Analysis of aerosol coagulation in a sound field

by

:Earl Gage

Cole

SUBMITTED IN PARTIAL FULFI

L

LMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

BACHELOR

OF SCIENCE

in the

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

(1951)

S

i

gnature

of

Au

th

o

r

• • • • • "' .... i • • •

Signature redacted

-.-• -• • • -. -• • • • • • • • • • • • • • • •• • • • •

Depar

t

m

e

nt

of

Physics

,

May 18, 1951

Signature redacted

Certified

by -

-

.

-

-_______..,

..

' - -

~--

----,---

--•

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

•

•

Professor Josephs. Newell

Cambridge, Massachusetts

May 18, 1951

secretary of the Faculty

Massachusetts Institute of Technology

Cambridge, Massachusetts

Dear Sir:

This thesis, entitled "Analysis of Aerosol

Co-agulation in a Sound Field", 1s respectfully submitted in

partial fulfillment of the requirements for the Bachelor

of science Degree.

Yours very truly,

Signature redacted

F.arl

G.

Cole

EGC :kr

A statistical analysis was made of the coagula-tion phenomenon of an aerosol in a sound field. The effect of changing important variables has been determined.

Quanta-tive calculations have been made and the results give trends that agree with the physical facts; however, the calculations have not been extended to a point where satisfactory com-parison with experimental data can be made.

tude and to extend due credit to Dr. Francois L. Giraud, who holds a Richard C. DuPont Post-Doctoral Memorial

Fellow-ship at the Gas Turbine Laboratory, for his assistance in

the development of the statistical analysis of this thesis.

His earlier work in this field established a general

founda-tion for constructing the analysis. To Professor E. P.

New-man, thesis supervisor, the author is grateful for his many

helpful suggestions and the backing that he provided.

The efforts of Mrs. Antonia Walker and Mrs. Eileen Gillis in making the computations is greatly appreciated.

Finally, the author wishes to thank Mrs. Kitty

Rathbun for typing this thesis.

Letter of Transmittal Abstract . * . . . .* Acknowledgements . . . List of Figures . . . List of Symbols ..*. Introduction . . . . Statistical Analysis Cbmputations. . Discussion of Results 10.

Bibliography

0 . 11. Figures . . . 0 12. Appendices . . . . . . . . . . . . .0 . . . . . . . . . . . . . . . 0 . . . 0 . . . . 0 0 . 0 . . . .* . . . . .0

ii

iv vi vii 1 4 1. 2. 3. 4. 5. 6. 7. 8. 9.

1. Variables Effecting Collision Efficiency

2. Derivation of Laws of Motion . .* . . . .

V 17 .. .. . . .. . . .21 .. . . . . 24 . ... . . . . . 25-32 33 37

Fig. Title Page

1 Initial Distribution Function . . . . . 25

2 Cyclone Efficiency Curve. . . .

.26

3

0*vs

x 0 0 40 e 0

.. ... ...

27

4 Volume of Particle per Volume Interval dx* vs x

Case

2

. .. . . . 28

5 Volume of Particle per Volume Interval dx* vs x*

Case

1

. . . 29

6 Rate of Change of Distribution Function . . . 30

7 Collection Efficiency vs Time . . . . . 31

8 Number of Particles Per Unit Volume vs Time . . . 32

A B C co d E f g (x, t) H(xIx₂₎

J

K

1

N

n(t)

Distribution amplitude Constant (Dimensionless)

Dimensionless Parame ter

Velocity of Sound Particle diameter Cyclone efficiency Frequency Distribution function Collision rate Sound intensity

Parameter (Define in text Eq.21)

Mean distance between particles

Collection efficiency

Number of particles/unit volume

- 7 6 M-w

em/sec

cm = 10+4 microns sec 1l cm-6 secm watts/cm2 cm3/sec cm --- a ---w Collision efficiency Radius of particle Particle amplitude

Mean particle amplitude Time Particle velocity Gas velocity

vii

Q

r

S

t

u p

'119

cm cm cm

sec

cm/sec

cm/sec

W Grain loading

x Volume of particle

y Volume of particle (variable of

integration)

Greek Letters

,P Density of paw ticle

J1

Density of gas

Viscocity of Gas

C(

Dimensionless velocity parameter

gm/cm3

cm

3 CM3

cm3

Superscripts

Dimensionless quantities

Change in variable

Subscripts g pertains to gas p pertains to particle 0 reference variable

1 denotes a particular volume

2 denotes a particular volume

viii

gm/cm3

gm/cm3

The object of this thesis is to present an analysis

of the phenomenon of aerosol coagulation in a sound field. An

attempt has been made to develop a theory that will predict the principle variables involved and determine their relationship to the coagulation rate. The main application of the phenomenon is as an intermediate step in the process of industrial

col-lection of aerosols. The large Brownian motion of extremely

small particles (less than one micron in diameter) cawaes these

particles to agglomerate; therefore, sizes less than one

mi-cron are seldom encountered in practice. There are centrifugal

machines which can collect particles larger than ten microns

in diameter with an efficiency of about 90 percent. Sonic

ap-plication is in the range of particle sizes from one to ten

microns where an attempt is made to make small particles grow into larger particles by agglomeration.

The term aerosol is used to define particles that

are dispersed in a gas. The term pertains to both solid and

liquid substances and covers the familiar types which are called

smoke, dust, fume, etc. If the aerosol is a liquid, the

parti-cles will be spheres; small solid partiparti-cles, however, will also behave essentially as a sphere.

-It was observed over fifty years ago that sound waves would cause particles of an aerosol to agglomerate. Numerous experiments were carried out in an attempt to deter-mine the Oause of the coagulation and to deterdeter-mine the forces

acting on the particles. The general conclusions arrived at

from the results of these early experiments are that there

are three main effects.that could cause agglomeration. They

are:

(1) If a distribution of small spherical jarticles

are introduced into a sound field, they will be caused to oscillate with different amplitudes

de-pending upon their size. This relatively

in-creased motion will result in more collisions per

unit time than by normal thermal agitation and

will greatly increase the, agglomeration rate. Ref. (5,7)

(2) A hydrodynamic analysis of the forces between

two spheres which are at rest in a vibrating medium results in both attractive and repulsive

forces which are inversely proportional to the four power of distance between particles.

Ref. (6,3)

(3) The third effect is the phenomenon of acoustical radiation pressure. An ana2ysis of the forces

waves shows that the sphere migrates toward the

node. Particles, therefore, will tend to collect

at nodes and agglomerate. This effect is easily

observed in a sound tube with standing waves using

a stationary aerosol. In the case of a moving

aerosol passing through a sound chamber with

standing waves, it is reasonable to assume that

this effect will only increase the turbulence of the flow and aid in the coagulation process by maintaining a random distribution of the particles in space.

It is the desire of the author to formulate a theory that will aid in the design of more efficient industrial col-w

lectors. In this analysis, a mathematical representation of

the coagulation phenomenon of an aerosol flowing through a

sound chamber with standing waves is formulated. The

stipula-tion of a fldwing aerosol is made since this condition tends to

maintain a more random distribution of the particles in space and supports one of the basic assumptions made in the analysis

STATISTICAL ANALYSIS

The present analysis is based on a simple statis-tical interpretation of the collision phenomenon of small

particles in a moving gas exposed to a sound field. The most

important' assumption which we have to make is that of the laws

of motion which the particles in .the sound field obey. In

this analysis, the assumption is made that the motion of

dif-ferent-sized particles relative to each other is the dominant

one in the range of particle sizes in which we are interested.

We will neglect hydrodynamic forces and the influence of radia-tion pressure, except to the extent that it will produce a

more random distribution of particle position. For these who feel that this may be too strong an assumption, let us point

out that the hydrodynamic forces could be included by consider-ing the effective cross-section of each particle as beconsider-ing

en-larged slightly, since the hydrodynamic forces are proportional

to the reciprocal of the fourth power of the distance between two particles and do not become significant except at small dis-tances.

We, therefore, make the following assumptions:

(1) Particles move with the root mean square value of

the following relation

-where

(up) max (ug)max

41Tfr2

_{+ 1}

L4Fg I

(1)

(See Appendix 2)

(2) The particles are essentially spherical

(3) The particles are independent, that is electri.

cal, gravitational, and magnetic forces are neg..

lected.

(4) Collisions result in particles whose volumes

are the sum of the volumes of the colliding par-ticles.

(5) The distribution of particles in space remains random.

We consider a non-homogenous aerosol with particles

of various sizes given by a distribution function.

Let x 3 volume of a particle

g(x,t) : distribution function : number of particle per

unit volume per volume element dx

n(t) : number of particles per unit volume at any time (t)

dn(x,t) * number of particles per unit volume having their

volume between x and x + dx

Therefore dn(x,t)

=

g(x,t) dx

.'. dn(x,t) n(t) a g(x,t)dx

0 0

(2)

The invariant in the problem is the total mass or rather the

volume since the density is a constant. Therefore

00

V .fx g (xt) dx =constant. (4) 0

The mechanism of coagulation is a collision

phenome-non; therefore, we must calculate a rate of collision between

the particles of different sizes. We assume that the particles can be considered as having a velocity given by equation (1). Equation (1) represents the velocity amplitude of a sinusoidal varying velocity, therefore we want to use its effective value.

(uP)eff, (Up)max x -V2

Brownian motion of particles, ineffective collisions, and tur-bulence of the flowing gas in the sound chamber will tend to

maintain a random distribution of particles in space.

We define the following quantities:

Q

z collision efficiency - It will probably be a function

of a nuxaber of particles per unit volume (n(t)]j, the maximum and minimum relative velocities, and finally a function

of any given aerosol.

H(x₁

,

X₂ ) = number of effective collisions per unit

time between a particle having volume x with all the particles having volumes

x₂ and x + dx

dn(x2 ,t) zg(x2 ,t) dx2 a number of particles per unit

volume having volumes between x2 and X2 +d2 (5)

We assume that the particles of volume x2 are

sta-tionary and let particles of volume x. move with velocity

ul - u2Ieff* Then the effective volume for collision

wiT

(r + r2)2 u - u2Ieff. (6) Therefore H(x1 , x2)

Q1T

(r1+ r2)2 u - u2 eff (7) using r

=X

(8)

We have H(x,,x2) =

L(X1

+ x2 )1 ul-u2 Jeff dh(x2tt) (9)

Let us examine more closely the types of phenomena

that the collision efficiency (Q) would have to represent. The

U &

amplitude of a vibrating particle is given by S where

2*w

ot depends on the mass of a particle and variesfrom 1 to zero

and is defined more explicitly later in the text. When the

mean distance between particles (1 T V_ ) increases to a

value larger than the vibrating amplitude of the particle, the

number of collisions per unit time will become small.

There-fore as the number of particles decreases, the collision

ef-ficiency (Q) will decrease as a function of time. If the

As the parameter (S - 1) approaches zero, the collision

ef-ficiency will decrease rapidly. If the intensity and

fre-quency are such that the quantity (S-1) is large at the initial

time (t

=

0), then Q ~%l and will decrease slowly with time (See

Appendix 1). The collision efficiency could also contain the

influence of a meximum and minimum velocity. There may be a

minimum velocity below which there will be no effective

col-lisions resulting in agglomeration. There may also be a

maxi-mum velocity above which collisions will result in disintegra-. tion instead of agglomeration. The exact nature of this

phe-nomenon is unpredictable.

Finally, the collision efficiency will depend upon the type of aerosol. Attractive forces that hold the particles

together will depend largely upon the physical and chemical properties of the aerosol.

It might seem at first sight that we have included

too many unknowns into a single parameter (Q), but if the

in-tensity, frequency, and grain loading are not too unfavorable, the collision efficiency should remain almost constant over a

time interval of the order of several seconds.

We now proceed to calculate the variation in the

num-ber of particles of volume x in a time St.

Let y = variable of integration

Therefore

X/ X max

Idn(x,t5 rj

H(y,x-y) dn(y,t) - H(x,y) dn(yt)j t (10)

min _{X min}

The first integral in equation (10) represents the collisions between particles that tend to increase dn(x,t), while the second integral represents the collisions between particles of volumes x with all of the other particles, thereby leading to a decrease in dn(xt). After combining equations 5, 9, 10, we

pass to the limit, and cancel common terms. Therefore, the

governing equation has the form

In order to generalize the theory and make the equa-w tions more simple for computation, we introduce a dimensionless volume ratio into our equation.

Let

x*

=

_x0

a

and1

*

Y

0 X0anywhere x0 will be the vol-.

ume of a particle with some reference diameter do.

rITd

0

(12)

It is also desirable to have the velocity dependence on a

dimen-sionless basis.

Therefore

(up)eff

=

.707 (ug)

max

41T f JP 43 II -tl i y.Lx where r x Now, if we let 2 -f2 d94 C 2 7 2

81

(14)

=

-- drrP,.

we can write

.707 (u m p eff*

₍₁₅₎

(13)

4hr where XO CC X*/ + 1

1-0

Since .707 (ug)ma is a constant for any given intensity, we can take it out of the integral sign. For computation purposes,

we then define

o

1/2

₍₁₆₎

IC + 1

The velocity then has form (up)eff .707 (ug)m

oL

Combining these dimensionless terms, we can transform equation

(11) into the following form:

2/3 Ig(xt)

Q

x (.707) (u

)M

(X*t)

12 (17) where x* max 2-( _{x--)g(y*,t)g(x*ey ,t)dy*} x min V3*/

x ma

and I2

TLx

+

y] OX)-- O((y)I g(y*, t)dy* (19)

x* min

By use of the well known relation between velocity

amplitude of particles in a gas and the intensity of the sound field, we can reduce the terms preceeding the bracket in equa-tion (17).

u4W2j(20)

where

j

=

sound intensity

fg

=

density of gas

Cog velocity of sound

Let K represent this collection of terms. After simplifica-tion, we get

Qd2-

Q

d

F

K

=(

2 (21)

4 C 0

where now we can see the dependence of the important variables.

Equation (14) can now be written in the simple form

g(x*, t) K I'

g(x,t)

1₂

(22)

The general equation is very difficult to solve

since it is a non-linear partial differential integral equation. However with certain assumptions it is possible to solve it

numerically.

we

can consider it essentially as a initial value

problem where we are given the distribution function g(x*,O) at

time t

=

0* Then we can proceed step by step to the new

dis-tribution by using the series expansion, neglecting terms of higher order.

g(x*,t

2)

g(x 'tl' +

At21 1 g

Lj...

₍₂₃₎

will allow us to follow the agglomeration process as a function

of the exposure time in the sound field.

The magnitude of the distribution function is

deter.-mined essentially by the grain loading of aerosol and the

ref-erence volume.

Let

W: grain loading (gm/am3)

then

W

fix

x* g(x*,O)dX* (24)

0

For the present calculations, g(x*,O) was assumed to be of the

following form:

g(x*,0) a A x*

e

₍₂₅₎

where x* a 1 was taken to be most probable size. Therefore

since x*M _ X 1 we can pick the reference volume xo to be

the probable particle size in the distribution. The constant

A is determined by equation (24) with a given WfP, x0.

The variation of grain loading will be reflected in

the following manner. Suppose we have the r esults for a given

initial function g(x*,O). Then we change the grain loading

keeping the density and xo constant. That is, g'(z,) - B g(x,O)

where

B;Z

const.

Then _g,

_(x*,t)

_{= gt(x*,0)}

g'(x*,tl) = Bg(x*,0) + At' B2 age,(X*0)]

where

we go(x*,0)

k[I

-

I(XO)

--

L

~

(Y0

since I, contains g(x *,0) squardd and 12 contain g(x*,O).

Therefore g' (x*,t)

=

Bg(x*,0) if we let At

B.

Then

the general effect of a change in grain loading is to change the value of the distribution function and the time scale by a constant. If we decrease the grain loading, that is B is less than one, then we increase the time scale and decrease

the distribution function g (x*,t).

The effect of an intensity change can be seen by

con-sidering equation (23) in a slightly different form.

(xt2)x tt)

+tt.Qdo'Il

- g(x*,t1) 12] (26)

_,C, 4

If we change the intensity, holding the other variables

con-stant, we can g et the same result by considering that the time

scale is changed by the square root of the intensity ratio.

Let J' z RJ

then

At'

-If we, therefore, increase the intensity, we increase the co-agulation rate and need less time to obtain any desired amount of agglomeration.

The effect of a frequency change is best represented by selecting different values of the parameter C, which is a

manor factor in determining the relative velocity of the

parti-cles. If we try to change the frequency, holding the parameter

C constant, we have to _{change the particle density,} the reference

volume, or the temperature of the gas. These changes would be

reflected in the parameter K and also in the distribution

func-tion through a change in grain loading. If we change the

par-ticle density to compensate for a frequency change, holding C

constant, we get a change in grain loading. The effect of a

change in grain loading has been discussed previously. Changing

the reference diameter to compensate for a frequency change will result in a change in K and the grain loading. Any change in the parameter K can be compensated by a change in time scale for results that have been calculated for a constant K.

Attempts to compensate for frequency change by a tem-perature change in the gas results in a variation in gas

vis-cosity, gas density, and velocity of sound. Neglecting the

change in particle density, these effects produce a change in

the parameter K and can be calculated by a tine scale shift.

Results for different particle densities can be

ob-tained in the following manner. An increase in particle density

and by a grain loading change if we hold the distribution

fune-tion constant. If we want to keep the grain loading constant,

then we must reduce the distribution function and shift the time scale accordingly.

A change in the reference volume is felt in all

para-meters, C, K, W. To obtain the results of a change in d0 with

given values for C, K, W, we can hold C constant by a frequency

variation, let the time scale shift for the K change and let

the magnitude of distribution function change with another time

scale shift to account for holding W _constant.

Thus we see that if we have the computations for a

particular case of a given C, K, and initial distribution

func-tion, we can determine the results of changing different vari-.

ables by shifting the time scale or making compensating changes

in other variables. The fact that a few computations will

lead to a large volume of results for different conditions

COMPUTATIONS

Since the general analysis predicted that the

various parameters varied qualitatively in agreement with

physical observations, some calculations were made for com-parison with experimental data. The initial distribution was assumed to have the form of Maxwell's thermal velocity

distri-bution.

*

g(x*,0)

=

Ax* e-bx*

If we take x* x 1 to be the most probable particle size, we

get b 1.

Next, we must specify the various parameters and in-dependent variables, letting the time be the in-dependent variable. Calculations were made using the following values.

do 3 microns (10 4cm)

,P particle density:= 1 gm/cm3

Jmax 2 watt/cm2 2 x 107 ergs/cm2

W = 3 grain/ft3 - 6.87 x 10-6 gm/cm3

.-

=

.001205 gm/cm3 T a 200 C, 1 atm.

fAg Z181 x 10 6 yAsec T a20 _C

Q a collision efficiency a 1 (100f.)

CO0

=

3.452 x 104 cm/sec T

=

200 C

With these values the parameter K has value K 5.44 x 10-6

cm3/sec. The coefficient A (the amplitude of distribution

f.P

o

kw

x

*3e

X*

&

d

0

where x* max : 3 was taken as the upper limit of integration.

X* a 3 corresponds to a diameter of d

=

3 do z 4.32 microns.

Since

fx*Zeix

.2

61*

X3.xdx*3, we get

A

=

9.72 x 105 1/cm3. Therefore,

g

(x*,0)

9.72 x 10

5

*2 .X

Calculations were made for two values of the parameter C, which is dimensionless.

Case C Frequency with above conditions

1 .0135 669 c.p.s.

2 1.35 6,690 c.p.s.

The initial plan was to extend both calculations to a point where they could be compared with experimental data. After the

calculations were begun, however, they proved to be more

com-plex than was anticipated and there was not sufficient time to

extend the calculations to the desired goal.

The first step in the calculations is to compute the

initial rate of change of the distribution function. Then

several values of g(x*,t1) are plotted with different time

The distribution function must not change too much in any one step or there will be too much distortion in the results. As the calculations proceed, it may be necessary to change the time interval for physical reasons. For example, if the

dis-tribution function should become negative at any time, it

in-dicates that too large a time interval was used.

The distribution function is then computed as a func-tion of time using equafunc-tion (23). The two integrals I, and I2

were computed by using the trapizoidal rule where computation

time dominated accuracy. Even with the approximate procedure,

it was estimated that each integral was accurate to . 3%;

there-fore the distribution functions were accurate to

:

6%.

Finally, the results must be put in a form comparable with experimental data. For this purpose, a theoretical

ef-ficiency was defined for a hypothetical collection system. The

system is composed of a sound chamber and a cyclone collector. We assume that the cyclone efficiency curve is that of the

3-inch cyclone efficiency presented in Fig. (2) and that the performance is essentially independent of grain loading.

SOUNO

IM OUT

We define the following collection efficiency.

.z(Mass)in - (Mass)out - (Nass)collected

N -(Mass)in (iass)ln

Since the density is constant, we have

(Volume)collected N a (Volume )in Now x*x z volume of particle at x* and z V* X0

x max

[

x*g(x*,t)dx*,

xmin

so that we have N V* collected

V* in x*max

V* collected E(x*) x*g(* ,t) d

x*min

where E(x*) is the cyclone efficiency.

Finally x*max

N

E ()

x*g(x*,t)

dx* i*min x g (x *,t) dk* X*Min

The collection efficiency was calculated for each distribution function and is plotted in Fig. 7.

DISCUSSION OF RESULTS

In Figures 4 and 5, the change of the distribution

function with time is presented. The actual parameter plotted

is x* g(x*,t) which represents the amount of the particle

vol-ume per unit volvol-ume between x* and x* + dx** The reason for

plotting tnis parameter is that the area under the curve

re-mains constant and equal to the initial volume V*. To get the

actual volume, multiply V* by x. Since the density is con-stant, these curves also represent the amount of mass at any

particular volume. By computing area ratios, we can

deter-mine what fraction of the total mass is contained in any

vol-ume interval.

The influence of the relative velocity of the

parti-cles,which is expressed by dimensionless parameter c'*, on the

rate of change of the distribution function is demonstrated in

Figure 6. The rate of change curve for case 2 (C = 1.35) is

roughly ten times in magnitude that for Case 1 (C = .0135).

The reason for this difference is easily seen by considering

the oc,* parameter plotted in Figure 3. In case 1, we are

operating on the flat portion of the curve and the relative velocity between particles of different sizes is small;

there-fore the rate of change of the distribution function is small.

large for case 2, it will decrease as the distribution function

moves toward larger volumes. With the present initial

distri-bution function, the agglomeration process would proceed more rapidly with time if a combination of two frequencies were

used. First, use a frequency corresponding to C 1.35 and

then when the mean volume has reached x* * 10, switch to a

frequency corresponding to the C

=

.0135 curve.

The result of taking too large a time interval is

demonstrated by considering Figures 5 and 6. If too large a

time interval is taken, the next distribution function will

have too many large particles and a hump will appear in the

distribution function. If we assume this to be physically

correct and calculate the next step using this function, we get the obvious result that the hump will move toward large

volumes. However, these results are distorted in that we are

apt to get too high an collection efficiency and too short a

time scale. Case 1 is an example of selecting too large a time

interval, while case 2 is closer to a continuous process. There

will always be some error becaise of a .finite time interval. Further evidence that case 2 has a more accurate time interval is shown in Figure 8, where the number of particles per unit

volume is plotted as a function of time. These results obey

from a statistical phenomenon of this kind.

Finally, the collection efficiencies presented in

Figure 7 are calculated by multiplying the curves in Figures

4 and 5 by the cyclone efficiency and integrating.

At the present time, these curves have not been ex-tended into the range where they can be compared with experi-mental data. However, they show the correct trends and cal-culations are being continued to extend their application.

BIBLIOGRAPHY

1, Andrade, E. N. deC., "Coagulation of Smoke by Supersonic

Vibrations", Transactions of Faraday Society, 1936,

vol. 322 p.*1111.

2.

... "On the Circulation by the Vibration of Air

in a Tube", ProcR Society of London, 1931,

vol. 134, p. 445.

3. Atomic Energy Commission, Handbook on Aerosals, Washington,

D. C., 1950.

4. Bergman, L., Ultrasonics, New York, 1938.

5. Brandt, O., and Hiedemann, "The Aggrigation of Suspended

Particles in Gases by Sonic and Supersonic Waves", Faraday Society Transactions, vol. 322, 1936, p. 1101.

6. Konig, W., Ann. Physik, vol. 42, 1891, p.353.

7. Neuman, E. P., and Norton, J. L., "Application of Sonic

Energy to Commercial Aersol Collection Problems" (Paper to be published soon.)

8. St. Clair, H. W., "Sonic Flocculator as a Fume Settler:

Theory and Practice", . Bureau of Mines, R. I.

3400, 1936, p.51,

9. ta41sir, H. W., Spendlone, M. J., Patter, E. V.,

"Floc-culation of aerosols by intense High Frequency Sound",

o8of Interior, Bureau of Mines, R. I. 4218,

4C

3c

20

to

x

FIGURE

I

2..

-

3.

E

-p-INITIAL DISTRIBUTION

FUNCTION

*2 52

X

5 p-

-G(X)=9.7,x10

X

e

I I

010

_4--- ---

--C

3

INCH CYCLONE

3

MICRONS

6

8

*

10

X

IL z

w

080

060 z

w

U-w

40

20

LL -t k4 # --- _IT - t

---2

4

12

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INITIAL

DISTRIBUTION

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TIME

(SEC.)

FIGURE

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INTENSITY

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APPENDIX I

VARIABLES EFFECTING COLLISION EFFICIENCY

The collision efficiency Q may depend upon the

type of aerosol and a possible maximum and minimum velocity

of the particles. The only effect considered here, however,

is the dependence on the relation between the vibrating am.-plitude and the mean distance between particles.

Let

mean distance between particles

0(

-particle

amplitude.

2I~4F,

CO

27f

If the particle amplitude is greater than 1, there will be many collisions which will maintain a random distribution of

the particles in space. As the parameter (S-1) approaches

zero, the effective number of collisions is reduced. It may

be possible to handle this effect with a variation in col-lision efficiency.

Consider a specific case. Let us take the following

conditions:

J 2 watt/cn2

g 1.205 x lO- (air T = 200 C) "/ms

c = 3.452 x 104 cm/sec

0(*=

n(o)

=

4.3 x 105 particles/cm3

W

=

3 grain/ft

3

where n(o) was computed from the distribution function used

in our calculations

fl (o)

J

O(xO)

uX'=

4.30

x (0 number/cm3

0

For this case we have the following table:

Frequency (S) (cm) n(t) (cm) (cps) max. 103 l56 n (o) .0132 5 x 103 .078 n(o)/2 .0167 104 .0156 n(o)/10 .0285 n(o)/100 .0614

If the frequency is low enough, and the intensity is high,

we have (S-1)~10. Under these conditions, we should have

ran-dom motion if n(t) does not decrease too rapidly. In case I,

(f = 669 cps) for which we have made calculations, n(t) has

not dropped to n(o)/2 in one and a half seconds; therefore

our assumption of random motion and a constant Q seem to be

If we had considered a case where n(t) was

chang-ing rapidly so that (S-i) was approaching zero, we would have

had to consider Q changing with time. It is reasonable to

as-sume that Q could be represented by some function (f) which

is functicn of parameter (3-1).

The function f could have the form

f

1 - He I. where H = constant and is practically

equal to one. 10 is the mean distance at time t =

o.

Then Q would have the form shown in plot below

1.o

For any specific case S and 1 1(t) N would be a

function of time. We would have to calculate a mean

compli-tude S since not all the particles have the same amplicompli-tude and the mean amplitude would decrease with time as the particles grow larger. We can calculate S in the following manners

14

This particular form of

Q

would seem to satisfy all

of the physical conditions in the problem but it would have to

be verified by actual experimental data and an attempt made to

determine the constant H.

APPENDIX II

DERIVATION OF LAW OF MOTION

We assume that a small sphere moving in a gas has a force on it given by Stakets Law

F ".6 1T. rV

where t viscosity of medium

V u relative velocity of particle and medium

r

= PARTICE RAdtuS

v has the form (ug - u) where ug is the gas velocity and u is

particle velocity.

tg a u sin wt uo a velocity amplitude

The equations of motion are

m

WE ti

r

3

J

:

6

'Y -gr(uo sin wt -u)

If we let

z

z9/2

-we get

du

.a _{(u0 sin wt - u)}

Solution of this equation is

By using the trigometric identity,

sin ( -3) A sino( cos /S - cosoc sin₁

,

we can transform the above equation into the following form:

u n ZUO

z2+w2

sin (wt -)

where

N

tannl

"

z

If we consider only the velocity amplitude, we get

P) max (uma)

+j

1~

4

lyfr2P]12

However, the important quantity is the effective value of the velocity or rather the root mean square velocity.

( (up)eff (up 2

It is a simple integration to show that

pe(uP)Max when the velocity is a sinusoidal