, 11 and the
size of the charge on a particle q = VE is obtained. Knowing VE
-and q the motion of a particle D is determined according to
-formula (15.1).
The particle motion B also itiy be found by its speed of fall
when pole VS is absent (see formula 15.3), and by the strength of a
counterbalancing pole BB , which characterizes the mass of the parti-
cle (15.2). The shape and density of the particle is excluded from
both of these methods.'
A very substantial effect is duo to the fact that the motion of
particles in an electric field, determined by the first method, and
, -
the motion in the gravitational field, determined by the second method,
may differ considerably because of the orientating action of the elec-
trical field tends to divert particles which are aligned along other
axes into alignment with the electric field (see paragraph 43). For
some reason or other this circumstance has not been taken into considera-
tion by any of the researchers working in this field, and explains many
instances of disagreeme&. between theery and experiment in investigation
of the motion of aerosol particles in a vertical electric field.4
C. Determining the Size of Particles
When determining the size of aerosol particles by their speed
of preeipitatioa under the attraction of gravity (i.e., according to
formula 15.4) two difficulties are encountered:
(1) For particles in an aggregated state the apparent density,
which is used in (15.4), is not known beforehand, as already has been
- mentioned.
(2) In the case of very small particles (r 410-4 cm) Brownian
movement causes very large fluctuations in the measurement results. These
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.4.19
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. ' difficulties may be eliminated to a considerable extent throdgh'tho.
.t
use of the following variant method [961. When VE and q are foUnd,
As,, described aboye, they are used for the determination?of r , u3Tnt
(15.1) or, in the case of non-spherical particles, re x is determAned,
; ..
%
This enablies elimination of particle density.,I The infivence of,.
; .
1.f
Brownian 'movement in thib case is lessened beause.pq speed VE (i.
Is
i ?,... ...
, ? .
clistinction to Vs):MayAel,mae suf4i.cientlyAarge thr6ugh TlipreAsitig.
1
the tensiou. of lield E .
The size of particles with unknown density also may be found by
measuring the sPeed of their precipitation under various pressures [89].
Since the length of the free path of gaseous molecules is inversely
? proportional to the pressure of the gas, equation (15.4) may be written
in the form
or, 1. pr j
(15.10)
Here A' = Apt = Apojo , where to is the length of the free
?
path of a molecule at pc, atmospheric pressure. Thus if V51 and
Vs2 are the speeds of precipitation at pressures pl and p2 , then
Vci 1'.2
,.-
1 1- (.1 /m)I HI- (A' 111211 ' (15.11)
from which we obtain
l'?P2-1',IPI A' (15.12)
S2 r= ;
/ il --V '' iT1-13;"-
Because similarly to this equation r is determined in relation
to the difference between fairly close values, this method cannot give
'precise results. In addition, the theory of the motion of small parti-
cles in a gaseous medium in the transition zone (see paragraph 8) was
;- developed only for particles of spherical shape, and in the case of
other shapes it is unknown, and a coefficient must be introduced which
allows for the different shapes, in the form of the correction factor
,4
1 -+ At/r ?
-
-33 -
r-
? - -
,
4:4.:1;s:?:4-7\ACZ7-2ia-
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I
4.
? -4
? 01.?
?
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'',-- ?
I
???
. . c
D. Determination.of'Apparont Density and the Dynamic Coefficient!,
of the'-Shapes of Particles
When the particle radius is found by the method described atieye
and the particle mass is determined by the strength of the counter-
balancing field, may be determined,, i.e., in the case of particles in
an aggregate state, the apparent density. Using this method Whiteloy-
Gray.[31] discovered the value of particle density in several smokes,
which is shown in Table 2. With this method correct results may be
obtained only for spherical particles. For particles with other shapes
the coefficient x must be introduced in (15.1) which, contrary to the
above methods, gives 'an increased value for the equivalent radius r
and depressed values for the apparent density.
In the case of non-spherical individual particles, the density
of which is known beforehand, their equivalent radius re may be deter-
mined by their charge and by the strength of the counterbalancing field
using (15.2). Furthermore, the value x may be found through the speed
VE in an electric field, and the same value may be determined through
the speed Vs under the influence of gravity. AS already mentioned,
various values may be obtained for x by this method because of the
orientation of particles in an electric field.
Determination Of particle density also is possible using an
original modification of the vertical 'field method, proposed by' Plachek
[24], which used a non-uniform electric field with divergent lines of
force. As the theory indicates, the force F acting on a non-charged
particle in this field Is equalto
FE ==.7K EV grad IP, (15.13)
where V is the volume of the particle, and XE is a coefficient "
which is determined by the shape and dielectric permeability Ek of: '
3 ck-1.3
the particle in the relationship for non-conducting and
8n ck1-2 1-
for conducting Spheres. It is very important in the latter case that
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XE Mayo the same value both for contiguous individual particles and
, for aggregates, if only the foremost particles of which they are com-
posed have electrical contact. If, in the site in which the particles
are located E and grad E are oriented vertically, then the force
,
'acting,?pn the particles is
FE----z2xEvEdEldi:
(15.1.4)
? .)
Ik the gradient is directed upward, then under the conditions
-FE = mg or EdE/dz. mg , (15A5)
2,2cEv 2.7,E
a particle will be counterbalanced in the air. As may be seen from
(15.15), all non-charged spherical particles having uniform density
- and cOntant dielectric, are counterbalanced at the same value' EdE/dz
independent of their size. This conclusion has been 'proven in experi-
ments with oil droplets and with individual mercury droplets. The up-
per face of the condenser was shaped in a form necessary for the pro-
duction of a non-uniform field. In this case EdE/dz is proportional
to the square of the tension on the condenser faces.
In these experiments the following important fact was observed:
comparison of the value EdE/dz corresponding to the counterbalancing
of oil and mercury droplets showed that the coefficien xE' has almost
the identical value for these and other drops. Thus, oil droplets are
?
practically polarized in a constant electric field and act as a conductor
of electricity, apparently because of the ionizing impurities, which they
contain. Because mineral oils belong to the class of very poor elec- ?
trical conductors, the same Probably is true of particles of the majority
of other substances. Because of this, for the sake of simplification- .
,aerosol particles will be considered as behaving like conductors in an
electrical field.
In accordance with (15.15) the value of EdE/dz may be deter-
mined, precisely through observation of individual droplets at variOus
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vero1.161....
?
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ar?
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;
. points of the electric field formed by the condenser. Through
similar observation of aggregate -particles of metallic aerosols the
apparent density Of these partiales was'determined by the method
,
described above. /n the given case the particles were counterbilanCed
at Various values of Haiii/dz 1 and thus had different apparent densities.
By this method it was established that the particles of mercury fogs ob-
tained by mechanical dispersion at law air pressure had normal density,
but the density of particles of fogs obtained by dispersion at high
pressure or by volatilization was 5- to 10-fold below normal (see (4.7)).
The vertical field method also was applied in research on the
kinetica of evaporation of droplets, photo-effect on droplets, Brownian
movement, movement in a temperature gradient field, and many other
aerosol problems.
The motion of aerosol particles in the earth's gravitational
field with the superposition of a horizontal electrical field also
may be used for the determination of and for several other tasks
mentioned above. In this case the motion of the particles describes
an inclined straight line, and is described by the same equations
-(15.2) and (15.3), the only difference being that Vs in this case
indicates vertical and V indicates the horizontal component of the
speed of the particle. Equation (15.5) and all the remaining. deduc-
tions remain in force. Both speed components are measured by photography
under intermittent lighting [100] because this method enables working
with much larger particles (up to r = 101) than can be usedin the
vertical field method, and is an advantage in the precise determination
of . On the other hand, multiple overcharging of particles in thia
experiment is excluded. Because this method has been used only for de-
termining more precisely the size of an elementary charge, the approxi-
mate value of which was considered as known, the number of elementary
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,
?
? I. ???
. . I..
,f0,-ohages"on'particle6 may be determined directly from the experiment
(for'yariant methods see Chapter IV, paragraph 25).
?
[PageS.1073]
'CHAPTER IV. CURVILINEAR MOTION OF AEROSOL PARTICLES'
. 2. General Theory of the Curvilinear Motion of Aerosol Particles.
precipitation of Aerosols in an Applied Horizontal Electric Field.
The theory of the curvilinear motion of aerosol particles is
relatively simple with respect to the proportionality of resistance of
the medium and the speed of particles, i.e., at small numerical
values of Re t Assuming that the non-inertia character of the re-
sistanco of the medium, which was practically observed above with re-
spect to the straight-line movement of particles, also holds true for
.curvilinear trajectory, we arrive at the following equation fpr the
motion of particles in vector form:
dV
inw==--67rtirCV--U)-17F,
where V and U are,the ',vectors of the speed of the particle and
-
of the medium and F ,ie the vector of external foree.
(25.1)
In coordinate form (25.1) is written as
dit x ? 6inir (V x ? I ,) + F x; m?dV
? dc7-- dtv== (25.2)
where Vx is the component of the speed of a particle along the
,5-axis, etc.
r-
As equation (25.2) indicates, the component speed of a particle
along any axis complies with ihe same eqUation that applies to recti-
. . ? - ,
iiiiear motion because motion along the various axes are independent
-
of each other. This circtimstarice ?greatly eases analysis of the curvi-
linear motion' of particles.
?
. The situation is different with a.large numerical value for Re .
In this case,, considering that
?
2T gr ? I (V ?u),
llo =
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.1
???
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?
.? -- ,', ? ? ,>?' ':,: ? . _. .
. . ? , . ,
ilq obtain.thedollowing'Vectorlai oquAtion for the movement ot:ph.rticleP:,:.:, .,
,...,.. . .
( 2gr I (V 7- U)I ) 1 g 2(V x ? ( ? 1 ) I (V ? U) 1+ Fx ''' est c
: ?
......,?: - :' ?
.,
- where i(V - ) I is the length pf the vector V - .
" 1'7
.. g? . I
4
010iitting equation (25.4) along the coordinate axis, wq obtain
'1)* ''' f i i' .? . 1.01. t . *; 2-egr?ItY11,11-41 nr1
,1,) . )---i--12`-- /6; ----,,, -(V ? 07 YU) ?1? ? U),1+ F, , -S25.4)
. ,a; ? A A - :, ? cd
--.--:..-.----
Tile tirst member of the right hald of thii equation depends ii'''
both on the tomponcht relative Speed 'along the x-axis, and on the
, 1
-absolute value of this speed. Thus the speeds along the various axes
.iire il'a. Ihdependent. The system of equation (25.5) is not generally
-solved, and analytic investigation of the curvilinear motion of parti-'
-cies at large values of Re is possible only in certain cases. Be-
cause of this we shall limit ourselves to the investigation of curvi-
linear motion at small values of Re .
The task of studying the curvilinear motion of aerosol particles
may be divided into two groups: motion in media at rest and in moving
media. In the first only particles moving in a horizontal direction
under the action of an external force will be considered, such as an
electrical field.' This is one of the few examples of experimental re-
search on the curvilinear motion of aerosol particles. Taking the
above into'consideration, we obtain the following type of equation for
. .
the settling motion of. particles: ,?.
_
V2.. ,mtg/3== Tg, (25.6)
dV .
zit -(7r = F ? 6TcrirV., , (25.7)",
,
Where x is the horizontal and z is the vertical direction belw
the axis.
If the ,force F is measured sinusoidalIy?with time, ir..Fosincot,
then tie horizontal speed of a particle is expressed by formulas
(19.6) and (19.11), and the horizontal displacement is i,ndicated by
the formula.
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?
?
'
? , ,
, c
hFurthetmord
time t' is equal io
? .?
? .
which giYes
the vertical
z 7
F cns (cot ? qi)
co -f- (42-T3
tg = 'ma V,co/g.
displacement of a particle OUI:ing
z = ?cgt.
(25.10)
"
' Thtis, thg ,trajeciOikoT'a imuticle is expressed
'A? '? ?"1- " ?
' ?FoB cos ?
? -
cos ? ?
, .
-1
*Abbot [170] tested these eon-elusions experimentally by photo-
by.the sinuSofd
(P.I.L11)
gtaphing water droplets of 302to !e10p.radius.falling through a hori-
zontal electric field of 66 hertz. The trajectory of the particles
r ;
described precise sinusoidal curves. Through the use of special in-
sttuments the moments of time corresponding to the value F = 0 which
enabled determinationof the displaCement of oscillation phase 91 .
. .
Thit value p founcCin this way agreed sufficiently with formula (25.9).
'
''rhe small variations (1 or 2o) apparently are explained by de-
viations of the horizontal and vertical components of the motion of
droplets in the formula of Stock.
This method doubtless may be used for the simultaneous,determina-
tion of the size and charge of aerosol particles. However, in this re-
spect it proved to be more advantageous to work with the constant of the
strength, and the variable of the direction of the field, instead of
working with the sinusoidal curve. In this case zigzag lines, composed
",of straight sections, are Obtained. The vertical displacement, of i
-particle pen single period of oscillation is determined by the distance
between zigzags, from which (25.10) PC and thence the particle radius
s'aredetermined. 'rhe horizontal speed of a-particle is equal to Vx = BEq
is thefield tensionj-and q is the charge of the particle). Thus
. ? ;'
? ? ? ?
the'tan_gent.of,the angle of inclination of trajectory sections from the
horizontal ate equal to V /V = gm/Eq and, knowing the particle radius,
.. "
-
? ? ? Z X
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!2,
4.4.4 ?
e, I
where ,D is the coefficient of molecu1dr diffusion:of-particles. Taking -.
. ? e
V As the volume, and S as the ..surface of the chamber side walls, then ,
?.vdn. - ' ' - '(0:17)
I ?
?
Thus, we obtain the following formula for the coefficient 0 In formula,
? SI SD
..-........
(44.15) p.a-
.
It must be remembered that S depends not,oh*:on the intensity sof"c6n-;
-....t..
,. . ? i .
. ? , , . ? 'OF :.1. ? ? -11T .
vection,frlalso'On d:' Actually, t!!ea4zHf,A,I.i.e dkerAlined by v
.. .
the condition that at distance S 'prop tie yall.:tihe cp4ficients of mc-
-
lecular and convective diffusion are equal. The greater the coefficient
of molecular diffusion, the greater is S . Because of this, there can
be no simple proportionality between the speed of precipitation on the
' walls and D , and the speed of precipitation must be proportional to
D?', 0 oe&l.
From the above the following conclusAon may be drawn: the
greater the degree of dispersion of an aerosol,,the greater is the num-
ber of particles precipitating on the side walls and the smaller the
number of particles precipitating on the bottom.
K. Shifrin and his associates (359] investigated (through measure-
ment of the transparency of a smoke) the kinetics of precipitation of a
smoke with r 0.5p. on the walls''of a cylindrical chamber 9m high
and 4m in diameter, and found. that equation (44.15) is Applicable in
this case, and that the concentration decreased two-fold during 2 to 3
hours. Because the starting concentration of the smoke was of the order
104 per cm3 the decrease in concentration due to coagulation of the
aerosol could be ignored. At the indicated degree of dispersion of the
aerosol and Chamber height precipitation on the bottom of the chamber
played no noticeable role in the results. From the curves proposed by
Shifrin it was possible to calculate that tiie ratio o the number I .
of particles precipitating per cm2 of wall surface per second, to the
-
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-
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e ?
- concentration of the latter was ;5AO--Ic 'cm.sec.
?
;
the value S pe 0,5p, was'obtained,.for .ih6 'thickness of the , , , , -
" .t?
a
,, .,... .,,.,-.
t.' In he expenlments of GiiI4s1Cii Langstroth [357], which ,
. ,, ... ., .
..
. Were conducted with alchambei. of 12 ma capacity and NH4Cl smoke with. .
? . ,
1.
pafticle radius in the. interval 0.3 to 2.0 , the, number of_ particles
Pi-ecipitating on the bottom, side walls and top of the chamber was
? - e
? ,
determined directly through-the use of plates of proper shape Which
were installed in the chamber. It was found that precipitation on the
top was negligible, and precipitation on the side' walls during the
- first 100 minutes was approximately one-third less than on the bottom,
after which precipitation on the siddvalls practically ceased. From
the graphs prepared by these researchers it may be roughly s,een that
the side of Tin in these experiments was on the order of 10-4, and .
that the thickness of the wall layer 2s201.1L, i.e., approximately two
orders larger than the results of Shifrin. Because the speed of de-
crease in concentration of an aerosol observed by Shifrin in such a
large chamber and in the absence of coagulation was in clear contradic-
tion to the data of numerous works dedicated to the study of the co-
agulation of aerosols, and because direct data on the precipitation of
aerosols on walls is much more promising than those obtained by cal-
culation of the decrease in concentration, apparently the values for
fin and S obtained by Gillepsie and Langstroth must be given
preference until new data are obtained on this problem.
The charge of particles has .a great influence on the speed of,
precipitation of aerosol particles on walls. In recent experiments
of Gillepsie [360] the total, number of S102 particles with .r = 0.4flo
precipitating during the first few minutes of the life of the aerosol
on the walls and bottom of a chamber If 0.2m3 'volume increased 2- to 3-
fold when the average number of elementary charges on the particles was
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?
,r- 1
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.,.?
increased from 7 to 14 (ti cased the maximum Charge f:rom ap-'
' .?
? ,?
? ,..? ? , .
' .
--- .-; ....
proximately 50 to 100). Proceeding on the assumption that ai the
,.,
speed of precipitation ,of the aerosof on the bottom of the?chamber
1,.?
' ? i
the charges could lordly have any:m en
arked influc*;?the increase of
. h := ? ":
tt,r y: '
. ' Z
I
precipitation on the side walls inthese experiments actually was Much
.i. t..
- isirrg'er.
'I- It is curious that this influence of the charges gradually .
q? ? . .
t'46WaS"ed,
,. O. ..
% 1.1* "
ie.,.-
,cause the charge of the aerosol was bipolar and fairly symmetrical,
,
,
and total4l-ry disappearde -after several score minutes. lje-
-
411151 electrostatic diffusion (see paragraph 24) did not play a marked
role here, it is beyond doubt that the indicated effect was caused
,by induction attraction of the charged particles to the walls.
The very great influence of this effect may be calculated, as-
suming that the particles which reach the wall layer move toward the
-wall under the influence of the force of induction. The speed of this
movement to the border of the wall layer according to formula (41.1)
is equal to q2b/482 , from which it is found that the number of parti-
cles precipitating per cm2, of wall surface per second is I = q2/ ,. Bnoa .
qe2
The number of particles precipitating due to molecular diffusion is found
by formula (44.16). We may establish the relationship
-1-1T?4 ==48k7"
- ?
-10
Taking q = Ev = 4.8 ? 10 ( v is the number of elementary
...????????????
(44.19)
charges), kT = 4 ? 10-14 , 29.? 1074 cM_, we have
= 0,7?.10-av2. (44.20)
For particles with maximal charges v = 100 , and I /I = 7 .
These particles precipitate much more rapidly 'than non-charged particles
because the induction effect is-insignificant for Particles with small
charges. Because.of this, after the particles with large charges have
precipitated .further precipitation proceeds "normally."
It has been established in many studies that under artificial
mixing f an aerosol the gravimetric concentration in a chamber,decreases
- 95 -
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44-
more rapidly than yithout mixing. This is partly explained by in-
creased coagulation caused by the mixing (see paragraph 55), and con-
sequently,-by an increase in the size of particles. However, the
bain cause appears to be without .doubt the inertial' Precipitation of
-)
particles on the chamber walls. This was directly indicated Ipt
! ?
Gil-
lcsie and Langsti?Oth [357] .who measured the ,speed of pregipktatip.n on
the walls of the chamber and 'derived the coagulation constant method
for NH4C1 smoke at various speeds of mixing, which is described in
paragraph 55. -It was found that 1/n increased linearly with the
average speed Of air flow in a chamber of one m3 capacity, and-at a
speed of 50 m.min-1, I/n was approximately 5-fold larger.tfian in the
absence of mixing; artificial mixing had an insignificant effect on
the size of the constant of coagulation. According to an experiment
of Ye. Vigdorchik [358] when air in a chamber 1.2 X 1.2 X 1.2 m was
mixed by a propeller at an average speed of flow of 4 msec'1 the con-
centration of quartz dust decreased 3 to 4-fold faster than in the
absence of mixing. The cause of this phenomenon is a decrease in the
thickness of the laminar wall layer S , caused by the mixing, and in-
ertial precipitation of particles. Actually, assuming that the chamber
used in the experiments of Ye. Vigdorchik was cylindrical, it may be
calculated that at the indicated air speed the centrifugal force at
the wall, mV2/R = m4002/60 , is, almost three times greater than the
force of gravity mg .
Actually, the picture of inertial precipitation with artificial
mixing is much more complicated: inertial .precipitation takes place
In individual places, where vortexes with axes parallel to the walls
formed by the mixing touch the latter,. The inertial force generated
in vortexes with small diameter may be considerably greater than the
calculated figure. The character of the mixing also is very important.
If the mixing is done with wide paddles which almost reach the walls
- 96 -
4
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??
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,
??
d
.44'
of the Chamber (361] the rate of decrCase of:,the-aerosol Concentratipn,
-
? .
4
-(NH4C1.smoke with r Att 0.5r04 increases 3- to 4-fold even at an average-
- - - , ?
airspeed of 50 cm'sec-1 This -type of mixing undoubtedly enables the
,
formation of intense vortexes with axes .parallel and.close.tc? the walls.
Calculation of the speed of inertial precipitation on t)pit walls
under artificial mixing is very difficult, but. the number.oflpar!iiicles
s ? , -:,?? ? ??4 ,.. ? . ??k ?
? ..
p7:ecipitating on one cm2. wall surace per secondjn any case must be
.r.??
proportional to n2 , and the decrease in aerosol concentration: with
time must satisfy the equation
dh ? bn (r) t
dt
where b is a coefficient depending upon the intensity and character
of the mixing. From this it follows that
Sb
n (r) = no (r) oxp ?v 1.
Since vs = g , the total change in aerosol concentration with
time (ignoring diffusion precipitation) is expressed by the equation
n (r) = no (r) oxp[? (5.7b? -I) (r) t], (44.21)
which has the same form as in the absence of inertial precipitation, .
Sb g
but with the substitution of the coefficient g/h for + . In
addition to equation (44.11) we now have
[__(Sb44)
(r) t] n no (r)oxp
0
As already has been mentioned, the change in in n with time
in the case of a polydispersed aerosol may be expressed by an upwards
convex curve. However, if noticeable coagulation of the aerosol takea
place simultaneously with precipitation the increase in particle size
(44.22)
. ?
. caused by the former somewhat compensates for the decrease in average
particle size caused by precipitation, which leads to rectification of
the curve (inn, t). Practically rectilinear curves sometimes 'are
.obtained [361].
The change in gravimetric concentration of a polydispersed
aerosol with time is expressed by the equation
' - 97 -
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ri ?
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?
where m(r) is the mass of a pariTTEE NT' in-Fadius r . In the case
i.e., the gravimetric' concentration
decreases faster than the calculated concentration,
'Actually we may write
d 111 c ic dc IM'cln
c dt dt ' (4424)
, dt
where, m is the average mass of particles contained in the aerosol
= .
and in is,the average mass of precipitating particles. Since irn;?in,
where the difference
Increases with the degree of polydispersion of the aerosol, and also
with the absolute value of dn/dt , i.e., with the speed of artificial
mixing. These conclusions have been verified by experiments [361]
In conclusion, reference is made to a phenomenon caused by
. natural convection, which is of interest in the treatment of rooms
with aerosols. If a closed vessel with holes in opposite vertical
walls is placed in the aerosol chamber many quite large (r"./ 7,0 par-
ticles penetrate these holes because of horizontal convection ("draft")
4248]. If there is only one hole there is no penetration. An analo-
gous situation is found in cracks in wooden walls: an aerosol may
penetrate quite deeply into holes which pass all the way through, but
. cannot penetrate blind holes.
45. Movement of Aerosol Particles in a Turbulent Stream
Theoretical and experimental research on the conduct of aerosols
in turbulent flow naturally is more difficult than in the case of laminar
flow. Regardless of the great successes which have been achieved in
recent time in the study of turbulence, there are very,fsw data dva11-7.
able on the movement of suspended particles in a turbulent stream.- I
? ,
particular,the,very important-problem of the degree of blOing%awayHo
- particles by turbulent pulsations has not been .clarified.-'
0,
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, ,Adcording,tp-contemporary, views turbarent-116W-may;,be
- ,
. .
sented as the result'oi the super'positiOn of a'continuous spectram.7Ot
, , - ,
, .'. , ,
pulsations of varying scales on the5ba6ic (aVerage) speed of flow. The '
_
,....z.?. , . _
:.?: , ?
. '..
A .' ' ' ?
. ,
'Y : . -Z. . . ? ' , ? =. t 'If?
first pulsation which-Arises A through-tthe breaking away of a'vortex irdm
i
. -: '....--Y?.'..,. r* , ? .....
., ' s
?._.
the wall as a scale equal to thediameter of the tube through whic the
:.? .. ,
- ''',,...?
liquid'tlows; the speeas correspondini to this pulsion ,depend' oriAfie
7.'''. ? ??? -
???????
- .
,v,
direction: in particular, the ppeeds of pulpations'in a direction
t
tO the walls are grdater than' in a Petpendicular direction.
0,'
. . .
The.epergy of large-scale pulsations gradually turns into
. ?
6mal4er and smaller scale pulsations', where for scales which are small
,
???.'
in comparison with the diameter of the tube the pulsations are isotropic.
According to the theory of A. Kelmogorov [362] the total energy, of such
pulsations of scale A is proportionate to )013 . The rule remains
valid because the transition of the energy to smaller-scale pulsations
is not accompanied by a notable dissipation of energy (conversion into
heat), i.e., for scales which are large in comparison with certain
critical values of Ao (the inner scale of turbulence). In the range
the decrease in energy in the transition to smaller-scale pulsa-
tions is much faster.
Only the spectrum of pulsations at an immobile point may be
studied experimentally, i.e., the value of function F(11)
indicating
which part of the turbulent energy falls within a pulsation with
velocity $t V sec-1 , recorded by a stationary observer (this is called
the Eller speed of pulsation). In the transition to a system of cd-
cordinates moving with a stream with neutralized velocity, U', we ob-
tain the "scale spectrum" of pulsations, i.e., the cl(A) functions,
indicating the fraction of the turbulent energy which is allotted per
length of pulsation of scale: A . However, the velocity spec-Crum of
pulsations in a moving system is unknown (this speed is- called the La-
- grange speed).
_
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? ".?
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?:?
.... , ?'. ' _?
,. . ,:, - ? , , ..-
, , , : ? -
Usinethd data of Simmons and Saltei-j363], in-which the spectrunt
_. .?
.-. '
. A
of pulsations is measured in a 120-cm abrodynamic tube 'with a grid with
. ..
- 75-mm aperturis, the degree of turbulence, i.e., the ratio u/U , where
????
, ?
U s the total average quadratic speed of puls'apon is -equal to 004
and hardly depends vpon Ref . Thus, turbulent energy constitutes ap-
.,...?1"
?
proximately 0.0009 of all' the energy of the stream. The present author
01
has constructed Table 24 from results obtained at U = 7.5 m-sec-1, in
which uA indicates the speed corresponding to .pulsations of the scale Q.!
"2. A ?
A., cm
37
19 .
7.5
3.7
3.0
2.0
1.5
1.0
0.75
0.50
TABLE 24
SPECTRUM OF TURBULENCE IN AN AERODYNAMIC TUBE
AT
fl(,)
U = 7.5 msec-1
* uA /U-102?
AND Ref = 600,000
-1
LTA, cm?see uA/ye3
'11/uA
2.6
0.74
19.5
5.9
2.0
0,)58
2.3
17.3
6.5
1.1
0.36
1.8
13.5
6.9
0.55
0.19
1.3
9.8
6.3
0.37
0.13
1.1
8.3'
5.8
0.36
0.07
0.8
6.0
4.8
0.33
0.05
0.67
5.0
4.4
0.30
0.019
0.41
3.1
3.1
0.32
0.008
0.26
1.9
1.7
0.40
0.0025
0.14
1.1
0.9
0.45
1/3
AS is seen from the data of Table 24 the ratio uA/A
according to the principle of A. Kolmogorov must remain constant, begins
to decrease sharply at approximately A = 2 C. A similar resUlt is ob-
tained at U = 10.5 m?sec-1. Thus the inner scale of pulsation Ao in
, which
an aerodynamic tube at' Re of the order 106 has a value on the order
-of one cm. Because considerable error is-possible in the experimental
study of turbulence spectra this derivation should be proved with other
experimental data. The following values ofo may be obtained from
the formulas and experimental data of A. Obukhov and 'A. Yaglom (3.64,]':
1.1 cm at U = 12.2 m?sec-1, 0.7 cm at U 24.4 m?sec , and
-r
0.6 cm at U = 30.5 m.sec-1, i.e., the same order of values 'of A
According to Obukhov 065)o = 0,5 cm in atmosphere at a height of
- 100 -
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1.p in, and-aCcokding to Taylor [366]
-
.cm ati. a heighe:of 2 in,
z,
?
and 13 cm.at a height of 30 in. According to Obukhov and YaglOm the
lower internal scale ii(A) is propor.tional to A 2 and according to
ix , ?
.,.. . .
Heilepberg [367] is proportional to AP
.,
,
rapidly with a decrease in the scale.
C ?
) 4
? i.e.,. it decreases very
As already has been mentioned, the distribution of turbulent
eller& in the function of Lagrange speed or pulsation periods cannot
be found directly from experimental data. The,former are necessary
for solution of the problem of the degree of increase in particles by
turbulent pulsations. This problem may be solved very roughly in the
following way. Assuming that pulsations with the scale di 'are developed
In motion in a stream with speed U of vortexial cords;, of diameter A
?
the axes of which are perpendicular to the direction of the stream. The.
average speed of pulsation u may be taken as the speed of circulation
at a distance Am from the axis of the cord. Then, the Lagrange period
of the corresponding pulsation tL is equal to 0.51t) /uA , and the Eller
period tE = 2A/U . The relationship tE/ti, is congruent to u/U ,
i.e., in the first approximation is equal to the degree of turbulence.
The complete (conventionally 99%) blowing away of particles by pulsations,
in conformance with Figure 20, takes place at 'CAL/4,0.02 , i.e.,
"t:i.0.01 according to Table 13, or at r/x.30/4.. for particles with den-
sity 1.
Using the value for t calculated under paragraph 18 for super-
Stock, particles at a typical average speed of pnlsation of 313-cm/sec, or
-T= 0.1 sec at r = 0.1 mm and 't = 6.3 sec at r = 1 mm and density
1; we find that in the first case the degree of"blowing away of particles
is 70%, and 2 % in the second case. Thus particles with size on the
order of 1 mm Ipractically are not involved in the pulsations of the.
medium.
; - 101 -
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. :
TWWOlier, these calculations "are'ry rought-even though they -.....,
. - .
. k
, apparently give the correct order of greatness of piw:iiclps vglich 9re
blown any and which are not blown away by the pulsationst Acctirate
: .
,
_
-.L
solution of the problem of the degree o1 blowing awar of particles. is
, .
possible either through study of the turbulence spectrum with an bi--
_ strumdnt which is moving at the speed of the stream, or through ultra-
.,,i
_ microscopic observation in a stream which contains both very small and
coarse particles.
0(- :
The above problem is very important also in the solution of
another basic task in th.e mechanics of aerosols, the vertical distriL
bution of particles in a horizontal turbulent stream, which was first
Investigated by W. Schmidt [30]. If in the turbulent flow of an
aerosol in a horizontal canal the precipitate is blown away and enters
a state of suspension (this occurs, for example, in the pneumatic
mixing of materials in a dust form), theft a determined stationary dis-
tribution of particles according to height must be established in the
canal. The number of particles passing down through a horizontal
area one cm2 per second under the influence of gravity is equal to
Vsn . The number of particles passing through this area in the op-
posite direction because of turbulent diffusion is equal to Dt dn/dz
where Dt is the coefficient of turbulent diffusion of particles.
'Equating these expressions, we have
dn Vs
= d;
from which we obtaiii
- -7
(45.1)
z
ito (45.2)
ds
3 '
Where- no is the concentration at the bottom of the canal, and z is
the distance from the bottom. Because Dt rapidly varies with z
close to the bottom, but varies elo'.vly at greater distances from the
bottem,t may be tahen as constant in the central region Of the
canal, and (45.2) takes the form
'
- 102 -
-? ?
no '
Vs
Di ?
(45.3)
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1_
-
'
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Mr144.4.4.44.,.
t
t
?;.--:, ? ,,_ ' It is 'emphasized-that'the distribUtfda eX0r4esca by these ?
-..
I
1: 0 l 1 , v ?
1. .....
-..
?:formulas occur only n.a stationary..state,,i..e;; if the particles-
c - i .
,
.
. .
which have precipitated on the bottom -cif the canal are!returned to the
. .
? -, -
????
e
- ?
suspended 'state by the action of the stream.
,
' I
'-ft4-may be readily assumed that a large part-of the aerosol sUcceeds in
precipitating until this distribution As attained.
4
In the opposite case, it
The theory of vertical distribution of particles in suspension
in a turbulent stream whiCh has been described herein was criticized
by several hydrologists [370],.the main objection being the usual as-
sumption that the coefficients of turbulent diffusion of particles and
the medium are equal.
aerosol particles with
It may be seen from the foregoing, that for
14. 30p. :this postulate apparently is justified,
Furthermore, in the general expression for the coefficients of diffu-
sion D 12t , where 1 is the length of the "path," and t is the
time consumed in traversing this path, in the case of turbulent dif-
fusion 2 is the scale and t is the period of pulsation. Because of
this, at incomplete blowing away of particles the relationship between
the coefficients of turbulence of the diffusion of particles and the
medium DI/D
t t
is equal to the relationship between the squares of the
amplitude of pulsation of particles ,and medium, or the "degree of
blowing away? of particles, calculated as described above. Formula
? (45.2) apparently remains applicable in the case of incomplete blowing
away when the coefficient Dt used in the formula has beem derived in
- ?
this manlier..
Mention also must be made of the work of B. Brounshteyn and
P. Todes [371]. Starting with the Bol't8man distribution of concen-
tration in a field with tension n = noexp. (? ri*I1c7) and taking an
analogy from the gas kinetics formula k2k MD212, where v2 is the
averafte square of the speed of pulsation of .a particle and in is its
. .
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,
-
mass and, consequently, n= no exp (-3gz/v2),. these researchers uied the,
formula [372] v==-(3 yeTig-1-2yRup where u is the speed of ptilsatipn
of the medium, for the derivation of a value which is correct
for the movement of a particle in an ideal liquid. In the opinion of
the present author the fermal application of as kinetics formulas to
?
the movement of particls suspended in a turbulent medium is not justi-
t:-
fled. Acckding to Brounshteyn and Todes the vertical distribution of
particles does not depend upon the size of the particles which, however,
contradicts experience. Furthermore, according te this method of cal-
culation a speed of pulsation of the air of the order 1 km.sec-1 would
be required by the condition in which the concentration of the aerosol
would decrease no more than two-fold at a height of 10 cm!
More or less detailed experimental research on the vertical dis-
tribution of particles in a turbulent stream has been done only in
water suspensions. In the experiments of Vanoni [373] Dt has been
calculated for the profile of the speed of flow (see following paragraph).
In the work of Kalinske [374] Dt was determined experimentally for the
dispersion of a water solution jet in a stream. The vertical distribu-
tion of sand particles with radius on the order of several score microns
(up to 701,b) which was found in these experiments more or less satis-
factorily agreed with formula (45.2)..,
The only measurements on aerosols were conduCted by M. Kalinushkin
" [375], who investigated. the distribution of saw dust and other gust par-
ticles in a 25-cm'horizontal round tube at IV= 10 to 17 wsec-1. It is
- noted that in round tubes turbulent flow usually is accompanied by ro-
tation of the gas around the axis of the tubr, with the result that the
? g
vertical distribution is transformed into radial distribution,, in Whiah
the concentration of particles is maximum at the periphery of the tube
and decreases in the direction of the tube's axis. To o'Jtain normal
- 104 -
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?
,Vertical distribution in a round tithe rotation must be eliminated
by mans of a screen. The data of M. Kalinushkin lie, well within a,
,
straight line graph, corresponding to formu1q..(45.3).
The data of 'Sherwood end Woertz [376] also may be inttoduced
here, in which D1 was measured in an air stream in a canal with
?.
right-Angle cross section, 5.3 cm in height and with a high ratio of
.?
Width to height. At Ref = 10,000, excluding the thin layers at the
Upper and lower wall surfaces, Dt was found to be Practically con-
stant, which is expressed by the empirical formula
Dg 0,044 v Be'
(45.4)
S.1%.? .0.... 0.:0
where V is the kinematic viscosity of the gas. For particles with .
r = 5fu and density 1 , at U = 13 m/sec (Ref = 40,000) the value of
Vs/Dt was found to be equal to 0.016, and the ratio-of concentration
at the lower and upper walls in this case was 1.1, i.e., we'have an
almost uniform distribution for all sections of the channel.
We have described the action of a turbulent stream on particles
. suspended in it. The opposite problem, of the action of the dispersed
.phase on its carrier turbulent stream also is of great interest. From
experiments it is known that the critical value of the Ref number in
clay suspensions is larger than in clear water [377, 378]. In the
experiments' of Vanoni [373] with sand suspensions in an open channel
a marked decrease in the coefficient of turbulent viscosity was ob-
served and, consequently, hydraulic resistance in comparison with cleat
water; this as also accompanied by a distinct drop in the degree of '
turbulence of the stream, and in the coefficient of turbulent diffusion.
The cause of these phenolnena is the following: as we hdve seen,
every particle of the dispersion phase partidipating in the pulsations
of the pedium is in continuous notion in respect to the layer of the
? medium adjacent to it due to the attraction of gravity. This is
- 105 -
? ts
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_
. ? ..rg.
accompanied, by the dispersion Of mechanical1energy)(cOnversien into-.
,
, .
heat) which may be removed only from.theenergy of pulsation.-- wThis
?
must result in a decrease in the-deg-ree of tU'rbulence::?rBecause.'the
, -
amount of' energy dissipated by the particles *pct47-, ,
? - '1
tionate to the product of the. weight of a particleiandrit re
4,sspeed of . ;0 .
cipitation, i.e., the mass Of a pt 10 to the degree 5/3
? . .? i
a constant gravimetric concentration e = nm the decrease in the de6.66
?
t
? - ' '
of turbulence muse increase with the height of the particles. The Ob-*
?
-
perya4on introdu? ced under paragraph
29, that the resistance of.a cy-
clone decreases with an increase in the concentration of. dust, explains !
;?
, ?
the decrease in turbulence under the influence of suspended dust.' This.:
already has been proposed by many. researchers [379]. (In an article
published in 1951 [380] the present author described a decrease-in tur-
bulence by the dissipation of energy caused by the relative motion be-
tween the particles and the medium caused by incomplete blowing away
of particles by pulsations. As is apparent from the above this effect
must be very small.)
The theory of this phenomenon was developed by G. Barenblatt
[381]. The size of this effect is determined by the value of the non-
dimensional expression, which in the case of an aerosol has the form
_
K gdc I dz
yg / dzr "
(45.5).
?
Where z is the distance from the bottom and U is the average speed "
of flow.
At K (41 the dispersion phase has no influence on the degree
of turbulence nor on the profile of the speed of flow: only in this
case is formula (45.2) applicable for the vertical distribution of
concentration. At K equal to unity the mean-square speed of,pulsa.
tions u is expressed by the formula u = 110(1. - , where uo Is
the value of u for a pure gaq. In this case formula (45.2) is not ap-
plicable and calculation of the distribution of concentration is complicated.'
-'106 -
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- 46. Precipitation of an Aerosol in?TurbuIcht Plow
In ternina' to the p'roblein of the precipitation of an ,aerOsoi in
. , ?
turbulent flow it is noted that almost all which has been ."Said"ulider
- paragraph 44 on gravitational precipitation in the presence Of 41vec-
tion and artificlal. mixing is valid also in respect to turbuleht
Because the mean-square pulsation speed in a direotion perPendiddlar to
the line of flow is appreximately equal to 0.03 to 0.f
- is iheAdUer.age.ppeed,of fellow [363. 382. 376. 3833, the
cipitation of particles with radius less than .10p. in'
tube is considerably less, even at a value of a of several meters
per second, than the vertical component of the average pulsation speed,
and these particles are more or less evenly distributed throughout all
.-.sections of the tube. The number of particles precipitating per second
per unit length of a tube with radius R is 2RnV 2Rgnv , Where n
. 4 '
trr , where IF
speed of pre-
a-horizontal .
is the concentration of particles in the stream. During the passage
of an aerosol layer one cm thick through a path dx , i.e., during
time dx/U 2Rgnv dx/ii particles precipitate from the layer. Since
the volume of this layer is equal to nR2 , we have.:the
equation
dn /Rpm bilw
=
n
dx nitz1.1 u R
7; ? (46.1)
From this, analogously to (44.21), we derive the formula '
_
2mgm
n= no exp
?
(46.2) .
which indicates the variation of the concentration of particles of a
given size in the function of an aerosol passing through the path x
The speed of precipitation of the aerosol, which is proPortiongte to
-n , decreases exponentially in the.direction of flow..
. ,
-
These considerations also are applicable to the theory of the
. .
mixing of the aerosol. also takes
in a cyclone and, in conformance with formula (29.1), that .the
.??
speed of flow at the external wall of the cyclone is equal to U/2
o.
Cyclone. Assuming that turbulent
place
the radial speed of particles is equal to Uo-t /4R
- 107 - ?
p and the num6er
2
?
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?
h?
. d.f.1 part.i.ales,precip-itation per cm2 of wal/ s.ur.face per second is
,
,
.
,/, nue0.1"
pill
.
,,,. , .1 ? .
., .
.._. ....rili-s? ?
,
a 4 8. the heighi-..f% the 'cyclone, and s
..1',n.d,ac,r!ting 'the height Of the ahliiii53-ath in he.
, g. particlee Willq-precipitate pe.r cm lengt ) of.. tills spiral
Is 'the number of t'u4i?n*s) P
...: sy ,.. 1 . ;...11.1 ,a. a , in .di,d, .0 . .
,The"nnizbe.a? of particie'sr.ivh ch enter the-cylcone per second, ascend the
. ,7 -
, , .
,
.$?, .
1."(iiiti. ta-te-alOn. g its length is inn% , where II
? ,?
t
iitc;etol;ai,. \tie !have
.xy/.. ;42 ..'ai.e he helfai cl width of th 'entrance -opening of the cyclone.
. .
_
...v-,
S., . ??????????? ..."
r I Cin nti2 Ti , per sec.
. ivhere
..,;?:
7 - x
! 4----i-j-4--12: '41?6,?314 '
(46.4) ?
,...s the :ciittereptial .of. the length of the,apiral. Thus, the
. ,, A . .
and leaying, the cy
latter. p: 2nR2s.integrating equation (
enttre length of
, ?
'we .have tor, the .r
(talsAng the height
the entran.Ce"-opening
,
-.i.e., considerably les
..,
, ?
tUrbvent mixing (formu
?
Although the -lack
? precipitation in cyclo
,
olationship between the number of particles entering
clone, the for/nu/a-
t
46.4),
n 1?1?27c1-121 ntioTt 71gite
-
?t he spiral path as? equal to the height of
"1-1 Thu, the efficiency of the cyclone Is
S than that calcu/ated without taking account of
/a (29.24)).
.of accurate experimerltal data on the
efficiency
. Mance with the latter formula-, partie/es With determined, large
r- to re&lit3, ti;an Orm,Va (29..14). In con-
. rall. dimensions must precipitate comple.teiy. A
ventS the- conduct of a qua
, ? _ ?
yoncl doubt- thatcioae
(46.5)
(46.6)
nes aS fufiction of' 4the size of particles p.re-
ntitative test of _formula (46.6.), :it is be-
pitation, even, of large _parties/0-r; 12
42 bserved in eye/ Ones, but _the
ffiCiency of Precipitation, does increase e '6.1111/qt/sly with
tl
.
? "
?
?-?
? ?
aettlal 100% preci-
n particle Size [384]; ; ina-;' .seen ?from formula erea
.
.?
?
Thy.
nce has a siar-, effect oJ) the preitation of illghly
zne
r
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?S.?
? ?'I.-,:?'idiatietrbiad'aerosOlilin si-oondenser'(paragraph 27):
?,... ?
at a transition
---,0i.fr?inlamihar to turbul6nt flow the strength of the cqrrezit%at a given
$ I
, ?. _ ?
tension is consf4eraOly;less than the strength df flow theoretically
i?
Amiculated for',41:aMi:riar- system [181].
-$.
e ,
t ,
4' e?
e' ?
".
Before discussing the diffusion precipitation of ah aerosol
'1 ?
. ?
, stroriz. aturbuleiri-litream several important facts will first
I
r
?
n. the. experimental
? ?
-Le
? .
investigation-.ofthe
speecf- of ?turbulent :flow in tubes and izi
.?
-.'tablished that, udder these donditions a
0
, 5.*
??.[
?
a?lt z.+ b., is obaerved, wherp 'U
? . ''' .
,
?tance z from,the wall. Proceeding on
be cited.
distribution of average
open channel it has been es-
logarithmic 'profile of speeds,
is the average speed at dis-
the ides of the "length of
the path'of mixture" (a. valuewhich plays the same role in the theory
of turbulent viscosity and diffusion as isplayed by the length of the
, , ,
? ?
free path-?of a- molephle. in the theory of molecular viscosity and
fusion)P Prandtl' [385] derived the formula
stf?
x v
(z>k,, 'see below) (46.7)
in. which 7t. is'the constant of Karman, which combines the length of ?
'
'.'ehe,.path of mixture / with the distance from the wall according to
,
:4
the equation. ?
I = Xi;
(46.8)
C. is a constant which eannot be determined, by theoretical means, and
is a Value with a'diniension of speed called the "dynamic speed,"
a
or the "speed of frictioni" and 'is linked through the equation
' with the impulses T:,transmitted per second by the turbulent pulsa-
. ?
. ,
?
tions-through'a parallel Wall of area one cm2. Close to the wall "c
- =1/7-,Fig
(46.9)
Mhs;>be,considered constant,. (independent
equal .0. the fOrte-o riction between
"
-V .1. 2 ? , .? .
4),,r cm, ?
of the latter:. The values Z = 0.4, C = 5.5 were found
"
throukh,eXperiinehts d
. , an formula (46.7) may be written in the form
-
of z ), and consequently
the floWiqg gas and the wall
,
zu* zUs
- UM*. ? 5,5 = 5,75 lg +5,5 (z > 80. (46.10)
-109 -
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..-',..,,,.70 ; ? ?. ',,f'..?;=:?: ,' .; '', :;-, .:;:i.,;' ? ' ,r, a'.:
-,, .
, It is: noted alsothat U,, has-the.same..Order. of-magnitude for
- , .,: ? , ..: ? a ?
. r ? ., a . , v .' A . .
h % ,
the mean ,square speed of turbulent pulsations u,.
. . ,
.?-. , --? ?
.:. .
, ..., ? , ''' _
.. , ...
, ...
u ,
This,formula,'is,noi aPpliOable,at?very-small distances from
' .
.? the wall because at, z --4 0 it lead to U -.4 7cx), sincO at the
5: '::
.,r ..,?
? .0 .?
.?.:* t- ,z ; ' 1.: .. ,v ...
' wall surface the S-Oed. pf flow 16 equal to zerO. Because of this a
,' .4 ? -,, - .- . ,
? .,',
. ? ? ?
. ,.... .
,-1 ',4 c;.1
laminar
1 .7-e 4, ? M .
6
thin layer of thiekges OT, must assumed at 11 surface,
-"?-3
ft.
- - ? ? , .
-..
-OrY1.4,611. VW gas moves in' accordance with the ordinary laws of the flow'
P
..- ?
of a viscous 'liquid, and the,iMpulse is transmitted th'rough molecular
. viscosity. Thus, .t = idlildz 'at z 1;14.,-, from. which it follows
?
?.; I ? ,
',-.that'in a laminar layer-? ,
,
, U/U..=-- viti. -,:-? um :.7, v
`...rs - W2 yes u?z.
---
(g 20 by formula (48.10),
.,...
% and' in the intermediate zone is expressed by a continuous transitional
?curv,e' [385]. Because of this the thickness Of the laminar layer is of
L
???
? ;
,
, ,?
? ??? ":" ? Tv t
the order of magnitude ar, 1.0v / (46.12) ,
Using the idea of "turbulent viscosity,", the equation
= 711 dU I clz (z>4). (46.13)
may be written.
Inserting in the latter the expression frem (46.7) for- dU/dz
and the.eXpression from (46.9) for t , we have 1,==7gU-pc. From
which -the following formula for the "kinematic turbulent viscosity"
is obtained:
vs xU* (46.14).
? Because the ,mechanism of-turbulent transthission of an impulse
and mass are identical, the coefficients of turbulent diffusion D
't
and the viscosity V. must have approximately the same value. Actually,
' according to Prandtr Dt/yt 1.4 to 2.6 , and according to the ex-
. ,
periments of Shorwpod and Woertz [370] pt/vt = 1.6 . Using these values,
rthq,coeffici:ent of turbulent diffusion may be eXpressed by the formula
Df =4,5.:? 2,0 ? Crz x 0,8 z (z >8a (46.15) '
v.
I.M0,010411=0101r-----
I I ..ve-Avvk
-
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!=4,
" J ,
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HOwevbv, this formula:id applicable OnlY at:z > SL , i.e., out-
.
-.
-'side oflaminar flow. There arevo points of.view concerning thb me-
- ?
,
,.chanisp of diffusion inside this
Taylor[387] turbulent pulsations are lacking inside a laminar
'
layer: accordini to Prandt1' and
layer,
Oe transmission of substance
-diffusf0 (the same as.
liada0,0,I,Oxich [3883
?
'flOW, dying 'out only at
occurs.soldly through molecular
the4rinsmiSsion
e
turbulent pulsations
the Wall surface, ,and
of momentum),
and according to
penetrate the laminar
in which Dt ispropor-
tional not to the first, but to the fourth degree of z , or
U'z4/84.8.
(46.16)
%
At the wall surface there is a "diffusion substratum" of thick-
ness SD in which molecular diffusion predominates over turbulent dif-
fusion. HereLD . Thus, according to Prandtli and Taylor the
border of the diffusionlayer, which in appearance coincides with the
border of the laminar layer, is determined by the fact that on the former
the coefficients of turbulent and molecular viscosity are equal, and
according to Landau and Levich the border is determined by the tact that
at this layer the coefficients of turbulent and molecular diffusion co-
incide.s In the cases in which 1J/tip 2e Dt/D , these borders actually do
. coincide and the hypothesis of Prandt1' and Taylor appears applicable.
:Assuming that .Vt./Dt =5, 1 , it is necessary that the Schmidt number
Sc = v/D also be of the order of unity, which actually occurs; for
example; in the diffusion pf gases.. However, in aerosols *Sc>> 1 ,
Vt/3.144. Dt/D . Because in the zone adjacent to the wall the in-
tensity of turbulent pulsations and the length of the mixing path and,
consequently,
,
proached. ,However, li and D are constant values, and from the fore-.
? .
,
,
-going it follows that Dt/D = 1 at a considerably ?:shorter distance-fi-om
. ,
,
,
. _
V and Dt in all cases decierle'as the wall is.?ap-
,t
?
the wall than ,does Vt /V ' ? i.e. .' SD for
r o'd o ?Mnst'be'condiderab
? -
, ?..
less-4hanS ? In addition, under agiVen-syste0
L ? -
given- ST ,the value of S
? ?
cally with D .
. ,
-
cannot be cpnataa u
. -
'
1
-',111
?,
...
2.MiiStrirar
..
yrnpatheti
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,
In particular, according to Landau.and.LeviCh,, frem .tho deter- ,
mination of SD it follows that l).=:Lc...:so'br''6LgL from which.
In calculating the speed of diffusion precipitation. of ah
(46.47)
- aerosol on the walls of A tube changes an the concentration of the
aerosol in the direction of flow are ignored, i.e., it is assumed that
the concentrattion does not depend upon, time, upon the x 'cdoTdinates
tukin in the direction of flow, nor even upon the distance from the
. wall z . Im view of the low speed of diffusion precipitation and the
comparatively high speeds of flow under.a turbulent system this simpli-
ncation is completely justified. In this case the same number of
particles I pass through every parallel wall of one cm2 area per second,
under which condition because of the great size of the coefficient of
turbulent diffusion at points far from the walls the concentration of
the aerosol may be taken as constant (no) throughout, with the ex-
clusion of the wall layer. Indicating the effective coefficient of
diffusion by Du , which includes both the molecular and turbulent
transmission of substance, the equation may be written in the general form
/ DE == const. (46.18)
In calculating I it is necessary to start from a definite hy-
pothesis on the size of DE in the function of the distance from the
wall. It is easiest to make this computation on the basis. of the
Prandtl'-Taylor hypothesis, i.e., assuming that at z 4SL only molecular
diffusion occurs, but at z >S only turbulent diffu-sion occurs. In
the laminar layer
=
dit(z(81),
I LI
dx
from which n.= Iz/D or, since at z = 0 n = 6
' n /z / D (z al).
whence
- 112 -
n -,-- In z + C2.
MU' .
(46.20)
6.15) and,
(46.21)
(46.22)
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? 4+1,
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? ? .
Indicating the distance; from,-,th.
e wall by h ,? *a which n has
,
. the Constant value no we obtain
,
h + C2 06.23), '
? .
eliminating. C2 through (46.22) and, (46.23),
? .1.
n.= no +a-T,/7
(46.24).
_a.
At z S the 4i4)Fessione (?.20) "4. (06.22) must coincide.
? L
-1? ? ?
t? .
1
tia"
Substitutink z for; nthel latter and equating, we obtain
'
. f ,
1== noD
D
' 81, ariz? In -17
1010,4 according to (46.12)
.
or, 'substituting U* with
?noD noD
D8 L 81. ? 8L
81' + OW I 1.1 7
. ? _ ?
(46.25)
(46.26)
Since in aerosols Sc>>.1- , the second member's in the parenthe-
ses must be ignored, and we finally obtain
8i. ? (46.27) .
-''Thus', according to Prandtl'-Taylor, the speed of diffusion pre-
cipitation is prqportional to the first degree of the coefficient of
thexmal diffusion of an aerosol in which, as je apparent from formula
(46.27), the concentration of an aerosol must be considered practically
constant right up to. the border of the laminar layer, which has been
done in (44.14)-and (44.15): An analogous computation based on the
hypothesis 'of Landau and Levich is considerably mbre complicated, and
only the final result is inclUded. here', in the form of the formula
flnob
4
,
(46.28)
...where A 'jisthe numerical coefficient of the order of unity, the. value
of which May not be determined theoretically. Since, according to
'(46.17) Sr; ,is proportiOnal td D'4, I is'proportional to D3/4 .
-This dependent, relationship is,explained'by the fact that when the
!,
coefficient Of-thermal diffusion is decreased,tUrbulent pulsations carry
particles' loser to the wall and-thuS partially compensate for the de-
cipase in D
? '
I.
- Although. the description of ,the prqcess.of diffusion brecipitatiou
- 113 -'
J
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? ?.
? ?11
of an aerosol from a turbuient'stream in accordance with the hypothesis
of Landau and Levich Undoubtedly gives a- better approximation of reality
1-, - - . I
than accol ,
-ding to the Prandtl'-Taylor theory, the mechanist of this
through experiment.
process in the final analsis May be explained only
?
It would be
iulfuric ac
facilitate
? Takin fogs
advantageous, i
t)
id produced by
,; i ? -
n the latter, to use isodispersion fog d Of'
_
the La Mer [6] generator, which would greatly
. ,
I the amount Of preci:pitate 'On the walls,
ii? .$, $ ? ' it
' :t.. ? ' .?.0;10$ v
on
determination o
With liarious-s
coulid beLf0.nd,
?
he distance
???
-,
ized partiples, the deiiendence of I
? #
s, ?, . 1-1?,
from which the law of itariation (;)
... ? ? a,
.o. ? .,':" ,
'from the wall coullsk be established.
. ? :I 1.? ? - ?
i; * 0
Dt. with diAnxes,
It is suggested that formulas .(4.6.27) and (46.28) are suitable
In this respect experimental data [389] obtained
/or: practical use.
in smooth tubes at Re 4L105
where
may be uSed, according. to which
us= 0,161/m? ? - 00
HO. Re% '
? /. . /
Um is the speed of flow at the axis of the
(116.29)
tube, and ii is
the average speed. From this we 'have
0r,==10v/ b" 1.-4.-f. 50v Roy' "I 77..... loo 11 / llo74
(R is, the radius of the tube), and-
Dno .
Dnollq. r
(from Prandt-Taylot)
r
Further, In accordance with (4613), .
and
------- 57 RD'I. / ;J.!
/"21"?11-(41svih
==.
(from Landau-Levich)
r.
No experimental data are,available on the non-evoked sedimenta-
l-
tion of the precipitation of an aeri5Sol on the walls of a tube under
[390] atomized water by means
(46.31)
(46.32)
(46.33)
turbulent flow. Alexander and Coldren
of .a jet through the axis of-a short horizontal tube and measured the
concentration,of fog At various points or the tube, from which they
calculated the Speed of precipitation of the fog on the walls,
in these experiments the drop in concentration from the axis of the
, tube to the periphery obvieusly has nothing in comMon with diffusion
- 114 -
I
However,
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?
e4.
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?
' e?
"
?
pio6cipitation, ind is related-tothe profile Of concentration
4"
in a free streaM.- .
-' Turning to the inertial precipitation of an 'aerosol from a
turbulent stream, it is possible to think that 'because of complete
blowing away of particles by the-turbulent pulsations inertial,pre-
. '
cipitation Must be absent in this .case.. 'However, the calculations in
lw
,
e?
and following Table 2i are related to the central particles in a tur-
P
_ .? en,e4 ? . -4,-
':bulent stream. ,Furthermore, observation 44.a.s.ahown that tbe,p.peed of
ooluasat.idns increases in proportAoa-to proximity to the wall, and begins
to decrease Only at a very small distance from the wall [382, 391, 392].
' On the other hand, the length of the miking path and, consequently, the
diameter of the vortexes, continuously decrease as the wall is approached,
but pulsations perpendicular to the wall observed by ultramicroscopic
methods are of a very small scale, even when the distance from the wall
is on the order of several microns [382].. Thus it is possible that near
the walls .a sufficiently large inertial force acts upon the particles
to enable precipitation.
Unfortunately there are no experimental data on this problem.
A copious precipitate of dust frequently is found on the walls of
vertical tubes, which possibly is caused by inertial precipitation from
? ? .
- -a 'turbulent stream. It Must be remembered that at a very high rate
of flow solid particles which have precipitated may be blown away by-
the stream, and therefore the lack of a precipitate does not necessarily
s indicate the absence of precipitation. The solution 'of this *problem
would require experiments with fogs, or the walls must be smeared with
. a-viscous liquid.
On the other hand, precipitation on walls caused by "constant"'
local vortexes which are caused by floW around various obstacles, etc,
Commonly known. Figure 66 .shows a photograph of a dust precipitate
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. r
a..
????
. !.: .
. iilimed-iinr.i.ng the cburse'-of one year on the wall:of a Ventilated room
. , .
,
in the vicinity of a telehhone wire, causbd by a thin stream of air
..
'
J.
4 . ,Lasuing from a vertical slit located to the left of the wire. The '
.., . . . ..
... :L .-
dark, narrow line of dust is in'front of the wire., and the thicker band
4
.-t1A behind the wire.. In addition,-there'is an Inertial precipitate On
.!
. . .
t, ?
? .-- .
.7
- the frontal 'surface 4A.the wire, wifich was forWed in the abedhce of .
t,
-1..
,:voitexes.ojalie,vertexestformed bylair.flowing'around an ods'tacle are
? 4
e ? . ? ?
71,1
shown diagrahmatidally in-Figure-6/. A very similar picture was ob-
?
derved in the theordtical investigations of N. Zhukovskiy [393] in the
!animation of snow banks near obstacles (fences), however, in this case
the main effect ad de to the sedimentation of arrested snow instead of
..?
?
inertial precipitation from vortexes.
Vdriexial precipitation also is observed under known conditions
by the-flow of freely distributed particles around an obstacle, in
which a precipitate is., fdrmed not only on the windward, or upstream
side, but also on the leeward side of the body. According to the ex-
periments of'Yeomans [248] at a speed of flow of 8 m?sec-1 2.5-fold
more droplets of fog with r = 5.6p, precipitated on the leeward side
of a- glas.s disc 7 cm in diameter placed perpendicularly to the stream,
. than on 'the windward-side. AcCording to Landdahl [236] more particles
precipitated on the leeward Side' of. glass plates 2.5 cm wide at a speed
of 0,5 m-secTi than at a_speed=of 1.5 m.sec-1. In the experiments of
Asset arid Pury [394j droplets of a-isodispersed fog with r = 6.514, did
'not precipitate'on the' leeivard side of glass cylinders of 7.5 cm di-.
.... -.
Z$-
ameter nor on the hair of the uncdvered forearms of persons, at-a speed.
1 , .
of'flow Of 2-.3 m!ge671. PreciPitation on the hairs of the- covered fore-
,
, ,. .., 4 '
arms of pe-rspns.waSci.mtely 4-fold less on the leeward side than
,
,
_ .
on the windward. Apparently definite hydrodynamic conditions are .
7. I
7,
necessary for precipitatien to take place on a leeward surface: on one
-
Of flbw'in the vortexes must be adequate for inertial,.
,
hand, the Speed
4 ke
?
,
4. ;I:
,
:
.} 1.
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,
? -
? :?1 ,
r
rirecipitdtien of particles, and'on-the other hand the vortexes must.
Of
not.be too rapid in the direction
- -
which' the stream 'is
4- ? '
leading away from
the obstacle
?!
1 . ,
,
This possibly is, -'related. to the formatiOil -of a dense dust
: . .,
- :?? a r, - , .
1 around
1-
'P recipitate on the walls of diffusors at the oil.e; becaue this is
, .. ? /
, ? , ,,,.....ti.. ! ).?
, the place.where,the,vorcte4ep'tear ary from the walls.
,. i,-)., :.1.?: , ?:.. . f.?.' ? . ,
47t Distribution of Aerosols in the Atmosphere
The problem of the movement of aerosols in the atmosphere
, (smoke issuing from factory chimneys, camouflage, and insecticide fogs
,
. ,
produced by special generators, etc) has great practical importance
from a technical and hygienic point of view, such as in the problem
of the prevention of pollution of inhabited localities from industrial
aerosols. This problem may be of great importance in connection with
atomic bomb devices, the explosions of which create enormous clouds of
;
radioactive aerosols which extend through the atmosphere, the concentra-
tion of 'which constitutes a danger to human beings at distances of several
hundred miles from the explosion site. Bewever, despite the fairly largec.
number of experimental and theoretical investigations which have been
;,.cOnducted, this problem, still has not been sufficiently studied, mainly
-:because of the extremely great complexity and variability of atmospheric
stre,ams,.'which give rise- to enormous fluctuations in variables which have
not been encompassed by mathematical analyses, Because of this the ex.-
?
?
perimental proving of various theories is extremely difficult.
The movement of aerosols in the atmosphere is composed of the
?
?
., movement of the ,air itself, and the relative, movement of particles and
?
air, which 'for:not very large particles bells down7to their precipita-
,
?
?
ytien under the
precipitation,
influence of gravity. At the outset we shall ignore this
,
i.e., we shall-consider the distribution of highly dis,-
persed-derosols ii the atmosphere. In this case 'the laws of the
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?
? s. ?
.?
s4.4.?
r-4
,,:,: ?" ..-?, .
"
distribution of-aerOsols and of gaseous polubnts,are the same. These
-r?f, ?
? ,?,:have been discussed in detail
. Andyeyev.[396] rind,utton [397
..,,
. . ? ,
.., -
,
. .. %'
,s .
,... W -
,. ??..??? ,?,,, .. . ; ,
attention on,the principal facet Of the problem,
A
. ? ?
in ar'ticles' by SheleykhovS4y. [395],
?3-98]; and?therefore we shall focus
C.
-ttentiOn to the pr&dtical details
,
46 ?
without 'special
The problem to be considered fifst
is.-(he mechanism of turbulent dissipation
1,... ?
.? .
- ?tills' problem As Ot covered'Atery'idully'in
kt,tp.in6O-ricily described,
,?,. IP'... - . ,a,,, .. .
:.!. .,
. ?
The general-'dharacter 9f the dAs3iyation of ah aerosol in the
in the atmosphere, because
e+
theIiterature, and occasionally
t r.i It?e
atmosphere is as followsl issuing from any point a cloud or a con-
,
?
tinuous aerosol stream moves along with the wind and simultaneously
is dissipated by the'actiodof atmospheric turbulence. Molecular dif-
fusion has practically no role in this process with the exception of
, -
a very. thin..air layer' at the surface of.a body in contact with the
aerosol.
Atmospheric turbulence has several specific traits which must
be mentioned briefly-here. In turbulent flow in tubes, channels,
r:iveis, etc, at each point of the stream there is an average speed of
7
/low U which has a practically constant value and direction and which
.May be easily determined experimentally, upon which are superimposed
, ':irregular turbulent pulsations: The changes in U , if they take
place at all- (such as daily or seasonal changes in the speed of flow
of rivers), haVe .a-Teriod several orders greater than the period of
the largest pulsations. Furthermore, U considerably surpasses the
mean-Square speed of pulsations-. Thus, in this case the border'be-
tween the neutral flow and pulsations iS .ireiy sharp.
?
In the'atmosithere the wind velocity, measured by wind vanes, etc,
continuously dhanges'hoth in direction and intensity, in which the size
of the variations monotonouslyincteases with the time during which it
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?
'is observed. Here we,have a full spectrUM of pulsations, beginning
with very small pulsations with periods measured in hundredths of a
second and endingin daily and annual variations in. the speed of the
wind. Because of this no border may be established between he average
and pulsations, the speeds ef which cannot be established in this case,
:P
4
and the average wind velocity measured by 'various meteorological instru-
ments for one-minute periods may be considered -the pulsation speed ih
?
?
s, ?
? Observations extending over a year.
? , ? .5
Considering here the mechanism of diffusion in a turbulent stream,
we shall begin with the simple case of flow in a tube.er channel. Taking
an aerosol stream issuing continuously through a long, narrow slit at
any point 0 (Figure 68) perpendicular to the stream, line 0 - 0 in
Figure 68 indicates the stream of neutral flow, and the curves OA and
OB indicate the contour of an aerosol stream fixed at a given moment,
determined by the condition that the concentration of the aerosol along
these curves is equal, for example, to 10% of the concentration at the
axis of the stream. The mean square of the deviation of particles from
the fixed aixs 0 - 0' , registered by a stationary observer (Eller dif-,
fusion), is expressed by the general eq11,atI9p_
(47A)
xl 2DI t,
(for simplicity, Dt is taken as the usual coefficient of turbulent dif-
fusion; constant for the entire field of flow). Thence, for the dis-
tribution of concentration of an aerosol in flow (curve II) in the case
of a continuously moving source the following equatien, which was intro-
? , -? ?s '
duced?above under (39.19), 'obtained
?
? ? 0 e of x
n = ?.
? 1,(470iUx
and from the foregoing it follows that this formula gives the average
, (47.2)
?
concentration per unit time; calculated according to general laws.
,
? The situation is different in the cape of mutual 'diffusion of
:particles in' a thrbulent stream, i.e.., movement 'of the,pariicles in
?
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?
:?
-respect to an observer moving at the sdille rate as one of the particles.
.? s.
(La&inge diffusion). Because Eller diffusion occurs in all pulsations,
only pulsationsof the scale A participate in Lagrange diffusion,
and A ip of same, or smaller order of greatness than that which
_ exists between particles x . These pulsations continuously increase
.. .
the distance between partiOtes, i.e.., thwlead to expansion of the
..
, b! .. ,. r. I ? /
strealnt44. pulsations with A *x 'cannot, obviously, change.
47i4,41- 4
. the dtisNanIce NetWeen particles, i.e., the width of the stream, but de-
.
M
crease its width and break up the-stream
into individual puffs. Mo-
mentary distributions of the concentration in the stream (curve I) ob-
viously are .determined by Lagrange diffusion. ,By way of illustratiori.
a photograph of smoke issuing from a factory chimney under various de-
grees of atmospheric turbulence [399] is included below (Figure 69).
'The scale of pulsations which are able to change this condition
increases with an increase in the width of the stream, i.e., an in-
crease in x . In Other words, the Lagrange coefficient of diffusion
DL increases continuously in proportion to the movement and widening
of the aerosol cloud, which also is a peculiarity of turbulent diffu-
sion., It is emphasized again, that this circumstance is of importance
, in the distribution of momentary concentrations, .but not in time-
average concentrations. By empirical procedures Richardson [400]
_ found, the follewing relatiOnship ,for DL:
,
(47.3)
- As A. Obukhov [364] indicates, formula (47.3). may be theoretically
4
derived for the range of pulsation scales to which the theory of.Kol=
_ inogorov may be applied.
It is not difficult to see that at x = 0 , Actually DL = 0 ,
becaiise particles distributed in a dense line cannot be easily broken
up by the action of turbulent pulsations. On the other hand, in
?
-several cases, such as a aide' river, turbulent pulsations at two points
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14.
fairly wiaely separated may become independent of each other. In this
I '
t, ?
?
pas() formula (47.3) is inapplicable, and the Lagrange coefficient of
- diffusion is equal to the sum of both (Eller) coefficients of diffusion
Dir,==41)1,--FA,
at these points
(47.4)
, in the same way that this occurs in molecular diffusion (see equation
Generally speaing At,'.3 minutes
?
? s'
t , during the course of
this decrease was hardly noticeable. According to formulas (47.15) and
(47.16) the maximum concentration is equal to
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ec assi le in art-' anitize
opy Approve, or e ease I /12/ :
? -<
1. ?
-in the-Case of ajihear, aad,
ir1C.the case of a pc44t:Sairce.
? ..
_
' .
,?? ? ? ? ?
riMaZ 2011
nmaX ??-? 20 / irCyCzUe
14 II 411 888
so`
(47.17): ?
(4 74 8)
. ? -
. ,
,A.tp:i gradient ciog t?,normal, which odcurs in these experiments
' d .? is equal to 1.75cas has been, mentioned above; i.e., -id-the case of
,
, ? - ?? -?
.. a linear'dodreq? n . -.is proportiOnal to ..x-Q.875' , and in the case of
max
_
'.'
, . a 'pointtoui"?ce is proportional to -x-1'7'q It was actu
. -ally found that '
, . .
first case, and to -x-1'76 Int
'n
max yirbportional to
? , ,
_-the decond case.
-0 9-
-in the
? Furthermore, from formula (47.15),'assuming the dependent re-
'
- lationship of C on 11 , it follows that nmax C-1U-1 U-0875
z-
' Thus at wind ratios of, 1:2:3 the corresponding value of n must
max
.'
'be taken at 1:2-?'875:3-.0.875 = 1:1/1.84:1/2.61. The actual relation-
,
_
ship.was found to. be 1:1/1.64:1/2.77 .
.,
. . - .,.
, :.? , r
t? ,e
The, absolute value Of nmax at x =.100 m, U = 5 me5ec-1,
-,.
43 : = 1 vsec1- , was found tO'be 2.0 mg'm3 in the case of d point
source (C = 0.21, Cz = 0.12), and in which nmax = 1..6 mg'm3 must
Y
_- 43. - -1 -1
be used. In thecase of a linear source,.at = 1 g.sec 'm the
theoretical value is n = 33 mrm-3 , and experimental value 31 mg'm-3.
, max . ,
,.
Assuming that at the "border" of h clolid the concentration is
?
-equal to 0.1 of the maximum concentration, taking H = 0 in formula (47.15),
we findthat the "height" of the ?Cloud zo is.duing from a linear source
determined by tshe equation
22
0,1.-= exp (? ---t- ,
? CL'2) ,
%.:. .
i.e:, at x = 100 m , zo = 10 m . The value'found expekmentally was
(47.19)
. 10 in.
9 '
Similarly, the theoretical value of the "width" of a cloud
. issuing from a -point source at the earth surface -was found to be 34 in'
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?
?
?
. ":t a ? , , - ? . - . - ?
"at tv'distance of 100 m, and'the'eXperimOtal,valne'obtained'was-35
. ,
?
4 ? t
?
5
,
According to the experiments of Ye. Teverovsk4y [407] formula
a I 4?
? 4; ' 1:i
r
, "147117Ywith coefficient' s if 2 ,is applicable over a fairly wide in-.
(;?
.. , .
, .
terval of t4e value of the teniperature gradientt Ye. Teverovskiy found
- an average Value of 0.027 for Cz in the case of areas with even,
'
"
abundant vegetation, and 0.086 for broken terrain with tall vegetation.
At this value of Cz was found to be practically independent of the
wind velocity. From this it is apParent that the data of Table 25 re-
,
r, ?
. ?
fer only to areas in which the experiments of Sutton were carried out.
A more satisfactory theoryOf the dissipation of a cloud issuing
-
fiom'a linear source at the earth surface'at a normal gradientis given
"'by Calder [408]. If 'formula (47.6) is taken for the wind P;',11.1e and
,?
? ,the coefficients of turbulent diffusion and viscosity in a vertical di-
'
'reciion are taken to be equal, then the expression
D12..xU*(z--d).'
*. is.4)btained'fOr Dtz according to formulas (46.11) through (46.16).
_ ?
? ?
. Since d usually has a value on the order of several centimeters
-it may be taken that d p . Table 26 which has been constructed on the
basis of the experiments of Deacon [409], is introduced by way of 11-
?
lustrat ion. In Table 26 h' indicates the height of the grass upon
-,
which ,the measurements were taken; U1 and U2 indicate wind velocity
at a height of 1 m and 2 m, for which the parameters d zo ,and
U*
were computed: To = rgU*2 indicates the force of friction of the wind.
per cm2 of earth surface (see (46.7) and ff). It is noted that direct .
'measurement of r.0 gave results close to these calculated by formula
(47.6) (410], Which indicates that this is a promising method of calcu-
lation.
From Table 26 the increase in turbulent diffusion with the force
of the wind and the degree of cparsenesS of the earth surface is clearly
. apparent.
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? WIND. P
????
?
,
TABLE 26 -
h. cm
I
CFI. cm.Se(rs
(1,1(1,
--
.?
d. cm
' .-
to' cm
.
5.?7,01?93e-t
-.Allem-.
. .
_
00 - 443e!"
Di :. -tho
at z- cm
?
?70
t 110
1,5
3,0
4,8 i
l
100
200
300
450
100--800
100-800
. 200
450
%
1,45 ,
1,35
1,32
1,211
1,112
1,140
1,191
i;170
15
16
21
32
0
0
0
0
15,9
,-8,8
5,6
3,0
0,20
0,71
? 2,65
1,74
23,9
35,5f
45,4
57,8
. 6,4-41
8--65
22,0
44,5.
0,68
1151
2,47
4,00 ?
0,05--3,2
0,08--5,0
0,58
2,38
.910 ,
1190
1430
1570
260-2190
320-2600
880
1780
Dtz
Taking U = const in the integration of equation (47.5), then
proportional to z , the following' solution is easily obtained:
n == IvUs (47.21)1
xu.x ?x1T-laTi3P
whence
limn=
CY,
xirs'
(47.22)1
Formulas (47.21) and (47.22) were tested [408] under conditions
. closely approximating those of the experiments of Sutton. The coef-
ficient zo was equal to 3 cm, wind velocity at a height of 2 in was
5 m.sec-1, d = 0 , and U* = 50 cm.sec-1. At = 1 gm-1 the
theoretical "altitude" of the cloud at a distance of 100 in was 16.4 in,
and the experimental value was 10.0 in. The maximum concentration was
(z = 0), theoretical 44 mg'm-3 , and experimental concentration was
36 mg'm-3 . Furthermore, it was found that n was proportional to
max
-1
and x-1 , in conformance with formula (47.22).
The great advantage of formula (47.21) over the formula of Sutton
consists of the fact that ip has-been derived fairly strictly: the only
?
inaccuracy in the, derivation of the former is the postulate that U is
constant in the integration of equation (47.5). In addition, formula
(47.21) contains only experimentally determined values, and not coef-
ficients, the izalUes of which may be selected by comparison with experi-
,
mental data.
In another work Calder [404] .approxilliated the logarithmic profile
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?
'
?.?????- ?".
. -
?
??v
: - -
.. .- ..
, .
-_-
_ with- the aid of a-graduated'function (7.8), from which4 he ,fOuad
-.
, . .. ,. , 4, , ' * tz
in by the method indicnted and solved equation (17.5),
- -
,-.
*till-not.., postulating 1.1 = const , i.e.., completely strict derivation.
i
Tho formula obtained in this way is not included herein because of its
-complexity, but it gives a better comParison with experimental data than
formula (47.21).
Deacon [409] gives an analogous derivation for non-adiabatic
gradients, starting from the profile
1
i rfill.11.7---13)[(10--1.1' ? (47.23)'
,
1
- in which p > 1 in the case of a .superadiabatic gradient, and P 1
,
in the case of inversion. Both cases give a satisfactory conformance
-,.,:With experimental :results. For the profile U = U1(z/z1)4' Frost [.411]
..,
, .
.,
derived a formula for n(x, z) . It is noted that the calculation of
,
- Dtz for the wind profile is possible only at a height at which the value
a?
of T may be taken as constant and equal to the force of friction of
the wind at the-earth surface To , i.e., up to the order of several tens
of meters. Thus, the formulas for concentration in a cloud, derived for
a normal gradient, are true only at a distance on the order of several
= , hundred meters.
;
In view of, the above mentioned peculiarities of horizontal dif-
lnsion in the atmosphere the case of a point source at the surface of-
fors, several fairly large,dffficulties-. Through semiempirical methods
Calder.[400 derived the following formula fora normal gradient:
n=s-cgriclexp [?xUU?x(f +.z)1'
whence'
(47.24)
;, mu
0725)
where is the output of the source, and (5 is the ratio of the mean
Squared speed of horizontal and verticnl pulsations, the value Of which-
may be determined easily by experiment. A test of this formula under
the above described meteorological conditions (in connection with
131 -
,
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?
.N...,:e tir , ?A ' -... ? ' " . ':'4, , ? 1, , , 9 k 1 -,.' ? c., ,..- ? - ., ,.,:s, f.,......!:. -.
: r ? .4; . ? %.,,-,....?..? :???????,,,T i?
?:%-11." ?-?17'": s'f di4111. (41 0211 ) ' a-e-.- 4gave,?the.rfolloWing ..results. `i' /lie tiltittide ? ;,... \i?
.1 *t. ? r ? r.-?
, 7. i. - ' ? .- :........ - .. . 7,..z, . .... . , .. ., f i 2. " : - .. ?
? of the "cloud 'At, x ? 100, rol theoretical 10.4 16.4 m; .end experimental 10:0-m ? , f
8 ! ? ... ? ??? . 8
-? 1. ' . ?? ? . ?
..4 , ?.. ?1) I . -, .! , ?.. v , . - ' A.4 .'.- ' ' ',?-",?;' "
-??(i.e., the same as .f or a linear'. sonrce);., the width'of the cloud:,
,.-. ! .
.. ,.4.,,, " .:-
? ..,' : -,
? 1 ? ,
. .`? ? theoreticaV 41z., i'n',- experimeint_a-1:.35', rd;' ' n - ??., at 0, a1 .g?seer.'..i.7 thee- ?
? N, ? ? ,... roax , ...,
4 -..
... ? ; -,. ...3 . , , ??
. ..are ? . > t --
? 2retidal.' 2.5 'rig's:If:tr., and experimental 2 .0 ? mg?m .,. , The law of clearease
t ? ;,1,
.,- ? 1, . ? e .
_r
- of,'. rtratt3i with x: ;?._ theoretical n?,? ptoportional to x ;?, and ex-
.. ? , fr. , - . I , , ? -? . 4 s,' ' ' . ' 44"'":
? "... T.' ,' ,r ..h ,?-
, - 4 ? , ? / - ,A e's 1, ' . ? 4 r, *. r
' Ilbi. ' 04/ . ? ? , f ??? , 1 . ? ? ....? 4..r. ?::
? l
* . ? . ? I ? .
8 . .... . , per-imental x ... "f DOpendehce o n:.: on ? 11-:/t theoretical-and ex--.
; i? -? ? , i
.. , .. ? , .
' '1 4: ,, t % ' ' ' C ' 41i '; ::t ? ? ... %It., 't 4.* - -1 -1!'i. 2.1.."7"
Peisx.iiiental. n pOportiotraeto U -. it lp neees,sary. again to note
, maA , 0. - - u . _ ; . ? ; ...
, ,
- -*, hi., t? ,..4. 1.,. ? :' ?
"!.th'a*:;yjAe formula f9r ,a 'ppipt soyrek, iptuie( PeAlcMdcd 10 the natOre of a
.
C ? ' r1:: 1';?. 1 , , ,). , -.1,. : , d ,
stibstantial factor of time, 'atirin* which the concentration iesaitremen-ts
. ? ; ' ?
? i.
?????
are 'made at
? ?
were 3 to 4
?
_
'carious points, which' in the experiments of Sutton. and Calder
minutes. The ,fact that the.formulas Of ,both authors give':,
,
more or less similar. reSultsis 'explained mainly Vthe "decay" of the
. , .
4 ; ,
. .
, .
*,
Sutton coefficients CY and Cz as a,iesult,of measurement. .
A reeentlyI!nblished workof Crozier and Seely [5731 describes
a study of the distribution of concentration in. an aerosol issuing, from
a point source on the earth surface, along the horizontal- axis, pre-
pendicular to the direction of the wind. The test took place in an air-.
plane flying through the aerosol stream at various altitudes. Thus a
. _
practically momentary: deterMination of the dIstributionof concentration
?
4
was attained. It was shOWn'tliat even as.far as several tens of kilometers
Ira) the source the concentration of-the aerosol was approximately .pro-
... ,
, portional to exp(By) , where, y Is the distance from the momentary -
-,? ;
axis of the stream, p 40 a constarit, and n z 1,7 , which is fairly
'
close to the Sutton. Index- Of- 2'.0-4 (see (47.13)).
? a 4
?
t '
,
In the opinion of the :present author, further ,progress in this
e ,
-
field will require;., ?
-. . , .
. .
1. per febtion by thgoretical and' experimental means,- of present
' . .
known, on the magnitude of vertAcaYEileriart diffusion as a function of
?t
??-?
132
s?-?
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1/
Cr.
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,)
-altitude
2. Conduct of similar
'-.fusion in the function of time A qb.Seryation.
A
..Thq .
-ARVal:e.nV.y.ot,king has been Opp.,along,.ti1s ii
?
-dissipatiow.flem A
. , ., I J?t 4 ?
( , %
4 " : ? ? . ? :4 ,i ? 10' ' '
r p
.;,
insertion of experimentally*terisided functions, Igq,:- ' ' - ?
..s. . ? "
v... - f.,? ? s-o ')y. q.,:Y.-- ' , , ,.
: .. . -
? i . 4 . .
and ,Dty(1, t) injtqbatie47P):or.W91,appr-Opiliat,e equation,'
7
. . K-, '..t- i ? *-, ' '. . -
solArce,and precise'solutiOn.of these'
?eepilar.tionwith the:aid , '1- * ,
.. . , , . ?
'
?. . . ,
of electronic computing machine for gitriesi-of characteristic conditions ?:
ag
? ..
- of weather and locality.
The dissipation Of smokes, gases, and aerosols discharged,info .the'-',
.r ? , ? ?
- ? . ., , .: . ., ,
. ,, .. . .
atmopphere%by factory smoke sticks is of special interest for:..c.ommneklhy- .
,
, - , ,?.: 1 , q , ... g
n.-
g#nei In. this Case the concentration of polluents in the air at the
.i ?? , .? - ..
earth surface is important; the following expression iCotita'iRpd through,
,
formulas (47.15) and (47.16) for these concentrations: .1- . .
?
..? WY , HS' ,
.., .
? r=.0 exp.(
il-- `--
.(0.26YC2T.) - , .
? . . ? ,
.. f
t '
2- . .
, ? .. 1 . .
. - , . I, ? . -,, i
for a linear source with altitude B , dnd .. ., .e
20' ' (Ir).
-...--..?.- "....---.-.?..27) -
.
? .
(47: '' ?'
' .. '`? '
.. ?
for 'a point source.
In this case the values
Bosanquet and Pearson [4131 obtained other expreasionsAfor; Con-
centration at the earth surface. In these derivations theYtookas - '
v .Zr f
their starting point the following formula for a linearT...96-urce-,at:the
?
earth surface:
?
where b.. is an
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. ? .61 .614.,i 64
? ? ? 3-0,
??? ?
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?
.4
?
;.? ? ? -
'thig'fiiiih1W,Is'deilvedmith the thatpostUlatiow, ,D . is pro-,
1
,? tZ??
? .7 i??? ? 4 1 ?
e
A - ? "
?
portional tb, ",and that U =!.'const ,, afiifogbusly to formula 47.2l),
. ' ''. . ? . , ? ? ? .1. ' : ? _
0 ,
and is -conveheOhtb thecformerlf.it is assumed that ;15:14!)ii
.-' i - '' -:- ? , . 14..? ';?3,i1 ,
?? .
1 ?-,
, , .? _ ? .
,
, FhAllermorej'Bosakhetandrear*oniproposea-theory of reciprobiiY"
.,
' ?? .f. - .
, ' , ,....? . , ., r . "., r 0 1
,
? 4 --aVeart.h.T.SU,rface"co*entration of a 'cloud issuing fiom,a linear sotirce
.3
?1
of Altithde If :11.p... brii -propOAional to z , .is equal .to concentration
.
. -
''J -)
Waltitude,. if,. of a dloyd.issuing from a linear source at the earth
4 , :
_
surface'. 4 FrOm:this 'formula.; ' :f- -.. ? .. ,, *
w =Wily,
.. . . 4 no...0= bizrze . ..
1. ? 7
, ? . . ? t ) I. ?;*
is obtained:.
. (47.29)
?
From i,poiht source these authors give the following' semiempirical'?
.?
, ,
3 0y.
-
formula: ,.. / )1 - Y2 1
,
..... ? Fir."? rgrbybiliti "P ...- ci ? 2bsaixsi ' (4.1.7
?
?
,
..... ...... ....._
,
- ,
,
where b ' is,a,coefficient analogous' to bz , and characterizes the
Y .
.6 4
?
horizontal diffusion.
1 ?
-
In the-experimental determinatron of the degree of pollhtion of
. ,
the atmosphere resulting from the die.charge from facibry chimneys,. there
? ,,- <
4.
aie very great fluctuations in the results, neutralized data are very
:inaccurate, and thereforeit is fairly difficult to judge which formulas
(Sutton, or Bosanquet,and Pearson)".conferwrthe best to reality. 4s a
result of numerous experiments, several authors [414, 415] arrive at
`'the conclusion-that'with'pfoper selection of coefficients these, formulas',
- ?
give approxiMateiy,the same degree of confordity with reality. '
,
, In the-iappkicatioh,of,these formulas it must be taken into account
-1 ..
,
also., that gasesissue from a g,iiloke stack - With a high Vertical speed
;and their temperature is',-muchhigher than'that,of the sUrroundingmedium.-
..
1 , . 1 : ' ? '1 ? , .,
,-
"The calculation of the risink of a smoke stream above the Smoke stack
, , ., ., . , , - ? ,
? , ,
. ?
because of these tactors is giyen.i.n,the monograph cited earlier in.
. ,.-
:f this section, anci,will not be included here. ?,.
....
'?????. ' ?
, t
e
? :
? , . - 134 -
s
? ,?
7 ?-?
? .?
? f'??. ?
3.?
3.4
-' ? ,,*(? ????
,
A, ? .3".,-? 4, 4
-
_
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Xmaz H / 2bz,
nn;ix =`4b0 I V-27thyll'H1e2. ? (41.34)
' because '2/s IS close to , we obtain the following. cOhclu-
... , ? , ?
siOn from both formulas, which has practical tmpOrtance: the distance
? ,
from the chimney to the point Of maximum pollution of the air is pro-
.,.,
portional tO.the:height,o; the chimney, and the,maximud. concentration
;.., , ..- : ?
. - ? , , , ., ; -,
of pollution. 1s inversely proportional to the, square of the height of
? ? ,
the chimney and the wind velocity. Por!further details the reader is
ts ? -
referred tO the monOgraphs of P. Andreyev [396] and G. Sheleykhovskiy? [395].
In conclusion, mention must be made of the external appearance of
;an aerosol stream'issUing from a smoke -stack [416]. ACVery bmallyalues
of Dtz which occurat'inversion,xayery narrow vertical stream' Is ob-
tamed whibh'widens very -gradually. The stream is sharply definedrat.a:
?
very great distanceliom,the dhimney and iS bounded by two straight lines
,
intersecting in an angle of only a few degrees. However, In a horizontal
.,
- , 4 '
? ? ' ' '
plane the stream is shaxply bent, which clearly indicates
? r f . ? -?- .
?
'between 'vertiCal and' hOtizontal, diffusiOn. ,
., v. ? ., ..
? .-' - ? -
? r,
., . .
'
stream_ is much2greater. Upon? daefuld?servation-it may be detedfed
that in vortexeswith horizontal axes rotation is'Such that they
. ? ? .
, .
, , .-?
almost touch the underlying layer of air. Actually; in this case tuve
- -,
?
buiende is caused mainly b'ilOcal:cohvedtion'currents which. ? arise
? 135.: - ?'
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4 . . .
any? source is at first determined by. the dAns.ity and siZe. of the clouds
, ,
' Moving as a whole .with the gas in which they are contained' (see para-
? . ?.
? . , : t.
-
,-.7.
graph_ 13). In particular, aerosols of -thermal origin Iiiive amuch higher'
'
- ? ..
. 1: ?t ? ? . ? :
, . . ? ? e ' + ? '
. ? , .., ; ? . ? 4
..- _temperature than the surrounding air, and because of this they hail() a
.- .
' ? ....:
?
, ? ?-.?
.? , ' -
contiiderable initial vertical yelocity. -. For ?aerosols isdiline?from verti-
cal tubes, the speed acquired by the aerosol in the tube ie added to
.,. , ? .
,-- :
?.
the latter At the time when the teinp'erlittire of the cloud levels out
. ?
with the temperature of the, surrounding izue to turbulent niixing the
. .
( ? ?,?? ? ?
density of the clOnd, and of the. air are practically equal and this type
? --, 4 _
of mo:ciement cea-ses .Here. the influenee of gravity" appears in, the pre-
-..
. , . ? _
dipitation of particles with relation to the 'medium. As mentioned above, ,
, ?:. ?
, . - ,, . . '?;-
for isodispetelen ? clouds a uniform precipitation of the ' cloud of speed ? .
.? ,...
. .
Vd" -
Is superimposed' on , its. dissipation, the ?former -corresponding to the. -
. . ,
. - ? , ? ,
? ' ' ''
?,. ? . , . ,_ , , .-. .
4 ,
. .
speed ,pf. precipitations ? of partiCles in stable. air. , A 'polydis,persiOn
? .
-
5 .? . 1,
. ? ,
. - ', , ? 1,
. ? , ), ,.
cloud may be _considoed as ,consisting of several isodispersion sclouds .,
_ . ?? 05-
. ,:, ' ??L' 1.
precipitating at variou,.s 'speeds . :":-.,..-f?... ? , -,-?. :. . ,?,
. .?
''.. 4 .. ? : , .1 / ' : '..
/ ... . t... , 2. ... f ? . .? A ? .1 4 J ..? .? 1
, ,, ..., ,.... 41'?:., c'" . ?-.4:1, ( . - ' - . : ... , . ,
? e
? infOrtunateiy;:.-oe calculation of etfie speed ''Of precipitation '
?
' ? ? : 1, , ?
- ? ,
,
? of aerbso. partidleS,;:(number of- 'particles ? pre4pitation per. second-z-', ?
k: _ 2 ,,1:4,?,?. .1?:: . - ?
, ? . ? .,
.. .
. '
per cm ): on thearth'syrface from -a.,,-n\pyint ?"cl'ohci?:;pr..aeroso.1 stream, "
... ?. ,? ..... -; : ?
-. ?and, which has.: gi,qa.wt - practical ,signif.iCance, is ' very difficult.... In tlie-
.? ,.. el..., i. ? ,
. .. , ? . ,. ._
? ??_? ?
_ ? .. .., .-, :.,., _. , ?
v? ? ' . ., -
? ., 3 . , .
.1 ' , . A ? , .. ce ? .. .''' . v ? .. .. 1",...'.. 4
' '''. .1 ' C.' . . V . ; V . . : ? . .4 s "t - I '
N-. - No? ".,' 's r
1
?-; 1:36., --? ? '-
',. , i..
. ? i .1 .. '..,?!,-(..:. ',,it;?.".:..?; , ,,'
... ' ., . ?)4.'? *- : . : . ?,- ,?:?
? - . - ' '' '
? ? ., . k ?., .... A, ,.. ..., ?, ,... 4
, ..,.....,....:. ,. . i z t ,..N,, ,v, ..t., ,- , ,
, n 4.. . ?
I k ? -.: ? ,...-t- . c ? 1,''' / k ? , -.f..4*--
11 I. ' .-,' ? % 1::"-",..-,' -"'
4- 5'5- '''554.. 5 ,' j''''r. Z45f4z.^.- . t 1.-4...r . 'at% ? ' . '1 2,,.. ,t_t ? .?'.1.1.,-:.' - -'27?&.:111.1-..., :: 3 ',141. - 't-? '..L -?. ?A_.: t::;-
;
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Declassified in Part - Sanitized Copy Approved for Release 2012/12/12 : CIA-RDP81-01043R001400130002-6
?tream ,(for eicanipiet?'?in- the 'cases ment,,totied following".tabip- 26 at
4 ? ,-
VS. Pi. 0.41U ^.-... 40 ?cm?s13P-, ,f 1 ; e?:;??a:t ? r ?;,., ' 0.I,, min) this aimplifi:cation:, ia?a.-
. ,. *? . " t " , ?
..; N