Research of the mechanism of capture of particles on the liquid droplet Текст научной статьи по специальности «Физика»

UDC 532.527

R. R. Usmanova, G. E. Zaikov

RESEARCH OF THE MECHANISM OF CAPTURE OF PARTICLES ON THE LIQUID DROPLET

Keywords: capture ratio; coagulation; turbulence; diffusion; particle settling; spherical particles; capture ratio.

The mechanism of coagulation monodisperse spherical particles on the liquid droplet. It was found that the calculations on them lead to low results. For practical calculations of particle coagulation of different sizes offered to use interpolation on a graph, showing the best results of convergence.

Ключевые слова: коэффициент захвата; турбулентность; осаждение частиц; сферические частицы; коэффициент

захвата.

Рассмотрен механизм коагуляции монодисперсных сферических частиц. Предложены эмпирические зависимости для расчета коэффициента захвата частиц. Установлено, что расчеты по ним приводят к заниженным результатам. Для практических расчетов коагуляции частиц разных размеров предложено использовать интерполяцию по графическим зависимостям, показывающую наилучшее согласование результатов.

1. Current state of a problem

Aerosols, as well as many other disperse systems, have the restricted life time. In them, there are the various processes leading to integration of primary corpuscles, to their aggregation, formation of offsprings, and the subsequent sedimentation. Some of these processes proceed spontaneously, others-under the influence of electric, hydrodynamic, or a gravitational field. In the course of integration of corpuscles, there is a decrease in superficial energy at the expense of decrease of their specific surface [1-4].

Integration of corpuscles can go two ways: as a result of isothermal distillation, if corpuscles liquid (Calvin's effect), and as a result collisions and an adhesion-Concretions. In case of merge of liquid corpuscles, this process name coalescence. So, generally as concretion is called decrease of degree of dispersion, i.e., decrease of number of corpuscles of a dispersoid at their integration.

The Brownian motion of corpuscles can be the concretion reason and in this case it is called as Brownian, either a spontaneous coagulation, or affecting of superposed forces-hydrodynamic, electric, gravitational, etc., corpuscles leading to collision, and in this case concretion is called as forced. The basic regularity of concretion of aerosols was installed for the first time by the Polish physicist M. Smoluhovsky [7]. For the account of efficiency of an adhesion of corpuscles at collision, it has injected two concepts: "racing" and "slow" concretion. In the first case of collision conduct to an adhesion of corpuscles, and in the another adhesion occurs not from the first collision or not all faced corpuscles coagulate.

In the first case of collision conduct to an adhesion of corpuscles, and in the another adhesion occurs not after the first collision or not all faced corpuscles coagulate.

2. Coagulation of equigranular spherical corpuscles

Most simple aspect of concretion is a thermal coagulation of equigranular spherical corpuscles. As are

observed, only the first some certificates of collision of corpuscles, the size of the formed corpuscles will not differ aloud from a size of initial corpuscles. This model is used many years for firm corpuscles and can form a basis for definition of a constant of concretion. It approaches and for the description of concretion of drops of a liquid as the size of drops after merge increases proportionally to a cube root from quantity of drops, its components. This approach was offered Smoluhovsky [8] for concretion in the diluted electrolytes, but it can be used for aerosols within the restrictions observed above [6]. In approach Smoluhovsky, it is supposed that in system of spherical corpuscles in diameter 2R distances between corpuscles are chaotically distributed, at t = 0. If corpuscles move also chaotically by thermal diffusion it is necessary to know probability of their collision during some time.

Smoluhovsky, the first has observed a case when one corpuscle fixed in space, is the center of concretion for other corpuscles. It has defined speed of diffusion of other corpuscles to this central corpuscle. The equation of an unsteady-state diffusion looks like:

— = Dd V2C 8t d

(1)

where C - concentration of corpuscles; Dd - factor of their diffusion. If r - the distance from the center of the fixed corpuscle assuming spherical symmetry of system, Eq. (1) can be written down the equation in an aspect:

8C

8t

Dr

(8 2C 8r2 '

2 <8C_ r 8r

\

(2)

or in more convenient form: 8( Cr )

= D.

82 (Cr)

(3)

dr a dr2 As corpuscles have the same size, we assume that they will face the central corpuscle when pass in distance limits 2R from it. Thus, concentration will be equal in this point to null, that is CU= 0 at r = 2R (for t =0). Besides, it was originally supposed that corpuscles are in regular intervals distributed on all volume with concentration C. Thus, CU= C at t = 0.

X

Fig. 1 - Change of concentration of corpuscles in a time

Then Eq. (3) it is possible to present the

equation:

C' = C

2R 2R

1--+-erf

r r

i TO ^

r - 2R

(4)

The number of corpuscles N, which diffuses within distance 2R from the central fixed corpuscle in unit of time, is equal to product of a diffused stream on the square of a surface of sphere in radius 2R. The diffused stream is defined from the equation

J = - Dd (dC'jdr), (5)

Where the derivative (dCD/dr) should be sized up at r = 2R.

- 2 DC' N = 16nR 2 Dd-

8r ()

at r = 2R. And from the Eq. (4) we will gain

8С 8r

C_ 2R

2R

(7)

Then the number of corpuscles, which attain a surface surrounding the central corpuscle, in time d t, makes

Ndx = 8%RDdC

1 +

2R

Dd x

dx.

(8)

Now, we will assume that the fixed corpuscle can diffuse the same as other corpuscles. Diffusion also should be considered. The general factor of diffusion of two corpuscles rather is each other equal to the sum of factors of diffusion of separate corpuscles so the propelled corpuscle faces with

2R

16kRDC

1 +

■sJftDx

c/x

Corpuscles in the interval d t, in view of that the Eq. (8) is applicable for equal sizes. In unit volume will occur C/2 collisions, if C - number of corpuscles two corpuscles participate in unit volume and in each collision. The number of collisions in unit volume, which occur in time d t, is presented by the equation:

dC dx

-—krd.c2 2

1 +

2

(9)

The another member in brackets, it is possible not to consider, as it will be much less unit if t it is great enough. The Fuchs [1] has shown that

2R/ JxDd =4 .

The probability of that a corpuscle originally is near to the fixed corpuscle. Thus, Z it is aimed to null as stationary speed increases. As, Z it is not enough for practical conditions, this magnitude neglect. However, it is necessary to mean that in some conditions Z, it is necessary to consider. This member leads to increase in speed of concretion. Defining a constant of concretion K0 as

K = 16nRDd = (10)

3I

We gain results which, at least, for large corpuscles will not depend on a size of corpuscles.

Using K0 as a constant of concretion and neglecting the another member in the Eq. (9), we will gain the usual form of the equation of concretion:

dC =-K_ c 2

dx 2 (ii)

Integrating the Eq. (11) at entry condition C = C0 when t = 0, we will gain:

(12)

1 - ± = ^L x

C C0 2

The Eq. (12) shows that the reciprocal quantity of concentration of corpuscles is a time linear function, and the straight line inclination gives a concretion constant. If tD assign as a time for which concentration decreases twice,

C = Coj (l + t/T*\ (13)

Expression (13) can be directly used for machining of experimental data on concretion.

3. Dust laying on drops at liquid spraying

Efficiency of sedimentation of corpuscles on drops of liquid (the kinematic concretion) depends first of all on magnitude of their relative traverse speed w. The kinematic (gravitational) concretion to proceed at free falling of drops through motionless an aerosol of count concentration of n. In this case, number of the small corpuscles entrained by one drop in 1s, it can is possible to define by formula:

Q = 1/4 n n dK vc (14)

If drops are precipitated in a moving stream of the aerosol, which speed it is impossible to neglect Eq. (14), it is necessary to inject a relative traverse speed into the formula w corpuscles concerning a drop instead of speed of subsidence ve*. The total factor of capture of corpuscles a spherical drop n depends from a flow regime.

Efficiency of trapping according to Fuchs calculations is defined first of all by a size of corpuscles. For example, for corpuscles in density p = 2,000 kg/m3 those are trapped only, which size > 0,5 |am. Thus for corpuscles of small sizes (0,5...0,7 |im), the more largely a drop, the efficiency of trapping above. It proves to be true only in a case small relative speeds.

r

If relative speeds are great, as it occurs at liquid injection in a gas stream efficiency of concretion of corpuscles grows with decrease of a size of drops. It is possible to be convinced of it, if expression (14) to refer to drop volume.

Capture of corpuscles by drops depends on the several reasons. Here, along with the kinematic proceeds and graded region concretion. The momentous role in sedimentation is played by turbulence of a stream.

In wet-type collectors of bubbling type motion of gas and liquid drops can be organized on one of three circuit designs: counterflow, direct-flow, or with a cross current [5]. We will carry out the analysis of process of sedimentation of a dust on drops of liquid depending on the circuit design of motion of streams and it is definable its efficiency.

Fig. 2 - The loading diagram of efficiency of the apparatus for motion: (a) - a backward flow; and (b) -a cross current

Let's gate out an element of space with a size dldbdh (Fig. 2). We will mark out a heading of material streams depending on the motion circuit design. In all three cases on the overhead basil (dldb), the stream of the drops, which volume flow rate arrives, m3/s,

V, = av (Mb ) (15)

where v - a traverse speed of drops; and - a share of the volume occupied with drops.

The dust-laden flow of gases depending on the motion circuit design arrives on overhead, bottom, or a lateral face (dbdh).

If mass concentration of a dust on an entry in a volume element is equal in gas C, kg/m3 the dust stream on an entry will make, kg/s:

For direct-flow and counterflow circuit designs

g1 = u(dldb) (16)

For the circuit design about a gas cross-flow

g1 = u (dbdh ^ (17)

where u - speed of a dust, equal to speed of gas.

On an exit from a volume element gas will contain a dust (C~dC), the kg/m3, and a dust stream will make accordingly, kg/s:

g2 = u (dldb X^ - dC ) (18) For the circuit design about a gas cross-flow

g2 = u(dbdh ))C - dC) (19)

Dust laying on drops occurs to relative speed w. In the circuit design about a forward flow this speed w=u - vt backward flow w=u -v9 and in case of cross motion in a gas stream heading

= u, vi = 0

According to the formula (15) on one drop in diameter dK at efficiency of capture ts, it is precipitated corpuscles in weight:

1!4CwTtd(20) In total in the gated out volume, there is a thaw 6(dldbdh)aa(Kd1'K) (21)

Thus, as a result of capture by drops a dust particle flux

g3 = {dldMh) =

3 aaC^Y. 2

(22)

(dldbdh)

JK

Let's write down the equation of a material balance of a volume element:

S - g 2 - g3 = 0 for direct- and the counter flow circuit design

uCdldb - u(C - dC)ldb -

3 araC^Y

2

d

dldbdh = 0

(23)

K

(24)

and for a cross current

uCdbdh - u(C - dC)bdh -

----^ dldbdh = 0

2 dK

After transformations, we will gain for the first two circuit designs, where ® = u ± v

dC 3 a „

A •—dh

C

2 d

K

And for a cross current, where w=u at u =0,

dC___ 3 ^Jn C _ 2 ' d

•d!

(25)

(26)

к

Let us inject a parameter of an irrigation of gases m, m3/m3, as the relation of volume flow rates of a liquid and gases: for the first two circuit designs Vl avdldb av

m =■

V.

g

=--(27)

udldb u

aa a

-= m —

u v

And for the circuit design about a cross-flow in terms of that u = v

Vl adl

m =-

Vg

ah

(28)

or adl _ mdh.

Let us substitute the gained expressions in the Eqs. (2526):

dC 3 a J

— _ —m--—dh

C 2 v dK

dC 3 tiy „ — = —m^^dh С 2 dK

(29)

After integration on all altitude of a zone of contact of a dust with drops of liquid (apparatus altitude) expression (29) can be written down in an aspect.

u

For circuit designs with line and counter flow motion and

(

q = 1 - ехр

3 а

--/77 — •

2 1/

н

'к

(30)

For the circuit design with a lateral motion of streams.

(

q = 1 - ехр

3

--/77 •

2

Н

'к

(31)

The analysis of efficiency of three circuit designs has been spent at following assumptions:

(1) drops are distributed in regular intervals by

volume;

(2) process isothermal, without changes of

phase;

(3) corpuscles have speed of gas and are trapped only by drops as a result of collisions in the presence of relative speed.

Real process of trapping of a dust by drops of liquid much more difficult also does not give in to the full account. Therefore, efficiency of the apparatus defines as a rule, by means of empirical dependences.

Conclusions

1. As appears from stated above the theory Smoluhovsky, it is fair for process of concretion of a monodisperse aerosol. In practice such aerosols meet rather seldom, the methods allowing at conservation of substantive provisions of rapid sweeping concretion to use of them for calculation of process of concretion of inequigranular aerosols therefore are necessary.

2. For calculation of process of concretion two approaches are observed. First, it is connected with correction (refinement) of formulas for definition of a constant of concretion or, more precisely, in selection of such expressions, which would allow, without penetrating

into details of disperse composition of an aerosol, to gain correct values of speed of concretion. The another approach assumes working out of mathematical model of process of concretion of inequigranular aerosols.

3. The case of concretion of an aerosol in a turbulent stream, which is characterized by origination of the inertia differences between corpuscles of different sizes is observed. Owing to turbulence of a corpuscle are sped up till the various speeds depending on a size, and can then face with each other. Empirical dependences for calculation of factor of capture of corpuscles calculations on which will well be coordinated with experimental data are offered.

References

1. Fuchs, N. A.; Mechanics of Aerosols. Publishing House: Academies of Sciences the USSR; 1955.

2. Fuchs, N. A.; To the theory of a sprinkler irrigation of "warm" clouds. The Rep Acad. Sci. USSR. 1951, 81(6), 10431045

3. Deitch, M. E.; and Fillipov, G. A.; Gas kinetics of two-phase medium. М: Energy. 1981.

4. Saltanov, G. A.; Supersonic Two-Phase Currents. Minsk: The Higher School; 1972.

5. Hidy, J. M.; On the theory of the coagulation of noninteracting particles in Brownian motion. J. Colloid. Sci. 1965, 20, 123-144

6. Herne, K; The classical computation of the aerodynamic capture by spheres. In: Aerodynamic Capture of Particles. New York: Pergamon Press; 1960.

7. Smoluhovsky, M.; Experience of the mathematical theory of kinetics of concretion of the colloid openings. Coagulation of Colloids. М: ONTI. 1936a, 7-39, (in Russian).

8. Smoluhovsky, M.; Three reports on Brownian molecular motion and concretion of colloidal particles Brownian motion. М: ONTI. 1936b, 332-417, (in Russian).

© R. R. Usmanova - She is currently Associate Professor of the Chair of Strength of Materials at the Ufa State Technical University of Aviation in Ufa, Bashkortostan, Russia, [email protected]; G. E. Zaikov - DSc. Professor of the Chair Plastics Technology Kazan National Research Technological University, [email protected].

© Р. Р. Усманова - канд. техн. наук, доц. каф. СМ Уфимского госуд. авиационного технич. ун-та, [email protected]; Г. Е. Заиков - д-р хим. наук, проф. каф. ТПМ КНИТУ, [email protected].